Efficient High-Order High-Resolution Methods And The Applications

In the development of computational fluid dynamic(CFD)techniques,numerical accuracy and efficiency have gradually become the focus of the research,hindering the wide application of large scale computing.The most commonly used CFD methods are usually 2nd-order only,and can hardly be extended to high-order due to some limitations.However,in the problems of turbulence,computational aeroacoustics(CAA),etc.,the amount of dispersion and dissipation should be strictly under control,indicating that the 2nd-order methods cannot be used.This paper focused on the high-order methods on structured multiblock grids for practical configurations.Specific research includes the following parts.(1)Firstly,several important topics in the application of high-order methods are studied and discussed,which are closely related to the accuracy,efficiency and robustness of the high-order methods developed in this paper.These topics include: geometric discontinuities induced by skew or abrupt change in grid,parallel computing,geometric conservation law,discretization order of turbulence model and characteristic projection in three-dimensional computations.(2)Secondly,the approximately high-order reconstruction-based finite volume method is developed and discussed in detail.This method has great tolerance to the highly skewed grids with poor quality.The easiness in implementation and efficiency in computation are two advantages of this method over other high-order methods.Numerical cases indicate that the approximately high-order finite volume method has superior accuracy than the second-order method.(3)Thirdly,the cell-centered finite difference method(CCFDM)and the cell-centered symmetric conservative metric method(CCSCMM)are proposed,which is the main contribution of this paper.By combing CCFDM with CCSCMM,we could achieve high-order computation on structure grid for practical configurations.The detailed research includes:a)Firstly,the discretization for flow variables in CCFDM is shown,which consists of three main steps: interpolation from cell to face,flux evaluation at face and the differencing from face to cell.The CCFDM is an interpolation-based method for the discretization of flow variables.Then,the properties and advantages of CCFDM are discussed,including: both flux differencing splitting schemes and flux vector splitting schemes could be applied at face-centers;characteristic-based interface condition(CBIC)is no longer needed for multiblock interface;the degrees-of-freedom(DOF)will not increase with block splitting;the discretization of geometric variables is in central form not in upwind form;etc.b)Secondly,CCSCMM is proposed in discretization form to obtain metrics and Jacobian for CCFDM under the satisfaction of the geometric conservation law.The designed CCSCMM is an extension of the node-centered symmetric conservative metric method from Deng and Abe.The differencing operators used in CCSCMM are rearranged into three categories: node-to-edge differencing operator ?3,edge-to-face differencing operator ?2 and face-to-cell differencing operator ?1.These differencing operators indicate that the geometric information is gradually transformed from grid nodes to edge-centers,from edge-centers to face-centers and from face-centers to cell-centers,respectively.c)Thirdly,when all schemes used for CCFDM and CCSCMM are 2nd-order,the whole method(CCFDM with CCSCMM)is proven to be the same with cell-centered finite volume method in discretized form.This conclusion makes the developed approximately high-order reconstruction-based finite volume method fully compatible with the CCFDM and CCSCMM.d)Finally,the Green-Gauss equation for the gradient at cell-centers in finite volume method is extended to its high-order version in finite difference method.This method is named as the high-order Green-Gauss equation in this paper,which offers a new way to calculate high-order gradient under the satisfaction of geometric conservation law.(4)Fourthly,efficient time-stepping methods are studied and discussed,including two major topics: the accurate construction of time-stepping Jacobian and the efficient solver for linear equations with large sparse matrix.In this paper,several approaches for the time-stepping Jacobian are developed,which include: the Roe Jacobian and the eigenvalue Jacobian.The solvers for large sparse linear equations include: LU-SGS,DADI,DDADI,D3 ADI and GMRES.The implicit boundary condition for time-stepping methods is also focused and discussed.In this paper,the sub-iteration technique proposed by Huang for DDADI and D3 ADI is combined with the iteration of implicit boundary condition to achieve Newton convergence in simple cases.The final method,which is DDADI or D3 ADI with sub-iteration and implicit boundary condition,could achieve much faster convergence rate than traditional implicit time-stepping methods.(5)Finally,the developed high-order high-resolution CCFDM with CCSCMM is applied to several cases with relatively complex flow structures.To be specific,the cases include: Hi Lift PW-1 case,the Taylor-Green vortex case,the dual-cylinder and triple-cylinder problems from the 4th CAA workshop and the delayed detached eddy simulation of the three-dimensional cylinder at Re D=3900.According to the analysis and numerical validations,the high-order high-resolution method proposed in this paper has relatively good accuracy,efficiency and robustness properties,and will be promising in further applications.