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谐波平衡法是一种高效周期性非定常流动频域计算方法. 本文研究可变周期谐波平衡法, 通过求解Navier-Stokes方程模拟低速不可压条件下的二维钝体绕流周期性非定常涡脱落问题. 对于这类流动变化周期未知的非定常问题, 将涡脱落周期T作为变量, 采用基于残差导数的可变周期计算方法推进求解. 以圆柱绕流和方柱绕流为例, 研究考察了谐波平衡法的计算精度和效率, 并分析研究了不同参数对计算结果的影响. 针对圆柱绕流问题, 采用三种不同优化方法进行周期T的迭代计算, 对比研究了它们的计算精度和效率. 计算结果表明: 谐波平衡法采用3个谐波数就可以准确模拟周期性非定常涡脱落问题, 辨识的涡脱落频率和阻力系数与实验值及其他数值结果一致, 与时域方法相比该方法具有较高的计算效率. 不同优化方法的计算结果相同, 共轭梯度法和牛顿法的收敛速度与最速下降法采用较大搜索步长时的收敛速度一致. 由于牛顿法没有参数问题, 因此该方法在工程计算中更有优势.The harmonic balance method (HBM) is an efficient frequency-domain approach to computing periodically unsteady flows. The basic principle of this method is to decompose the flow variables into a Fourier series, and transform the unsteady flow into several steady problems coupled by a spectral time-derivative operator, from which the whole time history of a complete unsteady periodic flow can be reconstructed. In the present work, we investigate the ability of the HBM to be used for modeling the periodic unsteady vortex shedding behind a bluff body at low Reynolds numbers via solving the unsteady incompressible Navier-Stokes equations. For the periodic problem where the time period T of the unsteadiness is unknown, a variable-time-period method based on residual gradients is used to compute the exact time period iteratively starting from an initial guess T0. By simulating the two-dimensional laminar flows over a circular cylinder and a square cylinder, the accuracy and efficiency of the HBM are investigated and the effects of different parameters on the final results are analyzed. Comparisons with the results of fixed-time-period HBM using a constant time period are also implemented. Three practical methods of optimization are used to iterate the time period, and the values of accuracy and efficiency of different methods are compared with each other. The results show that the HBM can accurately capture the complex nonlinear flow field physics with only three harmonics. The Strouhal frequency and mean drag coefficient each as a function of the Reynolds number agree well with existing experimental and computational data. For both test cases, the computational efficiency of HBM is higher than that from the traditional time-domain method. For the square cylinder test case, the HBM offers speedup rate up to nearly 18 times. The real time period of vortex shedding can be predicted by the gradient based variable-time-period method, and the final result is insensitive to search step λ. The calculation result is sensitive to the initial T0, and when such a variable is greater than a certain value, the result will converge to an approximate integer multiple of the real one. Therefore, it deserves further exploration on how to specify this initial condition. The shedding time periods computed by different optimization methods are converged to the same value. The computational efficiency from the FR conjugate gradient method and that from Newton method are both equivalent to that from the steepest descent method with the maximum search step λ = 100. Avoiding prescribing parameters such as the search step λ, the Newton method possesses higher application value in engineering calculation than the other two schemes.
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Keywords:
- periodic unsteady flows /
- numerical simulation /
- harmonic balance method /
- variable-time-period method
[1] McMullen M, Jameson A, Alonso J 2006 AIAA J. 44 1428Google Scholar
[2] Mosahebi A, Nadarajah S 2013 Comput. Fluids 75 140Google Scholar
[3] Hall K C, Grawley E F 1989 AIAA J. 27 777Google Scholar
[4] Zhang Z, Yang S, Chen P C 2012 J. Aircraft 49 922Google Scholar
[5] Ning W, He L 1998 J. Turbomach. 120 508Google Scholar
[6] Hall K C, Thomas J P, Clark W S 2002 AIAA J. 40 879Google Scholar
[7] Ekici K, Hall K C 2007 AIAA J. 45 1047Google Scholar
[8] McMullen M, Jameson A, Alonso J 2001 39th Aerospace Sciences Meeting and Exhibit Reno, NV, January 8−11, 2001 AIAA 2001-0152
[9] Gopinath A, Jameson A 2005 43rd AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, January 10−13, 2005 AIAA 2005-1220
[10] Rubino A, Pini M, Colonna P, Albring T, Nimmagadda S, Economon T, Alonso J 2018 J. Comput. Phys. 372 220Google Scholar
[11] Lindblad D, Montero Villar G, Andersson N, Capitao Patrao A, Courty-Audren S K, Napias G 2018 AIAA Aerospace Sciences Meeting Kissimmee, Florida, January 8−12, 2018 AIAA 2018-1004
[12] Reddy T S R, Bakhle M 2009 45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit Denver, Colorado, August 2−5, 2009 AIAA 2009-5420
[13] Cvijetic G, Jasak H 2018 AIAA Aerospace Sciences Meeting Kissimmee, Florida, January 8−12, 2018 AIAA 2018-0833
[14] Hall K C, Thomas J P, Ekici K, Voytovych D M 2003 33rd AIAA Fluid Dynamics Conference and Exhibit Orlando, Florida, June 23−26, 2003 AIAA 2003-3998
[15] Hall K C, Ekici K, Thomas J P, Dowell E H 2013 Int. J. Comput. Fluid Dyn. 27 54
[16] Lindblad D, Andersson N 2017 55th AIAA Aerospace Sciences Meeting Grapevine, Texas, January 9−13, 2017 AIAA 2017-1171
[17] 杜鹏程, 宁方飞 2017 航空动力学报 32 528
Du P C, Ning F F 2017 Journal of Aerospace Power 32 528
[18] Thomas J P, Custer C H, Dowell E H, Hall K C, Corre C 2013 AIAA J. 51 1374Google Scholar
[19] Thomas J P, Custer C H, Dowell E H, Hall K C 19th AIAA Computational Fluid Dynamics San Antonio, Texas, June 22−25, 2009 AIAA 2009-4270
[20] Guillaume D, Frédéric S, Guillaume P 2010 AIAA J. 48 788Google Scholar
[21] Ekici K, Hall K C, Dowell E H 2008 J. Comput. Phys. 227 6206Google Scholar
[22] Da Ronch A, Vallespin D, Ghoreyshi M, Badcock K J 2012 AIAA J. 50 470Google Scholar
[23] Da Ronch A, McCracken A J, Badcock K J, Widhalm M, Campobasso M S 2013 J. Aircraft 50 694Google Scholar
[24] Murman S M 2005 43rd AIAA Aerospace Sciences Meeting Reno, NV, January 10−13, 2005 AIAA 2005-0840
[25] Hassan D, Sicot F 2011 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition Orlando, Florida, January 4−7, 2011 AIAA 2011-1242
[26] 陈琦, 陈坚强, 袁先旭, 谢昱飞 2014 力学学报 46 183Google Scholar
Chen Q, Chen J Q, Yuan X X, Xie Y F 2014 Chinese Journal of Theoretical and Applied Mechanics 46 183Google Scholar
[27] 柴振霞, 刘伟, 刘绪, 杨小亮 2018 国防科技大学学报 40 30Google Scholar
Chai Z X, Liu W, Liu X, Yang X L 2018 J. Nat. Univ. Defense Technol. 40 30Google Scholar
[28] Clark E B, Ekici K, Beran P S 2014 44th AIAA Fluid Dynamics Conference Atlanta, GA, June 16−20, 2014 AIAA 2014-3323
[29] Cvijetic G, Jasak H, Vukcevic V 2016 54th AIAA Aerospace Sciences Meetin San Diego, California, USA, January 4−8, 2016 AIAA 2006-0070
[30] McMullen M, Jameson A, Alonso J J 2002 40th AIAA Aerospace Sciences Meeting & Exhibit Reno, NV, January 14−17, 2002 AIAA 2002-0120
[31] Gopinath A K, Jameson A 2006 44th AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, January 9−12, 2006 AIAA 2006-449
[32] Spiker M A, Thomas J P, Hall K C, Kielb R E, Dowell E H 2006 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference Newport, Rhode Island, May 1−4, 2006 AIAA 2006-1965
[33] Mosahebi A, Nadarajah S K 2010 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition Orlando, Florida, January 4−7, 2010 AIAA 2010-1267
[34] Yao W, Jaiman R K 2016 J. Fluids Struct. 65 313Google Scholar
[35] Yao W, Marques S 2015 AIAA J. 53 2040Google Scholar
[36] 张炜, 席光 2009 西安交通大学学报 43 114Google Scholar
Zhang W, Xi G 2009 Journal of Xi'an Jiaotong University 43 114Google Scholar
[37] Jameson A 1991 10th Computational Fluid Dynamics Conference Honolulu, HI, June 24−26, 1991 AlAA 1991-1596
[38] Landon R H 1982 NACA 0012 Oscillatory and Transient Pitching Tech. Rep. AGARD-R-702
[39] Batina J T 1990 AIAA J. 28 1381Google Scholar
[40] Henderson R D 1995 Phys. Fluids 7 2102Google Scholar
[41] Wieselsberger C 1922 Physik. Z. 22 321
[42] Roshko A 1954 On the Development of Turbulent Wakes From Vortex Streets (California Institute of Technology, NACA) Tech. Rep. 1191
[43] Williamson C H K 1988 Phys. Fluids 31 2742Google Scholar
[44] Williamson C H K 1998 J. Fluids Struct. 12 1073Google Scholar
[45] 张宝林 2005 最优化理论与算法 (北京: 清华大学出版社)
Zhang B L 2005 Theory and Algorithms of Optimization (Beijing: Tsinghua University Press) (in Chinese)
[46] Sohankar A, Davidson L, Norberg C 1995 Twelfth Australaian Fluid Mechanics Conference Sydney, Australia, December, 199 p517
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表 1 NACA0012翼型俯仰振荡AGARD CT5算例计算条件
Table 1. Computational conditions of the AGARD CT5 test case for the NACA0012 airfoil.
Parameter Value Ma 0.755 α0 0.016° αm 2.51° k 0.1628 表 2 时域计算结果与实验结果对比
Table 2. Time-averaged coefficient and Strouhal number compared with experiment data.
表 3 不同谐波数下的计算结果
Table 3. Strouhal number and time-averaged coefficient computed by different number of harmonics.
NH St CD0 1 0.1745 1.2817 2 0.188 1.3440 3 0.1856 1.3479 4 0.1857 1.3506 TDM 0.185 1.3457 Roshko[42] 0.185 表 4 时域计算结果
Table 4. Time-averaged coefficient and Strouhal number computed by time-domain solver using different physical time steps.
∆t St Cd, avg 0.1 0.134 1.443 0.01 0.1415 1.487 Sohankar[46] 0.142 1.466 表 5 Re = 100时不同谐波数下的计算结果对比
Table 5. Convergency of frequency and time-averaged coefficient with speedup estimates.
NH St Cd, avg Speedup 2 0.1419 1.4846 23.27 3 0.1414 1.4863 17.88 4 0.1414 1.4865 1.944 TDM 0.1415 1.487 1 -
[1] McMullen M, Jameson A, Alonso J 2006 AIAA J. 44 1428Google Scholar
[2] Mosahebi A, Nadarajah S 2013 Comput. Fluids 75 140Google Scholar
[3] Hall K C, Grawley E F 1989 AIAA J. 27 777Google Scholar
[4] Zhang Z, Yang S, Chen P C 2012 J. Aircraft 49 922Google Scholar
[5] Ning W, He L 1998 J. Turbomach. 120 508Google Scholar
[6] Hall K C, Thomas J P, Clark W S 2002 AIAA J. 40 879Google Scholar
[7] Ekici K, Hall K C 2007 AIAA J. 45 1047Google Scholar
[8] McMullen M, Jameson A, Alonso J 2001 39th Aerospace Sciences Meeting and Exhibit Reno, NV, January 8−11, 2001 AIAA 2001-0152
[9] Gopinath A, Jameson A 2005 43rd AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, January 10−13, 2005 AIAA 2005-1220
[10] Rubino A, Pini M, Colonna P, Albring T, Nimmagadda S, Economon T, Alonso J 2018 J. Comput. Phys. 372 220Google Scholar
[11] Lindblad D, Montero Villar G, Andersson N, Capitao Patrao A, Courty-Audren S K, Napias G 2018 AIAA Aerospace Sciences Meeting Kissimmee, Florida, January 8−12, 2018 AIAA 2018-1004
[12] Reddy T S R, Bakhle M 2009 45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit Denver, Colorado, August 2−5, 2009 AIAA 2009-5420
[13] Cvijetic G, Jasak H 2018 AIAA Aerospace Sciences Meeting Kissimmee, Florida, January 8−12, 2018 AIAA 2018-0833
[14] Hall K C, Thomas J P, Ekici K, Voytovych D M 2003 33rd AIAA Fluid Dynamics Conference and Exhibit Orlando, Florida, June 23−26, 2003 AIAA 2003-3998
[15] Hall K C, Ekici K, Thomas J P, Dowell E H 2013 Int. J. Comput. Fluid Dyn. 27 54
[16] Lindblad D, Andersson N 2017 55th AIAA Aerospace Sciences Meeting Grapevine, Texas, January 9−13, 2017 AIAA 2017-1171
[17] 杜鹏程, 宁方飞 2017 航空动力学报 32 528
Du P C, Ning F F 2017 Journal of Aerospace Power 32 528
[18] Thomas J P, Custer C H, Dowell E H, Hall K C, Corre C 2013 AIAA J. 51 1374Google Scholar
[19] Thomas J P, Custer C H, Dowell E H, Hall K C 19th AIAA Computational Fluid Dynamics San Antonio, Texas, June 22−25, 2009 AIAA 2009-4270
[20] Guillaume D, Frédéric S, Guillaume P 2010 AIAA J. 48 788Google Scholar
[21] Ekici K, Hall K C, Dowell E H 2008 J. Comput. Phys. 227 6206Google Scholar
[22] Da Ronch A, Vallespin D, Ghoreyshi M, Badcock K J 2012 AIAA J. 50 470Google Scholar
[23] Da Ronch A, McCracken A J, Badcock K J, Widhalm M, Campobasso M S 2013 J. Aircraft 50 694Google Scholar
[24] Murman S M 2005 43rd AIAA Aerospace Sciences Meeting Reno, NV, January 10−13, 2005 AIAA 2005-0840
[25] Hassan D, Sicot F 2011 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition Orlando, Florida, January 4−7, 2011 AIAA 2011-1242
[26] 陈琦, 陈坚强, 袁先旭, 谢昱飞 2014 力学学报 46 183Google Scholar
Chen Q, Chen J Q, Yuan X X, Xie Y F 2014 Chinese Journal of Theoretical and Applied Mechanics 46 183Google Scholar
[27] 柴振霞, 刘伟, 刘绪, 杨小亮 2018 国防科技大学学报 40 30Google Scholar
Chai Z X, Liu W, Liu X, Yang X L 2018 J. Nat. Univ. Defense Technol. 40 30Google Scholar
[28] Clark E B, Ekici K, Beran P S 2014 44th AIAA Fluid Dynamics Conference Atlanta, GA, June 16−20, 2014 AIAA 2014-3323
[29] Cvijetic G, Jasak H, Vukcevic V 2016 54th AIAA Aerospace Sciences Meetin San Diego, California, USA, January 4−8, 2016 AIAA 2006-0070
[30] McMullen M, Jameson A, Alonso J J 2002 40th AIAA Aerospace Sciences Meeting & Exhibit Reno, NV, January 14−17, 2002 AIAA 2002-0120
[31] Gopinath A K, Jameson A 2006 44th AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, January 9−12, 2006 AIAA 2006-449
[32] Spiker M A, Thomas J P, Hall K C, Kielb R E, Dowell E H 2006 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference Newport, Rhode Island, May 1−4, 2006 AIAA 2006-1965
[33] Mosahebi A, Nadarajah S K 2010 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition Orlando, Florida, January 4−7, 2010 AIAA 2010-1267
[34] Yao W, Jaiman R K 2016 J. Fluids Struct. 65 313Google Scholar
[35] Yao W, Marques S 2015 AIAA J. 53 2040Google Scholar
[36] 张炜, 席光 2009 西安交通大学学报 43 114Google Scholar
Zhang W, Xi G 2009 Journal of Xi'an Jiaotong University 43 114Google Scholar
[37] Jameson A 1991 10th Computational Fluid Dynamics Conference Honolulu, HI, June 24−26, 1991 AlAA 1991-1596
[38] Landon R H 1982 NACA 0012 Oscillatory and Transient Pitching Tech. Rep. AGARD-R-702
[39] Batina J T 1990 AIAA J. 28 1381Google Scholar
[40] Henderson R D 1995 Phys. Fluids 7 2102Google Scholar
[41] Wieselsberger C 1922 Physik. Z. 22 321
[42] Roshko A 1954 On the Development of Turbulent Wakes From Vortex Streets (California Institute of Technology, NACA) Tech. Rep. 1191
[43] Williamson C H K 1988 Phys. Fluids 31 2742Google Scholar
[44] Williamson C H K 1998 J. Fluids Struct. 12 1073Google Scholar
[45] 张宝林 2005 最优化理论与算法 (北京: 清华大学出版社)
Zhang B L 2005 Theory and Algorithms of Optimization (Beijing: Tsinghua University Press) (in Chinese)
[46] Sohankar A, Davidson L, Norberg C 1995 Twelfth Australaian Fluid Mechanics Conference Sydney, Australia, December, 199 p517
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