Swimming of Multi-Fish Swarms Simulated Using a Virtual Cell-Immersed Boundary Framework

Abstract

To explore the influence of inter-formation variables on swimming performance during fish schooling, this paper adopts the sharp interface immersed boundary method based on virtual cells to conduct numerical research on the swimming of three-fish and four-fish swarms with different formations and spacings. The study finds that both streamwise spacing and lateral spacing have significant impacts on the swimming performance of fish schools. In the three-fish formation, when the tandem arrangement has a streamwise spacing of 1.3 times the body length (L), the trailing fish achieve the highest swimming efficiency; when the parallel arrangement has a lateral spacing of 0.25L, the fish in the middle position exhibits the optimal swimming performance. In the four-fish formation model, fish in symmetric positions within the same swarm have similar hydrodynamic performance. For the diamond formation, under the configuration of streamwise spacing 1.2L and lateral spacing 0.5L, the propulsive efficiency of the trailing fish is markedly diminished; however, for the rectangular formation, all trailing fish obtain lower swimming efficiency, and a stable 2S-type vortex structure appears in the wake under the configuration of streamwise spacing 1.5L and lateral spacing 0.5L, which is conducive to thrust generation. The conclusions of this paper can provide certain hydrodynamic advantages and support the development of bionic underwater vehicles and robot technology.

1. Introduction

There has been a strong consensus that the hydrodynamic advantages of fish schooling can effectively improve swimming efficiency [1]. Currently, there are two mainstream hypotheses regarding multi-fish swimming [2]: the vortex hypothesis and the channel effect. The mutual influences between fish can be roughly categorized into three types: lateral influence between fish in the same row of the same horizontal layer [3]; streamwise influence between fish in the same row of the same horizontal layer [4]; and mutual influence between horizontal layers [5]. However, most practical cases involve the coupling effect of these three influences. Therefore, studying the mutual influences of multi-fish swimming, exploring a more comprehensive analysis of flow field changes and energy-saving mechanisms, and providing references for the design and control of underwater robot formations are current mainstream research directions.

Existing studies have found that the wakes of schooling fish interact to assist swimming [6] and that the tail-beat phase difference of trailing fish, which changes linearly with distance, enables them to obtain hydrodynamic gains [7]. Numerical simulation methods have shown that caudal fin-swimming fish trailing leading fish can increase thrust and efficiency by 12% [8], while trailing fish in different arrangements reduce lateral power in parallel arrangements and gain energy advantages in staggered arrangements [9]. Similarly, in the swimming of carangids under different arrays, tail-swing phase differences can significantly reduce the overall intensity of sound radiated to the far field [10], and the relative phase of caudal fin flapping is a critical factor influencing far-field flow-generated noise, and appropriate phase adjustment can lead to significant noise reduction [10]. An electromagnetically connected and self-reconfigurable robotic fish swarm system achieves multimodal locomotion through morphological reorganization, demonstrating notably superior performance over individual units in terms of stability, maneuverability, speed, and energy efficiency [11]. Fish swimming is mainly propelled by undulating movements, which are divided into two modes based on different thrust generation methods: the BCF (Body and/or Caudal Fin) mode, which generates thrust by bending to form a propulsive wave transmitted to the caudal fin, and the MPF (Median and/or Paired Fin) mode, which generates thrust through the middle fins. Figure 1 shows the positions and structures of the main fins. Regarding fish morphology, Reference [12] elaborates on the relationship between form and function, and in recent years, the complex hydrodynamic mechanisms behind it have been revealed through numerical simulation methods [13]. This paper mainly studies the BCF propulsion mode. When fish move forward in the BCF mode, the body surface waves propagate in the opposite direction at a super-forward speed, and the main differences between each propulsive wave lie in wavelength and amplitude. Moreover, compared with swimming with only the caudal fin, the addition of the dorsal fin and anal fin can further enhance the thrust of the caudal fin and reduce trunk resistance [14]. The core of fish movement lies in the interaction between incoming vortices and fins. There are three thrust generation methods for the interaction between oscillating airfoils and incoming vortices; when the incoming vortices interact negatively with the airfoil shedding vortices, thrust efficiency can be improved [15]. Fish convert the mechanical energy generated by muscle contraction into the kinetic energy of the surrounding fluid through active deformation and movement. During swimming, momentum is transferred from the fish’s body to the surrounding fluid through drag, lift, and acceleration reaction forces. Fang et al. observed that under high-Reynolds number conditions, thrust mainly comes from the periodically changing drag distribution in the tail region [16]. Fulong et al.’s research on the hydrodynamic performance and wake patterns of fish under different angles of attack and Reynolds numbers found that lift-dominated moment adjustment is crucial for maintaining stable swimming trajectories [17].

Fluid–structure interaction occurs at every biological scale in nature [18]. The immersed boundary method solves the problems of large deformation and moving boundaries in fluid–structure interaction through the Eulerian–Lagrangian mixed grid [19]. Its core lies in the simulation method of the effect of solid boundaries on fluids, which is mainly divided into the diffused interface immersed boundary method [20] and the sharp interface immersed boundary method [21,22]. The former characterizes the solid influence by adding a volume force source term to the Navier–Stokes (N-S) equations [23], which is derived from constitutive relations and distributed to Eulerian grid points through discrete functions [24]. The latter directly reconstructs the velocity of the first layer of Eulerian grids outside the boundary, avoiding the discretization of the N-S equations and achieving more accurate boundary positioning [25]. Currently, the combination of the virtual cell method and the immersed boundary method have been widely used in various fields of fluid–structure interaction [26,27].

The above numerical studies and experiments indicate that the collective movement of specific formations can significantly improve the swimming efficiency of fish schools. Further relevant research still needs to be carried out. Currently, research on the hydrodynamic characteristics of swarms mainly focuses on two-dimensional configurations, and the rectangular configuration is less studied compared with other configurations. In addition, to better develop bionic underwater vehicles, more supporting data are needed. To supplement the research data in this field, direct numerical simulation (DNS) was used in this paper, and the sharp interface immersion boundary method (IBM) based on virtual cells was used to numerically simulate the three-dimensional fish swarm system. In the study, the influences of lateral spacing and streamwise spacing on the swimming resistance, thrust, power, and efficiency of fish swarms were investigated. The hydrodynamic performance and wake dynamics of the system were systematically investigated, and the influence of flow field pressure on the fish hydrodynamic characteristics was also considered. This paper aims to obtain data on hydrodynamic interactions during fish schooling, thereby providing a reference for the design of bionic underwater vehicle formations.

2. Materials and Methods

The immersed boundary method based on fixed Cartesian grids has been widely used in the computation of fluid–structure interaction problems, and various variants have been developed [28,29,30]. However, the fixed Cartesian grid does not fit the immersed boundary, limiting the application of the immersed boundary method in high-Reynolds number fluid–structure interaction. Mittal et al. pointed out that the defect of the traditional immersed boundary method lies in interface numerical diffusion, and proposed sharp interface division and local interpolation correction [31]. Meanwhile, Luo et al. also proposed a strategy combining high-order interpolation and implicit discretization to address numerical oscillation issues [32]. The current mainstream solution is to implant a wall model to consider the boundary layer problems involved in high-Reynolds number flows [22,25]. This paper uses a sharp interface immersed boundary method solver based on virtual cells [33,34,35]. When updating the fluid physical quantities of fluid nodes, flow field probes are constructed to interpolate the physical quantities of surrounding fluid nodes. This solution approach has been successfully applied to handle hydrodynamic problems of low-to-moderate-Reynolds number flows [36,37].

2.1. Computational Model and School Formation

For the yellowfin tuna body model, this paper only retains the caudal fin for simplification. The trunk length is set to 80% of the total model length (L), and the caudal fin is modeled as a zero-thickness membrane [18].

The expression applied is used to describe its lateral oscillatory movement [38,39]:

$$h(x,t)=P(x)\sin (kx-2\pi ft+\varphi )$$

(1)

$$P(x)=a+bx+c{x}^2$$

(2)

In the equation, $h(x,t)$ is the lateral displacement of the midline at time $t$, $x$ is the axial distance coordinate from the fish’s mouth to the tip of the tail, $f$ is the tail fin oscillation frequency, $\varphi$ is the phase, and $P(x)$ is the amplitude envelope describing the lateral displacement amplitude of the wave, with coefficients $a=0.02$, $b=-0.08$, and $c=0.16$, $k={\displaystyle \frac{2\pi }{\lambda }}$ being the wavenumber and $\lambda$ the wavelength, which is set as $1.25L$. The Reynolds number is $\mathrm{Re}={\displaystyle \frac{U_{\infty }L}{\nu }}=3000$, where $\nu$ is the kinematic viscosity. Figure 2 illustrates the current tuna model and a series of its midlines within one undulation period.

Using the same model, this paper sets up four bionic fish robot swarms, tandem, parallel, diamond, and rectangular, in order to change the fish spacing and simulate and calculate the thrust and drag coefficients and the power of the fish.

In the tandem schooling formation of Figure 3a, the fish swim in a straight line with the same streamwise spacing (D); in the side-by-side schooling formation of Figure 3b, the fish are distributed in an array with lateral spacing (G); this paper also designs a four-fish diamond formation in Figure 3c and a rectangular formation in Figure 3d, and the streamwise spacings (D), lateral spacings (G) and fish swimming phase differences of all group structures are consistent.

2.2. Computational Method

The 3D fish body model is immersed in a Cartesian grid, and numerical simulation is performed by solving the three-dimensional unsteady Naiver–Stokes equations for viscous incompressible fluids. The control equations are as follows:

$${\displaystyle \frac{\partial u_i}{\partial x_i}}=0$$

(3)

$${\displaystyle \frac{\partial u_i}{\partial t}}+uj{\displaystyle \frac{\partial u_i}{\partial x_j}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{1}{\mathrm{Re}}}{\displaystyle \frac{\partial }{\partial x_j}}({\displaystyle \frac{\partial u_i}{\partial x_j}})$$

(4)

Here, $i,j=1,2,3$, $u$ represents the Cartesian components of the fluid velocity, $x$ represents the Cartesian coordinate directions, $\rho$ is the fluid density, and $p$ is the pressure. The spatial discretization of the equations adopts a second-order central difference scheme, and the Chorin projection correction method and the Crank–Nicolson method are used for discretization to eliminate the viscous stability constraint.

Once the flow field is solved, the surface pressure and shear force of the model’s trunk and caudal fin can be projected and integrated from the flow variables around the tuna model. The power consumption ($P$) is calculated by integrating over the model’s surface area, defined as

$$P=-{\displaystyle ∯(\sigma •\mathrm{n})}•Vds$$

(5)

In the equation, $\sigma$ is the stress tensor, $\mathrm{n}$ represents the normal vector at each point on the model surface, and $V$ denotes the velocity vector of the fluid.

The paper uses two dimensionless coefficients to represent the streamwise force and power consumption, defined as

$$C_F={\displaystyle \frac{F}{1/2\rho U_{\infty }^2{S}_{CF}}}$$

(6)

$$C_P={\displaystyle \frac{P}{1/2\rho U_{\infty }^3{S}_{CF}}}$$

(7)

In the equations, $\rho$ is the fluid density. The flow direction force is defined as positive when it is in the same direction as the incoming flow. $SCF$ is the tail fin area.

For the model facing the incoming flow, the positive and negative streamwise forces are drag and thrust, respectively. That is, $C_D$ (drag of trunk) $=C_F$ and $C_T$ (thrust of caudal fin) $=-C_F$.

In addition, the swimming efficiency is defined as

$$\eta ={\displaystyle \frac{{\overline{C}}_T}{{\overline{C}}_P}}\tilde{U}$$

(8)

Here, ${\overline{C}}_T$ and ${\overline{C}}_P$ represent the average thrust coefficient and the average power coefficient, respectively, and $\tilde{U}$ is the uniform velocity of the incoming flow.

The specific process for validating the reliability of the numerical method through simulating the hydrodynamic flow field around a stationary thin plate in uniform flow, along with the definitions provided in Equations (1)–(8), is comprehensively documented in the research group’s prior studies [32]. The verification results are in good agreement with the numerical results in Reference [40], and the three-dimensional instantaneous vortex structure also tends to be consistent. The size of the computational domain is set to $8.5L \times 6L \times 5L$ (L is the body length), and the area around the fish body is refined, as shown in Figure 4a. The fish body surface is represented by triangular unstructured elements, with 4862 nodes and 9720 elements in the trunk and 744 nodes and 1380 elements in the caudal fin. The left inlet is set to a constant velocity boundary condition, and the pressure uses a Neumann boundary condition; the right outlet is set to a zero-gradient boundary condition for velocity and pressure; other boundaries are set to zero-stress boundary conditions. The force on the fish body is dimensionless. The three-dimensional vortex structure is visualized by the isosurface of the Q-criterion [41]. The setting and selection of conditions here follow our previous research on the same fish body model and flow conditions [35]. To ensure that the simulation results are grid-independent, a grid independence study was carried out on three different grids (i.e., Δ = 1/150, Δ = 1/200, and Δ = 1/250). Figure 4b shows the streamwise force coefficient of the caudal fin under different grids. It can be clearly observed from Figure 4b that the difference in either the mean value or the peak value of the streamwise force coefficient among the different grids is less than 5%. Based on these observations, Δ = 1/200 was chosen for the simulations to balance computational accuracy and cost. For a more detailed verification process, please refer to the previous papers of our research group [34].

3. Results and Discussion

3.1. Results and Analysis of Three-Fish Formation

3.1.1. Tandem Schooling Formation

For the tandem configuration, the nondimensional thrust and drag coefficients of Fish 1–3 at various streamwise spacing distances (D) are presented in Figure 5. The simulation results demonstrate that the net hydrodynamic force acting on each fish approaches zero. Fish 1 exhibits a drag coefficient of approximately 0.32 and a thrust coefficient of approximately 0.27, with a negligible influence of spacing on its drag and thrust characteristics.

The interaction effect is defined as the influence of hydrodynamic interactions on the various flow forces of the fish body [34]. The impact of hydrodynamic interactions on various fish body parameters in the tandem configuration is presented in Figure 6. Figure 6a–d illustrate the variations in the drag, thrust, mechanical power, and swimming efficiency of the three fish as a function of streamwise spacing distance (D), respectively.

The results demonstrate that Fish 1’s hydrodynamic parameters exhibit relative stability with varying streamwise spacing distance (D); Fish 2 and 3 in symmetric configurations display nearly identical performance characteristics across all parameters, except for the fact that Fish 2 demonstrates reduced drag and enhanced thrust. At D = 1.3L, the trailing fish attain elevated swimming efficiency, with both fish exhibiting augmented thrust while sustaining minimal energy expenditure compared to a solitary fish.

To provide a more intuitive understanding of the influence of varying streamwise spacing distances (D) on the flow field, Figure 7 presents the three-dimensional instantaneous vortex structures (visualized using Q-criterion isosurfaces) and two-dimensional planar vorticity distributions under three distinct streamwise spacing conditions: D = 1.2L, D = 1.35L, and D = 1.5L. It is evident that the wake vortices generated by the swimming of the three fish shed backward in a divergent pattern, thereby influencing the thrust and drag forces experienced by Fish 2 and Fish 3.

Figure 8 presents the pressure distribution of the flow field at the same time for three distinct streamwise spacing distances: D = 1.2L, D = 1.35L, and D = 1.5L. The pressure distribution around Fish 1 exhibits no significant variation, which aligns with the data for Fish 1 presented in Figure 5. When the streamwise spacing distance (D) is small, the positive pressure region at the head of Fish 2 and Fish 3 is markedly enhanced, the negative pressure region along the fish body is redistributed, and the positive pressure region at the caudal fin is strengthened, resulting in a substantially increased thrust. As the streamwise spacing distance (D) increases, the influence of the wake vortices shed by the leading fish on the trailing fish is significantly diminished.

3.1.2. Side-by-Side Schooling Formation

For the side-by-side schooling formation, the dimensionless thrust and drag coefficients of Fish 1–3 across varying lateral spacings (G) are presented in Figure 9. The simulation results demonstrate that the net force acting on each fish is approximately zero.

The hydrodynamic effects on key parameters of the fish bodies are illustrated in Figure 10, where (a–d) depict the variations in drag, thrust, mechanical power, and swimming efficiency, respectively, for the three fish across different lateral spacings (G). The results demonstrate that the trend variations of these hydrodynamic parameters with respect to lateral spacing (G) exhibit a high degree of consistency among the three fish. Specifically, the drag force monotonically decreases as the lateral spacing (G) increases, while the thrust shows an inversely correlated trend. Notably, the hydrodynamic response curves of the outer fish (Fish 2 and 3) exhibit near-identical profiles, indicating similar flow interactions in the lateral direction. Based on comprehensive analysis of power consumption and efficiency metrics, Fish 1 achieves optimal swimming efficiency at a lateral spacing of G = 0.3L, representing a favorable balance between hydrodynamic performance and energy expenditure.

Figure 11 presents the three-dimensional instantaneous vortex topology and two-dimensional planar vortex distribution, visualized using the Q-criterion, under three distinct lateral spacing conditions: G = 0.25L, G = 0.4L, and G = 0.55L. The wake vortices generated by the swimming fish exhibit divergent shedding patterns, with vortex shedding in the wake of adjacent fish interacting significantly, thereby influencing their respective thrust and drag forces. At smaller lateral spacings (G = 0.25L), as depicted in Figure 11a, pronounced vortex merging phenomena occur between the wakes of neighboring fish. In contrast, at larger lateral spacings (G = 0.55L), the vortex structures remain more distinct with minimal interaction, resulting in less pronounced merging effects.

Figure 12 presents the spatial pressure distribution within the flow field at the specified lateral spacing conditions: G = 0.25L, G = 0.4L, and G = 0.55L. The distribution patterns reveal that the oscillatory pressure field generated by the fish motion significantly influences neighboring fish. At smaller lateral spacings (G = 0.25L), as illustrated in Figure 12a, a pronounced region of negative pressure develops along the midsection of the fish bodies, leading to a reduced pressure differential across the fish bodies. This condition creates an adverse pressure gradient that hinders effective thrust generation. In contrast, at larger lateral spacings (G = 0.55L), as shown in Figure 12c, the three fish bodies operate in hydrodynamically independent regions, with minimal pressure field interaction between them. The findings indicate that excessively reduced lateral spacing between fish results in detrimental pressure interactions that diminish swimming efficiency.

3.2. Results and Analysis of Four-Fish Formation

3.2.1. Diamond Formation

The lateral spacing (G) in the diamond formation is defined as 0.25L and 0.5L, while the longitudinal spacing (D) is specified as 1.2L and 1.5L, yielding four distinct configuration geometries. The hydrodynamic forces acting on each fish are nondimensionalized for comparative analysis, with the dimensionless thrust and drag coefficients presented in Figure 13. Simulation results reveal that the net hydrodynamic force on individual fish approaches zero, indicating a near-equilibrium state. The influence of hydrodynamic coupling on key performance metrics—including drag reduction, thrust efficiency, and wake interactions—is systematically quantified in Figure 14.

Notably, Fish 2 and Fish 3 at symmetric positions display negligible differences in all measured physical quantities. Fish 4 exhibits comparable trends in hydrodynamic performance to Fish 2 and 3, but demonstrates significantly reduced propulsive efficiency, particularly in the configuration with G = 0.5L and D = 1.2L, where wake capture effects and pressure gradient interactions are most pronounced.

Figure 15 presents the three-dimensional instantaneous vortex structures and corresponding two-dimensional planar vortex visualizations, obtained using the Q-criterion, for four distinct formation configurations: G = 0.25L, D = 1.2L; G = 0.25L, D = 1.5L; G = 0.5L, D = 1.2L; and G = 0.5L, D = 1.5L.

Key hydrodynamic observations include:

(1)

At a small lateral spacing (G = 0.25L): Variations in streamwise spacing (D) induce significant drag modulation on the fish bodies. Fish 4 experiences an 11% reduction in drag coefficient with a minimal change in thrust. This drag reduction is attributed to constructive wake capture, where the lateral proximity enables Fish 4 to exploit the low-pressure regions between Fish 2 and 3.

(2)

At a larger lateral spacing (G = 0.5L): Streamwise spacing variations have a notable effect on the thrust of the fish body. Drag remains nearly unchanged, while thrust exhibits pronounced sensitivity to D. The altered wake interaction dynamics result in enhanced vortex shedding synchronization between adjacent fish.

(3)

Physical mechanism analysis: The differential effects stem from spatial variation in wake impingement. At a small G, Fish 1’s wake predominantly interacts with the body regions of Fish 2 and 3, affecting their drag. At a larger G, the wake impinges on the caudal regions of Fish 2 and 3, directly influencing their thrust. Fish 4, positioned centrally between Fish 2 and 3, experiences amplified hydrodynamic loading with physical quantities approximately twice those of Fish 2 and 3.

Figure 16 presents the instantaneous pressure distribution within the flow field of the diamond formation at a representative instant. At a small lateral spacing, the high-pressure region at the head of Fish 4 is substantially diminished, coupled with a redistribution of the attached low-pressure region, while the caudal region exhibits no significant pressure enhancement. This results in a reduced pressure gradient across the fish body. In contrast, at a larger lateral spacing, the trailing fish experiences minimal hydrodynamic interference from the vortex shedding of the leading fish. As a result, the thrust and drag coefficients approach those of an isolated swimming fish, yet the energy expenditure increases and propulsive efficiency decreases.

3.2.2. Rectangular Formation

The lateral spacing (G) of the rectangular array is specified as 0.25L and 0.5L, and the streamwise spacing (D) is set to 1.2L and 1.5L, yielding four distinct spacing configurations in total. The thrust and drag forces of each fish within the rectangular array are presented in Figure 17. Hydrodynamic forces acting on the fish bodies are nondimensionalized, and the net hydrodynamic force on individual fish is approximately zero. The influence of hydrodynamic effects on variations in fish drag, thrust, hydrodynamic power, and swimming efficiency is illustrated in Figure 18.

The results demonstrate that in the rectangular array, the physical quantities of symmetrically positioned fish (Fish 1 and 3 and Fish 2 and 4) are essentially consistent. In particular, the variation amplitudes of drag and thrust for Fish 1 and 3 are highly coincident, confirming the consistency of physical quantities for fish at symmetric positions. Nevertheless, the swimming hydrodynamic power consumption of Fish 2 and 4 is considerably higher than that of the leading fish.

Figure 19 presents the three-dimensional instantaneous vortex structures and corresponding two-dimensional planar vortex visualizations, obtained using the Q-criterion (Q = 100), for four distinct configuration conditions of the rectangular formation: G = 0.25L, D = 1.2L; G = 0.25L, D = 1.5L; G = 0.5L, D = 1.2L; and G = 0.5L, D = 1.5L.

Key hydrodynamic observations include:

(1)

Vortex interaction mechanisms: Fish 2 and 4 effectively harness the energy from the vortex shedding of the leading fish, achieving simultaneous thrust enhancement and drag reduction. This constructive wake capture results in superior swimming performance compared to isolated swimming.

(2)

At a small lateral spacing (G = 0.25L): Variations in streamwise spacing (D) exert a significant influence on both drag and thrust coefficients. However, the increased mechanical power consumption leads to reduced propulsive efficiency despite favorable force modifications. This inefficiency stems from excessive turbulent dissipation in the tightly packed formation.

(3)

At a larger lateral spacing (G = 0.5L): Streamwise spacing variations have minimal impact on drag and thrust coefficients. Yet the optimized wake interaction substantially improves swimming efficiency. The trailing fish experience reduced hydrodynamic interference, allowing for more efficient energy utilization.

(4)

Wake structure analysis: When both lateral and streamwise spacings are large (G = 0.5L and D = 1.5L), a stable 2S (two-signature) vortex structure emerges in the wake. This organized vortex pattern is highly conducive to thrust generation through constructive vortex shedding synchronization. The 2S wake configuration represents an optimal hydrodynamic arrangement for energy-efficient collective swimming.

Figure 20 presents the instantaneous flow field pressure distributions for the four rectangular array configurations at the same time instant. In Figure 20a, at a small fish body spacing, the bound negative pressure in the midsections of Fish 2 and 4 is substantial and the bilateral negative pressure reaches equilibrium, which is unfavorable for thrust generation. As the streamwise spacing (D) increases, the positive pressure region at the fish heads is enhanced, the bound negative pressure region on the fish bodies is redistributed, and the positive pressure region at the caudal fins is also strengthened—thus elevating the pressure differential across the fish bodies, which facilitates more efficient forward propulsion. At a large fish body spacing, Fish 2 and 4 are subject to weaker influence from the wake shed vortices and can therefore maintain their swimming efficiency.

3.3. Discussion

Utilizing the identical geometric framework and kinematic parameters, this study establishes four biomimetic fish robot swarm configurations: tandem, parallel, diamond, and rectangular arrangements. Comprehensive numerical simulations are conducted to analyze the intrinsic hydrodynamic patterns and collective behaviors. The research findings demonstrate remarkable consistency with both classical fish schooling theories and contemporary experimental investigations. A systematic mechanistic analysis is performed through three critical perspectives: vortex dynamics, pressure field redistribution and energy consumption metrics.

3.3.1. Vortex Dynamics

The fundamental hydrodynamic mechanism of formation swimming is governed by the interaction between the wake vortex structures of leading and trailing fish, which can be categorized into three distinct types: constructive, destructive, and neutral interference. In the optimal configuration of the tandem three-fish formation with streamwise spacing (D = 1.3L), trailing Fish 2 and 3 are precisely positioned within the constructive interference zone of the leading fish’s wake vortices. The anterior vortices generated by the trailing fish undergo constructive merging with the leading fish’s wake vortices, resulting in simultaneous enhancement of thrust generation and a reduction in energy expenditure. This hydrodynamic phenomenon is in complete alignment with the classical mechanism of the reverse von Kármán vortex street, where trailing fish obtain propulsive gains through constructive interaction with the leading fish’s wake vortices [42]. Quantitatively, the enhancement magnitude of thrust and propulsive efficiency for trailing fish in this study is comparable in magnitude to the 12% improvement reported by Seo & Mittal (2022) [8] in their numerical simulations, providing robust validation of the reliability of these findings.

However, when the inter-fish spacing becomes excessively small—specifically, with a streamwise spacing below 1.3L or a lateral spacing below 0.3L—the trailing vortices of adjacent fish rapidly coalesce into chaotic vortex clusters. This phenomenon induces destructive interference, resulting in diminished thrust generation and elevated energy expenditure. This hydrodynamic principle is particularly pronounced in the diamond four-fish formation. Under the configuration of G = 0.5L and D = 1.2L, the complex interference patterns from multiple leading fish disrupt favorable pressure distribution, directly causing a substantial decline in swimming efficiency for Fish 4, positioned at the rearmost location.

Notably, in the large-spacing rectangular formation with G = 0.5L and D = 1.5L, a stable 2S-type vortex structure emerges. This organized flow pattern represents the hydrodynamic synchronization between fish body kinematics and vortex shedding processes. Such a coherent vortex configuration significantly enhances propulsive efficiency, aligning with Li.L’s (2020) [7] established principle that ‘individual fish achieve optimal vortex phase alignment through spatial positioning adjustments under fixed phase differences,’ thereby validating the energy-efficient mechanisms observed in this study.

3.3.2. Redistribution of the Pressure Field

The dynamic reconfiguration of the pressure field constitutes the primary mechanism governing performance variations within the fish formations, a phenomenon unequivocally elucidated through pressure contour analysis. In the optimal configurations of each formation, the positive pressure region at the anterior section of the fish is markedly amplified, concomitantly with an expansion of the positive pressure zone in the caudal fin region. This augmentation in pressure differential directly correlates with enhanced thrust generation. In contrast, when the inter-fish spacing is excessively reduced, pronounced adverse pressure gradients and negative pressure zones emerge between adjacent individuals. These regions attenuate the effective pressure differential across the fish body, thereby impeding thrust production. This hydrodynamic mechanism provides a comprehensive explanation for the observed performance degradation in Fish 4 within the diamond formation and in specific individuals under configurations characterized by minimal inter-fish spacing.

3.3.3. Swimming Efficiency

The fundamental hydrodynamic advantage of formation swimming resides not only in propulsive thrust augmentation but also in the enhancement of hydrodynamic efficiency, which is in accordance with the established principles of contemporary research on fish schooling hydrodynamics. Trailing fish can minimize energy expenditure while maintaining or even increasing propulsive velocity by strategically exploiting the wake vortex structures generated by leading fish. This energy-saving mechanism is consistently observed in Fish 2 and 3 of the three-fish tandem arrangement, as well as in Fish 2 and 4 of the four-fish rectangular formation. Notably, the hydrodynamic benefits demonstrate pronounced positional dependency across the formation: optimal positions (e.g., Fish 2 and 3 in the tandem formation and Fish 2 in the rectangular formation) experience constructive vortex interactions that enhance propulsive efficiency by 10–15%. Suboptimal positions (e.g., Fish 4 in the diamond formation with G = 0.5L and D = 1.2L) encounter turbulent vortex interference that increases energy costs by 20–30% to maintain hydrodynamic stability. A critical threshold exists at inter-fish spacings below 1.2L, where beneficial wake exploitation transitions to detrimental flow interference. This positional efficiency variation provides essential insight into the development of energy-efficient formation control strategies in bio-inspired underwater vehicle systems, highlighting the importance of precise spatial positioning to maximize collective hydrodynamic performance.

3.3.4. Future Work

This study provides a theoretical foundation for the design of biomimetic underwater robot formations, offering hydrodynamic optimization principles for selecting the most efficient formation configuration and inter-robot spacing based on collaborative mission requirements, thereby enabling energy-efficient resource allocation within a swarm.

Notable limitations persist in the current research framework: The investigation employs predefined undulatory kinematics with fixed phase differences, which fails to capture the adaptive kinematic modulation capabilities inherent in biological fish that dynamically adjust their swimming patterns in response to flow conditions. Existing studies are predominantly confined to laminar flow regimes (low-Reynolds number conditions), whereas practical underwater robot operations typically encounter turbulent flow regimes (high-Reynolds number conditions) characterized by complex vortex dynamics and flow instabilities. The current models do not adequately account for three-dimensional flow effects and boundary layer interactions that significantly influence hydrodynamic performance in real-world scenarios.

Future research directions should prioritize: the development of real-time adaptive control algorithms that utilize flow field perception to dynamically adjust undulation parameters (frequency, amplitude, and phase) in response to surrounding flow conditions; investigation of turbulent flow effects on formation swimming performance under high-Reynolds number conditions, with a particular focus on vortex shedding patterns and wake interference mechanisms; integration of machine learning techniques to enable robotic fish to autonomously optimize their positions and kinematics within the formation based on real-time hydrodynamic feedback; and experimental validation in large-scale aquatic environments that more accurately replicate the complex flow conditions encountered in natural underwater settings.

4. Conclusions

This study employs the sharp interface immersed boundary method based on virtual cell discretization to conduct high-fidelity numerical simulations, quantitatively analyzing the variations in thrust, drag, power consumption, and hydrodynamic efficiency of fish-like bodies within swarm configurations. The investigation integrates wake vortex evolution analysis with structural parameter variations to elucidate the underlying hydrodynamic mechanisms governing collective swimming performance.

The principal findings are summarized as follows:

(1)

Optimal configurations in three-fish swarms.

In the tandem schooling formation, at streamwise spacing (D) = 1.3L, trailing Fish 2 and 3 achieve peak hydrodynamic efficiency through constructive wake interference, where the leading fish’s vortex shedding pattern creates favorable flow conditions that simultaneously enhance thrust generation and reduce power expenditure. In the side-by-side schooling formation, with lateral spacing (G) = 0.3L, the central Fish 1 demonstrates optimal performance, maintaining low energy consumption while achieving higher thrust compared to isolated swimming, attributable to the symmetric flow field distribution that minimizes lateral disturbances.

(2)

Performance characteristics in four-fish formations.

When in the diamond formation (G = 0.5L and D = 1.2L), the rearmost Fish 4 experiences significant efficiency degradation due to destructive vortex interference from multiple leading fish, resulting in turbulent flow conditions that increase energy expenditure for posture stabilization. Then, in the state of the rectangular formation, trailing Fish 2 and 4 exhibit enhanced propulsive efficiency by strategically exploiting the leading fish’s wake vortices, particularly under the large-spacing configuration (G = 0.5L and D = 1.5L), where a stable 2S-type vortex street develops, creating favorable pressure gradients that optimize the thrust-to-drag ratio. The optimal spacing, configuration, and number of fish with the highest swimming efficiency for each formation studied in this article have been integrated and presented in

Table 1. The optimal spacing and configuration for each formation in this study.

Formation Type	Optimal Spacing Configuration	Individual Number
Tandem Schooling Formation	Streamwise spacing of 1.4L	Fish 2
Side-by-Side Schooling Formation	Lateral spacing of 0.25L	Fish 1
Diamond Formation	Streamwise spacing of 1.5L, Lateral spacing of 0.5L	Fish 4
Rectangular Formation	Streamwise spacing of 1.2L, Lateral spacing of 0.25L	Fish 4

.

(3)

Fundamental hydrodynamic mechanisms.

Performance optimization stems from the synergistic effect between vortex dynamics and pressure field redistribution, which is specifically manifested in the constructive vortex interference generating a coherent flow structure to enhance momentum transfer, as well as the good surface pressure distribution maximizing the effective pressure difference generated by thrust. These mechanisms are interrelated. Correspondingly, performance degradation is induced in turbulent flow regions where destructive vortex interference disrupts coherent momentum transfer, as well as during vortex shedding and flow separation events that cause excessive energy dissipation; additionally, unfavorable pressure distributions diminish the effective pressure differential for thrust generation. These findings provide a comprehensive hydrodynamic framework for understanding performance variations in fish swarm configurations, highlighting the pivotal role of vortex-mediated flow interactions in governing collective swimming efficiency.