<u>mw wrote</u>:

"Can incompressible fluid flow within confined domains express velocity jumps, or vortex sheets?"

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<u>Tom replied</u>:

Depends on the initial condition. If you have a continuous twice differentiable initial condition I would say no - it's believed that the viscous equations cannot spontaneously form a discontinuity (but there is no rigourous proof of this as yet in 3D).

<u>mw replied</u>:

Fair-enough, I'd say for an extremely smooth inlet condition (twice differentiable) & for no internal objects/singularities.

If you take a look at the CUMEC3 paper on <http://adthermtech.com/documents/CU-...A_22032007.pdf>, on pg 25/26 - 'flow over cylinder within a pipe' & pg & 27/28 - 'flow over spatial vibration mechanism within a pipe', what can be observed? What are the lines we see for the u2 plots?

Surely, the N-S should be able to evidence a discontinuity at positions where the jump-conditions are fulfilled? Now whether these are the result of a continuously-applied forcing function, or spontaneous is interesting.

<u>Tom wrote</u>:

However if you insert a discontinuity into the initial state in an inviscid fluid it will persist (i.e. a vortex sheet is a material surface). In a viscous fluid the vortex sheet will diffuse out which will smooth out the discontinuity (the analytical solution for the diffusion of a vortex line shows this).

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<u>mw replied</u>:

I would agree with this statement. An interesting example of the response in a duct can be seen in the CU-MEC5 paper at <http://adthermtech.com/documents/CU-...r_part2_m3.pdf>, for the modelling of a plug wave moving down a pipe. The wave launches off the inlet face (step input) & continues down the tube. The diffusive effect is evident in that the wave front begins to spread out with distance.

In this example, the inlet step velocity is certainly not twice differentiable (dirac)

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<u>Tom wrote:</u>

A simple example is the shock in Burgers equation - the invisid result requires jump conditions to define the discontinuity but the viscous solution does not (because the jump is smoothed out by the diffusion operator).

<u>mw replies</u>:

A nice example of this is modelling in Matlab (1D), for instance & then gradually varying parameters until the jump evidences itself as a spread-out viscous jump.

(Sorry about that, the post jumped up the page for some odd reason).

<u>mw wrote:</u>

"Can incompressible fluid flow within confined domains express velocity jumps, or vortex sheets?"

<cr />

<u>Tom replied:</u>

Depends on the initial condition. If you have a continuous twice differentiable initial condition I would say no - it's believed that the viscous equations cannot spontaneously form a discontinuity (but there is no rigourous proof of this as yet in 3D).

<u>mw replied:</u>

Fair-enough, I'd say for an extremely smooth inlet condition (twice differentiable) & for no internal objects/singularities.

If you take a look at the CUMEC3 paper on <http://adthermtech.com/documents/CU-...A_22032007.pdf>, on pg 25/26 - 'flow over cylinder within a pipe' & pg & 27/28 - 'flow over spatial vibration mechanism within a pipe', what can be observed? What are the lines we see for the u2 plots?

Surely, the N-S should be able to evidence a discontinuity at positions where the jump-conditions are fulfilled? Now whether these are the result of a continuously-applied forcing function, or spontaneous is interesting.

<u>Tom wrote:</u>

However if you insert a discontinuity into the initial state in an inviscid fluid it will persist (i.e. a vortex sheet is a material surface). In a viscous fluid the vortex sheet will diffuse out which will smooth out the discontinuity (the analytical solution for the diffusion of a vortex line shows this).

<u>mw replied:</u>

I would agree with this statement. An interesting example of the response in a duct can be seen in the CU-MEC5 paper at <http://adthermtech.com/documents/CU-...r_part2_m3.pdf>, for the modelling of a plug wave moving down a pipe. The wave launches off the inlet face (step input) & continues down the tube. The diffusive effect is evident in that the wave front begins to spread out with distance.

In this example, the inlet step velocity is certainly not twice differentiable (dirac)

<u>Tom wrote:</u>

A simple example is the shock in Burgers equation - the invisid result requires jump conditions to define the discontinuity but the viscous solution does not (because the jump is smoothed out by the diffusion operator).

<u>mw replies:</u>

A nice example of this is modelling in Matlab (1D), for instance & then gradually varying parameters until the jump evidences itself as a spread-out viscous jump.

mw...

Oh dear... a bad day Admin, can you please re-position this post to the back end of the long thread further down the page - if possible. Many thanks...

mw...

"Surely, the N-S should be able to evidence a discontinuity at positions where the jump-conditions are fulfilled? Now whether these are the result of a continuously-applied forcing function, or spontaneous is interesting."

You can generate a discontinuity in two-ways (1) have a density jump, or (2) a jump in viscosity. Both of these (two fluid) situations will require jump conditions to be maintained across the interface (assuming the fluids are imiscible). Actually a third may exist if you have some form of porous barrier fixed within the fluid.

As for the spontaneous formation I don't believe so (and you've effectively agreed with this in your later comments on my specific examples). However this is a million dollar question - my personal belief is that the incompressible viscous equations have smooth solutions for all time provided physically sensible conditions are applied (i.e. no infinite energy initial conditions) - the 3D invisid equations on the otherhand have probably got a finite time singularity.

<u>mw wrote:</u>

"Surely, the N-S should be able to evidence a discontinuity at positions where the jump-conditions are fulfilled? Now whether these are the result of a continuously-applied forcing function, or spontaneous is interesting."

<u>Tom replied:</u>

You can generate a discontinuity in two-ways (1) have a density jump, or (2) a jump in viscosity. Both of these (two fluid) situations will require jump conditions to be maintained across the interface (assuming the fluids are imiscible). Actually a third may exist if you have some form of porous barrier fixed within the fluid.

Fair enough. I'd like to add in a 4th possibility for a jump to occur - that of what I'm terming a <u>reflective jump</u>. This basically comes as a consequence of the [r(u - V). n] = 0 condition.

The [(r(u - V). n] = 0 condition basically has a number of possibilities inherent in its structure, & this can be seen from the expression, with V=0:

[(r(u - 0). n] = 0 <cr />

[(u . n] = 0 for r=const <cr />

(u+ . n+) - (u- . n-) = 0 <cr />

(u+ . n+) = (u- . n-) <cr />

|u+|*|n+|*cos(a+) = |u-|*|n-|*cos(a-) <cr />

|n+| = |n-| = 1

|u+|*cos(a+) = |u-|*cos(a-)

A number of alternatives can be explored with the different terms, due to evenness of cos():

1. a+ = (a-) (straight through)

2. a+ = -(a-) (reflection in one component)

This logic follows a fairly natural physical law of optics, & its relatives used for reflection, refraction & so on.

Extrapolating this logic back, & reviewing the series of jump conditions : continuity, momentum & energy, the null stress tensor can be used to predict where this reflective jump occurs.

Remember to lines behind the cylinder in the papers? These seem to be such reflective jumps. Neat.

mw...

<u>Tom wrote:</u>

As for the spontaneous formation I don't believe so (and you've effectively agreed with this in your later comments on my specific examples). However this is a million dollar question - my personal belief is that the incompressible viscous equations have smooth solutions for all time provided physically sensible conditions are applied (i.e. no infinite energy initial conditions) - the 3D invisid equations on the otherhand have probably got a finite time singularity.

<u>mw replies:</u>

I'd tend to agree with this view. I have always felt that the inviscid equations were extremely 'brittle' & that they could jump at the smallest provocation - especially with no compressibility. This can be simulated using a penalty-based scheme & is very interesting.

I wonder if this is possible to simulate experimentally? I'm moving my research focus towards high-speed flows for a while under the supervision of a leading Indian academic & a well-respected Japanese academic. Perhaps we could design a few tests along the way & try to answer this question?

mw...

"Fair enough. I'd like to add in a 4th possibility for a jump to occur - that of what I'm terming a reflective jump. This basically comes as a consequence of the [r(u - V). n] = 0 condition."

This sounds like a Kirchoff free-streamline in inviscid flow. It is not an option in a viscous flow due to the required jump in the normal pressure gradient. A high Rynolds number asymptotic analysis shows that near the free-streamlines boundary layers are required that "smooth out" the required jumps; i.e. the viscous solution cannot jump - it just creates thin shear layers!

<u>mw wrote:</u>

"Fair enough. I'd like to add in a 4th possibility for a jump to occur - that of what I'm terming a reflective jump. This basically comes as a consequence of the [r(u - V). n] = 0 condition."

<u> Tom wrote:</u>

This sounds like a Kirchoff free-streamline in inviscid flow. It is not an option in a viscous flow due to the required jump in the normal pressure gradient. A high Rynolds number asymptotic analysis shows that near the free-streamlines boundary layers are required that "smooth out" the required jumps; i.e. the viscous solution cannot jump - it just creates thin shear layers!

<u>mw replies:</u>

A consequence of this kind of jump, located at a null stress tensor for N-S is (in 2d, for instance) (wip):

p=0 (momentum-driven flow)

u,1=0 (fully-developed in x)

u,2-v,1=0 (irrotational)

v,1 = 0 (linked to irrotational flow condition)

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A point of interest in an enclosed domain with internal cylinder is the position preceding the cylinder, where the flow field goes from du,1>0 ; through du,1=0; to du,1<0 just prior to the cylinder. This can be clearly seen in the u2 (v) plots in the first paper.

The lines in the u2 (v) plots for the enclosed cylinder, behind the cylinder are also of this type. For a backward-facing step, with a sinuous flow distribution post the step, at each kink - the same reflective jumps occur - in this case v,1=0 is obvious at the apex of each wave crest. Basically, the solution represents a meandering wave which is composed of:

x-direction - carried along by the bulk u field; y-direction - oscillation in space.

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Under this jump type, the u field is typically unchanged, but the v field flips direction across the jumps, thus resulting in a reflection/kink/flip in direction. Intriguing. I think it's a new one.

mw...

I don't think you have a jump - nondimensionalize the equations using the inlet velocity and channel width and redo the simulation with a convergence criteria on the pressure iteration of the O(1e-6). I suspect the jumps are "numerical noise" due to rounding which will not show up in the nondimensional solution.

<u>Tom wrote:</u>

I don't think you have a jump - nondimensionalize the equations using the inlet velocity and channel width and redo the simulation with a convergence criteria on the pressure iteration of the O(1e-6). I suspect the jumps are "numerical noise" due to rounding which will not show up in the nondimensional solution.

<u>mw replies:</u>

No. A series of jump are most definitely there & it's also pretty easy to prove its action & mechanism. I've done all kinds of experiments on the pressure issue. I'll have a look at your suggestion, but I'm pretty convinced that this is a physical phenomenon - remember Cox/Hui's findings.

The +-1e-6 cut information is basically to reduce the information around the 'zero' line dividing the panels between jumps. Don't let that put you off - it was an early capturing technique. The u2 values in each panel between jumps increase in value as they move towards the panel centre - they can be rather large. Across the jump, the flow direction flip over completely, so that, in each panel, the flow path alternates (for u2 (v) field). This was an early visualisation technique which was superceeded by a more precise capturing method.

Using the new capturing method, these jumps are at the same position as found using the previous method. Basically, what you have is alternative panels in the u2 (v) direction.

Using the new method, with increasing flow velocity, eventually captures flip-jumps in the x-direction - with jump cells (enclosed on all 4 sides). In fact, there is an amazing amount of jump-type information in these flow fields, depending on the capture method used. In one study, the flow velocity is gradually raised & the jump actions observed. The onset of major activity to the rear of the cylinder becomes very, very clear. Fun & games begin when jump lines intersect - in fact, this allows a clear view of the onset of unsteady flow behind the cylinder.

In fact, with an array of cylinders in a wider duct, some pretty astounding jump cells are observed. You'd have a better picture when you see the study results. I'll put them up over the next months for folks to digest & kick around.

The thing that begins to over-ride the numeric error argument - my old nemesis - is that as the flow velocity increases, the acceleration field rises in value until it begins to become clearly dominant. This is very strongly seen in the wake region & around it. The generation of momentum wave phenomena is rather amazing using hi-res visualisation techniques.

With momentum-driven wave phenomena, or oscillations, the pressure gradients are almost incidental & are not the driving mechanism - the velocity & acceleration fields are where this phenomena are observed. This would concur with the findings of Hui & Cox on their oscillation-type findings.

I'm going to throw in another viewpoint here. There are two wave structures - one emits from the upper Hopf bifurcation line, & the other emits from the lower Hopf bifurcation line - each to the rear of the flow cylinder. These two waves then move into lock-step, couple (out of phase) & move down the flow-field. This accounts for the alternating panels in the u2 (v) plots. This is good old wave-mechanics & I'm working through Pierce's wave book at the moment. The degree of inter-wave coupling will increase with flow speed - when it has really locked in, is when the velocity field will begin showing signs of instability - under the surface - at acceleration level, all is seen very early on.

Wave structures are very evident in the papers. I'll prepare some further presentations & put them up for display.

Before I forget, I'd like to ask your view on the horseshoe pattern formation & destruction in section 15.2.2 (CU-MEC 3). This was the first example I cited a few moons back. Knock off the leading structure & it look very similar to a high-speed bow wave.

mw...