Advanced study for the numerical approximation of Navier-Stokes equations by implementation of a finite volume solver

Abstract

The objective of the thesis is to build different fluid solvers from scratch, whereby the following three cases are considered: Resolution of diffusion problems, convection-diffusion and Navier-Stokes equations. In the process, the conservation equations are approximated in order to convert them into an algebraic system of equations. The code will be implemented in C++ and the equations will be solved either directly or iteratively. The investigation will be applied to incompressible and laminar flows. Within this thesis the finite volume (FV) discretization method is used. Furthermore, a cartesian coordinate system is chosen and the numerical grid applied is an orthogonal cell center mesh. The influence of the refinement of the mesh as well as the time step influence will be analyzed. Moreover, Neumann and Dirichlet boundary conditions will be applied. A different solver will be implemented for each case: First, pure conduction heat transfer in 1D and 2D will be discussed regarding the resolution of diffusion problems. The implementation with the help of the TDMA and line-by-line method will be carried out. The results will be compared to analytical values to prove the accuracy of the solver. Second, a resolution of the convectiondiffusion equation for 2D flows will be investigated. For this purpose, the Smith-Hutton case and a diagonal flow in a square domain will be used to test the code as a benchmark problem. Upwind-, central- and smart-difference schemes will be discussed. The results will be compared to numerical studies on different Peclet numbers. Finally, the resolution of the Navier-Stokes equations will be studied with the fractional step method. As a benchmark problem the driven cavity case will be used, for a 2D case. Upwind- and central- schemes will be covered. Moreover, the effect of different Reynolds numbers on the solution will be examined.