###### Abstract  
Biological flows in swimming and flying bio-locomotion usually involve intricate flexible or rigid structures that undergo large deformations and displacements, as well as rich mechanisms of bio-fluid interactions. Immersed boundary methods (IBMs) have gained increasing prevalence in numerical investigations of such biological flow problems due to their simplicity and capability for simulating these problems on a Cartesian mesh, which does not require tedious grid-regeneration or mesh deformation processes. In recent years, the vigorous development of IBM variants has enriched numerical techniques for bionic simulations. This review focuses on the development of the IBM and its applications in the field of biological aerodynamics and hydrodynamics, including both diffuse and sharp interface IBMs. The fundamentals of the former are introduced in detail, and the hybrid Cartesian-IBM is briefly presented as one representative method of the latter. In particular, the velocity correction IBM is highlighted in the diffuse interface IBM due to its superiority in accurately satisfying no-slip boundary conditions. To shed light on the dynamic characteristics of flying and swimming behaviors with predefined or passive motion and deformation, some recent results from IBM applications are also presented. Finally, this review discusses some challenges and promising techniques in the research of bio-inspired motions based on the IBM. immersed boundary method; biological flows; swimming and flying; fluid-structure interaction +
Footnote †: journal: Applied sciences  
to match the moving structures, which causes a high computational cost. The immersed boundary method (IBM) is a representative approach for the other [8]. It adopts a non-conformal mesh, which is fixed during the whole simulation regardless of the moving and deforming interface, without any re-meshing process. The frequently used Cartesian mesh is easier to code and generate than body-fitted mesh and is even unaffected by the surface complexity, demonstrating its enormous potential in simulating sophisticated biological flows. The IBM was initially proposed by [PERSON] for blood flow in the heart [8]. To solve more problems, the original IBM has been improved, producing a series of variants. They have been successfully applied to simulate flow problems involving flexible or rigid bodies over the past few decades due to their simplicity and practicality. In the IBM, the fluid and solid domains are discretized by independent meshes and are described by Eulerian and Lagrangian coordinates, respectively. Therefore, the fluid field and solid motion/deformation can be solved by their own governing equations. Then, the IBM is applied to deal with the boundary conditions at their mutual interface through a so-called forcing term. The different treatments of this term motivate various IBMs. In the original IBM [8; 9; 10], the forcing term is computed by the elastic law of flexible structures, which restricts its application to rigid bodies. By introducing some free parameters to model the forcing term, different versions of the penalty forcing IBM are developed [11; 12; 13; 14; 15]. However, the values of the user-defined free parameters are hard to choose for different problems, and the rigidity restriction has not been completely overcome. A variety of direct forcing IBMs have been developed in the following years, aiming to solve the rigid particle-laden flows [16; 17; 18; 19; 20; 21; 22]. Since then, the limitations of IBMs in problems with rigid bodies have been lifted. Due to separate solvers for fluid and solid, IBMs can combine with most of the prevalent fluid solvers, such as Navier-Stokes solvers [8; 10; 23; 24] and the lattice Boltzmann method (LBM) [16; 25; 26]. More details will be discussed later. Among the developed IBMs, to our knowledge, a momentum exchange-based IBM is unique for LBM solvers [27; 28].