Extension of Kleiser and Schumann’s influence-matrix method for generalized velocity boundary conditions

In 1980, Kleiser and Schumann introduced a novel influence-matrix method to treat the incompressibility and no-slip boundary conditions when solving the Navier–Stokes equations. They also outlined the related “tau” error correction technique which is essential for the high accuracy direct numerical simulation (DNS) of turbulent flows. However, their method is not valid for Robin type velocity boundary conditions (i.e., B(u) = αu + βu′ − γ = 0). In this note, a new influence-matrix method is introduced where the boundary condition and “tau” correction are enforced in one step using an extended influence matrix. The new method is simple and easy to be implemented. It broadens the applicability of the Kleiser and Schumann method. Examples with the new method show excellent agreement with data in the literature and the velocity field is divergence free up to machine precision.