Zhen-Xia Chai, Wei Liu, Xiao-Liang Yang, Yun-Long Zhou. Application of variable-time-period harmonic balance method to periodic unsteady vortex shedding [J]. Acta Physica Sinica, 2019, 68(12): 124701-1
- Acta Physica Sinica
- Vol. 68, Issue 12, 124701-1 (2019)
Fig. 1. Mesh for the NACA0012 airfoil.NACA0012翼型计算网格
Fig. 2. (a) Lift and (b) pitching moment coefficients dynamic dependence of NACA0012 airfoil.NACA0012翼型的(a)升力系数和(b)俯仰力矩系数迟滞曲线
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.
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. 4. Pitching moment coefficient convergence history for the HBM with respect to the number of harmonics: (a) N H = 1; (b) N H = 3.
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. 5. CPU time speedup of the HBM with respect to the TDM.CPU时间加速比随谐波数的变化
Fig. 6. Computational grid for cylinder in cross flow.二维圆柱计算网格
Fig. 7. Time history of lift coefficient C L and drag C D.
升、阻力系数收敛曲线
Fig. 8. Convergence from initial guess to exact time period with varying number of harmonics.不同谐波数下的周期T 收敛曲线
Fig. 9. Time history of lift coefficient C L with N H = 3.
升力系数收敛曲线(N H = 3)
Fig. 10. Variation of C L over one period.
升力系数随时间的变化
Fig. 11. Variation of C D over one period.
阻力系数随时间的变化
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 .
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. 13. Comparison of instantaneous entropy contours: (a) TDM results; (b) HBM results (N H = 3).
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. 14. Strouhal number as a function of Reynolds number.Strouhal数随Re 的变化
Fig. 15. Mean coefficient of drag versus Reynolds number.平均阻力系数随Re 的变化
Fig. 16. Time period convergence computed with three different step sizes λ .
不同步长λ下的周期T 收敛曲线
Fig. 17. Time period convergence with various starting guesses T 0.
不同步长T 0下计算的周期T 收敛曲线
Fig. 18. Variation of C L over one period with converged time period T = 11.43.
T = 11.43 时重建的升力系数曲线
Fig. 19. Comparison of the HBM St data results with TDM results.
HBM计算的St 与TDM计算结果的对比
Fig. 20. CPU time speedup of various Reynolds number.不同雷诺下的加速比
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.
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. 22. Variation of C L over one period at different iterations: (a) Overall; (b) local.
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. 23. Time history of lift coefficient C L at various time instances over one period with T = 5.389 (N H = 3).
T = 5.389时各个时刻升力系数收敛曲线(N H = 3)
Fig. 24. Change in phase of unsteady lift versus time period for Re = 180 (N H = 3).
相位差随周期T 的变化(Re = 180, N H = 3)
Fig. 25. HBM solution residual versus time period for Re = 180 (N H = 3).
残差随周期T 的变化(Re = 180, N H = 3)
Fig. 26. Convergence of shedding time period computed by Newton method and SDM: (a) T 0 = 4; (b) T 0 = 5.41.
采用牛顿法和SDM计算的周期T 收敛曲线对比图 (a)初始T 0 = 4; (b)初始T 0 = 5.41
Fig. 27. Convergence of shedding time period computed by FR conjugate gradient method (a) and compared with the SDM results (b).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. 28. Convergence of shedding time period computed by three different methods of optimization.采用三种不同优化方法计算得到的周期T 收敛曲线图
Fig. 29. Computational grid for rectangular in cross flow.二维方柱绕流计算网格
Fig. 30. Comparison of lift coefficients of HBM and TDM at Re = 100.
升力系数随时间的变化
Fig. 31. Comparison of the instantaneous entropy contours: (a) TDM results; (b) HBM results (N H = 3).
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)
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Table 1. Computational conditions of the AGARD CT5 test case for the NACA0012 airfoil.
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Table 2.
Time-averaged coefficient and Strouhal number compared with experiment data.
时域计算结果与实验结果对比
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Table 3.
Strouhal number and time-averaged coefficient computed by different number of harmonics.
不同谐波数下的计算结果
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Table 4. Time-averaged coefficient and Strouhal number computed by time-domain solver using different physical time steps.
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Table 5. Convergency of frequency and time-averaged coefficient with speedup estimates.
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