[FMDenaro]

Originally Posted by Martin Hegedus

A truly incompressible fluid means that the fluid is actually at rest. So, no pressure or velocity gradients.

well, that's no fully correct...from the continuity v=0 make compatible having grad rho <> 0 (stratified conditions)

[Martin Hegedus]

The incompressibility assumption for the Euler and Navier Stokes is an assumption that the change in density becomes negligible. Also, the flip side of the speed of sound going to infinity is that the velocity goes to zero. However, the velocity never gets to zero, it only approaches zero. The Euler/Navier Stokes equations are actually discontinuous at V=0. The limit of the solution as the Mach number goes to zero is not the same as the solution when the Mach number is zero. The same is true for the speed of sound. The limit of the Euler/N.S. equations as the speed of sound approaches infinity is not the same as if the speed of sound was actually infinity. The same is true for compressibility. The limit of the Navier Stokes equations as they become truly incompressible is not the same as if the solution was truly incompressible.

[Martin Hegedus]

But, isn't true (for lack of a better term) stratification a discontinuity?

[FMDenaro]

Originally Posted by Martin Hegedus

But, isn't true (for lack of a better term) stratification a discontinuity?

just think the case of stratified marine water... immagine the sea at the rest but you have gradients of density, temperature, salinity, etc.

The continuity equation is satisfied and you have an "incompressible fluid"

[Martin Hegedus]

Sure, but I don't think we were talking about gravity gradients or issues about equations of state. Also, I gather the issue was if the fluid was truly 100% incompressible, not almost incompressible. Water is compressible. It does have a finite speed of sound which is about 4 times that of air.

All I'm saying is, in general, if the gradient of density is truly 100% zero, then it is very likely that changes of pressure and velocity are also truly 100% zero.

Also, I want to correct myself by saying that non-dimensional variables such as Cp are discontinuous when velocity gets to zero.

And if this statement "It has nothing to do with potential flow" is true, then I really don't understand what is going on here and I should bow out.

[Martin Hegedus]

Here is one of the original questions: "If not a surface or contact force, what force keeps the velocity divergence-free?"

OK, but "velocity divergence-free" is the same as saying conservation of mass if drho/dt is assumed zero.

So, the question is the same as "what keeps the conservation of mass"

Correct?

[Martin Hegedus]

I gather one of the issues might be that some might not realize that incompressible actually means small density perturbations. Also, one can not necessarily neglect the drho/dt in the conservation of mass equations since that drho/dt may represent a very large dp/dt for something that is very incompressible.

For example, take a block of steal and start pushing it (1-D problem). A pressure gradient will exist in it and since it is moving there will be a dp/dt. However, drho/dt will be very small.

[Martin Hegedus]

I'm going to disagree.

OK, lets make the variable p equal to pressure and assume one dimension.

Dp/Dt = dp/dt + v*dp/dx

Now take a bar of steal (a solenoid) and accelerate it by appling a force at one end, i.e. a pressure. If one's frame of reference follows the bar of steal Dp/Dt is zero and dp/dx is some value because one end of the bar has an applied pressure and the other does not. Therefore, there must be a dp/dt. This dp/dt is represented by the drho/dt in the conservation of mass equation. Sure, mathematically one can neglect it, but then you're stuck with dp/dx being zero.

[Martin Hegedus]

I'm just trying to figure out what the original poster was saying and I'm fumbling.

If I take the incompressible 1D Euler equation and apply it to a steal bar, I get

-(dp/dx)*A*L=rho*A*L*DV/dt. So -(dF/dx)*L = m*DV/dt. Seems OK to me. So, I don't understand what the issue is.

[FMDenaro]

If you assume Div v =0 you dont need to have constant (time-space) density.
Density will be constant only along the path-line as

drho/dt + v* Grad rho = 0

[Martin Hegedus]

One must still satisfy the conservation of energy equation and equation of state. A discontinuous density seems to suggest a discontinuous temperature and/or equation of state.

And the same is true if one assumes small density changes. Small density changes allows one to uncouple the equations of mass and momentum from the equations of energy and state. The chosen density distribution must also satisfy the equations of energy and state to be physically correct.

[Martin Hegedus]

Yes, one can have a bar of copper next to a bar of steal with zero relative motion and the N.S. equations are fine with that. However, the equation of state is discontinuous.

[Martin Hegedus]

But a continuous equation of state does not allow for it.

The incompressible Navier Stokes equations only allow for a continuous pressure and temperature distribution, correct? Or am I wrong? If the pressure, temperature, and equation of state are continuous, then I gather the density must also be continuous.

So give me an example where pressure and/or temperature is not continuous for the incompressible Navier Stokes equations.

[Martin Hegedus]

High speed flow and incompressible flow are two different things.

High speed, in the sense that the velocity is above M=0.3, means that the flow is compressible. V*drho/dx can no longer be neglected since it is on order of rho*dV/dx. (In this case V is the local mach number so it has been non-dimensionalized by the speed of sound).

But, OK I'll look it up. But, I hope it is straight forward because I'm not going to much, if any, mathematical manipulation to try go figure out whether your point is true or not. And I sure hope the flow is not irrotational, because then we have potential flow.

BTW, does this have anything to do with the original posters question?

[Martin Hegedus]

BTW, by "jump" equations are you talking about a shock wave. Geez, that's compressible. I give up.

[Martin Hegedus]

However, "tangential velocity discontinuities" = infinite viscous forces. Or are we talking about one of those super cooled fluids that display no viscous forces? Frankly, I'm totally unknowledgeable about that subject (for which the name completely escapes me now) so I can say nothing about it.

And vortex sheets are not singular since viscosity does not allow for it.

And, honestly, I would rather continue working on my next version of Aero Troll, http://www.hegedusaero.com/software.html, then continue with this discussion. And, I'm sure you would like to get back to whatever you were doing.

And yes, I got the point, you want to prove I'm wrong. Fine I'm wrong.

[Martin Hegedus]

BTW, with a tangential velocity discontinuity, pressure and temperature are still continuous across it!

[Martin Hegedus]

I'm not going to spend time to guess what your discussion points are. That tends to be a smoke and mirror game, as you have already done to me with the topic of discontinuities across shocks and shear layers for the incompressible Navier Stokes equation. I'm sure we both have something else to do. I know I do.

And yes, I get it, you want to prove that I am wrong. So be it. I'm wrong.

[FMDenaro]

Anyway, the original post is about physics of viscous incompressible flow (I don't like to consider a fluid incompressible).
In such model I don't see any real physical pressure....

[Jonas Holdeman]

Thank you Filippo and Desmond for your insightful comments. I posed the question to get reactions before the talk with a similar title that I am giving at a mathematics conference in two days, and am leaving home in a few hours.

An incompressible fluid is of course just a concept. Probably the closest thing to it in the universe would be the core of a neutron star, and that is certainly out of reach of our experiences. But there is a lot of loose thinking out there on the subject and this gives us a chance to critically examine the concepts.

Seriously, I think the idea of what passes for pressure deserves more thought, if not in the fluid, then in terms of the forces such a material might exert on structures.

Thank you again for your serious thoughtful comments.