Numerical Study on the Hydrodynamics of Fish Swimming with Different Morphologies in Oblique Flow

Abstract

In confined and intricate aquatic environments, fish frequently encounter the need to propel themselves under oblique flow conditions. This study employs a self-developed ghost-cell immersed boundary method coupled with GPU acceleration technology to numerically simulate the propulsion dynamics of flexible biomimetic fish swimming in oblique flow environments. This research scrutinizes diverse biomimetic fish fin morphologies, with particular emphasis on variations in the Strouhal number and angle of attack, to elucidate hydrodynamic performance and wake evolution. The results demonstrate that as the fin thickness increases, the propulsion efficiency decreases within the Strouhal number range of St = 0.2, 0.4. Conversely, within the range of St = 0.6 to 1.0, the efficiency variations stabilize. For all three fin morphologies, an increase in the Strouhal number significantly augmented both the lift-to-drag ratio and thrust, concomitant with a transition in the wake structure from smaller vortices to a larger alternating vortex shedding pattern. Furthermore, within the Strouhal number range of St = 0.2 to 0.4, the propulsion efficiency exhibits an increase, whereas in the range of St = 0.6 to 1.0, the propulsion efficiency stabilizes. As the angle of attack increases, the drag coefficient increases significantly, while the lift coefficient exhibits a diminishing rate of increase. An increased fin thickness adversely affects the hydrodynamic performance. However, this effect attenuates at higher Strouhal numbers. Conversely, variations in the angle of attack manifest a more pronounced effect on hydrodynamic performance. A thorough investigation and implementation of the hydrodynamic mechanisms demonstrated by swimming fish in complex flow environments enables the development of bio-inspired propulsion systems that not only accurately replicate natural swimming patterns, but also achieve superior locomotion performance and robust environmental adaptability.

1. Introduction

Fish locomotion, a ubiquitous phenomenon in nature, has inspired the development of biomimetic vehicles and underwater robots, owing to its exceptional swimming performance [1]. These biomimetic vehicles play a pivotal role in executing underwater tasks within complex flow environments, including monitoring submarine locations and developing marine energy resources. Such environments may encompass motion in uniform oblique currents or unsteady flows, which are conditions under which fish typically demonstrate reduced propulsion efficiency. Despite the prevalence of movement in oblique currents for both fish and biomimetic vehicles, our understanding of biomimetic locomotion in complex flow environments remains limited. Consequently, this study utilized numerical simulation methodologies to investigate the swimming behavior of biomimetic fish bodies exhibiting diverse morphological thickness variations in uniform oblique currents. The primary objectives were to elucidate the hydrodynamic mechanisms governing fish locomotion in complex flows and explore strategies for propulsion enhancement.

The hydrodynamic characteristics of fish locomotion have been extensively investigated using diverse methodologies encompassing both experimental research and numerical simulations. Initial research endeavors have predominantly focused on comparative experiments, exploring optimal motion states through the systematic modification of experimental parameters. Jia et al. [2] analyzed the swimming patterns of zebrafish larvae and Schizothorax prenanti under varying Reynolds number conditions. Wang et al. [3] performed experiments utilizing a mechanical stingray model, demonstrating that stingrays can generate sufficient thrust for forward propulsion. Their findings indicated that the linear swimming speed and impulsive force increased rapidly in correlation with the augmentation of the three driving parameters of the pectoral fins. Xue et al. [4] investigated the hydrodynamic performance of robotic fish swimming at inclined angles in confined spaces by employing an experimental apparatus comprising a solid fish-like robot, a hydrodynamic measurement platform, and a six-axis force sensor. Their findings showed that at specific pitch angles, the augmented difference between gravity and buoyancy compensates for the diminished force generated by the caudal fin during robotic fish swimming.

Extensive theoretical and numerical investigations have been conducted to elucidate the hydrodynamic characteristics of fish swimming in uniform flow. Wang [5] developed a theoretical estimation of biomimetic fish swimming characteristics by integrating a Newton–Euler method-based dynamic model with a central pattern generator network. Wei [6,7] conducted an examination of the transient and dynamic–steady propulsion performance of biomimetic fish, demonstrating that efficient propulsion necessitates elevated Strouhal and Reynolds numbers. Shua [8] and Zhang [9] investigated fish undulation behavior at higher Reynolds numbers and found that both lift and drag coefficients exhibit a decreasing trend with increasing Reynolds numbers, consequently resulting in an incremental effect on the propulsive force of fish motion. Macias [10] conducted a study on the hydrodynamic performance of two distinct fish morphologies, specifically tuna-like and lambari-like forms, by employing angular kinematics. Their findings revealed an association between the leading-edge vortices generated by the caudal fin and thrust peaks within the motion cycle. In the case of a two-dimensional undulating fish, thrust generation occurs when the conventional Kármán vortex street, initially aligned with the flow direction, undergoes a transition to a reverse configuration.

The hydrodynamic mechanisms underlying fish swimming have been extensively investigated from multiple perspectives, including three-dimensional effects, interactions between fish bodies and fins, and school formations. Fish fins serve as critical organs for locomotion and maintenance of body equilibrium in most fish species. Liu [11] elucidated that fish can generate high-performance propulsion through the utilization of complex interactions between fins and body. Xia [12,13] conducted computational analyses of the viscous flow field surrounding a three-dimensional biomimetic fish, revealing that the spanwise motion of the caudal fin enhances propulsive performance. Kopman [14] proposed a dynamic model for a robotic fish with a flexible caudal fin, integrating the vibration of an undulating compliant fin modeled using Kirchhoff’s equations of motion and the Euler–Bernoulli beam theory. Liu [15] and Ghommem [16] investigated the influence of Reynolds and Strouhal numbers on the propulsive performance of biomimetic undulating caudal fins. Their findings demonstrated that elevated Reynolds numbers result in higher thrust and propulsive efficiency, whereas increased Strouhal numbers lead to enhanced instantaneous thrust, but generally diminished or even deteriorated propulsive efficiency. More comprehensive reviews can be found in the literature [17,18,19,20,21,22]. In the context of fish school swimming, Deng [23] and Singh [24] observed that two fish arranged in tandem experience greater thrust than a single fish, with fish school swimming efficiency reaching its maximum at low wavelengths. Gao [25] and Li [26] elucidated that biomimetic fish bodies typically employ vortex phase matching as an energy conservation mechanism. Contemporary research methodologies involve recording fish motion using high-speed cameras, digitizing the data, and generalizing them by traveling waves from head to tail, ultimately representing the motion as Fourier coefficients using sine/cosine functions. Precise kinematic studies of diverse fish species, encompassing both steady and unsteady swimming conditions, have established a foundation for quantitative research on swimming mechanisms.

The existing literature predominantly focuses on simplified flow conditions, such as quiescent water or uniform flow aligned with the swimming direction. However, in natural environments fish may encounter steady oblique flows [27,28,29] or unsteady dynamic flows [30,31]. Yu et al. [32] proposed a novel sensor for determining the angle of attack through aquatic experiments facilitated by a global vision measurement system. Kang Ding [33] observed an inverse relationship between Reynolds numbers and fluid boundary layer thickness, accompanied by an increase in drag and lateral force coefficients. Shao and Li [34] conducted an investigation into the hydrodynamic performance and flow characteristics of a two-dimensional undulating foil subjected to oblique flow. Their study delineated three distinct flow regions: tail vortex shedding, head vortex shedding formation, and head vortex shedding. To quantitatively assess the influence of angle of attack on tuna swimming efficiency, Xia et al. [35] performed comprehensive numerical investigations on a three-dimensional tuna-like flexible model with prescribed kinematic parameters. Their numerical results demonstrated that optimal hydrodynamic performance, characterized by peak swimming velocity and propulsive efficiency, was achieved at caudal fin angles of attack ranging from 20° to 30°. Employing an integrated numerical–experimental approach, Matthews et al. [36] systematically investigated the hydrodynamic performance of a biomimetic robotic model under varying angles of attack. Their experimental observations revealed that elevated angles of attack resulted in decreased thrust efficiency, primarily attributed to flow separation and subsequent lift deterioration of the caudal fin. It is important to emphasize that the angle of attack generated in oblique flow conditions differs fundamentally from those reported by Xia et al. [35] and Matthews et al. [36]. In the present study, the angle of attack is induced by the oblique orientation of the fish body relative to the free stream, in contrast to the studies by Xia et al. [35] and Matthews et al. [36], where the angle of attack was governed by the oscillatory motion of the caudal fin.

Under identical kinematic conditions, airfoils of varying thicknesses exhibit significant variations in their hydrodynamic performance characteristics [37]. The airfoil thickness has been established as a fundamental parameter governing the lift-to-drag ratio. Sankarasubramanian [38] demonstrated through experimental analysis that the tangential force coefficient exhibited a non-monotonic relationship with airfoil thickness, manifesting as an initial increase followed by a subsequent decrease. Ahmed [39] demonstrated that optimal performance was achieved with a thickness-to-chord ratio of 0.8% at the trailing edge of the NACA4412 airfoil configuration. Experimental investigations have revealed that the relative thickness of airfoils substantially affects the stall onset angle of attack, stall severity, and dynamic stall model parameters under varying incidence angles [40]. In their investigation of thickness effects on fish swimming efficiency, Li et al. [41] reported that in uniform flow conditions, increasing airfoil thickness resulted in a sharp decrease in velocity, while propulsive efficiency displayed a non-monotonic trend, initially increasing before declining. Despite extensive research, the existing literature has predominantly focused on the aerodynamic implications of airfoil thickness, resulting in substantial knowledge gaps regarding hydrodynamic performance characteristics. There remains a notable absence of comprehensive studies examining the influence of key flow parameters in oblique flow conditions.

This study presents a numerical simulation of the propulsive motion of a biomimetic vehicle in oblique flow utilizing the ghost-cell immersed boundary method accelerated by GPU parallel computing technology. First, to investigate the underlying swimming mechanisms, detailed analyses were performed on time-averaged drag and lift forces, time-averaged tangential and normal forces, force profiles, spectral characteristics, and wake patterns under varying angles of attack, and geometric profiles. Second, the interrelationship between hydrodynamic performance and wake patterns was extensively examined across various angles of attack and airfoil configurations. It is known that the evolved fish shape includes social and genetic factors, reduced risk of predation, and advantages in predation, and is not solely due to hydrodynamic factors. This study intends to explore the optimal hydrodynamic performance of different geometric profiles. It can provide guidance for the design of advanced biomimetic propellers, which do not necessarily have a similarly direct relevance to ichthyology.

2. Problem Description and Numerical Model

2.1. Description of the Physical Problem

A biomimetic vehicle subjected to oblique flow can be conceptualized as an oscillating foil in a uniform flow oriented at an inclined angle relative to the flow direction. The body surface of the vehicle was characterized by NACA0024, NACA0018, and NACA0006 airfoil profiles. An oblique flow velocity U was imposed at the inlet boundary. The slip boundary conditions were implemented on the upper and lower walls of the computational domain. Fully developed free-stream boundary conditions were applied at the computational domain outlet. With respect to pressure, a Dirichlet boundary condition with zero reference pressure was prescribed at the outlet, whereas Neumann boundary conditions were imposed on the remaining walls. The aforementioned configuration is schematically shown in Figure 1.

The biomimetic fish body motion in this study was modeled through a traveling wave propagating from head to tail, utilizing NACA0024, NACA0018, and NACA0006 airfoil profiles. The chord length of the airfoils was considered the spine of the biomimetic fish. The undulatory motion amplitude along the fish body was modeled using a quadratic polynomial expressed as follows:

$$A(x)=C_0+C_{1}x+C_2{(x)}^2$$

(1)

where x represents the local axial coordinate from head to tail; A(x) denotes the amplitude envelope along the streamwise direction, with its specific control equation given in Equation (2); and C₀, C₁, and C₂ are constant coefficients [42]. From these data, we can derive A(0) = 0.02, A(0.2) = 0.01, and A(1) = 0.10. The lateral motion of the biomimetic fish body, simulating its undulatory movement, is defined as:

$$y(x,t)=A(x)\sin (kx-ct) 0\le x\le L$$

(2)

where, y represents the lateral coordinate relative to the centerline; the wave number k is defined as k = 2π/λ, where λ is the wavelength, and λ = L (L being the body length); t denotes time; and c is the phase velocity, expressed as c = 2πf, where f is the undulation frequency.

The Reynolds number (Re) and Strouhal number (St) are crucial flow parameters in the hydrodynamic characteristics of biomimetic fish propulsion. The Reynolds number was defined as Re = ρUL/μ, where μ is the dynamic viscosity of the fluid. The Strouhal number is defined as St = fL/U. In our study, the shape parameters of the biomimetic fish body were kept constant, whereas the Strouhal number was varied across different simulation cases by adjusting the undulation frequency. This approach allows for a systematic investigation of how propulsive performance and flow characteristics change with varying Strouhal numbers at a fixed Reynolds number.

This study investigated the dynamic characteristics and wake patterns resulting from the combined effects of the Strouhal number and angle of attack. In addition, it examines the relationship between different airfoil profiles and Strouhal numbers under oblique flow conditions.

Table 1. Experimental conditions for the locomotion of fish in inclined flow.

Test Condition	Wing Shape	Angle of Attack	Strouhal Number
Case#1	NACA0006	10°, 30°, 45°	0.2, 0.4, 0.6, 0.8, 1.0
Case#2	NACA0018	10°, 30°, 45°	0.2, 0.4, 0.6, 0.8, 1.0
Case#3	NACA0024	10°, 30°, 45°	0.2, 0.4, 0.6, 0.8, 1.0

presents the computational parameters for various test conditions.

2.2. Numerical Model

In this study, the GPU-accelerated ghost-cell immersed boundary method (GCIBM) [43,44] was employed to simulate the oblique flow around an undulating fish body. The boundary of the biomimetic fish was tracked using a Lagrangian approach, whereas the fluid flow was described using a Eulerian framework. The fluid dynamics are governed by two-dimensional, incompressible, laminar Navier–Stokes (N–S) equations. The N–S equations with immersed boundaries on a Cartesian grid are given in vector form as:

$$\nabla \cdot \mathit{u}=0$$

(3)

$${\displaystyle \frac{\mathsf{\partial }\mathit{u}}{\mathsf{\partial }t}}+\nabla \cdot (\mathit{u}\mathit{u})=-{\displaystyle \frac{1}{\rho }}\nabla p+{\displaystyle \frac{1}{\rho }}\nabla \cdot \left[\mu (\nabla \mathit{u}+\nabla {\mathit{u}}^T)\right]$$

(4)

$${\left.\mathit{u}\right|}_{\mathsf{\Gamma }}=V,{\left.{\displaystyle \frac{\mathsf{\partial }p}{\mathsf{\partial }\mathit{n}}}\right|}_{\mathsf{\Gamma }}=-\mathit{n}\cdot \mathsf{\alpha }$$

(5)

where, u represents the velocity vector in the grid system; p denotes the pressure field; ρ is the fluid density; μ is the dynamic viscosity of the fluid; ∇ is the Nabla operator, representing spatial derivatives; V and α are the velocity and acceleration vectors of the biomimetic fish body, respectively; Γ denotes the surface of the biomimetic fish body; and n is the unit normal vector pointing outward from the fish body surface.

The present numerical model is implemented based on a ghost-cell immersed boundary method (GCIBM) with GPU implementation. The present solver is programmed by a CUDA Fortran 90 language. For a detailed implementation, readers are referred to the studies of Xin et al. [45] and Shi et al. [44]. The governing Equations (3) and (4) were solved using an explicit finite difference method on a staggered Cartesian grid, accelerated by the GPU implementation. The TVD-MUSCL scheme proposed by van Leer was employed for the discretization of convective terms at the computational boundaries. Viscous terms were treated using a second-order central difference scheme. The no-slip boundary condition on the solid surface in Equation (5) states that the velocity of the fluid at the boundary of a solid object is equal to the velocity of the boundary itself. It is enforced through the current GCIBM where the local reconstruction is implemented on many ghost cells representing the solid boundary. This GCIBM [44,45] is advantageous for arbitrarily moving boundaries, because the grid generation is simple, and the numerical discretization is not affected by the geometric complexity. We employed a two-dimensional model primarily due to the significant computational demands and complexity associated with three-dimensional simulations. While we recognize that our two-dimensional approach may not capture all critical mechanisms of vortex dynamics, it allows us to focus on the fundamental flow characteristics relevant to biological swimming and flying. Furthermore, we wish to highlight that in both natural and engineering contexts, high aspect ratio airfoils are prevalent in many flying organisms and underwater propulsors. Our two-dimensional model is effective in describing the flow characteristics of these high aspect ratio configurations, which is a key aspect of our study.

To assess the flow characteristics, the instantaneous drag (C_D) and lift (C_L) coefficients were defined as follows [46]:

$$C_D={\displaystyle \frac{F_D}{\frac{1}{2}\rho U^{2}L}}, C_L={\displaystyle \frac{F_L}{\frac{1}{2}\rho U^{2}L}}$$

(6)

where the drag (F_D) and lift (F_L) forces are computed by integrating the pressure and viscous forces over the body surface using a bilinear interpolation method. The time-averaged drag ($C_D^{avg}$) and root-mean-square lift ($C_L^{rms}$) coefficients are calculated as follows:

$$C_D^{avg}={\displaystyle \frac{1}{nT}}{\displaystyle {\int }_0^{nT}{C}_D}dt, C_L^{rms}=\sqrt{{\displaystyle \frac{1}{nT}}{\displaystyle {\int }_0^{nT}{C}_L^{2}dt}}$$

(7)

where T represents the oscillation period of the biomimetic fish body, defined as T = 2πf; and n denotes the number of oscillation periods. The fluid forces F_D and F_L are defined within a global x − y coordinate system that aligns with the direction of the free stream. To facilitate the analysis of propulsion performance in oblique flows, a local T − N coordinate system is also introduced, where T indicates the direction of motion of the biomimetic fish body, and N represents the normal to the centerline, as illustrated in Figure 1. Additionally, fluid forces F_T and F_N, acting along the T and N directions, are introduced and expressed as follows:

$$F_T=F_D\cos \theta +F_L\sin \theta$$

(8)

$$F_N=F_L\cos \theta -F_D\sin \theta$$

(9)

In Equation (8), tangential force F_T represents the total force that propels or hinders the forward motion of the biomimetic fish body under the assumption that the numerical tether has been removed. A negative total force indicates thrust; conversely, a positive total force indicates drag. The normal force F_N contributes to negative work. The corresponding tangential drag coefficient (C_T) and normal lift coefficient (C_N) were obtained by substituting Equations (8) and (9) into Equation (6). The average tangential drag coefficient $C_T^{avg}$ and the mean square normal lift coefficient $C_N^{rms}$ were derived by substituting Equations (8) and (9) into Equation (7). These coefficients are expressed as:

$$C_T={\displaystyle \frac{F_D\cos \theta +F_L\sin \theta }{\frac{1}{2}\rho U^{2}L}}, C_N={\displaystyle \frac{F_L\cos \theta -F_D\sin \theta }{\frac{1}{2}\rho U^{2}L}}$$

(10)

$$C_T^{avg}={\displaystyle \frac{1}{nT}}{\displaystyle {\int }_0^{nT}{C}_T}dt, C_N^{rms}=\sqrt{{\displaystyle \frac{1}{nT}}{\displaystyle {\int }_0^{nT}{C}_N^{2}dt}}$$

(11)

The total power of oblique fish undulation can be decomposed into the power in x-direction (P_D) and y-direction (P_L), which can be expressed as:

$$P_L={\displaystyle {\int }_0^{nT}(F_D\cos \theta +F_L\sin \theta })/\sin \theta dt$$

(12)

$$P_D={\displaystyle {\int }_0^{nT}(-F_D}\sin \theta +F_L\cos \theta )/\cos \theta dt$$

(13)

Therefore, the total power can be obtained from P_T = P_L + P_D, where thrust contributes positive work and lateral forces contribute negative work. Consequently, the propulsion efficiency can be defined as:

$$\eta ={\displaystyle \frac{P_D}{P_T}}$$

(14)

2.3. Numerical Validation

The accuracy and capability of the current methodology have been comprehensively demonstrated in previous studies [44,45]. To further validate the model’s ability to handle flexible boundary flows, a simulation of uniform flow around an oscillating biomimetic fish body was conducted. The geometric profile and kinematic form of the biomimetic fish body are detailed in Section 2.1. In our work, the Reynolds number was set to Re = 500. The computational domain was configured as [−3L, 9L] × [−3L, 3L], with a non-uniform grid implemented in the region [−0.2L, 1.2L] × [−0.2L, 0.2L].

Table 2. Mesh information for convergence testing at St = 0.8.

Grid Notion	Grid Size	$\mathbf{\Delta }{\mathit{x}}_{\mathbf{min}}\mathbf{,}\mathbf{\Delta }{\mathit{y}}_{\mathbf{min}}$	${\mathit{C}}_{\mathit{L}}^{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{C}}_{\mathit{D}}^{\mathit{a}\mathit{v}\mathit{g}}$
G1	200 × 80	0.0065, 0.005	7.10	−0.35
G2	260 × 120	0.0065, 0.005	7.23	−0.39
G3	350 × 180	0.0037, 0.0022	7.29	−0.40
G4	440 × 260	0.0031, 0.0015	7.37	−0.41

presents the grid convergence study results, where $C_L^{max}$ represents the maximum lift coefficient, and $\Delta x_{\min },\Delta y_{\min }$ denote the minimum grid sizes in the x and y directions, respectively.

Table 2 shows the convergence of the force coefficients towards the values derived from the finest grid resolution. The maximum lift coefficient $C_L^{max}$ exhibited identical values for the two high-resolution grids. The average drag coefficient $C_D^{avg}$ demonstrated close concordance, with a relative error of 2.25%. These findings suggest the attainment of a grid convergence. Figure 2 depicts the temporal evolution of the lift and drag coefficients for the four grid systems at St = 0.8. The numerical results demonstrate satisfactory consistency with those reported by Khalid et al. [42]. The force coefficients obtained from the two higher-resolution grids exhibited a close correspondence with Khalid et al.’s findings. Specifically, in comparison to Khalid et al.’s results, the present study underestimated $C_L^{max}$ by approximately 4.2%, while overestimating the maximum drag coefficient by nearly 3%.

3. Numerical Results and Discussion

3.1. Effects of the Attack Angle

This section examines the influence of the angle of attack on the dynamic characteristics of the oscillating fish body and associated wake properties across various Strouhal numbers. The pertinent flow parameters are delineated in Case #1 in Table 1. A nonuniform Cartesian grid was employed, incorporating a consistently refined region encompassing the fish body, as depicted in Figure 3.

Initially, the dynamic characteristics of the oscillating fish body were examined. Figure 4 depicts the variations in the drag coefficient (C_D) and lift coefficient (C_L) at three distinct angles of attack across a range of Strouhal numbers. The drag coefficient exhibited a monotonic increase with respect to the angle of attack, accompanied by an accelerating rate of increase. Conversely, the lift coefficient increased at lower angles of attack. At St = 0.2 and 0.4, the force coefficient trends displayed minimal variation across the range of angles of attack. Throughout the oscillation cycle, the horizontal force consistently retained its drag configuration. With an increase in the Strouhal number, the amplitudes of both the horizontal and vertical forces exhibited significant augmentation. At St = 0.6, 0.8, and 1.0, the force coefficient oscillation patterns deviated marginally from a purely sinusoidal mode. As the angle of attack increased, a diminution in the amplitude of thrust-type regions was observed, concomitant with an expansion in the duration and magnitude of strokes within drag-type regions.

To investigate the dynamic transition characteristics, Figure 5, Figure 6 and Figure 7 illustrate the temporal evolution (a) and Fast Fourier Transform (FFT) spectra (b) of the tangential (left) and normal force (right) coefficients for a fish body with the NACA0006 airfoil at angles of attack (AOA) of 10°, 30°, and 45° under three Strouhal numbers. At an AOA of 10°, the temporal evolution of the normal force coefficient (C_N) demonstrated an approximately sinusoidal behavior, which can be attributed to the alternating vortex shedding in the wake of the fish tail. These observations were consistent with the validation results obtained at AOA = 0°. As illustrated in the FFT of C_N in Figure 5b, the dominant peak occurred at f = f₀, accompanied by a minimal secondary peak at f = 2f₀ when St = 0.2, where f₀ represents the oscillation frequency of the biomimetic fish body. The tangential force coefficient (C_T) exhibits period-doubling characteristics, with variations repeated every two oscillation cycles at different amplitudes. This inter-cycle amplitude disparity is significant at lower Strouhal numbers. The FFT spectrum of the C_T revealed harmonic components at both f₀ and 2f₀. At St = 0.2, two strong harmonic components of comparable magnitudes were observed, resulting in significant amplitude differences. However, at St = 0.6 and 1.0, the secondary peak at 2f₀ became dominant, substantially exceeding the secondary peak at f₀. This behavior may be attributed to the asymmetric flow induced by the nonzero angle of attack, and the primary peak of C_T at 2f₀ correlates with the alternating vortex shedding from the oscillating fish. The fundamental frequency of C_T is twice that of C_N because a tangential force is symmetrically generated when the tail fin oscillates through the centerline of the fish. This frequency-doubling phenomenon has been extensively documented in airfoil flapping [47] and vortex-induced vibration studies [48].

At an angle of attack of 30°, the tangential force coefficient demonstrated regular triple-periodic oscillations at St = 0.2, primarily owing to modulation effects. Figure 6a illustrates that within each triple period, a double-periodic oscillation with a plateau was followed by a minor peak in the tangential force coefficient, whereas the normal force coefficient exhibited three peaks of distinct amplitudes, as evidenced by the amplitude spectra in Figure 6b. When the Strouhal number was increased to St = 0.6 and 1.0, significant frequency modulation occurred in the tangential force coefficient, characterized by alternating major and minor peaks within each period. In contrast, the normal force coefficient exhibited quasiperiodic variations with different amplitudes. The amplitude spectra revealed that the tangential force coefficient was dominated by both the fundamental frequency f₀ and the secondary frequency 2f₀. The normal force coefficient was characterized by a large-amplitude component at the fundamental frequency f₀ and a small-amplitude component at the secondary frequency 2f₀. The fundamental frequency originates from the fish body oscillation, whereas the secondary frequency component results from the interaction between the leading-edge and trailing-edge vortices.

At an elevated angle of attack (AOA = 45°), as illustrated in Figure 7a, both the tangential and normal force coefficients exhibited multiperiodic characteristics owing to the frequency modulation. At St = 0.2, the tangential force coefficient manifested triple-periodic oscillations characterized by two local peaks and one relatively flat trough, whereas the normal force coefficient demonstrated a double-periodic pattern. These characteristics were clearly demonstrated in the amplitude spectra of the force coefficients. The amplitude spectra of both C_T and C_N revealed two primary peaks with several secondary frequency peaks of lower magnitude, indicating the dominance of flow patterns induced by the interaction between the leading-edge and trailing-edge vortices. At St = 0.6 and 1.0, multiple harmonic components were observed in the C_T spectra at frequencies f₀ and 2f₀. Furthermore, the amplitudes of the secondary frequencies attained magnitudes comparable to the fundamental frequency, resulting in a beat pattern, whereas the C_N spectra were predominantly characterized by the fundamental frequency f₀ and secondary frequency 2f₀. The results indicate that at high angles of attack, the peaks at the fundamental frequency, induced by the nonzero angle of attack, become more pronounced.

To quantitatively assess the swimming efficiency η of the biomimetic fish body, Figure 8 illustrates the relationship between the swimming efficiency η and St for three different airfoils at various angles of attack. The efficiency η of the fish decreased with increasing Strouhal number across all three angles of attack at $St\ge 0.4$. Notably, the variation in efficiency was more pronounced at smaller angles of attack and Strouhal numbers. As the angle of attack increased, the rate of efficiency growth at a lower St decreased. However, at an AOA of 45°, the swimming efficiency η gradually decreased with increasing St, indicating the absence of a critical Strouhal number, as the biomimetic fish could swim forward at any St.

To elucidate the physical mechanisms of fish swimming in oblique flow, Figure 9, Figure 10 and Figure 11 depict the instantaneous vorticity (left) and pressure profiles (right) of the NACA0006 fish body at angles of attack of 10°, 30°, and 45° for three Strouhal numbers.

For AOA = 10°, a classic von Kármán vortex street with extensive horizontal and lateral distances was observed at St = 0.2. The vortex street propagated along the central axis of the fish body and was oriented at an angle relative to the flow direction. The head region of the fish body experiences relatively high pressure, albeit modest, compared to the majority of the surface area, which is enveloped by lower pressures primarily concentrated at the head and tail. As the Strouhal number increased to St = 0.6, the von Kármán vortex street transitioned to a reversed configuration with alternating negative and positive vortices in the upper and lower rows, respectively, as illustrated in Figure 10. The shed vortices underwent a transformation from a crescent shape proximal to the foil to a compact triangular configuration, subsequently forming downstream vortex strips. Moreover, the mean flow velocity posterior to the fish increased substantially, thereby generating a jet along the central axis of the fish. This phenomenon results in a region of excess momentum posterior to the fish, facilitating the transition from drag to thrust, as shown in Figure 4. At St = 1.0, the shed vortices exhibited increased intensity and compactness, further augmenting the jet posterior to the fish. The enhanced pressure differential results in diminished spacing of the vortex street. Furthermore, a more robust and elongated jet formed along the central axis, further amplifying the thrust, as illustrated in Figure 4.

As the angle of attack increased to AOA = 30°, Figure 10 illustrates that the rapidly forming vortex street exhibited gradual dissipation, and the vorticity intensity diminished as it extended beyond 5L at St = 0.2. The positive vorticity generated at the head was weakened and dissipated. As the Strouhal number increased, the lateral separation between the positive and negative vorticity in the near-wake region decreased. At St = 0.6, the head-shed vorticity was influenced by the negative vorticity, initiating tail vortex bifurcation. The upper row coupled with negative vorticity, whereas the lower row bifurcated, ultimately coalescing into a large vortex downstream. At St = 1.0, this complex interaction intensified; high-intensity vortices shed approximately 4L posterior to the fish gradually coalesced with the positive vorticity from the surface of the fish, forming a large vortex. Concurrently, the negative vorticity from the upper surface bifurcated into two components of varying intensity, with the higher-intensity upper row coupling with the bifurcated positive vorticity, resulting in a reverse von Kármán vortex street pattern that generates the thrust. With respect to pressure, the extended low-pressure region interacted with the pressure generated by the tail vortices, collectively influencing the fish body. As the Strouhal number increased, the pressures experienced on the upper and lower surfaces of the fish body increased substantially.

At a larger angle of attack, AOA = 45°, the vorticity pattern and force characteristics at St = 0.2 resembled those at AOA = 30°, exhibiting alternating shedding vortices with increased spacing and a threefold periodic variation in force coefficients. The vortex street exhibited a more pronounced downward deflection. At St = 0.6 and 1.0, the shed vortices generated by the oscillating fish coalesced with the leading edge and trailing edge vortices, augmenting the vortex intensity. Simultaneously, smaller vortices shed from the trailing edge progressively dissipated along the central axis of the foil. With an increase in the Strouhal number, vortex shedding was delayed, leading to an extended shedding period between the two sides of the foil. Two substantial single vortices with extensive horizontal separation were observed, constituting the 2S model, as illustrated in Figure 11. Despite the evident thrust generation across various Strouhal numbers, no jet flow induced by the oscillating fish foil was observed. This phenomenon is attributed to the predominant interaction between the leading edge and trailing-edge vortices, wherein the shed vortices generated by the oscillating fish coalesce with the trailing-edge vortices, resulting in thrust formation.

3.2. Effects of the Wing Shape

To investigate the influence of the airfoil, this study simulated free flow around a flapping fish body at angles of attack of 10°, 30°, and 45°. The experimental conditions encompassed three airfoil profiles: NACA0006, NACA0018, and NACA0024. Section 2.1 provides a comprehensive description of the boundary conditions and the computational grid.

Figure 12 depicts the influence of the Strouhal number on the mean drag coefficient ($C_D^{avg}$) and root mean square lift coefficient ($C_L^{rms}$) for the three airfoil profiles at various angles of attack. The results demonstrate consistent trends in $C_D^{avg}$ and $C_L^{rms}$ across different airfoil profiles. For small angles of attack, $C_L^{rms}$ decreased with an increasing Strouhal number, whereas at an AOA of 45°, $C_D^{avg}$ exhibited a positive correlation with the Strouhal number. Furthermore, $C_L^{rms}$ demonstrated a positive correlation with the Strouhal number.

Moreover, as the airfoil thickness decreased, variations in $C_D^{avg}$ became more pronounced with increasing Strouhal numbers, which is attributable to the enhanced pressure differential generated across the upper and lower surfaces of thinner airfoils. The Strouhal number exerts a significant influence on $C_D^{avg}$ and $C_L^{rms}$, suggesting a substantial impact on the hydrodynamic performance of fish during locomotion, whereas the effect of airfoil thickness is comparatively minor.

At small angles of attack, a critical Strouhal number was observed, corresponding to the point at which the net force on the fish body became zero. A negative correlation was observed between the airfoil thickness and critical Strouhal number. Exceeding the critical Strouhal number results in negative $C_D^{avg}$ values, which are indicative of thrust generation.

The temporal evolution and amplitude spectra of tangential and normal coefficients for fish bodies equipped with NACA0018 and NACA0024 airfoils are illustrated in Figure 13 and Figure 14, analyzed at an AOA of 30° across the three Strouhal numbers. For St = 0.2, the NACA0006 configuration demonstrated a characteristic triple-period pattern, comprising a double-periodic oscillation with a plateau phase followed by a minor peak in the tangential force coefficient. The NACA0018 configuration exhibited pure double-periodic oscillation, whereas the NACA0024 profile exhibited double-periodic behavior with an additional secondary peak in the tangential force coefficient. An analysis of the amplitude spectra revealed that the normal force coefficients for all three airfoil configurations exhibited distinct trimodal peaks with varying magnitudes. For higher Strouhal numbers St = 0.6 and 1.0, the C_T spectrum of NACA0018 exhibited a dominant 2f₀ frequency component, with peak amplitudes substantially exceeding those of the secondary f₀ component. Conversely, the NACA0006 and NACA0024 configurations demonstrated strong harmonic modulation between the f₀ and 2f₀ frequencies. This behavior is attributed to the periodic interaction between the leading-edge and trailing-edge vortex shedding in the oscillating fish body, a mechanism that will be examined in detail in subsequent sections.

Figure 15 illustrates the relationship between the propulsion efficiency η of biomimetic fish bodies with varying airfoil profiles and the Strouhal number at three distinct attack angles. At lower Strouhal numbers, the variations in propulsion efficiency across all attack angles were more pronounced. As St increased, the efficiency trends for the three airfoil profiles converged. The data reveal an inverse relationship between the airfoil thickness and propulsion efficiency of the biomimetic fish body. This phenomenon can be attributed to the increased surface area perpendicular to the lateral forces as the thickness increased, resulting in greater lateral forces, while the vertical dimension remained constant. The biomimetic fish body incorporating the NACA0006 airfoil profile exhibited the highest propulsion efficiency.

The hydrodynamic interactions of the oscillating fish body were investigated through instantaneous vorticity and pressure contours, as illustrated in Figure 16 and Figure 17, for NACA0018 and NACA0024 airfoils at angles of attack of 30° across three Strouhal numbers. Figure 11 demonstrates that at AOA = 30°, both leading-edge and trailing-edge vortices of the thinner profile remained attached to the fish surface during swimming and subsequently merged in the tail region, thus suppressing oscillation-induced vortex shedding. As the airfoil thickness increased, partial leading-edge vortices began to detach and ceased interacting with the trailing-edge vortices. This phenomenon resulted in decreased forward propulsive force, while simultaneously the detached vortices formed a reverse Kármán vortex street and jet flow. The length of these structures increased proportionally with airfoil thickness, as evidenced in Figure 16 and Figure 17. At St = 1.0, the shed vortices behind the NACA0018 fish body failed to remerge into a larger vortex, thereby reducing the forward propulsion efficiency.

4. Conclusions

This study employed numerical simulations of an undulating fish-like body at attack angles of 10°, 30°, and 45° using a GPU-accelerated ghost-cell method within a high-performance GPU cluster parallel computing framework. Validation of the numerical simulations was achieved through the analysis of an NACA0012 hydrofoil in uniform flow, demonstrating satisfactory grid convergence. The investigation examined the influence of the angle of attack, Strouhal number, and airfoil profile on the hydrodynamic performance and wake patterns. Through a comprehensive analysis of the oblique flows and biomimetic geometric shape of the biomimetic fish, this study advances our understanding of propulsion mechanisms governing biomimetic fish locomotion. Also, this investigation provides guidance for the design of advanced biomimetic propellers. In the study, the following conclusions are yielded.

With an increasing angle of attack, the mean drag coefficient demonstrated a monotonic progression from negative to positive values, particularly pronounced at elevated Strouhal numbers. The propulsive performance evaluation revealed a critical St, defined as the point at which the net force on the swimmer vanishes. The magnitude of St exhibited an inverse relationship with increasing angles of attack. Elevated Strouhal numbers yielded enhanced thrust generation, although this was accompanied by a proportionally increased energy expenditure. The wake topology underwent a transformation from an aligned von Kármán vortex street along the body’s central axis to its inverse configuration. Higher angles of attack facilitated enhanced thrust production while simultaneously reducing the energy consumption. The force coefficients underwent a transition from quasi-periodic variations to triple-periodic and multi-periodic oscillatory modes. At an AOA of 30°, spatiotemporal multiscale vortical structures emerged, which are attributed to the coupled effects of body oscillation and leading-edge/trailing-edge vortex interactions. At an AOA of 45°, the flow field manifested a 2S vortex shedding mode, dominated by the dynamic interaction between the leading-edge and trailing-edge vortical structures.

The influence of the airfoil profile on the force coefficients was substantially less pronounced than that of the Strouhal number. With increasing airfoil thickness, C_D and C_L exhibited slight variations, with C_D initially increasing and subsequently decreasing. The influence of the airfoil became more significant at higher Strouhal numbers. Additionally, the propulsion efficiency η of the fish exhibited an inverse relationship with airfoil thickness. Irregular patterns of the force coefficients were observed for the NACA0024 airfoil profile. Moreover, the vortex pairing patterns and deflections were more pronounced at high Reynolds and Strouhal numbers. For the NACA0006 configuration, the leading-edge and trailing-edge vortices maintained surface attachment during the swimming motion and subsequently coalesced in the wake region, thereby inhibiting oscillation-induced vortex shedding. Progressive increases in airfoil thickness induced partial separation of the leading-edge vortices, precluding their interaction with trailing-edge vortices. This phenomenon resulted in the attenuation of the forward propulsive force. The separated the leading-edge vortices, interacting with trailing-edge vortices, facilitated the formation of an inverse von Kármán vortex street and the corresponding jet flow. The spatial extent of both the vortex street and jet flow exhibited a linear correlation with the airfoil thickness, culminating in enhanced forward propulsion.

This study investigated the two-dimensional propulsion characteristics of a biomimetic fish. However, it is important to note that three-dimensional effects may significantly influence the dynamic and wake characteristics. Future work will focus on implementing high-fidelity three-dimensional numerical simulations to investigate the complex flow phenomena associated with both single-body and multi-body fish configurations.