# Jellyfish and Fish Solve the Challenges of Turning Dynamics Similarly to Achieve High Maneuverability

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## Abstract

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## 1. Introduction

_{P}= $\sum _{i=1}^{N}m$

_{i}r

_{i}

^{2}where I

_{P}represents the sum moments of inertia for the constituent parts (i … N) of a swimmers body with m

_{i}denoting that body part’s mass (e.g., the head or tail of the body) and r

_{i}its distance from the whole-body center of rotation. There are straightforward means to minimize I

_{P}, e.g., the mass of the body can be re-arranged to place body components closer to the axis of whole-body rotation. This is commonly achieved by bending body parts closer to the axis during a turn. Flexible bodies that allow bending by animal swimmers permit dramatically greater angular velocities during turns than are possible for rigid animal bodies or rigid human-engineered structures [18]. However, it remains unclear how these flexible swimmers resolve the fundamentally conflicting demands of high torque production (expanded body configuration) with those of low moment of inertia (contracted body configuration) to achieve high turning performance. The results are important for understanding maneuverability by swimming animals, and potentially, human engineered vehicles.

## 2. Materials and Methods

#### 2.1. Animals and Imaging

#### 2.2. Particle Image Velocimetry (PIV)

#### 2.3. Pressure and Torque Measurement

#### 2.4. Turning Equations of Motion

#### 2.5. Moment of Inertia and Angular Velocity Measurements

_{i}having a centroid located at distance r

_{i}from the whole body centroid. The area moment of inertia for each frame p was then calculated as:

## 3. Results

^{−2}. This motion was transmitted to the adjacent water via a process known as the acceleration reaction or added-mass effect [25].

## 4. Discussion

^{−1}, where U and L are the nominal animal swimming speed and size, respectively, and ν is the kinematic viscosity of the water), angular momentum generated during periods of maximum torque would experience rapid viscous dissipation, leaving little remaining angular momentum to complete the turn during the subsequent period of major body bending. For large animals with body lengths on the order of tens of meters, power requirements for rapid body bending may exceed the available muscle capacity. In geometrically similar animals, angular acceleration scales to the −2/3 power of body mass [26], making it more difficult for large animals to generate the initial pressure transient or to alter their moment of inertia through body rearrangement to increase their angular velocity. Hence, very large swimmers such as whales may not bend as readily as smaller animal swimmers such as zebrafish [27]. However, the majority of animal swimmers exist within the millimeter to meter size range [28,29], in which a time-varying lever arm enabled by body bending would provide favorable performance advantages relative to rigid body turning mechanics.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Turning parameters for replicate medusae (Aurelia aurita) executing turns of different magnitude. Variable designations are same as in Figure 1: torque per unit depth (red line), angular velocity (blue line) and moment of inertia (green line). Bell diameter and total turn angle for each turn: (

**a**) 2.7 cm, 53°, (

**b**) 1.8 cm, 50°, (

**c**) 2.3 cm, 13°, (

**d**) 4.9 cm, 30°, (

**e**) 5.4 cm, 20°, (

**f**) 2.5 cm, 23°. Local peak in torque is indicated by vertical dashed line each panel.

**Figure A2.**Turning parameters for replicate zebrafish (Danio rerio) executing turns of different magnitude. Variable designations are same as in Figure 1: torque per unit depth (red line), angular velocity (blue line) and moment of inertia (green line). Fish body standard length and total turn angle for each turn: (

**a**) 4.4 cm, 17°, (

**b**) 3.5 cm, 95°, (

**c**) 3.2 cm, 62°, (

**d**) 3.3 cm, 24°. Local peak in torque is indicated by vertical dashed line each panel. The fluctuations in torque near the end of the turn cycle in panels A–C are within 0.001 mN/m

^{2}and are within the margin of error that includes zero (Figure 2d).

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**Figure 1.**Swimming turns require both translational and rotational components of motion. The mechanics of these components are described by parallel but different physical terms for translational force and rotational torque (F = thrust force, m = mass, a = acceleration; τ = torque, I = moment of inertia and α = rotational acceleration).

**Figure 2.**Turning kinematics and fluid pressure for representative medusa (Aurelia aurita, 30° rotation, profiled in Figure A1d in Appendix A) and zebrafish (Danio rerio, 62° rotation, profiled in Figure A2c) turns. The red line shows the midline of the medusa (

**a**–

**d**) and the fish (

**m**–

**p**) throughout the turn, along with PIV vector and vorticity fields. Pressure fields around the medusa (

**e**–

**h**) and the fish (

**q**–

**t**) demonstrate that both animals generate large, asymmetric pressure gradients around their bodies (panels (

**f**) and (

**r**), respectively) before major body orientation shifts (illustrated by the midline position). Force vectors exerted on the animal due to local fluid pressure at the medusa (

**i**–

**l**) and zebrafish (

**u**–

**x**) body surface indicated in red arrows. Note that force vectors, and hence torques, were not calculated on the central region of the oral surface of the jellyfish (the bottom of the bell), as the bell margin in this region protrudes outward from the 2D imaging plane and blocks the view of the subumbrellar surface within the bell cavity, the surface where forces and torques would actually act. Black circles represent the center of mass in each of the latter panels. Note that during peak torque periods, forces along the body stabilize the center of mass while causing rotation of extended body regions such as the bell margin of medusae (

**j**) and caudal fin of fish (

**v**). For jellyfish, the most rapid rotation occurs during bell contraction and bell relaxation may be accompanied by negative torque (

**l**) that brakes bell rotation.

**Figure 3.**Rapid fluid accelerations during turn initiation give rise to high torque forces along the bodies of jellyfish and fish. Fluid acceleration (positive values correspond to vertical motion toward bottom of page) along animal bodies during turn initiation by medusa (

**a**) Aurelia aurita and zebrafish (

**b**) Danio rerio. Fluid accelerations in both panels are for the same turning sequences as depicted in Figure 2, so that the acceleration field in panel (

**a**) corresponds to the high pressure state of Figure 2f, while panel (

**b**) corresponds to that of Figure 2r.

**Figure 4.**Normalized data for comparison of turning variables between jellyfish and fish. Patterns represent data for replicate individuals during variable turn excursions (medusa Aurelia aurita, panels (

**a**–

**c**), n = 6; bell diameters 1.8–5.4 cm, range in turn angles 13–53°; zebrafish Danio rerio, panels (

**d**–

**f**), n = 4, fish lengths 3.2–4.4 cm, range in turn angles 17–95°). Data for each replicate turn was divided into a uniform number of sample intervals and each variable (time, area moment of inertia, angular velocity and torque) was normalized by the highest value of each replicate sequence so that all variables could be expressed in dimensionless form with a maximum value of 1. Solid curves represent the mean value and dashed lines represent one standard deviation above or below the mean for each sample interval. Note that peak values do not always reach 1 because they are averages of all the turns and not all the peak values occurred in the same time interval for every turn. The original, non-normalized data for each individual replicate are displayed in Figure A1 (medusa) and Figure A2 (zebrafish).

**Figure 5.**Conceptual summary of turning dynamics by the medusa (Aurelia aurita) and the zebrafish (Danio rerio). Arrows for each axis represent increasing magnitude for that variable. A turn is initiated by a subtle body bend, which builds torque before the animal turns (changes heading). After peak torque production, the animal bends its body more radically to minimize its moment of inertia. This decreases the body’s resistance to rotational motion while increasing angular velocity and turning the animal. The turning sequence ends as negative torque brakes the turning rotation when the body returns to its extended configuration with high moment of inertia and low angular velocity. Black circles represent the center of mass for each body image.

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**MDPI and ACS Style**

Dabiri, J.O.; Colin, S.P.; Gemmell, B.J.; Lucas, K.N.; Leftwich, M.C.; Costello, J.H. Jellyfish and Fish Solve the Challenges of Turning Dynamics Similarly to Achieve High Maneuverability. *Fluids* **2020**, *5*, 106.
https://doi.org/10.3390/fluids5030106

**AMA Style**

Dabiri JO, Colin SP, Gemmell BJ, Lucas KN, Leftwich MC, Costello JH. Jellyfish and Fish Solve the Challenges of Turning Dynamics Similarly to Achieve High Maneuverability. *Fluids*. 2020; 5(3):106.
https://doi.org/10.3390/fluids5030106

**Chicago/Turabian Style**

Dabiri, John O., Sean P. Colin, Brad J. Gemmell, Kelsey N. Lucas, Megan C. Leftwich, and John H. Costello. 2020. "Jellyfish and Fish Solve the Challenges of Turning Dynamics Similarly to Achieve High Maneuverability" *Fluids* 5, no. 3: 106.
https://doi.org/10.3390/fluids5030106