Study On The High-Order Numerical Methods And Acceleration Techniques On Unstructured Grids

In recent years,the computational fluid dynamics(CFD)has been widely used in aircraft design,and has played a more and more important role in fluid mechanics.Unstructured grids have random data storage,disordered adjacent grid nodes,and thus they have a natural ability to adapt to complex shapes.Therefore,the method for flow solver based on unstructured grids has become the mainstream of CFD.Compared with the second-order accuracy methods,the high-order schemes have smaller numerical dissipation and dispersion,and can predict more elaborate and detailed results for the complex flows,such as wave propagation,unsteady flow and vortex-dominated flows.However,they also encounter many difficulties such as poor stability and low computational efficiency that have not been well solved,which limits their applications in the engineering field.Based on the finite volume method,the thesis developed the basic framework of high-order schemes on unstructured grids.With a view to improving the numerical stability and computational efficiency,we mainly research on the aspects of the high-order flow reconstruction method,accuracy preserved limiter design and flow solver acceleration method.The main research works are as follows:(1)A high-order reconstruction method based on radial basis function(RBF)has been proposed for finite volume method in the thesis.The multiquadric basis function has been used in reconstruction step based on the unstructured grids.And we also prove the accuracy of RBF interpolation method in theory by using multiquadric basis function in one-dimensional problem.Compared with the traditional reconstruction method based Taylor series polynomial,the RBF method has more flexibility,smaller numerical errors and lower dissipation.(2)Two kinds of accuracy preserved limiters,DWBAP(distance weighted BAP)and DDWENO(distance derivative weighted ENO),have been developed for high-order finite volume method.The DWBAP limiter improves the numerical accuracy when the traditional BAP limiter extends to high-order schemes on unstructured grids.The DDWENO limiter combines the advantages of the slope limiters and WENO-type limiters,and can achieve the similar effect of WENO schemes in the fixed stencil with high computational efficiency.Twoand three-dimensional numerical cases demonstrate that both the developed limiters can capture the shock waves clearly and steeply,and preserve the numerical accuracy in smooth regions as well.(3)A mode multigrid(MMG)method has been proposed and applied to accelerate the convergence of the steady state flow on unstructured grids.The method projects the flowfield solutions from the physical space into the modal space,and truncates the high-frequency modes in order to filter out the perturbations of the flow field.The MMG method is easy to implement,and ingeniously avoids the complicated process of the grid coarsening and data transfer between the fine grids and coarse grids.Furthermore,the method has no dependence on computational mesh,and can be suitable for explicit/implicit time-marching method and second/high-order schemes,which exhibits great potential in extensive applications on complex configurations and engineering.