﻿ Euler方程的分裂型通量分裂双时间步隐式方法
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Euler方程的分裂型通量分裂双时间步隐式方法

Split-type implicit scheme using flux splitting and dual-time step for Euler equations
DONG Haitao , CHEN Zh e, LIU Fujun
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:There are some shortcomings of the traditional implicit schemes such as complex forms and large amount of computations. Using the idea of operator splitting combining with implicit discrete schemes—flux vector splitting and dual-time step scheme—a simpler split-type implicit difference scheme for Euler equations was developed. The validity and reliability of the new implicit scheme were verified by performing numerical experiments on some typical problems in aerodynamics, and the properties of the new scheme were discussed in detail at the same time. The new scheme has common advantages of good stability and few constraints on time step just like other implicit schemes. In addition, the new scheme has the following advantages: it has simple formulas; it is easy for programming; it needs smaller amount of computations by avoiding solving systems of equations and doing inverse matrix operation compared with conventional implicit schemes in single time step; it has faster convergence rate compared with LU-SGS scheme.
Key words: Euler equations     operator splitting     flux vector splitting     dual-time step     implicit scheme

1 计算方法

1.1 按物理时间分裂

1) 2个方向分别进行虚拟时间合成,然后再做一次物理时间合成:

2) 2个方向混合进行虚拟时间合成:

1.2 按虚拟时间分裂

1.3 二阶时间精度 1.3.1 按虚拟时间分裂

1.3.2 按物理时间分裂

1.4 数值通量选择

NND格式通量为[11, 12]

MUSCL格式通量为[13, 14]

1.5 定常问题隐式

1.6 其他问题

2 算例验证 2.1 Sod激波管

 图 1 t=0.15s时密度分布计算结果Fig. 1 Results of density distribution when t=0.15s

 图 2 二阶时间精度计算结果Fig. 2 Results of second order time accuracy

2.2 Lax激波管

 图 3 均匀网格情况、相同计算量下不同CFL数的 计算结果Fig. 3 Results with the same computations and different CFL numbers for uniform grid

2.3 改进的Lax激波管

 图 4 非均匀网格情况、相同计算量下,不同CFL数的 计算结果Fig. 4 Results with the same computations and different CFL numbers for non-uniform grid

2.4 超声速流绕前台阶流动

 图 5 计算模型示意图Fig. 5 Schematic diagram of calculation model

2.4.1 计算条件1

 图 6 前台阶网格1Fig. 6 Grid 1 of forward-facing step

 图 7 网格1前台阶绕流等密度线分布(C=1.0)Fig. 7 Density contours of forward-facing step flow for Grid 1 (C=1.0)

2.4.2 计算条件2

 图 8 前台阶网格2Fig. 8 Grid 2 of forward-facing step

 图 9 网格2前台阶绕流等密度线分布(C=8.0)Fig. 9 Density contours of forward-facing step flow for Grid 2 (C=8.0)

2.5 NACA0012翼型绕流

NACA0012翼型网格如图 10所示,网格数为239×59.

 图 10 NACA0012翼型网格Fig. 10 Grid of NACA0012 airfoil

1) 来流马赫数为0.8,攻角为1.25°.

 图 11 流场压强等值线和翼型表面压强系数分布Fig. 11 Pressure contours and pressure coefficients on airfoil surface

2) 来流马赫数为1.2,攻角为7°.

2.6 双椭球无粘绕流

 图 12 计算网格Fig. 12 Grid for calculation
 图 13 实验激波纹理照片Fig. 13 Photo of experimental texture[17]

 图 14 流场压强等值线和对称面上下物面压强系数Fig. 14 Pressure contours and pressure coefficients on top and low surfaces

 图 15 Sod激波管2种方法的比较Fig. 15 Comparison of two schemes in Sod shock tube
3.2 超音速流绕前台阶流动

 图 16 前台阶绕流2种方法的比较Fig. 16 Comparison of two schemes in forward-facing step flow

LU-SGS方法为了提高计算效率,避免矩阵求逆,采用了简化的A±、B±、C±分解求法,故对求解的收敛性有了一定的影响,本文方法使用了严格的A±、B±、C±分解,所以收敛性能较好,也起到了减小计算量的作用.

1) 方法具有隐式格式的普遍优点,即稳定性好,CFL数可以放大;此外,方法格式简单,易于编程,单步时间推进不涉及方程组常规求解和矩阵求逆,计算量小,与LU-SGS方法相比,收敛更快.

2) 对于定常问题,方法可以在增大CFL数、提高计算效率的情况下保证精度.

3) 对于非定常问题,方法的计算精度主要由计算量决定,同等计算量下,不会随CFL数的增大而丧失精度.

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#### 文章信息

DONG Haitao, CHEN Zhe, LIU Fujun
Euler方程的分裂型通量分裂双时间步隐式方法
Split-type implicit scheme using flux splitting and dual-time step for Euler equations

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(5): 776-785.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0326