High-Order Flux Reconstruction on Stretched and Warped Meshes

High-order computational fluid dynamics is gathering a broadening interest as a future industrial tool, with one such approach being flux reconstruction (FR). However, due to the need to mesh complex geometries if FR is to displace current lower?order methods, FR will likely have to be applied to stretched and warped meshes. Therefore, it is proposed that the analytical and numerical behaviors of FR on deformed meshes for both the one-dimensional linear advection and the two-dimensional Euler equations are investigated. The analytical foundation of this work is based on a modified von Neumann analysis for linearly deformed grids, which is presented. The temporal stability limits for linear advection on such grids are also explored analytically and numerically, with Courant?Friedrichs?Lewy (CFL) limits set out for several Runge?Kutta schemes, with the primary trend being that contracting mesh regions give rise to higher CFL limits, whereas expansion leads to lower CFL limits. Lastly, the benchmarks of FR are compared to finite difference and finite volumes schemes, as are common in industry, with the comparison showing the increased wave propagating ability on warped and stretched meshes, and hence FR?s increased resilience to mesh deformation

Effect of Mesh Quality on Flux Reconstruction in Multi-dimensions

Theoretical methods are developed to understand the effect of non-uniform grids on Flux Reconstruction (FR) in multi-dimensions. The analysis reveals that the same effect of expanding and contracting grids is seen in two dimensions as in one dimension. Namely, that expansions cause instability and contractions cause excess dissipation. Subsequent numerical experiments on the Taylor-Green Vortex with jittered elements show the effect of localised regions of expansion and contraction, with an initial increase in the kinetic energy observed on non-uniform meshes. Some comparison is made between second-order FR and second-order finite volume (FV). FR is found to be more resilient to mesh deformation, however, FV is found to be more resolved when operated at second order on the same mesh. In both cases, it is recommended that a kinetic energy preserving/conservation formulation should be used as this can greatly increase resilience to mesh deformation

Temporal Stabilisation of Flux Reconstruction on Linear Problems

Filtering is often used in Large Eddy Simulation with a global filter width, instead here a filter width in the reference domain of high order Flux Reconstruction is considered. It is shown via Von Neumann analysis how filtering effects the dispersion and dissipation of the scheme when spatially and temporally discretised. With it being shown that filtering stabilises the scheme temporally by upto $25\%$ for forth order FR. The impact of filtering on error production is calculated, highlighting the reduction in convective velocity caused and showing numerically the impact on order of accuracy. Finally, the turbulent Taylor-Green case is used to understand the effect of reference domain filtering on the transition to turbulence, and a filter Reynolds number is defined that is shown to be useful in understanding the effect of filtering on simulations.Comment: AIAA Aviation Forum June 201

Artificial Compressibility Approaches in Flux Reconstruction for Incompressible Viscous Flow Simulations

Copyright © 2021 The Author(s). Several competing artificial compressibility methods for the incompressible flow equations are examined using the high-order flux reconstruction method. The established artificial compressibility method (ACM) of \citet{Chorin1967} is compared to the alternative entropically damped (EDAC) method of \citet{Clausen2013}, as well as an ACM formulation with hyperbolised diffusion. While the former requires the solution to be converged to a divergence free state at each physical time step through pseudo iterations, the latter can be applied explicitly. We examine the sensitivity of both methods to the parameterisation for a series of test cases over a range of Reynolds numbers. As the compressibility is reduced, EDAC is found to give linear improvements in divergence whereas ACM yields diminishing returns. For the Taylor--Green vortex, EDAC is found to perform well; however on the more challenging circular cylinder at $Re=3900$, EDAC gives rise to early transition of the free shear-layer and over-production of the turbulence kinetic energy. This is attributed to the spatial pressure fluctuations of the method. Similar behaviour is observed for an aerofoil at $Re=60,000$ with an attached transitional boundary layer. It is concluded that hyperbolic diffusion of ACM can be beneficial but at the cost of case setup time, and EDAC can be an efficient method for incompressible flow. However, care must be taken as pressure fluctuations can have a significant impact on physics and the remedy causes the governing equation to become overly stiff.https://arxiv.org/abs/2111.07915v

Generalised Sobolev Stable Flux Reconstruction

A new set of symmetric correction functions is presented for high-order flux reconstruction, that expands upon, while incorporating, all previous correction function sets and opens the possibility for improved performance. By considering FR applied to the linear advection equation, and through modification to the Sobolev norm, criteria are presented for a wider set of correction functions. Legendre polynomials are then used to fulfil these criterion and realise functions for third to fifth order FR. The sufficient conditions for the existence of the modified norms are also explored, before Fourier and Von Neumann analysis are applied to analytically find temporal stability limits for various Runge-Kutta temporal integration schemes. For all cases, correction functions are found that extend the temporal stability of FR. Two application-inspired investigations are performed that aim to explore the effect of aliasing and non-linear equations. In both cases unique correction functions could be found that give good performance, compared to previous FR schemes, while also improving upon the temporal stability limit

Generalised Lebesgue Stable Flux Reconstruction

A unique set of correction functions for Flux Reconstruction is presented, with there derivation stemming from proving the existence of energy stability in the Lebesgue norm. The set is shown to be incredibly arbitrary with the only union to existing correction function sets being show to be for DG. Von Neumann analysis of both advection and coupled advection-diffusion is used to show that once coupled to a temporal integration method, good CFL performance can be achieved and the correction function may have better dispersion and dissipation for application to implicit LES. Lastly, the turbulent Taylor-Green vortex test case is then used to show that correction functions can be found that improve the accuracy of the scheme when compared to the error levels of Discontinuous Galerkin

Inline vector compression for computational physics

A novel inline data compression method is presented for single-precision vectors in three dimensions. The primary application of the method is for accelerating computational physics calculations where the throughput is bound by memory bandwidth. The scheme employs spherical polar coordinates, angle quantisation, and a bespoke floating-point representation of the magnitude to achieve a fixed compression ratio of 1.5. The anisotropy of this method is considered, along with companding and fractional splitting techniques to improve the efficiency of the representation. We evaluate the scheme numerically within the context of high-order computational fluid dynamics. For both the isentropic convecting vortex and the Taylor–Green vortex test cases, the results are found to be comparable to those without compression. Performance is evaluated for a vector addition kernel on an NVIDIA Titan V GPU; it is demonstrated that a speedup of 1.5 can be achieved

A New Family of Weighted One-Parameter Flux Reconstruction Schemes

The flux reconstruction (FR) approach offers a flexible framework for describing a range of high-order numerical schemes; including nodal discontinuous Galerkin and spectral difference schemes. This is accomplished through the use of so-called correction functions. In this study we employ a weighted Sobolev norm to define a new extended family of FR correction functions, the stability of which is affirmed through Fourier analysis. Several of the schemes within this family are found to exhibit reduced dissipation and dispersion overshoot. Moreover, many of the new schemes possess higher CFL limits whilst maintaining the expected rate of convergence. Numerical experiments with homogeneous linear convection and Burgers turbulence are undertaken, and the results observed to be in agreement with the theoretical findings

On Fourier analysis of polynomial multigrid for arbitrary multi-stage cycles

The Fourier analysis of the \emph{p}-multigrid acceleration technique is considered for a dual-time scheme applied to the advection-diffusion equation with various cycle configurations. It is found that improved convergence can be achieved through \emph{V}-cycle asymmetry where additional prolongation smoothing is applied. Experiments conducted on the artificial compressibility formulation of the Navier--Stokes equations found that these analytic findings could be observed numerically in the pressure residual, whereas velocity terms---which are more hyperbolic in character---benefited primarily from increased pseudo-time steps