Lift and Drag Acting on the Shell of the American Horseshoe Crab (Limulus Polyphemus)

Bulletin of Mathematical Biology https://doi.org/10.1007/s11538-019-00657-2 ORIGINAL ARTICLE

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

Alexander L. Davis1,4 · Alexander P. Hoover2 · LauraA.Miller3,4

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

Abstract The intertidal zone is a turbulent landscape where organisms face numerous mechanical challenges from powerful waves. A model for understanding the solutions to these physical problems, the American horseshoe crab (Limulus polyphemus), is a marine arthropod that mates in the intertidal zone, where it must contend with strong ambient flows to maintain its orientation during locomotion and reproduction. Possible strategies to maintain position include either negative lift generation or the minimization of positive lift in flow. To quantify flow over the shell and the forces generated, we laser-scanned the 3D shape of a horseshoe crab, and the resulting digital reconstruction was used to 3D-print a physical model. We then recorded the movement of tracking particles around the shell model with high-speed video and analyzed the time-lapse series using particle image velocimetry (PIV). The velocity vector fields from PIV were used to validate numerical simulations performed with the immersed boundary (IB) method. IB simulations allowed us to resolve the forces acting on the shell, as well as the local three-dimensional flow velocities and pressures. Both IB simulations and PIV analysis of vorticity and velocity at a flow speed of 13cm/s show negative lift for negative and zero angles of attack, and positive lift for positive angles of attack in a free-stream environment. In shear flow simulations, we found near-zero lift for all orientations 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 orientation of the crab during locomotion and mating. This study provides a preliminary foundation for assessing the relationship between documented morphological variation and potential environmental variation for distinct populations of horseshoe crabs along the Atlantic Coast. It also motivates future studies which could consider the stability of the horseshoe crab in unsteady, oscillating flows.

Keywords Immersed boundary method · Computational ﬂuid dynamics · Pedestrian aquatic locomotion

B Alexander L. Davis [email protected] Extended author information available on the last page of the article 123 A. L. Davis et al.

1 Introduction

Ambient flow of air or water provides a challenge with which many organisms must contend. The intertidal zone is at the interface of these two mediums where organisms are exposed to powerful wave action and strong currents (Denny and Gaines 1990; Denny 1991). Intertidal inhabitants like limpets and other invertebrates have evolved morphologies that reduce the effects of hydrodynamic lift and drag, preventing waves from dislodging the shell (Denny et al. 1985; Denny 1989). Other organisms adopt postures that allow them to remain attached to the substrate by reducing drag (Maude and Williams 1983; Martinez 2001; Webb 1989). Although being swept away by the drag from crashing waves is important to consider, hydrodynamic lift can be just as dangerous (Trussell 1997). Because forces on intertidal animals may be large, reducing lift is important for maintaining attachment (Denny 1989; Bell and Gosline 1997). While many studies of hydrodynamic forces on organisms focus on sessile animals, far fewer have investigated intertidal forces acting on locomoting, legged organisms (Martinez 2001; Bill and Herrnkind 1976;Blake1985; Martinez 1996; Pond 1975). The American horseshoe crab (Limulus polyphemus) is a marine arthropod that primarily relies on its legs for locomotion. Horseshoe crabs have a highly conserved morphology that resembles fossils from the Mesozoic, indicating a suitable morphology for their environment (Selander et al. 1970; Walls et al. 2002). There is, however, documented morphological and genetic variation in shell curvature and the presence of spines between distinct populations along the Atlantic Coast of North America (Pierce et al. 2000;Riska1981; Saunders et al. 1986; Zaldívar-Rae et al. 2009; Sekiguchi and Shuster 2009). Additionally, there are three other extant species, T. gigas (Müller, 1785), T. tridentatus (Leach, 1819), and C. rotundicauda (Latreille, 1802), and multiple related fossil species with varying carapace morphologies (Stoermer 1952). The generally conserved shape of Limulus spp. and the inter- and intra-specific varia- tions of some features make horseshoe crabs a useful system for investigating the relationship between morphology and hydrodynamic forces in legged, aquatic organisms. Horseshoe crabs are particularly interesting for studying lift reduction because they are 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 beaches in the surf zone where they experience significant wave action and ambient currents speeds that are far larger than self-generated flows (Brockmann 1990). This poses a problem for the crab because once flipped over righting is a challenge. Remaining upside down can be fatal, particularly for older individuals, as horseshoe crabs are unable 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 or generating negative lift in this scenario would serve to maintain the organism’s position against the substrate and prevent flipping. Negative lift may also aid in righting by generating a force upward after a crab has been flipped over. Furthermore, pedestrian organisms 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 123 Lift and Drag Acting on the Shell of the American Horseshoe Crab…

Fig. 1 Temporal sequence of a horseshoe crab using its telson and arching motions to right itself after a ﬂip. (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 must swim upside down because their shell acts as a “hydroplane” that maintains buoyancy when upside down. In this case, “negative” lift would result in a force upward. It is not clear how this mechanism would be employed; however, because a 2D cross section of a horseshoe crab looks similar to an airfoil that would generate a positive lift. Of the relatively few studies on the flow around horseshoe crab shells, most rely on qualitative techniques like dye visualization and the use of hydrogen bubbles. Experiments performed with horseshoe crab models in “free-swimming” scenarios demonstrate a trapped vortex in the proximal end of the ventral surface of the carapace (Fisher 1975). Other investigations have used dye visualization to reveal small vortices over the top of horseshoe crab shells (Dietl et al. 2000). Additionally, spine length in the fossil species Euproops danae has been demonstrated to affect drag on the body and change passive settling rates (Fisher 1977). The only previous horseshoe 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 negative lift (defined here as force directed toward the underside of the carapace) during free swimming. This study, however, only considered a thin shell that did not include the structures on the ventral surface that would break up trapped vortices (Krummel et al. 2014). In this paper, we combine experimental and computational methods to investigate the hypothesis that horseshoe crab shells generate negative lift in flow. To create an accurate representation of the shell morphology, we digitally reconstruct a juvenile specimen using a laser scanner. We then use a 3D-printed model and particle image velocimetry experiments to reconstruct flow fields around the shell. These results were used to validate numerical simulations using the immersed boundary (IB) method (Peskin 2002). The IB method has been used to investigate other problems in biological fluid dynamics (Fish et al. 2016; Miller et al. 2012) and allows us to quantify flow around the shell at a variety of orientations. Understanding the interaction between shell morphology and force production may inform the engineering design of biologically inspired robots and will expand our understanding of horseshoe crab ecology and evolution. 123 A. L. Davis et al.

2 Methods

2.1 3-Dimensional Model and Finite Element Mesh

A juvenile horseshoe crab molt was painted with isopropyl alcohol and chalk dust to reduce the reflectance of the carapace. A tabletop laser scanner (NextEngine) was used to digitally reconstruct the shell. Eight scans were compiled in order to minimize the number of holes in the 3D reconstruction. The compiled scans were then cut and mirrored yielding a fully intact shell that retained much of the detail of the original molt. Physical models were printed on a filament 3D printer (Lulzbot) with 0.1-mm resolution at a scaled size of 5.5cm (Fig. 2). Before meshing, the model was simplified from 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 0.0464cm was generated using Bolt (2018, Csimsoft) for use in immersed boundary simulations.

2.2 Flow Tank

For PIV experiments, 3-D printed models were attached by fixing a rigid metal rod to the downstream end of the telson. The rod was clamped approximately 3cm downstream 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. All experiments were performed with a flow speed of 13cm/s unless otherwise noted.

2.3 Particle Image Velocimetry (PIV)

Flow speed measurements were made using 2-D planar time averaged PIV. A green 532nm shuttered CW laser was used to generate the laser sheet for PIV measurements. The laser beam was turned into a sheet using a set of focusing optics, and the laser sheet 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 1000 frames/second. The ﬂow tank was seeded with 10-micron hollow glass beads and mixed to a homogeneous distribution. Images were analyzed using a double-pass cross-correlation algorithm in DaVis 8.0.7 (LaVision) with interrogation and search

Fig. 2 3D-printed horseshoe crab reconstruction. Scale bar is 1 cm 123 Lift and Drag Acting on the Shell of the American Horseshoe Crab…

Fig. 3 (Color figure online) Schematic diagram of the PIV experimental setup. Note that the flow is from right 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-processing was used for the images. For each set of data, 300 images were taken, providing 150 velocity vector fields.

2.4 Immersed Boundary Simulations

Fluid–structure interaction problems are found in many biological systems, and a number of computational frameworks have been developed to examine them. The immersed boundary (IB) method (Mittal and Iaccarino 2005; Peskin 2002;Griffith2009)isan approach to fluid–structure interaction originally introduced by Peskin in the 1970’s to study the cardiovascular dynamics of blood flow in the heart (Peskin 1977). The IB method has been used to examine the fluid dynamics of animal locomotion at low to intermediate Reynolds numbers similar to those simulated here, including undulatory swimming (Fauci and Peskin 1988; Bhalla et al. 2013; Hoover et al. 2018), insect flight (Miller and Peskin 2004, 2005, 2009; Jones et al. 2015), lamprey swimming (Tytell et al. 2010, 2016;Hamletetal.2015, 2018), crustacean swimming (Zhang et al. 2014), and jellyfish swimming (Herschlag and Miller 2011; Hoover and Miller 2015;Hamletetal.2011; Hoover et al. 2017) [53]. IBAMR has been applied (Tytell et al. 2010) and validated (Griffith and Luo 2017) for FSI problems at Re on the order of 104 or less. This is well within the range of our numerical simulations as we are considering juvenile crabs in modest flows. Although we do not take full advantage of the FSI capabilities here since we are modeling a nearly rigid horseshoe crab, we chose this approach to take advantage of a freely available, parallelized immersed boundary software library with adaptive mesh refinement (IBAMR) (Griffith 2014). The IB formulation of fluid–structure interaction uses an Eulerian description of the momentum and incompressibility equations of the coupled fluid-structure system, and it uses a Lagrangian description of the structural deformations and stresses. Here, x = (x, y, z) ∈ denotes physical Cartesian coordinates, where is the physical region occupied by the fluid-structure system. Let X = (X, Y , Z) ∈ U denote Lagrangian material coordinates that are attached to the structure, with U denoting the Lagrangian 123 A. L. Davis et al. coordinate domain. The Lagrangian material coordinates are mapped to the physical position of material point X at time t by χ(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 ∂u(x, t) ρ + u(x, t) ·∇u(x, t) =−∇p(x, t) + μ∇2u(x, t) + f(x, t) (1) ∂t ∇· ( , ) = u x t 0(2) f(x, t) = F(X, t), δ(x − χ(X, t)) dX (3) U

F(X, t) · V(X) dX =− P(X, t) :∇XV(X) dX + G(X, t) · V dX U U U (4) ∂χ(X, t) = u(x, t)δ(x − χ(X, t)) dx. (5) ∂t

Here, ρ is the fluid density of water (1000 kg m−3), μ is the dynamic viscosity −2 of water (0.001 N s m ), u(x, t) = (ux, uy, uz) is the Eulerian material velocity field, and p(x, t) is the Eulerian pressure. Here, f(x, t) and F(X, t) are Eulerian and Lagrangian force densities. F is defined in terms of the first Piola-Kirchhoff stress tensor, P,inEq.(4) and an external force acting on the body, G(X, t), using a weak formulation, in which V(X) is an arbitrary Lagrangian test function. Another quantity of interest in this study is vorticity, ∇×u = ω = (ωx,ωy,ωz). The Eulerian and Lagrangian frames are connected using the Dirac delta function δ(x) as the kernel of the integral transforms of Eqs. (3) and (5). In this study, the structural stresses due to the material properties of the horseshoe crab molt are calculated from the first Piola-Kirchhoff stress tensor of Eq. 4.Themolt is described with a Neo-Hookean material model

− P = ηF + (λ log(J) − η)F T (6)

F = ∂χ F η where ∂X is the deformation gradient of the mesh, J is the Jacobian of , is the shear modulus, and λ is the bulk modulus. The shear and bulk moduli are defined, respectively, as E η = (7) 2(1 + ν) and Eν λ = (8) (1 + ν)(1 − 2ν) where E is the Young’s modulus (N m−2) and ν is the Poisson ratio. Here, the Young’s modulus, E,isfixedat1.4×103 Nm−2. In addition to the structural stress, this model uses the body force G(X, t) as a tethering force that holds the molt in its initial configuration. Here, the tethering force is

G(X, t) = κ(χ(X, 0) − χ(X, t)), (9) 123 Lift and Drag Acting on the Shell of the American Horseshoe Crab… 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 formulation for the Eulerian equations and a finite element formulation to describe the horseshoe crab structural equations. More details on the IB/FE method can be found in 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 domain was constructed to match the dimensions of the flow tank. The angles of attack relative to flow that were simulated were θ =−20◦,0◦,10◦,20◦,30◦, and 40◦. Eight rotation angles 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 Dirichlet boundary conditions given as u = [0, 13, 0] cm/s (Re = 5500) on all sides of the domain. The fluid was initially accelerated from rest. The fluid velocity of 13cm/s can be thought of as simulating a long duration wave encountering the horseshoe crab in the middle of the water column. From the IB simulations, we computed the time-resolved velocity 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 solved over. b A diagram of the angle of attack θ and the rotation angle φ 123 A. L. Davis et al.

2.6 Shear Flow Simulations

Immersed boundary (IB) simulations of a horseshoe crab at multiple rotation angles φ were also performed in shear flow with the organism positioned at the bottom of the domain. 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 solved over the same domain as the free-flow simulations; however, the boundary conditions and 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 ground effects. 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 model a horseshoe crab resting against the substrate.

2.7 Force Coefficients

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

2F C = lift , (10) L ρv2 A 2Fdrag C = , (11) D ρv2 A

where ρ is the density of water, A is the projected area normal to ﬂow, and v is the free-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 taking the 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 values represent a decrease in the angle of attack. The y-component of the torque represents roll perpendicular to ﬂow. All torques were divided by the x-component of the torque for the 0◦ free-stream example and plotted as normalized values.

123 Lift and Drag Acting on the Shell of the American Horseshoe Crab…

3 Results

3.1 Experimental Validation of Base Case

Particle image velocimetry shows a flow separation point approximately 2cm posterior to the front of the carapace. Velocity vector fields of the yz-plane from PIV show good qualitative agreement with those generated from the immersed boundary simulations (Fig. 5). As fluid begins to accelerate, it remains in an attached layer that is directed downward in the wake, generating lift. In a short amount of time, the separation point begins to move forward toward the anterior part of the shell, settling on a separation point between 1.5 and 2.5cm from the anterior end. In both the PIV and IB velocity fields, fluid on average moves upward in the wake, indicating that the shell is generating negative 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 for all angles of attack, similar to the base case. At a zero-degree angle, the separation point begins to move forward at 0.25s, but it takes approximately one second for the separation 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 wake width is dependent on the angle of attack, getting wider with increasing angle. Note that the case when θ =−20◦ has a notably larger wake and more vortex shedding than 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. The colormap corresponds to the magnitude of the velocity. Note the low flow region behind the crab is representative of its wake 123 A. L. Davis et al.

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

Fig. 7 (Color figure online) Representative velocity vector fields with arrows showing the magnitude and direction 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 for the negative angle of attack and above the carapace for positive and zero angles of attack

123 Lift and Drag Acting on the Shell of the American Horseshoe Crab…

Fig. 8 Lift (CL) and drag (CD) coefficients over time for a horseshoe crab at a zero-degree angle of attack in free-stream flow to flow. At θ =−20◦ and θ = 0◦ fluid moves upward in the wake, corresponding to negative lift. For all other angles, the fluid moves downward in the wake, indicating positive lift. For all angles of attack, there was an initial spike in drag acting on the shell as the fluid accelerated from rest (Fig. 8). For angles of attack between − 20◦ and 20◦, the drag decreased following the initial spike and then began to oscillate. Such oscillations in force are associated with strong vortex shedding in the wake of the crab. In some cases, the drag coefficient experienced large oscillations between − 2.0 and 2.0 dimensionless units. For simulations where θ =+30◦ and θ =+40◦, the drag remained approximately constant after the initial spike. The lift coefficient was −0.5 for θ =−20◦ and increased 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◦.Forθ =+20◦ to +40◦, the lift initially increased and then remained constant or decreased slightly. The average lift and drag coefficients for each simulation after the initial acceleration can be found in Fig. 9. The coefficients were averaged over a 1s interval at the end of the simulation. The drag coefficient increased with the magnitude of the angle of attack (and consequently with the projected area). In contrast, the lift on the shell increased with increasing angle of attack. Two orientations had negative lift, θ = − 20◦ ◦ ◦ and 0 .Atθ =10 the normalized lift was minimal (CL = 0.14) and then increased rapidly from θ =+20◦ to +40◦. The magnitude of the lift/drag ratio was highest at ◦ ◦ θ =+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),the lowest value we recorded. This is analogous to a horseshoe crab resting flat.

123 A. L. Davis et al.

Fig. 9 a Average lift (CL) and drag (CD) coefficients for each free-flow simulation. b Lift-to-drag ratio calculated 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 positive and then becomes negative (Fig. 10). The amount of time it takes for this transition to occur ranges between 0.5 s (φ = 100◦) and 5.8 s (φ = 160◦). The lift coefﬁcient was small for all angles (|CL| < 1) and experienced only small oscillations. Drag, however, experienced large oscillations and could be much higher than lift (|CD| > 7). These oscillations in drag most likely indicate powerful vortex shedding. Representative examples from φ =40◦ and φ = 100◦ are shown and represent cases with strong and weak oscillations, respectively (Fig. 11). When looking at the average lift and drag of the simulations, the magnitude of ◦ ◦ the 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 drag on 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 small ◦ oscillation case (120 ). Both examples are simulated in free-stream ﬂow 123 Lift and Drag Acting on the Shell of the American Horseshoe Crab…

Fig. 11 (Color figure online) Representative flow fields illustrated with velocity vectors and out-of-plane ◦ vorticity shown by the colormap for a rotation angle with large drag oscillations (40 ) and small drag ◦ oscillations (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 the carapace was facing into ﬂow than the posterior end.

3.4 Shear Flow

The shear flow simulation of φ =0◦ showed a spike in the magnitude of negative lift, followed by the lift coefficient approaching a near-zero positive value. For rotation angles 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 a relatively constant value, with L/D ratios less than 1 (Fig. 13). The major discrepancy between free-flow and shear flow examples is that the lift is positive. Additionally, there 123 A. L. Davis et al.

◦ ◦ 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 acceleration ◦ ◦ in shear flow for 0 and 180 rotation angles. Blue colors denote positive out-of-plane vorticity, and red denotes 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

123 Lift and Drag Acting on the Shell of the American Horseshoe Crab…

Fig. 16 (Color figure online) a The x-andy- components of the torque on a horseshoe crab shell with ◦ changing angles of attack (representing pitch and roll, respectively). Below θ =10 , the torque would act to stabilize the crab, but at larger angles the torque would cause the crab to pitch back and flip. b The same torque components as in a, plotted for both free-stream and shear flow cases with changing rotation angle. Torques are generally lower in shear flow cases which more closely approximate conditions near the substrate is a difference between head-on (φ =0◦) and tail-on ( φ = 180◦) forces. The spike in lift 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 calculated the torque on the shell where the x-component represents pitch up and down in the direction of flow and the y-component represents roll perpendicular to flow. We find that 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 direction because 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, there is little roll. When the crab is rotated between φ =0◦ and φ = 180◦,thex-component of the torque is positive, causing pitching of the crab toward the substrate as opposed to causing it to flip. This pattern is true for all but one free-stream case and half of the shear flow cases. Torque causing the crab to roll is relatively low; however, there is significant torque acting on the shell at φ = 120◦ and φ = 180◦. It should also be noted that generally, the torque is lower in shear flow cases where the flow is more similar to what would be found near the substrate (Fig. 16).

4 Discussion

Organisms that inhabit the surf zone and use legged locomotion must contend with hydrodynamic lift and drag forces from waves and self-generated propulsion. Mini- 123 A. L. Davis et al. mizing lift and drag is important for preventing flipping over, something that poses a particularly difficult challenge for horseshoe crabs. Morphology and posture dic- tate the forces that organisms experience in flow. In this study, we used computational fluid dynamics and experimental flow visualization to demonstrate that horseshoe crab shells generate negative lift in free-stream flow, and near-zero lift and low drag in shear flow at most relevant orientations. Experimental data acquired using 3D-printed horseshoe crab shells showed similar velocity profiles around the shell as those generated by immersed boundary simulations. Because the model is digitized but retains much of the detail of an actual horseshoe crab, we were able to manipulate the orientation of the shell to illuminate the full picture of the forces experienced by the organism at relevant postures that may occur while mating or during locomotion. Vortex generation and wake patterns on the 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 on horseshoe crab shells in a similar simulation to the base case considered here, but the model used in this study retains details of the ventral surface of the carapace that the authors acknowledge would disrupt the trapped vortex found in their work. When compared to other pedestrian aquatic organisms, the horseshoe crab is unique in 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 generate positive lift (Martinez 2001;Blake1985; Jacklyn and Ritz 1986). Some benthic or intertidal organisms like the mayfly (E. sylvicole) and stonefly larvae (P. Bipunctata) generate negative lift by modifying their postures (Weissenberger et al. 1991); however, this constant posture change would be less broadly applicable for generating negative lift and preventing dislodgement than a passive morphological mechanism would be. Instead of constantly expending energy to control posture, lift reduction via morphology does not require active control and prevents the organism from being flipped by a surprise wave. While lift on horseshoe crab shells stands out when compared to other organisms, the drag force experienced is similar to that of other pedestrian and benthic species (Denny and Gaylord 1996; Denny 2000;Blake1985; Jacklyn and Ritz 1986), discussed in Martinez (2001). When the crab is on the beach during mating and spawning, it is exposed to waves coming from the front, back, and side. To assess the effect direction has on the forces experienced by the organism, we quantified lift and drag across a sweep of rotation angles. As in the case of head-on flow, we found negative lift at every rotation angle with the exception of 180◦ in free-steam scenarios. In shear flow we found spikes in negative lift followed by near-zero lift. While lift was fairly constant across all angles, drag varied dramatically. Some angles, like 120◦ and 160◦, have near constant drag after an initial spike, but others, like 40◦ and 60◦, experience violent oscillations in drag. These oscillations are most likely the result of powerful vortex shedding at certain angles. Wild swings in drag pose a problem for horseshoe crabs that could get swept away if the drag is too high. This is especially true near 100 degrees relative to flow where CL/CD =−0.25. High drag forces combined with little negative lift may dislodge the animal. While the optimal orientations for the crabs to adopt are those with few oscillations in drag, it remains to be tested whether they adopt this positioning in the wild. 123 Lift and Drag Acting on the Shell of the American Horseshoe Crab…

It is also important to consider the effect of the substrate on the forces experienced by the organism. Here, we examine ground effect by placing the horseshoe crab on the bottom of the computational box and enforcing the no slip condition with a prescribed free-stream velocity at the top of the box. This created a linear fluid velocity profile that is a close, but imperfect match for a real boundary layer. Ground effect is typically responsible for an increase in lift and decrease in drag of aerodynamic objects as they are placed closer to a stationary surface. When considering objects that produce negative lift (also termed downforce), there is an increase in the magnitude of the negative lift produced until the object is very close to the substrate (Zerihan and Zhang 2000; Zhang et al. 2006). We likely did not find increased negative lift when including the ground because we set the distance between the crab and the substrate to 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 expands the limited body of work on legged aquatic locomotion. Furthermore, it provides a foundation for investigating how documented inter- and intra-specific morphological variation 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 that spine length impacts passive settling rates (Fisher 1977), so it is possible that other changes in spine number and shell curvature have a hydrodynamic effect. Additionally, horseshoe crabs have been used for biologically inspired design of intertidal robots (Krummeletal.2014). These engineering applications could also be expanded to other situations where negative lift would be desired such as sensors for tornados or waterways, car roof racks, or high-end backpacking tents that are exposed to high winds. Passive morphological mechanisms for generating negative or minimal lift at a zero angle of attack have rarely been found in benthic arthropods, but our findings demonstrate that horseshoe crabs have this capability. Given the limited work on pedestrian aquatic organisms, it is possible that other benthic arthropods employ this strategy to maintain position during locomotion or reproduction. This finding is also consistent with observations of horseshoe crabs swimming upside down, and the relatively high mortality rate of individuals that fail to right themselves.

Acknowledgements We would like to thank Brad Erickson and Jonathan Rader for help with laser scanning and 3D-reconstruction, and Miles Hackett for help with experiments. Additionally, we would like to thank the UNC Ofﬁce of Undergraduate Research and the William W. 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 · LauraA.Miller3,4

Alexander P. Hoover [email protected] Laura A. Miller [email protected]

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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