Scalable Dual-Servo Pectoral Fin Platform for Biomimetic Robotic Fish: Hydrodynamic Experiments and Quasi-Steady CFD

Abstract

Biomimetic pectoral fin propulsion offers a low-noise, highly maneuverable alternative to conventional propellers for next-generation underwater robotic systems. This study develops a manta ray-inspired dual-servo pectoral fin module with a CPG-based controller and employs it as a single-fin test article in a recirculating water tunnel to quantify its hydrodynamic performance. Controlled experiments demonstrate that the fin generates stable thrust over a range of flapping amplitudes, with mean thrust increasing markedly as the amplitude rises, while also revealing an optimal frequency band in which thrust and thrust work are maximized and beyond which efficiency saturates. To interpret these trends, a quasi-steady CFD analysis using the k–ω SST turbulence model is conducted for a series of static angles of attack representative of the instantaneous effective angles experienced during flapping. The simulations show a transition from attached flow with favorable lift-to-drag ratios at moderate angles of attack to massive separation, deep stall, and high drag at extreme angles, corresponding to high-amplitude fin motion. By linking the experimentally observed thrust saturation to the onset of deep stall in the numerical flow fields, this work establishes a unified experimental–numerical framework that clarifies the hydrodynamic limits of pectoral fin propulsion and provides guidance for the design and operation of low-noise, highly maneuverable biomimetic underwater robots.

1. Introduction

In recent years, biomimetic propulsion has become an attractive alternative to conventional underwater propulsion systems due to its low acoustic signature, high maneuverability, and potential for high energy efficiency [1]. While propeller-based platforms are technologically mature, they often suffer from high noise [2,3], complex mechanical structures [4,5], and limited agility—factors that can hinder stealth, precision operations [6,7], and safe interaction with the environment.

A wide range of biomimetic robotic fish prototypes has been developed to investigate these mechanisms from a design, hydrodynamic, and control perspective [8]. Li et al. [9] introduced a manta-ray-inspired underwater vehicle that achieves low flow disturbance and excellent stability. Other works have examined hybrid propulsion architectures: Huang et al. [10] integrated pectoral fin-mounted thrusters with caudal fin motion, Bai et al. [11] combined biomimetic fins with propellers for enhanced flexibility. These studies collectively highlight the broad design space and performance potential of bio-inspired fin mechanisms.

Hydrodynamic investigations have played a central role in understanding and optimizing these systems [12]. Advanced geometric integration methods have further enhanced the fidelity of such simulations [13,14]. Concurrently, isogeometric sensitivity analysis methods have been established to accurately evaluate the response of physical fields to geometric perturbations, offering a theoretical basis for structural refinement [15,16]. Further research has demonstrated that pectoral fins significantly influence maneuverability, braking, and turning performance, especially when combined with optimized control strategies and multi-fin coordination [17]. Advanced modeling techniques integrating novel view synthesis with CFD have been proposed [18,19], CAD-integrated reduced-order models have been introduced to achieve efficient shape optimization in complex physical environments [20].Complementary advances in Computational Fluid Dynamics (CFD) have provided deeper insight into the unsteady flow fields, vortex dynamics, pressure gradients, and wake patterns generated by flapping foils, gradually building a detailed hydrodynamic understanding of oscillatory-fin propulsion [21,22,23].

At the actuation and control level, numerous bio-inspired approaches have been proposed to reproduce compliant, adaptive, and coordinated fin motions [24]. Nie et al. [25] proposed a novel Twisting Water Hydraulic Artificial Muscle (TWHAM) for flexible robotic fish, offering a promising approach for pectoral fin propulsion. Sun et al. [26] formulated a virtual control law based on agent dynamics for tail fin-driven robotic fish, achieving smooth and adaptive motion control. Ding et al. [27] developed a tensegrity-based robotic fish with multiple adjustable joints and experimentally verified the influence of body stiffness on high-frequency swimming efficiency. Moreover, stochastic finite element methods have been applied to quantify uncertainties in mechanical properties, ensuring the reliability of flexible actuation systems [28]. In addition, Central Pattern Generator (CPG)-based locomotion models have been widely used to generate rhythmic fin motions and adaptive gaits [29,30].

Despite these advances, many existing studies focus on a single aspect—such as structural design, hydrodynamic testing, or numerical simulation [31]—without providing an integrated framework that links physical experiments with theoretical flow analysis. In this work, a modular biomimetic pectoral fin platform is constructed, incorporating a dual-servo actuation system and a CPG-based control interface. Systematic hydrodynamic experiments are carried out in a recirculating water tunnel to quantify the thrust performance under various kinematic parameters. Furthermore, to interpret the experimental results, quasi-steady CFD simulations are performed to examine the fundamental flow-field behavior, including vortex formation, pressure distribution, and separation characteristics at different effective angles of attack. By combining experimental measurements with physical flow analysis, this study establishes a unified framework for investigating the hydrodynamic performance of pectoral fin-based propulsion, thereby providing both theoretical insight and practical guidance [32,33] for the design of low-noise, highly maneuverable biomimetic underwater robotic systems.

2. The Design and Implementation of the Modular Biomimetic Pectoral Fin Platform

2.1. System Overview and Design Objectives

To investigate the hydrodynamic mechanisms of pectoral fin propulsion, a scalable biomimetic robotic fish platform was developed. Unlike traditional propeller-driven vehicles, this platform utilizes an oscillatory fin mechanism to achieve low-noise and high-maneuverability propulsion. The system is designed not only to support fundamental hydrodynamic research on bio-inspired propulsion but also to meet potential application requirements in underwater tasks such as reconnaissance [34], tracking, and precision engagement.

While the ultimate goal is a fully autonomous vehicle, this study focuses on the fundamental hydrodynamic performance of the pectoral fin. Therefore, the system is designed with a modular architecture, allowing the pectoral fin actuation unit to be isolated and mounted in a water tunnel for precise quantitative evaluation.

The design objectives focus on

• Bio-fidelity: Reproducing the coupled flap-and-twist motion of manta rays using a dual-servo mechanism;
• Control Precision: Ensuring accurate and repeatable generation of kinematic trajectories (frequency, amplitude, and phase) for experimental validation;
• Modularity: Enabling the actuation unit to function both as part of a free-swimming robot and as a standalone test article in hydrodynamic experiments.

2.2. Pectoral Fin Actuation Mechanism

The core of the propulsion system is the dual-servo pectoral fin actuation mechanism, designed to replicate the complex kinematics of biological fins. Similar two-degree-of-freedom pectoral fin mechanisms have been shown to enhance maneuverability and propulsion efficiency in previous robotic fish studies [35].

Each fin module is driven by two high-torque servomotors (rated at 70 kg·cm). The kinematic chain functions as follows:

• Flapping Motion: The first servo, located at the root, drives the primary vertical flapping motion (heaving), determining the flapping amplitude and frequency;
• Pitching/Twisting Motion: The second servo is coupled to the fin base to control the instantaneous angle of attack (pitching).

By coordinating these two degrees of freedom with a specific phase difference, the mechanism generates a “flap-and-twist” trajectory. This flexibility allows the system to simulate various swimming modes and, crucially for this study, allows the isolation of specific kinematic parameters (e.g., amplitude and frequency) to analyze their impact on thrust generation.

2.3. Mechanical Structure and Waterproofing

The mechanical structure adopts a modular design to ensure reliability during underwater operations. The pectoral fin is constructed from a semi-rigid frame covered with a flexible material to approximate the stiffness profile of biological fins. The central hull serves as a waterproof enclosure for the actuators and control electronics.

For the water tunnel experiments presented in this work, the actuation module is designed to be easily detached from the main body and mounted onto a force-measurement fixture. This design ensures that the geometric boundary conditions remain consistent between the full-robot design and the isolated hydrodynamic tests.

2.4. Motion Control Framework

To generate smooth and continuous rhythmic fin motions, a Central Pattern Generator (CPG)-based control strategy is implemented on an NVIDIA Jetson Orin NX embedded controller. Recent research on boxfish-inspired robotic platforms has demonstrated that CPG-based multi-fin coordination can effectively regulate thrust generation and body attitude by modulating the rhythmic properties of fin movements [36].

Unlike simple sinusoidal position control, the CPG network models the rhythmic signals as coupled non-linear oscillators. This framework offers two key advantages for this study:

• Stability: It ensures smooth transitions when changing flapping frequency or amplitude during experiments, preventing mechanical jerks that could introduce noise into force measurements.;
• Adjustability: It allows for real-time modulation of the motion parameters (frequency, amplitude, and phase offset), enabling a systematic sweep of the kinematic space during the hydrodynamic tests.

3. Experimental Investigation of Biomimetic Pectoral Fin Hydrodynamics

Hydrodynamic experiments are essential for evaluating the performance of biomimetic propulsion systems and understanding the mechanisms that govern thrust generation, maneuverability, and flow–structure interactions. Prior studies on flapping foils have demonstrated that controlled water- or wind-tunnel experiments play a crucial role in quantifying unsteady propulsion forces, vortex dynamics, and wake structures under well-defined kinematic parameters [37]. Compared with conventional propellers, oscillatory pectoral fins enable low-speed cruising, turning, and attitude stabilization using flexible motion patterns—capabilities that are valuable for stealthy and precise underwater operations. However, most previous experimental work has focused on whole-body propulsion or caudal fin dynamics, leaving the hydrodynamic behavior of individual pectoral fins relatively underexplored. This gap limits the broader application of pectoral fin propulsion in underwater robotic platforms.

To address this deficiency, the present study designs and fabricates a single-sided biomimetic pectoral fin prototype for controlled testing in a recirculating water tunnel. The experimental setup allows fine adjustment of flow velocity and precise regulation of fin kinematics. Swimming velocity is used as the primary performance metric, and the influence of flapping amplitude on thrust generation is systematically evaluated. The resulting data provide quantitative insights into the propulsive contribution of a single pectoral fin and lay the groundwork for analyzing its suitability for stealth cruising and near-field maneuvering.

3.1. Experimental System Configuration

The hydrodynamic experiments were conducted in a closed-circuit recirculating water tunnel (model GHSY-XH, Puyang Guanghe Environmental Technology Co., Ltd., Puyang, China), as illustrated in Figure 1. The facility features a fully enclosed loop comprising a test section, contraction section, diffuser, and four turning corners equipped with guide vanes to minimize secondary flow. The water tunnel measures 6.2 m × 1.05 m × 2.65 m and is constructed from 304 stainless steel, with CCC-certified tempered-glass windows installed in the test section for flow observation and high-speed imaging.

The system operates at a rated power of 15 kW (380 V, 40 A) and provides a uniform and stable inflow with a maximum velocity of 1.2 m/s. The test section has a cross-sectional area of 0.6 m × 0.6 m, and the turbulence intensity is maintained below 0.5%. Similar low-turbulence water tunnels have been widely used in flapping-foil studies to enable accurate measurement of unsteady hydrodynamic forces under controlled kinematics [38].

The experimental platform is equipped with a four-axis motion control system to allow precise multi-degree-of-freedom actuation of the prototype:

(1)

X-axis: Servo-driven linear motion, 1.20 m stroke, sinusoidal actuation.

(2)

Y-axis: Servo-driven linear motion, 0.55 m stroke, sinusoidal actuation.

(3)

Z-axis: Servo-driven linear motion, 0.35 m stroke, sinusoidal actuation.

(4)

C-axis: Continuous 360° rotation with adjustable forward and reverse speeds.

Figure 2 shows the experimental assembly, which consists of the biomimetic single-sided pectoral fin, a mechanical measurement module, and the four-axis motion platform. The fin is mounted to a six-axis force/torque sensor (model GLH91003AA0) via a custom fixture. The sensor measures hydrodynamic forces (Fx: thrust/drag, Fy: lateral force, Fz: vertical force) and torques (Tx: roll moment, Ty: pitch moment, Tz: yaw moment) along all three spatial axes and is connected to the motion platform to ensure high-precision alignment between the fin and the incoming flow.

A high-precision digital servo at the fin root, driven by a Jetson Orin NX controller, generates periodic flapping motions with prescribed amplitudes and frequencies. This configuration ensures strict geometric alignment, thus minimizing extraneous forces and measurement errors. The Jetson module, running a Linux-based ROS environment, issues synchronized control commands defining flapping frequency, amplitude, and rotational angle.

Force and torque data from the six-axis sensor are transmitted to a host computer via a high-speed data acquisition card at a sampling rate of 1–5 kHz. This high sampling frequency enables accurate capture of dynamic force variations within each flapping cycle. Flow velocity in the water tunnel is continuously adjustable from 0 to 1.2 m/s, providing an equivalent swimming speed environment for the prototype.

A dedicated data acquisition and monitoring system coordinates multi-axis actuation, sensor recording, and flow regulation, ensuring the repeatability and precision required for hydrodynamic experiments of oscillatory fin propulsion.

3.2. Biomimetic Pectoral Fin Actuation Device

In nature, fish pectoral fins generate thrust and regulate posture through coordinated pitching and flapping motions. Inspired by this mechanism, a dual-servo coordinated biomimetic pectoral fin actuation device was designed in this study, as shown in Figure 3. The core concept is to employ two degrees of freedom—flapping (up-and-down) and rotational (twisting) motion—to reproduce the natural trajectory of real fish pectoral fins, thereby generating hydrodynamic effects similar to those observed in biological locomotion. Comparable dual-degree-of-freedom pectoral fin modules with soft or flexible fin structures have been developed for ray-inspired robotic fish, and their hydrodynamic experiments confirm that such mechanisms can achieve efficient thrust and maneuverability over a wide range of operating conditions [39].

Each pectoral fin is actuated by two servomotors:

• Servo 1 (flapping actuator): Mounted at the fin root to drive the primary up–down flapping motion.
• Servo 2 (rotational actuator): Attached to the fin base to adjust the instantaneous rotational (twist) angle and modulate the angle of attack.

The two servomotors are coupled through a hinged mechanism, allowing the pectoral fin to transition among three key motion states: the initial position, maximum upward stroke, and maximum downward stroke. By appropriately configuring the phase difference and rotational speed of the servomotors, the fin can generate a typical sinusoidal flapping trajectory. In the experiments, the motion amplitude of the servos was adjustable to accommodate different testing requirements. For instance, at a nominal amplitude of ±30° and a flapping period of 0.7 s, the system was verified to produce stable thrust output. To further investigate the influence of kinematics on performance, a wider range of amplitudes (from 15° to 60°) was systematically evaluated, as discussed in Section 3.5.

The actuation system achieves synchronized multi-channel servo control through a PCA9685 module. PWM signals are generated by an embedded main controller—such as a Raspberry Pi, Arduino, or Jetson Orin NX—and converted by the PCA9685 into servo actuation commands. A time division activation mechanism is employed: the flapping servo is initiated first, followed by the rotational servo with a delay of approximately 0.2 s. This configuration introduces an appropriate phase difference, enabling coordinated motion between flapping and rotation of the pectoral fin.

To further enhance the biomimetic realism and smoothness of motion, the latest version incorporates a Central Pattern Generator (CPG)-based control strategy. The CPG is capable of generating stable sinusoidal output signals while allowing real-time modulation of both frequency and amplitude. This approach effectively prevents abrupt transitions during motion switching, ensuring continuous and convergent flapping behavior of the pectoral fin.

The biomimetic pectoral fin actuation device possesses the following characteristics:

• High biomimetic fidelity: The dual-degree-of-freedom design realistically reproduces the coupled flapping and rotational motions of natural pectoral fins.
• Flexible control: By adjusting the servo phase difference, amplitude, and frequency, the system can achieve multi-modal motions such as straight swimming, turning, ascending, and diving.
• Stable propulsion: Periodic sinusoidal control ensures continuous thrust output, which has been experimentally verified to produce stable propulsion in quiescent water.
• Scalability: The control system is based on standard PWM driving and the ROS framework, facilitating integration into larger-scale biomimetic robotic platforms.

This pectoral fin actuation device not only provides a controllable and repeatable platform for hydrodynamic experiments but also offers a potential propulsion solution for low-noise, highly maneuverable underwater robots and next-generation autonomous underwater vehicles (AUVs). With further optimization of its structure and control strategy, it is possible to enhance energy efficiency and motion stability while maintaining high biomimetic fidelity.

3.3. Hydrodynamic Experimental Measurement Methods of the Biomimetic Pectoral Fin

Hydrodynamic thrust generated by biomimetic flapping pectoral fin can be evaluated using several commonly adopted experimental methods, as summarized in

Table 1. Common hydrodynamic measurement methods table.

Method	Experimental Method	Advantages	Limitations
Fixed propulsion method	Fixed tank, active drive.	Direct thrust data acquisition.	Thrust is not equal to drag.
Autonomous propulsion method	Prototype on low-friction track, moves until stable.	Accurately reflects propulsion performance.	Thrust is hard to measure directly.
Towing force feedback method	Prototype towed at constant speed to measure resistance.	Directly acquire fluid resistance data.	Measured drag is not the same as actual pectoral fin drag.

. Each method differs in how the prototype is constrained and how thrust or drag are inferred.

Fixed-thrust method: In the fixed-thrust (fixed propulsion) method, the prototype is rigidly mounted at a specified position in the test section, and the propulsion mechanism is actively driven while the surrounding flow is controlled. The thrust generated by the fin or body is measured directly by a force sensor, without allowing the prototype to translate. Since the measured thrust does not necessarily equal the hydrodynamic drag that would be experienced during free swimming, this method is particularly suitable for verifying whether the propulsion system can overcome flow resistance and for characterizing thrust under well-defined kinematic conditions.

Free-swimming method: In the free-swimming (autonomous propulsion) method, the prototype is attached to a low-friction rail or carriage system, allowing it to move along a track until a steady swimming speed is reached. This configuration provides a more realistic representation of actual swimming performance, because the measured steady velocity reflects the balance between thrust and drag. However, accurate thrust quantification is challenging and often requires indirect estimation or additional instrumentation, such as load cells on the carriage or motion tracking combined with dynamic models.

Towing force feedback method: In the towing force feedback method, the prototype is fixed to an external towing platform and pulled at a prescribed constant speed. The measured force then corresponds to the hydrodynamic resistance of the prototype at that speed, which can be used as a reference for evaluating propulsive capability. Nevertheless, the resulting drag is not strictly equal to the net thrust produced by a flapping mechanism operating at the same nominal swimming speed, because the kinematics and flow conditions may differ between towing and self-propelled configurations.

In this study, the biomimetic hydrodynamic experimental system adopts the fixed-thrust method, with the test article restricted to a single pectoral fin of the manta ray-inspired prototype, as illustrated in Figure 4. The fin assembly is locked in position using an air-bearing rail mechanism so that no global translation occurs, and propulsion is generated exclusively by the flapping motion of the single pectoral fin.

The use of a single-fin configuration in this study is a deliberate simplification aimed at isolating the fundamental hydrodynamic mechanisms of the pectoral fin without the confounding interference of body-wave interactions or fin-to-fin flow coupling. For a manta ray-inspired robot operating in a steady, straight-line swimming gait, the propulsion system is assumed to be bilaterally symmetric. Under this symmetry assumption, the hydrodynamic data obtained from a single fin can be extrapolated to a full bilateral system by superimposing the force vectors. While this approach neglects potential interference effects in the wake-overlap region, it provides a high-fidelity baseline for the individual propulsive unit, which is essential for the modular design and control of the full vehicle.

A six-axis force/torque sensor is mounted at the fin root to capture the hydrodynamic loads. In the x-direction, the sensor records the net force composed of the axial thrust produced by the fin and the opposing flow resistance; in the y-direction, it measures lateral forces associated with sideward flapping; in the z-direction, no significant force component is expected due to the experimental configuration. This arrangement enables direct, high-precision measurement of the thrust generated by a single pectoral fin and provides a reliable basis for evaluating its hydrodynamic characteristics and propulsive efficiency.

Because the prototype is fixed, the instantaneous swimming speed does not directly follow from fin motion alone. To overcome this limitation, the inflow velocity of the recirculating water tunnel is gradually adjusted until the net axial force measured in the x-direction approaches zero, indicating that the thrust generated by the fin balances the hydrodynamic drag. The corresponding equilibrium flow speed is then defined as the effective swimming speed of the single pectoral fin under the given kinematic parameters, thereby establishing a consistent link between fixed-thrust measurements and the equivalent self-propelled condition.

3.4. Data Processing and Parameter Normalization

To quantitatively evaluate the hydrodynamic performance and ensure comparability across different scales and conditions, all measured forces and moments are normalized into non-dimensional coefficients. The reference parameters used for normalization are defined based on the fin’s geometry and flow conditions:

• Reference Area (S): The planform area of the triangular pectoral fin, calculated as $S=2.0 \times {10}^{-4} {\mathrm{m}}^2$.
• Characteristic Length (c): The root chord length of the fin, c = 0.0224 m.
• Reference Velocity ($U_{\infty }$): The steady inflow velocity of the water tunnel.
• Fluid Density ($\rho$): The density of water, $\rho =997.56 {\mathrm{k}\mathrm{g}/\mathrm{m}}^3$.

The non-dimensional force and moment coefficients are defined as follows:

$$C_i={\displaystyle \frac{F_i}{\frac{1}{2}\rho U_{\infty }^{2}S}}{\displaystyle (i=x,y,z)}$$

(1)

$$C_{Ti}={\displaystyle \frac{T_i}{\frac{1}{2}\rho U_{\mathrm{\infty }}^{2}Sc}}{\displaystyle (i=x,y,z)}$$

(2)

where Fx, Fy, Fz represent the measured thrust, lateral force, and vertical force, respectively, and Tx, Ty, Tz are the corresponding hydrodynamic torques (moments) recorded by the sensor. $C_{Ti}$ denotes the non-dimensional torque coefficients.

Furthermore, to address the energy output characteristics, the “thrust work” ($W_T$) mentioned in this study is defined as the time-integrated effective propulsive energy delivered by the pectoral fin against the incoming flow. It is computed as the integral of the product of instantaneous thrust and reference velocity over a specific duration T:

$$W_T={\int }_0^T F_x{\displaystyle \left(t\right)}\cdot U_{\infty }dt$$

(3)

This definition quantifies the cumulative useful work output of the propulsion system during the experimental window, distinguishing it from the mechanical power input of the actuators.

3.5. Results and Discussion

The thrust performance of the single pectoral fin was evaluated under four flapping amplitudes (15°, 30°, 45°, and 60°), at their respective effective swimming speeds where net axial force balances, as shown in Figure 5. The measurements reveal a clear increase in mean thrust with increasing flapping amplitude, accompanied by distinct changes in the temporal characteristics of the thrust coefficient (Cx), which is normalized according to Equation (1) using the instantaneous Fx values. To better characterize the thrust generation process, two primary peaks within each flapping cycle are identified and labeled as F1 and F2 in Figure 6, representing the thrust produced during the downstroke and upstroke phases (or the two symmetrical halves of a cycle), respectively.

At low amplitudes (15° and 30°), the thrust coefficient remains relatively small and exhibits irregular fluctuations with poorly defined F1 and F2 peaks. This suggests that the leading-edge vortices shed by the fin are weak and unstable, contributing limited net thrust. In contrast, medium and large amplitudes (45° and 60°) yield a pronounced increase in mean thrust, and the thrust coefficient oscillations become more regular and strongly periodic. This indicates the formation of a coherent wake vortex pattern that enhances propulsive performance. Similar dependencies of thrust and efficiency on oscillation amplitude, frequency, and phase have been reported in previous hydrodynamic studies of manta ray-inspired pectoral fins, further supporting these observations [40].

Although larger amplitudes can further augment thrust, they may also introduce challenges for attitude control and structural loading of the fin mechanism. The experimental results nevertheless demonstrate that the proposed setup provides reliable and repeatable thrust measurements, and the zero-crossing point of the net axial force can be used to identify the critical swimming speed at which the single pectoral fin achieves self-propulsion under a given set of kinematic parameters.

To facilitate comparison across different operating conditions, the raw force data are post-processed prior to analysis. A low-pass filter with a cutoff frequency of 20 Hz is applied to remove high-frequency noise, and the time histories are segmented by flapping cycle to extract cycle-averaged hydrodynamic forces (mean thrust and lift) and moments. These processed quantities are used to characterize the hydrodynamic performance of the pectoral fin motion in the subsequent analysis.

In addition to the swimming speed tests described above, a series of static-water fixed-thrust experiments was carried out to further quantify the combined effects of flapping frequency and amplitude on the thrust performance of the single pectoral fin, as shown in Figure 6. In these tests, 12 flapping frequencies (0.25–0.80 Hz with an increment of 0.05 Hz) and four flapping amplitudes (15°, 30°, 45°, and 60°) were considered, yielding a total of 48 combinations of motion parameters. Figure 6a presents the variation in the cycle-averaged streamwise thrust Fx with flapping frequency for different amplitudes, while Figure 6b shows the corresponding cumulative thrust work over a 6 s time window. This $W_T$ is calculated as the time integral of the product of the measured thrust and inflow velocity, following the definition in Equation (3). As the flapping frequency increases, the mean thrust first rises and then decreases, indicating the existence of an optimal frequency band. For the present prototype, the mean thrust reaches a relatively high level when the flapping frequency is approximately 0.55–0.65 Hz, and the maximum instantaneous static thrust of about 11 N occurs at a frequency of 0.70 Hz with a flapping amplitude of 60°. These results confirm that the pectoral fin can achieve efficient propulsion only within a specific combination of frequency and amplitude, which is crucial for determining the optimal operating conditions and designing appropriate control strategies.

To further elucidate the temporal characteristics of thrust generation, Figure 7 plots the time history of the streamwise thrust Fx in static water for a representative flapping case at a frequency of 0.5 Hz. It can be observed that two pronounced positive thrust peaks occur within each motion cycle, accompanied by two negative thrust valleys. This double-peaked thrust waveform arises from the two distinct rotational strokes of the pectoral fin in the positive and negative lateral directions within one complete cycle. Compared with a single-peaked thrust pattern, this feature enables the fin to deliver more continuous and sustained forward propulsion, which is beneficial for improving propulsion efficiency and reducing fluctuations in the swimming speed of the ray-inspired robotic fish.

To illustrate the typical behavior of the lateral and vertical force components, Figure 8 shows the time histories of the lateral force Fy and vertical force Fz in a static-water fixed-thrust test at a representative flapping frequency of 0.5 Hz. It can be observed that Fy exhibits an approximately symmetric oscillation about zero, with positive and negative values of similar magnitude within each motion cycle, so that the net lateral force over one cycle is very small. This feature is favorable for maintaining heading stability in straight-swimming gaits. In contrast, Fz remains predominantly negative during the flapping motion, indicating a downward-directed vertical force component, and its fluctuation amplitude is comparable to that of the streamwise thrust. These characteristics suggest that the single pectoral fin naturally produces only a small net side force in straight swimming, while providing a usable vertical control component. In a full bilateral configuration, the lateral forces (Fy) from the left and right fins would oscillate in opposite directions and cancel each other out, thereby enhancing heading stability. Conversely, the vertical forces (Fz) from both fins would superimpose, providing a significant collective force component that can be exploited, together with the thrust, for the depth and pitch control of the ray-inspired robotic fish.

Finally, the hydrodynamic moments generated by the single pectoral fin were also measured in the static-water fixed-thrust tests. These moments are non-dimensionalized into coefficients ($C_{Tx},C_{Ty},C_{Tz}$) using the root chord length (c = 0.0224 m) as the characteristic length, according to Equation (2) in Section 3.4. The cycle-averaged roll moment Tx, pitch moment Ty and yaw moment Tz under different flapping frequencies and amplitudes are summarized in Figure 9. The magnitudes of all three coefficients increase with flapping frequency and flapping amplitude, indicating that the fin can generate appreciable rolling, pitching and yawing moments. This confirms that, in addition to providing forward thrust and lateral/vertical forces, the biomimetic pectoral fin also has the potential to contribute to attitude control of the ray-inspired robotic fish.

While the experimental results successfully quantified the macroscopic propulsion performance (e.g., swimming speed and thrust trends), the underlying flow physics remains unresolved. Specifically, the force measurements revealed a saturation or decline in efficiency at high flapping amplitudes (e.g., 60°), but the force sensor alone cannot distinguish whether this is caused by flow separation, vortex breakdown, or other hydrodynamic phenomena.

Regarding the generalizability of these findings, the single-fin results presented here serve as a foundational hydrodynamic baseline for the modular design of the full vehicle. In a complete bilateral system, while the primary thrust is expected to scale linearly with the number of fins, potential hydrodynamic interference—such as wake overlap and the blockage effect of the central hull—may introduce secondary variations in efficiency. However, for the current manta ray-inspired design where the fins are positioned with a relatively wide span, these interference effects are secondary to the dominant flow separation mechanisms identified in this study. Thus, the single-fin performance metrics provide a robust and conservative estimate for the propulsion and maneuverability envelopes of the real robotic platform.

By defining these precise normalization parameters and coefficients, the experimental results are rendered independent of the specific scale of the prototype, providing a standardized hydrodynamic basis that can be directly compared with the numerical simulations in the following sections.

To elucidate the physical mechanism behind this thrust saturation, it is necessary to look inside the flow field—a capability inaccessible in the current experimental setup. Consequently, the following section employs Computational Fluid Dynamics (CFD) to visualize the microscopic flow structures. By isolating the effects of the angle of attack, the simulation aims to provide the missing physical link between the fin’s kinematic motion and the force limits measured in the water tunnel.

4. Mechanistic Investigation of Pectoral Fin Hydrodynamic

This section presents the numerical investigation of the hydrodynamic characteristics of the pectoral fin. To interpret the thrust generation and potential flow separation mechanisms observed in the dynamic experiments, a computational fluid dynamics (CFD) analysis was conducted. First, the theoretical link between dynamic flapping and static aerodynamics is established. Subsequently, the numerical model, boundary conditions, turbulence model, and mesh strategy are described. Finally, the flow field is analyzed to elucidate the physical origins of the forces measured in the experiments.

4.1. Quasi-Steady Assumption and Effective Angle of Attack

While the pectoral fin in the experiments undergoes complex unsteady flapping motion, its fundamental hydrodynamic performance limits can be effectively analyzed using the quasi-steady assumption. This approach posits that the instantaneous forces acting on a flapping foil are strongly correlated with its instantaneous effective angle of attack (AOA), particularly regarding the onset of flow separation.

During the dynamic flapping cycle, the fin experiences a heave velocity $V_{heave}\left(t\right)$ due to the flapping motion and a forward velocity $U_{\infty }$ from the incoming flow. The effective angle of attack ${\alpha }_{eff}\left(t\right)$ at any instant can be approximated as:

$${\alpha }_{eff}{\displaystyle \left(t\right)}\approx arctan\left({\displaystyle \frac{V_{heave}{\displaystyle \left(t\right)}}{U_{\infty }}}\right)+{\theta }_{pitch}{\displaystyle \left(t\right)}$$

(4)

In the water tunnel experiments, increasing the flapping amplitude to 60° significantly increases the heave velocity $V_{heave}$. According to Equation (4), a high heave velocity results in a large instantaneous effective angle of attack, which can periodically exceed the critical stall angle of the fin. Therefore, the static CFD simulation at an extreme angle of attack (90°) is employed as a representative limit state. This allows us to isolate and visualize the severe flow separation mechanisms (deep stall) that physically limit the thrust generation during high-amplitude dynamic flapping, serving as a quasi-steady proxy for the efficiency saturation observed in the experiments.

Therefore, this study employs static CFD simulations at representative angles of attack (0°, 15°, 45°, and 90°) to isolate the effects of flow separation. The 90° case serves as a limiting condition to capture the fully separated flow regime (“deep stall”). This static analysis offers a physical explanation for the drag penalties and efficiency saturation observed in high-amplitude dynamic flapping, without the computational cost of fully resolved unsteady simulations.

To further characterize the unsteady nature of the experimental regime and justify the methodological approach, two key nondimensional parameters—the reduced frequency (k) and the Strouhal number (St)—are reported. These are defined as

$$k={\displaystyle \frac{\pi fc}{U_{\infty }}}$$

(5)

$$St={\displaystyle \frac{2fA}{U_{\mathrm{\infty }}}}$$

(6)

where f is the flapping frequency, c is the characteristic chord length, and A is the flapping amplitude (heave displacement). For the experimental conditions investigated in this study, the reduced frequency ranges from 0.12 to 0.48, and the Strouhal number spans 0.25 to 0.55. Reporting these parameters provides a quantitative basis for evaluating the degree of unsteadiness in the pectoral fin’s hydrodynamic environment and sets the stage for discussing the validity of the quasi-steady approximation.

The use of this quasi-steady approach within the reported ranges of k and St is primarily justified by the sharp-edged geometry of the pectoral fin. Since separation points are geometrically fixed at the sharp edges rather than being sensitive to boundary layer transition details, the instantaneous effective AOA serves as the dominant driver for the separation-induced pressure distribution. This allows the static CFD simulations at discrete angles to effectively capture the “limit states” of the flow field—particularly the transition to deep stall at α = 90°—providing a direct mechanistic explanation for the thrust saturation observed in the dynamic experiments.

However, it is important to acknowledge that this simplified framework omits several complex unsteady phenomena. Specifically, the quasi-steady model does not account for: (1) Dynamic stall and LEV dynamics, where the onset of separation may be delayed, and the formation of a leading-edge vortex (LEV) could enhance instantaneous forces beyond static predictions; (2) Added mass effects, which represent the inertial resistance of the fluid and can become significant at higher reduced frequencies (k > 0.1); and (3) Vortex-body interactions from previous flapping cycles. Despite the omission of complex unsteady terms, the use of this quasi-steady static framework is methodologically justified as a “mechanistic limit-state probe.” While the experimental prototype undergoes dynamic flapping, the “thrust saturation” observed at high amplitudes (e.g., 60°) is fundamentally a consequence of the flow entering a separation-dominated regime.

By isolating the 90° angle of attack in a static CFD environment, we can evaluate the worst-case pressure drag penalty without the confounding influence of time-varying kinematics. This static limit serves as a physically accurate proxy for the peak instantaneous state of a high-amplitude stroke, where the effective AOA periodically exceeds the stall margin. Therefore, the qualitative consistency between the static lift-to-drag degradation and the time-averaged experimental thrust trends confirms that geometrically induced flow separation—rather than unsteady vortex-body interaction—is the primary physical constraint governing the performance envelope of the robotic fish. This alignment bridges the gap between the two approaches, ensuring that the CFD results provide a valid and logically consistent explanation for the macroscopic phenomena recorded in the water tunnel.

4.2. Governing Equations

In this study, the fluid flow around the pectoral fins is modeled as three-dimensional, viscous, and incompressible, assuming a constant density. The numerical simulations were performed using the STAR-CCM + 2306 software. The governing equations are based on the Reynolds-Averaged Navier–Stokes (RANS) formulation, which includes the continuity equation for mass conservation and the momentum equations. These equations can be expressed in Cartesian tensor notation as follows:

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

(7)

$$\rho {\displaystyle \frac{\partial u_i}{\partial t}}+\rho {\displaystyle \frac{\partial \left(u_i{u}_j\right)}{\partial x_j}}=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\mu \left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)-\overline{\rho u_i^{\prime }{u}_j^{\prime }}\right]+f_i$$

(8)

where $u_i$ represents the Reynolds-averaged velocity components, $p$ is the mean pressure, $\rho$ is the fluid density, $\mu$ is the dynamic viscosity, and $f_i$ denotes the body forces. The term $\overline{\rho u_i^{\prime }{u}_j^{\prime }}$ represents the Reynolds stresses, which account for the turbulent fluctuations in the flow.

To close the system of equations and resolve the Reynolds stress term, the k-omega turbulence model was employed. This model is chosen for its superior performance in predicting boundary layer behavior and flow separation under adverse pressure gradients compared to other models.

Numerically, the governing equations were solved using a segregated flow solver, which solves the integral conservation equations of mass and momentum sequentially. The pressure-velocity coupling was handled using a SIMPLE-type algorithm suitable for constant density flows. The hydrodynamic forces (lift and drag) acting on the fins were calculated by integrating the pressure and wall shear stress distributions over the fin surfaces.

4.3. Numerical Model and Boundary Conditions

To investigate the hydrodynamic characteristics of the pectoral fin, a three-dimensional incompressible flow model was established in STAR-CCM + 2306. The computational domain was defined as a rectangular prism designed to minimize blockage effects. The domain extends 0.20 m in the streamwise direction and 0.50 m in the transverse direction, with a spanwise depth of 0.20 m to fully accommodate the fin and its wake development. The main flow direction was aligned with the streamwise axis connecting the inlet and outlet boundaries.

The inlet boundary was specified as a velocity inlet. While the water tunnel experiments were constrained to low-speed conditions (<1.2 m/s) due to experimental facility limitations, the numerical simulation was intentionally designed to investigate the hydrodynamic limits of the pectoral fin under high-speed cruising conditions. The rationale for selecting a significantly higher inlet flow speed (6 m/s) is two-fold. First, it allows for an exploration of the “performance ceiling” of the current fin design, providing a conservative analysis of flow separation mechanisms and vortex dynamics that are critical for future high-speed iterations of the robot. Second, at this higher velocity, the flow enters a fully developed turbulent regime (Re ≈ 1.5 × 10⁵), where the hydrodynamic coefficients of sharp-edged bodies typically reach a plateau. This ensures that the captured “worst-case” separation scenarios define the ultimate performance envelope of the prototype, regardless of the specific experimental speed constraints. Therefore, the inlet flow speed was set to 6 m/s. This higher velocity ensures that the flow enters a fully developed turbulent regime (Re ≈ 1.5 × 10⁵), allowing for a conservative analysis of flow separation mechanisms and vortex dynamics that are critical for future high-performance iterations of the robot. The focus here is to capture the “worst-case” separation scenarios that define the performance envelope.

The pectoral fin was modeled as a rigid body with a planform shape approximated by an isosceles triangle. To ensure physical consistency with the experimental component, the geometric parameters of this triangular model—including the aspect ratio, sweep angle of the leading edge, and total planform area—were strictly matched to the mean projection of the bio-inspired prototype described in Section 3.2. By utilizing this equivalent lifting-surface model, the simulation filters out secondary flow disturbances caused by the complex biological surface textures, allowing for a focused interrogation of the fundamental pressure-driven separation mechanisms that govern the planform’s hydrodynamic limits.

The two equal sides had a length of 0.0224 m, and the base length was 0.020 m. Critically, this triangular planform was specifically designed to maintain the identical reference area and root chord length as the bio-inspired experimental prototype. This ensures that the non-dimensional hydrodynamic coefficients (C_L and C_D) derived from the simulation remain mathematically comparable to the experimental measurements. The isosceles triangle serves as an equivalent lifting surface that captures the primary aspect ratio and sweep characteristics of the actual fin.

Although the experimental fin consists of a semi-rigid frame and a flexible covering that undergoes passive deformation during flapping, a rigid-body simplification is adopted in this numerical model. This choice serves to isolate the fundamental geometric influence on the hydrodynamic performance and provides a “baseline” state for analyzing flow separation. Furthermore, the sharp-edged triangular geometry provides a conservative estimate of the flow separation limits, as the organic curved profiles of the real fin typically function to delay stall. By neglecting the complexities of fluid–structure interaction (FSI), the study focuses on the primary pressure-driven mechanisms that dictate the performance limits of the triangular planform.

The triangle altitude was oriented parallel to the upper and lower symmetry boundaries. The fin surface was subjected to a no-slip wall boundary condition in order to accurately represent viscous effects at the solid–fluid interface. The fin was positioned sufficiently far from both the inlet and outlet, on the order of several chord lengths, such that the boundaries did not noticeably influence the local flow around the fin. This was confirmed by monitoring that further extension of the domain in the streamwise direction caused negligible changes in the integral force coefficients.

The working fluid was taken as incompressible, Newtonian water with constant properties. The density was set to 997.561 kg/m³, and the dynamic viscosity was 8.8871 × 10⁻⁴ Pa·s, corresponding to water at room temperature. The resulting kinematic viscosity is 8.91 × 10⁻⁷ m²/s.

Using the root chord length L = 0.0224 m as the characteristic length and the inlet velocity $U_{\infty }$ as the characteristic velocity, the Reynolds number of the flow is 1.5 × 10⁵, indicating a turbulent, separated flow regime in which boundary-layer development, flow separation, and wake formation play a significant role, similar to those reported in previous computational studies of fish pectoral fins [41].

It is worth noting that the pectoral fin in this study features a thin profile with sharp leading and side edges. According to classical hydrodynamics, for such sharp-edged geometries, the points of flow separation are geometrically fixed at the edges rather than being sensitive to the boundary layer transition. Consequently, the non-dimensional hydrodynamic coefficients (C_L and C_D) and the global flow structures (such as leading-edge vortices) exhibit significant Reynolds number independence once the flow is fully turbulent (Re > 10⁴). This provides a robust physical basis for utilizing the high-speed numerical results ($U_{\mathrm{\infty }}$ = 6 m/s, Re ≈ 1.5 × 10⁵) to interpret the lower-speed experimental observations ($U_{\mathrm{\infty }}$ ≤ 1.2 m/s, Re ≤ 3.0 × 10⁴). Since both regimes reside well within the Reynolds-independent turbulent region, the underlying separation mechanisms remain fundamentally similar. Therefore, the high-speed CFD analysis serves as a physically accurate and representative proxy for identifying the performance limits and efficiency saturation observed in the water-tunnel experiments.

4.4. Turbulence Model and Solver Settings

The flow around the pectoral fin was simulated using a Reynolds-averaged Navier–Stokes (RANS) approach. Given the moderate-to-high Reynolds number and the presence of separated flow regions at higher angles of attack, the k–ω shear stress transport (SST) turbulence model was employed. This model combines the advantages of the standard k–ω formulation near walls with the k–ε behavior in the free stream and has been widely demonstrated to provide reliable predictions for adverse-pressure-gradient and separated flows over lifting surfaces [42].

The fluid was assumed to be of constant density (incompressible flow), and the governing equations were solved in an implicit unsteady (transient) formulation. An implicit time integration scheme was used to advance the solution in time. A constant physical time step of 5 × 10⁻⁵ s was adopted, which yields a maximum Courant number smaller than unity in the finely meshed regions near the fin, ensuring temporal accuracy and numerical stability. For each physical time step, up to five inner iterations were performed to update the flow variables and reduce the residuals of the momentum and turbulence equations.

The total physical simulation time was set to 0.20 s, corresponding to more than fifty characteristic convection times based on the fin chord and inlet velocity. This duration was sufficient for the flow to evolve from the initial condition to a statistically stable state. Convergence within each time step was assessed by monitoring the normalized residuals and the hydrodynamic force histories: the residuals were reduced below 10⁻⁴, and the variations in the instantaneous lift and drag coefficients between successive time steps remained within 1% once the transient start-up phase had passed.

Near-wall resolution was ensured by means of prism-layer cells generated along the fin surface, as detailed in Section 4.5. The height of the first prism layer was chosen such that the non-dimensional wall distance y+ around most of the fin surface remained in the range 0.5 ≲ y+ ≲ 2, which is appropriate for resolving the viscous sublayer when using the k–ω SST model without wall functions. This near-wall resolution, combined with the transient RANS formulation and the chosen time step, enables the numerical model to capture the separated, unsteady turbulent flow around the pectoral fin with an appropriate balance between computational cost and predictive accuracy.

As shown in Figure 10, all residuals decrease significantly from their initial values during the first iterations, with the specific dissipation rate (Sdr) residual dropping by several orders of magnitude to the 10⁻⁵ level. After the initial transient phase, the residuals of continuity, momentum, and turbulent kinetic energy oscillate around nearly constant levels, which is typical behavior for implicit unsteady RANS simulations. Combined with the stable time histories of the lift and drag coefficients, this behavior indicates that the numerical solution has reached a statistically converged state and that the transient flow field is adequately resolved for the subsequent analysis.

4.5. Grid Generation and Grid Independence Verification

The accuracy and robustness of the numerical solution strongly depend on the quality of the computational grid. In this study, an unstructured hybrid mesh combining polyhedral cells in the bulk flow and prism layers near the fin surface was employed. The polyhedral cells provide good numerical stability and efficiency, whereas the prism layers enable accurate resolution of the near-wall flow and boundary layers. To balance computational cost and solution fidelity, a systematic grid-independence study was performed prior to the production runs.

4.5.1. Grid Setup

A three-dimensional automated mesh generator in STAR-CCM + 2306 was used to construct the computational grid. A hybrid mesh strategy combining polyhedral cells for the bulk flow and prism-layer meshes for the boundary layer was adopted. Polyhedral cells were chosen for their ability to efficiently discretize the volumetric fluid domain with optimized convergence properties.

A medium grid configuration with a total of 55,810 cells was selected as the baseline, providing a good compromise between accuracy and efficiency. The base cell size was set to 0.001 m, which is fine enough to resolve the flow near the fin and throughout the domain, while avoiding the excessive computational cost associated with overly refined grids. The surface grid growth rate was set to 1.2, ensuring a gradual increase in cell size from the fin surface to the far field and avoiding abrupt changes in grid spacing.

Fifteen layers of prism cells were used, with a stretch ratio of 1.5 between adjacent layers, allowing the boundary layer to be adequately resolved and the near-wall velocity gradients to be captured accurately. To further improve accuracy in critical regions (such as the leading edge, side edges, and wake region), local grid refinement was applied. A cylindrical refinement zone was created around the pectoral fin, and additional boundary-layer refinement was introduced to ensure that the flow characteristics in these regions were properly resolved. Figure 11 illustrates the mesh distribution around the pectoral fin, highlighting the refined grid near the leading edge, side edges, and wake region.

To verify the influence of the grid on the simulation results, grid independence tests were conducted with different grid densities. Simulations were run with coarse, medium, and fine grid setups, and the results were compared to assess the grid sensitivity. The results of the grid independence verification are shown in

Table 2. Grid Independence Verification Results.

Grid Scheme	Total Grid Cells	Drag Force	Difference from Previous Grid
Coarse Grid	15,342	42.5	-
Medium Grid	55,810	43.3	1.88%
Fine Grid	179,865	43.4	0.23%

.

As shown in Table 2, the drag force gradually stabilizes as the grid is refined. The difference in results between the medium and fine grids is only 0.23%, indicating that the medium grid setup is sufficient to achieve a sufficiently accurate simulation result. Therefore, the 55,810 grid cells medium grid configuration was selected for the final simulation, balancing both result accuracy and computational resource efficiency.

Through the grid independence verification analysis, it was confirmed that the medium grid configuration (55,810 cells) yields sufficiently accurate results while optimizing computational efficiency. This grid setup will be used for the subsequent simulations, ensuring reliable and accurate outcomes. Furthermore, the grid refinement around the pectoral fin’s leading edge, side edges, and wake regions ensures the accuracy of boundary layer flow and wake structure simulation, providing a solid foundation for further water dynamics analysis.

4.5.2. Reynolds Number Sensitivity and Validation

To directly address the discrepancy between the high-speed numerical (Re ≈ 1.5 × 10⁵) and low-speed experimental (Re ≤ 3.0 × 10⁴) regimes, an additional sensitivity simulation was conducted at $U_{\infty }$ = 1.2 m/s (matching the experimental upper limit) at an angle of attack of 15°. This comparative study serves as a quantitative validation to ensure the comparability of results across different Reynolds numbers. As shown in

Table 3. Comparison of hydrodynamic coefficients at different Reynolds numbers (AOA = 15°).

Condition	$\mathbf{Velocity} \mathbf{\left(}{\mathit{U}}_{\mathit{\infty }}\mathbf{\right)}$	Re	CL	CD	Deviation
Experimental Match	1.2 m/s	3.0 × 104	0.821	0.158	-
Numerical Baseline	6.0 m/s	1.5 × 105	0.854	0.164	3.8%

, the deviation in C_L and C_D between the high-speed baseline and the experimental matching case remains below 5%. This close agreement directly bridges the gap between the two flow conditions, proving that the flow mechanisms and force trends identified at 6.0 m/s are fully representative of the physics governing the experimental regime. This result empirically confirms the Reynolds number independence discussed in Section 4.3 and provides a solid foundation for using high-Re CFD data to interpret low-Re prototype performance.

4.6. Flow Field Analysis and Correlation with Experimental Results

While the water tunnel experiments provided macroscopic measurements of the propulsion performance (e.g., thrust and efficiency), they could not fully reveal the microscopic flow details responsible for force generation. Therefore, to elucidate the hydrodynamic mechanisms underlying the experimental observations, numerical simulations were employed to analyze the pressure field, velocity distribution, vortex dynamics, and viscosity field. In this section, the numerical results are presented and correlated with the experimental findings to explain the lift generation mechanism and the performance degradation (stall) observed at high angles of attack.

4.6.1. Pressure and Velocity Fields from at Different Angles of Attack

To investigate the fluid dynamics governing the force production, the pressure and velocity distributions around the pectoral fin were analyzed at various angles of attack (0°, 15°, 45°, and 90°), as shown in Figure 12 and Figure 13.

Figure 12 illustrates the pressure coefficient contours. At low angles of attack (0° and 15°), a distinct pressure difference is established between the pressure and suction sides, which is the primary source of the lift force. As the angle of attack increases to 45°, the pressure differential intensifies, corresponding to the increased propulsive forces observed in the experimental trials. However, at 90° angle of attack (representing the peak effective angle during high-amplitude flapping), the pressure distribution becomes highly non-uniform and disordered. This flow deterioration explains the experimental observation where propulsion efficiency drops significantly under extreme kinematic parameters.

Similarly, the velocity contours in Figure 13 reveal the transition from stable attached flow to massive separation. At the angle of attack 0° and 15°, the flow remains attached, ensuring efficient hydrodynamic performance. At 90° angle of attack, a large-scale stall occurs, characterized by chaotic vortices and reverse flow. These simulation results provide visual evidence for the “performance inflection point” identified in the experiments, confirming that flow separation is the physical cause of the drag penalty at high angles of attack. The breakdown of the smooth flow structure at 90° directly corresponds to the deterioration of lift forces reported in morphological and hydrodynamic studies [43].

Figure 14 illustrates the instantaneous velocity field around the pectoral fin at a high angle of attack. A pronounced low-velocity region is observed upstream of the fin base, corresponding to the stagnation zone where the incoming flow is decelerated. The flow is strongly accelerated along the two lateral edges of the triangular fin, forming high-speed regions on both sides. Downstream of the fin, a large recirculation zone develops, characterized by low-velocity reverse flow and a pair of shear layers enclosing the wake. These shear layers, highlighted by the strong velocity gradients between the high-speed outer flow and the low-speed core, are associated with the formation and shedding of vortical structures in the wake. The far-field flow remains nearly uniform, indicating that the influence of the boundaries on the local flow around the fin is limited.

Overall, the pressure and velocity fields demonstrate that while moderate angles of attack promote favorable suction and attached flow, extreme angles lead to extensive separation. While dynamic flapping may delay the onset of this separation, the static simulation confirms that at high effective angles of attack (analogous to the peak of a high-amplitude stroke), the flow structure is inherently prone to breakdown. This provides a clear physical mechanism for the efficiency limits observed in the dynamic experiments.

4.6.2. Lift and Drag Characteristics and Lift-to-Drag Ratio

To validate the numerical framework against the experimental data, the time-history and statistical averages of the hydrodynamic forces were analyzed. Figure 15 and Figure 16 show the convergence history of the lift and drag coefficients. After the initial start-up transient, both coefficients settle into a quasi-steady or periodic state, indicating a converged numerical solution. As illustrated in Figure 15, the oscillation amplitude of C_L remains constant after t = 0.1 s, confirming that the unsteady vortex shedding has reached statistical stability.

Figure 15 presents the evolution of the lift coefficient as a function of iteration (time) in the transient simulation. During the early stage, the lift coefficient undergoes a pronounced transient adjustment and changes its sign as the flow around the pectoral fin develops from the initial condition. After this start-up phase, the lift coefficient exhibits a clear periodic oscillation with an almost constant amplitude, indicating that the unsteady flow field and the associated vortex shedding have reached a statistically periodic state. This behavior confirms that the lift response is physically consistent with the separated, vortical flow structures observed in the flow-field visualizations.

Figure 16 shows the time evolution of the drag coefficient. At the very beginning of the simulation, the drag coefficient exhibits a high peak due to the impulsive start of the flow and the rapid establishment of the boundary layer on the fin surface. Subsequently, the drag coefficient decreases quickly and then fluctuates slightly around a nearly constant mean value, indicating that the global resistance experienced by the fin has reached a quasi-steady state. Together with the residual histories and the periodic behavior of the lift coefficient, this trend demonstrates that the drag prediction is numerically converged and physically reliable for evaluating the hydrodynamic performance of the pectoral fin.

Figure 17 summarizes the integral hydrodynamic performance (mean C_L, C_D, and C_L/C_D). The curves clearly demonstrate that while C_L increases with the angle of attack up to 45°, the concurrent rise in C_D leads to a significant degradation of the lift-to-drag ratio, particularly in the deep stall regime at 90°.

To bridge the gap between static numerical coefficients and dynamic experimental performance, the relationship between these parameters is established via the quasi-steady projection. In the experimental coordinate system, the measured net thrust coefficient (Cx) represents the streamwise projection of the lift vector minus the profile drag. Therefore, the lift-to-drag ratio (C_L/C_D) derived from CFD acts as a direct indicator of propulsive efficiency.

The relationship can be understood as follows:

• Thrust Production Mechanism: In a flapping cycle, the forward thrust is primarily derived from the projection of the lift vector. An increase in the lift coefficient C_L at moderate angles (0° to 45°) directly supports the experimental finding that increasing flapping amplitude (and thus the effective angle of attack) enhances mean thrust.
• Efficiency Saturation: The simulation reveals a precipitous drop in the lift-to-drag ratio as the angle of attack approaches 90°. This numerical trend provides the physical evidence for the “thrust saturation” observed in Figure 5 of Section 3.5. In the dynamic context, when the fin flaps at excessive amplitudes (e.g., 60°), the effective angle of attack periodically enters this high-drag “deep stall” regime. As shown in Figure 17, the massive surge in pressure drag (C_D) at 90°—caused by the massive flow separation visualized in Figure 13d—effectively counteracts the propulsive component. This static aerodynamic characteristic elucidates why the thrust efficiency in experiments tends to saturate or decline under extreme kinematic parameters, confirming that flow separation is the governing constraint.

4.6.3. Vorticity Structure and Wake Dynamics

To further interpret the unsteady forces, the vorticity vector distribution was visualized (Figure 18). The simulation reveals that the thrust generation is intrinsically linked to the shedding of coherent vortex structures. At high angles of attack, strong shear layers separate from the leading and trailing edges, forming a complex wake. This vortex shedding process, which was difficult to capture in the standard experimental setup, is identified here as the primary mechanism for the periodic force fluctuations and flow instability.

Figure 18 reveals the occurrence of flow separation, especially in the trailing edge region of the pectoral fin. In the plot, the direction and intensity of the vorticity vectors change significantly in the wake region, indicating that the flow begins to separate and form vortices. In the region behind the plate, the intense vortices indicate the complexity and instability of the flow. These regions’ vorticity vectors show clear signs of flow separation.

4.6.4. Turbulent and Effective Viscosity Distributions

Figure 19 and Figure 20 present the spatial variation in turbulent and effective viscosity. High viscosity regions are concentrated in the separated shear layers and the wake, indicating regions of intense energy dissipation. This energy loss mechanism, quantified by the high turbulent viscosity in the CFD results, elucidates why the propulsive efficiency of the prototype decreases when forced to flap at high amplitudes.

Turbulent viscosity is a key parameter that describes the viscous effects in turbulent flow, particularly in turbulent flows where it reflects the energy dissipation and vortex intensity. The color bar in Figure 19 shows the range of turbulent viscosity values.

The turbulent viscosity is higher in the leading edge region, indicating greater flow instability and turbulence, while the turbulent viscosity is lower in the wake region, suggesting smoother flow. The formation of vortices and flow separation increases the turbulent viscosity, which further affects the flow stability and characteristics. These analyses show that the variation in turbulent viscosity in different regions is closely related to the complexity and stability of the flow.

The effective viscosity distribution of the pectoral fin model as shown in Figure 20. Effective viscosity is an important parameter for describing the viscous and turbulent effects in fluid flow. Particularly in turbulent flows, it reflects the energy dissipation and the stability of the flow. The color bar in the figure shows the range of effective viscosity values, with red areas representing high effective viscosity and blue areas indicating low effective viscosity.

The effective viscosity is higher in the leading edge region, indicating greater flow instability and turbulence. In the downstream region, as the flow stabilizes, the turbulence intensity decreases, and the effective viscosity gradually decreases. The formation of flow separation and vortices leads to an increase in turbulent viscosity, further affecting the flow stability. This analysis helps to better understand the impact of turbulent flow on flow performance, especially the flow instability at high angles of attack.

The distributions of turbulent and effective viscosity correlate well with the observed pressure, velocity, and vorticity fields. High-viscosity zones coincide with separated shear layers and intense vortical motion.

While a direct numerical comparison between static CFD forces and dynamic experimental thrust is not applicable due to the differing kinematic frames, the qualitative trends are highly consistent. The experimental results showed stable thrust growth at low-to-medium amplitudes and performance limits at high amplitudes. Correspondingly, the CFD results demonstrate a linear lift increase in the attached-flow regime and a sharp efficiency drop (stall) in the separated-flow regime. This consistency verifies that the quasi-steady CFD model successfully captures the dominant flow physics—specifically the transition from attached to separated flow—that dictates the performance of the biomimetic pectoral fin.

4.6.5. Validation of Experimental Thrust Trends

To bridge the gap between the static numerical model and the dynamic experimental measurements, this section validates the experimental thrust trends using the hydrodynamic coefficients derived from CFD. Despite the difference in absolute inflow velocities, the numerical framework is linked to the experiments through the non-dimensional coefficients (C_L, C_D), which were proven in Section 4.5.2 to be Reynolds-independent for this sharp-edged fin geometry.

In the water tunnel experiments (Section 3.4), a key observation was the non-linear behavior of the thrust. As shown in Figure 5, while increasing the flapping amplitude from 15° to 30° resulted in a proportional thrust increase, further increasing the amplitude to 60° yielded diminishing returns in propulsion efficiency (thrust saturation). This phenomenon can now be validated by the static CFD results shown in Figure 17.

The connection is established as follows:

• Linear Regime Validation: At low flapping amplitudes (corresponding to low effective AOAs, e.g., 0–15°), the CFD results show a linear increase in the lift coefficient (C_L) and a consistently low drag coefficient (C_D). This explains the stable and efficient thrust generation observed in the low-amplitude experimental trials.
• Saturation Regime Validation: At high flapping amplitudes (corresponding to high AOAs, e.g., >45°), the experimental thrust growth slows down. The CFD simulation provides the physical validation for this limit: at 90° AOA, the flow enters a deep stall regime where the drag coefficient (C_D) increases by an order of magnitude, drastically reducing the effective lift-to-drag ratio.

Therefore, the “efficiency drop” predicted by the CFD model at high angles serves as a direct mechanistic validation for the “thrust saturation” measured in the experiments. While the static CFD model intentionally neglects the secondary unsteady force enhancements (such as LEV dynamics and added mass) discussed in Section 4.1, the high degree of qualitative consistency with the experimental thrust trends indicates that separation-induced pressure drag is indeed the primary physical constraint governing the performance envelope. The consistency between the high-Re simulation and low-Re experiments confirms that the performance limits of the robotic fish are physically governed by geometrically induced flow separation rather than viscous transition effects. This justifies the use of high-speed CFD as a computationally efficient and physically accurate proxy for predicting the operating envelopes of the dynamic prototype.

4.7. Summary of Hydrodynamic Characteristics

In this section, a detailed numerical study of the hydrodynamics of a triangular pectoral fin was carried out using CFD. The numerical model, boundary conditions, turbulence closure, and mesh configuration were carefully designed and validated through a grid-independence study, ensuring that the predicted hydrodynamic coefficients are sufficiently accurate and robust [44]. The robustness of the numerical model across different Reynolds numbers ensures that the captured flow structures are representative of the actual operating conditions of the robotic fish.

The flow field analysis revealed that the angle of attack has a pronounced influence on both the global performance and the local flow structures. While the rigid-body assumption may slightly overestimate the intensity of flow separation compared to the semi-rigid experimental prototype, the transition from attached to separated flow captured by the simulations provides a robust physical framework for interpreting the performance trends. At small and moderate angles of attack, the pressure and velocity fields remain relatively smooth, and the wake is well organized, supporting the efficient thrust generation observed in the stable swimming modes of the experiments. Conversely, at high angles of attack (representing high-amplitude flapping), the flow becomes fully separated with strong vortex shedding and high drag. This transition from attached flow to separated flow, captured by the simulations, provides the fundamental hydrodynamic mechanism responsible for the complex force fluctuations and potential efficiency saturation recorded during the dynamic tests.

The distributions of vorticity, turbulent viscosity, and effective viscosity provided a consistent physical picture of the flow: regions with strong vortical motion and separation are associated with enhanced mixing and energy dissipation, which in turn explains the observed trends in lift and drag.

In conclusion, the close alignment between the non-dimensional force trends in the equivalent triangular CFD model and the macroscopic behavior of the bio-inspired prototype confirms that the study’s dual approach is logically consistent. The high-speed static simulation successfully identifies the fundamental geometric stall boundaries that dictate the operational envelope of the dynamic robotic platform. These findings offer valuable guidelines for the design and optimization of robotic pectoral fins and form the hydrodynamic basis for the subsequent analysis of propulsive performance and motion control of the underwater robotic fish.

5. Conclusions

In this study, a comprehensive biomimetic robotic fish platform driven by dual-servo pectoral fins was developed, experimentally validated, and numerically analyzed to investigate the hydrodynamic mechanisms of pectoral fin-based propulsion. The work integrates mechanical design, embedded control, recirculating-water-tunnel experiments, and high-fidelity CFD simulations, forming a unified research framework for advancing low-noise, high-maneuverability underwater robotic systems.

First, a scalable robotic fish prototype was designed, featuring a two-degree-of-freedom pectoral fin mechanism capable of reproducing the coordinated flap-and-twist motions observed in manta rays. The integration of a Jetson Orin NX controller, PCA9685-based servo actuation, low-frequency underwater communication, and a CPG-based control strategy enabled stable rhythmic fin movements and flexible locomotion modes. This hardware–software architecture provides an adaptable platform for future underwater robotic applications requiring stealth, precision, and agility.

Second, controlled hydrodynamic experiments were conducted using a single-sided pectoral fin installed in a recirculating water tunnel. The results demonstrated that the fin can generate stable forward propulsion across multiple kinematic conditions, with thrust increasing markedly with flapping amplitude. At medium and high amplitudes, the fin produced regular periodic thrust waveforms, indicating the formation of coherent wake vortex structures. The experimental approach—balancing thrust against flow resistance to determine effective swimming speed—proved reliable and reproducible, providing key quantitative insights into the propulsive contribution of individual pectoral fins. These findings further highlight the advantages of biomimetic flapping propulsion for low-speed, low-disturbance underwater operations.

Third, a quasi-steady CFD study provided the theoretical link between flow physics and the experimental observations. By mapping the dynamic flapping amplitudes to effective angles of attack, the simulation successfully reproduced the efficiency trends measured in the water tunnel. Specifically, the CFD results confirmed that the ‘thrust saturation’ at high flapping amplitudes is physically driven by the onset of deep stall and massive flow separation observed at high angles of attack.

Overall, the combined experimental and numerical findings verify both the feasibility and the hydrodynamic effectiveness of pectoral fin-based propulsion. The results provide new insights into vortex-driven thrust generation, kinematic parameter optimization, and the performance boundaries of biomimetic fin mechanisms. These outcomes hold significant implications for the development of next-generation underwater robotic platforms, particularly those requiring low acoustic signatures, high maneuverability, and fine-scale control in complex environments.

Future work will extend this research in several directions. First, full-body robotic fish experiments integrating bilateral pectoral fins and caudal fin interactions will be conducted to evaluate whole-system propulsion efficiency. Second, PIV-based flow-field measurements will be incorporated to experimentally validate wake topology and vortex dynamics predicted by the CFD simulations. Finally, optimization studies leveraging multi-objective algorithms and advanced control frameworks will be pursued to enhance energy efficiency and maneuvering agility under real-world operating conditions.

Together, this study establishes a coherent methodology—linking mechanical design, hydrodynamic experimentation, and numerical modeling—that advances the theoretical understanding and practical development of biomimetic underwater propulsion systems.