THE APPLICATION OF A FINITE-VOLUME MULTIGRID METHOD TO 3-DIMENSIONAL FLOW PROBLEMS IN A HIGHLY VISCOUS-FLUID WITH A VARIABLE VISCOSITY

In this paper we discuss the application of a finite-volume multigrid method to solve three-dimensional thermally driven convection in a hig hly viscous, incompressible fluid with a variable viscosity. The conse rvation laws are solved in the primitive variable formulation. A secon d-order control volume method is used as discretization. Two schemes a re used for time stepping, a second-order implicit-explicit scheme bas ed on the Crank-Nicolson and Adams-Bashforth method, and a fully-impli cit theta-method. The implicit system of nonlinear equations are solve d using multigrid iteration with the SIMPLER method as smoother. In th is paper, we describe the implemented multigrid method and investigate its efficiency and the robustness for different viscosity contrasts. fs. Convergence tests showed that with a small modification of the SIM PLER method, the multigrid method exhibits a satisfactory convergence rate even for viscosity contrasts up to 10(9). Three cases of time-dep endent thermally driven convection with viscosity contrasts up to 10(5 ) are considered and discussed and the multigrid method has demonstrat ed its robustness also for these cases. Further, we have also compared the computational efficiency of the two time stepping methods. It app eared that the fully-implicit scheme is a little more efficient than t he implicit-explicit scheme for a constant viscosity and it was consid erably more efficient for a viscosity contrast of 10(3).