| In this thesis a new h-p finite element method for the compressible Navier-Stokes equations is presented. The method is based on a discontinuous Galerkin formulation for both the advective and the diffusive contributions. High-order accuracy is achieved by using an orthogonal spectral hierarchical basis inside the triangles/tetrahedra. This basis is formed by combining Jacobi polynomials of high-order weights that retain a tensor product property, and accurate numerical quadrature. The formulation is conservative, and monotonicity is enforced by appropriately lowering the basis order and performing h-refinement around discontinuities. In order to solve the Navier-Stokes equations in moving or deforming domains, an efficient Arbitrary Lagrangian Eulerian (ALE) formulation in the discontinuous Galerkin framework is implemented. A new force-directed algorithm for updating the grid is developed. A number of simulations using the new method is presented including validation using published results, benchmark problems showing exponential convergence in space and third-order convergence in time, subsonic and supersonic flows around pitching airfoils. |
