High-Order Numerical Schemes for Compressible Flows

High-order numerical methods for Computational Fluid Dynamics have undergone significant fundamental developments over the last two decades owing to combined efforts from the applied mathematics and engineering communities. Even though low-order numerical methods are still the standard in industry, the increased requirements of engineering applications have led to significant scientific interest in developing efficient and robust numerical methods. Applications that would benefit from high-order numerical methods include Direct Numerical Simulations (DNS), Large Eddy simulations (LES), Computational Aero-Acoustics (CAA) and vortex dominated flows. The objective of this thesis is to successfully implement and validate a fifth order traditional WENO scheme in a finite volume framework, for a solver currently being developed in the Aerodynamics group of TU Delft. A detailed literature study of classical numerical schemes has been performed along with a study of the traditional WENO schemes. The quality of results using the fifth order scheme is studied for a variety of test cases to study the shock capturing ability of the scheme. Implementing the finite volume WENO schemes includes the calculation of numerical flux at cell faces using Gaussian quadrature formulas. The effect of varying the number of Gaussian quadrature points while calculating the numerical flux is investigated. Also, the effect of the approximate Riemann solvers on the quality of results is studied by implementing four different Riemann solvers and studying the results for different test cases using these Riemann solvers. The test cases are governed by the inviscid Euler equations and deals with flow in the compressible regime. They involve shocks, other discontinuities and often also complicated structures in the smooth part of the solution which tests the design of the schemes to be non-oscillatory at the discontinuities and still gives a high order of accuracy in the smooth parts of the flow. Convergence tests of the error for test cases using the linear advection equation is used to study the order of accuracy of the scheme using different number of Gaussian quadrature points. The tests clearly show that the order of accuracy remains the same irrespective of the number of quadrature points used. This result is important as it allows simulation run with just one quadrature point which is less expensive, and saves memory. This result is highly relevant while running test cases for LES where very fine grids have to be used. WENO schemes have been considered to be too dissipative for LES in their traditional form. This is indeed true as seen by Kelvin-Helmholtz type small scale vortices (which are characteristic of high Reynolds number flows), even in the test cases using the inviscid Euler equations, due to the inherent dissipation in the schemes. However, this could be seen as motivation for using the WENO schemes for Implicit LES where no explicit sub-grid scale models are used to represent the unresolved scales. The different Riemann solvers exhibit different levels of dissipation and recommendations are made for the choice of Riemann solvers according to the application.Aerospace EngineeringFlight Performance and Propulsio