Numerical solution of buoyancy-driven flow problems

Abstract

Numerical solution of buoyancy-driven flow problems in two spatial dimensions is presented. A high-order spectral method is applied for the spatial discretization, while the temporal discretization is done by operator splitting methods. By solving the convection-diffusion equation, which governs the temperature distribution, a thorough description of both the spatial and the temporal discretization methods is given. A fast direct solver for the arising system of algebraic equations is presented, and the expected convergence rates of both the spatial and the temporal discretizations are verified. As a step towards the Navier--Stokes equations, a solution of the Stokes problem is given, where a splitting scheme technique is introduced. An extension of this framework is used to solve the incompressible Navier--Stokes equations, which govern the fluid flow. By solving the Navier-Stokes equations and the convection-diffusion equation as a coupled system, two different buoyancy-driven flow problems in two-dimensional enclosures are studied numerically. In the first problem, emphasis is put on the arising fluid flow and the corresponding thermal distribution, while the second problem mainly consists of determining critical parameters for the onset of convection rolls.