A higher-order unstructured finite volume solver for three-dimensional turbulent compressible flows

This thesis presents the development of a higher-order unstructured finite volume solver for three-dimensional turbulent compressible flows. The solver’s higher-order accuracy offers potential reductions in computational costs compared to traditional second-order methods. Among various discretization approaches, we select the k-exact reconstruction finite volume scheme for its straightforward formulation, reduced degrees of freedom on the same mesh compared to finite element methods, and ease of integration with industrial computational fluid dynamics (CFD) codes, which primarily use second-order finite volume methods. Existing three-dimensional higher-order finite volume solvers are typically limited to solving the Euler and laminar Navier-Stokes equations. Since many industrial applications involve turbulent flows, this thesis aims to extend our in-house two-dimensional turbulent solver to three dimensions. We begin by reviewing the k-exact finite volume formulation for the compressible Reynolds-averaged Navier-Stokes equations, coupled with the negative Spalart-Allmaras turbulence model, and discuss the flux functions and boundary conditions used. Next, the preprocessing steps for three-dimensional meshes in higher-order simulations are outlined. Additionally, we introduce a curvilinear coordinate transformation to overcome the challenges of k-exact reconstruction on curved, highly anisotropic meshes in three dimensions. We also develop a new higher-order method for wall distance computation on 3-D, curved, unstructured meshes, as accurate wall distance is crucial for both the turbulence model and the curvilinear coordinate transformation. The algorithm solves an optimization problem using Newton’s method and achieves fourth order accuracy for wall distance on curved boundaries represented by cubic polynomials. To solve the discretized system of nonlinear equations resulting from the finite volume method, we employ the pseudo transient continuation method, which converts the nonlinear system into a series of linear systems. An efficient preconditioning method is introduced to solve these linear systems, along with strategies for faster convergence to higher-order solutions. Finally, we present and verify solutions to four three-dimensional turbulent flow problems: subsonic turbulent flow over a flat plate, an extruded NACA 0012 airfoil, a hemisphere-cylinder geometry, and a three-dimensional bump in a channel. The results demonstrate the advantages of higher-order methods for achieving a desired level of accuracy.