Optimized geometrical metrics satisfying free-stream preservation

TL;DR

This paper demonstrates that optimizing geometrical metric terms in computational fluid dynamics significantly improves accuracy, especially on distorted grids, by using entropy stable schemes for 3D inviscid and viscous flow simulations.

Contribution

It introduces an optimized metric computation method that enhances accuracy over traditional approaches in CFD simulations on complex grids.

Findings

Optimized metrics outperform Thomas and Lombard metrics in accuracy.

Entropy stable schemes effectively handle distorted high-order elements.

Improved solutions are observed in 3D compressible flow cases.

Abstract

Computational fluid dynamics and aerodynamics, which complement more expensive empirical approaches, are critical for developing aerospace vehicles. During the past three decades, computational aerodynamics capability has improved remarkably, following advances in computer hardware and algorithm development. However, for complex applications, the demands on computational fluid dynamics continue to increase in a quest to gain a few percent improvements in accuracy. Herein, we numerically demonstrate that optimizing the metric terms which arise from smoothly mapping each cell to a reference element, lead to a solution whose accuracy is practically never worse and often noticeably better than the one obtained using the widely adopted Thomas and Lombard metric terms computation (Geometric conservation law and its application to flow computations on moving grids, AIAA Journal, 1979). Low and high-order accurate entropy stable schemes on distorted, high-order tensor product elements are used to simulate three-dimensional inviscid and viscous compressible test cases for which an analytical solution is known.