Chapter 2 - Spectral numerical methods for turbulence simulation

Abstract

Spectral numerical methods are commonly used for the direct numerical simulation (DNS) and large eddy simulation (LES) of turbulence. The reason is that they exhibit very rapid convergence to smooth solutions and have excellent resolution properties, allowing the number of computational degrees of freedom required for a given accuracy to be minimized. This is particularly important for turbulence simulation in which a wide range of spatial scales must be represented. In spatial directions in which the turbulence is homogeneous so that periodic boundary conditions are generally used, spectral methods based on Fourier expansions are applicable. Fourier functions have the additional advantage of being eigenfunctions of the derivative operator, which simplifies solution algorithms. In other situations, spectral methods based on orthogonal polynomials are appropriate. This chapter is an introduction to the use of spectral methods for turbulence simulations, including discussion of the properties of Fourier and orthogonal polynomial expansions, and practical considerations in the use of spectral methods for this application.