Analysis of the Unsteady Navier-Stokes Equations Based on Green’s Function Approach

A computational method for the solutions of the time-dependent Navier-Stokes equations in their stream function and vorticity formulation, is developed. Integral representations of stream function and vorticity and an integral equation for the time-dependent boundary vorticity, are obtained by use of Green’s functions. A two-dimensional flow in a rectangular cavity is considered, assuming that the fluid is at rest initially and the upper wall is set in motion at t{=}0. The discretization procedure is applied to solving the integral equations. Numerical solutions of transient state reaching the steady one are presented for Reynolds numbers R of 0, 100 and 200.