Mathematical Modelling in
Swirling Flows:
a Hamiltonian perspective
Erik R. Fledderus
The research described in this thesis was carried out at the Department of Applied
Mathematics, University of Twente, Enschede, the Netherlands. The research was
subsidized by the Netherlands Organization for Scientific Research NWO, ‘Nonlinear
Systems’ project (NLS 620-61-249). The NLS-fund is gratefully acknowledged for
financial support.
ISBN 90 365 09807
MATHEMATICAL MODELLING IN
SWIRLING FLOWS:
A HAMILTONIAN PERSPECTIVE
MATHEMATICAL MODELLING IN
SWIRLING FLOWS:
A HAMILTONIAN PERSPECTIVE
PROEFSCHRIFT
ter verkrijging van
de graad van doctor aan de Universiteit Twente,
op gezag van de rector magnificus,
Prof. dr. F.A. van Vught,
volgens besluit van het College voor Promoties
in het openbaar te verdedigen
op vrijdag 22 augustus 1997 te 15.00 uur
door
Erik Richard Fledderus
geboren op 15 maart 1970
te Heerenveen
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. E.W.C. van Groesen
Voorwoord
De afgelopen vier jaar heb ik gewerkt als onderzoeker in opleiding binnen de vakgroep
Toegepaste Analyse van de faculteit Toegepaste Wiskunde aan de Universiteit Twente.
Gedurende deze tijd heb ik veel mogen leren van mijn collega’s, maar ook mogen delen
met studenten tijdens de verschillende colleges. Een deel van alle indrukken en ideeën
is terechtgekomen in dit proefschrift. Ik wil op deze plek een aantal mensen met name
noemen die deze periode iets extra’s voor mij betekend hebben.
Allereerst wil ik professor Brenny van Groesen bedanken voor zijn inspiratie. Hij
was niet alleen promotor maar bovenal een collega bij wie je (vrijwel) elk moment
kon binnenvallen met vragen of opmerkingen. Zijn kritische houding heeft mij erg
gemotiveerd en geholpen bij het schrijven van dit proefschrift en het geven van colleges
en voordrachten.
De rest van de vakgroep wil ik bedanken voor de gezellige sfeer tijdens de ‘koffiebreaks’, de werkbesprekingen, de vrijdagmiddag-projecten en andere momenten. Met
name wil ik noemen Frits van Beckum, Stephan van Gils, Ruud van Damme en mijn
kamergenoot Dejana Djokovic̀.
Ik heb de inzet van Kasper Valkering en Marit Steggink zeer op prijs gesteld in de
periode dat ze enkele aspecten van mijn onderzoek hebben uitgewerkt. Ook de discussies met Gianne Derks over met name de hoofdstukken 2 en 3 hebben verhelderend
gewerkt.
Ik wil Jim Kok bedanken voor zijn inspanningen en aandacht tijdens en tussen de
verschillende voortgangsbesprekingen.
Een deel van het onderzoek vond plaats aan de faculteit Applied Mathematics and
Theoretical Physics van de University of Cambridge, UK. Ik wil professor David
Crighton en Stephen Cowley bedanken voor de goede tijd die ik daar heb gehad. Met
name de inzichten en de kritische houding van Stephen Cowley heb ik erg gewaardeerd
en hebben veel bijgedragen tot mijn werk.
De belangstelling van vrienden, kennissen en familie heb ik erg op prijs gesteld; onvermoeibaar hoorden zij mij aan wanneer ik in enigszins begrijpelijke taal probeerde
te verwoorden wat mij bezig hield.
Als laatste wil ik mijn ouders en Gretha, Tom en Amarins bedanken voor hun geduld
en alle afwisseling in mijn leven.
Hengelo, juli 1997
Erik Fledderus
i
Contents
Contents
Contents
iii
Prologue
v
1 Introduction
1.1 Fluid dynamical aspects of swirling flows and helicity . . . . . . . . . .
1.1.1 Vortex breakdown: an overview . . . . . . . . . . . . . . . . . .
1.1.2 Helicity and swirling flows . . . . . . . . . . . . . . . . . . . . .
1.2 Structured equations and coherent structures . . . . . . . . . . . . . .
1.2.1 A Lagrangian approach to the Euler equations . . . . . . . . .
1.2.2 The Poisson structure for the Euler equations in a domain with
fixed boundary . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Conserved quantities and their dynamics . . . . . . . . . . . . .
1.3 Approximation methods: a different viewpoint . . . . . . . . . . . . .
1.3.1 Quasi-static approach . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 An example: the Duffing equation . . . . . . . . . . . . . . . .
1.4 The Navier-Stokes equations as a singular perturbed system . . . . . .
1.4.1 What is a singular perturbed problem? . . . . . . . . . . . . .
1.4.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Relevance of example for the Navier-Stokes equations . . . . .
1.5 Experiments and numerical simulation . . . . . . . . . . . . . . . . . .
A.1 Function spaces and their decompositions . . . . . . . . . . . . . . . .
1
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2
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2 A zoo of swirling flows
2.1 Swirling flows as constrained extremizers of the energy
2.2 The Beltrami flows in a pipe . . . . . . . . . . . . . .
2.2.1 The vector-Helmholtz equation in a pipe . . . .
2.2.2 Construction of Beltrami flows . . . . . . . . .
2.2.3 Properties of the Beltrami flows . . . . . . . . .
2.3 Ordering of the relative equilibria . . . . . . . . . . . .
2.3.1 Non-columnar relative equilibria . . . . . . . .
2.3.2 Columnar relative equilibria . . . . . . . . . . .
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ii
Contents
2.4
2.3.3 Multi-mode relative equilibria . . . . . . . . . . . . . . . . . . .
Axisymmetric flows and the Bragg-Hawthorne equation . . . . . . . .
3 Stability of swirling flows
3.1 Linear stability for columnar flows . . . . . . . . . . . . . . . . . . .
3.1.1 Axisymmetric disturbances . . . . . . . . . . . . . . . . . . .
3.1.2 Parallel disturbances . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Three-dimensional disturbances . . . . . . . . . . . . . . . . .
3.2 Nonlinear constrained stability . . . . . . . . . . . . . . . . . . . . .
3.2.1 Nonlinear constrained stability of Beltrami flows . . . . . . .
3.2.2 Nonlinear constrained stability of columnar relative equilibria
3.2.3 High-wave number cut-off (Szeri and Holmes (1988)) . . . . .
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4 Swirling flows in an expanding pipe
127
4.1 Re-introducing the radius as a length-scale . . . . . . . . . . . . . . . 133
4.1.1 The non-columnar relative equilibrium solutions . . . . . . . . 133
4.1.2 A precessing vortex core . . . . . . . . . . . . . . . . . . . . . . 135
4.1.3 The columnar relative equilibria and multi-mode relative equilibrium solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2 The quasi-homogeneous approximation . . . . . . . . . . . . . . . . . . 139
4.2.1 Consistent evolution and orthogonality conditions . . . . . . . 139
4.2.2 Consistent evolution: the parameter dynamics . . . . . . . . . 144
4.3 Analysis of the error equation . . . . . . . . . . . . . . . . . . . . . . . 153
4.3.1 The Kernel Theorem . . . . . . . . . . . . . . . . . . . . . . . . 154
4.3.2 Multi-mode relative equilibrium solutions and the degeneracy
of the linearised Euler equations . . . . . . . . . . . . . . . . . 155
4.3.3 The solvability condition at criticality . . . . . . . . . . . . . . 155
4.3.4 Embedding the solvability condition in the quasi-homogeneous
approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.4 Comparison and relation with other approaches . . . . . . . . . . . . . 162
4.4.1 The quasi-homogeneous approximation and the Bragg-Hawthorne
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.4.2 Vortex-breakdown: (slow) passage through bifurcation . . . . . 169
5 Navier-Stokes: viscosity as perturbation
5.1 Balances and boundary layers . . . . . . . . . . . . . . .
5.1.1 The general case . . . . . . . . . . . . . . . . . .
5.1.2 The mass balance . . . . . . . . . . . . . . . . .
5.1.3 The angular momentum balance . . . . . . . . .
5.1.4 The axial momentum balance . . . . . . . . . . .
5.1.5 The energy balance . . . . . . . . . . . . . . . . .
5.1.6 The helicity balance . . . . . . . . . . . . . . . .
5.2 Viscous swirling flows in a constant-diameter pipe . . .
5.2.1 Rigid-body rotation in a constant-diameter pipe
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iii
Contents
5.2.2
Viscous effects on relative equilibria in a
constant-diameter pipe . . . . . . . . . . . . . . . . . . .
5.3 Viscous swirling flows in an expanding pipe . . . . . . . . . . .
5.3.1 Coherent structures in the boundary layer . . . . . . . .
5.3.2 The evolution of relative equilibria for the viscous flow
expanding pipe . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Boundary layer theory beyond separation . . . . . . . .
A.1 The boundary layer equations . . . . . . . . . . . . . . . . . . .
A.1.1 Derivation of the boundary layer equations . . . . . . .
A.1.2 Transformation of the boundary layer equations . . . . .
A.1.3 Transformation of the boundary layer measures . . . . .
A.2 Viscous flow in a constant-diameter pipe . . . . . . . . . . . . .
A.3 Viscous flow in an expanding pipe . . . . . . . . . . . . . . . .
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in an
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Epilogue
217
Bibliography
219
Samenvatting
227
About the author
229
iv
Contents
Prologue
Rotation is an essential ingredient in many industrial processes: mixing, separation,
stabilisation, etc. On the other hand, in certain circumstances rotation is an (inevitable) by-product causing much damage and financial loss: temperature-differences
between ocean and atmosphere leading to thunderstorms and tornado’s, vortices generated by wings of large aero-planes leading to delay during landing-procedures.
In order to utilise the positive effects of rotation and to limit its harmful effects,
science tries to understand and describe the precise mechanisms at work. This thesis
focusses on swirling flows in (slowly expanding) pipes.
The phenomena that can occur in this particular geometry have served the last thirtyfive years as a model for a phenomenon that has its analogies in various branches of
physics: vortex-breakdown. Depending on interest/taste it can best be described as
either a sudden thickening of an otherwise thin ink-streak in a rotating fluid (see figure
1.3) or as the manifestation of a differential equation with a (slowly) varying parameter, passing through bifurcation. Associated with the breakdown the occurrence of
a recirculation-zone is observed and it is this zone that has its direct application in
industrial burners where it is used to stabilise flames and to separate different regions
in the combustion chamber in order to decrease the formation and emission of NOx .
We give an account of what has been stated and conjectured about this intriguing
phenomenon in the first part of chapter 1. We will use sometimes terminology that
is either assumed known to the reader but otherwise a footnote refers to a relevant
part of the thesis or the literature.
The rest of chapter 1 sets the mathematical background for the problem, starting of
course with (deriving and) stating the governing equations, the incompressible Euler
equations. However, the perspective in this thesis is such that we focus on a specific
property of the Euler equations, namely its Hamiltonian (or Poisson) structure. This
concept is well-known in the context of classical mechanics, but only with the work
of Arnol’d in the sixties one has tried to restate and understand various problems
in fluid dynamics in terms of the Hamiltonian nature of the underlying equations:
several examples are given in chapter 1.
Concerning swirling flows in a pipe, previous analytical investigations ended at columnar flows (velocity profiles only depending on the radial coordinate) or at most steady
axisymmetric flows. The resulting Poisson structure has infinitely many conserved
vi
Prologue
quantities (cf. the situation in 2D inviscid flow), while 3D Eulerian flow has at most
7. In section 1.2 we treat various topics in Hamiltonian theory, with direct application
to the Euler equations. As a result we formulate a constrained critical point problem that will act as a source of special swirling flows, the so-called relative equilibria.
These relative equilibria are intrinsically static, but they can be related to dynamic,
time-dependent swirling flows by exploiting the Hamiltonian structure of the Euler
equations!
The topic of non-steady, non-columnar swirling flows is one of main lines of this thesis.
Whereas travelling and rotating waves have their direct interpretations in the field of
channel flow and 2D flows, their (practical) relevance is not immediate clear in pipe
flow. Note that in a non-columnar, non-axisymmetric flow it is the velocity field that
rotates—the particles do already rotate for an arbitrary columnar swirling flow.
In chapter 2 we show that all these special solutions essentially consist of three parts,
namely (i) a rigid body rotation, (ii) a uniform translation and (iii) a so-called Beltrami flow v, with the property that v and curl v are aligned:
v = β curl v,
β a real number.
Beltrami flows have received quite some attention in the last few decades, in particular
in relation to a conserved quantity, the helicity:
Z
helicity =
u · curl u.
D
It was observed that the presence of Beltrami flows tends to decrease the nonlinear interactions between (Fourier) scales in turbulence, and that it slowed down the
energy-dissipation. This last property of Beltrami flows appears when we discuss the
temporal stability of swirling flows in chapter 3. A classical result shows that the
(inviscid) evolution of the (volume-averaged) energy of a perturbation u of a certain
columnar swirling flow U is given by
1 ∂
hh|u|2 ii = −hhU · (u × curl u)ii,
2 ∂t
(1)
where hh·ii denotes the volume-averaged quantity. A standard remark is that irrotational disturbances (curl u = 0) do not produce nor dissipate energy, but it is easily
observed that when u is a pure Beltrami flow the right-hand side of (1) vanishes as
well.
In chapter 3 we encounter the travelling waves as well, when we discuss the linear
stability of columnar swirling flows (section 3.1). It turns out that they represent the
slowest and fastest infinitesimal travelling wave that can be superposed on a columnar
relative equilibrium. Especially the slowest wave is interesting, and in particular its
speed. When this speed is larger than zero, every (infinitesimal) disturbance can only
travel downstream, but when this speed is smaller than zero, a disturbance can travel
in all directions and is ‘smeared out’. This situation is quite similar to open-channel
flow and high-speed compressible flow, and the two cases described above are referred
to as supercritical and subcritical, respectively. The transition from supercritical to
vii
subcritical is called hydraulic jump or shock, depending on the context, and in swirling
pipe flow it has been called vortex-breakdown. This transition occurs precisely when
the slowest travelling wave has speed zero, and since we know this speed explicitly we
know exactly when something interesting is happening.
In chapter 4 we finally discuss the non-axisymmetric rotating swirling flows. In particular, we are able to relate for these special flows the angular frequency to the radius
of the pipe and the (integrated) angular momentum:
angular frequency =
2 × angular momentum
.
(radius pipe)2
(2)
A non-axisymmetric velocity profile that slowly rotates is called a precessing vortex
core and it is believed to play a significant role in flame stabilisation processes. At
this moment we are aware of only two well-controlled experiments that can produce
a precessing vortex core (see section 1.5), but the resulting angular frequencies differ
by 4 orders of magnitude! However, we show in section 4.1 that the relation in (2),
applied to both experiments reproduces in both cases the measurements rather well.
This illustrates an important philosophy behind this thesis namely that one should try
to define parameters, but also flows, “coherent structures” in terms of global (read:
integrated) quantities that have a clear physical meaning.
Now we have discussed the non-steady non-columnar swirling flows, we pick up a
second main line namely (slow) deformation of columnar relative equilibria. This
deformation can be caused by a changing cross-section (chapter 4) or viscous effects
at the boundary (chapter 5). The goal is to describe the transition from supercritical
to (nearly) critical in a slowly expanding pipe and the influence of viscosity on this
process, and on swirling flows in general. Starting point is the collection of columnar
relative equilibria that we construct in chapter 2. The velocity profiles contain several
parameters, corresponding to physical (conserved) quantities like the integrated flux,
angular momentum and helicity. If we take the expanding pipe as an example, we
assume that the axial variation is slow (adiabatic), and that we can write at every
cross-section the velocity profile as a columnar relative equilibrium with parameters
suitably adjusted, plus an error. In this construction the axial variable plays the role
of dynamic (time-like) variable; the approximation is called quasi-homogeneous. In
section 1.3 we show that this is the cornerstone of many approximation methods.
The Hamiltonian structure of the Euler equations is used to construct the parameter
dynamics. To be precise, the conserved quantities correspond to local conservation
laws, and by integrating these conservation laws over a cross-section we obtain axially
conserved quantities, like the angular momentum flux, the flow-force, the mass-flux,
etc. The requirement that the quasi-homogeneous approximation satisfies these integrated conservation laws yields the desired parameter-dynamics.
But what about the error, i.e., the difference between the approximation and the exact
solution? At this point we have to address the issue of how the exact solution looks
like: is it steady, is it axisymmetric, both, or neither of the two? The problem is that
there are few (if any) statements that guarantee the existence of a unique solution
of the 3D Euler equations. But as already mentioned, until now most emphasis has
viii
Prologue
been put on steady, axisymmetric flows, and in section 4.4 we carry out a comparison between classical theory and the quasi-homogeneous approximation, under the
assumption that a steady, axisymmetric solution exists.
Finally, in chapter 5 we discuss viscous effects, leading to boundary layers along the
wall of the pipe. A boundary layer can be viewed as a thin region where the flow
is decelerated, and it is this region where viscous effects are important, whereas the
flow in the core of the pipe can be viewed as mainly inviscid. By means of a detailed
order-of-magnitude analysis, we are able to model the effect of the boundary layer on
the ‘conservation’ laws (from now on, balance laws). The observation that (turbulent)
flows in pipes are non-dissipative in the core region and that inlet conditions have a
large effect on the downstream flow in this region is a motivation to use the same
approach as in the (inviscid) expanding pipe problem. This time we require the
quasi-homogeneous approximation to satisfy the integrated balance laws, yielding a
description of how the core flow changes due to viscous effects at the boundary. Also
in this chapter the use of integrated quantities is advocated, thereby linking up with
existing ideas in turbulence theory. An interesting hypothesis in that field is the
influence of Beltrami flows on the decay of energy (the “helicity hypothesis”) and
we will show how in laminar pipe-flow this hypothesis gains credit. In addition the
combination of viscous effects and variations in cross-section is interesting.
The conclusions should remain limited, since the boundary layer grows rather quickly,
leading to separation. Hence, any progress in this case necessitates more refined
techniques, such as interactive boundary-layer coupling.
How to read this thesis
Every chapter, except the first one, starts with a relative long introduction that can
be read on its own. These introductions give a (short) preview of the contents of the
specific chapter and state the main results. When a section is rather involved, effort
has been made to present the spirit of the section at the beginning. Especially the
first chapter contains several of these “Intermezzo’s” that can serve as a refresher or
as an illustration of the main ideas.
1
Introduction
1.1
Fluid dynamical aspects of swirling flows and
helicity
Many examples in everyday life show that rotation or a swirling motion is an important mechanism with positive or negative/disastrous effects: the stirring in a cup of
tea (mixing), the uncorking of a bottle of wine (dynamic versus static resistance),
the cleaning of polluted air in hydrocyclon separators, tornados, etc. From the list
of numerous industrial applications of swirling flows, we explicitly mention the use
of swirling flows to stabilise flames in combustion chambers (Escudier (1988)). As a
side-effect, the formation of NOx , an important cause of acid rain, is greatly reduced
(figure 1.1). These applications of swirling flows but also their harmful effects in
combustion zone
reducing
of
volatile
matter
zone
producing zone of
reducing species
oxidising
zone
Figure 1.1: Low NOx combustion concept (Rosendal (1993)).
2
1 Introduction
e.g. aircraft aerodynamics has led to scientific interest to understand and control the
dynamics of swirling flows.
In the first part of this section we give an overview of a very intriguing phenomenon
that is often observed in swirling flows and that is called ‘vortex breakdown’. With
the term ‘vortex’ we refer to a rotating structure in general. This can be a swirling
flow in a tube but also an area of low pressure. A vortex-core is that part of the
vortex where most of the vorticity (= curl of the velocity field) is concentrated.
The overview is more or less chronologically ordered but sometimes we anticipate on
later developments. Moreover, we do not strive completeness but try to indicate the
important developments and questions that somehow relate to the questions that are
considered in this thesis. In the second part we introduce a quantity, the helicity,
that only recently has been ‘linked’ to the vortex breakdown phenomena (Moffatt
and Tsinober (1992)). The helicity measures more or less the complexity of the fluid,
and especially the knottedness of the vorticity lines. Its history and introduction in
the fluid dynamics world dates back to the sixties (Moreau (1961), Moffatt (1969)).
We will briefly review some of its properties in subsection 1.1.2.
1.1.1
Vortex breakdown: an overview
In 1957, Peckham and Atkinson did some tests on a wing in a wind tunnel and they
observed that when the angle of attack exceeded a certain critical value, vortices generated from the wing suddenly ‘bellied’ out (figure 1.2). Since then this phenomenon
Figure 1.2: ‘Bursting’ of vortices in water. The two streaks of dye are at the centre of
strong vortices formed through the roll-up of the shear layer separation from the leading
edges of a delta wing (Lambourne and Bryer (1961)).
is called (vortex) breakdown, referring to the apparent collapse of the vortex (core).
This did not mean that the phenomenon as such was understood. Related to that, the
terminology was based on observations; later on characterisations of the term ‘vortex
breakdown’ were based on a particular explanation of the phenomenon.
1.1 Fluid dynamical aspects of swirling flows and helicity
3
The first experiments
Harvey (1962) succeeded in isolating the vortex from the wing by studying rotating
air, pushed through a tube. And also in this bounded environment he observed
vortex breakdown. Moreover, the critical angle of attack in the wing-experiment had
as counterpart in the tube a critical swirl number Γ, sometimes referred to as the
Squire number (1960), being the ratio between the (maximal) azimuthal velocity and
the axial velocity: Γ ∼ uϕ /uz . When the swirl number exceeded Γcrit , a bubble was
formed, moving upstream when Γ was increased and eventually, when Γ was increased
even more, the whole tube was filled with a core region with reversed flow. Moreover,
it was observed that the flow did not change and was organised for a long time (see
figure 1.3). In the early seventies, Sarpkaya (1971a,b) published the results of his
Figure 1.3: Photograph of nearly axisymmetric vortex breakdown in a tube (Sarpkaya
(1971a)).
measurements that confirmed Harvey’s findings and showed in addition that there
exist several types of vortex breakdown. These ranged from double helix and spiral
(later on called S-type) to an axisymmetric bubble (called B-type). The occurrence
of one of these types depends on the swirl number as well as on the Reynolds number 1
Re, and for a wide range of combinations (Γ, Re) transitions between different types
were observed. These and other experiments formed a source of inspiration for the
theoreticians and at the same time confirmed or undermined earlier explanations of
the phenomenon.
The first models of vortex breakdown
Hall (1972) reviewed the explanations and models that were proposed to explain the
vortex breakdown phenomenon. The different ideas could be categorised as follows
(Hall (1972)):
(i) The phenomenon is intimitally related to the existence of a critical state (e.g.
Squire (1960), Benjamin (1962, 1967) and Bossel (1967)).
1 See
e.g. section 1.4.
4
1 Introduction
(ii) The phenomenon resembles the separation of a two-dimensional boundary layer
(e.g. Gartshore (1962), and Hall (1967)).
(iii) The phenomenon is caused by hydrodynamical instability (e.g. Ludwieg (1962)).
Hall subsequently treats the pros and cons of the different proposals, thereby focussing
on the degree of explanatory power of each explanation with respect to the observed
phenomena. Although (parts of) each proposal was a guide line in the later decades,
especially the first one involving the notion of critical state drew a lot of attention and
we will highlight this explanation. We will define the notion of critical state following
Benjamin.
Definition of critical state Let u = (ur , uϕ , uz ) be the velocity field of an incompressible swirling flow in a tube, with (r, ϕ, z) cylindrical coordinates and p the
pressure; the density ρ is set to unity. Assume the fluid to be inviscid, and the flow to
be steady and axisymmetric. The spatial coordinates are scaled with the radius of the
pipe, so e.g. 0 ≤ r ≤ 1. It can be shown (see e.g. Batchelor (1967) for a derivation)
that in the absence of external forces the equations of motion (Euler equations) may
be reduced to a single (elliptic partial differential) equation for the stream function
ψ,
dI
dH
∂ 2 ψ ∂ 2 ψ 1 ∂ψ
+
−
= r2
−
,
2
2
∂z
∂r
r ∂r
dψ
dψ
(BHE)
where ur and uz are related to ψ according to ur = −r−1 ∂ψ/∂z, uz = r−1 ∂ψ/∂r.
The equation connects the azimuthal component of the vorticity ω := curl u to the
total head H and the quadratic circulation I. BHE is called the Bragg-Hawthorne
(1950) or Long-Squire (1953, 1956) equation. In BHE, the total head H = p + 12 |u|2
and the quadratic circulation 2I = (ruϕ )2 are functions of ψ alone, determined from
upstream conditions. The replacement of the Euler equations by BHE is justifiable at
all points in a steady, inviscid and axisymmetric flow with the exception of stagnation
points (ur = uz = 0) and re-circulation zones. The pressure p has disappeared as a
dynamical variable, essentially because in the derivation the curl of the momentum
equation is taken, eliminating all gradients.
Consider a primary flow, independent of z, represented by the stream function Ψ(r)
and suppose that a small steady disturbance is superposed on it that does not disturb
the axial flux,
ψ(r, z) = Ψ(r) + εφ(r)eimz ,
|ε| 1,
(1.1.1)
with φ(0) = φ(1) = 0. Substituting (1.1.1) into BHE and truncating the result at
order ε leads to a linear eigenvalue problem for φ:
2Uϕ dUϕ
Uϕ
d2 φ 1 dφ
1 d2 Uz
1 dUz
2
−
+
−m
−
−
+
+
φ = 0, (1.1.2)
dr2
r dr
Uz
dr2
r dr
rUz2
dr
r
1.1 Fluid dynamical aspects of swirling flows and helicity
5
with Uz and Uϕ related to Ψ(r) according to
Uz =
1 dΨ
,
r dr
Uϕ =
1p
2I(Ψ(r)).
r
This is a classical Sturm-Liouville problem for φ with eigenvalue m2 . The theory
proves the existence of an infinite sequence of real eigenvalues, m20 > m21 > m22 >
. . . → −∞ (Ince (1926)).
Definition If m20 is positive, the primary flow is said to be subcritical : it can support
an infinitesimal steady disturbance in the form of a sinusoidal wave with wavelength
2π/m0 . If all eigenvalues m2 are negative the primary flow is called supercritical.
The condition m0 = 0 marks the boundary between the two and the corresponding
eigenfunction is called the critical state.
This definition clearly distinguishes between a supercritical and subcritical state based
on the ability of sustaining standing waves. It is also possible to give an alternative
definition, by defining a parameter, say N (like the Froude number for open-channel
flows), which will be greater or smaller than unity as the flow is supercritical or
subcritical (see e.g. Benjamin (1962)), respectively. This distinction is based on the
existence of infinitesimal travelling waves, superposed on the primary flow:
ψ(r, z, t) = Ψ(r) + εφ(r)eim(z−ct) .
The resulting eigenvalue problem for φ is (1.1.2) with Uz replaced by Uz − c. Then a
second way to discriminate between a supercritical and a subcritical flow is given in
the following definition (Benjamin (1962)).
Definition Let c+ and c− denote the absolute velocities (according to a laboratory
frame), measured positively in the direction of the flow, at which waves propagate
respectively with and against the flow. c+ is necessarily positive, but c− can have
either sign, being positive when the convective action of the flow on the waves outweighs the relative propagation (the supercritical case) and negative when the waves
can progress upstream against the flow (the subcritical case). Accordingly, we define
N=
c+ + c−
c+ − c−
as a universal characteristic parameter 2 , with N > 1 specifying supercritical conditions and N < 1 subcritical conditions:
c+ + c− > c+ − c−
⇔ ( c− > 0 ) .
Interpretations of the critical state One of the first interpretations of vortex
breakdown using the notion of critical state was proposed by Squire (1960). He
looked for subcritical velocity profiles (0, Uϕ , Uz ) because as he stated “the possible
2 In
the case of an open-channel flow this parameter coincides with the Froude number.
6
1 Introduction
existence of standing waves in the field of flow” (first definition) implies that “disturbances which are generally present downstream, will spread forward (upstream)
along the vortex” (second definition) “and cause breakdown”. With other words,
Squire considered the switch from supercritical to subcritical as a cause for vortex
breakdown. Comparing the computed velocity profiles that were just on the criticality boundary he proposed as a criterion for breakdown that the maximum azimuthal
velocity is “rather larger” than the axial velocity (resulting in the definition of the
Squire number Γcrit we mentioned earlier). Later on, the ratio between the axial and
azimuthal vorticity was used as an important number in relation with breakdown,
and more specific with the stability of swirling flows (see below).
On the other hand, Benjamin (1962, 1967) proposed an analogy with the hydraulic
jump in open-channel flow, where a transition occurs between two (steady) states.
This transition is from a supercritical to a subcritical flow. To support his ideas Benjamin introduced several concepts that played a role in the hydraulic jump, thereby
strengthening the analogy.
Definition A pair of z-independent flows ψA (r) and ψB (r) is called conjugate when
both are solutions of BHE:
dH
dI
∂ 2 ψ 1 ∂ψ
−
= r2
−
,
∂r2
r ∂r
dψ
dψ
for one and the same pair (H, I), being functions of ψ alone and determined from
upstream conditions, with boundary conditions ψA (0) = ψB (0) = 0, ψA (1) = ψB (1)
provided the solution curves ψA (r) and ψB (r) do not intersect in the (r, ψ)-plane
except at their endpoints.
Remark Fraenkel (1967) gave rigourous mathematical arguments concerning the
existence of conjugate flows, using the theory of ordinary differential equations. 3
Benjamin showed that when flow A is supercritical the conjugate flow B is subcritical.
Another difference is the value that the flow force or momentum flux,
Z 1
S = 2π
p + u2z rdr,
(1.1.3)
0
attains in both flows: S(ψA ) < S(ψB ). Since in a pipe (with constant diameter) S is
a conserved quantity in the axial direction, i.e.,
∂
S=0
∂z
when ∂t u = 0,
Benjamin postulated (1962) that a transition from ψA to ψB was possible when small
standing waves were superposed on the (subcritical!) solution ψB . These standing
waves decreased the value of the flow force S(ψB ) slightly, and hence, this scenario
makes only sense when the difference ∆ = S(ψB ) − S(ψA ) is small3 . In 1967 he
3 Meaning that the difference ∆ is small, requires the state of flow A to be close to critical, because
only then the conjugate flow B differs little from A. The distance from criticality is measured by
the distance of N from unity.
7
1.1 Fluid dynamical aspects of swirling flows and helicity
established such a transition from ψA via ψB back to ψA using the conservation of S,
without the formation of steady periodic waves. The perturbation of the supercritical
flow ψA turned out to be a standing solitary wave4;
√
ψ(r, z) = ψA (r) + εf ( εz)χ(r),
with
εχ(r) = ψB − ψA ,
f (Z) =
√
3/2
o , Z = εz
np
cosh2
3D/2CZ/2
and C and D determined by ψA and ψB . It was argued that a small amount of
dissipation would lead to a wave-train (radiation) instead of a soliton. Hence, unlike
Squire, Benjamin suggested that the existence of waves does not lead to breakdown
but that the waves merely represent breakdown.
A third interpretation of vortex breakdown, using the concept of critical state was
given by Bossel (1967, 1969). He suggested (1969) that vortex breakdown is inevitable
when the flow is supercritical but close to critical, and has a high swirl. To support
this, he solved BHE with a rigid body rotation as inlet condition (so H is linear, and I
quadratic in ψ; see section 2.4) and a suitable downstream distribution ψdown (r) that
provided information about H and I within the (necessarily) half-open re-circulation
bubble (see figure 1.4). However, as Hall (1972) remarks, Bossel (1967) gives exam-
ψup
re-circulation bubble
ψdown
Figure 1.4: Formation of bubble (schematically). In order to determine the flow inside the
re-circulation bubble, ψdown has to be prescribed on the dashed line, while ψup (prescribed
at the dotted line) determines the flow around the bubble and its contour.
ples showing that whether breakdown occurs or not, depends very sensitively on the
choice for ψdown (r), and hence Bossel’s original (1967) suggestion, stating that nearcriticality is a necessary condition for breakdown is much safer. But this is close to
Squire’s (1960) interpretation.
4 This situation can be compared with supercritical flows along horizontal open channels (Benjamin
and Lighthill (1954)).
8
1 Introduction
Shortcomings of the interpretations All interpretations had their strong points
and their weaknesses. Squire’s explanation is rather a posteriori, in the sense that no
suggestion is offered as to when and where the critical state is reached. Moreover, the
abrupt change in core structure is left unexplained. The proposal by Benjamin leaves
a gap between the small perturbations of the supercritical state (theory) and the
‘physical preference’ (Hall (1972)) for large non-dissipative perturbations. Something
that is lacking from the beginning is the possible existence of non-axisymmetric modes:
BHE is an axisymmetric model.
The other proposals The non-axisymmetric modes played a profound role in the
proposal put forward by Ludwieg (1962, 1965). This proposal emphasizes the stability properties of the swirling flow, especially with respect to non-axisymmetric
disturbances. Ludwieg’s criterion (1962), that provides a sufficient condition for instability of an inviscid swirling flow to non-axisymmetric disturbances, was derived
for flows in a narrow annulus. But unfortunately, the criterion turned out to be of no
importance for swirling flows in a pipe, although the idea as such remained important.
Finally, the analogy with boundary layer separation due to an adverse pressure gradient emphasizes the occurrence of a stagnation point in the velocity field. This
causes the failure of the underlying equations, being the quasi-cylindrical equations
(abbreviated with QCA) and Prandtl’s equations 5 The former are derived from the
Navier-Stokes equations6 for steady (axisymmetric) flow under the assumption that
the axial gradients are small compared with the radial ones (e.g. Hall (1966)). The
main (mathematical) resemblance between the QCA and Prandtl’s equations is that
both are of parabolic type. Hall (1967) performed numerical calculations and established a region of decelerating flow, which eventually lead to a stagnation point beyond
which the calculations could not be continued. By varying the inlet conditions Hall
was able to produce qualitative agreement with some of the experiments. However,
the existence of a certain critical value of the swirl number, and the abruptness of the
breakdown were left unexplained. Moreover, the description of the flow downstream
of the stagnation point was lacking.
Concluding, one can say that during the first decade since the ‘discovery’ of vortex breakdown, people were looking for the mechanism, thereby focussing mainly to
inviscid axisymmetric flows and were trying to identify special up and downstream
running flows, together with possible transitions.
Structure of vortex breakdown
The experiments by e.g. Sarpkaya (1974) and Faler and Leibovich (1978) revealed
more of the structure of the vortex breakdown, and the effects of several external
parameters were studied systematically. Since the first experiments by Sarpkaya
(1971a,b), most measurements were obtained within tubes, and very often using water
(with the danger of invoking cavitation in the core centre (Hoeijmakers (1997)). Adverse pressure gradients, especially important in the explanation by Hall (boundary
5 Chapter
6 Section
5.
1.4.
9
1.1 Fluid dynamical aspects of swirling flows and helicity
layer separation analogy) were imposed by slowly expanding walls, thereby decelerating (part of) the flow. Sarpkaya (1974) performed systematic measurements to study
the effect of a pressure gradient on the position of the breakdown, and considered also
variations in swirl and flow rate.
Extensive data-analysis (Faler and Leibovich (1978)) showed that the flows upstream
of the vortex breakdown were fitted well by a combination of Burgers’ vortex and a
jet-like profile (the combination was coined “Q-vortex” by Leibovich (1984); see figure
1.5):
−1/2
∼ α-
6
uϕ
uz
∼ Qα
-r
slope ∼ Qα
√
∼Q α
1
ωz
√
∼ w2 α
?
?
-r
ωϕ
1
Figure 1.5: The Q-vortex according to (1.1.4). The different ‘length’ scales are indicated
in the picture. At the left the azimuthal and axial velocity, at the right the corresponding
vorticity components.
uz = w1 + w2 e−αr ,
2
uϕ =
2
Q
(1 − e−αr ),
r
ur = 0.
(1.1.4)
Near the re-circulation zone, the axial velocity profile showed a rapid fall. Moreover, the profiles upstream of the stagnation point were laminar and steady (Faler
and Leibovich (1978)). However, the flow field in the re-circulation area itself was
always unsteady and low-frequency fluctuations were observed. Lessen et al. (1974)
suggested that these oscillations arose from an instability of the mean flow and they
investigated the stability of profiles of the form (1.1.4) (with arbitrary but fixed parameters) for non-axisymmetric normal modes of the form φ(r) exp[i(κz + mϕ − ωt)].
Howard and Gupta (1962) had investigated the stability with respect to axisymmetric
disturbances. All results appeared to depend on a single parameter q, being defined
for the vortex in (1.1.4) as
√
q = Q α/w2 ;
(1.1.5)
10
1 Introduction
q is a measure for the ratio of axial vorticity ωz to the azimuthal vorticity ωϕ in the
core of the vortex7 . Stability for axisymmetric disturbances was assured when q > 0.4
and for all infinitesimal disturbances provided q > 1.5.
(Linear) stability criteria for swirling flows
The experiments revealed the importance of non-axisymmetric descriptions of vortex
breakdown. And also the instability-question raised by Ludwieg (1962) started to
press. As for the stability results, in the early eighties a fairly complete list of stability results of general nature was available. It consisted of necessary and sufficient
conditions for instability and/or stability of inviscid pipe flows. We list the main
results below.
(i) Rayleigh (1880) proved that a pure swirling flow (no axial velocity) is linearly
stable to axisymmetric perturbations if (sufficient)
Rayleigh discriminant Φ := r−3
d
(rUϕ )2 > 0.
dr
(1.1.6)
Synge (1933) showed that (1.1.6) is also a necessary condition for axisymmetric
stability. This result can be extended in two ways.
(ii) Rayleigh (1880) and Fjortoft (1950) gave as a necessary condition for instability
of a swirling flow to non-axisymmetric two-dimensional perturbations (only ded
pending on r and ϕ) that the axial vorticity Ωz = r−1 dr
rUϕ has an extremum
in the interior of the fluid, i.e.,
dΩz
changes sign at least once in 0 < r < 1.
dr
(1.1.7)
Fjortoft refined (1.1.7) by proving that a necessary condition for instability of a
swirling flow to two-dimensional perturbations is that
!
Uϕ
Uϕ dUϕ
−
<0
r
r r=r
dr
∗
somewhere in the field of flow, where r∗ is a point at which
d
dr Ωz
vanishes.
(iii) Howard and Gupta (1962) extended the result in (1.1.6) to swirling flows with
an axial component. They showed that if
J :=
Φ
1
>
2
(dUz /dr)
4
then the flow was stable to axisymmetric disturbances.
7 Note
that w1 does not play a role in the definition of q, whereas it is important in the definition
of the Squire or swirl number. A proper way to define q is by taking the ratio of the integrated axial
vorticity to the integrated azimuthal vorticity:
Z
qglobal =
Z
1
0
ωz rdr
Z
1
0
ωϕ rdr
1
=
0
1 d(ruϕ )
rdr
r dr
Z 1 du
z
−
0
dr
rdr .
1.1 Fluid dynamical aspects of swirling flows and helicity
11
(iv) And the results were completed by Leibovich and Stewartson (1983), who stated
a sufficient condition for the instability of an arbitrary columnar (only depending
on r) swirling flow, being
" 2 #
dUz
d Uϕ
d Uϕ d
(rUϕ ) +
<0
Uϕ
dr
r
dr
r
dr
dr
at any point of the flow field.
(v) Another result, not related to any of the previous criteria, is given by Maslowe
and Stewartson (1982), stating that a sufficient condition for instability is
Uϕ (1) 6= 0,
dUz
(1) 6= 0.
dr
Wave models In a series of papers (Randall and Leibovich (1973), Leibovich and
Ma (1983)) Leibovich and coworkers considered perturbations of a primary flow Ψ(r)
(not necessarily a Q-vortex) that incorporated axisymmetric as well as non-axisymmetric travelling/rotating waves (Leibovich (1984)):
1 dΨ
kz − ωt
uz (r) =
+ εA(z, t)w0 (r) + εs S(z, t)w1 (r) exp i
exp [−iϕ] + c.c. ,
r dr
δ
where ε, εs and δ are small parameters and c.c. stands for complex conjugate. If εs = 0
it was shown (Randall and Leibovich (1973)) that A(z, t) satisfied the Korteweg-De
Vries equation with parameters determined by the unperturbed flow, Ψ. The function
S describes modulations of the non-axisymmetric mode, and is allowed to vary slowly
compared to the spatial scale O(δ/k) and the temporal scale O(δ/ω)8 . If ε = 0
soliton solutions for S are possible, described by the nonlinear Schrödinger equation
(Leibovich and Ma (1983)). When both A and S are present the stability properties of
the primary flow become increasingly important. Leibovich (1984) speculates about
possible scenarios, leading him to the conjecture that from the inviscid theory the Stype (non-axisymmetric) appears to be unstable at lower value of q (see (1.1.5)) and
that the B-type (axisymmetric) emerges from the S-type as a secondary bifurcation.
Concluding we can say that the results from experiments9 forced people to look at
non-axisymmetric models. The relevance of axisymmetric equations, like BHE, for
these non-axisymmetric models was difficult to decide. Leibovich (1984) suggested
that the stream function in BHE should be viewed as the stream function for the
time-averaged flows downstream of the re-circulation bubble.
Bifurcation diagrams
We have paid attention to two of the three original proposals that were described
in Hall (1972), namely critical state and stability. Of course, every new development
8 Note
9 We
that A(z, t) is given in the original, ‘fast’ variables z and t.
pay specific attention to non-axisymmetric swirling flows in experiments in section 1.5.
12
1 Introduction
tried to incorporate previous results; ingredients from the three proposals were mixed,
since “the phenomenon of vortex breakdown is unlikely to be attributed to only
one physical mechanism” (Leibovich and Kribus (1990)). During the last decade
much attention has been paid to the occurrence of a singularity in the (linearised)
axisymmetric Euler equations or BHE, and possible bifurcations c.q. non-uniqueness
of solutions. This point was already touched upon by Mager (1972) and Trigub
(1985), but the systematic study of the bifurcation diagram started with Leibovich
and Kribus (1990) and later on studies of e.g. Beran and Culick (1992), Lopez (1994),
Buntine and Saffman (1995) and Wang and Rusak (1997) contributed much to the
understanding and unification of the results.
Although the kind of columnar flow, around which the equations were linearised, differed from paper to paper, the common factor was that all velocity profiles were fixed
except the swirl, that contained a parameter, related to the swirl (Squire) number,
as a multiplicative bifurcation parameter:
uz = f (r),
uϕ = γg(r), γ variable, f (r) and g(r) fixed.
We will shortly describe some specific features of each of the studies mentioned above
and collect the results in figure 1.6. Note that the first important point that can be
drawn in every figure is γcrit where the flow turns from supercritical into subcritical.
Leibovich and Kribus (1990) linearised BHE (implying that only steady disturbances
were considered) around the Burgers’ vortex (w2 = 0 in (1.1.4)) and found four
types of branches. One class of bifurcating flows was again columnar, and these
branches were identified as the conjugate flows. A second class consisted of solitary
waves, connecting the primary (supercritical) state with itself (homoclinic orbit). The
third class contained the periodic wave-trains (the period was fixed), branching off a
subcritical flow and finally a fourth class consisted of solitary waves that connected
the primary flow with a conjugate flow.
Beran and Culick (1992) used the steady and axisymmetric Navier-Stokes equations
with the Q-vortex as inlet profile. They found for large Reynolds numbers (Re ≈ 2000)
two limit points of swirl that connect three branches of steady state solutions of
different type (see figure 1.6).
Computations by Lopez (1994) confirmed the results of Beran and Culick (1992).
Since he used the unsteady (axisymmetric) Navier-Stokes equations, he was able to
determine the stability of the different branches in figure 1.6 with respect to axisymmetric disturbances. Moreover, his results suggested that the hysteresis in figure 1.6
existed only when the Reynolds number exceeded some critical value.
Buntine and Saffman (1995) examined the development of a steady and axisymmetric
inviscid swirling flow in a slowly diverging pipe of finite length. Using the Q-vortex
as inlet condition they studied the influence of various outlet conditions and pipe
geometries on the resulting velocity field. When at the outlet a Neumann condition
was applied to the stream function, i.e., ∂ψ/∂z, they found a limit point of swirl
where the branch folded. Changing the various parameters, Buntine and Saffman
(1995) observed that at the fold the flow was critical, i.e., the slowest infinitesimal
travelling wave has a vanishing speed—a result that was proven by Trigub (1985).
Moreover, they showed that criticality does not necessarily imply stagnation.
13
1.1 Fluid dynamical aspects of swirling flows and helicity
min uz (0, z)
min uz (0, z)
6
I
6
A
(a)
A
III
II
(b)
IV
B
-γ
(c)
-γ
Figure 1.6: At the left: schematic solution diagram, showing (partially) the results of
Leibovich and Kribus (1990): (I) trivial branch (primary flow); (II) principal conjugate
branch (columnar); (III) soliton wave branch; (IV) wave-train branch. At (A) the primary
flow is critical. At the right: a representative solution diagram, showing (partially) the results
of Beran and Culick (1992). Branch (a) and (c) are stable to axisymmetric disturbances and
(b) is unstable. (A) and (B) are the limit points; —, non-reversed flow, - - -, reversed flow.
Finally, Wang and Rusak (1997) study the solutions of BHE in a straight pipe, corresponding to critical points of the following functional,
)
2
2
Z z0 Z 1 ( 1 1 ∂ψ
1
1 ∂ψ
I(ψ)
E(ψ) =
+
−
+ H(ψ) − 2
rdrdz,
2 r ∂r
2
r ∂z
r
0
0
with z0 arbitrary but fixed. More specific, they are interested in the global minimiser
and its behaviour depending on the swirl number. Using three different profiles they
prove that (i) when γ is small, the global minimiser is unique and does not depend
on z, (ii) at γ ∗ another (local) minimiser, completely different from the global one
can be found, (iii) for γ ∗ < γ < γ0 , the columnar flow is still a global minimiser, but
two other critical points can be found (a min-max and a local minimum), (iv) at γ0
there are two global minimisers, one of them being columnar while the other has a
stagnation point at z = z0 , (v) for γ0 < γ < γb the columnar state is not a global
minimiser and (vi) at γ = γb the columnar state is no longer a local minimum and is
critical in the sense of Benjamin (1962). See figure 1.7.
Concluding remarks; main motivation for thesis
We have seen how in the last forty years people have struggled with explaining and
understanding a phenomenon that is related to swirling flows. The analysis and
synthesis of different possible mechanisms resulted in simplified models that tried to
capture important details of vortex breakdown. The reason for this low-dimensional
14
1 Introduction
Ε(ψ)
γ
b
γ
0
γ∗
γ < γ∗
ψ
ψ
0
Figure 1.7: The bifurcation diagram of solutions to the Bragg-Hawthorne equation (reproduced from Wang and Rusak (1997)).
approach is that at the moment it is almost impossible to solve the unsteady NavierStokes equations for realistic Reynolds numbers (≈ 106 –107 ), without turning to
turbulence modelling; and even with turbulence modelling the problem cannot be
resolved: recent studies (Parchen (1993) and Steenbergen (1995)) show that standard
models like k − ε and Reynolds-differential stress give different results when applied
to swirling flows, especially when it concerns the stability of the flow and the size
and position of the re-circulation area. Moreover, contrary to the recommendations
(of e.g. Abujelala and Lilly (1984) and Kitoh (1991)) to use second-order turbulence
models (like ReStress) for (topologically) complex flows, Parchen (1993) observed
that the results obtained with the k − ε model have much more resemblance to the
experimental measurements than the more advanced (second order) models.
It is interesting to note how in certain models like BHE and QCA the axial variable
is used as a dynamical (time-like) variable. However, we did not come across a study
that tried to make this dynamical character explicit, i.e., to rewrite for instance the
steady Euler equations as
∂
W = K(W )
∂z
with W a state-variable and K(W ) a vector field, not containing axial derivatives.
This situation does occur in the theory of water-waves (e.g. Amick and Kirchgässner
(1989) and Baesens and McKay (1992)). The approach of Benjamin (1967) is related
to these remarks, in the sense that he explicitly uses a quantity that is conserved in
axial direction, in order to construct a (weak) transition between different states—the
flow force S (see (1.1.3)). This transition can be triggered by a (slowly) varying geometry (Randall and Leibovich (1973)). However, the flow force is not the only (axially)
1.1 Fluid dynamical aspects of swirling flows and helicity
15
conserved quantity as will be shown in section 1.2, and we will pursue Benjamin’s
idea, thereby exploiting the so-called Poisson structure of the Euler equations. This
structure will (implicitly) determine the (consistent) way in which the various quantities (= functionals) will evolve in the axial direction, even when they are not axially
conserved anymore, for instance in an expanding pipe;
∂
F (u) = K̃i (u).
∂z i
(1.1.8)
The use of integral quantities to define swirling flows will give rise to interesting
predictions, some of which are checked against experimental results. In addition,
since integrated quantities are robust they are frequently used in turbulence, both in
experiments and for modelling purposes (Holmes et al (1996)).
One of these quantities that we believe is important for describing vortex breakdown
is the helicity. In the next subsection we (briefly) describe the role that helicity plays
in fluid dynamics.
1.1.2
Helicity and swirling flows
Consider a fluid flow in a domain D ⊂ R3 , bounded or unbounded and denote as
before the velocity field by u = u(x, t) and the vorticity field by ω = ω(x, t). The
helicity of this flow is the integrated scalar product of u and ω,
Z
1
B(t) =
u · ω dV.
2 D
Moreau (1961) and Moffatt (1969) distinguished independently the importance of this
quantity in the sense that B(t) is a dynamical invariant for the Euler equations and
that B admits a simple, kinematical interpretation when the vorticity field consists
of infinitesimal vortex tubes Ci with strength κi . Using Kelvin’s circulation theorem
(Thomson (1868), Batchelor (1967)), Moffatt (1969) showed that
Z
X
u · ω dV =
αij κi κj
(1.1.9)
D
i,j
with αij the linking (or winding) number between Ci and Cj . Moffatt and Tsinober
(1992) gave a very elegant decomposition of a continuous vorticity field that resembles
(1.1.9). If u = [u1 (x, y, z), u2 (x, y, z), u3 (x, y, z)], then we define
ω 1 = (0, ∂u1 /∂z, −∂u1/∂y)
and by cyclic permutation ω 2 and ω 3 , so that ω = ω 1 + ω 2 + ω 3 . The integral
curves or vortex lines corresponding to e.g. ω 1 are determined by the two constraints
{x =constant, u1 =constant} and so on. The helicity B is (summation over repeated
indices is assumed)
Z
Z
X
1
1
B=
Bij , Bij =
(ui ei ) · ωj dV =
(u e ) · ω i dV,
2 D
2 D j j
i6=j
16
1 Introduction
z
ω3
ω2
ω1
y
x
Figure 1.8: Decomposition of arbitrary vorticity fields into three linked fields, each of which
has trivial topology. After Moffatt and Tsinober (1992).
with Bij the ‘degree of linkage’ of the two vorticity fields ωi and ω j (see figure 1.8).
The dynamical invariance of helicity has special importance in the generation of an
ordered structure in a slightly viscous flow. In a very influential paper Hasegawa
(1985) described how dissipation rates of global quantities, conserved in the inviscid flow, may lead to different time-scales in the evolution of the velocity field. A
well-known example is two-dimensional incompressible flow where the kinetic energy
and the enstrophy are conserved in the inviscid problem. When little viscosity is
introduced the enstrophy dissipates much faster than the kinetic energy (Hasegawa
(1985), Van Groesen (1989); see Derks and Ratiu (1997) for an example in magnetohydrodynamics). Then, the flow field may relax to the minimum enstrophy state at a
given kinetic energy—the planar Taylor vortices. Moreover, these states are connected
with exact viscous solutions, that have the same structure for all t > 0!
The situation is much less clear in 3D viscous flow, i.e., it is not known whether the
helicity dissipates faster than the kinetic energy, or not. An additional problem is the
fact that the helicity is indefinite whereas the enstrophy as well as the kinetic energy
are definite. Note that the enstrophy involves (higher order) spatial derivatives, just
like the helicity, and hence one might argue by analogy that the helicity will dissipate
more quickly than the kinetic energy. However, numerical results show that there
is not a cascade of helicity to the smaller scales, like the energy (Kida and Takaoka
(1994), and references therein).
In the seventies and eighties much attention was paid to the influence of helicity on
the energy cascade (André and Lesieur (1977), Pelz et al. (1985)). The significance of
helicity, and in particular the “helicity-hypothesis” (Lilly (1986)) in turbulence and
more specifically, its role in generating coherent (large-scale) structures divided part
of the fluid dynamics community in different camps (Levich (1987)).
Of course, much more can be said about the role of helicity in for instance topological
fluid dynamics. However, we refer the interested readers to one of the many review
1.2 Structured equations and coherent structures
17
papers that appeared in the last decade, e.g. Levich (1987) and Moffatt and Tsinober
(1992). We close this section with a quote from this latter study, that may partly
serve as an extra motivation for this thesis:
“Qualitative changes in the topology of flows resulting from vortex breakdown are (...) presumably closely related to large changes in helicity,
which may therefore be an appropriate parameter for the characterisation
and classification of such flows.”
1.2
Structured equations and coherent structures
In the previous section we described the history of a rather new fluid dynamical problem, and we have seen that there is much evidence that the onset of vortex-breakdown
and related phenomena can be understood in the framework of an incompressible and
inviscid fluid. Strictly spoken, these kind of fluids are a mathematical entity, being
sometimes a good approximation of a real fluid. The mathematical model consists of
an evolution equation for the velocity field of the fluid, the Euler equations. The Euler
equations can be and have been interpreted in many ways, e.g. (i) a (straightforward)
application of Newton’s laws, (ii) part of a hierarchy related to the Boltzmann equation, describing collisions of particles and (iii) the geodesic flow on the Lie-algebra
of volume-preserving maps. Every interpretation has its own merits and can bring
forth new insights that were hard or impossible to obtain otherwise. In subsection
1.2.1 we will choose for a rather geometrical approach, based on the work by Arnol’d
(1966a,b) and the clear exposition in Marchioro and Pulverenti (1994) and Van Groesen and De Jager (1994). This approach is related to the Principle of Stationary
Action, that is well known in classical mechanics and that was advocated first by
P.L.M. de Maupertuis in 1744, although Fermat used the principle avant la lettre in
his study of optics. It turns out that with the help of this principle many equations
in Mathematical Physics can be understood. These equations define evolutions Φt
for which a functional called the action A attends a critical value with respect to all
possible and admissible evolutions, i.e., A is stationary in Φt .
The different concepts of evolution and action are explained in subsection 1.2.1; at
this point we want to spend some time on the notion of the variational derivative,
being important throughout the thesis, and related to the idea of stationarity.
Intermezzo: the variational derivative
Define for functions u(x), x ∈ D ⊂ Rn a density functional L,
Z
L(u) =
L[u](x) dx,
D
where L[u] may depend on x, u and (finitely many) derivatives of u with respect to x.
Our aim is to define in a sensible way the derivative of the functional L with respect to
variations in u. Similar to the differentiation of functions of several variables, we start
with introducing the directional derivative. Let u be an element of some subspace H
18
1 Introduction
of a function space. Then we define a line m in H through u (when H is not linear or
affine we need a curve; see figure 1.9) in the direction v, parameterised by ε:
m : ε 7→ u + εv.
Restricting the functional L to this line we obtain a function of one variable only:
@
@
1 @
@
@ u •
@
@
@
•
H
@
@
1 @
@
@
•
@
@
@
Tu H
1 εv
•
O
Figure 1.9: A tangent vector v to the manifold H at a point u defines a line that is tangent
to a curve on H; the set of such tangent vectors defines the tangent space Tu H at u.
ε 7→ L(u + εv).
Definition The first variation of a functional L at u in the direction v is denoted
by δL(u; v) and defined as
d
δL(u; v) :=
.
L(u + εv)
dε
ε=0
Remarks (i) When H is a differentiable manifold, v is an element from the tangent
space, Tu H, defined by
 
 ∃ curve ϕ : I ⊂ R → X s.t. 
d
Tu H := v .
ϕ(ε)
=v 
 ϕ(0) = u,
dε
ε=0
In the Calculus of Variations, Tu H is also called the space of admissible variations.
(ii) When L(u; v) is linear in v, it is also known as the Gateaux derivative. Moreover,
in that case, δL(u; v) can be identified as a linear functional on the tangent space
Tu H, i.e., an element of the co-tangent space Tu∗ H.
(iii) Usually, δL(u; v) can be rewritten by means of an integration by parts. This leads
to a combination of (in our case) an L2 (D)-inner product of v with some function,
denoted by δL(u) and a boundary term (Σ denotes the boundary of D):
Z
Z
δL(u; v) =
δL(u) · v dV +
δLb (u; v) dA.
D
Σ
1.2 Structured equations and coherent structures
19
δL(u) is called the variational derivative of L at u, and δLb (u; v) is the variational
boundary contribution.
3
When a linear approximation of L(u + εv) is sought, we conclude directly, on using
the concept of first variation, that
L(u + εv) = L(u) + εδL(u; v) + o(ε).
Definition A point û is called a critical or stationary point of the functional L on
the set H if
δL(u; v) = 0 for all v ∈ Tu H.
4
In subsection 1.2.1 we define a so-called action A, being a functional on the space
of incompressible evolutions. The two issues to be solved are the computation of δA
and a representation of the space of admissible variations.
It will lead to the conclusion that the material or convective derivative of the velocity
field u,
Du
∂u
:=
+ u · ∇u,
Dt
∂t
is (L2 −)orthogonal to the space of divergence-free functions,
Z
Du
(x) · η(x) dV = 0, for all η(x) with div η(x) = 0,
D Dt
(1.2.1)
with η · n = 0 on Σ as an additional restriction on η (n denotes the unit (outer)
normal on Σ). Using that η is orthogonal to gradients, i.e.,
Z
Z
I
∇φ · η(x) dV =
div (φ η(x)) dV =
φ η · n dA = 0,
D
D
Σ
(1.2.1) is turned into the Euler equations:
Du
∂u
=
+ u · ∇u = −∇p.
Dt
∂t
The next step is identifying the Euler equations as a generalised Hamiltonian system,
a Poisson system. Before we turn to the question of what a Poisson system is, it is
better first to spend some time on the why.
An important property of a Poisson system is that it has a natural conserved quantity,
the Hamiltonian H, usually related to the (physical) energy of the system. This means
that trajectories or solutions remain on a level set of this Hamiltonian. If this would
be all, there was no need to spend so much time on the matter any further. But
an old result by Kelvin, reformulated by Lax (1968) states that when a dynamical
system has a first integral, the critical points of this integral, or any combination with
20
1 Introduction
H, form a set S that is dynamically invariant. This invariant set can still be quite
complicated, consisting of isolated points where no dynamics is possible (and hence,
being true equilibria) and possibly connected components. At this point two questions
come up: how relevant is this invariant set S for the rest of the dynamics and what
is (if any) the dynamics in these connected components. The first question is related
to the dimension of the set and its stability properties. Although the dimension
of S is low, its elements turn up in many practical circumstances being ‘coherent
structures’ such as travelling waves and vortices. Due to the conservative nature of a
Poisson system, every equilibrium or critical point is either a saddle or a (nonlinear)
centre, and (specialistic) techniques have been developed to determine the nonlinear
stability of critical points in Poisson systems (e.g. Arnol’d (1966b, 1969), Holm et
al. (1985), Marsden et al (1989)). The second question can, in all its generality, only
be answered for Poisson systems. Observe that a connected component in S points
to a degeneracy in the system, or better, a continuous symmetry. This symmetry
is in a Poisson system connected to a first integral, I, and a local conservation law.
Moreover, the first integral I defines the flow ΦI that determines the evolution in
the connected component. Finally, before going into a bit more detail, we want to
emphasize the point of consistent modelling.
Many equations in mathematical physics have originally been derived using physical
motivations. Later on, people started to discover that many equations had a special
structure10 , very often had a Poisson structure (or a perturbation of it). Examples
are Maxwell’s equations, Schrödinger’s equations, the sine-Gordon equation, etc. But
regarding developments of either numerical schemes or (low-dimensional) models describing the main mechanisms behind a certain phenomenon, what do we retain from
this structure? It is often desirable to have a conservative (symplectic) integrator and
some techniques have been devised in order to attempt to conserve for instance mass
or energy exactly (finite volume/elements, Strauss-Vazquez technique for nonlinear
Klein-Gordon and sine-Gordon; for references see e.g. Strauss and Vazquez (1978),
Marsden (1992), and Van Groesen, Van Beckum, and Valkering (1990)). It is one of
the main ideas behind consistent or mathematical modelling to transfer the structure
of the full set of (continuous) equations to an approximation of it or a simpler model
based on it. See for several conservative and dissipative examples Fledderus and Van
Groesen (1996a).
Intermezzo: Poisson systems
Now that we motivated our choice for putting emphasis on the Poisson structure in
general and of the Euler equations in particular, we can explain some notions related
to Poisson systems.
Let D denote the physical domain (finite, semi-infinite or infinite) and let H be a
subspace of some function space (in our case of the Euler equations it is the space
of divergence free velocity fields with Dirichlet conditions on its normal component
and/or periodic boundary conditions). Finally, let F(H) be the class of all functionals
10 The ‘story’ of the KdV-equation may serve as an archetypical example; see e.g. the conference
proceedings of the KdV ’95 symposium (Hazewinkel et al. (1995)).
21
1.2 Structured equations and coherent structures
on H,
F(H) := {F : H → R} .
The evolution of functionals is now specified by introducing the concept of Poisson
bracket. From this evolution equation for functionals, an evolution equation for a
state variable (Euler equations: the velocity field) can be derived.
Definition Let H be a manifold and let F(H) denote a class of functionals on H.
Consider a map (the Poisson bracket) from a pair of functionals to another functional,
{ , } : F(H) × F(H) → F(H).
The pair (H, { , }) is called a Poisson manifold if { , } satisfies the following properties.
(i) skew-symmetry for all F, G ∈ F(X):
{F, G} = −{G, F};
(ii) bilinearity for all F1 , F2 , G ∈ F(H) and numbers α1 , α2 ∈ R:
{α1 F1 + α2 F2 , G} = α1 {F1 , G} + α2 {F2 , G};
(iii) Jacobi’s identity for all F, G, H ∈ F(H):
{{F, G}, H} + {{H, F}, G} + {{G, H}, F} = 0;
(iv) Leibniz’ rule for all F, G, H ∈ F(H):
{FG, H} = F{G, H} + {F, H}G.
Remark When { , } satisfies only the first three properties, (H, { , }) is called a
Lie-algebra (Marsden (1992)).
3
Definition A Poisson system is a dynamical system on a Poisson manifold when for
some functional H ∈ F(H) the functionals evolve according to
∂t F(u) = {F, H}(u) for all F ∈ F(H).
(1.2.2)
Remarks (i) The functional H is called the Hamiltonian of the system.
(ii) The skew-symmetry of { , } implies that H is constant during the evolution:
∂t H(u) = {H, H}(u) = 0.
(iii) Again, due to the skew-symmetry of { , }, a functional I is a first integral or
constant of the motion iff I Poisson-commutes with H:
{I, H} = {H, I} = 0.
(iv) Functionals C that commute with any F ∈ F(H),
{C, F} = 0,
are called distinguished (Olver (1988)) or Casimir functionals. Existence of Casimirs
points at a degeneracy of the Poisson bracket.
3
22
1 Introduction
Using the concept of variational derivative, tangent space and the linear functionals
on a tangent space (the co-tangent space) we start again at the definition of a Poisson
manifold. With property (ii) and (iv) it can be proven for finite 11 dimensional Poisson
systems that there exists a map Γ(u) from the co-tangent space to the tangent space
such that the (canonical12 ) Poisson bracket can be written as
{F, G}(u) = hδF(u), Γ(u)δG(u)i,
(1.2.3)
where h , i can be viewed as the L2 -inner product or (more geometrically) as the
duality map between the tangent and the co-tangent space. This map inherits all
kind of properties related to those of the Poisson bracket; in particular Γ(u) is skewsymmetric. It goes under several names, like structure map and co-symplectic map.
Using the formulation in (1.2.3) and rewriting ∂t F(u) as
∂t F(u) = hδF(u), ∂t ui,
the evolution equation (1.2.2) leads to the important result that
∂t F(u) − {F, H} = hδF(u), ∂t u − Γ(u)δH(u)i = 0
(1.2.4)
for all functionals F. Recall that the variational derivatives δF(u) are in the co-tangent
space Tu∗ H, and hence the evolution equation for u is found from
∂t u − Γ(u)δH(u) ⊥ Tu∗ H.
4
Note that (1.2.4) resembles (1.2.1); in subsection 1.2.2 we actually start off with
(1.2.1), define the Poisson bracket and the Hamiltonian and pay specific attention to
the boundary contributions. This results in an alternative (‘total head’) formulation
of the Euler equations,
∂t u + ω × u = −∇h,
with vorticity ω = curl u and total head h = p + 12 |u|2 . Moreover, when restricting
the class of flows to axisymmetric ones, we can make a consistent reduction of the
Poisson structure and formulate alternative equations with two state variables instead
of three, resembling the Bragg-Hawthorne equations.
Although we have defined the concept of Poisson systems and conserved quantities,
we still lack a notion of ‘special’ solution and their corresponding interesting dynamics
as we promised in the why of Poisson systems. This notion is briefly explained in the
following paragraph, while the specific details are deferred to subsection 1.2.3.
11 Although
a formal proof is lacking for the infinite dimensional case, in many examples the map
can be (relatively easy) constructed.
12 In case of boundary contributions to δ F(u; ), this definition of Γ is much more involved. See
Lewis et al. (1986) for background and section 1.2.2 for the example of the Euler equations in a finite
domain.
1.2 Structured equations and coherent structures
23
Intermezzo: relative equilibrium solutions
When dealing with a Poisson system with Hamiltonian H, the critical points û of H,
satisfying
δH(û) = 0,
correspond to equilibrium solutions, cf. (1.2.4). The dynamical invariance of the
collection of equilibrium solutions is immediate. The next step is considering, besides
H, a first integral, I. Since the value of I is constant during the evolution, we can
prescribe it and look for critical points û of H for a given value of I, a so-called
constrained critical point or relative equilibrium:
Crit {H | I = γ} .
u
(1.2.5)
Standard techniques in the Calculus of Variations learn that û is a solution of (1.2.5)
only when δH(u) is a multiple of δI(u),
δH(û) = λδI(û),
(1.2.6)
with λ the Lagrange multiplier. When I is a nontrivial constant of the motion (so
not a Casimir), the set of critical points is (in general) degenerate and dynamically
invariant and related to one of these connected components in S. The next question
concerns the actual dynamics in this component. To that end we define the concept
of flow(-map) or evolution:
Definition Consider a dynamical system with evolution equation
∂t u = K(u),
u|t=0 = u0 with u, u0 ∈ H;
(1.2.7)
K(u) is the vector field. Assume that for each u0 ∈ H there is a unique solution u(t),
with t in some neighbourhood of 0. The solution u(t) will be denoted by
u(t) = Φt (u0 );
Φt is called the flow-map or evolution.
In particular, when (1.2.7) is a Poisson system with Hamiltonian H, the flow-map
is denoted by ΦH
t , and is known as the so-called H-flow. Moreover, for an arbitrary
functional G, the G-flow is defined via the solution u(t) of (modulo existence and
uniqueness)
0 = hδF(u), ∂t u − Γ(u)δG(u)i for all F ∈ F(H),
⇒ u(t) = ΦG
t (u0 ).
u|t=0 = u0 ,
Remark The statement that I is a conserved functional of (1.2.2,1.2.4) can be
rephrased as I being invariant under the H-flow:
I(u0 ) = I(ΦH
t (u0 )).
But due to the symmetry in the commuting relation between H and I, i.e., {I, H} =
{H, I} = 0, we obtain that H is a conserved functional of ∂t u = Γ(u)δI(u), and hence
H is invariant under the I-flow, H(u0 ) = H(ΦIt (u0 )).
3
24
1 Introduction
We have then the following important result, stated in (Van Groesen and De Jager
(1994))
Proposition 1.1 (Relative equilibrium solutions) Let γ in (1.2.5) be such that
there exists a solution û, satisfying equation (1.2.6) with multiplier λ. Then the
application of the I-flow with an amount λt yields a solution of the Poisson system
(1.2.4):
I
ΦH
t (û) = Φλt (û) =: v̂.
The solution v̂ is called a relative equilibrium solution. Note that if the action of the
I-flow is trivial, e.g. I is a Casimir, this solution is time-independent.
4
In subsection 1.2.3 we compute the conserved quantities and a Casimir for the Euler
equations and the corresponding flows and conservation laws. Chapter 2 is completely
devoted to the determination of the relative equilibria.
1.2.1
A Lagrangian approach to the Euler equations
The displacement of a continuum of particles in a domain D ⊂ R3 can be viewed as a
map or evolution from D into itself. An evolution is called incompressible or volume
preserving when it leaves the (Lebesgue) volume of any subset D ⊂ D invariant. Besides this invariance condition, there are three other properties that should be satisfied
by the collection of all volume preserving evolutions, G. Note that these properties
are automatically guaranteed for any well-posed ordinary differential equation.
Assign to any number t ∈ R such an evolution, Φt (t plays the role of time). Then it
should hold that
(i) transitivity Φs (Φt (x0 )) = Φs+t (x0 ), for all x0 ∈ D and for all s, t ∈ R.
(ii) invertibility Φ−t (Φt (x0 )) = Φ0 (x0 ) = x0 for all x0 ∈ D and for all t ∈ R.
(iii) differentiability Φt (x0 ) depends smooth on t and initial data x0 .
Remark These three properties turn G = {Φt | t ∈ R} into a Lie-group.
3
Associated with this motion is the Eulerian vector (velocity) field u:
u(x, t) =
d
Φ (x ),
dt t 0
with Φt (x0 ) = x.
u(x) is also called the infinitesimal generator. Figures 1.10 and 1.11 illustrate these
concepts.
From this point we pose some differentiability conditions on u, namely u(x, t) ∈
C 1 (D×R; R3 ). Moreover, it is assumed that it has a continuous limit on the boundary
Σ. On Σ, we pose the kinematic condition that particles cannot pass through it:
u(x) · n = 0 for x ∈ Σ.
1.2 Structured equations and coherent structures
25
Φ(A)
•
•
Φ(x0 )
A
•
x•
0
•
•
Figure 1.10: Φ is a volume preserving evolution, i.e., |A| = |Φ(A)|.
In the interior the particles will move freely with the only constraint that different particles cannot occupy the same position at the same time. The free moving
of particles suggests an application of the Principle of Stationary Action, with the
incompressibility constraint as a condition on the admissible variations.
To be precise, the Lagrangian13 of the system consists only of the kinetic energy,
2
Z 1
d
E(Φt ) =
Φt (x0 )
dV,
2 D dt
and according to the Principle of Stationary Action the action functional is then
defined for fixed but arbitrary t0 and t1 as
2
Z Z 1 t1
d
A(Φt ) =
Φ (x )
dV dt.
2 t0 D dt t 0
The action A is a functional on the space of volume preserving evolutions, G, and
assigns a number to any evolution that lasts from t0 to t1 . The space of admissible
variations of Φt , TΦt G, consists of elements ηt being tangent to a curve Φεt ⊂ G. The
Principle of Stationary Action then states that the actual motion is such that A is
stationary for all variations Φt + ηt of Φt :
δA(Φt ; ηt ) = 0.
Conditions on Φεt , and thereby on ηt , are the following:
• Φεt is a volume preserving map for all ε in a neighbourhood of 0: div ηt = 0.;
13 The reader can consult any proper textbook on classical mechanics for the precise definition of
a Lagrangian system, e.g. Goldstein (1950), or Arnol’d (1978).
26
1 Introduction
(s + t)
(s)
•
Φt (x0 )
Φs+t (x0 )
•
Φs (Φt (x0 ))
(t)
•
x0
Figure 1.11: Illustration of the transitivity property. Note that the velocity field is tangent
to the trajectory that is set out by t 7→ Φt (x0 ).
• Φεt fixes the end-points: Φεt0 = Φt0 , Φεt1 = Φt1 .
• the velocity field corresponding to Φεt is parallel to the boundary Σ.
It easily follows from the definition of A that
Z t1 Z
d
d
0 = δA(Φt ; ηt ) =
Φt (x0 ) · ηt (x0 ) dV dt,
dt
dt
t0
D
with
d ε ηt =
Φ
.
dε t ε=0
(1.2.8)
(1.2.9)
Differentiation with respect to ε in (1.2.9) of the family of volume preserving maps,
{Φεt | ε, t fixed}, leads to a velocity field as well, being divergence-free by virtue of
the Liouville Theorem (Arnol’d (1973)), div ηt = 0, and parallel to the boundary,
ηt · n = 0 on Σ. An integration by parts of (1.2.8) leads to
Z t1 Z 2
d
Φ (x0 ) · ηt (x0 ) dV dt = 0,
(1.2.10)
2 t
t0
D dt
where
d2
d
∂u
Du
Φ (x ) = u(x, t) =
+ u · ∇u =:
dt2 t 0
dt
∂t
Dt
with D/Dt the material derivative. Note that all temporal boundary contributions
vanish because ηt0 = ηt1 = 0. Equation (1.2.10) should hold for arbitrary t0 , t1 , and
hence
Z
Du
· ηt dV = 0.
(1.2.11)
D Dt
1.2 Structured equations and coherent structures
27
Equation (1.2.11) states that Du/Dt is orthogonal to the set of all divergence-free
vector fields with vanishing norm component at the boundary. Using the Weyl-HodgeDecomposition (appendix) this is equivalent to



 Du = −∇p, div u = 0 in D,
Dt
(1.2.12)


 u · n = 0 on Σ,
for some p(x, t).
Remarks (i) The interpretation of (1.2.12) is as follows (Marchioro and Pulverenti
(1994)). Dt u is the acceleration of a fluid particle when moving freely, but it is
balanced by a force −∇p that takes care that the motion remains incompressible.
−∇p plays a comparable role as the normal force that acts on a bead when it is moving
on a rotating circle (Arnol’d (1978)). Marchioro and Pulverenti give an example of a
free motion (so ∇p = 0) in (1.2.12) that violates the incompressibility constraint. p
is called the pressure.
(ii) When the domain D is not finite or when the boundary conditions are mixed
(partially Dirichlet on the normal velocity and periodic) the line of reasoning remains
valid but the bookkeeping is getting cumbersome. In case of an infinite domain we
must specify the asymptotic behaviour of u(x) as |x| → ∞.
3
1.2.2
The Poisson structure for the Euler equations in a domain with fixed boundary
Finding the Poisson structure corresponding to (1.2.12) implies that the evolution
of functionals needs to be considered. In the introduction to this section we already
defined the class F(H) of functionals on H. We will specify the space H and the concept
of variational derivative related to functionals on H. The precise function spaces are
most times omitted, to improve on the readability. See however the appendix for
details.
Definition Let H be the subspace of L2 (D)3 , consisting of all divergence-free velocity
fields with vanishing normal velocity at the boundary Σ. H is supplied with the
(vector) L2 -inner product.
A functional F : H → R has a variational derivative (δF(u), δFb (u; ·)) if for all u ∈ H
there exists (i) a vector field δF(u), being divergence-free and parallel to Σ, and (ii)
δFb (u; ·) a linear operator with range in R, satisfying
Z
Z
δF(u; η) =
δF(u) · η dV +
δFb (u; η)|Σ dA
D
Σ
for all η ∈ Tu H ∼
= H. Before we state the Poisson bracket for incompressible, inviscid
fluid flows, note that (this is proven in the appendix) an arbitrary vector field v can
be decomposed as
v = ∇φ + w.
28
1 Introduction
If we let φ be the solution of the elliptical problem
∆φ = div v, in D,
∂φ/∂n = v · n, on Σ.
then φ is unique up to a constant, and w ∈ H:
div w = 0 in D, w · n = 0 on Σ.
The assignment w to v defines a mapping14 P ,
P (v) = w,
with the property that P (∇φ) = 0.
The Poisson bracket for incompressible, inviscid fluid flows including boundary terms
is given by15 (Lewis et al. (1986))
Z
{F, G}Eul =
δF(u) · (−ω × δG(u)) dV +
D
Z
{δGb (u; P (ω × δF(u))) − δFb (u; P (ω × δG(u)))} |Σ dA. (1.2.13)
Σ
A natural candidate for the Hamiltonian is the kinetic energy,
Z
Z
1
2
|u| dV, δE(u; η) =
u · η dV.
E(u) =
2 D
D
With this choice for the Hamiltonian, the evolution equation for u can be obtained
from
0
= ∂t F(u) − {F, E}Eul (u)
Z
δF(u) · (∂t u + ω × u) dV +
=
D
Z
δFb (u; P (∂t u + ω × u))|Σ dA.
(1.2.14)
Σ
Since (1.2.14) should hold for all F ∈ F(H) we conclude that
∂t u + ω × u = −∇h,
for some h.
Remark The relation between h and the pressure p in (1.2.12) is obtained by using
the vector identity
1
ω × u = (u · ∇)u − ∇( |u|2 )
2
and hence
1
h = p + |u|2 .
2
h is called the total head.
3
14 P
is sometimes called the Leray operator.
bracket in Lewis et al. (1986) has been rewritten leading to the result in (1.2.13) which is
simpler than their result.
15 The
29
1.2 Structured equations and coherent structures
Axisymmetric flows
We will further specify the domain D to a pipe-geometry, while the velocity field in
this paragraph is assumed to be periodic16 in the axial variable:
D = x = (x, y, z) ∈ R3 | x2 + y 2 < 1, −L/2 < z < L/2 .
and
Σ = {x ∈ R3 | x2 + y 2 = 1, −L/2 < z < L/2}.
Let (r, ϕ, z) denote cylinder-coordinates. The space H is given by

div u = 0,



u · n = 0 on Σ,
H = u = u(r, ϕ, z) 

u regular at r = 0,

u periodic in z with period L







.
We use the shorthand hf , giD and hf , giΣ for the inner products
Z
Z
hf , giD =
f · g dV, hf , giΣ =
f |Σ · g|Σ dA.
D
Σ
In the last part of this subsection we will now pay some attention to axially symmetric
flows, i.e., velocity fields that do not depend on the angle ϕ, and specifically how the
Poisson structure changes under this assumption. This consistent reduction of the
bracket is favoured in Szeri and Holmes (1988) (with boundary contributions) and
Van Groesen and De Jager (1994) (without boundary contributions, but starting the
reduction at non-isentropic, compressible, fully 3D flows).
Due to the axial symmetry, the incompressibility condition is given by
div u =
1 ∂
∂uz
(rur ) +
= 0,
r ∂r
∂z
leading to the introduction of a stream function ψ,
rur = −
∂ψ
,
∂z
ruz =
∂ψ
.
∂r
This reduction in the number of dependent variables, i.e., (ur , uϕ , uz ) → (ψ, uϕ ) is a
first step towards a simpler Poisson structure. It is commonplace to use instead of
uϕ the azimuthal circulation, γ = ruϕ (it is no coincidence that γ is the density of a
conserved quantity: see subsection 1.2.3). The velocity and vorticity are expressed in
terms of γ and ψ like
∂ψ −1
∂ψ
u = (ur , uϕ , uz ) = −r−1
, r γ, r−1
,
∂z
∂r
∂γ
∂
∂ψ
∂2ψ
∂γ
ω = (ωr , ωϕ , ωz ) = −r−1 , −
r−1
− r−1 2 , r−1
.
∂z
∂r
∂r
∂z
∂r
16 Alternatively, we can work with almost periodic functions (Joseph (1976)), but the notation gets
more involved.
30
1 Introduction
Finally, introduce the meridional circulation, ζ = r−1 ωϕ =: Lψ, with
L = −r−1
∂
∂r
∂
∂2
r−1
− r−2 2
∂r
∂z
The variational derivatives with respect to γ and ζ may be expressed in terms of δur F,
δuϕ F and δuz F:
δγ F̂ = r−1 δuϕ F;
∂
∂
δ F̂ = rδuz F;
δ F̂ = −rδur F,
∂r ζ
∂z ζ
(1.2.15)
with F̂(γ, ζ) = F(u).
The reduction of the Poisson bracket boils down to substituting (1.2.15) into (1.2.13),
integrating by parts and using boundary conditions (Szeri and Holmes (1988)). The
resulting equations for γ and ζ are
∂t γ
∂t ζ
= {γ, ψ},
(1.2.16a)
= {ζ, ψ} + {γ, γ/2y},
(1.2.16b)
with
{f, g} =
∂f ∂g ∂f ∂g
−
,
∂y ∂z
∂z ∂y
y=
1 2
r .
2
The equations in (1.2.16a,b) can also be written with the help of the (transformed)
Hamiltonian,
Ê(γ, ζ) =
1
2
Z Z
γ2
Q L/2
ψζ +
u | 2πdz,
dV +
2y
4π −L/2 z Σ
D
Q = 2πψ|Σ .
Then
γ
0
∂t
=
ζ
{γ, · }
1.2.3
{γ, · }
{ζ, · }
!
δγ Ê
.
δζ Ê
(1.2.17)
Conserved quantities and their dynamics
The existence of conserved quantities in Poisson systems is both related to (physical) symmetry and degeneracy of the Poisson bracket. Olver (1982) has computed
all (local) symmetry groups of the 3D Euler equations in an infinite domain, the
corresponding conserved quantities and the Casimirs, related to the degeneracy of
the Poisson bracket. His conclusion is that only seven conserved quantities exist and
one Casimir. These numbers change when additional assumptions are made, for instance axisymmetry leads to an infinite number of conserved quantities as we will see.
When restricting the domain to a pipe geometry only three survive, and an additional
assumption is needed to have the Casimir.
1.2 Structured equations and coherent structures
31
Proposition 1.2 (according to Olver (1982)) The three-dimensional Euler equations in a pipe admit the following three conserved quantities (we normalise them with
the (fixed and finite) volume of D; this has advantages as will be seen in e.g. chapter
2):
(i) Axial momentum:
Fax (u) = |D|−1 hu, ez iD.
(ii) Angular momentum:
A(u) = |D|−1 hru, eϕ iD.
(iii) Energy:
E(u) =
1 −1
|D| hu, uiD.
2
Moreover, the helicity B,
B(u) =
1 −1
|D| hu, ωiD.
2
being the only Casimir for unbounded flows, remains a Casimir when the normal
component of the vorticity ω vanishes at Σ.
Proof We only prove the statement on the helicity, the others are proved in Olver
(1982). In order to determine whether the helicity Poisson commutes with any functional we compute the variational derivatives of B.
1 −1
1
|D| hu, curl ηiD + |D|−1 hη, ωiD.
2
2
1
= |D|−1 hω, ηiD + |D|−1 hu × n, ηiΣ ,
2
δB(u; η) =
and hence
δB(u) = ω,
δBb (u) =
1
u×n
2
with respect to the (normalised) L2 -inner product. Using the bracket in (1.2.13) we
conclude that the only term left is (remember G ∈ H)
{B, G} =
and hence the result.
1 −1
|D| hω · n, u · δG(u)iΣ
2
2
32
1 Introduction
In chapter 2 we will use these conserved quantities to compute relative equilibria that
are the solutions of
Crit {E(u) | B(u) = b, A(u) = a, Fax (u) = f } .
u
(1.2.18)
When the solutions of (1.2.18) are obtained, they can be used to construct full 3D
solutions of the Euler equations according to proposition 1.1. To that end we need to
find the A-flow and the Fax -flow (note that the B-flow is trivial (the identity) since
it is a Casimir).
Proposition 1.3 The A-flow, acting on a non-trivial, i.e., ϕ-dependent, initial condition, is a rotation. The Fax -flow, acting on a non-trivial, i.e., z-dependent, initial
condition, is a translation.
Proof The evolution equation that determines the A-flow is obtained from the condition that
0 = ∂t F(u) − {F, A}Eul(u),
for all F ∈ F(H).
The first variation of A is simply A itself because of its linearity. This leads to
∂t u + ω × reϕ = −∇h,
(1.2.19)
u|t=0 = u0 .
It is easily observed that up to a constant h = ruϕ and u(r, ϕ, z, t) = u0 (r, ϕ − t, z)
constitute the solution of (1.2.19). Along the same lines the Fax -flow is obtained from
∂t u + ω × ez = −∇h,
u|t=0 = u0 ,
yielding h = uz and u(r, ϕ, z, t) = u0 (r, ϕ, z − t).
2
Recall that every functional evolves according to
∂t F(u) = {F, E}Eul (u).
When a functional is conserved, the density of the functional is part of a local conservation law, i.e., a divergence-expression,
c(u) := ∂t e(u) + div f (u) = 0.
The results for the axial momentum, angular momentum, energy and helicity density
are obtained from straightforward manipulations with the Euler equations:
∂t uz + div (uz u + pez )
∂t ruϕ + div ruϕ u + peϕ
= 0,
(1.2.20a)
= 0,
(1.2.20b)
1
∂t |u|2 + div (hu) = 0,
2
∂t (u · ω) + div (u · ω)u + (h − |u|2 )ω = 0,
div (u)
= 0.
(1.2.20c)
(1.2.20d)
(1.2.20e)
1.3 Approximation methods: a different viewpoint
33
When the velocity field is time-independent, integration of (1.2.20a–e) over a crosssection leads to conserved cross-sectional quantities. We came already across one of
them, namely the flow force S, being the z-component of the axial momentum flux,
integrated over a cross-section. The axial conserved quantities, related to (1.2.20a-e)
are
Z 1 Z 2π
∂
∂
S :=
(u2z + p) rdrdϕ = 0,
(1.2.21a)
∂z
∂z 0 0
Z 1 Z 2π
∂
∂
A
:=
ruϕ uz rdrdϕ = 0,
(1.2.21b)
∂z cross
∂z 0 0
Z 1 Z 2π
∂
∂
Ecross :=
huz rdrdϕ = 0,
(1.2.21c)
∂z
∂z 0 0
Z 1 Z 2π
∂
∂
(u · ω)uz + (h − |u|2 )ωz rdrdϕ +
Bcross :=
∂z
∂z 0 0
Z 2π
∂
(h − |u|2 )ω|Σ · n dϕ = 0,
(1.2.21d)
∂z 0
Z 1 Z 2π
∂
∂
uz rdrdϕ = 0.
(1.2.21e)
Qcross :=
∂z
∂z 0 0
Relations (1.2.21a–e) will be used in chapter 4 when we deal with slowly varying
geometries; in that respect, compare the expressions with (1.1.8).
Axisymmetric flows
The bracket for the axisymmetric flows (or the corresponding structure map in (1.2.17))
is much more degenerate than the the parent bracket { , }Eul : it has an infinite number of Casimirs, arranged in two families (Szeri and Holmes (1988)):
Z
CI =
yζ dV (impulse);
ZD
CA =
a(γ) dV (generalised swirl );
ZD
ζb(γ)dV (generalised helicity),
CB =
D
with a and b arbitrary continuous functions.
1.3
Approximation methods: a different viewpoint
We have investigated in the previous section inviscid swirling flows in a pipe with
constant diameter, and more specific the Poisson structure of the 3D-Euler equations.
In the presence of a few conserved quantities (energy, helicity, angular and axial
34
1 Introduction
momentum) we could define special, exact solutions of the Euler equations, related
to constrained, critical points of the energy:
U ∈ arg Crit {E(u) | B(u) = b, A(u) = a, Fax (u) = f } .
u
In this section we will emphasize the fact that (i) U is related to an exact solution V
of the Euler equations in a pipe with constant diameter (accordingly, U is also called
the state and V the evolution) and (ii) U depends continuously on the parameters b,
a and f , i.e., we have a continuous family of exact solutions.
The main question to be answered is “What is the relevance of M = {U (p)} when we
consider a perturbation of the original problem?”. We will describe our approach how
we can construct the best possible approximation in M. We think that this approach is
the cornerstone for many classical perturbation techniques in ODE’s (see e.g. Nayfey
(1973) and Kevorkian and Cole (1996)), and PDE’s (e.g. Whitham (1974), McLaughlin and Scott (1978), Aceves et al. (1986) and Pudjaprasetya and Van Groesen (1996)),
although it is usually not presented in that way. This statement is illustrated with
the example of the Duffing equation.
Parts of the first subsection are taken from Fledderus and Van Groesen (1996a),
section 2.
1.3.1
Quasi-static approach
The starting point is an evolution equation that we describe in a general form as a
first order (in time) equation:
∂t u = K(u).
(1.3.1)
The main property that we use is that there exists a family of special solutions. That
is, we assume that there is a family of solutions depending smoothly on parameters p;
we will denote these evolutions by V (p, t). The corresponding states, U (p), obtained
as the profile of the solutions at each moment of time, will form some manifold M
(see figure 1.12):
M = {U (p) | p}.
By its construction, the set M is invariant for the flow: starting with an initial
condition in M, the whole future evolution remains in M. This set of states forms a
manifold of what will be in this thesis the coherent structures; the dimension is the
number of parameters, say n.
Analysis of the perturbed system, investigation of the error
Since the set M is invariant for the original flow, the vector field K, evaluated at M,
has only a tangential component. Let us therefore introduce explicitly a ‘perturbation’
of the equation:
∂t u = K(u) + εS(u).
(1.3.2)
1.3 Approximation methods: a different viewpoint
35
V (p1 , p2 )(t)
@
R
@
M
•
U (p1 , p2 )
@
@
@
@
@
@
@
•
@
p1 @
@
I
@
@ @
@ @ : p2
Figure 1.12: The manifold of relative equilibria (MRE) M. p1 and p2 act as local coordinates and U (p1 , p2 ) plays the role of mapping.
This perturbation may also be the effect of a local homogenisation of an inhomogeneous equation as we will see in chapter 4, but it is simpler to view it here as a
given vector field S. The parameter ε, assumed to be small, scales the perturbation.
Decomposing S in a tangential and transverse component (in a neighbourhood of M,
S
S⊥
Sk
p2
U
p1 M
Figure 1.13: Decomposing the perturbation S (or a general vector field) on M.
see figure 1.13)
∂t u = K(u) + εSk (u) + εS⊥ (u),
a special case is obtained if the transversal perturbation S⊥ vanishes on M. Then
clearly M remains invariant and a suitable parameter dynamics t 7→ p(t) will produce
36
1 Introduction
the solutions of
∂t u = K(u) + εSk (u)
on M. In the more interesting case, however, S is not tangent to M and M is not
invariant for the perturbed dynamics. Then the decomposition of S in a tangential
and transverse component has to be specified.
We will now indicate that the condition to obtain a uniformly valid quasi-static approximation in M will determine the decomposition, and therewith the quasi-static
evolution in M.
Denote the quasi-static evolution in M by û(t), and its difference with an exact
solution by η:
u(t) = û(t) + η,
û(t) ∈ M.
Starting with an initial condition in M, i.e., η(0) = 0, we look for evolutions for which
η(t) = O(ε)
on time intervals for which the evolution in M changes in order unity. For the investigation of the error, the linearised equation has to be studied. Therefore, introduce
for the evolution process the notation
E(u) ≡ E0 (u) − εS(u),
E0 (u) ≡ ∂t u − K(u)
and for the linearisation
E00 (u)η ≡ [∂t − K 0 (u)]η.
Here K 0 (u) is the formal Frechet derivative:
d
for any η: K (u)η ≡
.
K(u + µη)
dµ
µ=0
0
Then, as long as η is of the order ε, it holds that
E(û + η) = E(û) + E0 (û)η + O(ε2 ) = E(û) + E00 (û)η + O(ε2 ),
and so in lowest order the error η satisfies
E00 (û)η = −E(û).
(1.3.3)
In this equation, the residue E(û) is recognised: the implicit information how much
the approximation û as a trajectory in M deviates from being an exact solution of the
perturbed equation. In view of the definition of M and the order ε perturbation, this
residue will be of order ε. Then from the linearised error equation it should follow
that also η = O(ε). This is certainly the case if E00 (û) is bounded invertible. However,
we will show below that from the fact that the manifold M consists of solutions of the
unperturbed equation, the linearised operator is in fact degenerate. Then a solution
of O(ε) can only exist provided the residue lies in the range of E00 (û), which means
that certain necessary solvability conditions have to be satisfied.
1.3 Approximation methods: a different viewpoint
37
Degeneracy as a natural consequence
Let us investigate the degeneracy. Since we have an n-parameter family of solutions
in M, there is n-fold degeneracy. In the notation introduced above, the fact that
V (p, t) are solutions is expressed by
E0 (V (p, t)) = 0.
Differentiating with respect to each of the parameters results in
E00 (V (p, t))ξ k = 0 with ξ k ≡
∂V (p, t)
.
∂pk
Restricted to a specific time, at which V (p, t0 ) is denoted by U (p), the same restriction
leads to the tangent space to the manifold M at the point U :
E00 (U (p))ξ k = 0
where we can interpret the tangent vectors also as follows:
ξk ≡
∂U (p)
.
∂pk
Solvability conditions; the Fredholm alternative
Now that we have observed that degeneracy of the linearised evolution operator is a
natural consequence of the fact that the basic manifold consists of exact solutions,
we can investigate the solvability problem for the error equation (1.3.3). To that end
we simplify notation and write L ≡ E00 (û), and f ≡ −E(û), and the solvability is
investigated for the operator equation
Lη = f.
The necessity of certain solvability conditions can be seen without any additional
conditions and will now be derived. The starting point is to view the operator equation
in a weak sense as follows. With h , i some inner product, consider sufficiently many
appropriate elements ζ such that the original equation Lη = f is equivalent to the
condition that for all ζ it holds that
hLη, ζi = hf, ζi.
A next step is to introduce the (formal) adjoint of the operator L as the operator L∗
such that hLη, ζi ≡ hη, L∗ ζi. With the adjoint, the weak formulation of the operator
equation can be rewritten like
hη, L∗ ζi = hf, ζi
which should hold for all appropriate ζ. Now observe that if the adjoint L∗ is singular, there can be no solutions provided f satisfies the correct solvability conditions;
specifically, it follows that f should satisfy
hf, ζi = 0,
for all ζ for which L∗ ζ = 0.
(1.3.4)
38
1 Introduction
Briefly this can be stated that f should be orthogonal to the kernel of the adjoint:
f ⊥ ker(L∗ ).
This is (the necessity part of) the Fredholm alternative:
• either L∗ is regular, i.e,. ker(L∗ ) is trivial, and then no solvability conditions
for f are found in this way,
• or ker(L∗ ) is non-trivial, say it is a q-dimensional space; then f has to satisfy
q solvability conditions: hf, ζi = 0 for q elements ζ from ker(L∗ ) that form a
basis.
The determination of the kernel of the adjoint is in general difficult. In chapter 4 we
will show that, in contrast to the general case, for Poisson systems the kernel of the
adjoint linearised equation and that of the linearised operator are related in a simple
way.
The relation between the solvability conditions and the decomposition of the perturbation (being part of the residue) is as follows. First of all, the residue of û is built
up from (i) the perturbation εP (û) and (ii) a part that indicates how much û is not
a solution of the unperturbed equation—the residue of û related to E0 (u):
E(û) = εP (û) + E0 (û).
The interpretation of the solvability conditions is that, by means of suitable parameter
dynamics we remove the parallel part of the residue, E(û)k , i.e., the part of E(û) that
lies in the tangent space Tû M.
1.3.2
An example: the Duffing equation
Perturbation techniques in ordinary differential equations like strained coordinates,
multiple-scales, Krylov-Bogoliubov-Mitropolsky and and Lindstedt-Poincaré technique
may be presented as a composition of three steps:
(i) Obtain (explicitly or implicitly) a continuous family of exact solutions V (p, t)
of the unperturbed problem (1.3.1), and denote the collection of corresponding
states U (p) by M = {U (p)|p}.
(ii) Consider a perturbation of (1.3.1), equation (1.3.2) plus initial conditions, and
denote its solution by u(t). Let the O(1)-term in the approximation û(t) of u(t)
be an evolution in M (with possibly different time-scales), i.e.,
u = û + η = U (p(t)) + η,
|û| = O(|u|) = O(1), |η| = o(1),
with | · | a suitable norm. Use the error equation (1.3.3) and the corresponding
solvability conditions (1.3.4) to determine the parameter dynamics. The solvability conditions are related to the avoidance of secular (non-uniform) terms.
39
1.3 Approximation methods: a different viewpoint
(iii) Analyse the residue and improve on û by writing
ηi = η̂ + ηi+1 ,
|ηi+1 | = o(|ηi |), η0 = η.
The situation in the field of PDE’s is rather different. Asymptotic solution techniques are more recent, and very often the validity of the approximation is still open.
Examples are given (Kevorkian and Cole (1996)) by the WKB-method (nearly periodic waves), multiple-scale methods (weakly nonlinear conservation laws, GinzburgLandau equation) and averaging or homogenisation techniques.
Let us illustrate the ideas in this section with an example: the Duffing equation. It
is a prototype of a weakly nonlinear oscillator, but with a slight change it can also
serve as an example of a strictly nonlinear oscillator with slowly varying coefficients.
The weakly nonlinear Duffing equation
The weakly nonlinear Duffing equation is introduced and physically motivated in
Kevorkian and Cole (1996), page 307. When all physical quantities are scaled away,
we are left with a ‘normal’ form,
du
d2 u
+ εβ
+ u + εu3 = 0, |ε| 1, β = O(1),
dt2
dt
d2 u
+ u + εu3 = 0,
dt2
(DUF)
(DUF0 )
where β is related to the damping and ε to the nonlinear part of the oscillator.
Remark When β = 0 (no damping), DUF has a first integral of the form
I=
1
2
du
dt
2
1
ε
+ u2 + u4 .
2
4
The level sets of I are bounded for all ε > 0. Moreover, DUF0 has a Hamiltonian
structure with I as the Hamiltonian.
3
The unperturbed and perturbed equation, E0 (u) and E(u) are defined by
E0 (u) =
d2 u
+ u,
dt2
E(u) = E0 (u) + εβ
du
+ εu3 .
dt
The unperturbed equation has solutions V (t) = A cos(t + θ0 ) and the corresponding
states are,
U (A, θ) = A cos θ,
M = {U (A, θ) | A, θ}.
(1.3.5)
The solution of DUF, supplied with initial conditions {u(0) = a, u̇(0) = 0} is denoted
by u(t), with u̇ = du/dt. An approximation of u(t) is sought as an evolution in M,
t 7→ U (A(t), θ(t)) =: û(t).
40
1 Introduction
The condition that û is, at zeroth order, a solution of the unperturbed equation leads
to (we use the dots to denote differentiation with respect to time)
∂ 2 U 2 ∂U
∂U
∂2U
E0 (û) =
θ̇
+
θ̈
+
Ä
+
2
Ȧθ̇ + U = h.o.t.
∂θ2
∂θ
∂A
∂A∂θ
Since A = constant is a solution for DUF0 , we assume
Ȧ = o(1),
Ä = o(1).
Using (1.3.5) we obtain
θ̇ = 1 + o(1), θ̈ = o(1),
and hence θ(t) = t + Θ(t), Θ̇ = o(1).
The next step is the computation of the residue of û. This will determine the order
of for instance Ȧ in the sense that only one will give a balance between linear and
disturbance terms; sometimes this relation is called distinguished (Hinch (1991)).
More specific,
∂U
∂U
∂2U
∂U
∂U
E(û) =
θ̈ +
Ä + 2
Ȧθ̇ + εβ
Ȧ +
θ̇ + εU 3 ,
∂θ
∂A
∂A∂θ
∂A
∂θ
and it is observed that the required balance is obtained when we assume17 Ȧ = O(ε),
Ä = O(ε2 ), Θ̇ = O(ε) and Θ̈ = O(ε2 ). Then at first order, the residue is given by
3 3
dΘ
dA
1
E(û) = ε
A − 2A
cos θ − 2
+ βA sin θ + A3 cos 3θ + h.o.t.
4
dt
dt
4
(1.3.6)
Now we come to the solvability conditions. Differentiation of U with respect to the
parameters A and θ leads to the elements in the kernel of the linearised operator,
E00 (û):
(!)
E00 (û)ξ 1 = E0 (ξ 1 ) = 0, E00 (û)ξ 2 = 0,
with ξ 1 =
∂U
∂U
, ξ =
,
∂A 2
∂θ
and where we used at (!) the linearity of E0 (û). Since E0 (û) is self-adjoint, the kernel of
the adjoint linearised operator is immediately given by ker E00 (û)∗ = {ξ 1 , ξ 2 }. Because
∂U
∂U
= cos θ, ξ 2 =
= −A sin θ
∂A
∂θ
the solvability conditions require that the coefficients of cos θ and sin θ in the residue
(1.3.6) vanish:
dA
ε 2
+ βA
= O(ε2 ), A(0) = a,
(1.3.7a)
dt
dΘ
3 3
(1.3.7b)
ε
A − 2A
= O(ε2 ), Θ(0) = 0
4
dt
ξ1 =
17 Many standard books like Nayfey (1973) and Kevorkian and Cole (1996) use this example to
illustrate the multiple time-scale method.
41
1.3 Approximation methods: a different viewpoint
Undamped motion For this case, the damping β = 0. The system (1.3.7a,b)
reduces to
dA
= O(ε), A(0) = a,
dt
3 3
dΘ
A − 2A
= O(ε), Θ(0) = 0.
4
dt
The solution is at first order A = a = constant, i.e., the amplitude of the motion is
preserved and the phase Θ is
Θ=
3 2
a εt.
8
Turning to the approximation û, we find
3
û = a cos(t + a2 εt) + h.o.t.
8
Kevorkian and Cole (1996) show that the error between the exact solution u(t) and
the approximation û remains O(ε) on a long time-scale: 0 ≤ t ≤ T (ε) = O(ε−1 ).
Damped motion The solution of (1.3.7a,b) is given by
β
3a2
A(T ) = a exp − T , Θ(T ) =
(1 − exp{−βT })
2
8β
and the corresponding approximation by
εβ
3a2
û = a exp − t cos t +
(1 − exp{−εβt}) + h.o.t.
2
8β
The strictly nonlinear Duffing equation with slowly varying coefficients
This example is treated from the classical point of view rather thoroughly in Nayfey
(1973), page 286 ff. The equation is given by
d2 u
+ α(εt)u + β(εt)u3 = 0,
dt2
|ε| 1,
(1.3.8)
with α(0) and β(0) both O(1) and varying O(1) as t = O(ε−1 ). When α and β are
just constants, the solution of (1.3.8) can be expressed using Jacobi’s elliptic functions
(Nayfey (1973)), for instance,
u(t) = A sn (Kt, ν).
The parameters A, K and ν are related to α and β, leaving one free parameter, say K.
The corresponding states are collected in M = {U (η) = A(α, β, η)sn (η, ν(α, β, η)) | η}.
We refer to Nayfey (1973) for the technical details. Note that, although the unperturbed equation is nonlinear, the linearised operator is still self-adjoint, and hence
“differentiating U (η) with respect to η leads to (...) a solution of the homogeneous
part of the linearised equation. (...) the inhomogeneous part (i.e., residue) must be
orthogonal to the solution of the homogeneous part.” (Nayfey (1973)).
42
1.4
1 Introduction
The Navier-Stokes equations as a singular perturbed system
In this section certain concepts related to singular perturbation theory are introduced.
This is done at the hand of the monograph by Eckhaus (1973) but the reader can
consult other books like Van Dyke (1975), Hinch (1991) and Kevorkian and Cole
(1996). The definitions of these concepts are followed by an example that will illustrate
certain techniques. Finally, the relevance of this example for the viscous swirling flows
in a pipe is explained. Viscous effects on swirling flows will be studied in chapter 5.
1.4.1
What is a singular perturbed problem?
We discuss the concept of a singular perturbed problem in the context of differential
equations. That is, let x ∈ D ⊂ Rn and ε a (real) parameter, ε ∈ (0, ε], and study
the problem for y(x, ε) given by the equation
E(y, ε) = 0,
x∈D
(1.4.1)
with some conditions for y on the boundary ∂D. We imagine E a differential operator,
but this is not essential.
Definition A function y(x, ε), defined in x ∈ D and ε ∈ (0, ε], will be called an
approximation of y(x, ε) in D for the norm || · || if
lim ||y − y|| = 0.
ε→0
The terms ‘regular’ and ‘singular’ perturbed problem can now be explained at the
hand of only this definition.
Definition Given a norm, the perturbation problem (1.4.1) is called regular if there
exists an approximation y(x) in D of y(x, ε) for this norm, independent of ε and it is
called singular if no such approximation exists.
This definition is fore-shadowed when we can decompose the operator E(·, ε) like
E(·, ε) = E0 (·) + E1 (·, ε)
with E0 (·) independent of ε and E1 (·, ε) such that for every function z(x), independent
of ε,
lim ||E1 (z, ε)|| = 0.
ε→0
Very often the singular perturbed problem has such a structure that the differential
operator E1 (·, ε) is of higher order compared to E0 (·), since a possible approximation
of the form y(x), that is related to the (limit) problem
E0 (y) = 0,
cannot satisfy, in general, all boundary conditions. The implication is then that very
often y is not an approximation in the maximum norm (Eckhaus (1973)).
1.4 The Navier-Stokes equations as a singular perturbed system
43
Remark Note that in the definition of ‘approximation’ we explicitly included the
domain D on which the approximation should hold. Hence, although for a singular
perturbed problem there does not exist a regular approximation in D, there may exist
(different) sub-domains of D where the problem is regular. Singular perturbation
theory should resolve the behaviour of the solution y(x, ε) in the rest of the domain.
3
An approximation can be viewed as being part of a larger structure, a formal asymptotic approximation.
Definition Suppose that for any function z(x) in D it holds that18
||E(z, ε)|| = OS (1).
Then a function y(x, ε), satisfying
||E(y, ε)|| = o(1) for x ∈ Dr ⊂ D
is called a formal asymptotic approximation of the exact solution y, that is, we only
know that the residue is ‘small’, whereas the error ||y − y|| can be ‘big’ on D.
A formal asymptotic approximation can be set up as an asymptotic series,
y=
m
X
δi (ε)yi (x, ε)
(1.4.2)
i=0
with δi+1 = o(δi ) and yi = OS (1) in Dr . It is called a regular asymptotic series if the
variables ε and x in (1.4.2) separate completely:
y=
m
X
δi (ε)yi (x);
(1.4.3)
i=0
m is called the order of the approximation.
Remarks (i) When discussing the uniqueness of an asymptotic series, the answer
depends on the dimension of the space Rn . In the case of n = 0 (so y = y(ε), coming
from some algebraic equation) two different expansions of the function can only differ
by a quantity which is not analytic or, more precise, transcendentally small with
respect to the given order functions δn (Eckhaus (1969). In the case n > 0 uniqueness
of expansions is given in the equivalence class of the relevant order function (Eckhaus
(1973)).
(ii) The suggested structure of the asymptotic series in (1.4.3) need not to be the
right one. Eckhaus (1973) gives quite a few examples where this Ansatz fails.
3
18 The
notation OS (f (ε)) (Eckhaus (1973)) stands for strictly of order f (ε) as ε → 0:
g(ε) = OS (f (ε)) if g(ε) = O(f (ε)) ánd g(ε) 6= o(f (ε)).
Another notation for this case is ord (f (ε)) (Hinch (1991)).
44
1 Introduction
Substituting (1.4.3) in the differential equation and arranging terms we find equations for subsequent terms in the series, and by solving these, the order of the formal
asymptotic series is increased. Very often the domain of validity of the regular approximation, Dr , contains a part ∂Dr of the boundary ∂D, and hence boundary
conditions for y along ∂Dr can be imposed on the terms in the formal expansion.
In order to resolve the behaviour of y(x, ε) in Ds ⊂ D, being a ‘small’ neighbourhood19
of a set S of measure 0 where no regular approximation exists, we introduce variables
x0 such that say {x0 | x01 = . . . = x0p = 0} represents S. The local variables X are
defined according to
(
x0j
δ (j) = o(1) for j = 1, . . . , p,
Xj = (j) , with
δ
δ (j) = O(1) for j = p + 1, . . . , n.
In these variables equation (1.4.1) turns into
E∗ (Y, ε) = 0
on D∗s
(1.4.4)
with D∗s a bounded and closed domain in X-space and Y (X, ε) = y(x(X), ε). Let
Y =
m
X
δi (ε)Yi (X)
(1.4.5)
i=0
be a (local) asymptotic series defined for X ∈ D∗s . This series is substituted in (1.4.4)
and arranging terms will yield equations for the different terms in (1.4.5). Boundary
conditions will come from the remaining part ∂Ds = ∂D \ ∂Dr of the boundary.
Due to the lack of boundary conditions, in general this procedure does not lead to a
unique determination of the formal (local) approximation. Further conditions can be
obtained from so-called matching principles 20 , that relate the two approximations.
There are several matching principles, but we will describe briefly the Overlap Hypothesis: see Eckhaus (1973) and Fraenkel (1969) for a more extensive treatment.
The Overlap Hypothesis states that there exists a domain Di in which both the regular and the local expansions are valid. Since the expansions represent the same
function, each on a different domain, they should be identical on Di . Introduction of
intermediate coordinates
ηi = θ(i) Xi
with θ(i) = o(1) and rewriting both the regular and the local approximation in terms
of η will determine a unique formal (local) approximation.
19 See
Eckhaus (1973) for more precise information.
method, re-introduced in the late forties by Lighthill (1949), is the method of strained
coordinates. Indeed, it can be traced back to Stokes (1847) and Poincaré (1892). However, Stokes
did not get the credits and since the survey of Tsien (1956) many authors called it the PLK-method
(P for Poincaré, L for Lighthill, and K for Kuo, who applied it to a viscous flow problem in 1953 and
1956). See for a comparison with matched asymptotic expansions and/or and extensive treatment
of the PLK-method Van Dyke (1975) and Kevorkian and Cole (1996).
20 Another
1.4 The Navier-Stokes equations as a singular perturbed system
45
Remark Justification of matching rules is usually obtained when a certain structure
of the uniform expansion in D is assumed. See Kaplun (1957) and Fraenkel (1969)
for the justification of two different types of matching rules.
3
Finally, the two approximations can be combined to form a composite approximation
that usually turns out to be a uniform approximation on D. This composition is not
unique and this is reflected in the various ways one can combine the regular and the
local approximation:
• adding the regular and the local and subtracting their common form in the
overlap, yo :
y c = y + Y − yo ;
• multiplying the regular with the local and dividing by their common form in
the overlap:
yc =
yY
.
yo
Remark The regular and the local approximation are very often called the outer
and the inner approximation, respectively. Ds is usually called the boundary layer.
We will use these names in the sequel.
3
1.4.2
An example
Consider the problem

2

 ε d y + dy + 1 y = 0,
dx2
dx 4

 y(0) = 0, y(1) = 1
y(x) ∈ R, x ∈ [0, 1], 0 < ε 1
One can solve (1.4.6) explicitly, yielding
n
o
n
o
√
√
exp − (1− 2ε1−ε)x − exp − (1+ 2ε1−ε)x
n
o
n
o .
y(x) =
√
√
exp − 1− 2ε1−ε − exp − 1+ 2ε1−ε
(1.4.6)
(1.4.7)
The graph of y for different values of ε is drawn in figure 1.14.
The outer expansion
The problem as it stands can fall in the category ‘Singular Perturbed Problems’ since
the order of the ODE is decreased by one when setting ε = 0. However, let us proceed
by treating the problem as a regular perturbation problem, that is, all derivatives in
(1.4.6) are assumed to be of O(1), even in the limit ε ↓ 0, and write
y(x, ε) ∼
m
X
i=0
εi yi (x)
(order functions δi (ε) = εi )
(1.4.8)
46
1 Introduction
1.2
1
ε = 0.05
0.1
0.2
0.8
y
0.6
6
0.4
0.2
00
0.2
- x 0.6
0.4
0.8
1
Figure 1.14: The solution y(x, ε) versus x for various values of ε. Note the boundary layer
near x = 0, where y rapidly changes from zero to order unity.
with yi (x) of O(1) uniformly on [0, 1]. The series expansion in (1.4.8) is a formal
expansion: we cannot decide on fore-hand whether it converges or not, and in more
advanced problems this is usually very difficult to prove; moreover, it very often
occurs that the series is diverging but even then one can use this attempt to ‘solve’
or understand the problem.
Substituting (1.4.8) into (1.4.6) and rearranging powers of ε we obtain
at ε0 :
at εi :
dy0
1
+ y0 = 0,
dx
4
d2 yi−1
dyi
1
+ yi = −
,
dx
4
dx2
(1.4.9a)
for i ≥ 1.
(1.4.9b)
The boundary conditions for (1.4.9a,b) are left open on purpose. From figure 1.14
we expect only a regular approximation in an interval I including x = 1, so the only
possible boundary condition for (1.4.9a) would be y0 (1) = 1 and (1.4.9b) should be
supplied with the boundary condition yi (1) = 0, i ≥ 1. With this information yi (x)
is recursively defined by


 y (x)
i

 y (x)
0
= e
= e
−x/4
Z
1
x
1/4−x/4
eτ /4
d2 yi−1
dτ,
dτ 2
i≥1
.
Note that the exact solution (1.4.7) behaves in first order like y0 (x) for fixed x > 0
47
1.4 The Navier-Stokes equations as a singular perturbed system
and ε ↓ 0 (T.S.T. stands for Transcendentally Small Term):
√
(1 − (1 − 12 ε + O(ε2 )))
(1 − 1 − ε)x
= exp −
2ε
exp −
2ε
x
n x
o
n xo
= exp − + xO(ε) −→ exp −
as ε ↓ 0, x fixed;
4
4
√
n x
o
(1 + 1 − ε)x
exp −
= exp − + xO(1) = T.S.T. as ε ↓ 0, x > 0 fixed,
2ε
ε
and so forth.
The inner expansion
However, the naive approach did not result in a good approximation over the whole
domain: at x = 0 a region of non-uniformity is introduced:
max |y(x) − y0 (x)| 6= o(1),
x∈[0,1]
i.e., the error |y − y0 | is not uniformly small21 . In order to analyse the behaviour near
x = 0, a stretched co-ordinate is introduced,
X = x/εα ,
for x → 0
The governing equation then becomes
ε1−2α
d2 Y
dY
1
+ ε−α
+ Y = 0.
dX 2
dX
4
The potentially interesting scalings of an equation are the ones that produce a balance
between two or more terms in this equation. Such scalings are sometimes called
distinguished limits (Hinch (1991)). Only α = 0 (the outer expansion, for which
the second and third term balance) and α = 1 (for which the first and second term
balance) lead to a balance; for α = 1:
1 dY
1 d2 Y
1
+
+ Y = 0.
ε dX 2
ε dX
4
(1.4.10)
Again a formal expansion in powers of ε is sought:
Y (X, ε) ∼
m
X
εi Yi (X).
i=0
Substituting (1.4.11) into (1.4.10) and comparing coefficients of εi yields
at ε−1 :
at ε−1+i :
21 Note
d2 Y0 dY0
+
= 0, Y0 (0) = 0;
dX 2
dX
d2 Yi
dY
1
+ i = − Yi−1 , Yi (0) = 0, i ≥ 1,
dX 2 dX
4
however that the L2 -error is o(1).
(1.4.11)
48
1 Introduction
with solutions
Y0 (X) = B0 (1 − e−X );
Yi (X) = Bi (1 − e−X ) +
1
4
Z
X
eX̂−X Yi−1 (X̂) dX̂ −
0
1
4
Z
X
Yi−1 (X̂) dX̂,
i ≥ 1,
0
and constants of integration Bi . Note that the exact solution behaves in first order
like Y0 (X) for fixed x/ε as ε ↓ 0 when B0 = e1/4 , but this result can also be obtained
in a more precise and ‘legal’ way.
Matching the inner with the outer
At this point two asymptotic expansions for the exact solution have been obtained,
one for fixed x and one for fixed X as ε ↓ 0. It will be shown that these expansions
have a similar form in an overlap or intermediate region where both x is small and
X is large, that is, ε X · θinn = η = x/θout 1 with θinn = o(1) and θout = o(1).
Forcing the two expansions to be equal in this intermediate region will determine the
constants of integration, and thereby the two expansions.
First the inner and outer expansions are expressed in terms of the intermediate variable η,
η = x/εα = Xε1−α ,
with 0 < α < 1.
Then the limit ε ↓ 0 is taken with η fixed, that is, x → 0 and X → ∞ at the same
time. To fix ideas, the value α = 1/2 is used, but this is not essential. Then the outer
expansion turns into
2
1 1/4 √
1 1/4
η 1/4
1/4
e − e η ε+
e + e
ε + O(ε3/2 )
4
32
16
and the inner expansion is given in terms of η by
2
√
1
1
η
B0 − B0 η ε +
B + B − B1 + O(ε3/2 ) + T.S.T.
4
32 0 4 0
Comparing both expansions at every order of ε, they will be identical up to O(ε3/2 )
iff
e1/4 = B0 ,
3 1/4
e
= B1 .
16
Composite approximation
Using both asymptotic series we can construct a composite approximation that is
uniformly valid, and is correct of the order ε: This leads to the following statement:
1.4 The Navier-Stokes equations as a singular perturbed system
49
Proposition 1.4 An O(ε)-correct approximation on R+ of y(x) is given by
(
lim y0 (x)
x→0
= e1/4−x/4 + e1/4 (1 − e−x/ε) − e1/4 .
y0 (x) + Y0 (x/ε) −
lim Y0 (X)
X→∞
If we would have used the ‘multiplication-division’ approach, the result would have
differed by an exponentially small amount. Indeed, the difference can be computed
exactly,
y0 + Y0 − yc −
y0 Y0
= e1/4 e−x/ε (e−x/4 − 1)
yc
which is transcendentally small in the limit ε ↓ 0. In figure 1.15a we have drawn a
graph of y, together with the first term of both approximations, y0 and Y0 , and their
composition, y. The difference between y and y is shown in figure 1.15b.
1.2
1
y−y
0.008 6
y0
Y0
y, y
0.006
0.8
0.004
0.6
0.002
0.4
00
0.2
0.4
0.2
0.6
-x
0.8
1
-0.002
00
0.2
-x
0.4
0.6
(a)
0.8
1
(b)
Figure 1.15: (a) The exact solution y, the first term in the regular and singular approximation, y0 and Y0 , respectively, and the composition y versus x. Note that y hardly can
be distinguished from y. The value of ε is taken 0.1. (b) The difference between the exact
solution y and (the uniform approximation) y. Same value for ε as in plot (a).
1.4.3
Relevance of example for the Navier-Stokes equations
The original physical problem from which the ideas of mathematical boundary layer
theory originated was the problem of viscous, incompressible flow past an object. The
aim was to resolve D’Alembert’s paradox, stating that a fluid with zero viscosity cannot exert a force on a fixed rigid body, which is clearly in contrast with observation.
The resolution of this long-standing controversy was obtained by recognising the underlying inviscid-flow equations as a singular perturbed problem when the viscosity
50
1 Introduction
is very small. To be precise, these equations manifest their singular character when
the viscosity µ → 0 only in the case of solid boundaries present22 . In the rest of this
section we will introduce the Navier-Stokes equations for further reference and see
how the previous example, and in particular proposition 1.4, can be relevant when
studying the effects of viscosity on swirling flows in an expanding pipe.
Let u∗ = u∗ er∗ + v∗ eϕ + w∗ ez∗ . The Navier-Stokes equations in dimensional form
are given by (see e.g. Batchelor (1967)),
ρ
Du∗
= −∇p∗ + µ∇2 u∗ ,
Dt∗
div u∗ = 0,
(1.4.12)
describing the motion of for instance an incompressible swirling fluid in a pipe. In
order to non-dimensionalise (1.4.12), we choose a characteristic length and a characteristic velocity as follows. Let the amount of fluid flowing per second through a
cross-section be given by Q; it can be seen as a force that maintains the axial movement of the fluid particles. Since a swirling motion cannot sustain without input of
energy as well, there is some characteristic external angular velocity Ω given. In terms
of these quantities, the following characteristic scales can be defined:
scale
length
velocity
time
pressure
definition
pinlet radius R0
W = (ΩR0 )2 + (Q/(πR02 ))2
T = R0 /W
p0 = ρW 2
new variables
r = r∗ /R0 , z = z∗ /R0 ;
ur = u∗ /W, uϕ = v∗ /W, uz = w∗ /W ;
t = t∗ /T ;
p = p∗ /p0 .
In terms of these new non-dimensional variables, equation (1.4.12) turns into (Kevorkian and Cole (1996))
1
∂t u + u · ∇u ≡ ∂t u + ∇( |u|2 ) − u × ω = − ∇p − Re−1 curl ω, (1.4.13a)
2
|
{z
}
|{z}
|
{z
}
transport of inertia
pressure viscous force
div u = 0,
(1.4.13b)
including the ‘no-slip’ condition on the body:
u=0
on D.
Definition The Reynolds number Re is defined as ρR0 W/µ.
This shows very clearly that the order of the equations is lowered in the outer limit
(Re → ∞, resulting in the Euler equations), and at the same time part of the ‘noslip’ condition is lost. The region of non-uniformity (or boundary layer) near ∂D
corresponds to the inner limit.
22 When solid boundaries are absent and certain regularity conditions on forcing and initial state
are taken care of, one can prove that the vanishing viscosity limit of the function uµ , being a solution
of (1.4.12), is a solution of the Euler equations (see e.g. Constantin and Foias (1988), chapter 11).
1.5 Experiments and numerical simulation
51
Proposition 1.4 will be used as an assumption in the case of boundary layers in
swirling flows, chapter 5. Then at every cross-section the Navier-Stokes solution u of
(1.4.13a,b) will be thought of as
u = U + v − V + w,
with U the Eulerian flow, V its slip velocity, and v the first order boundary layer flow.
w is in general uniformly of the order O(Re−1/2 ), which is the order of non-uniformity
in U . The relevant equations for v are the Blasius equations (constant-diameter pipe)
and the generalised Falkner-Skan equations for an expanding pipe. The only property
of v that is needed here is its decay at infinity23 . This turns out to be exponentially
or transcendentally small:
v − V ∼ exp(−N ) as N = n/Re−1/2 → ∞.
Here n is a co-ordinate normal to the wall of the pipe. This suggests that when
considering orders of magnitude of for instance cross-sectional integrals involving v,
it may be useful to find a much more explicit analogy in the previous example. We
will come back to this subject in chapter 5.
1.5
Experiments and numerical simulation
In section 1.1 we dealt with axisymmetric vortex breakdown in some detail. We
noticed that the flows in experiments are sometimes nearly axisymmetric, but that
downstream of the stagnation point the flow very often had a non-axisymmetric component. The appearance of such a component is usually caused by instability and it
can be triggered by e.g. changes in geometry. This phenomenon of a rotating, nonaxisymmetric flow field has not received much attention, although linear instability
results (e.g. Batchelor and Gill (1962) and Pedley (1968)) showed its significance. The
group of Syred in Cardiff started in the seventies to work on swirling flows in combustion processes and coined the name Precessing Vortex Core (PVC) for the presence
of three dimensional time-dependent instabilities (Syred and Beer (1974), Syred et
al. (1975)). One of the main characteristics of such flows is its frequency, and although relations between frequency, flow rate and burner geometry where suggested
in Syred et al. (1994), the matter is far from solved especially since the appearance
of the experimental results of Kok et al. (1993): the frequencies in both experiments
differ by a factor of 5000. In section 4.1 we show how we can relate both results.
Attempts (e.g. Steggink (1996)) were made to realise a swirling flow in an expanding
pipe using commercial CFD packages like FLUENT or FLOW3D, but difficulties have
arisen in choosing an appropriate turbulence model. To illustrate this, two simulations
are given for a steady, axisymmetric swirling flow in a venturi, using FLOW3D. The
first one (figure 1.16) uses the ‘Low Reynolds number k−ε model’, whereas the second
one employs a Reynolds differential-stress model (figure 1.17): note the difference in
23 Van Dyke (1975) notes that certain mathematical arguments, suggested by Stewartson (1957)
and Chang (1961), indicate that this behaviour is universal for any boundary layer.
52
1 Introduction
the size of the recirculation bubble. The latter model is usually thought to be better
capable of describing ‘difficult’ flows like swirling flows, but recent studies (Parchen
(1993), Steenbergen (1995), Steggink (1996)) have shown that in certain cases the k−ε
model gives better correlations with experiment. This illustrates the need for better,
Figure 1.16: The axial velocity, uz in a venturi with swirl. The turbulence model is ‘Low
Reynolds k − ε’ (Steggink (1996)). The swirl-number at the entrance is approximately 2.
Figure 1.17: The axial velocity, uz in a venturi with swirl. The turbulence model is
Reynolds differential-stress (Steggink (1996)). The inlet profile is the same as in figure 1.16.
low-dimensional models, capturing the main characteristics of the problem.
A.1
Function spaces and their decompositions
In this appendix we provide some background on the functional analysis behind the
Euler equations. Most of the results are based on Temam (1977) and Mahalov et
al. (1990).
The domain is a pipe, denoted by
P = {x = (x, y, z) ∈ R3 | x2 + y 2 < 1}.
D is some finite part of P,
D = {x ∈ P | − L/2 < z < L/2}
and Σ its physical boundary (the wall of the pipe),
Σ = {x ∈ R3 | − L/2 < z < L/2, x2 + y 2 = 1},
A.1 Function spaces and their decompositions
53
with L some positive real number. Define
Z
L2 (D) = f : D → R f is measurable,
|f (x)|2 dx < ∞
D
and let
H m (D) = {f | Dβ f ∈ L2 (D), |β| ≤ m}
denote the Sobolev spaces of order m. H m (D) is a Hilbert space with inner product
Z X
hf, giH m =
Dβ f (x) · Dβ g(x) dx.
D |β|≤m
When the subscript H m is omitted the L2 -inner product is meant. The smooth
periodic functions live in
∞
Cper
(D) = {f ∈ C ∞ (P) | f (x, y, z) = f (x, y, z + L) for all x ∈ P}
and the periodic functions that vanish on the boundary Σ are collected in
∞
∞
(D) = {f ∈ Cper
(D) | supp (f ) ∩ (D \ Σ) is compact in D}
C0,per
where D denotes the closure of D. Finally, set
E(D) = {u ∈ L2 (D)3 | div u ∈ L2 (D)},
∞
V = {u ∈ C0,per
(D) | div u = 0},
H = the closure of V in L2 (D)3 ,
where the derivatives are taken in distributional sense whenever necessary.
A trace theorem (Temam (1977))
It is known (Lions (1962), and Lions and Magenes (1972)) that there exists a (bounded)
linear operator γ0 : H 1 (D) → L2 (Σ), called the trace operator, resembling the restriction operator f → fΣ for C 1 -functions on D. Obviously, the kernel of γ0 is the space
H01 (D). The image space γ0 (H 1 (D)) is denoted by H 1/2 (Σ) and it can be shown that
it lies dense in L2 (Σ). Let H −1/2 (Σ) denote the dual space of H 1/2 (Σ). Then the
following trace theorem holds for the normal component u · n with u ∈ E(D):
Theorem A.1.1 There exists a linear continuous operator γn : E(D) → H −1/2 (Σ)
such that
γn (u) = the restriction of u · n to Σ, for every u ∈ C0∞ (D)3 .
The Stokes formula,
Z
hu, grad pi + hdiv u, pi =
γ0 (p)γn (u) dA
Σ
holds for all u ∈ E(D) and p ∈ H 1 (D).
A proof can be found in Temam (1977).
(A.1.1)
54
1 Introduction
Characterisation of the space H
The following characterisation of H and its orthogonal complement H⊥ in L2 (D)3 can
be given.
Proposition A.1.2 (Weyl-Hodge Decomposition)
H⊥ =
u ∈ L2 (D)3 | u = grad p, p ∈ H 1 (D) ,
H =
u ∈ L2 (D)3 | div u = 0, γn (u) = 0 on Σ .
(A.1.2)
(A.1.3)
Proof of (A.1.2) If u = grad p with p ∈ H 1 (D), then according to (A.1.1)
Z
hu, vi = hdiv (pv), 1i =
γ0 (p)γn (v) dA = 0
Σ
for all v ∈ V . On the other hand, if hu, vi = 0 for all v ∈ V , a result by de
Rham (1960) assures that u = grad p for some distribution p. Using the fact that
u ∈ L2 (D)3 and hence all derivatives of p are in L2 (D), it can be proven (proposition
1.2 in Temam (1977)) that p ∈ H 1 (D).
5
Proof of (A.1.3) Let us first prove that H is contained in the right-hand side of
(A.1.3), H• . Since H is the closure of V in L2 (D), there is a sequence {um } ⊂ V with
u = limm→∞ um . Clearly div u = 0, and hence u ∈ E(D). Since γn is continuous we
conclude that γn (u) = γn (um ) = 0 and hence H ⊂ H• .
Suppose H 6= H• , and denote the orthogonal complement of H in H• by H•• . In virtue
of (A.1.2), every v ∈ H•• is the gradient of some p ∈ H 1 (D); but since p satisfies
∂p ∆p = div v = 0,
= γn (v) = 0,
∂n Σ
we conclude that p is constant and hence H•• = {0}: H = H• .
2
2
A zoo of swirling flows
The cornerstone of many approximation techniques is a family or collection of parametrised functions that is used as a basis for the function space on which the (infinitedimensional) problem is projected. Examples are the Fourier-modes, where the parameters are the (time-dependent) amplitudes of every mode, the nested function
spaces in wavelet theory (Daubechies (1992)) and empirical eigenfunctions (e.g. Lumley (1967), Sirovich (1987), and Holmes et al. (1996)) 1 . The first two techniques work
with functions that have, in principle, nothing to do with the underlying problem.
This may sound as a disadvantage, but who is not looking for a problem-independent
solver that solves every problem?!
This extreme viewpoint makes clear that we could try to use some knowledge of
the system or a priori properties of the solution in order to have a chance to solve
it in a reasonable time. An attempt in this direction is the method of empirical
eigenfunctions, which can be especially suited for very complicated dynamics.
However, we have seen in section 1.3 that many methods are based on one and the
same principle: take advantage of a family of exact solutions U (p) of the unperturbed
problem, p ∈ Rm , and exploit (the structure of) the perturbed equation in deriving
dynamics for the parameters p. As we illustrated in section 1.3 this one principle
got several different names, often historically motivated: WKB-method, multiple(time)scales, adiabatic dynamics and quasi-stationary approach. Again, the common
ground for this method is the presence of a (large) family of parameterised exact
solutions.
This thesis is concerned with Hamiltonian or Poisson systems, a type of system that
widely occurs in Mathematical Physics: surface waves, electro-magnetic theory, elasticity, etc. And as we noted in the previous chapter, section 1.2, the system of Euler
equations describing the motion of an (incompressible) inviscid fluid in a pipe has
a Poisson structure as well. Moreover, we have seen that besides the energy there
are several other conserved quantities that give rise to families of parameterised exact solutions, the relative equilibria (RE), solutions of the constrained extremization
1 This method is also known as the Proper Orthogonal Decomposition (POD) or the KarhunenLoève procedure.
56
2 A zoo of swirling flows
problem
Crit {E(u) | B(u) = b, A(u) = a, Fax (u) = f } .
u
(2.1)
The solutions of (2.1), specifying a family of exact solutions of the Euler equations,
will be the basis for the rest of the thesis. This chapter is meant to distinguish the
several branches of solutions of (2.1) corresponding to various combinations of the
parameters (a, b, f ). Transitions between branches and parameter dynamics can be
triggered by instability (chapter 3), geometric inhomogeneity (chapter 4) or viscous
effects (chapter 5).
Solutions of (2.1) are relevant for the Navier-Stokes equations as well. More specific,
in section 2.1 we show that a solution û of (2.1) may be decomposed as
û± = ũ± + αreϕ + Λez ,
with curl ũ± = ±β −1 ũ± .
(For this moment we do not bother about the precise meaning of the reals α, β and
Λ; assume they are given and fixed.) Then, in section 2.3 we show that û corresponds
to a solution v̂ of the Euler equations (compare with section 1.2):
v̂ ± = ṽ ± + αreϕ + Λez ,
with curl ṽ ± = ±β −1 ṽ ± .
Since
curl curl v̂ ± = curl ±β −1 ṽ ± + 2αez = β −2 ṽ ±
we conclude that
Proposition 2.1 The velocity field w, given by
νt
±
w = ṽ exp − 2 + αreϕ + Λez
β
satisfies
∂t w − w × curl w
div w
= −∇h − νcurl curl w,
= 0,
(NS)
the Navier-Stokes equations (without boundary conditions). The total head is up to a
constant given by
h = (Λ ∓ 2αβ)wz + αrwϕ .
Decomposition of the relative equilibria
The problem in (2.1) can be solved by applying the Lagrange Multiplier theorem (see
e.g. Van Groesen and De Jager (1994)). However, one of the requirements of this
theorem is that we should exclude singular points in the constrained set M := {u ∈
H | B(u) = b, A(u) = a, Fax (u) = f }.
57
Definition A point us ∈ M is called singular if the (linear) functionals δB(us ; ·),
δA(us ; ·) and δFax (us ; ·) are linearly dependent. A non-singular point is called regular.
The adjective ‘singular’ may seem worrying but the velocity field corresponding to us
is regular; in fact, us can be given explicitly:
(
1
0, 2ar, − a1 (b − 2af )r2 + 2a
(b − 2af ) + f
if a 6= 0,
us =
0, 0, 12 α̃r2 + f − 14 α̃
if a = b = 0.
Note the appearance of the special combination b − 2af . This combination will be
called the Beltrami helicity for reasons to be explained in a moment.
We show in section 2.1 that regular points û in M are solutions of (2.1) when
û = β curl û + αreϕ + λez .
(2.2)
with (β, α, λ) the so-called Lagrange multipliers. Apart from the rigid body rotation
αreϕ and the uniform translation λez , the relative equilibria are (pure) Beltrami
flows: exact steady solutions of the Euler equations so that the velocity and vorticity
are collinear everywhere. This fact is made more explicit by the definition of ũ =
û − αreϕ − (2αβ + λ)ez . Together with (2.2), and the fact that
curl αreϕ + λez = 2αez ,
it turns out that ũ is a pure Beltrami flow:
curl ũ = β −1 ũ.
(2.3)
The corresponding variational problem for ũ is
n
o
Crit E(u) | B(u) = b̃ ,
u
with β as Lagrange multiplier.
Hence, a relative equilibrium can be decomposed into a Beltrami component, ũ, a
rigid body rotation and a uniform translation:
û = ũ + αreϕ + Λez ,
with curl ũ = β −1 ũ and Λ = 2αβ + λ. Note that β −1 provides a measure of the
extent to which the velocity field is Beltrami. On the other hand, in section 2.3 it is
identified as a kind of generalised swirl number.
The dependence of this decomposition on the constraints differs in two cases. The simplest case is when û is a non-columnar flow, i.e., depends on at least two coordinates,
including the radial one. Then the amplitude or norm of the Beltrami component is
determined by the Beltrami helicity, b̃ = b − 2af , while α and Λ are determined by
a and f . However, when ũ is a columnar flow, all components are involved in every
functional, and hence the parameter dependence is not so clear.
58
2 A zoo of swirling flows
The Beltrami problem
The Beltrami problem (2.3) is much studied in the context of periodic flows, e.g.
Dombre et al. (1986) and references their in, and for a pipe geometry by Constantin
and Majda (1988), Mahalov et al. (1990) and Dritschel (1991). Landman (1990),
Waleffe (1992) and Takaoka (1996) investigated the (viscous) interaction between
Beltrami flows.
Concerning the classification of Beltrami flows, a very elegant description can be found
in the article by Constantin and Majda (referred to as CM88 in the following) where
the curl-operator is viewed as the square root of −∇2 on the space of divergence-free
functions:
(
(
u± = 12 v ± λ−1/2 curl v satisfies
∇2 v + λv = 0,
√
⇒
div v = 0,
curl u± = ± λu± , div u± = 0.
This viewpoint is taken up in section 2.2, where some technical details concerning
the (non-periodic) boundary conditions are solved. The main conclusions of CM88
apply in our case, one of them being the orthogonality of the Beltrami flows: this
property will be used in the nonlinear stability analysis in section 3.2. An important
new result concerns the case of the columnar Beltrami flows. They are identified as
being special in various respects, since the Lagrange multiplier β is not fixed by the
boundary condition.
From a jungle to a zoo
The jungle of solutions of (2.2) is ordered in section 2.3 where a characteristic numbers
are defined that will indicate to which branch a certain RE belongs. One of these
numbers is the Beltrami helicity b−2af , that we already met; another is a discriminant
∆0 that will indicate which combinations of columnar and non-columnar Beltrami
flows can coexist. Such a combination will be called a multi-mode RE and these are
particularly relevant for the linear stability theory in section 3.1 and the description
of a precessing vortex core in section 4.1. As Dritschel (1991) noted, special solutions
of the stability problem are not only infinitesimal solutions, but are part of exact
solutions of the Euler equations—the multi-mode relative equilibria.
Another remarkable result is that, when β −1 exceeds a certain threshold value such
that multi-mode relative equilibria can exist, the first one appearing is not an axisymmetric travelling wave but a travelling and rotating wave, where the axial wavelength
L is finite and nonzero, independent of the constraints, see figure 2.1. Note that
important models like BHE and QCA cannot capture such a wave and compare this
result with a conjecture of Leibovich (1984) that “from the inviscid theory the nonaxisymmetric breakdown may get earlier unstable (...) and that the bubble (axisymmetric) form emerges from the non-axisymmetric form as a secondary bifurcation”.
In the last section, 2.4, we mention another approach to finding a family of exact
solutions via the flow-force (Benjamin (1962)). This leads to a family of axisymmetric
flows with only axial harmonics. It turns out that the relative equilibria with a general
Beltrami component can only be found from a nonlinear Bragg-Hawthorn equation.
59
2.1 Swirling flows as constrained extremizers of the energy
6
β −1 6
isy
-ax
non
5
mm
etri
c
y
axis
ρ1,1
4
-
0
mm
etri
c
isy
-ax
non
1
2
mm
etr
ic
3
- axial frequency
4
Figure 2.1: The eigenvalue β −1 versus the axial frequency of a multi-mode relative equilibrium solution. The interpretation of β −1 is that it provides a measure of the extent to
which the velocity field is Beltrami. In section 2.3 β −1 is identified as a kind of generalised
swirl number.
2.1
Swirling flows as constrained extremizers of the
energy
We have shown in section 1.2 that special solutions of the Euler equations can be
identified as constrained extremizers of the kinetic energy, i.e., we prescribe the values
of a few (or all) dynamical invariants, define the set M of all possible flows that satisfy
these constraints and look for extremizers of the energy on this set. The present
section is rather involved in order to define this procedure properly. However, the
result, as presented in (2.2) is a simple looking equation, an equation that should be
satisfied by an extremizer.
Recall the function space H of periodic and divergence-free functions in a pipe (subsection 1.2.2, and define a subspace M, being the intersection of the angular momentum,
helicity and flux level sets,
M = {u ∈ H | B(u) = b, A(u) = a, Fax (u) = f }.
The condition that u is an extremizer of the energy E on this set M is expressed by
δE(u; η) = 0 for all variations η in the tangent space of M at u.
(2.1.1)
In section 1.2 we introduced the notion of tangent space,
n
o
Tu M := η ∈ Tu H ∼
= H | ∃ε0 >0 ∀|ε|<ε0 ∃w(ε,η) u + εη + w ∈ M ⇒ ||w||L2 = o(ε) .
(2.1.2)
60
2 A zoo of swirling flows
When M is given as the intersection of level sets of density functionals the formulation
in (2.1.2) is equivalent to
Tu M := {η ∈ Tu H | δB(u; η) = δA(u; η) = δFax (u; η) = 0} .
(2.1.3)
Note that the formulations in (2.1.2) and (2.1.3) give conditions on η in order to be
a proper variation, i.e., such that the distance between u + εη and M is o(ε). These
conditions are in the form of linear constraints, related to the (non-linear) constraints
that are involved in M.
The functionals A and Fax simply yield
Z
1
δA(u; η) =
rη dV = |D|−1 hreϕ , ηiD,
(2.1.4a)
|D| D ϕ
Z
1
δFax (u; η) =
η dV = |D|−1 hez , ηiD.
(2.1.4b)
|D| D z
The quadratic helicity functional is more interesting since it involves boundary contributions. Recalling the result from section 1.2, proposition 1.2,
δB(u; η) = |D|−1 hω, ηiD + |D|−1 hη × u, niΣ .
(2.1.5)
The boundary contribution in (2.1.5) does not vanish automatically and leads in view
of (2.1.3) to an extra condition on η, namely
hη × u, niΣ = 0,
⇔ η k u on Σ.
(2.1.6)
η ∈ H,
The interpretation of (2.1.6) is that changing the direction of the velocity field at the
wall changes the helicity.
The tangent space in (2.1.3) is spanned by the three linear functionals δA(u; ·),
δFax (u; ·) and δB(u; ·) if they are linearly independent. This condition is one of
the premises in the Lagrange Multiplier theorem, so it should be checked.
M and its singular points
Since eϕ and ez are linearly independent, so are δA and δFax , thus the question of
singularity, defined in the introduction to this chapter, boils down to exploring the
equation
δB(u; η) + α̃δA(u; η) + λ̃δFax (u; η) = 0
for all u ∈ M, η ∈ Tu H and α̃, λ̃ ∈ R, or
ω + α̃reϕ + λ̃ez = ∇φ.
(2.1.7)
Taking the divergence of (2.1.7) yields an equation for φ,
∆φ = 0.
(2.1.8)
61
2.1 Swirling flows as constrained extremizers of the energy
On the other hand, when taking the curl of (2.1.7) the gradient term vanishes:
curl ω + 2α̃ez = 0.
(2.1.9)
Equation (2.1.9) shows that ω and thereby u depends solely on r, and hence with
(2.1.7) ∇φ as well. In first instance this leads to
c
φ = c1 z + c2 ϕ + f (r), ∇φ = f 0 (r), 2 , c1 .
r
Combined with (2.1.8) we can determine the unknown function f (r),
f (r) = c3 + c4 log r,
f 0 (r) =
c4
.
r
Due to the regularity-condition at r = 0 we immediately have c2 = c4 =
finally the normalisation Fax (∇φ) = 0 cancels the last relevant coefficient:
Concluding, the singular points in M should satisfy (2.1.7) with ∇φ = 0. It
proven that
(
1
(b − 2af ) + f
if a 6= 0,
0, 2ar, − a1 (b − 2af )r2 + 2a
us =
1
1
2
0, 0, 2 α̃r + f − 4 α̃
if a = b = 0.
0. And
c1 = 0.
is easily
(2.1.10)
Only when a = 0 6= b there is no singular point.
Remarks (i) Note that the singular points correspond to perfectly normal, smooth
flows.
(ii) Note that the extra helicity b − 2af on top of the helicity 2af induced by the rigid
body rotation 2areϕ and the uniform translation f ez determines whether there is a
parabolic component or not.
(iii) Note that for almost every triple (b, a, f ) in parameter space there is a singular
flow. This shows that the problem is rather degenerate when flows are sought depending only on r. We will come back to this degeneracy more often in this chapter.
3
The singular solutions us are collected in Ms ,
Ms = {u ∈ M | u singular} = {us }.
When leaving out this set Ms in our considerations, we can formulate the main
conclusion of this section as follows:
Proposition 2.2 (Lagrange Multiplier theorem) A (regular) point û ∈ M \ Ms
satisfies (2.1.1) if and only if there are real numbers β, α and λ such that
δE(û; η) = βδB(û; η) + αδA(û; η) + λδFax (û; η)
for all η ∈ Tû H ∼
= H.
(2.1.11)
62
2 A zoo of swirling flows
Using the Weyl-Hodge Decomposition (appendix to chapter 1) we conclude with
(2.1.4a,b), (2.1.5) and
Z
1
δE(u; η) =
u · η = |D|−1 hu, ηi
|D| D
that condition (2.1.11) is equivalent to
u − βω − αreϕ − λez = ∇φ in D.
(2.1.12)
The following corollary merely extracts from (2.1.12) the main eigenvalue problem
and shows that ∇φ = 0.
Corollary An extremizer û of the energy on M \ Ms , being a solution of (2.1.12),
consists of a pure Beltrami flow ũ, a rigid body rotation and a uniform translation:
û = ũ + αreϕ + (2αβ + λ)ez ,
where
curl ũ = β −1 ũ.
(2.1.13)
Proof We merely have to show the vanishing of the gradient component. We already
determined the equation for φ, being the Laplace equation (2.1.8). Next, we determine
the boundary conditions for φ. This is based on the requirement that ∇φ does not
contribute to the constraints or, equivalent, that ∇φ is orthogonal to the rest of the
components of û. Evaluating the three different constraints in the RE, using the
specific partition of û mentioned in the corollary, we reach the following conclusions:
• Fax (∇φ) = 0: φ is periodic in z.
• A(∇φ) = 0: φ is periodic in ϕ.
• B(û) = B(û − ∇φ): the radial velocity of the Beltrami component ũ vanishes
at r = 1, being a boundary condition for ũ that will accompany (2.1.13).
• û|r=1 = ũ|r=1 = 0: ∇φ · er = 0 at r = 1.
This last boundary condition for the normal derivative of φ, together with ∆φ = 0 in
D, finally gives ∇φ = 0.
2
Remarks (i) When taking the inner product of (2.1.12) with u we obtain a special
relation between the energy, helicity, angular momentum and flux when evaluated in
a RE:
2E(û) = 2βB(û) + αA(û) + λFax (û).
(2.1.14)
63
2.2 The Beltrami flows in a pipe
(ii) The equation for ũ in (2.1.13) can be obtained from (2.1.12) by setting α = λ = 0.
This means that the angular momentum and flux are no longer constrained, but their
value is merely slaved to the value of the helicity constraint. So, the corresponding
extremalization problem for ũ is
n
o
Crit E(u) | B(u) = b̃ ,
u
and we already remarked that when û is non-columnar, b̃ = b − 2af .
2.2
3
The Beltrami flows in a pipe
We concluded the previous section with the observation that Beltrami flows can be
viewed as constraint extremizers of the energy on level sets of the helicity. In this
section we will start off with (2.1.13) and determine its solutions in H. The leading
thread for the following section is the special relation between the operators −∇2
and curl , the latter being the square root of the former. This is made explicit in the
implication
(
∇2 v + λv
= 0,
div v
= 0,
(
⇒
v ± λ−1/2 curl v satisfies
√
curl u = ± λu± , div u± = 0.
u± =
1
2
±
(2.2.1)
Note that any physical boundary condition should be posed only on u± ; v is merely a
(non-physical) help function. Relation (2.2.1) is used in CM88 as an elegant description of Beltrami flows; they work in Euclidean coordinates and take a periodic box
with sides 2π as their domain, where any square-integrable incompressible velocity
field u can be written as


X
X

u(x) =
uk eik·x  .
λ∈Λ
|k|2 =λ
The Fourier coefficients uk for k ∈ Z3 need to satisfy two conditions,
(i) u−k = uk ,
(ii) uk · k = 0,
in order to guarantee real-valued velocities and incompressibility, respectively. The
set Λ, being a subset of the numbers, contains only elements λ that can be written as
the sum of three squares. In this context of periodic Euclidean flows, ∇2 u simplifies
to −|k|2 u = −λu, and as a result Beltrami flows can be constructed rather easily by
manipulating the Fourier coefficients (CM88). However, this procedure shows clearly
that there is some work to be done when one wants to apply the ideas to swirling
flows in a pipe where the Laplacian does not separate completely. Since the domain of
64
2 A zoo of swirling flows
interest is a cylinder, we recall the curl-operator in cylindrical coordinates. Its action
on u = (ur , uϕ , uz ) is given by
1 ∂ur
∂uϕ
∂ur
∂uz
1 ∂
1 ∂uz
−
er +
−
eϕ +
ruϕ −
ez .
curl u =
r ∂ϕ
∂z
∂z
∂r
r ∂r
r ∂ϕ
In this section we show how the concept of the Fourier expansion is translated, including conditions guaranteeing incompressibility and real-valuedness. Most important,
we state and work out the conditions on the base flows such that u±
λ is a Beltrami
flow. It is rather involved but the main conclusions are gathered in proposition 2.4
(the Beltrami flows), proposition 2.5 and 2.6 (orthogonality result) and proposition 2.8
(energy, helicity, angular momentum and flux of a so-called Beltrami decomposition).
2.2.1
The vector-Helmholtz equation in a pipe
The equation for v in (2.2.1) is called the vector-Helmholtz equation. Since we wish
to find u± , we should apply the boundary condition ur (r = 1) = 0 to u± , and not
to v. Moreover, it implies that in most cases v will not be a solution of the Euler
equations, satisfying the boundary conditions since
ur = vr ± λ−1/2 (curl v) · er = 0 (at r = 1)
6⇒
vr = 0 (at r = 1),
and also the implication the other way around does not hold:
−1/2
(curl v) · er = 0 (at r = 1).
vr = 0 (at r = 1) 6⇒ u±
r = vr ± λ
Remark Note the difference with the case of a periodic box that is treated in Constantin and Majda (1988). There u± inherits the periodicity of v.
3
So let us write v k as


V1,k (r)
v k (r, ϕ, z) = V2,k (r) ei(`ϕ+2πmz/L) ,
V3,k (r)
k = (k, `, m) ∈ N × Z2
(2.2.2)
with no a priori conditions on V1,k at the point r = 1. The number k will play a role
when we determine the eigenvalues: it is an enumeration variable. We will drop the
subscript k as much as possible.
The functions {V1 (r), V2 (r), V3 (r)} ∈ C 2 ([0, 1]; C3 ) should satisfy
1 d
`
(rV1 ) + i V2 + im̃V3 = 0, m̃ = 2πm/L,
(2.2.3)
r dr
r
to guarantee incompressibility.
The functions V1 , V2 and V3 should satisfy the following system of coupled ODE’s


V
`V


(2.2.4a)
−λV1 = ∆V1 − 21 − 2i 22 ,


r
r


−λv k = ∇2 v k ⇔ −λV2 = ∆V2 − V2 + 2i `V1 ,
(2.2.4b)

r2
r2





−λV3 = ∆V3 ,
(2.2.4c)
65
2.2 The Beltrami flows in a pipe
with
1 d d
`2
r
− 2 − m̃2 ;
r dr dr r
do note the coupling between V1 and V2 in (2.2.4a) and (2.2.4b). Introduce
∆=
β 2 = 1/λ,
β > 0.
Equation (2.2.4c) is uncoupled and can be solved explicitly in terms of Bessel functions:
p
p
V3 (r) = AJ` (rβ −1 1 − β 2 m̃2 ) + ÃY` (rβ −1 1 − β 2 m̃2 ).
Since V3 should be regular at the origin we omit the term involving Y` . Using this
result for V3 , we free V2 from the incompressibility condition (2.2.3) and substitute
i d
rm̃
(rV1 ) −
V
` dr
` 3
into equation (2.2.4a). This leads to a second order, inhomogeneous ODE for V1 :
V2 (r) =
β 2 r2
d2 V1
dV
+ 3β 2 r 1 + V1 r2 (1 − β 2 m̃2 ) − β 2 (`2 − 1) =
2
dr
dr
p
− 2iβ 2 rm̃AJ` (rβ −1 1 − β 2 m̃2 ). (2.2.5)
Lemma 2.3 The regular solution of equation (2.2.5) is given by
V1 (r) =
with y = rβ −1
p
1 − β 2 m̃2 .
B
iβ m̃A
J`0 (y)
J` (y) + p
r
1 − β 2 m̃2
Proof First p
we cast (2.2.5) into standard form. Thereto we introduce q(r) = V1 r
and y = rβ −1 1 − β 2 m̃2 :
y2
dp
2iβ 2 y 2 m̃A
dp
+y
+ p(y 2 − `2 ) = −
J` (y).
2
dy
dy
1 − β 2 m̃2
(2.2.6)
with p(y) = q(r(y)). Two independent solutions of the homogeneous equation corresponding to (2.2.6) are given by
p1 (y) = J` (y),
p2 (y) = Y` (y).
The Wronskian of p1 and p2 is given by (Watson (1944))
J`+1 (y)Y` (y) − J` (y)Y`+1 (y) =
2
.
πy
(2.2.7)
Using standard techniques for solving (second order) ODE’s (see e.g. Kamke (1961)),
the general solution of (2.2.6) is given by
p(y) =BJ` (y) + B̃Y` (y)
Z y
Z y
iπβ 2 m̃A
0
0 0
0
0 2 0
0
+
J` (y)
Y` (y )J` (y )y dy − Y` (y)
J` (y ) y dy
1 − β 2 m̃2
0
0
The integrals can be evaluated exactly in terms of Bessel functions and using relation
66
2 A zoo of swirling flows
(2.2.7) once more this leads to
p(y) = BJ` (y) + B̃Y` (y) +
iβ 2 m̃A
(`J` (y) − yJ`+1 (y)) .
1 − β 2 m̃2
Re-introduction of r, q and finally V1 and remembering that Y` is singular at the
origin finishes the proof.
2
At this point we can compute V2 in two ways. First of all we can again use the
incompressibility condition, leading to
V2
=
=
i d
rm̃
(rV1 ) −
V
` dr
` 3
p
iB 1 − β 2 m̃2 0
β m̃A`J` (y)
J` (y) − p
.
`β
1 − β 2 m̃2 y
On the other hand, we could substitute (2.2.6) into (2.2.4b) and solve for V2 . It is
easy to show that both ways lead to the same answer.
We conclude that there are two linearly independent classes of solutions, one corresponding to the amplitude A,


β m̃
0
J` (y) 
 ip


1 − β 2 m̃2




β
m̃
`J
(y)
`
(2.2.8)
,
1 V = A − p


2
2
y
1
−
β
m̃




J` (y)
and one involving the amplitude B,
p

1 − β 2 m̃2
J
(y)
`


 p βy

 1 − β 2 m̃2


0
V
=
B
(2.2.9)
J` (y)
2
i
;
`β




0
p
y denotes as before rβ −1 1 − β 2 m̃2 . Since 1 V and 2 V are not the actual velocity
fields, but merely a source to construct the velocity fields in a nice way, we will not
apply the boundary condition yet. The formula (2.2.8) and (2.2.9) for 1 V and 2 V
are also valid for the case m = 0 (z-independent solutions). The case ` = 0 should be
solved separately, yielding




m̃β
p
−
i
J
(y)
1


 0 
1 − β 2 m̃2












1V = A 
 , 2 V = B J1 (y) .
0








0
J0 (y)
67
2.2 The Beltrami flows in a pipe
Remark It is easily observed that 1 V and 2 V are orthogonal. And it will not be a
surprise that curl 1 V and 2 V are linearly dependent,
curl 1 V =
i`
V.
1 − β 2 m2 2
But these two observations lead together to the remarkable fact that the helicity of
both 1 V and 2 V vanishes!
3
2.2.2
Construction of Beltrami flows
Now that we constructed the eigenfunctions of the Laplace operator ∇2 , it is time to
extract the Beltrami flows from them, by manipulating the components of 1 V . This
manipulation is (implicitly) defined in (2.2.1). More precise, let
L = v | ∇2 v + λv = 0 for some λ ∈ R, div v = 0 .
When decomposing an element v ∈ L like in (2.2.2), the operators ∇2 and curl are
‘projected’ to simpler operators, acting on a simpler space. The simpler space is
denoted by L`,m , and is defined by


V (r) satisfies (2.2.3) 

L`,m = V (r) = (V1 , V2 , V3 ) and (2.2.4a–c) for some


λ = λ`,m > 0
Remember the remark that in the case of a periodic box a Beltrami flow could be
found from an arbitrary (single-mode) flow by manipulating its Fourier coefficients
(CM88). Here we still have to deal with the vector function V , depending on r, and
hence the manipulation or projection operators will be thought as acting on L`,m . The
notation is a bit more involved than in CM88 because of the nature of the coordinates.
Note that we set β > 0; the negative eigenvalues are taken care of by switching the
sign in (2.2.1).
Definition Consider (`, m) ∈ Z2 , β = λ−1/2 > 0 fixed. The action of the map Q,
assigning to v ∈ L a new element 12 (v ± βcurl v) ∈ L reduces to a map Q±
(D,`,m),β :
L`,m → L`,m , defined by

i`
±
β
−
i
m̃V
V
V
2 
 1
r 3




1
±

Q(D,`,m),β (V ) :=  V2 ± β (im̃V1 − DV3 ) 

2



i`
V3 ± β D∗ V2 − V1
r

with
Df =
df
,
dr
D∗ f =
1
D(rf ).
r
68
2 A zoo of swirling flows
The maps Q± have several properties (see CM88, proposition 1.1). It follows that
Q±
(D,`,m),β (V ) exp{i(`ϕ + m̃z)} are the Beltrami flows.
Proposition 2.4 The dependence of a single-mode Beltrami flow, U ± (r) exp{i(`ϕ +
m̃z}, on the radial coordinate r is, up to a multiplicative constant, given by
U±
:= Q±
(D,`,m),β (1 V )


β m̃J`0 (y) ± `J` (y)/y
p
i




1 − β 2 m̃2

1  ±J 0 (y) + β m̃`J` (y)/y 

`
=
−

p


2
1 − β 2 m̃2




J` (y)
(2.2.10)
p
with y = rβ −1 1 − β 2 m̃2 ; ` and m = m̃L/2π denote the wave numbers in azimuthal
and axial direction respectively.
Remark The case β = m̃−1 is treated separately here. It is rather straightforward
to show that for that value of β the Helmholtz solution equals
(
Br`−1 if ` 6= 0,
1 m̃Ar`+1
`
V3 = Ar , V1 = −
+
2 `+1
0
if ` = 0,
and
1 mAr`+1
V2 = −
−
2 `+1
(
Br`−1
0
if ` 6= 0,
if ` = 0.
However, when computing the corresponding Beltrami field, it turns out that it cannot
satisfy the boundary condition, and hence β = m̃−1 is not an eigenvalue.
3
Fixing the eigenvalues β
±
Notation We will write U ±
(k,`,m) as a shorthand for U β ±
.
3
(k,`,m)
In order to determine β we apply the boundary condition at r = 1, namely that the
radial velocity vanishes.
When ` and m are not both zero, we get a nontrivial relation that determines β,
namely
p
p
`βJ
(
1 − β 2 m̃2 /β)
0
`
p
β m̃J` ( 1 − β 2 m̃2 /β) ±
= 0,
(2.2.11)
1 − β 2 m̃2
with the +-sign for the +-flow. We can distinguish four cases namely
69
2.2 The Beltrami flows in a pipe
(i) m = 0 (non-axisymmetric, non-columnar): J` (β −1 ) = 0, leading to
β(k,`,0) = ρ−1
`,k ,
the k-th zero of J` . Since J−` = (−1)` J` we have the relation β(k,−`,0) =
β(k,`,0) = β(k,|`|,0) .
(ii) ` = 0 (axisymmetric, non-columnar):
p
p
J00 (β −1 1 − β 2 m̃2 ) = −J1 (β −1 1 − β 2 m̃2 ) = 0,
leading to
2
β(k,0,
m̃) =
1
,
ρ21,k + m̃2
implying β(k,0,−m) = β(k,0,m) = β(k,0,|m|) .
Remark Note that the axial velocity of U ±
(k,l,0) (non-axisymmetric) vanishes at
the boundary, and similarly the tangential velocity of U ±
(k,0,m) (axisymmetric).
3
(iii) ` 6= 0 and m 6= 0 (non-columnar): (2.2.11) in full. Also in this case we are
able to introduce an enumeration variable k. In the previous two cases the
indefinite sign did not play a role but when ` and m are both non-zero we need
to distinguish between the two. The ones corresponding to solutions of (2.2.11)
+
and similarly with
with the plus-sign, (2.2.11)+ for short, are denoted by β(k,`,m)
−
the minus-sign. Note however that although β(k,`,m) > 0, the corresponding
−
eigenfunction has a negative eigenvalue, being −1/β(k,`,m)
. Moreover, we have
±
a lower-bound on β(k,`,m) , namely
±
1/β(1,`,
m̃) > m̃ = 2πm/L.
The counting is illustrated in figure 2.2. We have the following relations2
±
∓
±
β(k,−`,
m̃) = β(k,`,m̃) = β(k,`,−m̃)
±
±
β(k,−`,−
m̃) = β(k,`,m̃) .
(iv) ` = m = 0 (columnar): This case can be viewed as a limiting case of either the
axisymmetric one (` = 0), suggesting
β(k,0,0) =
1
ρ1,k
,
2 These relations play a role at the background when we take the complex conjugate of a (complex)
valued Beltrami flow; they are part of the conditions that guarantee real-valuedness.
70
2 A zoo of swirling flows
1
0.8
0.6
0.4
0.2
-0.2
+
- β −1
2
4
–
0.4
0.2
k=1
+
4
–
?
1
0.8
0.6
0.4
0.2
k=2
6
6
k=1
8
10
(a)
- β −1
2
-0.2
6
?
8@
I 10
@
@
+
4
-0.2
k=2
–
(b)
- β −1
6
8
10
(c)
Figure 2.2: Plots of the right-hand side of (2.2.11) versus β −1 for different values of ` and
m̃. (a) ` = 1, m̃ = 1; (b) ` = 2, m̃ = 1; (c) ` = 1, m̃ = 2. The counting is illustrated in the
(b)-plot. Note that β −1 = m̃ is a zero for all values of m̃ when ` > 1.
or of the non-axisymmetric one (m = 0 or L → ∞), suggesting
β(k,0,0) =
1
.
ρ0,k
On the other hand, since the concerning Beltrami flow is given by


U±
(k,0,0)
0





1

−1
= ±J1 (rβ )
,
2



−1
J0 (rβ )
(2.2.12)
any value of β will do, in view of the fact that the radial velocity of U ±
(k,0,0)
vanishes identically.
2.2.3
Properties of the Beltrami flows
Orthogonality
Using the preceding information on β we have the following important facts.
71
2.2 The Beltrami flows in a pipe
Proposition 2.5 Let (`, m) ∈ Z2 \ {0, 0} be the wave number of the Beltrami flow
(proposition 2.4) in azimuthal and axial direction. Let U and Û denote the radial
dependence of two of such Beltrami flows:
curl u = β −1 u,
curl û = β̂
−1
u = U (r) exp{i(`ϕ + m̃z)},
û,
û = Û (r) exp{i(`ϕ + m̃z)}.
Then u and û are orthonormal when β 6= β̂.
Proof Using the definitions of u and û and a vector identity we have
hu, ûiD = hu, β̂curl ûiD
= hβ curl u, ûiD
= hu, β curl ûiD + βhû × u, niΣ
and hence
1
1
−
β β̂
hu, ûiD = hû × u, niΣ .
The only non-vanishing part of the boundary contribution involves integration over
r:
1
Z L/2 Z 2π
hû × u, niΣ =
r(ûϕ uz − ûz uϕ ) dϕdz.
(2.2.13)
−L/2 0
0
With proposition 2.4 and the shorthands Y and Ŷ ,
Y = β −1
q
p
1 − β 2 m2 , Ŷ = β̂ −1 1 − β̂ 2 m2 ,
we can write the boundary contribution in the right-hand side of (2.2.13) as follows:
R.H.S. = −
±J`0 (Ŷ ) + β̂ m̃`J` (Ŷ )/Ŷ
±J`0 (Y ) + β m̃`J` (Y )/Y
q
p
· J` (Y ) + J` (Ŷ ) ·
1 − β 2 m̃2
1 − β̂ 2 m̃2
−
`J` (Ŷ )
β̂ m̃
+ β̂ m̃`J` (Ŷ )
−
`J` (Y )
β m̃
+ β m̃`J` (Y )
q
p
· J` (Y ) + J` (Ŷ ) ·
Y 1 − β 2 m̃2
Ŷ 1 − β̂ 2 m̃2
`J` (Y )J` (Ŷ )
1
`J` (Y )J` (Ŷ )
1
= q
·
− β̂ m̃ − p
·
− β m̃
β m̃
Y 1 − β 2 m̃2
β̂ m̃
Ŷ 1 − β̂ 2 m̃2
(!)
=−
= 0,
and hence the result. We used at (!) the fact that the normal velocities ur and ûr
vanish at r = 1 and by virtue of that we expressed J`0 in terms of J` , using (2.2.10).
2
72
2 A zoo of swirling flows
Proposition 2.6 Consider the set of columnar Beltrami flows in (2.2.12). The
columnar Beltrami flows are pairwise orthogonal iff either the azimuthal velocity or
the axial velocity at the boundary vanishes:
−1
β(k,0,0)
= ρ0,k
ór
−1
β(k,0,0)
= ρ1,k
(2.2.14)
Proof The proof is along the same lines as for the previous proposition. However,
since Ur vanishes identically, it cannot be used to rewrite the right-hand side of
(2.2.13). Hence, the boundary term itself should vanish directly, implying (2.2.14).
Note that requiring the quotient of the axial velocity and the azimuthal velocity at the
boundary to be independent of β is not enough. This will imply only orthogonality
between two different ‘plus’-flows or ‘minus’-flows.
2
Notation When it is not necessary to emphasize the vector k, we will use the
3
notation u±
β for a Beltrami flow.
Completeness
The completeness of the Beltrami flows is more than just another property. It will be
used to decompose a perturbation in section 3.2, where the nonlinear (constrained)
stability of a columnar Beltrami flow is studied. The adjective ‘constrained’ indicates
that the perturbation should not change the values of the constraints, with other
words,
Fax (base flow + perturbation) = Fax (base flow),
A(base flow + perturbation) = A(base flow),
B(base flow + perturbation) = B(base flow).
Note that since Fax and A are linear we have a direct condition on the perturbation,
namely that its axial flux and angular momentum vanish.
To obtain the completeness result, we note that a Beltrami flow can never be a
potential flow, since curl (∇φ) ≡ 0. Hence, in order to show the completeness of the
Beltrami flows in a subspace of H, we have to get rid of the potential flows in H.
Since we showed in section 2.1 that H contains no periodic potential flows, we only
focus on
c c
φ = c1 z + c2 ϕ + c3 + c4 log r, ∇φ = 4 , 2 , c1 .
r r
Due to the regularity, c2 = c4 = 0, and hence when we add the condition Fax (u) = 0
to H we have eliminated all potential flows: this is exactly one of the conditions on a
perturbation to start the proof of constrained stability!
Let H0 denote the subspace
H0 = {u ∈ H | Fax (u) = 0} .
All Beltrami flows are automatically in H0 , except the columnar flows:
73
2.2 The Beltrami flows in a pipe
−1
Lemma 2.7 A columnar Beltrami flow u±
β has vanishing axial flux iff β = ρ1,k .
It can be shown (Mahalov et al. (1990)), using the compactness of the (Stokes) operator A = P (∇ × (∇ × ·)) (with P the projection onto H; section 1.2) that the Beltrami
flows form a complete orthogonal basis for H0 .
Definition Let u ∈ H0 . The orthogonal expansion
X
+
− −
u=
(A+
k uβ + Ak uβ )
(2.2.15)
β=βk ∈B
with
A±
k =
hu, u±
β iD
±
hu±
β , uβ iD
is called the Beltrami decomposition of u. The set B consists of all eigenvalues βk± .
The convergence of the series in (2.2.15) takes place in L2 (D)3 ∩ H0 . Some facts on
the Beltrami decomposition are listed in the following proposition (CM88), which will
be of main importance in chapter 3 on (constrained) stability.
P
+
Proposition 2.8 Let u ∈ H and Fax (u) = 0, i.e., u ∈ H0 . Let u = β∈B(A+
k uβ +
− −
Ak uβ ) be its Beltrami decomposition (see definition above).
(i) The Beltrami decomposition of curl u is
X
+
− −
curl u =
β −1 (A+
k uβ − Ak uβ ).
β∈B
(ii) The energy and the helicity of u are given respectively by
X
+
− 2
−
2
E(u) =
(A+
)
E(u
)
+
(A
)
E(u
)
,
k
β
k
β
β∈B
B(u) =
X
+
− 2
−
2
β −1 (A+
k ) E(uβ ) − (Ak ) E(uβ ) .
β∈B
(iii) The angular momentum of u is
X J (ρ1,k ) 0
A+
− A−
.
A(u) = −
(k,0,0)
(k,0,0)
ρ1,k
k∈N
The proof is straightforward.
It is possible to express the energy of u±
β explicitly in terms of Bessel functions. One
immediate result is that the energy of the plus and minus flow are identical when the
same eigenvalue is considered. With (2.2.1) we have
2
1 |u± |2 = v ± λ−1/2 curl v 4
1 2
=
|v| + λ−1 |curl v|2 ± 2λ−1/2 v · curl v ,
4
74
2 A zoo of swirling flows
with ∇2 v + λv = 0. As we noted in the remark on page 67, the helicity-density
v · curl v corresponding to v vanishes and hence the result.
Lemma 2.9 The energy of a single-mode Beltrami flow is given by
Z
1 1
)
=
|u± |2
E(u±
β
|D| 2 D β
2J` (Y )J`+1 (Y )`
1 J`0 (Y )J` (Y ) J` (Y )2 + J`+1 (Y )2
+
=
−
8
Y
1 − β 2 m̃2
Y (1 − β 2 m̃2 )
p
with Y = β −1 1 − β 2 m̃2 .
(2.2.16)
Proof The only thing to determine is the radial integration. With the explicit
formula for U ±
β in (2.2.10), we find that the essential part of the integrand equals
2 2
2
1
β2
β m̃ + 1
`
2
0
2
2
yJ
|u±
|
rdr
=
(y)
+
yJ
(y)
−
1
+
`
`
β
4 1 − β 2 m̃2 1 − β 2 m̃2
y2
2
J` (y)J`0 (y)
2
yJ
(y)
±
4β
m̃`
dy
1 − β 2 m̃2 `
1 − β 2 m̃2
The first term between braces is solved by integration by parts and the substitution of
Bessel’s equation. The second is a bit more involved (Gradsteyn and Ryzhik (1980)):
Z
1
yJ` (y)2 dy = y 2 J` (y)2 + J`+1 (y)2 − y`J` (y)J`+1 (y).
2
Note that the right-hand side is symmetric for the change ` → −` as it should be
(not obvious!). Finally, the third term is straightforward. Altogether, this leads to
the (intermediate) result
1
β 2 m̃2 + 1
2
1 2
0
E(u±
)
=
J
(Y
)J
(Y
)
+
Y J` (Y )2 + J`+1 (Y )2 −
`
`
β
2
2
2
4
8 Y (1 − β m̃ )
β Y
2
2β m̃`
2
J
`Y J` (Y )J`+1 (Y ) ± 2
(Y
)
(2.2.17)
Y (1 − β 2 m̃2 ) `
When `, m 6= 0 we can get rid of the ±-term by using information on u±
β,r . Again
with (2.2.10) and the boundary condition on the radial velocity, the last term between
braces in (2.2.17) is replaced by
±
2β m̃`
2m̃2 β 2
2
J
(Y
)
=
−
J 0 (Y )J` (Y ),
`
Y 2 (1 − β 2 m̃2 )
Y (1 − β 2 m̃2 ) `
and hence the final result.
2
Remark Formula (2.2.16) holds as well for the case (`, m) = (0, 0) with arbitrary β
(columnar). This leads to following interesting observation. Note that using proposition 2.8 we obtain a relation between the energy and the helicity
±
± 2
±
± ±
E(A±
k uβ ) = (Ak ) E(uβ ) = ±βB(Ak uβ ) = ±β b̃.
75
2.3 Ordering of the relative equilibria
Restricting to the columnar flows (with arbitrary β) we observe that when the helicity
b̃ is fixed and nonzero, the energy is unbounded in the limit β −1 → 0. On the
other hand, when β → 0 (or β −1 → ∞), the energy vanishes, implying that the
corresponding Beltrami flow tends to the zero-profile in the energy (= L2 ) norm. 3
2.3
Ordering of the relative equilibria
Now that we have solved the Beltrami component of the RE, cf. (2.1.12) and (2.1.13),
it is time to clarify the role of the constraints. Recall that the RE is a solution of
Crit {E(u) | B(u) = b, A(u) = a, Fax (u) = f } ,
u
(2.3.1)
i.e.,
RE ∈ M = {u ∈ H | B(u) = b, A(u) = a, Fax (u) = f }.
Moreover, the RE can be decomposed in a Beltrami component, a rigid body rotation
and a uniform axial translation. To fix notation in the following, we write β = βk
and denote a non-columnar RE by
± ±
û±
k = Ak uβ + αreϕ + Λez ,
Λ = ±2αβ + λ,
(2.3.2)
and in case the RE is columnar we leave out the subscript k = (k, 0, 0):
û± = A± u±
β + αreϕ + Λez .
(2.3.3)
In this section we will order the jungle of solutions of (2.3.1). The first ordering is
obtained by making distinction between columnar and non-columnar RE.
• In the latter case, the Beltrami component u±
k in the RE plays no role in the
linear constraints A and Fax ; in that case α and λ are completely determined by
a and f . The value of the quadratic helicity constraint determines the amplitude
A±
k but only after subtracting the helicity generated by αreϕ + Λez from b: the
Beltrami helicity b − 2af .
• The results for the columnar RE (where β can be chosen arbitrarily) are not so
clear: all components play a role in every constraint. However, the fact that β
can be viewed as a continuous parameter gives us the possibility of investigating
the limits β → 0 and β −1 → 0. It turns out that, depending on the Beltrami
helicity, only one branch of columnar flows has finite energy for all β. Moreover,
when β −1 → 0, the finite-energy-limit flow is identified as an element of Ms
(the singular points—parabolic profiles—in the constrained set M). The special
choice β −1 = ρ1,k , that was needed in order to have completeness of the Beltrami
flows, and the values of β that follow from the condition that ûϕ vanishes at
the boundary connect the present work with previous results (Van Groesen et
al. (1995)), Fledderus and Van Groesen (1996a,b)).
76
2 A zoo of swirling flows
Finally, we investigate the possibility that a columnar RE can co-exist with a noncolumnar RE, the so-called multi-mode RE. The importance of these flows for chapter
3 on stability is already mentioned in the introduction to this chapter. An important
±
parameter in this case is a discriminant that relates the amplitudes A±
k and A .
2.3.1
Non-columnar relative equilibria
First we treat the case where flows depend on at least two coordinates. This may
seem the most involved one, but it turns out that only then the constraints Fax and
A on the one hand and B and the other hand do not mix. Hence, the algebra turns
out to be rather simple.
In the following lemma the values of α, Λ and the amplitude A±
k of the Beltrami
component are computed in terms of the values of the constraints b, a and f .
Lemma 2.10 Let û±
k be a single-mode RE according to (2.3.2), i.e., let (k, `, m) ∈
N × (Z2 \ {0, 0}) be fixed but arbitrary. Then û±
k ∈ M if
λ = f ∓ 4aβ,
b − 2af
2
(A±
.
k ) = ±β
E(u±
β)
α = 2a,
Λ = f,
Proof Direct computations show that
Z
Z
1
1 1
±
Fax (û±
)
=
û
=
(±2αβ + λ) 2πrdr = ±2αβ + λ =: f,
k
|D| D k,z
π 0
Z
Z
1
1 1 2
1
±
±
A(ûk ) =
rû
=
αr 2πrdr = α =: a,
|D| D k,ϕ π 0
2
and hence
α = 2a,
λ = f ∓ 4aβ,
Λ = f.
In order to determine the helicity, we use that curl (αreϕ + Λez ) = 2αez . Hence
Z
1 1
±
û± · ω̂ ±
B(ûk ) =
k
|D| 2 D k
Z 1 1
±
−1 ± ±
=
Ak uβ + 2αez
A±
k uβ + αreϕ + Λez · ±β
|D| 2 D
Z 1
1
± 2
±
−1
= ±β (Ak ) E(uβ ) +
2α(±2αβ + λ) 2πrdr
2π 0
±
2
= ±β −1 (A±
k ) E(uβ ) + 2af =: b,
implying the remaining relation.
2
Remarks (i) We observe that for given f , a and b we have either the plus-flow or
the minus-flow. Note that, due to the rigid body rotation and/or the net flux, this
77
2.3 Ordering of the relative equilibria
dichotomy does not coincide with positive or negative b, but with the sign of the
Beltrami helicity.
(ii) In figure 2.3 we have drawn a Beltrami vector field and the corresponding integral
curves in a cross-section, i.e., the axial velocity is omitted.
3
1
1
0.8
0.6
0.5
0.4
0.2
0
-1
-0.5
0.5
1
−0.2
−0.4
-0.5
−0.6
−0.8
−1
−1.5
−1
−0.5
0
0.5
1
1.5
-1
+
Figure 2.3: The velocity field (û+
r , ûϕ , 0) (at the left) and the corresponding integral curves
(at the right). The values of the wave numbers are ` = 1 and m̃ = 1 with corresponding
+
β(1,1,1)
≈ 0.31. We have not included a rigid body rotation (α = 0).
We have to determine what the status of û±
k is in relation to solutions of the Euler
equations: what is the RE solution corresponding to û±
k ? Here we touch upon the
dynamical character of the constraints, that we explained in section 1.2. There we
defined the notion of relative equilibrium solution, a dynamic solution of the Euler
equations that is obtained from the static relative equilibrium. The relation between
the two is given by the combination of the axial and angular momentum flow. Indeed,
the nontrivial Euler flow ΦE
t evaluated in a relative equilibrium simplifies to
(!)
F A
A F
ΦE
t = Φλt Φαt = Φαt Φλt = ‘translation + rotation’,
RE
where we used at (!) the fact that the fwo flows commute.
±
Proposition 2.11 The RE solution v̂ ±
k corresponding to ûk is given by
n
o
± ±
i(`(ϕ−αt)+m(z−λt))
v̂ ±
+ αreϕ + Λez .
k = < Ak U k (r)e
This exact solution of the Euler equations has total head (up to a constant)
±
±
+ αrv̂k,ϕ
.
h = λv̂k,z
Although the relevance of these rotating and translating swirling flows may not be
clear at the moment, do note that we have the speeds with which they rotate and
78
2 A zoo of swirling flows
translate explicitly at our disposal (lemma 2.10). This lemma reveals that a nonaxisymmetric multi-mode solution always rotates, but that it not necessarily translates: λ may be zero for non-trivial values of a and f . The latter observation will
play a role when we need a description of the critical state of a columnar relative
equilibrium (section 3.1 and 4.3).
2.3.2
Columnar relative equilibria
In the previous section we obtained the result that the condition of orthogonality pins
down the eigenvalues in the columnar case, namely
−1
= ρ0,k
β(k,0,0)
−1
ór β(k,0,0)
= ρ1,k .
In the following we will show another possibility for choosing β, by adding the condition that the energy has an extremum in û±
k.
Since this columnar case has been the subject of previous publications, we will relate
the exposition taken up in this thesis to the specific notation in Van Groesen et
al. (1995) and Fledderus and van Groesen (1996a,b).
The columnar RE with arbitrary β
Suppose β is arbitrary but fixed. In that case the expression for the RE in (2.3.3) is
given by
1 ±
1 ±
±
−1
−1
û±
) + αr, û±
) + Λ.
(2.3.4)
r ≡ 0, ûϕ (r) = ± A J1 (rβ
z (r) = A J0 (rβ
2
2
The proof of the following lemma is a tedious yet straightforward manipulation of
integrals and Bessel functions.
Lemma 2.12 The multipliers α and λ and the amplitude A± in (2.3.4) are related
to the values of the constraints as follows. The amplitude A± satisfies a quadratic
equation,
c2 (A± )2 + c1 A± + c0 = 0,
(2.3.5)
with
c0 = 2af − b,
1
c±
2aJ2 (β −1 ) ∓ f J3 (β −1 ) ,
1 =
2
1
±
c2 = ± β −1 J2 (β −1 )2 − 5βJ2 (β −1 )J3 (β −1 ) + J3 (β −1 )2 .
4
±
2
The discriminant of (2.3.5) is denoted by ∆± := (c±
1 ) − 4c0 c2 . The indefinite term
±
−4c0 c2 can be made positive whatever values of the constraints and β are chosen
±
by selecting the right sign of c±
2 . Given solutions A , the multipliers α and λ are
obtained from
α = 2a ∓ 2A± βJ2 (β −1 ), λ = f ∓ 4aβ + A± βJ3 (β −1 ), Λ = f − A± βJ1 (β −1 ).
79
2.3 Ordering of the relative equilibria
Remark Note the symmetry
(a, b, ±) −→ (−a, −b, ∓).
In the following the angular momentum will be assumed non-negative if not stated
otherwise.
3
Using this lemma, we can express the energy in terms of the values of the constraints,
and β.
1
1
1
1
E(û± ) = ±βB(û± ) + αA(û± ) + λFax (û± ) = ±βb + αa + λf.
2
2
2
2
(2.3.6)
Substituting the relevant relations for the multipliers, we obtain
1
E(û± ) =: E(a, b, f ; β) = a2 + f 2 + e(β −1 )(A± )2 ,
2
with
J (β −1 ) J (β −1 )
2
3
1
−1
e(β ) = 4 0 −1
J2 (β ) J30 (β −1 )
.
The value function
The energy, evaluated in a RE, leads to the so-called value function of the extremal
problem Crit{E | B = b, A = a, Fax = f }. Besides β, that was arbitrary but fixed
at the start, the value function depends on the constraints a, b and f . It has some
special properties, in general, and one of these is the relation between the value of
the constraint and its multiplier. In general, it holds that
∂(value function)
= multiplier.
∂(constraint)
Let us illustrate and prove this with the help of a generic example. Denote by U (γ)
the solutions of
Crit {H(u) | I(u) = γ}
u
and let λ be the corresponding Lagrange multiplier, i.e.,
U (γ) satisfies: δH(u) = λδI(u).
The value function will be denoted by H(γ),
H(γ) := H(U (γ)).
80
2 A zoo of swirling flows
Differentiating H(γ) with respect to γ leads to
dH
=
dγ
dU
dU
(!) dI(U )
= λ δI(U ),
=λ
δH(U ),
= λ.
dγ D
dγ D
dγ
Note that every step but the one marked with (!) applies even when the domain has
a fixed boundary. There is a problem when we need an integration by parts, as we
do for the helicity. To be precise, in computing ∂E/∂b, we arrive at
±
∂E
± ∂ û
−1
= ±|D| β ω̂ ,
∂b
∂b D
since
∂ û±
∂ û±
= λ ez ,
≡ 0.
α reϕ ,
∂b D
∂b D
Obviously we have
±
±
∂ 1 −1 ± ±
1
1
∂ ω̂
± ∂ û
±
−1
−1
|D| hω̂ , û iD = |D|
, û
1=
ω̂ ,
+ |D|
.
∂b 2
2
∂b D 2
∂b
D
Since curl and ∂/∂b commute, we can perform an integration by parts, leading to
−1
1 = |D|
∂ û±
ω̂ ,
∂b
±
1 −1
∂ û±
±
+ |D|
,
û × n,
2
∂b Σ
D
and hence
∂E
∂ û±
1
.
= ±β ∓ β |D|−1 û± × n,
∂b
2
∂b Σ
(2.3.7)
The interpretation of (2.3.7) is that only when the (infinitesimal) change in helicity,
induced by a variation of u perpendicular to M, is coming from the interior, the (infinitesimal) energy change is related to the multiplier β. In general, ∂u± /∂b changes
the direction of u± on Σ and hence there is a contribution of the boundary to the
helicity change as well. Only when we prescribe in the extremal problem the angle
between the axial and azimuthal velocity at the boundary Σ, say
û±
z
= tan θ,
û±
ϕ
the boundary contribution vanishes identically since
∂ û±
z
∂b
∂ û±
û±
ϕ
= z± = tan θ.
∂b
ûϕ
2.3 Ordering of the relative equilibria
81
The boundary term can be evaluated, yielding
1
∂ û±
1
∂A±
8β 2 f ∓ 4βa − f +
∓ β|D|−1 û± × n,
= β J1 (β −1 )
2
∂b Σ
2
∂b
±
−1 ∂A
− βJ0 (β )
(2βf ∓ a) .
∂b
It is easy to check that for the non-columnar RE it holds that ∂E/∂b = ±β, and
hence they can be treated as a generic case as far as the value function concerns (see
proposition 2.10) while the columnar RE can be marked ‘degenerate’.
Limit flows; recovering the singular points
In order to plot the energy versus β −1 , we have to determine A± from (2.3.5). This
equation gives in principle four solutions:
p
−c±
∆±
1 +T
A±
=
, T = 1 or T = −1;
T
±
2c2
(
p
p
∆+
∆+
−c+
−c+
1 +
1 −
+
+
explicitly:
A1 =
, A−1 =
,
+
+
2c2
2c2
)
p
p
−
−
−
−
+
∆
−
∆
−c
−c
1
1
A−
, A−
,
1 =
−1 =
2c−
2c−
2
2
but for most parameter combinations only two are real. Are their any special values
of β −1 that are ‘practically’ important and that can indicate which solution of (2.3.5)
is most relevant? Ignoring the amplitude A± we observe that, when β −1 → 0, the
velocity profiles tend to smooth out and the nonlinear character of the Beltrami
component disappears. Hence, we call this the long-wave limit. And at the other
extreme, when β −1 → ∞, the RE develops high-frequency waves, but only localised
near the axis since the amplitude of a Bessel function decreases as its argument tends
to infinity: see figure 2.4.
Long wave limit: β −1 → 0. Observe that the ‘long wave limit’ (β −1 → 0) is able
to produce a rather nice looking flow. Going back to e(β −1 ), we note that
e(β −1 ) = constant · (β −1 )4 + O((β −1 )6 ) as β −1 → 0.
To determine the behaviour of A± as a function of β −1 when β −1 → 0, we make series
expansions of the various coefficients in (2.3.5), leading to the following results.
Lemma 2.13 When the Beltrami helicity b − 2af is fixed, nonzero (and independent
of β), the solutions of (2.3.5) have the following behaviour when β −1 → 0:
(
∓192aβ 3 + O(β 2 )
when T = −1,
±
(i) a 6= 0: AT =
−1
2
8a (b − 2af )β + O(β) when T = 1;
82
2 A zoo of swirling flows
β −1 = 0.01
β −1 = ρ1,1 ≈ 3.8
β −1 = 100
Figure 2.4: The change from flat profiles (β −1 → 0) to confined high-frequency profiles
(β −1 → ∞). At each value of β −1 , the axial (top) and angular (bottom) velocity of the
Beltrami component are drawn, ignoring the amplitude A± .
√
5/2
(ii) a = 0: A±
+ O(β 2 ).
T = ∓16T ±6b β
When the Beltrami helicity vanishes, one of the solutions of (2.3.5) is the zero solution
and the others behave like
(iii) b − 2af = 0: A± = ∓192aβ 3 + 16f β 2 + O(β).
Observe that in the case a = 0 all (=two) solutions have the same behaviour. The
case a 6= 0 shows three different solutions, two of them having the same behaviour.
But most important, in all three cases, the solutions blow up! Combining lemma 2.13
with e(β −1 ) we conclude that
• b − 2af 6= 0, a 6= 0: energy finite when T = 1.
• b − 2af = 0, a = 0: energy finite.
For these two cases the energy versus β −1 is drawn in figure 2.5 (small β −1 ) and
figure 2.6 (β −1 up to 20). Especially the last figure shows clearly the wavy character
of e(β −1 ). When we plot for the same values of the constraints the angle between the
azimuthal and axial velocity at the boundary we observe that the extrema of the value
function coincide with the extrema of this quotient (see figure 2.7)! Moreover, the
angle between the velocity components at the boundary is restricted to some specific
interval, the interval depending on the combination of parameters.
A finite energy means finite velocity profiles as well. Let us determine the profiles
corresponding to the specific limits.
1
1
• b − 2af 6= 0, a 6= 0: ûlimit = 0, 2ar, − (b − 2af )r2 + (b − 2af ) + f ;
a
2a
• b − 2af = 0, a = 0: ûlimit = 0, 0, −2f r2 + 2f .
83
2.3 Ordering of the relative equilibria
0.16
0.35
−1
0.3
e(β
0.25
6
0.14
)
e(β −1 )
6
0.12
0.2
- β −1
+
0.15
1
2
5
4
3
1
2
1
2
0.08
-3β −1
4
5
4
5
0.06
0.05
–
0.8
0.6
∆
6
∆
6
0.04
–
0.4
0.03
0.2
0.02
1
2
4
3
- β −1
-0.2
-0.4
+
5
0.01
-0.6
3
- β −1
Figure 2.5: At the left: case b − 2af 6= 0 (a = b = f = 1). Top: the energy e(β −1 )
for both plus and minus flow with T = 1 (and hence energy remains finite in the limit
β −1 → 0). Observe that one (plus) of the solutions does not exist for β −1 > 2.4. Bottom:
the determinant for the same values of the constraints. When the determinant crosses the
axis the corresponding solution stops being real. At the right: case b − 2af = 0, a = 0
(a = b = 0 and f = 1). Top: the energy for the nonzero solution of (2.3.5) and at the
bottom the determinant.
The limit flows turn out to be the singular flows defined in section 2.1, equation
(2.1.10), with α̃ = −4f . Let us indicate the reason behind this.
From (2.3.6) we have the special relation between the energy, helicity, angular momentum and flux in a RE,
2E(û) = 2β B(û) + αA(û) + λFax (û)
= 2β b + αa + λf.
(2.3.8)
When we divide (2.3.8) by β, take the limit β −1 → 0 and assume the energy to be
finite for all β −1 and in this limit in particular, we find
0 = 2b + α̃a + λ̃f,
with
α̃ = −1
lim αβ −1 ,
β
→0
λ̃ = −1
lim λβ −1 ,
β
→0
precisely related to the the non-regularity condition (2.1.7).
84
2 A zoo of swirling flows
0.15
0.35
0.3
0.25
0.2
e(β −1 )
0.125
0.1
6
0.075
0.15
e(β −1 )
6
0.05
5
10
15
20
0.025
0.05
5
10
- β −1
15
20
- β −1
Figure 2.6: The energy for the two cases for which a finite limit exists when β −1 → 0:
b − 2af 6= 0, T = 1 and b − 2af = 0, a = 0. In the first case the minus flow is taken since it
exists for large values of β −1 as well. Observe the wavy character of the energy: the extrema
indicate an alternative choice for the eigenvalues. The values for the constraints are the same
as in figure 2.5.
Remark During the limit-process all integral quantities remain constant. However,
the limit flow need not to respect this fact when an arbitrary norm is taken like for
instance the L2 /(energy)-norm.
3
High frequency limit: β → 0.
e(β −1 ) tends to a constant,
e(β −1 ) =
Let us consider the limit β → 0. In that case
p
1
+ O( β) as β → 0,
2π
and the solutions of (2.3.5) remain finite:
p
2
(A±
T ) = ±2π(b − 2af ) + O( β) as β → 0.
The L2 -limit of J0 (rβ −1 ) and J1 (rβ −1 ) is the zero-function, since
Z 1
1
J0 (rβ −1 )2 rdr =
J0 (β −1 )2 + J1 (β −1 )2 ∼ β,
2
0
Z 1
1
J1 (rβ −1 )2 rdr =
J0 (β −1 )2 − 2βJ0 (β −1 )J1 (β −1 ) + J1 (β −1 )2 ∼ β,
2
0
and hence, the limit flow consists of just a rigid body rotation:
ûlimit = (0, 2ar, f ).
Note that the helicity of this limit flow is equal to 2af , and hence independent of
the value of the helicity constraint. Note that we used the L2 -topology to obtain the
limit flow. When we use a stronger topology, e.g. the H 1/2 -norm, all the values of the
constraints remain invariant. Let us explain this.
85
2.3 Ordering of the relative equilibria
1.4
0.02
0.015
0.01
0.005
1.2
1 angle
0.8
0.6
0.4
0.2 energy
5
10
15
- β −1
20
-0.005
-0.01
-0.015
-0.02
5
10
15
20
- β −1
Figure 2.7: At the left, the angle between the axial and azimuthal velocity at the boundary
(zero angle corresponds to vanishing azimuthal ‘slip’ velocity), together with the energy
value function, versus β −1 . The values of the parameters are the same as in the previous
two figures. At the right, the derivative of the energy value function with respect to β −1 ,
together with the derivative of the angle with respect to β −1 , focussing on the β −1 -axis.
Note that the extrema of the value function and the angle coincide.
First of all, note that
Z 1
0
2
Z 1
1
d
−1
J (rβ −1 )2 rdr,
J (rβ ) rdr = 2
dr 0
β 0 1
which blows up in the limit β → 0, and hence a H 1 -norm will not produce the desired
result. Instead, we ‘try’ the combination
||u||2E,B = ||u||2L2 + ||u||L2 ||ω||L2 ,
(2.3.9)
since the last term on the right-hand side of (2.3.9) reminds of the helicity:
p
p
− (u · u)(ω · ω) ≤ u · ω ≤ (u · u)(ω · ω).
Although || · ||E,B is not a norm (it does not satisfy the triangle inequality) it is
straightforward to show that, using || · ||E,B, the limit-flow is given by
1 p
1p
2π|b − 2af |J0 (rβ −1 ) + f ,
ûlimit = 0, σ 2π|b − 2af |J1 (rβ −1 ) + 2ar,
2
2
with σ = sign (b − 2af ) and that the limit flow satisfies all the constraints.
Conclusion When β is allowed to vary continuously we have two physical realistic
branches (with finite energy for β −1 → 0, corresponding to T = +1) and two singular
ones (energy blows up as β −1 → 0, corresponding to T = −1). Focussing on the
former, we need to distinguish between the plus and the minus flow. Depending on
the sign of the Beltrami helicity, b − 2af , either the plus (jet) flow (b − 2af < 0) or
86
2 A zoo of swirling flows
the minus (wake) flow (b − 2af > 0) cannot be obtained for all β −1 ≥ 0: the sign of
the discriminant changes at finite and nonzero β −1 , and related to that a singularity
in the energy. Hence, there is actually only one branch that couples finite energy to
existence for all β −1 :
b − 2af > 0 −→ T = +1, plus (jet) flow,
b − 2af < 0 −→ T = −1, minus (wake) flow.
However, note that at any nonzero value of β −1 we have at least two regular solutions.
This brings us to the following paragraph, on special values of β −1 .
Relation with previous work; special choices for β
In previous publications we also discussed the columnar flows but used other notation.
We want on the one hand clarify the differences, but on the other hand pay attention
to the specific choices made for β.
Most important, the ±-sign in this thesis does not correspond to the ±-sign used in
Van Groesen et al. (1995) and Fledderus and Van Groesen (1996a,b), but to the cases
β > 0 and β < 0. And the ±-sign in Van Groesen et al. (1995), etc., corresponds to
T = +1 and T = −1, i.e., indicating different solutions of (2.3.5).
We derived in section 2.2 two specific choices for β:
• β = ρ−1
0,k : as a ‘limit’ of non-axisymmetric flows.
• β = ρ−1
1,k : as a limit of axisymmetric flows when the wavelength L tends to
infinity, but also as a condition for the completeness of the Beltrami-flows. In
this case the axial flux of the Beltrami component vanishes, and moreover, the
angular velocity of the Beltrami component can be viewed as a true disturbance
on the rigid body rotation,
J1 |r=0 = J1 |r=1 = 0,
and hence when we would rotate the pipe with a certain fixed angular velocity,
say α, this choice for β would be natural.
Another choice for β would result from the condition that the azimuthal velocity
of the relative equilibrium vanishes at r = 1; this condition is part of the stability
criterion of Maslowe and Stewartson (1982), see section 1.1. Thus,
1
± AJ1 (β −1 ) + α = 0.
2
±
We will investigate the two cases β = ρ−1
1,k and ûϕ r=1 .
β = ρ−1
1,k
The quadratic equation for the amplitude A± simplifies to
2
±
±
A± J0 (ρ1,k ) · c̃±
2 + A J0 (ρ1,k ) · c̃1 + c0 = 0,
(2.3.10)
87
2.3 Ordering of the relative equilibria
with
±
1 −1 2
−1
c̃±
2 = ± ρ1,k ρ1,k − 4 , c̃1 = ρ1,k ±2f − aρ1,k ,
4
k = 1, 2, . . . ,
and −c0 the Beltrami helicity. When the Beltrami helicity is small enough we have
both a plus and a minus flow for small k. However, fixing c0 6= 0, and increasing k,
we eventually have to choose a sign:
2
− 4c̃±
∆ = c̃±
1
2 c0 ≥ 0.
Concluding, we can say that there exists for every k at least two solutions (T = ±1),
corresponding to two infinite branches, i.e., k ∈ N when ±(b − 2af ) ≥ 0, and to two
finite branches, i.e., k ∈ I ⊂ N, when ±(b − 2af ) < 0 (see figure 2.8). Moreover, note
∆
6
0 -
k=1
2
3
4
5
6
7
Figure 2.8: The discriminant ∆ versus the k-th positive root of J1 . The values of the
constraints are b = 2.12, a = f = 1 and thus the Beltrami helicity b − 2af is positive. The
light bars indicate the discriminant for the minus-flow, the dark bars for the plus-flow. Note
that for k = 1, 2, 3 there exist four solutions of (2.3.10), while for k > 3 only two solutions
exist, corresponding to the plus-flow.
that when ±(b − 2af ) ≥ 0, the solutions in the infinite branches (T = ±1) eventually
correspond to a positive and a negative amplitude.
± 12 A± J1 (β −1 ) + α = 0 First we can express a in terms of A± and β,
1
∓A± J1 (β −1 ) = α = 2a ∓ 2A± βJ2 (β −1 ) ⇒ a = ± A± J3 (β −1 ),
4
and using this in (2.3.5) the linear term in A± disappears:
2
1 ± β J0 (β −1 )2 + J1 (β −1 )2 β −2 − 4J1 (β −1 )2 (A± )2 − b = 0.
4
This can be turned into an expression for A± in terms of β and b, but we gain more
insight when we substitute a in favour of A± , leading to a relation that should be
88
2 A zoo of swirling flows
satisfied by β:
J3 (β −1 )2 β
4a2
.
=
±
J0 (β −1 )2 − 4β 2 J1 (β −1 )2 + J1 (β −1 )2
b
(2.3.11)
The left-hand side of (2.3.11), plotted in figure 2.9, defines an upper bound for the
quotient 4a2 /b. The interpretation of this upper-bound is that when e.g. the helicity
is too small, there does not exist a columnar RE with vanishing azimuthal velocity at
r = 1. Hence, prescribing the angle between the velocity components at the boundary
gives bounds on the parameter or combinations of parameters, a situation that is
comparable with the previous subsections where β was arbitrary and where a certain
parameter combination implied bounds on the angle.
S
6
0.25
0.2
0.15
0.1
0.05
5
10
15
- β −1
20
Figure 2.9: The swirl function S versus β −1 .
Remark The right-hand side of (2.3.11), 4a2 /b, can be interpreted as the swirl
number. Indeed, for a rigid body rotation Ωreϕ and a uniform translation W ez it is
easily shown that
4a2
Ω
=
= swirl number.
b
W
We will call S the swirl function. Moreover, since the left-hand side of (2.3.11) has
as Taylor series
J0
(β −1 )2
J3 (β −1 )2 β
1 −1
=
β + O(β −5 ),
− 4β 2 J1 (β −1 )2 + J1 (β −1 )2
12
as β −1 → 0,
β −1 may be viewed as a kind of generalised swirl number. The interpretation above
can be restated as when the swirl number is too high there does not exist a columnar
RE with vanishing azimuthal velocity at r = 1.
3
2.3 Ordering of the relative equilibria
2.3.3
89
Multi-mode relative equilibria
We have investigated single modes and columnar states. The columnar state had as a
distinguishing property that the eigenvalue β −1 was not determined by the boundary
condition. This freedom leads to the notion of multi-mode relative equilibria: the
co-existence of a single-mode (that fixes β) and a columnar state with the same β.
Let us denote such a multi-mode by
± ±
± ±
û±
+ αreϕ + Λez .
(2.3.12)
k = Ak uk + A u
We will analyse the behaviour of the parameters in (2.3.12) in relation with the values
of the constraints.
First of all, since the non-columnar and the columnar mode are orthogonal (this is
obvious), the effect on the helicity is that the helicity b in the formulae for c0 up to
c±
2 in (2.3.5) is changed to (see lemma 2.10)
±
2
b → b ∓ β −1 (A±
k ) E(uk ).
This change carries over to the discriminant ∆, which is lowered by an amount of
J (β −1 ) J (β −1 ) 3
± 2
2
, ∆0 = (c± )2 − 4c0 c± .
∆ = ∆0 − β −2 (A±
1
2
k ) E(uk ) J 0 (β −1 ) J 0 (β −1 ) 2
3
(the expression with the Bessel functions has a fixed sign, turning out to be positive,
±
for all β). Given a value of A±
k , we can determine A and with lemma 2.12 the values
of α and Λ; note that they are only coupled with A± . Let us fix now the sign of the
+
Beltrami component, say the plus sign; we will write Ak = A+
k and A = A , and
leave out the indefinite sign on all symbols.
When fixing some (positive) value of ∆0 , the amplitude Ak can vary between two
bounds, and related to that the amplitude of the columnar state, A,
√
−c1 + T ∆
.
A = AT =
2c2
When ∆0 is now decreased, the freedom for Ak is tightened, until ∆0 = 0, where
Ak = 0, and
c
AT = − 1 , T = ±1.
2c2
See figure 2.10.
Remarks (i) The decrease of ∆0 can be achieved in several ways, by changes in a,
b or f , or a combination. When a and f are kept fixed, ∆0 = 0 corresponds to some
threshold value of the helicity, bthreshold . Below this value, the combination in (2.3.12)
is not possible for the sign chosen.
(ii) The Beltrami helicity b − 2af is again very special. The vanishing of the Beltrami
helicity defines the only location in parameter space for which Ak and A can vanish
simultaneously, and at this line we can change continuously sign, or even modes! 3
90
2 A zoo of swirling flows
Ak
−c1 /2c2
6A
k,max- ?
•
0
A
1T
XXX
z ∆0
• Ak,min = −Ak,max
Figure 2.10: For a fixed, positive value of ∆0 , the amplitude Ak of the translating and/or
rotating Beltrami component can vary between Ak,max and Ak,min = −Ak,max . Given its
value, the amplitude of the columnar flow can be calculated.
Until now we varied the values of the constraints and kept β fixed. What if the roles
are interchanged, i.e., a, b and f are kept fixed and β can move freely. This problem
is basically studying the transcendental equation (2.2.11),
p
p
`βJ` ( 1 − β 2 m̃2 /β)
0
2
2
p
β m̃J` ( 1 − β m̃ /β) ±
= 0,
(2.3.13)
1 − β 2 m̃2
where ` and m = Lm̃/2π are the wave numbers in azimuthal and axial direction
respectively. The cases ` = 0 or m̃ = m = 0 were simple, but it turns out that a mixed
mode is the first one to turn up! In figure 2.11 we plotted the solutions of (2.3.13) for
` = 0, ` = 1 and ` = 2 versus m̃, with m̃ = 2πm/L, for both the plus as the minus
sign. The minimum value for which a mixed mode can exist, being approximately
3.2, corresponds to a critical wavelength L ≈ 1.2 in the z-direction. The behaviour
in figure 2.11 for small m̃ can be computed using a perturbation argument. Assume
m̃ = ε,
β −1 = B0 + εB1 + o(ε),
then rearranging powers of ε in (2.3.13) leads to
ε0 :
ε1 :
J` (B0 ) = 0 ⇒ B0 = ρ`,k ;
1 ± `B1 = 0 ⇒ B1 = ∓`−1 ,
where we used in the computation of the ε1 -term already the information on B0 of the
previous step. Hence, at ρ`,k , the non-axisymmetric plus flows bifurcate subcritical.
91
2.4 Axisymmetric flows and the Bragg-Hawthorne equation
ρ1,2
7
ρ3,1
-
β(2,0,1)
−
β(1,3,1)
+
β(2,1,1)
+
β(1,3,1)
6
β −1 6 β −
(1,2,1)
ρ2,1
-
ρ1,1
-
5
−
β(1,1,1)
+
β(1,2,1)
β(1,0,1)
4
+
β(1,1,1)
0
1
2
- m̃
3
4
Figure 2.11: The eigenvalue β −1 versus m̃ for different values of the angular mode `. The
subscripts indicate the specific value of k, ` and m. Recall the definition of m̃: m̃ = 2πm/L
and hence m̃ → 0 corresponds to long wave-lengths.
The precise relevance of the travelling and rotating wave solutions for the physics is
still unclear. In chapter 3 we try to relate the multi-mode RE to the status of the
columnar flow when it concerns its criticality. Moreover, in section 4.1 we re-introduce
the non-dimensional radius R of the pipe and then it turns out that e.g. the angular
velocity α scales with a · R−2 , a result that can be traced back in experiments (Syred
et al. (1994), and Kok et al. (1993)). This suggests that such an explicit number as
1.2 can be identified in experiments as well.
There may be a relation with the results of e.g. Batchelor and Gill (1962), who
discovered that the most (linearly) unstable mode of a round axisymmetric jet is
the mode with azimuthal wave number ` = 1. Similar results were obtained in the
seventies. Moreover, Crow and Champagne (1971) suggested that the most unstable,
axisymmetric mode on a turbulent jet has a preferred wavelength of approximately
2.38×diameter pipe. Note that all these results discuss instability, whereas the multimode RE solutions are unamplified.
2.4
Axisymmetric flows and the Bragg-Hawthorne
equation
In the previous three sections we took a Hamiltonian point of view when it came to
determining 3D-solutions of the Euler equations. This is of course not the only way
but it turns out to be rather difficult to find another systematic way of finding ‘a lot
of’ interesting solutions. With ‘interesting’ we mean solutions depending on at least
92
2 A zoo of swirling flows
two coordinates and with three nonzero velocity components.
The only other method known to us is related to the flow-force invariant that we
mentioned and described in section 1.1. Its formulation is due to Benjamin (1962)
who showed that it leads to a (in principal) nonlinear Bragg-Hawthorne (or LongSquire) equation for the (axisymmetric stream) function ψ,
∂ 2 ψ 1 ∂ψ ∂ 2 ψ
dH
dI
−
= r2
+
−
∂r2
r ∂r
∂z 2
dψ
dψ
(2.4.1)
(see section 1.1 for background). H is the total head and I (half) the quadratic
circulation. For example, we plotted in figure 2.12 the functions H and I for a RE.
Note that although the Lagrange multiplier equation (2.2) is linear, the corresponding
Bragg-Hawthorne equation is nonlinear. In order to be able to obtain explicitly a
12
0.035
0.03
I0.025
60.02
0.015
0.01
0.005
11
H
6 10
9
0.25 0.5 0.75
1
1.25 1.5
0.25 0.5 0.75
1
-ψ
1.25 1.5
-ψ
Figure 2.12: The total head H (left) and the circulation I (right) versus the stream function
ψ, evaluated in a columnar RE. The values of the parameters are α = 1, λ = 3, β = 0.25
and A− = 4 (we took the minus flow).
family of solutions of (2.4.1), we assume H and I quadratic at most, implying a linear
equation. Moreover, the choice
1
Ω2
Ω2
ψ + U 2 , I = 2 2 ψ2
(2.4.2)
U
2
U
related to a uniform axial velocity U ez and a rigid body rotation Ωreϕ is used in e.g.
Batchelor (1967) (and will be used here) to illustrate the basic mechanism how a flow
can turn from supercritical to subcritical by explicitly constructing the steady waves.
Splitting of a term 12 U r2 from ψ, we are left with
H=2
∂ 2 Ψ ∂ 2 Ψ 1 ∂Ψ
+
+
+ (κ2 − r−2 )Ψ = 0
∂z 2
∂r2
r ∂r
where ψ(r, z) = 12 U r2 + rΨ(r, z) and κ = 2Ω/U . Separation of variables leads to
d2 Ψ1
+ m̃2 Ψ1
dz 2
= 0,
Ψ1 = Ψ1 (z),
(2.4.3a)
d Ψ2
dΨ
+ r 2 + Ψ2 (r2 (κ2 − m̃2 ) − 1) = 0,
dr2
dr
Ψ2 = Ψ2 (r).
(2.4.3b)
2
r2
2.4 Axisymmetric flows and the Bragg-Hawthorne equation
93
The second equation (2.4.3b) is Bessel’s equation of order 1 when κ2 − m̃2 ≥ 0 and
the modified Bessel equation for κ2 − m̃2 < 0. Its regular3 solution is given by
(
√
AJ1 ( κ2 − m̃2 r), when κ2 − m̃2 ≥ 0;
√
Ψ2 (r) =
BI1 ( m̃2 − κ2 r), when κ2 − m̃2 < 0.
In order to obtain m̃ we apply the boundary condition to ur ,
1 ∂ψ ur |r=1 = −
= −Ψ01 (z)Ψ2 (1)
r ∂z r=1
(
√
−Ψ01 (z)AJ1 ( κ2 − m̃2 ), κ2 − m̃2 ≥ 0;
√
=
−Ψ01 (z)BI1 ( m̃2 − κ2 ), κ2 − m̃2 < 0.
Since I1 is strictly increasing, being non-negative everywhere on [0, ∞) we arrive at
the important conclusion that the rigid body rotation Ωreϕ plus the uniform axial
translation U ez can only sustain harmonics with frequency smaller than κ = 2Ω/U .
The frequency of these wavy flows has to satisfy
p
κ2 − m̃2 = ρ1,k ⇒ m̃2 = κ2 − ρ21,k ,
so in addition to the conclusion that ‘high’-frequency waves can not be sustained, the
parameter κ needs to exceed a critical value κcrit = ρ1,1 before ‘low’-frequency waves
are possible at all.
Proposition 2.14 The Bragg-Hawthorne equation (2.4.1) with H and I according to
(2.4.2) has solutions
(
2
2
ArJ
1 2
when κ < ρ1,1 ;
1 (κr),
q
ψ(r, z) = U r +
2
< ArJ1 (ρ1,k r)eim̃k z , when κ2 ≥ ρ21,1 , m̃k = κ2 − ρ21,k .
The corresponding velocity fields are given by
ur = 0, uϕ = Ωr + AκJ1 (κr), uz = U + AκJ0 (κr)
when κ2 < ρ21,1 and
ur = < −Aim̃k J1 (ρ1,k r)eim̃k z ,
uϕ = Ωr + < AκJ1 (ρ1,k r)eim̃k z ,
uz = U + < AκJ0 (ρ1,k r)eim̃k z ,
when κ2 ≥ ρ21,1 .
3 Note
that we have to differentiate Ψ2 in order to obtain the axial velocity:
uz =
1 ∂ψ
= U + Ψ1
r ∂r
Ψ
2
r
+
dΨ2
dr
.
94
2 A zoo of swirling flows
The parameter κ plays a similar role as β −1 in the Beltrami flows and the proposition is
particularly interesting when it is compared with subsection 2.3.3 on multi-mode RE,
where we changed amongst others the value of β. Our conclusion was that when β −1
−1
is below a certain critical value βcrit
no multi-mode RE can exist and moreover, when
−1
β −1 < βcrit the only RE are the columnar ones. At β −1 = ρ1,1 the first axisymmetric
non-columnar mode occurs (see figure 2.11). However, the main difference is that
already before β −1 = ρ1,1 a mixed mode can exist, that is not incorporated in the
axisymmetric model of the Bragg-Hawthorne equation; see figure 2.13.
non-axisymmetric
mode possible
only columnar RE
−1
βcrit
ρ1,1
axisymmetric and
non-axisymmetric
modes possible
β −1
κ=
κcrit
2Ω
U
only columnar flows
standing waves possible
Figure 2.13: Comparison between the role of β −1 in the multi-mode relative equilibria and
the role of (the swirl number) κ in the change from supercritical to subcritical.
3
Stability of swirling flows
In the previous chapter we have constructed many solutions to one and the same
extremalization problem,
Crit {E(u) | B(u) = b, A(u) = a, Fax (u) = f } .
u
Furthermore, we have found relations between different solutions and conditions when
certain solutions can co-exist. The aim of stability theory and of this chapter is
to deduce the critical values of the parameters in play which separate (families of)
solutions when considering long time dynamical behaviour: what states are preferred
by the Euler equations in the long run?
Part of the theory of this chapter is based on Sattinger (1970) and Joseph (1976).
However, since we are dealing with equations that have a Hamiltonian structure, we
can not pass by the stability methods developed by Arnol’d (1965, 1969) and later on
by e.g. Holm et al. (1985), and applied by e.g. Szeri and Holmes (1988).
(Linear) stability results on swirling flows were sketched in section 1.1. We will not
repeat this but cite related results when appropriate.
Concepts in stability theory: global methods
In order to determine the behaviour of solutions U after a long time, we study the
dynamics of an initial perturbation, superposed on U :
u = U + εũ,
p = P + εp̃,
(3.1)
with ε in first instance meant for bookkeeping, later on it can be vanishingly small.
Global methods tend to study the dynamics of some (preferably) coercive or a definite
functional L(u). An example is the energy-method, that can be traced back to
Reynolds (1895) and (modern approach) Serrin (1959a,b). Especially in the case
when viscous effects are included in the equations, this method has proven to be
very powerful and applicable in many different cases (see e.g. Joseph (1976) for many
examples). With the work of Arnol’d and Marsden and co-workers the range of
96
3 Stability of swirling flows
applications was extended to conservative systems; there the notion of constraint
stability was introduced (see section 3.2).
We will apply the energy-theory to our coherent flows and derive equations which
provide many insights in the dynamics of the energy of the perturbation. Let us
introduce the (volume-averaged) energy of a disturbance (Joseph (1976)),
Z
1
1 2
1
E(ũ) =
|ũ| dV =: hh |ũ|2 ii.
|D| D 2
2
We will call the basic solution U nonlinearly stable with respect to the energy-norm
if for every δ1 > 0 there is a δ2 > 0 such that if E(ũ)t=0 < δ2 , then E(ũ) < δ1 for
t > 0. It is called asymptotically nonlinear stable when in addition
lim E(ũ) = 0
t→∞
If there exists an upper bound for δ1 > 0, the solution U is said to be conditionally
stable. If there exists no upper bound, the basic solution is said to be unconditionally
or globally stable (Joseph (1976); Holm et al. (1985)). Finally we define instability as
not-conditionally stable.
Let U denote a coherent flow, depending only on r (we suppress all dependences
on parameters and coordinates) and P the corresponding pressure. Let the Euler
equations be written in operator form as
EEul (u, p) = ∂t u + u · ∇u + ∇p,
u ∈ H.
Let (ũ ∈ H, p̃) be a disturbance of this flow (set ε = 1 in (3.1)). The evolution of ũ
is given by
0
= EEul (U + ũ, P + p̃)
∂ ũ
= EEul (U , P ) +
+ (U · ∇)ũ + (ũ · ∇)U + (ũ · ∇)ũ + ∇p̃
∂t
dũ
=
+ (ũ · ∇)U + (ũ · ∇)ũ + ∇p̃,
dt
(3.2)
with
dũ
∂ ũ
=
+ (U · ∇)ũ.
dt
∂t
Multiplying equation (3.2) by ũ and integrating the result over the volume D we are
left with (Joseph (1976), §191 )
d
E(ũ) = −hhU · (ũ × curl ũ)ii.
dt
(3.3)
1 This holds (trivially) for periodic perturbations, but also in case (ũ, p̃) are almost periodic. Then
D = [z1 , z2 ] × S1 × [0, 1] and hhf, gii should be interpreted as
hhf, gii =
lim
1
|z2 −z1 |→∞ |D|
Z
D
f · g dV
97
Remark Joseph notes that, according to (3.3), irrotational disturbances do not
produce energy. But this conclusion can be strengthened by noting that the term
ũ × curl ũ
vanishes for every Beltrami flow, i.e.,
curl ũ = µũ,
µ ∈ R;
3
the case µ = 0 corresponds to an irrotational disturbance.
The right-hand side of (3.3) can be rewritten as
−hhU · (ũ × curl ũ)ii
= −hhũ · (ũ · ∇)U ii
Z dUz
d Uϕ
1
= −
ũ ũ
+ ũr ũϕ r
dV.
|D| D r z dr
dr
r
(3.4)
Remarks (i) A rigid-body rotation superposed on a uniform translation is energetically stable2 .
(ii) An arbitrary rigid-body rotation can be subtracted from Uϕ (r) without changing
the equations; the same holds true3 for an arbitrary uniform translation subtracted
from Uz (r).
3
Returning to (3.3), we conclude that, when U is a relative equilibrium solution, the
Beltrami component of U determines the essential structure of the energy equation
(Dritschel (1991) makes this observation too).
Concepts in stability theory: local methods
At the other extreme we have local methods and the crudest one in some sense is the
linear stability or spectral theory. In that case the quadratic term in the perturbation
ũ in equation (3.2) is neglected and we write
E0Eul (U , P )(u, p̃) :=
∂ ũ
+ (U · ∇)ũ + (ũ · ∇)U + ∇p̃ = 0,
∂t
ũ ∈ H.
(3.5)
Equation (3.5) is an autonomous linear system and there exist exponential solutions
of the form
ũ(x, t) = eiσt v(x),
p̃(x, t) = eiσt $(x)
(3.6)
for those numbers σ such that the spectral problem
E0Eul (U , P )(eiσt v, eiσt $) := iσv + (U · ∇)v + (v · ∇)U + ∇$ = 0,
v∈H
(3.7)
has nontrivial solutions. The special numbers iσ are the eigenvalues of (3.7) and
the (nontrivial) solutions (v, $) are the corresponding eigenfunctions. It is useful
2 See
also exercise 4.4 in Joseph (1976).
conclusion is also valid when
depends on ϕ and z as well as on r.
3 This
U
98
3 Stability of swirling flows
to translate the (Lyapunov) stability concepts into concepts related to the spectral
problem (Joseph (1976)). Using the representation for (ũ, p̃) in (3.6), the basic flow
U is called spectrally stable if there are no eigenvalues such that =(σ) ≤ 0; marginally
or neutrally stable if there is one eigenvalue with =(σ) = 0 and =(σ) > 0 for the other
eigenvalues; and (spectrally) unstable if at least one eigenvalue has =(σ) < 0. The
eigenvalues can be ordered according to their real part, while the σ’s can be ordered
by their imaginary part.
The relation between the nonlinear problem (3.3) and the spectral problem is put into
words by the principle of linearised stability (Sattinger (1970)):
Suppose that =(σmin ) > 0. Then U is conditionally stable. In contrast, if
=(σmin ) < 0, then U is unstable.
Depending on the domain we can proceed by assuming a certain series expansion of
(v, $). In case of a pipe P we write
Z
Z
X
X
i(`ϕ+µz)
ξ(r; µ, `)e
, $=
ρ(r; µ, `)ei(`ϕ+µz) .
v=
µ∈R `∈Z
µ∈R `∈Z
Since v is real,
ξ(r; µ, −`) = ξ(r; −µ, `),
ξ(r; −µ, −`) = ξ(r; µ, `),
and the same for ρ. Because (3.7) is linear in v, we can look at the stability of the
basic flow U with respect to a normal-mode disturbance:
(v, $) = <((ξ(r; µ, `)ei(`ϕ+µz) , ρ(r; µ, `)ei(`ϕ+µz) )).
Substituting such a normal mode in (3.7) and eliminating the pressure (take the curl),
ξϕ and ξz leads to the following (Orr-Sommerfeld-like) equation for ξr (Howard and
Gupta (1962)):
γ 2 D (SD∗ ξr ) − γ 2 + γa(r) + b(r) ξr = 0
(3.8)
where
df
1
, D∗ f = D(rf ),
dr
r
γ = σ + `r−1 Uϕ + µUz , σ = σr + iσi ,
Df =
r2
,
+ `2
Uϕ
γ
a(r) = rD S D∗
+ 2` 3
,
r
r
Uϕ b(r) = −2µ 2 S µrD∗ Uϕ − `DUz .
r
S=
µ2 r2
3.1 Linear stability for columnar flows
99
The boundary conditions for (3.8) are (Leibovich and Stewartson (1983))
ξr (r = 0) = 0 when |`| 6= 1,
Dξr (r = 0) = 0 when |`| = 1,
ξr (r = 1) = 0.
Recall that the stability of the base flow to the normal-mode disturbance is determined by the sign of σi : when σi > 0 the flow is stable and when σi < 0 the
flow is unstable. In section 1.1, we reviewed several stability criteria for base flows
(U , P ) = (0, Uϕ (r), Uz (r), P (r)), related to (3.8). The first section 3.1 is devoted to
the evaluation of these criteria in terms of the relevant parameters in the relative
equilibria: the ratio κ between the Beltrami component and the rigid-body rotation,
κ = A/α, and the radial frequency β −1 . The plus and minus flow are in this context
identified as ‘co-rotating’ and ‘counter-rotating’, respectively, thereby referring to the
corkscrew that is traced out near the axis by the Beltrami component. The results
are depicted in figure 3.1, figure 3.5 and figure 3.6. When discussing the axisymmetric disturbances in subsection 3.1.1, we are particularly interested in the minimum
and maximum wave-speeds that are possible. It turns out that the multi-mode RE
solutions play a profound role in this process: they describe the fastest or slowest
travelling wave that can be superposed on the corresponding columnar RE.
In section 3.2 we turn to nonlinear stability, using the Energy-Casimir method developed by Holm et al. (1985) and applied to axisymmetric flows by Szeri and Holmes
(1988). More specifically, we consider the constrained stability of the RE; the adjective ‘constrained’ implies certain conditions on the perturbation namely that the
perturbed flow satisfies the same constraints as the base flow. Moreover, the theory
allows for an investigation of the stability of non-columnar swirling flows c.q. RE
solutions.
This chapter elaborates and extends the results in Valkering (1996).
3.1
Linear stability for columnar flows
Results in linear stability theory applied to hydrodynamics have long been restricted
to general criteria, indicating sufficient or necessary conditions for the base flow to be
stable or unstable. Important advances (see also section 1.1) were made by Rayleigh
(1880), Synge (1933), Fjortoft (1950), Howard and Gupta (1962), Leibovich and Stewartson (1983) and very recently by Wang and Rusak (1996). We will state the relevant
criteria and evaluate them for the columnar RE obtained in chapter 2:
Uϕ
Uz
H
P
1
−1
:= û±
) + αr,
ϕ = ± AJ1 (rβ
2
1
−1
:= û±
) + Λ, Λ = ±2αβ + λ
z = AJ0 (rβ
2
±
:= λû±
z + αrûϕ ,
1
1 ±2
±
:= h − |û± |2 = λû±
z + αrûϕ − |û | .
2
2
(3.1.1a)
(3.1.1b)
(3.1.1c)
(3.1.1d)
100
3 Stability of swirling flows
Due to the fact that we have a family of exact solutions, depending continuously on
parameters, we are able to obtain special, explicit solutions of (3.8), that in fact turn
out to be part of exact solutions of the Euler equations (this remark is also made
in Dritschel (1991) where it concerns two-dimensional disturbances; we will show the
generalisation to three dimensions). To be precise, consider a multi-mode RE solution
(defined in section 2.3),
v̂ ±
k
± i(`(ϕ−αt)+µ(z−λt))
= A±
+ A± u± + αreϕ + Λez ,
k uk e
1
±
±
p = λv̂k,z
+ αrv̂k,ϕ
− |v̂ ±
|2 .
2 k
(3.1.2a)
(3.1.2b)
±
Substituting v̂ ±
k into the Euler equations, and differentiation with respect to Ak at
±
4
Ak = 0 shows that
E0Eul ((0, Uϕ , Uz , P ))(ũ, p̃) = 0,
with
ũ = u±
ei(`(ϕ−αt)+µ(z−λt)) ,
k
±
±
±
i(`(ϕ−αt)+µ(z−λt))
p̃ =
λu±
.
k,z + αruk,ϕ − Uϕ uk,ϕ − Uz uk,z e
The relevance of these special explicit solutions in the form of travelling waves is
considered in the case of a vanishing Beltrami component, i.e.,
1
1
Uϕ = ± εJ1 (rβ −1 ) + αr, Uz = εJ0 (rβ −1 ) + Λ,
2
2
|ε| 1.
We will look at the influence of ε and β on certain stability properties of the solid
body rotation αreϕ + Λez by means of the Froude number for axisymmetric flows
(Benjamin (1962); see also section 1.1)
N=
c+ + c−
c+ − c−
with c± the maximum/minimum absolute speed with which long waves travel.
3.1.1
Axisymmetric disturbances
We start off with equation (3.8) that governs the linear stability of a columnar swirling
flow. With the assumption ` = 0, this equation is reduced to
2Uϕ (Uϕ + rUϕ0 ) Uz00 r − Uz0
DD∗ ξr + −µ2 +
−
ξr = 0,
(3.1.3)
r2 (Uz − c)2
r(Uz − c)
with σ = −µc.
4 This
degeneracy is discussed and used in chapter 4.
101
3.1 Linear stability for columnar flows
Stability criteria
The relevant stability criterion for (3.1.3) is due to Howard and Gupta (1962). They
state:
Proposition 3.1 (Howard and Gupta (1962)) Consider a columnar swirling flow
(0, Uϕ (r), Uz (r)). If a so-called Richardson number F ,
F :=
Φ
,
(dUz /dr)2
with Φ = r−3
d
(rUϕ )2 ,
dr
exceeds a quarter in the whole flow field, the base flow is stable to axisymmetric
disturbances. Φ denotes the Rayleigh discriminant that determines the stability in
pure swirling flows (no axial velocity).
Remark Note that when dUz /dr → 0 (locally or globally) the number F → +∞ as
Φ > 0 (and hence stability) and F → −∞ as Φ < 0 (and hence no sufficient condition
for stability).
3
Evaluating F in the relative equilibrium (3.1.1a,b) yields
2β 12 AJ1 (rβ −1 ) ± αr 12 AJ0 (rβ −1 ) ± 2αβ
±
.
F =
2
r 12 AJ1 (rβ −1 )
Note that F is invariant under the following transformations:
(α, ±) → (−α, ∓),
(A, ±) → (−A, ∓).
If we relate A to α by means of A = κα, the number of variables is reduced by one:
2β 12 κJ1 (rβ −1 ) ± r 12 κJ0 (rβ −1 ) ± 2β
±
Fnew =
,
2
r 12 κJ1 (rβ −1 )
±
and Fnew
is invariant under (κ, ±) → (−κ, ∓). In the following we will denote the
±
by F ± .
minimum of Fnew
Remark In this section we will keep κ positive and investigate the plus and the
minus flow. A more appropriate name for the plus and minus flow is based on the
following facts. Consider the Beltrami component of the columnar flow,
1
1
Uϕ,Beltrami = ± καJ1 (rβ −1 ), Uz,Beltrami = καJ0 (rβ −1 ).
2
2
We already chose α to be non-negative in chapter 2—corresponding to a counterclockwise rigid-body rotation—but the reasoning that follows does not depend on the
sign of α. We observe that the indefinite sign determines whether the flow is ‘corotating’ (plus flow) or ‘counter-rotating’ (minus flow) (in the sense of a right-handed
corkscrew), irrespective of the sign of κ. Although this nomenclature only refers to the
Beltrami component itself, a ‘co-rotating’ flow will increase at the axis the rotation
rate due to the rigid-body rotation, and a ‘counter-rotating’ flow will decrease this
rotation rate, independent of the sign of α.
3
102
3 Stability of swirling flows
Note that the stability of the base flow does not depend on the value of Λ, and a
similar remark can be made for other (in)stability criteria. Although the general
stability equations (3.8) do depend on Λ one may conjecture that the stability of a
swirling flow does not depend on its specific value. Compare this with our observation
in the introduction, related to equation (3.4), where it was actually proved that an
additional uniform translation does not influence the nonlinear stability properties of
the flow.
In figure 3.1 we have drawn the ‘stability’-boundary in the (κ, β −1 )-plane: F ± = 14 .
We make a few observations.
15
14
12
14
F−>
1
4
13
κ
610
12
κ
6
11
10
8
9
4
2
- β −1
8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
F + < 14
6
F − < 14
F ± > 14
1
2
3
4
5
- β −1
6
7
8
9
10
Figure 3.1: The contour lines F ± = 1/4 in the (κ, β −1 )-plane. The dotted line marks the
boundary between F + < 1/4 and F + > 1/4, while the solid line indicates regions where
F − < 1/4 or F − > 1/4. The corner “small β −1 , large κ” is enlarged and it is observed
that the parabola and hyperbola are not connected (this is proven in the text). At the other
side, the parabola comes close to the F + = 1/4-contour, but the two remain separated (also
shown in the text).
±
• Fixing β, we make a series of Fnew
around κ = 0:
±
Fnew
= κ−2
16β 2
+ O(κ−1 ),
J1 (rβ −1 )2
as κ → 0,
(3.1.4)
and hence when κ is small the RE is linearly stable with respect to axisymmetric
disturbances. This can be interpreted as
Proposition 3.2 When the rigid-body rotation dominates the Beltrami component the flow is linearly stable with respect to axisymmetric disturbances.
3.1 Linear stability for columnar flows
103
We also observe from (3.1.4) that a high-frequency Beltrami component (β −1
large and thus β small) exerts much more influence on the stability properties
of the base flow, in the sense that we need relatively much more rigid-body
rotation to stabilise the flow.
On the other hand, the case that the Beltrami component dominates is covered
by the limit κ → ∞ (β fixed). Then,
±
=
Fnew
2βJ0 (rβ −1 )
+ O(κ−1 ) as κ → ∞.
rJ1 (rβ −1 )
Equating this limiting behaviour to 1/4 and solving for β −1 leads to an asymptote in the (κ, β −1 )-plane:
2βJ0 (rβ −1 )
1 (!)
=
⇒ 8J0 (β −1 ) = β −1 J1 (β −1 ) : β −1 = 2.128 . . .
0≤r≤1 rJ1 (rβ −1 )
4
min
where we used at (!) that the minimum is at r = 1. Hence,
Proposition 3.3 When β −1 < 2.128 . . . and fixed, the RE in (3.1.1a,b), consisting only of the Beltrami component is linearly stable to axisymmetric disturbances.
• When we fix κ and take the limit β −1 → 0 we obtain
±
Fnew
=
64 4
β + O(β 3 ) as β −1 → 0,
r2 κ2
implying again the linear stability of the base flow with respect to axisymmetric
disturbances.
• Finally, we will try to resolve the behaviour in the left upper corner in figure
3.1. To that end, we let β −1 → 0 but assume at the same time that κ may vary
as a function of β, κ = κ(β):
2β 14 κrβ −1 ± r + O(κβ −3 ) 12 κ ± 2β + O(κβ −2 )
Fnew =
2
r 12 κJ1 (rβ −1 )
4 14 κ ± β + O(κβ −2 ) 14 κ ± β + O(κβ −2 )
=
1
−1 ) 2
2 κJ1 (rβ
2
4 14 ± β + 14 ± β · O(κβ −2 ) + O(κ2 β −4 )
=
.
1
−1 ) 2
2 κJ1 (rβ
This shows that only when we deal with a counter-rotating flow, there can exist
a balance between κ and β such that we have no ‘sufficient stability’ as β −1 → 0.
This balance is given by
κ = 4β + O(1) as β −1 → 0,
104
3 Stability of swirling flows
and this corresponds exactly to the case when the net rotation rate at the axis,
being the sum of the Beltrami component and the rigid-body rotation, is of
higher order in r.
So, when κ = 4β ánd β −1 → 0 we have
−
Fnew
→ 0,
and hence no sufficient condition for linear stability. To prove that there exists
a band around κ = 4β outside which F − > 1/4 we write κ = 4β + δ, δ = O(1):
−
Fnew
=
1 2
4δ
+ O(β −1 )
2
(r + O(β −1 ))
→
1 δ2
1
≥ δ2
4 r2
4
as β −1 → 0,
and hence
Proposition 3.4 When β −1 tends to zero, while the ratio κ between the amplitude of the Beltrami component, A, and the rigid-body rotation, α, is such
that the angular velocity at the axis vanishes, i.e., κ = 4β, there exists a band
around the curve {(β −1 , κ) | κ = 4β} in the (β −1 , κ)-plane outside which the
sufficient condition for linear stability to axisymmetric disturbances does hold
for the counter-rotating flow.
Multi-mode RE and their relevance
As we noted in the introduction to this section, a special solution of the (axisymmetric) linearised Euler equations can be obtained when the temporal behaviour of the
perturbation is given by c = λ, describing a neutrally stable mode. Indeed, in that
case (3.1.3) even further reduces, independent of A, to
DD∗ ξr + −µ2 + β −2 ξr = 0,
with regular solutions given by
ξr = A1 J1 (rβ −1
p
1 − β 2 µ2 ).
A solution, periodic in z (µ ∈ R), can only exist when β −1 exceeds the first zero of
J1 as can be seen when we apply the boundary condition to ξr :
ξr (r = 1) = A1 J1 (β −1
p
1 − β 2 µ2 ) = 0 ⇔ β 2 =
µ2
1
1
≥ 2 ,
2
+ ρ1
ρ1,1
with ρ1 6= 0 a zero of J1 .
Remark Note that this condition on β was also found in chapter 2, where we studied
multi-mode RE. More specific, figure 2.11 shows exactly the lower-bounds for β −1 in
order to have a travelling wave solution with this particular speed λ.
3
105
3.1 Linear stability for columnar flows
One can ask what the relevance is of this special solution, especially since it exists for
all amplitudes A1 only when β −1 exceeds ρ1,1 . Indeed, it is quite possible that there
exist other travelling wave solutions and we will investigate this question when the
amplitude A of the Beltrami component is small, using a perturbation argument.
To be precise, let
1
1
Uϕ = ± εJ1 (rβ −1 ) + αr, Uz = εJ0 (rβ −1 ) + Λ,
2
2
|ε| 1.
(3.1.5)
This situation can be achieved when the value of the Beltrami helicity, b − 2af , is
close to zero (see chapter 2, section 2.3). The values of Λ and the multipliers λ and
α equal
Λ = f − εβJ1 (β −1 ),
λ = f ∓ 4aβ + εβJ3 (β −1 )
α = 2a ∓ 2εβJ2 (β −1 ).
(3.1.6a)
(3.1.6b)
(3.1.6c)
We will be interested in long waves, and hence µ2 → 0. Substitution of (3.1.5) into
(3.1.3), taking the limit µ2 → 0, writing
ξr = ξr(0) + εξr(1) + o(ε),
c = c(0) + εc(1) + o(ε),
and truncating the result at different orders of ε leads to the following equations:
• ε0 : At this order we obtain an equation that describes the stability of a rigidbody rotation to axisymmetric disturbances:
DD∗ ξr(0) +
16a2
ξ (0) = 0.
(f − c(0) )2 r
The eigenfunction, describing at first order the radial profile of the long wave,
is given by
!
√
r 16a2
(0)
ξr, long = A1 J1
.
f − c(0)
Using the boundary condition we obtain
(0)
cT = f + T
4|a|
,
ρ1,1
with T = ±1.
The Froude number related to these velocities equals
N=
f ρ1,1
c+1 + c−1
=
.
c+1 − c−1
4|a|
(3.1.7)
106
3 Stability of swirling flows
Recall that N = 1 marks the boundary between supercritical and subcritical,
i.e.,
|a| =
1
ρf.
4
Note that ξr, long corresponds to a Beltrami component at β −1 = ρ1,1 , exactly
the value of β −1 at which an axisymmetric travelling wave could start to exist.
Moreover, comparing (3.1.7) with (3.1.6a-c) we have the following remarkable
result.
(0)
Proposition 3.5 Consider the (axisymmetric) multi-mode relative equilibrium
solutions in (3.1.2a), with u± as in (3.1.1a,b). Set β −1 = ρ1,1 , A± = A = ε
and ` = 0. Then
– the counter-rotating flow corresponds in leading order to the fastest travelling wave that can be sustained by the rigid-body rotation plus a uniform
translation 2areϕ + f ez , and
– the co-rotating flow corresponds in leading order to the slowest travelling
wave that can be sustained.
• ε1 : Using the information from the previous step we obtain an inhomogeneous
(1)
eigenvalue equation for ξr :
ρ1,1
1 J1 (rβ −1 )
(1)
2
(1)
DD∗ ξr + ρ1,1 ξr =
T ρ21,1 c(1) −
T β −1 ± ρ1,1 −
2a
2
r
1
J0 (rβ −1 ) −T β −2 ± ρ1,1 β −1 + 2T ρ21,1 +
4
−1
−1
ρ1,1 βJ1 (β ) T ρ1,1 ± 8β ∓ 4ρ1,1 βJ0 (β ) ξr(0) .
(3.1.8)
It is possible to find a solution in closed form, expressed in terms of (integrals
of) Bessel functions. However, this is not the sort of information we are after.
Note that we want to determine c(1) , and just like in the previous step, this
follows when applying the boundary condition to the solution. If we consider
the equation
DD∗ f + ρ21,1 f = h
with f = f (r) and h = h(r), and take the inner product with g = J1 (ρ1,1 r), we
obtain after a few integrations by parts
Z
−ρ1,1 J0 (ρ1,1 )f (1) =
1
h(r)J1 (ρ1,1 r) rdr.
0
107
3.1 Linear stability for columnar flows
Applying this result to (3.1.8) we find
ρ1,1
1
0 =: −ρ1,1 J0 (ρ1,1 )ξr(1) (1) =
T ρ21,1 c(1) K0 − (T β −1 ± ρ1,1 )K1 −
2a
2
1
(−T β −2 ± ρ1,1 β −1 + 2T ρ21,1)K2 +
4
ρ1,1 βJ1 (β −1 ) T ρ1,1 ± 8β K0 ∓
4ρ1,1 βJ0 (β −1 )K0 ,
with
Z
Z
1
J1 (ρ1,1 r)2 rdr, K1 =
K0 =
0
Z
K2 =
1
J1 (rβ −1 )J1 (ρ1,1 r)2 dr,
0
1
J0 (rβ −1 )J1 (ρ1,1 r)2 rdr.
0
In figure 3.2 we have plotted K1 and K2 versus β −1 . This results in an explicit
0.08
0.08
K1
K2
0.06
0.06
0.04
0.04
0.02
0.02
2
2
4
6
8
10
12
4
6
8
10
14
- β −1
12
- β −1
14
Figure 3.2: K1 (left) and K2 (right) versus β −1 . Note that most of ‘mass’ is around
β −1 = ρ1,1 , related to the orthogonality of Bessel functions.
expression for c(1) :
c(1) =
2(β −1 ± T ρ1,1 )K1 + (−β −2 ± T ρ1,1 β −1 + 2ρ21,1 )K2
2ρ21,1 J0 (ρ1,1 )2
−1
2
−1
± 4T ρ−1
) − βJ1 (β −1 ) ∓ 8T ρ−1
).
1,1 βJ0 (β
1,1 β J1 (β
(3.1.9)
Note the symmetry (T, ±) → (−T, ∓) that leaves c(1) invariant.
When we investigate how the fastest travelling wave on the counter-rotating
flow (T = 1, ± → −) and the slowest travelling wave on the co-rotating flow
(T = −1, ± → +) behave for β −1 = ρ1,1 , we observe that the quotient involving
the integrals K1 and K2 vanishes and that the remaining terms evaluate to
108
3 Stability of swirling flows
βJ3 (β −1 ) = ρ−1
1,1 J3 (ρ1,1 ), being exactly the linear correction to λ according to
(3.1.6b).
Proposition 3.6 Consider the (axisymmetric) multi-mode relative equilibrium
solutions in (3.1.2a), with u± as in (3.1.1a,b). Set β −1 = ρ1,1 , A± = A = ε
and ` = 0. Then
– the counter-rotating flow corresponds to first order to the fastest travelling
wave that can be sustained by the columnar relative equilibrium (3.1.5),
and
– the co-rotating flow corresponds to first order to the slowest travelling wave
that can be sustained.
When we want to determine the Froude number corresponding to a columnar RE
with a small Beltrami component, we can use (3.1.7) and (3.1.9):
Nε
=
c+1 + c−1
c+1 − c−1
(0)
=
=
(1)
(0)
(1)
c+1 + εc+1 + c−1 + εc−1
+ O(ε2 )
(0)
(1)
(0)
(1)
c+1 + εc+1 − c−1 − εc−1
2β −1 K +(2ρ2 −β −2 )K
1
1,1
2
−1
−1
−
2β
J
(β
)
2f ρ1,1 + ε
2
1
ρ1,1 J0 (ρ1,1 )
+ O(ε2 ).
2K1 +β −1 K2
−1
−1 2
−1
−1
8a ± ε
+ 8ρ1,1 βJ0 (β ) − 16ρ1,1 β J1 (β )
J (ρ )2
0
1,1
This result shows the following interesting feature. We observed that when the base
flow (the columnar RE) has β −1 = ρ1,1 , the velocity λ coincides with either the fastest
or the slowest wave that can be superposed on a rigid-body rotation plus uniform
translation. When we investigate at this value of β −1 the influence of the Beltrami
component on the Froude number we discover an interesting difference between the
co- and counter-rotating flow. In figure 3.3 we plotted the linear corrections to the
Froude number of the ε = 0-flow versus the Froude number, i.e., γ versus N0 with
Nε = N0 + γε + O(ε2 ).
Note that the counter-rotating (minus) flow is much more sensitive to the presence
of a small Beltrami component than the co-rotating (plus) flow, especially when the
unperturbed flow is supercritical (N0 > 1).
A comparison theorem
The statements in proposition 3.5 and 3.6 might suggest that when the amplitude A of
the Beltrami component in the columnar flow (3.1.1a,b) is O(1), the slowest and fastest
travelling wave are related to the multi-mode RE solutions. Indeed, this is the case
and this result shows that the multi-mode RE solutions are not just a mathematical
curiosity but have a very specific physical interpretation. Moreover, these multi-mode
109
3.1 Linear stability for columnar flows
0.8
0.7
0.6
γ
6
0.5
–
0.4
0.3
0.2
+
0.1
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
- N (Froude number ε = 0-flow)
0
Figure 3.3: The linear correction to the Froude number of the unperturbed rigid-body
rotation plus uniform translation (= N0 ), caused by a weak Beltrami component, versus N0 .
The dotted line corresponds to the minus flow and the solid line to the plus flow.
RE solutions that connect chapter 2 and 3 with chapter 4: there we will investigate
the point where a modulated columnar RE changes from supercritical to subcritical,
i.e., where the slowest travelling wave has zero speed. This wave will determine socalled solvability conditions for the error-equation (see chapter 4 for details): it is
rather rare that this slowest travelling wave can be obtained so explicitly. The proof
of this result is based on Sturm’s Fundamental Theorem (e.g. Ince (1926)).
Proposition 3.7 (Sturm’s Fundamental Theorem) Consider the two problems



 DD∗ u − G1 (r)u = 0,
(P1 )


 u(0) = 0, u0 (0) = 1.



 DD∗ v − G2 (r)v = 0,
(P2 )


 v(0) = 0, v 0 (0) = 1.
with r ∈ [0, 1]. Further, let G1 and G2 be continuous, G1 ≥ G2 throughout (0, 1), and
G1 6= G2 at some points of the interval. Then v oscillates more rapidly than u. In
particular, if v(1) = 0 and v > 0 on (0, 1), than u > 0 on (0, 1].
For a proof see Ince (1926).
With proposition 3.7 we can prove one of the main theorems in this chapter and in
this thesis.
110
3 Stability of swirling flows
Theorem 3.8 Consider the multi-mode RE solution,
± ± iµ(z−λt)
v̂ ±
+ A± u± + αreϕ + Λez
k = < Ak uk e
(3.1.10)
with

µJ1 (rρ1,1 )
−i


ρ1,1

 q


2 + µ2 J (rρ


1
ρ
)
±
1
1,1
1,1
,

u k = ±

2

ρ1,1




J0 (rρ1,1 )

corresponding to βk2 =
1
ρ21,1 +µ2


0




q


±

u = ±J1 (r ρ2 + µ2 )

1,1


q


J0 (r ρ21,1 + µ2 )
(3.1.11)
q
, k = (1, 0, 1), and Λ = λ ± 2α/ ρ21,1 + µ2 ; A±
k is
arbitrary. Write A± = κα, α > 0. The solutions in (3.1.10) describe travelling
waves, superposed on a columnar swirling flow. The following holds true:
• for
−
4
4
≤κ≤−
,
ρ1,1
ρ1,1 J0 (ρ1,1 )
+ iµ(z−λt)
lim < A+
u
e
is the slowest travelling wave that can be sustained by
k k
µ→0
+
lim καu + αreϕ + Λez ;
µ→0
• for
4
4
,
≤κ≤
ρ1,1 J0 (ρ1,1 )
ρ1,1
− iµ(z−λt)
lim < A−
u
e
is the fastest travelling wave that can be sustained by
k k
µ→0
+
lim καu + αreϕ + Λez .
µ→0
Remark The bounds ∓4ρ−1
1,1 mark the point where the angular velocity vanishes at
the axis, i.e., the transition between counter-clockwise to clockwise rotation near the
axis. When κ is in the interval described in the theorem, the axial vorticity ωz of the
columnar flow,
µ→0
ωz = ±β −1 (Uz − λ) = ±ρ1,1 (Uz − λ)
is sign-definite on the interval [0, 1]: the bounds ∓4ρ−1
1,1 /J0 (ρ1,1 ) mark the point where
this axial vorticity starts to change sign.
3
111
3.1 Linear stability for columnar flows
Proof Since we are dealing with axisymmetric disturbances ξ, let us introduce the
stream function ψ,
ψ(r, z) = φ(r)eiµ(z−ct) ,
such that
µ
ξr = −i ψ,
r
ξz =
1 dφ iµ(z−ct)
,
e
r dr
ξϕ =
ψD∗ Uϕ
.
r(Uz − c)
Hence, in order to have regular solutions, we require Uz − c to be sign-definite. With
this definition, (3.1.3) can be written as
2Uϕ (Uϕ + rUϕ0 ) Uz00 r − Uz0
d 1 dφ
2
r
+ −µ +
−
φ = 0,
dr r dr
r2 (Uz − c)2
r(Uz − c)
(3.1.12)
φ(0) = φ(1) = 0,
±
with Uϕ = καu±
ϕ +αr, Uz = καuϕ +Λ (see (3.1.11)). We noticed already that c = λ in
(3.1.12) yields a solution: the non-columnar part (travelling wave) of the multi-mode
RE solution in (3.1.10): φ(r) = rρ−1
1,1 J1 (ρ1,1 r). For this value of c the term between
brackets in (3.1.12) reduces simply to −µ2 + β −2 = −µ2 + ρ21,1 + µ2 = ρ21,1 . The
statement that this non-columnar travelling wave corresponds in the limit µ → 0 to
an extremely long wave with smallest or largest speed can now be translated into:
when c = λ ∓ γ in (3.1.12) with µ → 0, there is no solution φ when γ > 0.
Using Sturm’s Fundamental Theorem we need to prove
2Uϕ (Uϕ + rUϕ0 )
Uz00 r − Uz0
−
≤ ρ21,1
2
2
r (Uz − (λ ∓ γ))
r(Uz − (λ ∓ γ))
(3.1.13)
for all γ > 0 and
1
Uϕ = ± καJ1 (rρ1,1 ) + αr,
2
Uz =
1
καJ0 (rρ1,1 ) ± 2α/ρ1,1 + λ.
2
Rearranging terms in (3.1.13) we obtain the condition
1
r2 ρ21,1 γ 2 + rρ1,1 α ±κ rρ1,1 J0 (rρ1,1 ) + 2J1 (rρ1,1 ) + 8r γ ≥ 0,
2
|
{z
}
g(r)
or
Krρ1,1 J0 (rρ1,1 ) + 2KJ1 (rρ1,1 ) + 8r ≥ 0
on [0, 1],
with K = ±κ. We plotted the function f (z) = zJ0 (z) + 2J1 (z), z ∈ [0, ρ1,1 ] in figure
3.4, and with this information the first bounds on K (and so on κ) are obtained from
112
3 Stability of swirling flows
f (z)
1.5
6
1
0.5
00
0.5
1
1.5
2
2.5
-z
3
3.5
ρJ0 (ρ)
-0.5
-1
?
-1.5
Figure 3.4: The function f (z) = zJ0 (z) + 2J1 (z) (solid line) and its tangent at zero,
`(z) = 2z (dashed line) versus z, for z ∈ [0, ρ], ρ = ρ1,1 .
g 0 (0) ≥ 0
→ K≥−
g(1) ≥ 0
→ K≤−
4
ρ1,1
,
8
.
ρ1,1 J0 (ρ1,1 )
Combining these bounds with the requirement that Uz − λ should not change sign on
(0, 1) leads to the result stated in the theorem.
2
3.1.2
Parallel disturbances
When the disturbance is such that the axial direction does not enter, the equation
that determines the stability of the base flow reduces to (set µ = 0 in (3.8))
rD(D∗ Uϕ )
2
D(rD(rξr )) − ` + −1
(3.1.14)
ξr = 0,
r Uϕ − c
with σ = −`c. The relevant instability criterion for (3.1.14) is due to Rayleigh (1880)
(and refined by Fjortoft (1950)):
Proposition 3.9 (Rayleigh (1880)) Consider a columnar swirling flow (0, Uϕ (r), Uz (r)).
A necessary condition for instability to parallel disturbances is that D(D∗ Uϕ ), being
the gradient of the axial vorticity, changes sign in 0 < r < 1.
This gradient of the axial vorticity evaluates to
D(D∗ Uϕ ) = β −2 J1 (rβ −1 )
when substituting a columnar relative equilibrium. Hence,
113
3.1 Linear stability for columnar flows
Proposition 3.10 The RE in (3.1.1a,b) is (neutrally) stable to parallel disturbances
when 0 ≤ β −1 ≤ ρ1,1 .
By substituting c = −`α in (3.1.14) we can obtain an exact solution, corresponding
to the parallel multi-mode RE solution.
3.1.3
Three-dimensional disturbances
Finally, we will investigate the stability of the columnar RE towards three-dimensional
disturbances. The governing equation is (3.8).
Instability criterion: ring modes
Leibovich and Stewartson (1983) formulated a sufficient condition for linear instability
of an arbitrary columnar flow.
Proposition 3.11 (Leibovich and Stewartson (1983)) Consider a columnar swirling
flow (0, Uϕ (r), Uz (r)). If
" 2 #
d Uϕ
d Uϕ d
dUz
G = Uϕ
(rUϕ ) +
<0
dr
r
dr
r
dr
dr
at any point of the flow field, then the flow is unstable.
Roughly, this result is obtained by looking for the most unstable mode as a function
of the quotient µ/` (ratio between axial and azimuthal frequency), keeping ` fixed.
The disturbances that cause instability can be thought of as “ring modes”, since they
are concentrated at some value of r in the interior of the flow field. Different from
these ring modes are the “wall modes”, discussed by Maslowe and Stewartson (1982),
that are concentrated near the wall. We will come back to the latter following the
discussion of the behaviour of G.
Evaluating G in a columnar RE, and writing A = κα, we obtain the following result:
1
1
J (rβ −1 )
∓ β −1 κ 2
G± =α4 ± κJ1 (rβ −1 ) + r
·
2
2
r
(
2 )
1 −1
1 −1
1 −1
−1
−1
−1
± β κJ0 (rβ ) + 2 +
∓ β κJ2 (rβ )
κβ J1 (rβ )
.
2
2
2
±
The maximum of G± on the unit interval [0, 1] is denoted by G , and in figure 3.5 we
±
indicated the contours G = 0 in the (κ, β −1 )-plane. We combined the F + = 1/4+
contour and the G = 0-contour in figure 3.6a and similarly with the minus flow in
figure 3.6b. Let us make some remarks about the results.
• Fixing β, we study the limit κ → 0:
G± =
1 4 J2 (rβ −1 )2 κ2
α
+ O(κ3 ),
2
β2
as κ → 0.
114
3 Stability of swirling flows
15
−
κ
610
G <0
±
G >0
+
G <0
5
±
G >0
0
0
2
4
6
- β −1
8
10
±
Figure 3.5: The contour lines G = 0 in the (κ, β −1 )-plane. The dotted line marks the
+
+
−
boundary between G < 0 and G > 0, while the solid line indicates regions where G < 0
−
or G > 0.
Hence, G± is larger than zero in (0, 1) and only zero in isolated points when the
Beltrami component is sufficiently (in comparison with the rigid-body rotation),
implying that the sufficient condition for instability is not satisfied.
On the other hand, when the Beltrami component dominates we need to examine
the leading order term of G± related to κ (we abbreviate J1 (rβ −1 ) with J1 and
so on):
α4 J1 J2 rJ12 − 2J1 J2 + rJ22
G±
=
−
+ O(κ−1 ),
κ4
16
r2 β 3
as κ → ∞.
The numerator in the fraction is positive for all combinations rβ −1 , and hence,
asymptotically G± is less than zero when β −1 < ρ1,1 .
Proposition 3.12 For fixed β −1 < ρ1,1 , the RE in (3.1.1a,b), consisting only
of the Beltrami component (κ → ∞) is linearly unstable to true three-dimensional
disturbances. The presence of rigid-body rotation (κ < ∞) tends to stabilise a
pure Beltrami flow.
+
• The abrupt change in the G = 0-contour at β −1 ≈ 5 can also be explained.
When we investigate the different terms multiplying an appearing power of κ in
115
3.1 Linear stability for columnar flows
15
15
III
κ
10
6
II
κ
10
6
IV
IV
III
5
5
IV
II
I
0
0
I
V
2
4
6
(a)
8
- β −1
10
0
0
0.5
1
1.5
(b)
2
- β −1
2.5
Figure 3.6: (a) (co-rotating flow) The dotted line is the F + = 1/4-contour, the solid
+
line is the G = 0-contour and the dashed line is β −1 = ρ1,1 and indicates the border
between parallel stability and parallel instability. Note that the three curves intersect at
(β −1 , κ) = (ρ1,1 , −4/J0 (ρ1,1 )/ρ1,1 ) (this is proven in the text). The regions I, II, III, IV and V
indicate regions in parameter space where we have (I) stability to axisymmetric and parallel
disturbances, (II) instability to three-dimensional disturbances, (III) instability to threedimensional and possibly axisymmetric disturbances, (IV) possible instability to parallel,
axisymmetric and three-dimensional disturbances and finally (V) stability to axisymmetric
disturbances but possible instability to parallel and three-dimensional disturbances.
−
(b) (counter-rotating flow) The dotted line is the G = 0-contour and the two solid lines
−
are the two branches of the F = 1/4-contour. Note that the whole plot lies at the left
of β −1 = ρ1,1 . The regions I, II, III and IV indicate regions in parameter space where we
have (I) stability to axisymmetric and parallel disturbances, (II) no sufficient condition for
either stability or instability, (III) instability to three-dimensional and possibly axisymmetric
disturbances and (IV) instability to three-dimensional disturbances.
G+ we obtain the following three terms:
κ4 :
κ3 :
κ2 :
1 4
rJ 2 − 2J1 J2 + rJ22
,
α J1 J2 1
16
r2 β 3
1
rJ 2 − 4J1 J2 + rJ22
− α4 J2 1
,
8
rβ 3
1 4 2 −2
α J2 β .
2
−
(3.1.15a)
(3.1.15b)
(3.1.15c)
The numerators in the fractions are positive for all combinations rβ −1 . Consider
now a narrow strip in the (κ, β −1 )-plane around β −1 = ρ2,1 ≈ 5.14. Looking at
figure 3.5 we can identify three different stages. For relative small κ the term in
(3.1.15c) dominates, implying G+ > 0. When κ increases, the term in (3.1.15b)
starts to dominate, and this term has a sharp change at β −1 = ρ2,1 where the
116
3 Stability of swirling flows
maximum changes sign—the straight part in figure 3.5. And finally, when κ
increases even more, the presence of the term in (3.1.15a) is perceptible, and we
will (slowly) move to the asymptote β −1 = ρ1,1 .
• Let us analyse the line β −1 = ρ1,1 for the co-rotating flow (plus-sign). To
be more precise, let us take a neighbourhood of κ = −4/J0 (ρ1,1 )/ρ1,1 ≈ 2.6.
This neighbourhood is close to the point where three important curves intersect
in figure 3.6a. It is easy to show that in this neighbourhood G+ has a local
maximum at r = 1, which can only be a global one (in this neighbourhood)
when G(r = 1) > 0. Substituting κ = −4/J0 (ρ1,1 )/ρ1,1 + ν and r = 1 into G+
we obtain
2
1
G+ (r = 1) = J0 (ρ1,1 )ρ1,1 −4 + νJ0 (ρ1,1 ) ν
8
and since J0 (ρ1,1 ) < 0 we observe that when ν > 0 we have a sufficient condition
+
for instability (G < 0) and when ν < 0 we do not have a sufficient condition
+
for instability (G > 0).
Let us finally show that the third curve, namely F + = 1/4 intersects the point
(β −1 , κ) = (ρ1,1 , −4/J0 (ρ1,1 )/ρ1,1 ) as well. To that end, we analyse the numerator and the denominator of F + separately. We concentrate on analysing r = 1,
since there the denominator vanishes, and depending on the sign of the numerator F sets off to plus or minus infinity. Substituting κ = −4/J0 (ρ1,1 )/ρ1,1 ) + ν
into denominator and numerator, and making a Taylor series around r = 1 we
obtain
numerator(F + ) = 4J0 (ρ1,1 )3 ρ21,1 ν + 8J0 (ρ1,1 )2 ρ31,1 (r − 1)2 +
O(ν(r − 1), (r − 1)3 ),
denominator(F + ) = 16J0 (ρ1,1 )2 ρ31,1 (r − 1)2 + O(ν(r − 1)2 , (r − 1)3 ).
This seems to indicate that the point in the (β −1 , κ)-plane corresponding to
ν = 0 does not belong to the F + = 1/4-contour. However, when ν 6= 0 and
fixed, we have a region near r = 1 where F + tends to either +∞ (ν < 0) or
to −∞ (ν > 0), and in the latter case the minimum of F + is definitely below
1/4. Hence, the point in the (β −1 , κ)-plane corresponding to ν = 0 marks the
boundary between two different regions and should be included in the F + = 1/4contour.
• When looking at figure 3.6b, we observe that the curve (β −1 , κ = 4β) plays
again an important role. Indeed, on substituting κ = 4β + ν into G± and
analysing the behaviour for β −1 → 0, we obtain
G± = −
α4 4 −5
r νβ + O(β −6 ),
16
as β −1 → 0
and hence a sufficient condition for instability above the curve (ν > 0) and not a
−
sufficient condition below the curve (ν < 0). This shows as well that the G = 0-
3.2 Nonlinear constrained stability
117
contour runs precisely between the two branches of the F − = 1/4-contour as
can be detected from figure 3.6b.
Instability criterion: wall modes
Maslowe and Stewartson (1982) formulated a rather awkward but very transparent
looking condition that is sufficient for instability. It only requires information at the
wall of the pipe.
Proposition 3.13 (Maslowe and Stewartson (1982)) A columnar swirling flow
(0, Uϕ (r), Uz (r)) is unstable if either
dUz Uϕ |wall 6= 0,
6= 0
dr wall
or Uϕ |wall = 0.
We noted before that the modes which induce the instability described by Maslowe
and Stewartson can be thought of as ‘wall modes’.
Their criterion is easy to evaluate for the columnar RE, yielding that only when
β −1 = ρ1 we do not have immediate instability (as long as α 6= 0).
3.2
Nonlinear constrained stability
In the introduction to this chapter we highlighted two types of stability concepts,
namely linear stability and nonlinear stability. The linearisation (and thus in general
simplification) in the former approach has its drawbacks in that (i) when (part of)
the spectrum lies on the imaginary axis no conclusions can be drawn, and (ii) that
certain symmetry gets lost: the nonlinear stability properties of a swirling flow do
not change when an (intrinsically stable) rigid-body rotation or uniform translation
is added, contrary to the linear theory. Moreover, note that most of our results in
section 3.1 were formulated as necessary or sufficient conditions for either stability or
instability. What then can we expect from the nonlinear theory?
Nonlinear stability of a base flow requires that small but finite perturbations remain
uniformly bounded for all time as they evolve subject to the full nonlinear equations.
A classical method in ordinary differential equations uses the concept of Lyapunov
function:
Definition A real-valued function V : N ⊂ Rn → R, where N is a neighbourhood
of (the equilibrium) ue ∈ Rn , is said to be a (weak) Lyapunov function in N if
• the partial derivatives of V exist and are continuous;
• V is positive definite, i.e.,
V (u) > 0 for u ∈ N \ {ue } and V (ue ) = 0;
118
3 Stability of swirling flows
• V is dynamically conserved (this condition may be replaced by the condition
that dV /dt is negative semi-definite when dissipation is present).
When we are able to construct a Lyapunov function for our dynamical system we have
formal stability (Holm et al. (1985)), and in finite dimensions this implies nonlinear
stability.
From the two conditions that V vanishes in the equilibrium point and that V is
dynamically invariant, we deduce that in Poisson systems a natural candidate for V
consists of a (linear) combination of conserved quantities, i.e.,
V (u) = (E − βB − αA − λFax ) (u) +
1
(αA + λFax ) (û)
2
(3.2.1)
which vanishes, together with its first variation, for our RE û, defined and constructed
in chapter 2. Definiteness of V (u) can be interpreted as definiteness of δ 2 V (RE).
However, in infinite dimensions, the definiteness of δ 2 V (RE) is necessary, but not
sufficient for nonlinear stability (Ball and Marsden (1984)). To obtain nonlinear
stability we need to prove the local convexity of V near the RE.
However, an additional complication is the indefinite character of the helicity (recall
the positive and negative eigenvalues of the curl-operator). Szeri and Holmes (1988)
overcome this problem by applying a high-wavenumber cut-off: this will be discussed
in subsection 3.2.3. Instead, we use the concept of constrained stability. The main idea
behind both approaches is to define an appropriate space of perturbations that can
be physically motivated such that the indefinite helicity becomes definite. The fewer
ad hoc conditions are placed on the perturbation, the more elegant the description of
this subspace will be.
To introduce the concept of constrained stability, let a perturbation of a RE û be given
by u = û + εη, |η| = O(1), such that u takes on the same values of the constraints
as û, i.e.,
Fax (u) = Fax (û) =: f,
A(u) = A(û) =: a,
B(u) = B(û) =: b.
(3.2.2a)
(3.2.2b)
(3.2.2c)
This leads to conditions on η, and two obvious ones are
Fax (η) = A(η) = 0.
The relation with unconstrained stability becomes clear when we reverse the order,
i.e., first choose the perturbation u, compute the corresponding values of the con˜ ã, b̃) and construct the RE to these values. Denote a particular RE,
straints, say (f,
corresponding to constraint values (a, b, f ) by RE(a, b, f ). Then (see also figure 3.7):
u = RE(ã, b̃, f˜) + η̃
˜ − RE(a, b, f ) + η̃
= RE(a, b, f ) + RE(ã, b̃, f)
˜ − û + η̃,
= û + RE(ã, b̃, f)
119
3.2 Nonlinear constrained stability
with RE(ã, b̃, f˜) of the same ‘topological’ type as û, i.e., the same spatial structure.
˜ and assuming that the distance beProving the constrained stability of RE(ã, b̃, f)
(a, b, f )
1û
0
0
1
= RE(a, b, f )
˜
(ã, b̃, f)
00
11
u
00
11
η
1
0
˜
0RE(ã, b̃, f)
1
0
1
Figure 3.7: Relation between constrained stability and unconstrained stability.
tween RE(ã, b̃, f˜) and RE(a, b, f ) can be controlled with a − ã, b − b̃ and f − f˜ leads
to unconstrained nonlinear stability.
In the first subsection we will study the constrained stability of pure Beltrami flows
in H0 (defined in section 2.2), so α = λ = 0 in (3.2.1). The reason for this is the
remark on page 97 in connection with (3.4), where we noted that subtracting a rigidbody rotation and a uniform translation does not change the evolution of the energy
of a perturbation. As a consequence, we only require (3.2.2c) in subsection 3.2.1.
In subsection 3.2.2 we will turn to the constrained stability of a columnar RE, and
we will observe that an extra rigid-body rotation does matter for the constrained
stability. Finally, in the last subsection we will discuss the high-wavenumber cut-off
in the context of RE solutions.
3.2.1
Nonlinear constrained stability of Beltrami flows
Let the base flow in this subsection be denoted by
û = Au± ,
curl u± = ±β0−1 u± ,
and let us scale u± such that E(u± ) = 1. In order to prove the definiteness of V1 (u),
V1 (u) = E(u) ∓ β0 B(u)
we compute the second variation of V1 (u):
hδ 2 V1 (u)η, ηiD :=
∂2
V
(u
+
εη)
.
1
2
∂ε
ε=0
120
3 Stability of swirling flows
Because δB(u; η) and δE(u; η) are linear in u, the second variation of B and E is
obtained by simply replacing u by η:
Z
Z
1
hδ 2 B(u)η, ηiD =|D|−1
η · curl η + |D|−1
(η × η) · n = 2B(η),
2 Σ
ZD
hδ 2 E(u)η, ηiD =|D|−1
|η|2 = 2E(η).
D
Remark Since the only components in V1 (u) are the energy functional and the
helicity functional—a Casimir—the resulting approach is an example of the energyCasimir method that is systematically described in Holm et al. (1985), and applied
to axisymmetric pipe flows by Szeri and Holmes (1988). They essentially work with
an infinity of Casimirs (see section 1.2) but can only obtain nonlinear axisymmetric
stability or instability. However, as we found in the previous section on linear stability, the non-axisymmetric, true three-dimensional disturbances are very important
and should be incorporated in the analysis. The consequence for allowing threedimensional perturbations is that only one Casimir can be used: the helicity.
3
Explicit conditions on η
First of all, we have that η ∈ H0 . This space is defined in section 2.2 and there we
proved that the Beltrami flows form a basis of H0 . The explicit Beltrami decomposition of an element u ∈ H0 that we introduced there will play a role on the background,
but the only important components of η are (i) the component that is aligned with
û and (ii) the component that is orthogonal to û:
εη = ε1 u± + ε2 η ⊥ ,
hu± , η ⊥ iD = 0;
(3.2.3)
η and η ⊥ are normalised to unity:
E(εη) = ε2 E(η) = ε2 = E(ε1 u± + ε2 η ⊥ )
= ε21 E(u± ) + ε22 E(η ⊥ )
= ε21 + ε22 .
Note that curl η ⊥ is still orthogonal to u± , as can be checked with proposition 2.8.
Using this decomposition, (3.2.2c) turns into
0 =: B(û + εη) − B(û) = ±β0−1 2ε1 A + ε21 E(u± ) + ε22 B(η ⊥ )
= ±β0−1 2ε1 A + ε21 + ε22 B(η ⊥ ).
(3.2.4)
Remark Let us interpret (3.2.4). Fixing η ⊥ , (3.2.4) determines another relation
between ε1 and ε2 . Combining this relation with ε21 + ε22 = ε2 , a quadratic equation
for ε1 is obtained:
−1
±β0 − B(η ⊥ ) ε21 ± 2Aβ0−1 ε1 + ε2 B(η ⊥ ) = 0.
(3.2.5)
121
3.2 Nonlinear constrained stability
Equation (3.2.5) has a solution iff
(±2Aβ0−1 )2 − 4ε2 B(η ⊥ )(±β0−1 − B(η ⊥ )) =
4
A2 ∓ β0 ε2 B(η ⊥ ) + ε2 β02 B(η ⊥ )2 ≥ 0.
β02
Hence,
• when the sign of the helicity of η ⊥ , B(η ⊥ ), is opposite compared to the helicity
of the base flow û, we always have a solution ε1 ;
• when the helicity of η ⊥ , B(η ⊥ ), and the helicity of the base flow have the same
sign, we have a bound on B(η ⊥ ), depending on β0 , A and ε.
3
The constrained perturbations are collected in N:
the Beltrami decomposition
.
N = η ∈ H0 of η in (3.2.3) satisfies (3.2.4)
The second variation of V1 (u) on N; a sufficient condition for constrained
stability
The conditions on εη will now be used in the analysis of δ 2 V1 (û), acting on εη.
2
δ E(û) ∓ β0 δ 2 B(û) εη, εη
= 2E(εη) ∓ 2β0 B(εη)
= 2ε22 E(η ⊥ ) ∓ β0 B(η ⊥ )
(3.2.6)
Combining (3.2.6) with (3.2.4) we obtain
2
δ E(û) ∓ β0 δ 2 B(û) εη, εη = 2ε22 E(η ⊥ ) + 2 2ε1 A + ε21 E(u± )
= 2ε2 + 4Aε1
(3.2.7)
Observe that the first term on the right-hand side in (3.2.7) is always positive definite,
and hence, a sufficient condition that V1 (u) is positive definite on N is that Aε1 > 0.
Proposition 3.14 A Beltrami flow û = Au± (columnar or non-columnar) is formally constrained stable, i.e., their exists a Lyapunov function V1 (u), when the space
of perturbations is restricted to
N≥ = {v ∈ N | hv, ûiD ≥ 0} .
Writing the perturbation εη = ε1 u± +ε2 η ⊥ with η ⊥ , u± D = 0, the second variation
of the Lyapunov function is given by
2
δ V1 (û)εη, εη = 2ε22 E(η ⊥ ) + 2 2ε1 A + ε21 E(u± ),
and hence its convexity on N≥ is obvious. Note that the condition Aε1 > 0 is a sufficient condition for positive definiteness. Using (3.2.5) we can restate this condition
by saying the helicity of η ⊥ should have the opposite sign compared with the helicity
of the base flow û. Let us find out whether this condition is necessary as well.
122
3 Stability of swirling flows
u±
1111
0000
0000
1111
N>
0000
1111
0000
1111
1111
0000
0000
1111
0000
1111
N<
0000
1111
N=
−u±
Figure 3.8: The partition of the function space caused by the manifold N= , containing all
constrained perturbations that are orthogonal to û: N= = {v ∈ N | v, u± D = 0}.
A necessary condition for constrained stability
Let us start with choosing η ⊥ = A± u±
β , β 6= β0 , such that
E(η ⊥ ) = A±
2
E(u±
β ) = 1.
Note that
±B(η ⊥ ) = β −1 E(η ⊥ ) = β −1 > 0,
and hence Aε1 < 0 (see (3.2.4)). For this choice of η ⊥ , we can simplify (3.2.6) to
2
β0
2
(3.2.8)
δ V1 (û)εη, εη = 2ε2 1 −
β
This expression is positive as long as β0 /β ≤ 1.
Proposition 3.15 Let the base flow u±
β0 be such that for all eigenvalues β 6= β0 ,
β0 /β > 1. A necessary and sufficient condition for constrained stability is that the
perturbations should be restricted to
N≥ = {v ∈ N | hv, ûiD ≥ 0} .
Proof The sufficient part is covered by proposition 3.14. Hence, choose η⊥ such
that ±B(η ⊥ ) > 0 but arbitrary otherwise, implying Aε1 < 0, η ∈ N< . In order to
123
3.2 Nonlinear constrained stability
make the helicity of η ⊥ of the same sign as B(û), η ⊥ contains necessarily a Beltrami
component u±
β∗ , β∗ 6= β0 (recall the Beltrami decomposition in chapter 2). In view of
the assumption on β0 we have β0 /β∗ > 1. Since η ⊥ is arbitrary, and E(η ⊥ ) = 1, we
can distribute the energy of η ⊥ in such a way that it accumulates in u±
β∗ . According
2
to (3.2.8) the second variation is negative and hence indefiniteness of V1 .
Depending on the period L in the axial direction, the β0 in proposition 3.15 corresponds to either a 3D Beltrami flow or a columnar Beltrami flow with β = ρ−1
1,1 or a
parallel Beltrami flow with ` = 1 and also β = ρ−1
(see
figure
2.11).
1,1
3.2.2
Nonlinear constrained stability of columnar relative equilibria
Let the base flow in this subsection be denoted by
û = Au± + αreϕ + Λez ,
u± columnar and curl u± = ±β0−1 u± .
In order to prove the definiteness of V2 (u),
V2 (u) = E(u) ∓ β0 B(u) − αA(u) − λFax (u) +
1
(αA(û) + λFax (û))
2
we compute the second variation of V2 (u). Note that the extra linear terms in V2
do not contribute to the second variation; they are necessary for a vanishing first
variation in the RE.
Explicit conditions on η
Because of the extra condition on η due to the angular momentum constraint (3.2.2b)
(the axial momentum constraint (3.2.2a) is automatically satisfied since η ∈ H0 ), we
decompose η ⊥ in a non-columnar part and a columnar part:
⊥
εη = ε1 u± + ε2 η ⊥
nc + ε3 η c ,
±
⊥
hu± , η ⊥
nc iD = hu , η c iD = 0.
(3.2.9)
⊥
η, η ⊥
c and η nc are normalised to unity:
⊥
E(εη) = ε2 E(η) = ε2 = E(ε1 u± + ε2 η ⊥
nc + ε3 η c )
2
⊥
= ε21 E(u± ) + ε22 E(η ⊥
nc ) + ε3 E(η c )
= ε21 + ε22 + ε23 .
The angular momentum constraint (3.2.2b) can now simply be written as
ε1 A(u± ) + ε3 A(η ⊥
c ) = 0.
(3.2.10)
124
3 Stability of swirling flows
The helicity constraint (3.2.2c) will have some additional terms compared with (3.2.4),
due to the columnar cross-terms:
0
=: B(û + εη) − B(û)
α
ε1 A(curl u± ) + ε3 A(curl η ⊥
= ±β0−1 2ε1 A + ε21 +
c ) +
2
2
⊥
ε22 B(η ⊥
)
+
ε
B(η
)
(3.2.11a)
nc
3
c
h
i
α
(!)
⊥
= ±β0−1 2ε1 A + ε21 − ε3 A(η ⊥
c ∓ β0 curl η c ) +
2
2
⊥
ε22 B(η ⊥
(3.2.11b)
nc ) + ε3 B(η c ),
where we used at (!) the fact that
ε1 A(curl u± ) = ±ε1 β0−1 A(u± ) = ∓ε3 β0−1 A(η ⊥
c ).
Remarks (i) Condition (3.2.10) shows that we need a columnar component in η ⊥ in
order to have a nontrivial problem: ε3 = 0 implies ε1 = 0 and hence, using (3.2.11b),
B(η ⊥
nc ) = 0.
(ii) Conditions (3.2.10) and (3.2.11b), together with ε21 + ε22 + ε23 = ε2 determine ε1 ,
3
ε2 and ε3 .
The constrained perturbations are collected in N:
the Beltrami decomposition of η in
N = η ∈ H0 .
(3.2.9) satisfies (3.2.10) and (3.2.11b)
The second variation of V2 (u) on N
Using (3.2.6) and (3.2.9) we obtain
⊥
3
⊥
⊥
hδ 2 V2 (û)εη, εηi = 2ε22 E(η ⊥
nc ) ∓ β0 B(η nc ) + 2ε2 E(η c ) ∓ β0 B(η c ) .
(3.2.12)
Multiplying (3.2.11b) by ±2β0 and adding it to (3.2.12) we obtain
⊥
hδ 2 V2 (û)εη, εηi = 2ε2 + 4Aε1 − ε3 αA(η ⊥
c ∓ β0 curl η c ).
(3.2.13)
Note that (3.2.13) is a generalisation of (3.2.7). We only give the sufficient condition
for stability.
Proposition 3.16 A columnar relative equilibrium û = Au± + αreϕ + Λez , with
curl u± = ±β0−1 u± , is formally constrained stable, i.e., their exists a Lyapunov function V2 (u), when the perturbations εη satisfy
hεη, ûiD ≥ 0
and
− αA(εη ∓ β0 curl (εη)) ≥ 0 .
Just like in the subsection on stability of a pure Beltrami flow the sufficient conditions
can be related to the sign of the helicity of η ⊥ . We observe that the term between
⊥
square brackets in (3.2.11b) is positive when the helicity of ε2 η ⊥
nc + ε3 η c has the
opposite sign compared to the helicity of the Beltrami component in the RE:
2
⊥
± ε22 B(η ⊥
nc ) + ε3 B(η c ) ≥ 0.
125
3.2 Nonlinear constrained stability
3.2.3
High-wave number cut-off (Szeri and Holmes (1988))
In this subsection we will analyse (3.2.6), without restrictions on η. This analysis has
been performed by Szeri and Holmes (1988) for axisymmetric flows, in which case the
Lyapunov function is much more involved due to the continuum of Casimirs, but as
a result it offers more freedom to investigate all kind of different columnar flows.
First of all, we analyse the effect of Beltrami components with helicity that have the
opposite sign compared with the base flow û; this picks out the definite term in (3.2.6)
namely, when we write
η⊥ =
X
+
− −
∓ ∓
A+
k uβ + Ak uβ + Ak uβ ,
0
β∈B\{β0 }
0
A±
k =
hη ⊥ , u±
β iD
±
hu±
β , uβ iD
,
(3.2.14)
(3.2.6) turns into
⊥
⊥
hδ V (û)η , η i = 2
2
|
X
β∈B
2
A∓
k
E(u∓
β)
{z
definite
β
1+ 0
β
+2
}
|
X
β∈B
2
A±
k
E(u±
β)
{z
indefinite
β
1− 0
β
.
}
To bound the indefinite term we can restrict the perturbations to those that have
components with higher wavenumbers then the base flow:
−1
A±
> β0−1 .
k = 0 in (3.2.14) when β
This is the high-wavenumber cut-off in the context of Beltrami flows and 3D perturbations.
Szeri and Holmes (1988) give physical arguments for these high-wavenumber cut-offs.
“In a real fluid, the arbitrarily large velocity gradients that accompany high-spatialwavenumber vortex density (and swirl) perturbations necessitate inclusion of viscous
effects, no matter how small the kinematic viscosity may be. Thus, inviscid stability
results can only be expected to yield physically reasonable predictions for perturbations of bounded wavenumber. (...) It is possible to show that high-wavenumber disturbances (...) decay exponentially with a time constant of order wavenumber squared
multiplied by viscosity (see also introduction of chapter 2). Thus we conclude that
dangerous disturbances are typically of low to moderate wavenumber, even in weakly
viscous flows.”
126
3 Stability of swirling flows
4
Swirling flows in an
expanding pipe
Until now we have investigated the flow in a pipe with a constant diameter by constructing several types of solutions, that were connected by varying (physical) parameters. Moreover, we obtained stability diagrams of the columnar flows and the stability
questions shed new light on the importance of special solutions that were identified
already in chapter 2 (the axisymmetric multi-mode relative equilibrium solutions).
In this chapter and in the next one we will use this ensemble of solutions to describe
the distortion of a particular member of it, the distortion caused by variations in the
diameter of the pipe (this chapter) or viscosity (next chapter). This description is set
up along the lines that were introduced in section 1.3. There we showed that several
approximation techniques are based on one and the same principle: quasi-stationary
or quasi-homogeneous approach (QHA for short; Fledderus and van Groesen (1996a)).
This technique is common practise in e.g. optics (Newell and Moloney (1992)). Let
us describe in this introduction how this technique can be applied to a swirling flow
in an expanding pipe.
First of all, we will use in this chapter and chapter 5 one particular value of β,
namely β = ρ−1
1,1 . This value of β is special in the sense that the sufficient condition
for instability, due to (Maslowe and Stewartson (1982)),
dUz Uϕ |wall 6= 0 and
6= 0 ⇒ instability,
dr wall
is not satisfied, since the radial derivative of the axial velocity vanishes:
Aρ1,1
d 1
AJ0 (rρ1,1 ) + Λ =−
J1 (ρ1,1 ) ≡ 0.
dr 2
2
wall
Since the radius of the pipe will play a profound role in the rest of the chapter we
take a close look (section 4.1) how this radius enters all quantities that we studied
until now. The effect of simply replacing r by r∗ = r · R with R = R(z) the radius of
128
4 Swirling flows in an expanding pipe
the pipe will give rise to interesting predictions about for instance the dependence of
the frequency of a precessing vortex core on the radius.
After having explicated the radius of the pipe we can denote a columnar RE in a
constant-diameter pipe, together with the corresponding total head, by
(4.1)
(U ± , H ± ) = U ± (R, a, b, f )(r), H ± (R, a, b, f )(r) .
Remarks (i) Since we are dealing with a columnar RE in (4.1), the a, b and f
are cross-sectional values of the angular momentum, helicity and axial momentum,
respectively.
(ii) The columnar RE in (4.1), being exact solutions in a constant-diameter pipe, will
serve as base functions when we are dealing with a pipe with varying diameter.
(iii) We will omit the ±-sign in the notation, and mostly work with the co-rotating
(plus) flow.
3
Since we will use in addition to the pressure formulation the total head formulation
of the Euler equations, we introduce its ‘operator’ form:
EEul (u, h) = ∂t u + ω × u + ∇h,
div u = 0
with ω = curl u and h = 12 |u|2 + p as before.
Consider a pipe with slowly varying radius and with adjustable ‘outlet’ radius (see
figure 4.1):
n
o
Pδ,ε = (x, y, z) ∈ R3 | x2 + y 2 < R̂(z) = R0 + δR1 (εz) ,
n
o
Σδ,ε = (x, y, z) ∈ R3 | x2 + y 2 = R̂(z) = R0 + δR1 (εz) .
We assume R1 to be monotone, 0 ≤ R1 ≤ 1. The z-interval of interest has length
O(ε−1 ). Now we pose two types of problems. The first one concerns an axisymmetric,
ε−1
6
δ
?
-z
Figure 4.1: The pipe geometry Pδ,ε . The meaning of δ and ε are indicated in the picture.
steady solution of the Euler equations in the expanding pipe Pδ,ε :
EEul (u, h) = 0 in Pδ,ε ,
u = u(r, z), h = h(r, z),
(Q1 )
129
with div u = 0 in Pδ,ε and u · n = 0 on Σδ,ε . The second problem widens the range
of solutions to arbitrary 3D, time-dependent solutions:
EEul (u, h) = 0 in Pδ,ε ,
u = u(r, ϕ, z, t), h = h(r, ϕ, z, t),
(Q2 )
with u divergence-free and the same condition on the boundary as in (Q1 ). We make
the following observations and assumptions:
• When δ = 0, (Q1 ) and (Q2 ) have a columnar solution. Moreover, the columnar
relative equilibrium in (4.1) is used as inlet condition at some reference point
z0 for certain values of the parameters.
• We assume that for δ small enough, (Q1 ) and (Q2 ) are regular when the columnar relative equilibrium is supercritical (we will come back on this last remark).
For example, when (u(δ, ε), h(δ, ε)) denotes the solution of (Q1 ) that coincides
at z0 with (U , H), then
u(δ, ε) = modulated columnar flow + O(ε),
h(δ, ε) = modulated columnar total head + O(ε),
for ε → 0. The ‘O(ε)’ stands for the error that is O(ε) when integrated over
a cross-section. This assumption is motivated by the results in Buntine and
Saffman (1995). They studied the axisymmetric Euler equations on geometries
like Pδ,ε for different values of ε and different swirl numbers. The swirl number
gives an indication whether the flow is super or subcritical.
We proceed in section 4.2 with considering (Q1 ) since this involves only axisymmetric
velocity fields. The regularity assumption invites us to try to develop for finite but
small δ its solution, u = u(δ, ε), h = h(δ, ε), into an ε-series, and we are interested in
the leading order term.
Remark Since we expect at some radius non-adiabatic behaviour, i.e., ‘vortexbreakdown’, we need to set up the analysis very careful and to investigate upper
bounds on δ. The relation between δ and ε is such that for every fixed value of δ the
value of ε can be made arbitrarily small, so we are in principle interested in slowly
varying geometries.
3
Comparable to quasi-static evolutions in systems with a slow time-dependence, we
will construct the leading-order term as a quasi-homogeneous succession of columnar
flow fields (4.1) (recall that we dropped the ±-sign),
v̂
= U (R(Z), a(Z), b(Z), f (Z))(r),
ĥ = H(R(Z), a(Z), b(Z), f (Z))(r) + h(Z),
(4.2a)
(4.2b)
with Z = εz and R(Z) = R̂(z). The incompressibility condition will provide the
radial velocity (since U in (4.1) is columnar, it has only a tangential and an axial
130
4 Swirling flows in an expanding pipe
component). This 3D approximation is denoted1 by û:
ε ∂Ψ
û = v̂ + −
er ,
r ∂Z
with Ψ such that
1 ∂Ψ
= ûz = v̂z = Uz (R(Z), a(Z), b(Z), f (Z))(r).
r ∂r
The evolution of the values of the cross-sectional angular momentum, helicity and
axial momentum, and of h(Z), will be determined from the requirement that the local
conservation law for the axial momentum, the axial flux, the angular momentum and
the helicity are satisfied. Let us recall two of these conservation laws from subsection
1.2.3 as an example: the angular momentum and the energy conservation law. In
(1.2.20b,c) the local conservation law were given as
∂t ruϕ + div ruϕ u + peϕ
= 0,
(4.3a)
1
(4.3b)
∂t |u|2 + div hu = 0.
2
When we consider a steady, axisymmetric velocity field u, and integrate (4.3a,b) over
R(Z)
1
0
0
1
ez
Figure 4.2: A pipe with slowly varying cross-sectional area. When a local conservation law
is integrated over a shaded region we obtain an axial conserved quantity.
a cross-section with radius R = R(Z) (see figure 4.2) we obtain
)
( Z
R(Z)
∂
ruϕ uz rdr
= 0,
2π
∂z
0
( Z
)
R(Z)
∂
2π
huz rdr
= 0,
∂z
0
(4.4a)
(4.4b)
Relations (4.4a,b) hold for every exact solution of the Euler equations and we require
that the QHA satisfies (4.4a,b) as well.
1 Observe that, in contrast with the previous two chapters, the RE is not anymore denoted by
but by .
U
û
131
This approach leads to system of coupled non-autonomous ODE’s for the amplitude
A of the Beltrami component and the rigid body rotation α. An important feature
of this system is that the solution breaks down at criticality, i.e., when the multiplier
λ in the RE, evolving with z, approaches zero.
Remark When we consider axisymmetric flows, we have the condition that the
circulation ruϕ and the total head h are constant on streamlines, and hence functions
of the stream function alone:
1
C = ruϕ = C(ψ), H = p + |u|2 = H(ψ).
(4.5)
2
Using (4.5) we observe that (4.4a,b) are satisfied automatically:
Z R(Z)
Z R(Z)
∂ψ
R(Z)
dr = C(ψ)|0
2π
ruϕ uz rdr = 2π
C(ψ)
≡ constant.
∂r
0
0
Z R(Z)
Z R(Z)
∂ψ
R(Z)
dr = H(ψ)|0
2π
huz rdr = 2π
H(ψ)
≡ constant.
∂r
0
0
with C(ψ) = dC/dψ and H(ψ) = dH/dψ. However, instead of requiring the point
wise conditions (4.5) we require the weaker (integrated) conditions (4.4a,b). Note
that there is no a priori reason to believe that an axisymmetric approximation cannot be O(ε)-correct if (4.5) is not satisfied. Moreover, it is even possible that the
approximation is close to a non-axisymmetric solution, for which (4.5) is not even
relevant.
3
This remark does raise the question concerning the higher-order terms in the expansion of u(δ, ε) and h(δ, ε). Define
η = u(δ, ε) − û,
h̃ = h(δ, ε) − ĥ.
Then
0 = EEul (û + η, ĥ + h̃) = EEul (û, ĥ) + E0Eul (û, ĥ)(η, h̃) + h.o.t.,
and hence
E0Eul (û, ĥ)(η, h̃) = −EEul(û, ĥ).
(4.6)
Remarks (i) The right-hand side of (4.6) denotes the residue of the approximation
and is O(ε). The higher order terms are O(ε|η|, |η|2 , ε|h̃|, |h̃|2 ).
(ii) The left-hand side of (4.6) is given by the linearised, axisymmetric, steady Euler
operator, with (û, ĥ) the modulated RE at some cross-section z, acting on (η, h̃).
Compare this with (3.1.12) with c = 0:
2Uϕ (Uϕ + rUϕ0 ) Uz00 r − Uz0
d 1 dφ
r
+ −µ2 +
−
φ = 0,
(4.7)
dr r dr
r2 Uz2
rUz
with ψ̃(r, z) = φ(r)eiµz the stream function for the error η and where φ(0) = φ(R) = 0.
(iii) Together with the steady, axisymmetric Euler equations we will frequently make
use of the time-dependent, axisymmetric Euler equations, especially when discussing
the linearised equations.
3
132
4 Swirling flows in an expanding pipe
Using the theory of chapter 3, we observe that the linearised operator in (4.6) is
invertible as long as (û, ĥ) is supercritical, i.e., as long as there are no standing waves
possible. When the flow is critical, there just exists a columnar solution φcrit of (4.7).
Another way to characterise φcrit is to embed it in a two-parameter family φ(µ, c),
being solutions of (4.7) with Uz replaced by Uz − c; µ denotes the axial frequency of
a travelling wave with speed c. Then φcrit is given as
lim
lim φ(µ, c)
= lim lim φ(µ, c) = φcrit .
c→0
µ→0
µ→0
c→0
|
|
{z
}
{z
}
long travelling waves
standing waves
These limits can be realised by letting the base flow change from supercritical to
subcritical or the other way around.
In section 1.3 we differentiated the parameterised solutions of the unperturbed problem, U (p), with respect to their parameters in order to obtain solutions of the linearised equation. What is the relevance of this procedure in the pipe flow? First of
all, note that the Euler equations for a columnar flow are degenerate in the sense that
any velocity field u(r) = (0, uϕ (r), uz (r)) is a solution of the Euler equations:
EEul (u(r), p(r)) =
u2ϕ dp
−
+
r
dr
!
er ,
and hence when we a priori assume that a perturbation ξ on a columnar base flow
depends only on r, we will get an equation that is not related to (4.7) with µ2 → 0.
This discrepancy between the two equations is known as the long wave singularity.
Hence, the only interesting and possible candidates that result from differentiation
with respect to one of the parameters in the RE are those that are non-columnar, i.e.,
the ones arising from the multi-mode RE solutions, as we investigated in chapter 3.
In section 4.3 we will investigate this matter more precisely.
Finally, in section 4.4 we investigate a particular example, being the rigid-body rotation plus a uniform translation entering an expansion. When we assume2 the existence
of a steady, axisymmetric solution of the Euler equations for a given (steady, axisymmetric) inlet profile, the results of the QHA can be compared with those using the
Bragg-Hawthorne equation, until a point of flow-reversal is reached. However, when
such a solution is not temporarily stable and/or spatially stable, the BHE have only
theoretical value and we should turn to the Euler equations instead. This observation may be compared with a suggestion of Leibovich (1984) when he states that the
stream function in BHE should be viewed as the stream function for the time-averaged
flows downstream the re-circulation bubble.
The result of the comparison between the QHA and the BHE can be stated as follows:
2 The
boundary conditions for a general steady Euler flow which will ensure existence and local
uniqueness are not known. Buntine and Saffman (1995) state that a reasonable procedure for axisymmetric flow in a finite part of the pipe Pδ,ε , say 0 ≤ z ≤ L, is to give uz on the inlet boundary
z = 0, and knowledge of uϕ and ωϕ for incoming fluid. Note that this procedure does not say
anything about temporal stability of the obtained flow.
133
4.1 Re-introducing the radius as a length-scale
(i) for arbitrary but fixed z, the cross-sectional error is O(ε):
Z
R̂(z)
|η|2 rdr = O(ε).
0
(ii) for arbitrary but fixed εz = Z (and hence for arbitrary but fixed radius), the
cross-sectional error is O(1):
Z
R(Z)
|η|2 rdr = O(1).
0
For the specific example the error is less then 6%.
This result is explained in section 4.4.
4.1
Re-introducing the radius as a length-scale
In this section we will look at the influence of the radius of the pipe on the different
parameters in the RE. This is basically recalculating lemma 2.10 and 2.12. The
integral quantities like angular momentum and helicity are given per unit volume (or
area) and by keeping them constant as we look at different constant-diameter pipes
(with different radii) we obtain interesting scaling laws.
In the first subsection we will look at the non-columnar RE solutions and as a corollary we make an attempt to explain the results of two experiments concerning a
precessing vortex core (PVC; Kok et al. (1993) and Syred et al. (1994)): a rotating non-axisymmetric velocity field. Especially the frequency of the precession is of
interest since the results in the aforementioned experiments differ by a factor of 5000;
frequency PVC
Kok et al. (1993)
Syred et al. (1994)
1/160 ≈ 6 · 10−3 [s−1 ]
100 [s−1 ]
In subsection 4.1.2 we study the columnar RE with β = ρ−1
1,1 as a special case. Moreover, we will identify the independent parameters that can be varied in both the
columnar and the multi-mode case.
4.1.1
The non-columnar relative equilibrium solutions
In chapter 2 we derived non-columnar RE solutions and denoted them as
±
v̂ ±
k = A1 v k + αreϕ + Λez
with
n
o
± i(`(ϕ−αt)+m̃(z−λt))
v±
,
k = < Uk e
(4.1.1)
134
4 Swirling flows in an expanding pipe
−1 ±
and where Λ = ±2αβ + λ and curl v ±
v k . We will leave out most sub and
k = ±β
superscripts in the following.
The eigenvalue β satisfied
β m̃J`0 (β −1
p
p
`βJ` (β −1 1 − β 2 m̃2 )
2
2
p
1 − β m̃ ) ±
= 0,
1 − β 2 m̃2
(4.1.2)
and (4.1.2) was obtained from the requirement that the radial velocity vanishes at
the wall; m̃ denotes the axial frequency, m̃ = 2πm/L, and ` denotes the angular
frequency. When we re-introduce the radius R, we have to replace r by r∗ = r · R and
z by z∗ = z · R, and consequently L by L · R: with the star as subscript we denote
the unscaled variables or parameters.
Substituting these changes into (4.1.2) we obtain
p
p
`β∗ J` (Rβ∗−1 1 − β∗2 m̃2 /R2 )
β∗ m̃ 0
−1
2
2
2
p
J (Rβ∗
1 − β∗ m̃ /R ) ±
= 0,
R `
R 1 − β∗2 m̃2 /R2
with β∗ = βR the unscaled eigenvalue.
The unscaled multipliers α∗ and λ∗ and the unscaled amplitude A1,∗ follow from
working out the constraints.
Lemma 4.1 Re-introduction of the radius R into the non-columnar RE solution in
(4.1.1) leads to
v̂ ∗ = v ±
∗ + α∗ r∗ eϕ + Λ∗ ez ,
−1 ±
with Λ∗ = ±2α∗ β∗ + λ∗ and curl v ±
∗ = ±β∗ v ∗ , with
α∗ =
2a
,
R2
λ∗ = f ∓
A21,∗ = ±Rβ
4aβ
,
R
Λ∗ = f,
b − 2af /R2
.
E(v ±
∗)
Proof Direct computations show that
Fax (v̂ ∗ ) =
1
|D∗ |
1
A(v̂ ∗ ) =
|D∗ |
Z
D∗
1
πR2
r∗ v̂∗,ϕ
1
=
πR2
Z
D∗
Z
v̂∗,z =
0
R
(±2α∗ β∗ + λ∗ ) 2πr∗ dr∗ = ±2α∗ βR + λ∗ =: f,
Z
0
R
α∗ r∗2 2πr∗ dr∗ =
1
α R2 =: a,
2 ∗
and hence
α∗ =
2a
,
R2
λ∗ = f ∓
4aβ
,
R
Λ∗ = f.
4.1 Re-introducing the radius as a length-scale
135
In order to determine A1,∗ we use the helicity constraint:
Z
1 1
B(v̂ ∗ ) =
v̂ · curl v̂ ∗
|D∗ | 2 D∗ ∗
Z
1 1
−1
±
=
A1,∗ v ±
∗ + α∗ r∗ eϕ + Λ∗ ez · ±β∗ A1,∗ v ∗ + 2α∗ ez
|D∗ | 2 D∗
Z R
1
= ±β∗−1 A21,∗ E(v ±
)
+
2α∗ (±2α∗ β∗ + λ∗ ) 2πr∗ dr∗
∗
2πR2 0
2
= ±β∗−1 A21,∗ E(v ±
∗ ) + 2af /R =: b,
implying the remaining relation.
4.1.2
2
A precessing vortex core
We noted in chapter 1 that, given certain conditions on the inlet flow, experiments
and numerical simulations of swirling flows in pipes reveal a region of flow-reversal.
Moreover, in many experiments it was observed that the velocity profile was not steady
but oscillated with a certain dominant frequency. When this oscillation is in fact a
rotation the resulting phenomenon is called a precessing vortex core (PVC) (Syred
and Beer (1974)). A PVC is by definition a rotating non-axisymmetric velocity field
and in that respect the non-axisymmetric, parallel RE solutions are of interest. The
inviscid interaction of various weak Beltrami flows, i.e., the amplitude of the Beltrami
component is small in the RE solution, is investigated in e.g. Dritschel (1991) and
Valkering (1996). We would like to emphasize at this point the rather explicit formula
for the angular velocity α∗ in lemma 4.1, α∗ = 2a/R2.
Note that in this relation only global or integrated quantities appear, and this can be
regarded as one of the messages of this thesis: whenever a parameter or concept is
introduced, try to define it in terms of global quantities, like integrals (see for instance
section 1.1 for an example, concerning a swirl number). The reason is that on the one
hand such quantities are less subject to inaccuracies when determined in experiments.
On the other hand, and this applies in particular to the angular velocity α∗ , when
such a parameter can be regarded as an eigenvalue, corresponding to some constrained
optimisation problem, we have the principle that a ‘small’ difference between the
actual profile at hand and an optimal profile has little effect on the corresponding
eigenvalue; see figure 4.3 for a graphical interpretation.
At the hand of two experiments we show how the formula α∗ = 2a/R2 can give an
order estimate of the frequency of the PVC. The two experiments are described in
Kok et al. (1993) and Syred et al. (1994). Below we will list the relevant parameters
of each experiment and try to deduce the frequency of the PVC at the hand of the
relation α∗ = 2a/R2 .
Kok et al. (1993) 3 The radius of the pipe is 22 [cm], the tangential velocity profile
has a maximum of 10 [cm · s−1 ] and we approximate it with a Rankine-type
3 The data correspond to an expansion rate of 0.23 and a swirl number of 11 in Kok et al. (1993);
the velocity profile is measured at the outlet of the swirler (their figure 5).
136
4 Swirling flows in an expanding pipe
λ ≡ Q(u)/N (u)
6
O(ε2 )
-u
O(ε)
Figure 4.3: Typical example (Courant and Hilbert (1953)) of the characterisation of eigenvalues λ as critical points of the quotient Q(u)/N(u). N(u) plays the role of constraint,
while Q(u) is the functional to be optimised.
of distribution (rigid body rotation in the vortex-core, power-law decay outside
the core) with exponent -1.0 (see figure 4.4a):
(
Ω(r/r0 ),
for 0 ≤ r ≤ r0 = 4[cm] and Ω = 9[cm · s−1 ]
uϕ =
Ω(r/r0 )−1.0 , for r0 ≤ r ≤ 22[cm]
The angular momentum per area for this profile is approximately 35[cm 2 · s−1 ]
and the frequency α∗ equals
α∗ =
2 · 35
≈ 1.4 · 10−1 [s−1 ],
222
whereas the measured frequency is approximately 1/160 = 6.3 · 10−3 [s−1 ].
Syred et al. (1994) 4 The radius of the pipe is 60 [mm], the tangential velocity
profile has a maximum of 18.5 [m · s−1 ] and is approximated with a Rankinetype of distribution with exponent -7.5 (see figure 4.4b):
(
Ω(r/r0 ),
for 0 ≤ r ≤ r0 = 37[mm] and Ω = 19[m · s−1 ]
uϕ =
Ω(r/r0 )−7.5 , for r0 ≤ r ≤ 60[mm]
The angular momentum per area for this profile is approximately 240 ·103[mm2 ·
s−1 ] and the frequency α∗ equals
α∗ =
2 · 240 · 103
≈ 130[s−1 ],
602
whereas the measured frequency is approximately 100[s−1 ].
137
8
6
∼ r−1.0
15
6
10
mean tangential
velocity [m · s−1 ]
10
mean tangential
velocity [cm · s−1 ]
4.1 Re-introducing the radius as a length-scale
∼ r−7.5
6
4
5
2
5
- r [cm]
10
15
20
10
20
30
40
- r50[mm]
60
-2
(a)
(b)
Figure 4.4: (a) The tangential velocity profile as described in Kok et al. (1993), approximated by a Rankine-type of vortex: rigid body in the core and power-law decay outside
the core. The dots indicate their measurements. (b) The (mean) tangential velocity profile
as described in Syred et al. (1994), approximated by a Rankine-type of vortex. The dots
indicate their measurements.
The results above indicate that, although the velocity profile corresponding to the RE
solution may not be the profile that is measured in an actual experiment, the formula
yielding the angular velocity can give a prediction of the frequency of a precessing
vortex core. Moreover, although this prediction may seem bad for each experiment
separately, that one and the same formula is applicable to two completely different
cases is rather remarkable.
The difference between prediction and measurement in the ’93-experiment deserves
some more attention. Note that the relative equilibrium solutions are over the whole
domain ‘vorticity-rich’, i.e., we cannot distinguish a core and a vorticity-free region.
The flow in Kok et al. (1993) differs in that respect in a structural way from the
relative equilibrium solutions since the core is rather narrow, in contrast to the ’94experiment. Indeed, when the data in figure 4.4a are confined to ‘the’ core, the
prediction will be of the order of the measurement: α∗ ≈ 2.5 · 10−3 [s−1 ].
4.1.3
The columnar relative equilibria and multi-mode relative
equilibrium solutions
As we showed in section 2.3, applying the constraints in the columnar case leads to a
more involved equation for the amplitude of the Beltrami component. Let us denote
4 The velocity profile is measured 8 mm above the outlet of the swirler and corresponds to figure
8 in Syred et al. (1994).
138
4 Swirling flows in an expanding pipe
the columnar RE by
1
1
û∗,r = 0, û∗,ϕ = ± A2,∗ J1 (r∗ β∗−1 ) + α∗ r∗ , û∗,z = A2,∗ J0 (r∗ β∗−1 ) + Λ∗ .
2
2
(4.1.3)
In this chapter we are interested in the case β = β∗ /R = ρ−1
1,1 , and hence we will
restrict our analysis in this section to this specific value of β.
Lemma 4.2 The multipliers α∗ and λ∗ and the amplitude A2,∗ in (4.1.3) are related
to the values of the constraints as follows. The amplitude A2,∗ satisfies a quadratic
equation,
c2 A22,∗ + c1 A2,∗ + c0 = 0,
(4.1.4)
with
c0 = 2af /R2 − b,
aρ ∓ 2f R
,
c1 = −J0 (ρ)
ρR2
1
ρ2 − 4
c2 = ± J0 (ρ)2
,
4
Rρ
where ρ = ρ1,1 . Given solutions A2,∗ , the multipliers α∗ and λ∗ are obtained from
α∗ =
A2,∗ J0 (ρ)
2a
±2
,
2
R
Rρ
λ∗ = f − 4
A2,∗ J0 (ρ)
a
±
ρ2
Rρ
.
The proof is straightforward.
In section 2.3 we also studied the effect of adding a non-columnar RE solution to
(4.1.3)—the multi-mode RE solution:
±
v̂ ∗ = A1,∗ v ±
∗ + A2,∗ u∗ + α∗ r∗ eϕ + Λ∗ ez ,
±
with v ±
∗ the non-columnar, rotating and translating Beltrami component, curl v ∗ =
−1 ±
±
±
−1 ±
±β∗ v ∗ , and u∗ the columnar Beltrami component, curl u∗ = ±β∗ u∗ ; β = β∗ /R
is not necessarily equal to ρ1,1 . We observed in section 2.3 that the equation for A2,∗ ,
i.e., equation (4.1.4), is slightly changed when fixing A1,∗ :
c0 → c0 ± β∗−1 A21,∗ E(v ±
∗ ).
So, the number of independent parameters in a multi-mode RE solution equals four:
the values of the constraints a, b and f and the amplitude A2,∗ of the non-columnar
part.
In section 4.2 we let the multipliers α∗ and λ∗ (or Λ∗ ) and amplitude A2,∗ depend
on Z since these appear explicitly in the velocity profiles. By inverting lemma 4.2 we
can express the constraints in terms of the multipliers and amplitude.
4.2 The quasi-homogeneous approximation
139
Lemma 4.3 The values of the constraints, a, b and f , are related to the values of
the multipliers and the amplitude in (4.1.3) as follows.
R
1
α∗ R2 ∓ J0 (ρ)A2,∗ ,
2
ρ
f = Λ∗ ,
1 ρJ0 (ρ)2 2
1
b = ±
A2,∗ − α∗ J0 (ρ)A2,∗ + α∗ Λ∗ .
4 R
2
a =
(4.1.5a)
(4.1.5b)
(4.1.5c)
The proof is again straightforward.
In the following we will omit the stars.
4.2
The quasi-homogeneous approximation
In this section we will obtain indirectly the evolution of the modulated parameters
a(Z), b(Z), f (Z) and h(Z). We showed in the previous section that from the evolution
of the multipliers α(Z) and λ(Z) and the amplitude A(Z) of the Beltrami component
we can obtain the dynamics of a, b and f , and since it are the multipliers and the
amplitude that appear in the velocity profiles we will determine
Z 7→ (α(Z), λ(Z), A(Z), h(Z)).
(4.2.1)
We showed in the introduction to this chapter that there are at least two ways to
determine (4.2.1). First we can require the consistent evolution of the conserved
functionals, and secondly we can analyse the equation for the error (4.6), and obtain
solvability conditions at criticality of the (modulated) base flow, i.e., orthogonality
conditions for the residue.
In subsection 4.2.1 we will show how consistent evolution is related to orthogonality
conditions for the residue, and subsequently in subsection 4.2.2 we give the evolution
(4.2.1) that follows from the requirement of consistent evolution; solvability questions
will be dealt with in section 4.3.
4.2.1
Consistent evolution and orthogonality conditions
Let us briefly recall the idea of consistent evolution, as described in section 1.2. There
we explained how dynamically conserved functionals are part of a local conservation
law,
c(u) := ∂t f (u) + div j(u) = 0.
(4.2.2)
When we restrict our attention to steady (axisymmetric) velocity fields, and integrate (4.2.2) over a cross-section with slowly varying radius R(Z) we obtain axially
conserved functionals:
(Z
)
R(Z)
∂
jz (u) 2πrdr + 2πR(Z)j · n|r=R(Z) = 0.
(4.2.3)
∂z
0
140
4 Swirling flows in an expanding pipe
When the fluxes j(u) corresponding to the energy, axial momentum, angular momentum, axial flux and helicity (see section 1.2) are substituted in (4.2.3) we define
Z
Z
R(Z)
huz rdr, S = 2π
Ecross = 2π
0
Z
R(Z)
(u2z
+ p) rdr, Across = 2π
0
Z
ruϕ uz rdr,
0
Z
R(Z)
Qcross = 2π
R(Z)
(u · ω)uz + (h − |u|2 )ωz rdr,
R(Z)
uz rdr, Bcross = 2π
0
0
and we obtain
dE
ε cross
dZ
0
dS
2πεR (Z)p ε
− p
dZ
1 + ε2 R0 (Z)2 = 0,
(4.2.4a)
= 0,
(4.2.4b)
= 0,
(4.2.4c)
= 0,
(4.2.4d)
= 0.
(4.2.4e)
r=R(Z)
dAcross
dZ
dQcross
ε
dZ
ε
ε
dBcross
+ 2π(h − |u|2 )ω · n|r=R(Z)
dZ
Remark The normal component of the vorticity of the QHA at r = R(Z) is given
by
−εR0 (Z)ωz + ωr
−ε(α0 R + 2R0 α)
ωn = p
= p
,
1 + ε2 R0 (Z)2
1 + ε2 (R0 )2
with 0 = d/dZ. Hence, only when α(Z) = c1 /R(Z)2 the vorticity surface corresponding to the QHA and the surface of the pipe coincide. We will return to this, later in
this section.
3
If we denote the residue of the QHA by R,
R = EEul (û, ĥ),
then we observe the following.
Proposition 4.4 The requirement that the QHA satisfies (4.2.4a–c) is equivalent to
the requirement that the residue R vanishes in the direction δE(û) = û, δFax (û) = ez
and δA(û) = reϕ . The requirement that the QHA satisfies (4.2.4d) is equivalent to
the requirement that the wall of the pipe r = R(Z) is a stream surface.
Remark The requirement that the QHA satisfies (4.2.4a) will be called E-consistency.
Similarly we have Fax -consistency, A-consistency, Q-consistency and B-consistency.
3
141
4.2 The quasi-homogeneous approximation
Proof We prove the proposition for (4.2.4a); (4.2.4b,c) follow along the same lines.
The conservation law for the energy can be obtained by multiplying EEul (u, h) with
u and a rearrangement of terms:
1
u · EEul (u, h) = ∂t |u|2 + div hu.
2
(4.2.5)
Integration of (4.2.5) over a cross-section, and the restriction to steady (axisymmetric)
flows yields
)
( Z
Z R(Z)
R(Z)
∂
u · EEul (u, h) rdr = ε
huz rdr .
2π
2π
∂Z
0
0
Recall that δE(u) = u. Substituting for (u, h) the QHA (û, ĥ) and recalling that
EEul (û, ĥ) = R 6= 0 we obtain the desired result.
Equation (4.2.4d) is readily seen to be
ε
∂
Ψ(R(Z), Z) = 0
∂Z
and hence Ψ(R(Z), Z) ≡ constant.
2
Remark It is easy to show that
Rr = O(ε2 ), Rϕ = O(ε), Rz = O(ε),
3
irrespective of a(Z), b(Z), f (Z) and h(Z).
Proposition 4.5 The requirement that the QHA satisfies (4.2.4e) is equivalent to
the requirement that
Z
R(Z)
[(curl û) · R + û · curl R] rdr = 0.
2π
(4.2.6)
0
Proof The proof is along the same lines as the previous one. The conservation law
for the helicity can be obtained by adding the R3 -inner product of ω with EEul (u, h)
and the R3 -inner product of u with curl EEul (u, h):
∂t (u · ω) + div (u · ω)u + (h − |u|2 )ω = ω · EEul (u, h) + u · curl EEul (u, h)
with
curl EEul (u, h) = ∂t ω + (u · ∇)ω − (ω · ∇)u
the vorticity equation. The rest of the proof is straightforward.
2
142
4 Swirling flows in an expanding pipe
Corollary An integration by parts in (4.2.6) leads to
Z
Z
R(Z)
û · curl R rdr
(curl û) · Rrdr +
=
0
I
R(Z)
Z
R(Z)
(curl û) · Rrdr + (R × û) · en |r=R(Z) R(Z) +
=
0
ε
(R × û) · n
0
∂
∂Z
Z
R(Z)
(R × û) · ez rdr,
0
and hence with (2.1.5) in chapter 2 we have that the requirement that the helicity of
the QHA evolves consistently is equivalent to requiring that
δB(û; R) = 0.
Remark The interpretation of the requirement that the residue R vanishes in a
certain direction can be illustrated by the Galerkin approximation (Fledderus and
Van Groesen (1996a)).
Consider the evolution equation
∂t u = K(u).
An approximation is sought as an evolution in the linear space spanned by the base
elements N ≡ span [φ1 , . . . , φN ], i.e., it is described as as a linear combination with
coefficients that depend on time:
t 7→ û(t) ≡
N
X
ck (t)φk .
1
It should be observed that in the linear Fourier method, in most cases no evolution
in N will produce an exact solution of the underlying system: N is not invariant for
the flow in general. This is related to the choice of the base elements in N, which are
in general independent of the equation.
Nevertheless, in many cases we use a space like N, and evolutions therein, to approximate an exact solution of an equation. Without further specification the choice of this
dynamics is not (uniquely) defined. This can be rephrased in more geometrical terms
as follows. If an evolution in N is to be viewed as a projection of some solution into
this set, the question is to determine this projection. Such a projection is explicitated
by specifying the parameter dynamics.
Let us recall how such a projection is obtainedP
in the Fourier-like methods. Then
N
often the residue is considered. That is, if û(t) = 1 ck (t)φk (x) is the approximation,
with coefficients ck (t) to be determined, one looks how well û satisfies the equation.
Defining the evolution operator E by
E(u) := ∂t u − K(u),
143
4.2 The quasi-homogeneous approximation
the residue of û is E(û(x, t)). The residue vanishes only for exact solutions and it
should be small for a good approximation. In order to achieve this, the method
of weighted residues requires the residue to vanish at each time in N independent
directions. Specifically, if h , i denotes some (spatial) inner product, and a set of N
‘weight-functions’ ψk is chosen, the vanishing of
hE(û), ψk i = 0
for k = 1, . . . , N leads to N equations from which, in principle, the N coefficients
ck can be determined. In the special case that ψk is a set dual to φk , meaning that
hφk , ψj i = δkj (Kronecker symbol), this is known as Galerkin approximation.
3
We have seen that consistent evolution of the helicity and orthogonality of the residue
with ω̂ = curl û are the same up to boundary contributions. Since in the construction
of the RE in section 2.1 the boundary term δB(u; ·) is neglected, we expect a relation
between E-, A-, Fax - and Q-consistency on the one hand and hω̂, R on the other hand.
Indeed, recall that
U = β curl U + αreϕ + λez ,
(4.2.7)
First of all, we have the following result that tells us how well a modulated RE is still
a true5 relative equilibrium.
Lemma 4.6 The modulated RE in (4.2a,b) satisfies (4.2.7) up to a residue RLM that
is O(ε) point-wise:
RLM = û − β curl û − α(Z)reϕ − λ(Z)ez
(4.2.8)
Proof The order of the residue RLM is mainly determined by the first two terms in
the right-hand side of (4.2.8). Let us compute its order, component by component:
RLM · er = ûr − (β curl û) · er = −
∂ ûϕ
ε ∂ ψ̂
+ εβ
= O(ε);
r ∂Z
∂Z
∂ û
β ∂ 2 ψ̂
RLM · eϕ = ûϕ − (β curl û) · eϕ − α(Z)r = ûϕ − α(Z)r + β z −ε2
= O(ε2 );
2
∂r
r
∂Z
|
{z
}
≡0
RLM · ez = ûz − (β curl û) · ez − λ(Z) = ûz − λ(Z) −
β ∂
rû ≡ 0.
r ∂r ϕ
2
On using lemma 4.6 we have the following relation between the results in proposition
4.4 and 4.5.
5 It is not clear at the moment whether the critical point problem that has been solved in chapter
2 for a constant-diameter pipe has any solutions in case of a slowly expanding pipe.
144
4 Swirling flows in an expanding pipe
Proposition 4.7 Suppose the evolution Z 7→ (α(Z), λ(Z), A(Z), h(Z)) is such that
the QHA satisfies (4.2.4a–d), i.e., the cross-sectional energy, angular and axial momentum and axial flux evolve consistently. Then
Z
R(Z)
ω̂ · R rdr = −β −1
0
Z
R(Z)
RLM · R rdr = O(ε3 ).
0
Proof E-consistency and lemma 4.6 yield
Z
Z
R(Z)
û · R rdr = 2π
0 = 2π
0
R(Z)
βcurl û + αreϕ + λez + RLM · R rdr.
0
Using the assumption on the consistent evolution of the cross-sectional angular and
axial momentum we obtain
Z R(Z)
Z R(Z)
0 = 2π
β(curl û) · R rdr + 2π
RLM · R rdr.
(4.2.9)
0
0
The order of the second term in (4.2.9) is obtained using lemma 4.6 and spelling out
R = EEul (û, ĥ).
2
We will construct the QHA based on a consistent evolution of the energy, angular
momentum, axial momentum and axial flux. The motivation to prefer a consistent
evolution of the cross-sectional energy to a consistent evolution of the helicity is that
E-consistency is a weak version of the requirement that the total head is constant on
stream lines as we showed in the introduction to this chapter.
4.2.2
Consistent evolution: the parameter dynamics
Let us now determine the parameter dynamics (4.2.1) from the requirement that QHA
satisfies (4.2.4a–d). The values of the axially conserved quantities Ecross , S, Across
and Qcross at Z = 0 are denoted by Ecross,0 , S0 , Across,0 and Qcross,0 respectively, with
Qcross,0 > 0 (mass flux is from left to right).
Proposition 4.8 Consider the modulated columnar RE,
ûϕ
ûz
1
= ± A(Z)J1 (rρ/R(Z)) + α(Z)r,
2
1
2α(Z)R(Z)
=
A(Z)J0 (rρ/R(Z)) ±
+ λ(Z),
2
ρ
ĥ = λ(Z)ûz + α(Z)rûϕ + h(Z),
1 2
p̂ = ĥ −
ûϕ + û2z ,
2
(4.2.10a)
(4.2.10b)
(4.2.10c)
(4.2.10d)
with ρ = ρ1,1 and R(Z) the slowly varying radius of the pipe. When we require
consistent evolution of the energy, axial momentum, angular momentum and axial
145
4.2 The quasi-homogeneous approximation
flux, α(Z) = ±α̃(Z)Qcross,0 and A(Z) = Ã(Z)Qcross,0 satisfy a coupled set of ordinary
non-autonomous differential equations,
dR
G11 G12
Ỹ1
dÃ/dZ
=0 G=
, Ỹ =
(4.2.11)
G
+ Ỹ
G21 G22
dα̃/dZ
Ỹ2
dZ
with
1
G11 = −R4 J0 (ρ)2 π Ãρα̃ + 2R5 J0 (ρ)π α̃2 − R2 J0 (ρ)ρα̃ + RJ0 (ρ)2 Ãρ2 ,
2
1 4 2
1
2
5
2
G12 = − R π Ã ρJ0 (ρ) + 4R π Ãα̃J0 (ρ) − R ÃρJ0 (ρ) + R3 ρ2 α̃,
2
2
G21 = 2R4 J0 (ρ)π α̃ − RJ0 (ρ)ρ,
1
G22 = 2R4 π ÃJ0 (ρ) + R2 ρ2 ,
2
3
1
Ỹ1 = − π Ã2 ρR3 α̃J0 (ρ)2 + 8π ÃR4 α̃2 J0 (ρ) − 3ÃρRα̃J0 (ρ) − Ã2 ρ2 J0 (ρ)2 +
2
4
+ ρ2 R2 α̃2 ,
Ỹ2 = 8π ÃR3 α̃J0 (ρ) − ÃρJ0 (ρ) + ρ2 Rα̃.
Λ(Z) = λ(Z) ± 2α(Z)R(Z)/ρ and h(Z) are given in terms of α(Z) and A(Z):
Λ(Z) =
Qcross,0
,
πR(Z)2
dh
(Z) = −Q2cross,0
dZ
1 2 dα̃2
Ã2 J0 (ρ)2 dR 2α̃ÃJ0 (ρ) dR
R
+
+
2
dZ
4R
dZ
ρ
dZ
!
.
Proof The proof follows from a straightforward substitution of the QHA in (4.2.4a–
d). The energy and axial momentum conservation law are used to eliminate the total
head and the conservation law for the axial flux yields the explicit expression for Λ(Z).
The first equation in (4.2.11) corresponds to the energy conservation law, whereas the
second corresponds to the angular momentum conservation law.
2
The tilde’s are omitted in the subsequent analysis.
Analysis of (4.2.11); the special case α(Z) ≡ 0
First of all, we connect the results in proposition 4.8 with the results obtained in
Van Groesen et al. (1995) by setting α(Z) ≡ 0. In that case we do not require
A-consistency, and study the dA/dZ-equation in (4.2.11) with α(Z) ≡ 0:
1
1
RJ0 (ρ)2 Aρ2 A0 − A2 ρ2 J0 (ρ)2 R0 = 0,
2
4
p
yielding the solution A(Z) = c1 R(Z). This shows that when the inlet flow is a jetflow, it remains a jet-flow and when it starts as a wake-flow it remains a wake-flow.
146
4 Swirling flows in an expanding pipe
In particular, flow-reversal at the axis can only occur when c1 < 0, i.e., the inlet flow
is a wake-flow (see figure 4.5):
p
p
1
∼
Q
c
R(Z)
axial velocity at axis = Qcross,0 c1 R(Z) +
cross,0
1
πR(Z)2
for R(Z) large.
When we substitute the result in (4.1.5c) we find that the cross-sectional helicity per
area is constant for this special case.
z
z
Figure 4.5: Axial velocity profiles of the jet-flow (at the left) and of the wake-flow (at the
right) at successive locations in an expanding pipe (Van Groesen et al. (1995)). The profiles
evolve according to the QHA with A(Z) = c1 R(Z) with c1 > 0 for the jet-flow and c1 < 0
for the wake flow. The dashed lines indicate the “uz = 0” line for the successive profiles.
p
Analysis of (4.2.11); the special case α(Z) ∼ R(Z)−2
Another case of interest is α(Z) · R(Z)2 = constant. As a consequence, the normal
component of the vorticity vanishes at the wall of the pipe (see remark related to
(4.2.4a–e)), i.e., we have dynamical conservation of helicity. Moreover, the point-wise
condition, stating that the circulation ruϕ is a function of the stream function alone,
implies that at the wall of the pipe
R(Z) · ûϕ (R(Z), Z) = α(Z)R(Z)2 = constant.
Hence, instead of the point-wise condition in the whole flow field, only the condition
at the wall of the pipe is satisfied.
The prescription of α(Z) and the relation between A-consistency and the angular
momentum conservation law serve as a motivation to require in this case Fax -, Econsistency and Q-consistency. This leads to a non-autonomous differential equation
for A(Z):
dA
1 Aρ 2πAR2 ΩJ0 (ρ) + AρQcross,0 J0 (ρ)R + 4ΩQcross,0 dR
=
(4.2.12)
dZ
2
dZ
R −2RπΩ + ρQcross,0 (AρJ0 (ρ)R − 2Ω)
147
4.2 The quasi-homogeneous approximation
with α(Z) = ΩR(Z)−2 . In figure 4.6 we plotted the contour lines where the numerator
2
A
6
k
1
%
&
0 1
-1
R=
P
• 1
P
• 2
1.2
1 ρQcross,0
2
πΩ
&
%
- R
1.4
%
&
&
%
A = − RJ
1.6
4ΩQcross,0
0 (ρ)(2πRΩ+ρQcross,0 )
1.8
2
R
-2
Figure 4.6: A contour plot of the vector field in (4.2.12) for Qcross,0 = 1 and Ω = 0.5.
The dashed lines indicate the zero-contours of its denominator and the solid lines the zerocontours of its numerator.
or the denominator vanish, together with the direction field. There are two special
points where the numerator and denominator vanish simultaneously.
P1 : When we substitute
R=
1 ρQcross,0
+ δR ,
2 Ωπ
A=−
4πΩ2
+ δA
ρ2 J0 (ρ)Qcross,0
into (4.2.12) we obtain a linear system that describes the behaviour of the
solution curves (R(Z), A(Z)) near P1 :
3
ρ J0 (ρ)Q2cross
δA
δ̇A
−12π 2 Ω3
=
, ˙ = d/dτ,
δR
0
−2ρ3 J0 (ρ)Q2cross
δ̇R
with τ a (non-physical) parameter to describe the trajectory. Hence P1 is a
saddle, implying that A(Z) either goes off to infinity or moves to P2 ;
P2 When we substitute
1 ρQcross,0
+ δR ,
2 Ωπ
into (4.2.12) we obtain a linear system that
solution curves (R(Z), A(Z)) near P2 :
δ̇A
1 0
=
0 1
δ̇R
R=
A = δA
describes the behaviour of the
δA
δR
,
148
4 Swirling flows in an expanding pipe
and hence P2 is a node.
Since R is monotonically increasing in an expanding pipe, P1 and P2 are not equilibria
and we should view the phase-portraits in figure 4.6 purely as static pictures: the fact
that for instance P2 is a repelling node is of no significance. Even more, a solution
curve (R(Z), A(Z)) will pass right through P2 .
At the dashed line Rcrit = 12 ρQcross,0 /πΩ the flow is critical:
λ=
Qcross,0
2Ω
−
= 0.
2
πRcrit
ρRcrit
When we start with a suitable jet-flow (positive A, below the drawn line in figure
4.6), we have a transition through criticality to a wake-flow. Only after this passage
it is possible that flow-reversal occurs, depending on whether the solution curve hits
the dashed (singularity) contour.
Analysis of (4.2.11); the general case
Equation (4.2.11) can be viewed as a system of non-autonomous differential equations,
with the (scaled) axial variable Z as time. To make this analogy even stronger, we
invert the matrix G and divide by dR/dZ (and hence the radius R is the essential
independent variable):
dA/dR
Y1
−1
= Y , Y = G Ỹ =
.
(4.2.13)
dα/dR
Y2
Y1 and Y2 are rather awful looking fractions, but the zero-contours at R = 1 of the
numerators and denominators are given in figure 4.7.
Remark Note that α = 0 is not an invariant subset for (4.2.11): when A > 0,
α0 < 0 and when A < 0, α0 > 0 in a neighbourhood of the origin (see figure 4.7);
0
= d/dR. This means that the reaction of the flow on the expansion is such that
a rigid-body rotation is generated with the opposite sign compared to the Beltrami
component. The reaction when starting with only rigid-body rotation (A = 0 at
R = 1) is completely different: A0 < 0 for all |α| small enough. This means that
immediately a wake-flow is generated (see also section 4.4 for a comparison of this
case with the quasi-cylindrical approach).
3
The contours move as a function of R as follows:
Contours related to Y1 : The dashed lines in figure 4.7 indicate the zero-contours
of the denominator. They are given by
①:
②:
ρ
,
2πR3
4R
ρ2 − 4
A=−
α−
.
ρJ0 (ρ)
2πR2 J0 (ρ)
α=
149
4.2 The quasi-homogeneous approximation
②
↓
↓3 ④
2
↑
↑
-2
↓
-1
↓
3
①
↓
A
6
00
-α
1
↓
←
↑
↓
-1
2
③
1
2
-2
A
6
⑥
←
1
→
00
-1
→
↑
→
②
→
1
←
-1
-2
-2
-3
-3
-α
⑤
Figure 4.7: A contour plot of the vector field Y at R = 1. At the left the lines where Y1
vanishes (solid lines) and where it has a singularity (dashed lines). At the right the lines
where Y2 vanishes (solid lines) and where it has a singularity (dashed lines). The arrows
indicate the initial behaviour of A(Z) and α(Z): the ↑ and ↓ in the left picture mean initial
increase or decrease of A(Z), while the → and ← in the right picture mean initial increase
or decrease of α(Z).
On ①we have that
λ = Λ ∓ 2αR/ρ = Qcross,0
1
2αR
−
πR2
ρ
= 0,
and hence for the co-rotating (plus) flow we have at ① criticality according to
theorem 3.8 in chapter 3. At the left the flow is subcritical and at the right the
flow is supercritical—the latter being a physical correct initial condition.
The formulae show that ①is (rapidly) moving to the A-axis while ②steepens
and is moving towards the origin.
The solid lines are the zero-contours of the numerator. The precise functional relationship does not reveal much, but the shape of the contours remains constant
and some aspects concerning the shape can be given rather explicitly:
horizontal asymptote of ③ :
minimum of ④ :
vertical asymptote of ④ :
ρ2
,
8πR2 J0 (ρ)
ρ2 + 8
A=−
as R → ∞,
8πR2 J0 (ρ)
ρ
α=
.
πR3
A=−
Together with the information that (A, α) = (0, 0) is always a zero of Y1 we
observe that the zero-contour ③ is collapsing as R grows.
2
150
4 Swirling flows in an expanding pipe
Contours related to Y2 : The zero-contour of the denominator is the same diagonal
as we discussed for Y1 . The contours of the numerator are given by the two
branches of
⑤and ⑥ :
vertical asymptote of ⑤and ⑥ :
horizontal asymptote of ⑤and ⑥ :
2α(ρ2 − 4)
;
J0 (ρ)(3ρ − 10πR3 α)
3ρ
α=
;
10πR3
2(ρ2 − 4)
A=−
.
10πR2 J0 (ρ)
A=
Hence, for large R the branches will more and more coincide with the α- and
A-axis. Moreover, since α0 is negative at ① we note that the transition between
supercritical to subcritical cannot be described.
The actual dynamics of A and α can be understood as an interplay between the
movement of the contours. In the rest of this section we will be interested in the
following types of solutions:
• bounded for all finite ‘time’ (radius);
• finite-time blow-up solutions;
• switch from positive to negative A (in view of flow-reversal).
More specific results for the case A0 = 0, α0 arbitrary are given in section 4.4.
We will analyse the behaviour of initial conditions starting in one of the four different
quadrants in the (A, α)-plane.
First quadrant When an initial condition is taken above ⑥and at the right of ①(and
hence subcricital; note that ①is at the right of the vertical asymptote of ⑥), α
will increase while A will decrease. Since ⑥and ③are collapsing this behaviour
will remain: A → 0 and α increasing but bounded for all finite time (figure
4.8a). When the initial condition is between ③and ⑥and at the right of ①, we
have a decrease of A and α. But since ⑥is collapsing there is a chance that it
will hit the solution curve (α(Z), A(Z)) such that α starts to grow again. The
possibility that the curve catches up with ③can be ruled out since near ③A
hardly changes while the contour has its own intrinsic dynamics—a collapse.
The interplay between the solution curve and ②(representing a singularity in
Y1 ) is more interesting and an initial condition near ②can serve as a worst-case:
the (initial) decrease of α can lead to a finite-time blow-up. However, it turns
out that ②is moving too fast to the left. A characterisation of the evolution
would be (α(Z), A(Z)) ∼ (α, AR(Z)−2 ) with α and A order unity, depending
on the precise initial conditions (figure 4.8b). This same characterisation can
be given when the initial condition is below ③(but A still positive) and at the
right of ②.
When we start supercritical, i.e., at the left of ②the situation changes dramatically. Since only the dynamics of A is driven by the presence of ②and not the
151
4.2 The quasi-homogeneous approximation
3•
2.5
2
•
2
1.8
A
6
1.6
A
6
1.4
1.2
1.5
-α
1.55 1.6 1.65 1.7 1.75 1.8 1.85
0.45 0.475
0.8
0.525 0.55 0.575 0.6
-α
0.6
(a)
1
•
α
1.02 1.04 1.06 1.08 1.1 1.12 1.14
1.02 1.04 1.06 1.08 1.1 1.12 1.14
-3
A
-2
-3
(c)
“
-R
-1
-2
-R
-1
-4
(b)
•
-4
uz at axis
6
-5
-6
(d)
“
Figure 4.8: (a) Evolution of (α, A) for initial condition (α, A) = (1.5, 3). (b) Evolution
of (α, A) for initial condition (α, A) = (0.61, 2). (c) α and A versus R for initial condition
(α, A) = (0.5, 0.5). The lightning flash indicates finite-time blow up of the solution. Note
the steep fall of A near R = 1.14. (d) The axial velocity at the axis versus R, for the same
data as in (c). Note the start of flow-reversal at R = 1.10.
dynamics of α, ②catches up with the solution curve and either A blows up to
+∞ (initial condition above ③) or A blows up to −∞ (initial condition below
③): see figure 4.8c,d. The corresponding change in velocity profiles is depicted
in figure 4.9.
Second quadrant When an initial condition is taken at the right of ②and below
③and ⑤we have basically two possibilities: (i) the solution curve enters the
third quadrant (and blows up in finite time; see next item) or (ii) contour
③and ⑤catch up with the solution curve, there is a turning from ‘south-east’
to ‘north-west’ and finally the solution curve hits ②: finite-time blow-up. This
last scenario occurs when the initial condition is ‘near’ the crossing of ②, ③and
⑤. See figure 4.10.
Third quadrant When we start in the third quadrant at the right of ②, α will
increase until the solution curve hits ⑤, ends up at the right of ⑤but at the left
of ①, and is finally caught by ①: finite-time blow-up. See figure 4.11a.
Fourth quadrant When we start supercritical (at the left of ①) we have the same
situation as in the third quadrant: finite-time blow-up. When we start subcrit-
152
4 Swirling flows in an expanding pipe
z
Figure 4.9: The axial velocity profile of the quasi-homogeneous approximation for the same
data as in figure 4.8c,d: starting supercritical at the inlet as a jet-flow, turning to a wake-flow
(R = 1.08), flow-reversal (R = 1.10) and criticality (R ≈ 1.14). The vertical lines are the
zero-level lines of the corresponding axial velocity profile, while the thick dash-dotted line
indicates the streamline ψ̂ = 0, marking the re-circulation zone.
ical (at the right of ①) A increases until the solution curve hits ③and then it
slowly starts to decay to zero: A ∼ AR−2 . α is decreasing until it hits ⑥, and
then it approaches some finite value α. See figure 4.11b.
Discussion of the results
An immediate observation is that when the initial condition is between ①and ②the
solution blows up in finite time. The precise time or radius Rblow-up where blow-up
occurs depends on the initial condition. This leads to a bound on the parameter δ in
the definition of the domain Pδ,ε :
“the QHA is applicable in Pδ,ε when 0 ≤ δ < Rblow-up .”
When we start (well) below ③(and still in between ①and ②) we observe a transition
from jet to wake flow. In all cases this transition leads to flow-reversal before the
solution blows up: contour ①catches up with the solution curve. In view of figure
4.9 and the special meaning of contour ①, being related with a singularity not only
in (4.2.11) but also in the linearised Euler equations, this blow-up is called vortexbreakdown.
Finally, when starting at the right of ①and below ⑥the solution remains finite, and
more specific (α, A) ∼ (α, AR−2 ) for α and A order unity.
153
4.3 Analysis of the error equation
•
-1
“
-α
-0.8
-0.6
-0.4
1.204
-0.2
•
A -4
6
-6
-8
-10
-α
1.202
-2
-1.002
1.198
1.196
“
-0.998
-0.996
-0.994
-0.992-0.99
A
6
1.194
1.192
(a)
(b)
Figure 4.10: (a) Evolution of (α, A) for initial condition (α, A) = (−1, 1). (b) Evolution of
(α, A) for initial condition (α, A) = (−1, 1.2).
-α
-0.06
-0.04
-0.02
-α
0.02
0.5
0.6
0.7
0.8
0.9
-10
-0.5
-20
-30
A
6
-1
“
A
6
-1.5
-2•
(a)
(b)
Figure 4.11: (a) Evolution of (α, A) for initial condition (α, A) = (−0.5, −0.5). (b) Evolution of (α, A) for initial condition (α, A) = (1, −2).
4.3
Analysis of the error equation
In this section we correlate the kernel of the linearised Euler equations to that of the
adjoint linearised Euler equations. When we write
E0 (u)η = ∂t η + ω × η − u × curl η,
Γ = ω×,
the result that will be obtained in subsection 4.3.1 can be stated as follows:
h(E0 (u) ◦ Γ + Γ ◦ E0 (u)∗ ) ξ, ζiD = 0,
(4.3.1)
for all divergence-free vector fields ξ, ζ and all solutions u of EEul (u, h) = 0. Γ is the
structure map for the Euler equations. In fact, note that (4.3.1) states that E0 (u) ◦ Γ
is symmetric. This result is a corollary of a much more general theorem on arbitrary
Poisson systems (Van Groesen (1994) and Fledderus and Van Groesen (1996a)).
154
4 Swirling flows in an expanding pipe
After that (subsection 4.3.2) we relate the kernel of the linearised Euler equations to
a degeneracy of the multi-mode RE, the same degeneracy we already met in chapter
3. Since the main features of the error equation are contained in the reduced linearised equations, i.e., equation (4.7) for the stream function corresponding to the
axisymmetric error, we will continue in subsection 4.3.3 with (4.7), and formulate
the solvability condition at criticality. This condition is worked out analytically, and
implemented in the QHA in subsection 4.3.4.
4.3.1
The Kernel Theorem
Consider a columnar base flow U in a constant-diameter pipe, and denote its vorticity
by
Ω = curl U = 0, −Uz0 , r−1 (rUϕ )0 ,
with 0 = d/dr. The linearised Euler equations for axisymmetric disturbances6 are, up
to a gradient, given by
E0 (u)η = ∂t η + ω × η − u × curl η,
E0Eul (u, h)(η, h̃) = ∂t η + ω × η − u × curl η + ∇h̃.
with η periodic in z, i.e., η ∈ H. Then the following holds true.
Proposition 4.9 The operator E0 (U ) ◦ Ω× is symmetric:
∗
(E0 (U ) ◦ Ω×) η − (E0 (U ) ◦ Ω×) η ≡ (E0 (U ) ◦ Ω×) η + (Ω × E0 (U )∗ η) = ∇(κ(r) · η)
with κ(r) = U × Ω, and for all η ∈ H, periodic in time.
Proof The proof is a straightforward application of vector-identities. First we compute the adjoint linearised equation.
h∂t η + Ω × η − U × curl η, ζiD = h−∂t ζ − Ω × ζ + curl (ζ × U ), ηiD;
the boundary terms vanish. To prove the symmetry of E0 (U ) ◦ Ω× we compute the
difference between E0 (U ) ◦ Ω× and its adjoint:
∗
(E0 (U ) ◦ Ω×) η − (E0 (U ) ◦ Ω×) η
≡ (E0 (U ) ◦ Ω×) η + (Ω × E0 (U )∗ η)
= −U × curl (Ω × η) +
Ω × curl (U × η).
(4.3.2)
It is easy to show by writing out that (4.3.2) is a gradient and hence the inner product
∗
of (E0 (U ) ◦ Ω×) η − (E0 (U ) ◦ Ω×) η with an arbitrary divergence free vector field
vanishes.
2
As we noted in the introduction to this section, proposition 4.9 is a special case of a
more general result on Poisson systems, known as the Kernel Theorem (Van Groesen
(1994), and Fledderus and Van Groesen (1996a)).
6 This assumption only simplifies the exposition of the proof. The general result, of which the
proposition is a corollary, handles arbitrary solutions and disturbances.
4.3 Analysis of the error equation
4.3.2
155
Multi-mode relative equilibrium solutions and the degeneracy of the linearised Euler equations
In the introduction to this chapter we mentioned the long wavelength singularity: the
discrepancy between the linearised equations for a columnar perturbation, depending
on the moment when the assumption on the perturbation is substituted. The main
idea is that a columnar perturbation should be embedded in an axisymmetric family
that is obtained by continuously changing the base flow. Moreover, this singularity
can be eliminated by means of the following argument:
Proposition 4.10 When η is a solution of the linearised Euler equations, then η +
η(r) is a solution as well for every η(r). Hence, the disturbance η is assumed to
depend a priori on r, ϕ and z, and any columnar part is assumed to be absent. Stated
more mathematically: η 1 is equivalent to η 2 , η1 ∼ η 2 , when η 1 − η 2 = η(r).
Remark In the following we will call a columnar solution of the linearised equations
non-trivial if it is obtained as a limit of non-columnar flows, i.e., it is a solution of
(4.7) with µ = 0.
3
We would like to link the fact that the multi-mode RE solutions, as discussed in
section 4.1,
v̂ = A1 v ± + A2 u± + αreϕ + Λez ,
(4.3.3)
are solutions of EEul (u, h) = 0 for every value of A1 , A2 , α and Λ, or equivalently, for
every value of a, b, f and A1 , to elements in the kernel of E0Eul (u, h). We established
this link for ∂ v̂/∂A1 |A1 =0 in chapter 3. However, it is easily observed that differentiation of (4.3.3) with respect to a, b or f , keeping in mind that A2 = A2 (a, b, f, A1 ),
α = α(a, b, f, A1 ) and Λ = Λ(a, b, f, A1 ), leads to the sum of a columnar and a noncolumnar part, while putting A1 = 0 leads to only a columnar flow. According to
proposition 4.10, ∂ v̂/∂a|A1 =0 ∼ 0, and likewise differentiation with respect to b and
f.
Hence, the only non-trivial solutions of the linearised Euler equations that can be
±
derived from the multi-mode RE solution is ∂ v̂/∂A1 |A1 =0 = v ± with β = ρ−1
1 in v .
4.3.3
The solvability condition at criticality
Let us return to the equation for the error, derived in (4.6):
E0Eul (û, ĥ)(η, h̃) = −EEul (û, ĥ) = −R,
(4.3.4)
with η = η(r, z), h̃ = h̃(r, z). Since (û, ĥ) evolves in the z-direction, the spectrum of
E0Eul (û, ĥ) evolves as well. Moreover, although the problem is posed in a pipe with
slowly varying radius, (4.3.4) can be thought of as an equation in a constant-diameter
pipe with local radius R(Z).
156
4 Swirling flows in an expanding pipe
Suppose (û, h̃) is such that 0 ∈ spectrum (E0Eul (û, ĥ)), i.e.,
E0Eul (û, ĥ)(η, h̃)
= 0,
ψD∗ ûϕ 1 dφ iµ(z−ct)
µ
,
e
,
η = −i ψ,
r r(ûz − c) r dr
(4.3.5)
with ψ = φ(r)eiµ(z−ct) has a non-trivial (columnar) solution for c = µ = 0. We want
to relate subsection 4.3.1 to 4.3.2, i.e., construct from an element in the kernel of
E0Eul (û, ĥ) an element in the kernel of E0Eul (û, ĥ)∗ .
Recall that we can rewrite (4.3.5) into an equation for φ,
Lφ + −µ2 + B φ = 0,
(4.3.6)
φ(0) = φ(R(Z)) = 0,
with
L=r
d 1 d
,
dr r dr
B=
2ûϕ (ûϕ + rû0ϕ )
û00 r − û0z
− z
2
2
r (ûz − c)
r(ûz − c)
and 0 = ∂/∂r. Hence, we can rephrase our question:
“Relate a solution φ of (4.3.6) to a solution φ∗ of the adjoint equation,
L∗ φ∗ + −µ2 + B φ∗ = 0,
∗
(4.3.7)
∗
φ (R(Z)) = 0, φ regular at r = 0,
with ψ ∗ = φ∗ (r)eiµ(z−ct) and L∗ the adjoint of L.”
Lemma 4.11 The adjoint operator L∗ for the standard L2 −inner product is given
by
L∗ g =
1 d 1 d 2
(r g).
r dr r dr
The proof is straightforward. The boundary conditions on φ∗ in (4.3.6) are such that
the boundary contributions that arise when determining L∗ vanish.
As an immediate solution of (4.3.7) we have
φ∗ =
φ
,
r2
(4.3.8)
and since φ = O(r2 ) as r → 0 (see e.g. chapter 3, theorem 3.8), φ∗ in (4.3.8) is regular
in r = 0.
157
4.3 Analysis of the error equation
In order to determine the solvability condition(s) at criticality, we need to rewrite
(4.3.4) such that the left-hand side turns into the left-hand side of (4.3.6). In that
case, equation (4.3.6), and in particular its right-hand side, is changed into
ûϕ
∂Rz
∂Rr
(ûz − c)
−
− 2 Rϕ
2
∂r
∂z
r
.
(4.3.9)
Lφ + −µ + B φ = irµ−1
2
(ûz − c)
Recall that R = O(ε).
It is the µ−1 -term that calls for the solvability condition.
Proposition 4.12 Equation (4.3.9) has a bounded solution φ iff the right-hand side
of (4.3.9) satisfies the following solvability (or orthogonality) condition at criticality,
i.e., for µ, c → 0 in (4.3.9), :
ûϕ
Rcrit
∂Rr
∂Rz
−
− 2 Rϕ ûz
∂r
∂z
r
· φcrit rdr = 0,
r
(4.3.10)
2
ûz
r2
0
Z
R=R
crit
where û is such that φcrit is a solution of (4.3.6) and with
1 ∂2Ψ
Rr = −ε2 ûz
,
2
r ∂Z
∂ ûϕ
1 ∂Ψ
Rϕ = ε ûz
−
D∗ ûϕ ,
∂Z
r ∂Z
Z
ûϕ ∂ ûϕ
1 ∂ ûz 1 ∂Ψ
1 ∂2Ψ
∂h
Rz = ε 2
dr −
+ ûz
+
r ∂Z
r ∂r r ∂Z
r ∂r∂Z
∂Z
2
1 ∂Ψ ∂ Ψ
− ε3 2
.
r ∂Z ∂Z 2
Remark The solvability condition in (4.3.10) can be compared with the solvability
condition in Trigub et al. (1994). They call the solution that satisfies the solvability
condition when passing the singularity the intermediate solution.
3
The integrand in (4.3.10) splits in an O(ε)-term and an O(ε3 )-term. Since higherorder corrections will deal with the O(ε3 )-term, we are especially interested in the
O(ε)-term. Substituting
1
rρ
R(Z)
A(Z)J0 (
) + 2α(Z)
,
2
R(Z)
ρ
1
rρ
= A(Z)J1 (
) + α(Z)r,
2
R(Z)
1
R(Z)
rρ
R(Z)
= A(Z)
rJ1 (
) + α(Z)r2
,
2
ρ
R(Z)
ρ
rρ
= rJ1 (
),
R(Z)
ûz,crit =
ûϕ,crit
Ψcrit
φcrit
158
4 Swirling flows in an expanding pipe
we obtain
ûϕ
∂Rr
∂Rz
−
− 2 Rϕ
φ
∂r
∂z
r
r
· crit
=
û2z
r2
rρ
rρ dR
ρ
dR
dα
J
(
(
ε
)
rα(Z)
+
rR(Z)
+
A(Z)J
)
+ O(ε3 ). (4.3.11)
1
R(Z)2 1 R(Z)
dZ
dZ
R(Z) dZ
ûz
Integration of (4.3.11) over a cross-section yields the following solvability condition:
at criticality:
(A(Z)ρJ0 (ρ) − 2α(Z)R(Z))
dR
dα
− 2R(Z)2
= 0.
dZ
dZ
(4.3.12)
Analysis of (4.3.12)
The only assumption in deriving (4.3.12) was that λ(Z) = 0, i.e., we are at criticality.
This implies that we can determine whether the various approximations in section 4.2
satisfy (4.3.12).
α(Z) ≡ 0: In this case λ(Z) ≡ Λ(Z) = Qcross,0 /πR(Z)2 , and hence the approximation remains supercritical.
α(Z) = Ω/R(Z)2 : With this choice for α(Z), equation (4.3.12) changes to
at criticality: R(Z) (A(Z)ρJ0 (ρ)R(Z) + 2Ω)
dR
=0
dZ
(4.3.12’)
Moreover, since
λ(Z) = Λ(Z) − 2α(Z)
Qcross,0
R(Z)
Ω
=
−2
,
ρ
πR(Z)2
ρR(Z)
the QHA turns critical at
R=
1 ρQcross,0
.
2 πΩ
Substituting this value of R into (4.3.12’) we obtain the solvability condition in
terms of a condition on A at criticality:
Acrit = −
4πΩ2
ρ2 J0 (ρ)Qcross,0
.
This corresponds exactly to P2 in figure 4.6, but as we noted there, this point
is a saddle and hence it cannot be reached in the QHA. Instead, we cross the
line of criticality at A = 0, and hence (4.3.12’) is not satisfied.
159
4.3 Analysis of the error equation
general: The QHA is critical at ①(see figure 4.7), i.e.,
α(Z) =
ρ
.
2πR(Z)3
Substituting this into (4.3.12) we obtain again a solvability condition in terms
of a condition on A at criticality:
at criticality: Acrit = −
2
.
πR(Z)2 J0 (ρ)
(4.3.12”)
When we compare this expression with the expression for the horizontal asymptote of ③,
horizontal asymptote of ③ :
A=−
ρ2
2
<−
,
8πR2 J0 (ρ)
πR(Z)2 J0 (ρ)
we conclude that, since the QHA, represented by a curve in the (α, A)-plane,
turns critical below ③, the solvalibity condition (4.3.12”) is again not satisfied.
4.3.4
Embedding the solvability condition in the quasi-homogeneous
approximation
We observed that none of the approximations in section 4.2 satisfied the solvability
condition (4.3.12). This leads to the attempt to embed the solvability condition in
the QHA. To be precise,
Proposition 4.13 Consider the modulated columnar RE in (4.2.10a–d). When we
require Q-, A- and E-consistency ánd, in addition, the solvability condition (4.3.12) at
every cross-section, α(Z) and A(Z) satisfy a coupled set of ordinary, non-autonomous
differential equations,
M
dÃ/dR
dα̃/dR
+ X̃ = 0
M=
M11
M21
M12
M22
, X̃ =
X̃1
X̃2
with
M11 = 0;
M12 = −2R2 ;
M21 = 2πR4 αJ0 (ρ) − ρRJ0 (ρ);
1
M22 = 2πAR4 J0 (ρ) + ρ2 R2 ;
2
X̃1 = AρJ0 (ρ) − 2Rα;
X̃2 = 8πAR3 αJ0 (ρ) + ρ2 Rα − AρJ0 (ρ).
The proof is straightforward.
In figure 4.12 we plotted the contour lines of the components of X = −M −1 X̃. The
160
4 Swirling flows in an expanding pipe
3
.
&
⑨
P2
•
-1
%
-2
-1.5
-1
2
-0.5
P1
•
①
A
6
.
0.5
.
⑧
-α
1
1.5
2
-1
&
&
-2
%
⑦
-3
Figure 4.12: The zero-contour lines of the numerator of X1 (solid; ⑧and ⑨), the denominator of X1 (dashed; ①) and the numerator of X2 (dot-dashed: ⑦). The arrows indicate the
initial dynamics of α and A. P1 (at the origin) and P2 are ‘equilibria’ (see the text).
behaviour of the different curves as a function of the radius R is as follows:
ρ
①:α=
,
2πR3
2αR
⑦:A=
,
ρJ0 (ρ)
1 (4πAR2 J0 (ρ) + ρ2 − 4)AρJ0 (ρ)
.
⑧and ⑨ : α = −
2
(12πAR2 J0 (ρ) + ρ2 )R
We make the following observations.
• First of all, it is easy to check that for every R, there are exactly two equilibria:
P1 = (0, 0) : a saddle,
1 ρ(ρ2 − 2) 1 ρ2 − 2
P2 = −
,
−
:
16 πR3
8 πR3 J0 (ρ)
an attracting node.
• We are merely interested in initial conditions at the left of ①, the line of criticality. When we start above the shaded region, we are in the basin of attraction
of P2 (figure 4.13a).
However, when we start in the shaded region, the solution curve is eventually
pushed away from the origin, towards the line of criticality ①, where the solution
blows up (figure 4.13b).
In so far as we expected a stable finite transition through criticality by incorporating
the solvability condition in the QHA, we observe that this is certainly not the case.
161
4.3 Analysis of the error equation
-α
0.46
2.5
0.47
0.48
0.49
2
-2
A
6
1.5
1
-4
“
0.5
-2
-1.5
-1
-0.5
-α
A 0.51
6
0.5
(a)
-6
“ ““
-8
(b)
Figure 4.13: (a) Evolution of (α, A) for initial conditions α0 = 0.5 and A0 = 0.8, 1, 1.5
and 2.5 (solid lines) and α0 = −2 and A0 = 0.5, 1, 1.5 and 2.5 (dashed lines). (b) Evolution
of (α, A) for initial conditions α0 = 0.5 and A0 = 0.4, 0.3, 0.2 and 0.1. At the lightning sign
the solution blows up.
Although (4.3.12) is satisfied at every cross-section, figure 4.14 shows that the blowingup of A is compensated by a blowing-up of α0 , that is due to the coupling in the vector
field X:
dα
AρJ0 − 2Rα
= X2 =
→ ∞ as A → −∞ and α bounded.
dZ
2R2
Observe that balance in X2 requires α to go off to infinity, which essentially means a
crossing of ①, i.e., a transition from supercritical to subcritical.
10
1.01
1.02
1.03
1.04
1.05
-R
-10
-20
Figure 4.14: A (solid line) and α0 (dashed line) versus R for initial condition (α0 , A0 ) =
(0.5, 0.1).
162
4.4
4 Swirling flows in an expanding pipe
Comparison and relation with other approaches
In this last section we start (subsection 4.4.1) to compare the QHA with the quasicylindrical approach (QCA for short) for (slowly) expanding pipes, using the BraggHawthorne equation. This comparison is relevant when we assume a steady axisymmetric solution of the Euler equations in a slowly expanding pipe, given a particular
inlet condition. It is shown that the QCA leads to an O(ε)-correct approximation as
R(z) varies O(1).
As a particular case we focus on the rigid body rotation plus a uniform translation
at the entrance of an expansion, and we will investigate various cross-sectional errors
as a function of R = R(Z):
(Z
R
||(∆uz )(R)|| =
u
2
2πrdr
z,QCA − ûz
)1/2
0
(Z
R
||(∆uϕ )(R)|| =
uϕ,QCA − ûϕ 2 2πrdr
0
(Z
||(∆ψ)(R)||
=
R
ψQCA − Ψ2 2πrdr
,
)1/2
,
)1/2
.
0
Moreover, it is to be expected that these errors are maximal when R is maximal,
i.e., when either one of the two approximations is critical or when the BHE-based
approximation shows flow-reversal. When the latter is the case the assumption, underlying the BHE, that all stream lines originate at −∞, or, alternatively, that the
relation ψ = ψ(r) is globally invertible, is violated. It is explicitly shown that for this
example the maximal error is O(1) instead of O(ε). The explanation of this result
boils down to the fact that the residue should not be restricted to the Euler equations
but rather—in case of steady, axisymmetric flows—should include the (point-wise) restriction that the circulation and the total head are functionally related to the stream
function; this relation is determined at some reference position. Moreover, we will
be able to point out the difference between the swirling flow problem and the other
examples in Fledderus and Van Groesen (1996a): it causes the triangle of equivalences
between consistent evolution and projection arguments on the one side and solvability
conditions for the error equation on the other side to fall apart (see figure 4.15).
We saw in chapter 1 how the phenomenon of vortex breakdown has been compared
with other phenomena like boundary layer separation (Hall (1972)) and hydraulic
jumps (Benjamin (1962)). Some analogs were physically motivated while others
leaned more on the mathematical background of the problem. In subsection 4.4.2
we state a more mathematically oriented analog, including related questions that
might have been answered for special cases. The key-word is ‘slow passage through
bifurcation’, being prompted to us by recent work on second-order nonlinear systems
(Marée (1996)). Systems with a slowly-varying bifurcation parameter, e.g. the radius
of the pipe in our case, can undergo rapid changes near a bifurcation point (see e.g.
Neishtadt (1987), and Marée (1996)), and we propose a more intensive study of this
4.4 Comparison and relation with other approaches
163
proposition 4.4 and 4.5
consistent evolution
projection arguments
solvability conditions
Figure 4.15: Three different approaches how to obtain parameter dynamics in perturbed
systems. Under certain conditions (Fledderus and Van Groesen (1996a)) these approaches
are equivalent for Poisson systems. One of these conditions is that the dynamical variable
is already present in the relative equilibria. If this is not the case, their is no equivalence
between the upper part in the triangle and the lower part.
mechanism in the vortex-breakdown phenomenon. The paper by Trigub et al. (1994)
is for instance in this direction; the authors show that near criticality, elliptical effects (recall that the quasi-cylindrical equations are parabolic) and viscous effects are
important.
4.4.1
The quasi-homogeneous approximation and the BraggHawthorne equation
Let us assume the existence of a steady axisymmetric solution of the Euler equations
in a slowly expanding pipe, with upstream or inlet condition the following profile:
Qcross,0
1 Qcross,0 2
uinlet = 0, α0 r,
, ψinlet =
r .
(4.4.1)
2
πR(0)
2 πR(0)2
R(0) is chosen equal to one. The corresponding BHE for the stream function, describing a steady, axisymmetric flow with (4.4.1) as upstream condition, is given by
(section 2.4 or Batchelor (1967))
α2 π
α2 π 2
∂ 2 ψ ∂ 2 ψ 1 ∂ψ
+
−
= 2 0 r2 − 4 20
ψ
2
2
∂z
∂r
r ∂r
Qcross,0
Qcross,0
ψ(0, z) = 0, ψ(R(εz), z) =
(4.4.2)
1
2π Qcross,0 .
The quasi-cylindrical approach
Let us turn for a moment to the general BHE,
dH
∂ 2 ψ ∂ 2 ψ 1 ∂ψ
dC
+
−
= r2
−C ,
∂z 2
∂r2
r ∂r
dψ
dψ
ψ(0, z) = 0,
ψ(R(εz), z) = 1.
(BHE)
164
4 Swirling flows in an expanding pipe
The QCA suggests to write a solution of BHE as a series,
ψ(r, z) = ψ0 (r; R(εz)) + ε2 ψ1 (r, z) + . . .
such that ψ0 satisfies the ‘columnar part’ of BHE,
∂ 2 ψ0
1 ∂ψ0
dC 2 dH =
r
−
−
C
,
∂r2
r ∂r
dψ ψ=ψ
dψ ψ=ψ
0
ψ0 (0) = 0,
0
ψ0 (R(εz)) = 1.
The correction ψ1 is then governed by the linear equation
!
2
2
∂ 2 ψ1
1 ∂ψ1
dC
d2 C ∂ 2 ψ0
∂ 2 ψ1
2d H
−
r
+
−
−
−
C
ψ
=
−
1
∂z 2
∂r2
r ∂r
dψ 2
dψ
dψ 2 ∂Z 2
ψ=ψ0
ψ1 (0) = 0,
(4.4.3)
ψ1 (R(εz), z) = 0.
As long as ψ0 is supercritical, the linear operator at the left-hand side of (4.4.3) is
invertible and hence ψ1 remains bounded.
The velocity field is given in terms of the stream function,
u0,z =
1 ∂ψ0
,
r ∂r
u0,r = −
1 ∂ψ0
,
r ∂z
u0,ϕ =
C(ψ0 )
.
r
(4.4.4)
However, (4.4.4) describes the velocity field as it is ‘transported’ along a streamline,
and when we are interested in the velocity field at a certain cross-section, say at
distance z = O(ε−1 ), we have to address the question how well the streamline related
to ψ0 approximates the exact streamline.
Determining a streamline can be regarded as integrating a system of coupled ODE’s
((r−1 ∂ψ/∂r, −r−1 ∂ψ/∂z) is the vector field) over a certain ‘time’-interval, and this
process is subject to accumulation of errors: to be precise,
resulting error = order of the approximation × length of integration path.
Hence, when the vector field is O(ε2 )-correct, and we integrate over a distance O(ε−1 ),
the resulting error in the position of a certain particle is O(ε). It is this error in the
position that determines the error in the Eulerian velocity field at a cross-section at
distance z = O(ε−1 ).
Returning to our specific example, the equation for ψ0 can be stated as
∂ 2 ψ0
1 ∂ψ0
α20 π 2
α20 π 2
−
=
2
r
−
4
ψ0
∂r2
r ∂r
Qcross,0
Q2cross,0
ψ0 (0) = 0, ψ0 (R(Z)) =
1
2π Qcross,0 .
(4.4.5)
4.4 Comparison and relation with other approaches
The solution of (4.4.5) is given by (Batchelor (1967))
RQcross,0
Qcross,0 2
1
1 − 2 rJ1 (κr) +
r ,
ψQCA := ψ0 = −
2πJ1 (κR)
R
2π
κRQcross,0
Qcross,0
1
uz,QCA = −
1 − 2 J0 (κr) +
,
2πJ1 (κR)
R
π
κRQcross,0
1
uϕ,QCA = −
1 − 2 J1 (κr) + α0 r,
2πJ1 (κR)
R
165
(4.4.6a)
(4.4.6b)
(4.4.6c)
with κ = 2α0 π/Qcross,0 and R = R(Z). Note the resemblance with a modulated
columnar RE with parameters A, β, α and λ according to
Qcross,0
κRQcross,0
1
A=−
1 − 2 , β = κ, α = α0 , Λ =
.
2πJ1 (κR)
R
π
The value of the quantity κR determines whether the flow is supercritical or subcritical:
κR < ρ1,1 : supercritical,
κR > ρ1,1 : subcritical.
In the following we fix Qcross,0 to unity and vary α0 between 0 and 12 ρ1,1 /π ≈ 0.6098.
Remark Note that flow-reversal starts when
1
1 2 1
uz,QCA r=0 =
−1
= 0.
RJ1 (κR) + κR
πRJ1 (κR)
2
R2
Near criticality we have
uz,QCA r=0 ∼
κR2
2πRJ1 (κR)
1
−1
R2
<0
and hence we have first flow-reversal and then criticality; the same ordering as in the
QHA.
3
Recall that up to the point of flow-reversal the Euler equations and BHE are equivalent. Hence, let us determine the residue of the QHA related to (4.4.2), and the
equation for the error.
Define
∂ 2 ψ ∂ 2 ψ 1 ∂ψ
α20 π 2
2
EBH (ψ) =
+
−
− 2
r −κ ψ .
∂z 2
∂r2
r ∂r
Qcross,0
The stream function corresponding to the QHA is given by
Ψ(r, Z) =
R(Z)
1
1
rρ
A(Z)
rJ1 (
) + Λ(Z)r2
2
ρ
R(Z)
2
(4.4.7)
166
4 Swirling flows in an expanding pipe
and with Qcross,0 /π = Λ0 we obtain
1
rρ
ρ2
α20 2 Λ(Z)
R(Z)
2
EBH (Ψ) = A(Z)
rJ1 (
) κ − 2 +2 r
− 1 + O(ε2 ).
2
ρ
R(Z)
R
Λ0
Λ0
(4.4.8)
Since EBH is affine in ψ, the equation that governs the error is linear and straightforward to solve. As long as κR < ρ1,1 the order of magnitude of the error is determined
by the first two terms on the right-hand side of (4.4.8):
E0BH (Ψ)φ :=
∂ 2 φ ∂ 2 φ 1 ∂φ
+ 2 −
+ κ2 φ = −EBH (Ψ)
∂z 2
∂r
r ∂r
with φ = φ(r, Z) leads to
φ=
1
1
R(Z)
rρ
1 (Λ0 − Λ(Z)) R(Z)
(Λ − Λ(Z)) r2 − A(Z)
rJ1 (
)−
rJ1 (κr)
2 0
2
ρ
R(Z)
2
J1 (κR(Z))
(4.4.9)
Using the evolution of A, α and Λ as constructed in subsection 4.2.2, proposition 4.8,
we will investigate several properties of both the QHA and of (4.4.6a–c):
• maximum cross-sectional error in the velocity profile and the stream function:
(∆uz )max := max ||(∆uz )(R)|| = ||(∆uz )(Rflow-reversal )||
R
with Rflow-reversal that value of R where flow-reversal starts (similarly for (∆uϕ )max
and (∆ψ)max ): figure 4.16;
• the magnitude of the different coefficients in (4.4.9), versus R(Z) for some fixed
value of α0 (see figure 4.17);
• position of flow-reversal at the axis for both QHA and (4.4.6b), and the difference between both: figure 4.18.
• the evolution of the constraints: figure 4.19.
The conclusion in this particular example—the QHA is not O(ε)-correct for O(1)variations in the pipe-radius—is not restricted to this case.
Consider a steady, incompressible and axisymmetric flow in a pipe. As long as there
are no internal stagnation points the flow can be described by the Bragg-Hawthorne
equation (see section 1.1):

∂ 2 ψ ∂ 2 ψ 1 ∂ψ
dC0

2 dH0
EBH (ψ) :=
+
−
− r
− C0
= 0,
(4.4.10)
∂z 2
∂r2
r ∂r
dψ
dψ

ψ(0, z) = 0, ψ(R(z), z) = 1.
with H0 = H0 (ψ) and C0 = C0 (ψ) the total head and circulation as function of the
stream function ψ. These functions are defined at some reference position z0 .
167
4.4 Comparison and relation with other approaches
(∆uϕ )max
(∆ψ)max
0.06
0.25
0.05
6
0.2
(∆uz )max
0.04
0.15
0.03
0.1
0.02
- α0
0.01
0.1
0.2
0.3
(a)
0.4
- α0
0.05
0.5
0.6
0.1
0.2
0.3
0.4
0.5
0.6
(b)
Figure 4.16: (a) The maximal error (∆uz )max (solid line) and (∆uϕ )max (dashed line),
defined as the maximum of (∆uz )(R) and (∆uϕ )(R) respectively over all R where the BraggHawthorne equation is defined. The maximum is attained when flow-reversal starts, so most
downstream. (b) The maximal error (∆ψ)max . Note the different behaviour near α0 = 0 for,
on the one hand, the error in the velocity profiles and on the other hand the stream function.
Moreover, let the profile at z0 be columnar and given by the RE
1
rρ
1
rρ
uinlet = 0, A0 J1 ( ) + α0 r, A0 J0 ( ) + Λ0 , uz,inlet > 0 on [0, R0 ],
2
R0
2
R0
1 R
rρ
1
ψinlet = A0 0 rJ1 ( ) + Λ0 r2 ,
2
ρ
R0
2
with corresponding H0 = H0 (ψ; A0 , α0 , Λ0 ) and C0 = C0 (ψ; A0 , α0 , Λ0 ).
Proposition 4.14 In a pipe with slowly varying radius the structure of the QHA Ψ
as defined in (4.4.7) has residue
dH dH0 dC
dC0 EBH (Ψ) = r2
−
−
C
−
C
+ O(ε2 ),
0
dψ
dψ ψ=Ψ
dψ
dψ ψ=Ψ
with H = H(ψ; A, α, Λ) and C = C(ψ; A, α, Λ); this residue is O(A−A0 , α−α0 , Λ−Λ0 ),
which is in its turn O(R−R0 ) and hence order unity for order unity changes in the pipe
radius. As long as the flow is supercritical, i.e., λ(Z) = Λ(Z) − 2α(Z)R(Z)/ρ > 0,
the error-equation is regular, and the order of magnitude of the error is O(R − R0 ) as
well.
The swirling flow problem as a perturbed Poisson system
Besides the explanation of the results, as stated in proposition 4.14, we like to interpret
our findings in the context of perturbed Poisson systems. More precisely, why does
the swirling flow problem not follow the general theory as developed in Fledderus and
Van Groesen (1996a) when we restrict ourselves to steady, axisymmetric flows?
168
4 Swirling flows in an expanding pipe
0.25
0.2
0.15
0.1
0.05
-R
1.1
1.2
1.3
1.4
1.5
-0.05
Figure 4.17: The different coefficients in (4.4.9) versus R(Z): 12 (Λ0 − Λ(Z)) (solid),
1
A(Z)R(Z)/ρ (dashed) and 12 (Λ0 − Λ(Z))R(Z)/J1 (κR(Z)) (dot-dashed).
2
One of the most important results in Fledderus and Van Groesen (1996a) is that it
connects consistent evolution with solvability conditions for the error-equation. It is
this connection that is absent in the swirling flow problem. The reason for this is
that the tangent space of the manifold of RE is trivially invariant for the linearised
Euler equations (cf. proposition 4.10)—it only consists of columnar flows. In addition,
proposition 4.10 shows that there is a continuous degeneracy, that cannot be removed
by a finite number of constraints. Moreover, the only non-trivial element of the
kernel of E0Eul follows from the travelling wave RE solution, containing essentially the
z-variable that plays the role of time in the QHA.
A non-axisymmetric QHA
We have seen that when the QHA is restricted to the axisymmetric framework, the
resulting approximation fails to be O(ε)-correct. In this paragraph we speculate on a
non-axisymmetric extension of the analysis given in the previous sections, guided by
the RE solutions constructed in chapter 2 and their application to the PVC in section
4.1.
Let v denote a non-axisymmetric Beltrami flow, corresponding to the eigenvalue β =
Rρ−1
1,1 , i.e., a rotating wave (see figure 2.11). Since β curl v = v, the Beltrami flow
itself can serve as a potential in order to define an incompressible, modulated nonaxisymmetric Beltrami flow:
R(Z)
v̂ = curl
v = v + O(ε)-correction.
ρ
Because of the identity div curl ≡ 0, v̂ is automatically divergence-free. Then a
modulated multi-mode RE solution is given by
R(Z)
1
ûmm = curl
v + Ψeϕ ,
ρ
r
169
9
7
relative
error ×100%
Rflow-reversal
4.4 Comparison and relation with other approaches
-0.1
6
-0.2
65
0.1
0.1
0.2
0.3
0.4
0.5
0.6
- α0
-0.3
3
-0.4
1
0.1
0.2
0.3
(a)
0.4
- α0
0.5
0.6
(b)
Figure 4.18: (a) The value of the radius where flow-reversal starts; the solid line is for the
approach based on the Bragg-Hawthorne equation and the dashed line is for QHA (remember
the radius at the inlet is scaled to unity). (b) The relative distance (error) between the two
curves in (a).
with Ψ as in (4.4.7). This gives us one new parameter being the amplitude of v.
Other Beltrami flows might be added. Dritschel (1991) has performed numerical
calculations of a rigid-body rotation perturbed by a combination of Beltrami flows. He
finds essentially a very non-trivial interaction of these flows and it would be interesting
to find out how these combinations act on slow variations of the pipe diameter.
4.4.2
Vortex-breakdown: (slow) passage through bifurcation
In the last part of this chapter we will briefly advocate an approach to the vortexbreakdown problem that has not been widely recognised.
We have seen that the radius R is the determining parameter in both the QHA and
the QCA: at some radius the flow turns critical, indicating singular behaviour of the
linearised operator, or in terms of dynamical systems, a bifurcation point. Hence, R
can be viewed as a bifurcation parameter, and when (4.4.10) is written as
 





 EBH (ψ) = 0,






(4.4.11)
 ψ(0, z) = 0, ψ(R(z), z) = 1,






 dR/dz = ε,
we have a prototype of a slowly varying dynamical system with slow passage through
bifurcation (Marée (1995)):
• for ε = 0 we have, depending on the specific value of R, various ‘equilibria’
(columnar flows), corresponding to the supercritical and subcritical states (Benjamin (1962); Fraenkel (1967));
170
4 Swirling flows in an expanding pipe
-R
1.1
1.2
1.3
1.4
1.5
0.095
0.06
0.05
a
0.09
6
b
0.04
6
0.08
0.02
0.075
0.01
0.085
0.03
0.07
-R
1.1
(a)
1.2
1.3
1.4
1.5
(b)
Figure 4.19: (a) The evolution of the cross-sectional angular momentum a, as a function
of R. The solid line is the value corresponding to the QPA, the dashed line is the value
corresponding to the QHA. (b) The evolution of the cross-sectional helicity b, as a function
of R. The solid line is the value corresponding to the QPA, the dashed line is the value
corresponding to the QHA.
• for 0 < ε 1 we have slowly varying behaviour, until we are near Rcrit (or until
the point where the equations are not valid anymore because of an internal
stagnation point); this dynamics is described by the quasi-parallel equation;
• near Rcrit we have rapid changes.
The papers by Trigub et al. (1994) and Sychev (1992) are particularly interesting for
the swirling flow problem, but also in other fields problems with a similar structure
are studied: e.g. Neishtadt (1987, 1988), Kevorkian and Yu (1989) and Marée (1996).
The essence of all these studies is a slow passage through bifurcation, and the idea
is to match the quasi-parallel equation, valid far away from the bifurcation point,
with an equation valid in a transition layer around this bifurcation point where rapid
changes can occur. This approach may also shed light on the stability of swirling
flows near criticality and the question of upstream influence.
5
Navier-Stokes: viscosity as
perturbation
This chapter is concerned with the application of classical boundary layer theory to
the problem of swirling flow in an expanding pipe. The development of this theory
started in 1904 with the paper “Über Flüssigkeitsbewegung bei sehr kleiner Reibung”
by Prandtl. He suggested that one could analyse practically important viscous flows
by dividing the flow around a solid body into two regions: a very thin layer in the
immediate neighbourhood of the body (the viscous or boundary layer ) where friction
and convection terms are equally important, and the rest of the domain, outside the
layer, where the dynamics is mainly driven by convection and where friction may be
neglected. Blasius (1908) applied this idea to a specific geometry, the finite flat plate
in an aligned, uniform oncoming external stream. Since then, many contributed to
and extended this theory, only a few of them can be mentioned in this thesis. For
a survey on the historical development of (classical) boundary layer theory, see e.g.
Tani (1977).
Although the idea of a boundary layer turned out to be the source for a major simplification of the general Navier-Stokes equations, the resulting problem of finding
the velocity distribution explicitly was in the pre-computer era still much too hard to
handle, especially in the first half of this century. Instead, one started to ask different
questions, related to integral quantities instead of point-wise quantities. Assuming
that these integral quantities were less sensitive to the precise velocity distribution
than the point-wise quantities, one proposed different types of distributions. Examples are the Pohlhausen and Mangler polynomials, Timman’s exponentials and the
self-similar profiles by Walz (see Gadd et al. (1963), for more information). As a
result, rules-of-thumb were derived for (the decay of) various integral quantities, like
the angular momentum flux (Kitoh (1991)). This can serve as an extra motivation
for using coherent flows, defined in terms of integral quantities: the dynamics of these
quantities will lead to a dynamics in the Manifold of Coherent Flows.
172
5 Navier-Stokes: viscosity as perturbation
How to link boundary layers to coherent flows?
We will pursue Prandtl’s idea, i.e., the attached flow strategy (Smith (1982)), in this
chapter, thereby concentrating on the integral quantities. Moreover, we will try to
couple both the inner and outer region by viewing the balance laws in the boundary
layer as the viscous correction to the Eulerian conservation laws in the inviscid region.
In this thesis the word “balance law” refers to an expression that can be viewed as a
perturbation of a conservation law, that is, it consists of a convective (conservative)
part and a perturbation (very often of dissipative nature):
b(u) := ∂t e(u) + div f (u) − d(u).
(5.1)
In chapter 1 a few examples of conservation laws for densities evolving according
to the Euler equations were given and it seems now appropriate to give the viscous
corrections for evolutions given by Navier-Stokes, leading to the following four balance
laws (we consider only Newtonian fluids, for which the viscosity is constant):
linear momenta:
∂t u + div (uu + pI) = Re−1 ∇2 u,
u = (ux , uy , uz );
(5.2)
angular momentum:
energy:
helicity:
∂t (u · ω) + div
uϕ ∂t ruϕ + div ruϕ u = Re−1 r ∇2 uϕ − 2 ;
r
(5.3)
1
1
∂t |u|2 + div p + |u|2 = Re−1 u · ∇2 u;
2
2
(5.4)
1 2
p − |u| ω + [u · ω]u = Re−1 ω · ∇2 u + u · ∇2 ω
2
(5.5)
We already identified Navier-Stokes as a singular perturbed problem, in the sense that
a solution U of the vanishing-viscosity problem, the Euler equations, needs in general
on O(1)-correction near a solid boundary when compared to the exact solution, u.
The region of non-uniformity or boundary layer has a width δ0 (Re), related to the
Reynolds number Re. The scaled equations lead to an asymptotic series of boundary
layer solutions, starting with, let say, v. It is shown in for instance Kevorkian and
Cole (1996) that δ0 (Re) = Re−1/2 .
Assumption 5.1 In the following it is assumed that the approximation
u := U + v − V
(5.6)
of u is O(Re−1/2 ) correct on a suitable spatial domain including (part of ) the boundary
of the pipe. Here V denotes the slip velocity of U . U will have derivatives all of O(1).
More specifically, it is assumed that the following (asymptotic) behaviour holds (in
certain cases this can be proved; T.S. stands for ‘Transcendentally Small’):
173
inner region
U=
v=
w
intermediate region
outer region
000000
111111
0000000011111
11111111
00000
000000
00000000
11111111
00000
11111
U111111
O(1), V ←
O(1)
k
000000
111111
00000000
11111111
00000
11111
O(1)
000000
111111
00000000
11111111
00000
O(n)
U111111
O(n)
⊥
000000
0000000011111
11111111
00000
11111
111111
000000
00000000
11111111
00000
000000
111111
0000000011111
11111111
00000
11111
000000
00000000
00000
v k 111111
O(1), |V − v k | T.S. 11111
O(1) 11111111
000000
111111
00000000
11111111
00000
11111
O(Re−1/2 )
−1/2
O(n)
)
O(Re
v ⊥111111
000000
00000000
11111111
00000
11111
000000
111111
00000000
11111111
00000
000000
111111
0000000011111
11111111
00000
11111
000000
111111
00000000
11111111
00000
11111
000000
111111
00000000
11111111
00000
11111
−1/2
00000
11111
000000
111111
00000000
O(n) 11111111
O(Re
)
O(Re−1/2 )
000000
0000000011111
111111
11111111
00000
111111
000000
11111111
0000000000000
11111
Define w := u − u: it contains higher order corrections on U as well as on v. The
idea is to minimise w, either by increasing its order or by requiring the residue to
vanish in certain directions.
To be more precise, denote the Navier-Stokes equations as
ENS (u) := ∂t u + u · ∇u + ∇p + Re−1 curl ω,
(t, x) ∈ R+ × D
with “no-slip” boundary conditions on u:
u(t, x) = 0
for x ∈ ∂D.
Then
0 = ENS (u) = ENS (U + v − V + w) = ENS (u) + E0NS (u)w + O(|w|2 ).
(5.7)
Taking the inner product of (5.7) with a vector function of u, say a(u), and integrating
the result in time and space,
Z
Z
0 = hENS (u), a(u)i dt + hE0NS (u)w, a(u)i + O(|w|2 ) dt,
suggests that, when the residue ENS (u) vanishes in the direction a(u), w vanishes
when integrated against a the density E0NS (u)∗ a(u).
Note that the balance laws in (5.2-5.5) can be viewed as the (R3 ) inner products of
ENS (u) with certain functions, for instance
u · ENS (u) = energy balance law,
and hence the requirement that the residue ENS (u) vanishes in certain (well chosen)
directions is equivalent to the demand that the approximation obeys the (integrated)
balance laws.
174
5 Navier-Stokes: viscosity as perturbation
The degree of freedom that is needed for this “optimisation” is found in the parametrised
coherent flows. That is, in general U in (5.6) is an exact solution of the Euler equations, and hence in our case, a coherent flow with parameters kept constant. However,
these parameters will be allowed to vary as function of z: in this way we can tune
them such that the approximation satisfies the integrated balance laws. But not only
the parameters in the coherent flows are allowed to vary: the boundary layer flow
v, depending on the slip velocity and hence indirectly on the parameters, is allowed
to anticipate gradual changes in the inviscid outer flow as well. Therefore we need
to make the dependence on the slip velocity as explicit as possible, leading to a second family of parametrized flows, this time in the boundary layer. It turns out that
the parameters will be solely driven by the viscous effects in the constant-diameter
pipe, and by a combination of these viscous effects and the expansion rate in a slowly
varying pipe.
So, U is replaced by U (µ̂(z), R̂(z)), where µ̂(z) are the Lagrange multipliers and other
variables related to the integral quantities like angular momentum and helicity, and
R̂(z) is a kind of effective radius, like the intermediate variable η in the introductory
section 1.4. This suggests that R(z) − R̂(z) = O(Reα ) with −1/2 < α < 0. On
the other hand, the physical interpretation of R̂(z) would be the representation of
the displacement of the outer flow by a certain amount, the displacement thickness,
being of the order Re−1/2 . A last argument for taking R̂(z) as a kind of intermediate
variable is suggested in Hinch (1991), which is explained as follows.
When integrating the balance laws over a volume as shown in figure 5.1 and letting
z− → z ← z+ we can split the range of integration into two at some point, and it is
tempting to use r = R̂(z). Hinch argues that R(z) − R̂(z) should be large compared
σ
θ(σ)
inner region
outer region
R̂(z)
z
•
Figure 5.1: Splitting of the fluid domain and illustration of the concept of effective radius,
R̂(z), and of boundary layer thickness, θ(σ). The shaded region represents the intermediate
region (see also the schematic drawing on page 173).
with the small region of nonuniformity but small compared with the whole interval,
i.e., Re−1/2 R(z) − R̂(z) R(z). The two parts can be treated separately and
when they are finally combined “the result should be independent of the artificially
introduced R̂(z)”. This ‘being independent’ suggests that the ultimate behaviour of
175
the integral quantities should not depend on R̂(z), that is, we should come up with a
pair {µ(z), R(z)} that gives the same behaviour. The idea of effective radius is then
just a tool to separate global contributions and local contributions to the complete
integral, making the (limiting or asymptotic) behaviour of v k explicit.
Relation with perturbed Poisson systems
How does this approach fit in the framework of perturbed Poisson systems? Compared
with a similar problem, namely the vortices in 2D-Navier-Stokes and 2D-RMHD (Van
Groesen (1989), Derks and Ratiu (1997)), a few differences should be mentioned. First
the decoupling between velocity components and vorticity component in 2D (or the
essential vanishing of the helicity). This simplifies the interactions with boundaries
and the structure of the equations of motion itself. Furthermore, we like to view
the z-variable as a kind of dynamic variable, but axial derivatives of cross-sectional
energy and so on are difficult, if not impossible, to obtain directly, whereas the timedependent behaviour of these (adiabatic) quantities is rather straightforward, even in
our case of 3D-swirling flows.
We have seen in the previous paragraph the following important idea, underlying this
thesis and related work: a curve in the MRE is takes as an approximation of the
evolution of the perturbed system, with initial condition on the MRE. Pursuing this
idea for the case of swirling flows, we want to approximate the solution in the inviscid
core or outer region at every cross-section by a suitable relative equilibrium; the
dynamics of the parameters should follow from the balance laws mentioned previously,
which will be turned into a set of implicit differential equations for the Lagrange
multipliers and amplitudes, like in chapter 4. Using the relations between, on the
one hand, the multipliers and amplitudes and on the other hand the constraints, we
obtain the desired curve in the MRE.
An extra motivation for using slowly varying ‘inviscid’ flows U as base flows in the
outer region is the observation by Kitoh (1991) that “in the core region (...) turbulent
motion with very low frequency prevails and the flow is non-dissipative. The inlet
conditions have a large effect on the downstream flow in the core region. Thus for
the core region there is a long history effect.”
Note that the evolution of the cross-sectional quantities itself will differ in principle
from the dynamics of the constraints. This is immediately clear from the angular
momentum example. Suppose the relative equilibrium is denoted by U(r; µ(z)), with
µ the collection of multipliers and amplitudes that are into play. Then the value of
the angular momentum that should be used to recover this relative equilibrium is
given by
ARE
1
=
πR(z)2
Z
R(z)
rUϕ (r; µ(z)) 2πrdr,
0
whereas the cross-sectional angular momentum for the Navier-Stokes solution u is
176
5 Navier-Stokes: viscosity as perturbation
given in first order by
ANS
1
=
πR(z)2
(Z
R(z)
rUϕ (r; µ(z)) 2πrdr −
0
−1/2
Z
2πRe
∞
R̃(σ)(Vϕ (µ(z)) − vϕ ) R̃(σ)dN
. (5.8)
0
The same principle as used for the angular momentum balance law will be applied
to the helicity, an intriguing quantity in the case of boundary layers (Moffatt (1969),
and Moffatt and Tsinober (1992)): its influence can be of order 1, to be compared
with e.g. the O(Re−1/2 )-corrections in (5.8).
In the first section we exploit assumption 5.1 and the concept of effective radius to
approximate the integrated balance laws: the range of integration is split and the two
parts are analysed separately, leading to a global contribution (integration over the
outer region) and a local contribution (integration over the thin boundary layer). This
idea is applied in a straightforward way to (5.2-5.4). However, the most interesting
from the point of order-of-magnitude-estimation is the helicity balance law in (5.5):
all kind of derivatives with respect to the small boundary layer variable play a major
role in the final result.
In section 5.2 we treat the case of a constant-diameter pipe. The boundary-layer equations (that are stated in the appendix) reduce to an ordinary differential equation.
The result is applied to the cases of a rigid-body rotation (Uϕ = Ωr) superposed on
a translation (Uz = W ) and a relative equilibrium. A particular result is the increase
of both the mass flux and the angular momentum related to the relative equilibrium.
This may be compared with an ice-skater that brings her arms to her body, thereby
increasing her rotation speed. Besides that, we pay attention to the decay of the
angular momentum flux. This quantity is known as swirl intensity as well, and investigators like Baker (1967) and Kitoh (1991) have reported exponential decay in the
axial direction for it, based on empirical results. However, the exponential behaviour
is restricted to a domain that lies downstream of the inlet region; this region cannot
be reached with the linear theory due to the violation of its assumptions, caused by
the growth of the boundary layer. For the inlet region, we are able to give an explicit
expression for the decay coefficient in terms of the Reynolds number, the azimuthal
skin friction and the meridional slip velocity. We also investigate the helicity hypothesis (Lilly (1986)), stating that the presence of a large Beltrami component decelerates
dissipation processes. This is done by computing the axial decay of the energy flux
in relation to the magnitude of the Beltrami component. The dissipation is clearly
retarded (see figure 5.8).
Finally in section 5.3 we make an attempt to get the dependence of the boundary layer
profiles on the slip velocity as explicit as possible in the case of a slowly expanding
pipe. We state the equations that describe the interplay between the viscous effects
and the expanding pipe as they act on the parameters in the relative equilibrium.
We obtain a balance between the two effects when ε ≈ Re−1 , a result that can be
compared with the conclusions of Abramowitz (1949) where he studied the case of a
177
5.1 Balances and boundary layers
Poiseuille flow in a diverging pipe (without rotation). From these equations both the
inviscid expanding pipe problem and the viscous constant-diameter pipe problem can
be derived by setting the corresponding parameters to zero.
5.1
Balances and boundary layers:
a splitting argument
We have seen in the introduction that one way of linking the viscous effects to the
coherent flows is by integrating the balance laws. Every integration will consist of
basically two parts: one related to the coherent flow, integrated over the outer region
(the global contribution) and one related to the boundary layer flow, integrated over
the inner region (the local contribution). In this section we want to find precise
estimates of the error, using assumption 5.1 and the archetypical example in section
1.4. Note that we do not yet need any equations for either the inviscid flow or the
boundary layer flow; we only use their asymptotic behaviour.
In order to analyse the integration over the inner region, we have to construct local
coordinates. Ideally we would use spherical coordinates, which can be used both
locally and globally. However, we have chosen for the following pair, that is used in
most standard works. Introduce coordinates n and σ, normal and along the surface,
respectively (see figure 5.2).
σ
σ=0
n
• (z, r)
α
z=0
Figure 5.2: Definition of coordinates n and σ.
Given n and σ, (r, z) are found from
r = σ sin α − n cos α + 1,
z = σ cos α + n sin α.
These relations can be inverted in order to compute n and σ for given (z, r):
n = z sin α − r cos α + cos α,
σ = z cos α + r sin α − sin α.
178
5 Navier-Stokes: viscosity as perturbation
The new components of the velocity vector, (vn , vϕ , vσ ), are given in terms of (ur , uϕ , uz )
by
vn = −ur cos α − uz sin α,
vσ = ur sin α + uz cos α,
vϕ = uϕ .
and conversely,
ur = −vn cos α + vσ sin α,
5.1.1
uz = vn sin α + vσ cos α.
The general case
Now let us apply the ideas of splitting the range of integration to the balance law in
(5.1). We will specifically look at the integration of e(u) and the behaviour of the
divergence term, div f (u), and we assume that all integrands depend (polynomially)
on at least one of the ‘slip-velocities’ (uϕ , uz ). In case of derivatives, we need to check
things separately. Further it is assumed that θ̂ = OS (Reα ) with −1/2 < α < 0.
The temporal part
A ‘straightforward’ splitting leads to
Z
Z
R
e(u) =
0
e(U + v ⊥ ) +
0
R
Z
Z
R̂
R̂
Z
Z
e(v k ) +
R̂
e0 (U + v ⊥ )(v k − V + w) + h.o.t. +
e0 (v k )(U + v ⊥ − V + w) + h.o.t.
R−θ̂
e(U + v ⊥ ) + O(w) + h.o.t. +
=
Z
R
0
R̂
0
R
Z
R−θ̂
e(v k ) +
R
R̂
e0 (v k )(U + v ⊥ − V + w) + h.o.t..
The upper limit of the integral near the boundary will tend to infinity when introducing a scaled coordinate N = n/Re−1/2 . This causes some problems related to the
convergence of the integral, since v k → V 6= 0 as N → ∞. This can be circumvented
by the following splitting:
Z
Z
R
e(u) =
0
0
R
Z
Z
R
R−θ̂
e(U + v ⊥ ) +
0
R−θ̂
e0 (U + v ⊥ )(v k − V + w) + h.o.t. +
e(v k ) − e(U + v ⊥ ) + e0 (v k )(U + v ⊥ − V + w) + h.o.t.
Comparing the second integral with the table in assumption 5.1, we immediately have
Z
0
R−θ̂
e0 (U + v ⊥ )(v k − V + w) + h.o.t. = O(|w|)
179
5.1 Balances and boundary layers
To approximate the integral over the remainder of the range R − θ̂ to R, we rewrite
it slightly:
Z R
Z R
e(v k ) − e(U + v ⊥ ) +
e0 (v k )(U + v ⊥ − V + w) + h.o.t.
R−θ̂
R̂
(!)
Z
θ̂
=
0
e(v k ) − e(V ) + [e0 (v k ) − e0 (V )](U − V ) +
Z
θ̂
0
[e0 (v k ) − e0 (V )](v ⊥ + w) + h.o.t.
where we introduced the normal coordinate n at (!). It is this splitting that is fit for
the boundary layer scaling. Writing n = Re−1/2 N , and letting Re → ∞ it can be
shown that1
Z θ̂
Z ∞
e(v k ) − e(V ) −→ Re−1/2
(e(v k ) − e(V )) R̃dN,
0
Z
Z
θ̂
0
0
θ̂
0
[e0 (v k ) − e0 (V )](U − V ) −→ O(Re−1 ),
[e0 (v k ) − e0 (V )](v ⊥ + w) −→ O(Re−1 , |w|),
with R̃ = R̃(σ) = 1 + σ sin α. Concluding we can say that
Z
Z
R
e(u) rdr =
0
0
R
e(U + v ⊥ ) rdr −
−1/2
Z
∞
Re
0
[e(V ) − e(v k )] R̃dN + O(Re−1 , |w|).
Remark Note that the ‘viscous’-correction appears at two places: the effect of a
non-vanishing normal velocity, that is perceptible over the whole domain (one can
speak of a ‘displacement’-effect, although this word is reserved for a different kind of
term) and the effect of the retardation of the flow near the boundary. The latter is
the effect that appears when the balance law in the boundary layer is studied.
3
The flux part
The integration of the divergence term seems to bring in something new, namely
a difference in fluxes through the artificial ‘boundary’ between the inner and outer
region. However, since this boundary is artificial, it should not introduce a new term,
related to it. Moreover, integration of the divergence term leads to
I
Z R
f (u) · n = ∂z
fz (u) + f (u) · n|r=R
(5.1.1)
0
1 One
can use the example in section 1.4, together with assumption 5.1.
180
5 Navier-Stokes: viscosity as perturbation
and since the last term in (5.1.1) vanishes for all balance laws (also for the helicity-law
as we will see), we conclude that the integration of the divergence part leads to
Z
R
0
Z R
div f (u) rdr = ∂z
fz (U + v ⊥ ) rdr −
0
Z ∞
−1/2
∂σ Re
(fσ (V ) − fσ (v k )) R̃dN + O(Re−1 , |w|).
(5.1.2)
0
Remark In most textbooks (e.g. Rosenhead (1963)) the momentum equations are
given for the boundary layer only. In that case the flux-term through the boundary
does not vanish. Since the balance formulation in this chapter integrates the momentum equations in the boundary layer together with those in the inviscid core, the two
corresponding flux-terms cancel each-other and we are left with (5.1.2).
3
We will treat the ‘dissipative’ terms on the right-hand sides of (5.2-5.5) separately.
5.1.2
The mass balance
We will first treat the mass conservation law that is not affected by viscous effects; it
remains
div u = 0.
(5.1.3)
Integration of (5.1.3) over a cross-section, and the subsequent splitting procedure
is completely along the lines of the previous paragraphs, and leads to (note that
v ⊥,σ = O(αRe−1/2 ))
(Z
∂z
)
R(z)
Uz 2πrdr
0
Z
− ∂σ 2πRe−1/2
∞
(Vσ − vσ ) R̃(σ)dN
=
0
O(αRe−1/2 , Re−1 , |w|).
(5.1.4)
It is commonplace to interpret terms in the integrated boundary layer equations that
are related to flux defects as a kind of measure for ‘the boundary layer thickness’2 .
In this case we define the following thickness:
Z ∞
Vσ θ1σ =
(Vσ − vσ ) R̃(σ)dN (displacement thickness.3 )
0
2 It is impossible to indicate a boundary-layer thickness, δ, in an unambiguous way, because the
influence of the viscosity in the boundary layer decreases asymptotically outwards. The component
of the velocity along the solid surface, v k , tends asymptotically to the corresponding value of the
outer flow, say V . In practical circumstances it is sometimes desired to define the boundary–layer
thickness as that distance for which the |v k | = 0.99|V | (Schlichting (1979)).
3 Recalling the asymptotic expansion for the velocity field in the outer region, the term displacement thickness has a straightforward interpretation. When matching the inner and outer expansion
it turns out that the outer stream function appears to vanish at θ1σ . Thus the boundary layer
displaces the outer inviscid flow like a solid wall (Van Dyke (1975), chapter 7).
181
5.1 Balances and boundary layers
5.1.3
The angular momentum balance
Since all the dissipative right-hand sides of the balance laws are multiplied by Re−1 ,
the only terms in the integrand that play a role are those that bring ‘down’ enough
powers of the Reynolds number Re. By virtue of assumption 5.1 we need derivatives
with respect to the normal coordinate to assure this. This leads to the investigation
of Re−1 r∇2 uϕ , integrated over a cross-section:
Re−1
Z
R
Z
r∇2 uϕ = Re−1
0
Z
−1
R̂
r∇2 Uϕ + r∇2 (vk,ϕ + v⊥,ϕ − Vϕ + wϕ ) +
0
R
r∇2 vk,ϕ + r∇2 (Uϕ + v⊥,ϕ − Vϕ + wϕ ).
Re
R̂
The integrals over the outer region are all O(Re−1 ). The ‘inner-integral’ is split in
two parts, with the second part clearly of the order O(Re−1 ) (even though the second
derivative of wϕ with respect to n brings down Re−1 ) and the first part of the order
Re−1/2 (note again that v ⊥,ϕ ≡ 0 and v k,ϕ = vϕ ):
−1
Z
2
Re
Z
∞
∂ 2 vϕ
R̃dN
∂N 2
0
∂vϕ + T.S.T.
= −Re−1/2 R̃2
∂N N =0
R
r∇ vϕ = Re
R̂
−1/2
R̃
Together with the results on the convective part it leads to the following balance law
for the angular momentum (note that v ⊥,σ = O(αRe−1/2 ))
(Z
R(z)
rUϕ 2πrdr − 2πRe
∂t
0
−1/2
Z
∞
)
R̃(σ)(Vϕ − vϕ ) R̃(σ)dN
0
(Z
)
R(z)
∂z
rUϕ Uz 2πrdr
Z
−1/2
∂σ 2πRe
0
∞
+
−
R̃(σ)(Vϕ Vσ − vϕ vσ ) R̃(σ)dN +
−1/2 ∂vϕ 2
2π R̃(σ) Re
= O(Re−1 , αRe−1/2 , |w|). (5.1.5)
∂N N =0
0
Remarks (i) Usually the balance laws are integrated only over the boundary layer.
This leads then to a dynamical equation for e.g. the angular momentum defect, R̃(Vϕ −
vϕ ): see Rosenhead (1963), chapter 8, equation (69). Moreover, the presence of the
inviscid or outer region has been accounted for through an extra term that represents
the flux of momentum into the boundary layer. When coupling the dynamics in the
boundary layer and the inviscid region this term naturally cancels.
182
5 Navier-Stokes: viscosity as perturbation
(ii) In this case the following thicknesses are defined:
Z ∞
Vϕ θ1ϕ =
Vϕ − vϕ R̃(σ)dN (θ1ϕ : displacement thickness),
Z0 ∞
Vϕ Vσ θA =
Vϕ Vσ − vϕ vσ R̃(σ)dN (θA : angular momentum thickness),
0
∂vϕ τϕ =
(τϕ : azimuthal skin friction).
∂N N =0
3
5.1.4
The axial momentum balance
The derivation of the dissipative term is similar to the one in the angular momentum
balance:
Z R
∂vσ −1
−1/2
2
r∇ uz = −Re
+ O(Re−1/2 α),
Re
R̃
∂N
0
N =0
and hence
(Z
R(z)
Uz 2πrdr − 2πRe−1/2
∂t
0
(Z
Z
∞
(Vσ − vσ ) R̃(σ)dN
0
(Uz2
0
−1/2
+
)
R(z)
∂z
)
Z
∞
∂σ 2πRe
0
2π R̃(σ)Re−1/2
+ p) 2πrdr
−
−
∂vσ ∂N (Vσ2
vσ2 ) R̃(σ)dN
+
= O(Re−1 , αRe−1/2 , |w|).
(5.1.6)
N =0
We define the following thicknesses:
Z ∞
2
Vσ θ2σ =
Vσ2 − vσ2 R̃(σ)dN
0
∂vσ τσ =
(τσ : meridional skin friction).
∂N N =0
5.1.5
The energy balance
Although the splitting of the integral of the quadratic dissipative right-hand side of
(5.4) is a bit more involved, the end result contains only normal derivatives of the
boundary layer velocity v k :
−1
Z
R(z)
−1/2
Z
u · ∇ u = −Re
2
Re
0
0
∞
∂v k 2
−1
∂N R̃(σ)dN + O(Re ).
5.1 Balances and boundary layers
183
Combining this with the convective part yields (note that |U + v ⊥ |2 = |U |2 + |v⊥ |2 +
O(αRe−1/2 ) = |U |2 + O(αRe−1/2 , Re−1 )):
(Z
∂t
)
Z ∞
1
1
−1/2
2
2
2
|U | 2πrdr − 2πRe
(|V | − |v k | ) R̃(σ)dN +
2
2
0
0
(Z
)
R(z) 1
∂z
p + |U |2 Uz 2πrdr −
2
0
Z ∞ 1
1
P + |V |2 Vσ − P + |v k |2 vσ R̃(σ)dN +
∂σ 2πRe−1/2
2
2
0
Z ∞ ∂v 2
k
−1
−1/2
2πRe−1/2
, |w|). (5.1.7)
∂N R̃(σ)dN = O(Re , αRe
0
R(z)
The remarks for (5.1.5) hold also for equation (5.1.7). In addition, the pressure in the
boundary layer does not vary across the layer and equals the ‘slip pressure’. This is
related to the fact that there are, in general, no boundary-conditions for the pressure
at solid boundaries. The thicknesses that can be defined are
Z ∞
(|V |2 − |v k |2 ) R̃(σ)dN
|V |2 θ2 =
0
Z ∞
Vσ |V |2 θE =
(|V |2 Vσ − |v k |2 vσ ) R(σ)dN (θE : energy momentum thickness),
0
Z
|V | DE =
2
0
5.1.6
∞
∂v k 2
∂N R̃(σ)dN
(DE : energy dissipation integral).
The helicity balance
All the balance laws we treated until now have an a priori small influence on the outer
flow since we integrated an O(1) quantity (see assumption 5.1) over the boundary
layer, which is by definition of the order δ0 (Re) = Re−1/2 . The only exception is the
helicity balance law that is treated in this paragraph.
One of the most explicit and early remarks on the fact that boundary contributions
can be essential for the (dynamic evolution of) helicity in a fluid was made by Moffatt
(1969). His argument for unbounded flows around a solid body at rest4 is that a
volume integration of (5.5) leads to
Z
Z
dB
= −2Re−1
ω · (∇ × ω) dV, B =
u · ω dV.
dt
V
V
Let L be the typical length scale on which u varies near Sb , let W be the scale of
|u| itself just outside the boundary layer5 . As we remarked, the thickness δ of the
4 When
the body is at rest,
!⊥ = det(@ k , vk ).
5 That
u = 0 on its boundary Sb , and hence ! · n = !⊥ · n = 0 on Sb , since
is, the scale of the slip velocity in our approach.
184
5 Navier-Stokes: viscosity as perturbation
boundary layer is in general of the order LRe−1/2 with Re the local Reynolds number,
LW/ν, and the normal and tangential components of the vorticity in the layer are of
the order6
ω⊥ = O(W/L),
ωk = O(W/δ).
2
/δ)
Therefore, the integrand Re−1 ω·∇×ω will be of the order O(Re−1 W 2 /δ 3 ) = O(ω⊥
and hence the contribution to dB/dt from a unit area of the boundary layer is of the
2
order ω⊥
, being independent of Re in the limit Re → ∞. This is the reason for the a
priori prime importance of the the structure of the boundary layer when we want to
determine dB/dt and (in our case) the rate of change of the helicity flux. A practical
consequence is that in the following we will only be interested in O(1)-effects due to
the boundary layer.
We will split the vorticity field as
curl u := ω = curl U + curl v − curl V + curl w
=: Ω + χ − X + ω̃.
Remarks (i) First of all, note that lim Ω 6= X, since X = curl V and V does not
r→R
depend on n. It turns out that
Vϕ ∂r̃
1
∂
(r̃(σ, n)Vϕ )en −
e
r̃(σ, n) ∂σ
r̃(σ, n) ∂n σ
dVϕ
Vϕ ∂r̃
=
+ O(α) en −
e ;
dσ
r̃(σ, n) ∂n σ
∂Uϕ
∂Uσ
lim Ω − X = lim
eϕ −
eσ .
n→0
n→0
∂n
∂n
X=
Both Ω and X are O(1) uniformly.
(ii) The well structured velocity hierarchy related to orders of magnitude is destroyed
when taking the curl. The tangential component χk of χ blows up like O(Re1/2 ) as
N ↓ 0, but on the other hand this order is not uniform over the whole boundary layer:


Re1/2 ∂vσ
as N ↓ 0,
1/2 ∂vσ
−1/2 ∂vN
∂N
χϕ = Re
− Re
→
∂v

∂N
∂σ
−Re−1/2 N as N → ∞;
∂σ

∂v

−Re1/2 ϕ as N ↓ 0,
Re1/2 ∂
∂N
χσ = −
(r̃(σ, N )vϕ ) →
Vϕ ∂r̃

r̃(σ, n) ∂N
−
as N → ∞.
r̃(σ, n) ∂n
6 It
is easy to check the orders of magnitude by means of the following determinant expression:
! = ∇×u=
{ek }
@k
uk
∂⊥ .
u⊥ e⊥
185
5.1 Balances and boundary layers
The normal component χ⊥ is O(1) uniformly over the whole boundary layer:
1
∂
(r̃(σ, n)vϕ )en .
r̃(σ, n) ∂σ
χ⊥ = χn en =
On the whole we conclude that
lim χ − X = −Re−1/2
N →∞
∂vN
e
∂σ ϕ
exponentially fast.
(iii) Finally, although |ω̃| is O(1) in the max-norm, its influence in the balance law is
still of higher order:
lim |ω̃|max = O(Re−1/2 ).
lim |ω̃|max = O(1),
N →0
N →∞
3
The temporal part A straightforward splitting of the first term in (5.5), integrated
over a cross-section gives
Z
Z
R
Z
R̂
u·ω =
0
0
R̂
U ·Ω+
U · (χ − X + ω̃)
Z
0
Z
Z
R̂
(v − V + w) · Ω +
+
0
R
v · (χ − X) +
R̂
Z
(v − V + w) · (χ − X + ω̃)
0
Z
R
+
R̂
v · (Ω + ω̃)
R̂
Z
R
R
(U − V + w) · (χ − X) +
+
R̂
(U − V + w) · (Ω + ω̃).
R̂
Writing n = Re−1/2 N , and letting Re → ∞ it can be shown that all but the first
local contributions are O(Re−1/2 , |w|). Hence,
Z
Z
R
u · ω rdr =
0
Z
R
U · Ω rdr −
0
0
∞
∂vϕ
∂v
vσ
− vϕ σ
∂N
∂N
R̃(σ)dN +
O(Re−1/2 , |w|). (5.1.8)
Remarks (i) Note that we took the essential terms from v · (χ − X), and put the
rest in the O(Re−1/2 , |w|)-term.
(ii) Note that the local contributions are O(1) as well. The consequence is that the
neglected terms are an order of magnitude lower than in the previous balances. 3
The flux part Since the normal component of the vorticity vanishes at the boundary when the velocity vanishes, the integration of the divergence term in (5.5) has no
186
5 Navier-Stokes: viscosity as perturbation
contribution from the wall of the pipe. Keeping in mind that only the O(1)-effects
from the boundary layer are taken into account, it is straightforward to show that
Z
R
0
1 2
div
p − |u| ω + [u · ω]u rdr =
2
(Z )
R
1
2
p − |U | Ωz + [U · Ω] Uz rdr −
∂z
2
0
Z ∞ ∂vϕ
∂vϕ
∂vσ
1
2
∂σ
P − |v k |
+ vσ
− vϕ
v
R̃(σ)dN +
2
∂N
∂N
∂N σ
0
O(Re−1/2 , |w|)
(5.1.9)
The dissipative part Integration of ω ·∇2 u+u·∇2 ω over a cross-section proceeds
along the same lines as in the previous paragraphs. Since we restrict ourselves to the
O(1)-contribution the following result is easily obtained:
Re−1
Z
R
ω · ∇2 u + u · ∇2 ω rdr = −2Re−1
0
Z
= −2
0
Z
R
(ω · curl ω) rdr
0
∞
∂vϕ ∂ 2 vσ
∂vσ ∂ 2 vϕ
−
∂N ∂N 2
∂N ∂N 2
!
R̃(σ)dN.
Combining this result with the convective part in (5.1.9), and rewriting the flux term
by means of a few integration by parts we obtain the helicity balance law that connects
the inviscid core with the boundary layer:
(Z
∂t
0
)
∂vϕ
∂vσ
U · Ω 2πrdr − 2π
vσ
− vϕ
R̃(σ)dN +
∂N
∂N
0
(Z
)
R(z) 1
2
∂z
p − |U | Ωz + [U · Ω] Uz 2πrdr −
2
0
Z ∞
∂vϕ
1
1
∂σ 2π
R̃(σ)dN + 2π R̃(σ)Vϕ P + Vϕ2 − Vσ2
vσ2
+
∂N
6
2
0
!
Z ∞
∂vϕ ∂ 2 vσ
∂vσ ∂ 2 vϕ
−
R̃(σ)dN = O(Re−1/2 , |w|). (5.1.10)
4π
∂N ∂N 2
∂N ∂N 2
0
R(z)
Z
∞
We have not come across (5.1.10) in the literature, not even the part that is only
concerned with the boundary layer itself.
5.2 Viscous swirling flows in a constant-diameter pipe
187
We can define three new thicknesses:
Z ∞
∂vϕ
∂vσ
2
− vϕ
R̃(σ)dN
|V | Θb =
vσ
∂N
∂N
0
Z ∞
∂vϕ
Vσ |V |2 ΘB =
R̃(σ)dN (ΘB : helicity momentum thickness),
vσ2
∂N
0
!
Z ∞
2
2
∂v
∂
v
v
∂
∂v
ϕ
ϕ
σ
R̃(σ)dN (DB : helicity ‘dissipation’ integral).
|V |2 DB =
− σ
∂N ∂N 2
∂N ∂N 2
0
Discussion of (5.1.10)
Until now we emphasized the O(1)-effect of the helicity contribution at the boundary
layer, but with (5.1.10) we can make a better appraisal of this statement. First of all,
when the variation of the slip-velocity and the pressure in the stream wise direction
vanishes or is small, say O(Re−1/2 ), the term
1
1
∂σ 2π R̃(σ)Vϕ P + Vϕ2 − Vσ2
(5.1.11)
= O(α, Re−1/2 ).
6
2
The small parameter that will determine the order of magnitude of the axial variation
of the flow in a constant-diameter pipe is Re−1/2 . Hence, the term in (5.1.11) will
not be order unity but O(Re−1/2 ).
The other terms measure in some sense the difference in development in the meridional
and the azimuthal boundary layer. To be precise, we show in the appendix that
the axial and tangential profile, vσ /Vσ and vϕ /Vϕ respectively, are initially (at the
entrance of the pipe) the same and only develop (slowly) under the influence of the
(slow) change in the geometry. This one-and-the-same profile is a self-similar profile,
and it easily follows that the other terms in (5.1.10), arising from the boundary layer,
vanish at the entrance of the pipe.
When the velocity profile in the boundary layer has a strong Beltrami character, i.e.,
∂vσ
+ h.o.t.,
∂N
∂vϕ
vσ = β(Re)χσ = −β(Re)Re1/2
+ h.o.t.,
∂N
vϕ = β(Re)χϕ = β(Re)Re1/2
the helicity thicknesses reveal their indefiniteness very clearly, measuring the difference
in the meridional and the azimuthal component.
5.2
Viscous swirling flows in a constant-diameter
pipe
Remark In this section we will assume that we are dealing with a steady (axisymmetric) flow in a constant-diameter pipe.
3
188
5 Navier-Stokes: viscosity as perturbation
Now that we have introduced integral quantities like θ1σ and so on, and equations
relating them, we would like to compute them. This will be done by explicitly solving
the underlying boundary-layer equations, thereby emphasizing the role of equations
(5.1.3), (5.1.7), (5.1.4) and (5.1.10) as corrections to the conservation laws and not as
equations from which the integral quantities should be solved. This last option would
imply some mathematical difficulties since there are fewer7 equations than unknowns!
In the appendix we formulate the system of boundary layer equations, transformed in
such a way that standard numerical routines are applicable. Since these equations are
fuelled by the outer solution, we have to supply the system with a kind of archetypal
flow. In this section we drew from the appendix the equations in case of a constantdiameter pipe—Blasius equation for the self-similar velocity profiles:
G000 + GG00 = 0,
G(0) = G0 (0) = 0,
on [0, ∞),
(5.2.1)
lim G0 (η) = 1,
η→∞
with G = G(η) and vσ /Vσ = vϕ /Vϕ = g(η) ≡ G0 (η). These profiles are used as
‘coherent structures’ in the boundary layer:
(
Nboundary layer =
r
v k = (0, Vϕ g(η), Vσ g(η)), η = N
Vσ
2σ
)
Vϕ , Vσ .
(5.2.2)
In this way we can concentrate on the relation between the slip velocities and the
−1/2
integral quantities: for instance the displacement thickness is linearly related to Vσ
in a uniform flow over a flat plate.
Note that (5.2.1) is obtained for constant Vϕ and Vσ . However, in the same way as
we let the RE solutions in chapter 4 adopt to a varying radius of the pipe, we let
them now adopt to viscous effects, and, at the same time, we let the boundary layer
profiles in (5.2.2) anticipate on the gradual changes in the slip-velocity by allowing
the parameters to vary as function of z. The resulting approximation is required to
satisfy the integrated balance laws for mass, angular momentum, axial momentum
and energy.
In the first subsection we try to get some feeling for the subject with an easy example—
rigid-body rotation plus uniform translation in a pipe. This example is extended to
columnar relative equilibria. We will look at the decay of the fluxes, and compare
this with experimental results on turbulent swirling flows.
7 Starting in 1921 with the work of Pohlhausen, this problem of underdeterminancy has been solved
in various ways but the main idea was the reduction of the number of quantities by interrelating
them by means of an assumption on the velocity distributions. Very often these assumptions gave
satisfying results but especially in the case of boundary layer separation, when we should turn to a
different description of the “boundary layer” anyway, the specific characteristics of quantities such
as the skin friction could not be resolved.
5.2 Viscous swirling flows in a constant-diameter pipe
5.2.1
189
Rigid-body rotation in a constant-diameter pipe
Let us apply (5.2.1) to a rigid-body rotation plus a uniform translation, entering a
pipe with constant radius R = 1:
Uϕ = Ω0 r, Uz = Λ0 ,
σ = z ≥ 0, 0 ≤ r ≤ 1.
The slip velocities are just constants, Vϕ = Ω0 , Vσ = Vz = Λ0 , and the displacement
thickness, angular momentum thickness and azimuthal skin friction are given by
p
p
p
θ1z = µ1 z/Vz ; θA = µA z/Vz ; τϕ = µτ Vϕ Vz /z.
(5.2.3)
with
√ Z ∞
µ1 = 2
(1 − g) dη = 1.721,
0
Z
∞
√
µA = 2
(1 − g 2 ) dη = 2.385,
0
√
µτ = g 0 (0)/ 2 = 0.332.
As we explained in the introduction, we will use these measures to compensate within
the conservation laws for the boundary layer effects, and we will end up with a matching principle for quantities like angular momentum, mass flux and helicity.
The mass-balance
Substituting Uz = Λ(z) into (5.1.7) we obtain
o
∂ n
πΛ(z) − 2πRe−1/2 Λ(z)θ1z = 0.
∂z
Using (5.2.3) and solving for Λ(z) leads to
Λ(z) = Λ0
1
1 − 2Re−1/2 θ1z
= Λ0 + 2Re−1/2 µ1
p
zΛ0 + O(zRe−1 ).
The angular momentum balance
In the same spirit we will write the tangential velocity in the outer region as Uϕ (r, z) =
Ω(z)r, Ω(0) = Ω0 , and we want to find Ω(z) from the angular momentum balance. As
we showed in the previous section, the matching principle for the angular momentum
is given by (steady flow is assumed)
Z 1
d
−1/2
rUϕ Uz rdr − Re
Vϕ Vz θA = −Re−1/2 τϕ .
(5.2.4)
dz
0
Using our result on Uz and the specific thicknesses, we get a differential equation for
Ω(z):
dΩ
= T (z)Ω(z),
dz
Ω(0) = Ω0 ,
(5.2.5)
190
5 Navier-Stokes: viscosity as perturbation
where T (z) equals
T (z) = −
Re−1/2 µτ
p
dp
dΛ
/4
Λ(z)/z − Re−1/2 µA
Λ(z)z +
dz
dz .
p
Λ(z)/4 − Re−1/2 µA Λ(z)z
Introducing the scaled variable ζ,
ζ = Re−1/2
p
Λ(z)z,
the differential equation (5.2.5) turns into its well-balanced form where all terms are
of the same order of magnitude:
dΩ̃
2 (4µτ − 2µA + µ1 )
= −Ω̃(ζ)
,
(5.2.6)
dζ
f − 2ζ(2µA − µ1 )
with solution
Ω̃(ζ) = Ω0
f − 2ζ(2µA − µ1 )
f
exponent
,
(5.2.7)
with
exponent =
4µτ − 2µA + µ1
≈ −0.564.
2µA − µ1
Solutions for different values of f are given in figure 5.3.
3
2.5
2 Ω̃(ζ; 1)
6
1.5
0.02
0.04
0.06
-ζ
0.08
Figure 5.3: The solution Ω̃(ζ; Ω0 = 1) according to (5.2.7) for different values of f : from
top to bottom, f = 0.5, 1 and 2.
Remarks (i) One immediately notices that the denominator of T vanishes at
ζ0 =
f
≈ 0.165f ;
2(µA − µ1 )
5.2 Viscous swirling flows in a constant-diameter pipe
191
hence, (5.2.5) is valid only on ζ ∈ [0, ζ0 ).
(ii) The effect of µ1 is a growing Λ̃(ζ). This can easily be understood, since the same
mass flux is squeezed through an, effectively, smaller pipe. However, when µA and
µτ would be zero, Ω̃(ζ) would decrease in order to keep the same momentum flux.
µτ , representing the dissipation at the wall, accelerates this process, while µA has
the same effect on Ω̃ as µ1 on Λ̃: an increase of Ω̃. The exponent shows the delicate
balance between the three processes.
3
Asymptotic analysis for ζ ↓ 0 shows the following results, already suggested by figure
5.3:
Proposition 5.2 (i) The swirl of a rigid-body rotation, superposed on a uniform
translation, increases linearly as a function of ζ as the result of a combination of
viscous forces:
Ω(z) ∼ Ω0 (1 + c1 ζ),
ζ ↓ 0,
c1 =
4µA − 2µ1 − 8µτ
≈ 3.43/f
f
(ii) The ‘real’ angular momentum is given by
ANS =
1
Ω(1 − 4µ1 ζ/Λ)
2
Using the results on Λ(z) and Ω(z) we find that ANS dissipates:
ANS =
1
Ω (1 − c2 ζ),
2 0
c2 =
6µ1 + 8µτ − 4µA
= 3.43/f.
f
Remark The speeding up of the rigid-body rotation in the inviscid core looks similar
to the speeding up of an ice-skater while she (or he) decreases her (his) moment of
inertia. The analog for the mass-concentration in the flow-problem is the peaking of
the mass-flux (or, alternatively, the axial velocity). The dissipative term at the righthand side of (5.2.4) is too small to cancel this effect, just as the air- and ice-resistance
is too small in the skate-example.
3
Decay of angular momentum flux
For engineering purposes it is important to understand the decay process of swirl
intensity along the pipe. One way to study this is to measure or calculate the decay
of the angular momentum flux, Across , or swirl intensity (Kitoh (1991)). Kitoh argues
that when Across is extremely small, an exponential decay formula can be obtained,
downstream of the inlet region as
z − z0
Across = Across,0 exp 2a1
,
(5.2.8)
2R
with Across,0 and z0 the swirl intensity and axial position of a suitable reference point,
respectively.
192
5 Navier-Stokes: viscosity as perturbation
Let us evaluate the corrected angular momentum flux, Across,NS
Z 1
Across,NS =
rUϕ Uz rdr − Re−1/2 Vϕ Vz θA
0
p
√
1
=
Ω(z)Λ(z) − Re−1/2 µA zΩ(z) Λ(z)
4
1
=
Ω f − 2µτ Ω0 ζ + O(ζ 2 ).
4 0
The result in (5.2.9) can be viewed as the leading-order terms of
√ 1
µτ z
Across,NS = Ω0 Λ0 exp −8 √
.
4
f Re
(5.2.9)
(5.2.10)
Note first of all that since we are working in the inlet region, i.e., the flow has not√
yet
adjusted itself to the no-slip condition, the spatial behaviour is governed by Re−1/2 z.
When we start with a fully developed (Hagen-Poiseuille) flow and work upstream, the
axial scale is given by Re−1 z = ζ 2 (Crabtree et al. (1963), chapter VIII.15). Most
theoretical studies are centred around the decay of swirl when a rigid-body rotation is
superposed on a Hagen-Poiseuille flow (see e.g. Talbot (1954) and Mackrodt (1976)).
Hence, the two formula in (5.2.8) and (5.2.10) have different domains of validity.
The decay coefficients are plotted in figure 5.4 where we give on the one hand the
results of Kitoh (1991) and Baker (1967), valid in the region downstream of the inlet,
and on the other hand (5.2.10) which is valid near the inlet, where ζ is still small, but
large compared to ζ 2 .
5.2.2
Viscous effects on relative equilibria in a
constant-diameter pipe
In this subsection we turn to the relative equilibria and study the effect of the boundary layer on the parameters α, Λ and A, and indirectly on the cross-sectional densities.
We use the same family of relative equilibria in a constant-diameter pipe as in chapter
4 (only the co-rotating flow):
1
(5.2.11a)
AJ (rρ) + αr,
2 1
1
Uz =
(5.2.11b)
AJ (rρ) + Λ,
2 0
1
H =
λAJ0 (rρ) + αrUϕ + h(z),
(5.2.11c)
2
with Λ = 2α/ρ + λ and ρ = ρ1,1 . We abbreviate J0 (ρ) to J0 . The values of the
constraints in terms of (α, Λ, A) are given by (see also section 4.1):
Uϕ
=
Fax,RE = πf
= Λ;
1
ARE = a =
α − AJ0 /ρ;
2
1 2 2 1
BRE = b =
ρA J0 − αJ0 A + αΛ.
4
2
193
5.2 Viscous swirling flows in a constant-diameter pipe
5
4
−2a1 × 102
3
6
2
1
- Re × 10−4
0
1
5
2
10
15
Figure 5.4: Decay coefficient a1 of Across ; the dashed lines are Kitoh’s (1991) results
for various values of Across,0 :
, 0 < Across,0 < 0.04; 2, 0.04 < Across,0 < 0.09; 4,
0.09 < Across,0 < 0.45; , 0.45 < Across,0 < 0.80. The dotted line is Baker’s (1967) result
and the solid line is the result from (5.2.10).
r
b
We will use (5.2.11a–c) as the profile in the inviscid core, with A, α, Λ and h functions
of z, to be determined, and ρ fixed. When we define the variable S as twice the square
root of the axial slip velocity,
r
1
S=2
AJ + Λ.
2 0
we can immediately use the proper axial scale,
ζ = Re−1/2
p
√
1
zVz = SRe−1/2 z.
2
In view of the remark in subsection 5.1.6, following equation (5.1.10), stating that
the O(1)-effects in the helicity balance vanish in a constant-diameter pipe, or are at
most of O(Re−1/2 ), we will use the mass, axial momentum, angular momentum and
energy balance to determine the unknowns in (5.2.11a–c). Hence, it may be possible
to compare the results with chapter 4 for the expanding pipe.
The mass balance
Conservation of mass-flux leads to the first algebraic relation between Λ, ζ and f :
f = Λ − 2µ1 ζ,
(5.2.12)
194
5 Navier-Stokes: viscosity as perturbation
with Λ = Λ(ζ) and
1
π
f=
Z
1
uz 2πrdr.
0
The angular momentum balance
The balance for the angular momentum leads to a differential equation:
d
{I − µA ζα} + 2µτ α = 0,
dζ A
with
1
1
IA = Λα −
4
ρ
1 2
S −Λ
4
(5.2.13)
α
Λ−2
.
ρ
The axial momentum balance
The axial momentum balance, as well as the energy balance, involve a few new terms,
related to the Blasius profile. To be precise, we need
p
p
p
p
θ2z = θA = µ2 z/Vz ; θE = µE z/Vz ; τσ = µτ Vz Vz /z; DE = µD |V |2 Vz /z,
with
µ2 = µA = 2.385,
√ Z ∞
µE = 2
(1 − g 3 ) dη = 2.766,
Z0 ∞
1
µD = √
(g 0 )2 dη = 0.261.
2 0
The axial momentum balance involves amongst others the total head:
1
d
2
2
IF − µ2 ζS + µτ S 2 = 0,
dζ
4
4
(5.2.14)
with
IF =
1 2 1 2 1
Λ + α + h.
4
8
2
The energy balance
Finally, the energy balance closes the system.
d
1 2
α
IE −
S −Λ
Λ−2
+ α2 + h µ1 ζ −
dζ
4
ρ
1
1
1
α2 + S 4 (µE − µ1 )ζ + 2µD α2 + S 4 = 0,
2
16
16
(5.2.15)
5.2 Viscous swirling flows in a constant-diameter pipe
195
with
IE
=
1
1 αS 4
1
1 αΛS 2
1
1
ΛS 4 −
− Λ2 S 2 +
+ α2 Λ + Λ3 +
32
16 ρ
4
4 ρ
4
2
α2 Λ
1 α2 S 2 1
+
Λh
−
2
+
2 ρ2
2
ρ2
The dynamics of Λ, α, S and h
We solve equation (5.2.12) for Λ(ζ), yielding
Λ(ζ) = f + ζµ1 .
(5.2.16)
This is substituted into (5.2.13), (5.2.14) and (5.2.15). After solving for dh/dζ we
obtain a system of non-autonomous, first order differential equations:
dα/dζ
A11 A12
B1
A
+ B = 0, A =
, B=
.
(5.2.17)
dS/dζ
A21 A22
B2
The matrix entries are given in the appendix (formula (A.2.1–A.2.4)) and (A.2.5,A.2.6).
8
Like in chapter 4, we draw contour plots of the vector field Y = −A−1 B at ζ = 0:
figure 5.5 and 5.6. Although the figures may look simple, the general picture for
ζ 6= 0 is very complicated: this is related to the strong deformation of the contours as
ζ grows.
√ We are interested in the development of a supercritical jet-flow (A ≥ 0 or
S ≤ 2 f; α < f ρ/2). Moreover, as soon as the core flow has a different orientation as
the slip velocity we have an inflexion-point and the flow turns unstable. This means
that besides S ≥ 0 we also have α ≥ 0 or α ≤ Aρ/4. In the latter case the rigid-body
rotation is negative enough to make the tangential velocity clockwise on the whole
domain.
We make a few observations.
• First of all the line α = 0, i.e., the points where the azimuthal slip-velocity
vanishes. From figure 5.5 and 5.6 we observe that initially S increases and α
decreases. Hence the tangential profile develops an inflexion-point when α0 is
near 0, and therefore an instability in the form of boundary layer separation
since τϕ passes through zero.
The line S = 0 marks the points where the meridional slip velocity vanishes.
When we are away from the line of criticality we have an initial increase of S.
However, when the rotation α is strong enough, S is driven to zero and we have
boundary layer separation since τσ ‘passes’ through zero.
• Recalling the symmetry, we observe from figure 5.5 that when α0 is positive but
near zero, the flow tends to weaken its Beltrami-component, i.e., α < 0 when the
8 When we perform the calculations with the indefinite sign still present, the results are invariant under the transformation (±, α) → (−∓, −α). This is related to a reflection of the complete
tangential velocity, and not only the tangential slip velocity. In the following we take the plus-sign.
196
5 Navier-Stokes: viscosity as perturbation
2.005
2.004
2.003
←
→
←
-α
→
S
6
2.002
4
2.001
3.5
S
6
2
1.999
←
3
1.998
→
1.997
1.996
2.5
1.995
−4
−3
−2
−1
0
1
2
3
4
2
1.5
←
1
0.5
0
−4
→
→
-α
−3
−2
−1
0
1
2
3
4
Figure 5.5: The zero-contours of the numerator (solid line) and denominator (dashed line)
of Y1 for ζ = 0 and f = 1. The region near S = 2 is enlarged in the right-upper corner.
√
The arrows indicate the initial dynamics of α. Note that the line S = 2 f = 2 separates
wake-flow (S > 2) from jet-flow (S < 2).
flow is co-rotating and α > 0 when the flow is counter-rotating. However, when
no separation occurs, we have eventually a growing α, and thus a strengthening
of the Beltrami component (see figure 5.7). The growing of α causes S to
decrease and eventually (for the initial conditions in figure 5.7, we need ζ ≈ 0.15)
we approach S = 0, leading to separation.
√
• The line S = 2 f , marking (initially) the boundary between a wake and a jet
flow seems hard to cross, since there
√ is a thin repelling buffer between a region
where S grows and the line S = 2 f . Starting near this buffer (but still as a
jet, so S < 2) seems to indicate that we are near the previous example in the
sense that we observe a ‘finite-time’ blow-up of the solution. This analogy is
strongest near α = 0 (nonlinearity in Y is small) since then the initial flow is
nearly a parallel flow with a rigid-body rotation superposed.
• We should emphasize that the validity of (5.2.17), regardless of any singularity,
is only for small ζ, since with increasing ζ the boundary layer thickens.
p A good
measure for the thickness of the boundary layer is Re−1/2 θ1z = µ1 ζ/ Vz , and
hence most plots will be from ζ = 0 to ζ = 0.1.
Finally, we want to investigate the influence of the Beltrami component on the dis-
197
5.2 Viscous swirling flows in a constant-diameter pipe
2.01
2.008
↓
2.006
S
↑ 6
↓
↑
↑
2.004
4
↓
2.002
2
3.5
1.998
3
2.5
S
6
↑
↓
↑
1.996
1.994
↓
1.99
−4
2
↑
−3
−2
−1
0
1
2
3
-α
4
↓
1.5
1
0.5
↓
1.992
↓
↑
↑
↑
0
−4
-α
−3
−2
−1
0
1
2
3
4
Figure 5.6: The zero-contours of the numerator (solid line) and denominator (dashed line)
of Y2 for ζ = 0 and f = 1. The region near S = 2 is enlarged in the right-upper corner. The
√
arrows indicate the initial dynamics of S. Again, the line S = 2 f = 2 separates wake-flow
(S > 2) from jet-flow (S < 2), whereas the dashed line at α ≈ 1.9 separates supercritical
(left) from subcritical (right).
sipation of the energy flux (note that the energy flux is not definite). It is easily
observed that a large Beltrami component suppresses the nonlinearity in the NavierStokes (and Euler) equations, implying that the vortex stretching and tilting, which
are crucial to the development of inertial ranges of turbulence, are balanced by advection. It is thought that this result suggests the reduction of diffusing and dissipating
effects of turbulence (see e.g. Lilly (1986), and Levich (1987)). This hypothesis has
been the subject of many investigations, especially in the late seventies and eighties,
but there is still a lot of controversy (see e.g. Pelz et al. (1985), Rogers and Moin
(1987), and Moffatt and Tsinober (1992)).
Kraichnan and Panda (1988) proposed a quantity to measure the “Eulerization” of a
flow, given by the ratio
Z=
|(u × ω)s |2
,
|u|2 |ω|2
with
(u × ω)s := u × ω − ∇s
198
5 Navier-Stokes: viscosity as perturbation
2
1.8
S
6
1.6
1.4
•
1.2
•
•
•
0.8
0.2
0.4
0.6
0.8
-α
1
1.2
1.4
Figure 5.7: Solution curves for different initial values: S0 = 0.8, 1, 1.2 and 1.4 with α0 = 0.1
(solid lines); α0 = 0.3, 0.5, 0.7 and 0.9 with S0 = 1 (dashed lines). ζ runs from 0 to 0.1.
the “solenoidal projection” of the (Lamb) vector u × ω. However, when we interpret
“Eulerization” as “non-dissipative”, it seems more useful to compute the evolution of
a quantity that is exactly conserved in the inviscid limit, like the energy flux Ecross .
So let us compute9 the axial evolution of Ecross,NS ,
Z
Ecross,NS =
0
1
1
hUz rdr − Re−1/2 hr=1 Vz θ1 − Re−1/2 (Vϕ2 + Vz2 )Vz (θE − θ1 ).
2
In figure 5.8 we give
d
d
log Ecross,NS =
E
dζ
dζ cross,NS
Ecross,NS ,
as a function of ζ, together with A(ζ), for two different starting values of A (or S);
α0 is kept fixed. The result clearly shows that the dissipation is retarded.
5.3
Viscous swirling flows in an expanding pipe
In the last section of this chapter we are going to combine the viscous effects with a
slowly expanding pipe. This expansion has influence on the boundary layer equations
as well. Although it turns out that these changes do not enter the equations that
govern the evolution of Λ, S, α and h, some aspects are interesting in themselves (see
subsection 5.3.1).
9 The
total head h is computed from (5.2.14), once we have α and S.
199
5.3 Viscous swirling flows in an expanding pipe
4
0.02
0.04
0.06
-5
2
-ζ
0.02
0.04
0.06
0.08
0.08
-ζ
0.1
-10
0.1
-15
-2
-20
-4
-25
-30
-6
-35
(a)
(b)
Figure 5.8: (a) The (scaled) dissipation of Ecross (solid line) versus ζ, together with A(ζ)
(dashed line). Initial values are: A(0) = 4.36, Ecross (0) = 0.43, α0 = 0.3. The constant f is
taken unity. (b) The (scaled) dissipation of Ecross (solid line) versus ζ, together with A(ζ)
(dashed line). Initial values are: A(0) = 0.94, Ecross (0) = 0.05, α0 = 0.3.
Subsequently, we show that we can discriminate between three different cases:
Re−1 ε : chapter 4: inviscid flow in an expanding pipe;
ε Re−1 : section 5.2: viscous flow in a constant-diameter pipe;
ε ≈ Re−1 : viscous flow in an expanding pipe.
For the special case ε ≈ Re−1 we derive the equations that combine all effects in the
form of two kinds of parameters: (i) the boundary layer quantities, like µ1 , etc., and
(ii) the linear expansion rate of the pipe, R0 (Z). When one of the two is set equal to
zero, the corresponding problem is obtained.
We are basically interested in three effects. First, the effect of viscosity on the vortex
breakdown. Note that this effect is studied in the case when the expansion rate and the
Reynolds number are of the same order of magnitude. Hence, for a real experiment,
it is probably the case that one of the two ‘perturbations’ dominates, and that we
better study a simpler problem. Second, the effect of the expansion on the viscous
decay of the energy flux. And finally, the way the boundary layer separates.
5.3.1
Coherent structures in the boundary layer
In the appendix to this chapter, section A.1, we derive the boundary-layer equations
inside a pipe with variable radius. The result can be compared with chapter VIII.3
in Rosenhead (1963), where the axisymmetric boundary layer equations around a
body of revolution are stated; inside or outside, the equations are the same10 . These
10 The formal difference between the two—a minus sign in the normal coordinate and the normal
velocity—cancel each other exactly. This may be regarded as a discrete symmetry.
200
5 Navier-Stokes: viscosity as perturbation
equations are stated in physical variables but they reveal their properties much better
when we introduce a self-similar coordinate η, and a scaled meridional coordinate, ξ:
∂f
∂2f
∂f
∂F ∂f
2
+
F
+
γ
(ξ)
1
−
f
=
2ξ
f
−
+ γ1 (ξ) 1 − g 2 , (5.3.1a)
2
2
∂η
∂η
∂ξ
∂ξ ∂η
2
∂ g
∂g
∂g ∂F ∂g
+
F
=
2ξ
f
−
+ γ3 (ξ)gf,
(5.3.1b)
∂η 2
∂η
∂ξ
∂ξ ∂η
together with appropriate boundary conditions. Here f denotes the derivative of F
with respect to η, and γ1 , γ2 and γ3 are given by
γ1 (ξ)
γ2 (ξ)
γ3 (ξ)
Vϕ (σ)2 R0 (σ)
V (x)2 R0 (σ)
ξ=2
ξ,
3
3
R(σ) W (x)
R(σ)3 Vσ (σ)3
W 0 (x)
Vσ0 (σ)
= 2
ξ=2
ξ,
2
W (x)
R(σ)2 Vσ (σ)2
2R0 (σ)
2V 0 (x)
2ξ
d
=
+
ξ=
(RVϕ ).
W (x)R(σ)3
V (x)W (x)
Vσ Vϕ R3 dσ
= 2
(5.3.2a)
(5.3.2b)
(5.3.2c)
Recall that R0 (σ) = O(ε), being externally given. Just as in the case of a constantdiameter pipe, where we first assumed that the slip velocity was constant in order to
determine the ‘coherent structures’ in the boundary layer, we assume here that Vσ
and Vϕ result from the inviscid problem, and hence (see chapter 4) the order of Vσ0
and Vϕ0 is O(ε) as well.
Remarks (i) When ε = 0, i.e., when we are dealing with a pipe with constant
diameter, the system decouples into two similar nonlinear ODE’s, Blasius equation:
F 000 + F 00 F = 0,
F (0) = F 0 (0) = 0 and F 0 → 1 as η → ∞.
Its solution is given in figure 5.9.
10
8
10
η
6
8
6
6
4
4
2
-F
2
4
(a)
6
8
η
6
2
-f
0.2
0.4
0.6
0.8
1
(b)
Figure 5.9: (a) The stream function F for the Blasius profile. (b) The Blasius profile f .
(ii) Equation (5.3.1a) can be viewed as a perturbed Blasius equation, but the left-hand
5.3 Viscous swirling flows in an expanding pipe
201
side has also strong resemblance with the Falkner-Skan equation for a flow around a
wedge with half-angle = 12 πγ2 . However, do note that γ2 in (5.3.1a) is not constant,
but is varying like εξ.
(iii) Observe that γ3 ≡ 0. This can be shown in several ways. First of all, when
applying the boundary condition at infinity to (5.3.1b) (recall that the asymptotics
are uniform, and independent of ξ), we obtain 0 = γ3 . Secondly, note that the
formula for γ3 in (5.3.2c) is exactly the tangential component of the Euler equations,
that govern the slip velocity. In fact, since we are dealing with a steady, axisymmetric
flow in the inviscid core, we can use the functional relationship between the circulation
ruϕ and the stream function, implying that ruϕ is constant on a stream line, and in
particular on the boundary.
(iv) Equation (5.3.1a) contains even another special case, namely when γ1 = γ2 =
0, we have the boundary layer equations for a uniform flow over a flat plate with
constant suction (see e.g. Rosenhead (1963), chapter VI.31). This shows that (5.3.1a)
is combination of several equations and effects: flow around a wedge, suction and a
coupling with the tangential component.
3
In order to harmonise the different terms in (5.3.1a,b), let us introduce the variable
Ξ = εξ. The relation between ξ and σ is given by
Z σ
Vσ (σ̃)R(σ̃)2 dσ̃,
ξ=
0
and hence, as long as dξ/dσ = Vσ (σ)R(σ)2 6= 0, i.e., as long as there is no meridional
stagnation point at the boundary, we can write any function of σ as a new function,
depending on ξ. With this observation, we write
γ1 = Ξγ˜1 (Ξ),
γ2 = Ξγ˜2 (Ξ),
and (5.3.1a,b) as
!
∂ 2 f˜ ∂ f˜
∂ f˜ ∂ F̃ ∂ f˜
2
˜
˜
+
F̃ + γ2 (Ξ) 1 − f = 2Ξ f
−
+ γ1 (Ξ) 1 − g 2 ,(5.3.3a)
2
∂η
∂η
∂Ξ
∂Ξ ∂η
!
∂g
∂g
∂
F̃
∂g
∂2g
+ F̃
= 2Ξ f˜
−
.
(5.3.3b)
∂η 2
∂η
∂Ξ
∂Ξ ∂η
In the following we omit the tildes.
Moreover, the previous remarks suggest that we can look for solutions of (5.3.3a,b)
of the form
F (η, Ξ)
f (η, Ξ)
g(η, Ξ)
= F0 (η) + Ξm1 (Ξ)F1 (η) + o(Ξ), m1 (Ξ) = O(1),
= f0 (η) + Ξm1 (Ξ)f1 (η) + o(Ξ),
= g0 (η) + Ξn1 (Ξ)g1 (η) + o(Ξ), n1 (Ξ) = O(1),
(5.3.4a)
(5.3.4b)
(5.3.4c)
202
5 Navier-Stokes: viscosity as perturbation
with f0 = F00 and f1 = F10 . Substitution of (5.3.4a–c) into (5.3.3a,b) and equating
coefficients of equal powers of Ξ leaves the following set of equations:
Ξ0 : • f000 + F0 f00 = 0 :
• g000 + F0 g00 = 0 :
Blasius equation for F0 ⇒ F0 = L, f0 = ` = L0
g0 = f0 = `;
Ξ1 : • (γ1 − γ2 )(f02 − 1) + m1 (f100 + F0 f10 − 2f0 f1 + 3f00 F1 ) = 0,
• n1 (g100 + F0 g10 − 2f0 g1 ) + 3m1 F1 f00 = 0.
(5.3.5)
The system does not decouple for higher order of Ξ. Equation (5.3.5) has a solution11
m1 = n1 = γ1 − γ2 with resulting equations for F1 , f1 = F10 and g1 :
f100 + F0 f10 − 2f0 f1 + 3f00 F1
g100 + F0 g10 − 2f0 g1 + 3f00 F1
with boundary conditions
= 1 − f02 ,
= 0,
F1 = f1 = g1 = 0 on η = 0;
f1 → 0, g1 → 0
as η → ∞
We plotted in figure 5.10 the graphs of F1 , f1 and g1 . Note that f1 and g1 are both
10
10
η
8
6
η
8
6
6
6
4
2
- F1
-1
-0.8
4
f1
-0.6
(a)
-0.4
-0.2
g1
-0.5
-0.4
-0.3
-0.2
2
-0.1
(b)
Figure 5.10: (a) The stream function F1 for the first-order correction f1 on the meridional boundary layer velocity. (b) The meridional (solid line) and tangential (dashed line)
corrections to the Blasius profile.
negative. The sign of m1 = n1 determines whether the flow heads for boundary-layer
separation (m1 > 0) or not (m1 < 0). m1 is a combination of two effects, as can be
seen from the relation m1 = γ1 − γ2 and (5.3.2a,b):
• γ1 is the effect of expanding or constricting the pipe; when we deal with an
expansion, γ1 > 0;
11 In fact, m can be any multiple of γ − γ , as well as n can be any multiple of m . This degree
1
1
2
1
1
of freedom cancels since a scaling of f1 and g1 annihilates the arbitrary constant.
203
5.3 Viscous swirling flows in an expanding pipe
• γ2 is the effect of acceleration or deceleration of the meridional slip velocity;
note that, due to the presence of swirl this effect is not clear a priori, since
flow-reversal at the axis can induce an acceleration near the boundary. This is
in contrast with a river entering a widening of the river bed; then the flow is
slowed down uniformly.
5.3.2
The evolution of relative equilibria for the viscous flow
in an expanding pipe
In this subsection we use the modulated columnar relative equilibria, as given in
(5.2.11a–c), section 5.2.
Balancing viscosity and inhomogeneity
Let us investigate how the effects of the expansion work through in the balance laws.
As an example we take the mass-balance:
(Z
)
Z ∞
R(Z)
−1/2
∂z
Uz rdr − ∂σ Re
Vσ
(1 − vσ /Vσ ) R̃(σ)dN = 0,
(5.3.6)
0
0
with Z = εz.
Let us first have a look at the second term in (5.3.6). Note that its order of magnitude
is at least O(Re−1/2 ). The evaluation of the integral yields (we use subsection A.1.2,
A.1.3 and table 5.14)
Z ∞
p √ Z ∞
Vσ
(1 − vσ /Vσ ) R̃(σ)dN =
ξ 2
(1 − f ) dη
0
0
Z
∞
p √
ξ 2
(1 − f0 ) dη −
=
0
Z ∞
p √
p
ξ 2 Ξ(γ1 − γ2 )
f1 dη + o(Ξ ξ). (5.3.7)
0
Whatever the scales on which the velocity profile changes, the second and higher order
terms in (5.3.7) are of minor importance. Therefore, the only viscous term that can
be used to balance the terms related to the slowly varying geometry is given by
p √ Z ∞
−1/2
∂σ Re
ξ 2
(1 − f0 ) dη .
0
Since σ = z + O(ε) we can safely replace ∂σ by ∂z , and hence
(Z
R(Z)
−1/2
Uz rdr − Re
∂z
0
∂z
p √ Z
ξ 2
∞
)
(1 − f0 )dη
=
0
p √
1
Λ(z)R(Z)2 − Re−1/2 ξ 2
2
Z
0
∞
(1 − f0 )dη
= 0. (5.3.8)
204
5 Navier-Stokes: viscosity as perturbation
The dynamics of Λ (and of α, h and S as well) are governed by two effects, each
having its own
√ spatial scale. The viscous effects are important on scales given by
ζ = Re−1/2 ξ, ζ = O(1), whereas the expansion of the pipe takes place in terms of
Z = εz, Z = O(1). Let us compare dR/dz and dζ/dz:
dR
= εR0 (Z), R0 (Z) = O(1),
dz
dζ
dξ dζ
dξ Re−1
dξ
=
=
,
= O(1),
dz
dz dξ
dz 2ζ
dz
and hence there is only a balance when ε = Re−1 .
Remarks (i) When Re−1 ε we can neglect the second term in (5.3.8), and Λ0 will
be O(ε); this leads eventually to the QHA in chapter 4.
(ii) When ε Re−1 , we can neglect differentiations of R and R̃; this leads eventually
to the viscous decay of a relative equilibrium in a constant-diameter pipe.
3
The balances
Now we have determined the distinguished limit, we can rewrite (5.3.8):
axial flux or mass balance:
n
o
∂ζ Λ(ζ)R̂(ζ)2 − 2ζµ1 = 0.
(5.3.9)
In the same way we can rewrite the balances for the angular momentum, the axial
momentum and the energy balance:
angular momentum balance:
n
o
0
∂ζ IA
− αR̂2 ζµA + 2αR̂2 µτ = 0;
axial momentum balance:
1 2
1
0
∂ζ IF − S ζµ2 + S 2 µτ = 0;
4
2
energy balance:
!
!
(
1 2
αR̂
0
2 2
S −Λ
Λ−2
+ α R̂ + h ζµ1 −
∂ζ IE −
4
ρ
)
1
1 4
1
2 2
α R̂ + S (µE − µ1 )ζ + 2 α2 R̂2 + S 4 µD = 0,
2
16
16
(5.3.10)
(5.3.11)
(5.3.12)
with α = α(ζ), h = h(ζ) and Λ = Λ(ζ). R̂ is defined according to R̂(ζ) = R(Z) and
S = S(ζ) is (up to a constant) the square root of the meridional slip velocity:
p
1 2
S = Vσ = 1 + ε2 R0 (Z)2 Uz |r=R(Z) .
4
205
5.3 Viscous swirling flows in an expanding pipe
0
0
The ‘unperturbed’ fluxes IA
, IF0 and IE
are similar to IA , IF and IE from section 5.2,
but corrected for the varying radius.
Before we derive the equations for α, etc. we investigate the relation between R0 (Z)
and R̂0 (ζ). By definition we have
R0 (Z) · δZ = R̂0 (ζ) · δζ,
but then
Re−1/2
Re−1
√ · δξ = R̂0 (ζ)
V (σ)R̃(σ)2 · δσ
2ζ σ
2 ξ
Re−1
Re−1
Vσ R̃2 cos α · δz = R̂0 (ζ)
V R̃2 cos α · δZ
= R̂0 (ζ)
2ζ
2εζ σ
R̂0 (ζ) · δζ = R̂0 (ζ)
= R̂0 (ζ)
S 2 R̂2
· δZ + h.o.t.,
8ζ
and hence
R̂0 (ζ) =
8ζ
S 2 R̂2
R0 (Z).
(5.3.13)
A prelude: rigid-body rotation plus uniform translation
As a prelude and transparent example, we apply (5.3.9), (5.3.10) and (5.3.13) to the
rigid-body rotation and the uniform translation as treated in section 5.2. We have
three unknowns, namely R̂(ζ), Λ(ζ) and Ω(ζ). It is straightforward to show that Ω(ζ)
is governed by
#
"
dΩ
2 (4µτ − 2µA + µ1 )
ζR0 (Z)
.
(5.3.14)
= −Ω(ζ)
+ 16
dζ
f − 2ζ(2µA − µ1 )
R̂(ζ)(f + 2µ1 ζ)
Without solving (5.3.14), we observe that it is clearly a generalisation of (5.2.6): when
R0 (Z) = 0 we retrieve the constant-diameter pipe problem. Moreover, the effect of
the expansion is such that it tends to decrease the tangential slip velocity (as well as
the meridional slip velocity). However, it has to compete with the dissipative effects,
in particular with µA in case of Ω(ζ) and with µ1 in case of Λ(ζ).
The dynamics of Λ, α, S and h
Now let us consider the relative equilibrium as we studied in subsection 5.3.2.
First (5.3.9) is solved for Λ(ζ), yielding
Λ(ζ) =
f
ζµ1
+2
.
2
R̂(ζ)
R̂(ζ)2
Note that when we formally set µ1 = 0 we retrieve Λ(Z) from chapter 4 where we
studied the inviscid flow in an expanding pipe. On the other hand, when R̂(ζ) ≡
206
5 Navier-Stokes: viscosity as perturbation
1 we obtain (5.2.16)—the solution for a viscous flow in a constant-diameter pipe.
Substituting Λ(ζ) in (5.3.10–5.3.12) leaves a system of non-autonomous, first-order
differential equations12 :






dα/dζ
C11 C12 0
D1
(5.3.15)
C  dS/dζ  + D = 0, C =  C21 C22 0  , D =  D2  .
C31 C32 1
D3
dh/dζ
The matrix entries are again given in the appendix. It is a combination of, on the
one hand, the entries in (A.2.1–A.2.4) and (A.2.5,A.2.6) and on the other hand the
entries in proposition 4.8, chapter 4: the latter is rather difficult to see at first sight
since we work with the slip velocity S instead of A.
We have to supply (5.3.15) with an equation for R̂(ζ), and obviously we take (5.3.13)
with R0 (Z) fixed and constant, say unity (linearly expanding pipe). At ζ = 0 the
contour plots of the first two components of T = −C −1 D are exactly the same as
for the non-expanding pipe: see figure 5.5 and 5.6. The reason for this is that R̂0 is
linear in ζ, and hence vanishes at ζ = 0. However, the evolution of the contour lines
as ζ grows does depend on the value of R0 (Z), and more specifically whether its zero
or not.
We are interested in how the results from chapter 4—vortex breakdown—are influenced by the viscous effects at the wall, and the other way around: how the change
in geometry modifies the results from section 5.2.
• Most of the initial conditions lead to separation. In order to compare results
with results of chapter 4, we have
√ to start rather close to a singularity of the
vector field, Y , namely S = 2 f . The behaviour of the contour lines is very
difficult, and it seems very hard to get any analytical result from it. In figure
5.11 we started at S0 = 1.9 and α0 = 0.5. For these initial values we plotted
A(ζ) and A0 (ζ) versus R̂(ζ); R0 (Z) was taken equal to 10. The blow-up of α0
and A0 are not related to a transition from supercritical to subcritical: λ(ζ) =
Λ(ζ) − 2α(ζ)R̂(ζ)/ρ > 0 at all values of ζ.
• Let us compute the axial decay of the energy flux for the expanding pipe problem. The flux is given in (5.3.12), and its evolution is depicted in figure 5.12.
Also in this case the presence of a relative large Beltrami component slows down
the dissipation process.
• Finally, we investigate the behaviour when the boundary layer starts to separate,
i.e., when S ↓ 0. It is known (see e.g. Goldstein (1948)) that near separation
the meridional skin friction τσ should behave like
τσ (ζ) ∼ κ(ζ∗ − ζ)1/2 ,
ζ ≤ ζ∗ ,
where ζ∗ denotes the point of separation. In figure 5.13 we choose p
S0 = 1,
α0 = 1.8, f = 1 and R0 (Z) = 10. We plotted τσ = 12 S(ζ)2 µτ versus ζ∗ − ζ
12 Note
that we cannot eliminate dh/dζ.
207
5.3 Viscous swirling flows in an expanding pipe
100
0.6
50
0.4
1.002 1.004 1.006 1.008 1.01
0.2
-50
1.002 1.004 1.006 1.008 1.01
- R̂(ζ)
-0.2
(a)
- R̂(ζ)
-100
-150
(b)
Figure 5.11: (a) The rigid-body rotation α (dashed line) and the amplitude of the Beltrami
component A (solid line) versus R̂(ζ). Note that the jet flow turns into a wake flow. (b) The
derivatives of α (dashed line) and A (solid line) versus R̂(ζ).
with ζ∗ = 0.007401106. The result is a straight line, as it should be. This
behaviour is typical near S = 0 as can be shown by making a Taylor series of
the vector field controlling S 0 near S = 0:
constant1 (ζ, α, h, R)
dS
=
+ h.o.t.,
dζ
S3
with solution
S(ζ)4 = 4ζ · constant1 (ζ, α, h, R) + constant2 (ζ, α, h, R).
5.3.3
Boundary layer theory beyond separation
It is difficult to draw general conclusions from the previous calculations. One remark
that can be made is that when the initial azimuthal slip velocity α(0) is small enough
the flow separates (finite-time blow-up of S 0 at S = 0). Moreover, when α(0) is
large enough, the system of differential equations in (5.3.15) cannot be continued
beyond a finite value of ζ as well, but at that point both S and α are nonzero, i.e., no
separation. This may be compared with the situation in section 5.2, where we studied
the development of a rigid-body rotation plus a uniform translation in a pipe with
constant diameter. However, the situation with the relative equilibrium is much more
complicated.
The problem of boundary layer separation is an interesting physical and mathematical problem. The assumption that the viscous effects are only important in a thin
layer near the wall is not valid anymore, and we have to find another ‘asymptotic’
description of the flow. Moreover, it is not even clear whether the information that is
obtained up to the point of separation is reliable. On the contrary, several examples
(see e.g. Stewartson (1970)) show that we have to consider the whole problem again,
but now using different techniques.
208
5 Navier-Stokes: viscosity as perturbation
4
-5
3
1.01 1.02 1.03 1.04 1.05
-ζ
-10
2
-15
-20
1
-ζ
1.0251.051.075 1.1 1.1251.151.175
-25
-30
(a)
(b)
Figure 5.12: (a) The (scaled) dissipation of Ecross (solid line) versus ζ, together with A(ζ)
(dashed line). Initial values are: A(0) = 4.17, Ecross (0) = 0.40, α0 = 0.5. The constant f
is taken unity, while R0 (Z) = 10. (b) The (scaled) dissipation of Ecross (solid line) versus
ζ, together with A(ζ) (dashed line). Initial values are: A(0) = 0.94, Ecross (0) = 0.094,
α0 = 0.5.
The main characteristic of these techniques is that the pressure ‘at the wall’ is no
longer prescribed, but rather part of the problem. The new set of equations are called
the interactive boundary-layer equations (see e.g. Smith (1982)). The combination
of boundary layer separation and vortex-breakdown has not received so much attention until now. However, as we already noted in the previous chapters, the vortexbreakdown problem bears strong similarity with the hydraulic jump in a channel.
This latter problem has been studied by Bowles (1995). When he considers standing
hydraulic jumps in liquid-layer flows on slopes, one of his results is that the effects
due to the slope and the viscosity balance when the slope is O(Re−1 ), in agreement
with our results for a pipe. It is likely that any progress on the swirling flow problem
should have close resemblance with the results in Bowles’ paper.
A.1
A.1.1
The boundary layer equations
Derivation of the boundary layer equations
Starting from the steady, axisymmetric13 Navier-Stokes equations,
u · ∇u
div u
u
u
=
=
=
=
−∇p − Re−1 curl ω, ω = curl u,
0, in D,
0 on ∂D,
u0 (r) at z = 0,
in D,
(A.1.1a)
(A.1.1b)
(A.1.1c)
(A.1.1d)
13 The steadiness and axisymmetry are for presentation purposes and because of the type of application in chapter 5.
209
A.1 The boundary layer equations
0.00175
0.0015
0.00125
1
2
0.0012 µτ S
0.000756
0.0005
-
0.00025
0.0002
0.0004
p
ζ∗ − ζ
0.0006
0.0008
0.001
Figure 5.13: The meridional skin friction, τσ near separation, as a function of the distance
to the separation point.
and with the theory exposed in section 1.4, we will discuss the classical ideas on
solutions of (A.1.1a–d) when Re 1. They form the attached flow strategy in which
it is assumed that viscous effects are mainly confined to a thin boundary layer along
the body surface. This strategy may be put forward in terms of the following two-part
scheme (Smith (1982)).
• First, for most of the flow field (the so-called outer region), R−r is of order unity
and we might expect ur , uϕ , uz and p to be of order unity as well. Therefore,
the following asymptotic expansion in the outer region is suggested:
u = U 0 + Re−1/2 U 1 + . . . ,
−1/2
p = P0 + Re
P1 + . . . .
(A.1.2a)
(A.1.2b)
The presence of the O(Re−1/2 )-terms in (A.1.2a,b) anticipates the result from
the next step below, suggesting an O(Re−1/2 ) correction to the O(1) terms
(U 0 , P0 ) is needed. Substituting (A.1.2a,b) into (A.1.1a,b), equating coefficients
of equal powers of Re−1 , leaves the inviscid Euler equations for (U 0 , P0 ) with
Re−1 ≡ 0 effectively in (A.1.1a).
Concerning the boundary conditions, we have still our inlet condition at z = 0
but the condition at ∂D, being the no-slip condition u = 0, cannot be imposed:
we can only require the normal velocity component to vanish as we approach
∂D (note that the order of (A.1.1a) drops by one if Re−1 ≡ 0; see section 1.4 for
some more general background information and e.g. Kevorkian and Cole (1996)
for more argumentation).
• Close to the surface ∂D, a region of non-uniformity is introduced. In this region
the axial and tangential velocity are decelerated from O(1) (just ‘outside’ this
region) to 0 at the surface. This deceleration is due to viscous forces, and
210
5 Navier-Stokes: viscosity as perturbation
any rescaling of the physical coordinates should lead to a balance between the
viscous term −Re−1 curl ω and the convective term u·∇u in (A.1.1a). Inspection
suggests that the normal coordinate n (defined in section 5.1, together with the
meridional coordinate σ) is of order Re−1/2 , therefore, n = Re−1/2 N say, and
that
vn
(vσ , vϕ )
= Re−1/2 v0,N + Re−1 v1,N + . . . ,
−1/2
= (v0,σ , v0,ϕ ) + Re
−1/2
p = p0 + Re
(v1,σ , v1,ϕ ) + . . . ,
p1 + . . . ,
(A.1.3a)
(A.1.3b)
(A.1.3c)
in the boundary layer where 0 ≤ N < ∞, σ > 0 (Smith (1982)). Substitution of
(A.1.3a–c) into (A.1.1a,b) now leaves the nonlinear boundary layer equations
2
∂v0,σ
∂v0,σ
v0,ϕ
∂ 2 v0,σ
dR
dp
+ v0,N
−
= −
+
, (A.1.4a)
∂σ
∂N
R(σ) dσ
dσ
∂N 2
∂v0,ϕ
∂v0,ϕ
v0,σ v0,ϕ dR
∂ 2 v0,ϕ
v0,σ
+ v0,N
+
=
,
(A.1.4b)
∂σ
∂N
R(σ) dσ
∂N 2
∂v0,N
1 ∂
+
R(σ)v0,σ
= 0.
(A.1.4c)
∂N
R(σ) ∂σ
v0,σ
Remark Note that p0 does not vary across the boundary layer: ∂p0 /∂N = 0.
3
We omit the subscript 0 in all what follows. The boundary conditions on
(A.1.4a–c) are
vσ = vN = vϕ = 0 on N = 0;
vσ → Vσ , vϕ → Vϕ as N → ∞
in order to match the boundary layer solution as N → ∞ with the inviscid
solution (U 0 , P0 ) as n ↓ 0.
(Much) more information and details can be found in standard books as e.g. Rosenhead (1963), Van Dyke (1975), Schlichting (1979) and Kevorkian and Cole (1996).
A.1.2
Transformation of the boundary layer equations
Starting with the classical boundary layer equations (A.1.4a–c) in the physical variables (apart from the scaled normal coordinate N ), we make subsequent transformations such as the introduction of similarity type of variables. The (computational)
advantages of this transformations are (Fanneloep (1979)):
“(a) the growth of the domain of calculation associated with the increasing
boundary layer is largely eliminated; and (b) the boundary layer profiles
are smoother and vary more slowly in the transformed plane allowing
larger step sizes to be used.”
211
A.1 The boundary layer equations
The first transformation shows the relationship between two-dimensional and axially
symmetric boundary layers without swirl. It is due to Mangler (1948) and it mostly
affects the incompressibility conditions which is ‘straightened’.
Introduce new coordinates,
Z σ
x=
R(σ̃)2 dσ̃, y = R(σ)N,
0
and new velocities,
1
R0
u(x, y) =
v (σ, N ) +
N vσ (σ, N ) , v(x, y) = vϕ (σ, N ), w(x, y) = vσ (σ, N ).
R N
R
Apart from the swirl v, the equations transform into the two-dimensional boundarylayer equations,
∂w
∂w
v2
+u
− 3 R0
∂x
∂y
R
∂v
∂v vw 0
w
+u
+ 3R
∂x
∂y
R
∂w ∂u
+
∂x
∂y
w
dW
∂2w
V2
,
− 3 R0 +
dx
R
∂y 2
∂2v
,
∂y 2
= W
=
= 0,
with W (x) = Vσ (N = 0, σ), V (x) = Vϕ (N = 0, σ) and R = R(σ), R0 = R0 (σ).
The second transformation is a similarity transformation:
Z x
Z σ
yW (x)
ξ=
.
W (x̃) dx̃ =
Vσ (σ̃)R(σ̃)2 dσ̃, η = √
2ξ
0
0
√
Moreover, we introduce a stream function, ψ(x, y) = 2ξF (ξ, η) such that
∂ψ
∂F
∂ψ
∂F
yW 0 (x) ηW ∂F
√
w=
= W (x)
, u=−
= W (x)
+
−
∂y
∂η
∂x
∂ξ
2ξ
∂η
2ξ
Finally, we scale the azimuthal velocity, g = v/V (x). Introducing all the scalings in
the PDE’s, it can be shown that F (ξ, η) and g(ξ, η) satisfy the following set of partial
differential equations:
∂ 3F
∂2F
+
F + γ2 (ξ) 1 −
3
∂η
∂η 2
2 !
∂F
=
∂η
∂F ∂ 2 F
∂F ∂ 2 F
+ γ1 (ξ) 1 − g 2 ,
2ξ
−
2
∂η ∂ξ∂η
∂ξ ∂η
∂g
∂2g
+F
= 2ξ
∂η 2
∂η
∂F ∂g ∂F ∂g
−
∂η ∂ξ
∂ξ ∂η
+ γ3 (ξ)g
∂F
∂η
(A.1.5a)
(A.1.5b)
212
5 Navier-Stokes: viscosity as perturbation
with
Vϕ (σ)2 R0 (σ)
V (x)2 R0 (σ)
ξ
=
2
ξ,
R(σ)3 W (x)3
R(σ)3 Vσ (σ)3
W 0 (x)
Vσ0 (σ)
ξ=2
ξ,
γ2 (ξ) = 2
2
W (x)
R(σ)2 Vσ (σ)2
2R0 (σ)
2V 0 (x)
d
2ξ
γ3 (ξ) =
+
ξ=
(RVϕ ).
W (x)R(σ)3
V (x)W (x)
Vσ Vϕ R3 dσ
γ1 (ξ) = 2
Since finite-difference equations of second order can be solved (by matrix inversion
routines) much more efficiently than third (or higher) order equations, it is of interest
to reduce the order of (A.1.5a). To that end we introduce the auxiliary variable
f = ∂F/∂η, and (A.1.5a,b) are rewritten as
∂2f
∂f
∂f
∂F ∂f
2
=
2ξ
f
,
+
F
+
γ
(ξ)
1
−
f
−
+ γ1 (ξ) 1 − g 2 (A.1.6a)
2
2
∂η
∂η
∂ξ
∂ξ ∂η
∂2g
∂g
∂g ∂F ∂g
+
F
(A.1.6b)
=
2ξ
f
−
+ γ3 (ξ)gf.
∂η 2
∂η
∂ξ
∂ξ ∂η
Equations (A.1.6a,b) can be compared with equation (9.64) in Schlichting (1979):
note however that because of the swirl we get a coupled system of PDE’s whereas
Schlichting treats a pure two-dimensional flow.
The boundary conditions for (A.1.6a,b) are

on η = 0;
 F =f =g=0
f → 1, g → 1
as η → ∞;

F = L(η), f = g = l(η) at ξ = 0,
with L(η) the solution of the Blasius equation,
L000 + LL00 = 0;
L(0) = L0 (0) = 0, lim L0 (η) = 1,
η→∞
and l(η) its derivative.
A.1.3
Transformation of the boundary layer measures
Since we will get all information about the velocity profiles in the viscous layer in
terms of the transformed variables F , f and g as a function of ξ and η, it seems wise
to transform all different boundary layer measures that we introduced in the first
section as well.
√ A few simple formula manipulations produce the list in table 5.14 (we
abbreviate 2ξ/W with θ).
213
A.2 Viscous flow in a constant-diameter pipe
densities, fluxes
Z
∞
θ1σ = θ
(1 − f ) dη
0
Z
∞
θ2σ = θ
dissipation
τσ =
RW ∂f θ ∂η η=0
(1 − f 2 ) dη
0
Z
∞
θ1ϕ = θ
(1 − g) dη
0
Z
∞
θA = θ
RV ∂g τϕ =
θ ∂η η=0
(1 − f g) dη
0
W 2f 2 + V 2 g2
θ2 = θ
1−
dη
V 2 + W2
0
Z ∞
W 2f 2 + V 2 g2
θE = θ
1−f
dη
V 2 + W2
0
Z
∞
Z
∞
Θb = R
0
0
(gf − f g ) dη
0
Z
∞
ΘB = R
0
g0
R2
DE =
θ
Z
∞
0
R3 W
DB =
θ2
Z
W 2 (f 0 )2 + V 2 (g 0 )2
dη
V 2 + W2
∞
(f 0 g 00 − g 0 f 00 ) dη
0
W 2f 2 + V 2 g2
dη
V 2 + W2
Table 5.14: The transformed integral quantities, arranged according to the equation in
which they appear (vertically) and their physical meaning (horizontally); f 0 denotes ∂f /∂η,
etc.
A.2
A system of coupled non-autonomous ODE’s;
viscous flow in a constant-diameter pipe
In this section we will give the matrix entries in (5.2.17).
The matrix A has the following entries.
A11
=
A12
=
1
1 S2
f
1 1
−2 2 +
µ1 ρ2 − 8µ1 − 2µA ρ2 ζ,
f+
2
2
4
2ρ
ρ
2ρ
αS
1 fS
µ1 S
−
−
ζ,
ρ2
2 ρ
ρ
(A.2.1)
(A.2.2)
214
5 Navier-Stokes: viscosity as perturbation
A21
A22
=
=
1
1 S4
f α 1 f S 2 αS 2
fα −
−4 2 +
+ 2 +
4
16 ρ
ρ
4 ρ
ρ
1
4µ2
− 2 2µ1 f ρ − µ1 S 2 ρ + 8µ1 α + µE αρ2 ζ − 1 ζ 2 ,
ρ
ρ
3
2
1 f αS
α S
1 3 1 2
1 αS
fS − f S −
+
+ 2 +
8
2
4 ρ
2 ρ
ρ
1
5
1
µ α 3
+ 2 1 + µ1 S 3 − µE S 3 − µ1 f S + µ2 f S ζ +
ρ
8
8
2
2
− 3µ21 Sζ 2 .
(A.2.3)
(A.2.4)
The vector B has the following entries:
A.3
B1
=
B2
=
1
1 µ1 S 2
µ2
µ f
µ α
µ1 α −
+ 4 1 − 4 12 − µA α + 2µτ α + 8 1 ζ,
(A.2.5)
2
2 ρ
ρ
ρ
ρ
µ1 αS 2
1
1
3
1
+ 2µD α2 − µE α2 + µD S 4 + µ1 S 4 − µE S 4 +
ρ
2
8
32
32
µ1 α2
µ1 f α 5
1
1
− µ1 f S 2 + µ2 f S 2 − µτ f S 2 + 3µ1 f 2 +
−4 2 −2
ρ
ρ
4
4
2
2
µ
(A.2.6)
+ 1 18f ρ − 3ρS 2 − 8α ζ + 24µ31 ζ 2 .
ρ
A system of coupled non-autonomous ODE’s;
viscous flow in an expanding pipe
In this section we will give the details of the matrix entries in (5.3.15).
The matrix C has the following entries.
C11
=
1 2 1 S 2 R̂4
f R̂2 1 R̂2
−
2
+
f R̂ +
µ1 ρ2 − 8µ1 − 2µA ρ2 ζ, (A.3.1)
2
2
2
4
2 ρ
ρ
2ρ
C12
=
αS R̂4 1 f S R̂ µ1 S R̂
−
−
ζ,
ρ2
2 ρ
ρ
C21
=
1
1 S 4 R̂3
f αR̂2 1 f S 2 R̂ αS 2 R̂4
f αR̂2 −
−4 2 +
+
+
4
16 ρ
ρ
4 ρ
ρ2
ζ
− 2µ1 f ρ − µ1 S 2 ρR̂2 + 8µ1 αR̂3 + µE αρ2 R̂3
+
ρ2 R̂
µ2
− 4 1 ζ2,
ρR̂
(A.3.2)
(A.3.3)
215
A.3 Viscous flow in an expanding pipe
C22
=
1 3 1 f 2S
1 αS 3 R̂3 1 f αS R̂ α2 S R̂4
fS −
+
+
−
+
8
2 R̂2
4 ρ
2 ρ
ρ2
1
5 µ1 f S
1 µ2 f S
µ αS R̂ 3
+ µ1 S 3 − µE S 3 −
+
+ 2 1
ρ
8
8
2 R̂2
2 R̂2
−3
C31
=
µ21 S
R̂2
1
αR̂2
2
!
ζ+
ζ2,
C32 = −
(A.3.4)
µ2 S
R̂2
ζ.
(A.3.5)
The vector D has the following entries:
D1
=
D2
=
1
1 µ1 S 2 R̂
µ αR̂2
µ2
µ f
µ1 αR̂2 −
+ 4 1 − 4 1 2 − µA αR̂2 + 2µτ αR̂2 + 8 1 ζ +
2
2 ρ
ρ
R̂ρ
R̂ρ
!
2
2
µ
1 µ1 S
µ αR̂
µ f
+ −4 1 ζ 2 + µ1 αR̂ −
− 2µA αR̂ − 8 1 2 − 4 1
ζ+
2
2
ρ
ρ
ρR̂
ρR̂2
!
1 f S2 1
f2
αS 2 R̂3
f αR̂
dR̂
+2
−
+ f αR̂ − 4 2 −
,
(A.3.6)
2
2
ρ
4 ρ
2
ρ
R̂ ρ dζ
µ1 αS 2 R̂
1
1
3
1
+ 2µD α2 R̂2 − µE α2 R̂2 + µD S 4 + µ1 S 4 − µE S 4 +
ρ
2
8
32
32
µ f α 5 µ1 f S 2 1 µ2 f S 2 1 µτ f S 2
µ f2
µ1 α2 R̂2
−2 1
−
+
−
+3 1 +
2
ρ
4 R̂2
4 R̂2
2 R̂2
ρR̂
R̂4
2 3
µ
µ
+ 1 18f ρ − 3ρS 2 R̂2 − 8αR̂3 ζ + 24 1 ζ 2 +
4
R̂4
ρR̂
−4
+ −
!
αS 2 µ1
R̂α2 µ1
f αµ1 5 f S 2 µ1
f 2 µ1
2
−
+
− R̂α µE − 8
+2
+
− 14
ζ+
ρ
ρ2
2 R̂3
R̂
R̂5
ρR̂2
S 2 µ21
f µ2
ζ 3 µ31 dR̂
αµ2
− 38 1 ζ 2 − 32
+ 4 1 +3
,
(A.3.7)
dζ
ρR̂2
R̂3
R̂5
R̂5
1 S 2 µ2
S 2 µτ
µ f
µ2
= −
+
+2 1 +4 1ζ +
2 R̂2
R̂2
R̂4
R̂4
2
h
f
µ1 f
µ21 2 dR
2
+ α R̂ + 2 −
−4
ζ −4 ζ
.
(A.3.8)
dζ
R̂ R̂5
R̂5
R̂5
fh
D3
3 R̂2 αS 4
R̂3 α2 S 2 1 f αS 2
1 f 2S2 3 f 3
f R̂α2
+2
+
+
−
−
4
+
16 ρ
ρ2
4 ρ
2 R̂3
2 R̂5
ρ2
216
5 Navier-Stokes: viscosity as perturbation
Epilogue
In this thesis we started with sketching the problem of vortex-breakdown in swirling
pipe flow. We discussed several topics and mechanisms that have been proposed to
explain this phenomenon: supercritical versus subcritical, (in)stability, (non-)uniqueness, etc. Moreover, numerical simulations of swirling pipe flows showed that choosing
the appropriate turbulence model is not an obvious matter at all, thereby suggesting
the need for low-dimensional models.
We have proposed such a model, by exploiting the Hamiltonian character of the
governing equations—the Euler equations. The solutions that played the role of base
functions were parametrised relative equilibria, consisting of a rigid body rotation, a
uniform translation and a Beltrami flow, while the parameters were easily related to
clear physical quantities like mass flux, angular momentum and helicity (chapter 2).
As a consequence, we were able to express for example the angular frequency of nonaxisymmetric swirling flows in terms of global quantities. This resulted in analytical
predictions which were checked against experimental data, with a rather high degree
of agreement.
We investigated the linear stability and the nonlinear (constrained) stability of the
relative equilibria (chapter 3). In the first problem we focused on necessary or sufficient conditions for (linear) stability; key parameters were the ratio κ, indicating the
relative strength of the rigid body rotation relative to the Beltrami component. Moreover, we showed the relevance of the travelling wave relative equilibrium solutions in
the context of the transition from supercritical to subcritical; exactly at the point
where the axial momentum is not constrained anymore, i.e., its corresponding multiplier vanishes, the swirling flow turns critical, and the eigensolution that is related to
the zero-eigenvalue is precisely the travelling wave relative equilibrium solution. Stability considerations in Hamiltonian theory can draw heavily on methods developed
by Arnol’d and researchers inspired by his work. One of these methods is the constrained stability and we discussed the application of it to the swirling flow problem.
This led to the result that the helicity of the base flow and that of the perturbation
should have opposite sign in order to have constrained stability, and vice versa.
In the last two chapters we discussed perturbations of the Euler equations in a pipe
of constant cross-sectional area. In chapter 4 we studied the dynamics of an inviscid flow in a pipe with slowly varying diameter, and more specifically in a slowly
218
Epilogue
expanding pipe. The parameter dynamics was obtained by requiring a consistent
evolution of conserved quantities and it was stated in the form of a coupled system of non-autonomous ODE’s, with the radius of the pipe playing the role of time.
This led to a qualitative picture that showed flow-reversal and vortex-breakdown;
the base flow turned critical. When we compared the model with existing steady,
axisymmetric models, we observed deviations that could be traced back to a degeneracy in the (linearised) Euler equations. In addition we proposed as a definition of
‘vortex-breakdown’ the slow passage through bifurcation, thereby referring to similar
studies in ODE’s. Moreover, this idea confirms recent work of Trigub and Sychev,
who studied vortex-breakdown in unbounded flows.
Finally, in chapter 5 we investigated the influence of viscosity. At the entrance of
an (expanding) pipe, the effects of viscosity are mainly felt near the wall, the socalled boundary layer. We modelled this boundary layer by splitting the velocity
field in two parts, each with its own range of validity. This splitting was introduced
in the integrated balance laws for the evolution of conserved quantities, and with
an order-of-magnitude analysis we were able to derive equations that describe the
leading-order behaviour of the solution. Using these equations, we formulated the
parameter dynamics in the two cases of (i) a viscous flow in a constant-diameter pipe
and (ii) a viscous flow in an expanding pipe, when the expansion rate is of the order
of one over the Reynolds number. One of the results was that the axial decay of the
energy flux was retarded when the amplitude of the Beltrami component was large;
this gives new arguments for the ‘helicity hypothesis’, stating that a Beltrami flow
(having ‘maximal’ helicity) decelerates the decay of e.g. the kinetic energy. However,
as we noted, the boundary layer is only thin near the entrance region, and especially
in the case of an expanding pipe we have to face the problem of boundary layer
separation. This phenomenon calls for new tools such as interactive boundary-layer
coupling that have been developed in the seventies and eighties, but until now these
have not been applied to the swirling pipe flow in an expanding pipe, having the
additional complication of vortex-breakdown in the core flow. Together with a direct
numerical simulation or large-eddy-simulation of the whole flow this seems a major
challenge for the (nearby) future.
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Samenvatting
De aanwezigheid van wervels is een essentieel onderdeel in veel industriële processen
zoals menging, scheiding, stabilisatie, etc. Maar het onbedoeld optreden van wervelingen, al dan niet door toedoen van de mens, kan ook grote schade aanrichten; tornado’s
zijn waarschijnlijk het meest bekend, maar ook de enorme vleugeltip-wervels die zich
achter grote vliegtuigen vormen kunnen op z’n minst tot vertraging maar ook tot
ongelukken leiden.
Het proefschrift begint met het schetsen van het fenomeen ‘vortex-breakdown’ in
roterende pijpstromingen. Hierna worden de wiskundige vergelijkingen ten tonele
gevoerd, en in het bijzonder haar Hamiltonse structuur. Deze structuur ligt ten grondslag aan een niet-lineaire spectraal methode welke gebruikt wordt om de invloed van
verstoringen op speciale stromingen te beschrijven. Meer specifiek, om een spectraal
methode te starten, is een (rijke) familie met geparametriseerde basisfuncties nodig,
eventueel gerelateerd aan het ongestoorde probleem: in ons geval de stroming van een
niet-visceus fluı̈dum door een rechte pijp. Deze familie wordt in hoofdstuk 2 geconstrueerd waarbij technieken uit de variatierekening centraal staan. Deze technieken
geven de mogelijkheid om stromingen te beschouwen die tot nu toe niet veel aandacht
hebben gekregen: de niet-axisymmetrische, tijdsafhankelijke roterende stromingen.
In hoofdstuk 3 wordt de lineaire en niet-lineaire stabiliteit van de corresponderende
stromingen onderzocht. Hierbij wordt gebruik gemaakt van ideëen die ontwikkeld zijn
door Arnol’d en mensen die door hem geïnspireerd zijn.
In de hoofdstukken 4 en 5 gebruiken we de geparametriseerde basisfuncties om twee
problemen te analyseren: een niet-visceus fluı̈dum in een (langzaam) expanderende
pijp en de effecten van viscositeit in zowel een rechte als een expanderende pijp. De
idee om de dynamica van de parameters te vinden is het best te beschrijven als we het
(abstracte) definitie van een Hamiltons of Poisson systeem voor ogen hebben: het is
een dynamisch systeem voor functionalen, met bepaalde eigenschappen. Van de benadering wordt geeist dat een aantal goed te kiezen functionalen een dynamica hebben
die consistent is. Deze functionalen zijn niet zomaar willekeurig maar corresponderen
met duidelijk fysische grootheden en zijn bovendien constanten van beweging. Dit
heeft als grote voordeel dat er een eenvoudige terugkoppeling naar experimenten kan
zijn. Bovendien is in het algemeen deze eis equivalent met oplosbaarheids voorwaarden (niet-resonantie condities) voor de vergelijking voor de fout.
228
Samenvatting
Als resultaat moet een stelsel gekoppelde, niet-autonome gewone differentiaal vergelijkingen worden opgelost, waarbij de dynamische variabele de straal van de pijp
(hoofdstuk 4) of een (geschaalde) coördinaat langs de wand van de pijp (hoofdstuk
5) is. Op deze manier wordt met een laag-dimensionaal model een goede beschrijving
gegeven van het snelheidsveld tot aan ‘breakdown’ en de loslating van de grenslaag in
een expansie. Bovendien worden er suggesties gedaan om modellen op te stellen voor
de beschrijving van het snelheidsveld vóórbij ‘breakdown’ en loslating.
About the author
Erik Fledderus werd op 15 maart 1970 geboren te Heerenveen. Vanaf 1982 bezocht
hij het Nassau College te Heerenveen en behaalde het VWO diploma in 1988.
Direct hierna begon hij met de studie Toegepaste Wiskunde aan de Universiteit
Twente wat een verhuizing naar Hengelo met zich mee bracht. Hij werd actief in het
Wiskundig Studie Genootschap Abacus als ondermeer materiaalcommissaris, penningmeester en bestuurslid van het Mathematisch Café. In 1993 studeerde hij af
bij Prof. Brenny van Groesen; zijn afstudeer-scriptie had de titel ‘Swirling Flows in
Expanding Pipes: Quasi-homogeneous Approximations with Relative Equilibria’ en
bezorgde hem een vliegende start voor het latere promotiewerk. Tijdens zijn studie
trouwde hij met Gretha Zijlstra en werd hij vader van twee kinderen, Tom en Amarins.
Daarnaast was (en is) hij vanaf 1991 begeleider van de jongerenclub Joining in Hengelo.
In juni 1993 startte hij als Onderzoeker in Opleiding in het ‘Niet-Lineaire Systemen’project dat gefinancierd werd door NWO. Dit onderzoek werd verricht in de vakgroep
Toegepaste Analyse, faculteit Toegepaste Wiskunde aan de Universiteit Twente, met
als directe begeleider Prof. Brenny van Groesen. Tijdens zijn onderzoek bracht hij
enkele maanden door in Engeland, als gast van Prof. David Crighton en Dr. Stephen
Cowley aan het Department of Applied Mathematics and Theoretical Physics in Cambridge.