Development of High-order Numerical Methods for Compressible Gas Dynamics and Turbulence Simulation

Abstract

For compressible flow simulations involving turbulence and discontinuities, the competing requirements for restoring high-order accuracy in smooth regions, capturing discontinuities sharply, and achieving high-spectral resolution across multiscale flow structures introduce significant challenges for numerical development. In this thesis, a series of advanced numerical methods tailored for compressible gas dynamics and turbulence simulation is proposed.

First, with the observation that the high-order polynomial reconstruction deployed in classical numerical methods is not suitable for representing the flow scales with sharp transitions, a new non-polynomial-based shock-capturing framework is proposed by integrating the infinitely differentiable RBF reconstruction and the jump-like non-polynomial interpolation. This innovative approach not only maintains the high-order accuracy in smooth regions but also delivers superior resolution for discontinuities.

Subsequently, the research addresses the challenges encountered in compressible turbulence simulations, where classical shock-capturing schemes typically fail to provide physically accurate solutions, especially for under-resolved large-eddy simulations deployed on coarse meshes. By recasting the newly optimized skew-symmetric-splitting method for smooth flows and invoking the nonlinear TENO scheme for non-smooth flows, the proposed TENO-S scheme predicts turbulence statistics significantly more accurately than other schemes when compared to the DNS reference.

Next, a novel perspective on the non-convergence property of high-order shock-capturing schemes is proposed for steady-state flow simulations. Besides, with the observation that compressible steady-state problems are typically characterized by the narrow shockwave regions of interest and other large regions with smooth flows, a new finite-volume converged ENO scheme is innovatively proposed for the multi-resolution framework with adaptive mesh resolution.

Finally, the research focuses on the numerical method for unstructured meshes, which can provide significant convenience for computational workflows that involve complex geometries. A class of high-order TENO-E schemes featuring high efficiency and extremely low dissipation is proposed. The superior performance demonstrates great potential for high-fidelity simulation of more challenging practical engineering flows.

Date of Award	2025
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology
Supervisor	Lin FU (Supervisor)