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...

"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.

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...

"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.

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...

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...

"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).

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...

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.

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!

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...

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

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...

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...

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

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)

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

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

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...

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)