Bulletin of Mathematical Biologyhttps://doi.org/10.1007/s11538-019-00657-2

ORIG INAL ART ICLE

Lift and Drag Acting on the Shell of the AmericanHorseshoe Crab (Limulus polyphemus)

Alexander L. Davis1,4 · Alexander P. Hoover2 · Laura A. Miller3,4

Received: 18 February 2019 / Accepted: 8 August 2019© Society for Mathematical Biology 2019

AbstractThe intertidal zone is a turbulent landscape where organisms face numerous mechani-cal challenges from powerful waves. A model for understanding the solutions to thesephysical problems, the American horseshoe crab (Limulus polyphemus), is a marinearthropod that mates in the intertidal zone, where it must contend with strong ambientflows to maintain its orientation during locomotion and reproduction. Possible strate-gies to maintain position include either negative lift generation or the minimizationof positive lift in flow. To quantify flow over the shell and the forces generated, welaser-scanned the 3D shape of a horseshoe crab, and the resulting digital reconstructionwas used to 3D-print a physical model. We then recorded the movement of trackingparticles around the shell model with high-speed video and analyzed the time-lapseseries using particle image velocimetry (PIV). The velocity vector fields from PIVwere used to validate numerical simulations performed with the immersed boundary(IB) method. IB simulations allowed us to resolve the forces acting on the shell, aswell as the local three-dimensional flow velocities and pressures. Both IB simulationsand PIV analysis of vorticity and velocity at a flow speed of 13cm/s show negative liftfor negative and zero angles of attack, and positive lift for positive angles of attack ina free-stream environment. In shear flow simulations, we found near-zero lift for allorientations tested. Because horseshoe crabs are likely to be found primarily at near-zero angles of attack, we suggest that this negative lift helps maintain the orientationof the crab during locomotion and mating. This study provides a preliminary founda-tion for assessing the relationship between documented morphological variation andpotential environmental variation for distinct populations of horseshoe crabs along theAtlantic Coast. It also motivates future studies which could consider the stability ofthe horseshoe crab in unsteady, oscillating flows.

Keywords Immersed boundary method · Computational fluid dynamics · Pedestrianaquatic locomotion

B Alexander L. Davisal.davis@duke.edu

Extended author information available on the last page of the article

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1 Introduction

Ambient flow of air or water provides a challenge with which many organisms mustcontend. The intertidal zone is at the interface of these two mediums where organismsare exposed to powerful wave action and strong currents (Denny and Gaines 1990;Denny 1991). Intertidal inhabitants like limpets and other invertebrates have evolvedmorphologies that reduce the effects of hydrodynamic lift and drag, preventing wavesfrom dislodging the shell (Denny et al. 1985; Denny 1989). Other organisms adoptpostures that allow them to remain attached to the substrate by reducing drag (Maudeand Williams 1983; Martinez 2001; Webb 1989). Although being swept away by thedrag from crashing waves is important to consider, hydrodynamic lift can be justas dangerous (Trussell 1997). Because forces on intertidal animals may be large,reducing lift is important for maintaining attachment (Denny 1989; Bell and Gosline1997). While many studies of hydrodynamic forces on organisms focus on sessileanimals, far fewer have investigated intertidal forces acting on locomoting, leggedorganisms (Martinez 2001; Bill and Herrnkind 1976; Blake 1985; Martinez 1996;Pond 1975).

The American horseshoe crab (Limulus polyphemus) is a marine arthropod thatprimarily relies on its legs for locomotion. Horseshoe crabs have a highly conservedmorphology that resembles fossils from the Mesozoic, indicating a suitable morphol-ogy for their environment (Selander et al. 1970; Walls et al. 2002). There is, however,documentedmorphological and genetic variation in shell curvature and the presence ofspines between distinct populations along the Atlantic Coast of North America (Pierceet al. 2000; Riska 1981; Saunders et al. 1986; Zaldívar-Rae et al. 2009; Sekiguchi andShuster 2009). Additionally, there are three other extant species, T. gigas (Müller,1785), T. tridentatus (Leach, 1819), and C. rotundicauda (Latreille, 1802), and mul-tiple related fossil species with varying carapace morphologies (Stoermer 1952). Thegenerally conserved shape of Limulus spp. and the inter- and intra-specific varia-tions of some features make horseshoe crabs a useful system for investigating therelationship between morphology and hydrodynamic forces in legged, aquatic organ-isms.

Horseshoe crabs are particularly interesting for studying lift reduction because theyare exposed to two different types of flows: (1) ambient flows from waves or tides and(2) self-generated flows from locomotion. Horseshoe crabs mate on sandy beachesin the surf zone where they experience significant wave action and ambient currentsspeeds that are far larger than self-generated flows (Brockmann 1990). This poses aproblem for the crab because once flipped over righting is a challenge. Remainingupside down can be fatal, particularly for older individuals, as horseshoe crabs areunable to use their walking legs for righting and must rely on their rigid telson (per-sonal observations) (Fig. 1) (Penn and Brockmann 1995). Minimizing positive lift orgenerating negative lift in this scenario would serve to maintain the organism’s posi-tion against the substrate and prevent flipping. Negative lift may also aid in righting bygenerating a force upward after a crab has been flipped over. Furthermore, pedestrianorganisms must maintain contact with the substrate for locomotion, another activ-ity in which negative or minimal positive lift would be beneficial (Martinez 2001,1996; Sekiguchi and Shuster 2009). Negative lift production has been demonstrated

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Fig. 1 Temporal sequence of a horseshoe crab using its telson and arching motions to right itself after aflip. (Note the entire attempt lasted over 30 s)

in another legged organism (Martinez 2001), and other benthic arthropods (Weis-senberger et al. 1991). Also, Vosatka 1970 states that juvenile horseshoe crabs mustswim upside down because their shell acts as a “hydroplane” that maintains buoyancywhen upside down. In this case, “negative” lift would result in a force upward. Itis not clear how this mechanism would be employed; however, because a 2D crosssection of a horseshoe crab looks similar to an airfoil that would generate a positivelift.

Of the relatively few studies on the flow around horseshoe crab shells, most relyon qualitative techniques like dye visualization and the use of hydrogen bubbles.Experiments performed with horseshoe crab models in “free-swimming” scenariosdemonstrate a trapped vortex in the proximal end of the ventral surface of the cara-pace (Fisher 1975). Other investigations have used dye visualization to reveal smallvortices over the top of horseshoe crab shells (Dietl et al. 2000). Additionally, spinelength in the fossil species Euproops danae has been demonstrated to affect drag onthe body and change passive settling rates (Fisher 1977). The only previous horse-shoe crab study that used computational fluid dynamics found minimal positive lift(2.86% of body weight) when the carapace was resting on a substrate, and negativelift (defined here as force directed toward the underside of the carapace) during freeswimming. This study, however, only considered a thin shell that did not include thestructures on the ventral surface that would break up trapped vortices (Krummel et al.2014).

In this paper, we combine experimental and computational methods to investigatethe hypothesis that horseshoe crab shells generate negative lift in flow. To create anaccurate representation of the shell morphology, we digitally reconstruct a juvenilespecimen using a laser scanner. We then use a 3D-printed model and particle imagevelocimetry experiments to reconstruct flow fields around the shell. These results wereused to validate numerical simulations using the immersed boundary (IB) method(Peskin 2002). The IB method has been used to investigate other problems in biologi-cal fluid dynamics (Fish et al. 2016; Miller et al. 2012) and allows us to quantify flowaround the shell at a variety of orientations. Understanding the interaction betweenshell morphology and force production may inform the engineering design of biolog-ically inspired robots and will expand our understanding of horseshoe crab ecologyand evolution.

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2 Methods

2.1 3-Dimensional Model and Finite Element Mesh

A juvenile horseshoe crab molt was painted with isopropyl alcohol and chalk dustto reduce the reflectance of the carapace. A tabletop laser scanner (NextEngine) wasused to digitally reconstruct the shell. Eight scans were compiled in order to minimizethe number of holes in the 3D reconstruction. The compiled scans were then cut andmirrored yielding a fully intact shell that retained much of the detail of the originalmolt. Physical models were printed on a filament 3D printer (Lulzbot) with 0.1-mmresolution at a scaled size of 5.5cm (Fig. 2). Before meshing, the model was simplifiedfrom over 500,000 faces to 10,000 faces using quadratic edge destruction in MeshLab(Cignoni et al. 2008). A finite element hex-mesh with an element size of 0.0464cmwas generated using Bolt (2018, Csimsoft) for use in immersed boundary simulations.

2.2 FlowTank

For PIV experiments, 3-D printed models were attached by fixing a rigid metal rod tothe downstream end of the telson. The rod was clamped approximately 3cm down-stream of the horseshoe crab. The flow tank had a working cross section of 7.9cm ×7.8cm, with collimators placed upstream and downstream of the working area. Allexperiments were performed with a flow speed of 13cm/s unless otherwise noted.

2.3 Particle ImageVelocimetry (PIV)

Flow speed measurements were made using 2-D planar time averaged PIV. A green532nm shuttered CW laser was used to generate the laser sheet for PIVmeasurements.The laser beam was turned into a sheet using a set of focusing optics, and the lasersheet was in the z−y plane at the midpoint of the horseshoe crab model (Fig. 3).A SA3 Photron high-speed camera (Fastcam) was used to capture images at 1000frames/second. The flow tank was seeded with 10-micron hollow glass beads andmixed to a homogeneous distribution. Images were analyzed using a double-passcross-correlation algorithm in DaVis 8.0.7 (LaVision) with interrogation and search

Fig. 2 3D-printed horseshoe crab reconstruction. Scale bar is 1cm

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Fig. 3 (Color figure online) Schematic diagram of the PIV experimental setup. Note that the flow is fromright to left. The green represents the laser sheet that illuminates the particles in the fluid

window sizes of 32×32 pixels and an overlap of 16 pixels. No pre- or post-processingwas used for the images. For each set of data, 300 images were taken, providing 150velocity vector fields.

2.4 Immersed Boundary Simulations

Fluid–structure interaction problems are found inmany biological systems, and a num-ber of computational frameworks havebeendeveloped to examine them.The immersedboundary (IB) method (Mittal and Iaccarino 2005; Peskin 2002; Griffith 2009) is anapproach to fluid–structure interaction originally introduced by Peskin in the 1970’sto study the cardiovascular dynamics of blood flow in the heart (Peskin 1977). The IBmethod has been used to examine the fluid dynamics of animal locomotion at low tointermediate Reynolds numbers similar to those simulated here, including undulatoryswimming (Fauci and Peskin 1988; Bhalla et al. 2013; Hoover et al. 2018), insectflight (Miller and Peskin 2004, 2005, 2009; Jones et al. 2015), lamprey swimming(Tytell et al. 2010, 2016; Hamlet et al. 2015, 2018), crustacean swimming (Zhanget al. 2014), and jellyfish swimming (Herschlag and Miller 2011; Hoover and Miller2015; Hamlet et al. 2011; Hoover et al. 2017) [53]. IBAMR has been applied (Tytellet al. 2010) and validated (Griffith and Luo 2017) for FSI problems at Re on the orderof 104 or less. This is well within the range of our numerical simulations as we areconsidering juvenile crabs in modest flows. Although we do not take full advantage ofthe FSI capabilities here since we aremodeling a nearly rigid horseshoe crab, we chosethis approach to take advantage of a freely available, parallelized immersed boundarysoftware library with adaptive mesh refinement (IBAMR) (Griffith 2014).

The IB formulation of fluid–structure interaction uses an Eulerian description of themomentum and incompressibility equations of the coupled fluid-structure system, andit uses a Lagrangian description of the structural deformations and stresses. Here, x =(x, y, z) ∈ � denotes physical Cartesian coordinates, where � is the physical regionoccupied by the fluid-structure system. Let X = (X ,Y , Z) ∈ U denote Lagrangianmaterial coordinates that are attached to the structure, withU denoting the Lagrangian

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coordinate domain. The Lagrangian material coordinates are mapped to the physicalposition ofmaterial pointX at time t byχ(X, t) = (χx(X, t), χy(X, t), χz(X, t)) ∈ �,so that the physical region occupied by the structure at time t is χ(U , t) ⊂ �.

The immersed boundary formulation of the coupled system is

ρ

(∂u(x, t)

∂t+ u(x, t) · ∇u(x, t)

)= −∇ p(x, t) + μ∇2u(x, t) + f(x, t) (1)

∇ · u(x, t) = 0 (2)

f(x, t) =∫UF(X, t), δ(x − χ(X, t)) dX (3)

∫UF(X, t) · V(X) dX = −

∫UP(X, t) : ∇XV(X) dX +

∫UG(X, t) · V dX

(4)∂χ(X, t)

∂t=

∫�

u(x, t) δ(x − χ(X, t)) dx. (5)

Here, ρ is the fluid density of water (1000 kgm−3), μ is the dynamic viscosityof water (0.001N sm−2), u(x, t) = (ux, uy, uz) is the Eulerian material velocityfield, and p(x, t) is the Eulerian pressure. Here, f(x, t) and F(X, t) are Eulerian andLagrangian force densities. F is defined in terms of the first Piola-Kirchhoff stresstensor, P, in Eq. (4) and an external force acting on the body, G(X, t), using a weakformulation, in whichV(X) is an arbitrary Lagrangian test function. Another quantityof interest in this study is vorticity, ∇ × u = ω = (ωx, ωy, ωz). The Eulerian andLagrangian frames are connected using the Dirac delta function δ(x) as the kernel ofthe integral transforms of Eqs. (3) and (5).

In this study, the structural stresses due to the material properties of the horseshoecrab molt are calculated from the first Piola-Kirchhoff stress tensor of Eq. 4. The moltis described with a Neo-Hookean material model

P = ηF + (λ log(J ) − η)F−T (6)

where F = ∂χ∂X is the deformation gradient of the mesh, J is the Jacobian of F, η is

the shear modulus, and λ is the bulk modulus. The shear and bulk moduli are defined,respectively, as

η = E

2(1 + ν)(7)

and

λ = Eν

(1 + ν)(1 − 2ν)(8)

where E is the Young’s modulus (Nm−2) and ν is the Poisson ratio. Here, the Young’smodulus, E , is fixed at 1.4×103 Nm−2. In addition to the structural stress, this modeluses the body force G(X, t) as a tethering force that holds the molt in its initialconfiguration. Here, the tethering force is

G(X, t) = κ(χ(X, 0) − χ(X, t)), (9)

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where κ is a spring constant (1 × 105 Nm−1).A hybrid finite difference/finite element version of the immersed boundary method

is used to approximate Eqs. (1)–(5). This IB/FE method uses a finite difference for-mulation for the Eulerian equations and a finite element formulation to describe thehorseshoe crab structural equations. More details on the IB/FE method can be foundin Griffith and Luo (2017).

2.5 Free-Flow Simulations

Immersed boundary (IB) simulations of a horseshoe crab at multiple angles of attack θ

and rotation angles φ were performed (Fig. 4) in free flow. The computational domainwas constructed to match the dimensions of the flow tank. The angles of attack relativeto flow that were simulated were θ = − 20◦, 0◦, 10◦, 20◦, 30◦, and 40◦. Eight rotationangles in the xy-plane were used for the simulations φ = 20◦, 40◦, 60◦, 100◦, 120◦,140◦, 160◦, and 180◦. The fluid was solved on a 16 × 32 × 16cm grid with Dirichletboundary conditions given as u = [0, 13, 0] cm/s (Re = 5500) on all sides of thedomain. The fluid was initially accelerated from rest. The fluid velocity of 13cm/s canbe thought of as simulating a long durationwave encountering the horseshoe crab in themiddle of the water column. From the IB simulations, we computed the time-resolvedvelocity vector fields, pressure, out-of-plane vorticity, and lift/drag coefficients.

Fig. 4 a The horseshoe crab model in the Lagrangian mesh that the Navier–Stokes equations are solvedover. b A diagram of the angle of attack θ and the rotation angle φ

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2.6 Shear Flow Simulations

Immersed boundary (IB) simulations of a horseshoe crab at multiple rotation anglesφ were also performed in shear flow with the organism positioned at the bottom of thedomain. The same eight rotation angles in the xy-plane were used for the simulations:φ = 20◦, 40◦, 60◦, 100◦, 120◦, 140◦, 160◦, and 180◦. The fluid equations are solvedover the same domain as the free-flow simulations; however, the boundary conditionsand the location of the horseshoe crab were changed. Fluid velocity was set to u = [0,30, 0] cm/s at the top of the domain and u = [0, 0, 0] cm/s at the bottom of the domain,creating a linear velocity profile that is a close, but not exact, representation of groundeffects. The flow velocities at the inlet, outlet, and sides of the domain were set to u =[0, 30y/H , 0] cm/s. The crab model was moved to the bottom of the domain to modela horseshoe crab resting against the substrate.

2.7 Force Coefficients

Hydrodynamic forces on the shell at different orientations were compared using thedimensionless lift coefficient (CL) and the dimensionless drag coefficient (CD). Bothcoefficients were calculated using the projected area of the base case model for thetime interval from 2.0 to 3.0 s to allow for comparison of the magnitude of forcesexperienced by the crab. The equations are as follows:

CL = 2Fliftρv2A

, (10)

CD = 2Fdragρv2A

, (11)

where ρ is the density of water, A is the projected area normal to flow, and v is thefree-stream fluid velocity.

2.8 Calculating Torque

To determine the pitch and roll caused by the lift and drag experienced by the crab,we calculated the average torque by taking the curl of the force in the x-, y-, and z-directions. Then, we considered each component of the torque individually by takingthe dot product of the torque and the appropriate unit vector. In this case, the x-component of the torque represents pitch up and down in flow, where positive valuesrepresent a decrease in the angle of attack. The y-component of the torque representsroll perpendicular to flow. All torques were divided by the x-component of the torquefor the 0◦ free-stream example and plotted as normalized values.

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3 Results

3.1 Experimental Validation of Base Case

Particle image velocimetry shows a flow separation point approximately 2cmposteriorto the front of the carapace. Velocity vector fields of the yz-plane from PIV show goodqualitative agreement with those generated from the immersed boundary simulations(Fig. 5). As fluid begins to accelerate, it remains in an attached layer that is directeddownward in the wake, generating lift. In a short amount of time, the separation pointbegins to move forward toward the anterior part of the shell, settling on a separationpoint between 1.5 and 2.5cm from the anterior end. In both the PIV and IB velocityfields, fluid on averagemoves upward in thewake, indicating that the shell is generatingnegative lift. Downstream of the separation point there is strong vortex shedding(Fig. 6).

3.2 Changing Angle of Attack

As the fluid accelerates from rest, it begins as an attached layer over the shell forall angles of attack, similar to the base case. At a zero-degree angle, the separationpoint begins to move forward at 0.25 s, but it takes approximately one second for theseparation point to form for other angles (Fig. 7). After the separation point settles,there is significant vortex shedding, particularly from the end of the telson. The wakewidth is dependent on the angle of attack, getting wider with increasing angle. Notethat the case when θ = − 20◦ has a notably larger wake and more vortex sheddingthan the case when θ = +20◦, despite having the same cross-sectional area normal

Fig. 5 (Color figure online) Comparison of flow fields from IB simulations (left) and PIV experiments(right). Arrows show the direction and magnitude of the flow (in mm/s) which is from left to right. Thecolormap corresponds to the magnitude of the velocity. Note the low flow region behind the crab is repre-sentative of its wake

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Fig. 6 (Color figure online) Representative flow fields with out-of-plane vorticity (s−1) shown by thecolormap and velocity direction and magnitude shown by the arrows. The temporal snapshots show howthe flow develops from rest for a zero-degree angle of attack. Red indicates negative out-of-plane vorticityand blue is positive vorticity

Fig. 7 (Color figure online) Representative velocity vector fields with arrows showing the magnitude anddirection of flow. The colormap describes the out-of-plane vorticity for three different angles of attack:− 20◦ (top), 0◦ (middle), and 20◦ (bottom). Red indicates negative vorticity and blue is positive vorticity.The temporal snapshots show how the flow develops in time. Note the strong wake below the carapace forthe negative angle of attack and above the carapace for positive and zero angles of attack

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Fig. 8 Lift (CL) and drag (CD) coefficients over time for a horseshoe crab at a zero-degree angle of attackin free-stream flow

to flow. At θ = − 20◦ and θ = 0◦ fluid moves upward in the wake, corresponding tonegative lift. For all other angles, the fluid moves downward in the wake, indicatingpositive lift.

For all angles of attack, there was an initial spike in drag acting on the shell as thefluid accelerated from rest (Fig. 8). For angles of attackbetween− 20◦ and20◦, the dragdecreased following the initial spike and then began to oscillate. Such oscillations inforce are associated with strong vortex shedding in the wake of the crab. In some cases,the drag coefficient experienced large oscillations between−2.0 and2.0 dimensionlessunits. For simulations where θ = +30◦ and θ = +40◦, the drag remained approximatelyconstant after the initial spike. The lift coefficient was −0.5 for θ = − 20◦ andincreased to 4.0 for +40◦. Lift was negative for part or all of the simulations for θ

= − 20◦ and 0◦, and near zero for θ = +10◦. For θ = +20◦ to +40◦, the lift initiallyincreased and then remained constant or decreased slightly.

The average lift and drag coefficients for each simulation after the initial acceler-ation can be found in Fig. 9. The coefficients were averaged over a 1 s interval at theend of the simulation. The drag coefficient increased with the magnitude of the angleof attack (and consequently with the projected area). In contrast, the lift on the shellincreased with increasing angle of attack. Two orientations had negative lift, θ =− 20◦and 0◦. At θ = 10◦ the normalized lift was minimal (CL = 0.14) and then increasedrapidly from θ = +20◦ to +40◦. The magnitude of the lift/drag ratio was highest atθ = +30◦ (CL/CD = 1.58) and lowest at θ = +10◦ (CL/CD = 0.22). Interestingly,the optimal angle for preventing dislodgement is θ = 0◦ because (CL/CD = −1), thelowest value we recorded. This is analogous to a horseshoe crab resting flat.

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Fig. 9 a Average lift (CL) and drag (CD) coefficients for each free-flow simulation. b Lift-to-drag ratiocalculated from the force coefficients in a

3.3 Changing Rotation Angle Relative to Flow

For all angles of rotation between φ = 20◦ and φ = 160◦, the lift is initially positiveand then becomes negative (Fig. 10). The amount of time it takes for this transition tooccur ranges between 0.5 s (φ = 100◦) and 5.8 s (φ = 160◦). The lift coefficient wassmall for all angles (|CL| < 1) and experienced only small oscillations. Drag, however,experienced large oscillations and could be much higher than lift (|CD| > 7). Theseoscillations in drag most likely indicate powerful vortex shedding. Representativeexamples from φ = 40◦ and φ = 100◦ are shown and represent cases with strong andweak oscillations, respectively (Fig. 11).

When looking at the average lift and drag of the simulations, the magnitude ofthe lift/drag ratio peaked at φ = 0◦ (CL/CD = −1) and was lowest at φ = 100◦(CL/CD = −0.25) (Fig. 12). Both the magnitude of negative lift and positive dragon the shell peaked at a rotation angle of φ = 100◦. The ratio was much lower at φ

= 100◦ than it was for the case with no rotation because drag depends more strongly

Fig. 10 Magnitude of drag (CD) and lift (CL) over time for a large oscillation case (40◦) and a smalloscillation case (120◦). Both examples are simulated in free-stream flow

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Fig. 11 (Color figure online) Representative flow fields illustrated with velocity vectors and out-of-planevorticity shown by the colormap for a rotation angle with large drag oscillations (40◦) and small dragoscillations (120◦). Red represents negative vorticity and blue represents positive vorticity

Fig. 12 a Mean +/− SD of lift (CL) and drag (CD) coefficients for rotation angles between 20◦ and 180◦.b Lift-to-drag ratio for rotation angles between 20◦ and 180◦

on projected area than lift. Furthermore, drag was higher when the anterior end of thecarapace was facing into flow than the posterior end.

3.4 Shear Flow

The shear flow simulation ofφ = 0◦ showed a spike in themagnitude of negative lift,followed by the lift coefficient approaching a near-zero positive value. For rotationangles between φ = 20◦ and 180◦, the same pattern was observed as for the free-flow simulations—there was an initial spike in lift and drag followed by a decline to arelatively constant value, with L/D ratios less than 1 (Fig. 13). The major discrepancybetween free-flowand shear flowexamples is that the lift is positive.Additionally, there

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Fig. 13 a Lift and drag coefficients for shear flow simulations and rotation angles between 0◦ and 180◦.b Lift/drag ratios for the same simulations from panel a

Fig. 14 (Color figure online) Temporal snapshots of the flow fields generated during the initial accelerationin shear flow for 0◦ and 180◦ rotation angles. Blue colors denote positive out-of-plane vorticity, and reddenotes negative. Vectors represent flow direction. Note the larger wake area for tail-on flow

Fig. 15 Lift and drag coefficients over time for head-on and tail-on shear flow simulations

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Fig. 16 (Color figure online) a The x- and y- components of the torque on a horseshoe crab shell withchanging angles of attack (representing pitch and roll, respectively). Below θ = 10◦, the torque would actto stabilize the crab, but at larger angles the torque would cause the crab to pitch back and flip. b Thesame torque components as in a, plotted for both free-stream and shear flow cases with changing rotationangle. Torques are generally lower in shear flow cases which more closely approximate conditions near thesubstrate

is a difference between head-on (φ = 0◦) and tail-on ( φ = 180◦) forces. The spike inlift is negative for head-on, but positive for tail-on (Figs. 14, 15).

3.5 Torque on the Shell

To further assess the postural stability of the horseshoe crab in flow, we calculatedthe torque on the shell where the x-component represents pitch up and down in thedirection of flow and the y-component represents roll perpendicular to flow. We findthat at θ = − 20◦ the torque on the shell would cause it to pitch back toward θ = 0◦,and at θ = 0◦ and θ = 10◦ the crab would pitch forward—the more stable directionbecause horseshoe crabs can use their legs to resist compression 16. Beyond θ = 10◦,the torque would cause the crab to pitch back even further, increasing lift and drag,and likely cause it to flip over. In all orientations with changing angle of attack, thereis little roll.

When the crab is rotated between φ = 0◦ and φ = 180◦, the x-component of thetorque is positive, causing pitching of the crab toward the substrate as opposed tocausing it to flip. This pattern is true for all but one free-stream case and half of theshear flow cases. Torque causing the crab to roll is relatively low; however, there issignificant torque acting on the shell at φ = 120◦ and φ = 180◦. It should also be notedthat generally, the torque is lower in shear flow cases where the flow is more similarto what would be found near the substrate (Fig. 16).

4 Discussion

Organisms that inhabit the surf zone and use legged locomotion must contend withhydrodynamic lift and drag forces from waves and self-generated propulsion. Mini-

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mizing lift and drag is important for preventing flipping over, something that posesa particularly difficult challenge for horseshoe crabs. Morphology and posture dic-tate the forces that organisms experience in flow. In this study, we used computationalfluid dynamics and experimental flow visualization to demonstrate that horseshoe crabshells generate negative lift in free-stream flow, and near-zero lift and low drag in shearflow at most relevant orientations.

Experimental data acquired using 3D-printed horseshoe crab shells showed similarvelocity profiles around the shell as those generated by immersed boundary simu-lations. Because the model is digitized but retains much of the detail of an actualhorseshoe crab, we were able to manipulate the orientation of the shell to illuminatethe full picture of the forces experienced by the organism at relevant postures that mayoccur while mating or during locomotion. Vortex generation and wake patterns onthe dorsal and ventral sides of the carapace are consistent with previous observations(Dietl et al. 2000; Fisher 1977). Additionally, Krummel et al. (2014) found lift onhorseshoe crab shells in a similar simulation to the base case considered here, but themodel used in this study retains details of the ventral surface of the carapace that theauthors acknowledge would disrupt the trapped vortex found in their work.

When compared to other pedestrian aquatic organisms, the horseshoe crab is uniquein its generation of negative lift. Other species like the shore crab G. tenuicrustatus,the portunid crab C. sapidus, and the lobsters I. peronei and T. orientalis all generatepositive lift (Martinez 2001; Blake 1985; Jacklyn and Ritz 1986). Some benthic orintertidal organisms like the mayfly (E. sylvicole) and stonefly larvae (P. Bipunctata)generate negative lift bymodifying their postures (Weissenberger et al. 1991); however,this constant posture change would be less broadly applicable for generating negativelift and preventing dislodgement than a passive morphological mechanism would be.Instead of constantly expending energy to control posture, lift reduction via morphol-ogy does not require active control and prevents the organism from being flipped by asurprise wave. While lift on horseshoe crab shells stands out when compared to otherorganisms, the drag force experienced is similar to that of other pedestrian and benthicspecies (Denny and Gaylord 1996; Denny 2000; Blake 1985; Jacklyn and Ritz 1986),discussed in Martinez (2001).

When the crab is on the beach during mating and spawning, it is exposed to wavescoming from the front, back, and side. To assess the effect direction has on the forcesexperienced by the organism, we quantified lift and drag across a sweep of rotationangles. As in the case of head-on flow, we found negative lift at every rotation anglewith the exception of 180◦ in free-steam scenarios. In shear flow we found spikesin negative lift followed by near-zero lift. While lift was fairly constant across allangles, drag varied dramatically. Some angles, like 120◦ and 160◦, have near constantdrag after an initial spike, but others, like 40◦ and 60◦, experience violent oscillationsin drag. These oscillations are most likely the result of powerful vortex shedding atcertain angles. Wild swings in drag pose a problem for horseshoe crabs that could getswept away if the drag is too high. This is especially true near 100 degrees relativeto flow where CL/CD = − 0.25. High drag forces combined with little negative liftmay dislodge the animal. While the optimal orientations for the crabs to adopt arethose with few oscillations in drag, it remains to be tested whether they adopt thispositioning in the wild.

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It is also important to consider the effect of the substrate on the forces experiencedby the organism. Here, we examine ground effect by placing the horseshoe crab on thebottom of the computational box and enforcing the no slip condition with a prescribedfree-stream velocity at the top of the box. This created a linear fluid velocity profile thatis a close, but imperfect match for a real boundary layer. Ground effect is typicallyresponsible for an increase in lift and decrease in drag of aerodynamic objects asthey are placed closer to a stationary surface. When considering objects that producenegative lift (also termed downforce), there is an increase in the magnitude of thenegative lift produced until the object is very close to the substrate (Zerihan andZhang 2000; Zhang et al. 2006). We likely did not find increased negative lift whenincluding the ground because we set the distance between the crab and the substrateto be zero, well below the height where the benefits of ground effect begin to reverse( 10% of the chord length).

An in depth understanding of the hydrodynamics of horseshoe crab shells expandsthe limited body of work on legged aquatic locomotion. Furthermore, it provides afoundation for investigating how documented inter- and intra-specific morphologicalvariation impacts hydrodynamic forces on the shell (Riska 1981; Zaldívar-Rae et al.2009). Previous work on the spines of fossil species of horseshoe crabs shows thatspine length impacts passive settling rates (Fisher 1977), so it is possible that otherchanges in spine number and shell curvature have a hydrodynamic effect. Additionally,horseshoe crabs have been used for biologically inspired design of intertidal robots(Krummel et al. 2014). These engineering applications could also be expanded toother situations where negative lift would be desired such as sensors for tornados orwaterways, car roof racks, or high-end backpacking tents that are exposed to highwinds.

Passive morphological mechanisms for generating negative or minimal lift at a zeroangle of attack have rarely been found in benthic arthropods, but our findings demon-strate that horseshoe crabs have this capability. Given the limited work on pedestrianaquatic organisms, it is possible that other benthic arthropods employ this strategy tomaintain position during locomotion or reproduction. This finding is also consistentwith observations of horseshoe crabs swimming upside down, and the relatively highmortality rate of individuals that fail to right themselves.

Acknowledgements Wewould like to thank Brad Erickson and Jonathan Rader for help with laser scanningand 3D-reconstruction, and Miles Hackett for help with experiments. Additionally, we would like to thankthe UNC Office of Undergraduate Research and the WilliamW. and Ida W. Taylor Foundation for funding.This work was also supported by NSF DMS Grant #1151478 (to L.A.M.).

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Affiliations

Alexander L. Davis1,4 · Alexander P. Hoover2 · Laura A. Miller3,4

Alexander P. Hooverhoover.alexander@gmail.com

Laura A. Millerlam9@email.unc.edu

1 Duke University, Room 137, Biological Sciences Building, 130 Science Drive, Durham,NC 27708, USA

2 Department of Mathematics, Buchtel College of Arts and Sciences, University of Akron, Akron,OH 44325-4002, USA

3 Department of Mathematics, University of North Carolina, Phillips Hall, CB 3250, Chapel Hill,NC 27599, USA

4 Department of Biology, Coker Hall, CB 3280, University of North Carolina, 120 South Road,Chapel Hill, NC 27599, USA

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