On the spectral solution of the three-dimensional Navier-Stokes equations in spherical and cylindrical regions

Abstract

This paper investigates the application of spectral methods to the simulation of three-dimensional incompressible viscous flows within spherical or cylindrical boundaries. The Navier-Stokes equations for the primitive variables are considered and a generalized unsteady Stokes problem is derived, using an explicit time discretization of the nonlinear term. A split formulation of the linearized problem is then chosen by introducing a separate Poisson equation for the pressure supplemented by conditions of an integral character which assure that the incompressibility and the velocity boundary condition are simultaneously and exactly satisfied. After expanding the variables in convenient orthogonal bases, these integral conditions assume the form of one-dimensional integrals over the radial variable for the expansion coefficients of pressure, and are shown to involve the modified Bessel functions of half-odd order, in spherical coordinates, and of integer order, in the case of cylindrical regions with periodic boundary conditions along the axis. Such integral conditions represent the counterpart for pressure of the vorticity integral conditions introduced by Dennis for studying plane and axisymmetric flows and reduce the solution of the three-dimensional unsteady Stokes equations within spherical and cylindrical boundaries to a sequence of uncoupled second-order ordinary differential equations for only scalar unknowns. A Chebyshev spectral approximation is then considered to resolve the radial structure of the flow field. Numerical results are given to illustrate the convergence properties of the discrete equations obtained by the tau projection method. The problem of the efficient evaluation of the nonlinear term is not examined in the present paper. Finally, for the sake of completeness, the treatment of coordinate singularity in regions bounded by a single spherical or cylindrical surface is also discussed.