The Fractional Step Navier–Stokes Solver: Preconditioning and Application to Conjugate Natural Convection Boundary Layers

Abstract

Viscous, incompressible flow is governed by the Navier–Stokes equations. The simulation of this flow is difficult to solve due to the coupling between the pressure and velocity in the momentum equations. Fractional step method is applied to the Navier–Stokes equations to decouple the pressure and velocity, which results in two segregate calculations, one of which is the momentum equations and the second is the pressure Poisson equation. In the first part of the thesis, preconditioning methods for the pressure solver in the fractional step method, in order to accelerate the convergence of the Navier–Stokes solution, are examined. The preconditioning methods investigated are the simple methods, the incomplete Lower-Upper variants and the sparse approximate inverses. The sparse approximate inverses are modified to have a defined sparsity patterns. Modified sparse approximate inverses give comparable performance to the adaptive one and the incomplete Lower-Upper variants. The method is also combined with domain decomposition method for parallel computing, in which the preconditioners are built locally, but this has been shown not to degrade the performance. The second part of the thesis contains the implementation of the Navier–Stokes solution for conjugate natural convection boundary layers. This type of boundary layer occurs when a conducting vertical plate is placed between two fluids at different temperatures. A scaling analysis for the unsteady growth of the boundary layers, including the start-up, transition and fully developed stages, is presented. The universal scalings, in relations to the Prandtl number, Rayleigh number and height location, are confirmed via numerical solutions and applicable over the full range of Prandtl numbers.