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提出了一种径向基函数虚拟网格法浸入边界法以模拟复杂或多体浸入边界黏性绕流问题.在该方法中，采用有限差分法离散固定笛卡尔交错网格上的不可压缩Navier-Stokes方程，以分步法结合三阶Runge-Kutta格式进行时间积分，高阶TVDMUSCL（total variation diminishing monotonic upstream-centered scheme for conservation laws）格式离散对流项；一个边界连续的虚拟网格法施加物面边界条件以考虑尖锐边界对流场的影响；引入具多项式基的径向基函数描述和重构任意复杂浸入界面，并识别背景网格属性状态.采用Fortran 90语言开发相应的求解器，模拟了绕圆柱、机翼和交错布置圆柱群的黏性绕流问题，验证了本文方法的正确性、可靠性和对复杂边界流动问题的适用性.A radial basis function ghost cell immersed boundary method of simulating flows around arbitrary complex or multiple immersed boundaries is proposed in this paper. In this method, incompressible Navier-Stokes equations are discretized on fixed Cartesian staggered gridby the finite difference method. A fractional step method is used for time integration, together with third order Runge-Kutta scheme. A high-order TVD MUSCL (total variation diminishing monotonic upstream-centered scheme for conservation law) scheme is used to discretize convective terms. Two salient features are emphasized in the present study. First, boundary conditions at the immersed interface are enforced by a continuous ghost cell method to consider the influence of immersed boundary on the flow field. The immersed bodies are treated as virtual boundaries immersed in the flow. And Navier-Stokes equations are solved in the entire computation domain, including solid domain. Therefore, programming complexity is greatly reduced and the treatment of immersed boundaries is simplified. Second, a polynomial and radial basis function is introduced to implicitly represent and reconstruct arbitrary complex immersed boundaries. Iso-surface distance functions about interface geometries are fitted with some sampling points of body surfaces. It is flexible and robust. Moreover, the information about interface positions on the background grid can be easily identified by the signed distance functions. Based on our in-house developed immersed boundary method solver, typical test cases are simulated to validate the proposed method. The flows around a cylinder at Reynolds numbers of 40, 100 and 200 are first simulated and a grid resolution study is carried out. Good agreement is achieved by comparing with previous numerical results, which shows that this method is accurate and reliable. In the second case of flow around airfoil, the good agreement with previous study shows that the present method has the ability to simulate complex immersed boundary flow. In the last case of flow around array of thirteen cylinders, the ability of present method for multiple immersed boundaries is well proved. And hydrodynamic interaction among multiple bodies is briefly analyzed.
[1] Li Q, Li W M 2016 Acta Phys. Sin. 65 064601 (in Chinese)[李强, 李五明 2016 65 064601]
[2] Sotiropoulos F, Yang X 2014 Prog. Aerosp. Sci. 65 1
[3] Lee J, You D 2013 J. Comput. Phys. 233 295
[4] Dechristé D, Mieussens L 2015 J. Comput. Phys. 314 465
[5] Yang J, Stern F 2012 J. Comput. Phys. 231 5029
[6] Majumdar S, Iaccarino G, Durbin P 2001 Annual Research Briefs 30 353
[7] Tseng Y H, Ferziger J H 2003 J. Comput. Phys. 192 593
[8] Xia J, Luo K, Fan J 2015 Int. J. Heat Mass Transfer 89 856
[9] Yang Q, Cao S Y, Liu S Y 2014 Acta Phys. Sin. 63 214702 (in Chinese)[杨青, 曹曙阳, 刘十一 2014 63 214702]
[10] Monasse L, Daru V, Mariotti C, Piperno S, Tenaud C 2012 J. Comput. Phys. 231 2977
[11] Schneiders L, Günther C, Meinke M, Schröder W 2016 J. Comput. Phys. 311 62
[12] Ghias R, Mittal R, Dong H 2007 J. Comput. Phys. 225 528
[13] Berthelsen P A, Faltinsen O M 2008 J. Comput. Phys. 227 4354
[14] Xia J, Luo K, Fan J 2014 Int. J. Heat Mass Transfer 75 302
[15] Mittal R, Dong H, Bozkurttas M, Najjar F M, Vargas A, von Loebbecke A 2008 J. Comput. Phys. 227 4825
[16] Hu O, Zhao N, Lin J M, Wang D H 2011 Acta Aerodyn. Sin. 29 491 (in Chinese)[胡偶, 赵宁, 刘剑明, 王东红 2011 空气动力学学报 29 491]
[17] Zhang X, Liu Y, Ma S 2009 Adv. Mech. 39 1 (in Chinese)[张雄, 刘岩, 马上 2009 力学进展 39 1]
[18] Versteeg H K, Malalasekera W 2000 An Introduction to Computational Fluiddynamics:the Finite Volume Method (Second edition) (Beijing:World Book Inc) p24
[19] Lee J, Kim J, Choi H, Yang K S 2011 J. Comput. Phys. 230 2677
[20] Shin B R 2000 European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, September 2-6, 2000 p1
[21] Kershaw D S 1978 J. Comput. Phys. 26 1
[22] Pan D, Shen T T 2009 Int. J. Numer. Method Fluid 60 1378
[23] Rendall T C S, Allen C B 2007 Int. J. Numer. Method Eng. 74 10
[24] Ye J J, Li T Q 2013 Proceedings of National Congress on Hydrodynamics Zhoushan, China, September 20-25, 2013 p343 (in Chinese)[叶俊杰, 李廷秋 2013 全国水动力学学术会议 舟山, 中国, 2013年9月20-25日 第343 页]
[25] Frisani A, Hassan Y A 2015 Comput. Fluids 121 51
[26] Imamura T, Suzuki K, Nakamura T, Yoshida Masahiro 2005 J. Comput. Phys. 202 645
[27] Li Q 2010 M. S. Thesis (Nanjing:Nanjing University of Aeronautics and Astronautics) (in Chinese)[李秋2010 硕士学位论文(南京:南京航空航天大学)]
[28] Gao T, Tseng Y H, Lu X Y 2007 Int. J. Numer. Method Fluid 55 1189
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[1] Li Q, Li W M 2016 Acta Phys. Sin. 65 064601 (in Chinese)[李强, 李五明 2016 65 064601]
[2] Sotiropoulos F, Yang X 2014 Prog. Aerosp. Sci. 65 1
[3] Lee J, You D 2013 J. Comput. Phys. 233 295
[4] Dechristé D, Mieussens L 2015 J. Comput. Phys. 314 465
[5] Yang J, Stern F 2012 J. Comput. Phys. 231 5029
[6] Majumdar S, Iaccarino G, Durbin P 2001 Annual Research Briefs 30 353
[7] Tseng Y H, Ferziger J H 2003 J. Comput. Phys. 192 593
[8] Xia J, Luo K, Fan J 2015 Int. J. Heat Mass Transfer 89 856
[9] Yang Q, Cao S Y, Liu S Y 2014 Acta Phys. Sin. 63 214702 (in Chinese)[杨青, 曹曙阳, 刘十一 2014 63 214702]
[10] Monasse L, Daru V, Mariotti C, Piperno S, Tenaud C 2012 J. Comput. Phys. 231 2977
[11] Schneiders L, Günther C, Meinke M, Schröder W 2016 J. Comput. Phys. 311 62
[12] Ghias R, Mittal R, Dong H 2007 J. Comput. Phys. 225 528
[13] Berthelsen P A, Faltinsen O M 2008 J. Comput. Phys. 227 4354
[14] Xia J, Luo K, Fan J 2014 Int. J. Heat Mass Transfer 75 302
[15] Mittal R, Dong H, Bozkurttas M, Najjar F M, Vargas A, von Loebbecke A 2008 J. Comput. Phys. 227 4825
[16] Hu O, Zhao N, Lin J M, Wang D H 2011 Acta Aerodyn. Sin. 29 491 (in Chinese)[胡偶, 赵宁, 刘剑明, 王东红 2011 空气动力学学报 29 491]
[17] Zhang X, Liu Y, Ma S 2009 Adv. Mech. 39 1 (in Chinese)[张雄, 刘岩, 马上 2009 力学进展 39 1]
[18] Versteeg H K, Malalasekera W 2000 An Introduction to Computational Fluiddynamics:the Finite Volume Method (Second edition) (Beijing:World Book Inc) p24
[19] Lee J, Kim J, Choi H, Yang K S 2011 J. Comput. Phys. 230 2677
[20] Shin B R 2000 European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, September 2-6, 2000 p1
[21] Kershaw D S 1978 J. Comput. Phys. 26 1
[22] Pan D, Shen T T 2009 Int. J. Numer. Method Fluid 60 1378
[23] Rendall T C S, Allen C B 2007 Int. J. Numer. Method Eng. 74 10
[24] Ye J J, Li T Q 2013 Proceedings of National Congress on Hydrodynamics Zhoushan, China, September 20-25, 2013 p343 (in Chinese)[叶俊杰, 李廷秋 2013 全国水动力学学术会议 舟山, 中国, 2013年9月20-25日 第343 页]
[25] Frisani A, Hassan Y A 2015 Comput. Fluids 121 51
[26] Imamura T, Suzuki K, Nakamura T, Yoshida Masahiro 2005 J. Comput. Phys. 202 645
[27] Li Q 2010 M. S. Thesis (Nanjing:Nanjing University of Aeronautics and Astronautics) (in Chinese)[李秋2010 硕士学位论文(南京:南京航空航天大学)]
[28] Gao T, Tseng Y H, Lu X Y 2007 Int. J. Numer. Method Fluid 55 1189
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