A Galerkin Spectral Method Based On Helical-Wave Decomposition For The Incompressible Navier-Stokes Equations

This paper presents a global Galerkin spectral method for solving the incompressible Navier-Stokes equations in three-dimensional bounded domains. The method is based on helical-wave decomposition(HWD), which uses the vector eigenfunctions of the curl operator as orthogonal basis functions. We shall first review the general theory of HWD in an arbitrary simply connected domain, along with some new developments. We then employ the HWD to construct a Galerkin spectral method. The current method innovates the existing HWD-based spectral method by (a) adding a series of auxiliary fields to the HWD of the velocity field to fulfill the no-slip boundary condition and to settle the convergence problem of the HWD of the curl fields, and (b) providing a pseudo-spectral method that utilizes a fast spherical harmonic transform algorithm and Gaussian quadrature to calculate the nonlinear term in the Navier-Stokes equations. The auxiliary fields are uniquely determined by solving the Stokes and Stokes-like equations under adequate boundary conditions. The implementation of the method under the spherical geometry is presented in detail. Several numerical examples are provided to validate the proposed method. The method can be easily extended to other domains once the helical-wave bases, which depend only on the geometry of the domains, are available. Copyright (c) 2015 John Wiley & Sons, Ltd.