Splitting methods for high order solution of the incompressible Navier–Stokes equations in 3D

Abstract

SUMMARYThe incompressible Navier-Stokes equations are discretized in space by a hybrid method and integrated in time by the method of lines. The solution is determined on a staggered curvilinear grid in two space dimensions and by a Fourier expansion in the third dimension. The space derivatives are approximated by a compact ÿnite di erence scheme of fourth-order on the grid. The solution is advanced in time by a semi-implicit method. In each time step, systems of linear equations have to be solved for the velocity and the pressure. The iterations are split into one outer iteration and three inner iterations. The accuracy and e ciency of the method are demonstrated in a numerical experiment with rotated Poiseuille ow perturbed by Orr-Sommerfeld modes in a channel.

Numerical solution of incompressible flow equations over irregular geometry, with application to coastal hydrodynamics

“…Grid convergence tests were carried out for all cases, with evidently good convergence behavior with grid refinement. With further development, the methodology will include: (a) expansion of the transformed N-S equations in three dimensions [38], (b) an implicit temporal scheme [12], [13], in order to alleviate time-step restrictions and (c) the application of acceleration techniques for the solution of the Poisson equation for pressure [71]. These are considered necessary steps to allow for high-order accurate solutions of increased Reynolds number flows.…”