Study on the Hydrodynamics of a Cownose Ray’s Flapping Pectoral Fin Model near the Ground

Abstract

Cownose rays typically swim close to the ocean floor, and the nearby substrate inevitably influences their swimming performance. In this research, we numerically investigate the propulsive capability of cownose rays swimming near the ground by resolving three-dimensional viscous unsteady Navier–Stokes equations. The ground effect generally has a favorable impact on swimming. The thrust and lift increase as the near-substrate distance decreases. Nevertheless, a body length is the recommended distance from the ground, at which the flapping efficiency and swimming stability obtain a good trade-off. The increase in lift is due to the pressure difference between the dorsal and ventral surfaces of the ray, and the thrust boost is due to the enhanced shear vortex at the fin’s leading edge when swimming near the substrate. Our results indicate that the ground effect is more noticeable when the fin flaps are symmetrical compared to asymmetrical. In asymmetric flapping, the hydrodynamic performance improves at a smaller value than the half-amplitude ratio (HAR). The frequency of flapping also significantly affects swimming performance. We find that a superior flapping frequency, at which maximum efficiency is reached, occurs when flapping close to the substrate, and this superior frequency is consistent with the behavior of our model’s biological counterpart.

1. Introduction

The ocean, as the largest geographical unit on Earth, possesses rich natural resources and hydrological characteristics that merit further exploration [1]. To explore the ocean deeply, humans design and manufacture autonomous underwater vehicles (AUVs) for underwater research. AUVs have broad applications and great potential value in research on the marine environment, aquatic mineral exploration, analysis of oceanic biology, and military fields [2]. Nevertheless, the screw thrusters used by AUVs for propulsion produce much noise; thus, they inevitably cause disturbance to the marine ecology. In comparison, fish swimming underwater are quiet and efficient, and they provide ample prototypes for humans to design high-performance bio-inspired underwater vehicles/robots. For this reason, research on aquatic animal locomotion has attracted much attention.

The swimming modes of fish are categorized into two types: Body Caudal Fin (BCF) and median and/or paired fin (MPF) [3]. Research on the BCF swimming mode can be found in the literature [4,5,6,7]. In comparison, the MPF swimming mode, represented by the cownose ray, has also become a research hotspot recently. As a member of the ray family, the cownose ray has a flat and streamlined, flexible appearance. It also achieves excellent maneuverability by flapping the paired pectoral fins and can even spin at null speed [8]. High maneuverability is achieved through the flapping of the pectoral fins, including spanwise, chordwise, and through twist deformation. The muscle controlling the flexible fin rays achieves this deformation. The cownose ray can achieve high swimming efficiency by combining gliding and fin-flapping to achieve long-distance cruising. As for the depth range of cownose rays, they can go across all depths of the ocean. They swim near the surface and even leap out of the water [9], as shown in Figure 1. They also swim in groups with fish in deep water.

A deep understanding of the hydrodynamic mechanism of cownose ray swimming is the key to developing high-performance biomimetic underwater vehicles. Cownose rays have benthic behavior and have been frequently seen to swim near substrates [10]. Because cownose rays’ skin color is close to that of the seabed, they can hide easily, as shown in Figure 1a. Webb [11] found that flatfish have a high acceleration rate when they start from the ground. The ground effect enhances the fast-start performance of flatfish. They also found that cownose rays often exhibit high speeds when gliding near the ground, which is influenced by the ground effect [12]. Consequently, the ground effect significantly enhances the hydrodynamics of rays when they swim close to the substrate compared to deeper water.

In addition to boosting the swimming of rays, the ground effect also significantly affects predation. Nauwelaerts et al. [13] found that feeding near the substrate extends the distance over which suction is effective, and a predator strike can be effective further from the prey. Thus, ground effects can extend prey efficiency.

Research in this area started with experimental studies of flapping plates near the sea floor. For example, Fernández-Prats et al. [14] utilized the DPIV and waterproof force sensors to measure the flexible flapping of an undulating foil. When swimming close to the ground, they found the foil increased both the speed and the thrust. To further research the generation of thrust when swimming using a pectoral fin, Piskur [15] used the PIV method to investigate a bionic robot with an MPF flapping mode. They explained the main aspects affecting thrust generation in their research. Nevertheless, some research based on numerical simulations indicated the ground effect is not beneficial for swimming. For instance, Blevins and Lauder [16] discovered that an undulating fin did not significantly increase swimming speeds near the substrate. More thrust was needed for each flapping period. A propulsion system with additional joints was designed by Piskur et al. [17]. In their swimming experiments, equipment at different flapping frequencies revealed that low-speed swimming has a high swimming efficiency.

In several numerical simulations, some researchers have indicated that foil flapping near the substrate has favorable effects [18,19]. An adverse influence of the ground effect was also found by Shi et al. [20]. In their study, the thrust and lift coefficients decreased as the distance decreased, and propulsive efficiency remained less influenced by the ground effect.

The above numerical simulations are self-propelled. The model is fixed in place, and the uniform flow flows to the model to imitate the swimming of fish. This simulation method does not faithfully illustrate the swimming of real fish. In comparison, the simulation of self-propelled swimming close to the substrate is more similar to actual fish behavior. Dai et al. [21] simulated the self-propulsion of 3D foil near the substrate. Flapping heights and rigidity were found to influence the shed vortex. The distance between the adjacent vortexes in the near-substrate zone becomes larger and the ground effect yields asymmetric shed vortices. Zhao et al. [22] simulated a self-propelled flexible plate’s shedding vortex when flapping near the substrate and found that it gradually rose as the plate moved further away from the ground. The hydrodynamics of the flapping wing improved as the near-ground height decreased.

The above studies are based on flapping plates near the ground whose shape and movement are different from those of real fish. The aforementioned MPF propulsion mode has some special kinematic features that may not be replicated by a simplified foil or plate model. Research on the swimming performance of MPF-based swimmers has been reported in the literature. For example, Ren and Yu [23] simplified the 3D stingray model as a 2D model and extracted the 3D pectoral fin kinematics equation to calculate the hydrodynamic properties of flapping near the substrate. Their results revealed that producing more trailing-edge vortex enhances thrust production but results in a negative lift near the substrate. Yi [24] found that increasing the flutter frequency and amplitude results in a higher swimming speed, while the propulsion efficiency does not improve. Similar conclusions were drawn by other researchers [23,25].

The flapping of two-dimensional fins near the ground may not apply to three-dimensional undulating pectoral fins. The sectional profile of pectoral fins in benthic hydrophones continuously varies along both the spanwise and chordwise directions. As a result, the vortex structure between the three-dimensional model and the substrate becomes increasingly complex when swimming near the ground, and the impact of near-ground effects on swimming is uncertain.

A three-dimensional laser scanner used to precisely produce models of live fish helps researchers obtain the true-to-life numerical geometry of real fish. For example, a high-fidelity numerical model of MPF-based fish-based swimming near the ground was reported by Su et al. [26], who used a commercial fluid code to calculate the hydrodynamics of a self-propelled 3D stingray. They found that the average swimming speed at a certain distance from the sea ground was about 3% higher and the velocity fluctuation amplitude was about 30% lower than those achieved while swimming far from the ground. Moreover, stingrays swimming near the ocean floor have improved flapping efficiency. This improvement is attributed to the decrease in the shed vortex as the height decreases, which stabilizes the stingrays during flapping.

While some researchers have calculated the hydrodynamics and vortex structure of underwater creatures when flapping close to the seafloor, recent studies have overlooked the asymmetric flapping of rays. Therefore, the interaction between rays’ asymmetric flutter and the seafloor needs further research. This paper hopes to find the impacts of flapping frequencies and spatially asymmetric flapping on the hydrodynamic performance of rays when swimming close to the seafloor. This study on rays flapping near the ground also seeks to provide valuable insights for path planning and efficient long-distance cruising for bio-rays, particularly when the bio-rays swim close to the ground.

2. Problem Description

In the present work, a cownose ray model acquired from a real Rhinoptera javanicas swimming in freestream [27] and a coordinate system established during cownose ray swimming are shown in Figure 2. The body length of the cownose is defined as BL, and the span of the cownose’s pectoral fin is denoted by SL.

Two principal motions of the pectoral fins, namely, spanwise deformation and chordwise traveling, are observed during cownose ray swimming. The number of waves in chordwise (W_n) is related to the production of thrust, which is defined as W_n = BL/λ, where λ is the chordwise travel wavelength. It describes the extent of torsional deformation during the pectoral fin flap. The maximum flapping distances reached by the pectoral fins upward and downward of the horizontal plane are defined as A_up and A_down. The ratio between A_up and A_down is defined as the half-amplitude ratio (HAR). When flapping up and down, the distal of the pectoral fin amplitude is defined as A (A = A_up + A_down). A_r and A_l denote the displacement on the right and left sides of the pectoral fin, respectively.

Previous research showed the kinetics of ray flapping in deep water and focused on the deformation of the pectoral fins of rays in both directions during swimming. Then, we observed cownose rays swimming near the substrate. Cownose ray flapping near the ground is consistent with swimming in deep water; therefore, we established Equations (1) and (2).

For the left pectoral fin with y₀ ≤ 0,

$$\begin{array}{l}x\left(x_0,y_0,t\right)=x_0,\\ y\left(x_0,y_0,t\right)=-\left(-y_0\right)\left(1-\left(1-k_l\right)\left|{\theta }_l\left(x_0,t\right)\right|\left(-y_0\right)/BL\right)\cos \left[{\theta }_{\max ,l}\left(-y_0\right)/BL\cdot s_l\left(x_0,t\right)\right],\\ z\left(x_0,y_0,t\right)=z_0+\left(-y_0\right)\left(1-\left(1-k_l\right)\left|{\theta }_l\left(x_0,t\right)\right|\left(-y_0\right)/BL\right)\sin \left[{\theta }_{\max ,l}\left(-y_0\right)/BL\cdot s_l\left(x_0,t\right)\right],\\ s_l\left(x_0,t\right)=\left[\sin \left(2\pi f_{l}t-2\pi W_{n,l}{x}_0/BL\right)+C_s\right]/(1+\left|C_s\right|).\end{array}$$

(1)

For the right pectoral fin with y₀ > 0,

$$\begin{array}{l}x\left(x_0,y_0,t\right)=x_0,\\ y\left(x_0,y_0,t\right)=y_0\left(1-\left(1-k_r\right)\left|{\theta }_r\left(x_0,t\right)\right|\left(-y_0\right)/BL\right)\cos \left[{\theta }_{\max ,r}{y}_0/BL\cdot s_r\left(x_0,t\right)\right],\\ z\left(x_0,y_0,t\right)=z_0+y_0\left(1-\left(1-k_r\right)\left|{\theta }_r\left(x_0,t\right)\right|\left(-y_0\right)/BL\right)\sin \left[{\theta }_{\max ,r}{y}_0/BL\cdot s_r\left(x_0,t\right)\right],\\ s_r\left(x_0,t\right)=\left[\sin \left(2\pi f_{r}t-2\pi W_{n,r}{x}_0/BL+\phi \right)+C_s\right]/(1+\left|C_s\right|).\end{array}$$

(2)

x₀, y₀, and z₀ are the positions of each point on the cownose ray, and the points are substituted into the flapping equation to obtain the flapping trajectory of the pectoral fin in a given period of time. C_s is the spatial asymmetry describing the pectoral fin’s upward and downward flaps. φ defines the phase difference between the two sides of the pectoral fins. θ_max and k represent the spanwise deformation capacity of the pectoral fins. The periodicity and asymmetry of the flapping pectoral fin are defined by s_r in the motion equation. The pectoral fin’s dimensionless flutter frequency f* is shown in

$$f^{*}=f\cdot BL/U,$$

(3)

where f is the actual flutter frequency of the ray swimming near the ground. U is the inflow velocity of the flow field computing domain.

The Reynolds number is defined as

$$Re=U\cdot BL/\nu ,$$

(4)

where ν is fluid kinematic viscosity.

The near-ground height $\stackrel{\mathrm{-}}{H}$ is defined as

$$\overline{H}=h/BL,$$

(5)

where h is the height between the origin of the ray’s coordinates and the seafloor.

The thrust and lift coefficient (C_T, C_L) are in the positive directions of the x and z axes, which are defined as

$$C_T(t)={\displaystyle \frac{F_x(t)}{0.5\rho U^2\cdot B{L}^2}},$$

(6)

$$C_L(t)={\displaystyle \frac{F_z(t)}{0.5\rho U^2\cdot B{L}^2}},$$

(7)

where F_x(t) and F_z(t) are created by the swimming ray. The fluid density, ρ, corresponds to water.

The total power coefficient (C_total) is defined as

$$C_{total}={\displaystyle \frac{P_{total}(t)}{0.5\rho U^3\cdot B{L}^2}},$$

(8)

where P_total(t) is the instantaneous power consumption of the cownose ray while it is flapping near the ground.

η is the efficiency of ray swimming near the ground, which means the proportion of thrust in the total power, expressed by

$$\eta ={\displaystyle \frac{\overline{C_T}}{\overline{C_{total}}}}.$$

(9)

We calculated the efficiency of ray swimming near the substrate using this equation. In the following section, we will explain the numerical approach to cownose ray swimming near the substrate.

3. Numerical Method and Self-Consistency Study

3.1. Numerical Method

The in-house fluid solver uses the URANS equation as the governing equation [28]. The density of rays is close to that of water; thus, the buoyant force generated by rays is equal to their weight. By neglecting the weight of the ray in the numerical simulation, the governing equation is

$${\displaystyle \frac{\partial }{\partial t}}{\displaystyle {\iiint }_{\Phi }\mathit{C}}d\Phi +{\displaystyle {\iint }_S(\mathit{W}-\mathit{U})\cdot \mathit{n}\mathbf{d}S}=0,$$

(10)

where Φ represents the fluid domain enclosed by boundary S, with n denoting the unit outward normal vector on the surface. The vector W signifies the convective flux, while the diffusion flux resulting from viscous shear stresses is represented by U. Moreover, C corresponds to the vector of conservative variables, defined as

$$\mathit{C}={\left(\rho ,\rho u,\rho v,\rho w,\rho E\right)}^T,$$

(11)

where u, v, and w are the velocities in the x, y and z-axis directions, respectively. E is the total energy.

To prevent significant grid distortion when the ray flaps near the ground, we use an overset grid, which makes it easier to reset the distance between the ray and the seafloor within the computational domain. The overset, multi-block structured grid used in the simulations of cownose ray swimming near the substrate is shown in Figure 3.

In prior studies, we utilized this in-house fluid solver to investigate the swimming dynamics of the cownose ray with asymmetric pectoral fin flapping [29]. Concurrently, this compressible fluid solver was employed for numerical simulations examining various incompressible fluids subjected to boundary motions on the cownose ray. The quantitative validation of this solver includes the vortex strength of the panel at a high Re number [20] and hydrodynamic research on jet propulsion by the flexible deformation of a squid cavity [30]. Consequently, the successful validation and applicability of this fluid solver have been demonstrated across a series of simulations involving incompressible flows with movement boundaries.

3.2. Computational Domain and Self-Consistency Study

3.2.1. Computational Domain

The size and division of the computational domain for cownose ray swimming near the substrate are shown in Figure 3. The flow domain measures 16 BL in length, 12 BL in width, and 12 BL in height. The ground and the surface of the ray are designated as no-slip walls, while the remaining boundaries are treated as far-field. The computational domain uses a structured grid with refinements near the ray’s surface to accurately capture flow information.

3.2.2. Self-Consistency Study

The swimming parameters we used in the validation and results are as follows: Re = 6000 with laminar flow, Re = 6000, A_r = A_l = 0.7 BL, W_n,l = W_n,r = 0.4, φ = 0, and f_l* = f_r* = 1.857. The “r” and “l” above the parameters represent the left and right sides of the fins. The sensitivity of the grid numbers and time steps was calculated using the in-house solver. We conducted a grid validation employing three distinct grid resolutions: a coarse grid containing 3.80 million cells, a medium grid consisting of 4.30 million cells, and a fine grid comprising 5.40 million cells, with the first grid height set to 4.1 × 10⁻⁴ BL. The result of thrust coefficients for the three grid configurations is illustrated in Figure 4a, revealing that the values from the medium grid closely approximate those from the fine grid. Subsequently, a sensitivity analysis of the dimensionless timestep sizes $\stackrel{\mathrm{-}}{∆t }$(defined as $\stackrel{\mathrm{-}}{∆t }=∆tU/ BL$) for the medium grid was conducted, as depicted in Figure 4b. The results of this analysis indicate that the medium-size grid and time step are calculated with high accuracy and efficiency.

4. Result

4.1. Cownose Ray Flapping near the Substrate

In this section, to research ray swimming near the ground, we calculate the hydrodynamic properties of symmetrical flapping at a height of 0.6 BL during a period of time. The flapping parameters in this section are the same as in the above section.

In Figure 5, we find that t/T = 0.5 is the time when the pectoral fin is flapping at the highest point, and t/T = 1.0 is the lowest position of the flapping. The fin flaps downward between t/T = 0.5 and 1.0, while the pectoral fin flaps upward from t/T = 0.0 to 0.5. There are two maximum and minimum points in the thrust coefficient (C_T) curve and one peak and valley value in the lift coefficient (C_L) and pitching moment (M_y) curves shown in Figure 5. By observing the thrust curve, we find that the second peak’s height surpasses the initial peak. In a flapping period, the average thrust coefficient remains positive, indicating sustained thrust generation. Notably, the maximum lift coefficient occurs at approximately t/T = 0.7, coinciding with the second thrust coefficient peak. At this moment, the pectoral fin flaps near the plane of symmetry during the descending process. Conversely, the lowest lift coefficient is recorded when the fin flaps upward towards a symmetric plane position. The lift coefficient obtains a positive value over the flapping period. Moreover, in the upward flapping symmetric plane, the pitching moment attains its minimum at t/T = 0.2, while it reaches the maximum at t/T = 0.7 when the downward fin flapping aligns with the horizontal plane. In contrast to the findings of Zhang et al. [31], who calculated ray flapping in deeper water, our analysis indicates a positive lift coefficient when the ray is swimming near the substrate. It is clear from these results that the ground effects have a significant impact on the hydrodynamics of rays.

The heave velocity, as shown in Figure 6, is defined as dz/dt at approximately the mid-chord point of the fin. The pectoral fin flapping at the horizontal position exhibits the highest flapping velocity, and the velocity is zero when the pectoral fin flaps at the top or bottom.

To present the angle of attack in flapping, we plotted the instantaneous curve of the y/SL = 0.5 cross-section of the fin in Figure 7b, and the effective angle of attack was defined as follows:

$${\alpha }_{\mathrm{e}}=\arctan {\displaystyle \frac{dz/dt}{U}}-{\theta }_p,$$

(12)

where the pitching angle θ_p is the angle between the central axis dotted line and the inflow speed U, as shown in Figure 7a. We observe similarities between the attack angle curve in Figure 7b and the velocity curve in Figure 6. A comparison of Figure 5 and Figure 7b reveals that the thrust coefficient reaches its maximum when the attack angle reaches its highest positive and negative values, while the minimum thrust coefficient occurs when the fin flaps in the horizontal position.

The pressure on the surface of the cownose ray model reveals the forces on the swimming ray. In Figure 8, we depict the normalized pressure contours (C_p = (P − P_∞)/0.5ρU²) on the cownose ray’s surface when it is flapping near the ground at different time instants in a period. The pressure on the surface of the swimming cownose ray directly contributes to its hydrodynamics. A significant difference in surface pressure distribution is noted on the dorsal and ventral surfaces during flapping at time instants t/T = 0.50 and 0.75. The dorsal surface is covered by lower pressure in the anterior area, whereas the ventral surface has a higher pressure in the anterior area when the ray flaps near the substrate, and the pectoral fin produces greater lift force during this flapping phase. This pressure distribution is also associated with the sustained positive pitching moment seen in the instantaneous pitching moment curve in Figure 5.

The dorsal surface of the cownose ray shows a positive pressure distribution across both the head and trunk regions of the body at t/T = 0.25. In contrast, the ventral surface experiences lower pressure, which results in a negative pitching moment. Meanwhile, at time instant t/T = 1.0, the dorsal surface of the cownose ray’s pectoral fin is covered by positive pressure and the ventral surface has negative pressure. The pressure difference concentrated on the rear side of the body of the ray model yields a negative pitching moment that tends to drive the cownose ray model to swim towards the ground.

To further show the flow structure around the ray fin flapping model, we present the instantaneous vortex contours visualized by normalized non-dimensional Q-criterion vorticity Q* (Q* = Q·BL²/U²) in Figure 9. When the cownose ray model swims far from the substrate, the wake structure generated by the pectoral fin’s upstroke moves the tilt downwards. As shown in Figure 9a, during the downstroke, the shed vortices V-A shift upward and merge with the wake structure, influenced by the ground effect. The strength of the tip vortex reaches the maximum as the pectoral fin flap reaches the peak at t/T = 0.50, while the size of the attached vortex on the cownose ray’s dorsal surface is the most limited, as seen in Figure 9b. The strength of the tubular vortex shed by the root of the trailing edge vortex V-B is greater than that of other shed vortices at this moment. At t/T = 0.75, we observe a noticeable leading-edge vortex V-C on the pectoral fin. This vortex is believed to benefit thrust production according to Han, Lauder, and Dong [5], and it may contribute to the thrust peak reached at this instant, as depicted in Figure 5. At this moment, a unique tip vortex V-D composed of a vortex ring which is absent at other time instants detaches from the fin tip.

To research the vortex structure produced by the rays, we captured the shed vortex generated by the pectoral fin shown in Figure 10. When the cownose ray flaps close to the substrate, the vortices generated by the pectoral fin display less regularity in comparison to those produced during flapping in deeper water. The current spanwise vortex appears to be directed upward, away from the seafloor, and exhibits a marked asymmetry relative to the horizontal axis, as depicted in Figure 10. A leading-edge vortex is observed at slice-2, located near the fin tip, while this feature is absent at slice-1, which is situated closer to the midsection of the fin.

4.2. Effect of Cownose Ray Swimming at Different Near-Ground Distances

As shown in the following results, we can observe that the wall significantly impacts the propulsive performance and vortex structure of the cownose ray fin flapping model. In this section, we systematically examine its impact by varying the model’s near-substrate heights. Specifically, the up-and-down asymmetric flapping of real cownose rays is considered. Nevertheless, symmetrical flapping, a special case of asymmetrical flapping when HAR = 0.0, is also included for comparison. For asymmetrical flapping, we select HAR = 1.7 and 3.0 in the following simulations. These two flapping modes are observed by the rays swimming near the seabed.

The lift and thrust coefficients of different flapping modes at different heights are depicted in Figure 11. Our simulation results show little difference when comparing cownose ray flapping in deep water and at $\stackrel{\mathrm{-}}{\mathrm{H}}$ = 3.0 BL; thus, the latter is used here to represent further distance from the ground. When the near-ground distance falls below 0.75 BL, the ground effect significantly impacts the lift coefficient. The lift coefficient has a sharp increase when the height is reduced to a certain value, and this height depends on the value of the HAR.

When the value of the HAR decreases, the corresponding critical height for rapid lift increase also decreases. This is due to the fact that as the HAR decreases, the amplitude of the pectoral fin’s downward flapping increases; thus, the distance between the tip of the pectoral fin and the ground is reduced. In this way, the influence of the ground on lift force would become more notable. In comparison, the distance from the ground has a smaller influence on thrust. The thrust coefficient generated by symmetric flapping is greater than that generated by asymmetric flapping, implying that thrust production does not synchronize with the lift increase from asymmetric flapping.

The data in Figure 11 also present a comparative analysis of lift and thrust generation at various heights relative to the ground. At a height of 0.6 BL and HAR = 0.0, the cownose ray experiences a significant increase in $\overline{C_L}$, specifically by 503%, due to near-ground flapping, as shown in

Table 1. Lift coefficient comparison at different HARs.

HAR	The Lowest Calculation Height	Far Field	Difference
0.0	0.338 (0.6 BL)	0.056	503%
1.7	0.360 (0.4 BL)	0.081	344%
3.0	0.232 (0.4 BL)	0.077	201%

. The ground effect also improves $\overline{{\mathrm{C}}_T}$, with the maximum thrust coefficient potentially increasing by up to 10%. More details are shown in

Table 2. Thrust coefficient comparison at different HARs.

HAR	The Lowest Calculation Height	Far Field	Difference
0.0	0.487 (0.6 BL)	0.465	4.73%
1.7	0.271 (0.4 BL)	0.246	10.16%
3.0	0.132 (0.4 BL)	0.123	7.32%

.

The results of the pitching moment and propulsion efficiency are displayed in Figure 12. They indicate that the pitching moment escalates when the ray swims close to the seabed. However, this increased pitching moment could pose challenges for the rays in maintaining a stable swimming posture. We also observe a maximum flapping efficiency at a height of 0.4 BL, as shown in Figure 12b. Generally, symmetrical fin flapping yields a higher efficiency compared to asymmetrical flapping. Although a higher propulsive efficiency is obtained at a height of 0.4 BL, the ray’s swimming may not be stable because of the considerable pitching moment. In comparison, at a height of 1.0 BL, the swimming efficiency and thrust production are the largest, other than at a height of 0.4 BL, but with a much smaller mean pitching moment coefficient. Therefore, swimming at 1.0 BL may be more sustainable for cownose rays.

Figure 13 shows the C_T and C_L of cownose ray flapping at varying heights and asymmetric modes. Notable differences are observed in the C_T and C_L curves across these parameters. Specifically, in C_T, the second peak is markedly higher when the ray swims close to the substrate compared to when it flaps at greater heights. Furthermore, the thrust peaks generated by symmetrical flapping are significantly greater than those produced by asymmetrical modes, a trend that is similarly reflected in the lift coefficient analysis. The large amplitude oscillatory variance of C_L with symmetrical flapping may affect the stability of swimming.

Finally, we visualize the vortex contours at $\stackrel{\mathrm{-}}{\mathrm{H}}$ = 0.4 BL and HAR = 3.0 in Figure 14. Compared with the symmetrical flapping shown in Figure 9, asymmetrical flapping produces fewer broken shed vortices, and the trailing edge of the pectoral fin generates a regularly tubular vortex. The negative pressure generated by the vortex shedding on the upper surface of the fin tip is attenuated. The pressure magnitude of the ring tip vortices is positive. Figure 14 indicates that the strength of the trailing-edge vortex is greater when the ray flaps close to the seabed, and the broken vortices on the trailing-edge fin are smaller at a larger height. Due to the ground effect, the angle of the shed vortex created by flapping upward and downward becomes smaller (α > β), as shown in Figure 14a.

In summary, a detailed examination of symmetric and asymmetric flapping close to the substrate reveals that the ground effect is conducive to enhancing the lift and thrust coefficients. Moreover, an optimal near-ground height of 1.0 BL has been identified as particularly beneficial for performance, which may explain why less asymmetrical flapping is chosen by rays while swimming close to the substrate. Symmetrical flapping is beneficial to reducing energy consumption and improving flapping efficiency.

4.3. Ray Flapping near the Ground with Different Frequencies

The results indicate that swimming at 1.0 BL generates significant thrust. We performed simulations at this height with different flapping frequencies at this height. We plot the $\overline{C_T}$, $\overline{C_L}$, and efficiency at various swimming frequencies in Figure 15. $\overline{C_T}$ increases as the flapping frequency increases. At a lower flapping frequency, the $\overline{C_T}$ generated by the pectoral fin cannot overcome the drag force to drive the cownose ray model to swim forward. The lift coefficient has little improvement as flapping frequency increases. Flapping efficiency reaches a maximum at f* = 2.0. At higher frequencies, the thrust coefficient improves significantly. Nevertheless, as the frequency continues to rise, the efficiency drops quickly. In light of this, the cownose ray could swim at a higher frequency in an emergency to speed up and escape from danger, but it is not able to sustain this while cruising.

The instantaneous thrust, total force, and lift at different flapping frequencies in a given time period are shown in Figure 16. As flapping frequency increases, the value of C_T on the peak and valley rises. To find how the optimal swimming frequency is reached, we plot the instantaneous energy consumption coefficient C_total in Figure 16b. The energy expenditure increases faster at a higher flapping frequency compared to the thrust. As a result, the propulsion efficiency does not improve with a higher flapping frequency and reaches a “peak” at f* = 2.0. When it comes to the lift coefficient, both the peak and valley values of the instantaneous lift curve rise with the increased frequency. The lift curve at a higher flapping frequency is smoother than that at a lower frequency within a given period.

To investigate the vortex structure around the ray, we plot the vorticity contours at different swimming frequencies in Figure 17. At a lower flapping frequency (f* = 0.5), there is a small vortex produced by the pectoral fin; thus, the thrust could not overcome the resistance generated by the incoming flow. The broken vortex V-A is generated by the trailing edge, and its strength is small at a low flapping frequency. The leading edge and tip vortex are barely visible. In this way, the C_T produced by the pectoral fin is not sufficient to swim forward. At a higher flapping efficiency f* = 2.0, the shed trailing-edge vortices become relatively regular, and the tip vortex is also generated at this suitable frequency. The upper and lower vortex rings are generally symmetrically distributed, as seen from the lateral view. For larger frequencies (e.g., f* = 3.0), the intensity of the shedding vortex is stronger on the trail-edge fin. A tubular vortex is shed from the tip fin, and a vortex ring is also generated at the tip of the trailing edge. The complex interaction of the vortices V-B shed near the root of the pectoral fin may consume additional energy at a higher flapping frequency; thus, we see a large increase in C_total in Figure 16b. This would cause a drop in flapping efficiency. Seen from the lateral side, the vortices generated near the ventral side are affected by the substrate, and the trajectory of the vortex is almost parallel to the ground wall. The increased lift coefficient may be associated with the forced upward movement of the vortices parallel to the surface.

5. Conclusions

In this work, we simulate a model of cownose ray flapping near the substrate using an in-house solver to resolve the N-S equations. The cownose ray model and the pectoral fin flapping kinematics equations are obtained using the kinematics of real-life rays. Firstly, we study the pectoral fin flapping within one period. Observing the instantaneous thrust curve, we notice the thrust produced near the substrate is larger than flapping far from the ground. As for the lift, the minimum lift coefficient is created when the fin flaps at the lowest position, and the largest lift coefficient is produced at the horizontal middle position while the pectoral fin is flapping downward. By visualizing the surface pressure contours, we find that the pressure distributed on the surface of the ray model causes an increase in lift and pitching moment when swimming near the ground. From the vorticity structure around the ray model, it is found that the shear vortex on the leading edge of the pectoral fin when the fins are flapping near the substrate would be enhanced, and this would increase the thrust production according to Zhang, Huang, Pan, Yang, and Huang [31], as shown in Figure 11b.

Therefore, asymmetric flapping close to the seabed has great effects on the hydrodynamic performance of rays. Cownose rays swimming near the substrate can obtain a boost to thrust (more than 10%) and lift (more than 500%). Ground effects have a greater impact on the lift coefficient than thrust. This is consistent with the findings of Xin and Wu [32]. However, swimming at a lower height has bad stability in the vertical direction due to the large oscillation of lift force. The most efficient swimming height is 1.0 BL, where the lift and thrust coefficients reach the maximum. This height has good stability in the vertical direction and relatively high propulsion efficiency.

Our simulation results also indicate that C_T and C_L increase as the flapping frequency becomes larger, with varying flapping frequencies at 1.0 BL. Nevertheless, propulsion efficiency reaches the maximum at the flapping frequency f* = 2.0. This is in line with the observation of actual cownose ray flapping.

This research may offer insights into the near-surface swimming behavior of biomimetic underwater robots. For instance, swimming at a specific depth may be advantageous. At this height, the bionic ray can obtain optimal thrust and higher swimming efficiency, and it can maintain stable swimming in the vertical direction. Additionally, improving the propulsive efficiency of ray-like bionic underwater robots is conducive to long-time and long-distance swimming. This work may also lay a foundation for future study of the landing, take-off, and other benthic behaviors of rays in free swimming mode.