diaw: (previous note)

>>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.

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diaw: (new)

A few additinal points of clarification.

The first form (1D) mentioned considers the position on the 'singularity line' itself - at the 'steady limiting velocity'. An example of this case would be Reynold's pipe-flow experiment. The (dx/dy) ratio is dominant & the 'v' component is ~ zero. (Where dx reads scaling dimension in x-direction, dy ~ in y-direction). Reynolds observed that flow responded to scaling on the dy dimension.

The second form basically introduces a geometric scaling into the 'u' + (scale)*'v' term of the '(1/Re)' term, in addition to the ((dx/dy)^2+1) geometric scaling. Under these conditions, simple 1D Reynolds scaling can become a problem.

When the full N-S scaling is applied - including time term - then some interesting effects emerge. The equation can be further manipulated into a symmetric shape which responds to 'u', 'v', 'dx', 'dy' & 'dt' scales. The edge of this shape corresponds to 'singularity index' of Rs=1 - all very convenient. This scaling gives insights to both large & small-scale efects & applies across both natures of the N-S.

This concept also then allows a simple concept of the 'shape' of the N-S than the conventional 'elliptic', 'parabolic' & 'hyperbolic' explanations offered in many texts. These never did rest too well with me, as many seems to skip over terms in the b^2-4ac view.

This concept also explains how the modification of time-scales can aid solution convergence, by moving the 'singularity line' into a more favourable position relative to the element dimension - for numeric simulations.

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I hope that some of these thoughts & findings can contribute towards a paradigm shift in our view of the flow of fluids. I have personally found these concepts to be of use & trust that they can be of benefit to others.

diaw...

This is a bit long

"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."

For the rhs write p = P(z) + q where P(z) is the background mean density which is a function of height only. the density equation then becomes q_t + u.grad(q) = -wP' ('=d/dz). It's the P' and the buoyancy that gives the waves (this is in Lighthills book if I recall)

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.

This is one of the problems with the incompressible limit (I believe Lighthill alludes to this in his chapters in Rosenheads "Laminar boundary layer theory"). Basically pressure signals propagate at the speed of sound in a real (compressible) fluid. In the incompressible limit the speed of sound tends to infinity - hence the infinite speed of signal propagation (the Poisson equation for pressure is elliptic). As an example consider two tanks of fluid connected by a long pipe and filled with water (or more precisely a fictious incompressible fluid). Now assume there is a plunger in one tank to push the water down (i.e. to try to compress it) with the other tank open to the atmosphere so that the water level can be measured. What happens when you push the plunger down (all walls are rigid and non deformable)? The constraint div(u)=0 requires that the volume of fluid is constant (this constraint must be upheld pointwise) so if the plunger is pushed down (reducing the volume of the tank) the fluid in the second tank must rise instantaneously to accomadate the change in volume. This is true no matter how long the pipe joining the tanks is; i.e. there is an infinite speed of signal propagation or "action at a distance" as Newton would have put it.

"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."

vortcity w=curl(u), div(u)=0 are related to the equations for electro-magnetic fields - div(u)=0 being often referred to as the "solenoidal condition" gives a hint at this. The introduction of vorticity as an auxillary quantity then allows the velocity to be obtained from the Biot-Savat law if w is known (see vortex dynamics by P. G. Saffman for example).

"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."

The communication is through the pressure - or, following the electro-magnetism argument, the Biot-Savat law (action at a distance). The waves you mention are probably the "Kelvin-Helmholtz" type instability of the detached shear layer; i.e. the flow has a local inflection point and so a high-frequency/wavenumber analysis (based upon the Reynolds number) results in Rayleigh instability (sometimes Rayleigh waves or vortex/wave interaction - the word wave is rather unfortunate here).

I'm not too sure what you're getting at with this "new" scaling law (I pretty much agree with Adrin's comment below) - the Reynolds number is nothing more than the pi-theorem (discussed at length in "Hydrodynamics: A study in logic,fact and similitude" by Garrett Birkhoff). I would have thought all the other exact symmetry properties of the NS equations were known (e.g. Applications of Lie Groups to Differential Equations by Peter Olver). All the other scalings I know about arise from approximations at high-Reynolds number, such as triple-deck theory, which involves expansions in various powers and logs of the Reynolds number.

Hope this clarifies a few things (even if it's a bit long), Tom

Thanks so much Tom for your very kind contribution. Your insights & experience are very much appreciated. You certainly are a true goldmine of knowledge.

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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: This is one of the problems with the incompressible limit (I believe Lighthill alludes to this in his chapters in Rosenheads "Laminar boundary layer theory"). Basically pressure signals propagate at the speed of sound in a real (compressible) fluid. In the incompressible limit the speed of sound tends to infinity - hence the infinite speed of signal propagation (the Poisson equation for pressure is elliptic). As an example consider two tanks of fluid connected by a long pipe and filled with water (or more precisely a fictious incompressible fluid). Now assume there is a plunger in one tank to push the water down (i.e. to try to compress it) with the other tank open to the atmosphere so that the water level can be measured. What happens when you push the plunger down (all walls are rigid and non deformable)? The constraint div(u)=0 requires that the volume of fluid is constant (this constraint must be upheld pointwise) so if the plunger is pushed down (reducing the volume of the tank) the fluid in the second tank must rise instantaneously to accomadate the change in volume. This is true no matter how long the pipe joining the tanks is; i.e. there is an infinite speed of signal propagation or "action at a distance" as Newton would have put it.

diaw: Let's replace the 'incompressible liquid' by a truly incompressible object ie. a solid. If a disturbance is introduced on one end - what happens? Does the information travel at infinite velocity?

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diaw: "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."

Tom: vortcity w=curl(u), div(u)=0 are related to the equations for electro-magnetic fields - div(u)=0 being often referred to as the "solenoidal condition" gives a hint at this. The introduction of vorticity as an auxillary quantity then allows the velocity to be obtained from the Biot-Savat law if w is known (see vortex dynamics by P. G. Saffman for example).

diaw: Thanks for that insight. Excellent.

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diaw: "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."

Tom: The communication is through the pressure - or, following the electro-magnetism argument, the Biot-Savat law (action at a distance).

diaw: I would most definitely agree with the pressure-wave connection. In terms of the N-S wave nature, in a particular direction (say x-direction), longitudinal pressure waveforms are predicted in addition to a transverse waveform. This arises directly from the dispersion terms. These waves are theoretically time-decaying & tend to lose the high-frequency components first.

The waveform that I find rather interesting is the one that gives rise to the sinusoid often observed in long ducts, or say downstream of a backward-facing step. Bejan has a go at the origin of this, but it ends up being a lot simpler than all that. It amounts to the transverse waveform for the y-direction - wave speed moving in the x-direction - with y-velocity component.

Tom: The waves you mention are probably the "Kelvin-Helmholtz" type instability of the detached shear layer; i.e. the flow has a local inflection point and so a high-frequency/wavenumber analysis (based upon the Reynolds number) results in Rayleigh instability (sometimes Rayleigh waves or vortex/wave interaction - the word wave is rather unfortunate here).

diaw: Tom, from what I have been observing in my research, I would say that you are almost certainly 'bang on the money' - in terms of relationships to shear layers. The local inflection points are clearly seen. I see this effect for instance at the rear section of a backward-facing step. There exists a sub-scale velocity field - usually smothered by the bulk-flow vectors, which comes up nicely when cut to the correct scale. Fantastic.

Tom: I'm not too sure what you're getting at with this "new" scaling law (I pretty much agree with Adrin's comment below) - the Reynolds number is nothing more than the pi-theorem (discussed at length in "Hydrodynamics: A study in logic,fact and similitude" by Garrett Birkhoff). I would have thought all the other exact symmetry properties of the NS equations were known (e.g. Applications of Lie Groups to Differential Equations by Peter Olver). All the other scalings I know about arise from approximations at high-Reynolds number, such as triple-deck theory, which involves expansions in various powers and logs of the Reynolds number.

diaw: Thanks for those thoughts. I'll read up your sources & report back. I see the pi-theorem as being more of a 1D affair. I think that my approach may turn out to be little different - in terms of scaling the complete N-S, but let's see what it turns out like.

Tom: Hope this clarifies a few things (even if it's a bit long),

diaw: Thank you so much for your deep insights & kind sharing of your time. You have indeed given me an immense amount of food for thought.

diaw: Let's replace the 'incompressible liquid' by a truly incompressible object ie. a solid. If a disturbance is introduced on one end - what happens? Does the information travel at infinite velocity?

Yes (or at least some of it does - the bit that requires the volume to remain fixed).

diaw: Tom, from what I have been observing in my research, I would say that you are almost certainly 'bang on the money' - in terms of relationships to shear layers. The local inflection points are clearly seen. I see this effect for instance at the rear section of a backward-facing step. There exists a sub-scale velocity field - usually smothered by the bulk-flow vectors, which comes up nicely when cut to the correct scale. Fantastic.

Although not strictly about dettached shear layers you could have a look at the paper(s)

"The evolution of free-disturbances in a two-dimensional nonlinear critical layer" in Eur. J. of Mech. B/Fluids vol 23 No.6.

and/or

"An Initial-Value Problem for Fully Three-Dimensional Inflectional Boundary Layer flows" in Theor. and Comp. Fluid dyn. vol 12 No. 3.

for a perspective on this idea - these are the more readable of the "rigourous" papers on this subject (I'm using the words rigourous and readable rather loosely here

These papers show how small scale structures are imbedded and interact with a larger scale flow through the Rayleigh instability mechanism.

Your interest in scaling reminded me of the book "Similarity, Self-Similarity, and Intermediate Asymptotics" by G.I. Barenblatt (he's also written a new(er) one called "Scaling" which I haven't read) which discusses dimensional analysis and how it can be used to solve various types of physical problems.

diaw: Let's replace the 'incompressible liquid' by a truly incompressible object ie. a solid. If a disturbance is introduced on one end - what happens? Does the information travel at infinite velocity?

Tom: Yes (or at least some of it does - the bit that requires the volume to remain fixed).

diaw: So, then, this would surely have to be a non-physical mechanism?

I'm aluding to elastic waves in solids eg. earthquakes, which travel at sometimes rather modest wave-speeds. By which mechanism would the infinite speed information travel? I can envisage surface waves & so forth, but if the information is transfered by molecular vibration, then it could not physically be infinite. We know that even light is constrained to electro-magnetic wavespeeds.

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For your additonal information & references on waves & scaling - thank so much for your kindness.

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Further thoughts on 'shock waves' in viscous flow fields.

Perhaps the term should be softened to 'bumps'? By their nature, if such waves were to exist, they would need to be visco-elastic in nature, rather than the inviscid-elastic waves seen at high speed. They would need to be slightly diffuse in nature - due to the viscous spreading effects. They may be a series of 'bump events' rather than the sudden inviscid-elastic shock waves we know so well. They would need to allow information to propagate upstream, since the fluid flow speeds are not constrained by sonic speed limits.

I understand nature to be a consistent continuum, in the sense that if low speed & high-speed wave-fields exist, that there must be definite activity in the regimes inbetween. This should surely be reflected directly by the equations used to model such flows & should produce such solutions directly without having to develop turbulence models dictating to the physics. I have observed that theoretical approaches seem to lie at two extremes: low-speed incompresible viscous flows & high-speed inviscid flows. There seems to be little bridging theory between the two.

I propose that this 'bridging region' could be a duality of bulk-flow & wave fields, with a particle-field interaction mechanism, perhaps similar to the electromagnetic wave mechanism. Bulk flow generates deviatoric field -> deviatoric (acoustic?) pressure, which in turn modifies bulk-flow... & so on.

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A few insights on the u.du/dx term (possibly well-worn & simplistic, but has given me a good feel for the physics):

set u.du/dx + c = 0

solution of form: u = +- sqrt[(u,o)^2-2.c.x]

where u,o is entry velocity, x is distance from inlet. This provides an interesting solution with +-u,o at x=0 & with a 'jump term' when x reaches the distance where the sqrt() becomes imaginary. This effect is also seen for the v.dv/dy term in the y-direction.

The 'c' term can be taken to be the instantaneous values of surrounding terms in the N-S, Burger eqn etc. This jump term can be seen to be the mechanism for say convergent-divergent nozzle solution jumps.

The v.du/dy & u.dv/dx cross-terms create a 'turn-over mechanism' which moves the vibration mode from say x-direction dominated compression-rarefaction eg. water hammer at tube inlet, to y-direction dominated towards tube exit. The central section rotates the vibration mechanism in discrete steps. Another intriguing 'mixing mechanism'. This can be seen from flow-simulation deviatoric velocity fields.

Thanks again, Tom, for your very kind input.

diaw: So, then, this would surely have to be a non-physical mechanism?

Yes and No. Based upon pure physical reasoning it is obviously incorrect. However when viewed as part of an approximation process it is seen to be the lead order contribution (in some region of the domain) to the flow field. In fluids this occurs through the low-Mach number expansion which assumes that the flow is everywhere slow compared to the speed of sound.

In reality all materials are compressible if you apply enough force - otherwise the intermolecular bonds would be capable of balancing arbitrarily high forces rather than collapsing as for example in a neutron star.

You must always remember that incompressibilty is an approximation and, like most approximations, can give what appear to be paradoxical results when viewed out of context.

diaw: So, then, this would surely have to be a non-physical mechanism?

Tom: Yes and No. Based upon pure physical reasoning it is obviously incorrect. However when viewed as part of an approximation process it is seen to be the lead order contribution (in some region of the domain) to the flow field. In fluids this occurs through the low-Mach number expansion which assumes that the flow is everywhere slow compared to the speed of sound.

In reality all materials are compressible if you apply enough force - otherwise the intermolecular bonds would be capable of balancing arbitrarily high forces rather than collapsing as for example in a neutron star.

You must always remember that incompressibilty is an approximation and, like most approximations, can give what appear to be paradoxical results when viewed out of context.

diaw: I have never really been comfortable with the concept of a truly incompressible fluid as used in most texts & was not surprised that such fluids presented tough issues for numeric simulation.

What would be the implications of not assuming infinite communication velocity? What minumum necessary 'slight adjustments' would need to be made to incompressible theory to match this physical requirement?

This is why I have tried to work with the restrictions as I originally stated them: 1. D(dm)/Dt=0 (mass conservation => no mass source); 2. Constant fluid properties.

If we could then 'soften' the second restriction slightly to accommodate slight-compressibility to pressure, what implications would this have? Could a simple linear expression for density change with pressure - for instance - allow a departure point into a more physically-compatible solution?

I'll dig deeper into this logic & see what effects it turns up.

Thanks again for your insights.

diaw (Des Aubery)

The address to which the message has not yet been delivered is:

des@xxxxxxx.com (I have replaced the correct address with these x's)

Delay reason: mailbox is full

Hi Ahmed,

I apologise the the server, it seems clear from my side. I'm not sure why it is acting strangely. I'll let my technician follow it up on Monday. (Today is an anti-government protest day & so things may be a little tense around here for a while).

I'll email you a few alternative addresses. You are bound to get through on one of those.

Sorry about that - I had been waiting for your e-mail.

diaw... (Des Aubery)

Would the contributors to this thread be in agreement that I collect the thought flow/s into a Wiki, or equivalent document & transfer it to the Wiki?

I feel that a lot of valuable dicsussion has ensued, with some excellent contributions by a number of folks, with specific thanks to both Adrin Gharakhani & Tom.

I thought that some of the thoughts could be easily tracked & refered to in the wiki. This could facilitate an information exchange from the forum to the wiki & vicaversa. It would be sad to see it disappear off the bottom of the forum listing, with the march of time. The discussion seemed to grow as we moved along.

Tom's very, very kind reference contributions & thought processes are profound, to say the least.

There are still a number of areas that need consolidation, & certain issues will certainly need further dicsussion, with time.

Regards, diaw... (Des Aubery)

Tom:

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.

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diaw:

I have been working through the previous logic arguments for later discussion purposes, I would like to signpost this apparent logic bifurcation point.

It appears that we have two scenarios:

1. Constant fluid properties => density=constant

or,

2. Incompressible fluid cell.

It appears, at this point, that the arguments moved towards 'incompressible' fluids. It may be useful to fully explore both arguments & see if they are compatible, or not.

diaw...

(As remarked in the prologue, it is a balance between the compressibility and the inertia of a fluid that governs the propagation of sound waves through it).

The above lines form the openning statement of "Waves in Fluids" by James Lighthill.

In an Incompressible flow the compressibility (Beta) is equal to zero.

Diaw, you have performed a simulation and came across a flow pattern that you are trying to attribute to a wave phenomnun, I would suggest that you load some of your plots on the web, and let other eyes look at them.

Ahmed:

(As remarked in the prologue, it is a balance between the compressibility and the inertia of a fluid that governs the propagation of sound waves through it).

The above lines form the openning statement of "Waves in Fluids" by James Lighthill.

In an Incompressible flow the compressibility (Beta) is equal to zero.

Diaw, you have performed a simulation and came across a flow pattern that you are trying to attribute to a wave phenomnun, I would suggest that you load some of your plots on the web, and let other eyes look at them.

diaw:

After reading & re-reading some of the early discussion, I have become more convinced that there seems to actually be two logic paths - as stated earlier:

1. Constant fluid properties; density=constant

2. Incompressible fluid.

For case 1, we are dealing with a fluid which has fixed properties - as determined on a macro-scale. Density = mass/volume taken for a large fluid sample. Measure the mass on a scale & measure the container dimensions - leading to a volume calculation. This bulk density (inverse specific volume) is a property of the fluid itself. We choose to maintain it constant for our numeric experiment.

The bulk density definition says nothing about the micro-distributions within the fluid continuum itself - but, it is assumed isotropic.

For case 2, we seem to have a mathematical definition based on div(u) as the starting point.

In my mind, I would like to see a physical definition of 'incompressibility' - perhaps something of the form - element resistant to deformation. In other words, proximity to a true solid.

The fact that the mathematical definition of incompressibility leads to an infinite communication velocity - in violation of physics, leaves me with some serious logical closure problems.

These two arguments seem to leave a few questions - in my mind at least.

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Simulation plots:

I would certainly like to place a few case studies on the web for discussion purposes. Perhaps on the wiki? I can also upload a few pictures to my webserver.

The case that shows up the 'low-speed viscous shock/bump' can be seen nicely in two high-resolution runs of water-flow inside a tube with internal cylindrical singularity (circle). This study has been performed using a FEM approach using: 1. CBS scheme - 'time-soft' - 3 node triangular elements 2. Galerkin scheme - steady - 6 node triangular elements.

The 'shock/bump lines' show up as very bright lines on a high-resolution velocity-magnitude plot (1024 colours). When a grayscale plot of this same velocity-magnitude plot shows up a number of details not clearly evident whan observing the colour plot (something to do with how our eyes interpret colour versus grayscale - an old trick to expose hidden details).

These lines actually move with velocity increase away from the initial position after the 'mode change'. Immediately after 'mode change', these lines look almost exactly like the high-speed bow-shock, with internal ?mach? lines.

For the cavity flows, I'll also put up a few interesting high-resolution contour plots (1024 contours).

Oh, if you really want to see what is really going on in the flow field set up high-resoltion contour plots (1024 not 16) - the contour sampling allows your eyes to catch data that a fringe plot will never, ever show you...

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The main issue to note when considering the possibility of wave activity in fluids is that the dispersive terms & structure of the 'wave form' of N-S, predict a number of primitive wave-forms. These encompass convection waves (from inertia terms), elastic waves & viscous (shear) waves - in various directions.

So the physical mechanisms seem to exist to provide 'shock waves' - which, after all, come from the combination of convection wave & elastic dispersion term. In viscous fluids, we add a few viscous (shear) terms which should exert a spreading effect on the elastic shock.

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Test geometry:

Tube 250x25 mm Circle (cylinder) : centre (12.5, 100) Circle (radius) : 0.09 mm (for instance)

Fluid: Water with properties Density = 1000 kg/m3 Viscosity 855e-6 N.s/m2

Fluid entry velocity: ~ 0.029 m/s

Do not use any Reynolds scaling, but simulate directly at full scale.

Give it a whirl & tell me what you find. It is best if a transient scheme is used. If you want to see some interesting wave activity, try capturing the simulation data every few runs, or so. It could be interesting to explain whythe peak pressure often ends up at the wrong end of the tube... The answer is simple, but needs some thought.

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Please note, that Matlab simulations in 1D, of the N-S have predicte the 'viscous shock/bump' event at tiny velocities. There is a trick to understanding what the partial(d2u/dy2) term means in the context of a 1d x-direction study. This has a physical meaning.

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Let me work on getting a few picturs available for download. You, in turn, try the simulation. Good luck.

diaw... (Des Aubery)

"What would be the implications of not assuming infinite communication velocity? What minumum necessary 'slight adjustments' would need to be made to incompressible theory to match this physical requirement?"

For the pressure signal you could make the flow compressible; i.e. study Low Mach number flows.

However if you are really concerned about any infinite speeds of propagation then you have a further problem with the Navier-Stokes equations in that the "viscous signal" also moves with infinite velocity. As an example of this consider the problem of an (infinitely long) flat wall y=0 in an unbound fluid which is at rest. Now set the wall in motion with (horizontal) velocity u=sin(t). At any instant in time t>0 the value of u is nonzero above the wall - the viscous signal has moved through the fluid at infinite velocity. This is actually the reason why explicit methods have stability problems with the heat equation while implicit ones do not.

This is basically the difference between Newton (N-S equations follow from this) and Einstein. The first has "action at a distance" while the second requires that no signal can move faster than the speed of light.

Basically (1) implies (2) but (2) does not imply (1); i.e. (1) is a special case of (2).

diaw:

I have been working through the previous logic arguments for later discussion purposes, I would like to signpost this apparent logic bifurcation point.

It appears that we have two scenarios: 1. Constant fluid properties => density=constant or, 2. Incompressible fluid cell.

It appears, at this point, that the arguments moved towards 'incompressible' fluids. It may be useful to fully explore both arguments & see if they are compatible, or not.

Tom:

Basically (1) implies (2) but (2) does not imply (1); i.e. (1) is a special case of (2).

diaw: (drawn from a reply to Ahmed)

For case 1, we are dealing with a fluid which has fixed properties - as determined on a macro-scale. Density = mass/volume taken for a large fluid sample. Measure the mass on a scale & measure the container dimensions - leading to a volume calculation. This bulk density (inverse specific volume) is a property of the fluid itself. We choose to maintain it constant for our numeric experiment.

The bulk density definition says nothing about the micro-distributions within the fluid continuum itself - but, it is assumed isotropic.

(New)

As I understand things, from an earlier argument, is that (1) really only implies that (2d) partial(du/dx) = partial(-dv/dy), once all the constant density terms (dp/dt, dp/dx, dp/dy, dp/dz) drop out.

Can we really imply anything other than that statement at this point?

I see this gradient trade-off as being a measure of the 'flexibility' of the element - as a velocity trade-off, nothing else. (I know we briefly addressed this in terms of 'stiffness', but feel that we perhaps did not complete the full argument).

Observations of Numeric solutions: When, for instance, we build a numeric code, we set up the two N2x & N2y (momentum-x & momentum-y) equations & strap in the partial(du/dx) + partial(dv/dy) = 0 to give the closure equation - fullfilling the 'mass conservation' requirement. Then, in order to solve the non-linearities due to the inertial terms, we discretise into say Newton-Raphson, develop Jacobian & required relationships. That's it. No other special magic, no convection stabilisation, or tricks, other than the assumption of density, viscosity = constant.

The numeric solution of this code results in wave-effects in the solution field, as evidenced by looking at various snapshots during the simulation itself. The way that for instance, inlet velocity is applied, can set off some serious wave activity within the flow domain. The crucial issue is whether this 'wave-activity' is purely numerical, or represents something inherent in the nature of fluids themselves. In moany cases, this wave activity causes the solver to diverge - no matter how much you massage it.

I have seen these effects in each & every solver I have run - reasearch & commercial, over the past few years. In some cases, the solver can be massaged past a 'pressure hump' & then persuaded to keep happy. Observing the solution just before & during this 'pressure hump' will show the return of the pressure peak from eg the far end of a tube, to strike the inlet face. This pressure-wave activity is perfectly explainable, but why is it there in the first place?

Very often, this is the very reason that the solutions suddenly 'blow-up'

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For case 2, we seem to have a mathematical definition (of incompressibility) based on div(u) as the starting point.

In my mind, I would like to see a physical definition of 'incompressibility' - perhaps something of the form - element resistant to deformation. In other words, proximity to a true solid.

Would it be possible to define 'incompressibility' in more solid terms, & then find our way back to the relationship partial(du/dx) = partial(-dv/dy), from that original definition?

The fact that the mathematical definition of incompressibility leads to an infinite communication velocity - in violation of physics, leaves me with some serious logical closure problems.

diaw... (Des Aubery)

diaw: "What would be the implications of not assuming infinite communication velocity? What minumum necessary 'slight adjustments' would need to be made to incompressible theory to match this physical requirement?"

Tom:

For the pressure signal you could make the flow compressible; i.e. study Low Mach number flows.

diaw: I must say that I'm still not comfortable with exactly why the pressure signal must travel at infinite speed? Perhaps I'm just missing something fundamental, but I cannot see why partial(du/dx) + partial(dv/dy) = 0 forces the infinite pressure velocity condition. I wonder if we may be perhaps be creating a rod for our own backs - so to speak.

Tom: However if you are really concerned about any infinite speeds of propagation then you have a further problem with the Navier-Stokes equations in that the "viscous signal" also moves with infinite velocity. As an example of this consider the problem of an (infinitely long) flat wall y=0 in an unbound fluid which is at rest. Now set the wall in motion with (horizontal) velocity u=sin(t). At any instant in time t>0 the value of u is nonzero above the wall - the viscous signal has moved through the fluid at infinite velocity.

diaw: Does the movement of a boundary imply instantaneous motion of the fluid layer above it? This could only apply if the fluid cell were infinitely stiff. This may be a mathematical consequence of the mathematical 'incompressiblity' requirement.

In practice - even in a constant density fluid, I would see the viscous wave information moving into the layer above it over a finite time - it simply cannot move at infinite velocity. Physically, this wave front will reach upwards through the fluid layer until the information transfer, reflection, re-bound cycles have settled throughout the domain.

Note: I would say that this is perhaps a crucial difference of understanding in what part the partial(du/dx) + partial(dv/dy) = 0 relationship provides. I see the relationship as not implying an infinitely stiff fluid cell, but a flexible cell, where the velocity gradients are constrained to move together - in opposite directions.

When the 'wave-form of the N-S' is used, there is no apparent requirement for infinite wave-speeds, at all - even with constant density. In fact, the partial(du/dx) + partial(dv/dy) = 0 relationship, when differentiated up in dx & dy & then substituted back into the momentum (N2x, N2y) equations (2d) actually simplifies the dispersion terms very nicely.

Tom: This is actually the reason why explicit methods have stability problems with the heat equation while implicit ones do not.

diaw: A question of thermal wave-front motion - solution speed versus physical wave speed?

Tom: This is basically the difference between Newton (N-S equations follow from this) and Einstein. The first has "action at a distance" while the second requires that no signal can move faster than the speed of light.

diaw: Newton's action at a distance, when understood in a wave context, will merely need to convey the information signal to the next closest atoms/molecules etc... No need for too much 'distance' in this case, since waves are an evidence of molecular vibrations.

Newton vs Einstein: But, sure, we may very well have something of similar ilk, with a governing wave speed - in the fluid - postulate speed of sound?

Thanks so much again Tom for your very kind debate. We seem to be sitting on two opposite sides of the same fence.

diaw: Does the movement of a boundary imply instantaneous motion of the fluid layer above it? This could only apply if the fluid cell were infinitely stiff. This may be a mathematical consequence of the mathematical 'incompressiblity' requirement.

This is a consequence of the heat equation - I picked this problem because you can write down the exact analytic solution . If you were to replace u by temperature you would see the same effect.

You need to think of the equations by there type (see an introductory text book on pdes):

(i) Hyperbolic - all characteristics real => signals move at finite velocities along the characteristics. The canonical example is the wave equation u_tt = u_xx.

(ii) Elliptic - no real characteristics => solution is everywhere determined by points on the boundary. This is infinite speed of propagation. Canonical example is the Laplaces equation u_xx + u_yy = 0.

(iii) Parabolic - one real characteristic (time). This is the middle ground between (i) and (ii) and is where the NS equations sit;i.e. information set a one instant propagates everywhere at the next. Standard example is the heat equation u_t = u_xx.

diaw: Newton's action at a distance, when understood in a wave context, will merely need to convey the information signal to the next closest atoms/molecules etc... No need for too much 'distance' in this case, since waves are an evidence of molecular vibrations.

Classical mechanics disagrees with this - think about the motion of planets under the inverse square law. Moving one planet instantaneously effects the other. In a vacuum there are no molecules to vibrate and send this information.

diaw: Does the movement of a boundary imply instantaneous motion of the fluid layer above it? This could only apply if the fluid cell were infinitely stiff. This may be a mathematical consequence of the mathematical 'incompressiblity' requirement.

Tom: This is a consequence of the heat equation - I picked this problem because you can write down the exact analytic solution . If you were to replace u by temperature you would see the same effect.

diaw: Very valid point in terms of the 'heat-equation' as you quote it. I'll comment a little later on what I believe is missing in this subset of the energy equation.

Tom:

You need to think of the equations by there type (see an introductory text book on pdes):

(i) Hyperbolic - all characteristics real => signals move at finite velocities along the characteristics. The canonical example is the wave equation u_tt = u_xx.

(ii) Elliptic - no real characteristics => solution is everywhere determined by points on the boundary. This is infinite speed of propagation. Canonical example is the Laplaces equation u_xx + u_yy = 0.

(iii) Parabolic - one real characteristic (time). This is the middle ground between (i) and (ii) and is where the NS equations sit;i.e. information set a one instant propagates everywhere at the next. Standard example is the heat equation u_t = u_xx.

diaw: Thanks again Tom. This is the classical approach used to work to & from energy equation/ N-S.

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

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

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Heat equation re-visited:

I believe that the heat-equation as we choose to use it, is not complete. The main reason is the terms we most often choose to discard - 'viscous dissipation'. When these non-linear terms are unfolded into linear partials in energy powers, we end up with beautiful 'wave terms' & singularity relationships. IMHO, these are the missing ingredients which may provide a more complete picture of the 'energy equation' & ovecome the instantaneous information velocity dillema.

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diaw: Newton's action at a distance, when understood in a wave context, will merely need to convey the information signal to the next closest atoms/molecules etc... No need for too much 'distance' in this case, since waves are an evidence of molecular vibrations.

Tom:

Classical mechanics disagrees with this - think about the motion of planets under the inverse square law. Moving one planet instantaneously effects the other. In a vacuum there are no molecules to vibrate and send this information.

diaw: You are perfectly correct in terms of instantaneous movement of the planet affecting the other - but at what transmission speed? Would not electromagnetic wavespeed still apply here? I can envisage this information wave moving the instant the planet get moving - tavelling outwardsat light-speed, until it reaches the planet it is affecting. This planet responds & sends back a wave response - & so forth until the whole field settles itself into 'equilibrium'.

BTW, I'd hate to try & move the planet instantaneously...

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

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

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

"Heat equation re-visited:

I believe that the heat-equation as we choose to use it, is not complete. The main reason is the terms we most often choose to discard - 'viscous dissipation'. When these non-linear terms are unfolded into linear partials in energy powers, we end up with beautiful 'wave terms' & singularity relationships. IMHO, these are the missing ingredients which may provide a more complete picture of the 'energy equation' & ovecome the instantaneous information velocity dillema. "

The easiest way to overcome the problem is to replace the heat equation with, if memory serves me correctly, Heavisides telegraph equation ( l.u_tt + u_t = u_xx with l<<1 a constamt). I don't think anyone has tried to modify the NS equations in this manner (although I've thought about it).

diaw: You are perfectly correct in terms of instantaneous movement of the planet affecting the other - but at what transmission speed? Would not electromagnetic wavespeed still apply here? I can envisage this information wave moving the instant the planet get moving - tavelling outwardsat light-speed, until it reaches the planet it is affecting. This planet responds & sends back a wave response - & so forth until the whole field settles itself into 'equilibrium'.

You need General relativity for this - Newtonian mechanics says the signal has infinite velocity (i.e. Newtons first law and the "fixed stars")