Research On Discontinuous Galerkin Finite Element Method Based On Unstructured Hybrid Grids

Unstructured grid eliminates the structural constraints of grid nodes and is more suitable for solving complex flow problems.High order accuracy numerical methods have less numerical dispersion and dissipation,and can simulate complex flowfield precisely.It is one of the main means to improve the efficiency of numerical simulation and is required for numerical simulation method such as large eddy simulation and direct numerical simulation.Therefore,it is of great significance to develop high-order accurate numerical methods based on unstructured hybrid grids.Discontinuous galerkin finite element method(DG)is a typical high-order accurate compact numerical methods.It is suitable for large-scale parallel computing of complex flow problems and is widely used in CFD,but the low computational efficiency and the instability of shock capture severely restrict the application of this method.The main goal of this thesis was to develop high-order accuracy discontinuous Galerkin finite element method based on unstructured hybrid grids,focusing on solving computational efficiency and shock-capturing problem of DG by developing high efficiency implicit iterative method,high efficiency parallel computing technology and shock-capturing technology.It will provide a computing platform for the step of establishing DES,ILES.Finally,the method is applied to the numerical simulation of hypersonic aerodynamic force/thermal,areoengine internal flow,helicopter rotor flowfield,aerodynamic noise and other peoblems.The thesis mainly included the following aspects:1.For unstructured hybrid grids,a discontinuous Galerkin finite element method was established based on the Taylor basis function.In order to improve the stability of turbulence simulation,the loose coupling method of the Navier-Stokes equations and the turbulence model equations was adopted,and a hybrid DG/FV method for solving RANS equations was established.The accuracy of the developed method was verified by many typical examples such as isentropic vortices,laminar flow over a flat plate,couette flow,and turbulent flow over a flat plate.2.In order to improve computational efficiency of DG,an implicit time iterative method based on jacobian matrix exact calculation technology and a large-scale parallel computing technology were developed.Combined with the PETSc library(Portable,Extensible Toolkit for Scientific Computation),GMRES(Generalized Minimum Residual)implicit time discretization method was developed to acclerate convergence of steady problem and LU-SGS(Lower-Upper Symmetric-Gauss-Seidel)was also developed.In order to further improve the convergence speed of GMRES,for the inviscous flux Roe's scheme and the viscous flux BR2 scheme(Bassi-Rebay 2),the jacobian matrix exact solution technology was developed by the chain rule.Based on this technology,the computational efficiency of GMRES was greatly improved.The parallel computing method was developed based on MPI(Message Passing Interface),which solved the memory requirement and computational efficiency problem for large scale grid.The computational efficiency of development method was analyzed by several typical examples.The results show that GMRES based on the Jacobi matrix exact solution technique can increase the CFL number of numerical simulation,greatly improve the computational efficiency and have better stability.3.In order to deal with complex flow with strong shocks,the thesis developed the limiter method in shock capture technology,including slope and reconstruction limiter,discontinuous detector,negative pressure and negative density correction technology.For weak discontinuity problem,a limiting method without affecting the convergence of residuals was established.Combined with the kxrcf discontinuous detector,a new class of kxrcf-grad discontinuous detector was proposed based on variables gradient of mesh and its adjacent surface element,which more accurately detected the ‘problem element' and improved simulation accuracy and efficiency.Two classes of limiter technologies including HWENO limiter and Venkatakrishnan limiter were summarized,the influence law of free parameter K in Venkatakrishnan limiter was obtained.The reason for the limter affected the residual convergence was discussed,anda limiter method without affecting the convergence of residusl was established for weak discontinuity.For negative density and negative pressure problems may occure in strong discontinuity,the correction method that was compatible with DG method was established and improved the stability of the calculation.The developed shock capture technology is verified by a typical example such as double Mach reflection,two-dimensional airfoil,and three-dimensional wing.4.Based on the established high-order accurate DG computation platform,numerical simulation of various typical flow problems was carried out,and the computational accuracy and numerical robustness of the method were further verified.The application covered typical examples included two-dimensional laminar flow,shock boundary layer interference,30P30 N multi-element airfoil,DPW-W1 wing,65° swept Delta wing with tip leading and a low aspect ratio flying-wing model.The results show that the developed DG method can be used to numerical simulate complex shapes,turbulence,boundary layers,and large separation flows,etc.