On the performance of WENO/TENO schemes to resolve turbulence in DNS/LES of high-speed compressible flows

High‐speed compressible turbulent flows typically contain discontinuities and have been widely modelled using Weighted Essentially Non‐Oscillatory (WENO) schemes due to their high‐order accuracy and sharp shock capturing capability. However, such schemes may damp the small scales of turbulence, and result in inaccurate solutions in the context of turbulence‐resolving simulations. In this connection, the recently‐developed Targeted Essentially Non‐Oscillatory (TENO) schemes, including adaptive variants, may offer significant improvements. The present study aims to quantify the potential of these new schemes for a fully‐turbulent supersonic flow. Specifically, DNS of a compressible turbulent channel flow with M = 1: 5 and Re τ = 222 is conducted using OpenSBLI, a high‐order finite difference CFD framework. This flow configuration is chosen to decouple the effect of flow discontinuities and turbulence and focus on the capability of the aforementioned high‐order schemes to resolve turbulent structures. The effect of the spatial resolution in different directions and coarse grid implicit LES are also evaluated against theWALE LES model. The TENO schemes are found to exhibit significant performance improvements over the WENO schemes in terms of the accuracy of the statistics and the resolution of the three‐dimensional vortical structures. The 6th order adaptive TENO scheme is found to produce comparable results to those obtained with non‐dissipative 4th and 6th order central schemes and reference data obtained with spectral methods. Although the most computationally expensive scheme, it is shown that this adaptive scheme can produce satisfactory results if used as an implicit LES model

Investigation of finite-volume methods to capture shocks and turbulence spectra in compressible flows

The aim of the present paper is to provide a comparison between several finite-volume methods of different numerical accuracy: second-order Godunov method with PPM interpolation and high-order finite-volume WENO method. The results show that while on a smooth problem the high-order method perform better than the second-order one, when the solution contains a shock all the methods collapse to first-order accuracy. In the context of the decay of compressible homogeneous isotropic turbulence with shocklets, the actual overall order of accuracy of the methods reduces to second-order, despite the use of fifth-order reconstruction schemes at cell interfaces. Most important, results in terms of turbulent spectra are similar regardless of the numerical methods employed, except that the PPM method fails to provide an accurate representation in the high-frequency range of the spectra. It is found that this specific issue comes from the slope-limiting procedure and a novel hybrid PPM/WENO method is developed that has the ability to capture the turbulent spectra with the accuracy of a high-order method, but at the cost of the second-order Godunov method. Overall, it is shown that virtually the same physical solution can be obtained much faster by refining a simulation with the second-order method and carefully chosen numerical procedures, rather than running a coarse high-order simulation. Our results demonstrate the importance of evaluating the accuracy of a numerical method in terms of its actual spectral dissipation and dispersion properties on mixed smooth/shock cases, rather than by the theoretical formal order of convergence rate.Comment: This paper was previously composed of 2 parts, and this submission was part 1. It is now replaced by the combined pape

An Adaptive Characteristic-wise Reconstruction WENOZ scheme for Gas Dynamic Euler Equations

Due to its excellent shock-capturing capability and high resolution, the WENO scheme family has been widely used in varieties of compressive flow simulation. However, for problems containing strong shocks and contact discontinuities, such as the Lax shock tube problem, the WENO scheme still produces numerical oscillations. To avoid such numerical oscillations, the characteristic-wise construction method should be applied. Compared to component-wise reconstruction, characteristic-wise reconstruction leads to much more computational cost and thus is not suite for large scale simulation such as direct numeric simulation of turbulence. In this paper, an adaptive characteristic-wise reconstruction WENO scheme, i.e. the AdaWENO scheme, is proposed to improve the computational efficiency of the characteristic-wise reconstruction method. The new scheme performs characteristic-wise reconstruction near discontinuities while switching to component-wise reconstruction for smooth regions. Meanwhile, a new calculation strategy for the WENO smoothness indicators is implemented to reduce over-all computational cost. Several one dimensional and two dimensional numerical tests are performed to validate and evaluate the AdaWENO scheme. Numerical results show that AdaWENO maintains essentially non-oscillatory flow field near discontinuities as the characteristic-wise reconstruction method. Besieds, compared to the component-wise reconstruction, AdaWENO is about 40\% faster which indicates its excellent efficiency

Bayesian Optimization on Fifth-Order Targeted ENO Scheme for Compressible Flows

Targeted ENO (TENO) has been proposed to overcome the shortcomings of WENO schemes, namely excessive dissipation of lower-order upwind-biased and degenerated schemes, and limited robustness of central-biased schemes. TENO offers a set of free parameters to shape the inherent effective local dissipation and dispersion. In the original formulation of TENO, these free parameters have been adjusting by means of the approximate dissipation-dispersion relation. Hence, the TENO formulation may be superior in this aspect, yet, it does not necessarily outperform other schemes in flows involving non-linear interaction of a broad range of scales. Data-driven methods enable optimizing these free parameters instead of adjusting them. In this work, we demonstrate the application of an iterative Bayesian optimization approach on designing fifth-order TENO (TENO5) schemes. Exploiting that Bayesian optimization efficiently and robustly finds an optimum of an expensive function with a low number of trials, we construct specific TENO5-schemes for compressible flows with gas dynamic discontinuities as well as for implicit large eddy simulation (ILES). For the former, we measure the error between under-resolved simulations of the Sod shock tube and its analytical solution for automatically generated TENO5 formulations as the objective. For the latter, under-resolved inviscid Taylor-Green vortex flows are evolved to their turbulent state, in which their kinetic energy spectrum in the inertial subrange is compared to the theoretical Kolmogorov-scaling solution to formulate its objective. We show that these two TENO5 formulations perform superior to the original formulation of TENO5 relevant to the specific types of flows. Also, a variety of benchmark test flows show that both specific TENO5 formulations outperform the original one in terms of phase speed, shock-preservation, as well as physical consistency of fluid-dynamic instabilities and turbulent flows