Dynamics of Atmospheres and Oceans
40 (2005) 3–21
No-slip walls as vorticity sources in
two-dimensional bounded turbulence
H.J.H. Clercxa,∗ , G.J.F. van Heijsta , D. Molenaara , M.G. Wellsb
a
b
Department of Physics, J.M. Burgers Centre for Fluid Dynamics, Eindhoven University of Technology, P.O.
Box 513, 5600 MB Eindhoven, The Netherlands
Department of Geology and Geophysics, Yale University, P.O. Box 208109, New Haven, CT 06520-8109, USA
Received 26 January 2004; accepted 4 October 2004
Available online 2 February 2005
Abstract
This paper addresses the role of no-slip boundaries as vorticity sources in two-dimensional turbulence on bounded domains. In recent years, numerical evidence from decaying two-dimensional turbulence simulations in circular and square domains has been provided showing that vorticity produced
at solid boundaries enters the flow at scales comparable with the average boundary-layer thickness.
The thin detached boundary layers contain high-amplitude vorticity, and these boundary layers roll up
to form small strong vortices which are advected by the background flow into the interior of the flow
domain. As a result, the vortex statistics of the flow are modified: while the merging process active
in the interior leads to a decrease of the number of vortices, the solid walls act as sources of vortices.
We will briefly review numerically obtained results for decaying two-dimensional turbulence, and
provide supporting experimental evidence from forced quasi-two-dimensional turbulence in rotating
containers. Additionally, numerical simulations of forced two-dimensional turbulence are reported
illustrating the dynamical impact of the presence of solid boundaries: a large-scale vortex is gradually
built up and later on suddenly destroyed by the destabilizing effect of strong small-scale vortices
produced near the solid walls. After the collapse, the self-organization process may start anew, and
most strikingly, the circulation of the central vortex may even show sign reversal.
© 2004 Elsevier B.V. All rights reserved.
PACS: 47.27.Eq; 47.32.Cc; 47.32.−y
Keywords: 2D turbulence; Boundary layers; No-slip walls; Vortex statistics
∗
Corresponding author. Tel.: +31 40 247 26 80; fax: +31 40 246 41 51.
E-mail address: [email protected] (H.J.H. Clercx).
0377-0265/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.dynatmoce.2004.10.002
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H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
1. Introduction
During the last decades many theoretical and numerical studies have been carried out
to improve the understanding of two-dimensional (2D) turbulence. The first attempts to
formulate a phenomenological theory of 2D turbulence was put forward 35 years ago by
Kraichnan (1967, 1971), Leith (1968), and Batchelor (1969). According to this theory, which
is actually an adaptation to 2D flows of the theory of self-similarity in three-dimensional
turbulence (as proposed by Kolmogorov), virtually all the energy injected in the flow at
wave number ki is transferred to the largest scales available, a process known as the inverse
energy cascade. The energy spectrum E(k) for wave numbers smaller than ki scales then
as k−5/3 . Virtually all the enstrophy injected at ki is transferred to the smallest scales, i.e.
to the dissipation wave number kd , and will be dissipated. In case of equilibrium, when all
the injected enstrophy is eventually removed by viscous dissipation, the energy spectrum
should scale as k−3 (with a logarithmic correction, see Ref. Kraichnan, 1971). This combined
inverse energy cascade and direct enstrophy cascade is known as the dual cascade, and is
formally derived for forced 2D turbulence on an unbounded domain.1
Direct numerical simulations (DNS) of forced 2D turbulence with periodic boundary
conditions confirmed the existence of the dual cascade, provided the forcing satisfies certain
properties. A forcing should be applied which is spectrally localized to a wave number
interval (ki , ki + k) with k ki . Moreover, the domain size, forcing scale, and the
dissipative scales should be sufficiently separated (Frisch and Sulem, 1984; Legras et al.,
1988; Borue, 1994; Smith and Yakhot, 1993). It should be mentioned that Legras et al.
(1988) observed steeper spectra for large wave numbers (small scales). The energy spectra
obtained in DNS studies of decaying 2D turbulence (with periodic boundary conditions)
show indications of the presence of the inverse energy cascade, whereas the direct enstrophy
cascade is often only established as a transient state, and is often steeper than predicted by
Kraichnan and Batchelor (Brachet et al., 1988; Santangelo et al., 1989). This is attributed
to the appearance of coherent vortices. (Note that the Kraichnan–Batchelor theory also
predicts a k−3 energy spectrum for decaying 2D turbulence.)
Experiments (Sommeria, 1986; Paret and Tabeling, 1997; Kellay et al., 1995; Rutgers,
1998) to confirm the presence of the inverse energy cascade, the direct energy cascade,
or both cascades simultaneously are even more complicated. This is due to the restriction
to flows with intermediate or low integral-scale Reynolds numbers (Re ≤ 2000) in 2D
turbulence experiments in thin, magnetically forced, fluid layers (Sommeria, 1986; Paret
and Tabeling, 1997), or due to the lack of 2D incompressibility in soap-film experiments
which is a consequence of thickness fluctuations (Kellay et al., 1995; Rutgers, 1998). In
these soap-film experiments higher integral-scale Reynolds numbers can be achieved. All
experimental setups mentioned above disregard the role of (no-slip) boundaries. For a more
extensive review of some recent experiments, see Tabeling (2002) and Kellay and Goldburg
(2002), and references therein.
The emergence and temporal evolution of a hierarchy of coherent vortices in decaying
2D turbulence has been subject to many theoretical, numerical, and even experimental
1
For a comprehensive overview of the developments until 1980, the review on hydrodynamic and plasma
applications by Kraichnan and Montgomery (1980) can be consulted.
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
5
studies (Leith, 1984; Carnevale et al., 1991, 1992; Cardoso et al., 1994; Hansen et al., 1998;
Weiss and McWilliams, 1993). An interesting result is the scaling theory as put forward by
Carnevale et al. (1991). In their theory, it is assumed that the total kinetic energy, defined as
1
E=
u2 dA,
(1)
2 D
with u the flow velocity and D the computational flow domain (and dA an infinitesimal
part of it), and the vorticity extremum ωext of the dominant vortices are conserved in freely
evolving 2D turbulence. By supposing an algebraic decay of the average number density of
vortices, ρ(t) ∝ t −ζ , with ζ undetermined (McWilliams, 1990), it is possible by applying
dimensional analysis to express the time evolution of the average enstrophy of the flow,
which is defined as
1
Ω=
ω2 dA,
(2)
2 D
with ω the vorticity of the flow, in terms of a power-law, Ω(t) ∝ t −ζ/2 . Similarly, power-law
exponents were derived for the average vortex radius and the average vortex separation,
and these power-law exponents could also be expressed as some fraction of ζ. Numerical
simulations yielded the estimate ζ ∼ (0.72–0.75) (McWilliams, 1990; Carnevale et al.,
1992; Weiss and McWilliams, 1993). These results are not based on simulations of the
Navier–Stokes equations, but either on a punctuated Hamiltonian dynamics of point vortices
(Carnevale et al., 1992), or on a modified set of equations with some high power of the
Laplacian operator representing the dissipation term (using hyperviscosity) (McWilliams,
1990; Weiss and McWilliams, 1993).
The boundedness of the computational domain needs some special attention, even when
periodic boundary conditions are applied. In a bounded domain, the upscale flow of energy
is halted at wave numbers representing the domain size, and condensation of the kinetic
energy of the flow in the lowest accessible mode of the system can occur (Hossain et al.,
1983; Matthaeus et al., 1991; Smith and Yakhot, 1993, 1994). The inverse energy cascade
can be approximately restored by applying dissipation that removes the kinetic energy at the
domain-sized scales, for example by a linear Ekman drag. Nevertheless, still some energy
might be reflected and get trapped in the inertial ranges (where dissipation is inactive),
which probably modifies the scaling of the spectrum. An elaborate discussion of this issue
is provided by Tran and Bowman (2003).
These (and many subsequent) flow simulations were carried out on a double-periodic,
square domain. It was shown by Li and Montgomery (1996) in a numerical study of decaying
2D turbulence on a bounded, circular domain that the decay scenario is essentially different
from that on a double-periodic domain, somewhat depending on the imposed boundary
condition (stress-free or no-slip) (Li et al., 1996, 1997). Experiments on decaying stratified
turbulence in circular containers by Maassen et al. (1999, 2002) support many of the observations reported by Li and Montgomery. A similar influence by the domain boundaries
was observed by Clercx et al. (1999) in a numerical study of decaying 2D turbulence on
a square domain with either stress-free or no-slip boundary conditions. In particular, the
no-slip calculations revealed the crucial role played by the boundaries, in the sense that
they act as sources of large-amplitude vorticity: whenever a vortex structure approaches a
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H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
wall, a boundary layer is formed with oppositely signed vorticity, part of which is usually
scraped off by the vortex flow, in the form of a thin filament of vorticity, which is hence
advected into the interior of the flow domain.
In this paper, we will show how no-slip boundaries, which serve as sources of highamplitude vorticity, will also affect forced 2D turbulence. The definition of the angular
momentum of the flow, and its time-derivative calculated from the Navier–Stokes equations, will be introduced in the following section. We will briefly recall some of our results
on one-dimensional (1D) spectra nearby no-slip walls, the modification of vortex statistics
by the presence of boundary layers, the spontaneous spin-up of the flow (i.e. a mean non-zero
rotation of the fluid) and the scaling of the enstrophy contained in the boundary layers in
order to illustrate the influence of boundary-layer vorticity on decaying 2D turbulence. We
will provide supporting experimental evidence from forced quasi-two-dimensional turbulence in rotating containers. Additionally, numerical simulations of forced two-dimensional
turbulence are reported illustrating the role of continuous injection of small-scale vorticity
from boundary layers near no-slip walls into the interior of the flow domain. Moreover,
the dynamical impact of the presence of solid boundaries is shown: a large-scale vortex
is gradually built up and later on suddenly destroyed by the destabilizing effect of strong
small-scale vortices produced near the solid walls. After the collapse, the self-organization
process may start anew, and most strikingly, the circulation of the central vortex may even
show sign reversal.
2. Angular momentum of the flow
Consider the 2D motion of a viscous fluid on a square bounded domain D with boundary
∂D (in dimensionless form the square [−1, 1] × [−1, 1]). Cartesian coordinates in a frame of
reference are denoted by x and y. Let the (horizontal) flow field be given by v = (u, v, 0) with
∂v ∂u
the vorticity ω = ∇ × v = (0, 0, ω), where ω =
− . This fluid motion is governed
∂x
∂y
by the Navier–Stokes equation, which reads in non-dimensional form:
∂v
1 2
(3)
+ (v · ∇)v = −∇p +
∇ v + Fsf ,
∂t
Re
with t the time, p the pressure, and Re = UW/ν the Reynolds number based on the rms
velocity U, the length scale W (the half-width of the container), and ν the kinematic fluid
viscosity. The quantity Fsf represents a stochastic external forcing of the flow. By taking
the curl, one obtains the vorticity equation:
∂ω
1 2
+ (v · ∇)ω =
∇ ω + Q,
(4)
∂t
Re
with Q = ∇ × Fsf = (0, 0, Q). Either equation should be solved subject to conditions at the
boundary ∂D. Impermeability of this boundary implies that the stream function ψ satisfies
ψ = constant on ∂D. In the case of a (physically realistic) no-slip boundary, the additional
condition is v|| = 0 on ∂D, with v|| the velocity component parallel to the boundary, or
taking the two relations together: v = 0 on ∂D. The boundary value for the vorticity is not
provided a priori for flows in bounded domains with no-slip walls.
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
7
A useful and relevant global quantity is the angular momentum L, defined with respect
to the origin in the domain centre as
L(t) =
k̂ · (r × v(r, t)) dA,
(5)
D
with k̂ the unit vector perpendicular to the flow plane, and r = (x, y) the position vector. It
is easy to verify that Eq. (5) can be written as:
1
L = (xv − yu) dx dy = −
r 2 ω dx dy.
(6)
2 D
D
The rate
of the total angular momentum of the fluid on domain D is given by
of change
dL
∂v
k̂ · r ×
=
dA and becomes, after substitution of the Navier–Stokes Eq. (3)
dt
∂t
D
and some rearrangements:
dL
= − v · ∇[k̂ · (r × v)] dA +
k̂ · [∇ × pr] dA
dt
D D
(7)
1
+
r · ∇ω dA +
k̂ · (r × Fsf ) dA.
Re D
D
The last contribution is proportional to the externally applied random torque: M(t) = D k̂ ·
(r × Fsf (t)) dA. The area integrals are now rewritten as contour integrals, and
using v · n̂ = 0
on the impermeable boundary of the domain and that the circulation Γ = D ω dA = 0 (the
no-slip condition applies and the boundaries are stationary), yields
dL
1
pr · ds +
ω(r · n̂) ds + M(t).
(8)
=
dt
Re ∂D
∂D
An analogous equation can be derived from the vorticity Eq. (4):
dL
∂ω
1
1
r2
ω(r · n̂) ds + M(t)
=−
ds +
dt
2Re ∂D ∂n
Re ∂D
(9)
∂
representing the normal derivative with respect to the boundary). In this equation,
∂n
an alternative definition for the externally applied random torque has been used,
1
M(t) = −
r2 Q(r, t) dA,
(10)
2 D
(with
which is fully equivalent to the one introduced below Eq. (7). As can be seen in Eq. (8), a
pressure contribution will also modify the time rate of change of L(t). It is clear from Eq. (9)
that this contribution should be proportional to the normal vorticity gradient integrated over
1 ∂ω
the boundary.2 It is interesting to note that the product Re
∂n should be finite for vanishing
viscosity, or equivalently, Re → ∞ (thereby assuming a finite pressure contribution in the
expression for dL(t)
dt in this limiting case).
2
Equivalence of both terms can also be shown by expressing the pressure boundary condition for the present
problem in terms of the normal vorticity gradient at the boundary.
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H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
The results shown in Eqs. (8) and (9) express the (rather trivial) fact that the change in
angular momentum is due to torques of wall forces, associated with the inviscid pressure
(normal stress), and viscous stresses (normal and shear stresses).
The flow evolution of decaying 2D turbulence on a square domain with no-slip walls is
characterized by a sudden growth of the angular momentum, a phenomenon totally absent
when a circular domain is used. This remarkable difference needs some clarification. An
important difference between the bounded square and circular case with regard to the time
evolution of L(t) (disregarding therole of forcing for the moment) is the absence of a
pressure-induced contribution (i.e. ∂D pr · ds) to the time-rate of change of the angular
momentum for flows in bounded circular domains. For a circular domain r · ds = 0, thus
dL
R
1
ω(r · n̂) ds =
ω ds,
(11)
=
dt
Re ∂D
Re ∂D
dL(t)
depends only on the vorticity produced at
dt
the no-slip wall, provided no external forcing is present, and this contribution is found to be
dL(t)
rather unimportant. In a square domain, where r · ds = 0,
is completely dominated
dt
by the pressure distribution integrated over the boundary (see Eq. (8)), whereas the total
shear stress along the boundary is of minor importance. In the present simulations on a
square domain it appeared that the shear stress contribution is approximately two orders of
magnitude smaller than the pressure contribution.
with R the radius of the circular domain.
3. Enstrophy production near no-slip boundaries
No-slip boundaries act as generators of high-amplitude vorticity and associated high
vorticity gradients, which eventually modifies the classical picture of the evolution of vortex
statistics as derived for decaying 2D turbulence in unbounded domains. More precisely,
this modification is due to the production of many small-scale vortices near the no-slip
walls as a result of boundary-layer detachment and subsequent roll-up of the vorticity
filaments. Moreover, the presence of filaments containing high-amplitude vorticity enhances
the dissipation of the kinetic energy of the flow. In a series of numerical simulations of
the dipole-wall collision with Reynolds numbers (Re) ranging from 625 to 160,000 we
were able to quantify the amount of enstrophy production during the collision, and the
associated dissipation of kinetic energy. Details of the numerical simulations and analysis of
the data can be found elsewhere (Clercx and van Heijst, 2002). The scaling behaviour of the
enstrophy in the large Reynolds number limit can reasonably well be understood on the basis
of the flat-plate boundary-layer theory (Schlichting, 1999; Clercx and van Heijst, 2002).
Consider the following schematic picture of a snapshot of the dipole-wall collision: a vortex
with circulation Γ and diameter D is situated near a no-slip wall where it induces a boundary
layer with thickness δ and width D. The circulation in the boundary layer, Γb , is assumed
to be independent of Re, with |Γb | ≈ |Γ |. Assuming a finite pressure distribution along the
∂ω
boundary, it can be shown (see discussion below Eq. (9)) that limRe→∞
| ∝ Re. The
∂n ∂D
boundary layer thickness scales like δ ∝ Re−1/2 . Combination of the large Reynolds number
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
9
√
∂ω
∂ω
| and δ yields for the vorticity at the boundary: |ωwall | ≈ |( | )|δ ∝ Re.
∂n ∂D
∂n ∂D
2
The enstrophy of the boundary layer induced by the dipole then scales like: Ω ∝ Dδωwall
∝
√
D Re. The amount of vorticity in the boundary layers in the present numerical simulations
of dipole-wall collisions (with Re 160,000) is one-to-two orders of magnitude larger
than the enstrophy contained by the dipole itself and the predicted scaling of the enstrophy
is indeed found (Clercx and van Heijst, 2002).
Similar studies should be carried out for 2D turbulent flows in containers with no-slip
boundaries. Lack of sufficient computer power prevents such numerical investigations for
the time being. Extrapolation of the results obtained with the present dipole-wall collision
experiments to bounded 2D turbulent flows is, however, tempting. Suppose we have a
decaying turbulent flow field consisting of a large number (N) of vortices. Without losing
generality we can assume that the vortex diameters D and the extremum vorticity values
ωext are all the same. Additionally, the vortex diameters are much smaller than the container
dimension, i.e. √
D W. With a homogeneous distribution of vortices it is reasonable to
assume that O( N) vortices will interact with the no-slip boundaries. The enstrophy of
the turbulent flow consists of two contributions. The first contribution comes from the N
vortices in the interior,
scaling of
2
Ωi ∝ Nωext
D2 .
(12)
For unbounded domains (and for turbulent flows in domains with periodic boundaries)
the enstrophy is fully determined by Ωi . The second contribution comes from the boundary layers. Introduce the boundary-layer thickness δ and the circulation contained by the
boundary layer, Γb . We assume that the circulation of the vortex core that collides with the
boundary and the circulation contained by the boundary layer induced during the collision
approximately cancel, i.e. |Γb | |ωwall |Dδ |ωext |D2 . By using the following estimate
for the enstrophy associated with the√boundary layer induced by one vortex–wall collision,
Ω ∝ Γb2 /Dδ, together with δ ∝ D/ Re, the estimate for Γb given above, and the fact that
√
on average N vortex–wall collisions are expected, we obtain
√ 2 2√
Ωb ∝ Nωext
D Re.
(13)
The dissipation of kinetic energy in bounded domains with no-slip walls is governed by the
classical relation
dE(t)
2
= − Ω(t).
(14)
dt
Re
Substitution of Ωi and Ωb in Eq. (14) yields
√ 2 D2
dE(t)
Re
2Nωext
2
= − (Ωi (t) + Ωb (t)) ∝ −
1+ √
.
(15)
dt
Re
Re
N
Note that the additional dissipation due to the no-slip walls is thus related with the enstrophy production only, a process totally absent in decaying 2D turbulence on unbounded or
periodic domains (where the enstrophy is bounded by its initial value). Obviously, Ωb will
dominate Ωi for sufficiently large Reynolds numbers as N is finite and tends to decrease
due to subsequent mergings.
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H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
The production of high-amplitude vorticity in the boundary layers, and the subsequent
formation of many small and strong vortices undoubtedly will modify the vortex statistics.
This is supported by numerical studies of decaying 2D Navier–Stokes turbulence in bounded
domains with no-slip walls (Clercx and Nielsen, 2000) that show an evolution of vortex
statistics that is distinctly different from that predicted on the basis of the theory proposed by
Carnevale et al. (1991, 1992). Moreover, a comparison of our results with simulations where
similar initial conditions have been used, but now with periodic boundary conditions, clearly
indicates the importance of the no-slip walls as sources of small-scale vortices. In particular,
the vortex density ρ(t) shows a much smaller algebraic power-law decay: ρns (t) ∝ t −0.75
and ρper (t) ∝ t −1.03 for the runs with no-slip (ns) and periodic (per) boundary conditions,
respectively (Re = 10,000 in these runs). Additionally, the mean vortex separation increases
significantly less in the decaying turbulence runs with no-slip boundaries compared to the
runs with periodic boundary conditions, and the vorticity extremum of the strongest (and
longest surviving) vortices decays substantially in course of time (even if normalized by
√
ωext
E). For example, we found for simulations with Re = 10,000 that √ ∝ t −0.3 (ensemble
E
average of eight runs) (Clercx and Nielsen, 2000). In our view, these data strongly suggest
that application of the scaling theory proposed by Carnevale and coworkers to decaying 2D
Navier–Stokes turbulence might be questionable in general, and that this theory cannot be
applied to 2D decaying turbulence in a square domain with no-slip boundaries in particular.
For fairness, we should mention that it was never claimed by Carnevale and coworkers
that their theory might be valid for the evolution of the vortex population in bounded decaying 2D turbulence simulations with no-slip walls. However, many validation attempts of
the scaling theory were based on laboratory experiments conducted in square or rectangular
containers, for example, the studies in thin, stratified electrolyte solutions by Cardoso et al.
(1994) and Hansen et al. (1998). We believe that any agreement between experimentally
obtained power-law exponents and the Carnevale approach is accidental. The experimental vortex statistics data also agree with the late-time viscosity-dominated decay stage of
our numerical simulations with no-slip boundaries (Clercx et al., 1999, 2003; Clercx and
Nielsen, 2000; Maassen et al., 2002).
4. Forced 2D turbulence in square domains with no-slip walls
The production of small-scale vorticity near no-slip walls is also reflected in onedimensional kinetic energy spectra (Clercx and van Heijst, 2000). The 1D energy spectra
measured close to the walls in numerical simulations with Re = 20,000 revealed during
the initial decay stage a k−5/3 inertial range, instead of a k−3 direct enstrophy cascade,
due to the production of small-scale vorticity near no-slip boundaries. The direct enstrophy cascade is virtually absent at early times during the decay process. Similar runs with
periodic boundary conditions were characterized by the absence of a k−5/3 inertial range
(the observed inertial range appears to represent a direct enstrophy cascade). As an illustration we have plotted in Fig. 1 the averaged 1D energy spectrum for a few runs with freely
evolving 2D turbulence with Re = 20,000. The spectrum shown in Fig. 1a was computed
after approximately four initial eddy turnover times. At later times the energy spectrum
shows the build-up of a direct enstrophy cascade with a k−3 inertial range together with
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
11
Fig. 1. (a) The ensemble-averaged 1D energy spectra for runs with no-slip walls and with periodic boundary
conditions (the steeper spectrum in the inertial range). The initial Reynolds number is 20,000 and the spectra are
measured at τ = 4. (b) The time evolution of 1D spectra for runs with no-slip walls (Re = 20,000) and (c) the
growth of the average boundary-layer thickness δav (16) compared to the position of the kink between k−5/3 and k−3
behavior. The symbols denote the position of the kink in the spectra ((- + -) for Re = 5000, (- × -) for Re = 10,000,
and (- ∗ -) for Re = 20,000), and the drawn (spiky) lines represent the computed average boundary-layer thickness
δav .
the inverse energy cascade for smaller wave numbers. The energy spectrum shows a kink,
which slowly moves to smaller wave numbers as time proceeds (see Fig. 1b). The position
of the kink can clearly be associated with the growth of an averaged local boundary-layer
thickness δav which is proportional to the ratio between the boundary-averaged vorticity
and the boundary-averaged normal gradient of the vorticity,
ω2 ds
δav ∝ ∂D 2 .
(16)
∂ω
ds
∂D ∂n
It is expected that δav grows in course of time. The strong correlation between the position
of the kink in the spectrum and the average boundary-layer thickness is obvious from the
data shown in Fig. 1c (for clearness, we did not plot the data on top of each other; the spiky
curves represent actually 21 δav ). The data displayed in Fig. 1c for Re = 20,000 is much more
spiky than for Re = 10,000 and 5000 because only 2 runs were available for averageing
(and 8 and 12 runs for Re = 10,000 and 5000, respectively). Further technical details such
as initial conditions, applied resolution, and the method to compute the spectra is provided
in Clercx and van Heijst (2000).
New observations from laboratory experiments and numerical simulations of continuously forced quasi-2D turbulence have revealed similar spectra. The laboratory experiments
were performed in a Perspex tank of square cross-section, filled with water, mounted on a
rotating table. The flow is made quasi-2D by a steady background rotation Ω0 . A small sinusoidal perturbation with amplitude AΩ0 and frequency f, added to the background rotation
results in Ω(t) = Ω0 [1 + A sin(ft)], and hence in a time-periodic forcing of the flow. This
forcing leads to the periodic formation of eddies in the corners of the tank by the roll-up
of vorticity generated along the side walls. When the oscillation period is greater than the
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H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
time scale needed to advect a full-grown corner vortex along a side wall to a neighbouring
corner, dipole structures are observed to form. These dipoles migrate away from the walls,
and the interior of the tank is continually filled with new vortices.
It is found that in this type of experiments no organization into larger eddy scales is
observed. The explanation is most likely related with the role of Ekman pumping at the
bottom of the tank, resulting in a decay of the vortex amplitudes, and from interactions of
vortices with other stronger vortices that strip off filaments of vorticity. Subsequent interactions between the weaker “old” vortices and the “young” vortices result in the straining
and, ultimately, the destruction of older vortices.
In laboratory experiments with Re = 15,000, where the Reynolds number is defined
UW
AΩ0 W 2
as Re =
=
, measurements of the energy spectrum obtained from particle
ν
ν
tracking velocimetry revealed a k−5/3 power-law for wave numbers k 50 m−1 . This is
consistent with an inverse cascade of energy in the 2D turbulent flow field. We were not
able to measure the k−3 power-law for k 50 m−1 due to lack of resolution of the particle
tracking method. Indications of the presence of the direct enstrophy cascade were indirectly
obtained by analysis of digital images of passive tracer distributions. Measurements of the
intensity spectrum of a passive scalar dispersed in the same experiment were consistent
with the Batchelor prediction (Batchelor, 1959) of a k−1 power-law at wave numbers k 50 m−1 . Theoretically, a k−1 power-law of the scalar spectrum should occur in the same
spectral range as the k−3 enstrophy cascade of the energy spectrum (Nam et al., 1999).
The numerical simulations were performed with a pseudo-spectral numerical code developed by Clercx (1997). The numerical scheme is based on an expansion of the flow
variables in Chebyshev polynomials rather than the more commonly used Fourier functions. This choice allows the application of no-slip conditions at the boundaries of the
computational domain. The computational domain D is the square [−1, 1] × [−1, 1] with
boundary ∂D. The Cartesian coordinates in a frame of reference are denoted by x and y. On
this domain, the vorticity equation in the co-rotating frame of reference can be written in
the dimensionless form
∂ω
1 2
+ (v · ∇)ω =
∇ ω − 2AΩ0 f cos(ft),
∂t
Re
(17)
where the last term on the R.H.S. represents the forcing. The Reynolds number is defined
AΩ0 W 2
as before: Re =
. The vorticity Eq. (17) has to be solved in combination with the
ν
relations
∇ 2 v = êz × ∇ω ,
on D,
(18)
êz · ∇ × v = ω ,
on ∂D,
(19)
v = 0,
on ∂D.
(20)
The time discretization of the vorticity equation consists of a second-order explicit Adams–
Bashforth scheme for the advection term and an implicit Crank–Nicolson procedure for the
diffusion term.
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
13
Fig. 2. Vorticity distributions obtained by numerical simulations representing the flow after 20 oscillation periods,
showing the turbulent vorticity field for Re = 5000 (a) and 25,000 (b). Note the smaller vortex size at larger Re
and the absence of any large-scale structure, and the presence of filaments detached from the boundary or from
strained vortices.
In Fig. 2, typical vorticity fields, obtained by numerical simulations of forced 2D turbulence with Re = 5000 and 25,000 (AΩ0 = 0.75 and f = 1), are displayed. The resolution
is 1612 Chebyshev modes for the run with Re = 5000 (Fig. 2a) and 2572 modes for the
run with Re = 25,000 (Fig. 2b). The vorticity fields are seeded with initial noise needed to
remove the fourfold rotation symmetry present due to the forcing protocol. This was done
by applying random noise of amplitude 0.01 to the initial vorticity field in the first 10 × 10
Chebyshev modes. In both cases, there are regions near the boundaries where new vortices
are formed by the separation and roll-up of the boundary layer. The difference in size of
the emerging vortices is then directly related to the difference in boundary layer thickness –
hence, the larger vortices are seen in the simulation with Re = 5000. The maximum values
of vorticity in Fig. 2b (where Re = 25,000) are approximately twice those seen in Fig. 2a
(where Re = 5000).
Fig. 3 shows kinetic energy spectra from similar runs as shown in Fig. 2 with Re = 5000
and 25,000 (AΩ0 = 0.75 and f = 1). The 2D energy spectrum is calculated from the
coefficients of the Chebyshev polynomials used in the pseudospectral code. For this purpose,
data from a 2D FFT of the kinetic energy of the flow is collapsed onto a 1D graph by
computing the energy spectrum according to
k+1
E(k) =
E(κx , κy ) dκ,
(21)
k
where κ = (κx2 + κy2 )1/2 . One can show that spectral slopes expressed using Chebyshev
wave numbers are the same as using Fourier components. √
The non-dimensional boundary-layer thickness scales as Re, so it is expected
that the
√
ratio of the injection scale thickness from both simulations should scale as Re2 /Re1 =
14
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
Fig. 3. A log–log graph of the energy spectrum for Re = 5000 and 25,000 at the same phase in the forcing cycle.
For both values of Re a k−5/3 power-law for low wave numbers, and a k−3 power-law fit exists for high wave
numbers. These power-laws are consistent with an inverse energy cascade, and an enstrophy cascade, respectively.
Also note the shift of the “knee” in these two spectra as the Reynolds number changes, consistent with changes in
boundary-layer thickness.
2.25. The displayed spectra in Fig. 3 clearly indicate that the “knee” where the slope changes
from k−5/3 to k−3 occurs at roughly k = 100 and 50, i.e. at a ratio similar to that predicted.
The magnitude of the wave numbers, where the
√ “knee” occurs at, is consistent with the
predicted boundary layer thickness δ via δ ≈ 5 νW/U. This prediction was based on the
assumption of maximum flow velocity occuring at the boundaries, so the spectra shown
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
15
here are both taken at the phase of the forcing where these boundary layers are thinnest.
The observed k−5/3 spectrum indicates the presence of an inverse energy cascade without
the eddy size necessarily evolving to fill the whole domain, as is usually seen in decaying
2D turbulence experiments.
Summarizing, we can conclude that the experimentally measured spectra are in agreement with the data from numerical simulations on continually forced 2D turbulence in a
square domain with no-slip walls. Moreover, the present results provide additional evidence
supporting our earlier observation (see Clercx and van Heijst, 2000) that the thin boundary
layers inject small-scale vorticity into the 2D turbulent flow.
5. Spontaneous sign reversal in self-organized states
The “final state” of decaying 2D turbulence on a square domain with no-slip boundaries consists of a large central cell with either positive or negative circulation, surrounded
by a shielding ring of opposite-signed vorticity, such that the total circulation of the flow
is zero (as dictated by the no-slip condition at the domain boundary). This long-time behaviour has been found both in laboratory experiments (Maassen et al., 2002) and in highresolution numerical flow simulation (Clercx et al., 1998, 2001). A remarkable observation
was that in many cases the total angular momentum L of the flow (being randomly initialised, with L(t = 0) ≈ 0) shows a sudden change to non-zero values – a feature termed
“spontaneous spin-up” (Clercx et al., 1998). This total spin-up of the fluid is directly associated with the self-organization of the flow into a single larger vortex structure that
fills the domain almost completely. In the next stage of the flow evolution, the angular
momentum |L(t)| shows a very slow decay to zero for very late times. It is important to
note that the no-slip boundary condition is a prerequisite for the spin-up, as the angular
momentum L(t) is an irrelevant quantity for the flow evolution on a double-periodic domain. Also, the square domain geometry is important, spin-up being absent on a circular
domain (Maassen et al., 2002; Li et al., 1997) (see the brief discussion in the last paragraph
of Section 2).
Spontaneous mean global rotation was also observed by Sommeria (1986) during his
experimental study of the two-dimensional inverse energy cascade in a square box. In these
experiments, a turbulent flow was generated in a thin horizontal layer of mercury by steady
forcing, and the flow was subject to bottom friction. Random reversals of the rotation
sense of the mean global rotation was observed, and by decreasing the bottom friction sign
reversals became more and more sparse. A numerical study of two-dimensional turbulence
in a square box with stress-free boundaries and bottom friction by Veron and Sommeria
(1987) supported the experimental observations by Sommeria. In order to investigate the
role of lateral no-slip boundaries on spontaneous spin-up of the flow we have carried out
a series of numerical simulations of unsteady forced 2D turbulence in square containers
without bottom friction.
In the direct numerical simulation of stochastically forced 2D turbulent flow on a square
domain with no-slip boundary conditions we also observed spontaneous spin-up behaviour,
although remarkably different from the decaying case. The numerical simulations are based
on the discretized versions of Eqs. (4) and (18)–(20), with Q(t, r) generated by a stochas-
16
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
tic process. The discrete-time counterpart Q(n, r) of Q(t, r) is assumed to be a first-order
Markov chain, a model that was first used in a DNS setting under periodic boundary conditions by Lilly (1969), and discussed in some detail by Maltrud and Vallis (1991). For a
physical analogue, Q(n, r) can be thought of as several mechanical stirring devices, moving
slowly through the fluid in random order. The Markov chain consists of a random phase
and a history term and is applied in Fourier space to all wave vectors k = (k1 , k2 ) in the
shell |k| ∈ [7, 9],
(22)
Q̂(n, k) = (1 − σ 2 )A0 ei2πθ(n,k) + σ Q̂(n − 1, k),
where θ(n, k) ∈ [0, 1] is a random variable, which is drawn from a Gaussian distribution for
each time step and wave number. Two free parameters are introduced with the Markovian
model, being the phase amplitude A0 and the correlation coefficient σ, determining the
relative strength of the two terms in the chain. For σ → 0, the Markov model evolves to
a pure white-noise process, whereas the limit σ → 1 yields a static forcing scheme with
an infinitely long time correlation. The forcing amplitude A0 strongly affects the flow
behaviour, as it determines the amount of energy added and hence influences the integralscale Reynolds number. For the simulations discussed in this section we used σ = 0.98
and A0 = 6.0. To apply the forcing scheme within the computational model, Q̂(n, k) is
converted to Chebyshev coefficients Qmn (n), using the Jacobi–Anger expansion3
Qmn (n) = cm cn
(23)
Q̂(n, k)im+n Jm (k1 )Jn (k2 ).
|k|∈[7,9]
Here cm , cn denote some constants and Jm are Bessel functions of the first kind of order m
(Courant and Hilbert, 1967).
A second set of parameters in the computational model determines the Reynolds number,
Re = UW/ν, which represents the degrees of freedom in the flow. In the definition used
here the natural length scale is taken as W = 1 and the kinematic viscosity in the present
simulation is set to ν = 5.0 × 10−4 . In forced turbulence, however, there is no readily
available estimate on the characteristic velocity scale U, defined as the root-mean-square
1/2
2 T
U = lim
E(t) dt
.
(24)
T →∞ T 0
Henceforth, an approximate integral-scale Re is used, which is determined a posteriori.
For the computation with ν = 5.0 × 10−4 it was established that U 1.5, achieving an
approximate Reynolds number of Re 3000.
In the forced turbulence simulations one observes several consecutive events of rapid
increase and decrease of |L(t)|, often with sign reversal of L between neighbouring peaks
in |L(t)|. A typical example of the evolution of the normalised angular momentum L̃(t) =
L(t)/Lu (t), where Lu (t) is the angular momentum associated with a uniform rotation with
kinetic energy E(t) – during a simulation with Re = 3000 is shown in Fig. 4. It is clearly
seen that – although starting at a zero initial value – the angular momentum tends to reach a
non-zero value, i.e. the flow exhibits spontaneous spin-up. Moreover, several sign changes
3
Note that Q(n, r) = Q(n, x, y) =
m
n
Qmn (n)Tm (x)Tn (y), with Tm (x) and Tn (y) Chebyshev polynomials.
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
17
Fig. 4. Evolution of the normalized angular momentum L̃(t) = L(t)/Lu (t), showing distinct phases of spin-up.
For this run, Re = 3000, A0 = 6.0, and σ = 0.98.
in L are observed to occur a couple of times: apparently the flow reverses abruptly in
those cases. This phenomenon is illustrated by the snapshots of the vorticity distribution
at t = 800, 900, and 1000 (see Fig. 5), corresponding with a well-developed central cell
with a positive circulation (with L > 0), a highly irregular flow state (with L 0), and
a central cell with negative circulation (with L < 0), respectively. In fact, a few different
stages can be distinguished in the flow evolution: first the flow is showing evidence of
self-organization or spin-up, as can be observed from the build-up of a larger circulation
cell and the associated angular momentum, followed by a relatively rapid destruction of the
cell and a dramatic decrease of the flow’s total angular momentum. Subsequently, the flow
becomes organized again into a larger circulation cell – either in the same or in the opposite
direction as the previous cell.
From these and other computational results the source term M(t) appearing in Eq. (9)
and defined in (10) was found to be at least an order of magnitude smaller than the leading
term in Eq. (9), and the correlation between M(t) and these other terms was found to be
insignificant. Henceforth, the evolution of L(t) is assumed to be independent of M(t) and
any occurring spin-up is indeed spontaneous.
Although the spontaneous spin-up stage is the same as in the case of decaying turbulence
in a square domain with no-slip walls, the subsequent destruction of the large circulation
Fig. 5. Snapshots of the vorticity evolution during the sign reversal of a large monopolar vortex structure for
Re 3000, with vorticity levels ranging from ω < −5 (black) to ω > 5 (white). For this run, A0 = 6.0 and
σ = 0.98.
18
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
pattern and the associated decrease of angular momentum is typical for forced 2D turbulence – at least for the values of the forcing and viscosity parameters used here. The
no-slip boundaries enclosing the flow domain play a crucial role in the repeated destruction of the organized flow state. The viscous layers at the walls contain oppositely-signed
vorticity, as a consequence of the no-slip condition. Once the larger cell is established, the
boundary layers detach, giving rise to the formation of smaller vortices in the corners of
the domain (see Fig. 5a and c). The corner vortices gradually grow in size and strength,
and subsequently start to interact with the flow in the interior. The central cell is thus progressively eroded, it soon becomes unstable, and then breaks down rapidly. At that stage,
the flow has become irregular, consisting of filamentary structures without any overall
coherence (see Fig. 5b).
The phenomenon of formation and subsequent break-down of larger cells was observed earlier by Sommeria (1986) in a laboratory experiment on steadily forced flows
in a thin layer of mercury, under the influence of viscosity. Due to the very limited experimental resolution, the mechanism explaining the large-scale reversals remained unclear. Numerical simulations of two-dimensional turbulence with bottom friction in a
bounded domain with stress-free lateral boundaries, i.e. ωwall = 0, were performed by
Veron and Sommeria (1987). Their observations were in agreement with the experimental
data by Sommeria (1986). Obviously, boundary layers near the lateral walls, that might
destabilize the large central vortex, are absent in these simulations. Similar phenomena were recently observed in Rayleigh–Bénard convection experiments (Niemela et al.,
2001), where the global circulation changed sign at irregular intervals. Although speculated upon, the detailed mechanism of the sign reversals was not given. The present
numerical simulations of forced 2D turbulence in bounded domains with no-slip walls
indicate that the sign-reversals are attributed to vortex–wall interactions, and this might
be the central mechanism explaining the experimental observations by Sommeria (1986)
and Niemela et al. (2001).
6. Conclusions
We have underlined the importance of the no-slip boundaries as vorticity sources in
2D turbulence on a bounded domain by two illustrative examples of forced 2D turbulent flows. These studies were motivated by observations from numerical simulations of decaying 2D turbulence, where it was shown that the high-amplitude vorticity produced at solid boundaries enters the flow at scales comparable with the average boundary-layer thickness δ. We have highlighted some of our earlier observations
of the role of no-slip walls on the dynamics and properties of decaying 2D turbulence,
such as the modified spectra, the evolution of vortex statistics and the spontaneous spinup phenomenon, and focused our investigation of forced 2D turbulence on some of
these aspects.
Both experiments and direct numerical simulations of forced 2D turbulence provide
additional evidence supporting our previous observations. The experiments and simulations
on continually forced 2D turbulence by oscillating spin-up underlines the role of the thin
boundary layers as sources of small-scale vorticity (which is visible in the energy spectra)
H.J.H. Clercx et al. / Dynamics of Atmospheres and Oceans 40 (2005) 3–21
19
and small vortices. The boundaries clearly act as sources of vortices, hence modifying the
vortex statistics. Long-time simulations of stochastically forced 2D turbulence provided the
clue to experimental observations on sudden sign changes of circulating flows in thin fluid
layers and in Rayleigh–Bénard convection: the destabilizing effect of the boundary layers
results in the sudden collapse of the large-scale circulation cell after which the flow spins
up again. This process goes on indefinitely.
Acknowledgements
One of the authors (D.M.) gratefully acknowledges financial support from the Dutch
Foundation for Fundamental Research on Matter (FOM). This work was sponsored by
the Stichting Nationale Computerfaciliteiten (National Computing Facilities Foundation,
NCF) for the use of supercomputer facilities, with financial support from the Netherlands
Organization for Scientific Research (NWO).
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