Generation of unstructured curvilinear grids and high‐order discontinuous Galerkin discretization applied to a 3D high‐lift configuration

Abstract

SUMMARYDiscontinuous Galerkin (DG) methods allow high-order flow solutions on unstructured or locally refined meshes by increasing the polynomial degree and using curved instead of straight-sided elements. However, one of the currently largest obstacles to applying these methods to aerodynamic configurations of medium to high complexity is the availability of appropriate higher-order curved meshes.In this article, we describe a complete chain of higher-order unstructured grid generation and higherorder DG flow solution applied to a turbulent flow around a three-dimensional high-lift configuration. This includes (i) the generation of an appropriately coarse straight-sided mesh; (ii) the evaluation of additional points on the computer-aided design geometry of the curved-wall boundary for defining a piecewise polynomial boundary representation; (iii) a higher order mesh deformation to translate the curvature from the wall boundary into the interior of the computational domain; and (iv) the description of a DG discretization, which is sufficiently stable to allow a flow computation on the resulting curved mesh. Finally, a fourth-order flow solution of the Reynolds-averaged Navier-Stokes and k-! turbulence model equations is computed on a fourth-order unstructured hybrid mesh around the three-dimensional high-lift simulation of wing-flow noise generation configuration.

A study of properties of adaptive quadratic grids for three‐dimensional near‐surface field computations

“…There have been a few efforts to generate high-order grids for flow field computations. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] Two major approaches can be considered for the generation of high-order meshes: the direct and the indirect. The direct creation of high-order meshes employs a high-order version of a classical meshing algorithm.…”

“…There have been a few efforts to generate high‐order grids for flow field computations . Two major approaches can be considered for the generation of high‐order meshes: the direct and the indirect.…”

Connectivity-change moving mesh methods for high-order meshes: Toward closed advancing-layer high-order boundary layer mesh generation

“…It would be harder to implement a directly high-order mesh generator. To curve this mesh, the used models are numerous : PDE or variational models, 1,15,22,23 smoothing and/or optimization procedures, 17,24,27 etc ... Our choice here is to use the linear elasticity equation as a model for the motion of the vertices to generate a P k mesh from a P 1 mesh. For this purpose, let us consider Equation (7) with f = 0 and Dirichlet boundary conditions.…”

“…To generate high-order meshes, several approaches exist. Some are using a PDE or variational approach to curve a P 1 mesh into a P k mesh, 1,15,22,23 others are based on optimization and smoothing operations and start from a P 1 mesh with a constrained P k curved boundary in order to generate a suitable P k mesh. 17,24,27 In all these approaches, the key feature is to find the best deformation to apply to the initial P 1 mesh.…”

A Higher-Order Unstructured Finite Volume Solver for Three-Dimensional Compressible Flows

“…For unstructured meshes, various approaches have been devised for achieving higher-order accuracy, most notably: continuous [5] and discontinuous [29] Galerkin finite element methods, the correction procedure via reconstruction formulations of the discontinuous Galerkin and spectral volume methods [31], and finite volume schemes [32]. (See the work of Andren et al [8] for a detailed comparison of numerical results).…”