SIMPLE pressure correction in compressible flows
I have been using CFD on incompressible flows for quite a while now, and have recently began studying compressible flows. One thing about these puzzles me considerably. In some of the books and codes I have studied (for example, 'Computational Methods for Fluid Dynamics' by Ferziger and Peric), a modified SIMPLEpressure correction method is used. I was under the impression that you need SIMPLE or its brethren in incompressible flows because you do not have an expression to compute the pressure field (and subsequently, its gradient). In compressible flows, you have an equation of state which relates density to pressure. (See for example on page 159 of the aforementioned book, and compare to page 293.)
Can someone please explain why you would want to use SIMPLElike pressure correction in compressible flows? 
Re: SIMPLE pressure correction in compressible flo
Not sure about the SIMPLE family, but the classic work at Los Alamos used a pressure correction field in incompressible problems to make sure the advancedtime velocities solved the continuity equation.
As you are aware, the pressure can sometimes be eliminated altogether from an incompressible problem, solving instead for the streamfunction vorticity or even the stream function only [via a 4th order PDE]. For good reason, these tactics have been more popular in twodimensional (space) than threedimensional space. 
Re: SIMPLE pressure correction in compressible flo
1) As you wrote, In compressible flows, you have an equation of state which relates density to pressure and temperature. Hence you do not need any special algorithm (or projection step) to compute the pressure field.
2) But, there can be three advantages using a projection step (such as in the simple family of schemes or in Chorin's projection's methods), even for compressible flows: a) the projection step tends to stabilize the scheme; b) with a projection step and cell centered FVM methods, you can make sure that the velocity field near walls agrees with the boundary conditions right at the begining of the solution process; c) the scheme can be used for both compressible (not hypersonic) and incompressible flows. 3) There might be one desanvantage: I'm not sure that projections methods are efficient when a entropic solution is sought (i.e. get the right speed of displacement of a shock). Good luck. 
Re: SIMPLE pressure correction in compressible flo
Sir i am working on research project "revemping of Francis Turbine applying CFD". I have got book "An introduction to computational fluid dynamics (finite volume) by malalasakra" it is very good book and i have understand SIMPLE, SIMPLER AND SIMPLEC method of solving Navier Stock equation but in this book just presentation that how Navier Stock equation can be solved but there is no solve example. Do you know any book or article in which this type of example is present. I mean that for any given problem(good if turbine rotor)firstly assume pressure field and then find velocity field then check pressure correction equation and repeat this procedure still convergence occure. waiting for your prompt reply thanks

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u better refer patankar for problem on SIMPLE alogrithm...because he was d guy who actually gave this therory...

You may want to use the SIMPLE family of algorithms for compressible flows, if:
[1] you already have a well tested and reliable flow solver for low Mach number flows that you now want to extend to high speed flows. [2] bulk of the flow domain has flow at low Mach number. 
See 2nd edition of Veersteg and Malasekhera. It has examples for the SIMPLE algorithm.

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