Is it theoretically possible that shock-waves ('bumps') - perhaps of extended spatial dimension - could exist in viscous, low-speed fluid flows?

In other words - reasonably abrupt changes of a fluid property - specifically pressure, or velocity - in the flow domain.

Does anyone have experience with 'solution wave phenomena' in flow simulations of viscous, low-speed fluid flows? Are these purely simulation abberations, or is there a physical explanation?

I am looking for some solid academic references to either totally disprove, or potentially support/explain this phenomenon. I would really appreciate candid constructive comments around these issues. Thanks so much.

diaw...

The basic answer is no if you mean classical shock waves - solutions at low Mach number should be relatively smooth.

On the otherhand if the Reynolds number is relatively high you can obtain regions of high gradients internal to the flow; e.g. detached shear layers - in the limit of infinite Reynolds number this type of solution is a "weak-solution" to the Euler equations. These Kirchhoff type solutions are discusssed in Batchelor's book on fluid mechanics and further elucidated in "Asymptotic theory of separated flow" by Sychev et al. - these "discontinuities" are not waves though!

You should probably look at the two books by Milne-Thompson "Theoretical hydrodynamics" and "Theoretical aerodynamics".

Continuum Mechanics by Peter Chadwick - this is a general introduction to the foundations of solid and fluid mechanics and discusses the various types of "wave of discontinuity".

and

High speed flow by C. J. Chapman.

Hope this helps, Tom

I would say yes, shock waves can also exist in low-speed fluid flows. Take a look at the water hammer phenomenon--you have both an abrupt change of pressure and velocity. You may want to take a look at Wylie and Streeter, "Fluid transients in systems", just to mention one. Indeed, this kind of shock wave has been already pretty long researched. Hope I could help you with this comment!

Thanks very much Tom for your extremely constructive contributions. I'll try & get hold of the books you cited.

diaw...

Thanks Philipp for that angle.

The water-hammer phenomenon seems to occur in many of my pipe-flow simulations, even at very low entry velocities - using both FVM & FEM numerical techniques.

----------- My issue is that, in certain simulations I have obtained what looks like sharp gradients - pressure, or velocity. The similarity to the observed high-speed phenomenon is so striking that it cannot be purely coincidental. My classic is flow within a tube, over a cylindrical obstacle.

A lot has to do with how the simulation data is displayed - certain modes can shown 'hidden details' that are not seen at first glance.

All academics that I have shown my work to instantly fall back to the position of 'numeric abberation'. This is frustrating, as I have, during the past few years perform many thousands of simulations - with differnt teams, in different countries - some results better than others. I have 'chased' this phenomenon from a theoretical basis, through Matlab 1D simulations, eventually onto detailed 2D & 3D simulations.

Sometimes it is like banging my head against a wall, so I'm back into a deep research phase to get to the bottom of this.

Thanks for your contribution.

diaw...

Hello Diaw The correct parameter to use is the Mach Number. And when the temperature is low, you get this number greater than 1 When a flow with a Mach greater than one is deflected you have to expect Shock waves. You can check this by computing the entropy generation in your flow field, most commercial CFD programmes can do this calculation. For your information I met this type of phenomenun analyzing steam turbines exits and micro channel flows.

Two comments. First, are you running a compressible code or purely incompressible code? That should be a starting point in making sure that your observed phenomenon is not due to numerics. Even "wrong" numerics do tend to tell you certain things about the flow if you're able to make sense of the "wrong" results.

Second, my own first reaction was that it could be due to numerics (but then I don't know the details of the flow). The flow could be slow but you could still have high Mach (just as with the watter hammer example). BTW, you don't need too high a Mach to observe your phenomenon. That's why I was asking whether you're solving a compressible flow problem (you should not be seeing this effect with an incompressible flow code). Having said all this, I empathize with your situation - I have had a similar situation where my simulation results for a shock tube experiment were waaaaay off from experimental data. Everyone blamed numerics, and it took me a while to get to the bottom of it and to prove that the experimental data was wrong (and why). So, perhaps you've stumbled onto something quite interesting and you should think about the physics more methodically and carefully. Obviously, we can't be of much help beyond this...

Adrin Gharakhani

Hi Adrin,

Thanks so much for your wise comments. I'll explain my current position below - without giving away too much fine detail over public communication media...

The flows are all incompressible, at very, very low velocities. No scaling is applied whatsoever (important). Most of the simulation has been for water at the correct physical density & viscosity values as tabulated in published literature. Air does also show similar effects.

Theoretical background: BSc Physics was my base degree... I have re-derived the N-S equations from a kinematic perspective to provide a deeper understanding of each term. The 'dual-character' of the N-S has been exposed from a theoretical perspective. The significance & character of each & every term in the N-S under the 'dual-nature' is explained. Full scaling rules for the N-S have been developed - for micro-to-macro scales. Reynolds number scaling can be simply explained by this scaling - & its range of applicability determined. The dual nature of N-S can also be explained by this work. The full meaning & implications of 'steady' flows has been explored. The deprecated solution form for the 'steady assumption' used incorrectly has been isolated - this explains solution divergence for this case mathematically. The Reynolds experiment has been simulated fully - interesting phenomena have been isolated, which explain his observations.

A theoretical model of potential numeric 'wave generation' has been explored. The role of the non-linear loop methodology in this phenomenon is explained.

Simulations: Various codes have been used: 1. FVM commercial code - in steady & transient modes; 2. FEM research code - CBS solver (time-soft) & Galerkin (steady).

For some flow geometries with obstructions, some solutions are impossible, as the solution diverges, no matter what mesh size is used. In others, mesh size needs to alter dramatically as certain 'mode changes' occur.

No convection stabilisation is used (for FVM, under-relaxation is used on u, v & p, where necessary).

Simulation 'snapshots' at various iterations are taken throughout the runs & the data analysed in fine detail. The emergence of certain sub-scale phenomena is very clearly seen in the plots. This is what I meant by 'solution waves'. ---------

Could you further explain the comment ?

"The flow could be slow but you could still have high Mach (just as with the watter hammer example). BTW, you don't need too high a Mach to observe your phenomenon."

---------

I have been extremely thorough in my theoretical development - with some thousands of pages of hand-written research notes. I have performed literally thousands of simulations probing & exploring my findings as deeply as possible. The 'dual-nature of N-S' demands certain boundary-condition settings in order for solution stability to be ensured. In mmany cases, the way B/C's are applied 'arouses' the 'dual nature' of the N-S in a way that causes problems.

----------- Personal note: I am deeply convinced of what I am 'seeing', but, the local academics do not seem able to get their minds around what I have found - sure, it's not easy at first - but, most choose to simply put up a barrier & try to re-focus me on trivial research.

Perhaps it is time to take my work to an alternative learning institution where I can tap into brains of a higher level? I believe that I have uncovered a 'paradigm change' of significance which will help us to bridge the vast theoretical gap between low-speed incompressible flows & high-speed flows.

So far, my project has been completely self-funded - but...

---------

So, there it is... I would value comments on the theoretical side... personal comments & interested senior research colleagues can contact me on my email address

diaw...

Hi Ahmed...

'micro-channel flows'... is exactly where I am working

Under what conditions could one experience Mach numbers greater than one, in low-speed flows in micro-channels?

If waves occur naturally in water at very low velocities (eg. in rivers & oceans) & at high-speed flows... what happens inbetween?

diaw...

Hello Diaw You need to a have a good thermodynamics book (Stay away of Engineering Thermodynamics books, they are intended to teach applications of thermodynamics not understanding the physical side of the subject) and search for the throttling process or the Joule-Kelvin (Thompson) effect.Basically, under certain pressure and temperature conditions, and in an isenthalpic process the flow cools down and as a result the speed of sound is reduced to the extent that the Mach number passes the sonic limit. As you see, even at moderate flow speeds you can get Mach numbers greater than 1 very easily. Now be aware, if that flow is not deflected by a solid wall, you do not get waves of any type, weak, expansion, Shock, etc. I have your email address, so expect one with some of the plots I obtained some time ago and more details about the thermodynamics of throttling, Cheers and Good Luck.

Thanks so much Ahmed... you have made some very interesting points there... that is intriguing. A thought, out loud... would the inclusion of the energy (temperature) equation be able to pick up some of the physics? I may have to add some source terms to bring out the correct physics... mmhh...

Your comment about the wall-deflection effect on showing waves is excellent. I've noticed a similar effect the moment I place a tiny singularity in the flow field... then the wave activity begins to show up...

I look forward to your plots.

Regards, diaw...

How do you calculate Mach number, or better how do you calculate speed of sound?

Angen

Angen: 1- If you are looking for the theory behind these calculations the following book is a good reference: Waves in Fluids By James Lighthill 2- If you are looking for the exact details then you have to contact ANSYS as I have used their programme (Flotran).

>>> Waves in Fluids By James Lighthill

Now we are going in the right direction... the 2nd 'nature' of the N-S

diaw...

In your incompressible problem there are only two ways in which you can obtain waves (both due to gravity - see the referenced book by Sir James Lighthill). The first way is if you have a free surface and the second is if you have density stratification. The first instance gives rise to the classic water waves problem while the second represents internal gravity waves (e.g. lee waves behind mountains).

A third possibility, which is of no relevance to you, are Rossby waves which occur in the beta-plane approximation of a rotating sphere (or the rotating sliced cylinder).

If you don't have one of these cases you don't have waves (the TS and Rayleigh wavs within boundary layers aren't actually waves in the above sense).

Tom commented:
:>>The first way is if you have a free surface and the second is if you have density stratification. The first instance gives rise to the classic water waves problem while the second represents internal gravity waves (e.g. lee waves behind mountains).

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diaw's comment:

Thanks very Tom much for engaging the debate...

I particularly like the 'free surface' example, & am very glad you brought it up. I have spent a while studying this phenomenon & engaging in a few mind experiments & simulations. I will try & paint a perspective that may not have been considered before - let's see.

Scenario: We have water flowing within a river channel, at a low speed. Suddenly a stranger comes along & inserts a stick vertically into the water flow field. The following occurs - the water flow adjusts itself to accomodate the singularity (stick) by creating waves on the free surace - both in front of & behind the stick. This is what we all know - nothing new here.

Now lets expand our thought process by moving downwards to a cut-plane part-way between the free surface & bottom of the stream. Does the pressure-field (& velocity field) have any relationship to the surface-wave field positioned vertically above it?

Well - lets again, for an instant, consider the peaks & troughs of the surface wave-field, positioned above our cut-plane. Let's draw vertical 'silo' divisions positioned at the original pre-singularity river level. We now have columns of different heights positioned above the mid-plane cut-plane we took earlier. The 'peak' columns have larger mass than the 'trough' columns & hence their 'static pressure' contribution at the cut-plane will alternate - high-low-high-low-... & so on. We can, by all means add in the velocity head ala Bernoulli.

We now sit with regions of varying pressure on our cut-plane - high-low-high-low... Let us now re-phrase matters a little... compression-rarefaction-compression-r-c-r-c...

Try a cfd computer simulation of the cut plane & take a look at the output. Use real water properties & realistic river flow-rates. See how well your solver converges. Do not use Reynolds scaling, but the real dimensions & properties.

Waves cannot maintain the peak-&-trough shape on the surface without exerting varying pressures down below them. Only forces cause water molecules to deviate in direction. Forces come from pressures... etc.

---------

There is an excellent French website for a researcher who studies texture changes of advecting surfaces. His work on flowing streams is refreshing indeed. He refers to 'shock waves' on simple river surface flows. A completely different perspective.

Thanks again Tom for your input.

diaw...

I would also like to offer a few thoughts on the concept of 'incompressibility'.

Let us take the Mass Conservation Equation & apply the condition of CONSTANT FLUID PROPERTIES (in 2d for now).

=> Continuity eqn becomes the simple form

partial(du/dx) + partial(dv/dy) = 0

or,

partial(du/dx) = - partial(dv/dy)

What is the implication of the last statement? Well, it simply states that if p(du/dx) increases, p(dv/dy) must decrease & vica versa. Basically a 'squishy deformable fluid cell'.

This too is a consequence of the simple view of the mass conservation equation:

D(dm)/Dt = 0

=> no mass source into/out of the 'squishy fluid cell' control mass.

In other words, a deformation of the fluid cell in one direction causes a deformation in the other directions to compensate for the unchanging fluid properties. This concept accomodates the possibility for compression-rarefaction to exist in 'constant fluid property' environments.

diaw...

The strict definition of incompressibility does not assume constant density - a flow can be incompressible but have variable a density (which is constant on a particle path).

The definition of incompressibility is the div(u)=0 which translates to a statement about the conservation of volume;i.e. volumes are preserved not mass (unless the density is constant). It is this fact that allows internal gravity waves to propagate.

This conservation of volume is effectively what you describe in your post. However there is no actual compression (when the fluid is squashed in one direction it expands in the other) and hence no possibility for a "ompression-rarefaction wave". The best/worst you could hope for is for the fluid element to be squashed to zero size in one direction => infinite velocity in the normal direction (this is what happens in the Goldstein and van Dommelen singularities of the boundary layer equations). Since most mathematicians, me included, believe that the NS equations are well-posed and so have regular solutions provided the initial data is sufficiently smooth this behaviour cannot occur.

"There is an excellent French website for a researcher who studies texture changes of advecting surfaces. His work on flowing streams is refreshing indeed. He refers to 'shock waves' on simple river surface flows. A completely different perspective."

He probably means caustics or hydraulic jumps (depending on what he's studying). Hydraulic jumps are very similar to shock waves (the shallow water equations are a special case of the 2D compressible Euler equations).

On the otherhand he may be talking about wave-breaking in which case the word shock is misleading/wrong - a surface gravity wave can overturn with no need to introduce a shock condition (the initial singularity is related to representation of the surface as y=f(x,t) instead of parametric form x=x(s,t), y=y(s,t) ).

Tom wrote: The strict definition of incompressibility does not assume constant density - a flow can be incompressible but have variable a density (which is constant on a particle path).

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diaw's reply:

Thanks Tom - excellent points indeed. Thank you very much. This is a critical concept in understanding of the physical phenomena we are dealing with. It is also very misunderstood by many folks. Let's explore the concept further.

Let me perhaps re-phrase the point/s I am trying to make. This is obviously open to interpretation.

The typical approach taken by, for instance Kays (Convection Heat & Mass Transfer, 4ed, pg 20/21) is one of 'constant-density flow', arriving at the form of the mass-conservation eqn I quoted. Kays refers to the two cases under which the time term disappears - incompressible, or constant density. He uses the latter to derive div(V)=0, then goes further to restrict to constant density.

Panton (Incompressible Flow, 3 ed, pg 72-) The continuity equation: "The time rate of change of mass of a material region is zero". dM,mr/dt = 0 (This view is also shared by various physicists => no mass source term).

If we extend Panton's view to the full form of the continuity equation, then apply the following additional restriction: density=constant (wrt x,y,t), after dropping the time-rate term & density variation in x,y , we arrive at

=> partial(du/dx) + partial(dv/dy) = 0 (as stated)

So, the restrictions are, in order of application: 1. dM,mr/dt = 0 (no mass source) 2. density=constant.

This translates to a deformable element, with mass conservation + constant density, does it not? No mention is made of 'constant volume'.

In addition, even if we elected to call it 'constant volume' - constant volume does not necessarily mean 'fixed shape in all directions' - or no?

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Tom wrote:

The definition of incompressibility is the div(u)=0 which translates to a statement about the conservation of volume;i.e. volumes are preserved not mass (unless the density is constant). It is this fact that allows internal gravity waves to propagate.

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diaw's reply:

I think that I have spoken to this concept in my first point. Volume conservation may not translate directly to 'shape conservation'.

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Tom wrote:

This conservation of volume is effectively what you describe in your post. However there is no actual compression (when the fluid is squashed in one direction it expands in the other) and hence no possibility for a "ompression-rarefaction wave". The best/worst you could hope for is for the fluid element to be squashed to zero size in one direction => infinite velocity in the normal direction (this is what happens in the Goldstein and van Dommelen singularities of the boundary layer equations). Since most mathematicians, me included, believe that the NS equations are well-posed and so have regular solutions provided the initial data is sufficiently smooth this behaviour cannot occur.

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diaw' reply:

In free space, there is no constraint to lateral cell deformation - but, within confined space, this is most definitely not the case, at all. This is a major departure point for an object positioned far away from any solid boundaries versus flow within a container. As an aside, this 'container effect' is what effectively can contribute to the second viscosity coefficient & bulk modulus. No container => no bulk modulus.

The edge containment stiffens the domain & affects the shape/deformation of the fluid cell - slightly. This results in the observed effects we see in tube computations at low speeds - the entry water-hammer effect. Close investigation will show compression-rarefaction effects in the region of water-hammer.

In terms of the smoothness, or otherwise of the N-S, this will certainly depend on the vantage-point. N-S can be shown to also accomodate a complete non-linear dispersion waveform. When the non-linearity is constrained to resolution in discrete steps a form results for which I have developed complete mathematical solutions - x, y, & p wave-forms. In fact, both bulk & deviatoric solutions happily co-exist in the same flow space...

I hope that I've answered adequately. This debate is very, very useful. Thanks Tom for your input.

diaw...