Research Of High Order Numerical Methods In Computational Fluid Dynamics

In recent years,high order methods have received considerable attention in the compu-tational fluid dynamics(CFD)communitity because of their attractive advantages.Recently,there are two popular ideas to construct efficient high order schemes.One is construction of the high order algorithms for unstructured grid based on a differential formulation to provide high efficiency,such as spectral difference(SD)method and correction procedure via recon-struction(CPR)algorithm.Another is the hybrid reconstruction algorithms.The high order discontinuous Galerkin(DG)method evolves some degree of freedoms(DOFs)in one cell to provide a polynomial with high degree and the finite volume(FV)method only solves one DOF in one cell but reconstruct a high degree polynomial from the information of neighboring cells.The hybrid reconstruction methods combine the two strategies to provide a compromise per-formance which alleviates the huge memory and central processing unit(CPU)requirements of the DG method.Following the two methodologies,the present thesis concentrates on the researches list below.1.Research on the SD method based on the Taylor expansion defined in the reference space.The Taylor expansion has been introduced into the SD scheme to express solution in the reference space.To evolve the DOFs,the governing equation and its derivatives are used to solve the unkowns in the expansion.In the context of the new proposed method,a p-multigrid algorithm has been implemented to accelerate the convergence.2.Research on the CPR method based on the Taylor expansion defined in the physical space.The idea behind the above new SD scheme has been imported into the CPR framework,which offers a new scheme facilitates the implemention of mixed mesh because the new method has a uniform formulation for arbitrary cell.Theoretical analysis and numerical results demon-strate that the proposed method is efficient,conservative and suitable for the implementation of limiting algorithm or reconstuction procedure for the shock-capturing.3.Based on the above new CPR algorithm,a PnPm-CPR method has been constructed by the introduction of hybrid reconstruction idea like the PnPm scheme.Theoretical analysis shows that this scheme is not stable on triangular element but stable on quadrilateral mesh.Numerical cases indicate that the new prodecure obtains the expect order of accuracy,and it is more efficient than the pure CPR algorithm for both the CPU costs and the memory require-ments.4.The method proposed by Bassi and Rebay(BR2 scheme)has been used to discretize the viscosity terms in the Navie-Stokes equations.Some numerical tests demonstrate that the high order CPR method prescribed in the present thesis performs very well than the second order algorithm.5.The radial basis functions(RBF)have been introduced for the generation of curvilinear mesh.Beside the algorithms described above which are used for the discretization of governing equation,high order curvilinear grid is essential for the approximation of boundary for the high order methods because linear mesh would cause unphysical solution.However,when the meshes on the boundary have been curved,the volume cells would overlap.Therefore,the RBF method is used to adjust grid nodes in the volume to avoid the overlapping mesh after the surface meshes have been curved.