Numerical Aspects of Gradient Reconstruction Schemes Applied to Complex Geometries

TL;DR

This paper evaluates three gradient reconstruction methods for viscous term calculation in unstructured grids, introduces a novel convergence acceleration technique, and compares results across complex flow cases, demonstrating improved stability and accuracy.

Contribution

It introduces a new convergence acceleration method and compares gradient reconstruction techniques for viscous terms in complex geometries.

Findings

Sophisticated gradient reconstructions outperform simple ones in stability and accuracy.

The new convergence method rapidly reduces residuals to machine precision.

Results are consistent with experimental and literature data across test cases.

Abstract

This work primarily focuses on the study of three gradient reconstruction techniques applied to the calculation of viscous terms in a cell-centered, finite volume formulation for general unstructured grids. The work also addresses different ways of formulating the limiter functions necessary to maintain stability in the presence of flow discontinuities. The flows of interest are simulated using the compressible Reynolds-averaged Navier-Stokes equations, and the negative Spalart-Allmaras model is used for turbulence closure. Definition of interface inviscid terms uses the Roe approximate Riemann solver, whereas the interface viscous terms are calculated with a standard centered scheme together with appropriate definitions of the interface gradients. Steady state solutions are obtained using an implicit time-integration method, together with a novel convergence acceleration technique. This new approach defines a set of three simple rules for controlling the global CFL number based on the residue evolution. The work considers three test cases, namely, the subsonic bump-in-channel flow, the subsonic NASA high-lift Common Research Model multielement airfoil and the transonic ONERA M6 wing. Present results are compared to experimental and numerical data available in the literature. Severe numerical instabilities are observed when the simplest gradient reconstruction technique is used, while more sophisticated formulations are able to provide excellent agreement with the existing literature. Current results are demonstrated to be highly insensitive to modifications made to the numerical flux entropy fix terms. Integrated aerodynamic forces are shown to be mildly dependent on the limiter formulation used, even in the absence of shock waves. The proposed convergence acceleration procedure manages to quickly drive the residue terms to machine zero, provided no major instabilities are present.