Spectral Calculations of Convection in Low-Pr Fluids

Abstract

The computations are done using the Navier-Stokes equations within the vorticity-stream function formulation. These equations associated (in the case Pr=0.015) with the temperature equations are solved using the spectral method proposed in [1]. The time-differencing makes use of the semi-implicit Adams-Bashforth / second-order backward Euler scheme [1],[2]: the diffusive terms are considered implicitely while the nonlinear convective terms are evaluated explicitely. In this way, at each time step, we have to solve a Stokes-type problem for the vorticity and the streamfunction, and a Helmholtz equation for the temperature. The spatial approximation makes use of Chebyshev polynomial expansions in both directions. The boundary conditions in the Stokes-type problem are handled by means of an influence matrix technique, leading to the solution of Helmholtz equations only. These equations are solved by the Tau method associated with the matrix diagonalization technique leaving the major part of the calculations in a preprocessing stage.