摘要
时空混沌和湍流可以在各种非线性系统中出现,例如流体力学系统、等离子体系统、激光系统、化学反应、约瑟夫森结阵列和生物网络系统等,在许多实际情况下这种行为是有害的。例如在磁约束等离子体中由压强驱动不稳定导致的漂移波湍流会产生异常横越磁场的粒子输运并导致不希望的能量损失,流动湍流(flow turbulence)同样也会产生有害的结果,它增加了管道、飞机、船和汽车的能量损耗,是飞行安全所必须考虑的因数。因此在这些系统中时空混沌和湍流控制具有很重要的意义。
流体和磁约束等离子体中的湍流可以认为是一种高维时空混沌,人们相信,湍流控制可能借鉴高维混沌系统中的混沌控制方法,但是基于非线性动力学和混沌控制思想的湍流控制才刚刚开始不久,人们已经获得了一些结果。管曙光等人使用时空混沌的全局和局域反馈去控制由二维Navier-Stokes方程描述的流动湍流,将控制信号作用在湍流速度场的两个分量上,他们发现湍流可以被控制到随时间变化的空间周期目标态上,但是当将控制信号只作用在湍流速度场的一个分量上时,湍流不能被完全控制到目标态;吴顺光等人研究了用一个小外力去控制由一维漂移波方程描述的时空混沌;Gravier等人通过实验研究了在实验室中产生的圆柱形磁化等离子体中的非线性漂移波并用延迟自同步反馈去控制它;但是至今在湍流控制中仍然存在许多困难和难解决的问题。
在这篇博士论文里,我们研究了在湍流控制中出现的控制效率、优化控制和控制后系统运动的表现形态等问题。我们提出了一些新的控制方法,并且讨论了这些控制方法的物理机制。
第一章简要地介绍了基本的混沌现象、混沌控制方法、流动湍流、漂移波湍流和湍流的直接数值模拟方法等内容。
第二章研究了流动湍流的控制问题,使用二维Navier-Stokes方程模型,完成了两个工作。首先我们考虑当反馈信号同时加在速度场的x和y分量上时如何提高湍流的控制效率的问题,提出通过移动控制器的方式来提高局域反馈控制(称为运动控制)的效率。结果表明:在控制器数量和注入的能量都相同的情况下,使用运动控制器的局域反馈比使用静止控制器的局域反馈(称为静止控制)更有效地将湍流控制到有序的目
Spatiotemporal chaos and turbulence occur in a variety of nonlinear dynamic systems such as hydrodynamics, plasma systems, laser systems, chemical reactions, Josephson junction arrays and biological networks. In many practical situations such behaviors are considered to be harmful. For instance, drift-wave turbulence which is caused by pressure-driven instability in magnetized plasma is generally believed to be responsible for anomalous cross-field particle transport that causes undesirable energy loss. Flow turbulence also has undesirable consequences: it enhances energy consumption of pipelines, aircrafts, ships, and automobiles;it is an element to be reckoned with in air-travel safety;and so forth. Therefore, spatiotemporal chaos and turbulence control in these systems is of crucial importance.Turbulence in fluid and magnetized plasma can be considered to be a type of high-dimensional spatiotemporal chaos. It is expected that flow turbulence control may be benefited from those strategies developed in controlling high-dimensional chaotic systems. However, flow turbulence control based on the applications of nonlinear dynamics analysis and chaos control methods is just at its beginning. Some results have been already achieved. Guan Shuguang et al have applied global and local feedback control method developed in spatiotemporal chaos control to control flow turbulence described by two-dimensional incompressible Navier-Stokes equation. By applying control signals to all components of flow velocity field, they found that turbulence can be controlled to desirable ordered target states. However, turbulence cannot be completely controlled to the target by global feedback when only one component of the velocity vector is uni-directionally coupled to a target state, while the other component is uncoupled. Wu Shunguang et al have studied controlling spatiotemporal chaos in a system described by a one-dimensional nonlinear drift-wave equation by applying small external periodic force. Gravier et al have studied experimentally nonlinear drift-wave in a cylindrical magnetized laboratory plasma and applied the time-delay auto-synchronization method to control drift-waves chaos. However, many difficult and challenging problems are still
remaining in turbulence control so far.In this paper, we study turbulence and spatiotemporal chaos control, focusing on the problems of control efficiency, optimization of control results and the properties of the asymptotic states of the systems under control. We suggest some new methods for controlling turbulence. The physical mechanisms underlying those control schemes are heuristically analyzed.In chapter 1, the basic chaotic phenomena, chaos control methods, flow turbulence, drift-wave turbulence and direct numerical simulation methods of turbulence are briefly introduced.In Chapter 2, we investigate flow turbulence control. The model studied is the two-dimensional incompressible Navier-Stokes equation. Two works have been done. We firstly consider how to improve the efficiency of flow turbulence control with the feedback signals being applied to both x and y components of the velocity field. We suggest a control strategy which applies local feedback injections with moving controllers (called moving control). It is shown that with the moving controllers, flow turbulence can be controlled more efficiently than the usual pinning strategy with static controllers (called static control). The moving control method can entrain the system from turbulence to ordered targets faster than the static control. In particular, the moving control can successfully suppress turbulence with a small number of controllers in a certain control time, with which the static control fails to suppress turbulence. The physical mechanism underlying this high control efficiency is heuristically analyzed. The advantages and difficulties of the proposed control strategy in practical applications are discussed.It is generally accepted that in experiments controlling a single component of velocity field could be easier than controlling the whole velocity vector. In this regard, we investigate how to enhance the efficiency of flow turbulence control with applying global feedback signals only to a single component of flow velocity field. We suggest a control strategy which applies global feedback sporadically. It is found that this control strategy can significantly enhance the control precision as the optimal fraction
of control period is chosen, both larger and smaller control time fractions may reduce the control precision. We further investigate optimization and controllability of flow turbulence control by applying local pinning feedback signals to control a single velocity component. It is found that with a certain number of controllers there exists an optimal control strength at which the control error reduces to the minimum value, and larger and smaller control strengths give worse control effects. Moreover, given a fixed control strength there may exist an optimal number of controllers achieving the best result, and larger and smaller numbers of controllers again provide worse control effects. We analyse this strange and interesting feature based on the mode-mode interactions of the turbulent systems.In Chapter 3, we study spatiotemporal chaos control in drift-wave systems. We consider a one-dimensional nonlinear drift-wave equation driven by a sinusoidal wave. We firstly apply time-delay and space-shift feedback signals to suppress spatiotemporal chaos. By using global and local feedback strategies, we show numerically that the spatiotemporally chaotic state can be effectively controlled to periodic states if suitable time delay length and space shift distance are chosen. It is the first time, to our knowledge, to show that spatiotemporal chaos can be suppressed by space-shift feedback only. The obvious advantage for this space-shift feedback over the time-delay feedback is that the former needs much smaller storage of data than the latter. Furthermore, we obtain an analytical expression of the controllability condition by combining partly analytical and partly numerical computations. It is found that an optimal combination of time-delay length r and space-shift distance Ia can distribute the phases of the modes with large amplitude such that the control signal can effectively drives the system to the asymptotic state of minimum energy, and thus successfully suppress turbulence. We further investigate the influence of system size on the efficiency of spatiotemporal chaos control. It is found that the efficiency of the control approach is practically not influenced by changing the system size. This result indirectly shows that our control method can work for different boundary condition.The variable measurement in magnetized plasma is usually difficult. The non-
feedback control is more convenient for drift-wave turbulence control. We apply an additional sinusoidal wave to suppress the drift-wave spatiotemporal chaos, and the system becomes a double-sinusoidal-wave (the original driving wave and the additional controlling wave) driven system. It is shown that with some proper choices of the frequency Q, and control strength g of the controlling wave spatiotemporal chaos can be successfully suppressed by the controlling wave. By transforming the drift-wave equation to a set of amplitude and phase equations of space modes, we find an interesting phenomenon of frequency entrainment. The quasi-periodic motion of the system can be decomposed such that the phases of modes rotate at high frequency with small modulation of frequency equal to the difference of the two frequencies of the diving and controlling waves. The amplitudes of modes and the energy of the system however oscillate periodically at this difference frequency. This frequency entrainment directly associated with the controllability transition of the system from spatiotemporal chaos to quasi-periodicity.
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