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von Karman Institute for Fluid Dynamics
Aerospace Department
Universitá ”La Sapienza” di Roma
PhD. Dissertation
Numerical study of stability of
flows from low to high Mach
number
Fabio Pinna
Promoters: Prof. Paolo Orlandi
Prof. Renato Paciorri
Supervisor: Prof. Patrick Rambaud
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Contact information:
Fabio Pinna
Von Karman Institute for Fluid Dynamics
Chaussée de Waterloo 72
1640 Rhode-Saint-Genése
Belgium
email: [email protected]
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Invictus
Out of the night that covers me,
Black as the pit from pole to pole,
I thank whatever gods may be
For my unconquerable soul.
In the fell clutch of circumstance
I have not winced nor cried aloud.
Under the bludgeonings of chance
My head is bloody, but unbowed.
Beyond this place of wrath and tears
Looms but the Horror of the shade,
And yet the menace of the years
Finds and shall find me unafraid.
It matters not how strait the gate,
How charged with punishments the scroll,
I am the master of my fate:
I am the captain of my soul.
William Ernest Henley (1849 - 1903)
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To my family.
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Acknowledgement
During these last four years a lot of people crossed my way and many of
them gave a direct or indirect contribution to this thesis. At the moment
of writing this page, this long process reached its natural end and therefore it is the right occasion to acknowledge all those contributions.
Among these people I would like to mention my supervisor at VKI Prof.
Rambaud, who pushed me to do my best, trusting my work and supporting me even in the darkest moments of my research. I would like to thank
also my promoters at the University ”La Sapienza” Prof. Orlandi and
Prof. Paciorri who allowed the collaboration with VKI, by accepting me
in the doctoral program. I want also to acknowledge Prof. Özgen for his
precious advices during his visits at the Institute.
After my Diploma Course a lot of friends moved away for new opportunities so I shared my daily troubles, both technical and personal, with
the remaining ones like Ben, Ian, Memo, Vasilis and one year later also
with Delphine: tank you for making this place feel like a new home. I
cannot forget other VKI mates who became friends even without sharing
the same DC year: Alessandro, Flora, Alessandro, Aurora, Filippo and
Marta. I would like to acknowledge also my office mates Anna and specially Lilla, with whom I spent a lot of time discussing about the most
diverse topics, both personal and technical. Last but not least the new
generation of PhDs like Alessandro (both), Antonino, Erik, Francesco,
Laura, Sergio and Michelangelo (technically not a PhD).
Luckily I was not alone while working on transition and I had the luck
to speak about it with Sandy, Davide, and Henny who shared part of my
research path. To all of you guys I wish good luck with your PhD and future career. At this regard I would like to mention Olaf who deeply helped
me clarifying many concepts about transition: thank you very much for
your kindness and expertise. On a similar note I cannot forget the first
student who worked with me: Gennaro, it was a real pleasure to have you
as a student I wish you all the best for your PhD.
The moment I enrolled in the doctoral program, me and my future wife
needed to move all our stuff: we had the help of our friends Francesco and
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Monica. I still have that long trip in mind as one of the funniest trip ever,
and when we were done with our sixteen hours driving through Europe,
we slept at my place with no furniture whatsoever: there are no words to
express my gratitude and love, and without your help this would not have
been possible at all. I also would like to thank my friend Federico who
helped me, from time to time with the little big problems I encountered
during my PhD.
My life expanded outside the border of VKI and I had the chance to
meet, ironically, two people coming from my city: Alberto and Marta, I
cannot count anymore the hours we spent playing board games but we
had definitely a good time. Later my friend Francesco moved to Bruxelles increasing my Italian community in Belgium together with Magda,
Riccardo, Enrico, Federica. It has been a pleasure to spend my time with
you, I thank you for the beautiful moments we had together.
Being away from the place where I grew up was hard and even harder
to be away from my family, nevertheless I never missed support for my
undertakings. I would like to thank my parents and my grand parents: I
hope this achievement will make you realize how good was your influence
on me. On the other hand the family concept become a bit wider during
these years, as I got married and I became part of the wonderful and big
family of my wife: to all of them a big ”thank you” for their support and
faith in me.
Not to miss anybody I would like to thank Fabiana, my wife. We started
this big adventure of moving abroad and I do not even know when it will
end. If I achieved this goal is due also to her contribution and to her
support during these long four years. I am looking forward to start a new
adventure with you in the coming years.
I want to mention, in the end, my daughter Flaminia (but a thought also
goes to her sister who we hope to meet soon), who literally wanted to sit
on my laps to code, prepare the graphs and touch the keyboard: if that
did not help at least it cheered me up during the hours spent in front of
my laptop.
Thank you all.
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Abstract
Interest in the study of stability lies in the strong link with the laminar-toturbulent transition with several implications in the design of new vehicles
in the aerospace community. If on one side aeronautics would like to predict and control transition to limit drag, boost efficiency and reduce the
fuel consumption on the other hand, space design faces reentry problem
where an accurate transition prediction could lead to a better sizing of
the thermal protection system, thus enhancing the overall performance of
the spacecraft.
Nevertheless one has to keep in mind that transition can occur because of
several causes, which make the link between stability and transition not
as direct as one would it like to be. On an engineering point of view the
use of the eN method has been selected as our preferred way of estimating
the onset of transition.
Despite the great amount of software available to study stability of incompressible flows and, to a lesser extent, low supersonic flows, there is
a restricted number of codes dealing specifically with hypersonic flows.
The objective of this work is then to write a consistent toolkit to be able
to study stability of flows at different regimes, from low to high Mach
numbers.
The VESTA toolkit (VKI Extensible Stability and Transition Analysis
toolkit) gathers a number of codes for different regimes which are based on
Chebyshev pseudo-spectral methods. It compares well against literature
for the incompressible and compressible flows while hypersonic cases are
more difficult to test, because of the narrower body of literature treating
them. Nevertheless some comparisons allow us to estimate a reasonable
good matching with existing results.
The incompressible study was not limited to the reproduction of standard cases and techniques but included also an original expansion of the
τ -method capable of treating boundary layer flows. Results have been
verified against the ones obtained by other methods even if its complexity makes the τ -method an unideal candidate for the development of the
compressible solver.
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The compressible linear stability solver is able to cope with both subsonic
and supersonic regime and it has been used to verify and expand the current database of neutral stability curves at different Mach, by plotting
results for adiabatic flows with different free stream static temperatures.
The compressible solver served as a base for the implementation of the
specialized hypersonic code. For this case, when temperature is high
enough, air molecules start dissociating and chemical reactions happen
between the different species. For this reason air should be considered as
a mixture of gases. In the present work this effect is taken into account
by means of the Local Thermodynamic Equilibrium (LTE) assumption.
A deep investigation has been carried on the effect of free stream temperature and pressure. It turned out that in the investigated range, pressure
plays a minor role, while temperature results to be the driving parameter, even more at lower Reynolds numbers. It has been observed that the
critical Reynolds number decreases when temperature increases. As the
free stream temperature increase the neutral stability curves show also a
larger instability area. Another important aspect of hypersonic flows is
the strong shock in front of the body. Its influence on the boundary layer
has been modeled as a boundary condition for the calorically perfect gas
(CPG) and LTE solver. Nevertheless the latter implementations is new
and only simple verifications against a calorically perfect gas was possible.
It is already known in literature that the shock stabilizes the flow at low
wave numbers. Our computations on chemically reacting flows found out
that LTE tends to stabilize it even more, with growth rates as low as three
times the respective one for CPG.
Future works will take advantage of the modularity of the VESTA toolkit
to add other stability feature like Parabolized Stability Equations and
BiGlobal stability. On the numerical side some work will be devoted to
the implementation of more general and faster algorithm retaining the
same level of accuracy of the present solvers. On a different level the
results found in this work could be readily used together with an eN code
for the prediction of transition for flight condition and ground testing
while the software will be used as the founding brick of an Uncertainty
Quantification for transition prediction.
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Contents
Nomenclature
1. Introduction
1.1. Historical perspective . . . . . . .
1.2. Paths to transition . . . . . . . . .
1.3. Different approaches for transition
1.4. Objective of the thesis . . . . . . .
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2
4
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2. Linear stability for incompressible flows
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Governing equations . . . . . . . . . . . . . . . . . . . .
2.3. Numerical methods . . . . . . . . . . . . . . . . . . . . .
2.3.1. Temporal and spatial case . . . . . . . . . . . . .
2.3.2. Definition Chebyshev polynomials . . . . . . . .
2.3.3. Mapping techniques . . . . . . . . . . . . . . . .
2.3.4. D2 method . . . . . . . . . . . . . . . . . . . . .
2.3.5. τ -Method . . . . . . . . . . . . . . . . . . . . . .
2.3.6. Results of τ -method . . . . . . . . . . . . . . . .
2.3.7. Chebyshev Collocation Method . . . . . . . . . .
2.3.8. Results of collocation method . . . . . . . . . . .
2.4. Summary on the incompressible linear stability problem
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3. Linear stability for compressible flows
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Remarkable features of compressible flows . . . . . . .
3.3. Governing equations . . . . . . . . . . . . . . . . . . .
3.4. Numerical method . . . . . . . . . . . . . . . . . . . .
3.5. Boundary conditions . . . . . . . . . . . . . . . . . . .
3.6. Incompressible computation . . . . . . . . . . . . . . .
3.7. Compressible computations . . . . . . . . . . . . . . .
3.8. Neutral stability curve computation . . . . . . . . . .
3.9. Summary on the compressible linear stability problem
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Contents
4. Linear stability for chemical equilibrium flows
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Governing equations . . . . . . . . . . . . . . . . . . . . .
4.3. Properties interpolation . . . . . . . . . . . . . . . . . . .
4.4. Comparison between LTE gas calorically perfect gas flows
4.4.1. Flat Plate flow at Mach 2.5, adiabatic wall . . . .
4.4.2. Flat plate flow at Mach 10 cold wall . . . . . . . .
4.4.3. Flat plate flow at Mach 10 adiabatic wall . . . . .
4.4.4. Pressure and temperature effects . . . . . . . . . .
4.5. Summary for lst of lte flows . . . . . . . . . . . . . . . .
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5. Shock as a boundary condition
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Mean flow . . . . . . . . . . . . . . . . . . . . . . . . .
5.3. Linearized shock boundary conditons . . . . . . . . . .
5.3.1. Numerical treatment of the boundary condition
5.3.2. Results . . . . . . . . . . . . . . . . . . . . . .
5.4. Summary on the shock boundary condition . . . . . .
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6. Conclusions
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6.1. Future works . . . . . . . . . . . . . . . . . . . . . . . . . 127
A. Computer Algebra Software and Stability Equations
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B. Computation of the Mean Flow in case of
B.1. Blasius profile . . . . . . . . . . . . . .
B.2. Compressible boundary layer . . . . .
B.2.1. Dimensional flow . . . . . . . .
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self similar profiles
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C. Local Eigenvalue algorithm
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List of Figures
139
List of Tables
143
Bibliography
145
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Nomenclature
Greek symbols
α
Non-dimensional stream wise wave number [-]
β
Non-dimensional span wise wave number [-]
δ
Boundary layer displacement thickness [m]
φ
Amplitude function vector
φx,y,z
Projected Rankine-Hugoniot on the respective axes
λ
Second viscosity coefficient [kg/(m · s)]
µ
Dynamic viscosity [kg/(m · s)]
ν
Kinematic viscosity [m2 · s]
ω
Non-dimensional frequency [-]
ψ
Generic basis function
Υ
Conservative variables vector
ρ
Density [kg/m3 ]
θ
Boundary layer momentum thickness [m]
ξ
Mapping function
Roman symbols
an
Generic expansion coefficient for Chebyshev series
b
Velocity vector [m/s]
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Contents
cp
specific heat at constant pressure [J/( kg · K)]
DN
Derivative operator or derivative matrix of the nth
order
I
identity matrix
k
Thermal conductivity [W/(m·K)]
l
Blasius length scale [m]
p
Pressure [Pa]
Q, q
Generic unsteady flow variable
Re
Reynolds number
T
Temperature [K]
Tn
nth Chebyshev polynomials
U, u
x-component velocity [m/s]
V, v
y-component velocity [m/s]
W, w
z-component velocity [m/s]
x
Stream wise coordinate [m]
y
Normal to the wall coordinate [m]
z
Span wise coordinate [m]
Ul
ν
[-]
Sub- and superscripts
e
Free stream quantities (or at boundary layer edge)
·
Generic mean flow variable
·0
Generic perturbed flow variable
ã
Amplitude function for perturbations
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Chapter 1.
Introduction
1.1. Historical perspective
Transition is still an active research topic these days. Despite the effort
undergone to study it, its nature is still far from being completely understood, being the number of causes that might affect it extremely large.
Without going back to the first observations on the nature of the fluid
flows, study on transition is usually dated back to the work of O. Reynolds
(1883) who injected an ink trace in a pipe with water flowing in it. He
could show clearly that two distinct flow regimes are possible: laminar
and turbulent. The introduction of the non-dimensional number carrying
his name was an early attempt to describe and possibly predict the behavior of the flow. Later experiments highlighted the fact that, though
Re is an important parameter it is not universal and other causes play a
role in determining the transition to turbulence.
In 1887 Lord Rayleigh made the hypothesis that the onset on transition is
depending on the development of unstable waves. A further assumption
was that the viscosity played a dumping role in the instability phenomena
and therefore it was, at first, neglected. Rayleigh found that a velocity
profile with an inflection point is unstable in nature and he concluded that
only flows with inflectional velocity profiles could lead to turbulence. He
could prove that this was a necessary condition in order to have unstable
waves while only with W.Tollmien several decades later (1935) the inflection point was demonstrated as sufficient to have amplified perturbations.
Nevertheless there were evidences of some type of flows, like Poiseuille’s
one, where transition happens despite the inflection-free velocity profile.
This paradox was explained by Prandtl who showed in 1921 the destabilizing role of viscosity. In the following years, the development of new
techniques will lead to the solution of the Orr-Sommerfeld equation (see
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Chapter 1. Introduction
chapter 2) in a more refined way.
In 1946 Lees and Lin [52] developed an inviscid theory for compressible
flows. This allowed to get some important results, though it will be the
work of Mack ([55], [56], [57], [58], [59], [54])l to highlight the main differences between the stability of compressible and incompressible flows.
First outcome of this research was that it is no longer possible to neglect
the role of 3D waves when the flows is supersonic. The second important
result states that there is an infinite sequence of wavenumbers for each
phase velocity whenever the mean flow is supersonic with respect to the
phase velocity.
1.2. Paths to transition
The main reason for the elusiveness of transition has to be sought in the
number of causes that influence it. Among them we should count as many
as body geometry, curvature, roughness, temperature, free stream turbulence, noise and many more. All of them may induce transition but the
way boundary layer reacts to such an input is still unclear even though
many advances have been made through the years. Apparently there is
not only a mechanism, which takes initial small disturbances leading them
to turbulence. Nevertheless without such an understanding it is impossible to have a clear mathematical modeling of the transition behavior.
What we need to understand is then how disturbances are entrained inside
the boundary layer (the receptivity problem) and then how those initial
disturbances grow (or decay) eventually breaking down into turbulence.
A general transition process could be described by a simplified scenario,
where the initial disturbances as, for instance, sound waves, hit the boundary layer which, according to some other parameters, like temperature
curvature, roughness and so forth could grow or die out. When perturbation are very small compared to the mean flow their behavior could
be described by linear stability theory. When the perturbation amplitude grows secondary instabilities appear. If initial perturbations are big
enough the linear phase could be by-passed causing the secondary instabilities to appear sooner and the flow to become turbulent quickly. In this
case linear stability computation does not apply and predictions based on
the well know eN method could not be formulated.
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1.2. Paths to transition
Increasing disturbance level
Forcing Environmental Disurbances
Receptivity mechanisms
A
Transient growth
E
B
Eigenmode growth
C
Parametric Instabilties
& Mode Interactions
D
Bypass
Mechanisms
Breakdown
Turbulence
Figure 1.1.: Paths to transition for wall bounded flows ([87])
In Reshotko [87]) five paths are identified related to the the introduction of the transient growth theory as depicted in Fig. 1.1. The path A
correspond to the traditional Tollmien-Schlichting Görtler, or crossflow
mechanisms. So disturbances are entrained in the boundary layer then
we have the linear phase the non linear instability and finally the the
breakdown to turbulence. This is the kind of instability that could be
looked at using tools like linear stability theory or parabolized stability
equations coupled with eN method. Path B described a scenario with no
obvious examples in literature where transient growth cause higher initial
perturbation amplitude with respect to the modal growth until the modal
instability takes over. In path C the eigenmode growth is not present
and is the most studied transient growth scenario. This is also related to
the blunt body paradox, which is a well known phenomenon in the hypersonic/supersonic transition community. The path D shows a turbulent
spectrum while basic flow is laminar and there are experimental evidences
for turbulence intensity greater than 1%. If the perturbation amplitude
is even higher there is no linear regime and the free stream spectrum does
not resemble what is found in wind tunnels.
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Chapter 1. Introduction
1.3. Different approaches for transition
With the increase of the computational power over the years, the overwhelming stability problem could be solved more easily than before and
new techniques were applied to gain a better insight into the physics of
transition. These new points of view are needed as transition is not always
predictable by traditional methods and later stages of transition are non
linear in nature, making it impossible to understand the physics behind
them with linear methods.
Traditional work on stability and transition started by linearizing Euler
equations, looking for the inviscid flow stability, nevertheless the number
of approaches has increased over the years and nowadays several techniques are available, according to the type of research one wants to pursue.
With the availability of more powerful computers, direct numerical simulation of Navier-Stokes equations became a more feasible way to study
turbulence and laminar to turbulence transition. The terms direct numerical simulation dates back to Orszag (1970), 1 where he referred to
the possibilities of getting result via turbulence theories or directly from
the Navier-Stokes equations.
dns and linear stability proved to be excellent also in comparison with
experiments, nevertheless linear stability has been the only way to study
transition at higher Reynolds number. Needless to say at the age of the
first computers also Orr-Sommerfeld equation was a formidable task to be
performed and coded; for this reason the temporal problem (see §2.3.1)
was the preferred one by the investigators as it required less resources.
One of the first spatial case is the one of Jordinson [45]. Still in the
nineties solution to the spatial linear stability problem for compressible
flow was quite expensive, such that approximate solution were used (see
[62]).
In supersonic flows the presence of the shock introduces another parameter to be coped with and its interaction with the boundary layer is quite
a broad topic in itself Though shock capturing is possible, another kind
of simulation, very common in the analysis of supersonic and hypersonic
flows, excludes the presence of the shock in front of the body to provide
the inflow condition after the shock as coming by a less refined simulation
1 ”Analytical
Theories of Turbulence”. Journal of Fluid Mechanics (see, e.g., page
385) 41 (1970): 363386)
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1.3. Different approaches for transition
or a self similar profile. This way, all the numerical problems related to
the treatment of discontinuities are discarded, making easier and more
efficient to use higher order methods (see for instance [6] and [48] ). A
lively community grew also around the roughness induced transition in
compressible flows: example could be seen in [53], [86] or [40], each with
a different numerical approach. The non linear behavior generated by the
roughness makes dns the perfect candidate to study this kind of problems,
anyway this is limited to very simple geometries and therefore only fundamental studies. Recently the simulation of hypersonic flows by means
of dns begins to include also the effect of chemical reactions. Flat plate
boundary flows have been studied, as in [93] or [66]. The intrinsic challenge of this kind of simulations lies in the number of equations that should
be solved at each iteration; “usual” Navier-Stokes should be modified to
treat the different species appearing in the flow (a number of mass balance equations equal to the number of species should be considered) and
also all the finite rate reactions should be computed. In case the flow is
considered in equilibrium, the computation could be made lighter by using a lookup table which is anyway more expensive than standard viscous
calculations. On the other hand this approach, even if it is not introducing any modeling in the Navier-Stokes itself, is actually using a series of
assumptions which do not always describe properly the thermo-chemical
behavior of such a flow.
A less expensive approach is represented by Large Eddy Simulation (les)
whose first results were presented by Deardrof in [28], [29], [27]. It aims
to resolve bigger turbulent scales (the so called large eddies), modeling
the remaining smaller ones. Nevertheless quite a large portion (around
80%) of kinetic energy should be resolved in order to have a reliable les
(see [81]). Moreover the mesh size requirements for wall bounded flows, or
regions close to the wall, are extremely close to dns ones, thus throwing
away part of the les appeal. The smaller scale modeling (sgs) is the way
part of the not resolved spectrum is accounted for within the simulation.
Of course this affects the final results and a different modeling leads to
different skin friction values (see [89]). Nevertheless les has been applied
to study laminar to turbulence transition and flow stucture belonging to a
transitional flow could be identified especially by using some specific sgs
model (see [90] and [89]) .
A good modeling capability of the flow stability and transition predictions
could be achieved by using the parabolized stability equations. They were
first introduced by Bertolotti and Herbert in [10] and [36]. It contains non
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Chapter 1. Introduction
parallel terms and it can consistently consider curvature. It needs a basic
flow which should be accurately computed before running it. Computation
of a supersonic flow has been achieved already by Bertolotti & Herbert
[11] and Chang [20]. Stability simulation with pse proved to be reliable
and some of the non linear behavior can be retrieved by the non-linear
formulation (npse).
A lot of work has been done to extend this technique to flow with chemical
reactions. Examples could be found in the work of Malik [63] or Johnson
& Candler [43] [42] [44]. Nevertheless the use of pse poses some problems
with respect to receptivity of the disturbance and some work has been
dedicated to this issue. There are not conclusive results and this is still
an open field of research. It is worth to remember that the system of
equation is valid up to the onset, while downstream of it the code simply
blows up. Another technique is the BiGlobal stability, reviewed recently
by Theofilis in [95]. Unlike the pse, there is no requirement to have a
convective instability and it can deal with non parallel boundary layer.
It has been used also for compressible cases, like on a cone at Mach 4,
nevertheless its main drawback is related to the huge computational power
required to solve the eigensystems. So far there has been no application
at hypersonic speeds.
Despite the fact that transition is an inherently unsteady phenomenon, a
number of researches proposes transition models where a number of relations should return the behavior of a transitional boundary layer, feeding
this results in a standard cfd code. Of course, as it happens in rans,
their validity is limited, nevertheless this approach could lead to some
results, especially for quantities like heat flux and skin friction, at least
from an engineering point of view. In the work of McKeel [70] a series of existing rans models is discussed and implemented, ranging from
Baldwin-Lomax to standard Wilcox’s k −ω. Also the Schmidt & Patankar
modification to low-Re k − method was investigated, along with the one
equation turbulence model accounting for first and second mode developed by Warren, Harris and Hassan. Anyway, since some years, it has
been certified that low-Re models cannot satisfactorily predict the transition process, being their behavior due to an incidental similarity between
the viscous sublayer and the developing laminar boundary layer. Also the
algebraic model of onera/cert and the one from Dey and Narashima,
which linearly blends the laminar and turbulent field through intermittency to simulate the transition region, have been analyzed in [70]. All
these models showed some sort of drawbacks, relying on a dedicated tun-
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1.3. Different approaches for transition
ing for a correct prediction of transition.
Other approaches, like the one of Sieger et al. [92], were based on empirical correlations to determine the start and ending transition Reynolds
numbers. These were used to compute, together with the Reynolds number based on the local boundary layer thickness, the intermittency value.
The development of Langtry ([51]), in 2006 used a transport equation for
the intermittency. This is not a novelty in itself as a series of models
have been developed around this idea; the strong point of this approach
is the use of local variables together with correlations, resulting in a good
match for many cases. It has to be noted that, as the model is relying on
correlations, it still lacks of general validity.
Another interesting recent approach has been propose in the work of Mayle
& Schulz [68] where the laminar kinetic energy is transported. The breakdown to turbulence happens, according to this model once the laminar
fluctuations reach a threshold level. This method has been recently reformulated by Walters & Leylek [100] to transport this quantities locally,
making the model ready to be used along with normal cfd applications.
Both models in [51] and [100] are originated in the turbo machinery community and application to hypersonic flows is somehow not recommended
without tweaking the model. Anyway, the special features of hypersonic
testing, make difficult the access to the correlations data, needed to recalibrate these models, and more work will be needed in the future.
As it is easy to imagine, the demand for a transition prediction tool is high
and, because of the lack of a universal description of the transition phenomena, empirical correlations have been created to fill this gap. A large
number of empirical criteria has been created over the years for many cases
ranging from incompressible to hypersonic flows, for natural transition or
roughness induced (distributed, isolated, three or two-dimensional). For
incompressible flows there are three largely accepted criteria: Michel’s one
([72]), Granville’s [35] and the so called H − Rx proposed by Wazzan et
al. in [102]. The first two have been designed with airfoil transition in
mind, while H − Rx can be applied to a wider class of problems (heating,
suction, axis symmetric). As it is typical for these methods, some boundary layer parameters should be computed and then compared against a
curve, which is supposed to return the point of transition.
In supersonic/hypersonic flows a well known criterion is the Shuttle criterion which is basically a ratio of Reynolds based on momentum thickness and Mach number outside the boundary layer. It was proposed by
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Chapter 1. Introduction
Goodrich et al. [34] but it has been criticized a lot since then, particularly
because it is mainly governed by the free-stream density and therefore
not very representative of a boundary layer behavior. The Van Driest
and Blumer, proposed in [31], has been the summary of a series of experiment on sharp cones with three-dimensional spherical roughness. The
Potter-Whitfield [82] has been prepared by using different geometries in
the experiment. None of them include the pressure gradient effect, so
they should be used with caution in case it is present. The work of Reda
([83], [84], [85]) produced an interesting review of existing criteria while
proposing new ones with successive refinement. Once again this relations
lack of generality ans therefore they should be used for a first estimation
of transition for design purposes, rather than scientific investigations.
1.4. Objective of the thesis
By the quick review of the previous section it is apparent that transition
and stability need a set of specifc tools designed to face different aspects
of the phenomenon. It is therefore absolutely out of question to propose a
general cheap method that suits all the needs. If a good general method
is undoubtedly dns, its use on a regular basis for engineering calculations
is still to come. The method proposed by the author aims at providing a
lightweight technique based on solid scientific grounds.
The most natural way to face this problem is the development of different codes based on different approaches and theories taking advantages
of each method. The way the author exploits this idea in the following
chapters of this works is directed to the implementation of a toolkit which
will be referred to from here on as vesta (Vki Extensible Stability and
Transition Analysis toolkit).
This work tries to tackle different goals at the same time, at different levels. The most important one is, as already stated, the development of a
consistent group of softwares performing different tasks but unified by a
common underlying philosophy in the implementation. These codes shall
be able to interact as much as possible with each other and one should
be capable of writing new scripts to solve specific problems, by re-using
what has been already implemented.
The second one is to allow vesta to run an analysis taking as input, the
data coming from several techniques and methods. Moreover we want to
be able to use the code for different flow regimes, according to the specific
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1.4. Objective of the thesis
situations.
A long term aim is the parallel use of vesta along with real facilities, comparing with experimental data. Very often the design of an experiment
and the analysis of the data is strongly linked to numerical computations.
This can be useful to the experimenter while designing the experiment
itself, focusing only on the interesting part of its signals. The pairing
computation-experiment is stronger than never in this field and the author strongly feels that vesta should move to this direction.
The objectives of this work can be summarized as: implementation of a
highly modular and extensible solver and its use in the analysis of different kind of flows.
In the second chapter you will find a detailed explanation of the basis of
linear stability theory, a discussion on the proposed numerical methods
and the details of the original boundary layer D2 τ -method implementation in addition to some general remarks and results. This chapter will
serve as a basis for all the remaining work.
In the third chapter you will find a detailed explanation of the theory of
stability of compressible flows and its findings, along with a study on the
parameters influencing stability
In the fourth chapter you will find the development of a set of stability
equation for hypersonic flows in chemical equilibrium with comparison
against the calorically perfect gas lst. Effects of chemistry will be presented in this chapter.
In the fifth chapter the role of the shock and its influence on boundary
layer stability for hypersonic flows, with or without chemical reactions,
are discussed.
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Chapter 2.
Linear stability for incompressible flows
2.1. Introduction
The first studies concerning stability dates back to Reynolds regarding
experimental evidences, while theoretical studies were performed first independently by Orr and Sommerfeld. The main theory they developed is
still widely used today for incompressible flows and has been extended to
compressible flows, and more recently to hypersonics.
Given the original aim of our work the following chapter will describe the
theoretical background and the numerical techniques used to solve the
linear stability problem for incompressible flows.
2.2. Governing equations
After Reynolds (1883) the evidence of the turbulent behavior of flows
under special circumstances gave momentum to the study of the turbulent phenomena and transition from laminar to turbulent. A common
idea at the time linked this change to small disturbances acting on the
mean flow. For this reason the main point of the whole phenomenon was
whether these disturbances would die out or get amplified. In this sense it
is possible to call a flow respectively stable or unstable. Reynolds (1894)
proposed the idea that a flow, initially laminar, would become turbulent
once a certain threshold for those disturbances has been exceeded.
To this regard the original aim is to find the value of an critical1 Reynolds
1 This
is sometimes called also indifference Reymolds number letting the term critical
to represent the point at which the transition is complete. In this work the term
“critical” refers to the Reynolds number value below which tall the perturbations
are dumped
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Chapter 2. Linear stability for incompressible flows
number: such a task was first accomplished by W.Tollmien and H.Schlichting.
It is important to note that unstable flows allow laminar solution but any
perturbation, arbitrarily small, will get amplified, leading eventually to
chaotic behavior and turbulence. As seen in the introductory chapter
there are several paths to transition and in this chapter we will investigate mainly the modal growth.
As the early investigators guessed, transition flows could be imagined as
a sum of a laminar mean flow and perturbations. Following the usual
approach used in the study of stability of physical systems one assumes
the disturbances to be small (in comparison with the mean flow) and linearizes the problem around the base solution i.e. the laminar mean flow.
Starting from the incompressible Navier-Stokes equation is then possible
to introduce the decomposition of the instantaneous flow as
Q(x, y, z, t) = Q(x, y, z) + q 0 (x, y, z, t)
(2.1)
where Q is a generic flow variable, like velocity or pressure. For our purpose we assume the flow to be particularly simple, namely not depending
on the stream-wise coordinate but only on the wall-normal direction y.
Additionally the wall-normal velocity component V is assumed to be 0.
This is consistent with the conservation of mass. In a two-dimensional
flow asking for dU / dx = 0 implies the derivative of V with respect to y
to be null, which is forcing the vertical velocity profile to be zero because
of the non penetrability condition
While channel or pipe flow correspond exactly to this representation
(established flow), boundary layer is only approximated by them and this
is justified only by the smaller dependence of the mean flow on y with respect to x leading to the introduction, in case of a boundary layer study,
of the parallel flow assumption.
After plugging eq. (2.1) into the Navier Stokes equations, all the products
of two perturbations quantity are dropped (considering them negligible),
thus linearizing around the mean flow (or basic state). The result is a set
of equations that could be solved directly once the mean flow is known,
making no hypotheses on the kind of perturbation but requiring a large
computational effort.
After defining the generic vector for the mean flow variable as Q =
[U , V , W , P ] and its counterpart for the disturbances q 0 = [u0 , v 0 , w0 , p0 ],
a truly parallel mean flow where U = U (y), V = 0 and W = W (y) is
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2.2. Governing equations
described by the following equations:
∂u0
dU
∂u0
∂u0
+U
+ v0
+W
∂t
∂x
dy
∂z
0
0
∂v
∂v
∂v 0
+U
+W
∂t
∂x
∂z
0
0
∂w
∂w
dW
∂w0
+U
+ v0
+W
∂t
∂x
dy
∂z
∂u0
∂v 0
∂w0
+
+
∂x
∂y
∂z
∂p0
1 2 0
+
∇ u
∂x
Re
∂p0
1 2 0
=−
+
∇ v
∂y
Re
∂p0
1 2 0
=−
+
∇ w
∂z
Re
=−
=0
(2.2)
(2.3)
(2.4)
(2.5)
where Re = U Lref /ν is the Reynolds number. Another key assumption
in Linear Stability Theory (hereafter contracted in lst) concerns the assumption of perturbations each one propagating independently. These are
called modes and for a general disturbance are represented by:
q 0 (x, y, z, t) = q̃(y) · exp(iαx + iβz − iωt) + c.c.
(2.6)
where “c.c” stands for complex conjugate.
Substituting eq. (2.6) in eqs. (2.2)-(2.5) we obtain a 6th order system of
equations that could be combined in the so called Orr-Sommerfeld (after
W.M.F. Orr (1907) and A.Sommerfeld (1908)) equation:
d2 ṽ
d2 ṽ
d4 ṽ
− 2k 2 2 +k 4 ṽ − iR[(αU + βW − ω)( 2 − k 2 ṽ)+
4
dy
dy
dy
2
d U
d2 W
− α 2 +β
ṽ = 0
dy
dy 2
(2.7)
where k 2 = α2 + β 2 . Substituting kU = αU + βW eq. 2.7 is identical to
its 2D counterpart. Impenetrability boundary condition gives ṽ = 0 while
the no-slip condition gives ũ = 0 that, after the linearized conservation of
mass, turns into dṽ/ dy = 0.
From Squire’s transformation it is known that a 3D Orr-Sommerfeld mode
correspond to a 2D Orr-Sommerfeld mode at lower Reynolds number.
This means that an incompressible parallel shear flow becomes unstable
to two dimensional modes or in other words that if a 3D Orr-Sommerfeld
mode is unstable a 2D mode is unstable at a lower Reynolds number.
Therefore for an incompressible flow it is enough to study only twodimensional cases.
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Chapter 2. Linear stability for incompressible flows
In case of boundary layer flows, all the previously stated assumptions
are valid only locally. This implies that at each stream-wise location the
problem should be rescaled according to a length Lr (defined later) which
takes into account the exact position of the computation. Eq. (2.6) needs
to be changed accordingly to this observation and a new phase function
Θ is introduced with eq. (2.6) that now reads
q 0 (x, y, z, t) = q̃ · exp(iΘ) + c.c.
(2.8)
where
Θ = Θ(x, z, t)
∂Θ
=α
∂x
∂Θ
=β
∂z
∂Θ
= −ω
∂t
In order to completely describe the three-dimensional flow field a second
equation should be provided andeee this is usually chosen to be the normal
vorticity equation
∂u ∂w
η=
−
(2.9)
∂z
∂x
From eq.(2.9) the Squire’s equations is retrieved which is depending directly on eq.(2.7), giving always stable modes. As transient growth phenomena will not be treated in this work Squire’s equation will not be
mentioned any further.
The wavenumbers α and β and the frequency ω are generally belonging
to C. By fixing ω ∈ R the desired values are the complex wavenumbers
α = αr + iαi and β = βr + iβi . This is called the solution of the spatial problem while if one fixes α ∈ R the problem is defined as temporal.
The equation are non-dimensionalized and the value outside the boundary
layer have been used for the velocity (Ue ), density (ρe ) and viscosity (µe ).
Length is non-dimensionalized with respect to a reference length (Lref )
and the non-dimensional time is t = Lref /Ue . For an incompressible flow
it is intuitive to take as reference length the boundary layer thickness.
Still, this is ambiguous because even for a simple Blasius boundary layer
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2.3. Numerical methods
there are multiple possible choices. One could take, for instance, the
displacement thickness:
r
νe x
δ = 1.7208
(2.10)
Ue
the momentum thickness
r
θ = 0.664
νe x
Ue
or even the simple Blasius length scale
r
νe x
l=
Ue
(2.11)
(2.12)
For our computations, if not otherwise specified, we will choose the last
definition. Nevertheless this choice does not affect the numerical result
provided that Re, α, β, ω are scaled accordingly
2.3. Numerical methods
Several methods have been developed over the years to compute linear
stability of boundary layer flows. It is worth to note that for several years
of the past century the fully viscous problem was numerically prohibitive
to solve and usually simplified version of the stability equation was solved
instead. The first numerical computation of the Orr-Sommerfeld equation has been done by E.F. Kurtz and S.H. Candrall in 1962 ([49]), three
decades after the first publication of the analytical effort of W.Tollmien
in 1929 ([96]). The papers of R. Jordinson in 1970 and 1971 ([45] , [46] )
should also be mentioned for the spatial solution of the Orr-Sommerfeld
equation. Nevertheless, it has been known for years that the stability of
the pipe (or channel) flow is an easier task, mainly due to the fact that
the laminar velocity profile is an analytical solution of the Navier-Stokes
equations, and therefore much of the early attempts studied the stability
of a Poiseuille flow.
The beauty and ease of having just one equation ready to describe the
stability feature of a parallel flow was also the cause of some of the well
known problem of its solutions. An historical way of solving the problem
is by finite difference and shooting method as proposed in [19]. Indeed
the problem is split in a four time bigger one. From [23] it is known that
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Chapter 2. Linear stability for incompressible flows
splitting the solution is not sufficient to solve the problem, as a matter of
fact to have an accurate enough solution a procedure called orthonormalization has to be executed from iteration to iteration in order to maintain
the independence of the solutions in which the problem has been split. In
[75] the Gram-Schmidt orthonormalization has been used with successful
results.
Though finite difference methods have been investigated early during this
work, they have been abandoned in favor of spectral methods. This is
due to the higher accuracy of spectral methods which need, in general,
a smaller number of points to achieve the solution. Mainly two pseudospectral techniques have been applied: Chebyshev τ -method (§2.3.5) and
Chebyshev collocation (§2.3.7). The collocation method will also bring
some simplifications in the implementation of the code and the manipulation of the boundary conditions, as it will be explained in the following
chapters.
2.3.1. Temporal and spatial case
Perturbations in boundary layer have been modeled according a wave-like
decomposition and this standard approach corresponds to the evidence
that a perturbation could propagate in time and/or in space. In this
sense the Orr-Sommerfeld equation eq. (2.7) is just a dispersion relation
in the form
F (α, β, ω, Re) = 0
(2.13)
This implies the possibility of solving the Orr-Sommerfeld equation for a
spatial case if the wave propagation happens in space or a temporal case
if the perturbation evolves in time.
The parameters α, β, ω belong to C and solve a temporal problem it
simply means to fix α as R and to solve for ω which in this case will be
complex. The imaginary part will be the one expressing the growth or
decay in time of the perturbation amplitude.
Though theoretically there is no difference between them it is worth to
note that the temporal case is much easier to solve than the spatial one.
This is due to the fact that the temporal case is a linear eigenvalue problem while the spatial one is non-linear as the corresponding eigenvalue α
appears up to the fourth power.
Our aim is always to find an algorithm that allows the simultaneous solution of the whole spectrum because iterative methods require a good
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2.3. Numerical methods
starting guess which is not always available. With this in mind we used
the algorithm presented in [14] and [15] which is reducing the non-linear
problem
Dn (α) = C0 λn + . . . C1 λn−1 + . . . Cn
(2.14)
Dn (α) · X = 0
(2.15)
to a standard eigenvalue problem of the form
A · X = λB · X
(2.16)
As the non-linearity is only of polynomial type, it is possible to write a
new eigenvector which reads
Y = [λn−1 X, λn−2 X, · · · , λX, X]tr
(2.17)
which enables us to write, for a D4 method:
ÃY = λB̃Y
where
and
C1
I
.
à = ..
..
.
0
C2
0
I
0
0
C0
0
.
B̃ = ..
..
.
0
(2.18)
. . . . . . Cn
... ... 0
..
.. ..
.
.
.
..
.. ..
.
.
.
... I
0
(2.19)
... ... 0
... ... 0
.
.. ..
.
. ..
.. ..
.
. 0
... 0 I
(2.20)
0
I
..
.
0
0
A similar layout could be found for the D2 method of §. 2.3.4 Because of
the reduced order in Eq. (2.14) n = 2, therefore we need to build only
three matrices from the system in Eq. (2.39). These are
C0 =
0
−I
−I
0
(2.21)
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Chapter 2. Linear stability for incompressible flows
0
−iReU
C1 =
0
iReU 00
D2
0
C2 =
−I
D2 + iωRe
Composing these matrices we have
−C1 −C2
C
−α 0
I
0
0
0
I
(2.22)
(2.23)
Y =0;
(2.24)
which could be solved with standard techniques.
2.3.2. Definition Chebyshev polynomials
Before starting the description of the methods used in this work, it is
appropriate to introduce Chebyshev polynomials, which are one of the
main building blocks. It is worth to note that there are four kind of
Chebyshev polynomials, the one presented here (see Fig. 2.1) is by far the
most used 2
Tn (η) = cos(n arccos(η)) ∀ n ∈ N0
−1 ≤ η ≤ 1
(2.25)
η = cos(θ)
After Moivre’s Theorem cos nθ is shown to be a polynomial with cos θ as
variable and degree n. It is then straightforward to compute few Chebyshev polynomials
T0 (η) = 1, T1 (η) = η, T2 (η) = 2η 2 − 1 . . .
(2.26)
A quicker way to compute Tn (η) is provided by the trigonometric relation
cos(nθ) + cos(n − 2)θ = 2 cos θ cos(n − 1)θ
(2.27)
By eq.(2.25) we find
Tn (η) = 2ηTn−1 (η) − Tn−2 (η),
n = 2, 3, . . .
(2.28)
2 Concerning
the implementation it is worth to remind that θ = π corresponds to
x = −1 and θ = 0 to x = 1 i.e. these two ranges are traversed in opposite
directions.
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2.3. Numerical methods
1
n=1
n=2
n=3
n=4
n=5
n=6
n=7
n=8
Tn(x)
0.5
0
−0.5
−1
−1
−0.5
0
x
0.5
1
Figure 2.1.: First eight Chebyshev polynomial
Of paramount
pimportance is their orthogonality property when a weighting function (1 − η 2 ) is considered
Z
1
−1
1
Tn (η)Tm (η)(1 − η 2 )− 2 dη =
where
(
2
1
n=0
n>0
(
0
1
n 6= m
n=m
cn =
δnm =
π
cn δnm
2
(2.29)
Another interesting property concerns the interpolation of a function that
will be used later in the collocation method. It is well known that Lagrange
interpolation, with evenly spaced sample points, fails to converge for N →
∞; the approximation get worse and worse as the number of points grows
as it oscillates close to the border of the domain up to values very far
from the ones to be interpolated. This is called the Runge phenomenon
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Chapter 2. Linear stability for incompressible flows
and it happens even for smooth function. Cure of this problem relies
on point clustering at the domain boundaries and this task is excellently
accomplished by Chebyshev polynomials. Such a good behavior could be
explained but it is far from the purpose of this work (see [97]).
2.3.3. Mapping techniques
The issue related to the fixed computational domain and the physical
domain has been mentioned in §2.3.2. This is of course independent of
the method and it will examined here in detail in a general way. The
small definition interval of the Chebyshev polynomials does not satisfy all
the computational needs of a generic problem. For this reason it is worth
to distinguish between a computational domain (the Chebyshev interval)
and a physical domain. In order to pass from one to the other a mapping
function is found. There is no universal choice for the mapping between
the two spaces and this depends strongly on the problem to be solved.
Taking the function y = ξ(η) where a ≤ y ≤ b and −1 ≤ η ≤ 1 it is
straightforward to find out that an equation f 0 (y) = m correspond to
∂f (ξ (η)) ∂η
df (y)
=m⇒
·
= m (η)
dy
∂η
∂y
(2.30)
which means f 0 (y) = f 0 (η)·(1/ξ 0 (η)). Going back to the matrix collocation
framework already presented eq. 2.30 becomes:
1
D1 f = diag
· D1 f ;
(2.31)
ξ0
where D1 represents the ”stretched” Chebyshev derivative matrix. No
other modification is required to transform the physical domain into the
computational one. Basically this operation turns out to be a simple coordinates transformation to be applied to the equation to be solved. The
interesting part is that this transformation could be directly applied to
the derivative matrix. It is possible to obtain a n-th order derivative in the
physical domain by multiplying the derivative matrix N times and this
approach remains valid also when mapping is included. Unfortunately
this method, although ideally correct, is not the best one with respect to
the matrix conditioning.
A more effective implementation is obtained (see [13]) by applying explicitly the coordinate transformation to the n-th order derivative. This
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2.3. Numerical methods
leads to a much more complicated transformation but preserves (or at
least does not worsen) the matrix condition number. For instance a 2-nd
derivative could be computed as follows:
00 f
1
·
D2
−
diag
· D1;
(2.32)
D2 = diag
f 02
f 03
For sake of completeness we report also the expression for the mapped
forth order derivative
1
D4 + diag1f 07 −6f 00 f 02 ) D3+
04
f
1
+ diag 07 −4f 000 f 02 + 15f 002 f 0 D2)+
f
1
+ diag 07 −f 0000 f 02 + 10f 000 f 00 f 0 − 15f 003 D1
f
D4 =diag
(2.33)
(2.34)
(2.35)
The expressions for other derivatives could be found again in [13]. This
technique stays the same regardless of the method used, though it could
be more easily programmed in collocation. Different mappings have been
tried particularly a linear one,
y = (x + 1) · L/2
(2.36)
where L is the maximum length of the domain, a logarithmic one (see
[25])
yj = −K · ln(xj )
(2.37)
where j = 0, . . . , N/2 and the one proposed in [62]
y=a
1+x
b−x
(2.38)
where b = 1+2a/ymax and a = yi ym ax/(ymax −2yi ) with yi corresponding
to the location x = 0.
This mapping clusters half of the total number of points in a portion
of the domain yi chosen by the user. Choice of the yi is not so crucial
for incompressible flows as the only care one should take concerns the
clustering close to the wall. Anyway if the clustering is not correct an
increasing number of point will be needed to solve properly the stability
equation while for a correct mapping a number of 40 points is sufficient
in most of the cases.
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Chapter 2. Linear stability for incompressible flows
2.3.4. D2 method
Independently from the two Chebyshev discretization techniques a manipulation of the equations could be introduced to solve the problem of
ill-conditioned matrices (more on ill-conditioning in §2.3.6 and §2.3.8).
This technique has been conceived specifically for the Orr-Sommerfeld
problem as this equation is a fourth order ode. As the conditioning depends on the matrix size and on the order of the derivative to improve the
accuracy a different approach has also been used following the example
of [30]. The main idea is about avoiding the use of a fourth order Chebyshev derivation matrix. Indeed the very first solution method with finite
difference relies on a reduction to a system of equation of first order as
suggested also in [18]. We name consistently this approach as D method,
while the one solving directly the Orr-Sommerfeld equation is called D4.
It is worth to note that while reducing the order of equation leads to
a smaller round-off error the memory requirement increase considerably
(keep in mind also that Chebyshev matrix are full and so it is not possible
to take advantage of sparse matrix manipulation algorithm). This method
split the usual Orr-Sommerfeld equation in two of second order
(D2 − α2 )φ − χ = 0
(2.39)
(D2 − α2 )χ − iαRe(U − c)χ + iαReU 00 φ = 0
The variable χ in this case is totally fictitious even if in special cases
could be given physical meanings. Of course the values we are interested in are the ones concerning φ. It is worth to underline that this is
a general method and then different numerical techniques could be used
to discretize this new set of equations. The solution to this new set of
equations has been computed both with collocation method and τ -method.
The temporal problem could be solved building this matrix from eq.(2.39)


I
D2 − α2 I


0
BC1




0
BC2


(2.40)
A=
1
1
2

iαRe
A4
−
iαReA5
D2
−
α
I
−
2
2




0
BC3
0
BC4
where the terms A4 A5 read respectively
A4 = U χ
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2.3. Numerical methods
00
A5 = U φ
The other matrix is




B=



0
0
0
−iαReI
0
0
0
0
0
0
0
0








(2.41)
Boundary conditions are expressed instead by the relations (proper mapping has to be considered also for boundary conditions)
X
BC1 =
(−1)n
(2.42)
X
BC2 =
(−1)(n+1) n2
(2.43)
X
BC3 =
1
(2.44)
X
BC4 =
n2
(2.45)
(2.46)
2.3.5. τ -Method
After the influential paper of Orszag [74] the τ -method technique has been
programmed. It belongs to the big family of weighted residual methods.
The peculiarity of the τ -method lyes in the possibility of choosing a basis
function which does not satisfy the boundary conditions exactly. Instead
of trying to satisfy the boundary conditions like other methods do (such
as the Galerkin Method, another flavor of spectral techniques) it adds
more terms to the expansion. So, having N point in the domain and k
boundary conditions, the general expansion for this method is
uN (x, t) =
N
+k
X
an (t)φn (x)
(2.47)
n=1
and the remaining k equations are obtained from the boundary conditions
N
+k
X
an Bφn = 0
(2.48)
n=1
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Chapter 2. Linear stability for incompressible flows
if B is a linear boundary operator.
Application of Chebyshev polynomials is simply to set the φn (x) = Tn (x).
One of the features making Chebyshev polynomials so appealing is the
possibility to establish some recurrence relation based on their properties.
Let’s imagine one wants to interpolate a function
u(x) ' uN (x) =
N
X
an Tn (x)
(2.49)
n=0
and compute its derivative.
u0N (x) =
N
−1
X
a(1)
n Tn (x)
(2.50)
n=0
(1)
where the set of coefficients an is different from the usual an . Another
way of expressing the derivative is also
u0N (x) =
N
X
n=1
an
dTn (x)
dx
(2.51)
The interesting point lies in the possibility of finding a relation between the
(1)
an coefficients of the u expansion and the an . From the prosthaphaeresis
formula we have
2 sin(θ) cos(nθ) = sin ((n + 1)θ) − sin ((n − 1)θ)
(2.52)
which by eq.2.25 becomes
2Tn (x) =
0
0
Tn+1
(x) Tn−1
(x)
−
n+1
n−1
(2.53)
Substituting the latter in eq.(2.50), one can compare it with eq.(2.51)
N
−1
X
n=0
X
N
0
0
Tn+1
(x) Tn−1
(x)
−
=
an Tn0 (x)
2
n+1
n−1
n=1
1
a(1)
n
(2.54)
Developing the previous finding one comes to the conclusion that the
following relation exists 3
(1)
(1)
ak − ak+2 = 2(k + 1)ak+1
(2.55)
3 derivation
of the relations among expansion coefficients could be found in many
source as, for instance, Johnson [41]
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2.3. Numerical methods
(1)
Finally an expression for the an coefficient is found
N
X
cn a(1)
n =2
pap
(2.56)
p=n+1
p+n odd
Applying eq.(2.56) recursively it is possible to find the relations for the
second derivative coefficient
cn a(2)
n
N
X
=
p=n+2
p+n even
p p2 − n2 ap
(2.57)
the third derivative coefficient,
cn a(3)
n =
1
4
N
X
m=n+3
m+n odd
m (m − n)2 − 1 (m + n)2 − 1 am
(2.58)
and the fourth derivative coefficient,
cn a(4)
n =
1
24
N
X
p=n+4
p+n even
p p2 (p2 − 4)2 − 3n2 p4 +
+3n4 p2 − n2 (n2 − 4)
2
(2.59)
ap
Upon substitution of the aforementioned formulas in a generic differential
equation, it is then possible to express everything as a function of the
coefficient an . The next step is the ”projection” onto a basis which is
made of Chebyshev polynomials.
Defining
1
2
ψm (x) =
Tm (x)(1 − x2 )− 2
(2.60)
πcm
it is possible to take the inner product of the equation to be solved in
the form hTn (x), ψm (x)i where
Z 1
2
hTn (x), ψm (x)i =
Tn (x)Tm (x)(1 − x2 )−1/2 dx = δnm (2.61)
πcn −1
Recalling eq. (2.29) one has
hTn (x), ψm (x)i = δnm
(2.62)
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Chapter 2. Linear stability for incompressible flows
where δnm is the Kronecker delta.
Boundary conditions are also projected and the original equation has been
reduced to a simple linear algebra problem. Due to the orthogonality of
Chebyshev polynomials is possible to achieve a very easy expression of a
discretized equation in the form
Lij aij = Fij ;
i, j ∈ (1; N + k)
(2.63)
where F is the forcing part, L is a matrix of coefficients and a is the vector
coming from the expansion coefficient.
Algebraic methods could then be used to solve the system according to
the kind of problem
It must be noticed that the original paper [74] deals with O-S for a
Poiseuille flow because this does not need any kind of mapping between
the physical and computational space: it suffices to set the boundary of
the channel as [−1, 1]. Our implementation was tested against the results
of [74] and this comparison is reported in §2.3.6.
A ”brand new” development of the method has been carried out in order
here to take also into account correctly the mapping for boundary layer
flows.
Once the equation has been mapped, the new terms should be also expanded in series with Chebyshev polynomials. The multiplication of the
expanded terms must be grouped and treated accordingly to the formulae
present in the appendix of [74] .
Before showing the development of the τ -method for a Blasius flow one
should first note that the main problem is the mismatch between the computational domain η ∈ [−1, 1] and the physical domain y ∈ [0, ∞[. One
can circumvent this limitation by using a mapping function ξ (undefined
at the moment).
Based on the above consideration one can rewrite a mapped version of
Eq. (2.7) that could be directly discretized with a Chebyshev τ -method.
The procedure is cumbersome for the 4th order equation so it will be applied to the set of eqs. (2.39) .
For sake of clarity each single term will be treated separately. Let’s take
the last term of the second equation in eq. (2.39)
iαReU 00 φ
(2.64)
In order to solve it in the computational space η ∈ [−1, 1] a change of
variable is applied that maps the domain in sucha way that U (y) → U (η).
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2.3. Numerical methods
First order derivatives then get transformed as
dU (y)
dU (η) dy
=
dy
dy dη
.
(2.65)
As a consequence the term U 00 (x)φ(x) becomes
!
2 2 2
d y dU (η)
dy
d U (η)
00
+
U (y)φ(y) =
φ(η)
dη
dy 2
dη 2
dy
Expanding it in Chebyshev series we get then
2 X
dy
=
dm Tm
dη
X
U=
bk Tk
X
φ=
an Tn
(2.66)
(2.67)
(2.68)
(2.69)
Reminding from [74] that
X
lm Tm
X
where
es =
gn Tn =
1X
es Ts
2
(2.70)
1 X
ls−r g r
cs
|r|≤N
and all the · quantities are a short notation for gn = c|n| g|n| . The first
term on the RHS of 2.66 reads
! N
2 2
N
N
X
X
X
dy
d U (η)
(2)
φ(eta) =
dm Tm
bk Tk
al Tl
(2.71)
2
dη
dη
m=0
k=0
l=0
By applying iteratively the eq. (2.70) the final Chebyhev representation
of the first term is

N
X
1 X 1 X
a|r|
d|m|
4 n=0 cn
|r|≤N
|m|≤N

X
p=||n−r|−m|+2
p≡||n−r|−m| (mod 2)

p[p2 − (n − r − m)2 )]bp 
 Tn
(2.72)
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Chapter 2. Linear stability for incompressible flows
The second term on the RHS of eq. (2.66)
2 X
X (1) X
d y dU (η)
φ(η) =
hm Tm
bk Tk
al Tl
2
dη
dy
(2.73)
where
d2 y
dη 2
=
X
dm Tm
(2.74)
(2.75)
One can group the first two terms using again eq. (2.70) and obtain
X
1 X (1)
1 X
es =
2pbp (2.76)
b|s−m| h|m| =
h|m|
cs
cs
m≤N
|m|≤N
p=|s−m|+1
p+(s−m)≡ (mod 2)
. Eq. (2.76) can then be combined with the last term of eq. (2.73)
X
1X
1X
es Ts
al Tl =
fn Tn
2
4
where
fn =
1 X
e|n−r| a|r| =
cn
|r|≤N
X
1 X
ar
h|m|
cn
|r|≤N
|m|≤N
X
2pbp
(2.77)
p=|n−r−m|+1
p+(n−r−m)≡1 (mod 2)
Looking back to the Orr-Sommerfeld as in the form of eq. (2.39) we see
that vertical velocity perturbation is derived two times both in the first
and the second equation,
D2 φ ,
orD2 χ
Another term of the equation is
2 2
dφ(y)
dy
d φ d2 y dφ
=
+ 2
dη
dη
dy 2
dη dy
(2.78)
(2.79)
The first part of Eq.(2.79) could be discretized in the following way
2 2
X (2)
dy
1X
d φ X
es Ts
(2.80)
=
dm Tm
al Tl =
2
dη
dy
2
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2.3. Numerical methods
en =
1
cn
X
p=n+2
p≡n (mod 2)
p(p2 − n2 )ap
X
d|n−m|
(2.81)
|m|≤N
|n−m|≤N
while the second part could be discretized
X (1)
d2 y dφ X
1X
=
d
T
es Ts
a
T
=
m
m
l
l
dη 2 dy
2
es =
1 X
ap
cs p=1
X
2pb|s−m|
(2.82)
(2.83)
|m|≤p−1
|s−m|≤N
m+p−1=mod(2)
2.3.6. Results of τ -method
Poiseuille flow
The τ -method has been first implemented to solve a simple plane Poiseuille
flow. Assuming the channel height is 2, going from −1 to 1 the computational domain overlaps exactly with the physical one and therefore there
is no need to use any kind of mapping. In this case the Reynolds number
is base on half channel height h
Re =
ρU h
2µ
(2.84)
Considering for the present case a maximum velocity equal to 1m/s, the
Reynolds nuber is simply expressed by Re = 1/ν
Considering the velocity profile as U (y) = 1 − y 2 , it could is decomposed
according to the Chebyshev expansion as U (y) = 0.5T0 (y) − 0.5T2 (y) with
T0 (y) and T2 (y) defined by eq. (2.26), so that bn = 0 for n 6= 0 and n 6= 2.
This observation can eventually encourage the implementation of a specific method as done in [74]. That is not the goal of the the present thesis,
as a general implementation is targeted; for this reason we implement a
non-specific method, potentially re-usable for other kind of flows. Hereafter some results are reported to show the performance of the software
and its agreement with the literature.
The standard case for Re = 10000 and α = 1, for which Orszag in
[74] inferred a correct result up to the eighth digit of λ = 0.23752649 +
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Chapter 2. Linear stability for incompressible flows
M
21
41
61
81
101
121
141
161
181
201
0.2772580424102508
0.2375273014059497
0.2375264886467045
0.2375264898500112
0.2375264872881557
0.2375265115241138
0.2375264105743754
0.2375265470710931
0.2375269044270349
0.2375259284479085
λ
+i 0.007497707127414019
+i 0.003741875925566369
+i 0.003739670406954323
+i 0.003739671541773028
+i 0.003739668684901756
+i 0.003739657380919045
+i 0.003739664735717313
+i 0.003739716935416854
+i 0.003739772491015945
+i 0.003739912969626702
Table 2.1.: Most unstable eigenvaluefor a subsonic Poiseuille flow Re =
10000, α = 1
i0.00373967, has been solved. Looking at Tab. 2.1 we could see that we
agree very well with this result. Anyway it should be noted that as the
number of discretization points keeps on growing the most unstable eigenvalue is changing because of the growing ill-conditioning of the Chebyshev
derivative matrices. This is a well known problem for this kind of methods, nevertheless we can state that even at the highest number of points
tested the solution is still good at the sixth digit, which is a remarkable
result.
What was not displayed in [74] it is how the spectrum changes while
increasing the resolution. In this case the spectrum improves for higher
number of points and, even if one looses a bit of resolution for a specific
mode, the overall description of the spectrum becomes more accurate.
In Fig. 2.2 a set of six different number of polynomials is tested, ranging
from under to completely resolved4 . For M = 21 (Fig. 2.2a), although
the most unstable mode is not so far from the reality we observe in the
spectrum other two eigenvalues which are potentially unstable. Of course
this is not true and the overall spectrum is poorly resolved and far from the
reality (see Fig. 2.3). Slightly increasing the number of polynomials taken
into account improves the situation and all the branches appear already
4 In
Tab. 2.1 the entry M represents the total number of polynomials that have been
used for the expansion, from T0 to TN . For Orr-Sommerfeld having four boundary
condition, in the τ -method k = 4 equations come from the boundary conditions
and M − k from Orr-Sommerfeld equation itself. Please note that this is slightly
different from the N used to described a simple Chebyshev expansion
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2.3. Numerical methods
for M = 41, nevertheless the spectrum reaches its actual shape only for
M = 121. Once it is converged the shape does not change anymore
(except the few variations experienced by each eigenvalue similarly to
what is shown in Tab. 2.1) and it could be seen in Fig. 2.3. All the
different branches are completely resolved and the shape is in agreement
with literature (see for instance [91] pag.64).
This flow does not have any inflection point nevertheless is unstable and
this is a classic example of viscous instability.
From the computation with M = 141 we extract the shapes of the most
unstable mode along with some stable ones. For sake of completeness in
Fig. 2.4 we also plot the mode of the vertical leg of the spectrum. The
Fig. 2.4a shows the most unstable mode. None of them has been refined
by a local algorithm but all of them came out directly of the QZ algorithm
(see [73]).
Boundary layer flow
temporal case Boundary layer flow is more challenging than the Poiseuille
flow because of the mapping technique to be applied. Thanks to our implementation and the development shown in § 2.3.5 it is possible to compute
the stability of this kind of flow. To simplify the implementation the D2
approach of § 2.3.4 is used to lower the number of terms appearing in a
classic D4 approach. So despite the fact that this technique will be explained with more in detail later in this chapter the main results will be
shown here to conclude the study and implementation of the τ -method
for stability of flows.
The most unstable eigenvalue for a temporal problem as reported in Mack
[60] for α = 0.179, β = 0, Re = 580 is c = ωα = 0.3641 + i0.0080.
In Tab. 2.2 the convergence of the code is presented starting from M = 21
to M = 201 for the most unstable eigenvalue. Excluding the case with the
smallest number of points all the remaining results agree very well with
the ones proposed by Mack [60] and Malik [62], proving the consistency
of the present derivation. Of course we experience the same growing illconditioning for increasing number of discretization points N.
Mapping also alter the matrix ill-conditioning and therefore the mapping
may be designed specifically to cure this problem [26], nevertheless the
introduction of another mapping will result in a more complicated structure than the one presented in § 2.3.5, with the consequence of making a
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0
0
−0.2
−0.2
−0.4
−0.4
λi
λi
Chapter 2. Linear stability for incompressible flows
−0.6
−0.6
−0.8
−0.8
−1
0
0.2
0.4
λr
0.6
0.8
−1
1
0
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
0
0.2
0.4
λr
0.6
0.8
−1
1
0
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
0
0.2
0.4
λr
0.6
0.8
1
0.2
0.4
λr
0.6
0.8
1
0.8
1
(d) M=81
λi
λi
(c) M=61
−1
0.4
λr
(b) M=41
λi
λi
(a) M=21
0.2
0.6
(e) M=101
0.8
1
−1
0
0.2
0.4
λr
0.6
(f) M=121
Figure 2.2.: Subsonic Poiseuille flow spectrum for Re = 10000, α = 1 for
different number of Chebyshev polynomials in the expansion
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2.3. Numerical methods
UNSTABLE
0
λi
−0.2
−0.4
STABLE
−0.6
−0.8
−1
0
0.2
0.4
λr
0.6
0.8
1
Figure 2.3.: Comparison of spectrum computation for Poiseuille flow at
Re = 10000 and α = 1: M = 121 black dot, M = 141 red
cross
better conditioning less appealing. It is worth to note that the doubling
of the problem size related to the D2 method causes an increase in computational cost of about eight time when using a QZ algorithm as done
for the results presented in Tab. 2.2
In Tab. 2.3 the most unstable mode and the least stable modes are reported against two references. We should note that the three methods
are different, nevertheless there is a good agreement among them. Comparison with the other discrete modes is less and less satisfactory as we
approach the continuous spectrum, while the other values are correctly
captured.
We decided then to compare the full spectrum with the picture published
in Schmid & Hennigson [91]. They make all the distances non-dimensional
with respect to the displacement thickness and with respect to our input
value we have then Re = 290.5625, α = 0.1162. From Fig. 2.5 we confirm
the least unstable eigenvalue is correctly captured with only M = 41 and
yi = 6. Nevertheless as for the Poiseuille case the remaining part of the
spectrum is not completely resolved, especially the continuous one which
looks very differently from a straight continuous line while the other two
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Chapter 2. Linear stability for incompressible flows
α =0.23753+0.0037397i
α =0.96463−0.035167i
1
1
0
0.5
real
imag
absolute
y
y
0.5
−0.5
−1
−0.5
real
imag
absolute
0
−0.5
0
0.5
Re(φ),Im(φ),|φ|
1
−1
−4
1.5
(a) 0.23753+i0.0037397
−2
α =0.96464−0.035187i
α =0.2772−0.050899i
0
y
y
0.5
−0.5
real
imag
absolute
0
−0.5
0
2
Re(φ),Im(φ),|φ|
4
−1
−2
6
(c) 0.96464-i0.0035187
−1
0
Re(φ), Im(φ),|φ|
1
2
(d) 0.2772-i0.050899
α =0.93632−0.063201i
α =0.67457−0.38966i
1
1
real
imag
absolute
0.5
0.5
0
y
y
6
1
real
imag
absolute
0.5
−0.5
−1
−4
4
(b) 0.96463-i0.0035167
1
−1
−2
0
2
Re(φ),Im(φ),|φ|
real
imag
absolute
0
−0.5
−2
0
Re(φ),Im(φ),|φ|
2
(e) 0.93632-i0.063201
4
−1
−5
0
Re(φ),Im(φ),|φ|
5
(f) 0.67457+i0.38966
Figure 2.4.: Subsonic Poiseuille flow most unstable and less unstable mode
eigenfunctions for Re = 10000, α = 1 for M = 141
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2.3. Numerical methods
M
21
31
41
61
81
101
121
141
161
181
201
0.3628461931939606
0.3640659641900133
0.3641347403253813
0.3641018455046314
0.3640858440869612
0.3640935834308552
0.3640849504715294
0.3641038579360132
0.3641019316336477
0.3641079150280655
0.3641015141240021
+
+
+
+
+
+
+
+
+
+
+
c
i
i
i
i
i
i
i
i
i
i
i
0.009147320004875724
0.007942096600889044
0.007945221158097492
0.007969637560926444
0.007949217409618523
0.007933792747852763
0.007940209156330168
0.007941046026256764
0.007945179560424785
0.007943059467471187
0.007957921137839670
Table 2.2.: Most unstable eigenvalue for a subsonic boundary layer flow
at Re = 580, α = 0.179
c
0.36410 + i 0.00796
n/a
0.28972 - i 0.27683
0.48440 - i 0.19218
0.56117 - i 0.35433
Malik [62](MDSP)
0.3641 +i 0.0079
0.2329 - i 0.1343
0.2897 - i 0.2768
0.4839 - i 0.1921
0.5571 - i 0.3655
Mack [60]
0.3641 + i 0.0080
n/a
0.2897 - i 0.2769
0.4839 - i 0.1921
0.5572 - i 0.3653
Table 2.3.: Most unstable eigenvalue and first four less unstable eigenvalues for a subsonic boundary layer flow at Re = 580, α = 0.179
for M=201
discrete modes are more or less in the right position. Anyway to get a
correct positioning of all the four modes while matching the continuous
part of the spectrum we need to use around 240 polynomials. The fourth
discrete eigenvalue is very close to the one shown in [91], although not
exactly the same and some spurious eigenvalue are present. The source of
this spurious eigenvalues has not been investigated but it is most probably related to the fact that the standard rewriting of the Orr-Sommerfeld
equation for D2 method is not providing directly new boundary conditions
for the newly introduced variable. This problem has been first presented
by McFadden et al. [69] and also investigated more recently in Bourne
[12]. Anyway in [12] the proposed method has been tested in the paper
only for Poiseuille flow and we have not extended it to a Blasius bound-
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Chapter 2. Linear stability for incompressible flows
ary layer. Spurious modes could be confused sometimes with unconverged
modes, nevertheless increasing the resolution usually allows to rule them
out, by picking up the only ones which remain unchanged after several
refinements, while spurious eigenvalue could change their location consistently.
Having available, from [91], the eigenfunctions we made a test to verify the
shape of the perturbations. In Fig. 2.6 and Fig. 2.7 we plot respectively
the normalized value of the stream wise and normal to the wall velocity
perturbations. The good agreement between the present study and the
literature is satisfactory, verifying the implementation of τ -method for
the Blasius boundary layer flow along with the mapping development as
provided in the previous chapter.
Spatial case This technique could be applied to the spatial case as well.
We report here the results related to this calculations to complete the
discussion on our implementation and extension of the τ -method for the
boundary layer flow. In Jordinson [45] a series of calculations is presented
for the computation of the spatial boundary layer stability case while
Danabasoglu & Biringen in [25] report some results from their Chebyshev collocation method. Our goal is just to show the behavior of the
implementation for a standard case and to draw some conclusion on the
method itself. We should stress here that in [25] the domain has been
mapped to the half space so the number of nodes is technically doubled
with respect to our calculations. Nevertheless we compare those results
with the same number of point N=50 in Tab. 2.5 showing that there is no
need of halving the space to have an efficient discretization. As expected
the results match quite well despite the fact there is no exact matching
neither with [45] nor [25] anyway a perfect match is very unlikely to be
achieved and also in literature such a difference could be observed. For
these reasons we could consider the comparison as satisfactory.
In Tab. 2.6 the convergence of the most unstable eigenvalue for the Reδ =
336 and ω = 0.1297 case is reported. Similarly to what we observed in
Tab. 2.2, a insignificant number of digits could be achieved directly with
just two refinement, but with the increase of the number of discretization
points the solution does not really converge to a defined value but it rather
moves around a supposed “asymptotic” value. The ill-conditioning of the
matrices is not allowing a perfect convergence.
The spectrum for the case at Reδ = 336 and ω = 0.1297 is displayed
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2.3. Numerical methods
Re=290.5625, α=0.1162
Re=290.5625, α=0.1162
−0.2
−0.2
ci
0
ci
0
−0.4
−0.4
−0.6
−0.6
−0.8
0
0.2
0.4
0.6
0.8
−0.8
0
1
0.2
0.4
cr
(a) M = 41
1
Re = 290.5625, α=0.1162
0
−0.2
−0.2
ci
ci
0
−0.4
−0.4
−0.6
−0.6
0.2
0.4
0.6
0.8
−0.8
0
1
0.2
0.4
cr
0.6
0.8
1
cr
(c) M = 121
(d) M = 161
Re 290.5625, α=0.1162
Re=290.5625, α=0.1162
0
−0.2
−0.2
ci
ci
0
−0.4
−0.4
−0.6
−0.8
0
0.8
(b) M = 81
Re=290.5625, α=0.1162
−0.8
0
0.6
cr
−0.6
0.2
0.4
0.6
cr
(e) M = 201
0.8
1
−0.8
0
0.2
0.4
0.6
0.8
1
cr
(f) M = 241
Figure 2.5.: convergence of the whole spectrum for a Blasius boundary
layer at Re = 290.5625, α = 0.1162 computed by a mapped
Chebyshev τ -method: comparison with Henningson [91] (red
circle)
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Chapter 2. Linear stability for incompressible flows
10
9
8
7
y/lref
6
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
kũk
Figure 2.6.: Stream wise velocity perturbation normalized: VESTA results (solid line) and Henningson results [91] (red circles)
10
9
8
7
y/lref
6
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
kṽk
Figure 2.7.: Wall normal velocity perturbation normalized: VESTA results (solid line) and Henningson results [91] (red circles)
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2.3. Numerical methods
Case
Re
ω
1
336
0.1297
2
598
0.1201
3
998
0.1122
Table 2.4.: Computed case for spatial stability computations on Blasius
boundary layer
1
2
3
τ -method
0.308368 + i 0.007965
0.307874 - i 0.001885
0.308616 - i 0.005715
Danabasoglu & al. [25]
0.30864 + i 0.00799
0.30801 - i 0.00184
0.30870 - i 0.00564
Jordinson [45]
0.3084 + i 0.0079
0.3079 + i 0.0019
0.3086 - i 0.0057
Table 2.5.: Most unstable/ less unstable eigenvalues for a subsonic boundary layer flow at different Reynolds and wave number for N =
50 (cases of Tab. 2.4)
N+1
21
31
41
51
61
81
101
121
141
161
181
201
0.3087433907582560
0.3083767768729003
0.3083638625418336
0.3083683133737977
0.3083658759005218
0.3083503447711690
0.3083340526272674
0.3083481354663898
0.3083471626580147
0.3083544507843028
0.3083460759014448
0.3083531518228103
+
+
+
+
+
+
+
+
+
+
+
+
c
i
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i
i
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i
0.008388056543142284
0.007938204741073049
0.007954017479429651
0.007964771262706057
0.007962100622817573
0.007945589665066210
0.007951352315709840
0.007950760990788178
0.007973692308920997
0.007952928745291231
0.007954879859455226
0.007955545112629298
Table 2.6.: Less stable eigenvalue for a subsonic boundary layer flow at
Reδ = 336, ω = 0.1297
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Chapter 2. Linear stability for incompressible flows
1
0.8
αi
0.6
0.4
0.2
0
0
0.2
0.4
αr
0.6
0.8
1
Figure 2.8.: Spatial spectrum for an incompressible boundary layer flow
at Reδ = 336 and ω = 0.1297 for N=50
in Fig. 2.8. It looks very clean and no spurious value could be found in
the solution. The stream wise and normal to the wall velocity perturbations for the least stable eigenvalue for the case 1 of Tab. 2.4 are shown
respectively in Fig. 2.9 and Fig. 2.10
In analogy to the temporal case study, we compare the spectrum coming
out of our code also with the one proposed in [91] in Fig 2.11. It appears
immediately evident that the continuous spectrum increases its resolution
along with the number of points; on the contrary, the less stable value remains approximately at the same location, as few nodes are needed in
order to capture it correctly. In Fig. 2.12 a whole set of physical eigenvalue shows up and this affect the resolution of one of the discrete modes
that should appear. This problem has not been further investigated as it
does not affect the most unstable/less stable eigenvalue and because the
method has not been further used in this work.
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2.3. Numerical methods
6
5
y/lref
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
|ũ|
Figure 2.9.: Stream wise velocity perturbation normalized Reδ = 336 and
ω = 0.1297 : VESTA results (solid line)
6
5
y/lref
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
|ṽ|
Figure 2.10.: Wall normal velocity perturbation normalized Reδ = 336
and ω = 0.1297: VESTA results (solid line) and Henningson
results [91] (red circles)
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1
1
0.8
0.8
0.6
0.6
αi
αi
Chapter 2. Linear stability for incompressible flows
0.4
0.2
0.4
0.2
0
0
0
0.2
0.4
αr
0.6
0.8
1
0
1
1
0.8
0.8
0.6
0.6
0.4
0.2
0.6
0.8
1
0.8
1
0.8
1
0.4
0.2
0
0
0
0.2
0.4
αr
0.6
0.8
1
0
(c) M = 101
1
1
0.8
0.8
0.6
0.6
0.4
0.2
0
0
0.2
0.4
αr
0.6
(e) M = 181
0.4
αr
0.6
0.4
0.2
0
0.2
(d) M = 141
αi
αi
0.4
αr
(b) M = 51
αi
αi
(a) M = 31
0.2
0.8
1
0
0.2
0.4
αr
0.6
(f) M = 201
Figure 2.11.: Convergence of the whole spectrum for a Blasius boundary layer at Reδ = 1000, α = 0.26 computed by a mapped
Chebyshev τ -method: comparison with Henningson [91](red
circle)
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2.3. Numerical methods
0.9
0.8
0.7
0.6
αi
0.5
0.4
0.3
0.2
0.1
0
−0.1
0
0.2
0.4
αr
0.6
0.8
1
Figure 2.12.: Comparison of the whole spectrum for a Blasius boundary
layer at Reδ = 1000, ω = 0.26 as computed by a the present
mapped Chebyshev τ -method with N = 240 points (black
dot) with [91] (red circle)
2.3.7. Chebyshev Collocation Method
The second method implemented to study the stability of incompressible
flows is the Chebyshev collocation. This proved to be flexible, reusable
and accurate with a programming effort smaller than others. Detailed
explanation about this and other spectral method could be found in [17];
only few highlights will be recalled here in order to establish a basic knowledge.
The main idea has been taken from [97] in which the author describes a
straightforward procedure to implement a Chebyshev collocation method.
First we choose the set of Gauss-Chebyshev-Lobatto points
jπ
xj = cos
.
(2.85)
N
Given a function v defined on the afore mentioned points it is possible
to obtain its derivative first defining a polynomial p respecting the usual
condition for an interpolation problem p(xj ) = vj for j = 0, . . . , N and
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Chapter 2. Linear stability for incompressible flows
then computing its derivative wj = p0 (xj ).
Interpolating a function with N points and taking its derivative is a linear
operation, so it could be represented by a simple multiplication by an
(N + 1) × (N + 1) matrix
w = DN · v
(2.86)
It is not intention of this paragraph to explain how the terms in DN could
be obtained for a general size matrix. These details could be found easily
in [17] as well as in [97], the key point is that it is possible to compute
DN by means of a general algorithm. A Chebyshev differentiation matrix
could be described by the coefficients

2
+ 2N6 +1 i = j = 0




 − 2N 2 +1 i = j = N
6
(2.87)
DNij =
xj
i = j = 1...N − 1
−

2(1−x2j )



i+j
 ci (−1)
i 6= j i, j = 1 . . . N − 1
cj (xi −xj )
(
ci =
2
1
i = 0 or N,
otherwise
(2.88)
Best performance are achieved for smooth function but this is not usually
a problem for our cases. One important issue is that a M-th order derivaM
tive could be easily computed by (DN ) .
It is worth to note that Chebyshev spectral method suffers, from a typical
ill-conditioning problem with increasing N. Several improvements could
be applied to this basic concept bringing to better conditioned matrices
but we found this to be not mandatory as we could achieve good solutions
with a relatively small number of points thanks to the spectral convergence property.
Boundary conditions
After discretizing the equation, one needs to take into account the boundary conditions to solve correctly the given set of equations. Misrepresentation of b.c. leads undoubtedly to a not reliable solution and to the
introduction of spurious results (see [13], [30] and [12]). Homogeneous
Dirichlet b.c. can be easily plugged into the collocation method. As the
first and last value in the approximated derivative have to be zero, the
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2.3. Numerical methods
N
20
30
40
60
80
100
120
140
160
180
200
0.3648253799203074
0.3641518911390230
0.3641262711375833
0.3641174768766077
0.3641239438626184
0.3641205528616733
0.3641205631646640
0.3641239690286770
0.3641225824401141
0.3641208874737341
0.3641131482508822
+
+
+
+
+
+
+
+
+
+
+
c
i
i
i
i
i
i
i
i
i
i
i
0.006680428410158767
0.007942648972622340
0.007948578713939723
0.007952365534196048
0.007951093633888630
0.007954293000615157
0.007953138762983193
0.007945759748491524
0.007955050739767676
0.007941527895669733
0.007952494068035934
Table 2.7.: Most unstable eigenvalue for a subsonic boundary layer flow
at Re = 580, α = 0.179
differentiation matrix stays unvaried and computations could be carried
out by skipping the first and the last rows and columns (DN becomes a
(N − 1) × (N − 1) matrix). Two zeros at the extremities of the resulting w
should be appended so that the final vector has the correct dimension. In
case of a Neumann boundary conditions the problem could be overcome
by extracting the corresponding row from the matrix D1 . This row substitution technique can be generally applied to more complicated boundary
conditions (see par. 3.5).
2.3.8. Results of collocation method
Temporal Case Similarly to what has been done for the τ -method we
propose a convergence study on the most unstable eigenvalue of a Blasius
boundary layer computed by means of a Chebyshev collocation method.
In Tab. 2.7. The algorithm immediately reach a convergence to the fourth
digit but it oscillates with a higher number of points preventing the convergence of more digits. The results here available have been computed
for a D4 method which is more ill-conditioned with respect to a D2 one
because the derivative matrix ill-conditioning increase with the order of
the discretized derivative.
In Tab. 2.8 a comparison of the collocation results for N=200 points is
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Chapter 2. Linear stability for incompressible flows
c
0.364113 + i 0.007952
0.289712 - i 0.276858
0.483876 - i 0.192098
0.557921 - i 0.368070
Malik [62](MDSP)
0.3641 +i 0.0079
0.2329 - i 0.1343
0.2897 - i 0.2768
0.4839 - i 0.1921
0.5571 - i 0.3655
Mack [60]
0.3641 + i 0.0080
0.2897 - i 0.2769
0.4839 - i 0.1921
0.5572 - i 0.3653
Table 2.8.: Most unstable eigenvalue and first four less stable eigenvalues
for a subsonic boundary layer flow at Re = 580, α = 0.179 for
N=200
reported. The last eigenvalue is slightly more accurate than the respective
one in Tab. 2.3. This happens even if both methods are formally spectrally
accurate. Boundary condition implementation details play a significant
role in the appearance of spurious eigenvalues and also on the pollution
of value close to the continuous spectrum. The collocation method is
less affected by this problem resulting in a better accuracy of the overall
spectrum.
A comparison of the most unstable mode eigenfunction against the eigenfunctions reported in [91] for the Re = 290, 5625 α = 0, 1162 case is
shown in Fig. 2.14 and Fig. 2.13. Similarly to what we have seen for
the τ -method the present results match perfectly. Looking at the convergence of the full spectrum in Fig. 2.15 we see that the first three modes
are correctly captured already with N=40 even though the third one can
be hardly distinguished from the under resolved continuous part of the
spectrum. At N=80 the three discrete modes are completely detached
from the continuous spectrum which anyway needs more than 240 points
to be resolved. The τ -method seems to be slightly superior in term of
resolution of the continuous part of the spectrum.
Spatial case We report hereafter the spatial computations obtained by a
D4 method implementation. We follow the theory explained in §2.3.1. A
direct D4 method has been used with satisfactory results for the most unstable/less stable eigenvalues. In Tab. 2.9 three cases have been computed
with 50 discretization points and compared against literature. In Tab. 2.5
the τ -method gave different results for a similar number of points. From a
computational point of view there is not a direct advantage in the spatial
computation of a D4 over a D2 method as the final size of the matrices is
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2.3. Numerical methods
10
9
8
7
y/lref
6
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
|ũ|
Figure 2.13.: Stream wise velocity perturbation normalized: VESTA results for collocation method (solid line) and Henningson results (red circles) N =200
10
9
8
7
y/lref
6
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
|ṽ|
Figure 2.14.: Wall normal velocity perturbation normalized: VESTA results for collocation method (solid line) and Henningson results (red circles) N = 200
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Chapter 2. Linear stability for incompressible flows
0.2
0
0
−0.2
−0.2
−0.4
−0.4
ci
ci
0.2
−0.6
−0.6
−0.8
−0.8
−1
0.4
0.6
0.8
−1
1
0.4
0.6
cr
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
0.4
0.6
0.8
−1
1
0.4
0.6
cr
0
0
−0.2
−0.2
−0.4
−0.4
ci
ci
0.2
−0.6
−0.6
−0.8
−0.8
0.6
0.8
c
r
(e) N = 200
1
0.8
1
(d) N = 160
0.2
0.4
0.8
cr
(c) N = 120
−1
1
(b) N = 80
ci
ci
(a) N = 40
−1
0.8
cr
1
−1
0.4
0.6
c
r
(f) N = 240
Figure 2.15.: Convergence of the whole spectrum for a Blasius boundary
layer at Re = 290.5625, α = 0.1162 computed by a mapped
Chebyshev collocation method: comparison with Henningson (red circle)
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2.4. Summary on the incompressible linear stability problem
1
2
3
τ -method
0.308357 + i 0.008004
0.307868 - i 0.001799
0.308623 - i 0.005584
Danabasoglu [25]
0.30864 + i 0.00799
0.30801 - i 0.00184
0.30870 - i 0.00564
Jordinson [45]
0.3084 + i 0.0079
0.3079 + i 0.0019
0.3086 - i 0.0057
Table 2.9.: Most unstable/ less unstable eigenvalues for a subsonic boundary layer flow at different Reynolds and wave number for N =
50 for cases in Tab. 2.4
the same, nevertheless the ill-conditioning problem is more evident for the
former. From Fig. 2.16 the overall spectrum convergence is shown. This
pseudo-spectral method is able to capture the discrete modes quite easily
even if spurious modes appear in the solution. Similarly to the results
in Fig. 2.5 a family of unwanted modes rise in the spectrum. Differently
from the τ -method the continuous mode convergence improves with the
increase in number of points only up to a certain extent. The spectrum
will be almost not recognizable for a 240 points. This is comparable to
the behavior of the collocation implementation of the temporal stability
problem (see Fig. 2.15) and it is clearly related to the D4 implementation,
as it is one of the main difference with respect to the implementation of
in § 2.3.6. While a D2 method could cure these problem we saw already
the potential ill-posedness of the boundary condition in such an artificial
splitting. These difficulties come mainly from the way the Orr-Sommerfeld
problem is built and in the next chapter we will observe a better posed
system of equations.
2.4. Summary on the incompressible linear
stability problem
In this chapter we explained the fundamental implementations and developments underlying much of the current work ongoing in linear stability.
Basic test were performed on Poiseuille flow to verify the behavior of the
toolkit. Original contribution, in this framework, is the expansion of the
τ -method to a Blasius boundary layer.
Codes have been tested, where possible, against literature results showing that our implementation is correct and works within the range of the
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Chapter 2. Linear stability for incompressible flows
0.4
0.3
0.3
αi
0.5
0.4
αi
0.5
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
0.2
0.4
0.6
αr
0.8
1
0.2
(a) N = 30
0.4
0.6
αr
0.8
1
0.8
1
0.8
1
(b) N = 50
0.4
0.3
0.3
αi
0.5
0.4
αi
0.5
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
0.2
0.4
0.6
αr
0.8
1
0.2
(c) N = 100
0.4
0.6
αr
(d) N = 140
0.4
0.4
0.3
0.3
αi
0.5
αi
0.5
0.2
0.2
0.1
0.1
0
0
−0.1
0.2
0.4
0.6
αr
(e) N = 180
0.8
1
−0.1
0.2
0.4
0.6
αr
(f) N = 200
Figure 2.16.: Convergence of the whole spectrum for a Blasius boundary
layer at Re = 1000, ω = 0.26 computed by a mapped Chebyshev collocation method: comparison with Henningson (red
circle)
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2.4. Summary on the incompressible linear stability problem
expected performance. The τ -method has been compared against the collocation method for the same set of boundary layer cases showing the
problems and the benefits associated with each of them.
The Chebyshev collocation method offers a great flexibility to the researcher with an easy implementation. The solution is also directly accessible in terms of physical value unlike the τ -method where a further
post processing should be applied in order to get the final perturbation
shapes. The ease of implementation is self evident when dealing with the
necessary mapping from the physical space to the Chebyshev domain of
definition.
Though the D4 and D2 methods provided very similar results for the discontinuous part of the spectrum the direct technique suffered from the
ill-conditioning of the higher order derivative discretization. This results
in a less resolved continuous spectrum. On the other side the τ -method
performance shown problem due to the application of the boundary conditions in the D2 method. Some theoretical work is present in the literature
to relieve the technique from this flaw but it has been only applied to a
Poiseuille flow.
In terms of future development the collocation method proved more effective and ready to be the basis of a toolkit development, and its main
problem related to the discretization of a high order derivative can be
solved taking into account a compressible physical model. Nevertheless
in terms of Orr-Sommerfeld stability computation there is not a clear
preference on the kind of method to be used as they both achieve good
accuracy in the calculation of the most unstable (or less stable) eigenvalues with a similar number of polynomials (or points). The good match
with literature is proved by the computation of the stability curve of an
incompressible boundary layer on a flat plate without pressure gradient
(Blasius’ profile) in Fig. 2.17. The flow is compared to a portion of the results presented in [75]. The comparison is interesting as those calculations
are performed by means of finite difference method with a Gram-Schmidt
orthonormalization procedure. The critical Reynolds number is around
519. These neutral stability curve has been confirmed also by the work of
Baines [7]
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0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
500
1000
1500
2000
2500
Reδ
3000
3500
4000
4500
5000
Chapter 2. Linear stability for incompressible flows
α
52
Figure 2.17.: Neutral Stability curve for a flat plate boundary layer (Blasius’ profile) computed by means of a local spatial solver
(solid line) and by a finite difference method [75]
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Chapter 3.
Linear stability for compressible flows
3.1. Introduction
With the rising interest in supersonic vehicles, study of stability has been
extended to compressible flows already from the forties. The first theoretical investigation on the subject was performed by Lees and Lin [52] in
1946 for a two-dimensional perfect gas temporal problem. They extended
the Rayleigh’s theorem to compressible flows and they discussed, first,
the supersonic waves moving in the free stream by means of an energy
method. They also discovered the role of the quantity
d
dU
ρ
=0,
(3.1)
dy
dy
which is the compressible counterpart of the d2 U/ dy 2 = 0 for the incompressible flows.
Nevertheless the community had to wait the extensive work of Mack ([55],
[56], [57], [58], [59], [54]) before the stability in compressible flows could
be completely characterized.
Problems were due also to the complexity of the equation to be solved
and the limited numerical resources of the time. Reduced set of equations
(Dunn-Linn) were found in order to study the significant phenomena at
limited cost. Anyway a deeper insight in compressible boundary layer
physics was reached only when direct calculations of the full compressible
stability equations could be resolved.
With the increase in the computational power, direct numerical simulation came into the field extending the acquired knowledge to the nonlinear transition mechanism and breakdown to turbulence. Compressibility
effect was first introduced for subsonic flow and it proved to have a stabilizing effect in these case; anyway when the Mach number grows stability
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Chapter 3. Linear stability for compressible flows
can dramatically change. In the following part of the chapter the reader
will find a description of the chosen method to solve the stability equations for compressible flows together with an explanation of the numerical
techniques used to solve it.
3.2. Remarkable features of compressible flows
Depending on the topic one could find some common points between the
incompressible and compressible stability as well as several differences.
These differences becomes more evident as soon as higher Mach numbers
are investigated.
A flat plate (one of the most studied configurations along with cone and
wedges) compressible boundary layer with no pressure gradient is unstable
to purely inviscid waves, unlikely its incompressible counterpart where the
instability mechanisms are originated by viscosity. In supersonic flows
the instability have been classified in three classic ways according to their
phase speed cr :
• subsonic: Ue − cr < ae
• sonic: Ue − cr = ae
• supersonic: Ue − cr > ae
where ae and Ue are the speed of sound and the velocity at the boundary
layer edge.
A necessary and sufficient condition for a neutral subsonic solution to exist is that a generalized inflection point condition, expressed by Eq. 3.1,
is fulfilled at a point where the phase velocity equals the the mean velocity. So for increasing Mach number the generalized inflection point moves
away from the wall toward the boundary layer edge and it is possible to see
from a neutral stability curve computations (see [5] and [98] for an outline
of supersonic neutral stability curve or the more recent work by Özgen &
Kircali in [76]) that at higher Reynolds number there is a wider instability
range. When the disturbance is supersonic the equations change in nature
and switch from elliptic (for subsonic disturbances) to a wave equation
which can have multiple neutral wave satisfying the same boundary conditions (more details in [5]) and having the same phase velocity.
This discovery was first made by Mack ([59]) along with the one of a
family of non inflectional waves appearing above Mach 4. These are not
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3.3. Governing equations
minor differences as the new families of waves gives the most unstable
eigenmodes.
3.3. Governing equations
Everything originates from the Navier-Stokes equation written for compressible flows in Cartesian coordinates
ρ,t + (ρbi ),i = 0
(3.2)
ρ(bi,t + bj bi,j ) = −(pδij ),j + τij ,j
(3.3)
ρcp (T,t + bj T,j ) = (kT,j ),j + p,t + bj p,j + τij bi,j
(3.4)
where the shear stress tensor is τij = (λbk,k ),j + µ ((bi,j ) + bj,i ),j and b is
the velocity vector [U, V, W ].
Of course also the gas state equation
p = ρRT
(3.5)
has to be included.
The basic hypotheses under the linear stability theory have been already
underlined and will not be discussed here again. We will focus instead on
the subtle differences.
First of all, among the variables to be perturbed one has to include also
the transport properties (unless being explicitly considered constant)
µ = µ + µ0
(3.6)
0
λ=λ+λ
(3.7)
0
(3.8)
k =k+k
and of course recalling eq.(2.1) we have to note that our vector of variables is now Q = [U, V, P, T, W ].
As for the incompressible case we will non-dimensionalize the equations.
1/2
Lengths are scaled by the Blasius length scale lref = (νx/Ue ) , veloci2
ties by Ue , density by ρe , pressure by ρe Ue , time by l/Ue and temperature
by Te . For viscosity (both first and second coefficient) we take as reference the value corresponding to the temperature Te and we name it µe ,
for sake of consistency.
It is worth to note that the current implementation is general enough to
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Chapter 3. Linear stability for compressible flows
allow arbitrary value of λ though, if not otherwise explicitly stated, the
Stokes’ hypothesis has been used for our computations, that is λ = − 23 µ.
Prandtl number in the boundary layer could be kept constant; this simplification has not a large impact on the final stability results and it is
widely used by other authors (as for instance Malik [62]). Viscosity is
computed by Sutherland’s law whose expression reads as:
µ = µref
T
Tref
32
·
Tref + S
T +S
(3.9)
where
µref = 1.7894 × 10−5 Kg/(s · m)
Tref = 288.16K
S = 110.4K
Thermal conductivity k could be prescribed by a similar formula, and in
our case as the specific heat at constant pressure is kept constant as well
as the Prandtl number P r, the value of k is automatically defined.
It is still to be noted that our code has been kept general enough to
accept specific and different values of k and µ allowing computations with
variable Prandtl number. In this case the local P r should not be confused
with the one obtained by the reference value and used as non dimensional
parameter in the equation.
Another important observation is that by perturbing all the variables and
transport properties one should solve for nine variables, having only six
equations (for a 3D case). Other three relations are needed in order to
close the problem. By considering the perturbed transport properties as
a small variation around the mean value (i.e. it is possible to linearize
around the mean value) one could write for instance
µ0 = µ − µ = µ +
dµ
dµ
dµ 0
∆T − µ =
T −T =
T
dT
dT
dT
(3.10)
This linearization is somehow contained in the small perturbation hypothesis and therefore perfectly coherent with the theory.
In our solver also the density perturbation ρ̃ is directly substituted by the
perfect gas state equation
ρ̃ = γM 2
p̃
ρ
− T̃
T
T
(3.11)
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3.3. Governing equations
Our variable vector now looks
h
i
φ = ũ, ṽ, p̃, T̃ , w̃
(3.12)
The final system of equation as implemented reads
x-mom eq.
i β ũ W
ṽ U y
i α ũ U
i ω ũ
+
+
+
=
T
T
T
T
T̃ µT U y y + T̃ µT T T y U y + T̃y µT U y + i α ṽ µT T y + ũy µT T y
Re
−α β w̃ µ + i α ṽy µ + ũy y µ + −β 2 − 2 α2 ũ µ
+
Re
−α β w̃ λ + i α ṽy λ − α2 ũ λ
+
− i α p̃
Re
(3.13)
y-mom eq.
i β ṽ W
i α ṽ U
i ω ṽ
+
+
=
T
T
T
T̃ µT i β W y + i α U y + 2 ṽy µT T y
+
Re
i β w̃ λT T y + ṽy λT T y + i α ũ λT T y + i β w̃y µ + 2 ṽy y µ
+
+
Re
−β 2 − α2 ṽ µ + i α ũy µ + i β w̃y λ + ṽy y λ + i α ũy λ
+
− p̃y
Re
(3.14)
z-mom eq.
i β w̃ W
i α w̃ U
i ω w̃
ṽ W y
+
+
+
=
T
T
T
T
T̃ µT W y y + T̃ µT T T y W y + T̃y µT W y + w̃y µT T y + i β ṽ µT T y
+
Re
w̃y y µ + −2 β 2 − α2 w̃ µ + i β ṽy µ − α β ũ µ
+
+
Re
β 2 w̃ λ + i β ṽy λ − α β ũ λ
−
− i β p̃
Re
(3.15)
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Chapter 3. Linear stability for compressible flows
mass equation
i α γ M 2 p̃ U
ṽ T y
i β γ M 2 p̃ W
i β T̃ W
i α T̃ U
+
− 2 +
−
−
2
2
T
T
T
T
T
(3.16)
2
i γ ω M p̃ i β w̃ ṽy
i α ũ i ω T̃
+
+
+
+
−
=
0
2
T
T
T
T
T
energy eq.
i β T̃ W
i α T̃ U
ṽ T y
i ω T̃
+
+
+
=
T
T
T
T
!
2
2
2
2
γ Uy
γ Wy
Wy
Uy
2
−
+
−
+
T̃ M µT
Re
Re
Re
Re
2iβ Wy
2 i α γ Uy
2 i α Uy
2iβ γ Wy
2
−
+
−
+
+ṽ M µ
Re
Re
Re
Re
2Wy
2γ Wy
+w̃y M 2 µ
−
+ Ec p̃ i β W + i α U + i ω +
Re
Re
2 Uy
2 γ Uy
−
+
ũy M 2 µ
Re
Re
!
2
kT T T y
kT T y y
β2 k
α2 k
+T̃
+
−
−
+
Pr Re
Pr Re
Pr Re
Pr Re
+
2 T̃y kT T y
T̃y y k
+
Pr Re
Pr Re
(3.17)
which could be compactly written in the form
AD2 + BD + C φ = 0
being D the derivative with respect to y

I 0 0
0 I 0

A=
0 0 0
0 0 0
0 0 0
. The matrix A is

0 0
0 0

0 0

I 0
0 I
(3.18)
(3.19)
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3.4. Numerical method
h
i
Our boundary conditions are ũ, ṽ, T̃ , w̃ = 0 for y = 0 and y → ∞. If
for the velocity this seems pretty obvious, for temperature we implicitly
imposed that the perturbations do not exchange heat with the wall, i.e.
they are adiabatic. Boundary conditions on pressure has been specifically
left aside as it will be treated separately in §3.5 The final arrangement
of the coefficients depends on the choice of performing a temporal or a
spatial computation. Anyway in both cases it is possible to retrieve a
general eigenvalue problem formulation
Aφ = ωBφ,
or,
Aφ = αBφ
(3.20)
.
3.4. Numerical method
Studying the stability of compressible flows is implying the computation
of a big number of terms and the application of the extension of the τ method for boundary layer analyzed in section 2.3.5 is not convenient.
Only the Chebyshev collocation method is used with some modifications
with respect to what shown in section 2.3.7.
The prominent feature of a compressible stability problem is the appearance of several equations which are the compressible counterpart of
Eq.(2.7). The 3D stability problem for a compressible flow is five time
bigger than the Orr-Sommerfeld requiring considerably more resources to
be solved. For a spatial problem with eigenvalues the matrix size is further
multiplied by a factor two. Such a demanding setup is compensated by
a better conditioning of the system. Comparing it with the fourth order
Orr-Sommerfeld equation the compact system in Eq. (3.18) is only second
order. Because of this reason, there is no need to test a reduced order system, though possible, as the matrix conditioning is acceptable. One can
infer that, because of its easy implementation, the Chebyshev collocation
method is a good choice for the compressible stability problem.
Being the complexity of the QZ algorithm O(N 3 ) the computationally
more demanding size refers only to the global solver (computing the whole
spectrum at once) while the local solver does not suffer from this problem
as it is not using the QZ algorithm (see Appendix C).
We should mention here that other methods are possible. Many codes,
for instance, implement the method introduced by Mack, which assumes
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Chapter 3. Linear stability for compressible flows
that every perturbation has a decaying behavior. If this is set, then we
could describe the outer perturbation with an analytically predetermined
decay leaving only a smaller part of the domain to be solved. This approach is extremely efficient for standard perturbation nevertheless is not
general enough. With the extreme flexibility of the Chebyshev collocation
method for the boundary conditions implementation we prefer not to fix
the trend of the perturbation decay but just its value.
3.5. Boundary conditions
The biggest advantage in the implementation of the collocation method
lies in the manipulation of the boundary conditions. The incompressible
case has four boundary conditions for the fourth order problem, all of
them on one variable; the compressible problem has eight homogeneous
boundary condition for ũ, ṽ, w̃, T̃ but nothing prescribed for pressure.
Two options are then available: the use of a staggered grid as done in
[47] or the introduction of a somehow fictitious boundary condition on
pressure. We follow the latter by re-writing the y-momentum equation at
the boundaries and appending it to the system as a new equation to be
solved. The procedure leads to two Neumann boundary conditions
∂ p̂ = X0
(3.21)
∂y y=0
∂ p̂ = Xm
(3.22)
∂y y=ymax
which are explicitly written in the form
1 Re ∂ p̂ l1
∂ ŵ ∂ û
0=−
+i
α
+β
+
l2 µ ∂y y=0,max
l2
∂y
∂y y=0,max
∂ 2 v̂ 1 dµ
0 ∂v̂
+
T
l
2
∂y 2 l2 µ dT
∂y
(3.23)
y=0,max
where
λ
µ
λ
l2 = 2 +
µ
l1 = 1 +
(3.24)
(3.25)
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3.6. Incompressible computation
The variable vector sees two more variables: p̃0 and p̃N .
As already explained due to the homogeneous boundary conditions the
first and last rows and columns in each discretized equation for each variable could be discarded. Of course the “additional variables” p̃0 and
p̃N discarded by this operations are introduced by appending the two
Neumann boundary conditions. At the same time two columns should be
added, reaching a final dimension of (5N − 3). The term of the discretized
equation which multiply p̃0 and p̃N should also be considered. For this
reason the matrix appear in the form
...
 ...

 ...
A=
 ...

 P0u
PN u

...
...
...
...
P0v
PN v
...
...
...
...
P0p
PN p
...
...
...
...
P0T
PN T
...
...
...
...
P0w
PNw
aip0
ai+1
p0
ai+2
p0
ai+3
p0
P0p0
PNp0

aipN

ai+1
pN 

ai+2
pN 

ai+3
pN 
P0pN 
PNpN
(3.26)
where the term aipX refers to the i-th equation and the last two rows
represent the matrix form of the 3.23. This formulation is enough to
ensure a good representation of our problem.
3.6. Incompressible computation
There is nothing preventing us from using the compressible code to study
the stability of the incompressible flow. This is actually an interesting step
mainly intended to verify the implementation. Moreover we can deduce
some further conclusions about the effect of the discretization method on
the continuous spectrum.
Analyzing the time problem we use one of the test cases proposed before,
that is Re = 580 and α = 0.179 to compare the compressible solver to
the Orr-Sommerfeld one. Of course in this case, in order to mimic the
incompressible flow, we set M = 10− 6, Te = 300K and P r = 0.7. The
eigenvalue (phase velocity) in Tab. 3.1 converges to ten digits for both the
real and complex part and this happen for N = 80 points. Comparing this
convergence rate with the result in Tab. 2.2, where a four digit convergence
was achieved, one could be easily convinced about the good conditioning
of this method. It should also be noted that the Orr-Sommerfeld case
in Tab. 2.2 was solved with a D2 method therefore, nominally the same
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Chapter 3. Linear stability for compressible flows
N
20
30
40
60
80
100
120
140
160
180
200
0.3631951891357532
0.3641013810880590
0.3641232276919413
0.3641228677053959
0.3641228675575113
0.3641228675621993
0.3641228675601516
0.3641228675589933
0.3641228675644979
0.3641228675689038
0.3641228675432092
+
+
+
+
+
+
+
+
+
+
+
c
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0.008523178234188513
0.007954809717663916
0.007959723050681329
0.007959720138660947
0.007959720363653333
0.007959720348680047
0.007959720366522901
0.007959720369855150
0.007959720374417459
0.007959720376732555
0.007959720369848397
Table 3.1.: Convergence of the most unstable eigenvalue for a subsonic
boundary layer flow at Re = 580, α = 0.179
level of ill-conditioning is present. The real difference here is played by
the boundary conditions where each unknown has its own. Recalling
the same case computed by means of the collocation method (shown in
Tab. 2.7), one observes a substantial improvement as the Orr-Sommerfeld
computation reached only four digits convergence. Comparison of most
unstable/less stable eigenvalues for Re = 580 and α = 0.179 as computed
in the present work and in literature is shown on Tab. 3.2. Present results
compare very well with literature ([62](MDSP), Mack [60]).
The convergence of the spectrum with respect to the case shown in [91]
for Re = 290.5625 α = 0.1162 in Fig. 3.1 it is similar to the one of the
τ -method in Fig. 2.5, nevertheless it is better than the Orr-Sommerfeld
collocation results (Fig. 2.15). This is mostly related to the bad conditioning of the D4 method. The compressible solver matches perfectly all
the discrete modes and it also adds a new family of modes not present in
the original Orr-Sommerfeld computation. This is evident in Fig.3.1 and
it is due to the role of the energy equation, which has been discarded in
the Orr-Sommerfeld equation, contrarily to the present chapter computations. A comparison of the stream wise and normal to the wall velocity
perturbation is reported respectively in Fig. 3.2 and Fig. 3.3.
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i
i
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
ci
ci
3.6. Incompressible computation
−0.6
−0.6
−0.8
−0.8
−1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−1
0.3
1
0.4
0.5
0.6
cr
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
0.4
0.5
0.6
0.7
0.8
0.9
−1
0.3
1
0.4
0.5
0.6
cr
0
0
−0.2
−0.2
−0.4
−0.4
ci
ci
0.2
−0.6
−0.6
−0.8
−0.8
0.5
0.6
1
0.7
0.8
0.9
1
0.8
0.9
1
(d) N = 160
0.2
0.4
0.9
cr
(c) N = 120
−1
0.3
0.8
(b) N = 80
ci
ci
(a) N = 40
−1
0.3
0.7
cr
0.7
c
r
(e) N = 201
0.8
0.9
1
−1
0.3
0.4
0.5
0.6
0.7
c
r
(f) N = 240
Figure 3.1.: convergence of the whole spectrum for a Blasius boundary
layer at Re = 290.5625, α = 0.1162 computed by a mapped
Chebyshev collocation method: comparison with Henningson
[91] (red circle)
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Chapter 3. Linear stability for compressible flows
c
0.364123 + i 0.007960
0.232921 - i 0.134343
0.289714 - i 0.276866
0.483929 - i 0.192068
0.557197 - i 0.365342
Malik [62](MDSP)
0.3641 +i 0.0079
0.2329 - i 0.1343
0.2897 - i 0.2768
0.4839 - i 0.1921
0.5571 - i 0.3655
Mack [60]
0.3641 + i 0.0080
0.2897 - i 0.2769
0.4839 - i 0.1921
0.5572 - i 0.3653
Table 3.2.: Most unstable eigenvalue and first four less unstable eigenvalues for a subsonic boundary layer flow at Re = 580, α = 0.179
for N=200
10
8
y/lref
6
4
2
0
0
0.2
0.4
0.6
0.8
1
kũk
Figure 3.2.: Stream wise velocity perturbation normalized for Re =
290.5625, α = 0.1162: VESTA results (solid line) for 200
points and Henningson [91] results (red circles)
1
2
3
collocation
0.308349 + i 0.007940
0.307846 - i 0.001897
0.308591 - i 0.005709
Danabasoglu & al. [25]
0.30864 + i 0.00799
0.30801 - i 0.00184
0.30870 - i 0.00564
Jordinson [45]
0.3084 + i 0.0079
0.3079 + i 0.0019
0.3086 - i 0.0057
Table 3.3.: Most unstable/ less unstable eigenvalues for a subsonic boundary layer flow for the cases of Tab. 2.4, for N = 50 solved by
means of the compressible stability solver
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3.7. Compressible computations
10
8
y/lref
6
4
2
0
0
0.2
0.4
0.6
0.8
1
kṽk
Figure 3.3.: Wall normal velocity perturbation normalized for Re =
290.5625, α = 0.1162: VESTA results (solid line) for 200
points and Henningson [91] results (red circles)
Me
0.5
2.5
10
T0 [K]
277.78
333.33
2333.33
Re
2000
3000
1000
yi
3
6
32
Table 3.4.: Computed case from Malik [62]
3.7. Compressible computations
A set of compressible cases has been evaluated to verify the code at higher
speeds. The test cases are compared with [62] and they range from subsonic to hypersonic. All the mean profile considered have been computed
with adiabatic wall condition. The first case is a subsonic compressible
flow at Mach 0.5 with a free stream temperature of Te = 264.55K and
Re = 2000. As the critical layer for this case is near the wall, our method,
which features a clustering close to the wall, performs well and reaches a
five digit convergence with only 33 points and six digit already with 41
points. The final result in Tab. 3.5 differs slightly from the one of Malik
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Chapter 3. Linear stability for compressible flows
N
17
25
33
41
51
61
81
0.029532804
0.029071021
0.029077325
0.029076128
0.029076171
0.029076177
0.029076174
ω
+i
+i
+i
+i
+i
+i
+i
0.0025747892
0.0022501874
0.0022440225
0.0022436985
0.0022437082
0.0022437073
0.0022437073
SDSP [62]
0.02917687 +i 0.00230046
0.02908247 +i 0.00224387
0.02908189 +i 0.00224427
0.02908185 +i 0.00224419
0.02908180 +i 0.00224419
-
Table 3.5.: Most unstable eigenvalue for a subsonic boundary layer flow
at M ach = 0.5, α = 0.1 β = 0.0
N
25
33
41
51
61
0.036831602
0.036607731
0.036702102
0.036678810
0.036679483
ω
+
+
+
+
+
0.000521447
0.000638316
0.000581119
0.000571159
0.000572772
SDSP [62]
0.0368934 +i 0.0004199
0.0366808 +i 0.0006584
0.0367521 +i 0.0005865
0.0367332 +i 0.0005832
0.0367339 +i 0.0005840
Table 3.6.: Most unstable eigenvalue for a subsonic boundary layer flow
at M ach = 2.5, α = 0.06 β = 0.1
but differences could come easily from a different specific heat coefficient
at constant pressure of slightly different constants in the Sutherland’s
law. Despite this minor difference, the final rate of convergence shown is
the same in both computation and our solution is accurate at the eight
digit with only 41 points. The second case treats a flow at M = 2.5 and
Re = 3000. Being a first mode the most unstable eigenvalue is obtained
for oblique waves, that is when β 6= 0. We do not sweep the wavenumber
β looking for the most amplified eigenvector but we consider only one
value β = 0.1 for verification purposes. As we observe from the results in
Tab. 3.6 the result is satisfactory also in this case
The result of the analysis performed for Mach 10 is in Tab. 3.7. At this
condition the temperature peak is produced close to the boundary layer
edge. The second order finite difference method proposed by Malik converges slowly, the fourth-order scheme converges only with a good initial
guess (at least 33 points from the second order FD method). The single
domain (similar to our method) and the multi-domain spectral method
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3.8. Neutral stability curve computation
N
61
81
101
161
181
201
ω
0.115230975 + i 0.000333674
0.115681803 - i 0.000215220
0.116049357 + i 0.000085308
0.115843745 + i 0.000122474
0.115841574 + i 0.000136168
0.115837311 + i 0.000133927
MDSP [62]
0.1158529 +i 0.0001999
0.1158519 +i 0.0001357
-
Table 3.7.: Most unstable eigenvalue for a subsonic boundary layer flow
at M ach = 10, α = 0.12 β = 0
oscillate with the latter converging with 61 points. This controversial case
is solved by our code with at least 181 points. The high number of points
is not a big problem as we saw that the code is not heavily affected by
ill-conditioning. We can conclude that even for ”badly posed” cases our
solver is robust enough to retrieve the correct eigenvalue.
3.8. Neutral stability curve computation
A set of neutral stability curves is computed to explain the effect of the
Mach number on the stability of supersonic boundary layer. Many parameters affect its final shape and considerations can be drawn only qualitatively as the actual values change with the conditions. A first set of
computations ranges from Mach 2 up to Mach 10 with adiabatic wall. The
free stream temperature is fixed for all these cases at 288K. The comparison in the Figs. 3.4-3.10 is against the computations performed by Özgen
and Kırcalı in [77], while an example of the velocity and temperature profiles is given in Fig. 3.11. Those curves are obtained for profiles which
consider a gas whose properties depend on temperature according to the
laws:
T 1 /2
k = S2
1 + S3 /T · 10−S4 /T
2 γperf − 1
θ
exp(θ/T )
cp = cpperf 1 +
γperf
T
(exp(θ/T ) − 1)2
while our calculations are performed for a calorically perfect gas with
fixed Prandtl number P r = 0.7. Only the use of the Sutherland’s law for
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Chapter 3. Linear stability for compressible flows
viscosity is common between the two computations. As expected different
properties lead to different velocity and temperature profiles nevertheless,
similar behavior are recovered for both cases. Bigger differences seem to
appear mainly in the upper branches.
Fig. 3.4 for a Mach 2 flow is in excellent agreement with the reference
literature, even if the solution published in [77] is obtained by means of a
different method. Despite the different property modeling the two curves
almost overlap. At this Mach number, differences in temperature are most
probably quite small, thus implying small variations in the mean flow.
For the Mach 3 case the shape of the curve changes because the viscous
instability merge with the inflectional one and this part of the graph is
the one showing most of the differences. The inflectional stability depends
more on the values of the properties than the viscous one, as it is related
to the position of the generalized inflection point and this influence is
clearly visible in the upper branch of Fig. 3.5. A second pocket shows up
in Fig. 3.6 and Fig. 3.7: this is linked to the appearance of the second
Mack’s mode when the Mach number is sufficiently high. With free stream
temperature fixed at 228 K we observe that the second mode requires a
higher Mach number to actually merge with the first mode curve (around
Mach 7, see Fig. 3.8).
Mach 10 computation by Özgen (Fig. 3.10) presents a little finger in the
upper branch plot. This is not noticeable in other stability curves the
author is aware of and it is likely linked to the thermally perfect gas
approximation (private communication).
Not many neutral stability curves for compressible flows are available in
literature for this reason we compare our code also with Arnal’s work [4]
(Figs. 3.12-3.17). Those have been computed considering a P r = 0.725
and a total temperature of 300K as long as the static temperature is above
50K (see Tab. 3.8). This choice is strongly affecting the shape of the mean
flow profiles and therefore the one of the stability curve. Changing the
free stream temperature has a deep effect even in our simplified calorically
perfect gas model where only viscosity and thermal conductivity changes
with temperature in such a way to keep the Prandtl number constant.
The Prandtl number value is slightly unusual carrying some effects on
stability: the main reason is that the ratio between the thermal and the
mechanical boundary layer changes moving consequently the generalized
inflection point. Calculations shown in Figs. 3.12-3.17 report the Reδ =
Ue δ/νe on the x-axis and the αδ = α∗ δ on the y-axis where α∗ is the
dimensional angular wavenumber. Our curves have been calculated for
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3.8. Neutral stability curve computation
0.08
Present calculation
Özgen
0.07
0.06
α
0.05
0.04
0.03
0.02
0.01
0
0
1000
2000
3000
4000
5000
Re
Figure 3.4.: Neutral Stability curve at Mach 2
0.06
Present calculation
Özgen
0.05
α
0.04
0.03
0.02
0.01
0
0
1000
2000
3000
4000
5000
Re
Figure 3.5.: Neutral Stability curve at Mach 3
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Chapter 3. Linear stability for compressible flows
0.4
0.35
0.3
α
0.25
Present calculation
Özgen
0.2
0.15
0.1
0.05
0
0
1000
2000
3000
4000
5000
Re
Figure 3.6.: Neutral Stability curve at Mach 4
0.25
Present calculation
Özgen
0.2
α
0.15
0.1
0.05
0
0
1000
2000
3000
4000
5000
Re
Figure 3.7.: Neutral Stability curve at Mach 6
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3.8. Neutral stability curve computation
0.2
Present calculation
Özgen
0.18
0.16
0.14
α
0.12
0.1
0.08
0.06
0.04
0.02
0
0
1000
2000
3000
4000
5000
Re
Figure 3.8.: Neutral Stability curve at Mach 7
0.2
0.18
0.16
0.14
α
0.12
Present calculation
Özgen
0.1
0.08
0.06
0.04
0.02
0
0
1000
2000
3000
4000
5000
Re
Figure 3.9.: Neutral Stability curve at Mach 8
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Chapter 3. Linear stability for compressible flows
0.25
0.2
0.15
α
Present calculation
Özgen
0.1
0.05
0
0
1000
2000
3000
4000
5000
Re
Figure 3.10.: Neutral Stability curve at Mach 10
our usual range, that is up to Re = 5000. The conversion to Reδ provides
a different scale for each case as the boundary layer thickness changes. We
plot the results starting from Mach 2.2, because there is little difference
at lower Mach with respect to the incompressible case. At low supersonic
speed the main instability is viscous (physically resembling the TollmienSchlichting mechanism of an incompressible boundary layer). Starting
from Mach 2.2 the inflectional instability is slightly influencing the curve
at high Reynolds number. Interestingly, each of these curves has a critical
Reynolds number below which any disturbance is damped. For Mach 3 in
Fig. 3.13 (analogously to Fig. 3.5 ) the viscous and inviscid instabilities
merge around Reδ1 = 5000. Differences are present in the curves specially
in the upper branch nevertheless the overall behavior is correct. Those
differences could be related to the inflectional instabilities and therefore to
the position of the generalized inflection point. At Mach 4.5 (Fig. 3.14) we
observe a second pocket in the neutral stability curve due to the so called
second mode. For these conditions the two pockets merge around Mach
4.8 and at Mach 5.8 (Fig. 3.15) just one curve is visible. The agreement
is satisfactory for all the following computations up to Mach 10.
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3.8. Neutral stability curve computation
40
30
y/lref
M10
20
M8
M6 M7
10
M4
M2
M3
0
0
0.2
0.4
0.6
0.8
1
U/Ue
(a) non dimensional velocity profiles
40
y/lref
30
20
10
M4
M2
0
0
M3
5
M6
M7
10
T/Te
M8
M10
15
20
(b) non dimensional temperature profiles
Figure 3.11.: Non dimensional velocity and temperature mean profile for
different Mach number at Re = 1000 for adiabatic wall and
free stream temperature T = 288K
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Chapter 3. Linear stability for compressible flows
0.35
Present calculation
Arnal
0.3
0.25
α
0.2
0.15
0.1
0.05
0
0
2000 4000 6000 8000 10000 12000 14000 16000 18000
Re
Figure 3.12.: Neutral Stability curve at Mach 2.2 and Pr 0.725
0.5
Present calculation
Arnal
0.45
0.4
0.35
α
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1000
2000
3000
4000
Re
5000
6000
7000
8000
Figure 3.13.: Neutral Stability curve at Mach 3 and Pr 0.725
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3.8. Neutral stability curve computation
3
Present calculation
Arnal
2.5
α
2
1.5
1
0.5
0
0
1
2
3
4
Re
5
4
x 10
Figure 3.14.: Neutral Stability curve at Mach 4.5 and Pr 0.725
4
Present calculation
Arnal
3.5
3
α
2.5
2
1.5
1
0.5
0
0
2
4
6
Re
8
10
4
x 10
Figure 3.15.: Neutral Stability curve at Mach 5.8 and Pr 0.725
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Chapter 3. Linear stability for compressible flows
6
Present calculation
Arnal
5
α
4
3
2
1
0
0
2
4
6
8
Re
10
4
x 10
Figure 3.16.: Neutral Stability curve at Mach 7 and Pr 0.725
9
Present calculation
Arnal
8
7
6
α
5
4
3
2
1
0
0
2
4
6
Re
8
10
4
x 10
Figure 3.17.: Neutral Stability curve at Mach 10 and Pr 0.725
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3.9. Summary on the compressible linear stability problem
Me
Te
2.2
152
3
107
4.5
59
5.8
50
7
50
10
50
Table 3.8.: Computed case from 3.17
3.9. Summary on the compressible linear
stability problem
The stability analysis of a compressible flow requires a bigger effort with
respect to the incompressible case. The cause of this difficulty lies in its
special features that include a set of phenomena without a corresponding
counterpart in the incompressible flow. Furthermore, because of compressibility, a compact notation such as the one achieved by the OrrSommerfeld equation, is impossible to achieve, leading to a bigger and
more complex system of equations to solve.
For high subsonic cases there are not major differences with the stability
of an incompressible flow and also in the low supersonic regime differences in the neutral stability curves are note qualitatively evident. This
is due to the persistence of the viscous instability up to low supersonic
Mach numbers, which has its incompressible counterpart in the TollmienSchlichting waves. The main difference is the instability of oblique waves
which are the most unstable for this kind of modes while, according to
the Squire’s transformation, in the incompressible case two-dimensional
waves are predominant.
At higher Mach numbers, compressible stability differs remarkably from
Orr-Sommerfeld. Around Mach 3 (the exact value depends on the conditions) inflectional instabilities come into play and around Mach 4 the
so called second mode is present also at low Reynolds numbers, becoming
the most unstable mode.
Our code is capable of computing the whole spectrum of a compressible
boundary layer retrieving good results for all these cases and verification
with literature proved to be successful. All kinds of flow regimes could
be studied regardless of their condition, type of instability, position of the
critical layer.
On top of this basic functionality, other functions have been added to
the toolkit allowing us to compute the neutral stability curve for different flow regimes. The comparison with literature is satisfactory; correct
curve shapes could be obtained despite small differences. In the work of
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Chapter 3. Linear stability for compressible flows
Malik [62], computations by means of different methods slightly disagree
on the final eigenvalue by a small amount, furthermore slightly different
properties have an effect on the flow stability, therefore we can confidently
think that the presented results are in the correct ball park.
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Chapter 4.
Linear stability for chemical equilibrium
flows
4.1. Introduction
Recently, a renewed interest for hypersonic flows is growing in the transition community. Projects as the nasp, x31 and recently lapcat and
expert are related to hypersonic vehicle design where transition is a fundamental part of the physical models involved in the design process. It is
clear that those are just examples of the kind of projects undergoing in
the main research institutions and space agency around the world, nevertheless it gives a rough idea of what we should expect in the near future.
Our effort is devoted to provide a tool able to cope with flows at different
Mach numbers up to extreme conditions where air dissociates and chemically reacts. For this reason the compressible linear stability solver (clst)
described in chapter 3 has been extended to study such a flow.
The derivation of a new set of equations, based on the hypotheses presented in the previous chapters, and under the Local Thermodynamic
Equilibrium (lte) assumption, is needed.
So while we retain the perfect gas nomenclature (consistently to what is
done currently in hypersonics for this kind of flows) we have to specify
that specific heats are no longer constant, internal energy and transport
properties are function of two thermodynamic variables.
Among the first to include chemistry effect in stability computations there
were Malik & Anderson [64] and Stuckert [94]. In the work of Hudson et
al. [39] also thermodynamic and chemical non equilibrium is considered.
More recently the work of Massa [67] further investigates boundary layer
stability with thermo-chemical non equilibrium.
It is important to note that for flows at very high temperature, where
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Chapter 4. Linear stability for chemical equilibrium flows
dissociation and ionization play a fundamental role, transition may not
be always an issue, at least from a practical point of view; on the contrary
flows in thermal equilibrium (chemical equilibrium or non equilibrium)
are a relevant problem that should be addressed properly. Keeping this in
mind, one can see how the LTE extension is useful and satisfactory from
the physical modeling stand point.
Chemical properties and their values affect enormously the final outcome
of this kind of computations both in mean flow and stability. Reaction
rates are scattered and comparison is difficult. A key point for a linear stability analysis is the use of consistent transport properties shared
between the mean flow computation and the chemical equilibrium lst.
4.2. Governing equations
Hypersonic flows at high temperature display a particular behavior that
could not be neglected in the derivation of the linear stability equations.
It should be kept in mind that above 2000K the air is better described by
the behavior of single species as, at this temperature, oxygen molecules
start dissociating. The fluid should reach 4000K to make nitrogen to
dissociate. At higher temperature also ionization occurs, increasing the
number of species to be solved for. Depending on the chemical activity
ongoing into the flow around the hypersonic vehicle (and also on its altitude) different regimes could be found, ranging from chemical equilibrium
to thermo-chemical non equilibrium.
The choice of one model over the others depends on the flow conditions.
Excited molecules interact among each other by means of collisions and
both the exchange of energy and the chemical reactions happen because
of them. Thermodynamic conditions affect this process quite intuitively;
high pressure, for instance, packs more molecules and atoms in the same
volume, causing more collision, to happen, on the same line high temperature increases the level of excitation of the molecule resulting as well in
a higher number of collisions.
An accurate representation of the physics involved can become rapidly
overwhelming, nevertheless the underlying mechanism is easily understandable.
We stated in the introductory section that the Local Thermodynamic
Equilibrium modeling is used. The flow is intended to be at the same
time in thermodynamic equilibrium and chemical equilibrium conditions.
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4.2. Governing equations
The former demands that a local Boltzmann distribution happens at each
point. This means that a Boltzmann distribution of molecules appears
over all the energy level of the system. From quantum mechanics a set
of quantized energy levels are known to exist and the molecules are distributed onto those energy levels. Boltzmann distribution represents, by
definition, the most probable molecule arrangement (more details could
be found in [99] and [2]) and therefore the state of thermodynamic equilibrium.
Speaking about chemical reactions, the flow is considered, locally, in chemical equilibrium with respect to the local value of pressure and temperature. Equilibrium constant could be retrieved by experiments or statistical thermodynamics and they are used in the procedure to determine the
mixture composition together with the pressure and temperature input.
Once the composition is known all the properties could be found.
Dealing with air it is possible to compute the composition for a set of pressure and temperature and to store the results in a look-up table. This
approach is not always applicable as the mixture is not always predetermined. Anyway for air it is possible to find already pre-compiled tables
(as the janaf tables). In this work we will use the vki mutation library
(it will be explained in §4.3 and more details could be found in [61]) to
find the properties needed.
In chemical equilibrium reactions happen in the flow at such a high rate
(eventually infinite, in comparison with the flow speed) that mixture could
be considered everywhere at anytime in equilibrium. This is often expressed by the Damköhler numbers,
Da τf low
τchem
If the characteristic reaction time τchem is comparable with the charachteristic flow time τf low , we have what is called finite rate chemistry. At
very high temperatures, where also this assumption is too restrictive, more
than one temperature is required to describe all the relaxations process
happening at molecular and atomic level. Anyway this model is needed
only at high speeds and altitudes, where transition is much less of an issue than at moderate speed and altitude, where flow can be modeled in
chemical equilibrium.
We use air-5 mixture (N2 ,O2 ,N ,O,N O) in the mutation library to compute the rates and then the flow properties. The reactions considered in
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Chapter 4. Linear stability for chemical equilibrium flows
this model are the following ones
N2 → 2N
O2 → 2O
NO → N + O
As the flow is a mixture of species, by applying the Dalton’s law of partial
pressure one could re-write the state equation
"n #
s
X
Ci
p = ρR
ρT
(4.1)
Mi
i=1
where R is the is universal gas constant R = 8314J/ (Kg mol K), Ci and
Mi are respectively the mass fraction and the molecular weight of the
i−th species . This could be finally simplified in
p = ρRT ζ
(4.2)
where R is the gas constant for the undissociated gas N2 − O2 mixture
R = 287J/(Kg K) and ζ = MMundiss
is the so called compressibility factor.
diss
Of course for the calorically perfect gas case Mundiss = Mdiss where
!−1
ns
X
Ci
(4.3)
Mdiss =
Mi
i=1
is the molecular weight of the dissociated gas. The compressibility factor
ζ could be expressed as
R
R
0.21MO2 + 0.79MN2
= Rζ → R =
→ ζ=
M
0.21MO2 + 0.79MN2
M (p, T )
(4.4)
By non dimensionalizing the state equation we arrive to
H p̂ = ρ̂T̂ ζ
(4.5)
where all the ˆ variables are non dimensional and H = Ec · cp /R.
Not only pressure, but also other values, such as internal energy, could be
represented as a weighted sum of the property of the respective species
similarly to (4.1). Enthalpy for instance is expressed as
h=
N
X
ρi
j=i
ρ
hi .
(4.6)
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4.2. Governing equations
For lte flows, the Naviers-Stokes equations are still formally valid in their
usual formulation without any addition, nevertheless the non dimensional
form is slightly different, especially for the energy equation that is reported
here as
ρ
Dp Ec
1
Dh
= Ec
+
∇·q .
∇ · τ u − u∇ · τ +
Dt
Dt
Re
Re · P r
(4.7)
The changes are formally related only to the introduction of a new nondimensional number, the Eckert number
Ec =
Ue2
.
cp Te
(4.8)
A second, yet fundamental, difference is in the way enthalpy is written.
In eq.(4.7) we did not express it directly as a temperature times a specific
heat at constant pressure. This would be plain wrong because enthalpy
is now a function of two thermodynamic variables. In order to remain
consistent with the previous chapter we choose these two variables to be
pressure and temperature.
Furthermore, the specific heat at constant pressure, for LTE flows, consists
of a part due to the chemical reactions summed to another one representing the frozen specific heat. Regardless of the source of this enthalpy we
use, for instance for the time derivative, equations of the following kind
∂h
∂h ∂T
∂h ∂p
=
+
.
(4.9)
∂t
∂T p ∂t
∂p T ∂t
computing the derivatives with respect to pressure and temperature by
means of mutation, according the procedure specified in §4.3.
The heat flux, neglecting radiation and the Dufour’s effect, is written as
q = −kf r ∇T +
N
X
hi J i
(4.10)
j=1
where kf r is the “frozen” thermal conductivity and J i is the diffusion
mass flux of the ith species. For our purposes eq. (4.10) is rewritten like
a Fourier’s law using an equivalent thermal conductivity keq
q = keq ∇T .
(4.11)
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Chapter 4. Linear stability for chemical equilibrium flows
For stability it is not relevant the way this is computed, it suffices to
consider h, k, λ, µ as function of pressure and temperature and to develop
the system of equations accordingly. Given this assumption one still has
more variable than equations and in this case the development done for
the eq. (3.10) leads to
µ0 =
dµ 0 dµ 0
T +
p
dT
dp
(4.12)
This first order simplification has been applied not only to the transport
properties but also to enthalpy (similarly to eq. (4.9)) and the derivative
have been computed outside of the mutation library. Reference values
are fixed the same way as it was done for the calorically perfect gas and
the final set of equations was obtained by substituting directly density
perturbation with pressure and temperature perturbations accordingly to
non dimensional perturbed equation of state for a perfect gas in lte
ρ̃ =
Gρ
Fρ
p̃ −
T̃
P
T
(4.13)
Therefore the final set of equations for linear stability in LTE is:
continuity eq.
−
ṽ H P T y ζT
ṽ H P P y ζP
−
+
2
Tζ
T ζ2
i β T̃ G H P W
i α F H p̃ U
i β F H p̃ W
−
+
+
2
Tζ
Tζ
T ζ
−
i α T̃ G H P U
2
T ζ
−
ṽ H P T y
2
T ζ
+
i ω F H p̃ ṽ H P y
+
+
Tζ
Tζ
(4.14)
i β w̃ H P
ṽy H P
i α ũ H P
i ω T̃ G H P
+
+
−
=0
2
Tζ
Tζ
Tζ
T ζ
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4.2. Governing equations
x-mom eq.
i β ũ H P W
ṽ H P U y
i α ũ H P U
+
+
+
Tζ
Tζ
Tζ
−
µ p̃ U y y
T̃ µT U y y
i ω ũ H P
= P
+
+
Re
Re
Tζ
T̃ µT T T y U y
µP T p̃ T y U y
+
+
Re
Re
µP p̃y U y
µ
P y p̃ U y
T̃ µP T P y U y
+ PP
+
+
Re
Re
Re
T̃y µT U y
i α ṽ µT T y
ũy µT T y
+
+
+
Re
Re
Re
i α ṽ µP P y
ũy µP P y
α β w̃ µ
−i α p̃ +
+
−
+
Re
Re
Re
i α ṽy µ ũy y µ β 2 ũ µ
+
−
+
+
Re
Re
Re
2 α2 ũ µ α β w̃ λ i α ṽy λ α2 ũ λ
−
−
+
−
Re
Re
Re
Re
+
(4.15)
y-mom eq.
i β ṽ H P W
i α ṽ H P U
+
+
Tζ
Tζ
−
i β T̃ µT W y
i α µP p̃ U y
i β µP p̃ W y
i ω ṽ H P
+
+
+
=
Re
Re
Re
Tζ
i α T̃ µT U y
2 ṽy µT T y
i β w̃ λT T y
+
+
+
Re
Re
Re
ṽy λT T y
i α ũ λT T y
2 ṽy µP P y
i β w̃ λP P y
+
+
− p̃y +
+
+
Re
Re
Re
Re
ṽy λP P y
i α ũ λP P y
i β w̃y µ 2 ṽy y µ
+
+
+
+
+
Re
Re
Re
Re
β 2 ṽ µ α2 ṽ µ i α ũy µ i β w̃y λ ṽy y λ i α ũy λ
−
−
+
+
+
+
Re
Re
Re
Re
Re
Re
(4.16)
+
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Chapter 4. Linear stability for chemical equilibrium flows
z-mom eq.
ṽ H P W y
i β w̃ H P W
i α w̃ H P U
i ω w̃ H P
+
+
−
=
Tζ
Tζ
Tζ
Tζ
p̃ T y W y
µP p̃ W y y
T̃ µT W y y
µ
T̃ µT T T y W y
+
+ PT
+
+
Re
Re
Re
Re
µ p̃y W y
µ
P y p̃ W y
T̃ µP T P y W y
+ P
+ PP
+
+
Re
Re
Re
(4.17)
T̃y µT W y
w̃y µT T y
i β ṽ µT T y
+
+
+
− i β p̃+
Re
Re
Re
i β ṽ µP P y
w̃y µP P y
w̃y y µ 2 β 2 w̃ µ α2 w̃ µ
+
+
−
−
+
+
Re
Re
Re
Re
Re
i β ṽy µ α β ũ µ β 2 w̃ λ i β ṽy λ α β ũ λ
−
−
+
−
+
Re
Re
Re
Re
Re
energy eq.
HP
iβW + iαU − iω hP p̃ + hT T̃ +
Tζ
HP hP P y p̃ + hT Ty T̃ =
Tζ
2
2 !
2
2 !
Wy
Uy
Wy
Uy
+
+ T̃ µT
+
Ec µP p̃
Re
Re
Re
Re
2 i α Uy
2 w̃y µ W y
2iβ Wy
+ṽ µ
+
+
+
Re
Re
Re
2 ũy µ U y
p̃ i β W + i α U − i ω +
+ ṽ P y +
Re
!
2
kT T T y
kT T y y
P y kP T T y
α2 + β 2 k
+
+
−
+
+T̃
Pr Re
Pr Re
Pr Re
Pr Re
!
2
kP T T y
kP T y y
P y kP P T y
+
+
+
+p̃
Pr Re
Pr Re
Pr Re
!
2 kT T y
P y kP
p̃y k P T y
T̃y y k
+T̃y
+
+
+
Pr Re
Pr Re
Pr Re
Pr Re
(4.18)
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4.3. Properties interpolation
4.3. Properties interpolation
The interface we decided to use for this chemical equilibrium flow is very
simple. In the first phase we extract the mean profile we want to analyze
and estimate the minimum and maximum temperature and pressure value.
The mixture at equilibrium is completely defined by these fixed ranges
of temperature and pressure, therefore it is possible to pre-compute the
interesting value for a specific thermodynamic condition.
The reason this approach has been chosen lyes in the greater efficiency
compared to a hard link with the library used for the computation of every
single point in the profile. Moreover, as it is evident from Eqs. (4.14)(4.18), the second derivative of those fluid properties should be computed
and at each location in the profile, in case of a direct link with mutation,
more evaluations should be done to estimate the derivative with respect
to pressure and temperature
With our approach there is only one library evaluation per point. Indeed
we use once again the power of Chebyshev polynomial and we use the data
coming from mutation as a function to be interpolated. The variables
to be derived are computed on a range of p and T spaced according
Chebyshev points
y = cos(
π(2j + 1)
),
2(N + 1)
j = 1, . . . , N ,
(4.19)
where N is the total number of points, and in a second step, the interpolants are derived
0
u (x)interp =
N
−1
X
a(1)
n Tn (x) ;
(4.20)
n=0
where
a(1)
n
2
=
cn
N
X
p=n+1
p+n odd
(
pap
where
cn =
2
1
n=0
.
n>0
(4.21)
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Chapter 4. Linear stability for chemical equilibrium flows
4.4. Comparison between LTE gas calorically
perfect gas flows
Reaction rates play an important role in determining the final stability
result, therefore it is not always easy to compare against pre-existing literature.
It is somehow easier to simulate calorically perfect cases looking for differences between the two calculations. This is indeed a straightforward way
of checking the implementation consistency. Nothing prevents us from using the lte solver also for lower Mach number cases, with flow properties
computed on physical principles instead of empirical laws like Sutherland’s
one. In this sense the lte solver is a superset of what explained in chapter
3.
4.4.1. Flat Plate flow at Mach 2.5, adiabatic wall
The first case we solve is the Mach 2.5 with adiabatic wall, already described in §3.7, and whose convergence is outlined in Tab. 3.6. The mean
flow (Fig. 4.1) computed by the lte solver shows only minor differences
with respect to the calorically perfect gas (cpg) solution and the corresponding eigenvalues are also similar to each other. Eigenvectors computed by both methods look alike, in Fig. 4.2 temperature and velocity
are shown and they match as expected. A comparison between the two
eigenvalues is reported in Tab. 4.1, showing a good, even if not perfect,
match. All those subtle differences could be explained by means of a varying specific heat (instead of constant, as it is assumed in chapter 3) and
the difference of viscosity and thermal conductivity as shown in Fig. 4.3.
Those differences are not related to a chemistry effect but merely to the use
of a different set of properties. Nevertheless values are matching closely
and we conclude that the lte solver is able to replicate the result of the
compressible solver.
4.4.2. Flat plate flow at Mach 10 cold wall
This case has been reported in Hudson et al. [39] and studied also in
Marxen et al. [66]. In [66] the flow has been modeled as a thermally
perfect gas.
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4.4. Comparison between LTE gas calorically perfect gas flows
CPG
0.036679483 + i 0.000572772
LTE
0.036392080 + i 0.000511932
Table 4.1.: Most unstable eigenvalue for a subsonic boundary layer flow
at M ach = 2.5, α = 0.06 β = 0.1 computed by lte and
calorically perfect gas solvers
2.2
1
CPG
LTE
CPG
LTE
0.9
2
0.8
1.8
0.7
0.6
T/Te
U/Ue
1.6
1.4
0.5
0.4
0.3
1.2
0.2
1
0.1
0.8
0
5
10
0
0
15
5
10
y/lref
15
y/lref
(a) Non-dimensional temperature profiles
(b) Non-dimensional velocity profiles
Figure 4.1.: Mean flow for a flat plate at Mach 2.5 with adiabatic wall in
lte and calorically perfect gas (Malik’s [62] case 3)
0.8
0.7
CPG
LTE
0.7
CPG
LTE
0.6
0.6
0.5
U/Ue
T/Te
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
2
4
6
8
10
y/lref
(a) Non-dimensional temperature profiles
0
0
2
4
6
8
10
y/lref
(b) Non-dimensional velocity profiles
Figure 4.2.: Most unstable mode eigenfunction for a flat plate at Mach
2.5 with adiabatic wall (lte and cpg)
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Chapter 4. Linear stability for chemical equilibrium flows
1.9
1.9
CPG
LTE
1.7
1.7
1.6
1.6
1.5
1.5
1.4
1.4
1.3
1.3
1.2
1.2
1.1
1.1
1
0
CPG
LTE
1.8
k/kref
µ/µref
1.8
5
10
y/lref
(a) non-dimensional viscosity profile
15
1
0
5
10
15
y/lref
(b) non-dimensional thermal conductivity
profiles
Figure 4.3.: Transport properties at Mach 2.5 with adiabatic wall (lte
and cpg)
We computed the mean flow by using both the cpg assumption and the
lte modeling. This is a Mach 10 flow with cold wall at 1200K with
P r∞ = 0.691 and a free stream temperature of T∞ = 278. The interest
of this configuration lies in the predominance of the cooling effect with
respect to the chemistry reactions. Nevertheless we can observe few differences in the temperature mean profiles in Fig. 4.4b. We compare our
lte and cpg solutions against Hudson’s solution and, although we see
a general agreement, the maximum temperature and maximum temperature location are slightly different. This results in a dissimilar stability
behavior. In Fig. 4.5 a comparison against chemical non equilibrium and
thermo chemical non equilibrium results of Hudson’s work is reported.
While the growth rate peak is substantially the same there is a shift in
the most amplified frequency for the lte calculations. On the contrary
cpg growth rates maintain approximately the same frequency for the
growth rate peak.
It is difficult to understand how the differences in the mean profile play
a role in this, anyway, as the results in Hudson’s paper come from a cfd
code where equilibrium is simulated by adjustment of the chemical reaction rates instead of a self similar solution, we could confidently state that
our solver is able to provide a reasonably good solution.
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4.4. Comparison between LTE gas calorically perfect gas flows
3500
2000
3000
CPG
LTE
CNEQ
2500
1500
T [K]
U [m/s]
2000
1500
1000
1000
500
500
CPG
LTE
CNEQ
0
−500
0
1
2
3
4
y [m]
0
0
5
x 10
(a) dimensional velocity profiles
1
2
3
4
y [m]
−3
5
−3
x 10
(b) dimensional temperature profiles
Figure 4.4.: Mean flow for a flat plate at Mach 10 with cold wall in chemical equilibrium, thermically perfect gas and calorically perfect
gas
−3
3.5
x 10
PG
LTE
CNEQ
TCNEQ
3
2.5
−αi
2
1.5
1
0.5
0
−0.5
0.08
0.09
ω
0.1
0.11
0.12
Figure 4.5.: Growth rate comparison for a Mach 10 flat plate at Re 2000
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Chapter 4. Linear stability for chemical equilibrium flows
4.4.3. Flat plate flow at Mach 10 adiabatic wall
The cold wall is a condition that represent well the behavior of a flow in
wind tunnel conditions especially if these are blow down or shot facilities
(as respectively the H3 and Longshot at vki). Adiabatic wall in fact could
be reached only if the vehicle (or the model) has been exposed to the flow
for a time long enough. Huge facilities would be required to ensure this
long time. From a practical point of view during flight, condition on the
temperature at the wall depends more on the kind of material available.
For instance in Barry et al. [9] the temperature at the wall is 1600 K
at Mach 10. Catalicity at the wall, ablation and other phenomena could
affect transition but they have not been extensively studied so far. Despite this, adiabatic wall condition remains an important case to study
as it allows to exclude from the simulation all those effect related to wall
cooling or heating.
This specific case has been introduced by Malik & Anderson [64] and recently studied also in the work of Marxen et al. [66]. All the calculations
hereafter have been made for ReL = 2000.
In case of adiabatic wall, velocity and temperature profiles differ quite a
lot (Fig. 4.6): boundary layer thickness decreases and maximum temperature at the wall is approximately half of the calorically perfect gas. In
lte, specific heat coefficients are no longer constants (varying in a wide
range for a chemical equilibrium boundary layer) and they increase with
temperature. This phenomenon is causing the temperature to decrease at
the wall.
We first present the comparison for the most unstable wavenumber for
the frequency ω = 0.068 at Re = 2000. The two pictures in Fig. 4.7
show the comparison of the dns results presented in [65] against the lst
code proposed here. The agreement is quite satisfactory and the biggest
discrepancy is the temperature perturbation. The mode that has been
chosen is the so called second Mack’s mode.
In Malik & Anderson [64] the eigenvalue corresponding to this case has
not been directly reported but a growth rate study has been carried on. In
Fig. 4.8 comparison of such a study is shown where also the computation
of Malik and the one done by castet code at onera (see [78]) are reported. Simulations with our lte solver seem to have a shift in frequency
slightly bigger than what shown in literature nevertheless the growth rate
peak is consistent with previous results. From this comparison the compressible code has a very good agreement with literature and this is due
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4.4. Comparison between LTE gas calorically perfect gas flows
(a) non dimensional velocity profile
(b) non dimensional temperature profile
Figure 4.6.: Non dimensional profile for a Mach 10 flow Re = 2000, [64]
(a) temperature and pressure
(b) stream wise and wall normal velocity
Figure 4.7.: Most unstable eigenmode perturbation for a Mach 10 flow
with adiabatic wall: line vesta, DNS Marxen et al. [65]
to more standard property values. An interesting phenomenon happening
in all the computations not performed under the calorically perfect gas
assumption is the raise of another peak, called the third mode. This mode
is quiescent in cpg but if temperature rises and the chemical reactions
take place then it get excited. These third mode peaks appear, with a
greater scatter than the ones related to the second mode, but still confined
in a small area.
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Chapter 4. Linear stability for chemical equilibrium flows
2.5
LTE VKI
LTE Malik
PG Malik
PG VKI
LTE ONERA
TPG ONERA
2
1.5
−αi ⋅ 103
1
0.5
0
−0.5
−1
−1.5
0
0.2
0.4
0.6
0.8
ω /Re ⋅ 10000
1
1.2
1.4
Figure 4.8.: Growth rate for a Mach 10 flow over a flat plate with no pressure gradient. Comparison with the computation performed
by Malik [64] and Perraud et al. [78]
4.4.4. Pressure and temperature effects
As the underlying physics behind an lte simulation is inherently more
complex than a cpg one we want to investigate in more detail the effects
driven by the two chosen thermodynamic variables, pressure and temperature. This is different from what usually done for compressible and
incompressible flows because it includes also the effect of chemical reaction. Also pressure gradient is usually not considered and neglected in
these computations because of the boundary layer assumption. Anyway,
for an lte flow, pressure is fundamental to obtain the mixture composition.
Our stability solver could relax the boundary layer assumption and consider the pressure varying along the normal to the wall coordinate anyway
this will not be applied in the following results, so we limit this study to
the effect induced by the derivative of the transport properties to the
thermodynamics variables as written in the eq. (4.12). We consider a set
of three different conditions for each thermodynamic variable as listed on
Tab. 4.2. For each pressure all the three temperatures have been com-
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4.4. Comparison between LTE gas calorically perfect gas flows
P[Pa]
T[K]
2000
111
4000
235
6000
350
Table 4.2.: Test matrix for Mach 10
puted and vice versa so that we finally have nine different cases.
The mean flow has been computed using the self-similar boundary layer
solver slightly modified to take in account the properties coming from
the mutation library. It is possible to have a simple self similar boundary layer because lte assumptions leave formally unmodified the NavierStokes equation (without any source term, for instance). Such a simulation can give some first quantitative results retaining one of the simplest
flow model that involve chemical reactions. In Fig. 4.9 and Fig. 4.10 all
the velocity and temperature profiles for all the cases are shown. It is
not difficult to spot that for the investigated range of temperature and
pressure the latter have a minor role. It is also found that the role of pressure increases with temperature so these differences are expected to get
larger once our interval of investigation is extended. As we are using self
similar profiles it is straightforward to compute a neutral stability curve
and therefore to assess the stability of each profile in comparison with the
others. Given the cases in Tab. 4.2 we gather all the data in two ways:
first gathering cases with same temperature, letting the pressure to vary
and then holding the pressure but varying the temperature.
We expect that, as pressure does not affect so much the mean profiles, the
stability property should not change much. This is confirmed by our computations, as seen in Fig. 4.11. The highest temperature case in Fig. 4.11c
is the one with the most visible effect of pressure. From the Figs. 4.11a4.11c it is already possible to spot a difference in the shape of the neutral
stability curve which is more clearly understandable in Fig. 4.12. We
can first observe that the instability area (i.e. inside the neutral stability
curve) increases along with temperature and also the curve shape changes.
As we infer from Figs. 4.9 and 4.10, at 111K chemistry is almost not playing a role and this is also the main reason for the change of shape among
this case and the other two. Also the critical Reynolds number appears to
change and the upper branch of the case at higher temperature appears
to be flatter than the low temperature counterpart.
In order to understand better this phenomena the growth rate has been
computed keeping the Reynolds number fixed for the three different value
reported in Tab. 4.3. Those computations could be interpreted as three
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1
1
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
5
10
15
20
y/lref
25
30
35
0
0
40
(a) p=2000 Pa T=111 K
0.1
5
10
15
20
y/lref
25
30
35
0
0
40
(b) p=2000 Pa T=230 K
1
1
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
U/Uref
1
0.5
0.5
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
5
10
15
20
y/lref
25
30
35
0
0
40
(d) p=4000 Pa T=111 K
10
15
20
y/lref
25
30
35
0
0
40
1
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
U/Uref
1
0.9
U/Uref
1
0.5
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
10
15
20
y/lref
25
30
35
(g) p=6000 Pa T=111 K
0
0
40
25
30
35
40
5
10
15
20
y/lref
25
30
35
40
0.5
0.4
5
20
y/lref
(f) p=4000 Pa T=350 K
0.9
0
0
15
0.1
5
(e) p=4000 Pa T=230 K
0.5
10
0.5
0.4
0
0
5
(c) p=2000 Pa T=350 K
0.9
U/Uref
U/Uref
0.5
0.4
0
0
U/Uref
U/Uref
1
0.9
U/Uref
U/Uref
Chapter 4. Linear stability for chemical equilibrium flows
0.1
5
10
15
20
y/lref
25
30
35
(h) p=6000 Pa T=230 K
40
0
0
5
10
15
20
y/lref
25
30
35
40
(i) p=6000 Pa T=350 K
Figure 4.9.: Self similar velocity profile for Mach 10 flow at Re 5000
Re
2000
1000
500
Table 4.3.: Reynolds for the growth rate computation for 111k and 350 K
cases at Mach 10
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4.4. Comparison between LTE gas calorically perfect gas flows
16
9
12
8
14
10
7
12
8
6
T/Tref
T/Tref
T/Tref
6
8
10
6
4
3
4
4
2
2
2
0
0
5
1
5
10
15
20
y/lref
25
30
35
0
0
40
(a) p=2000 Pa T=111 K
5
10
15
20
y/lref
25
30
35
(b) p=2000 Pa T=230 K
16
0
0
40
5
10
15
20
y/lref
25
30
35
40
(c) p=2000 Pa T=350 K
9
12
8
14
10
7
12
8
6
T/Tref
T/Tref
T/Tref
6
8
10
6
4
3
4
4
2
2
2
0
0
5
1
5
10
15
20
y/lref
25
30
35
0
0
40
(d) p=4000 Pa T=111 K
5
10
15
20
y/lref
25
30
35
(e) p=4000 Pa T=230 K
16
0
0
40
5
10
15
20
y/lref
25
30
35
40
(f) p=4000 Pa T=350 K
9
12
8
14
10
7
12
8
6
T/Tref
T/Tref
T/Tref
6
8
10
6
4
3
4
4
2
2
2
0
0
5
1
5
10
15
20
y/lref
25
30
35
(g) p=6000 Pa T=111 K
40
0
0
5
10
15
20
y/lref
25
30
35
(h) p=6000 Pa T=230 K
40
0
0
5
10
15
20
y/lref
25
30
35
40
(i) p=6000 Pa T=350 K
Figure 4.10.: Self similar temperature profile for Mach 10 flow at Re 5000
cut in the plot of Fig. 4.12 and the results are in displayed in Fig. 4.13.
For sake of clarity only the case at the lowest and highest temperature
are shown because the neutral stability curve at T = 230K does not differ
much in shape from the high temperature one.
Looking at the dash-dotted line representing Re 2000 (please note that
it is a similar case to what plotted in [64]) we note the presence of two
peaks, as expected. The peak at higher wavenumber is once again activated by the chemistry and, though at low temperature it plays a minor
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Chapter 4. Linear stability for chemical equilibrium flows
2000 Pa 230 K
4000 Pa 230 K
6000 Pa 230 K
0.25
0.2
0.2
α
α
2000 Pa 111 K
4000 Pa 111 K
6000 Pa 111 K
0.25
0.15
0.15
0.1
0.1
0.05
0.05
0
0
1000
2000
3000
4000
0
0
5000
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Re
Re
(a) T 111 K
(b) T 230 K
2000 Pa 350 K
4000 Pa 350 K
6000 Pa 350 K
0.25
α
0.2
0.15
0.1
0.05
0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Re
(c) T 350 K
Figure 4.11.: Neutral stability curve for varying pressure grouped by temperature at Mach 10.
role, the peak is visible also at 111K (black dash-dot line). High temperature just causes the peak to be sharper and higher, slightly increasing
its wavenumber. The 2nd mode peak is shifted at higher wavenumbers as
well, nevertheless the two curves are essentially comparable and they are
closely following each another.
At Re = 1000 differences between the two cases are remarkably bigger; at
low temperature the second peak disappears and, after the second mode
peak, growth rates decay, while at 350 K there is a kind of plateau taking
place after the second mode peak (dashed blue line). This effect is entirely
due to temperature. At the intermediate Reynolds, the second peak disappears completely at the highest temperature; growth rates increase and
the plateau make the ωi values computed at Re = 1000 comparable with
the ones at Re = 2000 around α = 0.15. This wavenumber corresponds to
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4.4. Comparison between LTE gas calorically perfect gas flows
4000 Pa 111 K
4000 Pa 230 K
4000 Pa 350 K
0.25
α
0.2
0.15
0.1
0.05
0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Re
Figure 4.12.: Neutral stability curve at 4000 Pa and varying temperature
at Mach 10
−3
1.5
x 10
1
ωi
0.5
0
−0.5
−1
−1.5
0.04
Re 500 T350
Re 1000 T350
Re 2000 T350
Re 500 T111
Re 1000 T111
Re 2000 T111
0.06
0.08
0.1
0.12
α
0.14
0.16
0.18
0.2
Figure 4.13.: Temporal growth rate for Mach 10 flow at 111K and 350K
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Chapter 4. Linear stability for chemical equilibrium flows
150
Re=500 α =0.1074
Re 500 α 0.1599
100
pressure phase
50
0
−50
−100
−150
−200
−250
−300
0
5
10
15
20
y/lref
25
30
35
40
Figure 4.14.: Phase for eigenfunction at the maximum growth rate outside
the second mode peak and for a wavenumebr corresponding
for the second peak for the Re 2000 case
the second peak of the Re = 2000 curve. Moreover the neutral stability
wavenumber differs quite a lot for the two temperature at Re = 1000.
The biggest difference is found at Re = 500: the first peak height is
similar for both temperatures but contrarily to the previous cases, the
biggest peak growth is reached for 111K. The wavenumber shift between
the respective peaks is negligible. The second peak is no longer visible in
both cases, but at 111K, after the 2nd mode peak, there is a strong decay
leading almost immediately to stable ωi , while at higher temperature the
first peak is still visible but almost completely absorbed by the following
plateau. At 350K the Re = 500 case shows a slight growth after the peak
instead of a decay, reaching almost the same growth rate as the 2nd mode
peak. The wavenumbers corresponding to the growth rate peaks decrease
along with the Reynolds number. This observation is true for both temperatures, therefore it is not explicitly related to chemical reactions.
On the Re = 500 case, we focus our attention at two different wavenumbers, one around the maximum growth outside the 2nd mode peak (namely
α = 0.1074) and one at the same location of the second peak in the
Re = 2000 case (α = 0.1599), observing the same number of phase changes
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4.4. Comparison between LTE gas calorically perfect gas flows
in the boundary layer pressure perturbation (see Fig. 4.14). This number
of phase changes corresponds to a third mode, so we can state that, even
if the peak is not visible as it is at higher Reynolds, the plateau owes
its shape to the excitement of the third mode brought in by both chemistry and high temperature. We stress the fact that not only chemistry
is causing the third mode excitement (as it could be evident at higher
Reynolds) but also temperature plays an important role. This statement
appears clearer once the difference in the growth rates is observed; at lower
Reynolds, the high temperature case shows higher growth rates than the
low temperature calculations, while, at higher Reynolds, differences are
less evident. In the Figs. 4.15-4.16 all the eigenfunctions for the values
around the growth rate peaks have been plotted: it is worth to note that
the shape of the perturbation changes pretty a lot at Re = 500, especially
comparing the results related to the third mode, while differences between
the second modes are relatively small.
Going back to Fig. 4.12, it appears evident that temperature is affecting
the neutral stability curve at lower Reynolds number, so a zoom in of
the region is required to understand better that behavior. In Fig. 4.17
the neutral stability curves are plotted for low Reynolds numbers. The
black solid line shows the last portion of the curve close to Recr for the
low temperature case. We observe that as temperature is increasing the
curves are pushed closer to the axes origin, thus enhancing the instability
area described by the curve. They do not show a monotonic behavior as
the 111K case, because of the peak below Re 10.
Between the case at 230 K and 350 K there are differences other than
just the growth rate. What has been found out is that, while for the
intermediate temperature case the upper and lower branch meet in one
point (the so-called critical Reynolds number Recr ) this is not happening
anymore at the higher temperature case. As a matter of fact the lower
branch follows its own trajectory bending back to higher Re, creating a
sort of local minimum as it is visible on the lower red curve in Fig. 4.17.
At the higher temperature we basically do not observe a critical Reynolds
anymore as the curve is completely open. We can then speak of critical
temperature at which the flow is always unstable, at least theoretically.
Anyway this finding was unexpected and regardless the absence of Recr
at low Reynolds the flows becomes more and more stable as the temperature is growing. Of course while Re → 0 the approach itself becomes
questionable as for very low Reynolds number, for instance, it is hard to
respect the quasi-parallel approximation, nevertheless, even fixing a range
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Chapter 4. Linear stability for chemical equilibrium flows
80
70
60
T,P
50
T Re 500
P × 5000 Re 500
T Re 1000
P × 500 Re 1000
T Re 2000
P × 500 Re 2000
40
30
20
10
0
0
5
10
15
20
y/lref
25
30
35
40
(a) kT̃ k kp̃k
1.4
U Re 500
V Re 500
U Re 1000
V Re 1000
U Re 2000
V Re 2000
1.2
1
U,V
0.8
0.6
0.4
0.2
0
0
5
10
15
20
y/lref
25
30
35
40
(b) kT̃ k kp̃k
Figure 4.15.: Eigenfunction amplitudes for the second mode peak for the
different Reynolds
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4.4. Comparison between LTE gas calorically perfect gas flows
140
120
100
T Re 500
P × 5000 Re 500
T Re 1000
P × 500 Re 1000
T Re 2000
P × 500 Re 2000
T,P
80
60
40
20
0
0
5
10
15
20
y/lref
25
30
35
40
(a) kT̃ k kp̃k
1.4
U Re 500
V Re 500
U Re 1000
V Re 1000
U Re 2000
V Re 2000
1.2
1
U,V
0.8
0.6
0.4
0.2
0
0
5
10
15
20
y/lref
25
30
35
40
(b) kũk kṽk
Figure 4.16.: Eigenfunction amplitudes for the third mode peak for the
different Reynolds
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Chapter 4. Linear stability for chemical equilibrium flows
4000 Pa 111 K
4000 Pa 230 K
4000 Pa 350 K
α
0.1
0.05
0
0
10
20
30
40
50
60
70
80
90
100
Re
Figure 4.17.: Neutral stability curve at 4000 Pa and varying temperature
at Mach 10: zoom in at low Reynolds
of validity of our computation, the main observation remains unchanged:
at higher temperatures the critical Reynolds number decreases till it is
virtually impossible to distinguish if it still exists.
4.5. Summary on the linear stability problem for
flows in thermodynamic and chemical
equilibrium
When analyzing a flow at hypersonic speeds one needs to keep in mind
that the simple calorically perfect gas assumptions as outlined in chapter 3 are no longer valid in most of the cases. This is because air start
dissociating and chemical reactions take place. Because of this consideration a set of equations has been derived independently to take into
account the theoretical feature underlying a flow with chemical reactions.
Our implementation uses the lte assumptions which provides a good first
approximation to the stability phenomena at high speed.
The present code has been verified against low Mach solutions obtained
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4.5. Summary for lst of lte flows
with the calorically perfect gas stability solver with good results. At
higher Mach number a slight shift of frequencies toward lower values has
been observed, nevertheless we could find a reasonable agreement with existing literature. Tests verified both a cold wall case, where cooling effect
is predominant with respect to chemistry, and an adiabatic wall where
the chemistry effect is better recognizable.
As the typical lte flow depends on pressure and temperature the role of
these two parameters have been investigated more deeply. It has been
found out that pressure has very limited importance on the flow stability
even if it is a fundamental parameter to determine the air mixture. On the
contrary temperature results to be the driving parameter, even more at
lower Reynolds numbers. If the free stream temperature is increased the
flow becomes more unstable, that is the inner area of the neutral stability
curve tends to increase. On a similar trend we observed that the critical Reynolds number decreases when temperature increases. The drop of
Recr is so abrupt that for a certain temperature it is virtually impossible
to estimate it, making the flow possibly unstable for every Reynolds. This
is a different finding with respect to what computed with our calorically
perfect gas solver. Interestingly, being neutral stability curve for lte flow
not available in literature, this behavior has not been documented before,
at least to the author’s knowledge.
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Chapter 5.
Shock as a boundary condition
5.1. Introduction
An important feature of supersonic and hypersonic flows is the shock in
front of the body. This affects heavily the development of the boundary layer from several points of view and therefore also stability analysis
should take it into account.
Shock waves act directly on flow stability as they react to upwind perturbations modifying the so called receptivity problem (see [38]). It is known
that any kind of wave interacting with the shock boundary generates all
three different types of modes, which are the acoustic mode, the vorticity
mode and the entropy mode (see [88], [71], [8], [33] ). Even when the
way perturbations penetrate through the shock layer is not considered,
the fact that the shock vibrates with a certain frequency affects the stability of the boundary layer. This aspect has been addressed, probably
for the first time, by Petrov [79] using a linearized Rankine-Hugoniot (RH) relation as the shock boundary condition. In Cowley & Hall [22] the
triple deck theory was used also with a R-H boundary condition. Chang
[21] used the unsteady R-H and a similar approach was proposed also by
Herbert & Esfahanian [37]. Stuckert [94] formulated a slightly different
approach, although starting from unsteady R-H. In many cases, approximated self-similar boundary layer profiles were used and cropped at the
shock location as predicted by a simple Euler computation.
This approach is efficient yet it neglects the shock boundary layer interactions, deformation of the mean flow profiles and the displacement of the
shock away from the wall.
Importance of the shock grows with the Mach number, not only because
of its strength, but also because it becomes shallower, widening the area
of the so called “viscous interaction”.
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Chapter 5. Shock as a boundary condition
5.2. Mean flow
We obtain the base flow either by solving a full Navier-Stokes simulation
or by using a self similar profile as explained in appendix B. In the former
case the properties behind and ahead of the shock and its position are
taken directly from the simulation and used as explained in §5.3. Some
care should be taken while meshing the shock in order to introduce as
little error as possible, considering that an eventual displacement of its
position could result in a different stability result. Fortunately, from our
computations, stability analyses seems not to be extremely sensitive to
it. When we use instead a self similar profile we compute beforehand
the shock slope and position and the variable values behind and ahead of
it. Standard tables could be used, introducing a negligible error on the
shock distance as they are obtained for inviscid flows, in case of calorically
perfect gas simulation. Dealing with flows in chemical equilibrium the
jump relations should be iteratively solved in order to obtain the correct
value for all the variables, nevertheless only negligible differences were
noted, with respect to the use of standard tables, in our test case.
5.3. Linearized shock boundary conditons
Shock is an unsteady phenomena and it oscillates according to the perturbations in the free stream, nevertheless an average can be drawn so that it
is possible to describe the linearized flow around the mean position. The
sketch in Fig. 5.1 depicts our assumptions in a clear way. From this point
of view the problem is not far from the linear stability hypotheses already
adopted in the earlier chapters.
Recalling our previous computations one reminds that perturbation amplitudes decrease exponentially above a certain distance from the wall to
reach asymptotically zero at an infinite distance from the wall. The main
effect of the shock is that perturbations have no longer the space to decay asymptotically but they are, in general, cut at the shock position as
depicted in Fig. 5.2. At that point, the shock vibration interacts with
boundary layer mode altering the most unstable/less stable frequencies.
It is natural to model the interaction of the shock as a boundary condition
as it does happen at the boundary of the stability computational domain,
therefore the usual boundary condition, as described in §3.5, should be
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5.3. Linearized shock boundary conditons
envelope
average
position
Shock
Boundary Layer
Figure 5.1.: Shock oscillation around the mean value
abandoned far from the wall and replaced by something representing the
shock forcing.
Classically the relations that links properties behind and ahead of the
discontinuities are governed by the so called Rankine-Hugoniot jump relations. The linearization of these relations leads to the desired description
of the shock motion to be used as a boundary condition. Part of the following derivation comes from [32] and [21].
Let’s take a generic volume (Fig. 5.3), split in two parts V 1 and V 2 by
a moving surface representing the discontinuity S, each one limited, respectively, by the surfaces S1 and S2 (which do not include S). Let’s
define the velocity of the discontinuity surface US as positive if moving
toward the volume V1 . From the V2 standpoint, the velocity is then −US .
Because of the transport theorem we can write, for the volume V1 (and
the volume V2 ),
d
dt
ZZZ
ZZZ
ρΥdV =
V1 (t)
V1
d
ρΥdV +
dt
ZZ
ZZ
ρΥU ndS +
S1
ρΥUs ndS (5.1)
S
where Υ is a generic conservative variable per unit mass. Conservation
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Chapter 5. Shock as a boundary condition
shock
mean flow
perturbation
Figure 5.2.: Sketch of shock influence on boundary layer mode
V2
V1
Us
S1
S
S2
Figure 5.3.: Control volume definition for jump relation derivation
laws for an inviscid fluid reads generally as
ZZZ
ZZ
d
ρΥ + P ndS = 0.
dt
V
(5.2)
∂V
where P is [0; pI; pU ]. By merging eq. (5.2) with eq. (5.1) and its equation for the volume V2 , and by keeping in mind the different sign of the
discontinuity surface velocity with respect to the second volume, one can
write
ZZZ
ZZ
ZZ
∂ρΥ
dV +
φU ndS −
(ρ2 Υ2 − ρ1 Υ1 )US ndS = 0 . (5.3)
∂t
V1 ∪V2
S1 ∪S2
S
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5.3. Linearized shock boundary conditons
where φ represents the flux of the Υ variables plus the P term of eq. (5.2).
As no particular assumption has been made on the volumes, the equations
derived so far are valid for a volume of arbitrary size. In the limit case
where the size of V1 and V2 tends to zero (that is S1 → S and S2 → S)
we obtain the usual Rankine-Hugoniot jump relations:
[φ]n = USn [ρΥ]
(5.4)
where USn is the component of the discontinuity surface velocity normal
to the surface itself, and the [ ] notation represents the difference (·2 −
·1 ). Using body coordinates, the projected Rankine-Hugoniot look as the
following
− US [Υ] + nx [φx ] + ny [φy ] + nz [φz ] = 0
(5.5)
where Υ, φx , φy , φz are
Υ = [ρ, ρU, ρV, ρW, E]
φx = ρU, ρU 2 + p, ρU V, ρU W, (E + p)U
φy = ρV, ρU V, ρV 2 + p, ρV W, (E + p)V
φz = ρW, ρW U, ρW V, ρW 2 + p, (E + p)W
Symbols have the usual meaning and E is the total energy
1
E = ρe + ρ U 2 + V 2
2
(5.6)
where e is the energy per unit of mass. The instantaneous shock position
ys = f (x, y, t) is linearized around the mean shock position so it reads
ys = f (x, y, t) = f (x) + f 0 (x, z, t)
(5.7)
Taking the time derivative of the distance of shock from the wall, one
obtains
∂f dx ∂f dz
∂f
dys
=
+
+
dt
∂x dt
∂z dt
∂t
which is the same as
US ny =
∂f
∂f
∂f
US nx +
+
∂x
∂z
∂t
The direction cosine of ∇f are expressed as
nx = −
∂f 1
;
∂x |ns |
ny =
1
;
|ns |
nz = −
∂f 1
;
∂x |ns |
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Chapter 5. Shock as a boundary condition
where
s
|ns | =
1+
∂f 2 ∂f 2
+
.
∂x
∂z
As a consequence of the above manipulation, eq. (5.5) could be rewritten
as
∂f
∂f
∂f
[Υ] +
[φx ] − [φy ] +
[φz ] = 0
(5.8)
∂t
∂x
∂z
Shock position in eq. (5.7) is treated similarly to other variables (see
eq. (2.1)), and, as a consequence, one could apply to the shock position
perturbation f 0 the decomposition of (2.6), which results in:
f 0 = f˜ · exp(iαx + iβz − iωt) + c.c. .
(5.9)
The linearization procedure is consistent with the lst development. We
decompose first all variables following eq. (2.1) and eq. (5.9), then we
substitute them in eq. (5.8). After discarding the mean-flow equations we
obtain the set of shock linearized perturbation equations. It is worth to
remind that in the whole process we do not consider perturbations ahead
of the shock (so no ũ1 or similar will appear). The general form of these
equations is
h i h i
iα φx + iβ φz − iω Υ f˜ + a φ̃x − φ̃y = 0
(5.10)
The relation coming from the mass conservation Rankine-Hugoniot is
solved for f˜ and then substituted in all the other equations to simplify
the computations. Bar operators are pretty obvious so we just write the
disturbance operator for the reader convenience


ρ2 ũ + ρ̃U2


ρ̃U22 + 2ρ2U2 ũ + p̃
h i



ρ̃U
V
+
ρ2V
ũ
+
ρ
U
ṽ
(5.11)
φ̃x = − 
2
2
2
2
2


ρ̃U2 W2 + ρ2W2 ũ + ρ2 U2 w̃
(E2 + p2 )ũ + U2 (Ẽ + p̃)


ρ2 ṽ + ρ̃V2
 ρ̃U2 V2 + ρ2V2 ũ + ρ2U2 ṽ 
h i



ρ̃V22 + 2ρ2V2 ṽ + p̃
φ̃y = − 


ρ̃V2 W2 + ρ2W2 ṽ + ρ2 V2 w̃
(E2 + p2 )ṽ + V2 (Ẽ + p̃)
(5.12)
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5.3. Linearized shock boundary conditons
where
U22 + V22 + W22
E2 = ρ2 e2 + ρ2
,
2
(5.13)
and
2
U2 + V22 + W22
Ẽ = ρ̃e2 + ẽρ2 + ρ̃
+ ρ2 (U2 ũ + V2 ṽ + W2 w̃) (5.14)
2
So far we reported the equations in dimensional form, which is valid for
both the cpg and lte approximations anyway non-dimensionalization
is changing their final appearance in a major way especially for what
concerns the energy related terms.
Calorically perfect gas
x-mom. eq.
ũ (ρ2 aΛx + ρ2 (V2 − 2aU2 ) Λ) + ṽ (−ρ2 Λx + ρ2 U2 Λ) +
γM22
+
+p̃ −aΛ + (aU2 − V2 ) (Λx − U2 Λ)
T2
ρ2
+T̃ −(aU2 − V2 ) (Λx − U2 Λen )
T2
(5.15)
y-mom. eq.
ũρ2 a (Λy − V2 Λ) + ṽρ2 (−Λy + Λ (2V2 − aU2 )) +
γM22
+p̃ Λ +
(aU2 − V2 ) (Λy − V2 Λ) +
T2
ρ2
+T̃ − (aU2 − V2 ) (Λy − V2 Λ)
T2
(5.16)
z-mom. eq.
ũaρ2 (Λz − W2 Λ) + ṽ(−ρ2 Λz + W2 Λ)+
γM22
(Λz − W2 Λ) (aU2 − V2 ) +
T2
ρ2
T̃ − (Λz − W2 Λ) (aU2 − V2 ) − w̃ρ2 Λ (aU2 − V2 )
T2
p̃
(5.17)
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Chapter 5. Shock as a boundary condition
energy eq.
ũ (−aρ2 Λen − a (e2 + γEc p2 ) Λ) + γEc Λρ2 U2 (aU2 − V2 ) +
ṽ (ρ2 Λen + ΛKρ2 V2 (aU2 − V2 ) − Λ (e2 + Kp2 )) +
γM 2
2
2
p̃ Λ(γM + K)(U2 − V2 ) +
(−Λe n + ΛK∆U )(V2 − aU2 ) +
T
ρ2
2
T̃ − (−Λe n + K∆U )(aU2 − V2 ) +
T2
w̃ΛKρ2 W2 (aU2 − W2 )
(5.18)
where
Λx = iα[ρU 2 + p] + iβ[ρU V ] − iω[ρU ] ;
Λy = iα[ρU V ] + iβ[ρV W ] − iω[ρV ] ;
Λz = iα[ρU W ] + iβ[ρW 2 + p] − iω[ρW ] ;
Λen = iα[(E + K p) U ] + iβ[(E + K p) W ] − iω[E] ;
Λ = iα[ρU ] + iβ[ρW ] − iω[ρ]
K = γ (γ − 1) M 2
Local thermodynamic equilibrium
x-momentum eq.
ũ (ρ2 aΛx + ρ2 Λ (V2 − 2aU2 )) + ṽ (−ρ2 aΛx + ρ2 U2 Λ) +
F ρ2
+
p̃ −aΛ + (aU2 − V2 ) (Λx − U2 Λ)
p2
−Gρ2
+ T̃ (aU2 − V2 ) (Λx − U2 Λ)
T2
(5.19)
y-momentum eq.
ũ (−ρ2 a (Λy + V2 Λ)) + ṽ (−ρ2 Λy − ρ2 Λ (aU2 − V2 )) +
F ρ2
p̃ Λ + (Λy − V2 Λ) (aU2 − V2 )
p2
−Gρ2
+ T̃ (aU2 − V2 ) (Λy − V2 Λ)
T2
(5.20)
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5.3. Linearized shock boundary conditons
z-mom. eq.
ũ (ρ2 a (Λz − W2 Λ)) + ṽ (−ρ2 Λz + ρ2 aW2 Λ) +
w̃ (λρ2 (−aU2 + V2 )) +
F ρ2
p2
Gρ2
+ T̃ ((Λz − W2 Λ) (aU2 − V2 ))
T2
p̃ ((Λz − W2 Λ) (aU2 − V2 ))
(5.21)
energy eq.
ũ (−aρ2 Λen + aΛ (E2 + Kp2 ) + Λρ2 KU2 (aU2 − V2 )) +
+ṽ ρ2 Λen + aΛ (ρ2 KV2 U2 ) − Λ ρ2 KV22 + E2 + Kp2 +
F ρ2
2
(−Λen + Λ(e2 + K∆U ))(aU2 − V2 ) + ep Σ +
+p̃ ΛK (aU2 − V2 ) +
p2
Gρ2
2
+T̃ −
(−Λen + Λ(e2 + K∆U ))(aU2 − V2 ) + eT Σ +
T2
+w̃Λρ2 KW2 (aU2 − V2 )
(5.22)
where
K = γ∞ Ec
5.3.1. Numerical treatment of the boundary condition
The implementation of the equations is, as usual, a paramount aspect.
The development we described so far could be applied to a variety of
methods and no particular limitation is present in the basic assumptions.
Nevertheless, as we already mentioned, one big advantage of our implementation is the ability to manipulate boundary conditions. These four
equations have to be applied only at the upper boundary as they have
been derived across the shock. The way to do it in our Chebyshev collocation implementation is to add four lines in the system to be solved. As
the standard boundary condition implementation of §3.5 drops the lines
and columns related to the variables at boundaries, all we have to do is
to add four rows (and columns) as in §3.5, eq. (3.26). For this case the
columns should contain the coefficient u(∞), v(∞) T (∞), w(∞).
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Chapter 5. Shock as a boundary condition
−3
2
x 10
1.5
1
ωi
0.5
0
Asymptotic b.c.
−0.5
Shock b.c.
−1
Esfahanian shock b.c.
Esfahanian asympt. b.c.
−1.5
0
0.05
0.1
α
0.15
0.2
Figure 5.4.: Growth rate for a Mach 8 flow on a wedge at different
Reynolds numbers Re = 590.91
5.3.2. Results
Wedges present a good test case as they produce a relatively simple flow
field, they are two-dimensional, and therefore fast to compute, and also a
self-similar boundary layer solution can be found. The first test is the one
presented by Esfahanian [32] where a Mach 8 flow with adiabatic wall is
computed on a wedge with a half angle of 5o . It should be noted that the
actual self similar profile is computed with the condition behind the shock
that is M = 6.86 and T = 71.88K while the wall is adiabatic. This computation is intended for the verification of the computer code. In Fig. 5.4,
the case for Re = 590.91 shows a similar behavior for the two calculations
highlighting the stabilizing role of the shock at low frequencies. Minor
differences could be found near the minimum of the shock boundary condition growth rate, nevertheless the computation is satisfactorily close to
the one presented in [32]
The same comparison is carried out for a different Reynolds at the same
condition. At Re = 1557.77 the influence of the shock is lower and the implementation matches perfectly with the reference solution of Esfahanian
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5.3. Linearized shock boundary conditons
−3
2.5
x 10
2
1.5
ωi
1
0.5
0
Shock b.c.
Asymptotic b.c.
−0.5
Esfahanian shock b.c.
−1
0
0.05
0.1
α
0.15
0.2
Figure 5.5.: Growth rate for a Mach 8 flow on a wedge at different
Reynolds numbers Re = 1557.77
(Fig. 5.5). The lte shock boundary condition has been assessed against
this case, expecting a similar result with the one given in Fig. 5.4, though
the boundary layer profiles in lte and cpg slightly differ from each other
as clearly seen in Fig. 5.6
Similar behavior of the growth rate is observed in Fig. 5.7. The small difference in the mean velocity and temperature profiles leads to a slightly
higher second mode peak and a shift in the wavenumber. This phenomenon has been already shown in [80] and in chapter 4 . Interesting enough, the wavenumber shift is less evident when the shock boundary condition plays a role, though a difference in peak amplitude is still
clearly visible in Fig. 5.7a. It is also evident from Fig. 5.5 and Fig. 5.7b
that considering a downstream location on the wedge, i.e. increasing the
Reynolds number, the shock gets farther away from the wall, affecting
less the boundary layer. Also while increasing the shock distance from
the wall, the wave numbers affected by the shock progressively decrease.
It is reasonable to think that the flow computed by means of a full Navier-
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Chapter 5. Shock as a boundary condition
9
1
CPG
LTE
8
0.9
0.8
7
0.7
6
U/Uref
T/Tref
0.6
5
4
0.5
0.4
3
0.3
2
0.2
1
CPG
LTE
0.1
0
0
5
10
15
20
y/lref
25
30
35
0
0
40
(a) Non-dimensional temperature profile
5
10
15
20
y/lref
25
30
35
40
(b) Non-dimensional velocity profile
Figure 5.6.: Non-dimensional velocity and temperature profiles for a Mach
8 flow at Re 1557.77
−3
2
−3
x 10
3
2.5
1
2
0.5
1.5
ωi
ωi
1.5
0
−0.5
1
0.5
−1
0
Shock b.c. LTE
Asymptotic b.c. LTE
Shock b.c. CPG
Asymptotic b.c. CPG
−1.5
−2
0
x 10
0.05
0.1
α
Shock b.c. LTE
Asymptotic b.c. LTE
Shock b.c. CPG
Asymptotic b.c. CPG
−0.5
0.15
0.2
(a) Re=590.91
−1
0
0.05
0.1
α
0.15
0.2
(b) Re=1557.77
Figure 5.7.: Growth rate for a Mach 8 flow in cpg or lte
M
8
8
14
14
Re
590
1557.77
1557.77
3760
ys /lsc
60.50
159.49
84.42
203.78
δ ∗ /lsc
19.44
19.44
29.16
29.16
ys /δ ∗
3.1
8.2
2.9
7.0
Table 5.1.: Wall distance to boundary layer thickness for calorically perfect gas
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5.3. Linearized shock boundary conditons
9
CFD
Self similar
1
8
7
0.8
U/Ue
5
T/T
e
6
4
0.6
0.4
3
2
0.2
1
CFD
Self similar
0
0
5
10
15
20
25
30
y/lref
(a) temperature profile
35
40
0
0
5
10
15
20
25
30
35
40
y/lref
(b) velocity profile
Figure 5.8.: Temperature and velocity profile comparison for a calorically
perfect gas flow at Re = 1557.77 M = 8
Stokes code is different from a self-similar one. We report in Fig. 5.8 the
comparison at Re = 1557.77 for a Mach 8 calorically perfect fluid flow.
Differences are small and more evident on the temperature profile. Nevertheless, even if slightly visible, we should remind the non dimensional
velocity plot for the Navier-Stokes computation does not reach directly
the value of one, just outside of the boundary layer but it slowly grows,
reaching it at the end of the domain. This happens because the flow has
been non-dimensionalized with respect to the speed behind the shock, and
because the velocity profile is not constant outside the boundary layer.
Not many publications, at least within the knowledge of the author, have
quantified the error introduced by considering the mean flow as self similar. In Fig. 5.10 such a difference is shown and the cfd profile results
more stable than the self-similar one. It should be noted that, despite the
fact that in a Navier-Stokes computation the shock moves further away
from the wall with respect to a simple Euler calculation, the wave number
at which the shock boundary condition departs from the asymptotic one
is not sensibly different. In this regard the shock wave influence on the
boundary layer is relevant for wave numbers with a wavelength which is
comparable or greater than the distance of the shock from the wall. This
wave number is approximated by the relation α0 = 2πcos(σ)/ys where σ
is the wave angle.
The use of similar profile, instead of a full Navier-Stokes solution, overestimates the perturbation growth which is very different from one case to
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Chapter 5. Shock as a boundary condition
the other. We should also note that this overestimation is not something
general but it depends on the computed cases, as we will see later in this
section.
A second case at higher Mach number has been investigated on the same
wedge with the assumption that the inlet flow at Mach 14 corresponds
to an after shock Mach of 10.755. Temperature and pressure behind the
shock are 350K and 17502.45P a respectively and the wall is adiabatic as
in the previous case.
One immediately understands that not only Reynolds number and shock
height are strictly related but also the Mach number is, for a given geometry. With an increasing Mach number, the shock becomes shallower so
that this new boundary condition is potentially more useful. With these
conditions, the difference between the cpg mean flow and the lte one is
remarkable, as seen in Fig. 5.11. The lte flow has a thinner boundary
layer and its maximum temperature at the wall is half the one of the cpg.
In order to understand the effects of the Mach number we keep the shock
approximately at the same distance from the wall (see Tab. 5.1). For this
reason, at Mach 14 the case at Re = 3760 is introduced and the profile
at Re = 590.91 is discarded. As one would expect, the heavy difference
in mean profile affects the growth rate and this is evident in Fig. 5.12b
and Fig. 5.12a. The lte flow shows two peaks due to the second and
third mode and a shift in the wave number. The shock, in these cases,
does not behave qualitatively different from the previous one, but only
the quantitative value of the growth rate is different.
The cfd computation and the self-similar profile (Fig. 5.13) are very close
to each other, in terms of the first peak, while at higher wavenumbers the
self similar boundary layer becomes progressively more stable. The effect
of the shock boundary condition is limited as the shock is pretty far from
the wall (shock position to displacement thickness ratio is (ys /δ ∗ ≈ 7).
Differences in the mean flow profile between the cfd and self similar computation are shown in Fig. 5.9: as usual velocity profiles are closer to each
other while temperature shows some discrepancies.
An interesting point is that frequencies of growth rate peak are consistently captured by both method. This means that in case this calculations
are meant to design an experiment where the most amplified frequency
should be captured even a simple self similar computation is useful in
providing that information. On the other hand the growth rate is not
comparable and the difference will reflect in an eventual estimation of the
transition onset.
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5.3. Linearized shock boundary conditons
10
CFD
Self similar
1
9
8
0.8
6
U/Ue
T/Te
7
5
0.6
0.4
4
3
0.2
2
CFD
Self similar
1
0
5
10
15
20
25
30
35
0
40
0
5
10
15
y/lref
20
25
30
35
40
y/lref
(a) temperature profile
(b) velocity profile
Figure 5.9.: Temperature and velocity profile comparison for a lte flow
at Re = 3760 M = 14
−3
3
x 10
Comparison Re = 1557.77, M= 8
2
1
ωi
0
−1
−2
BL Shock BC
BL Asymptotic BC
CFD Shock BC
CFD Asymptotic BC
−3
−4
−5
0
0.05
0.1
α
0.15
0.2
Figure 5.10.: Comparison of self similar boundary layer flow against full
Navier-Stokes solution
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Chapter 5. Shock as a boundary condition
20
1
CPG
LTE
0.9
16
0.8
14
0.7
12
0.6
U/Uref
T/T
ref
18
10
0.5
8
0.4
6
0.3
4
0.2
2
CPG
LTE
0.1
0
0
5
10
15
20
y/lref
25
30
35
0
0
40
(a) non dimensional temperature profile
5
10
15
20
y/lref
25
30
35
40
(b) non dimensional velocity profile
Figure 5.11.: Non-dimensional velocity and temperature profiles for a
Mach 14 flow Re 1557.77
−3
1.5
−3
x 10
1.5
x 10
1
1
0.5
0
0.5
ωi
ωi
−0.5
−1
0
−1.5
−2
−3
0
−0.5
LTE Shock b.c.
LTE Asymptotic b.c.
CPG Shock b.c.
CPG Asymptotic b.c.
−2.5
0.05
0.1
α
(a) Re 1557
0.15
0.2
−1
0
LTE Shock b.c.
LTE Asymptotic b.c.
CPG Shock b.c.
CPG Asymptotic b.c.
0.05
0.1
α
0.15
0.2
(b) Re 3760
Figure 5.12.: Growth rate for a Mach 14 flow at Re 1557.77 and Re 3760
5.4. Summary on the shock boundary condition
Real flows at hypersonic speed present a strong shock in front of the body
and this has an effect on the stability of boundary layers. We modeled
the presence of the shock as a sort of vibrating wall which is forcing the
perturbation on the opposite side to the wall. This is brought into the
original set of stability equations as a boundary conditions at distance L
from the wall. The shock becomes more relevant with increasing Mach
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5.4. Summary on the shock boundary condition
−3
1.5
x 10
1
0.5
ωi
0
−0.5
−1
−1.5
−2
0
BL Shock b.c.
BL Asymptotic b.c.
CFD Shock b.c.
CFD Asymptotic b.c.
0.05
0.1
α
0.15
0.2
Figure 5.13.: Comparison of self similar boundary layer flow against full
Navier-Stokes solution for a Mach 14 flow at Re 3760
number, because of its higher slope, and with decreasing Reynolds which
means closer to the leading edge where usually the shock is close to the
body.
In this chapter we showed the development of such a boundary condition after previous work found in literature. This has been applied to
a compressible flow at hypersonic speed with cpg and lte assumptions.
Comparison of the standard compressible flow solver against literature is
good and the main features of the linearized shock boundary condition
has been completely retrieved qualitatively and quantitatively. The shock
has a stabilizing effect at small wave number and this is related to the
distance of the shock to the wall. Wave length much smaller than this
distance are not really affected.
The lte implementation has been tested against the same calorically perfect gas test found in literature. Being our implementation and application
new there is no direct source in the published literature where this data
could be compared. While the use of this case is not entirely accurate,
the low temperature involved are likely to have a limited impact on the
stability of the flows. Results confirm this first impression and our implementation could be considered verified. The application to a higher
temperature case shows indeed more differences though the effect of the
shock is always stabilizing. Eventually, when chemistry plays a role, the
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Chapter 5. Shock as a boundary condition
shock stabilizes the flow even more than the calorically perfect gas case.
Effects are seen both on stability and on mean flow. For this reason full
Navier-Stokes computations were performed to understand the changes in
the mean flow. We found out that mean flow modification are not negligible, and depending on the case they could result in a more stable or
unstable flow. Nevertheless the maximum amplification rate peak happens at the same frequency of the simple self similar solution flow. From
our results flows from cfd computations seem to be more affected by the
shock boundary condition (always in a similar wave length range)
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Chapter 6.
Conclusions
This chapter will draw some conclusions about the work done and the
main results achieved, proposing future work to integrate the current implementation and results.
A toolkit for the stability analysis of flows at different regimes has been
implemented. A set of core functionalities has been used as a basic layer,
in order to make further development easier and consistent. The underlying discretization method was chosen with future development in mind
and the Chebyshev collocation method proved to be very flexible accurate
dealing with all the kind of flows. Implementation was focused on linear
stability theory, but also few preliminary tests were performed on parabolized stability equations with good results, confirming the high modularity
of the toolkit. The actual solvers were build upon a certain number of
functionalities, directly usable by the researcher once a set of condition is
passed to them. No object oriented coding technique was used to encourage portability toward other platforms.
The outer set of functionalities is obtained by simple script that uses the
lower function which determine the basic stability property of the flow.
A set of scripts is provided directly in the toolkit to compute neutral stability curves or growth rates plot, as these are by far the most common
needs of a user. Nevertheless they can be combined together with ease,
targeting different objectives. In order to complete the toolkit functions to
compute the incompressible and compressible self similar boundary layer
profile are provided along with the standard set of functions.
The implementation of the incompressible solver has been used as a test
to select the computational method. The choice narrowed to pseudospectral method and the final implementation relies only on the Chebyshev τ -method and the collocation method.
An extensive analysis of these two methods has been performed for a
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Chapter 6. Conclusions
Poiseuille flow comparing their advantages and disadvantages. Application to boundary layer flows required an original extension of the τ -method
to be performed: computations were compared against classic results but
the application of this method to flat plate boundary layer seems to be
totally absent in literature. Classic literature used this method only to
compute Poiseuille flow, because of the perfect matching of the computational domain with the physical domain.
Techniques to relief the effect of ill-conditioned matrices were adopted but
anyway the high order of the Orr-somerfeld equation proved to be quite
tough, specially for the resolution of the continuum spectrum. Some tests
have been run to retrieve few literature findings and both methods proved
to give satisfactory results, with very similar features. Only collocation
method was chosen for the implementation of the other solvers in vesta.
The compressible solver copes with high subsonic flows and supersonic
flows. It could be used also for hypersonic speed but it only features the
calorically perfect gas assumptions therefore its validity is limited to fairly
low hypersonic speed. At this stage a comprehensive set of computations
has been performed, highlighting important feature of compressible flow
stability. The effect of free stream temperature has been assessed comparing our result with the available literature.
Implementation of the Chebyshev collocation method has been tested
against simulation of incompressible flows and well know cases of supersonic and hypersonic flows.
Stability equations have been rewritten to compute hypersonic cases which
include chemical reactions. Comparison with literature is more daunting
than for the other case as fluid properties have a bigger uncertainty. Small
changes in the mean flow modify the final stability properties, therefore
growth rates could slightly differ.
Performed test at supersonic speeds compare satisfactorily against calorically perfect gas computations. This is a simple way to verify that the
code is able to retrieve the classic behavior where no chemical reactions
are involved. Flow under the lte assumptions are functions of pressure
and temperature and for this reason their effect has been studied and very
little literature is available about it. It turned out that pressure plays a
minor role with respect to temperature. This is not linked to the constant
pressure approximation of a boundary layer but with its role in determining the composition of the flow.
Temperature, on the other hand, has important effects on stability; the instability region in a classic neutral curve diagram increases, telling us that
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6.1. Future works
the flow is generally more unstable. Also effects on the critical Reynolds
number (below which all perturbation are damped) are reported in our
study: it decreases along with temperature up to small values where linear approximations can be hardly considered as valid. Moreover as the
Reynolds number decreases the temperature plays even a bigger role by
a greater excitation of the so called third mode, due to chemistry.
Vehicles flying at supersonic and hypersonic speeds see a shock and we
studied its effect on boundary layer stability while considering chemical
reactions. Implementation of the calorically perfect gas gave results comparable with literature while the software running under the lte assumptions is an original implementation and no direct comparison is available
in literature. Comparisons against the non reacting case present in literature verified that our implementation can deliver comparable results.
The shock has a stabilizing effect for a low range of wave numbers while is
not effective at high wave numbers. The separation between high and low
wave number is linked to the shock distance from the wall. When considering lte flows the shock makes the flow even stabler with a growth rate
twice as damped.
Effects of shock is also linked to the change in the mean flow profile. We
observed that a flow from a Navier-Stokes solution, even though close to
the self similar solution, has considerably different growth rates. Depending on the case, it is possible to find a more unstable or stable flows but
differences could be quite high. Instead, growth rate peaks, appear usually at very similar frequencies for both self similar and cfd solutions.
Also cfd solutions see a bigger effect (i.e. more stabilization occurs) than
the one obtained for a self similar profile.
6.1. Future works
Following the achievement of modularity, one of the first idea is the expansion of the number of tools available. A set of tests has been performed
on linear parabolized stability equations (pse) for incompressible flow and
some preliminary work was started on compressible linear pse. Another
useful tool worth adding would be the so called bi-global stability analysis. Our development and implementation tool would allow to implement
them with considerably less effort. Moreover there are few softwares featuring this variety of methods for all the flow regimes analyzed in the
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Chapter 6. Conclusions
present work, and this addition would be very precious.
An implementation of a general two/three dimensional eN method should
be included to link the stability study with transition prediction.
Our basic tool for self similar profile should be extended by a boundary
layer solver for non similar profiles and possibly a basic two-dimensional
Navier-Stokes code.
The presence of a dedicated three-dimensional Navier-Stokes code is out
of the scope of the toolkit but it is fundamental for further research in
stability and transition, especially in the non-linear phase.
Influence of shock shall be better characterized by means of unsteady cfd
solver and compared with our current result for flows with chemical reactions. Along with this activity, uncertainty in the mean flow data shall be
addressed in order to improve the computation of stability and transition
while comparing with wind tunnel test and real flight data.
From a numerical stand point a multi-domain scheme could be implemented to lower the number of point needed in hypersonic computations,
and some preliminary work is starting on novel technique for grid adaptation for stability calculations.
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Appendix A.
Computer Algebra Software and
Stability Equations
As it appears evident in the chapters 2, 3 and 4 including more and more
physics increases the number of terms involved in the equations and the
overall level of complexity.
This raises important issues in the development of equations for higher
temperature. Furthermore one would like to have a safer method of retrieving the stability equations after the common set of simplification and
hypotheses have been made.
For this reason a series of scripts has been implemented to manipulate
symbolically the Navier-Stokes equation exploiting the power of Computer Algebra Software.
In this appendix we will highlight the main features and advantages of
this approach. For our purpose maxima has been used. It has been chosen because of its license (gnu-gpl), reliability, long history, and active
community. On top of that it can cope quite well with tensors out-of-thebox.
As a first step Navier-Stokes equation are described in maxima language,
anyway to avoid a case-by-case treatment the equation have been written in invariant form using tensor calculus. This is the most general
description of a fluid flow (more detail could be found in [3]) and it is not
depending on the kind of coordinate system. Furthermore it is easy, once
the equation have been written using tensor notation, to convert them
from a, for instance, cartesian coordinate system, to a cylindrical coordinate system.
This description makes the covariant derivative to appear in the NavierStokes equations and it is worth to highlight that we have been using only space coordinate system in the computation of the covariant
derivative while time has been not included. This is consistent with the
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Appendix A. Computer Algebra Software and Stability Equations
non-relativistic representation of the physical phenomena studied in the
present work. Nevertheless time dependency of the flow variables is still
explicitly considered. For sake of simplicity we address the reader to Aris
[3] or Brillouin [16] to understand the details behind the use of covariant, contravariant, metric tensors and all the related tensor calculus tools;
here we just want to remind that covariant derivatives bring some more
terms, due to the variation of the base vectors, that need to be correctly
accounted for. Once the equations have been written in tensorial form is
possible to obtain our coordinate dependent system of equations by letting the development of the tenth of terms to be done by the cas and
substituting the general velocity vector by its component.
The second phase is related to the application of the hypotheses mentioned in linear stability chapters. As first step, one creates of a set of
variables following the template proposed in eq.(2.1), then those are substituted in the previously defined Navier-Stokes equations to create mean
flow and perturbation-plus-mean equations. From the latter all the terms
solution of the mean Navier-Stokes are discarded.
The following step concerns the simplification of all the higher order terms
by a specific routine that contains all the possible higher order terms appearing in the equation. This step allows to keep or discard terms depending on the kind of hypotheses that have been considered.
The final phase is related to the application of the wave-like decomposition and simplification of the exponential terms, bringing the equations
to the form that has been shown in this work.
The scripts have been tested for the compressible (chapter 3) and incompressible (chapter 2) linear stability equations and it has been applied to
the LTE linear stability equations (chapter 4) as well as incompressible
and compressible Parabolized Stability Equations.
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Appendix B.
Computation of the Mean Flow in case
of self similar profiles
As many stability and transition works, we study very often the property
of analytical or semi-analytical profiles. This allows us to draw some
general conclusions on the behavior of the flows and they could computed
more accurately with less resources than the simplest cfd computation.
So, together with the set of stability solvers shown in this work, some
routines have been implemented to compute self similar boundary layer
profiles.
B.1. Blasius profile
Given the Blasius equation
f 000 + f f 00 = 0
(B.1)
with boundary condtions
f 0 (∞) = 1
f 0 (0) = 0
f (0) = 0
(B.2)
the most straight forward way to find a solution is the application of the
shooting method. We use it in combination with a standard Runge-Kutta
method at the fourth order. The idea behind the shooting method is
as simple as using a solver for an initial value problem with a fictitious
boundary condition. We first reduce Eq. (B.1) to a system of first order
equations
 0

f = g
g0 = h

 0
h = −hf
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Appendix B. Computation of the Mean Flow in case of self similar profiles
with boundary conditions f (0) = 0, g(0) = 0. The third condition is
f (∞) = 0 is replaced by h(0) = s where s is a new unknown that needs to
be initialized. Now it is possible to apply to this system an ordinary ivp
method. At this point one uses a Newton’s iteration algorithm to converge
on the good initial condition s which satisfy also the substituted boundary
condition. Then we will iterate on a function of s, let’say P (s) = 0,
imposing, by mean of that function, the fulfillment of the third boundary
condition,
P (s) = f (∞, s) − 1 = 0
(B.3)
Please note that now f as well as g and h are also function of s.
By using the Newton’s method on P (s), the update for the new variable
s is
f (∞, s) − 1
P (s)
(B.4)
sit+1 = sit − dP = sit −
df ds
ds x=∞
To compute df
we need another set of equations. By calling
ds x=∞
F =
∂f
;
∂s
G=
∂g
;
∂s
H=
∂h
;
∂s
(B.5)
It is then possible to write another system of equations with new boundary
conditions

∂F

=G



∂x


∂G
(B.6)
=H

∂x




 ∂H = −(f H + hF )
∂x
with
boundary conditions F = 0, G = 0, H = 1. After this integration
df is available and could be used to estimate a new value for the
ds x=∞
variable s.
B.2. Compressible boundary layer
In the compressible case self similar profile could also be found, nevertheless they are described by two equations where the additional one, with
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B.2. Compressible boundary layer
respect to the Blasius case, comes from the energy equation. For a self
similar profile on a flat plate one can write
(cf 00 ) + f f 00 = 0
(B.7)
0
0
(a1 g + a2 f 0 f 00 ) + f g 0 = 0
(B.8)
where
u
ρµ
c=
ue
ρe µe
g = H/He a1 = c/P r
1
(γ − 1)M 2
1−
a2 =
c
Pr
1 + γ−1
M2
2
f0 =
For our purposes we will consider
µcp
= 0.7
kp
M = ue / γRTe
(B.9)
Pr =
(B.10)
T 3/2
µ = CSuth
T +S
where
√
CSuth = 1.458 · 10−6 kg(m s K)−1
(B.11)
S = 110.4K
Thermal conductivity could be computed with an expression similar to
the one of the viscosity. Nevertheless, keeping the value of P r constant,
k could be compute in a straightforward way.
Solution of each equation is let to the shooting method described before
for Blasius’ equations. According to the suggestion of [19] an iterative
procedure has been set, first guessing a temperature distribution and using Blasius profile as first guess of eq.B.7. An estimation of the transport
properties follows and then a computation of the momentum equation B.7
is performed. With the new velocity profile the energy equation could be
updated, then the transport properties and so on up to satisfying a previously defined tolerance.
Boundary conditions for eq.B.7 are f (0) = 0, g(0) = 0 and f (∞) = 0
while the energy equation could be solved with an adiabatic or isothermal
wall boundary condition.
Please note that the lower the Mach number the easier it is to converge
toward the solution, so, very often, for high Mach number flows the solution is at first computed for M = 1.5 and then the Mach number is
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Appendix B. Computation of the Mean Flow in case of self similar profiles
increased till matching the desired condition. This is due to the guessed
initial temperature profile proposed by [19] which is not very well suited
for a hypersonics.
B.2.1. Dimensional flow
Results coming from B.2 give a solution in their special transformed coordinates ξ and η which should be scaled in such a way to obtain a dimensional velocity and temperature profile. This is not a difficult task
but a fundamental one; stability computations depend also on the correct input of the profile given and of course the non-dimensionalization in
the stability equation is different from the one provide for B.7 and B.8.
Then to find the y-coordinate one should apply backward the Lewis-Lee
transformation
Z η√
2ξ
dη̄
(B.12)
y = y0 +
η0 ρue
The x coordinate could be retrieved in a similar fashion. Of course also
the derivatives with respect to y have an expression in the new coordinate
system which reads
ρue
∂
∂
=√
(B.13)
∂y x
2ξ ∂η ξ
By manipulating the self-similar profiles using these expressions is possible
to retrieve both the dimensional velocity and the enthalpy profile.
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Appendix C.
Local Eigenvalue algorithm
Dealing with eigenvalue problems an adequate solver is very important,
fortunately a number of robust algorithms is readily available and already
implemented in many libraries like, for instance, the well known LAPACK
[1]. The standard algorithm we used for our global computation is the QZ
algorithm [73] as implemented in LAPACK. No major problem arises in
case one or more eigenvalues of the system are infinite as stated in [101].
This gives us a good confidence in its use.
The global algorithm can provide an overview of the whole spectrum anyway it is quite heavy with a complexity of O(N 3 ) where N is the number
of the discretization points. This provides us with very useful information especially when we do not have any clue of the expected range of
frequency to be investigated. In different situations could be more favorable to look for eigenvalue individually making the search more efficient.
There are several algorithm to perform this task as well and all of them
require a starting guess. We will describe two algorithms, both implemented specifically for our purposes. The first is the generalized version
of the Rayleigh’s algorithm and the second one an application of the Newton’s root finding algorithm.
The Rayleigh’s algorithm is an iterative method extending the inverse
power method where at each iteration the estimated eigenvalue is computed by a Rayleigh quotient. For a generalized eigenvalue problem
(A − λB)X = 0 the Rayleigh’s quotient (see [24] and [50] ) is
R=
(X T AX)
.
(X T BX)
(C.1)
This could be seen as a minimization problem, if the search for the value
λ should minimize
kAX − λBXk
(C.2)
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Appendix C. Local Eigenvalue algorithm
This is actually an overcostrained problem which could be solved resorting
to the associated normal equation which leads us directly to Eq. (C.1).
By computing the vector Φ ∈ C
Φk+1 =
and Ψ ∈ C
Ψk+1 =
(BΦk )
,
(A − λB)
(C.3)
(B H Ψk )
,
(A − λB)H
(C.4)
we can compute the update of the eigenvalue at each iteration as
H
λk+1 = (ΨH
k AΦk )/(Ψk BΦk );
(C.5)
At each iteration the two vectors should be normalized. The sequence
{(Φ, ΨH )}∞
k=0 has also be shown to converge to the corresponding right
and left eigenvectors {(Φ, ΨH )} so that the algorithm actually provides
us with complete information. The algorithm converges cubically, anyway
the system is more and more ill-conditioned as we are approaching the
correct eigenvalue. Errors are usually assumed to be on amplitude more
than on the eigenvalue, nevertheless during standard operation the use
of such an algorithm can generate a number of ill-conditioned matrix
warning.
For this reason we implemented a simple Newton’s iterative algorithm
to compute the eigenvalue. The method is converging quadratically so
is slower but it does not become ill-conditioned while approaching the
correct eigenvalue. With both these methods we can choose the algorithm which fits best our needs and eventually implement some conditions
through which we can choose one over the other exchanging speed for robustness or vice versa.
Considering the eigenvalue problem as a a function f1 = (A − ωB) · X)
and imposing also a normalization condition f2 = XX H = 1 it is possible
to recast the problem into a root search of the function
(A − λB)X
(C.6)
F (x, λ) =
XX H − 1
This function has a root only for normalized eigenvectors and their corresponding eigenvalues therefore, many potential solutions are available.
The jacobian matrix of this function can be computed analytically and
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therefore once and for all, avoiding expensive numerical derivative operations
A − λB −Bx
JF (x, λ) =
(C.7)
2xT
0
After this definition the usual Newton’s method applies
Ξk+1 = Ξk −
F (xk , λk )
;
JF (xk , λk )
(C.8)
where Ξ = [x, λ] So while F is still badly conditioned close to the real
eigenvalue the Jacobian is causing no problem when dividing by it.
We ran few test to verify the correct implementations of Rayleigh’s algorithm and Newton’s method routine against the well tested LAPACK QZ
algorithm.
We give an easy example with 3x3 matrix

1 −1
A(x, λ) =  0 1 + i
0
0

1
1−i 
i
(C.9)
Of course more complex case could be tested and a more refined list of
the performances could be drawn, anyway this is a simple test in which
eigenvectors could be found exactly by hand and compared to the numerical results. The test has been chosen randomly and it is not meant
to represent in any sense a “best” or “worst” case for the performance
of any of the presented algorithm. Eigenvalues are obviously λ1 = 1,
λ2 = 1 + i, λ3 = i. Focusing on the λ3 the corresponding eigenvector is
X3 = [− 23 , − 2i , −1 + i1]
The three eigenvalue as computed by the QZ algorithm correspond exactly
to the numerical one. The computed eigenvectors are


1.0 0.70710678 + 0.00000000i 0.67419986 + 0.00000000i
R =  0.0 0.00000000 − 0.70710678i 0.26967994 − 0.539359890i 
0.0 0.00000000 + 0.00000000i −0.4045199 + 0.134839972i
(C.10)
The third column is the one corresponding to the already computed eigenvalue, and it is possible to verify after few manipulations that the value
correspond exactly to X3 computed analytically.
If the Newton algorithm is fed with the correct eigenvalue and eigenvector, it reports the same value as result. Using a standard eigenvector
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Appendix C. Local Eigenvalue algorithm
X0 = [1; 1; 1] and a λ30 = 1.1i it converges to the correct result for both
eigenvalue and eigenvector. Keeping the same eigenvalue as a start the
solution converge to λ3 even if the starting guess is λ30 = 1.4i.
For the Rayleigh’s algorithm trying with the same simple test we find
out that at λ30 = 1.4i the matrix becomes singular at working precision
while iterating and it does not converge to any solution. The system is
ill-conditioned already at λ30 = 1.2i but still providing a correct estimate
of the eigenvalue/eigenvector pair.
This basic test does not allow to see eventual benefit and flaws of these
two algorithms and this topic it is far beyond the scope of this work
nevertheless it highlights in a simple way that another algorithm different
from Rayleigh’s one could be an advantage for our computations.
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List of Figures
1.1. Paths to transition for wall bounded flows ([87]) . . . . .
3
2.1. First eight Chebyshev polynomial . . . . . . . . . . . . . .
2.2. Subsonic Poiseuille flow spectrum for Re = 10000, α =
1 for different number of Chebyshev polynomials in the
expansion . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Comparison of spectrum computation for Poiseuille flow at
Re = 10000 and α = 1: M = 121 black dot, M = 141 red
cross . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Subsonic Poiseuille flow most unstable and less unstable
mode eigenfunctions for Re = 10000, α = 1 for M = 141 .
2.5. convergence of the whole spectrum for a Blasius boundary layer at Re = 290.5625, α = 0.1162 computed by a
mapped Chebyshev τ -method: comparison with Henningson [91] (red circle) . . . . . . . . . . . . . . . . . . . . . .
2.6. Stream wise velocity perturbation normalized: VESTA results (solid line) and Henningson results [91] (red circles) .
2.7. Wall normal velocity perturbation normalized: VESTA results (solid line) and Henningson results [91] (red circles) .
2.8. Spatial spectrum for an incompressible boundary layer flow
at Reδ = 336 and ω = 0.1297 for N=50 . . . . . . . . . . .
2.9. Stream wise velocity perturbation normalized Reδ = 336
and ω = 0.1297 : VESTA results (solid line) . . . . . . . .
2.10. Wall normal velocity perturbation normalized Reδ = 336
and ω = 0.1297: VESTA results (solid line) and Henningson results [91] (red circles) . . . . . . . . . . . . . . . . .
2.11. Convergence of the whole spectrum for a Blasius boundary layer at Reδ = 1000, α = 0.26 computed by a mapped
Chebyshev τ -method: comparison with Henningson [91](red
circle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
32
33
34
37
38
38
40
41
41
42
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List of Figures
2.12. Comparison of the whole spectrum for a Blasius boundary
layer at Reδ = 1000, ω = 0.26 as computed by a the present
mapped Chebyshev τ -method with N = 240 points (black
dot) with [91] (red circle) . . . . . . . . . . . . . . . . . .
2.13. Stream wise velocity perturbation normalized: VESTA results for collocation method (solid line) and Henningson
results (red circles) N =200 . . . . . . . . . . . . . . . . .
2.14. Wall normal velocity perturbation normalized: VESTA results for collocation method (solid line) and Henningson
results (red circles) N = 200 . . . . . . . . . . . . . . . . .
2.15. Convergence of the whole spectrum for a Blasius boundary layer at Re = 290.5625, α = 0.1162 computed by a
mapped Chebyshev collocation method: comparison with
Henningson (red circle) . . . . . . . . . . . . . . . . . . .
2.16. Convergence of the whole spectrum for a Blasius boundary layer at Re = 1000, ω = 0.26 computed by a mapped
Chebyshev collocation method: comparison with Henningson (red circle) . . . . . . . . . . . . . . . . . . . . . . . .
2.17. Neutral Stability curve for a flat plate boundary layer (Blasius’ profile) computed by means of a local spatial solver
(solid line) and by a finite difference method [75] . . . . .
3.1. convergence of the whole spectrum for a Blasius boundary layer at Re = 290.5625, α = 0.1162 computed by a
mapped Chebyshev collocation method: comparison with
Henningson [91] (red circle) . . . . . . . . . . . . . . . . .
3.2. Stream wise velocity perturbation normalized for Re =
290.5625, α = 0.1162: VESTA results (solid line) for 200
points and Henningson [91] results (red circles) . . . . . .
3.3. Wall normal velocity perturbation normalized for Re =
290.5625, α = 0.1162: VESTA results (solid line) for 200
points and Henningson [91] results (red circles) . . . . . .
3.4. Neutral Stability curve at Mach 2 . . . . . . . . . . . . . .
3.5. Neutral Stability curve at Mach 3 . . . . . . . . . . . . . .
3.6. Neutral Stability curve at Mach 4 . . . . . . . . . . . . . .
3.7. Neutral Stability curve at Mach 6 . . . . . . . . . . . . . .
3.8. Neutral Stability curve at Mach 7 . . . . . . . . . . . . . .
3.9. Neutral Stability curve at Mach 8 . . . . . . . . . . . . . .
3.10. Neutral Stability curve at Mach 10 . . . . . . . . . . . . .
43
47
47
48
50
52
63
64
65
69
69
70
70
71
71
72
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List of Figures
3.11. Non dimensional velocity and temperature mean profile for
different Mach number at Re = 1000 for adiabatic wall and
free stream temperature T = 288K . . . . . . . . . . . . .
3.12. Neutral Stability curve at Mach 2.2 and Pr 0.725 . . . . .
3.13. Neutral Stability curve at Mach 3 and Pr 0.725 . . . . . .
3.14. Neutral Stability curve at Mach 4.5 and Pr 0.725 . . . . .
3.15. Neutral Stability curve at Mach 5.8 and Pr 0.725 . . . . .
3.16. Neutral Stability curve at Mach 7 and Pr 0.725 . . . . . .
3.17. Neutral Stability curve at Mach 10 and Pr 0.725 . . . . .
73
74
74
75
75
76
76
4.1. Mean flow for a flat plate at Mach 2.5 with adiabatic wall
in lte and calorically perfect gas (Malik’s [62] case 3) . .
89
4.2. Most unstable mode eigenfunction for a flat plate at Mach
2.5 with adiabatic wall (lte and cpg) . . . . . . . . . . .
89
4.3. Transport properties at Mach 2.5 with adiabatic wall (lte
and cpg) . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.4. Mean flow for a flat plate at Mach 10 with cold wall in
chemical equilibrium, thermically perfect gas and calorically perfect gas . . . . . . . . . . . . . . . . . . . . . . .
91
4.5. Growth rate comparison for a Mach 10 flat plate at Re 2000 91
4.6. Non dimensional profile for a Mach 10 flow Re = 2000, [64] 93
4.7. Most unstable eigenmode perturbation for a Mach 10 flow
with adiabatic wall: line vesta, DNS Marxen et al. [65] .
93
4.8. Growth rate for a Mach 10 flow over a flat plate with no
pressure gradient. Comparison with the computation performed by Malik [64] and Perraud et al. [78] . . . . . . . .
94
4.9. Self similar velocity profile for Mach 10 flow at Re 5000 .
96
4.10. Self similar temperature profile for Mach 10 flow at Re 5000 97
4.11. Neutral stability curve for varying pressure grouped by
temperature at Mach 10. . . . . . . . . . . . . . . . . . . .
98
4.12. Neutral stability curve at 4000 Pa and varying temperature
at Mach 10 . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.13. Temporal growth rate for Mach 10 flow at 111K and 350K 99
4.14. Phase for eigenfunction at the maximum growth rate outside the second mode peak and for a wavenumebr corresponding for the second peak for the Re 2000 case . . . . 100
4.15. Eigenfunction amplitudes for the second mode peak for the
different Reynolds . . . . . . . . . . . . . . . . . . . . . . 102
4.16. Eigenfunction amplitudes for the third mode peak for the
different Reynolds . . . . . . . . . . . . . . . . . . . . . . 103
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List of Figures
4.17. Neutral stability curve at 4000 Pa and varying temperature
at Mach 10: zoom in at low Reynolds . . . . . . . . . . .
5.1.
5.2.
5.3.
5.4.
Shock oscillation around the mean value . . . . . . . . . .
Sketch of shock influence on boundary layer mode . . . .
Control volume definition for jump relation derivation . .
Growth rate for a Mach 8 flow on a wedge at different
Reynolds numbers Re = 590.91 . . . . . . . . . . . . . . .
5.5. Growth rate for a Mach 8 flow on a wedge at different
Reynolds numbers Re = 1557.77 . . . . . . . . . . . . . .
5.6. Non-dimensional velocity and temperature profiles for a
Mach 8 flow at Re 1557.77 . . . . . . . . . . . . . . . . . .
5.7. Growth rate for a Mach 8 flow in cpg or lte . . . . . . .
5.8. Temperature and velocity profile comparison for a calorically perfect gas flow at Re = 1557.77 M = 8 . . . . . . .
5.9. Temperature and velocity profile comparison for a lte flow
at Re = 3760 M = 14 . . . . . . . . . . . . . . . . . . . .
5.10. Comparison of self similar boundary layer flow against full
Navier-Stokes solution . . . . . . . . . . . . . . . . . . . .
5.11. Non-dimensional velocity and temperature profiles for a
Mach 14 flow Re 1557.77 . . . . . . . . . . . . . . . . . . .
5.12. Growth rate for a Mach 14 flow at Re 1557.77 and Re 3760
5.13. Comparison of self similar boundary layer flow against full
Navier-Stokes solution for a Mach 14 flow at Re 3760 . . .
104
109
110
110
116
117
118
118
119
121
121
122
122
123
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List of Tables
2.1. Most unstable eigenvaluefor a subsonic Poiseuille flow Re =
10000, α = 1 . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Most unstable eigenvalue for a subsonic boundary layer flow
at Re = 580, α = 0.179 . . . . . . . . . . . . . . . . . . . .
2.3. Most unstable eigenvalue and first four less unstable eigenvalues for a subsonic boundary layer flow at Re = 580,
α = 0.179 for M=201 . . . . . . . . . . . . . . . . . . . . .
2.4. Computed case for spatial stability computations on Blasius boundary layer . . . . . . . . . . . . . . . . . . . . .
2.5. Most unstable/ less unstable eigenvalues for a subsonic
boundary layer flow at different Reynolds and wave number
for N = 50 (cases of Tab. 2.4) . . . . . . . . . . . . . . . .
2.6. Less stable eigenvalue for a subsonic boundary layer flow
at Reδ = 336, ω = 0.1297 . . . . . . . . . . . . . . . . . .
2.7. Most unstable eigenvalue for a subsonic boundary layer flow
at Re = 580, α = 0.179 . . . . . . . . . . . . . . . . . . . .
2.8. Most unstable eigenvalue and first four less stable eigenvalues for a subsonic boundary layer flow at Re = 580,
α = 0.179 for N=200 . . . . . . . . . . . . . . . . . . . . .
2.9. Most unstable/ less unstable eigenvalues for a subsonic
boundary layer flow at different Reynolds and wave number
for N = 50 for cases in Tab. 2.4 . . . . . . . . . . . . . . .
3.1. Convergence of the most unstable eigenvalue for a subsonic
boundary layer flow at Re = 580, α = 0.179 . . . . . . . .
3.2. Most unstable eigenvalue and first four less unstable eigenvalues for a subsonic boundary layer flow at Re = 580,
α = 0.179 for N=200 . . . . . . . . . . . . . . . . . . . . .
3.3. Most unstable/ less unstable eigenvalues for a subsonic
boundary layer flow for the cases of Tab. 2.4, for N = 50
solved by means of the compressible stability solver . . . .
3.4. Computed case from Malik [62] . . . . . . . . . . . . . . .
30
35
35
39
39
39
45
46
49
62
64
64
65
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List of Tables
3.5. Most unstable eigenvalue for a subsonic boundary layer flow
at M ach = 0.5, α = 0.1 β = 0.0 . . . . . . . . . . . . . . .
3.6. Most unstable eigenvalue for a subsonic boundary layer flow
at M ach = 2.5, α = 0.06 β = 0.1 . . . . . . . . . . . . . .
3.7. Most unstable eigenvalue for a subsonic boundary layer flow
at M ach = 10, α = 0.12 β = 0 . . . . . . . . . . . . . . .
3.8. Computed case from 3.17 . . . . . . . . . . . . . . . . . .
66
4.1. Most unstable eigenvalue for a subsonic boundary layer flow
at M ach = 2.5, α = 0.06 β = 0.1 computed by lte and
calorically perfect gas solvers . . . . . . . . . . . . . . . .
4.2. Test matrix for Mach 10 . . . . . . . . . . . . . . . . . . .
4.3. Reynolds for the growth rate computation for 111k and 350
K cases at Mach 10 . . . . . . . . . . . . . . . . . . . . . .
89
95
96
5.1. Wall distance to boundary layer thickness for calorically
perfect gas . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
66
67
77
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Appendix C. Bibliography
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[98] F. Vigneau. Etude thorique et esprimentale de la transition en
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[99] W. Vincenti and C. Kruger. Introduction to physical gas dynamics.
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[100] D. K. Walters and J. H. Leylek. A new model for boundary layer
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[102] A. R. Wazzan, C. J. Gazley, and A. Smith. H − Rx method for
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