发展并验证了一种新的支持多面体网格的欧拉方程离散算法,采用Fortran 95编写了支持任意网格拓扑及三维复杂外形的通用求解器。在空间离散上,基于径向基函数理论提出了一种新的梯度计算方法,并采用动能通量分裂格式来得到准确并且稳定的二阶精度重构。该方法不依赖于网格几何形状,因此对网格质量不敏感。由于在时间方向采用了点隐松弛推进方法,使得该求解器在大时间步长上仍能保持稳定性。最后通过若干数值算例对本文所发展的算法进行了验证,证明了本文的算法及求解器具有稳定、准确的特性及宽广的网格类型适应性。
A new algorithm for solving Euler equations on polyhedral grids is developed and validated in this paper. A general solver which supports arbitrary mesh topology and three-dimensional complex geometry is constructed by using Fortran 95 language. For spatial discretization, a new improved radial basis function method is proposed for gradient calculation. An accurate and robust second-order reconstruction is achieved by using the kinetic flux vector splitting scheme. The new method does not depend on the geometry of the grid. Thus it is much less sensitive to grid quality. With a point implicit relaxation time marching strategy, the solver remains stable at large time steps. The test cases indicate that the algorithm and the solver developed in this paper are stable, accurate while exhibiting good flexibility on mesh universality.
[1] Jameson A, Schmidt W, Turkel E. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA-1981-1259, 1981.
[2] Barth T J, Jespersen D C. The design and application of upwind schemes on unstructured meshes. AIAA-1989-366, 1989.
[3] Lhner R, Parikh P. Generation of three-dimensional unstructured grids by the advancing-front method[J]. International Journal for Numerical Methods in Fluids, 1988, 8(10): 1135-1149.
[4] Mavriplis D J. Multi-grid techniques for unstructured meshes, VKI lecture series VKI-LS 1995-02[M]. Belgium: von Karman Institute for Fluid Dynamics, 1995: 1-57.
[5] Frink N T. Upwind scheme for solving the Euler equations on unstructured tetrahedral meshes[J]. AIAA Journal, 1992, 1(1): 70-77.
[6] Venkatakrishnan V, Mavriplis D J. Implicit solvers for unstructured meshes. AIAA-1991-1537, 1991.
[7] Ahn H T, Carey G F. An enhanced polygonal finite-volume method for unstructured hybrid meshes[J]. International Journal for Numerical Methods in Fluids, 2007, 54(1): 29-46.
[8] Mandal J C, Deshpande S M. Kinetic flux vector splitting for Euler equations[J]. Computers and Fluids, 1994, 23(2): 447-478.
[9] van Leer B, Thomas J, Roe P et al. A comparison of numerical flux formulas for the Euler and Navier-Stokes equations. AIAA-1987-1104, 1987.
[10] Xu K. A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method[J]. Journal of Computational Physics, 2001, 171(1): 289-335.
[11] Gnoffo P A. An upwind-biased, point-implicit relaxation algorithm for viscous, compressible perfect-gas flows. NASA TP-2953, 1990.
[12] Buhmann M D. Radial basis functions: theory and implementations[M]. Cabridge: Cambridge University Press. 2003: 99-136.
[13] Wu Y L, Shu C. Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli[J]. Computational Mechanics, 2002, 29(6): 477-485.
[14] van Leer B. Towards the ultimate conservative difference scheme. V-A second-order sequel to Godunov's method[J]. Journal of Computational Physics, 1979, 32: 101-136.
[15] Venkatakrishnan V. On the accuracy of limiters and convergence to steady state solutions. AIAA-1993-880, 1993.
[16] Lhner R. CFD3: sample problems. (1999-08-24) . http: //lcp. nrl. navy. mil/cfd-cta/CFD3/
[17] Lhner R. Applied computational fluid dynamics techniques: an introduction based on finite element methods[M].2nd ed. Chichester, England: John Wiley & Sons Ltd, 2008: 360-361.
[18] Schmitt V, Charpin F. Pressure distributions on the ONERA-M6-Wing at transonic mach numbers. AGARD AR-138, 1979.
[19] Townsend J C, Howell D T, Collins I K, et al. Surface pressure data on a series of analytic forebodies at mach numbers from 1.70 to 4.50 and combined angles of attack and sideslip. NASA TM-80062, 1978.
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