# Hydrodynamic Performance of Aquatic Flapping: Efficiency of Underwater Flight in the Manta

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Manta Kinematics

^{®}(3-D Professional Ed., v1.5.4.8, Xcitex Inc., Cambridge, MA, USA, 2010) software. Single and multiple stroke cycles were analyzed by manually tracking the manta’s rostrum, tail base, and pectoral fin tip. As each manta was free to swim anywhere in its tank. As no scale could be placed on the animal and the various study sites did not have measurements on the individual mantas, linear distances of each digitized point was scaled relative to the animal’s body length (BL). The digitized data were used to measure relative swimming speed (U

_{BL}; BL/s), stroke frequency (f; Hz; 1/period of stroke cycle) and heave amplitude (A; BL; maximum vertical displacement of pectoral fin tips).

#### 2.2. Potential Flow Model

^{9}” method. The turbulent boundary layer properties were then determined via a momentum integral analysis assuming a 1/7-power velocity profile [22]. Using this boundary layer solver gave the skin friction drag over the body from both the laminar and turbulent portions of the boundary layer; however, this did not include the effect of mild trailing-edge separation that led to form drag. Form drag was estimated by determining the local angle of attack on each strip and the local frontal area, which was used to modify the skin friction drag coefficient for each strip.

#### 2.3. Viscous Model

^{®}Core™ i7-3770 CPU@ 3.4 GHz computer node were generally needed. Results presented here were obtained by simulating the flow over five fin strokes.

## 3. Results

#### 3.1. Manta Kinematics

_{BL}

_{BL}(Figure 9b; r = 0.154; p < 0.0579; d.f. = 150). The mean ± one standard deviation (SD) for A was 0.824 ± 0.225 BL.

_{BL}of 0.2 to 2.25 BL/s, the calculated U ranged from 0.37 to 4.16 m/s, based on a typical manta with BL = 1.85 m. Based on this BL, the amplitude of heave averaged 1.43 m. This range in speeds indicated that mantas would be swimming with a Reynolds number of 685,240 to 7,708,952. The Strouhal number (St) decreased asymptotically with increasing U

_{BL}(Figure 10). At U

_{BL}of 0.2 BL/s, St was 0.96, but decreased to 0.25 at U

_{BL}of 2.25 BL/s. U

_{BL}of approximately 0.72 BL/s represented the crossover point at which St moved into the optimal range of 0.2 to 0.4, where propulsive efficiency is considered to be maximal. All values >U

_{BL}of 0.72 BL/s were within the optimal range of St.

#### 3.2. Potential Flow Model

#### 3.3. Viscous Model

_{T}, opposite to x-direction) and drag (F

_{D}, along x-direction), respectively. The coefficients C

_{T}and C

_{D}were obtained using (C

_{T}, C

_{D}) = (T, D)/0.5ρU

^{2}BL

^{2}with ρ as the density of water. The coefficient of the resultant force along the forward direction of manta was denoted by C

_{X}= C

_{T}– C

_{D}. There was temporal variation of C

_{T}, C

_{D}and C

_{X}over one cycle when the simulations reached a periodic state, and a number of observations were made regarding these plots (Figure 18). Table 1 shows the mean values ($\overline{{C}_{T}}$, $\overline{{C}_{D}}$, and $\overline{{C}_{X}}$) of force coefficients over the last two flapping cycles.

_{T}happened when the pectoral fin flapping speed was relatively high (see the slope of the tip displacement), while negative C

_{T}appeared during the stroke reversals. In addition, the peak in C

_{T}during downstroke appeared near the moment as the pitching angle exhibit peak value. The above analysis suggested that the thrust production was correlated with both the flapping and the pitching of the pectoral fin. The viscous drag (C

_{D}) seemed to be minimally affected by the fin flapping, because variation of C

_{D}was much smaller than that of C

_{T}. The cycle-averaged value of C

_{T}is 0.111, which was slightly higher than the absolute value of cycle-averaged C

_{D}(0.097). This indicated that the simulation setup was close to steady swimming.

## 4. Discussion

#### 4.1. Comparisons among the Biological Data, Potential Flow, and Viscous Models

#### 4.2. Manta Efficiency

#### 4.3. Applications of Research (MantaBot)

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic showing the orientation of a pectoral fin relative to a ray and the coordinate system for the kinematic model; (

**b**) Image of a manta maximum upstroke and the normalized data extracted from the image compared to the at different time steps; (

**c**) Lateral image of a manta at half-cycle and the model twist angle at 3/4 span over a flapping cycle.

**Figure 2.**Boundary element method schematic showing the body elements and the wake elements. A body-fixed frame of reference is attached at the leading-edge of a swimmer. The inertial frame of reference is attached to the undisturbed fluid.

**Figure 3.**Left vertical axis is the percent change of a performance metric when the number of elements or time steps is doubled; Right vertical axis is the performance metric. (

**a**) Cruise velocity convergence as the number of body elements is increased; (

**b**) Economy convergence as the number of body elements is increased; (

**c**) Cruise velocity convergence as the number of time steps is increased; (

**d**) Economy convergence as the number of time steps is increased.

**Figure 4.**(

**a**) Lateral view image of manta swimming; (

**b**) rear view image of manta swimming; (

**c**) virtual skeletons and the manta model; (

**d**) the reconstructed model.

**Figure 5.**Schematic of the computational mesh and boundary conditions employed in the current simulation. The domain in this schematic was smaller than that of the simulation and one-fourth grid points are shown here.

**Figure 6.**Sequential frames of video of propulsive movements of the pectoral fins of a swimming manta. The time elapsed between each frame is indicated in the bottom right corner of each frame. Video courtesy of Julie Morton, Yap Divers, Yap, Micronesia.

**Figure 7.**Pathways following the movements of the fin tip (green), tail base (red), and eye (blue) of a swimming manta.

**Figure 8.**Comparison of vertical displacements of fin tip and swimming speeds for mantas over time. The average swimming speeds displayed for individual mantas were 0.29 (blue), 0.36 (black) and 0.76 (red) BL/s.

**Figure 9.**Relationships of pectoral fin oscillatory kinematics (frequency, heave amplitude) with swimming speed for all mantas measured. (

**a**) Frequency increased with increasing swimming speed. The black line shows the regression for the data; (

**b**) Heave amplitude remained constant over the range of swimming speeds. The solid black line indicates the mean heave amplitude and the black dashed lines show the mean heave amplitude ± one standard deviation.

**Figure 10.**Strouhal number (St) as a function of swimming speed. The shaded blue area indicates the region of St = 0.2–0.4, where maximum propulsive efficiency occurs. The line is a fitted curve based on a manta assumed to have a BL = 1.85 m, a flapping frequency from Equation (3), and a heave amplitude of 1.43 m.

**Figure 11.**(

**a**) Time-varying swimming speed for manta rays swimming at six different frequencies. From blue to red the oscillation frequency is increasing by increments of 0.1 Hz from 0.1 to 0.6 Hz; (

**b**) Cycle-averaged speed at as a function of frequency (left axis). Also, on the left axis is the biological data from Figure 7a marked by the blue points. On the right axis the cycle-averaged speed is normalized by fL.

**Figure 12.**(

**a**) St as a function of swimming speed per unit body length (left axis). The right axis shows the corresponding propulsive efficiency; (

**b**) The left axis shows the cost of transport as a function of swimming speed per unit body length. A non-dimensional cost of transport CoT* is also plotted on the left axis. The right axis shows the Froude efficiency.

**Figure 13.**(

**a**) Isosurfaces of the ${\lambda}_{2}$ criteria are shown to mark the vortex system (top view). Going from pink to purple the isovalues are ${\lambda}_{2}=-0.02,-0.03,$ and $-0.05,$ respectively for the vortices in (

**a**,

**b**,

**d**); (

**b**) Side view of vortex structures; (

**c**) Isosurfaces of the time-averaged x-component of velocity. Going from pink to purple the isovalues are 1%, 1.5% and 2% of the steady-state of the cycle-averaged velocity; (

**d**) Perspective view of vortex structures.

**Figure 14.**(

**a**) Time-averaged pressure coefficient distribution over the manta ray; (

**b**) Time-averaged skin friction coefficient distribution over the manta ray; (

**c**) Time-varying forces decomposed into pressure forces acting on the fins and body and shear forces acting on the fins and body. The form drag (pressure force) acting on the fins is also included; (

**d**) Time-averaged net force coefficient integrated over chordwise strips.

**Figure 15.**(

**a**) w component of velocity in wake of the ray on the z = 0 plane; (

**b**) Root-mean-squared circulation distribution along the span of the ray.

**Figure 16.**Sequential frames of pectoral fins’ movement of the reconstructed manta model in perspective view. The time elapsed between each frame is indicated in the bottom right corner of each frame.

**Figure 17.**Lateral view of a manta’s flapping motion during (

**a1**) downstroke and (

**a2**) upstroke. The red line with an arrow is the tip trajectory; (

**b1**) Five marker points in the leading edge of the pectoral fin; (

**b2**) up and down displacement (Y, normalized by body length) of the five marker points during a flapping cycle; (

**c1**) three chord lines at 0.3 (red), 0.6 (green) and 0.85 (blue) half-span on the pectoral fin; the pithing angle of these three chord lines during a flapping cycle.

**Figure 18.**(

**a**) Time history of force coefficients during a flapping cycle; (

**b**) Distribution of $\overline{{C}_{X}}$ on the manta surface.

**Figure 19.**(

**a**) Ventral; (

**b**) lateral; and (

**c**) perspective view of the 3D flow structures at t/T = 0.21; (

**d**) vorticity and velocity field on a vertical slice-cut 0.55 half-span away from the middle of the body, which is located in the thrust producing region of the fin. The rotating directions of an upstroke generated vortex ring (V3) and a downstroke generated vortex ring (V4) are labeled by black arrows (

**a**,

**c**). Red and green arrows in (

**d**) represent backward and forward jets, respectively.

**Figure 20.**Sequential images of MantaBot swimming in a pool from right to left. The red line traces the pathway of the fin tip in a sinusoidal motion and the green line traces the trajectory of MantaBot.

$\overline{{\mathit{C}}_{\mathit{T}}}$ | $\overline{{\mathit{C}}_{\mathit{D}}}$ | $\overline{{\mathit{C}}_{\mathit{X}}}$ |
---|---|---|

0.111 | 0.097 | 0.014 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Fish, F.E.; Schreiber, C.M.; Moored, K.W.; Liu, G.; Dong, H.; Bart-Smith, H.
Hydrodynamic Performance of Aquatic Flapping: Efficiency of Underwater Flight in the Manta. *Aerospace* **2016**, *3*, 20.
https://doi.org/10.3390/aerospace3030020

**AMA Style**

Fish FE, Schreiber CM, Moored KW, Liu G, Dong H, Bart-Smith H.
Hydrodynamic Performance of Aquatic Flapping: Efficiency of Underwater Flight in the Manta. *Aerospace*. 2016; 3(3):20.
https://doi.org/10.3390/aerospace3030020

**Chicago/Turabian Style**

Fish, Frank E., Christian M. Schreiber, Keith W. Moored, Geng Liu, Haibo Dong, and Hilary Bart-Smith.
2016. "Hydrodynamic Performance of Aquatic Flapping: Efficiency of Underwater Flight in the Manta" *Aerospace* 3, no. 3: 20.
https://doi.org/10.3390/aerospace3030020