# Naut Your Everyday Jellyfish Model: Exploring How Tentacles and Oral Arms Impact Locomotion

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## Abstract

**:**

## 1. Introduction

`IB2d → matIB2d → Examples → Example_Jellyfish_Swimming → Tentacle_Jelly`(see Appendix A.3 on how to run a simulation). It is worthwhile to comment that this is a generalized study of how tentacles/oral arms may effect forward swimming performance in jellyfish, and that we are not modeling one particular species.

## 2. Mathematical Methods

#### 2.1. Computational Parameters

#### 2.2. Jellyfish Computational Model

- Position of Lagrangian points.
- Forces on each lagrangian point (horizontal/vertical and normal/tangential forces).
- Fluid velocity.
- Fluid vorticity.
- Forces spread from the Lagrangian mesh onto the Eulerian grid (jellyfish forces onto fluid).

## 3. Results

- Is it only the outer placed tentacles that affect its swimming?
- How does tentacle density affect its swimming?
- If the placement of interior tentacles is varied, will it affect its swimming?

#### 3.1. Results: Varying $Re$ and Number of Tentacles/Oral Arms

#### 3.2. Results: Varying the Length of Tentacles/Oral Arms

#### 3.3. Results: Tentacle/Oral Arm Placement and Density

- Is the placement of the outermost tentacles/oral arms what affects swimming performance? We will hold the location of the outermost tentacles/oral arms constant and change the location of the inner tentacles/oral arms, see Figure 19.
- How does density of tentacles affect swimming performance? We will again hold the location of the outermost tentacles/oral arms constant, and vary the amount of other tentacles inside that region; however, in contrast to the question above, the spacing between tentacles/oral arms will change as more or less tentacles/oral arms are considered within that region, see Figure 23.
- How does stacking tentacle/oral arms towards the outermost ones affect swimming performance? We will hold the location of the outermost tentacles constant and place more tentacles towards the outermost tentacles/oral arms and observe how swimming performance is affected. In addition, we will explore if there are clusters of tentacles/oral arms and how they may affect forward swimming performance, see Figure 27.

#### 3.3.1. Is the Placement of the Outermost Tentacles/Oral Arms What Affects Swimming Performance?

#### 3.3.2. How Does the Density of Tentacles Affect Swimming Performance?

#### 3.3.3. How Does Stacking Tentacle/Oral Arms towards the Outermost Ones Affect Swimming Performance?

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$Re$ | Reynolds Number |

$IB$ | Immersed Boundary Method |

$COT$ | Cost of Transport |

## Appendix A. Details on IB

#### Appendix A.1. Governing Equations of IB

#### Appendix A.2. Numerical Algorithm

**Step 1:**Compute the deformation force density, ${\mathbf{F}}^{n}$, on each point along the immersed boundary, from the current boundary configuration, ${\mathbf{X}}^{n}$.

**Step 2:**Use Equation (A3) to spread these boundary forces from the Lagrangian grid (immersed boundary) to the Eulerian grid (fluid grid).

**Step 3:**Update the fluid velocity and pressure, e.g., solve the Navier–Stokes equations, Equations (A1) and (A2), on the Eulerian grid. This updates ${\mathbf{u}}^{n+1}$ and ${p}^{n+1}$ from the previous time step’s data, e.g., ${\mathbf{u}}^{n}$, ${p}^{n}$, and ${\mathbf{f}}^{n}$.

**Step 4:**Update the immersed boundary’s positions, ${\mathbf{X}}^{n+1}$, using the local fluid velocities, ${\mathbf{U}}^{n+1}$, via Equation (A4) and the newly updated fluid velocity field ${\mathbf{u}}^{n+1}$.

#### Appendix A.3. Running the Simulation in MATLAB & Visualizing in VisIt

**To run the simulation, one would need to do the following**:

- Either clone the IB2d repository or download the IB2d zip file at https://github.com/nickabattista/ib2d to your local machine. Note you can download or clone this repository to any directory on your local machine.
- Open MATLAB and go to the appropriate sub-directory within the IB2d software for the Tentacle_Jelly example. The path to this example is:
`IB2d → matIB2d → Examples → Example_Jellyfish_Swimming → Tentacle_Jelly` - To run the example as is (case: $Re=150$, 6 tentacles/oral arms, $\alpha =5.0e5$), type
`main2d`into the MATLAB command window and click`enter`. - Wait… it will produce two folders
`viz_IB2d`and`hier_IB2d_data`containing the simulation data in the form of $.vtk$ formatted files. As the simulation runs, it will print more data into these folders. Note that these simulations will take on the order of days.

**To visualize the Lagrangian or Eulerian Data, one would need to do the following**:

- Open VisIt
- Open the desired data (Lagrangian Points, Vorticity, Velocity Vectors, etc.)
- To Visualize the Lagrangian Points:
- (a)
- Click
`Open` - (b)
- Go to the
`viz_IB2d`data folder that the simulation produced - (c)
- Click on the grouping of
`lagsPts`, click`OK` - (d)
- In VisIt, click on
`Add`then`Mesh`→`mesh`. - (e)
- Then click
`Draw` - (f)
- You can elect to change the color of boundary or size by double clicking on the Mesh in the VisIt data listing window.

- To Visualize the Eulerian scalar data (e.g., Vorticity, Magnitude of Velocity, etc.):
- (a)
- Click
`Open` - (b)
- Go to the
`viz_IB2d`data folder that the simulation produced - (c)
- Click on the grouping of the desired Eulerian scalar data, for example,
`Omega`(for Vorticity), click`OK` - (d)
- In VisIt, click on
`Add`then`Pseudocolor`→`Omega`. - (e)
- Then click
`Draw` - (f)
- You can elect to change the colormap and/or colormap scaling by double clicking on Omega in the VisIt data listing window.

`IB2d`→

`data_analysis`→

`analysis_in_matlab`→

`Example_For_Data_Analysis`→

`Example_Flow_In_Channel`.

## Appendix B. Varying the Poroelastic Coefficient, α

**Figure A1.**Illustrating forward swimming speeds for different Poroelasticity Coefficients, $\alpha $, at $Re=150$.

## Appendix C. Varying the Reynolds Number, Re

**Figure A3.**Visualization comparing Lagrangian Coherent Structures (LCS) using finite-time Lyanpunov exponents (FTLE) for the case with 6 total tentacles/oral arms (3 symmetrically placed per side) and $Re=\{37.5,75,150,300\}$ between the 4th and 5th contraction cycle.

**Figure A4.**Visualization comparing Lagrangian Coherent Structures (LCS) using finite-time Lyanpunov exponents (FTLE) for cases with either $0,1,2,3$ or 4 symmetrically placed tentacles/oral arms per side for $Re=150$ between the 4th and 5th contraction cycle.

## Appendix D. Varying the Tentacle/Oral Arm Length

**Figure A5.**Visualization comparing Lagrangian Coherent Structures (LCS) using finite-time Lyanpunov exponents (FTLE) for the case with 6 total tentacles/oral arms (3 symmetrically placed per side) of varying lengths (in multiples of the bell radius, a, between the 4th and 5th contraction cycle.

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**Figure 1.**Anatomy of a “true” jellyfish (class Scyphozoa). Courtesy of the National Science Foundation [5].

**Figure 2.**Illustrating the diversity of tentacles/oral arms among different jellyfish species, including (

**a**) Moon jellyfish photos courtesy of the Two Oceans Aquarium [10] (left) and Audubon Aquarium of the Americas [11] (right), (

**b**) Australian Spotted jellyfish courtesy of the Aquarium of Niagara [12], (

**c**) Blue Blubber jellyfish courtesy of H. Steiger [13], (

**d**) Flame jellyfish courtesy of B. Abbott (juvenile, top) [14], the Osaka Aquarium Kaiyukan (adult, bottom) [15], (

**e**) Japanese Sea Nettle courtesy of the National Aquarium (USA) [16], (

**f**) upside-down jellyfish courtesy of the Key Largo Marine Research Lab [17] (

**g**), Fried Egg jellyfish courtesy of Fredski (2013) (left) [18] and A. Sontuoso (right) [19], (

**h**) Cannonball jellyfish courtesy of the National Aquarium (USA) [20], (

**i**) Lion’s Mane jellyfish courtesy of D. Hershman [21], (

**j**) Sea Wasp courtesy G. Gautsch take at the Port of Nagoya Public Aquarium [22], (

**k**) Purple-striped jellyfish courtesy of B. Spragg [23], and (

**l**) Clapper Hydroid courtesy of A. Hosia [24]. Moon jellyfish, Australian Spotted jellyfish, Adult Flame jellyfish, Japanese Sea Nettle, Upside-Down jellyfish, and Cannonball jellyfish photos taken by nickabattista, licensed under CC-BY-SA-4.0. Blue Blubber jellyfish by HaSt licensed under CC-BY-SA-4.0. Juvenile Flame jellyfish by Bill Abbott licensed under CC BY-SA 2.0. Fried Egg jellyfish photos by Fredski and Antonio Sontuoso licensed under CC-BY-SA-3.0 and CC-BY-SA-2.0, respectively. Lion’s Mane jellyfish by Dan Hershman licensed under CC-BY-2.0. Sea Wasp photo by Guido Gautsch licensed under CC-BY-SA-2.0. Purple-striped jellyfish by B. Spragg licensed under CC-0. Clapper Hydroid by A. Hosia licensed under CC BY-NC-SA 2.5.

**Figure 3.**Jellyfish model geometry composed of discrete points is a semi-elliptical configuration with tentacles/oral arms. The points along the bell are connected by virtual springs and virtual beams and the tentacles/oral arms are modeled as poroelastic structures, which include virtual springs and beams tethering adjacent points in the IB2d software.

**Figure 4.**A snapshot of a jellyfish simulation with 8 tentacles/oral arms swimming at $Re=150$ during its 5th contraction cycle, illustrating some of the simulation data obtained at each time step, e.g., positions of Lagrangian points as well as forces on them, magnitude of velocity, the velocity vector field, and vorticity. Note other data not visualized is the fluid pressure and Lagrangian forces spread from the jellyfish onto the Eulerian (fluid) grid. Lagrangian coherent structures (LCS) via finite time Lyapunov exponents (FTLE) are also illustrated, although they are computed during the postprocessing data stage.

**Figure 5.**Geometric model considered in Section 3.1 to determine how the presence of tentacles/oral arms affects forward swimming speed. This same geometry is used in Section 3.2 but with different tentacle/oral arms lengths, given in multiples of the bell radius, a.

**Figure 6.**Visualization comparing jellyfish swimming for a variety of different number of symmetric tentacles/oral arms, for $Re=150$ with a contraction frequency of f = 0.8 Hz. As the number of tentacles increases, forward swimming progress is more limited.

**Figure 7.**Visualization comparing a jellyfish with no tentacles/oral arms to the case with 6 tentacles/ oral arms (3 symmetric per side) at $Re=150$. The colormap is represents vorticity and uses the same scaling across all images.

**Figure 8.**Plots detailing (

**a**) distance swam and (

**b**) velocity over 8 bell contraction periods at $Re=150$ for differing numbers of symmetric tentacles.

**Figure 9.**Illustrating average forward swimming speed against Reynolds number, $Re$, for different number of symmetric tentacles/oral arms. Swimming speed is measured in nondimensional units (body lengths/contraction) in normal form (

**a**) and logarithmic form (

**b**). It is clear that the addition of tentacles/oral arms decreases forward swimming performance.

**Figure 10.**Plot depicting the relationship between Strouhal number, $St$, and Reynolds number, $Re$, for different numbers of symmetric tentacles/oral arms. $St$ is the inverse of nondimensional swimming speed.

**Figure 11.**Illustrating the relationship between cost of transport (COT) and Reynolds number, $Re$, for different numbers of symmetric tentacles/oral arms, when COT is computed using (

**a**) average power and (

**b**) average work.

**Figure 12.**Visualization comparing Lagrangian coherent structures (LCS) using finite-time Lyanpunov exponents (FTLE) for the case with 6 total tentacles/oral arms (3 symmetrically placed per side) and $Re=\{37.5,75,150,300\}$ at the beginning of the 4th contraction cycle. Note that the case of $Re=150$ with no tentacles is given to provide a comparison.

**Figure 13.**Visualization comparing Lagrangian coherent structures (LCS) using finite-time Lyanpunov exponents (FTLE) for cases with either 0, 1, 2, 3, or 4 symmetrically placed tentacles/oral arms per side for $Re=150$ at the start of the 4th contraction cycle.

**Figure 14.**Plot detailing distance swam against bell contractions performed for differing tentacle/oral arm lengths at $Re=150$. Tentacle/oral arm length is given in multiples of the bell radius, a.

**Figure 15.**Illustrating (

**a**) average forward swimming speed and (

**b**) Strouhal number for different lengths of symmetrically placed tentacles/oral arms at $Re=150$. Swimming speed is measured in nondimensional units (body lengths/contraction) and tentacle length is measured in multiples of the bell radius, a. As tentacle/oral length increases, forward swimming speed decreases, until it appears to steady out.

**Figure 16.**Illustrating the relationship between cost of transport (COT) and tentacle/oral arm length for different numbers of symmetric tentacles/oral arms at $Re=150$, when COT is computed using (

**a**) average power and (

**b**) average work. Tentacle/oral arm length is given in multiples of the bell radius, a.

**Figure 17.**Visualization of jellyfish position and a colormap of vorticity at the end of the 5th contraction cycle for each case of differing number of tentacles/oral arms of specified length, at $Re=150$. Note that length is given in multiples of the bell radius, a. Note that the colormap uses the same scaling across all images.

**Figure 18.**Visualization comparing LCS using finite-time Lyanpunov exponents (FTLE) for the case with 6 total tentacles/oral arms (3 symmetrically placed per side) of varying lengths (in multiples of the bell radius, a, at the start of the 4th contraction cycle).

**Figure 19.**Geometric setup for all cases considered in Section 3.3.1 to determine if the placement of the outermost tentacles/oral arms dictates forward swimming speed.

**Figure 20.**Visualization comparing the positions of the jellyfish across the first 5 contraction cycles for all cases considered in Section 3.3.1 for $Re=150$.

**Figure 21.**(

**a**) Forward swimming speed and (

**b**) power-based cost of transport for each simulation in Section 3.3.1. A nonlinear relationship between forward swimming speed, tentacle/oral arm number density, and placement emerges.

**Figure 22.**Visualization of jellyfish position and a colormap of vorticity across the 4th to 5th contraction cycle for each case considered at $Re=150$. Note that the colormap uses the same scaling across all images.

**Figure 23.**Geometric setup for all cases considered in Section 3.3.2 to determine how density of the tentacles/oral arms affects forward swimming speed.

**Figure 24.**Visualization comparing the positions of the jellyfish across the first 5 contraction cycles for all cases considered in Section 3.3.2 for $Re=150$.

**Figure 25.**(

**a**) Forward swimming speed and (

**b**) power-based cost of transport for each simulation in Section 3.3.2. A nonlinear relationship between forward swimming speed, tentacle/oral arm density and placement is observed again.

**Figure 26.**Visualization of jellyfish position and a colormap of vorticity across its 5th contraction cycle for each case considered at $Re=150$. Note that the colormap uses the same scaling across all images.

**Figure 27.**Geometric setup for all cases considered in Section 3.3.3 to determine how placing more tentacles/oral arms towards the outermost ones affect forward swimming speed.

**Figure 28.**Visualization comparing the positions of the jellyfish across the first 5 contraction cycles for all cases considered in Section 3.3.3 for $Re=150$.

**Figure 29.**(

**a**) Forward swimming speed and (

**b**) power-based cost of transport for each simulation in Section 3.3.3. A nonlinear relationship between number and density of tentacles/oral arms and forward swimming speed is observed.

**Figure 30.**Visualization of jellyfish position and a colormap of vorticity across its 5th contraction cycle for each case considered at $Re=150$. Note that the colormap uses the same scaling across all images.

**Table 1.**Morphological properties and range of the various jellyfish species shown in Figure 2.

Name | Scientific Name | Max. Bell Diameter (cm) | Tentacle/Oral Arm Length (cm) | Range | References |
---|---|---|---|---|---|

Moon Jellyfish | Aurelia aurita | 38 | $7.6$ | Along the East & West Coast, Europe, Japan, and the Gulf of Mexico | [25] |

Australian Spotted Jellyfish | Phyllorhiza punctata | 60 | ≳60 | Western Pacific (From Australia to Japan) | [26,27] |

Blue Blubber Jellyfish | Catostylus mosaicus | 45 | ∼45 | Along the east and north coasts of Australia | [28] |

Flame Jellyfish | Rhopilema esculentum | 70 | ≳70 | Warm temperate waters of the Pacific Ocean | [27,29] |

Japanese Sea Nettle | Chrysaora melanaster | 60 | 300 | Northern Pacific Ocean & adjacent parts of the Arctic Ocean | [30] |

Upside-Down Jellyfish | Cassiopea | 36 | 36 | Western Atlantic, including the Gulf of Mexico, Bermuda, and the Caribbean | [31] |

Fried Egg Jellyfish | Cotylorhiza tuberculata | 40 | ≳40 | Mediterranean Sea, coastal lagoons | [32] |

Cannonball Jellyfish | Stomolophus meleagris | 25 | ≳25 | Pacific Ocean and the mid-west of the Atlantic Ocean | [33] |

Lion’s Mane Jellyfish | Cyanea capillata | 250 | 3600 | Cold waters of the Arctic, Northern Atlantic, and Northern Pacific | [9,34] |

Sea Wasp (Box Jellyfish) | Chironex fleckeri | 30 | 300 | Australia and Indo-West Pacific Ocean | [35,36] |

Purple Striped Jellyfish | Chrysaora colorata | 70 | 800 | Eastern Pacific Ocean primarily off the coast of California | [37] |

Clapper Hydroid | Sarsia tubulosa | 0.5 | 3–4 | Central California to the Bering Sea | [7,8] |

Parameter | Variable | Units | Value |
---|---|---|---|

Domain Size | $[{L}_{x},{L}_{y}]$ | m | $[5,12]$ |

Spatial Grid Size | $dx=dy$ | m | ${L}_{x}/320={L}_{y}/768$ |

Lagrangian Grid Size | $ds$ | m | $dx/2$ |

Time Step Size | $dt$ | s | ${10}^{-5}$ |

Total Simulation Time | T | pulses | 8 |

Fluid Density | $\rho $ | kg/m${}^{3}$ | 1000 |

Fluid Dynamic Viscosity | $\mu $ | kg/(ms) | varied |

Bell Radius | a | m | $0.5$ (and varied) |

Bell Diameter | D ($2a$) | m | $1.0$ (and varied) |

Bell Height | b | m | $0.75$ |

Contraction Frequency | f | 1/s | $0.8$ |

Spring Stiffness | ${k}_{spr}$ | kg·m/s${}^{2}$ | $1\times {10}^{7}$ |

Beam Stiffness | ${k}_{beam}$ | kg·m/s${}^{2}$ | $2.5\times {10}^{5}$ |

Tentacle Spring Stiffness ${k}_{{T}_{spr}}$ | kg·m/s${}^{2}$ | $1\times {10}^{7}$ | |

Tentacle Beam Stiffness | ${k}_{{T}_{beam}}$ | kg·m/s${}^{2}$ | $2.5\times {10}^{-3}$ |

Muscle Spring Stiffness | ${k}_{muscle}$ | kg·m/s${}^{2}$ | $1\times {10}^{5}$ |

Poroelasticity Coefficient | $\alpha $ | m${}^{-2}$ | varied |

**Table 3.**Table giving the percentage difference in forward swimming speed when compared to the case with no tentacles/oral arms. Note that all cases with tentacles/oral arms are significantly slower.

Total # Tentacles/Oral Arms | $\mathit{Re}=37.5$ | $\mathit{Re}=75$ | $\mathit{Re}=150$ | $\mathit{Re}=300$ |
---|---|---|---|---|

2 | −55.9% | −45.5% | −43.6% | −44.3% |

4 | −64.3% | −58.3% | −55.1% | −52.7% |

6 | −79.5% | −68.0% | −67.0% | −60.7% |

8 | −91.8% | −92.7% | −84.6% | −77.3% |

**Table 4.**Table giving the percentage difference in forward swimming speed when compared to the case with no tentacles/oral arms. Note that all cases containing tentacles/oral arms are significantly slower.

$\mathit{Re}=37.5$ | $\mathit{Re}=75$ | $\mathit{Re}=150$ | $\mathit{Re}=300$ | |
---|---|---|---|---|

ABCDEF | −71.1% | −64.1% | −57.4% | −57.6% |

ABEF | −67.5% | −62.5% | −49.8% | −49.8% |

ACDF | −74.7% | −65.5% | −51.1% | −52.6% |

AF | −64.6% | −49.5% | −42.7% | −40.6% |

**Table 5.**Table giving the percentage difference in forward swimming speed when compared to the case with no tentacles/oral arms. Note that all cases with tentacles/oral arms are significantly slower.

$\mathit{Re}=37.5$ | $\mathit{Re}=75$ | $\mathit{Re}=150$ | $\mathit{Re}=300$ | |
---|---|---|---|---|

2 Per Side | −69.3% | −65.5% | −54.6% | −49.3% |

4 Per Side | −77.6% | −71.0% | −56.4% | −64.6% |

5 Per Side | −78.5% | −74.9% | −59.4% | −49.5% |

10 Per Side | −78.1% | −83.3% | −74.4% | −61.1% |

**Table 6.**Table giving the percentage difference in forward swimming speed when compared to the case with no tentacles/oral arms. Note that all cases with tentacles/oral arms are significantly slower.

$\mathit{Re}=37.5$ | $\mathit{Re}=75$ | $\mathit{Re}=150$ | $\mathit{Re}=300$ | |
---|---|---|---|---|

Outer 2 | −67.4% | −55.3% | −45.2% | −48.4% |

Outer 3 | −67.7% | −62.7% | −51.6% | −48.6% |

Outer 4 | −70.1% | −58.8% | −51.2% | −50.3% |

Outer 5 | −78.5% | −74.9% | −59.4% | −49.5% |

Outer/Inner | −75.4% | −65.9% | −55.9% | −51.0% |

Inner Unequal | −78.8% | −81.3% | −70.8% | −55.6% |

© 2019 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**

Miles, J.G.; Battista, N.A. Naut Your Everyday Jellyfish Model: Exploring How Tentacles and Oral Arms Impact Locomotion. *Fluids* **2019**, *4*, 169.
https://doi.org/10.3390/fluids4030169

**AMA Style**

Miles JG, Battista NA. Naut Your Everyday Jellyfish Model: Exploring How Tentacles and Oral Arms Impact Locomotion. *Fluids*. 2019; 4(3):169.
https://doi.org/10.3390/fluids4030169

**Chicago/Turabian Style**

Miles, Jason G., and Nicholas A. Battista. 2019. "Naut Your Everyday Jellyfish Model: Exploring How Tentacles and Oral Arms Impact Locomotion" *Fluids* 4, no. 3: 169.
https://doi.org/10.3390/fluids4030169