A high‐order finite difference method for incompressible fluid turbulence simulations

A Hermitian-Fourier numerical method for solving the Navier-Stokes equations with one non-homogeneous direction had been presented by Schiestel and Viazzo. An extension of the method is devised for solving problems with two non-homogeneous directions. This extension is indeed not trivial since new algorithms will be necessary, in particular for pressure calculation. The method uses Hermitian finite differences in the non-periodic directions whereas Fourier pseudo-spectral developments are used in the remaining periodic direction. Pressure-velocity coupling is solved by a simplified Poisson equation for the pressure correction using direct method of solution that preserves Hermitian accuracy for pressure. The turbulent flow after a backward facing step has been used as a test case to show the capabilities of the method. The applications in view are mainly concerning the numerical simulation of turbulent and transitional flows

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