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

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...

After the excellent inputs from both Tom & Adrin, amongst others, I would like to try to consolidate the core questions still remaining in my mind at this point & leave them with you. Some will be simple, but, I expect that many of these will not have direct, clear-cut answers. (Room for further research).

Numeric wave vs real wave activity: ----------------------------------- Does numeric wave activity tell us something about the physics of the fluid itself & its ability to sustain sub-scale waves? Is this purely due to the incompressible assumption - or due to the 'spring nature' of fluids themselves (refer chaos concepts - cubic springs, with the fluid now being the spring)? How to distinguish physical reality from numeric fiction?

If 'numeric fluids' can sustain 'numeric waves' within a common flow domain, what does it tell us about the ability of 'real fluids' to sustain 'real waves' in the flow domain? In other words, if nature is able to allow both bulk-flows & wave-fields to happily co-exist, & the N-S equations predict that this cannot happen & our numeric cfd simulations, can at best, support numeric waves - what does this say about the validity of the N-S equations themselves to adequately model the full spectrum from low-speed viscous flows, right through the high-speed inviscid flows? Do the N-S perhaps require a re-think with a few new additional terms? How do we fill the apparent no-man's land between low-speed, viscious incompressible flows & inviscid high-speed flows? What major numerical, &/or physical breakthroughs do we require to take us where we need to go?

Scaling & Reynolds: ------------------- What is the physical explanation for Reynolds experiment? What causes the fluid particle to move?

Scaling (conceptually): Reynolds scaling - long, slender tube - u velocity at entry - which dimension to scale on? Gradually reduce tube length - which scaling dimension applies? Further reduce to square - which dimension applies? Further reduce, until height exceeds length - which dimension to scale on? Further reduce, until height evry large, & length very small - which dimension to use?

On what basis did Reynolds develop his scaling rule?

Now repeat thought process, but with both u & v inlet velocities. Would there be any changes?

Scaling (complex geometries): Lets move to a complex flow geometry eg. automotive radiator. Essentially an inclined plate problem - with an array of some 16-30 deflectors per fin, with multiple fin rows. (example link <www.adtherm.com> no plug intended - go to Gallery section). Fin pitch (Fp) ~ 1 - 1.5 mm (distance between fin rows), louver pitch (Lp) ~ 0.8 - 1.2 mm, louver angle (La) ~ variable 18 - 39 degrees. Flow gap between fin rows a function of Fp, Lp, La. Reynolds scaling on Fp can be low laminar - on Lp can be much larger, but this does not truly represent the physics of the flow domain since the 2 dominant flow areas are: 1. that between fin-tips (duct flow) & 2. between fin louvers (louver-directed flow).

Will scaling valid for one flow region cause problems with the other flow region? Can we develop a scaling more appropriate to both the flow areas, or for the domain as a whole?

Cyclic (?periodic?) boundary conditions: ---------------------------------------- Effects of cyclic boundary conditions, versus 'brute force' approach (many fin rows ~ 26 - 50)? Effects of cyclic boundaries on numeric waves?

----------

I would like to thank everyone who participated both actively & passively in this debate. Special thanks go both Tom both & Adrin for their wisdom & depth of experience. It is indeed a rare privilege to have such respected input. I have personally learned a tremendous amount from this debate & it has helped to place a number of burning issues squarely in focus, in my mind, at least. Some issues will probably remain open for many years to come. I have a year's worth of reading to do...

Again, thanks so much.

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

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...

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)

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

>> Does numeric wave activity tell us something about the physics of the fluid itself & its ability to sustain sub-scale waves? Is this purely due to the incompressible assumption.

If you have an accurate and stable code - to the accuracy and precision of the simulation (and the model assumptions) numerics do replicate physics. One word of caution: more often than not we have problems with finding/applying correct boundary conditions. In compressible flow problems, the pressure boundary condition has been a major source of problem and the waves have been seen to go back and forth in the computational domain. The solver is faithfully solving the problem for _that_ particular boundary condition, which in this case is not necessarily physical. There has been a lot of work on non-reflecting/radiation boundary conditions to force the waves out of the computational domain (I haven't worked with compressible flow for a while, so I don't know what the current status is)

I am, however, concerned with the notion "purely due to the incompressible assumption", which you've often brought up in this discussion. Based on what you've mentioned, you're actually using a compressible flow code but tweaking a parameter to emulate incompressibility. Therefore, strictly speaking you're not assuming incompressible flow - the algorithm is still based on a compressible flow formulation. If you'd started from a purely incompressible flow formulation, _then_ your claim would be correct. In your case, you're "driving" the solution toward Ma~0 limit but it will never be Ma=0 and you will see background (noise level) waves as you do. (add to that bad boundary conditions and I can see how you can see numerical waves where none should exist)

>>Reynolds scaling - long, slender tube - u velocity at entry - which dimension to scale on? Gradually reduce tube length - which scaling dimension applies?

It "doesn't matter" which scaling you use, so long as you know the scaling parameters and are able to backtrack information. One error I see, often in papers is "Reynolds number is ...". This is meaningless. It should be "Reynolds number based on U and L scales is ...". Now you can choose L to be the diameter, the radius, the channel length, U to be the max velocity, the mean velocity, etc. etc. Scaling is used to allow you to make an order of magnitude analysis to examine which terms are important relative to others, and to make relative comparisons of performance when you make parametric changes (at least, that's what I think it's useful for). Having said that, obviously you want to scale the problem in a way that makes the most physical sense (so that you can make sound judgements). If the flow is characteristically dominant in the y direction compared to x, then you want to normalize in the y direction. If you have a sinusoidal inlet flow, then perhaps rather than length you want to focus on the frequency of oscillation (or perhaps both). It's useful to read up on similitude and dimensional analysis.

>>Flow gap between fin rows a function of Fp, Lp, La. Reynolds scaling on Fp can be low laminar - on Lp can be much larger,

This is an indication of the misunderstanding of scaling and its role in the analysis. As I mentioned above, you should always indicate the scaling parameter. By just changing the scaling parameters, obviously you don't expect the physics to change - therefore, what must change is the bar/criterion for when a flow is laminar or turbulent. For example, flow in a channel is laminar for Re_H<2000. It is very important to recognize that this is true ONLY when flow is normalized based on the channel height and mean velocity (and the limit was determined via experimental observation). However, change the Re to make it based on the shear stress effects and all of a sudden the flow is turbulent at much lower Reynolds numbers, but that's only a matter of the choice of scaling (if you were to write this new Re_tau based on H and U you'd still end up with Re_H>2000, so everything is self-consistent)

>Will scaling valid for one flow region cause problems with the other flow region? Can we develop a scaling more appropriate to both the flow areas, or for the domain as a whole?

Once again, scaling does not change the flow physics. It's like saying "this ruler is 1 foot or 12 inches". The bottom line/physics is the same. Computatioinally you cannot scale different regions of the flow differently - unless you take care to communiate this difference between the two regimes at the boundary of the two regimes (but why create problems). You cannot hope/expect to convert a turbulent flow into a laminar, or compressible into incompressible by just playing around with scales - if that's what you mean by your question. See above for more details.

Adrin Gharakhani

Hi Adrin,

Thank you for your excellent comments. Your comments regarding scaling selection of suitable scales for U & L are very useful. This seems to have been a difficult area for engineers.

From my research, I have become aware that the essential issue seems to be one of maintaining 'correct timing' between the flow physics & the boundary conditions. If the scaling results in a change to the 'timing' in a particular direction that is say different to that in another direction, then distortions can result. The 'timings' are all linked by the full-form scaling rule extracted from N-S. The terms of the 'scaling rule' are in terms of (1/time). Reynolds only used the concept in 1D-dominated flows. When flows are modelled, then there are dimensionless geometric scaling terms which should be applied. This basically links the Reynolds scaling to the physical scaling. When 2D flows are considered, then there are geometric modifiers linking the u & v velocity terms. As long as everything is selected correctly, then the 2d scaling will be correct, if not, then geometric distortions are predicted to distort the velocity scaling. To keep the physics happy, then, we must ensure that the scaling maintains the 'timing' of the physics.

Clear 'information waves' can be seen, for instance, during the solution of the solver in question, by observing the velocity-magnitude vector field, for instance. This seems to be the movement of the field correction information as the solver begins to settle the field values. Clear examples of grid vibration can also be observed.

Bow-wave effect: In terms of solvers, can I state, for the record, that the solver originally mentioned - with accompanying pictures, was only one of a group of solvers used. I have applied a steady Galerkin-type FEM & FVM commercial code (steady & transient).

To place the 'bow-wave effect' in perspective, this is a formation seen at a particular velocity, in a slender flow domain containing a singularity. If the singularity is too small then the effect is not so obvious. A lot has to do with the visual-effect of the velocity information we see & how we interpret it. The effect is seen under both steady & unsteady solvers.

The effect seems to be caused by the interaction between the walls, singularity & fluid. When the wall effect is moved further out, or singularity dimension reduced, then the bow-effect does not show up too clearly. From my research, this effect seems to show up slightly after what appears to be the 'onset of instability' in the flow field (as evidenced by the start of flow reversal behind the singularity - moved from potential-type flow pattern into flow reversal). Any further increase of flow velocity begins to move the velocity 'lobes' forward until eventually the forward bow-effect is observed - upon the convergence of 4 velocity 'lobes'. A further increase in flow velocity further alters the velocity 'lobe' positions & the forward 'nose' of the bow-effect begins to detach & move slightly upstream.

I have altered flow scales, fluids & solvers & the same 'pattern' persists. I would say, that this effect could, & should, emerge with almost any solver (in the example shown, the fact that 1024 resolution had been used, merely helped to show up the contrast over the velocity gradients & create a neat visual effect).

What it seems to boil down to, is that a natural pattern emerges in the flow, which looks remarkably like the high-speed bow wave - except for leading nose. This information is shown up in a velocity-magnitude plot. The information showign in the pressure plot seems not to be in-phase with the velocity magnitude information.

I believe that once we understand this bow-effect & its link to the onset of instability, that we may begin to obtain a more clear understanding of the physical processes occuring in the flow-field just beyond the onset of intability. I am convinced that with this knowledge, that we will be able to understand the turbulence phenomenon in a more physical light.

Across the bow-wave effect, the velocity magnitude values change reasonably abruptly - but, it should be said, not nearly as abruptly as for high-speed flows - since we are operating at very small velocity values.

It would be interesting to observe what effects other folks observe in similar simulations. The effect will be there, but you will need to work up to the 'sweet spot' event.

------

Thanks again Adrin for your excellent comments. They are very much appreciated.

Des Aubery...

diaw:
:> Does numeric wave activity tell us something about the physics of the fluid itself & its ability to sustain sub-scale waves? Is this purely due to the incompressible assumption.

Adrin:

If you have an accurate and stable code - to the accuracy and precision of the simulation (and the model assumptions) numerics do replicate physics. One word of caution: more often than not we have problems with finding/applying correct boundary conditions. In compressible flow problems, the pressure boundary condition has been a major source of problem and the waves have been seen to go back and forth in the computational domain. The solver is faithfully solving the problem for _that_ particular boundary condition, which in this case is not necessarily physical. There has been a lot of work on non-reflecting/radiation boundary conditions to force the waves out of the computational domain (I haven't worked with compressible flow for a while, so I don't know what the current status is)

diaw:

I would agree entirely with the pressure boundary condition creating wave-effects & that these may, or may not be physical in origin. Let's investigate what happens in nature. Take a wind-instrument - closing, or opening the far end will affect the response of the instrument. The phase (sign) of the returning pressure wave will change according to the selected boundary condition. When this returning pressure wave encounters the entry, it is forced to return with positive sign. Various standing waves will also be set up within the body of the instrument... musical notes. So, yes, this will apply to both incompressible viscous fields & compressible inviscid fields. The problem with inviscid flows is that there is no way to damp out the wave in time & it will persist. If a small amount of viscosity (as in natural flows within the atmosphere) is present, then I would expect many of the the high wave-number pressure components to damp out, but, this could still leave persistent lower wave-number components.

Let's now go back to Reynold's experiment. He set up stable inlet flow conditions, & then opened an exit_valve. What physical effect does this exit_valve exert on the flow? Is it not a physical back-pressure mechanism? When a slight back-pressure is applied (simulating the valve), interesting effects are observed. Are these physical? Why not? The correct choice of boundary conditions that reflect the physics is very important.

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.

---------

Adrin:

I am, however, concerned with the notion "purely due to the incompressible assumption", which you've often brought up in this discussion. Based on what you've mentioned, you're actually using a compressible flow code but tweaking a parameter to emulate incompressibility. Therefore, strictly speaking you're not assuming incompressible flow - the algorithm is still based on a compressible flow formulation. If you'd started from a purely incompressible flow formulation, _then_ your claim would be correct. In your case, you're "driving" the solution toward Ma~0 limit but it will never be Ma=0 and you will see background (noise level) waves as you do. (add to that bad boundary conditions and I can see how you can see numerical waves where none should exist)

diaw:

I do agree with you totally with respect to the issues raised by that particular solver. The artificial compressibility assumption will most certainly bring in some spurious effects. The other concern I have is using Taylor expansion in time for u & v velocities (Taylor-Galerkin). The time terms would surely bring in a few odd effects of their own?

In my other post, I did state that the particular solver mentioned was only one of a group of solvers used. It had a neat graphics frontend which allowed very high-resoultion plots of the flow fields. The high-contrast regions is what first drew my attention to a few effects. Some of this could very well be attributable to the solver itself - to be sure, but, the general effect is observed in other solvers. (See other message).

diaw:

>>Reynolds scaling - long, slender tube - u velocity at entry - which dimension to scale on? Gradually reduce tube length - which scaling dimension applies?

Adrin:

It "doesn't matter" which scaling you use, so long as you know the scaling parameters and are able to backtrack information. One error I see, often in papers is "Reynolds number is ...". This is meaningless. It should be "Reynolds number based on U and L scales is ...". Now you can choose L to be the diameter, the radius, the channel length, U to be the max velocity, the mean velocity, etc. etc. Scaling is used to allow you to make an order of magnitude analysis to examine which terms are important relative to others, and to make relative comparisons of performance when you make parametric changes (at least, that's what I think it's useful for). Having said that, obviously you want to scale the problem in a way that makes the most physical sense (so that you can make sound judgements). If the flow is characteristically dominant in the y direction compared to x, then you want to normalize in the y direction. If you have a sinusoidal inlet flow, then perhaps rather than length you want to focus on the frequency of oscillation (or perhaps both). It's useful to read up on similitude and dimensional analysis.

diaw:

I have made a few comments regarding the importance of maintaining the correct 'timing' in the previous posting. I believe that these considerations are particularly important when more than one flow direction is active, or where no clear flow dominance exists.

diaw:

>>Flow gap between fin rows a function of Fp, Lp, La. Reynolds scaling on Fp can be low laminar - on Lp can be much larger,

This is an indication of the misunderstanding of scaling and its role in the analysis. As I mentioned above, you should always indicate the scaling parameter. By just changing the scaling parameters, obviously you don't expect the physics to change - therefore, what must change is the bar/criterion for when a flow is laminar or turbulent. For example, flow in a channel is laminar for Re_H<2000. It is very important to recognize that this is true ONLY when flow is normalized based on the channel height and mean velocity (and the limit was determined via experimental observation). However, change the Re to make it based on the shear stress effects and all of a sudden the flow is turbulent at much lower Reynolds numbers, but that's only a matter of the choice of scaling (if you were to write this new Re_tau based on H and U you'd still end up with Re_H>2000, so everything is self-consistent)

diaw:

You are correctly refering to the qualification of the Reynolds scaling relative to the dominant U, L scales. I agree wholeheartedly. I raised the issue, becasue, I too, have heard the same things & always ask the question "What Reynolds? or, Reynolds relative to which dimensions?" In many cases, the correct scales are perfectly obvious, but in others, things can get a little more tricky. I agree with your comments.

diaw:

>Will scaling valid for one flow region cause problems with the other flow region? Can we develop a scaling more appropriate to both the flow areas, or for the domain as a whole?

Adrin:

Once again, scaling does not change the flow physics. It's like saying "this ruler is 1 foot or 12 inches". The bottom line/physics is the same. Computatioinally you cannot scale different regions of the flow differently - unless you take care to communiate this difference between the two regimes at the boundary of the two regimes (but why create problems). You cannot hope/expect to convert a turbulent flow into a laminar, or compressible into incompressible by just playing around with scales - if that's what you mean by your question. See above for more details.

diaw:

The confusion creeps in when scaling is performed using density & viscosity, in addition to L, U scaling - in the name of maintaining 'dynamic scaling'. Engineers are generally taught to scale as follows: (1) Geometric scaling - ie. prototype dimensions must scale to physical dimensions; then (2) Dynamic scaling - via the Reynolds number. If Reynolds number is maintained, then the physics will all scale correctly. In other words, selecting alternative test fluid properties to maintain 'dynamic similarity' gives correct answers. This thought process carries over to flow simulations & can create some very _odd_ effects.

I have found the follwing scaling philosophy to be useful: 1. Scale on geometry; 2. Scale on flow 'timing'; 3. Scale on physical properties (via Re) - as a last resort.

If (2) is not maintained, then strange effects emerge. I have not often seen this aspect emphasised. Oddly-enough, if you leave out (2), then you can still scale on properties, with different local values, but same Re, but the simulation will not be in the correct time scale & will miss the dynamic effects of the flow.

In my research work I try to move away from geometric_scaling & velocity_scaling completely & simulate the problem in its entirety. I would prefer to adjust large & small terms (scale) within the solver itself in order to control numeric over-under-shoot, round-off etc - to me, this makes sense.

Question: Can we use higher-resolution numerics than double-precision? Does say quad-precision exist? Are math libraries around for such things? I would rather try this approach & take the pain on numeric computation time, but minimise numeric round-off issues.

Thanks so much for your very kind & wise comments. This is a very, very interesting field & we need to get it right.

Des Aubery

"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.

Greetings Tom, wonderful to have your input. Thanks very much.

diaw:

"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."

Tom:

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.

diaw:

I'm sorry Tom, are you refering to the maths paper, or to the Reynolds experiment? Could you clarify?

Tom:

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.

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.

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.

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.

Their presence & structure can also be well-explained in terms of the wave-nature of the N-S equations. The dispersive terms each represent a particular wave-primitive in terms of type (direction of particle vibration) & direction of wave motion. The observations are in line with wave descriptions for solid & gaseous work in the excellent reference by Pain H.J. - "The Physics of Vibrations & Waves", Wiley, 6 ed. The main trick is to recognise the different primitives.

In numeric solvers, we induce disturbances through a number of sources - the inlet velocity is seen as an impulse by any wavefield (if present); the techniques used to bring in the results of each non-linear loop calculation onto the u, v, w & p solution fields - can excite the wave-nature - in my view, that's why under-relaxation helps a little in some tough cases. Obviously, artificial equation-induced wave-forms & numeric-perturbations will add their own influences.

Another issue is that of 'grid vibration'. With the correct set of wave boundary conditions, the computational grid can be made to vibrate in an amazing way - using a time-soft fluids solver. It appears that this grid vibration is also active during N-S simulations, although often to a lesser degree.

Tom:

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.

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.

Once again, thanks so much for your deep insights - they are very much appreciated. (Would you believe that this debate is now currently close to 100 pages?)

Des Aubery...

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.

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...

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...

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

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).

"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.

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...

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.