Fish robotics: multi-fin propulsion and the coupling of fin phase, spacing, and compliance

A fundamental challenge when designing underwater robots that are propelled by flapping fish-like fins is how to arrange and coordinate the fins to create the forces required for effective propulsion and agile maneuvers. This basic locomotory requirement is difficult, in part, because flapping fins do not create a constant thrust, but generate forces that vary in magnitude and direction throughout the fin's undulatory stroke. The forces created by fins are also highly dependent on the fin-fluid interactions that occur as wakes are passed between fins and flow moves along the body [1, 2]. The challenges associated with coordinating multiple propulsors and tailoring forces to the dynamics of the body are not unique to systems driven by fins, but are an underlying issue faced by many classes of robots and vehicles. As examples, the coordination of robot limbs to produce smooth gaits for walking and to dynamically stabilize against external disturbances, and the ways in which forces from multiple driven wheels are used to help maneuver a moving car are enduring areas of research [3–5]. Thus, in underwater vehicles, and more specifically in biologically inspired swimming robots, understanding how to design multiple, interacting fins to produce and shape propulsive forces is key to producing good locomotion and agile maneuvers.

Ray-finned fishes (Actinopterygii) have solved this challenge using a variety of fin morphologies and gaits. Rarely is the entire force that a fish generates to swim created by a single fin, flapping alone. Rather, fin and body motions are coordinated so that component forces add constructively, and so that downstream fins engage beneficially with the wakes shed by the upstream fins and body [6–8]. There is not a single solution to accomplishing this and across fish species different fin coordination strategies have been observed. Nor is there necessarily a single strategy used by an individual fish across its range of swimming speeds. There is great diversity in the sizes and locations of the median fins, and with this, differences in the timing and amplitudes of how each fin is flapped [9–11]. Freshwater sunfish (genus Lepomis), for example, have a symmetrical arrangement of the anal and dorsal fin, whereas the dorsal fins of trout (genus Salvelinus) and salmon (genus Salmo) are located farther forward on the body than the anal fins. Many fish that swim at higher speeds, such as tuna (tribe Thunnini), have caudal fins that are stiff, dorsal and anal fins that are elongated but thin, and finlets along the tail that help guide flow [12]. Additionally, fish can vary the morphology and kinematics of fins depending on the locomotory task, and even modulate a fin's stiffnesses as swimming speeds change [6, 13–16]. Sailfish (genus Istiophoridae), for example, retract much of the dorsal fin nearly flat against the body when swimming over distance, and extend the fin when maneuvering to catch prey so that the fin acts as a large, flexible control surface [2, 17]. Understanding how different fin arrangements, structural characteristics, and kinematic relations affect the flows that move between fins and the resultant forces is critical to adapting these biological strategies to robots.

When fins interact with the time-varying wakes produced by other flapping fins (figure 1(D)), the phasing between fins is among the most critical factors that affect the magnitude, direction and time-course of forces, and the propulsive efficiency of fins. Geder et al [18], Boschitsch et al [19] and Matthews and Lauder [20] found that tuning the phase relationship between a pair of fins can greatly affect thrust production and propulsive efficiency. Mignano et al [21] found, using robotic and 2D numerical models of fins, that the magnitudes of both the mean thrust and the root mean square (RMS) lateral force varied significantly as the phase difference between the dorsal/anal fin and caudal fin was altered. The changes in force were associated with changes in the way the downstream fin interacted with the wake shed by the upstream fins and body. The appropriate phase relationship between fins enabled the downstream fin to align with the wake flow and to entrain vortices produced by the upstream fin, leading to increased thrust production [21–25]. Flammang et al [26] demonstrated, using 3D volumetric flow visualization, that the vortex wake shed by the dorsal and anal fins can be entrained by the tail within a single fin beat. Using 3D computational fluid dynamic (CFD) simulations, Han et al [24] have shown that changing the dorsal/anal fin flapping phase affects the timing of the interaction between the vortices shed by the dorsal/anal fins and the leading-edge vortex of the caudal fin, with appropriate phasing increasing thrust by 25.6%. Similarly, Kurt and Moored [27] have shown with a set of in-line arranged propulsors, that fin spacing and phasing between the fins can significantly affect the timing of vortex impingement and net thrust.

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Based on the importance that the phase-based timing of fins has on the propulsive force; it is reasonable to expect that the kinematic and physical characteristics of fins that impact the timing of the wake will also affect the propulsive forces. These factors include the relative location and compliance of the fins, as well as fin size, shape, speed, and trajectories. Thus, the objective of this study was to understand how the horizontal (d) and vertical (h) spacing of paired, compliant flapping fins affected the propulsive forces produced as the phase difference between fins ( Φ ) was varied from 0 to 360 degrees. These studies contribute to understanding how the median fins of robotic swimmers should be located on the vehicle body and driven to create the forces desired for effective locomotion.

The study was conducted using two robotic swimmers–the 'HVAC' and 'PDAC' robots–and a series of CFD simulations (figure 1) to characterize the effect of varying 2D fin spacing and compliance on the relationship between fin phasing and propulsive forces. This work aimed to assess whether the identified trends in propulsive forces due to varying parameters were consistent and applicable to a broader spectrum of engineered systems. The use of multiple robotic platforms allowed for a more extensive exploration of the parameter space. Numerical CFD simulations were used to explore fin spacing with a finer resolution and provide additional insight into the fluidic fin-wake interaction, augmenting the results gathered with robotic experiments. The HVAC robot (figure 1(A)) has two fins (dorsal and caudal), each actuated at its base. The horizontal ( d ) and vertical ( h ) position of the dorsal fin can be adjusted. Studies using the HVAC robot were conducted using two sets of fins of different compliance (c). The PDAC robot has a dorsal, anal, and caudal fin, and an actuated joint at the peduncle or base of the tail (figure 1(B)). In this study the anal fin and peduncle joint were not actuated, being fixed to align with the dorsoventral plane during all trials. The position of the dorsal fin can be varied rostro-caudally to change the horizontal spacing between the dorsal and caudal fin ( d ). Numerical CFD simulations (COMSOL Multiphysics, Comsol Inc., Sweden) modelled the flapping fins as 2D rigid foils and enabled forces to be predicted and flows to be visualized as the phase difference and horizontal separation between fins were changed (figure 1(C)). For all pairs of fins, the horizontal distance between the fins ( d ) was measured in the horizontal plane, from the aft-most edge of the dorsal fin to the leading edge of the caudal fin. The vertical distance between the fins ( h ) for both robots was measured from the midline of the dorsal fin to the midline of the caudal fin.

The HVAC is a two-finned system that enables both the horizontal ( d ) and vertical spacing ( h ) of the compliant fins to be adjusted (figure 2(A)). It was designed as a complementary platform to the PDAC, having fins of similar size, actuated to produce similar fin motions, and attachments that allow the system to be tested using the same air-bearing carriage as the PDAC. The HVAC is comprised of three main subsystems: (1) a fish-shaped shell, (2) two actuated fins, the dorsal and the caudal, and (3) a control and power module. The external shell smooths and directs oncoming flow around the internal support structure and towards the median fins. It is fabricated from polyethylene terephthalate glycol sheets, and its size and shape can be adjusted by adding or removing panels as fin spacing is changed. This allows the location of the caudal fin to be adjusted by adjusting the length, and attachment point of removable tail beams. Each of the median fins is made using five compliant fin rays enclosed in an 84% polyester/16% elastane webbing (Under Armour, Inc., Baltimore, MD, USA) (figure 2(C)). The fin rays are manufactured using SLA (Durable Resin, FormLabs, Somerville, MA, USA) and have rectangular cross sections that taper from base to tip so that the fins are more flexible near the tip than at the base. Details of fin ray shapes and taper are described in [21] and were tuned such that the fins of both robots exhibited the same curvature when flapped. For the HVAC system, two distinct sets of fin rays were developed: one mirroring the fin compliance of the other robotic system (PDAC), and a second set designed for greater compliance. The augmentation of compliance in the latter set was achieved by reducing the thickness and taper of the fin rays. This adjustment's effectiveness was experimentally validated by measuring the tip deflection of each set when a calibrated load was applied at their tips. The results showed a 30% increase in tip deflection for the more flexible set relative to the baseline fin. Each fin is driven at its base using a waterproof servomotor (WR-4401, Xpert-RC USA, Bellevue, WA, USA). Servomotor trajectories are defined using an external PC and commanded using an 8-bit AVR microcontroller (Arduino UNO, Arduino LLC, Italy).

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The PDAC system is a robotic experimental platform developed to investigate multi-fin and body propulsion (figure 1(B)). The PDAC robot is comprised of three main subsystems: (1) a fish-shaped body with an actuated peduncle; (2) three fins (dorsal, anal and caudal); and (3) a support structure that suspends the system in a flow tank and that houses the actuators and control electronics. A detailed description of the PDAC robot is presented in [21].

The 2D CFD simulation of a pair of interacting rigid fins was developed (COMSOL

Multiphysics, Comsol Inc., Sweden). The two, 2-dimensional rectangular fins were 90 mm and 107.5 mm long and 2 mm wide and flapped in a simulated flow tank. A turbulent k-ε model with Reynolds-averaged Navier–Stokes formulation was used to calculate flows and forces [28]. Details of model approach and development are described in [21].

Experimental trials were conducted using the robotic and numerical systems to investigate the impact of fin spacing and compliance had on the net forces produced by and flows surrounding the fins (table 1).

In the PDAC robot, the horizontal spacing between the dorsal and caudal fins ( d ) can be varied continuously between 48 and 128 mm, with tests conducted for d = 48, 88 and 128 mm (table 1).

The HVAC robot was tested with three horizontal fin spacings ( d = 30, 80, 120 mm) and two vertical fin spacings ( h = 50, 100) for a total of six geometric configurations (figure 2(D)). To test the impact of fin compliance, forces were measured and flows were recorded using the HVAC robot using two sets of fins of different compliance, described in the previous section.

Numerical simulations were conducted as the distance between the dorsal fin and the caudal fin was increased incrementally by 20 mm from 40 mm to 240 mm (table 1).

For all tests, with both robots and the numerical simulations, the fins were flapped at 1.0 Hz with an amplitude of ±22°, with an oncoming flow velocity of 200 mm s⁻¹. These flow conditions correspond to a Strouhal number of approximately 0.5 with a Reynolds number of 105 660 and consistent with conditions observed with swimming fish [29]. For each configuration of different fin spacings, net forces and flows were measured for all systems as the phase relationship between the fins was varied from 0° to 360° (table 1).

The robotic systems were attached to an air bearing carriage (New Way Air Bearings, Aston, PA, USA) that suspended the robots underwater in a recirculating flow tank for testing. Net thrust and lateral forces produced by the robots were measured with a sample frequency of 250 Hz using two load cells (LSB200, Futek Inc., Irvine, CA, USA). High-speed videos (fps = 500, resolution = 1024 × 1024) of the ventral and lateral views were captured simultaneously with forces measurements. Digital particle image velocimetry was used to image and analyze the wakes produced by the robotic fins, as described in previous research [21, 30–32]. The laser-excited light sheet was projected orthogonally to the plane of the fins and aligned with the region where the wake shed by the upstream fin would interact with the downstream fin (figure 2(B)).

From each experimental trial, the force measurements from ten consecutive fin beat cycles were processed to produce force traces, force-phase curves, and averages for further analysis. After discarding the measurements from the first three fin beat cycles to minimize the inclusion of transients, measurements of the subsequent ten cycles in each trial were extracted and aligned using a synchronization signal generated by each robot's microcontroller. The raw force recordings were processed with a median filter with a five-sample window, followed by a low-pass filter which used a Kaiser window with a passband frequency of 9 Hz, as stopband frequency of 12 Hz and a peak error of 10⁻³ [33]. Mean net forces were then calculated from these force recordings for each tested condition. Since the mean of the lateral force over each fin beat was approximately zero, the RMS rather than the mean is used to quantify the net lateral forces. A 2nd order sinusoidal model was fit to the force-phase relationship data to facilitate comparison between data from each system and the various fin configurations and calculate phases at which the maximum mean thrust (Φ_THR) and minimum RMS lateral forces (Φ_LAT) occurred.

The 2D computational fluid dynamic model was used to better understand flow dynamics with analysis focusing on characteristics of the flow in the region at and around the leading edge of the downstream fin. Specifically, the vorticity of the wake shed by the upstream fin and the angle-of-attack (AoA) of the downstream fin were computed across a fin beat. At every tested distance, the amount of time spent by the downstream fin at low angles-of-attack, specifically [−25°, 25°], were computed. Estimates of net force were calculated by integrating the total stress of both fins in each direction. Similar to the experimentation using robots, each numerical simulation was run for ten consecutive fin beat cycles. To reduce the effect of transience, the initial three fin beat cycles were disregarded.

The range of forces that could be achieved by adjusting the phase difference between fins was highly dependent on fin spacing ( d , h ) and compliance (c) . The maximum and minimum mean thrust and the RMS lateral force that could be achieved were altered, and the phase differences at which the maximum and minimum mean forces occurred were shifted as these experimental factors were varied. Importantly, the underlying relationship between force and phase difference–where the mean thrust, and the RMS lateral force moved between a peak and trough as the phase difference between fins was modified–was preserved across all experimental conditions. No matter the combination of physical settings, a wide range of propulsive forces could be created by adjusting the phase between fins.

In conjunction with affecting the propulsive force, changes made to the experimental factors ( d, h , and c ) altered the timing, direction, and strength of the time-varying wake that was encountered by the downstream fin. None of the factors had a strong effect on the undulatory structure of the wake as it left the upstream fin, but the upstream fin's compliance did affect the direction and vorticity of the wake as the wake was shed. The major impacts of changes in fin spacing were on the engagement of the downstream fin with the oncoming flow.

The impact that each factor had on thrust (as it varied with phase difference) will be described using the maximum mean thrust ( Max _THR ) that was attained, the phase difference at which the maximum mean thrust occurred (Φ_THR), and the range over which the magnitude of the mean thrust varied as the phase difference between fins was cycled from 0° to 360° ( R _THR) (figure 3(B), left). Similarly, the impact each factor had on the lateral force will be described using the minimum RMS lateral forces (Min_LAT), the phase difference at which Min_LAT was observed (Φ_LAT), and the range over which the RMS lateral forces varied ( R _LAT) (figure 3(B), right). RMS lateral force, rather than mean (figure 3(A), left), is used because the lateral force exhibits two opposing peaks during a fin beat, with a mean net of approximately zero (figure 3(A), right).

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The engagement of the downstream fin with the undulatory wake shed by the upstream fin will be characterized using two parameters; the peak vorticity ( ω ) of the flow measured at the leading edge of the downstream fin (figure 4(A)), and by the proportion of the downstream fin's flapping period, as a percentage, during which the AoA between the fin's leading edge and the oncoming flow was less than 25° ( τ_<25 ) (figure 4(B)). Angle-of-attack(AoA) provides an insight into how the flow interacts with the fin. In this study, we consider the AoA to be low when it is under 25 degrees. τ_<25 served as a good indicator of a favorable interaction of the downstream fin with the wake shed by the upstream fin. Different values of τ_<25 was calculated when the phase difference between the fins was varied at each fin spacing. These values followed a similar trend as net mean thrust forces when the phase was altered (figure 4(B)). For example, at a particular distance, when net mean thrust was high, the angle of attack of the downstream fin remained low for majority of the fin beat ( τ_<25 = 64%) (figures 4(B) and (D)). In contrast, when net mean thrust was low, the angle of attack of the downstream fin remained high for majority of the fin beat ( τ_<25 = 19%, figures 4(B) and (C)).

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3.3.1. On net thrust

As the horizontal distance between fins ( d ) was increased, the range of forces that could be achieved by modulating phase changed steadily, albeit not monotonically (figures 5 and 6(A)) and the phase differences at which the maximum and minimum net thrust occurred increased nearly linearly (figures 5 and 6(B)). Patterns in the force-phase curves were consistent between the numerical and robotic models (figure 5). Magnitudes differed between systems, but the ranges and percent changes in force were similar. In the simulation of the two-fin system, the notable escalation in net force magnitude and range is likely attributed to the employment of rigid fins. Rigid fins, even when sharing identical areas and kinematics, outperform compliant fins in total force generation. Although the simulated forces are more pronounced, the fundamental trends, especially in relation to fin spacing, exhibit remarkable consistency with those observed in robotic implementations. When the fins were positioned closely ( d ∼ 40 mm), the numerical and robotic models produced relatively low maximum mean net thrusts (Max_THR) and could achieve a small range of net thrust ( R _THR) when the phase difference between the fins was modulated (figure 6(A)). The numerical model produced a maximum mean thrust of Max_THR = 93 mN with R _THR= 142 mN. The PDAC produced a Max_THR = 77 mN and the HVAC produced Max_THR = 74 mN with R _THR of 42 mN and 16 mN, respectively. In all tests, the HVAC system produced the smallest range of forces.

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As the horizontal separation between fins was increased, the maximum mean net thrust and range of net thrust that could be produced increased until reaching their highest values at a fin separation of d = 80–120 mm (figure 6(A)). The numerical model reached Max_THR of 128 mN and a R_THR of 176 mN at d = 120 mm. This was a 27% increase over MAX_THR when the fins were at their smallest separation ( d = 40 mm). The improvements for the two robot systems were similar. Maximum mean net thrust for the PDAC and HVAC reached 90 mN and 87 mN, a 17% and 15% improvement, respectively. The R _THR produced by the PDAC remained nearly the same but R _THR doubled to 23 mN for the HVAC robot.

Max_THR and R _THR for each system decreased steadily after their peaks until reaching their smallest values when fin separation was greatest. At d = 240 mm, the numerical model produced a maximum mean net thrust Max_THR= 88 mN and a range R _THR = 116 mN. Although Max_THR was similar to the values produced when d ∼ 40 mm, the range of net thrust was smaller by approximately 30 mN, and much smaller than at the peak. At a separation of d ∼ 130 mm, the PDAC system produced Max_THR = 64 mN with R _THR = 27 mN, and the HVAC robot produced a Max_THR = 75 mN with R _THR = 20 mN. These values were similar to, but smaller than, the R _THR and Max_THR that occurred when the fins were at their closest spacing.

Coupled to the changes that occurred in mean net thrust were changes to the phase difference between fins that was required to produce a desired net thrust. Essentially, as fin separation increased the curve representing mean net thrust versus phase difference shifted to the right (figure 5 top row). Specifically, there was a linear increase in the phase difference at which the maximum mean net thrust forces occurred as fin separation increased (figure 6(B)). For the HVAC and PDAC robots, the phase at which the maximum mean net thrust (Max_THR) occurred changed at rates of 0.64 °/mm (R² = 0.83) and 1.24° mm (R² = 0.95), respectively. For the numerical simulation, Max_THR increased linearly with distance at a rate of 1.64 °/mm (R² = 0.99). The increase in Max_THR with distance occurred for all distances tested with no indication that the change in phase would stop until phase difference had completed a full cycle.

3.3.2. On lateral force

3.3.3. On flows

Direction and AoA-An increase to the horizontal distance between fins, with no change to the fins' phase difference, changed the time-varying alignment of the downstream fin with the wake that was shed by the upstream fin. The new alignment altered how the AoA of the flow relative to the fin varied through the flapping cycle (figures 7(A) and (B)). The highest mean net thrust forces were produced when the downstream fin maintained a low AoA (<25°) with the oncoming flow during most of the fin beat (figure 7(D)). This resulted in minimal flow separation around the fin's leading edge. In contrast, low mean net thrust forces were produced when the AoA between the flow and fin remained high (>25°), and flow separation was significant, for much of the fin beat (figure 7(D)). In both good and bad cases, the AoA often exceeded 40° when the fin was at its most lateral position in the fin beat. In high net thrust cases, the AoA dropped quickly, while in low net thrust cases the high AoA was sustained.

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Across all distances, when the fins were flapped with a phase difference that maximized mean net thrust, the percentage of the period during which the fin's angle-of-attack remained below 25° ( τ _{< 25}) was maximized. For example, when the fins were 100 mm apart and flapped at Φ_THR = 195°, the phase at which maximum mean net thrust was produced, the τ _{< 25}= 55% (figure 7(D)). When the distance between the fins was increased from 100 mm to 200 mm while keeping the phase relationship constant (Φ= 195°), the lowest mean net thrust forces were produced and the τ _{< 25} decreased by approximately 2.4 times to 22% (figures 7(B) and (D)). When the phase relationship between the fins was increased ( d = 200 mm, Φ= 360°) such that maximum mean net thrust forces were produced, the τ _{< 25} increased to 66% (figure 7(D)).

Speed and vorticity-As horizontal spacing increased, there was an exponential decrease (R² = 0.93) in the peak vorticity of the wake ( ω ) where it reached the leading edge of the downstream fin (figure 7(C)). For the numerical model, when d = 100 mm at Φ_THR, the peak vorticity of the flow at the leading edge of the downstream fin was approximately 19 s⁻¹. When the horizontal distance of the fins was increased to 200 mm, the vorticity of the flow nearly halved to 11 s⁻¹. Even though vorticity of the wake at the leading edge of the downstream fin always decreased as the horizontal distance between the fins increased, mean net thrust force may increase or decrease with the same increase in d , depending on initial force-phase condition.

Experiments in which the vertical fin spacing was varied were conducted exclusively with the HVAC robot, the only system with the ability to alter vertical fin separation. Wake flows were visualized using particle image velocimetry (PIV).

3.4.1. On net thrust

Increasing the vertical separation ( h ) between the fins resulted in changes to net thrust and phase that were qualitatively similar to the changes that occurred when the horizontal separation ( d ) was changed (figure 8(A)). The maximums of Max _THR and R _THR did not occur when the fins were closest ( d = 30 mm, h = 50 mm) and nearly aligned, but occurred when the vertical separation was increased by approximately 50 mm ( d = 30 mm, h = 100 mm). At this separation, mean net thrust reached a maximum of Max_THR = 87 mN and the range of achievable forces R _THR = 30 mN (figure 8(B)). These values were similar to those that occurred when the fins were more closely aligned ( h = 50 mm) but separated horizontally by an additional 50 mm ( d = 80 mm, h = 50 mm), Max_THR = 85 mN and R _THR = 35 mN (figure 8(A)). When either the vertical or horizontal separation was increased further, Max_THR and R_THR generally decreased. This suggests that there is a 2D separation between interacting fins that maximizes the achievable net thrust.

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Similar to the observed relationship between phase and horizontal separation, the phase at which the maximum net thrust occurred (Φ_THR) increased consistently with vertical separation. The total change was not the same at all horizontal positions, nor did it vary consistently. For example, when the fins were close to each other horizontally (d = 30 mm) Φ_THR increased by 15°, at the middle location Φ_THR ( d = 80 mm) increased by 36° and at the furthest distance ( d = 150 mm) there was an increase of only 3°.

3.4.2. On lateral force

3.4.3. On flows

Experiments were conducted using the HVAC robot and two pairs of fins. One pair had a baseline compliance and the other was 30% more compliant. Sets of comparative trials were conducted at three fin separations: ( d , h ) = (30 mm, 50 mm), (80, 50), and (30, 100).

3.5.1. On net thrust

Increasing fin compliance decreased the overall magnitude and range of net thrust forces that were produced (figure 9(A) top row). The decrease in net thrust was significant at all phase differences and was largest at, or very near, the phase at which Max _THR occurred. For the closest configuration, ( d , h ) = (30 mm, 50 mm), mean net thrust produced by the more complaint fins was 20%–32% lower than for the stiffer fins, and the range of net thrust ( R _THR) was 59% lower. The difference was even greater when fin separation was increased. When the fins were separated by an additional 50 mm horizontally and vertically, ( d , h ) = (80 mm, 100 mm), Max_THR decreased by 42% and 43%, and R _THR by 60% and 69%, respectively. Taken together, the decreased maximum and range of net thrust resulted in the net thrust produced by the more compliant fins to change less with horizontal ( d ) and vertical distance ( h ), than did the net thrust produced by the stiffer fins.

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Increasing compliance increased the phase differences at which the maximum and minimum mean net thrusts were created, albeit not uniformly for the three fin configurations for separation. The increase in fin compliance produced a 20° increase in Φ_THR for the closest fin configuration. Increasing the fin separation by 50 mm in either the horizontal or vertical separation increased Φ_THR, by 50° and by only 7°, respectively with the increase in fin compliance.

3.5.2. On lateral force

3.5.3. On flows