Flux reconstruction for high-accuracy aerospace design tools

Abstract

Demand for high accuracy simulations of complex turbulent flows has steadily increased in recent years, for which conventional low-order numerical methods are generally unsuitable. High-order methods (HOMs) like the flux reconstruction (FR) approach have been widely proposed as a superior alternative, since the level of error can be substantially reduced for similar computational expense. While there have been many advances with respect to high accuracy spatial discretisation methods, less research effort has been devoted to complementing these with equally effective integration schemes for unsteady flow problems. For HOMs, it can be difficult to balance differences in space and time accuracy and time steps must often be substantially smaller to maintain numerical stability, particularly when viscosity cannot be neglected. Both these factors are detrimental to the performance of HOMs, so this thesis presents a novel space time approach with reformulation of diffusive terms such that they behave as advective fluxes during computation. This guarantees equally high-order accuracy with respect to space and time, no restriction on physical time step and the stability characteristics of viscous schemes are significantly improved relative to conventional approaches. This procedure was further developed with local adaptation of the pseudo-time step size throughout the space-time domain. This is not normally possible during the classical integration of unsteady solvers in space-only domains, since the global (physical ) time step must be consistent across all elements in the mesh. Sparse matrix-matrix multiplication was also used to substantially reduce execution time for schemes of higher order accuracy and dimensionality. It was verified that the new approach achieved the target order accuracy and stability properties for linear diffusion solvers in up to three spatial dimensions. The methodology was successfully applied to non linear equations including the compressible Navier-Stokes equations, which results in additional field variables corresponding to velocity gradients and heat fluxes. For all new schemes, eigendecomposition of the flux Jacobian confirmed that the transition in character from a parabolic or mixed-type partial differential equation (or equations) to an equivalent hyperbolic system had occurred. HOMs based on the new approach to the Navier-Stokes equations were capable of accurately resolving the cascade of turbulent energy from the integral scale until close to the point of dissipation by viscous action. The use of locally adaptive pseudo-time stepping (LAPTS) was shown to be very effective for enhancing the convergence of space-time schemes compared to typical time stepping schemes, even for the uniform meshes considered where there was little disparity in element size. Possible future research topics that may further improve the competitiveness of the space-time approach are summarised along with conclusions of the current work.

Date of Award	Jul 2024
Original language	English
Awarding Institution	Queen's University Belfast
Sponsors	Engineering and Physical Sciences Research Council
Supervisor	Marco Geron (Supervisor), Adrian Murphy (Supervisor), Declan Nolan (Supervisor), Rob Watson (Supervisor) & Sung in Kim (Supervisor)