可变周期谐波平衡法求解周期性非定常涡脱落问题

柴振霞; 刘伟; 杨小亮; 周云龙

doi:10.7498/aps.68.20190126

摘要

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.

关键词:

Abstract

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.

Keywords:

作者及机构信息

国防科技大学空天科学学院, 长沙　410073

通信作者: 刘伟, fishfather6525@sina.com

基金项目: 国家自然科学基金(批准号: 11502292, 11572348)资助的课题.

Authors and contacts

College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China

Corresponding author: Liu Wei, fishfather6525@sina.com

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11502292, 11572348).

文章全文 : translate this paragraph

参考文献

[1]	McMullen M, Jameson A, Alonso J 2006 AIAA J. 44 1428 Google Scholar
[2]	Mosahebi A, Nadarajah S 2013 Comput. Fluids 75 140 Google Scholar
[3]	Hall K C, Grawley E F 1989 AIAA J. 27 777 Google Scholar
[4]	Zhang Z, Yang S, Chen P C 2012 J. Aircraft 49 922 Google Scholar
[5]	Ning W, He L 1998 J. Turbomach. 120 508 Google Scholar
[6]	Hall K C, Thomas J P, Clark W S 2002 AIAA J. 40 879 Google Scholar
[7]	Ekici K, Hall K C 2007 AIAA J. 45 1047 Google 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. 372 220 Google 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 1374 Google 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. 48 788 Google Scholar
[21]	Ekici K, Hall K C, Dowell E H 2008 J. Comput. Phys. 227 6206 Google Scholar
[22]	Da Ronch A, Vallespin D, Ghoreyshi M, Badcock K J 2012 AIAA J. 50 470 Google Scholar
[23]	Da Ronch A, McCracken A J, Badcock K J, Widhalm M, Campobasso M S 2013 J. Aircraft 50 694 Google 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 46 183 Google Scholar Chen Q, Chen J Q, Yuan X X, Xie Y F 2014 Chinese Journal of Theoretical and Applied Mechanics 46 183 Google Scholar
[27]	柴振霞, 刘伟, 刘绪, 杨小亮 2018 国防科技大学学报 40 30 Google Scholar Chai Z X, Liu W, Liu X, Yang X L 2018 J. Nat. Univ. Defense Technol. 40 30 Google 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. 65 313 Google Scholar
[35]	Yao W, Marques S 2015 AIAA J. 53 2040 Google Scholar
[36]	张炜, 席光 2009 西安交通大学学报 43 114 Google Scholar Zhang W, Xi G 2009 Journal of Xi'an Jiaotong University 43 114 Google 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 1381 Google Scholar
[40]	Henderson R D 1995 Phys. Fluids 7 2102 Google 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 2742 Google Scholar
[44]	Williamson C H K 1998 J. Fluids Struct. 12 1073 Google 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

施引文献

图 1 NACA0012翼型计算网格

Fig. 1. Mesh for the NACA0012 airfoil.

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图 2 NACA0012翼型的(a)升力系数和(b)俯仰力矩系数迟滞曲线

Fig. 2. (a) Lift and (b) pitching moment coefficients dynamic dependence of NACA0012 airfoil.

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图 3 NACA0012翼型俯仰振荡过程中的瞬时压力系数分布　(a) 攻角减小过程中α = –2.41°; (b) 攻角增大过程中α = –2.00°

Fig. 3. Instantaneous pressure coefficient distribution compared to experimental data of NACA0012 airfoil: (a) α = –2.41° for decreasing angle; (b) α = –2.00° for increasing angle.

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图 4 HBM取不同谐波数时俯仰力矩系数收敛曲线　(a) N_H = 1; (b) N_H = 3

Fig. 4. Pitching moment coefficient convergence history for the HBM with respect to the number of harmonics: (a) N_H = 1; (b) N_H = 3.

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图 5 CPU时间加速比随谐波数的变化

Fig. 5. CPU time speedup of the HBM with respect to the TDM.

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图 6 二维圆柱计算网格

Fig. 6. Computational grid for cylinder in cross flow.

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图 7 升、阻力系数收敛曲线

Fig. 7. Time history of lift coefficient C_L and drag C_D.

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图 8 不同谐波数下的周期T收敛曲线

Fig. 8. Convergence from initial guess to exact time period with varying number of harmonics.

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图 9 升力系数收敛曲线(N_H = 3)

Fig. 9. Time history of lift coefficient C_L with N_H = 3.

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图 10 升力系数随时间的变化

Fig. 10. Variation of C_L over one period.

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图 11 阻力系数随时间的变化

Fig. 11. Variation of C_D over one period.

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图 12 Re = 180, N_H = 3条件下不同时刻的流线图　(a) t = T/3; (b) t = 2T/3; (c) t = T

Fig. 12. Streamlines at various time instances over one period (Re = 180, N_H = 3): (a) t = T/3; (b) t = 2T/3; (c) t = T.

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图 13 熵等值线图(C_L最小时刻)　(a) TDM计算结果; (b) HBM计算结果（N_H = 3）

Fig. 13. Comparison of instantaneous entropy contours: (a) TDM results; (b) HBM results (N_H = 3).

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图 14 Strouhal数随Re的变化

Fig. 14. Strouhal number as a function of Reynolds number.

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图 15 平均阻力系数随Re的变化

Fig. 15. Mean coefficient of drag versus Reynolds number.

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图 16 不同步长λ下的周期T收敛曲线

Fig. 16. Time period convergence computed with three different step sizes λ.

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图 17 不同步长T₀下计算的周期T收敛曲线

Fig. 17. Time period convergence with various starting guesses T₀.

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图 18 T = 11.43 时重建的升力系数曲线

Fig. 18. Variation of C_L over one period with converged time period T = 11.43.

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图 19 HBM计算的St与TDM计算结果的对比

Fig. 19. Comparison of the HBM St data results with TDM results.

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图 20 不同雷诺下的加速比

Fig. 20. CPU time speedup of various Reynolds number.

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图 21 升力系数和t = T时刻的残差收敛曲线(Re = 180, T = 4, N_H = 3)　(a)升力系数; (b)残差

Fig. 21. Time history of lift coefficient C_L at various time instances over one period and residual at t = T (Re = 180, T = 4, N_H = 3): (a) Lift coefficient; (b) residual.

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图 22 不同迭代步重建的升力系数随时间的变化(Re = 180, T = 4, N_H = 3)　(a)整体; (b)局部

Fig. 22. Variation of C_L over one period at different iterations: (a) Overall; (b) local.

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图 23 T = 5.389时各个时刻升力系数收敛曲线(N_H = 3)

Fig. 23. Time history of lift coefficient C_L at various time instances over one period with T = 5.389 (N_H = 3).

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图 24 相位差随周期T的变化(Re = 180, N_H = 3)

Fig. 24. Change in phase of unsteady lift versus time period for Re = 180 (N_H = 3).

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图 25 残差随周期T的变化(Re = 180, N_H = 3)

Fig. 25. HBM solution residual versus time period for Re = 180 (N_H = 3).

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图 26 采用牛顿法和SDM计算的周期T收敛曲线对比图　(a)初始T₀ = 4; (b)初始T₀ = 5.41

Fig. 26. Convergence of shedding time period computed by Newton method and SDM: (a) T₀ = 4; (b) T₀ = 5.41.

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图 27 采用FR法计算的周期T收敛曲线(a)及其与SDM计算结果的比较(b)

Fig. 27. Convergence of shedding time period computed by FR conjugate gradient method (a) and compared with the SDM results (b).

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图 28 采用三种不同优化方法计算得到的周期T收敛曲线图

Fig. 28. Convergence of shedding time period computed by three different methods of optimization.

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图 29 二维方柱绕流计算网格

Fig. 29. Computational grid for rectangular in cross flow.

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图 30 升力系数随时间的变化

Fig. 30. Comparison of lift coefficients of HBM and TDM at Re = 100.

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图 31 熵等值线图(C_L最小时刻)　(a) TDM计算结果; (b) HBM计算结果(N_H = 3)

Fig. 31. Comparison of the instantaneous entropy contours: (a) TDM results; (b) HBM results (N_H = 3).

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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°
α_m	2.51°
k	0.1628

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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.

Experiment	C_D0	St
Henderson^[40]	1.336
Wieselsberge^[41]	1.3
Roshko^[42]		0.185
Williamson^[43]		0.1919
Present	1.3457	0.185

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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.

N_H	St	C_D0
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

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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	C_{d, avg}
0.1	0.134	1.443
0.01	0.1415	1.487
Sohankar^[46]	0.142	1.466

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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.

N_H	St	C_{d, 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

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map

[1]	McMullen M, Jameson A, Alonso J 2006 AIAA J. 44 1428 Google Scholar
[2]	Mosahebi A, Nadarajah S 2013 Comput. Fluids 75 140 Google Scholar
[3]	Hall K C, Grawley E F 1989 AIAA J. 27 777 Google Scholar
[4]	Zhang Z, Yang S, Chen P C 2012 J. Aircraft 49 922 Google Scholar
[5]	Ning W, He L 1998 J. Turbomach. 120 508 Google Scholar
[6]	Hall K C, Thomas J P, Clark W S 2002 AIAA J. 40 879 Google Scholar
[7]	Ekici K, Hall K C 2007 AIAA J. 45 1047 Google 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. 372 220 Google 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 1374 Google 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. 48 788 Google Scholar
[21]	Ekici K, Hall K C, Dowell E H 2008 J. Comput. Phys. 227 6206 Google Scholar
[22]	Da Ronch A, Vallespin D, Ghoreyshi M, Badcock K J 2012 AIAA J. 50 470 Google Scholar
[23]	Da Ronch A, McCracken A J, Badcock K J, Widhalm M, Campobasso M S 2013 J. Aircraft 50 694 Google 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 46 183 Google Scholar Chen Q, Chen J Q, Yuan X X, Xie Y F 2014 Chinese Journal of Theoretical and Applied Mechanics 46 183 Google Scholar
[27]	柴振霞, 刘伟, 刘绪, 杨小亮 2018 国防科技大学学报 40 30 Google Scholar Chai Z X, Liu W, Liu X, Yang X L 2018 J. Nat. Univ. Defense Technol. 40 30 Google 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. 65 313 Google Scholar
[35]	Yao W, Marques S 2015 AIAA J. 53 2040 Google Scholar
[36]	张炜, 席光 2009 西安交通大学学报 43 114 Google Scholar Zhang W, Xi G 2009 Journal of Xi'an Jiaotong University 43 114 Google 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 1381 Google Scholar
[40]	Henderson R D 1995 Phys. Fluids 7 2102 Google 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 2742 Google Scholar
[44]	Williamson C H K 1998 J. Fluids Struct. 12 1073 Google 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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