## 4 Applications of IBM for Biological Flows
### Swimming Behaviors  
It is meaningful for biomimetics to investigate swimming behaviors, which can give inspiration in underwater swimmer designs [127; 128; 129]. Both sharp interface and diffuse interface IBMs have been tested for aquatic animal locomotion [52; 55]. Different species of underwater animals have their own shapes and swimming mechanisms, such as a mackerel-like body [52; 55; 63; 130; 128], oblate body [6; 131], anguilliform body [132; 133], and jellyfish [134; 135; 136]. Figure 12 presents the numerical results of two types of fishes with significantly different shapes, where [PERSON] et al. [6] revealed the role of vortices in the force enhancement mechanism in aquatic swimming using the derivative-moment trans  
Figure 11: Vortical structures for the single flapping wing: (**a**) Reproduced from [65], sharp interface IB-NS: the left column is for rigid wing, the middle and right columns are for elastic wing with different parameters, the schematic is only for elucidating the hovering flapping motion; (**b**) Our in-house diffuse interface IB-LBFS: the schematic of forward flight flapping motion matches the evolution of vortical structures. Figure 10: A series of instantaneous wake structures for the wing-body and wing-only models: (**a**) Reproduced from [97]: hummingbird; (**b**) Reproduced from [91]: cirada. formation (DMT) theory, and [PERSON] et al. [63] investigated the flow patterns and analyzed the periodic effects caused by tail paddling using dynamic mode decomposition (DMD). Both fishes undergo prescribed deformation, where the batioid fish deforms corresponding to different wavenumbers and longitudinal amplitudes while the tuna is governed by Equation (5). Underwater animals often have rich swimming mechanisms, such as self-propulsion [54; 57; 58] and power extraction [66; 113]. Figure 13 shows the self-propulsion process of an underwater swimmer, which is simulated by our in-house code of IB-LBFS. The geometry of the swimmer is the same as that in Figure 1 for the passive motion of rigid bodies, and relative physical parameters can refer to [54]. According to [PERSON] et al. [54], an unconstrained swimmer has two degrees of freedom to move, which are governed by Equation (4). A harmonic pitching motion, Equation (3), can drive its self-propelled motion. The reversed Karmann vortex is well-known for generating thrust in self-propulsion. The propulsion velocities are shown in Figure 13a, as well as the results from the work of [PERSON] et al. [54] and [PERSON] et al. [58], where the U velocity is always positive. [PERSON] et al. [58] proved that with a flexible fin clamped at the swimmer's tail, the propulsion performance can be improved. The flexible fin also takes on the role of power extraction from the ambient fluids, which is sensitive to the flexibility of the tail [66]. When it comes to multibody, the collective locomotion [112; 113; 114] and wake-body interaction [62; 64; 137] appear and are worth studying. Due to the self-propelled motion, the collective locomotion appears as a stable system with multiple swimmers in forward swimming together [113], as shown in Figure 14.  
Figure 12: Vortical structures around different swimming fishes: (**a**) Batoid fish, Reproduced from [6]; (**b**) Tuna, Reproduced from [63]. It is attractive to explore the fishes with flexible bodies governed by Equation (5) in special underwater environments. [PERSON] and [PERSON] [60] investigated the fish swimming in oblique flows and reported that the amplitude of undulation was dominant in thrust generation. [PERSON] et al. [61] considered the wall effects in their study on the power extraction efficiency of a fish with carangiform swimming in a narrow channel. Moreover, a group of these undulating fishes has complex wake-body interactions. High-density fish school is studied by [PERSON] and [PERSON] [62, 137], where the fishes have various phase differences that improve the thrust efficiency of the fish school.