On the numerical accuracy in finite-volume methods to accurately capture turbulence in compressible flows

TL;DR

This paper investigates how different numerical schemes and quadrature rules in finite-volume methods affect the accuracy of turbulence simulations in compressible flows, highlighting the importance of reconstruction methods for capturing turbulent spectra.

Contribution

It provides a detailed analysis of the interaction between reconstruction schemes and quadrature rules, offering practical guidelines for improving turbulence simulation accuracy.

Findings

High-order quadrature does not improve spectral accuracy in turbulence decay.

Reconstruction at cell faces critically affects turbulence spectrum capture.

All methods show second-order convergence in turbulence decay simulations.

Abstract

The goal of the present paper is to understand the impact of numerical schemes for the reconstruction of data at cell faces in finite-volume methods, and to assess their interaction with the quadrature rule used to compute the average over the cell volume. Here, third-, fifth- and seventh-order WENO-Z schemes are investigated. On a problem with a smooth solution, the theoretical order of convergence rate for each method is retrieved, and changing the order of the reconstruction at cell faces does not impact the results, whereas for a shock-driven problem all the methods collapse to first-order. Study of the decay of compressible homogeneous isotropic turbulence reveals that using a high-order quadrature rule to compute the average over a finite volume cell does not improve the spectral accuracy and that all methods present a second-order convergence rate. However the choice of the numerical method to reconstruct data at cell faces is found to be critical to correctly capture turbulent spectra. In the context of simulations with finite-volume methods of practical flows encountered in engineering applications, it becomes apparent that an efficient strategy is to perform the average integration with a low-order quadrature rule on a fine mesh resolution, whereas high-order schemes should be used to reconstruct data at cell faces.