自适应网格局部加密方法在可压缩流场模拟中的应用

THE APPLICATION OF ADAPTIVE MESH REFINEMENT TO THE NUMERICAL SIMULATION OF COMPRESSIBLE FLOW

本文基于M.J.Berger和J.Oliger发展的自适应网格局部加密算法,它采用多个组合网格的思想,用于求解双曲型方程组。它结合显式有限差分格式,采用Richardson外推技术自动进行局部截断误差估计,对精度低的区域产生新细网格局部加密,或去除无需再加密的旧细网格,以最小的运算量达到给定的精度要求。网格可一层层加密下去,按层次覆盖,每个网格是有任意方向的矩形均匀网格。这套算法独立于求解所用的差分格式,很容易和各种格式结合。 我们从M.J.Berger和J.Oliger程序出发,参考M.J.Berger和P.Colella,对守恒型方程组,实现了粗细网格交接面上数值通量守恒,从而能计算间断解。对网格边界处理、误差估计和网格重新生成等做了一些改进,并且可以灵活地在任何感兴趣区域指定加密。 我们用不定常Euler方程组计算了带前台阶的二维管道中Mach-3流动,采用MacCormack显式附加人工粘性差分格式。共用了四层网格结构:Δ=1/10、1/20、1/80和1/320。其中基本网格是Δ=1/10;到台阶上第一次激波反射区域从Δ=1/20加密至Δ=1/80,计算到定常所需运算量仅是同等均匀网格△=1/80的28％;我们还用Δ=1/320网格对台阶拐角和切向间断处指定加密,结果表明可明显改善台阶上的激波Mach反射,切向间断清晰可辨,所需的计算量是上述均匀网格Δ=1/

This work is based on the adaptive mesh refinement (AMR) algorithm developed by Berger and Oliger . The AMR method is based on the idea of using multiple, component grids on which the hyperbolic partial differential equations are solved by explicit finite difference scheme . Based upon Richardson-type estimates of the local truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. In addition, this pproacha is recursive in that fine grids can themselves contain even finer subgr-ids. High level refinement grids overlap the low level ones. Each grid is rectangular with arbitrary orientation and uniform spacing. It is assumed that the finite difference scheme involved is explicit but the algorithm is independent of the particular scheme, hence it can be easily combined with any difference scheme of the user's choice.Starting from the algorithm of Berger and Oliger. and following Berger and Colella , we have realized numerical fluxes conservation on the interfaces between the coarse and fine grids, so that discontinuous solutions can be computed. Also, we have made some improvement on boundary treatment, error estimation and regridding, and have added the option of refinement of any subdomain specified by the user.For unsteady Euler equations, a Mach-3 wind tunnel flowfield with a forward-facing step is computed, using the MacCormack explicit scheme with artificial viscosity. Four level grids are used: △=1/10, 1/20, 1/80 and 1/320, in which the △ = 1/10 grids are the basic grids and the subdomain from the left boundary to the right of first shock reflection on the step is refined adaptivly from 2 = 1/20 to A=1/80. The total CPU time to steady state of AMR is only 28% of that of the uniform fine grid △= 1/80. Also the subdomain with the corner of the step and the subdomain with the tangential discontinuity are refined by specification from △=1/80 to △ = 1/320. The results show that the shock reflection on the step is greatly improved and the resolution of the tangential discontinuity is good, while the total CPU time is 45% of that of the uniform grid △ = 1/80.