"The N-S equations are merely a 'disguised form' of an underlying wave equation..."

- Actually they are a "Heat equation" since the NS equations are Parabolic rather than hyperbolic like the Euler equations.

Tom wrote: "The N-S equations are merely a 'disguised form' of an underlying wave equation..."

- Actually they are a "Heat equation" since the NS equations are Parabolic rather than hyperbolic like the Euler equations. ---------

Hi Tom,

Go down another layer & tell me what you see... It is more of a wave-machine than you may ever want to imagine.

diaw...

There is no outlet.I expect that the flow field will have the shape of a mushroom. My final goal is to simulate the flow in an internal combustion engine.

Thank you for your time! I will check my code and maybe I will return to you.

> There is no outlet. I expect that the flow field will have the shape of a mushroom.

Why? Do you have analytical, experimental or other peoples numerical results against which to compare?

If you have incompressible flow (i.e. contant background pressure) the flow in must balance the flow out. If you have flow in but no flow out then you have specifed a problem that cannot be solved.

If you have compressible flow you will need proper boundary conditions to handle the waves at your inlet. How have you specified your boundary conditions?

I would suggest predicting a very simple flow for which you know the answer as a first test case (and a second and a third).

Andy wrote: If you have compressible flow you will need proper boundary conditions to handle the waves at your inlet. How have you specified your boundary conditions?

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Very, very good points there.

What would you suggest for appropriate boundary-conditions where wave phenomena show themselves?

Do commercial solvers allow such boundary-conditions?

diaw...

I have compressible flow and I treat the boundary conditions without any special treatment.The solid walls are solid walls and the inlet is like any other usual inlet.I did not know that I should do something different with the inlet.

George wrote: I have compressible flow and I treat the boundary conditions without any special treatment.The solid walls are solid walls and the inlet is like any other usual inlet.I did not know that I should do something different with the inlet.

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I have the impression that many solvers do not like acoustic effects. In my experience, the ones that don't diverge are generally very heavily damped anyway.

Acoustic effects can be driven upwards by pressure fluctuations & this can loop back into the bulk-flow calculation & cause solver divergence. Waves also have a nasty habit of reflecting & bouncing around your model space, causing unwanted problems.

The guys who could really be of assistance are those who solve high-speed problems, where shock waves appear. I seem to think that they treat supersonic boundaries in a special way.

Your 'rate of inflow' application can cause artificial 'acoustic effects' due to the shock effect it exerts... try using a ramped input velocity, or ramped applied pressure & see what happens. Also, make sure you are not using a 'steady solver', which, by definition, is not designed to handle flow unsteadiness - of any nature - bulk, or wave.

Apply an 'under-relaxation factor' say 0.3 to your pressure update at the end of each non-linear loop & see if it helps to damp out excess pressure swings.

Other than that, you are in 'wave mode' & will have to develop cunning methods to both understand & deal with the phenomena you see.

Feel free to e-mail me privately if you need more information.

diaw...

Tom wrote: "The N-S equations are merely a 'disguised form' of an underlying wave equation..."

- Actually they are a "Heat equation" since the NS equations are Parabolic rather than hyperbolic like the Euler equations. ---------

Hi Tom,

Go down another layer & tell me what you see... It is more of a wave-machine than you may ever want to imagine.

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A reference from 'Viscous Fluid Flow', White F.M., 3ed, 2006, page 78:

"...the full viscous-flow equations... , are too complicated & display a variable mixture of elliptic, parabolic, & hyperbolic behavior."

So, there we have the answer. Life is not as simple as we imagined at first blush...

diaw...

Hello Diaw How are you, as long as "P" is considered the "static pressure" ........... "P" is the thermodynamic equilibrium pressure.

Ahmed wrote:

Hello Diaw How are you, as long as "P" is considered the "static pressure" ........... "P" is the thermodynamic equilibrium pressure.

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Greetings Ahmed... I'm good, thank you... Good to hear from you again...

Can you elaborate a little more on your thoughts around "static pressure" & "thermodynamic pressure"?

As far as I understand wave theory, so far at least, is that when an 'excess pressure' field & a consequent pressure gradient is available, wave phenomena can be induced.

In George's case, the swings in his calculated 'pressure' value may be inducing acoustic waves. But, in all likelihood, the very act of rapid compression of his gas volume, is likely to produce wave phenomena anyway.

I would value your thoughts... Take care...

diaw...

A reference from 'Viscous Fluid Flow', White F.M., 3ed, 2006, page 78:

"...the full viscous-flow equations... , are too complicated & display a variable mixture of elliptic, parabolic, & hyperbolic behavior."

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This is just wrong (ask any mathematician) the Navier-Stokes equations are parabolic - researchers working on the theory of inertial manifolds rely on this since if the equations were not parabolic then the search for an existence proof of an inertial manifold for the NS equations would fail at the first hurdle rather than at the more complicated step of "all time existence". See the book "Infinite Dimensional Dynamical Systems" by J.C. Robinson.

The "variable mixture" of elliptic and hyperbolic is actually the definition of parabolic! i.e. parabolic is the dividing line between hyperbolic and elliptic problems.

You could argue that in practice knowing whether the equations are parabolic or not is not of much use in general - although I have some sympathy for this point of view you must remember that you use parabolicity of the equations when you pose the Initial Boundary Value Problem.

Tom wrote: A reference from 'Viscous Fluid Flow', White F.M., 3ed, 2006, page 78:

"...the full viscous-flow equations... , are too complicated & display a variable mixture of elliptic, parabolic, & hyperbolic behavior."

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This is just wrong (ask any mathematician) the Navier-Stokes equations are parabolic - researchers working on the theory of inertial manifolds rely on this since if the equations were not parabolic then the search for an existence proof of an inertial manifold for the NS equations would fail at the first hurdle rather than at the more complicated step of "all time existence". See the book "Infinite Dimensional Dynamical Systems" by J.C. Robinson.

The "variable mixture" of elliptic and hyperbolic is actually the definition of parabolic! i.e. parabolic is the dividing line between hyperbolic and elliptic problems.

You could argue that in practice knowing whether the equations are parabolic or not is not of much use in general - although I have some sympathy for this point of view you must remember that you use parabolicity of the equations when you pose the Initial Boundary Value Problem.

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Thanks for the excellent points - very well made.

As a point of departure, a question: On which 'layer' of N-S are you working - the 'bulk' flow layer, or the 'wave layer'?

diaw...

Thanks for the excellent points - very well made.

As a point of departure, a question: On which 'layer' of N-S are you working - the 'bulk' flow layer, or the 'wave layer'?

diaw...

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I'm not sure what you mean by the "bulk flow layer" and the "wave layer"? As far as I'm concerned there is just the Navier-Stokes equations as a whole (except when I'm doing an asymptotic expansion in which case the equations change type due to the approximations).

By "bulk" do you mean the full characteristics of the NS equations (time being the only real characteristic direction in this case) and "wave" the so-called bi-characteristics (they may have called them sub-characteristics -these are effectively the characteristics of the Euler equations) introduced by Kevorkian & Cole?

Tom wrote: I'm not sure what you mean by the "bulk flow layer" and the "wave layer"? As far as I'm concerned there is just the Navier-Stokes equations as a whole (except when I'm doing an asymptotic expansion in which case the equations change type due to the approximations).

By "bulk" do you mean the full characteristics of the NS equations (time being the only real characteristic direction in this case) and "wave" the so-called bi-characteristics (they may have called them sub-characteristics -these are effectively the characteristics of the Euler equations) introduced by Kevorkian & Cole?

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Hi again Tom,

Sorry if I was being a little sparse on words. What I meant is the following:

- substitute partial(dx/dt) for each occurrence of 'u' in the x-Momentum equation. Write it out in detail & then think about the significance of what you are seeing & its potential implications. Think of which are true 'forces' - motive & resistive - as distinct from fictitious 'equilibrium forces'. Look for 'wave terms' - they are there.

Of course this is a little simplistic in the full interpretation - this applies in all spatial dimensions, but it will give a glimpse of the 'lower level'. Think about what implication waves may play in our quest for resolving the next layer up - the 'bulk flow layer'. It is a deep, deep topic.

diaw...

Basically, as I mentioned in the previous post, it depends upon what I'm doing - however I always keep in mind the nature of the full unapproximated problem.

An example of what I meant (and appears what you are suggesting) is the "steady" high-Reynolds number (low Mach number) flow over, for example, a flat plate/thin aerofoil. Here the full problem is elliptic while the boundary layer equations are parabolic. The boudary layer thus violates the "upstream" influence requirement of the fuller elliptic problem (in supersonic flow this is a serious problem as originally noted by Lighthill in the 1950's). This discrepancy is resolved by triple-deck theory (or equivalently strong viscous-inviscid interaction theory) which reinstates the elliptic nature of the problem within the boundary layer.

The fact that the equations are of the wrong type within the boundary layer yields the Goldstein singularity indicating a failure of the existence of the boundary layer solution in contradiction to the observed fact that the full system does have a solution!

I think this example shows that, in your terminology, that you have to be careful with your interpretation of the "wave layer" since it may contradict the "bulk layer".

Hope this makes some sense,

Tom.

Tom wrote: Basically, as I mentioned in the previous post, it depends upon what I'm doing - however I always keep in mind the nature of the full unapproximated problem.

An example of what I meant (and appears what you are suggesting) is the "steady" high-Reynolds number (low Mach number) flow over, for example, a flat plate/thin aerofoil. Here the full problem is elliptic while the boundary layer equations are parabolic. The boudary layer thus violates the "upstream" influence requirement of the fuller elliptic problem (in supersonic flow this is a serious problem as originally noted by Lighthill in the 1950's). This discrepancy is resolved by triple-deck theory (or equivalently strong viscous-inviscid interaction theory) which reinstates the elliptic nature of the problem within the boundary layer.

The fact that the equations are of the wrong type within the boundary layer yields the Goldstein singularity indicating a failure of the existence of the boundary layer solution in contradiction to the observed fact that the full system does have a solution!

I think this example shows that, in your terminology, that you have to be careful with your interpretation of the "wave layer" since it may contradict the "bulk layer".

Hope this makes some sense,

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Thanks Tom for your deep insights. Excellent. Thank you very much.

Engineers seem to prefer the 'bulk flow' paradigm, whereas my physics training has pulled me into a 'wave' paradigm. To each his own.

One small thought: The 'wave layer concept' has allowed me to isolate the singularity in the N-S eqns & to understand the physical phenomenon it represents. (I hope to publish some of this in the not-too-distant future). The results may be enlightening to some, but utterly frustrating to others. It also explains why our solvers often misbehave, especially for incompressible, viscous flows. As I said earlier, it is a deep world at 'wave level'...

diaw...

I have changed the underrelaxation factor (from 0.001 up to 0.7) but nothing changes actually.I also added some artificial viscosity,but the main characteristics of the results remain the same. Nevermind,thank you once again and when I have something new,I will let you know.

George, under-relaxation does sometimes have 'golden numbers', especially when waves seem to be involved. Try a small range around 0.495-0.55 I have found that this sometimes does the trick where abolutely no other value will work.

diaw...

What I have figured is that when the wall temperature is higher than the temperature of the air in the duct, then there is some air movement away from the walls , and when the wall temperature is lower than thos of the air, then the opposite happens. But this movement is much more intensive than that of the air coming into the domain.I checked the boundary conditions of the temperature equation and they are OK,but something must not be going right,because when for example the temperature of the air at t=0 sec is 300K and the air coming inside has the same T and the walls have Tw=400K,in most of the cells the temperature is about 295K....The time step is very small (dt=0.0002 sec), but this is not normal.