自然界中的生物为了能够适应不断变化的外界环境，在经过漫长的自然选择之后，发展出了各种各样的运动方式、结构形态以及生存策略。例如，鸟类和鱼类在社会活动中逐渐发展出高效的推进方式以及群聚策略。陆生与水生植被进化出适应性的叶片结构并被动地进行形态重构。这些行为现象的背后通常涉及典型的流固耦合问题。深入理解这些流固耦合系统中复杂的相互作用以及潜在的力学机制，能够帮助设计各种性能优越的仿生飞行器和水下航行器。此外，还能够为工业领域中的诸多应用设计提供理论参考。另一方面，典型的流固耦合系统在一些场景下会涉及两种不相容的流体介质，即两相流的流固耦合。由于这类耦合系统会涉及到两种性质完全不同的界面，即刚性或柔性的流-固界面和具有复杂拓扑变化的流-流界面。这使得两相流的流固耦合问题更为复杂，为计算流体力学的建模计算带来了更多的挑战。深刻理解这类耦合系统背后复杂的物理机制对科学进步以及技术创新来说同样具有重要意义。

基于上述内容，本文的主要研究内容和创新性工作包括：

(一) 二维和三维柔性薄板的流固耦合

本文通过基于局部流场重构的浸没边界法对二维和三维柔性薄板的流固耦合问题进行了研究。对于二维情况，本文提出了一种新的串联柔性板系统。其中，上游柔性板进行由升沉和俯仰组合而成的主动运动。而下游柔性板则处于尾流区，受上游柔性板尾涡的影响而发生被动的涡激振动。并重点考察了上游柔性板的主动运动频率和两板间的流向间距对涡-结构和涡-涡相互作用的影响。部分研究结论能够帮助探索飞行或游动生物集群中的最佳能量利用机制，并为能量收集器的设计提供一定的参考。针对三维柔性薄板的流固耦合问题，考虑到现有对垂直固定柔性板的研究主要集中在二维或大展弦比的三维情况，关键的边缘效应的影响有待进一步的研究。因此，本文数值研究了均匀来流中垂直固定的三维小展弦比柔性板(展弦比为$0.1$)，并系统性研究了一些重要的无量纲参数，如弯曲刚度、雷诺数和质量比对柔性板动力学及水动力学行为的影响。识别出柔性板的两种不同动力学行为模式：偏转振动模式和完全偏转模式，并确定了两种模式发生转变的临界弯曲刚度。此外，还探究了柔性板偏转振动的频率与其自然频率的同步性。并对系统中的二维和三维尾迹结构以及边缘效应进行了分析讨论。

(二) 流-流界面的水平集方法

在两相流的流固耦合问题中，需要精确地表示其中具有复杂拓扑变化的流-流界面。因此，本文使用隐式的水平集方法对不相容两相流的流-流界面进行精确捕捉。并详细描述了使用水平集函数计算界面的相关几何物理量、水平集函数的重新初始化方程以及相关的空间和时间离散、近似符号函数的选择以及水平集函数的运动方程。并通过一个基准算例，即圆形界面的曲率驱动收缩，验证了本文所使用的水平集方法以及其程序的可靠性。最后，将水平集方法应用于不可压缩无粘流动中，并给出了三个具体的数值算例，包括涡流层卷曲、涡流层偶极子以及点涡运动。这些数值算例的计算结果与前人的结果非常吻合。

(三) 两相流的流固耦合

本文将流-固界面的浸没边界法与流-流界面的水平集方法相结合，发展出了一套能够计算两相流流固耦合的程序。并对程序中不可压缩两相流控制方程的离散格式和时间推进、流-流界面表面张力的计算方法、粘性应力张量的分解以及水平集方法质量守恒问题的修正方案进行了详尽阐述。最后，将这套两相流的流固耦合程序用于研究平直通道内粘性液滴与固体薄板的相互作用。并系统性探究了固体薄板与流动方向之间的倾角以及柔性板无量纲弯曲刚度的影响。

In nature, organisms have evolved various locomotion modes, morphological structures, and survival strategies to adapt to the ever-changing external environment through prolonged natural selection. For example, birds and fish have gradually evolved efficient propulsion modes and schooling behaviors for social activities, while terrestrial and aquatic vegetation have adapted their leaf structures and undergone passive morphological reconfiguration.Behind these behavioral patterns are the typical problems of fluid-structure interaction. Understanding the complex interactions and potential mechanical mechanisms in these fluid-structure interaction systems can facilitate the design of various high-performance bio-inspired aerial and underwater vehicles. Moreover, it can also provide theoretical guidances for numerous industrial applications. On the other hand, typical fluid-structure interaction systems may involve two immiscible fluids in certain scenarios, i.e. fluid-structure interaction of two-phase flow. Such coupling systems involve two interfaces with different properties, namely a rigid or flexible fluid-structure interface and a fluid-fluid interface with complicated topological variations. This complexity poses significant challenges to modeling and simulating of computational fluid dynamics. Therefore, a deeper comprehension of the complex physical mechanisms underlying such coupling systems is crucial for both scientific progress and technological innovation.

Based on the above, the main research contents and innovative works of this thesis are as follows:

1. Fluid-structure interactions of two- and three-dimensional flexible thin plates

In this thesis, the fluid-structure interactions of two- and three-dimensional flexible thin plates are numerically investigated using immersed boundary method based on local flow field reconstruction. For the two-dimensional case, a novel system of tandem flexible plates is proposed, in which the upstream flexible plate undergoes an active motion consisting of a combination of heaving and pitching. The downstream flexible plate, located in the wake region, undergoes passive vortex-induced vibration. The effects of the active motion frequency of the upstream flexible plate and the streamwise spacing between the two plates on the vortex-structure and vortex-vortex interactions are studied. The results have implications for exploring optimal energy utilization mechanisms and provide guidances for the design of energy harvesters.

For the three-dimensional case, considering that the existing researches on the vertically clamped flexible plate mainly focus on the two-dimensional cases or three-dimensional cases with large aspect ratios, the critical side edge effects need to be further investigated. Therefore, in this thesis, a vertically clamped three-dimensional flexible plate of very low aspect ratio 0.1 in a uniform flow is considered, and the effects of important dimensionless control parameters, such as bending stiffness, Reynolds number, and mass ratio, on the dynamic and hydrodynamic behaviors of the flexible plate are systematically examined.

Two distinct dynamic modes of the flexible plate are identified: deflected-flapping mode and fully deflected mode, and a critical bending stiffness between the two modes has been determined. Besides, the synchronization of the oscillation frequencies with the natural frequencies of the flexible plate is explored. The two- and three-dimensional wake structures and the side edge effects in this system are also analyzed and discussed.

2. Level set method for fluid-fluid interface

For the fluid-structure interaction problems of two-phase flow, an accurate representation of the fluid-fluid interface with complex topological variations is essential. Therefore, an implicit level set method is employed in this thesis to accurately capture the fluid-fluid interface. The calculation of the relevant geometric physical quantities of the interface using the level set function, the reinitialization equations and the associated spatial and temporal discretization, the selection of the approximate sign function, and the motion equations of the level set function are described in detail. Then a benchmark case of a shrinking unit circle with its mean curvature is used to validate the level set procedure of this thesis. Finally, the level set procedure is applied to incompressible inviscid flow, and three specific numerical cases are given, including vortex sheet, vortex sheet dipole, and point vortex motion. The computational results of these numerical cases are in good agreement with the previous ones.

3. Fluid-structure interaction of two-phase flow

In this work, the immersed boundary method of the fluid-structure interface is combined with the level set method of the fluid-fluid interface to develop a procedure that can simulate the fluid-structure interaction of two-phase flow. The discretization scheme and time advance of the incompressible two-phase flow governing equations, the calculation of the surface tension, the decomposition of the viscous stress tensor, and the correction method for the mass conservation problem of the level set method are elaborated in detail. Then, this novel procedure is used to study the interactions between a viscous droplet and a thin plate in a straight channel. And the effects of the inclination angle and the dimensionless bending stiffness of the flexible plate are systematically investigated.