Can 'shock waves' occur in viscous fluid flows?
 

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...

Re: Can 'shock waves' occur in viscous fluid flows
 

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

Re: Can 'shock waves' occur in viscous fluid flows
 

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!
:)

Re: Can 'shock waves' occur in viscous fluid flows
 

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

diaw...

Re: Can 'shock waves' occur in viscous fluid flows
 

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...

Re: Can 'shock waves' occur in viscous fluid flows
 

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.

Re: Can 'shock waves' occur in viscous fluid 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

Re: Can 'shock waves' occur in viscous fluid flows
 

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...

Re: Can 'shock waves' occur in viscous fluid flows
 

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...

Re: Can 'shock waves' occur in viscous fluid flows
 

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.

Re: Can 'shock waves' occur in viscous fluid flows
 

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...

Re: Can 'shock waves' occur in viscous fluid flows
 

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

Angen

Re: Can 'shock waves' occur in viscous fluid flows
 

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).

Re: Can 'shock waves' occur in viscous fluid flows
 

>>> Waves in Fluids By James Lighthill

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

diaw...

Re: Can 'shock waves' occur in viscous fluid flows
 

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).

Re: Can 'shock waves' occur in viscous fluid flows
 

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).

-----------

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...

Re-phrase 'incompressibility'...
 

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...

Re: Re-phrase 'incompressibility'...
 

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.

Re: Can 'shock waves' occur in viscous fluid flows
 

"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) ).

Re: Re-phrase 'incompressibility'...
 

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).

------

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?

------------

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.

----------

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'.

----------

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.

------------

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...

Re: Can 'shock waves' occur in viscous fluid flows
 

Thanks again Tom,

A few thoughts.

What keeps the surface waves from equalising to the original surface level? What holds them up? Remember, waves are the communication evidence of molecular vibrations - not bulk movement.

What effect is the singularity (stick) introducing into the complete flow cross-section? What does the pressure field look like down the complete length of the stick?

When a boat starts moving, does the wave activity only occur on the surface, with no local pressure-field variation down the height of the bow? Could this be possible?

diaw...

Re: Can 'shock waves' occur in viscous fluid flows
 

"What keeps the surface waves from equalising to the original surface level? What holds them up? Remember, waves are the communication evidence of molecular vibrations - not bulk movement."

Gravity - when you deform the surface you change the potential energy of particles on the surface from there equilibrium "flat value". when the surface is then let to evolve this potential energy is converted into kinetic energy and the surface wobbles. The surface will actually return to the flat equilibrium, assuming no external forcing, after a sufficiently long period of time - through a mixture of diffusion and dispersion.

"What effect is the singularity (stick) introducing into the complete flow cross-section? What does the pressure field look like down the complete length of the stick?"

You can solve this problem analytically - it's worth doing since it will give you a better understanding of what's going on!

"When a boat starts moving, does the wave activity only occur on the surface, with no local pressure-field variation down the height of the bow? Could this be possible? "

The waves appear on the surface (hence the name surface gravity waves) and there is a resulting motion in the rest of the flow in order to accomadate these waves.

You should look at the book "Water waves: the mathematical theory with applications" by J.J. Stoker.

Re: Can 'shock waves' occur in viscous fluid flows
 

diaw: "What keeps the surface waves from equalising to the original surface level? What holds them up? Remember, waves are the communication evidence of molecular vibrations - not bulk movement."

Tom: Gravity - when you deform the surface you change the potential energy of particles on the surface from there equilibrium "flat value". when the surface is then let to evolve this potential energy is converted into kinetic energy and the surface wobbles. The surface will actually return to the flat equilibrium, assuming no external forcing, after a sufficiently long period of time - through a mixture of diffusion and dispersion.

diaw: How would the free-body diagram look - at various points through the flow field? We should still have an equilibrium, no? Where are the forces? How do these forces project down further in the flow field?

Since there are forces in play, let's take their effect all the way down to the plane I previously mentioned - since gravity is involved, we have weight - which changes in response to the surface changes. What does this planar pressure distribution look like?

Tom: "When a boat starts moving, does the wave activity only occur on the surface, with no local pressure-field variation down the height of the bow? Could this be possible? "

The waves appear on the surface (hence the name surface gravity waves) and there is a resulting motion in the rest of the flow in order to accomadate these waves

diaw: So, good, we have motion in the rest of the flow to accomodate the free-surface effect/wobble... Let's take a cut plane as I suggested - what does the pressure & velocity distribution look like?

Physics (N2 & N3) will not allow the surface to act in isolation. Its forces must be transferred down thoughout the fluid. In fact, I propose that this wobbly free surface merely provides a wobbly boundary condition to the flow field - with the lower edge being fixed (river floor).

diaw... :)

Re: Re-phrase 'incompressibility'...
 

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

The full continuity equation is (read rho for p) p_t + div(pu) = 0 which states that, for a sufficiently smooth flow, the rate of change of mass within a fixed volume is equal to the flux of mass entering/leaving said volume. We can write this equation as p_t + u.grad(p) = -p.div(u). If div(u) = 0 then the vector field u is volume preserving (the phase space for the particle paths have zero e-folding and so volumes in phase space are constant). This leaves p_t + u.grad(p)=0 which states that p must be constant on a particle path (or a streamline if d/dt=0). This fact says nothing about what shape a volume (a sphere say at t=0) will look like for t>0. It only says that its volume will be unchanged! In particular the volume could become horrendously deformed if the particle path equations are chaotic. Try simulating Kelvin-Helmholtz instability with a passive tracer to see what I mean.

"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."

There is a constraint to the deformation due to the fact that the velocity is finite - a particle can only move a finite distance in a giving time interval.

I don't know what your comment "No container => no bulk modulus" means - the bulk viscosity is the term div(u) in the stress tensor which is zero in an incompressible flow irrespective of whether it is an internal or external fluid problem.

Derivation of linear gravity waves
 

A further thought on gravity waves.

The derivation of the gravity waves begins with 3 equations. The first two are essentially constructed from the substantial (total) derivative, with relevant dp/dx, dp/dy on rhs.

The third equation is a pressure balance which allows 3 eqn, 3 variable closure. The linearised form makes various simplifying assumptions about the bulk & deviatoric fields. The proposed wave solution is then substituted & the appropriate terms developed from the equations.

Of interest: In essence, the left-side of the gravity wave equations, before simplification, look astoundingly similar to the N-S. They lack dispersion terms.

diaw...

Re: Re-phrase 'incompressibility'...
 

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

Tom: The full continuity equation is (read rho for p) p_t + div(pu) = 0 which states that, for a sufficiently smooth flow, the rate of change of mass within a fixed volume is equal to the flux of mass entering/leaving said volume.

diaw: Agree. In other words - 'no mass source'. In = out & no storage.

Tom: We can write this equation as p_t + u.grad(p) = -p.div(u).

diaw: Agree.

Tom: If div(u) = 0 then the vector field u is volume preserving (the phase space for the particle paths have zero e-folding and so volumes in phase space are constant).

diaw: Certainly. Since we basically have a density-volume trade-off. If density changes, then volume must shrink, or swell to accomodate its change. If density is fixed, then volume would, by intimation, remain fixed. (Sorry for no comment on e-folding - I'm a physicist :)

Tom: This leaves p_t + u.grad(p)=0 which states that p must be constant on a particle path (or a streamline if d/dt=0).

diaw: Under the assumption of fixed fluid properties - in space (& time), surely these terms of no use? grad(p)->0, p_t->0. (Time restraint not totally necessary as fixed-property fluids anyway).

Tom: This fact says nothing about what shape a volume (a sphere say at t=0) will look like for t>0. It only says that its volume will be unchanged! In particular the volume could become horrendously deformed if the particle path equations are chaotic. Try simulating Kelvin-Helmholtz instability with a passive tracer to see what I mean.

diaw: Agreed.

diaw: "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."

Tom: There is a constraint to the deformation due to the fact that the velocity is finite - a particle can only move a finite distance in a giving time interval.

diaw: Think about the first fluid cell, closest to the wall. The wall pushes on the cell, it pushes against its neighbours, & so the 'stiffness effect' propagates some distance into the fluid field. This logic also corresponds to Bejan's observations of how boundary proximity 'stiffens' to flow field & effects the Reynolds number at which flow instability is observed.

Tom: I don't know what your comment "No container => no bulk modulus" means - the bulk viscosity is the term div(u) in the stress tensor which is zero in an incompressible flow irrespective of whether it is an internal or external fluid problem.

diaw: The wall-effect I mentioned in the point above will hopefully clarify the thought process. In practice, a fluid only has a measurable bulk modulus if it is compressed within a container. If the sides of the container were able to deform slightly, then the measured bulk modulus would become lower. Think about the equivalent bulk-modulus concept used in piping systems. If there is no container, then the fluid would end up on the floor... :)

Another caveat, the units of fluid bulk modulus differ by a time term when compared to elastic wave speed derivations. Thus, the rate of compression is also important.

diaw...

Re: Can 'shock waves' occur in viscous fluid flows
 

"diaw: How would the free-body diagram look - at various points through the flow field? We should still have an equilibrium, no? Where are the forces? How do these forces project down further in the flow field?"

The force comes from the dynamic boundary condition - the surface pressure has been set to a constant value (or its jump has been specified if you have surface tension).

"Physics (N2 & N3) will not allow the surface to act in isolation. Its forces must be transferred down thoughout the fluid. In fact, I propose that this wobbly free surface merely provides a wobbly boundary condition to the flow field - with the lower edge being fixed (river floor)."

Take away gravity and you nolonger have a wobbly surface. The wobbly boundary condition is called the "kinematic boundary condition"; i.e. there is no flow through the free surface.

I'm not sure what you're trying to get at with the cut-plane argument.

You should really try to read a Mathematics fluids book such as George Batchelor, Horace Lamb or David Acheson rather than one written by a physicist - In the UK fluids is the domain of applied maths and not physics (just look at JFM).

Re: Can 'shock waves' occur in viscous fluid flows
 

Tom: You should really try to read a Mathematics fluids book such as George Batchelor, Horace Lamb or David Acheson rather than one written by a physicist - In the UK fluids is the domain of applied maths and not physics (just look at JFM).

diaw: Thanks again Tom, & for your extreme patience with my reasoning & approach. You are very knowledgable indeed.

In my humble (not always :) opinion, the fact that CFD has generally resided in the realm of Mathematics, is precisely what is missing in the mix. This is actually a point I have seen for some time.

Physicists understand & model nature. Physicists are trained to see patterns & symmetry in nature. Very often the equations for completely different physical events end up with similar equations. This says much about the problem at hand.

Mathematicians invent innovative tools to assist in the solution of the physical models. (Time to run, I guess :) Sometimes, unfortunately, mathematics alone is not enough for physical processes. I propose that most major new historical breakthroughs will have come from physical-engineeers, rather than mathematical-engineers.

I see this each & every day around me. I was trained thoroughly as a BSc Eng - with a solid physics foundation. For this I am eternally grateful. It had carried e through some 29 countries during my time as a consultant.

I presently live in a country far from my birthplace, which has the mathematical-engineering philosophy. It seems to have modeled itself predominantly after the USA. The unfortunate end result is that the engineers cannot reason adequately, or develop an inner-feel for basic things like static & dynamic equilibrium. They have no 'feel' for the physics. I have had to sit with them & basically re-train their logic processes. Without the physics 'feel', they are basically useless as practicing engineers.

I have applied my thought processes to the fluid & cfd field precisely because this has been my passion for many, many years. I have probably run many, many thousands of cfd simulations of various shapes & form. This has left many, many unanswered questions in my mind. This is precisely why I began the search for the answers & re-entered academic life.

Hopefully I have come a little closer to explaining nature as it was originally designed.

Thanks so much for your very kind debate.

diaw...

Re: Re-phrase 'incompressibility'...
 

diaw: Under the assumption of fixed fluid properties - in space (& time), surely these terms of no use? grad(p)->0, p_t->0. (Time restraint not totally necessary as fixed-property fluids anyway).

No - in the atmosphere internal waves propagate precisely through this mechanism - div(u) is essentially zero but the air density varies with height and so the equation p_t + u.grad(p) =0 is essential since the background stratification provides a restoring force for any induced vertical motion.

diaw: Think about the first fluid cell, closest to the wall. The wall pushes on the cell, it pushes against its neighbours, & so the 'stiffness effect' propagates some distance into the fluid field. This logic also corresponds to Bejan's observations of how boundary proximity 'stiffens' to flow field & effects the Reynolds number at which flow instability is observed.

Yes but this signal propagates at "infinite velocity" to every point within the domain in an incompressible flow. The instability within the boundary layer occurs because of the interaction of the inertial terms with those of the viscosity (which is also the source of the stiffness - see the book hydrodynamic stability be Drazin and Reid). The "bulk of the flow" just wants to flow along the wall while the viscosity tries to slow the flow down in order to satisfy the no-slip condition. When the Reynolds number is sufficiently high this competition results in an instability. The three and five layer structures of the upper and lower branches of the neutral stability curve show how this occurs. For example on the lower branch the instability appears to be confined to wall (near the critical level) but the induced pressure signal reaches the flow outside of the boundary layer which then drives the instability through a feedback loop.

Re: Re-phrase 'incompressibility'...
 

Sure would be interesting to see you two standing at one white board going back and forth on this.

Visiting lecture anyone?

And thanks for the civilized nature of the discourse!

Re: Re-phrase 'incompressibility'...
 

Jim_Park: Sure would be interesting to see you two standing at one white board going back and forth on this.

Visiting lecture anyone?

And thanks for the civilized nature of the discourse!

---------

diaw:

Thanks so much Jim. I hold Tom in extremely high regard. His depth in Mathematics is very refreshing.

Visiting lecture... I'd welcome the opportunity to debate this topic more fully. Any place, any time...

Re: Can 'shock waves' occur in viscous fluid flows
 

No Ahmed, I am not looking for the theory behind this calculations. I know Lightill's book. I know how to calculate speed of sound in different conditions. My question was pointing to the problem that you have to be very careful in interpretation of your results for speed of sound especially when you are talking about Joule-Thompson effect. It means that your are close to the so-called inversion line (i.e. line dT/dp=0). Regardless if your fluid is liquid or gas your speed of sound may be wrong by an order of magnitude if you are not using right equation of state. Reproducing inversion curve is considered to be one of the most severe test for equations of states. If you are saying that you used a formula that is coded in Flotran and do not know what it is I would check it very carefully before drawing any definitive conclusion.

Angen

Re: Re-phrase 'incompressibility'...
 

diaw: Under the assumption of fixed fluid properties - in space (& time), surely these terms of no use? grad(p)->0, p_t->0. (Time restraint not totally necessary as fixed-property fluids anyway).

Tom: No - in the atmosphere internal waves propagate precisely through this mechanism - div(u) is essentially zero but the air density varies with height and so the equation p_t + u.grad(p) =0 is essential since the background stratification provides a restoring force for any induced vertical motion.

diaw: An excellent point. I had been refering to fixed-property, constant density fluids.

For atmospheric waves, the p_t + u.grad(p) = rhs, would be essential to forming a wave solution. The 'rhs' term, I would imagine, would probably be in the form of (1/p).nabla(P) where P=pressure, p=density.

diaw: Think about the first fluid cell, closest to the wall. The wall pushes on the cell, it pushes against its neighbours, & so the 'stiffness effect' propagates some distance into the fluid field. This logic also corresponds to Bejan's observations of how boundary proximity 'stiffens' to flow field & effects the Reynolds number at which flow instability is observed.

Tom: Yes but this signal propagates at "infinite velocity" to every point within the domain in an incompressible flow.

diaw: No, I believe that the information flow speed is governed by the properties of the medium - most likely limited to velocity of sound (wip). The wave speed could never exceed a certain 'natural speed', governed by the ability of the fluid elements to vibrate fast enough to propagate the wave. It could never be infinite - not even light has infinite wave velocity.

Tom: The instability within the boundary layer occurs because of the interaction of the inertial terms with those of the viscosity (which is also the source of the stiffness - see the book hydrodynamic stability be Drazin and Reid).

diaw: I've got Drazin & Reid. I'll certainly work through your suggestion. Thank you very much.

Tom: (cont) The "bulk of the flow" just wants to flow along the wall while the viscosity tries to slow the flow down in order to satisfy the no-slip condition. When the Reynolds number is sufficiently high this competition results in an instability. The three and five layer structures of the upper and lower branches of the neutral stability curve show how this occurs. For example on the lower branch the instability appears to be confined to wall (near the critical level) but the induced pressure signal reaches the flow outside of the boundary layer which then drives the instability through a feedback loop.

diaw: Again, thanks for those insights. I'll work through Drazin & collect my thoughts on his understanding.

I understand that you seem to be referring to the wall-no-slip effect on the boundary-side of elements closest to the wall & the effect of the fluid flow velocity on the other face - shear & direct stress, pressure gradient balance. This force-balance (imbalance) situation will certainly alter as we move further away from the wall. Most definitely.

Most of my work has been more focused on the onset of instability in the free-stream itself. In simulations (FVM & FEM), the wall circulations 'communicate' with each other - strangely-enough - even across a flow channel. The communication mechanism is seen at sub-scale velocity dimensions & is intriguing, to say the least. In fact, flow over the well explored backward-facing-step produces some very interesting sub-scale 'patterns', with clear evidence of wave-field activity.

Even the well-worn cavity-flow problem has rendered some rather amazing sub-scale flow patterns. Vortices communicating with vortices, wave field patterns - very reminiscent of electric & magnetic fields - with the vortices (singularities) acting as the 'poles'. Opposite-rotation vortices show attractive fields, same-rotation vortices show repulsive fields. It sheds a totally new light on the mechanisms at work.

The net effect is a duality which seems to interact & re-inforce itself - a bulk (particle) & vibration (wave) field. Both seem to work simultaneously in the same flow-space. The fact that this duality is reflected in the dual-nature of the N-S, is not, in my opinion, by chance.

I am constantly reminded of Maxwell's work in electromagnetic wave theory. The exception here is that N-S dispersive wave-forms decay in time - with a decay rate dependent on the wave-number squared. A little more complex that Maxwell's rather simple waveforms. :)

I have found the dividing line between the dual-natures to be my old friend - 'the N-S singularity/ies' (sure, 'him' again :)- in fact, a whole locus of them. This concept has ended up with a complete scaling law suitable for both the N-S natures. The ultimate & symmetric elegance of this scaling is amazing, to say the least.

diaw...

Re: Re-phrase 'incompressibility'...
 

Tom: The instability within the boundary layer occurs because of the interaction of the inertial terms with those of the viscosity (which is also the source of the stiffness - see the book hydrodynamic stability be Drazin and Reid). The "bulk of the flow" just wants to flow along the wall while the viscosity tries to slow the flow down in order to satisfy the no-slip condition. When the Reynolds number is sufficiently high this competition results in an instability. The three and five layer structures of the upper and lower branches of the neutral stability curve show how this occurs. For example on the lower branch the instability appears to be confined to wall (near the critical level) but the induced pressure signal reaches the flow outside of the boundary layer which then drives the instability through a feedback loop.

diaw:
:>>I understand that you seem to be referring to the wall-no-slip effect on the boundary-side of elements closest to the wall & the effect of the fluid flow velocity on the other face - shear & direct stress, pressure gradient balance. This force-balance (imbalance) situation will certainly alter as we move further away from the wall. Most definitely.

diaw: (cont) A kinematics approach (need a white-board here :) - Newton's 2nd law - results in a free-body force diagram for the fluid element as follows: (consider x-direction as an example)

1. Motive forces - act in direction of motion => dV.(-dP/dx) + p.dV.ax where: p= density; P=pressure; dV=element volume; ax=externally-applied acceleration acting at centre-of-mass

2. Retarding forces - act in direction opposing motion - direct stress & shear stress components * p.dV (to provide forces) (These have negative signs as per standard derivation - Anderson etc). I will denote these as minus 'elasto-viscous forces'.

3. Equilibrium (inertial forces) - acting in direction opposite to direction of motion (D'Alembert's principle) => (partials) [du/dt + u.du/dx + v.du/dy] * p.dV

Equation layout: Inertial forces + Retarding forces = Motive forces

or, Inertial forces - elasto-viscous forces = externally-applied forces

Singularity/ies occur when the lhs (system equation) goes to zero under any combination of circumstances. This is the basis of what Reynolds tried to develop with his form of:

inertia forces/viscous forces

When the full singularity concept is applied, then an elegant scaling results. It encompasses the special case of the Reynolds scaling, & allows it to be placed in the correct perspective. Reynolds scaling as defined works when a slender flow path is considered, or where the distance to lateral boundary is large. In other geometries, a dimensionless scaling factor emerges from the scaling rule.

This is why I am a little concerned to tie arguments regarding stability too tightly to the Reynolds-scaling. In simulation work, I noted that the scaling did not always work, in practice & did often result in scaling distortions.

I prefer to instead define a 'singularity index' which is 1 on the singularity locus.

diaw...

Re: Can 'shock waves' occur in viscous fluid flows
 

Angen: First of all I would like to thank you for your constructive comments.

Secondly I also wish to tell you or any reader of this comment that I have no business relationship of any kind with ANSYS, that is to say I am not promoting the programme in any way, explicitly or implicitly, I just mentioned the programme I used when I conducted that analysis. But honesty abides, the programme is one that I like to use whenever it is possible and for varies technical reasons (This is not the main reason of this discussion).

I have no doubt in my mind that any one conducting a compressible flow analysis using the ideal gas equation of state knows how to calculate the speed of sound and the Mach number, and I mean any one including the minds behind the ANSYS programme, nevertheless, I have emailed the technical support about your comments and hope one of these days you will see their reply on this forum.

Now to your comments: I am one analyist convinced that a physical analysis of a problem is an essential part of the solution, I mean running a CFD programme is not a big deal, but understanding the physics really needs a lot of hard work and study. In the case we are talking about, Diaw asks about the possibility of Shock waves being generated in normal velocity situations, and if we look to the physics we know that any supersonic flow that is perturbed (deflected even by the presence of a boundary layer) will generate this type of singularity. Now a supersonic flow has a Mach number greater than one, and as you know the Mach number is just the result of dividing the flow speed by the speed of sound. The speed of sound is a function of some physical properties of the medium and among them the temperature. Thermodynamics teach us that a throttling process (an isenthalpic process) leads to a drop in temperature and that drop in temperature decreases the speed of sound and hence increases the Mach number calculated for these low temperature flows, and it is possible to reach the sonic limit and beyond. The Joule-Kelvin effect has a lot of applications in real life, just to mention one, look at the thermodynamics of a refrigerator, a common kitchen appliance. If in the analysis of a micro channel I see that there is a severe drop in temperature I have to ask myself Why? and here comes the Joule-Kelvin effect to explain why. If you like to see my plots drop me an email and sure I will send some of these plots in return.

Diaw and Tom:

An interesting discussion, thank you both, but I have got the feeling that something is missing, here it is: Incompressible Flow is an Engineering design hypothesis. It is not based on any physical law, Engineers as you know have to come up with numerical answers to design problems and in the case of fluid flow problems they observed that a change in density values less than 1% can be neglected,...etc. In the real physical world Compressibility is always there to a different degrees yes, but it is part of matter. That is to say we cannot neglect its effects. Cheers and Good Luck

Re: Can 'shock waves' occur in viscous fluid flows
 

Ahmed:

An interesting discussion, thank you both, but I have got the feeling that something is missing, here it is: Incompressible Flow is an Engineering design hypothesis. It is not based on any physical law, Engineers as you know have to come up with numerical answers to design problems and in the case of fluid flow problems they observed that a change in density values less than 1% can be neglected,...etc. In the real physical world Compressibility is always there to a different degrees yes, but it is part of matter. That is to say we cannot neglect its effects. Cheers and Good Luck

diaw:

Thanks for your perspective, Ahmed.

The 'incompressible' concept is surely a mathematical abstraction, but it is useful in the extreme limit for flows. If we can fully understand the phenomena at work in this simplified flow field, then we have a reasonable departure point to launch into the slight-compressibility issues. If waves can exist in this form, then, surely they can exist in slightly-compressible flows.

Slight-compressibility (pressure-related) brings with it additional wave-effects, in addition to those in the 'constant fluid property' field. These effects become further complex when temperature-induced effects are introduced. We now have a temperature-field, in addition to a pressure-field - with velocity, pressure & temperature interactions.

To look into the full form of the Energy Equation, with its family of singularities, is an exercise that I have begun working on. These equation structures contain forms reminiscent of solitons - but in energy & I propose that these may, in fact, possibly offer the link between the wave & particle nature of light. To get to this point, the non-linear terms (the ones we usually discard) need to be unfolded into forms more easily identified with waves - heady stuff... :)

The form of the Energy Equation we use in most solvers really seems to be only a 'temperature equation'.

We have a long, long way to go before we truly expose all the secrets of nature, but the 'Conservation Equations' are an excellent starting point.

diaw... (Des_Aubery)

Re: Re-phrase 'incompressibility'...
 

This has been one of the most interesting/refreshing set of discussions on this list in quite a long time.

I have a bit of a problem with the issue of observed differences when simulating NS using its dimensional form as compared to its nondimensional/scaled form. Of course, there are various ways to scale the same equation, but I fail to see how nondimensionalization, which only has the effect of "normalizing" the equation could possibly produce results different from its dimensional version (in an infinite precision sense) - other than the possibility of numerical noise (and all sorts of associated perturbations in the simulation) in the dimensional form of the simulation. Afterall, the simulations are finite precision and it's possible that the dimensional simulation may be operating in a range with reduced level of accuracy. The normalized version scales pretty much everything back to O(1). You should see the same physics in both cases!

Adrin Gharakhani

Re: Can 'shock waves' occur in viscous fluid flows
 

Diaw 1- Your mail box is full, please try to delete, or better, use an excellent mail programme, I have been using the Mozilla Thunderbird programme for some time now, it is free, you can download it from www.mozilla.org 2- Very hard to me to accept what you are mentioning, just one example, put Beta (The compressibility factor) equal to zero in the equation for determinig the inversion temperature (The equation that is below fig 5.6) the inversion temperature is equal to infinity !!! i.e. there is no inversion at all, that is to say the whole Joule-Kelvin observations are just fantasies. For design purposes the incompressible assumption is fine, but to understand the physics we need to count on the real behaviour of real fluids. Remember, the mathematics that you are talking about came well after the engineers have developed the assumption of incompressible flow

Re: Re-phrase 'incompressibility'...
 

Adrin Gharakhani:

I have a bit of a problem with the issue of observed differences when simulating NS using its dimensional form as compared to its nondimensional/scaled form. Of course, there are various ways to scale the same equation, but I fail to see how nondimensionalization, which only has the effect of "normalizing" the equation could possibly produce results different from its dimensional version (in an infinite precision sense) - other than the possibility of numerical noise (and all sorts of associated perturbations in the simulation) in the dimensional form of the simulation. Afterall, the simulations are finite precision and it's possible that the dimensional simulation may be operating in a range with reduced level of accuracy. The normalized version scales pretty much everything back to O(1). You should see the same physics in both cases!

diaw:

Greetings Adrin - your kind contribution is very much appreciated.

If the 'full-scaling' form of the N-S is used - based on the 'singularity index', & not the Reynolds case alone, we end up essentially with a 'frequency' scaling relationship. Upon further manipulation of this relationship, the 'singularity index' (for 1d dominated flow field) ends up in the form of:

Rs = (1/Re)*((dx/dy)^2+1)

It is this additional dimensionless term that essentially puts the Reynolds number in its correct scale.

If a 2d case is considered, then the (1/Re) term is modified to include the 'v' velocity, with its associated geometric multiplier. The ((dx/dy)^2+1) term still exists.

What is the magic number of Rs=1?

It represents the singularity condition, which under the kinematic viewpoint represents the rupture of equilibrium. As we cross this point, the du/dt term must become active to maintain equilibrium, or the motive force term must be modified to restore equilibrium.

Our previous perspective has forced us to not cross the 'singularity line'. Thus we are most careful to stay on the 'correct side' of this singularity line. The moment we cross into that domain, then we have a very active wave field.

Of interest, is that under this viewpoint, for a square domain, the critical Reynolds number reduces to... 2!!!

The 'singularity index' rules are simple. Rs>1 is a dispersion-dominated region in the scaling plane. Rs<1 is a 'wave-dominated' region.

I have found these scaling rules to become important when trying to cross the singularity line. Suddenly, tiny elements with large time-step can cause solution divergence - the elements have to be enlarged dramatically & time-step decreased orders of magnitude. This represents a 'mode change' in the flow physics. Physically, it represents the 'touching' of two - what looks like - shock lines. At that point, a tiny, tiny change in inlet flow velocity causes the solution to simply diverge - no matter how small you make the elements. Just across the mode change ends up with a beautiful shock pattern (flow over a cylinder contained within a tube). It looks very much like the high-speed shock pattern, but has a leading 'nose' - representing information flow upstream.

Of interest is that some steady solvers can take you slightly across the 'singularity line' - often based on inclusion of dP/dx on the lhs - giving some 'springiness'- but predictably blow up within a small velocity increase.

diaw...

Re: Can 'shock waves' occur in viscous fluid flows
 

Hi Ahmed,

1. Let me try & sort out my mailbox... it does funny things sometimes - something to do with lots of incoming mail :)

2. In terms of incompressible fluids, you are surely correct in the strict sense, but please don't say that too close to some of our Mathematical colleagues - you may need to find the escape quickly.. :)

diaw... (Des Aubery)