Naut your everyday jellyfish model: Exploring how tentacles and oral arms impact locomotion

TL;DR

This study uses computational experiments to analyze how jellyfish tentacles and oral arms influence their swimming efficiency, revealing that these features can significantly inhibit speed and alter vortex formation.

Contribution

It is the first detailed in silico investigation into how tentacle and oral arm morphology affects jellyfish locomotion and fluid dynamics.

Findings

Tentacles/oral arms inhibit forward swimming speeds.

Morphological changes can reduce speed by up to 400%.

Tentacles suppress vortex formation during swimming.

Abstract

Jellyfish - majestic, energy efficient, and one of the oldest species that inhabits the oceans. It is perhaps the second item, their efficiency, that has captivated scientists for decades into investigating their locomotive behavior. Yet, no one has specifically explored the role that their tentacles and oral arms may have on their potential swimming performance, arguably the very features that give jellyfish their beauty while instilling fear into their prey (and beach-goers). We perform comparative in silico experiments to study how tentacle/oral arm number, length, placement, and density affect forward swimming speeds, cost of transport, and fluid mixing. An open source implementation of the immersed boundary method was used (IB2d) to solve the fully coupled fluid-structure interaction problem of an idealized flexible jellyfish bell with poroelastic tentacles/oral arms in a viscous, incompressible fluid. Overall tentacles/oral arms inhibit forward swimming speeds, by appearing to suppress vortex formation. Non-linear relationships between length and fluid scale (Reynolds Number) as well as tentacle/oral arm number, density, and placement are observed, illustrating that small changes in morphology could result in significant decreases in swimming speeds, in some cases by downwards of 400% between cases with to without tentacles/oral arms.

Tables

Name	Scientific Name	Max. Bell Diameter (cm)	Tentacle/Oral Arm Length (cm)
Moon Jellyfish	Aurelia aurita	38	$7.6$
Australian Spotted Jellyfish	Phyllorhiza punctata	60	$\gtrsim 60$
Blue Blubber Jellyfish	Catostylus mosaicus	45	$\sim 45$
Flame Jellyfish	Rhopilema esculentum	70	$\gtrsim 70$
Japanese Sea Nettle	Chrysaora melanaster	60	$300$
Upside Down Jellyfish	Cassiopea	36	$36$
Fried Egg Jellyfish	Cotylorhiza tuberculata	40	$\gtrsim 40$
Cannonball Jellyfish	Stomolophus meleagris	25	$\gtrsim 25$
Lion’s Mane Jellyfish	Cyanea capillata	250	$3600$
Sea Wasp (Box Jellyfish)	Chironex fleckeri	$30$	$300$
Purple Striped Jellyfish	Chrysaora colorata	70	$800$
Clapper Hydroid	Sarsia tubulosa	0.5	$3-4$

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

Name	Range	References
Moon Jellyfish	Slong the East & West Coast, Europe, Japan and the Gulf of Mexico	Monterey Bay Aquarium (2019)
Australian Spotted Jellyfish	Western Pacific (From Australia to Japan)	Haddad and Júnior (2006); Boon (2017)
Blue Blubber Jellyfish	Along the east and north coasts of Australia	Monterey Bay Aquarium (2019)
Flame Jellyfish	Warm temperate waters of the Pacific Ocean	Pitt and Purcell (2009); Boon (2017)
Japanese Sea Nettle	Northern Pacific Ocean & adjacent parts of the Arctic Ocean	ArcticOceanDiversity (2010)
Upside Down Jellyfish	Western Atlantic, including the Gulf of Mexico, Bermuda and the Caribbean	Georgia Aquarium (2019)
Fried Egg Jellyfish	Mediterranean Sea, coastal lagoons	Kikinger (1992)
Cannonball Jellyfish	Pacific Ocean and the mid-west Atlantic Ocean	Griffin and Murphy (2011)
Lion’s Mane Jellyfish	Cold waters of the Arctic, Northern Atlantic, Northern Pacific	National Aquarium in Baltimore (2019); McClain et al. (2015)
Sea Wasp	Australia and Indo-West Pacific Ocean	Barnes (1965); Lewis and Bentlage (2009)
Purple Striped Jellyfish	Eastern Pacific Ocean primarily off the coast of California	Georgia Aquarium (2019)
Clapper Hydroid	Central California to the Bering Sea	Wrobel and Mills (2003); Katija et al. (2015)

Table 3. Table 2: Numerical parameters used in the two-dimensional simulations.

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	$\mu$	$kg/\left(ms\right)$	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	$k_{spr}$	$kg\cdot m/s^2$	$1\times {10}^7$
Beam Stiffness	$k_{beam}$	$kg\cdot m/s^2$	$2.5\times {10}^5$
Muscle Spring Stiffness	$k_{muscle}$	$kg\cdot m/s^2$	$1\times {10}^5$
Poroelasticity Coefficient	$\alpha$	$m^{-2}$	varied

Table 4. Table 3: Table giving the percentage decrease in forward swimming speed when compared to the case with no tentacles/oral arms.

	$Re=37.5$	$Re=75$	$Re=150$	$Re=300$
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 4: Table giving the percentage difference in forward swimming speed when compared to the case with no tentacles/oral arms.

	$Re=37.5$	$Re=75$	$Re=150$	$Re=300$
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 5: Table giving the percentage difference in forward swimming speed when compared to the case with no tentacles/oral arms.

	$Re=37.5$	$Re=75$	$Re=150$	$Re=300$
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%

Equations

$$Re=\frac{\rho f_{jelly}{D}_{jelly}^2}{\mu }.$$

$$\textbf{F}_{spr}=k_{spr}\left(1-\frac{R_{L}(t)}{||{\bf{X}}_{A}(t)-{\bf{X}}_{B}(t)||}\right)\cdot\left(\begin{array}[]{c}x_{A}(t)-x_{B}(t)\\ y_{A}(t)-y_{B}(t)\end{array}\right)$$

$$\displaystyle\textbf{F}_{beam}=k_{beam}\frac{\partial^{4}}{\partial s^{4}}\Big{(}\textbf{X}_{C}(t)-\textbf{X}^{con}(t)\Big{)},$$

$${\text{f}}_{brink}$$

$${\text{U}}_b(\text{x},t)$$

$$RL(t)=2a\left(1-\left|\sin (\pi tf)\right|\right).$$

$$St=\frac{fD}{U_{avg}}$$

$$CO{T}_{work}$$

$$CO{T}_{power}$$

$$\rho\Big{[}\frac{\partial{{\bf u}}}{\partial t}({\bf x},t)+{{\bf u}}({\bf x},t)\cdot\nabla{{\bf u}}({\bf x},t)\Big{]}=\nabla p({\bf x},t)+\mu\Delta{{\bf u}}({\bf x},t)+{{\bf F}}({\bf x},t)$$

$$\nabla \cdot \mathbf{u}(\mathbf{x},t)=0$$

$$\mathbf{F}(\mathbf{x},t)$$

$$\mathbf{U}(\mathbf{X}(s,t))$$

$${\delta }_h(\mathbf{x})=\frac{1}{h^3}\varphi \left(\frac{x}{h}\right)\varphi \left(\frac{y}{h}\right)\varphi \left(\frac{z}{h}\right),$$

$$\phi(r)=\left\{\begin{array}[]{l}\frac{1}{8}(3-2|r|+\sqrt{1+4|r|-4r^{2}}),\ \ \ 0\leq|r|<1\\ \frac{1}{8}(5-2|r|+\sqrt{-7+12|r|-4r^{2}}),1\leq|r|<2\\ 0\hskip 151.76744pt2\leq|r|.\\ \end{array}\right.$$

Full text

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xx \issuenum1 \articlenumber1

\externaleditorAcademic Editor: name \historyReceived: date; Accepted: date; Published: date

\TitleNaut your everyday jellyfish model: Exploring how tentacles and oral arms impact locomotion \AuthorJason G. Miles1 and Nicholas A. Battista 1* \AuthorNamesJason G. Miles and Nicholas A. Battista

\corresbattistn@tcnj.edu; Tel.: +1-609-771-2445

1 Introduction

A notably distinct (and arguably most beautiful) feature of a jellyfish are its tentacles and oral arms. Unfortunately these appendages are what ruin many summer days at the beach, as they’re chalk full of their stinging cells, called nematocysts Higgins et al. (2008); Beckmann and Özbek (2012). The movement of these stinging cells are one of the fastest movements in the animal kingdom, with discharges as fast as 700 nanoseconds Nüchter et al. (2006). Jellyfish use these cells to subdue their prey for sustenance Strychalski et al. (2018), hence their tentacles and oral arms play a vital role in survival. Figure 1 illustrate the anatomy of a “true" jellyfish’s tenatcles and oral arms.

A “true jellyfish" is one of a specific class of jellyfish - Scyphozoa. Scyphozoans tend to be the jellyfish that are familiar to aquarium-goers, identifiable by the cup shape of their bell. Another class of Medusozoa are Cubozoa (box jellyfish), denoted by their cube-shaped medusae. Both of these jellyfish classes are known to ruin beach-goers relaxed, fun day at beaches around the world Cegolon et al. (2013). There are notable visual differences in tentacle and oral arm morphology among different classes and species of jellyfish in general, see Figure 2. Figure 2 presents 12 different jellyfish species illustrating different tentacle/oral arm morphologies- different numbers, lengths, configurations, and densities. Table 1 gives specific morphological data pertaining to the species shown in Figure 2. These jellyfish range on the smaller size, like the Clapper Hydroid (Sarsia tubulosa), which has a bell diameter $\sim 0.5\ cm$ with tentacles of length $\sim 3-4\ cm$ Wrobel and Mills (2003); Katija et al. (2015), to the Lion’s Mane Jellyfish (Cyanea capillata), which can have a bell of diameter $2.5\ m$ and tentacles/oral arms that can grow to be $36\ m$ long National Aquarium in Baltimore (2019)! Note that there are many more jellyfish species than those discussed above with different morphologies, like the Immortal Jellyfish (Turritopsis dohrnii) or Blue Jellyfish (Cyanea lamarckii).

The large diversity of jellyfish tentacle/oral arm morphology, leads to a variety of different modes of predation among different jellyfish species. For example, some jellyfish actively hunt for food, like box jellyfish Kinsey (1988), while others are more passive and opportunist predators, hoping that their food (prey) gets caught in their tentacles/oral arms, like a Lion’s Mane jellyfish Linnaeus (1758); Powell and Atland (2018). Some species even use a combination of photosynthesis and passive filter-feeding for sustenance, e.g., the upside-down jellyfish Hamlet and Miller (2012); Santhanakrishnan et al. (2011). A jellyfish’s foraging strategy may depend on its swimming performance potential, which may be limited by its morphology, not only by its bell but its tentacles and oral arms. If it is an effective/efficient swimmer, it display a more active predation strategy, while if it is not, it might display more passive, opportunistic foraging characteristics.

Some jellyfish, such as the Lion’s Mane Jellyfish (Cyanea capillata), predominantly move horizontally with the direction of prevailing currents Bastian (2011). They also exhibit a range of vertical migrations depending on tidal stage Bastian et al. (2012); Moriarty et al. (2012). Costello et al. 1998 Costello et al. (1998) observed Lion’s Mane jellyfish sometimes active swimming upwards towards the surface for extended periods of time, to whom once at the surface would cease swimming and passively sink towards the bottom. Other times, they would reorient themselves and actively swim towards the bottom Costello et al. (1998). Using an acoustic tagging system, Bastian et al. Bastian (2011); Bastian et al. (2012) quantified their vertical speeds as approximately $3.75\ bell\ diameters/min$, or $0.06\ bell\ diameters/s$. Moreover, Moriarty et al. 2012 Moriarty et al. (2012) found that Lion’s Mane (and Fried Egg jellyfish) active swimming speeds varied during the tidal stage, with deviations of approximately 95% from their maximal swimming speeds during the flooding stage. Although these jellyfish are known to be opportunist predators, they are effective carnivores Purell and Arai (2001); Purell (2003); Crawford (2016); they are not fast swimmers (see above), but rather their lengthy, numerous dense tentacles compensate as an appropriate foraging advantage.

Other jellyfish species’ migratory patterns are also influenced by the tidal stage, such as the sea wasp (Chironex fleckeri), one kind of box jellyfish Gordon and Seymour (2009). Large box jellyfish tend to swim against the current Kinsey (1986), while smaller individuals swim in either direction Rifkin (1996); Bastian (2014). Box jellyfish have been observed achieving swimming speeds of up to $\sim 1-2.5\ bodylengths\ per\ second$ and moreover that smaller individuals were observed swimming faster Colin et al. (2013). Compared to the Lion’s Mane jellyfish, box jellyfish have a much more stiff bell, which allows them to achieve more rapid swimming Shorten et al. (2005). The effects of varying bell stiffness on propulsion in medusae have been quantified in computational models Hoover and Miller (2015); Hoover et al. (2017). Box jellyfish are able use this to their advantage when foraging, in tandem with their vision to hunt prey and relocate to densely populated prey environments Courtney et al. (2015); Garm et al. (2016). As previously shown in Figure 2, Lion’s Mane and box jellyfish have distinct morphological differences, not only in their bell size and shape, but also their tentacle number, length, and density. Their bell kinematics and swimming behavior and performance are also different Grzimek (1972); Shorten et al. (2005), which supports the notion of that their foraging strategies depend on its morphology and potential swimming ability.

There are jellyfish that do a mixture of active and passive hunting. One such species is the Sarsia tubulosa, who are ambush predators. They wait for its unsuspecting prey to come into its capture range, near its 4 tentacles, while it floats almost motionless in the water. They will actively swim to different locations to forage for food, but do so in irregular swim bouts, even in aquaria Leonard (1978). Moreover, individual Sarsia are known to each use a unique set of contraction frequencies while swimming Leonard (1980). Katija et al. 2015 Katija et al. (2015) found that Sarsia tubulosa tend to swim about $2\ bodylengths\ per\ contraction$, with a range between [̃1,3.5] bodylengths per contraction. Similar results were obtained previously by Colin et al. 2002 Colin and Costello (2002) and Sahin et al. 2009 Sahin et al. (2009). Hoover et al. 2015 Hoover and Miller (2015) and Miles et al. 2019 Miles and Battista (2019) used an idealized jellyfish fluid-structure interaction computational model similar to the Sarsia tubulosa and predicted swimming speeds within that range for biologically relevant $Re$ and quantified how different contraction frequencies may affect forward swimming performance. However, neither considered any tentacles/oral arms in their model.

Many computational scientists have developed sophisticated computational models of jellyfish that produce forward propulsion Sahin et al. (2009); Dular et al. (2009); Wilson et al. (2009); Hershlag and Miller (2011); Alben et al. (2013); Yuan et al. (2014); Hoover and Miller (2015); Hoover et al. (2017); Porras and Miller (2019); Miles and Battista (2019); PAllasdies et al. (2019) and have compared swimming performance over a large mechanospace, including bell stiffness and flexibility, muscular contraction strength and amplitude, and contraction frequencies. While many in situ and laboratory studies have been performed that quantify bell kinematics, velocity, acceleration, feeding (predation), and/or vortex wake structure Costello et al. (1998); Colin and Costello (2002); Dabiri et al. (2005a); Shorten et al. (2005); Dabiri et al. (2005b, 2006); Bajcar et al. (2008); Peng and Dabiri (2009); Colin et al. (2013); Gemmell et al. (2013, 2014); Katija et al. (2015); Courtney et al. (2015); Gemmell et al. (2015); Costello et al. (2019); Kim and Chamorro (2019), computational studies are becoming increasingly more of an attractive alternative and/or complement for scientists, as they is easier (e.g., more cost and time efficient) to do widespread parameter studies. Yet, with all of the data available no one has specifically explored the role that tentacles/oral arms may have on jellyfish locomotion. The only study the authors are aware of was performed by K. Katija in 2015 Katija (2015), where they observed that an Australian Spotted Jellyfish (Phyllorhiza punctata) swam 360% faster without its oral arms; however, any further investigations have yet to have been performed.

In this paper, we will use an open-source implementation of the immersed boundary method, IB2d Battista et al. (2015, 2017, 2018), to study the swimming performance of a $2D$ idealized jellyfish with tentacles/oral arms within a fully-coupled fluid-structure interaction framework. We expand upon the work of Hoover et al. 2015 Hoover and Miller (2015) and Miles et al. 2019 Miles and Battista (2019), using their geometric bell morphology as a base model, but with the addition of poroelastic tentacles/oral arms. In particular studied how the addition of tentacles/oral arms will effect its potential forward swimming ability, across numerous fluid scales, given by the Reynolds Number, $Re$, and lengths, number, density, and placement of tentacles/oral arms. We also show that intricate relationships exist between vortex interactions, wake structure, fluid mixing, and ultimately their forward propulsive speed, highlighting that there appears to be no one-to-one connection between swimming efficiency and wake structure Smits (2019); ad T. V. Buren and Smits (2019).

In addition, we offer the science community the first open-source jellyfish locomotion model with poroelastic tentacles/oral arms in a fluid-structure interaction framework. It can be found at https://github.com/nickabattista/IB2d/tree/master/matIB2d/Examples/Example_Jellyfish_Swimming/Tentacle_Jelly. It is worthwhile to comment that this is a generalized study of how tentacles/oral arms may effect forward swimming performance in jellyfish and are not modeling one particular species.

2 Mathematical Methods

To model the interactions between a flexible jellyfish bell with tentacles/oral arms and a viscous, incompressible fluid, computational methods were used. Our goal was to explore forward swimming performance over a wide parameter space, including fluid scale ($Re$) and tentacle/oral arm length, number, density, and placement. While fluid scale effects have been previously studied for jellyfish locomotion Hershlag and Miller (2011); Hoover et al. (2017); Miles and Battista (2019), how tentacles/oral arms may impact forward swimming has yet to been investigated.

The mathematical framework used is in the vein of fluid-structure interaction systems (FSI), which couples the motion of a deformable object and the fluid to which it is immersed. The first robust numerical method developed to solve such problems was developed by Charles Peskin in the 1970s Peskin (1972, 1977). It is called the immersed boundary method (IB) Peskin (2002). Due to its elegance, the immersed boundary method has been continually used and improved upon Fauci and Fogelson (1993); Lai and Peskin (2000); Cortez and Minion (2000); Griffith and Peskin (2005); Mittal and Iaccarino (2005); Griffith et al. (2007); Griffith (2014); Griffith and Luo (2012); Huang and Tian (2019). It is still a leading numerical framework for studying problems in FSI due to its robustness Battista et al. (2017, 2018) that extends well beyond the physiological (or biological) applications it was originally developed for, and has permeated across all fields of engineering Mittal and Iaccarino (2005); Battista et al. (2018); Huang and Tian (2019) .

The IB has been successfully applied to numerous problems including cardiac fluid dynamics Miller (2011); Griffith (2012); Battista et al. (2017a, b); Battista (2018), aquatic locomotion Bhalla et al. (2013a, b); Hamlet et al. (2015), insect flight Miller and Peskin (2004, 2009); Jones et al. (2015), and even dating and relationships Battista et al. (2017). Additional details on the IB method can be found in the Appendix A.

We will now dive into the details regarding our jellyfish model’s implementation into the IB2d framework, e.g., the computational geometry, geometric and fluid parameters, and modeling assumptions. Our model is based upon the $2D$ jellyfish locomotion model of Hoover et al. 2015 Hoover and Miller (2015), which originally did not include tentacles/oral arms. It was originally implemented in the IB software called IBAMR Griffith (2014), which is parallelized IB software with adaptive mesh refinement Berger and Oliger (1984); Roma et al. (1999); Griffith et al. (2007).

2.1 Computational Parameters

In this study, we use the frequency-based Reynolds number, $Re_{f}$, to describe the locomotive processes of prolate jellyfish with tentacles/oral arms. The characteristic length, $D_{jelly}$, is set to the bell diameter at rest and the characteristic frequency, $f_{jelly}$, is set to the contraction (stroke) frequency. Therefore our characteristic velocity scale is set to $V_{jelly}=f_{jelly}D_{jelly}$, as in Eq.(1),

[TABLE]

Fluid parameters (density and viscosity) can be found in Table 2. Note that for our study in Section 3.1, we only vary the viscosity, $\mu$, which effectively changes the Reynolds Number, $Re$ (see Eq.(1)), and the number, length, placement, and density of (poroelastic) tentacles/oral arms. For our study in Section 3.2 we will vary the tentacle/oral arm length at $Re=150$. Finally in Section 3.3 we will vary the number, placement, and density of tentacles/oral arms.

For all studies, the computational width was kept constant as $L_{x}=5.$ Convergence studies were previously conducted in Miles et al. 2019 Miles and Battista (2019) that demonstrated low relative errors in forward swimming speeds for domain sizes from $L_{x}\in[3,8]$ for $Re=\{37.5,75,150,300\}$. In all cases, a similar trend was observed where narrower computational domains lead to slightly decreased forward swimming speeds while qualitative differences vortex formation were minimal. Additional grid resolution convergence studies were performed in Battista et al. 2019 Battista and Mizuhara (2019) using the same computational model of a $2D$ jellyfish (but without tentacles/oral arms) that demonstrate appropriate error tolerances for the numerical parameters described in Table 2.

2.2 Jellyfish Computational Model

The geometry is composed of a semi-ellipse with semi-major axis, $b=0.75$, and semi-minor axis, $a=0.5$, along with tentacles/oral hands that hang within the interior of the bell about its center, see Figure 3. We refer to the bell height and radius as the semi-major axis, $b$, and semi-minor axis, $a$, respectively. As shown, it is composed of discrete Lagrangian points that are equally spaced a distance $ds$ apart. We note that this Lagrangian mesh is twice as resolved as the background fluid grid, e.g., $ds=0.5dx$. Also, as the tentacle number, length, location, and density were varied through Sections 3.1-3.3, additional model geometry figures will be presented to distinguish between different geometrical setups when appropriate.

Although we view the jellyfish as being immersed in the fluid, the jellyfish (Lagrangian grid) and fluid (Eulerian mesh) only communicate through integral equations with delta function kernels (see Eqs. (14)-(15) in Appendix A for more details.) In a nutshell, the Lagrangian mesh is allowed to move and deform. As an outside observer, we can witness this during the simulation. On the other hand, the Eulerian mesh is constrained upon a specific discrete rectangular lattice. On this mesh we are only measuring fluid related quantities, e.g., its velocity, pressure, or external forces upon it. One can envision the Eulerian mesh as though we have placed a number of measuring devices at the lattice points and can only track fluid quantities at those points; we are not tracking individual fluid blobs.

The elegance of IB lies with how the Lagrangian grid and Eulerian mesh communicate to one another - through integral equations with delta function kernels (see Eqs. (14)-(15) in Appendix A). Simply put, these integrals say that the fluid points (on the rectangular) nearest the jellyfish (on the moving Lagrangian mesh) are influenced most by the jellyfish’s movement (via a force), while the fluid grid points further away feel substantially less (Eq. (14)). A similar analogy can be made that says the fluid motion that influences the deformations/movement of the jellyfish the most are the fluid grid points nearest the jellyfish (Eq. (15)).

Successive points along the jellyfish bell are tethered together by virtual springs and virtual (non-invariant) beams, as well as along each tentacle/oral arm, in the IB2d framework, as illustrated in Figure 3. Virtual springs allow the geometrical configuration to either stretch or compress, while virtual beams allow bending between three successive points. When the geometry stretches/compresses or bends there is an elastic deformation force that arises from the configuration not being in its preferred energy state, e.g., its initial configuration.

These deformations forces can be computed as below,

[TABLE]

where $k_{spr}$ and $k_{beam}$ are the spring and beam stiffnesses, respectively, $R_{L}(t)$ are the springs resting lengths (set to $ds$, the distance between successive points), $\textbf{X}_{A}=\langle x_{A},y_{A}\rangle$ and $\textbf{B}=\langle x_{B},y_{B}\rangle$ are the Lagrangian nodes tethered by the spring. Note that $R_{L}(t)$ indicates that the resting lengths could be time dependent. $\textbf{X}_{C}(t)$ is a Lagrangian point on the interior of the jellyfish bell and $\textbf{X}^{con}(t)$ is the corresponding initial (preferred) configuration of that particular Lagrangian point on the jellyfish bell. The spring stiffness are large to ensure minimal stretching or compression of the jellyfish bell itself, although via the beam formulation it is capable of bending and hence contracting. The $4^{th}$-order derivative discretization for the non-invariant beams is given in Battista et al. (2018). Note that the beams are deemed non-invariant because the preferred configuration is non-invariant under rotations, so if the jellyfish were to turn, the model dynamics would undergo undesired motion due to these artifacts.

In addition to the tentacles/oral arms being modeled using springs and beams, they are also modeled as poroelastic structures. As the tentacles/oral arms deform, fluid is then allowed to slip through them, based upon the magnitude of their deformation. The poroelasticity is based upon the Brinkman flow model which is extended to include a slip velocity based on the deformation of a Lagrangian structure Battista et al. (2018). The Brinkman force term is recognized as $\mu\alpha(\textbf{x})\textbf{u}(\textbf{x},t)$, where $\alpha(\textbf{x})$ is the inverse of the hydraulic permeability. This term is added onto the right-hand side of the momentum equation in the Navier-Stokes equations (see Eq.(12) in Appendix A) and is traditionally used to model flows through porous media Brinkman (1949); Nield and Bejan (1992).

Once the deformation forces along the tentacle/oral arms are calculated, they are equated to the Brinkman force term, which is then incorporated into a slip velocity, e.g.,

[TABLE]

In our model $\alpha(\textit{x})=\alpha$ is spatially independent and was chosen as $\alpha=500,000,10,000$ and $25,000$ for all simulations in Sections 3.1, 3.2, and 3.3, respectively. Note that varying the poroelastic coefficient, $alpha$, did not significantly affect forward swimming speeds (see Appendix B). Moreover, note that our model does not distinguish between tentacles or oral arms structurally and will from now on refer to them as tentacles/oral arms.

Lastly to mimic the subumbrellar and coronal muscles that induce bell contractions, virtual springs. These springs dynamically change their resting lengths in a sinusoidal fashion, see Eq.(8). Rather than tether neighboring points with virtual springs to model the muscles, we tether points across the jellyfish bell using all Lagrangian points that are below the top hemi-ellipse. The deformation force equation does not change from Eq.(4) besides using a different spring stiffness, which we call $k_{muscle}$, and time-dependent spring resting lengths. The equation that governs the time-dependent resting lengths, $R_{L}(t)$, is given as follows

[TABLE]

Upon running all simulations, we stored the following data in increments of 4% of each contraction cycle:

Position of Lagrangian Points 2. 2.

Horizontal/Vertical/Normal/Tangential Forces on each Lagrangian Point 3. 3.

Fluid Velocity 4. 4.

Fluid Vorticity 5. 5.

Fluid Pressure 6. 6.

Forces spread onto the Fluid (Eulerian) grid from the Jellyfish (Lagrangian) mesh

We then used the open-source software called VisIt Childs et al. (2012), created and maintained by Lawrence Livermore National Laboratory for visualization, see Figure 4, and the data analysis package software within IB2d Battista et al. (2017) for data analysis. Figure 4 provides a visualization of some of the data produced from a single jellyfish locomotion at a single snapshot in time during its $5^{th}$ contraction cycle, for a jellyfish with $8$ tentacles (see Section 3.1) at a Reynolds Number of $150$.

We stored $25$ time points per contraction cycle in each jellyfish simulation and temporally-averaged data over the 7th and 8th (last two) contraction cycles to compute average swimming speed for a particular simulation, $U_{avg}$. Note that forward swimming speed steadied out by $6$ contraction cycles (see Figure 8 in Section 3.1). Moreover we the computed the Strouhal Number, $St$,

[TABLE]

where $f$ is the contraction frequency, $D$ is the maximum bell diameter during a contraction cycle, and $U_{avg}$ is the temporally-averaged forward swimming speed. By taking the inverse of $St$ we can get a normalized forward swimming speed based on driving frequency. Previous studies on animal locomotion have hypothesized that propulsive efficiency is high in a narrow band of $St$, peaking within the interval $0.2<St<0.4$ Taylor et al. (2003). We are particularly interested in exploring the any relationships between $Re$, $St$, and tentacle/oral arm number density and placement. Furthermore, we will also compute the cost of transport (COT) and investigate its relationship to the aforementioned parameters. The cost of transport has been used as a measure of aquatic locomotion efficiency and is a measure of energy (or power) spent per unit distance traveled Schmidt-Nielsen (1972); Bale et al. (2014). We computed both a power-based COT and energy-based (work) COT, e.g.,

[TABLE]

where $F_{j}$ is the applied contraction force at the $j^{th}$ time step by the jellyfish, $U_{r_{j}}$ is is the bell contraction velocity at the $j^{th}$ time step, $d_{j}$ is the lateral distance the bell contracted (or expanded) during the $j^{th}$ time step, $N$ is the total number of time steps considered, and $d_{S}$ is the distance swam by the jellyfish traveled during during this period of time across all time steps considered.

Lastly, we computed the finite-time Lyapunov exponents (FTLE) which is used as a metric for instantaneous Lagrangian Coherent Structures (LCS) Haller (2000); Shadden et al. (2005); Haller (2015). In a nutshell, LCSs provide a systematic way to untangle the intricate, complex, and hidden dynamics of the system in a way that can be clearly visualized and interpreted. Within fluid flows, LCSs help reveal particle transport patterns that are of particular importance in biology. They can be used to understand various processes for jellyfish, including feeding and prey-capture Peng and Dabiri (2009); Sapsis et al. (2011) and locomotion Franco et al. (2007); Zhang (2008); Wilson et al. (2009); Lipinski and Mohseni (2009); Haller and Sapsis (2011); Taheri (2018); Miles and Battista (2019). FTLEs were computed using VisIt Childs et al. (2012), where trajectories were computed using instantaneous snapshots of the 2D fluid velocity vector field on the entire computational domain using a forward/backward Dormand-Prince (Runge-Kutta) time-integrator with a relative and absolute tolerance of $10^{-4}$ and $10^{-5}$, respectively, and a maximum advection time of $0.02s$ (1.6% of a contraction cycle) with a maximum number of steps of 250. They were visualized using a colormap corresponding to the FTLE value on the background grid as well as FTLE contours.

3 Results

Using an extension of an idealized jellyfish model by Hoover et al. 2015 Hoover and Miller (2015), and later modified by Miles et al. 2019 Miles and Battista (2019), we incorporated poroelastsic structures that mimic tentacles/oral arms and investigated how such complex tentacle/oral arm structure affects forward swimming performance, the cost of transport (COT), and differences in Lagrangian Coherent Structures (LCS). We first sought out to explore the relationship between $Re$ and the number of symmetric tentacles/oral arms. Next we chose four specific $Re$ (37.5, 75, 150, and 300), and investigated effects of tentacle/oral arm length using the same symmetric tentacle/oral arm geometry, but with differing lengths of the tentacles/oral arms, see Figure 5.

We then explored the how the placement and density of tentacles/oral arms affects forward swimming performance and cost of transport across $Re=\{37.5,75,150,300\}$. Thus we performed three sub-studies to investigate the following questions:

Is it only the outer placed tentacles that affect its swimming? 2. 2.

How does tentacle density affect its swimming? 3. 3.

If the placement of interior tentacles is varied, will it affect its swimming?

For each study we focused on quantifying differences in forward swimming speed (and Strouhal Number, $St$) and COT. We also investigated differences in Lagrangian Coherent Structures (LCS) formation using finite-time Lyanunov Exponents (FTLE) analysis to decipher regions of mixing and where fluid is being pulled/pushed in Sections 3.1-3.2 to explore dynamical differences due to including tentacles/oral arms.

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

Previous studies of jellyfish locomotion have considered forward swimming performance over a range of $Re$ Hershlag and Miller (2011); Yuan et al. (2014); Miles and Battista (2019); however, all have only modeled the jellyfish bell without any additional complex morphology, such as tentacles/oral arms. Here we consider the jellyfish geometries presented in Figure 5 to investigate how tentacles/oral arms may affect forward swimming speeds. To explore further, we placed the in silico jellyfish in increasingly more viscous fluids while holding all other parameters constant to effectively decrease the $Re$. We observed that forward swimming performance decreases, even with the addition of tentacles/oral arms.

This phenomenon is visualized in Figures 6, 7, and 8, where the first illustrates that at $Re=150$ including more tentacles/oral arms decreases forward swimming distance, the second shows dynamical differences in the vortex wake at for a case with 0 and 6 tentacles/oral arms at $Re=150$, and the last visualizes the distance swam and swimming velocity for the first 5 bell contractions performed for differing numbers of tentacles at $Re=150$. This data suggests that jellyfish with lesser numbers of tentacles/oral arms have higher forward swimming speeds and also that the presence of tentacles/oral arms suppresses vortex formation. Note that in this case here including more tentacles/oral arms means placing more further out within the bell, see Figure 5.

Figure 9 explores these relationships further by quantifying forward swimming speeds across different $Re$, ranging from $0.25$ to $1000$, for varying numbers of symmetrically placed tentacles. Both Figure 9$a$ and $b$ presents the same data, but $b$ uses a semi-logarithmic $Re$ axis. In agreement with previous models, forward swimming is negligible for $Re\lesssim 1$ and significant forward swimming begins around $Re\gtrsim 10$, when inertial effects become greater than viscous dampening. See Figure 32 in Appendix C to more clearly observe the speeds at lower $Re$.

Moreover, as viscosity decreases (and $Re$ increases) forward swimming performance, e.g., swimming speed, increases for $10\lesssim Re\lesssim 300$, similar to Miles and Battista (2019). Note that the lower end of this range various slighty for each case of differing tentacle number. For $Re\gtrsim 150$, forward swimming speed steadies out regardless of the number of symmetrically placed poroelastic tentacles/oral arms, see Figure 9. The addition of tentacles/oral arms monotonically decrease forward swimming performance. At higher $Re$ we see similar behavior to the case with no tentacles/oral arms, in which swimming speed asymptotically steadies out. Upon comparing the case with [math] tentacles/oral arms to $8$ tentacles (4 symmetrically placed per side), the case with [math] is $\sim 400\%$ faster than the $8$ tentacle case!

Next we investigated the relationship between $Re$, tentacle/oral arm number, and Strouhal Number, $St$. By reporting the forward swimming speed in non-dimensional terms, given in bodylengths per bell contraction, it is the inverse of $St$. The data illustrates that the inclusion of poroelastic tentacle/oral arms increases $St$ (since speed is slower), where for a given $Re$ more tentacles/oral arms increases $St$. The only case here that falls within the biological regime of $0.2<St<0.4$ Taylor et al. (2003) is the case with no tentacles/oral arms when $Re\gtrsim 50$. Furthermore, in terms of scaling relations, even with the addition or tentacles/oral arms, the $St$ is a monotonically decreasing function of $Re$ before steadying out around $Re\gtrsim 150$, suggesting that increasing $Re$ maximizes propulsive efficiency Nudds et al. (2014) agreeing with previous jellyfish studies of Miles and Battista (2019).

Experimental studies of jellyfish have concluded that the cost of transport (COT) for jellyfish is much lower than other metazoans Gemmell et al. (2013). They described this phenomenon by suggesting that jellyfish use passive energy recapture to decrease COT. It was later studied computationally by Hoover et al. 2019 Porras and Miller (2019), confirming this as the reason for lower COT in a model. We hypothesized that the scaling relationship between cost of transport, number of tentacles/oral arms, and $Re$ would be similar to those shown in Miles et al. 2019 Miles and Battista (2019), where higher $Re$ lead to decreases COT for a variety of contraction frequencies; however, where for a specified number of tentacles/oral arms the COT would increase due to slower forward swimming speeds, where the kinematics of contraction remain uniform between all tentacle/oral arm cases. Figure 11 highlights the COT data for both an average work-based and average power-based COT over multiple orders of magnitude of $Re$ and varying numbers of tentacles/oral arms. Generally, COT decreases as $Re$ increases. For a given $Re$, the highest COT corresponds to the case with the most tentacles/oral arms (similar to Figure 10). Note that for a particular jellyfish morphology (with a specified number of tentacles/oral arms), optimal swimming performance occurs near $Re\gtrsim 100$, where COT appears to be minimal and forward swimming speed is maximized.

We then performed Lagrangian Coherent Structure (LCS) analysis by computing the finite-time Lyapunov exponent (FTLE) Haller (2000). Small values of the FTLE highlight regions where flow is attractive while large values indicate areas in which the flow is repelling. For jellyfish, LCSs can be used to highlight the regions of fluid in which the jellyfish is pulling towards or pushing away from its bell during contraction and expansion, respectively Wilson et al. (2009). Figure 12 compares the FTLE LCS analysis at the start of the $4^{th}$ contraction cycle between $Re=\{37.5,75,150,300\}$ for the case of $6$ total tentacles/oral arms (3 symmetrically placed on each side of the bell). To see a comparison over the entire contraction cycle, see Figure 33 in Appendix C. During bell contraction, high FTLE values are seen near the tips of the jellyfish bell, indicating regions of high fluid mixing in all cases. For cases with $Re\gtrsim 75$, those regions on high fluid mixing are expelled downward by the time the contraction phase ends; however, in comparison to the case with no tentacles (see Miles and Battista (2019)), there is less mixing in the vortex wake due to suppressed vortex formation. Moreover, there appears to be further horizontal mixing in the wake in comparison. In general as $Re$ increases, the size of the regions with high FTLE also increase in the vortex wake.

In each case, the low FTLE values above the bell suggest that fluid is being pulled downwards towards the end of the bell during contraction, rather than the jellyfish horizontally pulling in fluid, even with the addition of tentacles. Within the bell, similar to Miles and Battista (2019), fluid is pulled inwards and towards the top during contraction and expansion; however, there is much less pronounced mixing within the center of the bell due to the presence of tentacles/oral arms. The high FTLE values near the bottom of the bell that suggest highly-attractive flow regions could be vital for allowing jellyfish to expel wastes from its mouth and out of their bells during a contraction cycle. Furthermore, more tentacles/oral arms appear to decrease the amount of mixing in the vortex wake, see Figure 13. There are still higher levels of mixing near the bottom of the bell with the addition of more tentacles/oral arms. To observe the effect of tentacle number on fluid mixing see Figure 34 in Appendix C.

For this particular jellyfish geometry, these results suggest that the jellyfish would ideally want its prey to be in front (a top) of its bell so that during a contraction cycle, the prey would be pulled downwards towards its bell and into its tentacles. These results could be specific to this jellyfish bell geometry. It is possible that changes in bell diameter or contraction kinematics could lead to differences in fluid mixing patterns, suggesting prey are captured more horizontally rather than vertically, like in the case of Cassiopea Santhanakrishnan et al. (2011); Hamlet and Miller (2012).

Next we explored how the length of tentacles/oral arms affects forward swimming.

3.2 Results: Varying the Length of Tentacles/Oral Arms

While in Section 3.1 we observed that the number of tentacles significantly affects locomotive processes at various fluid scales ($Re$), we did not vary the length of the tentacles/oral arms. In this section we will fix the Reynolds Number at $Re=150$, which is approximately the $Re$ of Sarsia and vary the number of tentacles/oral arms and their respective length. Note that we use the model geometry given in Figure 5 but with varying lengths of the tentacles/oral arms. As this is a first study of tentacles/oral arms we will still use uniform tentacle/oral arm length in each respective case. Throughout this section the length of a tentacle is given in multiples of the bell radius, $a$.

First we observed that longer tentacle/oral arms leads to decreased forward distance swam, see Figure 14. Figure 14 gives the distance swam against number of bell contractions for the case of $8$ tentacles/oral arms ($4$ per side). As tentacle/oral arm length increases, the jellyfish does not travel as far, thus it appears if they are short enough, they may not significantly change swimming performance from the case in which there are none. As length increases, forward swimming gets decreasingly less pronounced; however, when the lengths get long enough ($\sim 4a$), it appears elongating tentacles/oral arms further will not significantly decrease forward swimming performance, as the data appears to asymptotically steady out (see Figure 14 and compare with the $8a$ case).

This last idea is further solidified when quantifying forward swimming speeds, see Figure 15a. Figure 15a provides the forward swimming speed for multiple tentacle/oral arm lengths (given in multiples of the bell radius, $a$) for cases involving different numbers of tentacle/oral arms. If length is short enough, even in cases of $8$ total tentacles/oral arms ($4$ per side), swimming speeds are not significantly different from the case with no tentacles/oral arms. In fact, the data appears to converge towards the case with none as length gets shorter.

As length increases, the number of equally-spaced and symmetrically-placed tentacles/oral arms begins to matter. That is, the swimming speeds between all cases begins to diverge around $\sim 1.5a$. While swimming speed dramatically decreases in the case of $8$ tentacles/oral arms, for the case with $2$ tentacles/oral arms the forward swimming speed is not too different from the case with no tentacles/oral arms. This may be due due to the $8$ tentacles/oral arms taking up more space within the bell, which seems to suppress vortex formation and thus reduces the size of the vortex ring expelled during expansion. Thus, each number of tentacles/oral arms the forward swimming speed decreases at different rates as a function of the tentacle/oral arm length. As length increases, eventually swimming speeds appear to steady out, and elongating them further will not significantly decrease swimming speed.

Figure 15b gives the Strouhal Number, $St$, vs tentacle/oral arm length (quantified in multiples of the bell radius, $a$). For shorter tentacles/oral arms the $St$ converges to approximately 0.5, near the biological range of $0.2<St<0.4$; however for lengths greater than $\sim 2a$, $St$ grows of this range in all cases except case with $2$ tentacles/oral arms ($1$ per side), which grows after $\sim 3a$. As length increases the $St$ continues to increases in each case, but appears to asymptotically taper out. A similar trend is seen when investigating the cost of transport ($COT$), see Figure 16. Shorter tentacles/oral arms have lower $COT$, while longer have higher. As the number of the tentacles/oral arms increases, so does the $COT$.

In each case of different numbers of tentacles/oral arms, there appears to be 3 different regimes of tentacle/oral arm length. (1) If the length is short enough ($\lesssim 1.5$ bell radii), forward swimming speeds are not significantly different than the case of no tentacles/oral arms. (2) Within a certain length range ($\sim 1-2$ bell diameters), swimming speeds significantly drop off (3) For long enough tentacles/oral arms, swimming speed begins to steady out ($\gtrsim 2$ bell diameters). Coupling these ideas with the $COT$ data, lesser numbers of and shorter tentacles/oral arms suggest that these jellyfish may be better and more efficient active hunters (predators) than its jellyfish counterparts with more and longer tentacles/oral arms. One example of this could be box jellyfish (Carukia barnesi) rather than a jellyfish with longer tentacles, such as the lion’s mane (Cyanea capillata) Linnaeus (1758); Kinsey (1988); Powell and Atland (2018).

The decreases in forward swimming speeds may be attributed to suppressed vortex formation and ring dynamics, as suggested previously in Figure 7 and now Figure 17. Figure 17 provides a colormap of vorticity at the end of the $5^{th}$ contraction cycle for cases of varying tentacles/oral arm number and lengths. For the case of having lengths $1a$, it is clear that the addition of more equally-spaced and symmetrically-placed tentacles/oral arms decreases the size of the downstream vortex wake. In particular, for cases of more tentacles/oral arms, vortex rings dissipate closer to the jellyfish than in the cases with lesser numbers. As the tentacle/oral arm length increases, the topology of the vortex wake is significantly altered. Rather than vortex rings being pushed directly downward, they begin to expel more horizontally, more laterally. Higher number and longer tentacles/oral arms appear to act, in essence, as an inelastic structure bouncing vortices back off of them, not allowing them to move directly downward.

These ideas are further explored while performing LCS analysis (via computing the finite time Lyapnuov exponents (FTLE)). We explored how regions of fluid mixing varied due to variations of tentacle/oral arm length, see Figure 18. Figure 18 gives the FTLE as a colormap at the start of the $4^{th}$ contraction cycle for variety of tentacle/oral arm lengths. If lengths are short enough $(\lesssim 2a)$ there is still significant mixing downstream in a vertically longer vortex wake, as suggested by Figure 17. However, as lengths increases there is more horizontal mixing near the bottom of the bell, than downstream. Interestingly, in cases for lengths $\sim 8a$ the ends of the tentacles/oral arms do not experience much mixing at all. This further suggests when there long enough and enough tentacles/oral arms in number, they act as an inelastic pseudo-wall bouncing vortices back off of them, which in turn causes them to move laterally, rather than in the opposite direction of swimming motion. The increase of lateral mixing within the tentacle/oral arm region would help the jellyfish capture prey in its tentacles for feeding. To observe fluid mixing over the $4^{th}$ to $5^{th}$ contraction cycle see Figure 35 in Appendix D.

Underlying the above results and discussion was the assumption that all tentacles/oral arms in Sections 3.1 and 3.2 were equally-spaced and symmetrically-placed on each side of the jellyfish bell. That is, if we considered a jellyfish with $8$ tentacles/oral arms ($4$ per side), the geometry was exactly the same as the jellyfish with $6$, but with the addition of one more tentacle/oral arm per side, equally-spaced from the previously outermost. We will now relax this, and consider how changes in tentacle/oral density affects the swimming performance metrics studied above.

3.3 Results: Tentacle/Oral Arm Placement and Density

In Sections 3.1 and 3.2 we explored an idealized jellyfish model where all tentacles/oral arms were equally spaced apart from one another. For example, if a jellyfish had $6$ tentacles, four of its tentacles/oral arms would be placed exactly where the $4$-Tentacle/oral arm jellyfish had theirs placed, and then in addition to those four, it would have two additional tentacles/oral arms equally-spaced outside of those $4$, one per side. Here we will relax those assumptions and consider different placements and densities of the tentacles/oral arms inside a particular tentacle/oral arm containing region to investigate how placement and density affect potential forward swimming performance.

The density and locations of tentacles/oral arms are extremely diverse among different species of jellyfish, recall Figure 2. This is a preliminary study investigating how density and placement may play a role in differing swimming behavior for an idealized jellyfish model. In particular, we will address the following three questions to probe the surface of how different placements/densities of tentacles/oral arms may affect forward locomotion:

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. 2. 2.

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. 3. 3.

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?

For this study, we will include the tentacles/oral arms labeled $A$ and $F$ in our jellyfish model across all simulations, see Figure 19. We will then consider four cases: (1) $ABCDEF$ (2) $ABEF$ (3) $ACDF$ and (4) $AF$. These cases vary in the number of total tentacles/oral arms, either 2, 4 or 6, as well as placement of inner tentacles/oral arms for $Re=37.5,75,150$ and $300$. Note that the case $ABCDEF$ is the same case as in the $6$ tentacle case in Section 3.1.

Qualitative analysis of forward swimming performance is given in Figure 20, where positions of the Lagrangian points are illustrated across the first $5$ contraction cycles for the case of $Re=150$. It appears that the $AF$ case with only two tentacles/oral arms is able to swim faster than the other cases. Interestingly, it does not appear that less tentacles always leads to faster forward swimming as in Sections 3.1 and 3.2; the case of $ABCDEF$ (6 tentacles/oral arms) seems to swim slightly faster than either $ABEF$ or $ACDF$ (4 tentacle/oral arm cases); however, these were only the first $5$ contraction cycles.

Upon computing the forward swimming speed for each case of $Re$ and tentacle/oral arm configuration considered, the case of $AF$ appears the fastest for each $Re$, see Figure 21a. Figures 21a and 21b give the forward swimming speed and cost of transport (average power/total distance swam), respectively. For all $Re$ the $ABEF$ case was the second fastest; however, the third fastest case is the $ABCDEF$ for $Re=37.5$ and $75$, and $ACDF$ for $Re=150$ and $300$. Table 3 gives the percentage decrease in swimming speeds across all cases when comparing to the case with no tentacles/oral arms. More tentacles/oral arms do not always cause lower swimming speeds, see $Re=37.5$ and $75$ where $ABCDEF$ swims faster than $ACDF$. Note that higher $Re$ generally shows a decrease in the difference between the case with no tentacles/oral arms to those with. The cost of transport is lowest in the $AF$ case across all $Re$ considered, followed by the $ABEF$ case.

It is not the case that only the placement of the outermost tentacles/oral arms solely affects forward swimming speed. There appears to be a non-linear relationship between forward swimming speed and density of tentacles/oral arms packed into the same region within a bell. We will probe this relationship further in Section 3.3.2. Moreover, this section suggests that placing more tentacles/oral arms closer to the outermost ones may be beneficial for forward swimming performance; we will explore this idea further in Section 3.3.3.

Figure 22 illustrates a colormap of vorticity for each different tentacle/oral arm case across the $4^{th}$ to $5^{th}$ contraction cycle for $Re=150$. In the $AF$ case, there is still a pronounced vertically-aligned vortex wake, where as in other cases of more tentacles/oral arms, the vortices begin to disperse more horizontally, as similarly suggested by previous vorticity plots (Figures 7 and 17) as well as LCS plots (Figures 13 and 18). We hypothesize that the $AF$ case is the fastest because vortex formation is not as suppressed as in the other cases. However, this would not explain why the $ABCDEF$ case is faster than the $ACDF$ case for $Re\lesssim 75$. This behavior may be attributed to that in the $ABCDEF$ case as there are more densely packed tentacles/oral arms, they may be acting as more of a “rigid" wall, which allows for more elastic vortex interactions, while in the $ACDF$ cases, vortices are being “cushioned" into the tentacles/oral arms, giving rise to more inelastic interactions, as there is more open fluid-filled space. However, for $Re\gtrsim 150$, it appears this phenomena no longer suffices to provide any elastic/inelastic advantage.

3.3.2 How does density of tentacles affect swimming performance?

For this study, we will use the same placement of the outermost tentacles/oral arms as in Section 3.3.1 but place a different number of equally-spaced, symmetric tentacles/oral arms within the region, see Figure 23. Constructing such geometry will allow us to study effects of density of tentacles/oral arms. The even spacing was to reduce possible artifacts caused by unequal weighing of the tentacles, as in Section 3.3.1’s $ABEF$ and $ACDF$ cases. We will consider four cases of differing number of tentacles/oral arms (including the outermost): (1) 2 per side (2) 4 per side (3) 5 per side and (4) 10 per side. These numbers were chosen to keep equal spacing of the tentacles/oral arms in each subsequent case mandating that the outermost were attached to a particular Lagrangian point on the bell. Note that the previous case $ABCDEF$ in Section 3.3.1 (or the $6$ tentacle/oral arm case in Section 3.1) is different as the middle two tentacles/oral arms are not twice the distance apart, see Figures 19 and 23 for comparison. These cases were studied for $Re=37.5,75,150$ and $300$.

A qualitative analysis of forward swimming progress is given in Figure 24, where positions of the Lagrangian points are illustrated across the first $5$ contraction cycles for the case of $Re=150$. It appears that 4 tentacles/oral arms per side case is able to swim faster than the other cases, although this is only over the first $5$ contraction cycles. Note that less tentacles/oral arms per side do not always appear to contribute to higher swimming accelerations, as the $5$ tentacles/oral arms per side case accelerates faster than the $2$ and $10$.

Upon quantifying swimming speeds we saw that the case with $2$ tentacles swims faster than all other cases, although barely in the $Re=300$ case (see Table 4). This data is provided in Figure 25a. Only in the cases of $Re=75$ and $150$ do more tentacles always lead to decreased swimming speeds. In the $Re=37.5$ case, the $10$ tentacle per side case is slightly faster than the $5$ per side, while in the $Re=300$ case, the $5$ tentacles per side case is the second fastest (barely slower than the $2$ per side case), followed by the $10$ per side case, and finally the $4$ per side case. From Figure 25b, more tentacles per side does not always attribute to greater cost of transport, e.g., the $5$ and $10$ per side case for $Re=300$.

To compare the cases further, we compute the relative percentage differences in swimming speeds for each respective $Re$ by comparing to case with no tentacles. This data is presented in Table 4. There is no general linear relationship between forward swimming speed and density of uniformly spaced tentacles among all cases. Such relationship only appears to manifest for $Re=75$ and $Re=150$ only; the $Re=37.5$ and $300$ cases show a nonlinear relationship. Moreover, as generally as $Re$ increases the percentage difference between the case with no tentacles and cases with tentacles/oral arms decreases here, with two exceptions - for $4$ per side between $Re=150$ and $Re=300$ and for $10$ per side between $Re=37.5$ and $Re=75$.

Figure 26 gives a vorticity colormap from the $4^{th}$ to $5^{th}$ contraction cycle across each density case for $Re=150$. For every time-point shown, variations in vortex topology is observed, particularly between the $2$, $4$, and $5$ or $10$ case. The case with $5$ or $10$ tentacles/oral arms per side show a similar vortex wake structure, while vortex wake in the case with $4$ is qualitatively different. Recall that for $Re=150$, the case with $4$ per side was the fastest swimmer. It looks as though that this configuration was able to produce the most vertically-aligned vortex wake, comparatively; however, it is not understood why Smits (2019).

From Sections 3.3.1 and 3.3.2, it is evident that small changes in tentacle/oral arm morphology, e.g., their placement and density, may significantly affects forward swimming speed. By observing differences among fluid scale ($Re$) in Section 3.3.2, we hypothesize that fluid scale and density of tentacles/oral arms couple to vary energy (and vortex) absorption into the tentacle/oral arms during each contraction cycle, making some vortex-tentacle/oral arm interactions more elastic than others, which may enhance forward swimming speeds. This was also seen in Section 3.3.1 as well between cases of $ABCDEF$ and $ACDF$ for $Re\lesssim 75$.

The last study we performed was biasing the placement of tentacle/oral arms towards the outermost ones. We present these results in Section 3.3.3. It was motivated by the cases $ACDF$ and $ABEF$ in Section 3.3.1, which showed significant differences in swimming speed for the same number of tentacles, where the case with more further placed tentacles was faster ($ABEF$).

3.3.3 How does stacking tentacle/oral arms towards the outermost ones affect swimming performance?

For this study, we will use the same placement of the outermost tentacles/oral arms as in Sections 3.3.1 and 3.3.2 but place more tentacles/oral arms towards the outermost ones, rather than equally-spacing them within the bell, as in Section 3.3.2. This essentially makes clusters of the tentacles/oral arms towards the outermost ones. We will also include two other cases, not addressed in previous sections, where we include clusters of $2$ tentacles/oral arms at other locations, “Outer/Inner", and cluster tentacles/oral arms at the midpoint, “Inner Unequal Spacing", see Figure 27 for all cases considered here. These cases were studied for $Re=37.5,75,150$ and $300$.

A qualitative analysis of forward swimming progress is given in Figure 28, where positions of the Lagrangian points are illustrated across the first $5$ contraction cycles in the $Re=150$ case. The Outer 2 case looks to have swam the furthest after $5$ contraction cycles, followed closely by the Outer/Inner case and then the Outer 3 case. Note that these cases have 4, 8, and 6 tentacles/oral arms, respectively. This again illustrates that a nonlinear relationship exists between forward swimming speed and tentacle/oral arm number and density, as in Sections 3.3.1 and 3.3.2; less tentacles/oral arms per side do not appear to always contribute to faster swimming. However, recall that during these 5 contraction cycles, steady swimming has only started to have been achieved, and so we will now quantify steady swimming speeds across the seventh and eighth contraction cycle.

Swimming speeds and cost of transport for $Re=37.5,75,150$, and $300$, are given in Figures 29a and 29b, respectively. There is a nonlinear relationship with number and density of tentacles/oral arms and forward swimming speed, even when placing them closer to the outermost ones. The Outer 2 case is the fastest case across all $Re$, although barely for $Re=37.5$ and $300$. Furthermore, the second fastest case changes between the Outer 3 and Outer 4 case; Outer 3 is second fastest for $Re=37.5$ and $300$, while Outer 4 is second fastest for $Re=75$ and $150$. For $Re=37.5$, a linear relationship between number of outer placed of tentacles/oral arms and forward swimming speed appears to emerge, though; however, it does not exist in any other cases.

Moreover, density and placement affects swimming speed as we see that the Outer 4 and Outer/Inner cases, both which have $8$ total tentacles/oral arms (4 per side), do not swim with the same speed. Only in the $Re=300$ case do they swim at similar speeds, see Table 5, which gives the percent difference between each case described above the case of no tentacles/oral arms. Similarly to Sections 3.3.1 and 3.3.2, percent differences generally decrease as $Re$ increases, but with two explicit exceptions - Outer 2 between $Re=150$ and $Re=300$ as well as Inner Unequal between $Re=37.5$ and $Re=75$.

Similar to Sections 3.3.1 and 3.3.2, Section 3.3.3 highlights the existence of a complex relationship between fluid scale ($Re$), placement, number, and density of tentacles/oral arms in regards to potential forward swimming performance. Different variations of these parameters give rise to differing vortex wakes (see Figure 30), which could possibly be used to predict enhanced or inhibited swimming performance Miles and Battista (2019); however, such relationships are nontrivial or may be impossible to discern accurately Smits (2019); ad T. V. Buren and Smits (2019). In particular, different fluid scales ($Re$) and tentacle/oral arm densities may determine whether the tentacles/oral arms act as energy absorbing entities or otherwise (elastic/inelastic vortex collisions). There are further intricate relationships to decrypt on how this either enhances or inhibits forward swimming.

4 Discussion and Conclusion

Previous fluid-structure interaction models have shown that a jellyfish’s bell morphology, material properties, and kinematics affect its potential forward swimming speeds Hoover and Miller (2015); Hoover et al. (2017); Miles and Battista (2019). This is the first computational study to reveal that including tentacles/oral arms will also significantly affect forward swimming performance in jellyfish. Using an idealized $2D$ computational model, we illustrated a complex relationship between tentacle/oral arm morphology (number, length, placement, and density) and fluid scale ($Re$) on forward swimming speed.

In particular we discovered that including more symmetrically and equally spaced tentacles/oral arms within its bell, will greatly inhibit forward swimming speed (Section 3.1). Similarly to previous studies Hershlag and Miller (2011); Miles and Battista (2019), forward swimming speed steadies out as $Re$ increases, even in cases with tentacles/oral arms; the tentacles/oral arms simply lowers the maximal swimming forward speed achievable (Section 3.1). K. Katija in 2015 Katija (2015) saw a similar percentage decrease in swimming speed for Australian Spotted Jellyfish with and without oral arms. Moreover, including more symmetrically/equally spaced tentacles/oral arms seems to increase horizontal mixing by the jellyfish bell, which decreases vertical mixing as well. The vertical-size of the vortex wake also decreases. Thus jellyfish morphology not only can limits its intrinsic swimming ability, but overall fluid transport and mixing Costello et al. (2008); Katija et al. (2015).

For $Re=150$, which is the approximate scale of a Sarsia tubulosaa, varying the length of the tentacles/oral arms showed the existence of three possible swimming performance states. (1) If their length is short enough, forward swimming speeds are not significantly different than the case of no tentacles/oral arms. (2) For long tentacles/oral arms, swimming speed asymptotically steadies out, and longer tentacles/oral arms will not significantly affect forward swimming speeds. (3) There is a middle range in which swimming speed drops off from approximately the case with no tentacles/oral arms to where swimming speed asymptotically steadies out. It does not appear that longer tentacles will drop forward swimming speed to zero. Tentacles/oral arms of varying lengths lengths have different positional dynamics change during a contraction, where short enough tentacles/oral arms are pulled upwards into the bell with minimal amounts exposed out of the bell, while long enough tentacles are pulled upwards into the bell as well, but with the majority of the tentacle/oral arm still hanging outside the bell (Section 3.2). Thus, as tentacles/oral arms get longer, more fluid mixing occurs near the jellyfish bell, rather than downstream in the vortex wake.

A nonlinear relationship between tentacle/oral arm placement and density and swimming speed was also observed (Section 3.3). It is not that case that only placement of the outermost tentacles/oral arms dictates the inhibition of forward swimming speed (Section 3.3.1). Less dense tentacle/oral arm configurations (within the same region) did not always lead to increased swimming performance (Section 3.3.2). Stacking more tentacles/oral arms closer to the outermost ones, did not always lead to lower forward swimming speeds (Section 3.3.3). Furthermore, small changes in placement of the same number of tentacles/oral arms may results in non-linear differences in swimming speeds (Sections 3.3.1 and 3.3.3).

This study showed that small changes in tentacle/oral number, placement, density, or length significantly affected forward swimming performance in an $2D$ idealized jellyfish model. Any effects of tentacle/oral arm stiffness were not thoroughly investigated. Investigating the effect of tentacle/oral arms for a variety of bell morphologies and kinematics would provide further insight into jellyfish ecology. In particular, modeling a specific jellyfish species’ bell and tentacle/oral morphology and bell kinematics could highlight why it uses a particular predation strategy over others.

Even certain jellyfish species (Turritopsis nutricula) that have been deemed to have the potential for immortality Piraino et al. (1996), need to eat to sustain themselves. Under an evolutionary lens, locomotive mechanisms are largely believed to be based on the size of an organism Vogel (1996). Jellyfish have evolved and adapted to use a diverse variety of foraging strategies across a variety of sizes and morphologies. Some are active hunters, such as the sea wasp (box jellyfish, Chironex fleckeri), while others are more opportunist predators, who passively drift and wait for prey, such as the Lions Mane jellyfish (Cyanea capillata). Both species have significantly different bell, tentacle, oral arm morphologies, bell kinematics, as well as, forward swimming speeds Purell and Arai (2001); Purell (2003); Bastian (2011); Colin et al. (2013); Crawford (2016). Therefore, a foraging advantage for the Lions Mane jellyfish are its longer, numerous, and dense tentacles, as it does not actively hunt, possibly due to a constraint between such morphology and forward swimming performance. On the other hand, the box jellyfish may not benefit from more dense or numerous tentacles, as it could potentially inhibit its swimming performance and thereby its predation strategy. The evolution of jellyfish may detail an interesting story between shape and size (bell morphology, tentacle/oral arm number, placement, density, and length) and function (active hunting or passive foraging strategies), where evolutionary bifurcations and adaptations gave rise to some of the oldest (most successful) but possibly laziest (passive, drift eating), or fear-inducing (active hunting), efficient organisms.

\authorcontributions

Conceptualization, J.G.M. and N.A.B.; Data curation, N.A.B.; Formal analysis, N.A.B.; Funding acquisition, J.G.M. and N.A.B.; Investigation, J.G.M. and N.A.B.; Methodology, J.G.M. and N.A.B.; Project administration, N.A.B.; Resources, N.A.B.; Software, N.A.B.; Supervision, N.A.B.; Validation, N.A.B.; Visualization, J.G.M. and N.A.B.; Writing - original draft, J.G.M. and N.A.B.; Writing - review & editing, J.G.M. and N.A.B..

\funding

J.G.M. was partially funded by the Bonner Community Scholars Program and Innovative Projects in Computational Science Program (NSF DUE #1356235) at TCNJ. N.A.B. was funded and supported the NSF OAC-1828163, TCNJ Support of Scholarly Activity (SOSA) Grant, the Department of Mathematics and Statistics, and the School of Science at TCNJ.

Acknowledgements.

The authors would like to thank Laura Miller and Alexander Hoover for sharing their knowledge and passion of jellyfish locomotion and Yoshiko Battista for introducing N.A.B. to the world of marine life. We would also like to thank Christina Battista, Robert Booth, Christina Hamlet, Matthew Mizuhara, Arvind Santhanakrishnan, Emily Slesinger, and Lindsay Waldrop for comments and discussion. \conflictsofinterestThe authors declare no conflict of interest. \abbreviationsThe following abbreviations are used in this manuscript:

[TABLE]

\appendixtitlesyes \appendixsectionsmultiple

Appendix A Details on IB

A two-dimensional formulation of the immersed boundary (IB) method is discussed below. The IB software used was IB2d Battista et al. (2015, 2017, 2018). The software has been validated Battista et al. (2017) with specific convergence tests performed for the jellyfish model in Battista and Mizuhara (2019) and Miles and Battista (2019). For a full review of the immersed boundary method, please see Peskin 2002 Peskin (2002) or Mittal et al. 2005 Mittal and Iaccarino (2005).

A.1 Governing Equations of IB

The conservation of momentum and mass equations that govern an incompressible and viscous fluid are listed below:

[TABLE]

[TABLE]

where ${{\bf u}}({\bf x},t)$ is the fluid velocity, $p({\bf x},t)$ is the pressure, ${{\bf F}}({\bf x},t)$ is the force per unit area applied to the fluid by the immersed boundary, $\rho$ and $\mu$ are the fluid’s density and dynamic viscosity, respectively. The independent variables are the time $t$ and the position ${\bf x}$. The variables ${{\bf u}},p$, and ${{\bf F}}$ are all written in an Eulerian frame on the fixed Cartesian mesh, x.

The interaction equations, which handle all communication between the fluid (Eulerian) grid and immersed boundary (Lagrangian grid) are the following two integral equations:

[TABLE]

where ${\bf f}(s,t)$ is the force per unit length applied by the boundary to the fluid as a function of Lagrangian position, $s$, and time, $t$, $\delta({\bf x})$ is a three-dimensional delta function, and ${\bf X}(s,t)$ gives the Cartesian coordinates at time $t$ of the material point labeled by the Lagrangian parameter, $s$. The Lagrangian forcing term, ${\bf f}(s,t)$, gives the deformation forces along the boundary at the Lagrangian parameter, $s$. Equation (14) applies this force from the immersed boundary to the fluid through the external forcing term in Equation (12). Equation (15) moves the boundary at the local fluid velocity. This enforces the no-slip condition. Each integral transformation uses a three-dimensional Dirac delta function kernel, $\delta$, to convert Lagrangian variables to Eulerian variables and vice versa.

Using delta functions as the kernel in Eqs.(14-15) is what gives IB its power. To approximate these integrals, discretized (and regularized) delta functions are used. We use the ones given from Peskin (2002), e.g., $\delta_{h}(\mathbf{x})$,

[TABLE]

where $\phi(r)$ is defined as

[TABLE]

A.2 Numerical Algorithm

As stated in the main text, we impose periodic and no slip boundary conditions on the rectangular domain. To solve Equations (12), (13),(14) and (15) we need to update the velocity, pressure, position of the boundary, and force acting on the boundary at time $n+1$ using data from time $n$. The IB does this in the following steps Peskin (2002).

Step 1: Find the force density, ${\bf{F}}^{n}$ on the immersed boundary, from the current boundary configuration, ${\bf{X}}^{n}$.

Step 2: Use Equation (14) to spread this boundary force from the Lagrangian boundary mesh to the Eulerian fluid lattice points.

Step 3: Solve the Navier-Stokes equations, Equations (12) and (13), on the Eulerian grid. Upon doing so, we are updating ${\bf{u}}^{n+1}$ and $p^{n+1}$ from ${\bf{u}}^{n}$, $p^{n}$, and ${\bf{f}}^{n}$. Note that a staggered grid projection scheme is used to perform this update.

Step 4: Update the material positions, ${\bf{X}}^{n+1}$, using the local fluid velocities, ${\bf{U}}^{n+1}$, computed from ${\bf{u}}^{n+1}$ and Equation (15).

Appendix B Varying the Poroelastic Coefficient, $\alpha$

Varying $\alpha$ did not significantly affect forward swimming speeds for different numbers of tentacles/oral arms. Figure 31 shows that for $Re=150$ varying $\alpha$ did not drastically affect forward swimming speeds. Numerical stability issues were encountered for $\alpha<10^{4}$. The scope of this work was to explore general inhibitions on swimming performance by the addition of tentacles/oral arms, rather than focusing on effects of different poroelasticities.

Appendix C Varying the Reynolds Number, $Re$

This data was previously presented in Section 3.1; however, here we present the data in logarithmic form to more clearly illustrate swimming speeds at lower $Re$, see Figure 32. Figure 32 illustrates that over a particular range of $Re$ that swimming speed geometrically increases for a specified tentacle number. Interestingly, the geometric increase appears uniform across every case of differing tentacle number, as the linear slope on this logarithmic plot is approximately the same.

Figure 33 compares the FTLE LCS analysis over one contraction cycle (between the $4^{th}$ and $5^{th}$) between $Re=\{37.5,75,150,300\}$ for the case of $6$ total tentacles/oral arms (3 symmetrically placed on each side of the bell).

Figure 34 compares the FTLE LCS analysis over one contraction cycle (between the $4^{th}$ and $5^{th}$) between cases of differing number of tentacles at $Re=150$.

Appendix D Varying the Tentacle/Oral Arm Length

Figure 35 compares the FTLE LCS analysis over one contraction cycle (between the $4^{th}$ and $5^{th}$) between cases of differing tentacle/oral arm lengths at $Re=150$.

\reftitle

References