Numerical solution of the incompressible Navier–Stokes equations in primitive variables and velocity–vorticity formulation

A finite-difference method is presented for the numerical solution of the Navier–Stokes equations of motion of a viscous incompressible fluid in two dimensions in primitive-variables and velocity–vorticity formulation. For the case of primitive-variables, using a staggered grid and introducing an auxiliary function of the coordinate system and considering the form of the initial equation on lines passing through the nodal point (x0, y0) and parallel to the coordinate axes, we can separate it into two parts that are finally reduced to ordinary linear differential equations, one for each dimension. Discretization of these equations leads to a system of linear equations in n-unknowns which is solved by an iterative technique and the method converges rapidly giving satisfactory results. For the pressure variable we consider a pressure Poisson equation with suitable Neumann type boundary conditions. In case of velocity–vorticity formulation similar procedure is used, for a collocated grid. Numerical results up to Reynolds number 5000, confirming the accuracy of the proposed method, are presented for configurations of interest, such as Poiseuille flow, lid-driven cavity flow in square domain and flow in a backward-facing step in the presence of a pressure gradient.