Numerical Simulation of Incompressible Flows Within Simple Boundaries. I. Galerkin (Spectral) Representations

Galerkin (spectral) methods are explored for the numerical simulation of incompressible flows within simple boundaries. A major part of the paper is devoted to the development of transform methods for efficient simulation of flows in box geometries with periodic and free‐slip boundary conditions. Techniques for incorporating known symmetries and invariances into transform methods are illustrated for the Taylor‐Green vortex. Galerkin methods for accurate and efficient representation of rigid no‐slip boundary conditions are also explained. A class of pseudospectral approximations is introduced in order to handle more complicated dynamical interactions in more complicated geometries. Later papers in this series will demonstrate the important advantages of spectral methods over finite‐difference methods for simulation of many of the flows of current interest and will present specific numerical results for various transition and turbulent flows.