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针对使用可压缩流动数值方法求解不可压缩流动存在的刚性问题, 基于虚拟压缩法思想, 构造了一种以Mach数、速度、密度、温度等变量为元素的预处理矩阵, 改变了控制方程组的特征根并使其量级更接近. 通过理论推导与分析, 证明新方法相比Weiss, Pletcher, Dailey和Choi的方法而言, 不仅能降低方程组的刚性, 提高了数值求解效率, 而且拥有更好的稳定性, 此外还能实现低速流动和高速流动之间的光滑过渡. 采用有限差分格式进行离散, 对流项的Roe格式作为基本加权无振荡(WENO)格式的求解器, 黏性项则使用中心型紧致差分格式来计算, 与预处理矩阵相结合展开数值实验, 结果表明新预处理方法可以实现对无黏和有黏不可压缩流动问题的高精度模拟, 且拥有比Weiss和Pletcher等提出的方法更好的收敛性和稳定性.Low velocity flows often exhibit incompressible properties, and one of the most prohibitive aspects of these problems is a large number of computer resources required, including both CPU time and memory. Various numerical schemes used to calculate incompressible flow are constantly updated to accelerate convergence and reduce resource occupation, but incompressible flow is an ideal model for studying theoretical problems after all. In addition, it is a common phenomenon that high-speed and low-speed flow regions exist in the same system, and the influence of heat and volume force cannot be ignored in some cases. The artificial compressibility method is based on the idea that the numerical algorithms for compressible flows are used to solve incompressible flow. The system of compressible flow governing equations at very low Mach numbers is stiff due to the large disparity in acoustic wave speed, u+ c, and the waves convecting at fluid speed, u. The preconditioning algorithm is effective to change the eigenvalues of the compressible flow equations system so as to remove the large disparity in wave speed, and the essence is to multiply the time derivatives with a suitable matrix. A function in low growth rate with Mach number as a variable is used to construct another new preconditioning matrix. Compared with other matrices of Dailey, Weiss, Choi and Pletcher, the new matrix can well improve the stiffness of the governing equations and the smoothness of eigenvalues in all-speed domain. A one-dimensional numerical example shows that the preconditioning matrix has ability to improve the efficiency of solving low-speed flow problems. These preconditioning matrices are extended to two-dimensional problems to simulate inviscid flow passing through a pipe with bulge and viscous flows passing through a flat and cavity. The results indicate that the new matrix has not only better accuracy but also higher efficiency than Weiss’s and Pletcher’s.
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Keywords:
- preconditioning/
- compressible flow/
- astringency/
- numerical simulation
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k t x= 0.2 x= 0.4 x= 0.6 x= 0.8 x= 1.0 0.5 exact solution 0.89253 1.0425 1.1925 1.3050 1.4925 errors of no
preconditioning100 3.9029×10–5 1.5695×10–5 2.5871×10–5 3.4728×10–5 4.2505×10–5 150 1.1399×10–5 9.3383×10–6 1.3694×10–5 1.8359×10–5 2.2459×10–5 200 6.7822×10–6 4.6281×10–6 7.3170×10–6 5.8126×10–6 6.0973×10–6 errors of
preconditioning100 3.0175×10–6 9.8264×10–7 2.2705×10–6 1.8797×10–6 2.8140×10–6 0.04 exact solution 0.52790 0.53622 0.54454 0.55286 0.56118 errors of no preconditioning 100 9.3060×10–5 9.3411×10–5 9.3782×10–5 9.4174×10–5 9.4586×10–5 150 5.5757×10–5 5.5289×10–5 5.4841×10–5 5.4415×10–5 5.4010×10–5 200 4.5265×10–5 4.4483×10–5 4.3722×10–5 4.2983×10–5 4.2267×10–5 errors of
preconditioning100 3.9460×10–6 3.1288×10–6 5.5530×10–7 2.5512×10–6 3.1288×10–6 0.01 exact solution 0.50692 0.50894 0.51096 0.51298 0.51500 errors of no
preconditioning100 4.0176×10–4 4.0335×10–4 4.0494×10–4 4.0652×10–4 4.0811×10–4 150 2.6008×10–4 2.6108×10–4 2.6208×10–4 2.6308×10–4 2.6407×10–4 200 1.9108×10–4 1.9178×10–4 1.9249×10–4 1.9319×10–4 1.9389×10–4 100 3.3569×10–6 2.2214×10–6 7.5880×10–6 4.8149×10–6 6.3518×10–6 
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