Figure 1.
The sandfish lizard and the numerical simulation.
A: A sandfish lizard (Scincus scincus) resting on 3 mm diameter glass particles. B: A simulated sandfish with a uniform body resting on simulated 3 mm particles. A and λ represent the amplitude and the wavelength of the single period sinusoidal traveling wave. Dashed purple line shows the outline of a tapered body. C: The elements of the sandfish model in the Working Model multibody simulation environment. The cuboid body segments are connected by actuators and β is the angle between two segments. D: Diagram of the empirical force relations used for particle interaction. The normal (Fn) and tangential (Fs) forces between two particles are calculated based on the relative speed v = v1−v2 and the virtual overlap δ between these two particles. Panels B, C, and D are reproduced from [21].
Table 1.
Particle properties in simulation and in experiment.
Figure 2.
The empirical resistive force relations for the granular medium.
The empirical force relations were obtained by dragging a rod with square cross-section (width = height = 16 mm, length = 40 mm) through 3 mm glass particles in simulation, at constant depth of 7.6 cm. The perpendicular () and parallel (
) components of the surface forces are plotted as a function of the angle between the velocity direction and the rod axis (
), see inset. See text for the analytical expressions for
and
. For comparison,
and
are calculated for a long thin ellipsoid in a low Re fluid by choosing a viscosity that gives comparable magnitude of
; the low Re forces are plotted as dashed gray lines. Figure adapted from [21].
Table 2.
Fitting parameters for the analytical functions approximating and
.
Figure 3.
Comparison of swimming performance in experiment and models.
A: Average forward swimming speed versus undulation frequency in 3 mm particles. Solid symbols correspond to biological measurements, and the solid and dashed lines correspond to the RFT (for a uniform body) and simulation (for a tapered body) predictions, respectively. B: Wave efficiency (), defined as the ratio of the forward swimming speed to the wave speed, measured from biological experiment (the slope of
versus
in (A)), simulation and RFT. For the RFT (solid bar), the lower and upper limits of the
deviation correspond to maximum (flat head) and 30% of the maximum head drag, while the simulation (hatched) corresponds to the uniform body and tapered body shapes, respectively. In simulation,
and
Hz. Figure adapted from [21].
Figure 4.
Body kinematics of the sandfish in the simulation.
A: CoM trajectory. Gray region shows the configuration of the uniform body at s (not to scale). The (B) forward velocity, (C) lateral velocity and (D) body orientation as a function of time. Inset: Diagram showing calculation of the center of mass (CoM) and yaw angle (
). Black circles represent uniform body and blue triangles represent tapered body. Actuation began at t = 0 sec.
,
Hz.
Figure 5.
Segment kinematics of the model sandfish.
Trajectories of segments near the head, middle of the body, and the tail from both experiment (A) and simulation (B) are represented by black, magenta, and green lines, respectively. The markers in experiment are located at 0.13, 0.50, and 0.87 of the effective body length and the segments at 0.16, 0.50 and 0.84 of the total body length (defined as the length from snout to tail tip) are chosen as counterparts in the simulation with a tapered body. The light gray regions indicate the body position at an earlier time and the dark gray regions indicate the body positions at cycles later. C: The RMS of the lateral displacement of a segment normalized to the effective body length as a function of position on the body from experiments (colored lines and symbols) and a simulation with a tapered body, an entry angle of
, and an amplitude of
(thick gray line). The data is from two animals with contributions of 1 (green) and 3 (other color) runs. D: The RMS of the lateral displacement of a segment normalized to the total body length as a function of position on the body of the model sandfish. E: The correlation between the lateral motion of segments and the lateral motion induced by the CoM motion (the blue curve and the left inset) and yaw motion (the red curve and the right inset) of the body. The data is from the same simulation for panel A. The dashed black line is the sum of the blue and red lines. In (D) and (E), the model sandfish swims in the horizontal plane with a uniform body at
and
.
Figure 6.
Snapshots of reaction forces on the model sandfish during swimming.
Black, green, and blue arrows represent forces measured in uniform body simulation, forces predicted by the empirical force relations, and velocities, respectively. For visibility, only every 3rd segment velocities are shown for and the head drag is scaled by a factor of 0.25 and drawn in thick lines. Snapshots in the middle column were taken at
,
and
; snapshots with the same phases were chosen for the other two columns. The values below the arrows in the legend indicate the magnitudes of force and velocity corresponding to the length of their respective arrows. All diagrams show forces for a uniform body.
Figure 7.
Comparison of head drag and net force on the model sandfish from the simulation and empirical force relations.
A: Head drag ( on the head) as a function of time for
(green represents the empirical force relations and black represents simulation) and
(cyan represents the empirical force relations and blue represents simulation). The forward speed for
is re-plotted from Fig. 3B as the gray line to show relative phasing. B: The net force on the body (including head) in the forward direction as a function of time for
(magenta) and
(red). Bars between (A) & (B): The average values of the net forces and the head drag are given in corresponding colors. Uniform bodies were used in the simulations.
.
Figure 8.
Drag forces on an oscillating rod.
A: Schematic diagram of the simulation. The rod oscillates horizontally and normal to its axis. Rod width = 1.58 cm. B: The lateral displacement of the rod as a function of time. C: The resistive force in the lateral direction as a function of time. Blue, green and red lines represent the data from simulation with parameter sets (), (
), and (
and
), respectively. The black line represents the prediction from the empirical force law.
Figure 9.
The effect of body taper on the force on a segment.
A: A snapshot of the resistive forces on segments (black arrows). The blue arrow represents the head drag (net force on the tapered portion) with a different scale. B: Diagram of the forces on the segments in the tapered body regions near the tail (left) and near the head (right) when the velocities are nearly aligned with the mid-line of the segments. Corresponding segments are highlighted with yellow color on the body in panel (A). ,
.
Figure 10.
The torque generated by actuators of the sandfish model in simulation.
A: The torque generated by actuators at 0.25, 0.5, and 0.75 of the body (represented by green, red, and dark blue symbols and lines, respectively) as a function of time. B: RMS magnitude of the torque of an actuator as a function of position on the body. Large filled circles indicate the RMS of the torque curves in panel (A) with the same color scheme. The solid black line in panel B and data in panel A are from simulation with a uniform body and the blue dashed line in panel B is from a simulation with a tapered body. ,
.
Figure 11.
Spatial distribution of actuator power and power delivered to the granular medium.
A & B: Each bar represents a cross-sectional area (half cross-sectional area of the head) along the body or on the head (head area =
). The black bars represent areas on the uniform body and red bars represent areas from the blunt head. The green bars represent areas on the tapered body. C: The granular temperature (see text for the detail) calculated from particles within cells with dimensions of 0.3 cm (W) by 0.3 cm (L) by 1.6 cm (H) for a uniform body.
,
.
Figure 12.
The total power generated by the actuators as a function of undulation frequency.
Black circles represent uniform body and blue triangles represent tapered body. Dashed lines indicate a linear relationship between power and frequency. Fits are constrained to go through the origin and have slopes of 0.39 J (uniform) and 0.32 J (tapered). Averaging was done over an integer number of cycles and approximately 1 s of time. Error bars indicate the standard deviations of the fluctuations. .
Table 3.
Comparison of cost of transport.