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Method of lines for the incompressible Navier-Stokes equations in the stream-function formulation

Kosma Z.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

2005

Vol. 9, No 1

17-35

The aim of this paper is to simulate the laminar motion of viscous incompressible fluid and the transition between the laminar and the turbulent state in simply connected domains. The developed numerical algorithms are based on the solution of an initial-boundary value problem for the full incompressible Navier-Stokes equations, written in the form of a fourth-order equation for the stream function. The spatial derivatives and the boundary conditions are discretized on uniform grids by means of sixth-order compact schemes together with fourth-order finite-difference formulas, while the continuity of the time variable is preserved. The resulting system of ordinary differential equations has been integrated using the backward-differentiation predictor-corrector method. The efficiency of the numerical algorithms is demonstrated by solving two problems of viscous liquid plane flows in a square driven cavity and a backward-facing step. Calculations for the cavity flow configuration have been obtained for Reynolds numbers ranging from Re=100 to Re=30000 on uniform 50×50 and 100×100 grids. Calculations for the backward-facing step have been made for Re ≤ 3000 with channel lengths, L, within the range 10-30, on 30L×30 uniform grids. The computed stream-function contours and velocity fields have been compared with numerical results reported in the literature.

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.