High Order DG/FV Hybrid Method On Hybrid Grid

The use of unstructured/hybrid meshes for computational fluid dynamics problemshas become widespread due to their ability to discretize arbitrarily complex geometriesand the ease of adaptation in enhancing the solution accuracy and efficiency through theuse of adaptive refinement techniques. In recent years, significant progress has been madein developing numerical algorithms for the solution of the Euler and Navier-Stokesequations on unstructured/hybrid grids. Nearly all production flow solvers are based onsecond-order numerical methods, either finite volume method (FVM), finite differencemethod (FDM) or finite element method (FEM). Nevertheless, many types of problems,such as computational aeroacoustics (CAA), vortex-dominant flows, large eddysimulation (LES) and direct numerical simulation (DNS) of turbulent flows, call for highorder accuracy (third-order and higher). The main deficiency of widely available,second-order methods for the numerical simulations of the above-mentioned flows is itsexcessive numerical diffusion and dissipation of vorticity. Applications of high-orderaccurate, low-diffusion and low dissipation numerical methods can significantly alleviatethis deficiency of the traditional second order methods, and improve predictions of verticaland other complex, separated, unsteady flows. Therefore, high-order methods onunstructured/hybrid grids have been paid more and more attention in recent years.At present, most numerical methods on unstructured/hybrid meshes are originatedfrom finite element methods or finite volume methods. As the leader of high-ordernumerical methods for compressible flow computations on unstructured/hybrid grid, thediscontinuous Galerkin (DG) method has become more and more popular for problemswith complex physics and/or geometry. However, the DG method does have someweaknesses, including the high computational cost and huge memory requirement.Comparing with the classical second order DG method, the second order finite volumemethod need fewer memory requirement and computational cost on the same mesh,because they don’t need to compute the volume and surface integrals and the additionalequations for the degree-of-freedoms (DOFs) due to the high order basis functions. But forhigher-order FV methods, multi-dimensional re-construction needs larger grid stencil. Dueto the large stencil, these high-order FV methods can not keep the ‘compact’ feature of DGmethods. In addition, the matrix computations increase the computational costtremendously, especially for3D cases.To overcome these problems, Zhang Hanxin presented a three-order accuracy hybridmethod that is learnt from FV and FE methods. A concept of ‘static reconstruction’ and‘dynamic reconstruction’ had been introduced by Zhang Laiping, Liu Wei, et al. for higher-order numerical methods. Based on this concept, a class of DG/FV hybrid methodsfor triangular and Cartesian/triangular hybrid grids had been developed for conservationlaws using a ‘hybrid reconstruction’ idea. In the DG/FV hybrid scheme, the lower-orderderivatives (or DOFs) of the piecewise polynomial are computed locally in a cell by thetraditional DG method (called as ‘dynamic reconstruction’), while the higher-orderderivatives (or DOFs) are reconstructed by the ‘static reconstruction’ of the idea of FVmethod, using the known lower-order derivatives (or DOFs) in the cell itself and in itsneighboring cells. This method can be achieved easily on unstructured meshes forcomplex geometries. Not only obtains it high-order result, but also it uses so less stencilthat it facilitates the achievement of its programming. Furthermore, with this method, dataexchanges between cells only occur in neighboring cells, so there are a little datacommunication, thus it is very suitable for parallel computing. This hybrid method isverified efficiently and robustly by lots of typical test cases in this dissertation.In this dissertation, the high-order DG/FV hybrid methods are extended toNavier-Stokes equation on two-dimensional hybrid grids. The dissipation and dispersion,as well as the stability condition, are analyzed. Meanwhile, in order to improve theconvergence history, an implicit algorithm based on Newton/Gauss-Seidel iteration ispresented. The numerical results demonstrate that the DG/FV hybrid methods reach thedesired order of accuracy, and the convergence history of implicit algorithm is acceleratedby about one or two orders.This dissertation is divided into six chapters.In Chapter1, we review briefly the current progress of high-order, high-resolutionschemes and numerical methods on unstructured/hybrid mesh in computational fluiddynamics regime. The advantages and disvantages of several well-known methods arediscussed. And finally, the work of this dissertation is introduced in briefness.In Chapter2, the DG/FV hybrid method is presented in detail, such as the basic ideas,re-construction procedure, explicit/implicit time discretization method, numerical fluxscheme, gauss numerical integration, limiter, boundary conditions and the treatment ofcurve boundary etc..In Chapter3，we discuss the numerical properties of the DG/FV hybrid method,including the numerical dissipation and dispersion of the method, and its stabilityconditions.The numerical accuracy of the hybrid method has been verified using linearand nonlinear model equations. The costs of some typical order of DGM and DG/FVmethod have been qualitative analyzed and numerical verified.In Chapter4and Chapter5, some numerical examples are tested to validate theaccuracy and efficiency of present DG/FV hybrid methods. In Chapter4, some inviscidtest cases are simulated, including the isotropic vertex flow, subsonic flow over a cylinder,Bump flow problem, Sod’s shock tube problem, Shu-Osher’s problem, and supersonic flow in a tube with forward step. Meanwhile, in order to improve the convergence history(especially for viscous flows), an implicit algorithm based on Newton/Gauss-Seideliteration is presented. In Chapter5, some typical viscous flows are simulated, includingthe Couette flow between two plates, laminar flow over a plate, and viscous flow over acylinder, viscous flow in a cavity and over a NACA0012airfoil. The accuracy andefficiency are compared with the same order DG methods. The efficiency between explicitmethod and implicit method are compared and the influences of parameters in implicitmethod are studied.Finally, the conclusions are drwan and some possibilities for future work are given inChapter6.