A Compact-Difference Scheme for the Navier–Stokes Equations in Vorticity–Velocity Formulation

This paper presents a new numerical method for solving the incompressible, unsteady Navier–Stokes equations in vorticity–velocity formulation. The method is applicable to spatial simulations of transitional and turbulent boundary layer flows. It is based on a compact-difference discretization of the streamwise and wall-normal derivatives in Cartesian coordinates. A Fourier collocation approach is used for the spanwise derivatives. Important new features of the numerical method are the use of nonequidistant differences in the wall-normal direction; the use of split-compact differences in the streamwise direction; a new, fast iteration for a semi-implicit time integration of the wall-normal diffusion terms; and an improvement of the buffer domain technique to prevent reflections of waves at the outflow boundary. Results of test calculations are presented to verify the improvements obtained by the use of these new techniques.