I wonder, if there are any simple principles that allow one to figure out if given steady-state solution of Euler or Navier-Stokes equations is stable? The general approach is to linearize equations around the given solution and solve eigenvalues problem -- looks much too complex. My intuition says, these equations are local, i.e. temporal evolution in each point depends only on the close vicinity of that point, so, solving eigenvalue problem in whole space is overkill. There must be some simple approaches that allow to check stability in a given point locally, so that if all points are stable -- the solution is stable in general.
Can someone advice good review of works on fluid dynamics stability?
For incompressible (or low Mach number) flows the solutions are not local. The pressure is elliptic and couples even the velocity solutions at long distances.
But I agree completely that setting up and solving an eigenvalue problem is tedious. And in a sense it is definitely overkill because all you really want to know is the worst eigenvalue. And an eigenvalue set up has provided you with ALL the eigenvalues, which is much more than you really needed/wanted.
The only simple shortcut I know is to do it numerically. Solve the pseudo-linearized equation numerically with a random initial condition. If the input grows, then the mean flow is unstable.
In a tiny more detail. (1) Start with the Navier-Stokes equations. Decompose the velocity into a known (whatever flow you want to impose) and a fluctuating part. Plug that decomposition into N.S. and get an equation for the unknown fluctuating part. It looks roughly like N.S. with an extra stretching term and some extra non-linear terms. You can keep the non-linear parts. So this is pseudo-linearization (pulling the mean out). It is just rearranging the problem - no approximation.
Then start with a random initial condition for the unknown fluctuation. Noise. If there are unstable modes, then they will grow in time and you will see them. After long times, the solution is the most unstable mode. And the amount it increases each timestep is the most unstable eigenvalue.
If it is stable, then all modes decay and the norm of your initial condition heads towards zero (rather than infinity for the unstable case).
for steady state problems the best criteria to judge the solution's stability is to check the upper and the lower limit of the output values .also, does the solution goes into any undefined points or regions.
Stability researchers are always looking for simple criteria to determine stability without solving the equations. An example is Rayleigh's inflection point theorem. There are others like it, but only for simple, idealized flows. If you find a new one, please tell me! A good, quick introduction to the topic is Chapter 12 of Kundu's book "Fluid Mechanics".
For incompressible (or low Mach number) flows the solutions are not local. The pressure is elliptic and couples even the velocity solutions at long distances.
But I agree completely that setting up and solving an eigenvalue problem is tedious. And in a sense it is definitely overkill because all you really want to know is the worst eigenvalue. And an eigenvalue set up has provided you with ALL the eigenvalues, which is much more than you really needed/wanted.
The only simple shortcut I know is to do it numerically. Solve the pseudo-linearized equation numerically with a random initial condition. If the input grows, then the mean flow is unstable.
In a tiny more detail. (1) Start with the Navier-Stokes equations. Decompose the velocity into a known (whatever flow you want to impose) and a fluctuating part. Plug that decomposition into N.S. and get an equation for the unknown fluctuating part. It looks roughly like N.S. with an extra stretching term and some extra non-linear terms. You can keep the non-linear parts. So this is pseudo-linearization (pulling the mean out). It is just rearranging the problem - no approximation.
Then start with a random initial condition for the unknown fluctuation. Noise. If there are unstable modes, then they will grow in time and you will see them. After long times, the solution is the most unstable mode. And the amount it increases each timestep is the most unstable eigenvalue.
If it is stable, then all modes decay and the norm of your initial condition heads towards zero (rather than infinity for the unstable case).
J.Blair,
Why low Mach number solutions are not local? I agree that totally incompressible flow is not local. Yet, as soon as sonic speed is limited, it should be still local. Thus, it should be possible to solve the problem for very small compressibility, then direct it to zero. For Euler equations.
However, I'm not sure about locality of Navier-Stokes. Dispersion properties of dissipation terms is different, dissipative perturbations may travel at any speed. That's why some smooth shock structure is formed -- high-gradient perturbations can travel upstream of shock wave.
Anyway, even for Euler case, having local stability criteria would be very helpful.
J. Blair is right. The characteristic speed for disturbances to propagate (not to grow) is the speed of sound. Only in cases where the system size is large compared to this speed times a characteristic time, you have a local problem. But the above is just a vague description of the the criterium using the Mach number. In other the words the fastest growing mode of disturbances might be a long wave length mode. For this reason you need the entire system.
Günter,
I don't get it.
As soon as we have locality, and the solution is stable, the size of the system is not important. If solution did not change at all for millisecond (consider it a small time compared to characteristic time), it should not change for the next millisecond too.
Of course, even if all internal points are locally stable, global unstability may be caused by boundary conditions. Yet, the locality principle should work here as well, as soon as we have a good model of that conditions, and this model alone does not involve long characteristic times.
Roman, I'd be interested to know the flow geometry you have in mind, and the kind of process that you imagine causing instability at a point.
William, I don't have exact flow in mind that should be checked for stability. I'm interested in a general approach.
Just find the Lyapunov function. Now, find the function that is difficult!
Advantage, choose the condition of global stability.
Roman,
I might have been unclear. You have a stable solution, this is one solution. The question is whether there is an unstable solution that might grow spontaneously (i.e. due to fluctuations) out of this solution (as J. Blair suggested). The tricky point is that you do not know whether this unstable solution is a localised mode or wether is "lives" on the entire system size.
You need to look into super-critical and sub-critical flows, and laminar flows,
and turbulent flows, and the transition velocities where fluids switch from
one flow type to another flow type.
for non-linear equations it is more complex as far as there might be energy transformation among the different modes of oscillations.
As an experienced researcher in Liapunov stability problems I think there is no other way of checking stability by the first approximation but by computing the spectrum of the linearized system. But since this is a partial differential system, it appears to me that the spectrum of the linearised system does not contain eigenvalues only but continuous spectrum also. The numerical problem may look simpler by such considerations as yours but the result will not be convincing "beyond any reasonable doubt". In fact a good theory must provide you with tools to solve with simple means a problem that may have complicate statement.
Linear stability is not difficult. It is based on (uncoupled) normal modes and many times can be set up in the dual space, like the Fourier space. It then becomes a matrix problem for the eigenvalues and the complex increase rate (frequency).
A good choice of the perturbation function can even lead you to a single equation in terms of the wave-number and complex growth rates.
Non-linear stability is a pain in ... and does not bring to much information (only near by the transition threshold). In this case it pays off direct numerical simulations.
Note that even for simple flows amenable mathematically to linear stability analysis, the eigenvalue approach can be deceptive and/or not very useful.
Couette flow for example (which is hard to get much simpler) is formally stable at all Reynolds numbers using eigenvalue analysis. In practice, it is always turbulent at high Re number. All my big fat books on stability analysis are missing any section/chapter on Couette flow stability. I find that to be a big hint.
This problem appears whenever the stability matrix is not a normal matrix. This will happen in every shear flow and lots of other flow cases of practical interest. This is why B.L. eigenvalue stability predictions are off by about 100%. In the non-normal matrix case, the eigenmodes are not orthogonal. In fact, at high Re, they are highly aligned. So small perturbations (that are nearly orthogonal to the many aligned eigenmodes) actually require LARGE (and nearly cancelling) amplitudes on the eigenmodes (to form them). And the basic assumption of small perturbations implying small amplitude initial eigenmodes modes is incorrect. And the mathematical theory will be mathematically totally correct, but is entirely useless for reflecting the actual situation (which is that Couette flow goes turbulent).
Real fluids have small, but finite, perturbations. In practice, you need to know the response to very very small perturbations, not infinitesimal ones (and they are not the same). In which case, probably the most useful and cost efficient way to do stability analysis of fluid flow is usually to compute the response to small noise.
Dear Roman.
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As I am a mere user of stability conditions (case of surge tanks), I made a small remark hoping that others submit more elaborate solutions.
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In the absence of more elaborate solutions will make a suggestion to you about an easier way to find find dynamic stability by plane phase methods. I note that this method has limitations caused by the linearization of the equations around the equilibrium point, but it gives more consistent results than others.
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Can you just describe your movement around the equilibrium point by plane phase methods, and search the convergence or not around this point.
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In terms of linear stability is correct, but with a little knowledge you can try to find the limit cycles that will move away or get closer to the singular point. These cycles do not have correct value, because as you stray from the singular solution the perverse effects of the linearization of the equations will taking its toll.
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The advantage of this method is that nowadays there are softwares online for the stroke phase plans, and something that was extremely arduous in the past is extremely simple with these softwares.
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The implementation of this method is simple:
1) Find the equilibrium points (if they do not exist the problem is solved!).
2) Linearize the equations around the equilibrium points.
3) Calculate the coefficients of equations at each point and test separately.
4) Put the equations for each point in any software-tracing phase plans.
5) Check the type of the phase plane that forms (vortex, saddle, ....) and set the type of stability.
6) Test the limits of stability around the equilibrium point by checking the presence or absence of limit cycles.
7) Return some points inside this limit cycle the initial variables and check the difference between the solutions of the linearized system and the complete (this item is important to check for global stability).
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This is a engineer solution, not a mathematician, and as I am an engineer I try to solve problems and not create them.
When computer calculations were expensive there was a motivation to try to find only that eigenvalue which indicated that the flow was unstable to small disturbances. Today there is little reason to do so as computers are very powerful and good algorithms exist for determining all the eigenvalues accurately.
There does not seem to be a simple answer to why some seemingly simple flows are linearly stable, but again only after computers became powerful could the eigenvalues be solved and seen that their real parts did not change sign as Reynolds number for some other governing parameter was increased. Of course Rayleigh showed that profiles with inflection points yield to "cat's eye" patterns and for them he showed why this happens mathematically, namely the wave speed matches the flow speed at some point which removes the singularity of the reduced equations (once viscous forces are neglected).
Subsequently people became convinced that the linearly stable flows are unstable to nonlinear disturbances. One of the first convincing calculation of this for plane Poiseuille flow was by my late colleague Thorvald Herbert for his Habilitation.
In boundary layer flows W. Tolmien and C. C. Lin showed the existence of a critical layer where the Reynolds stresses cause the energy transfer from the base flow to the disturbance and gives rise to Tolmien-Schlichting waves. So in this problem the transfer takes place in a very thin region and from this point of view the stability is "local". Below is a link to one of the original papers by C. C. Lin
http://books.google.com/books?hl=en&lr=&id=HkOYt8w7ilgC&oi=fnd&pg=PA117&dq=C.C.+Lin+and+hydrodynamic+stability&ots=uKvy_Tykk6&sig=zTUjedbkgVr3bnifQdmGdIonDm8#v=onepage&q=C.C.%20Lin%20and%20hydrodynamic%20stability&f=false
Once the eigenvalues are found, it is a simple matter to set up energy integrals across the entire flow domain and see how the transfer of energy takes place between the modes and where in the flow domain this takes place. Typically this is not limited to thin layers.
I hope this gives some clarification to the issues discussed above, but of course, much more could be said.
I would also touch a point of interaction between physical and numerical instabilities, as far as the numerical simulation goes.
The experience shows that when a flow (of a Newtonian or viscoelastic fluid) is simulated near the stability threshold, for example, the critical Reynolds number may be really sensitive to the computational parameters (the mesh finesse).
So it may be challenging to figure out a real contribution of numerical or natural instability to the loss of regular solution in a particular case.
Igor, I agree with you if one uses finite difference or elements methods. of perhaps even spectral methods. This would not happen if one uses spectral methods to solve the stability equations. However, spurious eigenvalues sometimes appear and one has to be on guard for them.
Seppo, I also believe that an implementation of a spectral method may involve an "abridged" spectrum of eigenvalues compared to the real one. This would miss "abridged" potentially growing modes to bring about fake stability.
Probably this might be especially the case when a flow is complex and highly deformable (near boundary layers, stagnation points etc.)
Igor, I am not sure what you mean by an "abridged" spectrum. The high spatial frequency modes are certainly missing as a result of truncation. But I was thinking about are eigenvalues and functions that arise from aliasing. This can certainly bring spurious eigenvalues into the spectrum of most interest. They can be discarded by noting how they behave as the grid is refined.
I have to confess that I do not know how to carry out a stability analysis for "complex flows". I have only studied parallel and quasi-parallel flow and then secondary instabilities of flows for which the base flow is periodic as a result of its transition to some steady cellular pattern, or those in which there are waves of the Tolmien-Schlichting type.
By the way, I retired 6 years ago and do not think about these things much these days.
William David Smyth
Yes, We have found some. You may search the following titles: General Temporal Instability Criteria For Stably Stratified Inviscid Flow, Stabilities of Parallel Flow and Horizontal Convection.General stability criteria for inviscid rotating flow General stability criterion of inviscid parallel flow
introduction to Hydrodynamic Stability by Drazin will help you to get some general information o stability
"You may consider that your intuition is wrong." That should be on a t-shirt.
That said, my book "Instability in geophysical flows" summarizes some useful "rules of thumb" for predicting instability. I also have a community stability code for stratified shear flows: http://salty.oce.orst.edu/wave_analysis/SSF_index.html. Have fun.
Eigenvalue problems have their own baggage. Let me just point out some recent results why early instability studies started from Rayleigh, Klevin, Helmholtz and the German schools have deviated from the actual problem. Please have a look at this short letter:
Article Is Tollmien-Schlichting wave necessary for transition of zer...
Also, this will show no questions are shallow. Therefore there is no need to print t-shirts! RG should encourage exchange of ideas. There are some excellent comments! Keep it on!
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