Multiple-correction hybrid k-exact schemes for high-order compressible RANS-LES simulations on fully unstructured grids

Abstract

Primitive variable reconstruction Recentering process General grid Hybrid RANS/LES A Godunov's type unstructured finite volume method suitable for highly compressible turbulent scale-resolving simulations around complex geometries is constructed by using a successive correction technique. First, a family of k-exact Godunov schemes is developed by recursively correcting the truncation error of the piecewise polynomial representation of the primitive variables. The keystone of the proposed approach is a quasi-Green gradient operator which ensures consistency on general meshes. In addition, a high-order singlepoint quadrature formula, based on high-order approximations of the successive derivatives of the solution, is developed for flux integration along cell faces. The proposed family of schemes is compact in the algorithmic sense, since it only involves communications between direct neighbors of the mesh cells. The numerical properties of the schemes up to fifth-order are investigated, with focus on their resolvability in terms of number of mesh points required to resolve a given wavelength accurately. Afterwards, in the aim of achieving the best possible trade-off between accuracy, computational cost and robustness in view of industrial flow computations, we focus more specifically on the third-order accurate scheme of the family, and modify locally its numerical flux in order to reduce the amount of numerical dissipation in vortex-dominated regions. This is achieved by switching from the upwind scheme, mostly applied in highly compressible regions, to a fourth-order centered one in vortex-dominated regions. An analytical switch function based on the local grid Reynolds number is adopted in order to warrant numerical stability of the recentering process. Numerical applications demonstrate the accuracy and robustness of the proposed methodology for compressible scale-resolving computations. In particular, supersonic RANS/LES computations of the flow over a cavity are presented to show the capability of the scheme to predict flows with shocks, vortical structures and complex geometries.

A coupled implicit-explicit time integration method for compressible unsteady flows

“…Actually, the residual is defined as the sum of the flux over the whole boundary of a cell. Here, convection and diffusion fluxes are discretized by means of a k-exact formulation coupled with successive corrections [22,40]. Indeed, any high-order scheme needs a polynomial representation of the unknowns locally around a cell of interest.…”

“…This solver, developed by ArianeGroup, is used in the certification process of launchers during take-off and reentry, when complex physical phenomena occur. During the last years, the solver received attention and was the subject of many improvements [21,22,23,24,25,26] .…”

A high-order interpolation for the finite volume method: The Coupled Least Squares reconstruction

“…We also refer to the studies [19,31], with interest into application oriented simulations, where reconstructions beyond linear are presented. Finally we mention the recent study [23] where a variant of compact k-exact reconstruction is considered. In this study the authors use an upwinded scheme only up to third order and switch to a fourth order centered scheme in vorticity dominated regions.…”

High-Order k-Exact Finite Volume Scheme for Vertex-Centered Unstructured Grids

“…2 This is realized by relating successive derivatives of the solution to the polynomial coefficients, which are generally calculated with least-squares approximations of volume-averaged quantities in the vicinity of a cell. 3 Haider et al [4][5][6] presented a general procedure for the k-exact reconstruction on unstructured grids, based on recursive corrections of the approximate successive derivatives, which requires only exchange between adjacent cells. Pont et al 3 adopted this approach where required derivatives were calculated with a Green-Gauss formulation, ensuring consistency on highly deformed grids.…”

“…3 Haider et al [4][5][6] presented a general procedure for the k-exact reconstruction on unstructured grids, based on recursive corrections of the approximate successive derivatives, which requires only exchange between adjacent cells. Pont et al 3 adopted this approach where required derivatives were calculated with a Green-Gauss formulation, ensuring consistency on highly deformed grids. Within this work, the multiplecorrection hybrid k-exact scheme by Pont et al is extended for a vertex-centered median dual tesselation of arbitrary grids, which offers a higher number of direct neighbors for each cell and therefore leads to a higher accuracy for the polynomial reconstruction.…”