Numerical Techniques Of Spectral Difference Method In Computational Fluid Dynamics

Vortex dominated flows caused by fluid-structure interaction are ubiquitous and have important research significance in aerospace engineering applications.To simulate these flows,there are four major challenges: accurate capturing of vortical flow structures that are sensitive to numerical dissipation;accurate and efficient incorporation of moving geometries into a flow solver;robustness to complex flow;acceleration of the solver,for better engineering applications.Compared with the second-order scheme,high-order scheme can predict the complex separated flow,unsteady flow and vortex dominated flow more accurately,which has a great advantage in depicting the flow details.Among those high-order methods,the spectral difference(SD)method which is based on a differential formulation is simple and efficient.Therefore,we will meet the above challenges by developing high-order spectral difference method and its program framework.The main contents of the present thesis are as follows.(1)Research on the 2D SD method based on unstructured grids with mixed triangular/quadrilateral elements.For triangular element,inviscid flux and viscous flux are reconstructed in Raviart-Thomas space to get stable scheme.For quadrilateral element,the original SD method is rewritten for easy implementation on mixed elements.Simulation results show that the method achieves the designed accuracy.A stability analysis is also performed for mixed elements.We found our current SD scheme is stable for up to fourth-order accurate spatial discretizations.However,a stable fifth-order SDRT scheme remains to be identified.(2)Research on the 3D SD method based on unstructured grids with mixed-element meshes.A mixed tri-prism and tetrahedral grid is first refined using one-level hrefinement to generate a hexahedral grid while keeping the curvature of wall boundaries.The SD method designed for hexahedral elements can subsequently be applied to the refined unstructured grid.Through a series of numerical tests,we demonstrate that the present method is high-order accurate for both inviscid and viscous flows.Compared with the original grid,the refined grid has larger degree of freedoms but less computation cost.As the method order increases,this advantage becomes more and more obvious.(3)To deal with the translation or rotation of objects in fluid,the moving grid method is studied by sliding-mesh method and arbitrary Lagrangian-Eulerian method.By introducing nonuniform sliding-mesh method,the computational domain can be split into nonoverlapping subdomains,where each interior subdomain can move freely with respect to its neighbors.The arbitrary Lagrangian-Eulerian method enhances the current moving grid method.Parallelization strategy is studied and achieved by message passing interface implementation.The accuracy test shows the current method can preserve the order of SD method.Through a series of numerical tests,we demonstrate that the current method is excellent for rotational objects and translational objects with large relative displacements or small spacings.(4)Research of RANS simulation and shock capturing on hybird gird based on the SD method.The RANS equations are discretized by the SD method and the eddy viscosity is modeled by Spalart-Allmaras model.A filter is designed for hybird triangular/quadrilateral grid,then the artificial viscosity is developed successfully for flow field discontinuity.For the numerical tests with shocks,the current method can stabilize the computation,suppress the non-physical oscillation and capture the discontinuous boundary clearly.For the flow with high Reynolds number,the RANS simulation results satisfy well with existing results.(5)Research on implicit time marching method based on the SD method.The singletime LUSGS method and dual-time LUSGS method are deveploed.The SD method is adopted for space discretization of the residual,then the backward differentiation formulas is used to do the time discretization on the residual governing equations,the final form of full discretization equation is derived after some linearizations and simplifications.The parallelization strategy is studied to accelerate computation.The accuracy tests show that the implicit methods satisfy the designed time order.Other numerical tests show that the implicit methods have the advantage of fast convergence.