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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
[1] Yin P, Liandrat J, Shen W 2021 J. Comput. Phys. 434 110207Google Scholar
[2] Fernandez Fidalgo J, Nogueira X, Ramirez L, Colominas I 2018 Comput. Methods Appl. Mech. Eng. 335 91Google Scholar
[3] Ahmed T, Rehman A, Ali A, Qamar S 2021 Results Phys. 23 104078Google Scholar
[4] Zhang Y, Liu Y, Zhang Y, Wang W R, Han Y F 2021 Int. Commun. Heat. Mass Transfer 129 105688Google Scholar
[5] Xie W J, Tian Z Y, Zhang Y, Yu H 2022 Comput. Fluids 233 105215Google Scholar
[6] Ma Chao, Wu J, Yu H, Yang L 2022 Comput. Math. Appl. 105 13Google Scholar
[7] Lehel F, Hona J 2021 Chin. J. Phys. 73 360Google Scholar
[8] 张庆海, 李阳 2021 物理学报 70 130201Google Scholar
Zhang Q H, Li Y 2021 Acta Phys. Sin. 70 130201Google Scholar
[9] Xie W, Luo Z B, Zhou Y, Gao T X, Wu Y, Wang Q 2021 Acta. Astronaut. 188 416Google Scholar
[10] Barhoumi B, Bessrour J 2017 J. Comput. Phys. 21 86Google Scholar
[11] Bykerk T, Verstraete D, Steelant J 2020 Aerosp. Sci. Technol. 103 105883Google Scholar
[12] Huber S E, Trummer M R 2016 Comput. Math. Appl. 71 319Google Scholar
[13] 胡嘉懿, 张文欢, 柴振华, 施保昌, 汪一航 2019 物理学报 68 234701Google Scholar
Hu J Y, Zhang W H, Chai Z H, Shi B C, Wang Y H 2019 Acta Phys. Sin. 68 234701Google Scholar
[14] 辛建建, 陈振雷, 石凡, 石伏龙 2020 物理学报 69 044702Google Scholar
Xin J J, Chen Z L, Shi F, Shi F L 2020 Acta Phys. Sin. 69 044702Google Scholar
[15] Imran M A, Shaheen A, Sherif E M, Rahimi-Gorji M, Seikh A H 2020 Chin. J. Phys. 66 60Google Scholar
[16] 尹纪富, 尤云祥, 李巍, 胡天群 2014 物理学报 63 044701Google Scholar
Yin J F, You Y X, Li W, Hu T Q 2014 Acta Phys. Sin. 63 044701Google Scholar
[17] 牛海波, 易仕和, 刘小林, 霍俊杰, 冈敦殿 2021 物理学报 70 134701Google Scholar
Niu H B, Yi S H, Liu X L, Huo J J, Gang D D 2021 Acta Phys. Sin. 70 134701Google Scholar
[18] 李高华, 王福新 2018 物理学报 67 054701Google Scholar
Li G H, Wang F X 2018 Acta Phys. Sin. 67 054701Google Scholar
[19] Fasel M 1976 J. Fluid. Mech. 78 355Google Scholar
[20] Aziz K 1967 Phys. Fluids 10 314Google Scholar
[21] Harlow F H, Welch J E 1965 Phys. Fluids 8 2182Google Scholar
[22] Hirt C W, Nichols B D, Romero N C 1974 SOLA: A Numerical Solution Algorithm for Transient Fluid Flows (Los Alamos: Los Alamos Scientific Laboratory) Report5852
[23] Patankar S V, Spalding D B 1972 Int. J. Heat. Mass Transfer 15 1787Google Scholar
[24] Patankar S V 1981 Numer. Heat Transfer 4 409Google Scholar
[25] Van Doormaal J P, Raithby G D 1984 Numer. Heat Transfer 7 147Google Scholar
[26] Minkowycz W J, Sparrow E M, Pletcher R H, Schneider G E 1988 Handbook of Numerical Heat Transfer (New York: John Wiley & Sons) pp241–289
[27] Issa I R, Gosman A D, Watkins A P 1986 J. Comput. Phys. 62 66Google Scholar
[28] Davis G D V 1983 Int. J. Numer. Methods Fluids 3 249Google Scholar
[29] Chorin A J 1967 J. Comput. Phys. 2 12Google Scholar
[30] Turkel E 1993 Appl. Numer. Math. 12 257Google Scholar
[31] Choi D, Merkel C L 1987 AIAA J. 23 581
[32] Viviand H 1985 Numerical Methods for the Euler Equations of Fluid Dynamics (Philadelphia: SIAM) p519
[33] Choi Y H, Merkle C L 1993 J. Comput. Phys. 105 205
[34] Buelow P E O, Venkateswaran S, Charles L 1993 AIAA J. 32 2401Google Scholar
[35] Godfrey A G, Walters R W, Leer B V 1993 31st Aerospace Sciences Meeting Reno, NV, USA, January 11–14, 1993 p535
[36] Weiss J M, Smith W A 1994 Fluid Dynamics Conference Colorado Springs, CO, USA, June 20–23, 1994 p2209
[37] Pletcher R H, Chen K H 1993 11th Computational Fluid Dynamics Conference Orlando, FL, USA, July 6–9, 1993 p3368
[38] Dailey L D, Pletcher R H 1996 Comput. Fluids 25 791Google Scholar
[39] 成小磊 2007 硕士学位论文 (南京: 南京航空航天大学)
Cheng X L 2007 M. S. Thesis (Nanjing: Nanjing University of Aeronautics and Astronautics) (in Chinese)
[40] 谭勤学, 任静, 蒋洪德 2015 清华大学学报(自然科学版) 55 134Google Scholar
Tan Q X, Ren J, Jiang H D 2015 J. Tsinghua Univ. (Science and Technology) 55 134Google Scholar
[41] Buelow P E G, Schwer D A, Feng F J, Merkle C L 1997 13th Computational Fluid Dynamics Conference Snowmass Village, CO, USA, June 29–July 2, 1997 p120
[42] Roe P 1981 J. Comput. Phys. 43 357Google Scholar
[43] Jiang G S, Shu C W 1996 J. Comput. Phys. 126 202Google Scholar
[44] Sun Z S, Yu H U, Luo L, Yang Z W 2014 Sci. Chin. Phys. Mech. 57 971Google Scholar
[45] Shu C W, Osher S 1988 J. Comput. Phys. 77 439Google Scholar
[46] 胡丽燕 2011 硕士学位论文 (南京: 南京航空航天大学)
Hu L Y 2011 M. S. Thesis (Nanjing: Nanjing University of Aeronautics and Astronautics) (in Chinese)
[47] Ghia U, Ghia K N, Shin C T 1982 J. Comput. Phys. 48 387Google Scholar
[48] Delery J 1981 14th Fluid and Plasma Dynamics Conference Palo Alto, CA, USA, June 23–25, 1981 p1245
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图 1 经不同预处理方法后的特征系统(对于每种预处理后的速度有, U' = (λ3+λ4)/2, C' = |λ3–λ4|/2, 均为无量纲化速) (a) 条件数CN; (b) 伪速度U'; (c) 伪声速C'; (d) B函数; (e) 本文预处理法得到的特征根
Fig. 1. Characteristic system of the governing equations obtained after different preconditioning: (a) Condition number; (b) pseudo-speed; (c) pseudo-sound speed; (d) function B; (e) the eigenvalues obtained after preconditioning in this paper
表 1 数值解与精确解之间的误差
Table 1. Errors between numerical solutions and exact solutions.
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 -
[1] Yin P, Liandrat J, Shen W 2021 J. Comput. Phys. 434 110207Google Scholar
[2] Fernandez Fidalgo J, Nogueira X, Ramirez L, Colominas I 2018 Comput. Methods Appl. Mech. Eng. 335 91Google Scholar
[3] Ahmed T, Rehman A, Ali A, Qamar S 2021 Results Phys. 23 104078Google Scholar
[4] Zhang Y, Liu Y, Zhang Y, Wang W R, Han Y F 2021 Int. Commun. Heat. Mass Transfer 129 105688Google Scholar
[5] Xie W J, Tian Z Y, Zhang Y, Yu H 2022 Comput. Fluids 233 105215Google Scholar
[6] Ma Chao, Wu J, Yu H, Yang L 2022 Comput. Math. Appl. 105 13Google Scholar
[7] Lehel F, Hona J 2021 Chin. J. Phys. 73 360Google Scholar
[8] 张庆海, 李阳 2021 物理学报 70 130201Google Scholar
Zhang Q H, Li Y 2021 Acta Phys. Sin. 70 130201Google Scholar
[9] Xie W, Luo Z B, Zhou Y, Gao T X, Wu Y, Wang Q 2021 Acta. Astronaut. 188 416Google Scholar
[10] Barhoumi B, Bessrour J 2017 J. Comput. Phys. 21 86Google Scholar
[11] Bykerk T, Verstraete D, Steelant J 2020 Aerosp. Sci. Technol. 103 105883Google Scholar
[12] Huber S E, Trummer M R 2016 Comput. Math. Appl. 71 319Google Scholar
[13] 胡嘉懿, 张文欢, 柴振华, 施保昌, 汪一航 2019 物理学报 68 234701Google Scholar
Hu J Y, Zhang W H, Chai Z H, Shi B C, Wang Y H 2019 Acta Phys. Sin. 68 234701Google Scholar
[14] 辛建建, 陈振雷, 石凡, 石伏龙 2020 物理学报 69 044702Google Scholar
Xin J J, Chen Z L, Shi F, Shi F L 2020 Acta Phys. Sin. 69 044702Google Scholar
[15] Imran M A, Shaheen A, Sherif E M, Rahimi-Gorji M, Seikh A H 2020 Chin. J. Phys. 66 60Google Scholar
[16] 尹纪富, 尤云祥, 李巍, 胡天群 2014 物理学报 63 044701Google Scholar
Yin J F, You Y X, Li W, Hu T Q 2014 Acta Phys. Sin. 63 044701Google Scholar
[17] 牛海波, 易仕和, 刘小林, 霍俊杰, 冈敦殿 2021 物理学报 70 134701Google Scholar
Niu H B, Yi S H, Liu X L, Huo J J, Gang D D 2021 Acta Phys. Sin. 70 134701Google Scholar
[18] 李高华, 王福新 2018 物理学报 67 054701Google Scholar
Li G H, Wang F X 2018 Acta Phys. Sin. 67 054701Google Scholar
[19] Fasel M 1976 J. Fluid. Mech. 78 355Google Scholar
[20] Aziz K 1967 Phys. Fluids 10 314Google Scholar
[21] Harlow F H, Welch J E 1965 Phys. Fluids 8 2182Google Scholar
[22] Hirt C W, Nichols B D, Romero N C 1974 SOLA: A Numerical Solution Algorithm for Transient Fluid Flows (Los Alamos: Los Alamos Scientific Laboratory) Report5852
[23] Patankar S V, Spalding D B 1972 Int. J. Heat. Mass Transfer 15 1787Google Scholar
[24] Patankar S V 1981 Numer. Heat Transfer 4 409Google Scholar
[25] Van Doormaal J P, Raithby G D 1984 Numer. Heat Transfer 7 147Google Scholar
[26] Minkowycz W J, Sparrow E M, Pletcher R H, Schneider G E 1988 Handbook of Numerical Heat Transfer (New York: John Wiley & Sons) pp241–289
[27] Issa I R, Gosman A D, Watkins A P 1986 J. Comput. Phys. 62 66Google Scholar
[28] Davis G D V 1983 Int. J. Numer. Methods Fluids 3 249Google Scholar
[29] Chorin A J 1967 J. Comput. Phys. 2 12Google Scholar
[30] Turkel E 1993 Appl. Numer. Math. 12 257Google Scholar
[31] Choi D, Merkel C L 1987 AIAA J. 23 581
[32] Viviand H 1985 Numerical Methods for the Euler Equations of Fluid Dynamics (Philadelphia: SIAM) p519
[33] Choi Y H, Merkle C L 1993 J. Comput. Phys. 105 205
[34] Buelow P E O, Venkateswaran S, Charles L 1993 AIAA J. 32 2401Google Scholar
[35] Godfrey A G, Walters R W, Leer B V 1993 31st Aerospace Sciences Meeting Reno, NV, USA, January 11–14, 1993 p535
[36] Weiss J M, Smith W A 1994 Fluid Dynamics Conference Colorado Springs, CO, USA, June 20–23, 1994 p2209
[37] Pletcher R H, Chen K H 1993 11th Computational Fluid Dynamics Conference Orlando, FL, USA, July 6–9, 1993 p3368
[38] Dailey L D, Pletcher R H 1996 Comput. Fluids 25 791Google Scholar
[39] 成小磊 2007 硕士学位论文 (南京: 南京航空航天大学)
Cheng X L 2007 M. S. Thesis (Nanjing: Nanjing University of Aeronautics and Astronautics) (in Chinese)
[40] 谭勤学, 任静, 蒋洪德 2015 清华大学学报(自然科学版) 55 134Google Scholar
Tan Q X, Ren J, Jiang H D 2015 J. Tsinghua Univ. (Science and Technology) 55 134Google Scholar
[41] Buelow P E G, Schwer D A, Feng F J, Merkle C L 1997 13th Computational Fluid Dynamics Conference Snowmass Village, CO, USA, June 29–July 2, 1997 p120
[42] Roe P 1981 J. Comput. Phys. 43 357Google Scholar
[43] Jiang G S, Shu C W 1996 J. Comput. Phys. 126 202Google Scholar
[44] Sun Z S, Yu H U, Luo L, Yang Z W 2014 Sci. Chin. Phys. Mech. 57 971Google Scholar
[45] Shu C W, Osher S 1988 J. Comput. Phys. 77 439Google Scholar
[46] 胡丽燕 2011 硕士学位论文 (南京: 南京航空航天大学)
Hu L Y 2011 M. S. Thesis (Nanjing: Nanjing University of Aeronautics and Astronautics) (in Chinese)
[47] Ghia U, Ghia K N, Shin C T 1982 J. Comput. Phys. 48 387Google Scholar
[48] Delery J 1981 14th Fluid and Plasma Dynamics Conference Palo Alto, CA, USA, June 23–25, 1981 p1245
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