Solving unsteady compressible low speed flow
Dear sir, I want to write the program that can solve both incompressible and compressible flow. So I treat all flow as compressible flow that mean the incompressible flow is the compressible flow at low speed. However I wonder that how do I manage with the density in the unsteady term. It will be approximately zero at every time step, I think. I will get the problem from this or I won't? How do I solve this problem? Please suggest me? Thank you very much.
Best regards. Atit Koonsrisuk 
Re: Solving unsteady compressible low speed flow
You will not have any problem using the compressible flow solver at very low speeds. But the flow will never ever be fully incompressible though it might be close. Also, the efficiency of a compressible code at very low speeds would be very low compared to that of the incompressible flow solvers.
However, there are implicit schemes that can be used for compressible flow solver to enable use of large time steps. But the implicit scheme would take a lot of iterations and CPU time to converge at each time step if a good convergence acceleration technique such as multigrid or preconditioning are not used. The problem is that multigrid and preconditioning are tricks that do not always work well. You just try various of these techniques till you find one that you are happy with. Sometimes, if you know something about the physics of the flow, choosing an apropriate preconditioner may become easier. 
Re: Solving unsteady compressible low speed flow
Dear sir, Thank you very much for your answer. You said "if you know something about the physics of the flow, choosing an apropriate preconditioner may become easier." Could you please explain to me more about this with some example? Thanks.
Regards. Atit Koonsrisuk 
Re: Solving unsteady compressible low speed flow
Convergence is difficult when a systems of equations have a wide spectrumi, i.e., widely varying eigen values. Preconditioners, in an approximate sense, try to collapse the spectrum (bring the eigen values closer) to facilitate faster convergence.
As far as physics based conditioners are concerned, I am no expert. Here a couple of examples. If you have a flow in a pipe of length L (or a dump combustor of length L), then the acoustics most likely have wavenumbers which are integral multiples of (2 * pi)/L. You can used the socalled sine or cosine transform based (or Fourier) preconditioners to convergence the acoustics part of the flow. In some CFD solvers, you use what is called dual time stepping. It involves integration a system of equation with very large time steps. The iterative scheme is used to converge the solution at each time step. The iterations are done by pseudotimestepping. During this pseudotimestepping (inner iterations), you can use different pseudotime steps at different locations. Since the grid spacing may be different at different locations, the pseudo time stepping scheme has different stability restrictions at different locations. These are only simple examples. You can read a lot more about preconditioning in papers where steady state methods are discussed. 
Re: Solving unsteady compressible low speed flow
Although physically, impcompressible flow is just compressible flow that is moving slowly. Mathematically they are not always the same! A lot of compressible solvers treat the continuity equation as a transport equation for density with pressure being obtained from an equation of state. At low Mach numbers (below 0.3) with neglible changes in density, this approach becomes inaprropriate. To solve for incompressible flow you need to use a pressure correction algorithm like SIMPLE (Splading & Patankar 1972) or PISO (Issa 1986). These use the continuity equation as a constraint equation for pressure, ie guess the pressure field and then solve for velocites to satisfy continuity.

Re: Solving unsteady compressible low speed flow
You can use compressible solver for impressible problem with preconditioning. Please check some preconditioning methods of C. Merkle's or Turkle's.
Ding 
Re: Solving unsteady compressible low speed flow
Hi,
Although, Kalyan may be right in saying that there is no universal preconditioning technique, it is the only effective way of using a compressible formulation for Mach numbers tending to zero. For unsteady situations, a pseudo time step can be used. My previous group at Iowa State University use a preconditioning technique quite effectively for Mach numbers as low as 0.001. We also solve for the primitive variables (p,u,v,w,T) by substituting the ideal gas law everywhere for the density (p/RT). chidu... 
Re: Solving unsteady compressible low speed flow
you're quite correct. the unsteady compressible formulation can always be used even for slow flows. a combination of preconditioners, multigrid and dual time stepping is probably the best choice for unsteady flows as Atit desires. this atleast is position of my professor in CFD who is a colleague of Dr. Merkle. the use of dual time stepping (aka pseudo time, inner iterations etc) will improve the time accuracy which is lost by using preconditioning and/or (some) implicit solvers. multigrid can also be used and should be quite effective since it was originally invented for use with diffusion problems which this is. i haven't tried using multigrid myself but many people seem to have problems with it and i don't know why. the literature is filled with examples of very successful multigrid codes. search for Jameson or Periaux for references.

Re: Solving unsteady compressible low speed flow
Hi,
Though I have not worked on lowspeed solver, I am aware of few pressure based method, which are used for both compressible and incompressible flow, one of such paper is by Peric (year i don't exactly remember) in Int. Jou of Num. method in fluids) may be 199192. second in by prof. AW Date ( a finite difference solver) in Numerical Heat Transfer (1996 & 1998). You can refer that. One of my friend has worked on such solvers, he has used Turkel's preconditioning method (JCP, 1987 p.327 (no so sure of page no.)) and has obtained fairely decent result's, so I would suggest to take a look on it too. Regards Apurv 
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