ON THE SPECTRAL SOLUTION OF THE 3-DIMENSIONAL NAVIER-STOKES EQUATIONSIN SPHERICAL AND CYLINDRICAL REGIONS

This paper investigates the application of spectral methods to the sim ulation of three-dimensional incompressible viscous flows within spher ical or cylindrical boundaries. The Navier-Stokes equations for the pr imitive variables are considered and a generalized unsteady Stokes pro blem is derived, using an explicit time discretization of the nonlinea r term. A split formulation of the linearized problem is then chosen b y introducing a separate Poisson equation for the pressure supplemente d by conditions of an integral character which assure that the incompr essibility and the velocity boundary condition are simultaneously and exactly satisfied. After expanding the variables in convenient orthogo nal bases, these integral conditions assume the form of one-dimensiona l integrals over the radial variable for the expansion coefficients of pressure, and are shown to involve the modified Bessel functions of h alf-odd order, in spherical coordinates, and of integer order, in the case of cylindrical regions with periodic boundary conditions along th e axis. Such integral conditions represent the counterpart for pressur e of the vorticity integral conditions introduced by Dennis for studyi ng plane and axisymmetric flows and reduce the solution of the three-d imensional unsteady Stokes equations within spherical and cylindrical boundaries to a sequence of uncoupled second-order ordinary differenti al equations for only scalar unknowns. A Chebyshev spectral approximat ion is then considered to resolve the radial structure of the flow fie ld. Numerical results are given to illustrate the convergence properti es of the discrete equations obtained by the tau projection method. Th e problem of the efficient evaluation of the nonlinear term is not exa mined in the present paper. Finally, for the sake of completeness, the treatment of coordinate singularity in regions bounded by a single sp herical or cylindrical surface is also discussed.