A Spectral Algorithm for the Numerical Simulation of Turbulent Flows over Complex Geometries

The primitive variable formulation of the unsteady incompressible Navier-Stokes equations in three space dimensions is discretized with a combined Fourier-Legendre spectral method. A semi-implicit pressure correction scheme is applied to decouple the velocity from the pressure. The arising elliptic scaler problems are first diagonalized in the periodic Fourier direction and then solved by a multidomain Legendre collocation method in the two remaining space coordinates. In particular, both an iterative and a direct version of the so-called projection decomposition method (PDM) are introduced to separate the equations for the internal nodes from the ones governing the interface unknowns. The PDM method, first introduced by V. Agoshkov and E. Ovtchinnikov and later applied to spectral methods by P. Gervasio, E. Ovtchinnikov, and A. Quarteroni is a domain decomposition technique for elliptic boundary value problems, which is based on a Galerkin approximation of the Steklov-Poincare equation for the unknown variables associated to the grid points lying on the interface between subdomains. After having shown the exponential convergence of the proposed discretization technique, some issues on the efficient implementation of the method are given. Finally, as an illustration of the potentialities of the algorithm for the numerical simulation of turbulent flows, the results of a direct numerical simulation (DNS) of a fully turbulent plane channel flow are presented.