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Problems with convergence and initial guessed fields in compressible steady flows 

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April 19, 2000, 15:51 
Problems with convergence and initial guessed fields in compressible steady flows

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I try to solve a supersonic steady problem neglecting the temporal term, in the compresible NS ecuations with all terms term (i.e time step =inf). The geomtry is the typical test problem (channel with a circular arc "bump") My boundary conditions are density,pressure,end velocity prescribed at inlet , and extrapolate at outlet. I start with a uniform guessed velocity field with the same value that the inlet velocity. idem for temperature and idem for pressure. I solve Momentum and pressure correction (simple) since convergence in velocity is reached. (criterium is mean((unewuold)/uold)). I use strong underrelaxation in momentum and pressure correction. (p=p+alpha*p'); The two reflecting shock waves begin to develop in the channel and when iterations converge , I solve energy ecuation and actualize properties. Then begin to iterate again. When I do this two or three times the fiels diverge and the method shut down. Why this happens?.Is the grid incapable of support great magnitude gradient values? ,Might I regrid for more acuratte?. The problem ussually ocurrs in the cross of two shock waves. If anyone has any Solution or explanation to the Problem please contact me!. You Can help me a lot with my studiesend proyect!!!!. 

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April 19, 2000, 17:58 
Re: Problems with convergence and initial guessed fields in compressible steady flows

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(1). You need to find out first whether your method works for supersonic flow with shocks. (2). Try simple test cases to address this issue. (3). What is your Mach number? (4). Is the flow transonic or supersonic? These will affect your solutions. good luck.


April 20, 2000, 00:02 
Re: Problems with convergence and initial guessed fields in compressible steady flows

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Compressible supersonic flows at highspeed are much stiffer and more 'nonlinear' that incompressible flows. In order to obtain a solution, you must march in time slowly, by retaining as much of the time accuracy of the scheme as possible. A CFL constant not much greater than one is generally required to obtain a solution when strong shocks are present, unless the solution is already close to being converged. Just write your scheme so that it preserves time accuracy and your problem should converge more smoothly.
Also, when dealing with shocks, a TVD scheme based on a monotonic first order scheme is generally a must to discretize the convective terms. If the shocks are weak (transonic flow for eg) you might get away with a Jamesontype artificial dissipation scheme but such a method will fail at higher shock Mach numbers.. In short, compressible flows cannot generally be solved using an extension of methods developped for incompressible flows, the opposite being also true: you need different schemes for different problems. bernard 

April 25, 2000, 11:34 
Re: Problems with convergence and initial guessed fields in compressible steady flows

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Dear Petro,
From your email I can see you are using the SIMPLEalgorithm to solve the problem. Basically, this approach (originally developed for incompressible flow) should be extended for its use to solve high Mach number flows with shocks. So you have to take into account that a variation in pressure (p'in SIMPLE) leads to a variation/correction in both velocity and density. The influence upon temperature is achieved via the equation of state in the outer iterations process. How this can be done as well as the derivation of a pressure correction equation in its compressible formulation is given in 'Karki and Patankar (1989)ressure based ......, AIAA Journal, Vol 27, pp. 11671174' Another important fact which strongly influences convergence of the method is the treatment of density at the surfaces of the control volumes. UPWIND scheme provides sufficient numerical diffusion to get a stabil algorithm but leads to bad shock resolution. Here, a mixed approach (e.g. 80% higher order scheme plus 20% UPWIND) improves resolution of the shock significantly while retaining stability. To derive the compressible pressure correction equation the correction of density at the surfaces can be approximated by using the UPWIND scheme. This has no influence upon the accuracy of the final solution since corrections tend to zero then. As far as I know from my own calculations one can start to solve the whole set of equations (momentum, pressurecorrection, turbulence, energy etc...) right from the beginning. But the assumption of a constant total enthalpy should be justified at a first step. Good luck, Dietmar 

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