The hydrodynamics of swimming at intermediate Reynolds numbers in the water boatman (Corixidae)

The speed of swimming and its energetic requirements are determined by hydrodynamics that vary with the speed and size of an animal. The Reynolds number is a fundamental metric of hydrodynamic scaling that approximates the ratio of inertial to viscous forces. At intermediate Reynolds numbers (1<Re<1000), all viscous and inertial forces have the potential to contribute to thrust and drag. It is in this domain that a tremendous diversity of aquatic animals operate while swimming. The aim of the present study was to resolve which forces play a role in the paddling of the water boatman (Corixidae), an animal that operates in the intermediate regime (10<Re<200) (Blake, 1986).

A swimming animal must generate thrust to overcome drag. Drag acts in the direction of flow velocity, which is in opposition to the velocity of a body moving through a still pool of water. A paddling swimmer creates thrust by rapidly moving its appendages posteriorly during a power stroke (Fig. 1). This is considered ‘drag-based’ thrust because it is drag on the appendages that creates thrust for the body (Alexander, 1968). At the scale of the paddles or the body, drag may be created by a combination of skin friction and pressure drag. Skin friction emerges from the viscous resistance of water to shearing and it varies in direct proportion to flow velocity. It is the dominant force in the low Reynolds number (Re<1) propulsion of microorganisms. Pressure drag is created by stream-wise differences in velocity over a body's surface. Its magnitude varies in proportion the square of flow velocity (Lamb, 1945) and is dominant at high Reynolds numbers (Re>1000). For unsteady motion, such as that generated by an oscillating appendage, additional force may be generated by the added mass of water carried by a structure as it accelerates and decelerates. This acceleration reaction force varies in proportion to the volume of water entrained by the structure (Lamb, 1945). Therefore, skin friction, pressure drag and acceleration reaction could all play a role in swimming at intermediate Reynolds numbers.

The present study examined the relative magnitude of each of these forces in the swimming of water boatmen. Paddling by this aquatic insect is achieved by the motion of the metathoracic hindlegs. The tarsus (i.e. the most-distal) segment of this limb functions as a paddle (Blake, 1986). This paddle includes an array of setae that splay during the power stroke and collapse during the recovery stroke. This motion is similar to swimming in water beetles (Dytiscidae), which was classically investigated by Nachtigall (Nachtigall and Bilo, 1965). Both have provided a seemingly simple subject for hydrodynamic investigation (Daniel, 1984; Blake, 1986) and a textbook case for introducing biological propulsion (Alexander, 2003). Through mathematical modeling and kinematic measurements, these studies suggest that the acceleration reaction and pressure drag are important contributors to propulsion. We first tested this idea by measuring the thrust generated by tethered water boatmen (Figs 1, 2). These measurements were compared against the predictions of models that differed in their inclusion of the three fluid forces under consideration. We found the force coefficients for each model that best fitted the force measurements using a non-linear optimization routine (Fig. 3). The results of the tethered swimming experiments then provided the basis of a forward dynamic model of the hydrodynamics of free swimming. By additionally simulating drag on the body, this model predicted the center-of-mass changes in speed through time. Upon verifying these predictions against measurements of body kinematics, we were able to infer the major forces during swimming and their energetic implications.

We successfully tested mathematical models of thrust with measurements in tethered water boatmen. High-speed video recordings of the power stroke (e.g. Fig. 4A) showed how the extended hindlimbs moved smoothly from the anterior-most position toward the posterior (45 deg<ϕ<165 deg, where ϕ is wrist angle). The hindlegs reversed direction during the recovery stroke, as the paddles flexed to align with the long axis of the body (Fig. 4B). The angle (Fig. 4C) and velocity (u_P, Fig. 4D) of the paddle, in combination with morphometrics (Fig. 2F, Table 1), provided the necessary inputs to model thrust. Models for both pressure drag and skin friction predicted peak thrust with a timing that was coincident with measured thrust (Fig. 4E), which occurred approximately when the speed of the power stroke was greatest (Fig. 4D). However, only the pressure drag model was found to increase rapidly enough to match the magnitude of our measured changes in force (red curve in Fig. 4E). The rapid change in thrust by pressure drag was due to the dependency of this force on the square of paddle velocity. Skin friction, which varies in direct proportion to velocity, failed to predict these rapid changes in thrust (blue curve in Fig. 4E).

Because of the superior matching of the pressure drag model, we calculated the difference between its prediction and our measurements to consider any additional force that could play a role in thrust generation. The signal of this residual thrust (Fig. 4F) was not substantially greater than the high-frequency electrical noise of our measurements. Some sequences exhibited occasions where force measurements appeared to coincide with predictions for thrust from acceleration reaction (e.g. the first green curve in Fig. 4F). These occasions offer an indication that the acceleration reaction could be a minor contributor to thrust at the beginning of a power stroke (Fig. 4D).

We tested the accuracy of each model by comparing predictions for the momentum generated during the power stroke against measurements (Fig. 5). Accuracy was evaluated by calculating the coefficient of determination for model predictions against the measurements. Over the range of our momentum measurements (from 0.4 to 9.5 g mm s⁻¹), the skin friction model (Eqn 4) predicted only about one-quarter of the variation in measured momentum (R²=0.25, N=9). This was the result of an inability of the skin friction model to match the rapid changes in thrust that we observed (Fig. 4E). In contrast, the pressure drag model (Eqn 5) predicted almost all variation in measured momentum (R²=0.98, N=9). Note that this result does not preclude a role for skin friction as a contributor to propulsion, which is an issue we addressed by examining variation in the drag coefficient with Reynolds number (discussed below). Adding acceleration reaction (Eqn 5) to the pressure drag model increased the predicted momentum a small amount (~0.2 g mm s⁻¹), but this did not enhance the accuracy of the predictions (R²=0.97, N=9). Therefore, pressure drag on the paddle alone was sufficient to predict the momentum generated by a tethered water boatman.

We applied our findings on the hydrodynamics of tethered swimming to a model of a freely swimming water boatman. When untethered, the paddling of the hindlimbs served to propel the water boatmen forward (Fig. 6A,B). During the power stroke, the posterior-directed motion of the paddle was reflected in an increase in the wrist and paddle angles (Fig. 6C) and a rapid negative velocity (Fig. 6D). Simulations of our forward-dynamic model predicted oscillations in swimming speed in a manner similar to our kinematic measurements (Fig. 6E). Based on the results of our tethering experiments (Fig. 5), this model (Eqn 8) assumed that thrust was generated solely by drag on the paddle. Therefore, its predicted thrust (Fig. 6F) roughly coincided with the large changes in paddle velocity during a power stroke (Fig. 6D). As thrust was imparted momentum to the body, the body increased in speed and thereby encountered greater drag. This drag caused the body to decelerate as the velocity of the paddle declined at the end of the power stroke and increased through the recovery stroke.

The energetics of swimming were considered by analyzing the results of each simulation. When a water boatman was accelerating (e.g. the first power stroke in Fig. 6G), the total power generated by the paddle necessarily exceeded the thrust power. As the body moved forward at higher speed, the flow created by body motion enhanced thrust power (the final term in Eqn 13), but adversely affected the power generated by the paddle (Eqn 12). As a consequence, thrust power could exceed that generated by the paddle as the body gained speed (e.g. the second and third power strokes in Fig. 6G). However, the total energy generated by thrust could not exceed that generated by the paddle (Fig. 6H) for a water boatman starting from rest. The kinetic energy, and hence the speed, increased when thrust exceeded drag during the power stroke. During the recovery stroke, drag succeeded in dissipating the energy of propulsion (Fig. 6H) and thereby reducing the speed of the body (Fig. 6E).

The accuracy of our forward dynamic model was tested by comparing its predictions of body speed against our measurements. Similar to our tethered measurements of momentum (Fig. 5), we tested the model predictions of swimming speed against measurements by calculating a coefficient of determination with respect to unity. Variation in the mean speed of the body over the duration of a swimming cycle was well predicted by the model (R²=0.90, N=13; Fig. 7A). Variation in the maximum speed was similarly predicted by the model (R²=0.85, N=13; Fig. 7B). Therefore, our forward dynamic model (Eqn 8) accurately predicted the swimming speed of water boatmen.

Our kinematic measurements verified that water boatmen operate at intermediate Reynolds numbers. The Reynolds number values for both the paddle (Re_P, Fig. 8A) and whole body (Re_B, Fig. 8B) were positively correlated with body length. For free-swimming experiments, we used water boatmen that spanned a nearly threefold range in body length (2.7 mm≤l_B≤7.7 mm, N=12), where the Reynolds numbers of the paddle (3≤Re_P≤70) were lower than those for the body (24≤Re_B≤280). The tethered water boatmen spanned a more narrow range of body lengths (5.0 mm≤l_B≤8.8 mm, N=9) and Reynolds numbers (31≤Re_P≤130) that overlapped with the values of free-swimming animals. Therefore, the paddling of tethered and free-swimming animals operated in a similar hydrodynamic regime.

The relationship between drag coefficient and Reynolds number can offer valuable insight into the mechanisms of force generation. These coefficients were found by using optimization algorithms that minimized error between the model and measurements (Fig. 3). The greatest values for the drag coefficient for both the body and paddle were achieved at the lowest Reynolds numbers, which were generated by the smallest individuals (Fig. 8A). Furthermore, the range of the drag coefficient was substantially greater at lower Reynolds numbers. For example, the drag coefficient for the paddle (C_P) varied by about an order of magnitude for Re<30 (1.2<C_P<11.6, N=7), but showed not even a threefold range of variation at Re>75 (0.6<C_P<1.6, N=2; Fig. 9A). A similar trend was observed for the body of a water boatman, though with a more narrow range of variation in the drag coefficient (1.2<C_B<5.6, N=7), over a wider range of variation in Reynolds number (Fig. 9B).

Our modeling allowed for a consideration of the energetics of propulsion. We found that the energetic cost of generating thrust (8.7±7.3 J kg⁻¹ m⁻¹, N=13) accounted for about two-thirds of the total cost of transport (γ=13.0±10.4 J kg⁻¹ m⁻¹, N=13). The remaining one-third of this cost was lost to the generation of lateral forces. Although the mass of the body and surrounding water resists acceleration during the power stroke, this mass also acts to maintain an elevated speed during the recovery stroke. Therefore, mass has a neutral effect on the cost of transport. Because of the generation of lateral forces, the thrust efficiency (Eqn 17) was 72% (η_T=0.72±0.49, N=13), whereas the propulsive efficiency, which reflects the overall efficiency of generating kinetic energy (Eqn 17), was merely 10% (η_K=0.10±0.09, N=13). Therefore, inefficiency generated by drag far exceeds the losses due to lateral forces.

Our findings establish the major forces that govern the swimming hydrodynamics of the water boatman. Through experiments on tethered and free-swimming animals, we found that thrust is principally created by pressure drag on the paddle. However, viscous forces play a role by both contributing to propulsion and enhancing drag on the body. This role for viscosity is particularly important for the smaller water boatmen that operate at lower Reynolds numbers. At this scale, drag generated on the paddle and body greatly exceeds the quasi-steady values expected for each in isolation. Therefore, unsteady viscous interactions between the paddles and the body affect propulsion at the lower end of the intermediate Reynolds number regime. These principles may be applied toward understanding swimming in a broad diversity of animals.

The acceleration reaction has the potential to contribute to thrust at the start of a power stroke, as the mass of water entrained by the paddle is maximally accelerated toward the posterior of the body (Daniel, 1984). Previous studies on dytiscid beetles (Nachtigall, 1960) and water boatmen (Blake, 1986) appear to support this as a source of propulsion. Blake estimated the momentum from the acceleration reaction to be roughly equivalent to that generated by drag (Blake, 1986). His model employed the conventional assumption that the paddle accelerates a volume of water equal to a cylinder with the same dimensions. However, measurements of thrust in a dynamically scaled model of Artemia paddles found no evidence for an early thrust peak that would be expected from this added mass (Williams, 1994). These model measurements were performed at relatively low Reynolds numbers (Re ~5) compared with those of water boatmen (Fig. 9), but they suggested that force generated by the added mass may be smaller than previously considered. Our results agree with Williams' findings, as we found no consistent peak in thrust early in the power stroke (Fig. 4E). Furthermore, the inclusion of the acceleration reaction in our model did not enhance the accuracy of momentum predictions for tethered swimming (Fig. 5).

The acceleration reaction has similarly been shown to play only a minor role in other modes of locomotion at intermediate Reynolds numbers. This force is not an important source of propulsion in tethered and free-swimming ascidian larvae, which swim by tail undulation at Re ~100 (McHenry et al., 2003). It is perhaps not surprising, given the low density of air, that the acceleration reaction is also negligible in the lift generation of flies, which operate in a similar regime (Re ~140) (Dickinson et al., 1999). The acceleration reaction does appear to generate thrust in the impulsive accelerations of the chaetognath (Sagitta elegans), which attain relatively high intermediate Reynolds numbers (Re ~1000) (Jordan, 1992). At the larger scales of adult fish, the acceleration reaction also contributes to propulsion for steady swimming (Lighthill, 1971; Tytell, 2004; Wu, 1971) and accelerations (Frith and Blake, 1991; Tytell and Lauder, 2008; Weihs, 1972). Therefore, the acceleration reaction can act as an important source of propulsion for swimming in relatively large animals and perhaps in rapid accelerations at the high end of the intermediate Reynolds number regime.

Our findings allow for a nuanced interpretation of the role of viscosity in propulsion at this scale. We found that skin friction was insufficient to explain the rapid rise in thrust that was generated during the power stroke (Fig. 4E). As a consequence, this force alone failed to predict the momentum generated by tethered swimming (Fig. 5). However, this result does not preclude a role for viscosity in the generation of thrust. The drag coefficient represents force that is normalized by parameters that determine pressure drag. Therefore, the drag coefficient is predicted to be invariant with respect to Reynolds number where drag is generated entirely from the pressure differences that emerge from laminar flow. These conditions are violated at lower Reynolds numbers, where viscosity significantly contributes to drag. This occurs on a sphere, where skin friction causes the drag coefficient to elevate with decreasing Reynolds number (Fig. 9B). Williams found a similar trend for drag on the appendages of Artemia, which include an array of setae that are similar to the hindlegs of a water boatman (Williams, 1994). It is therefore unsurprising that we found a similar pattern for both free-swimming and tethered water boatmen for the drag on the paddle (Fig. 9A) and on the body (Fig. 9B). We consequently conclude that viscosity contributes to the generation of both thrust and drag, especially in small water boatmen. However, even when accounting for the effect of skin friction, small water boatmen generated larger forces than expected.

The range and maximum values for the drag coefficient that we measured for the body of a water boatman exceed the precedents in the literature. Jumping terrestrial insects span the intermediate regime and exhibit drag coefficient values that approximate unity. Blake found similar values by fitting a model of drag to the free swimming of a single water boatman (Blake, 1986). Lower values (C_B ~0.4) have been reported for water beetles (Dytiscidae), which paddle like water boatmen but at higher Reynolds numbers (up to Re ~10⁵) (Nachtigall, 1960). These values for drag coefficient approximate those of ellipsoidal bodies at comparable Reynolds numbers (Hoerner, 1965). In contrast, we found drag coefficients for the body that were up to an order of magnitude greater than in water beetles and at least fivefold greater than other insects (Fig. 9B). These values originated by fitting our forward-dynamic model to measurements of body speed (Fig. 3B). Therefore, lower drag coefficient values would have caused the model to fail to replicate the rapid decline in speed that we observed during the recovery stroke (Fig. 6E) and thereby not achieve the same accuracy in our predictions (Fig. 7).

We recognize a few possible explanations for the discrepancy between our values for drag coefficient and prior studies. First, the Reynolds numbers are lower in water boatmen than in jumping insects and water beetles. For reasons described above, skin friction should elevate the drag coefficient at low Re (Fig. 9B). In addition, the body of a water boatman is likely exposed to flow above what is explicitly considered by our model (Jiang and Kiørboe, 2011). The paddles induce flow past the body that should act to augment drag during the power stroke. Vortex shedding by the paddles should create flow, and consequently low pressure, at the posterior of the body. This would function to enhance drag at the transition between power and recovery strokes. Finally, although we observed no force generation during the recovery stroke in our tethering experiments (Fig. 4E), it remains possible that the recovery stroke motion of the paddles could contribute to drag in free swimming (Blake, 1986).

Our measurements for the drag coefficient of the paddle are also greater than values reported in prior studies. The drag for an Artemia appendage was found to approximate that of a cylinder of comparable dimensions (Williams, 1994). As in our measurements, these values increased at lower Reynolds numbers, presumably due to skin friction. However, we found the drag coefficient to be twofold or threefold greater than that of a cylinder in this regime (Fig. 9A) for reasons that are unclear. These large values are not due to the porosity of the paddle, because a semi-porous appendage does not exceed the force-generating capacity of a solid cylinder (Cheer and Koehl, 1987a; Cheer and Koehl, 1987b). It is possible that a combination of unsteady mechanisms and fluid–structure interactions account for the high thrust generated by water boatmen. For example, the paddles should create a ‘starting vortex’ at the initiation of a power stroke that establishes the circulation for thrust (Dickinson, 1996). Each paddle may interact with the paddle on the contralateral side to create a vortex loop that enhances the circulation, and therefore thrust, of both paddles. Such a mechanism could break down at higher Re, as a result of less viscous cohesion in the flow field. Although unsteady mechanisms of this sort have been shown to be ineffective at lower Re values in insect flight (Miller and Peskin, 2009), the geometry and motion of water boatmen may offer a contrast. Future flow visualization and mathematical modeling studies could resolve how a water boatman creates the very large force coefficients that we report here.

Hydrodynamics greatly determine the energetic cost of swimming. Owing to the relatively large drag coefficients that we measured, the cost of transport that we determined for water boatmen is relatively high. By comparison, a 5 g larval fish that swims at its most efficient speed requires a metabolic cost of transport of ~9.4 J kg⁻¹ m⁻¹ (Kaufmann, 1990; Videler, 1993). If one assumes the maximal efficiency of 50% for the red muscle of fish (Curtin and Woledge, 1993), then a mechanical cost of transport of 4.7 J kg⁻¹ m⁻¹ is predicted. This conservative estimate is comparable to the most energetically economical water boatmen that we have modeled (Fig. 10A). However, because of the many instances of higher values, we found a mean cost of transport that was about threefold greater than this approximated value for larval fish (γ=13.0±10.4 J kg⁻¹ m⁻¹, N=13). Although fast swimming may elevate the cost of transport by a factor of 2 or 3 in fish larvae (Kaufmann, 1990), we found some water boatmen exceeded this estimate by an order of magnitude (Fig. 10A). This may be due to the high inefficiency of paddling. Larval fish swim with a propulsive efficiency of around 0.3 (Li et al., 2012), which is about threefold greater than what we have found in most water boatmen (Fig. 10B). Furthermore, our experiments examined short bouts of swimming from a standstill, where swimming commences with relatively low efficiency. In contrast, measurements of metabolic efficiency generally report average values for sustained swimming (Kaufmann, 1990). Therefore, water boatmen appear capable of achieving a cost of locomotion that is comparable to that of fish of similar size under ideal conditions. However, water boatmen rarely exhibit this ideal because of their intermittent behavior and the inefficiency inherent to their mode of locomotion.

Despite the large influence that the Reynolds number has upon the generation of drag (Fig. 9), it has a complex relationship with swimming energetics. Neither the cost of transport nor the efficiency of swimming exhibits a clear dependency on Reynolds number (Fig. 10). Although these measures of energetic performance depend on Re-dependent hydrodynamics (Eqns 15, 16, 17, 18 and 19), fluid forces also affect the behavior of a swimmer. For instance, the relatively large drag coefficient encountered by a smaller water boatman may cause the animal to paddle more slowly than a larger animal (supplementary material Fig. S1) and consequently move with greater energetic economy. Therefore, the fluid forces that determine energetics also affect behavioral decisions and the force-generating capacity of muscle, which affect the kinematics that create fluid forces. Given this feedback loop, it is perhaps unsurprising that swimming energetics do not clearly vary with Reynolds number.

Our results illustrate the high sensitivity of propulsion to differences in size and speed at intermediate Reynolds numbers. In this domain, viscous interactions mediate the generation of body drag and propulsion to create force coefficients that may vary by an order of magnitude (Fig. 9) within a relatively narrow range of Reynolds numbers (10<Re<100). It is therefore interesting to consider how other strategies for generating propulsion may also vary with size for other animals within this regime. For example, copepods (100<Re<1000) propagate a wake from their antennae to abdominal paddles during impulsive ‘hopping’ (Borazjani et al., 2010). This mechanism could fail as a result of viscous attenuation at lower Reynolds numbers, or could lack the necessary cohesion at larger scales. Therefore, this mechanism of propulsion is likely to function only within a finite range of Reynolds numbers. Pteropods generate lift and thrust (1<Re<1000) with flexible wing-like appendages, which employ a variety of 3D motions (Chang and Yen, 2012). Given the scale dependency of propulsion, a kinematic pattern exhibited by a small pteropod may generate forces that contrast with those of a large pteropod that moves in a similar way. As exhibited by the water boatmen (Fig. 10), the relationship between the hydrodynamics and energetics of swimming is complicated by the influence of fluid forces on behavior. Nonetheless, this fascinating regime illustrates how the performance of locomotion critically hinges on the highly scale-dependent properties of fluid dynamics.

Insects were collected between June 2012 and August 2013 from the University of California National Reserve Rancho San Joaquin Marsh (RAMS App. no. 25020, Reservation no. 28126, Irvine, CA, USA). Collected insects were held at 21–24°C in pond water until used in an experiment. Tethered force measurements on nine individuals were successfully completed, each within 1 week of collection. The animals used in the tethering experiments ranged in body mass between 4.98 and 8.78 g (6.17±1.23 g, mean ± 1 s.d., N=9). Video recordings of free swimming were completed for 13 individuals, each within 2 weeks of collection. The animals used in the free-swimming experiments ranged in body mass between 2.66 and 7.69 g (4.42±1.72 g, N=13). Prior to all experiments, the wet mass of each water boatman was recorded on a fine scale (Denver Instrument Company, Bohemia, NY, USA) by extracting the animal from the water, gently drying the surface of the body with a Kimwipe (Kimtech Science, Roswell, GA, USA) and then anesthetizing it by perfusion with CO₂.

We measured the thrust generated by tethered water boatmen as a basis for testing hydrodynamic models. The tether was composed of twisted Teflon-coated stainless steel wire (0.1143 mm diameter, ~11 mm in length, A-M Systems, Inc., Everett, WA, USA) that was attached to the first dorsal segment of hardened cuticle. This was achieved with a flowable, light-cured dental composite (Prime-Dent, Prime Dental Manufacturing, Chicago, IL, USA), which was also used to anchor the tether to a surface that was positioned with a micro-manipulator. The micro-manipulator translated the animal to the center of a water-filled well (25×50 mm in width), submerged in deionized water at a depth of 6 mm at 22°C. The resonant frequency of the tether with an attached water boatman exceeded 300 Hz, which was sufficiently high to exceed the domain of our measurements and thereby not present resonance artifacts. Tether deflections during experiments did not exceed 1% of the length of the tether, which was well within the assumptions of the low-deflection beam theory that we employed in our calibrations (detailed below).

The thrust measurements were based on recordings of tether deflections that were created during swimming. This was achieved with an optical technique that used a laser with integrated sheet-generating optics (5 mW, 670 nm, Lasiris, StockerYale Canada Inc., Quebec, Canada), which was focused onto the mound of glue that affixed the insect to the tether (Fig. 1). The laser sheet was then focused in the vertical direction with a cylindrical lens (6.25 mm diameter, 100 mm focal length, Edmond Industrial Optics, Barrington, NJ, USA). The focused light and shadow cast by the mound were aligned with a photosensor (OSI Optoelectronics, Camarillo, CA, USA), which detected the position of the edge of the shadow. This was achieved with a custom-built amplifier designed for an earlier study (Lehmann and Dickinson, 1997). As the insect generated thrust during an experiment, the voltage changes inducted by the motion of the tether were recorded at 100 kHz by a National Instruments DAQ (NI USB-6009, Trabuco Canyon, CA, USA) and custom-written MATLAB software (v 2012a, MathWorks, Natick, MA, USA). After completing a series of successful recordings, the water boatman was anesthetized and killed through heat exposure by replacing the water within the well with 60°C water until limb movements ceased.

We performed a calibration in two parts to determine thrust from the recordings of the optical sensor. The first part determined the relationship between the reading of the optical sensor and the tether deflection. This was achieved by recording the sensor voltage as we moved the tethered insect, and hence the shadow cast on the sensor, with a digital actuator (Mitutoyo Corp., Kawasaki, Kanagawa, Japan) in increments of 2–10 μm over the range of recorded voltage. The voltage reported for each tether position was recorded as the mean value of a 1 s recording at 100 kHz. We characterized how voltage varied with the position of the tether by a non-linear least-squares fit of a smoothing spline (the ‘spaps’ function in MATLAB, Fig. 2A) to our recorded values. This spline provided the means to transform sensor voltage recordings into values of tether deflection.

The tethering experiments were recorded with high-speed video to measure the kinematics of paddling, which provided variables for our mathematical models of thrust. During the experiments, tethered insects were backlit with an array of white LED lights and viewed from the ventral perspective using a front-surface mirror (Edmond Industrial Optics, Barrington, NJ, USA) positioned beneath the chamber and oriented at 45 deg. This mirror allowed for a camera directed toward the side of the well to view the water boatman from below. Recordings were executed at 3000 frames s⁻¹ with a resolution of 512×512 pixels using a 1024 Photron camera (Photron Inc., San Diego, CA, USA) that was fitted with a macro lens (Micro-NIKKOR P.C Auto 55 mm with two Nikon PK-13 27.5 mm extension tubes, Nikon US, La Jolla, CA, USA) to record a ~16 mm square field of view. Three swimming cycles (i.e. power and recovery strokes) were recorded and analyzed for each individual.

We manually tracked landmarks on the body and a single hindleg to acquire paddling kinematics from video recordings using custom-written MATLAB software. The body was described by two coordinates for its anterior and posterior margins along the mid-sagittal plane (points 1 and 2, Fig. 2C). The paddle was tracked by coordinates for the proximal and distal ends (points 3 and 4 respectively, Fig. 2C) of the tarsus (i.e. paddle), the most distal segment of the metathoracic hindleg. We defined the wrist angle (ϕ) as the angular position of point 3 with respect to the body midpoint. Changes in the direction of the wrist angle were used to identify the initiation and completion of power strokes. The speed and orientation of the paddle were variables in our modeling (described below). Orientation was measured with the paddle angle (θ), which was defined with respect to the body midline (Fig. 2C) and factored into the calculation of thrust (described below, Fig. 2D,E). The paddle speed was calculated discretely as the product of the frame rate and the displacement of the paddle midpoint (i.e. spatial mean of points 3 and 4). These values were filtered with a least-squared fit of a smoothing spline (the ‘spaps’ function in MATLAB).

Morphometrics for each water boatman were measured after tethering experiments. Images were captured with a high-resolution camera (Zeiss AxioCam HRc) mounted onto a dissecting microscope (Zeiss SteREO Discovery V20, Thornwood, NY, USA). Linear dimensions and area measurements were performed with software (AxioVision Release 4.8.2, Zeiss) that autoscaled images according to the magnification of the microscope. The two lengths (l_T and l_L) used in our tethering calibration were measured from a photograph of the insect while still attached to the tether. The body of each individual was additionally photographed from dorsal, lateral and ventral perspectives. The maximum width (l_w) and depth (l_d) of the body were used to calculate its projected area (s_B), assuming an elliptical shape (i.e. s_B=1/4 π l_wl_d) and body length (l_B) provided a general metric of size. Using a photograph at higher magnification, the hindlimb was measured for the length of each segment: the femur, tibia and tarsus. We measured the area encompassed by the setae of the paddle (s_P) by excising the tarsus and splaying it on a drop of water under a coverslip (Fig. 2F). The area was measured by tracing the peripheral shape of all setae in a digital photograph of the splayed tarsus.

For both models, we used an optimization algorithm to determine the values for the force coefficients that best matched the thrust measurements (Fig. 3A). This worked by separately calculating the thrust generated by pressure drag (Eqn 3) and skin friction (Eqn 4) iteratively with varying values for the force coefficient (either C_P or C_V). Using the least-squares method, the optimization selected the value for each coefficient that minimized the error between predicted and measured thrust, defined as the sum of squared differences over the duration of the power stroke (implemented with the ‘lsqcurvefit’ function in the optimization toolbox in MATLAB, v R2013a). Therefore, our test of each form of drag used the ideal force coefficient that could be found for each drag equation.

We compared the accuracy of different models of thrust by testing their ability to predict the momentum generated during a power stroke. Momentum was determined by integrating thrust over the duration of a power stroke using the trapezoidal rule (implemented by the ‘trapz’ function in MATLAB). Predictions of momentum were tested for three models of thrust. The first treated thrust as being generated entirely by pressure drag on the paddle. For the second model, thrust was calculated as the sum of pressure drag and the acceleration reaction. The third model considered drag to be entirely generated by skin friction on the paddle. We calculated the coefficient of determination for the predictions of each model against our momentum measurements to describe the proportion of variation in the measurements that were predicted by each model and thereby provide a basis for comparing their accuracy.

We solved for the force coefficients using an optimization algorithm that was based on our forward dynamic model (Fig. 3B). We allowed the drag coefficient on the paddle to vary among the power strokes within a swimming sequence, because of the possible differences in stroke kinematics. However, the optimization did assume a constant value for the drag coefficient of the body because its geometry did not appreciably vary during a sequence. Similar to the analysis of our tethering experiments, the non-linear optimization used a least-squares approach to solve for the drag coefficient values for the body and paddle that minimized deviation between predicted and measured body speed over a series of simulations (e.g. Fig. 3C).

H. Jiang (Woods Hole Oceanographic Institution) and Tim Bradley (UC Irvine) and the members of the UCI Comparative Physiology Group offered valuable suggestions. Two anonymous reviewers offered excellent suggestions for improving the manuscript.