METHOD OF LINES FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN THE STREAM-FUNCTION FORMULATION

Abstract

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 R_e=100 to R_e = 30 000 on uniform 50×50 and 100×100 grids. Calculations for the backward-facing step have been made for R_e ≤ 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.