Finite-difference methods for calculating steady incompressible flows in three dimensions

Abstract A new stable numerical method is described for solving the Navier-Stokes equations for the steady motion of an incompressible fluid in three dimensions. The basic governing equations are expressed in terms of three equations for the velocity components together with three equations for the vorticity components. This gives six simultaneous coupled second-order partial differential equations to be solved. A finite-difference scheme with second-order accuracy is described in which the associated matrices are diagonally dominant. Numerical results are presented for the flow inside a cubical box due to the motion of one of its sides moving parallel to itself for Reynolds numbers up to 100. Several methods of approximation are considered and the effect of different discretizations of the boundary conditions is also investigated. The main method employed is stable for Reynolds numbers greater than 100 but a finer grid size would be required in order to obtain accurate results.