Compressible vs. Incompressible formulations
 

First, thanks to Mr. Chien for the advice on validation cases for my code to solve axisymmetric compressible flow.

My formation in CFD began with incompressible algorithms based in finite volume discretization and now my thesis deals with the extension of incompressible algorithms to compressible flows (you should understand that only recently we began to introduce in CFD in this part of the world). However, examining the lately works on compressible flow, it seems that nobody uses incompressible formulations in this field anymore.

My questions are: What are the principal disadvantages of using incompressible formulations for compressible flow? What are the advantages of the compressible formulations over the incompressible ones? Is there compressible flows that can't be solved with incompressible algorithms?

Actually, I am using the PRIME algorithm (by the way, anybody has ever heard about this algorithm?). With it I can easily obtain subsonic and supersonic solutions for the Euler equations.

Ahead of time, thanks for your cooperation.

Fernando Velasco Hurtado

Oruro-Bolivia

Re: Compressible vs. Incompressible formulations
 

Compressibility effects become important when the speed of the flow becomes comparable to the speed of sound in the flow. If you denote the Mach number M=v/c, where v is the speed of the propagation of the flow and c is the sound speed, then for small M the flow can be modeled with an incompressible code, while for large M it has to be modeled with a compressible flow. The limit between the two different modeling is for M around 0.3, but it can certainly vary from problem to problem. So for transonic and supersonic flows, you must use a compressible code, while for strongly subsonic flows you can use an incompressible code. The main thing that you are missing when you use an incompressible code for a compressible flow is the propagation of sound waves and shock waves. And that can be actually the most important thing.

Cheers, Patrick

Re: Compressible vs. Incompressible formulations
 

(1). Based on the form of governing equations (Navier-Stokes equations) used in the formulation, there are several types available. (2). The most general one is the transient, compressible flow equations, where the continuity equation is written as: dRHO/dt + d(RHO*U)/dx + d(RHO*V)/dy + d(RHO*W)/dz =0, this provide the density evolution, and is commonly called density-based approach. Most aerospace related applications are based on this type of formulation. It can handle shock waves, and supersonic flows. (3). But, since it is transient approach, it is very slow when the time step must be small. And methods used to solve this thpe of Navier-Stokes equations are sometimes derived from the Euler solver side. (4). For incompressible flows, the continuity equation becomes dU/dx+dV/dy+dW/dz=0 for both transient and steady-state flows. The direct use of the continuity equation in the numerical solution algorithm becomes a problem. Normally, a pressure equation of various form is derived from the continuity equation and the momentum equations. And the pressure equation is solved with the momentum equations in certain ways. (SIMPLE approach is an example). (5). When the continuity equation is replaced by the pressure equation, it is normally called the pressure-based approach or formulation. This approach was originally used for the incompressible flows. It was later relaxed to have density variation.(subsonic, variable density flow. combustor flow is a good example) (6). The pressure-based approach has been extended to supersonic flows also. But sometimes, special algorithms are borrowed from the density-based method, so that the shock waves can be captured more accurately. Since the density is constant or slow-varing function, the convergence rate of this formulation is normally many times faster than the density-based formulation. (this type of formulation sometimes is called incompressible, primitive variable formulation) (7). The third school of formulation is the so-called vorticity-based formulation, where the pressure terms are eliminated explicitly from the momentum equations. the resultant equations are the vector vorticity equation, the second-order velocity equation, and a pressure recovery procedure or equation. In this vorticity formulation, the pressure term is de-coupled from the momentum equations. This formulation can also be used for variable density flows. (8). Under the vorticity-based formulation, there is this very famous 2-D stream function-vorticity formulation, which was very popular in the early days of CFD development. Because of the use of the stream-function in the formulation, the conservation of mass throughout the computational domain is automatically satisfied. This is the only formulation which can satisfy the continuity equation automatically. (you are not so lucky using other types of formulations). (9). Traditionally, it is common practice to select or develop solution algorithms based on whether it is 2-D or 3-D, incompressible or compressible flows. It is very similar to the kitchen environment, where you have more than one kitchen knives to handle different tasks. (10). In the case of the density-based formulation, it can not function at the incompressible flow limit, where the density is nearly constant and the velocity field can be obtained independent of the pressure field. it becomes very hard to synchronize two twin solution of the velocity and the pressure. And there will be oscillations and extremely slow convergence (if possible). (11). With the addition of reacting flows, turbulent flows, multi-phase flows, moving boundary flows, CFD solution is extremely non-linear and very hard to find.

Re: Compressible vs. Incompressible formulations
 

A reference about the PRIME algorithm is "A solution method for three-dimensional parabolic flow problems in nonorthogonal coordinates", C.R. Maliska, PhD thesis, University of Waterloo, Canada, 1981. Other reference would be "Numerical methods for the solution of Incompressible and Compressible fluid flows", J P Van Doormal, PhD thesis, University of Waterloo, Canada, 1985.