Abstract

This study focuses on the dispersion and diffusion characteristics of high-order energy-stable flux reconstruction (ESFR) schemes via the spatial eigensolution analysis framework proposed in [1]. The analysis is performed for five ESFR schemes, where the parameter ‘c’ dictating the properties of the specific scheme recovered is chosen such that it spans the entire class of ESFR methods, also referred to as VCJH schemes, proposed in [2]. In particular, we used five values of ‘c’, two that correspond to its lower and upper bounds and the others that identify three schemes that are linked to common high-order methods, namely the ESFR recovering two versions of discontinuous Galerkin methods and one recovering the spectral difference scheme. The performance of each scheme is assessed when using different numerical intercell fluxes (e.g. different levels of upwinding), ranging from “under-” to “over-upwinding”. In contrast to the more common temporal analysis, the spatial eigensolution analysis framework adopted here allows one to grasp crucial insights into the diffusion and dispersion properties of FR schemes for problems involving non-periodic boundary conditions, typically found in open-flow problems, including turbulence, unsteady aerodynamics and aeroacoustics.

Highlights

  • Computational fluid dynamics (CFD) is facing the challenge to expand its current capabilities to flow problems that can be only marginally described by the prevailing numerical methodologies adopted today [3]

  • We present the results of the spatial eigensolution analysis for different values of the upwind parameter β and for various polynomial orders, taking into account the five energystable flux reconstruction (ESFR) schemes considered

  • The test cases taken into account span the five ESFR schemes considered in the spatial eigensolution analysis

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Summary

Introduction

Computational fluid dynamics (CFD) is facing the challenge to expand its current capabilities to flow problems that can be only marginally described by the prevailing numerical methodologies adopted today [3]. It is essential to explore alternative ways to enhance the predictive skills of CFD and facilitate their adoption in the broader industrial community working in the field [3,4] From this perspective, high-fidelity computations relying on high-order numerical methods, namely spectral element methods including discontinuous Galerkin and flux reconstruction approaches [5,6,7,8,9,10], are attractive, especially when used for large-eddy simulation (LES) and under-resolved direct numerical simulation (uDNS) of high Reynolds number turbulent flows — e.g. This work applies the spatial eigensolution framework proposed in [1] to the FR approach and highlights the diffusion and dispersion properties of energy-stable FR schemes when using different numerical fluxes — e.g. Riemann solvers — for under-resolved flow simulations relevant to real-world problems.

The flux reconstruction approach
Energy-stable FR schemes
Eigensolution analysis framework
Results
Numerical experiments for the linear advection equation
Numerical experiments in under-resolved vortical flows
Governing equations and flow configuration
Conclusions
Methods

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