High-Order Flux Correction/Finite Difference Schemes for Strand Grids

Abstract

A novel high-order method combining unstructured flux correction along body surfaces and highorder finite differences normal to surfaces is formulated for unsteady viscous flows on strand grids. The flux correction algorithm is applied in each unstructured layer of the strand grid, and the layers are then coupled together via a source term containing derivatives in the strand direction. Strand-direction derivatives are approximated to high-order via summation-by-parts operators for first derivatives and second derivatives with variable coefficients. We show how this procedure allows for the proper truncation error cancelling properties required for the flux correction scheme. The resulting scheme possesses third-order design accuracy, but often exhibits fourth-order accuracy when higher-order derivatives are employed in the strand direction, especially for highly viscous flows. We prove discrete conservation for the new scheme and time stability in the absence of the flux correction terms. Results in two dimensions are presented that demonstrate improvements in accuracy with minimal computational and algorithmic overhead over traditional second-order algorithms.

Implementation of the Flux Correction method on hybrid unstructured meshes

“…Её решение возможно путём использования расчётных сеток, по-лученных из поверхностной сетки экструдией вдоль нормали к поверхности. Такая технология описана в [14] [15]. Аналогичный результат может быть до-стигнут использованием прямых контрольных объёмов [8].…”

High‐order strand grid methods for low‐speed and incompressible flows

“…Interpreting the mesh in this manner leads to our discretization strategy, which is to solve a series of unstructured discretizations in each layer of the strand grid, while coupling the layers through a source term containing derivatives along the strands. This approach was first explored by Katz and Work [17] for compressible flows. Here, we provide a brief explanation of this discretization scheme and then extend the method to low-speed and incompressible flows through a preconditioner.…”

“…Such gradient approximations may be formulated in a variety of ways such as least squares procedures or Green-Gauss integration [12]. Following our previous work [17], we use high-order finite elementtype approximations to construct the required flux and solution gradients. We refer interested readers to our previous work for a detailed description of the high-order gradient procedure with equations and figures [17].…”

“…Importantly, compact strand meshing techniques offer self-satisfying domain connectivity, whereby each processor in a multi-processor computation has access to the global mesh data, increasing the efficiency of overset domain connectivity algorithms. Previous work has investigated our high-order approach for strand grids [17], but the present work represents the first extension of such methods to a much wider class of flows via a preconditioner.…”

Third-Order Inviscid and Second-Order Hyperbolic Navier-Stokes Schemes for Three-Dimensional Inviscid and Viscous Flows

“…[12] extended the third-order scheme to the viscous terms in 2D, and Refs. [13,14] extended it to 3D prismatic strand grids by combining it with a high-order finite-difference scheme applied in the direction normal to a surface. Our interest, on the other hand, is to extend the thirdorder scheme to Navier-Stokes formulations on purely tetrahedral grids for improving practical unstructuredgrid turbulent-flow solvers, allowing accurate and robust simulations with arbitrary isotropic/anisotropic grid adaptation for complex geometries [15,16,17,18,19].…”