I have a problem which is very variable. In early time, it behaves in compressible phase but as time goes, the state becomes converted into incompressible phase. The time step size is computed by CFL condition with the speed of sound. Thus, in incompressible state, the speed of sound is increased subsequently the stability problems are occured.

My question is to ask whether there is a unified method but more attention on compressible flows.

My current code is formulated by just Euler equations but it does not work for late time simulations, incompressible states.

Is there?

I am not sure another go is worthwhile but...

> In early time, it behaves in compressible phase but as time goes, the state
: becomes converted into incompressible phase.

What does this sentence mean? Phase is normally taken to mean gas, liquid, solid. Do you mean, for example, an initial high speed gas phase becomes a low speed liquid phase?

> The time step size is computed by CFL condition with the speed of sound.
: Thus, in incompressible state, the speed of sound is increased
: subsequently the stability problems are occured.

This also makes little sense. If your time step is adjusted for the speed of sound then it cannot be the cause of the instability. The speed of sound does not increase at low Mach numbers, it is determined in the same way as at high Mach numbers. If you assume an incompressible fluid then there are no sound waves because the thermodynamic state is fixed in space. People often refer to an infinite speed of sound because pressure perturbations are felt instantly throughout the solution region but this is an imprecise statement because it is not possible to have sound waves in an incompressible fluid.

What does incompressible phase mean? Do you mean low Mach number compressible flow. Do you want acoustic waves or not?

> My question is to ask whether there is a unified method but more attention
: on compressible flows.

Without knowing what you mean by incompressible and phase it is hard to answer with certainty but yes there are methods that work from zero Mach number upto about a Mach number of 2. There are methods that work from about 0.001 Mach number through to hypersonic. There are few that are arranged to work over the whole range.

In creating a code there are a wide range of design decisions to be made which determine how well it will work for certain flows. For example, if you want accurate acoustic waves at low Mach numbers then the choices made to optimise this would almost certainly make shocks less accurate and, possibly, prevent the simulation of hypersonics. Whereas a code targeted at accurate shocks and hypersonics is likely to predict low Mach number flows poorly.

> My current code is formulated by just Euler equations but it does not work
: for late time simulations, incompressible states.

The Euler equations are compressible not incompressible and so this sentence does not make much sense. If you mean Euler equations at low Mach numbers integrated over a long time period then yes this an area of active research.

If you include some dissipation then most codes are fine but without it things are much more interesting. So if the focus of your work involves turbulence, viscous forces and/or heat conduction then including these terms may be wise since it will avoid your addressing problems which are going to disappear later.

If you really want to integrate the Euler equations for long time periods without dissipation then, as far as I am aware, there are no codes that can do this in general in the presence of significant time and space errors but there are one or two that get close or work under reduced conditions. However, it is rare that real world problems do not include a bc that pins the state or includes some dissipation such as filtering to remove the undesirable and inaccurate high frequency motion. It is not really a significant problem in the real world but it is an interesting academic problem which is often used to test and benchmark different schemes.

The low Mach number setting is a singular limiting situation in compressible flows. As Mach number approaches zero, compressible (density-based) flow solvers suffer severe deficiencies, both in efficiency and accuracy. There are two main approaches advocated in the development of algorithms for the computation of low Mach number flows; first, There is the modification of compressible solvers (density-based) downward to low Mach numbers; second, extending incompressible solvers (pressurebased) towards this regime.

You can read full article in http://www.cs.swan.ac.uk/reports/yr2004/CSR2-2004.pdf

I suggest to forget compressible solver and insited, extend a pressure based to compressible regime.

The major problems here are the huge gaps in the available theory.

The progression, as I see it, at the moment, goes something along these lines (others may feel free to contribute in the gaps):

1. Low speed, viscous-dominated flows (incompressible) - Re<1; (viscous, oozy waves)

2. Viscous-convection balance flows (incompressible) - Re=1; (singularity situation)

3. Convection-dominated flows (incompressible) - Re>1; (emerge, the 'momentum wave')

...

...

No theory

...

Incompressibility limit ; M~0.3

...

...

M~0.7

4. High-speed flows, sub-sonic, inviscid (compressible);

5. Trans-sonic flows, inviscid (compressible);

6. Super-sonic flows, inviscid (compressible);

7. Hypersonic flows, inviscid (compressible).

--------------

Your problem is that you are trying to use a compressible solver down through a region of no theory, down into an area of murky theory (convection-dominated, incompressible flows).

Until the physical phenomena in the 'no theory' zone is better understood, you should use solvers from both ends. In the middle is anyone's guess.

mw...

<www.adthermtech.com/wordpress3>

Yes. Your comments are throughly right. But I cannot understand through effects. I am solving an explosion. Thus, the initial conditions is vastly different.

In high pressure region, the pressure 10^9Pa while 10^5 in low pressure region. Thus, early time effects can be simulated by compressible Euler equations. However, as time goes I suffered from stability problems. In these times, I found that the density variation is small as in incompressible flows.

My questions are that in such case, whether there is an unified solution method. My current code formulated by Euler equations was aborted in the computation.

> In high pressure region, the pressure 10^9Pa while 10^5 in low pressure
: region. Thus, early time effects can be simulated by compressible Euler
: equations. However, as time goes I suffered from stability problems. In
: these times, I found that the density variation is small as in incompressible
: flows.

This is low Mach number flow not incompressible flow. If you read the reference I gave in an earlier posting, they are simulating explosions and discuss what they had to do to get their code to behave down to low Mach numbers. To the degree I understand your problem, you would seem to be having the same one.

> My questions are that in such case, whether there is an unified solution
: method. My current code formulated by Euler equations was aborted in the
: computation.

Why do you need a unified method? It would seem to me all you need to do is understand the source of your instability and fix it.

> My questions are that in such case, whether there is an unified solution
: method.

Yes. But perhaps more relevant is what is required to make your density based approach behave at low Mach numbers. This is a long studied and widely understood issue.

> I suggest to forget compressible solver and insited, extend a pressure
: based to compressible regime.

I am not sure this is wise advice because a pressure based scheme would not work well in the early stages of the explosion and this is likely to be the more relevant part of the simulation to get right.

> The major problems here are the huge gaps in the available theory.

How to handle the limit of zero Mach number is understood both in theory and in practical numerical codes. Theory supporting the Navier-Stokes equations is indeed incomplete but this is a far cry from huge gaps.

The gaps are so huge & fundamental that you could surf between the two extremes - literally.

To try & draw/merge assymptotic solutions across as yet unclear physics is foolishness in the extreme.

mw...

<www.adthermtech.com/wordpress3>

As I suggested in my previous reply, read my review about compressible solvers at low Mach numbers. You need to modify incompressible pressure based method to accommodate compressibility. This is not very difficult. The good point about compressible pressure based methods is switching between incompressible and compressible based on local Mach number.

I refer you to this article: http://www3.interscience.wiley.com/c...TRY=1&SRETRY=0

Apologies if the last line sounded a bit harsh.

It was meant to convey the thought that there even though we have a set of equations - the Navier-Stokes, for instance - we have a long way to go before we really understand the fluid flow environment from low speed right up to supersonic flow - with all terms included. The Euler equations are an asymptotic solution assuming no viscous effects.

In my experience, much of what we find in solution blow-ups & problems at lowish speeds can be attributed more to a lack of understanding the physical mechanisms work. Once these are are better understood, we may be in a better position to apply the N-S across the whole spectrum.

For instance, understanding the physics & using the correct simulation timing & mesh sizes - with no solution stabilisation - lowish speed transient flows can be simulated successfully time after time without blowup. For instance I've not had a blow-up for years, if the parameters are set correctly.

The physics reigns supreme & is our master - not the other way around.

mw...

<www.adthermtech.com/wordpress3>

The main problem here is that if you use density based codes for an incompressible case/region, density doesn't change so the solver fails unless you do some fancy things (preconditioning etc). Pressure based codes (most commercial codes are) do not have this limitation. Via the equation of state the density is linked to the pressure so both will show the proper behaviour in the compressible case/region.

When it comes to discontinuities (shocks) density based codes can perform better (using flux splitting, limiters etc) than pressure based codes. The latter will always show under/over shoots in the region of the shocks.

In my opinion: When your problem is largely compressible and you expect shocks (and high Mach numbers), density based codes will perform better (with the proper numerics). Pressure based codes are more widely applicable (hence the use in commercial codes), but lack some accuracy in handling shockwaves.

For engineering purposes I (would) use pressure based codes when maximum Machnumbers do not extend 2-3. Less hassle in the case of imcompressible (low Mach number) regions

> You need to modify incompressible pressure based method to
: accommodate compressibility. This is not very difficult.

Indeed it is not difficult but pressure correction schemes do not work well at high Mach numbers where solving a Poisson equation for a pressure correction is not really following the physics of what pressure does at high Mach numbers. Kludges like "retarded pressure" can help a bit but a better solution is to adapt the numerical scheme to the physical behaviour.

> The good point about compressible pressure based methods is switching
: between incompressible and compressible based on local Mach number.

This depends on whether an incompressible flow is an acceptable assumption. For example, those interested in acoustics at low Mach numbers (or at least acoustics that is coupled to the flow) have to solve the compressible equations.

> In my experience, much of what we find in solution blow-ups & problems
: at lowish speeds can be attributed more to a lack of understanding the
: physical mechanisms work. Once these are are better understood, we may
: be in a better position to apply the N-S across the whole spectrum.

This is counter to my experience in that it is usually straightforward to determine why a scheme has failed since everything is easily interogated unlike in an experiment. It is also straightforward to see what physical conditions have been violated. The challenges lie in the choices of what to enforce strictly whether locally or globally and how to do it while going about things in an efficient manner.

>> I suggest to forget compressible solver and insited, extend a pressure
: based to compressible regime.

>I am not sure this is wise advice because a pressure based scheme would not work well in the early stages of the explosion and this is likely to be the more relevant part of the simulation to get right.

I did not say, use incompressible solution. I told adopt incompressible solver to accomodate compressibility. I refer to the Karki and Patankar paper as excellent paper. With a SIMPLEC code, I managed to get solution in a nozzle up to Mach number of 3.5.

Karki, K. and Patankar, S. V. (1989) "Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations", AIAA journal, Vol 27, No 9, pp. 1167-1174.

andy wrote:

This is counter to my experience in that it is usually straightforward to determine why a scheme has failed since everything is easily interogated unlike in an experiment. It is also straightforward to see what physical conditions have been violated. The challenges lie in the choices of what to enforce strictly whether locally or globally and how to do it while going about things in an efficient manner.

mw replies:

The issue of importance that allows decent, repeatable simulation is to understand the underlying physics. The wrong constraints at the wrong position/situation in the physics tends to over-constrain the solution & can create problems.

Many simulation folks think they can control everything as the solution progresses - this often tends to over-constrain the problem at hand. In my experience it is better to work in concert with the physics & simulate in a relatively relaxed environment. Remember, the momentum equations are non-linear in nature.

At this juncture, the progression from momentum-driven waves to compression-dominated waves is not understood - if at all. Until you gain a better understanding of this, it is very much like shooting in the dark. Adding 'squeezers' eg. flow-stabilisation techniques, creates problems & inaccuracy.

mw...

<www.adthermtech.com/wordpress3>

> I did not say, use incompressible solution.

Nobody said you did.

> I told adopt incompressible solver to accomodate compressibility.

What does this mean?

You appeared to be advocating the use of pressure correction methods which can be a wise move at low Mach numbers depending on exactly what is of interest but they do not work well at high Mach numbers and indeed usually fail. Since accurate high Mach number flow would appear to be more important to Jinwon's problem of explosions than low Mach number flow getting an accurate high Mach number method to work at low Mach numbers would seem more sensible than the reverse.

> I refer to the Karki and Patankar paper as excellent paper. With a SIMPLEC
: code, I managed to get solution in a nozzle up to Mach number of 3.5.

I suspect Jinwon's objective is not to get his code to "manage to work" upto a Mach number of 3.5 particularly if his explosions are generating Mach numbers substantially larger than this. You do understand why solving for a pressure correction which allows a disturbance to be felt throughout the solution region is physically wrong at high Mach numbers and organising the solution procedure around a stiffness that is not present is, at best, not particularly useful at high Mach numbers.

Pressure correction schemes are a robust and useful approach to a wide range of CFD problems but they are not suitable for all problems unless the traditional schemes are modified to more accurately handle, for example, the physics of high Mach number flows and the physics of some compressible effects at low Mach numbers.

> At this juncture, the progression from momentum-driven waves to
: compression-dominated waves is not understood - if at all. Until you gain
: a better understanding of this, it is very much like shooting in the dark.
: Adding 'squeezers' eg. flow-stabilisation techniques, creates problems &
: inaccuracy.

Again, I would consider this to be far too strongly overstated to be reasonable. That we do not know everything about everything to do with fluids is not being disputed but to claim that the more knowledgeable researchers studying fluid mechanics are shooting in the dark is utter nonsense.

Flow stabilisation techniques do not necessarily cause problems or significant inaccuracies depending on the interests of the research. For example, low Mach number compressible methods often heavily distort and dissipate acoustic waves and this is usually considered a good thing if the waves themselves are not of interest and a more efficient method to get to what is of interest is obtained.