Low-Mach number treatment for Finite-Volume schemes on unstructured meshes

Abstract

The paper presents a low-Mach number (LM) treatment technique for high-order, Finite-Volume (FV) schemes for the Euler and the compressible Navier-Stokes equations. We concentrate our efforts on the implementation of the LM treatment for the unstructured mesh framework, both in two and three spatial dimensions, and highlight the key differences compared with the method for structured grids. The main scope of the LM technique is to at least maintain the accuracy of low speed regions without introducing artefacts and hampering the global solution and stability of the numerical scheme. Two families of spatial schemes are considered within the k-exact FV framework: the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and the Weighted Essentially Non-Oscillatory (WENO). The simulations are advanced in time with an explicit third-order Strong Stability Preserving (SSP) Runge-Kutta method. Several flow problems are considered for inviscid and turbulent flows where the obtained solutions are compared with referenced data. The associated benefits of the method are analysed in terms of overall accuracy, dissipation characteristics, order of scheme, spatial resolution and grid composition.

Stencil selection algorithms for WENO schemes on unstructured meshes

“…The reconstruction algorithm used is based upon the approaches [2,3,11,35], that have been applied to smooth and discontinuous flow problems [22,[38][39][40][41][42][43][44][45][46][47] and only key-components will be presented herein, since the focus is towards the stencil selection algorithms and the reader is referred to [2,3,35] for further details.…”

“…For the scope of the present study only unique stencils are considered since they have provided a more robust WENO framework so far for various set of applications [2,3,22,[38][39][40][41][42]44,46]. The Big Stencil from the central stencil algorithm is employed here, and all the elements from the Big Stencil that satisfy a set of geometrical conditions are added to the directional stencils.…”

“…This test problem when employed at relative coarse-"under-resolved" meshes, reveals the dissipation and dispersion characteristics of non-linear methods at resolutions typically found in regions close to the boundary layer for larger Reynolds number wall bounded flows. It has been widely used by many researchers to assess the behaviour of various numerical schemes [3,46,[76][77][78][79][80][81][82], in the context of iLES. Furthermore there are well-documented DNS results of Brachet et al [83] to compare against and are used as a reference for this study.…”

Arbitrary high order central non-oscillatory schemes on mixed-element unstructured meshes

“…The present polynomial reconstruction is based upon the approaches of [5,6,13,52], that have been applied to smooth and discontinuous flow problems [38,47,72,[75][76][77][78][79][80][81][82][83][84][85][86] and only the key-components will be presented herein and the reader is referred to [5,6,52] for further details. Unstructured meshes involve the presence of various elements shapes and sizes, and a transformation from physical space to a reference space is performed to reduce the scaling effects as introduced by Dumbser et al [13,14,87].…”

“…The ILES of the 3D viscous Taylor-Green vortex test problem at Re = 1600 is employed, for assessing the performance of all the schemes. It is a widely used test problem for the validation of numerical methods, and in particularly at relative coarse-"under-resolved" meshes within the LES context [6,27,29,85,[106][107][108][109][110] due to the pronounced dissipation and dispersion characteristics of non-linear methods. The computational domain is defined as Ω = [0, 2π] 3 with periodic boundary conditions.…”