A discontinuous Galerkin method for the solution of compressible flows

Abstract

This thesis presents a methodology for the numerical solution of one-dimensional (1D) and two-dimensional (2D) compressible flows via a discontinuous Galerkin (DG) formulation. The 1D Euler equations are used to assess the performance and stability of the discretisation. The explicit time restriction is derived and it is established that the optimal polynomial degree, p, in terms of efficiency and accuracy of the simulation is p = 5. Since the method is characterised by minimal diffusion, it is particularly well suited for the simulation of the pressure wave generated by train entering a tunnel. A novel treatment of the area-averaged Euler equations is proposed to eliminate oscillations generated by the projection of a moving area on a fixed mesh and the computational results are validated against experimental data. Attention is then focussed on the development of a 2D DG method implemented using the high-order library Nektar++. An Euler and a laminar Navier–Stokes solvers are presented and benchmark tests are used to assess their accuracy and performance. An artificial diffusion term is implemented to stabilise the solution of the Euler equations in transonic flow with discontinuities. To speed up the convergence of the explicit method, a new automatic polynomial adaptive procedure (p-adaption) and a new zonal solver are proposed. The p-adaptive procedure uses a discontinuity sensor, originally developed as an artificial diffusion sensor, to assign appropriate polynomial degrees to each element of the domain. The zonal solver uses a modification of a method for matching viscous subdomains to set the interface conditions between viscous and inviscid subdomains that ensures stability of the flow computation. Both the p-adaption and the zonal solver maintain the high-order accuracy of the DG method while reducing the computational cost of the simulation.