激励Stuart-Landau方程的研究--周期解、稳定性及流动控制

解析得出了有外部激励的Stuart-Landau（S-L）方程的频率锁定周期解,对这些解与外部激励振幅和频率的依赖关系做了详细研究,并用周期系统稳定性理论确定了解的稳定性边界.还对S-L方程所描述的流动控制效果进行了研究,发现由于外部激励的作用,稳定的锁频解可能比原来的饱和解能量减少了,外部的控制最多能使扰动能量减少为原来的一半.

STUDIES ON THE STUART-LANDAU EQUATION WITH FORCING: PERIODIC SOLUTIONS, STABILITIES AND FLOW CONTROL

Oscillation of a circular cylinder is an important way for the active control of its wakes. Understanding the dynamical characteristics of the flows around an oscillating circular cylinder will be helpful for us to explore mechanisms of active control of fluid flows. Previous studies have proved that the Stuart-Landau equation can well describe the dynamics of the wake behind a circular cylinder at Reynolds numbers near the critical value for the onset of instability. The coefficients of the Stuart-Landau equation have been measured by experiments. Further experimental studies have showed that the Stuart-Landau equation with forcing is a good model for the dynamics of the wake behind an oscillating circular cylinder. Nevertheless a detail study is still lacking on this model equation.In present study, lock-in periodic solutions of the Stuart-Landau equation with active forcing are obtained analytically, and the dependencies of these solutions on the forcing amplitude and frequency are investigated in detail. The results clearly reveal the frequency selection phenom-ena existing in the lock-in periodic solutions. The optimum frequency to obtain the maximum amplitude of lock-in periodic solution is different from the linear instability frequency due to the nonlinear effect.The stability theory for a linear system of periodic coefficient is applied to determine the stability boundary of the periodic solutions. The results show that there is a stability boundary of V-shape (Arnold tongue) in the parameter plane of forcing amplitude versus frequency. When the forcing amplitude is below the threshold, there is no stable lock-in periodic solution, no mater how much the forcing frequency is. This V-shape stability boundary is in qualitatively agreement with available experimental results.The efficiency of active control of flows described by the Stuart-Landau equation is also studied. It is found that, due to the effect of active forcing, the energy of the stable lock-in solutions may decrease comparing to that of saturated solutions of unforced system, and the active control which results in lock-in periodic solution can reduce the perturbation energy by 50 percent at most. Those results suggest a more effective control must result in a non-periodic solution.