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 mechani- cal 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 strate- gies 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 founda- tion 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 morphol- ogy 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 mul- tiple 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 organ- isms. 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 posi- tion 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 cara- pace (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 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 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 biologi- cal 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 biolog- ically 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 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. All experiments were performed with a flow speed of 13cm/s unless otherwise noted.