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Numerical simulation of shock wave patterns in supersonic divergent symmetric nozzles

Namieśnik K., Doerffer P.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

2005

Vol. 9, No 1

53-63

This paper presents the results of numerical simulations of supersonic flows with shock waves in a divergent symmetric nozzle of an opening angle ranging from 2 degrees to 6 degrees. At certain Mach number values the shock pattern becomes asymmetric. This asymmetry is analysed here for different values of velocity upstream of the shock wave and for different nozzle divergence angles. Only the divergent part of the nozzle is considered. Supersonic conditions at the nozzle inlet were prescribed with a chosen Mach number value Ma>1. The inlet velocity profile included a turbulent boundary layer profile on side walls. The steady flow simulation was applied for nozzle opening angles, α , of 1.877 degrees, 2.5 degrees and 3 degrees, whereas the unsteady approach was necessary for a nozzle of the divergence angle α =6.54 degrees to obtain a converged solution. The asymmetry of the shock structure is visible in the unevenness of the heights of both λ-feet. It happens at the same Mach number, at the same boundary layer and with the same geometrical constraints. This is in contradiction with our current understanding of the parameters affecting λ-foot size. The paper provides an explanation of this problem.

Pseudospectral approximation compared with control volume formulation and finite differences- some test cases

Tyliszczak A., Bogusławski A.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

2000

Vol. 4, No 2

193-202

The paper presents comparison of the pseudospectral method with low-order approximation for two test cases. The first test case is quasi one-dimensional supersonic flow through converging-diverging nozzle for which exact solution exists. The comparison of the error of pseudospectral approximation and upwind finite-differences using Steger-Warming flux splitting method shows high accuracy of the pseudospectral method even for few collocation points. The same conclusion is formulated for the second test case, namely incompressible flow in two-dimensional driven cavity solved by control volume formulation with modified QUICK upwinding scheme and SIMPLEC algorithm for pressure correction. As usually conclusions concerning accuracy of numerical methods are flow case dependent, but the two examples shown give some idea about the accuracy and resolution of spectral approximation versus standard CFD schemes.