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Application of variable-time-period harmonic balance method to periodic unsteady vortex shedding

## Application of variable-time-period harmonic balance method to periodic unsteady vortex shedding

Chai Zhen-Xia, Liu Wei, Yang Xiao-Liang, Zhou Yun-Long
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• 图 1  NACA0012翼型计算网格

Figure 1.  Mesh for the NACA0012 airfoil.

图 2  NACA0012翼型的(a)升力系数和(b)俯仰力矩系数迟滞曲线

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

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°

Figure 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.

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

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

图 5  CPU时间加速比随谐波数的变化

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

图 6  二维圆柱计算网格

Figure 6.  Computational grid for cylinder in cross flow.

图 7  升、阻力系数收敛曲线

Figure 7.  Time history of lift coefficient CL and drag CD.

图 8  不同谐波数下的周期T收敛曲线

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

图 9  升力系数收敛曲线(NH = 3)

Figure 9.  Time history of lift coefficient CL with NH = 3.

图 10  升力系数随时间的变化

Figure 10.  Variation of CL over one period.

图 11  阻力系数随时间的变化

Figure 11.  Variation of CD over one period.

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

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

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

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

图 14  Strouhal数随Re的变化

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

图 15  平均阻力系数随Re的变化

Figure 15.  Mean coefficient of drag versus Reynolds number.

图 16  不同步长λ下的周期T收敛曲线

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

图 17  不同步长T0下计算的周期T收敛曲线

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

图 18  T = 11.43 时重建的升力系数曲线

Figure 18.  Variation of CL over one period with converged time period T = 11.43.

图 19  HBM计算的St与TDM计算结果的对比

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

图 20  不同雷诺下的加速比

Figure 20.  CPU time speedup of various Reynolds number.

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

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

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

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

图 23  T = 5.389时各个时刻升力系数收敛曲线(NH = 3)

Figure 23.  Time history of lift coefficient CL at various time instances over one period with T = 5.389 (NH = 3).

图 24  相位差随周期T的变化(Re = 180, NH = 3)

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

图 25  残差随周期T的变化(Re = 180, NH = 3)

Figure 25.  HBM solution residual versus time period for Re = 180 (NH = 3).

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

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

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)

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

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

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

图 29  二维方柱绕流计算网格

Figure 29.  Computational grid for rectangular in cross flow.

图 30  升力系数随时间的变化

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

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

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

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##### Publishing process
• Received Date:  22 January 2019
• Accepted Date:  28 March 2019
• Available Online:  16 August 2019
• Published Online:  01 June 2019

## Application of variable-time-period harmonic balance method to periodic unsteady vortex shedding

###### Corresponding author: Liu Wei, fishfather6525@sina.com;
• College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China