A Gas-Kinetic BGK Scheme for Parallel Solution of 3-D Viscous Flows on Unstructured Hybrid Grids (original) (raw)
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A discontinuous Galerkin Method based on a Bhatnagar-Gross-Krook (BGK) formulation is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. The idea behind this approach is to combine the robustness of the BGK scheme with the accuracy of the DG methods in an effort to develop a more accurate, efficient, and robust method for numerical simulations of viscous flows in a wide range of flow regimes. Unlike the traditional discontinuous Galerkin methods, where a Local Discontinuous Galerkin (LDG) formulation is usually used to discretize the viscous fluxes in the Navier-Stokes equations, this DG method uses a BGK scheme to compute the fluxes which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function. The developed method is used to compute a variety of viscous flow problems on arbitrary grids. The numerical results obtained by this BGKDG method are extremely promising and encouraging in terms of both accuracy and robustness, indicating its ability and potential to become not just a competitive but simply a superior approach than the current available numerical methods.
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This paper presents an extension of the gas kinetic BGK scheme to simulate inviscous high temperature equilibrium gas flows on unstructured meshes. In the present scheme, the internal degree of freedom of the gas distribution function is variable according to an effective c, which is obtained through polynomial curve fits of thermodynamic properties of high temperature equilibrium air. With this method, the proper Euler equations for high temperature equilibrium airflows are recovered from the Boltzmann BGK equation. To accelerate the convergence of the present scheme, the matrix-free LU-SGS implicit time marching scheme is also developed. In addition,for demonstrating the effectiveness and accuracy of the present method, several inviscous airflow examples are also provided. The numerical results agree well with the results of previous studies and show that the present scheme is accurate and robust.
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The goal of this paper is to investigate and develop fast and robust solution techniques for high-order accurate Discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured meshes. Previous work was focused on the development of hp-multigrid techniques for inviscid flows and the current work concentrates on the extension of these solvers to steady-state viscous flows including the effects of highly anisotropic hybrid meshes. Efficiency and robustness are improved through the use of mixed triangular and quadrilateral mesh elements, the formulation of local order-reduction techniques, the development of a line-implicit Jacobi smoother, and the implementation of a Newton-GMRES solution technique. The methodology is developed for the two-and three-dimensional Navier-Stokes equations on unstructured anisotropic grids, using linear multigrid schemes. Results are presented for a flat plate boundary layer and for flow over a NACA0012 airfoil and a two-element airfoil. Current results demonstrate convergence rates which are independent of the degree of mesh anisotropy, order of accuracy (p) of the discretization and level of mesh resolution (h). Additionally, preliminary results of on-going work for the extension to the Reynolds Averaged Navier-Stokes(RANS) equations and the extension to three dimensions are given. agglomeration multigrid technique (h-multigrid) for Discontinuous Galerkin discretizations. This paper represents an extension to the Navier-Stokes equations of the previous work originally presented in references. The extension includes the DG discretization of the viscous fluxes using the Symmetric Interior Penalty Method (SIP) together with the capability of using hybrid meshes on two-dimensional configurations. In order to maintain the convergence rate of the hp-multigrid solver on hybrid anisotropic meshes, a line creation algorithm and line-implicit Jacobi smoother have been devised and implemented . Additional solver efficiency is obtained through the implementation of a preconditioned Newton-GMRES Krylov method, which represents an alternative method for solving the linear system given by Newton's method. The methodology is developed for the two-and three-dimensional Navier-Stokes equations on unstructured anisotropic grids, using linear hp-multigrid schemes as preconditioners. While results presented here pertain only to steady-state problems, the methods are easily extended to time-dependent problems. 5 Extensions to the Reynolds-averaged Navier-Stokes equations, involving the solution of an additional turbulence modeling equation are investigated.
A coupled finite volume solver for the solution of incompressible flows on unstructured grids
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This paper reports on a newly developed fully coupled pressure-based algorithm for the solution of laminar incompressible flow problems on collocated unstructured grids. The implicit pressure-velocity coupling is accomplished by deriving a pressure equation in a procedure similar to a segregated SIMPLE algorithm using the Rhie-Chow interpolation technique and assembling the coefficients of the momentum and continuity equations into one diagonally dominant matrix. The extended systems of continuity and momentum equations are solved simultaneously and their convergence is accelerated by using an algebraic multigrid solver. The performance of the coupled approach as compared to the segregated approach, exemplified by SIMPLE, is tested by solving five laminar flow problems using both methodologies and comparing their computational costs. Results indicate that the number of iterations needed by the coupled solver for the solution to converge to a desired level on both structured and unstructured meshes is grid independent. For relatively coarse meshes, the CPU time required by the coupled solver on structured grid is lower than the CPU time required on unstructured grid. On dense meshes however, this is no longer true. For low and moderate values of the grid aspect ratio, the number of iterations required by the coupled solver remains unchanged, while the computational cost slightly increases. For structured and unstructured grid systems, the required number of iterations is almost independent of the grid size at any value of the grid expansion ratio. Recorded CPU time values show that the coupled approach substantially reduces the computational cost as compared to the segregated approach with the reduction rate increasing as the grid size increases.
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A set of implicit methods are proposed for a third-order hierarchical WENO reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids. An attractive feature in these methods are the application of the Jacobian matrix based on the P 1 element approximation, resulting in a huge reduction of memory requirement compared with DG (P 2 ). Also, three approaches -analytical derivation, divided differencing, and automatic differentiation (AD) are presented to construct the Jacobian matrix respectively, where the AD approach shows the best robustness. A variety of compressible flow problems are computed to demonstrate the fast convergence property of the implemented flow solver. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results on complex geometries indicate that this low-storage implicit method can provide a viable and attractive DG solution for complicated flows of practical importance.