# Can 'shock waves' occur in viscous fluid flows?

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 February 8, 2006, 19:28 Re: Explanation for 'bow-wave' effect in simulatio #81 Adrin Gharakhani Guest   Posts: n/a I concur with Tom entirely with regards to the results. My immediate reaction when I saw the plot and the range of values - O(10^-4 ~10^-6) if I remember correctly - was: (1) this is _obviously_ numerical due to lack of precision (2) the algorithm must be compressible at heart - not purely incompressible. Tom already brought up fine examples as regards to summing large and small numbers, as well as the issue of compressibility in the solution. You are essentially/computationally solving a compressible fluid problem (not incompressible) - thus the observed waves. If you were to scale the solution, which does not change the characteristics of the equation at all, you would eliminate most, but I bet not all, the waves. In the scaled version, your range of numbers might be O(1) and if you were to plot the field in O(1) you would most probably not see these waves, but if you were to narrow your visualization range to smaller scales you probably would still see (some of) them. But, these are numerical perturbations that you cannot get rid of in finite precision computing (you can reduce them at higher and higher precisions, but not eliminate them). When you perform an O(1)/scaled computation the perturbations are probably so small that they don't cause too much problem (they get damped out numericaly due to numerical diffusion), but with O(<1)/nondimensional computations the perturbation amplitudes now become large relative to the primary flow (due to reasons explained by Tom). In _any_ numerical analysis one has to be careful how to interpret the results by keeping in mind the order of accuracy and precision of the simulations. If the precision of your experimental equipment is say 1+/- 0.05, you know it is meaningless to talk about measured values such as 1.000x. The same goes for your "virtual experimental equipment" - the CFD code. Numbers below the precision of the computation are meaningless. Adrin Gharakhani

 February 8, 2006, 20:10 Re: Explanation for 'bow-wave' effect in simulatio #82 diaw (Des Aubery) Guest   Posts: n/a Hi Adrin, Thanks very much for your kind input to the debate. I think that your summary is very clear & puts matters in a clear perspective in terms of numeric precision & numerical 'waves'. Excellent. Thank you very much. Now, if we were to say have a real physical mechanism with both a large bulk-flow field & small deviatoric field co-existing in the same flow domain - how would we be able to distinguish this deviatoric field from purely numeric issues? For instance, if there is a bulk flow field at say 10 m/s, but a deviatoric field at say 0.001 m/s (possibly smaller), both co-existing in the same flow domain - how would we be able to resolve these fields? What kind of scaling would be appropriate? How would we work within the constraints of our numeric machine? diaw...

 February 8, 2006, 22:49 Re: Can 'shock waves' occur in viscous fluid flows #84 Ahmed Guest   Posts: n/a Dyaw wrote: 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). Ahmed:- Engineers have never defined that incompressible liquid. Engineers define incompressible flow situation and compressible flow, the only difference is that in the latter density changes are allowed while in the former density changes less than 1% are ignored. This is a handy way to get numerical design parameters. Could you please check that link again, I am not able to log on that site Cheers and good luck, always remember that there is nothing called incopressible fluid It is incompressible flow

 February 8, 2006, 23:55 Re: Can 'shock waves' occur in viscous fluid flows #85 diaw (Des Aubery) Guest   Posts: n/a Diaw wrote: 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). Ahmed:- Engineers have never defined that incompressible liquid. Engineers define incompressible flow situation and compressible flow, the only difference is that in the latter density changes are allowed while in the former density changes less than 1% are ignored. This is a handy way to get numerical design parameters. Could you please check that link again, I am not able to log on that site Cheers and good luck, always remember that there is nothing called incopressible fluid It is incompressible flow diaw: Greetings Ahmed. Some good observations about how engineers think. I'll quote from a very well respected expert in the field of Gas Dynamics. "... compressibility is the phenomenon by virtue of which the flow changes its density with change in speed. Now it may be asked as to what are the precise conditions under which density changes must be considered." (Pg 3) (then specifically for gases) "It is widely accepted that compressibility can be neglected when dp/p,i <= 0.05 ie. when M<=0.3" (Pg 5) "Gas Dynamics", E. Rathakrishnan, Prentice-Hall India, 1995. Would I be too bold so as to infer that this as a 'constant density' criterion, rather than a div(V) criterion? -------- (Try to copy the entire link to the address line of your browser - you will then be able to connect). Thanks for your input. diaw...

 February 9, 2006, 09:09 'Bow-wave' effect - an update #86 diaw (Des Aubery) Guest   Posts: n/a Just a short note to say that today, I ran up a few different cases of a singularity inside a pipe - using a commercial FVM solver. The dimensions are different to the earlier case, & air (instead of water) is used as the fluid. Both incompressible & compressible (perfect gas ~ function of pressure) models were set up (very little observed difference seems to occur - details still to be thoroughly checked). No Reynolds scaling is used. Here, a structured rectangular mesh is used, with accomodation for the circular singularity. A much larger inlet velocity was used than before. The Reynolds number, based on duct width is Re=56.37 I have isolated the same 'bow phenomenon' as that observed in the graphic I posted previously. I'll clean things up a little during the next while & post the pics up for review - together with the test details. During the next few weeks, I'll write a conversion routine & export the data to a high-resolution mesh viewer so that the velocity gradients can be observed in better detail. Numerics again? Perhaps other folks could give similar cases a try & see what they come up with? It would be nice to compare notes. Based on the previous discussions, I wanted to be certain in my own mind that I had not made some sort of glaring error. It seems that some physical effect may just be at work. At this point the 'singularity index' is very close to Rs=1. diaw... (Des Aubery)

 February 12, 2006, 19:30 Re: Explanation for 'bow-wave' effect in simulatio #87 Adrin Gharakhani Guest   Posts: n/a This is too general a question. It depends on what you want out of a simulation, and whether it is realistic. For example, in a turbulent flow simulation you don't surely expect to resolve down to the last possible scale, do you? The same is true in your case. If it so happens that an initial order of magnitude analysis leads to an asymptotic cancellation of some terms, then why not do that, realizing the limits of the underlying assumption? You're basically saying that below/above a certain value you don't know what happens. Does it matter? I don't know. But in a global sense, I don't think so. You need to use algorithms that are computationally stable - sometimes just a change in the order of implementation leads to stable vs. unstable algorithms! (see for example, the interesting book by N. J. Higham "accuracy and stability of numerical algorithms", which is about general (not necessarily CFD) algorithms, but it clarifies my point). Beyond this, I don't I can give a solid answer Adrin Gharakhani

 February 13, 2006, 08:00 Re: Concept & idea consolidation #91 Tom Guest   Posts: n/a "I read a paper last year where a team of Dutch Mathematicians had isolated a wave mechanism for flow in a tube - its velocity was related to (1/Re) (New Scientist, if my memory serves me correctly. Article available on-line.) I agree with their (1/Re) observation in the context of my wave research - for 1d-dominated flows." Are you talking about the experiment or the theory? The experiment showed that, as the Reynolds number R increased, that the amplitude of the disturbance required to cause transition scaled as 1/R; i.e. the higher the Reynolds number the smaller the disturbance needed. In the theory (the numerical nonlinear solutions) the "waves" so far found have all been, to my understanding, unstable and so will not be observed as final states in a numerical solution. I think I can see your problem with scaling - You need to sit down and perform a few nondimensionalizations of simple problems in order to be more comfortable with what you're doing (I don't think you can explain all the details of it easily in a forum such as this - I'd need a blackboard). In the case of you're problem you have two length scales - the radius of the pipe and that of your "singularity" (a true singularity has no size). You also have two velocity scales, namely that of the inlet and that induced by the singularity. When you nondimensionalize your equations, using for example the radius of the pipe and the maximum inlet velocity, you will find that your problem contains a number of dimensionless parameters - namely the Reynolds number, the ratio of the singularity size to that of the radius of the pipe and a parameter descrbing the relative strength of the singularity to the inflow velocity. Geomtetric similarity requires that all these parameters remain fixed under a change in the apparatus/fluid and not just the Reynolds number! Solutions at the same value of R will be different for different choices of the other parameters.

 February 13, 2006, 11:17 Re: Concept & idea consolidation #93 Tom Guest   Posts: n/a diaw: I'm sorry Tom, are you refering to the maths paper, or to the Reynolds experiment? Could you clarify? It was an actual experiment identical to that of Reynolds where they tried to control the source/strength of the background noise. diaw: This would seem to be a reasonable finding, as, over time, the dispersion terms would force the wave amplitudes to decay. In my work, I found the decay term to be of the form exp(-K^2*t) where, K refers to the group wave-number (2d & 3d). This would appear to show that large wave-number components drop out of the solution more rapidly than small wave-number terms. In the steady limit, all waveforms should decay to zero amplitude - under the proviso that no additional disturbance enters from the boundary during this decay. Because these "wave-like" solutions are actual full nonlinear solutions you can't think about them in this way - you don't have superposition in nonlinear problems. This means that these "waves" cannot coexist. diaw: The problem is that, as I see it, in real life, no experiment can ever truly be free from boundary disturbances. The slightest perturbation in the entry, or exit flows will trigger off another round of transient wave-forms. This would tend to maintain a wave presence, even though theoretically they should not be there. Yes - the main point is that, for example, transients introduced at the inlet decay as you move along the pipe. The interesting problem is why, in the pipe case, the disturbances grow when linear-stability theory suggests that they should decay? diaw: Obviously, artificial equation-induced wave-forms & numeric-perturbations will add their own influences. This was effectively my point about not trying to look at contour differences below or close to the numerical error. diaw: When I observe typical pictures of the Reynolds experiment, I cannot help but notice the ever-so-slight wavy die trace in the first portion of the tube. It basically follows a low-amplitude sinusoidal path. As the flow moves towards the transition zone, it begins to exhibit larger-amplitude wave activity. The presence of these patterns are, to my mind at least, indicators of wave activity in the flow-space. Calling this "wave-activity" is a dangerous (mis)use of terminology - "wave-activity" actually has a standard definition related to Whitham's concept of wave-action (as opposed to the so-called wave-energy). Another view of this would be the instability creating vorticity anomally which causes a wrapping up of the streamlines. I think Rayleigh instability is explained in this way in Lin's book on hydrodynamic stability (although my memory might be playing tricks on me). diaw: Thanks for those insights. You are correct in terms of the two length-scales & two-velocity scales beign active. I will do exactly as you suggest, & sit down with a few simple cases & work my way up to the pipe/'singularity' scaling - to try & settle this issue in my mind. It's worth thinking about this - the existence of more than one nondimensional parameter is often a problem when scaling up a lab experiment. A good, rather extreme, example of this is flow over a bump in a stratified fluid where you have two controlling parameters (the Reynolds number and the Froude number). If you try to scale this up to the atmosphere with the same Froude number then you will find that the Reynolds number is not the same as in the experiment. It's almost impossible to do the experiment in the same flow regime that the atmosphere sits.

 February 13, 2006, 12:16 Re: Concept & idea consolidation #94 diaw (Des Aubery) Guest   Posts: n/a diaw: I'm sorry Tom, are you refering to the maths paper, or to the Reynolds experiment? Could you clarify? Tom: It was an actual experiment identical to that of Reynolds where they tried to control the source/strength of the background noise. diaw: Thanks for clarifying the experimental work - very interesting result. I had mentioned a paper published (last year?) by a Dutch Mathematical team regarding wave solutions in the N-S for pipe-flow. (I'll do a search of my library & post the link). ------- diaw: This would seem to be a reasonable finding, as, over time, the dispersion terms would force the wave amplitudes to decay. In my work, I found the decay term to be of the form exp(-K^2*t) where, K refers to the group wave-number (2d & 3d). This would appear to show that large wave-number components drop out of the solution more rapidly than small wave-number terms. In the steady limit, all waveforms should decay to zero amplitude - under the proviso that no additional disturbance enters from the boundary during this decay. Tom: Because these "wave-like" solutions are actual full nonlinear solutions you can't think about them in this way - you don't have superposition in nonlinear problems. This means that these "waves" cannot coexist. diaw: The problem is that in nature, these wave-bulk flow effects happily coexist. Take a look at flows in rivers, into water tanks - basically anywhere where fluids are flowing. Perturb the fluid & wave evidence will show itself, then gradually disperse itself. In many cases, this wave phenomenon persists for as long as the fluid is flowing. Thus, flowing fluid induces wave field which in turn affects flow field & we cycle from there. What I have built conceptually to test out the dual field concept is a set of equations derived from the mean+deviatoric velocity components - for u,m+u', v,m+v', P.m + P'. Obviously these split reasonably neatly - except for the non-linear terms. The non-linear velocity inter-linking equations are maintained & not simplified - it would be vital to the velocity-linking. I plan to code up a test solver & see where this goes. The motive mechanism is the common pressure field => bulk + deviatoric. Concept is that the locally-modified pressure field than influences the particle trajectory. -------- diaw: The problem is that, as I see it, in real life, no experiment can ever truly be free from boundary disturbances. The slightest perturbation in the entry, or exit flows will trigger off another round of transient wave-forms. This would tend to maintain a wave presence, even though theoretically they should not be there. Tom: Yes - the main point is that, for example, transients introduced at the inlet decay as you move along the pipe. The interesting problem is why, in the pipe case, the disturbances grow when linear-stability theory suggests that they should decay? diaw: The explanation of a coexistent wave-field allows for pressure-field modification due to wave activity. The wave field possesses energy & 'pressure'. This deviatoric pressure will cause a local momentum transfer to the fluid particle & cause its path to deviate. (See my concept earlier in this post). Sometimes, waves set themselves up in certain 'standing wave' structures. Nodes & anti-nodes will position themselves at various places in the flow domain. The influence on the bulk flow is perceived to be via a pressure-coupling mechanism. I hope that I have stated the co-existent wave concept clearly. This flows naturally from the dual-nature form of the N-S. Des Aubery...

 February 13, 2006, 13:18 Linear & non-linear wave forms #95 diaw (Des Aubery) Guest   Posts: n/a diaw: This would seem to be a reasonable finding, as, over time, the dispersion terms would force the wave amplitudes to decay. In my work, I found the decay term to be of the form exp(-K^2*t) where, K refers to the group wave-number (2d & 3d). This would appear to show that large wave-number components drop out of the solution more rapidly than small wave-number terms. In the steady limit, all waveforms should decay to zero amplitude - under the proviso that no additional disturbance enters from the boundary during this decay. Tom: Because these "wave-like" solutions are actual full nonlinear solutions you can't think about them in this way - you don't have superposition in nonlinear problems. This means that these "waves" cannot coexist. diaw: If I may clarify a little further on the wave solution I was mentioning. The N-S lends itself naturally to the non-linear form, when derived directly from the N-S in terms of mean+deviatoric components. A slightly simplified form, for initial investigation can be constructed directly from an interpretation of the substantial derivative. (2d) D()/Dt = lim{ d()/dt + d()/dx.^x/^t + d()/dy.^y/^t } where : all derivative are partials ^x, ^y, ^t are delta-x, delta-y, delta-t In wave terminology define: ^x/^t = c,x = phase velocity in x-direction = k/w ^y/^t = c,y = phase velocity in y-direction = l/w ending up with, D()/Dt = d()/dt + c,x.d()/dx + c,y.d()/dy Plug this back in with the usual terms in N-S, interpret the dispersion terms in wave terminology & we are well on our wave to a 'linear' form of the N-S wave equations. (Actually I can have also investigated the non-linear solution, but the inertial wave-terms were non-physical in that they grew exponentially). This allows some insights into the workings of the wave-form. Rather like the concept of not considering the effect of the 'u' in the u.du/dx portion as not affecting the nature of the N-S (hyperbolic, parabolic, elliptic). In fact, coupled with the conceptual model I outlined in the previous post, would make the start of a pretty decent turbulence model, when linked to the bulk field. Des Aubery...

 February 13, 2006, 22:52 Errata: Linear & non-linear wave forms #96 diaw (Des Aubery) Guest   Posts: n/a Section should read: ------- In wave terminology define: ^x/^t = c,x = phase velocity in x-direction = **w/k** ^y/^t = c,y = phase velocity in y-direction = **w/l** ending up with, D()/Dt = d()/dt + c,x.d()/dx + c,y.d()/dy ------ Des Aubery

 February 14, 2006, 06:44 Re: Concept & idea consolidation #97 Tom Guest   Posts: n/a diaw: The problem is that in nature, these wave-bulk flow effects happily coexist. Take a look at flows in rivers, into water tanks - basically anywhere where fluids are flowing. Perturb the fluid & wave evidence will show itself, then gradually disperse itself. In many cases, this wave phenomenon persists for as long as the fluid is flowing. Thus, flowing fluid induces wave field which in turn affects flow field & we cycle from there. In the examples you give this is indeed so - all waves can be traced back to the linearized problem through a small parameter expansion. In the pipe flow problem this is not the case for the nonlinear finite amplitude solutions you have mentioned - these solutions are isolated and so there is always a "finite distance" between the solutions. This makes it impossible for the solutions to coexist as a superposition - the best you can hope for is for one solution to morph into another as time evolves. What you are talking about is transients and not waves. As a simple counter example consider the flow of an inviscid parallel flow in a 2D pipe which has no inflection point and so is stable by Rayleighs theorem. If you perturb the initial state locally what happens - if you look for a solution of the linearized problem you will not find any waves since there are no normal modes in the linear problem (the transients arise from the brach cut in the inversion of the Laplace transorm).

 February 14, 2006, 07:01 Re: Linear & non-linear wave forms #98 Tom Guest   Posts: n/a "Plug this back in with the usual terms in N-S, interpret the dispersion terms in wave terminology & we are well on our wave to a 'linear' form of the N-S wave equations. (Actually I can have also investigated the non-linear solution, but the inertial wave-terms were non-physical in that they grew exponentially)." There is a serious flaw to this "linear form of the NS equations" argument. Firstly what you really mean is that you are interested in the Lagrangian form of the equations and not the usual Eulerian one. In the Lagrangian form the all the nonlinear terms drop out of the Material derivative and so this operator becomes linear. This however comes at the cost of making the linear continuity equation horrendously nonlinear (not to mention viscous terms). Smoothness of the mapping between the Eulerian and Lagrangian frames in the nonlinear problem are seriously problematic. Actually an interesting property of the inviscid Lagrangian equations is that the pressure should be interpreted as the Lagrange multiplier of the mass conservation constraint in the minimization of the kinetic energy.

 February 14, 2006, 07:36 Re: Concept & idea consolidation #99 diaw (Des Aubery) Guest   Posts: n/a Greetings Tom. Thanks again for your input. diaw: The problem is that in nature, these wave-bulk flow effects happily coexist. Take a look at flows in rivers, into water tanks - basically anywhere where fluids are flowing. Perturb the fluid & wave evidence will show itself, then gradually disperse itself. In many cases, this wave phenomenon persists for as long as the fluid is flowing. Thus, flowing fluid induces wave field which in turn affects flow field & we cycle from there. Tom: In the examples you give this is indeed so - all waves can be traced back to the linearized problem through a small parameter expansion. In the pipe flow problem this is not the case for the nonlinear finite amplitude solutions you have mentioned - these solutions are isolated and so there is always a "finite distance" between the solutions. This makes it impossible for the solutions to coexist as a superposition - the best you can hope for is for one solution to morph into another as time evolves. diaw: I'm trying to understand why, in the pipe case, the solutions are isolated. This can certainly be said for traditional development of say gravity waves, where the perturbation is introduced, 3 sets of equations are developed - one for bulk flow, one for deviatoric & the final - for the non-linear linkages. The approach then makes simplifying assumptions which essentially reduce the non-linear linkage equation to zero - leaving two independent solutions operating in the same space - or drops the bulk solution completely - as in the case of gravity waves. I am proposing that no simplifying assumptions are made regarding the interlink equations & thus the two solutions are not independent. To maintain the transient wave solution, suitable physical boundary conditions would have to be applied eg. inlet noise, otherwise it would probably decay to zero - unless flow-generated perturbations exist - flow ?noise?. Tom: What you are talking about is transients and not waves. As a simple counter example consider the flow of an inviscid parallel flow in a 2D pipe which has no inflection point and so is stable by Rayleighs theorem. If you perturb the initial state locally what happens - if you look for a solution of the linearized problem you will not find any waves since there are no normal modes in the linear problem (the transients arise from the brach cut in the inversion of the Laplace transorm). diaw: Granted, I am certainly referring to transient wave-forms in the context of a transient wave-field. given a consistent boundary disturbance, would there be no possibility of non-linear 'normal modes' forming? I think that perhaps the difference in viewpoints may be one of considering a reasonably consistent source of input 'noise' as a boundary-condition, or of flow-generated noise/perturbations. I would love to know if sound measurements of the onset of instability have been conducted - I'm sure they have. Does flow in a pipe make a noise? In some cases, it certainly can. -------- If I may also introduce a slightly different perspective at this point - at the bulk-flow level: The typical velocity profile in a tube - has a velocity component in the x-direction. Take the velocity profile, mirror it through 180' about vertical axis, then mirror 180' about horizontal axis. What shape do you see? In which direction does the velocity move, or oscillate? Is this a familiar shape? The typical flow 'shape' seen in the downstream flow from a rear-facing-step. What shape is it? Is there a relationship between the centreline-distance between re-circulation zones located top & bottom of pipe? 'Push' flow into a tube. Reach a certain velocity & then tell me what shape we observe in the pipe. (Refer to Bejan's observations.) What happens when we 'pull' flow out of a pipe - by setting a slight suction on the outlet boundary? (This one can be rather fun to observe, although some solvers will definitely not want to oblige.) What would a pressure-wave look like in a pipe? All of these observations can be related to a 'velocity wave' interpretation of the N-S - at bulk flow level. Would these shapes persist of die out - as long a the fluid is flowing? At this point, we still have the standard substantial derivative in place - which is used in standard wave derivations - no tricks... Des Aubery...

 February 14, 2006, 08:45 Re: Linear & non-linear wave forms #100 diaw (Des Aubery) Guest   Posts: n/a Tom: Actually an interesting property of the inviscid Lagrangian equations is that the pressure should be interpreted as the Lagrange multiplier of the mass conservation constraint in the minimization of the kinetic energy. diaw: I think thatI've seen this concept used for chaos in cubic springs. I worked it back to find the pressure, as you say. It makes sense, as pressure is a potential. ------- diaw: "Plug this back in with the usual terms in N-S, interpret the dispersion terms in wave terminology & we are well on our wave to a 'linear' form of the N-S wave equations. (Actually I can have also investigated the non-linear solution, but the inertial wave-terms were non-physical in that they grew exponentially)." Tom: There is a serious flaw to this "linear form of the NS equations" argument. Firstly what you really mean is that you are interested in the Lagrangian form of the equations and not the usual Eulerian one. In the Lagrangian form the all the nonlinear terms drop out of the Material derivative and so this operator becomes linear. This however comes at the cost of making the linear continuity equation horrendously nonlinear (not to mention viscous terms). Smoothness of the mapping between the Eulerian and Lagrangian frames in the nonlinear problem are seriously problematic. diaw: I'm not sure I follow your logic. I never considered the Lagrangian form at this point. In principle, can you explain why the 'linear' form would be any different to the way the non-linear terms are treated in discrete calculation steps? In this sense (roughly), the c,x merely becomes u' held over from the previous time step. It was initially meant merely to provide a feel for how the solutions would develop for the transient wave-forms & to provide some means of comparison to general wave-type solutions. In the discrete sense - & in the way we compute - it seems to make sense. (I maintain a physics viewpoint). Practically, I would simulate the full non-linear forms, rather than waste too much time on the linear forms. Considering how many variety of turbulence models exist, though, I think that this approach could turn out, in end, to be of some use. Des Aubery.

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