Finite-element simulation of buoyancy-driven turbulent flows

Turbulent flows driven or significantly affected by buoyancy occur in a variety of problems including building ventilation, cooling of electrical equipment, and environmental science. The fundamental mathematical model are the non-isothermal, incompressible Navier-Stokes equations. Their solution can become turbulent (and hence computationally infeasible), if a critical parameter, e.g., the Reynolds number or the Rayleigh number, becomes too large. The aim of turbulence modelling is to devise mathematical models whose solutions are close to those of the Navier-Stokes equations, but can be computed at reasonable costs.In this thesis both a statistical turbulence model, i.e., the k-epsilon model, and three LES models, viz., the Smagorinsky model, the Iliescu-Layton model (both including a modification devised by Eidson), and the Galdi-Layton model, are considered. Near solid walls the solution often exhibits sharp gradients, called boundary layers. An appropriate near-wall grid refinement is computationally often infeasible, in particuler for most 3D problems of practical relevance. As a remedy an improved wall function concept is applied. This approach can be viewed as a fully-overlapping domain decomposition method: The flow problem is divided into a global problem and a problem in the near-wall region, called boundary-layer problem. The boundary-layer solution satisfies the correct boundary conditions at the wall for velocity and temperature resp. and is matched with the global solution on an artificial inner boundary. For the global problem modified boundary conditions are imposed, e.g., a boundary condition for the tangential stresses and for the heat transfer across the wall, with right hand sides being determined by the boundary-layer solution.This approach covers a variety of coupling schemes, depending on both the boundary-layer problem and the boundary conditions for the global problem. Existence and uniqueness of a solution can be proved for a certain scheme coupling LES as a global model with the Navier-Stokes equations in the near-wall region. In order to reduce computational costs and to facilitate the implementation, a much simpler model for the near-wall region is derived. This approach can be formulated as an improved wall-function model, which accounts for effects of stratification in the boundary layer. The iterative solution process requires the fast solution of linearized Navier-Stokes problems and of advection-diffusion-reaction problems. These subproblems are discretized using stabilized FEM. In order to parallelise the scheme, a non-overlapping domain decomposition method is employed. Finally, the accuracy of the approach is investigated. As a benchmark test case the natural convection flow in an air-filled closed square cavity with differentially heated side-walls is chosen. The numerical results are assessed by reference with experimental data.