Numerical Perturbation Algorithm and Its CFD Schemes
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摘要: 作者提出的数值摄动算法把流体动力学效应耦合进NS方程组和对流扩散(CD)方程离散的数学基本格式(MBS),特别是耦合进最简单的MBS即一阶迎风和二阶中心格式之中,由此构建成一系列新格式,称呼方便和强调耦合流体动力学起见,称它们为流体力学基本格式(FMBS)。构建FMBS的主要步骤是把MBS中的通量摄动重构为步长的幂级数,利用空间分裂和导出的高阶流体动力学关系式,把结点变量展开成Taylor级数,通过消除重构格式修正微分方程的截断误差诸项求出幂级数的待定系数,由此获得非线性FMBS。FMBS的公式是MBS与 (及 )之简单多项式的乘积, 和 分别是网格Reynolds数和网格CFL数。FMBS和MBS使用相同结点,简单性彼此相当,但FMBS精度高稳定范围大,例如FMBS包含了许多绝对稳定和绝对正型、高阶迎风和中心有限差分(FD)格式和有限体积(FV)格式,这些格式对网格Reynolds数的任意值均为不振荡格式。可见对不振荡CFD格式的构建,数值摄动算法提供了不同于调节数值耗散等常见的人为构建方法,而利用流体力学自身关系以及把迎风机制通过上、下游摄动重构引入中心MBS的解析构建方法,FMBS除了直接应用于流体计算外;对于通过调节数值耗散、色散和数值群速度特性重构高分辨率格式的研究,最简单FMBS提供了比最简单MBS更精确、但同样简单的基础和起步格式。FMBS用于计算不可压缩流,可压缩流,液滴萃取传质,微通道两相流等,均获得良好数值结果或与已有Benchmark解一致的数值结果。已有文献称数值摄动算法为新型高精度格式和高的算法和高的格式;本文FMBS比数值摄动格式的称呼可更好反映FMBS的物理内容。文中也讨论了值得进一步研究的一些课题,该法亦可用于其它一些数学物理方程(例如,简化Boltzmann方程、磁流体方程、KdV-Burgers方程等)MBS耦合物理动力学效应的重构。Abstract: The numerical perturbation algorithm presented by the author is to couple fluid dynamics effects with mathematical basic schemes (MBS), especially with the most simplest MBS, i.e. first order upwind and second order central schemes for the Navier-Stokes(NS) equations and convective diffusion equation. As a result, many new schemes are obtained and called fluid mechanics basic scheme (FMBS), for convenience and emphasizing MBS coupling with fluid dynamics effects. The main steps of constructing FMBS are that flux and coefficient of convective derivative in MBS are reconstructed as power-series of grid interval and that by splitting resultant scheme above and operating the splitted scheme, the high-order fluid mechanics relation is obtained and that the variables at upstream and downstream nodes are expanded in Taylor series and that by eliminating truncated error terms in the modified differential equation of the reconstructed scheme the undetermined coefficients in the power-series are determined and then the FMBS is obtained. Formulations of FMBS is product of MBS and the numerical perturbation reconstruction functions, that are simple polynomial of (or ), where and are grid Reynolds number and grid CFL number, respectively.FMBS and original MBS utlize the same nodes and are nearly identical simple. But FMBS have higher accurate and larger stable-range than MBS. For example, the most simplest and the most important six FMBS for the convection diffusion (CD) equation are : sixth-order upwind finite-difference (FD) FMBS, dual perturbation (DP) fourth-and eighth-order central FD-FMBS, dual perturbation third- and fifth-order finite volume (FV) central FMBS and sixth upwind FV-FMBS. This six schemes are absolute stability or absolute positive and are non-oscillatory schemes for any values of grid Reynolds number. In the case of one dimension, this six schemes are TVD scheme for any values of grid Reynolds number. However, the same order MBS must use multi-nodes and oscillate on coarse grids, Besides FMBS being directly used to calculate flow, FMBS is also acted as a basic or starting scheme for reconstructing high resolution scheme by self-adjust numerical dissipation. The above six FMBS and others have been used to calculate incompressible flows, compressible flows, mass transfer and Marangoni convection in a falling drop, two phase flows etc., some excellent numerical results are given. For example, FMBS solve lib-driven and buoyancy-driven cavity flows and result in several new Benchmark solutions. The numerical perturbation algorithm and corresponding schemes are also called Gao's algorithm and Gao's schemes. However, the name of FMBS is more reasonable than the numerical perturbation scheme, this is because that FMBS may depict exactly physical connotation of MBS coupling with fluid dynamics effects. Several subjects being worthy of study further are discussed. The present method is also suitable for reconstruction of MBS of other mathematical physics equation (such as the simplified Boltzmann equation, magnetic fluid mechanic equations, KdV-Burgers equation etc.) coupling with physics dynamics effects.
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