diaw: "If I may add a few thoughts on N-S wave-form: Take u_t = u_xx Split u= u,m+u' (u,mean & u',vibration) Set u' = partial(dn/dt) (n=vibration distance) Plug back into u_t = u_xx, to obtain n_tt = n_xxt as a rather primitive vibration equation.

What category does this fall into? Add a full set of non-linear & dispersion terms & what do we have?"

Tom: The problem is still parabolic (all you've done is differentiate the original equation with respect to time to make it appear higher order) - you have to be careful with tricks like this since there are examples of reducing a single equation to a system of first order equations using such manipulations. The resulting equations sometimes have solutions/waves which are not present in the original equation. An example of this is your new equation requires more initial conditions than the original. In the NS equations the "nonlinear terms" actually play no role in the classification.

diaw: Fair-enough, I follow your logic. Under the full N-S scaling that results from this manipulation, the geometric relationship is a circle - not centred at the origin - the perimeter of which has value Rs=1 (singularity index). This applies to both bulk & deviatoric scaling, ending up with the same final form. Region outside circle is dispersion-dominated, region inside circle is inertia-wave dominated.

I would suggest that the non-linearities do actually play a role in the position of the centre of the singularity circle & intersection of the geometric scaling axes. (wish I had a whiteboard).

The interesting final form of such 'manipulation' as I mentioned before is rather elegant. We end up back at the vector form of the N-S - same in each direction - so no additional terms are in fact added, or subtracted. No real tricks

I agree when too many levels of differentiation are taken eg. on non-linear terms, differentiate up to arrive at wave-forms - spurious additional terms & cross-wave terms will cause havoc. I have been careful at that point.

Thanks again Tom for your wise input & depth of experience.

In terms of the Heavyside inclusion, I saw a paper early last year on a wave-form in a nano-channel, using the telegraph eqn. Let me look it up - I'm not sure if it is widely published as it was at an International Visualisation Conference in Asia. I can't seem to get you on e-mail via CFD-online - perhaps you could e-mail me privately?

diaw...

Tom, I would like to publicly thank you for your amazing input to this debate. Your depth of experience in your field is obvious.

I consider it a privilege to have been able to discuss with you so many aspects of our obvious common passion - computational fluid mechanics - from different ends of the spectrum. You have been so extremely generous with your perceptions & information links.

I trust that this kind of healthy debate will be of benefit to the wider cfd community & that we can continue to challenge & persuade nature to provide us more of her secrets.

Thank you once again for your extreme kindness.

Sincere regards, diaw (Des Auebry)

div u= 0 This is a mathematical statement of conservation of matter (we engineers call it the continuity equation) as a result of applying the condition of zero compressibility to the general statement of conservation of matter. Books on thermodynamics will give the definition of compressibility as Kappa = -(1/V)(dV/dp) at constant temperature. If this is set equal to zero then the volume changes are zero and consequently the density is constant assuming there is no creation or destruction of mass in the volume under consideration. In engineering terminology, constant properties flows refer to properties such as viscocity, thermal conductivity and specific heat...etc. I do not know why, but seems to be the traditions but strictly speaking, incompressible referes to kappa equal zero. It is true that constant properties implies that there is no change in density too, i.e. kappa=0 but traditions are traditions my friend. Cheers and Good Luck. The following book is an excellent one:- "An introduction to Thermodynamics, The Kinetic Theory of Gases and Statistical Mechanics" By Francis Weston Sears. This is a good oldy, The edition I have dates back to 1956. Finally your server returned my email

Thanks for your input, Ahmed.

Based on your points, a few thoughts from my side - & a few unanswered questions.

Could you explain how 'incompressibility' leads to div(V)=0 & the effect, or other wise, on 'shape change'? Hint: Try to do this without the handy kappa relation.

What exactly is 'incompressibility' measuring? For instance, for a fluid element, if one face is deformed, where do the other faces move to? What are their inter-relationships. How does this relate to the div(V) term? What effect would be observed in solid element deformation? How does this affect the velocity gradients? Hint: if one increases, what happens to the others - in which directions - what does this mean physically?

Is Kappa perhaps a handy definition which goes part-way to trying to explain a physical phenomenon - &/or to maintain a certain mathematical assumption? How does kappa relate to bulk modulus - for a solid, & then for a fluid? Hint: the units are different.

Where does second viscosity fit in? Hint: is has a clear place in solid stress theory - upon which Stokes basically modelled his closure (constitutive) equations.

Funnily-enough for the ultimate 'incompressible fluid' => a solid, the issues are less problematic & waves happily travel in, on & 'through' these media. Volume deformation results in shear-waves providing lateral stiffening to elastic waves etc. Waves also happily travel through inviscid gaseous media & we have high-speed shock waves.

What are we missing for liquids? Are the issues merely perceptions, or are they more fundamental than that?

Why do we try so hard, in numeric solutions for N-S, to prevent 'touching' the singularity? What happens if we go across the 'singularity divide'? Is it impossible for the N-S to have singularities? If so, then what happens for the combination when all terms on the lhs, combine to tend towards zero? What physical event does this singularity represent?

This then begs a final question: How valid, if at all, are the Navier-Stokes equations in the first place? Has it all just been a huge waste of time & energy? Are we merely drawing pretty pictures?

-------

Lastly, you were e-mailed a list of suitable e-mail addresses. Try them. Perhaps my server doesn't 'like' your server

>Funnily-enough for the ultimate 'incompressible fluid' => a solid, the issues are less problematic & waves happily travel in, on & 'through' these media. Volume deformation results in shear-waves providing lateral stiffening to elastic waves etc. Waves also happily travel through inviscid gaseous media & we have high-speed shock waves.

>What are we missing for liquids? Are the issues merely perceptions, or are they more fundamental than that?

Just a comment. You keep on claiming that solids are incompressible, which is WRONG unless the Poisson's ratio is 0.5. In other cases they are compressible, with finite bulk modulus and shock waves may occur under intense loading. Even for lower loads, there are volume changes (i.e., compressibility) in solids, in addition to change of shape (deviatoric deformation). The exception, i.e., Poisson's ratio = 0.5 (i.e., rubber-like materials) is singular (infinite bulk modulus) and requires special techniques to deal with, as may be found in many references (e.g., the classical book of Zienkiewicz).

Rami:

Just a comment. You keep on claiming that solids are incompressible, which is WRONG unless the Poisson's ratio is 0.5. In other cases they are compressible, with finite bulk modulus and shock waves may occur under intense loading. Even for lower loads, there are volume changes (i.e., compressibility) in solids, in addition to change of shape (deviatoric deformation). The exception, i.e., Poisson's ratio = 0.5 (i.e., rubber-like materials) is singular (infinite bulk modulus) and requires special techniques to deal with, as may be found in many references (e.g., the classical book of Zienkiewicz).

diaw:

Thanks Rami for your very wise observation.

Indeed, you are completely correct - in that a solid element is deformable - of course it is. We would never dare to say that a solid is completely 'incompressible', or rigid. We know well - visually - that a direct stress in one direction - affects other directions. Every 1st year engineer is taught this.

On the other hand, we are happy to say that liquids are 'incompressible'. I used the solid 'incompresibility' - liquid 'incompressibility' to try & highlight an apparent logic problem. How does this 'incompressibility' relate to a solid, or to a unit fluid cell? In other words, how exactly are the div(V) terms related to the 'incompressibility' criterion, other than through the kappa relationship (apparently self-serving).

Could we we have somehow perhaps become mixed up in fluid logic between deformable fluid cells & fully-rigid cells.

****** This viewpoint, in my view, is crucial to ever accepting that wave phenomena could ever co-exist in our fluid models & that N-S actually allows for this to occur. ******

You see, when we can merely 'wave away' the possible co-existence of wave activity in low-speed viscous fluid domains - with the 'incompressible' assumption & then try to justify our position that it means the same as constant fluid properties, then I believe that we may have overlooked something. Sometimes logic becomes circular & self-serving.

Or, perhaps I have missed the boat completely?

--------

I saw a beautiful example of this bulk-flow/wave co-existence today. I was looking at a picture taken of flow over an inclined plate - using the hydrogen-bubble technique to show seperation behind the plate. The light seemed have caught another simultaneous phenomenon... it looks like a perfect set of two Kelvin-Helmholtz waves... it simply took my breath away. The interesting thing was that the K-H effect is reasonably lazy, whilst the bubble motion appears to be a little more energetic & does not appear in perfect synchronous to the K-H phenomenon.

I had seen this photo a few times in the past & always wondered what the 'background effects' were. I had basically written them off as a light aberations. Sometimes things seem to 'pop out' with re-looking. Perhaps there are simple explanations, but the effect was amazing.

diaw... (Des Aubery)

"I would suggest that the non-linearities do actually play a role in the position of the centre of the singularity circle & intersection of the geometric scaling axes. (wish I had a whiteboard)".

As far as the classification goes the nonlinearities play no role - strictly speaking the NS equations are quasi-linear and not nonlinear. As an example consider the generalized Burger equation

u_t - u_xx = f(u,u_x),

where f is an arbitrary function of u and u_x. This equation is always parabolic! If the "nonlinearities" were important I would not be able to make such a statement.

For the telegraph equation see "The effects of weak hyperbolicity on the diffusion of heat" by King et al.

Royal Society

This is the paper that originally made me consider the generalizing the NS equations some years ago.

"I can't seem to get you on e-mail via CFD-online - perhaps you could e-mail me privately?"

I don't tend to put my e-mail address on the posts for a number of reasons - I work for a government organization and so any posts should checked and a disclaimer added, Spam e-mail, students asking for jobs and/or help with there projects, etc.

diaw:

"I would suggest that the non-linearities do actually play a role in the position of the centre of the singularity circle & intersection of the geometric scaling axes. (wish I had a whiteboard)".

Tom:

As far as the classification goes the nonlinearities play no role - strictly speaking the NS equations are quasi-linear and not nonlinear. As an example consider the generalized Burger equation

u_t - u_xx = f(u,u_x),

where f is an arbitrary function of u and u_x. This equation is always parabolic! If the "nonlinearities" were important I would not be able to make such a statement.

diaw:

Greetings Tom.

The generalised Burger equation is a very interesting example in itself, I'm so glad that you brought it up. The unsteady Burger equation is capable of producing 'velocity waves' & is often used as a shock equation. Trying to set up a simple FEM solution, for instance, can have some interesting 'side-effects' & produces different solutions depending on initial guess - unless some of the boundary conditions are brought back into the solution as unknowns. [There is a very nice website for Japanese physicist showing a few interesting simulations (I can't for the life of me find that link today ]

u_t - u_xx = f(u,u_x)

set, for example f(u,u_x) = u.partial(du/dx) as a case in point,

u_t + u.du/dx - u_xx = 0

seems to produce a velocity wave effect. The wave primitive terms are 'convection wave' (from inertial term) & 'elastic wave' (from u_xx term). Actually, even for the steady-state case, the numeric solution can be a little 'picky'.

An extremely interesting case results with:

u.du/dx - u_xx = c

This is the case I initially simulated in 1D (Matlab) to test for the possibility of 'low-speed viscous waves'. Leaving c=0 will allow a very tight velocity swing as we encounter the critical velocity. Begin to manipulate the 'c' parameter & the critical velocity can be 'chased' all the way down to almost zero.

This concept basically allows simulation gradually increased u values until a critical velocity is reached at which point the lhs becomes zero... This critical u value is very sensitive to the 'c' parameter.

Conceptually this simulation represents the following: - steady flow; - raise velocity very, very slowly ie. no accelaration of note; - a critical velocity will be reached at which point the whole lhs goes to zero... - This corresponds conceptually to the rupture of 'steady' equilibrium, whereafter the acceleration term must become active.

--------

Does this mean that parabolic pde's can allow wave phenomena?

------

Tom:

For the telegraph equation see "The effects of weak hyperbolicity on the diffusion of heat" by King et al.

Royal Society

This is the paper that originally made me consider the generalizing the NS equations some years ago.

diaw:

Thanks so much for that link.

The paper I mentioned is: "Visualization of momentum waves in a flat nano-channel", V.V. Kilish, W.K. Chan & A. Sourin, The 8th Asian Symposium on Visualization, 2005.

Tom:

"I can't seem to get you on e-mail via CFD-online - perhaps you could e-mail me privately?"

I don't tend to put my e-mail address on the posts for a number of reasons - I work for a government organization and so any posts should checked and a disclaimer added, Spam e-mail, students asking for jobs and/or help with there projects, etc.

diaw:

I apologise if I seemed to have put you in a tough spot. I understand entirely. Disclaimers & spam can be a pain. So much for the internet giving us freedom

Des Aubery...

A link to a few pictures similar to what I mentioned above.

<http://www.eng.chula.ac.th/~fmeabj/Flow%20Visualization/Flow%20Images/Vortex%20Shedding/FMRL%20Vortex%20Shedding%20Frame.htm>

An excellent site, with a number of useful links on the homepage.

"set, for example f(u,u_x) = u.partial(du/dx) as a case in point,

u_t + u.du/dx - u_xx = 0

seems to produce a velocity wave effect. The wave primitive terms are 'convection wave' (from inertial term) & 'elastic wave' (from u_xx term). Actually, even for the steady-state case, the numeric solution can be a little 'picky'. "

There are two things happening (this needs uu_x >> u_xx). Firstly the balance u_t+uu_x = 0 results in a hyperbolic equation whose solution will fail to be single valued in a finite time (u remains constant on the characteristic). The second effect comes from the u_xx term (which destroys the characteristic so that u is nolonger constant along the particle path and there is communication between points). If for example you have right propagating wave in the "inviscid" system (u_xx term zero) then there is no signal ahead of the front of the wave and the wave eventually overturns due to the piling up of particles behind the front. The viscous term removes this problem by allowing "upstream influence" and smears out the front. This upstream influence corresponds to data being transmitted from places where the "inviscid wave" has not yet reached.

The best description, in my opinion, of this behaviour is diffusion with a nonlinear source flux. The Burger equation is actually the heat equation in disguise (through the Hopf-Cole transformation).

In multiple dimensions this is sometimes refered to as "frontogenesis" (there's a review article on this in the Ann. Rev. Fluid Mech. by Hoskins sometime in the early 80's).

There's an excellent discussion of the Burger equation (and waves in general) in the book "linear and nonlinear waves" by G.B. Whitham.

"u.du/dx - u_xx = c"

Wouldn't (u-c)u_x - u_xx be a better choice here; i.e. make it look like the Rayleigh/Orr-Sommerfeld equation? For a wave d/dt -> -c.d/dx.

"Does this mean that parabolic pde's can allow wave phenomena?"

High order/dimension parabolic equations can exhibit wave-like behaviour due to approximations etc. Whether these can be truely called waves is another question. The fact that these waves must have either complex wavenumbers or frequencies really means they are not true waves. In practice the smallness of the imaginary parts to the frequency and/or wavenumber means that the solutions are "approximate waves" and hence the terminology I mentioned earlier concerning Rayleigh and TS waves in boundary layers.

The distinction between the full characteristics and "sub-characteristics" which may have wavelike behaviour is covered in the book on perturbation methods by Kevorkian and Cole.

At the end of the day I think this really comes down to what you are calling a wave and is a matter of terminology. I would argue that a wave must be a persistant free solution of the equations like, for example, inviscid linear gravity waves on a free surface. Any wave-like disturbance which either decays with time or requires boundary forcing to maintain it does not count as a strict wave; e.g. exp(-t)sin(x-ct) is not strictly a wave (although you could term it a decaying-wave to highlight the wave-like x-ct contribution).

Thanks so much for that information. It is exactly what I was looking for. This has answered quite a few nagging questions I've had. (I've got enough reading to do for a year

If I may, I'll jump directly to the last portion.

Tom:

At the end of the day I think this really comes down to what you are calling a wave and is a matter of terminology. I would argue that a wave must be a persistant free solution of the equations like, for example, inviscid linear gravity waves on a free surface. Any wave-like disturbance which either decays with time or requires boundary forcing to maintain it does not count as a strict wave; e.g. exp(-t)sin(x-ct) is not strictly a wave (although you could term it a decaying-wave to highlight the wave-like x-ct contribution).

diaw:

I would heartily agree that this is really what we end up with - to be sure. In my research work, it is precisely these exp(-t)sin(x-ct) type waveforms that have interested me... perfect... I have developed solutions for these in 3d for the N-S 'wave form' (for u' & v') & pressure. The expressions become a little lengthy, but are actually rather straightforward.

What I am constantly reminded of, is that, in physical experimentation, with carefully-controlled inlet conditions, small pertubations, are being constantly created. I would venture to say, that in most real flow situations, these perturbations constantly re-introduce, or re-inforce, the 'transient wavefield'.

I have noted in simulations, that small sub-scale wave effects are clearly observed eg. flow within a straight pipe - & these phenomena build into what appears to be 'normal-mode'-like shapes - backing up against the inlet boundary. The outlet boundary has a different pattern depending on the boundary condition itself. The temporary flow structures are persistent even for tens of minutes into the simulations - but, most eventually do succumb until a more simple wave pattern is left in place - but, a definite mode pattern seems to persist above a certain threshold velocity - albeit very, very low. (All calculations are performed in double-precsion to convergences of around 1e-16)

In practice, it may be that the way that the boundary condition is introduced at each iteration can cause a small inlet disturbance. A sudden inlet velocity condition for a 'wave field' represents an 'impulse'... nasty.

In real life flows eg. Reynolds experiment - there is a consistent inlet perturbation entering the flow domain & establishing a wave pattern within said space - with some components gradually dying out, only to be replaced as the next perturbation enters the flow domain.

These sub-scale wave patterns are actually rather amazing to observe - especially when a small exit pressure condition is applied. An interesting wave interaction is clearly evident - clash of the titans, sloshing, you name it.

A further thought on what Reynolds observed. If a tracer dye is used in the experiment, the dye particle has mass & thus a certain threshold energy level (say acoustic pressure) needs to be reached before we can 'see' the particle movement. (Particles only move through an external force).

I am deeply convinced that what Reynolds observed was the effects of a co-existent wave-field.

Tom, again, thanks for your wisdom & insight. It has been a rare privilege to obtain your insights & contributions. You have answered many questions. This is something that would genuinely have been impossible without the internet.

(BTW, I have performed all my research 'in the dark' so to say, with some reference to physics-based references. This also forced me to 'invent' some fairly interesting - perhaps crude - approaches to interpreting what I was seeing.) Now, with your excellent references & thought process, I have a lot more to set my path straight. Thank you again).

Des Aubery...

If I could impose for just a little longer, I would like to still try & get to the bottom of why I am observing a consistent 'bow-wave' type effect in a viscous constant density solution field for flow inside a pipe, over a cylindrical singularity. (Test case quoted in a post answering Ahmed's queries).

I'll try to upload a few pics to a link during the next few days which should hopefully make the point more much clearly. (The joys of misbehaving web-servers).

The simulation progresses from very small entry velocity values, with large step sizes & small element sizes. Up to a certain speed, the velocity magnitude field looks rather viscous. The flow velocity is then increased still further until a certain 'critical velocity' is reached, whereafter, solution divergence occurs within a few iterations - pressure-wave-related. Simulation results at this point show a very 'unstable' flow field with rapid pressure swings up & done the pipe.

At this point, the strategy is then reversed with large element sizes & tiny time-step sizes. This allows this 'critical point' to be crossed successfully, whereupon, immediately, the bow-wave phenomenon emerges, 'mach' lines & leading bow 'shock' (bright line of large velocity gradient) leads the 'shock'.

If the inlet velocity is progressively increased, the bow shock begins to part & move gradually rearwards. As velocity is then further increased, the flow field become more unsettled & wavy in nature, with a very different appearance to traditional 'viscous-type' academic solutions.

The schemes used are also mentioned in the previous post. In these schemes no use is made of upwinding at all, or any form of convection stabilsation. The solvers are FEM CBS & Galerkin-types & have been tested in research for a number of years - by others.

diaw... (Des Aubery)

A few pics of wave effects in pipe simulations. The first one shows the 'bow-wave' effect.

<http://i48.photobucket.com/albums/f2...01_0p002_2.gif>

<http://i48.photobucket.com/albums/f2...03_Yvel_r1.gif>

<http://i48.photobucket.com/albums/f2...24gs_r8_pm.gif>

<http://i48.photobucket.com/albums/f2...24_r3_pm1p.gif>

I hope that the links work. I'll add a few more interesting cases in a few days.

diaw... (Des Aubery)

Rather try these - they are png's & seem to have come out a lot better.

<http://i48.photobucket.com/albums/f2...24_r3_pm1p.png>

<http://i48.photobucket.com/albums/f2...24gs_r8_pm.png>

<http://i48.photobucket.com/albums/f2...03_Yvel_r1.png>

<http://i48.photobucket.com/albums/f2...01_0p002_2.png>

diaw... (Des Aubery)

Some thoughts - I think you can safely say this is numerical.

Things to check are

(1) Are you using an artificial compressibility type FEM code. These used to be quite common 10-12 years ago.

(2) What is div(u) if you plot it (it's unlikely to be zero - see also point 1).

and most importantly

(3) Looking at the scale of numbers on your graphs - you really need to nondimensionalize this problem! This will reduce differencing/aliasing errors in your code (computers using double precision only have a finite degree of accuracy). Your problem with convergence suggests this is the main cause of your problem.

I suspect if you do (3) and only plot contours > discretization error of your code then you will not see this behaviour.

Tom: Some thoughts - I think you can safely say this is numerical.

Things to check are (1) Are you using an artificial compressibility type FEM code. These used to be quite common 10-12 years ago. (2) What is div(u) if you plot it (it's unlikely to be zero - see also point 1).

and most importantly (3) Looking at the scale of numbers on your graphs - you really need to nondimensionalize this problem! This will reduce differencing/aliasing errors in your code (computers using double precision only have a finite degree of accuracy). Your problem with convergence suggests this is the main cause of your problem.

I suspect if you do (3) and only plot contours > discretization error of your code then you will not see this behaviour.

diaw:

Greetings again Tom.

For all four plots shown, these were simulated using the same solver.

(1) This code in based on a recent papers coverign the CBS FEM scheme (Zienkiewicz, O.C. & Taylor, R.L.). I've been looking through the code & thesis. We have present in this particular solver: (a) Artificial compressibility; (b) Taylor expansion in time for u, v.

(2) I'll extract the data & compute the div(u) values. Good point.

(3) There is a specific reason for the tiny inlet velocities for the 1st three plots. This was to try & almost eliminate the bulk field effect & concentrate on the 'wave field' due to small disturbances.

The last plot has an entry velocity of 0.029 m/s & this is within the order of magnitude for the scaling predicted by Reynolds for his experiment. (At Re~2350, u,in ~ 0.0807 m/s). I have purposely tried to steer away from standard Reynolds scaling as I have specific experience with this not always being appropriate & did not want to introduce scaling errors.

With this particular solver, I believe that only single-precision may have been used.

-----------

Now that has been said, I will put up a few more plots, showing 'wave effects' using different solvers - one a steady Galerkin & the other a commercial FVM code (no names).

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Would you be so kind so as to explain the aliasing error concept? Would this be responsible for the sudden need to change spatial resolution & time-step? The time-step change is some 2 orders of magnitude.

Could you comment on the issues with CBS schemes, artificial compressibility, Taylor-Galerkin FEM schemes?

Thanks so much for your kind input.

Des Aubery...

"The last plot has an entry velocity of 0.029 m/s & this is within the order of magnitude for the scaling predicted by Reynolds for his experiment. (At Re~2350, u,in ~ 0.0807 m/s). I have purposely tried to steer away from standard Reynolds scaling as I have specific experience with this not always being appropriate & did not want to introduce scaling errors."

Correct nondimensionalization will never introduce scaling errors - in fact in CFD it actually reduces the computational error.

"With this particular solver, I believe that only single-precision may have been used."

You should never run in single precision.

"Would you be so kind so as to explain the aliasing error concept?"

The basic idea is that if you have a large number and a small number and you difference (or add) then accuracy is lost; e.g. in single precision what is 100 + 1.0e-6 and 100 - 4.0e-6?

"Would this be responsible for the sudden need to change spatial resolution & time-step? The time-step change is some 2 orders of magnitude."

Yes it can (you're trying to get a greater number decimal places than the computer stores). A good example of this is using an SOR routine to solve Laplaces equation in single precision with a tolerance of 10^(-5). The solver will work at coarse resolution but will stop converging as the resolution is increased. Changing to double precision fixes the problem (by doubling the number of decimal places) . In CG type solver you see the same thing in double precision if you set the dimensionless convergence tolerance too low (e.g. 10^(-9)).

"Could you comment on the issues with CBS schemes, artificial compressibility, Taylor-Galerkin FEM schemes?"

Not really. Artificial compressibility was designed to correct the stiffness problem with the incompressible NS equations. If you write out the discrete equations as one big system you find, because p is determined via continuity, that the matrix is badly conditioned (zeroes along the diagonal corresponding to the p equation - p does not appear in the continuity equation). In the artificial compressibility method you add a p dependant term to the continuity; e.g. solve something like div(u) = a.p_t and relax the RHS to zero. Note that p_t may simply be the difference in p between iterations at a fixed time. (It also has something to do with the positioning of nodes, for p, on the element but I can't remember exactly what - it was ~12 years ago that I last looked at this). With artificial compressibility the error in the unconverged solution will behave like acoustic waves.

Hope this helps, Tom.

diaw: "With this particular solver, I believe that only single-precision may have been used." REAL*8

Tom: You should never run in single precision.

diaw: You are correct - this is my standard programming style.

I would like to give a few observations, regarding under-precision numerics. This is based on many, many observations of solution dynamics, in many N-S flavours of N-S solvers. Generally, when the numerics begin to saturate, the answers line out at a certain value, with an ongoing slight oscillation. When numerics really begin to get themselves 'knotted' then often a small region will begin to move to either +- 'computer numeric infinity' - with an emerging solution blow-up - seen as an unphysical region spreading out, until the whole domain eventually becomes absurd. Very often, this 'solution spot' runs back from the flow outlet boundary. (I do not let a solver get away without proving to me why it cannot do the task at hand. I go in at regular iteration counts & observe the solution progress).

With this particular solver, I have been able to take the residuals down to around 1e-14 on pressure & velocity at long-iteration runs. At this point, small oscilations do appear but the solution basically 'lines out'. Odd for such low-precision numerics.

------

Tom: Correct nondimensionalization will never introduce scaling errors - in fact in CFD it actually reduces the computational error.

diaw: This is a very, very important issue - to be sure. The crucial question is "what is the correct scaling to use - that will provide the correct scaling of the physical phenomena"?

This is an issue very close to my heart & some of it goes back to some earlier posts on this thread. I'll pull them back up & re-visit what I was trying to get at - if we move into that area. This may turn out to be rather interesting in terms of Re in 2d domains. Which one to choose, if at all. Does Reynolds scaling in one direction imply that scaling in all other direction are correct - in every geometry?

--------

diaw: "Could you comment on the issues with CBS schemes, artificial compressibility, Taylor-Galerkin FEM schemes?"

Tom: Not really. Artificial compressibility was designed to correct the stiffness problem with the incompressible NS equations. If you write out the discrete equations as one big system you find, because p is determined via continuity, that the matrix is badly conditioned (zeroes along the diagonal corresponding to the p equation - p does not appear in the continuity equation). In the artificial compressibility method you add a p dependant term to the continuity; e.g. solve something like div(u) = a.p_t and relax the RHS to zero. Note that p_t may simply be the difference in p between iterations at a fixed time. (It also has something to do with the positioning of nodes, for p, on the element but I can't remember exactly what - it was ~12 years ago that I last looked at this). With artificial compressibility the error in the unconverged solution will behave like acoustic waves.

diaw: I can see where you are going with this. The issue with taking the pressure-gradient terms to the left-side of the system equations certainly leaves 'big holes' in the system matrix. I have never felt truly comfortable with this. Some Galerkin-type solvers go for this form of the Jacobian.

Looking over the code for the CBS solver mentioned, the pressure terms are brought in as 'load terms' on the rhs - basically in typical FVM style. The lhs is used to solve for u & v update terms.

Another issue to consider is if the N-S are constructed more in line with the 'force-balance' physical understanding. Here it is more natural to place the forcing functions on the rhs:

1/p.(-dP/dx) + a,x

This then allows an iterative solution to refine the initial p-field guess - very similar to FVM techniques. There should then be less 'holes' in the system matrices.

Wave-effects: I have noted the 'wave effects' in solutions that are not fully converged. In some cases, I am very reminded of Taylor-Couette flow fields. I'm not entirely convinced that these effects are all purely attributable to numerics.

An interesting observation: If I may direct you towards the 'other link I set up regarding the inclined plate under hydrogen-bubble addition flow field, in water. Look below the plate at the gradual 'roping'/movement of some of the bubble streamlines. Also look above the 'K-H' waveforms. The field is ever-so-wavy. There seems to be a small level of communication, even in this experiment. It could be tunnel-entry related, but interesting, never-the-less. Very often, the 'smoke remnants' in wind-tunnels have very clear wave traces. This could be due to slight compressibility-wave effects, but the runs would surely be considered incompressible in the sense of M < 0.3.

Tom:

Hope this helps, Tom.

diaw:

Your very kind assistance & debate is wonderful. I have missed this level of depth for a long, long time.

Sincere thanks, diaw... Des Aubery

"With this particular solver, I have been able to take the residuals down to around 1e-14 on pressure & velocity at long-iteration runs. At this point, small oscilations do appear but the solution basically 'lines out'. Odd for such low-precision numerics."

I assume this is a dimensional residual - it is imposible to reduce the nondimensional rms residual below 10^(-12) (double precision has only 12 digits). In practice you'll be lucky to get the error below 10^(-9).

"diaw: This is a very, very important issue - to be sure. The crucial question is "what is the correct scaling to use - that will provide the correct scaling of the physical phenomena"?"

For incompressible flows it doesn't really matter as long as you're consistent - pick a velocity scale that gives a maximum flow velocity of around 1 or 2 and choose a length scale which is typical of the flow domain (width of the channel, length of the plate, etc). You then need to ensure that your resolution is good enough to capture the physics you are studying; i.e. for the steady boundary layer you need points within the R^(-1/2) thick layer normal to the surface. If you want to study the inflectional instability of such a boundary layer then you will need to have comparable resolution in both the streamwise and normal directions plus a timestep that resolvles the R^(-1/2) time scale of the instability. For the TS instability there is another set of scalings (I think these are in Drazin & Reid). Actually ensuring you get these scales right for a engineering type problem is beyond currrent CFD capabilities (hence the use of turbulence models).

I often think engineers are thrown into CFD before they have had time to appreciate the fundamentals of fluid mechanics and so rely too heavily on the CFD package giving the "right" answer (and often believe it's right when it's wrong). Basically it's a good idea to know what you are doing and why before you perform the simulation - this will also give you an idea of what you expect to see in the simulation.

"An interesting observation: If I may direct you towards the 'other link I set up regarding the inclined plate under hydrogen-bubble addition flow field, in water. Look below the plate at the gradual 'roping'/movement of some of the bubble streamlines. Also look above the 'K-H' waveforms. The field is ever-so-wavy. There seems to be a small level of communication, even in this experiment. It could be tunnel-entry related, but interesting, never-the-less. Very often, the 'smoke remnants' in wind-tunnels have very clear wave traces. This could be due to slight compressibility-wave effects, but the runs would surely be considered incompressible in the sense of M < 0.3."

What you are seeing below the plate is the "deflection" of the flow by the plate plus contributions from the unsteady pressure field induced by the shedding on the otherside. (There may also be a small amount of unsteadiness is the background/inflow which may be amplifide by the flow - there appears to be separation on this side so the boundary layer is unlikely to be stable).

Above the plate "wavelike" behaviour is nothing more than the shedding of vorticity by the separating boudary layer - the separating (free) streamline is highly unstable and attempts to roll-up on itself. The flow above this free-stream sees an undulating surface and so undulates in response. Are these waves or are they just part of the unsteady solution? I tend to favour the latter - I think these pictures make perfect sense without the need to invoke a wave type argument.

Tom:

For incompressible flows it doesn't really matter as long as you're consistent - pick a velocity scale that gives a maximum flow velocity of around 1 or 2 and choose a length scale which is typical of the flow domain (width of the channel, length of the plate, etc).

diaw:

Let's take as an example, a thick-tube - or rectangular flow geometry - with length-to-width ratio of say 3. We have an inlet velocity of u & v into the mouth of the tube. What Reynolds scaling should we use?

Another more complex shape could be a projectile with the fluid-space above the projectile having a ratio say 1:6. The surface of interest on the projectile may be inclined & positioned a short distance into the flow field - placed on base of fluid domain. (I'm thinking of the typical projectile example often shown in texts explaining Tollmien-Schlichting waves). What scaling would be appropriate?

What I'm really trying to get at is what dimension we 'should' scale against. Engineers are typically told to 'scale on Reynolds number'. Very often there are a number of suitable Reynolds numbers to select from. Which one is appropriate, or can any one be selected. Would it be possible to scale on a 2d Reynolds number, if this exists, for instance?

Tom:

You then need to ensure that your resolution is good enough to capture the physics you are studying; i.e. for the steady boundary layer you need points within the R^(-1/2) thick layer normal to the surface. If you want to study the inflectional instability of such a boundary layer then you will need to have comparable resolution in both the streamwise and normal directions plus a timestep that resolvles the R^(-1/2) time scale of the instability. For the TS instability there is another set of scalings (I think these are in Drazin & Reid). Actually ensuring you get these scales right for a engineering type problem is beyond currrent CFD capabilities (hence the use of turbulence models).

diaw:

Thanks very much for those points. This is very valuable for a lot of researchers. I have an inbuilt mistrust of turbulence models. I feel that often, we are trying to impose rules onto nature, instead of observing nature & adapting to suit.

Tom:

I often think engineers are thrown into CFD before they have had time to appreciate the fundamentals of fluid mechanics and so rely too heavily on the CFD package giving the "right" answer (and often believe it's right when it's wrong). Basically it's a good idea to know what you are doing and why before you perform the simulation - this will also give you an idea of what you expect to see in the simulation.

diaw:

I would agree with this 100%. I've been guilty of this. I've often heard the term 'God's equations', or similar ilk. The impression often given is that if one can write out the N-S equations, then the person has an impressive IQ. There often seems to be little deep understanding, other than a simplification of the N-S into very reduced forms amenable to solution in the form of an infinite series. We are then said to have an 'exact solution'.

Another thing I have also noticed is that typical academic problems most often seem to have amazingly unrealistic fluid properties - very often vastly different from say air, or water. When these properties are used in the simulations, then the solvers begin misbehaving. This very often shocks the new engineer when he tries to model real fluids.

diaw: "An interesting observation: If I may direct you towards the 'other link I set up regarding the inclined plate under hydrogen-bubble addition flow field, in water. Look below the plate at the gradual 'roping'/movement of some of the bubble streamlines. Also look above the 'K-H' waveforms. The field is ever-so-wavy. There seems to be a small level of communication, even in this experiment. It could be tunnel-entry related, but interesting, never-the-less. Very often, the 'smoke remnants' in wind-tunnels have very clear wave traces. This could be due to slight compressibility-wave effects, but the runs would surely be considered incompressible in the sense of M < 0.3."

Tom: What you are seeing below the plate is the "deflection" of the flow by the plate plus contributions from the unsteady pressure field induced by the shedding on the otherside. (There may also be a small amount of unsteadiness is the background/inflow which may be amplifide by the flow - there appears to be separation on this side so the boundary layer is unlikely to be stable).

Above the plate "wavelike" behaviour is nothing more than the shedding of vorticity by the separating boudary layer - the separating (free) streamline is highly unstable and attempts to roll-up on itself. The flow above this free-stream sees an undulating surface and so undulates in response. Are these waves or are they just part of the unsteady solution? I tend to favour the latter - I think these pictures make perfect sense without the need to invoke a wave type argument.

diaw:

I can see what you are saying. Let's try to get to the mechanism/s active in the field. Now, would this pressure field be as the result of the flow, or would it be both as a result of & in turn influencing the flow?

In other words, the flow enters the flow section, begins to react to the plate, moving into suitable flow paths. This causes local pressure gradients, which, in add to (modify) the bulk pressure field. The flow field then compensates to accomodate the modified pressure field - by direction, or velocity change, which in turn sets off another pressure fluctuation - with the cycle repeating itself over & over. In other words, a 'bulk' phenomenon, causing a small pressure change (deviatoric presure component), which in turn interacts with the 'bulk' component & so forth. (In some cases, standing wave-forms may establish themselves - given suitable geometry & flow conditions.) The information transfer mechanism is pressure fluctuations - could this be deviatoric wave activity superposed onto a background pressure field?

I guess, what I'm saying is a mechanism similar to that of the electro-magnetic field. In our case the interaction between a 'bulk' & 'deviatoric' field. I see the pressure as the linking mechanism in fluid phenomena. (I think that we've both said as much along the course of the debate).

The issue would be how to experiment for such co-existence? The deviatoric velocity components are small, and it would be difficult to be able to detect them. If wave phenomena are really present, then perhaps frequency measurements could be taken? The problem is then, that a number of frequencies could be present.

I have seen experiments where a sound frequency is input into the flow field & it responds. What is the mechanism for this response? If it is sounds waves, then surely we are on track for a wave co-existence?

Back to Reynolds classic experiment. What is the mechanism causing turbulence in the pipeflow? The velocity is tiny ~ 0.0807 m/s. Why does the fluid react the way it does? It is a property of the fluid, of the environment, or both? It takes energy to deflect a fluid particle - where does the energy come from?

I hope that my arguments have come across reasonably coherently?

diaw...