Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions

Abstract

Two numerical methods were designed to solve the time-dependent, three-dimensional, incompressible Navier-Stokes equations in boundary layers (method A, semi-infinite domain) and mixing layers or wakes (method B, fully-infinite domain). Their originality lies in the use of rapidly-decaying spectral basis functions to approximate the vertical dependence of the solutions, combined with one (method A) or two (method B) slowly-decaying “extra functions” for each wave-vector that exactly represent the irrotational component of the solution at large distances. Both methods eliminate the pressure term as part of the formulation, thus avoiding fractional-step time integration. They yield rapid convergence and are free of spurious modes in the Orr-Sommerfeld spectra. They are also efficient, although the operation count is of order N 2 ( N is the number of modes in the infinite direction). These methods have been used for extensive direct numerical simulations of transition and turbulence. A new time-integration scheme, with low storage requirements and good stability properties, is also described.