Fig 1.
Observed St and SW ranges from meta–analysis data of body/caudal fin swimmers.
The Strouhal number (top) and specific wavelength (bottom) for the species considered in this study. Data points represent average values, where x ∈ {St, SW}, of individual species and error bars indicate ± one s.d. from the species mean. Dashed lines represent St = 0.3 and SW = 10. Distinction is made between anguilliform and non–anguilliform swimmers. Data labeled as Ambystoma mexicanum and Ambystoma mexicanum young are from adult and larval axolotl, respectively. All observations of Clupea harengus and Danio rerio came from anguilliform larvae. For some species, the error bars are not visible at this scale, while for others, only one observation is recorded and no error bars are available; see Table 1 for more details. The data are available in S1 Data.
Fig 2.
Variability in St as a function of Relat and AR.
Intraspecies mean Strouhal number vs. (a) lateral Reynolds number, and (b) aspect ratio for observed non-anguilliform(⚫) and anguilliform(▲) swimmers. Of these, orange (green) points represent swimmers with (without) well–defined caudal fins. Error bars indicate ± one s.d. from the species mean. For some species, the error bars are not visible at this scale, while for others, only one observation is recorded and no error bars are available; see Table 1 for more details. Data are available in S1 Data and Table 1.
Fig 3.
Variability in SW and OSW as a function of Relat and AR.
Intraspecies mean specific wavelength vs. (a) lateral Reynolds number, (b) aspect ratio for observed non-anguilliform(⚫) and anguilliform(▲) swimmers. Of these, orange (green) points represent swimmers with (without) well–defined caudal fins. Red crosses (×) represent the optimal specific wavelength for simulations done in the present study. Blue asterisks (*) represent the optimal specific wavelengths for robotic undulating sheets reported in [11, 12]. Error bars indicate ± one s.d. from the species mean. For some species, the error bars are not visible at this scale, while for others, only one observation is recorded and no error bars are available; see Table 1 for more details. Data are available in S1 Data and Table 1.
Fig 4.
Measured swimming speed and force from undulating sheet simulations.
(a) Axial swimming speed and propulsive force computed from free–swimming (green) and translation–locked (black) simulations of rectangular sheets plotted against the specific wavelength. In free–swimming simulations, the forward swimming speed of the undulating plate was an output parameter of the simulation. Simulations were carried out at a lateral Reynolds number Relat = 4.49, with corresponding swimming–speed Reynolds number range 1.8 × 101 < Re < 1.51 × 102. Data are available in S2 Data. (b) Evolution of axial Fx and heave forces Fz over time for a translation–locked, undulating sheet simulation with SW = 13.33. The oscillation in Fz is a signature of the linear recoil effect on the swimmer. The sway force Fy is not shown because the kinematics of the swimmer’s undulation lead to large Fy values, although it also oscillates about a mean value. (c) Evolution of U in each coordinate direction over time for a self–propelled, undulating sheet simulation with SW = 13.33. The oscillation of the heave velocity Uz about 0 is not easily visible at this scale.
Fig 5.
Measured propulsive force from low Relat undulating sheet simulations.
The axial propulsive force generated by a stationary undulating sheet plotted against specific wavelength. In both cases (a) & (b), plate span was varied. These data represent cases where Relat < 1 × 102. Data are available in S2 Data.
Fig 6.
Measured propulsive force from high Relat undulating sheet simulations.
The axial propulsive force generated by a stationary undulating sheet plotted against specific wavelength. In both cases (a) & (b), plate length was varied. These represent cases where Relat > 1 × 102. Data are available in S2 Data.
Fig 7.
Measured propulsive force from low Relat sheets with varying frequency and amplitude.
Axial propulsive force vs. specific wavelength for additional small–sheet simulations in which (a) frequency and (b) amplitude were varied. Data are available in S2 Data.
Fig 8.
Measured propulsive force from low Relat sheets with varying span and length.
Axial propulsive force vs. specific wavelength for additional small–sheet simulations with varying aspect ratios. In case (a), plate span was varied while in case (b), plate length was varied. Data are available in S2 Data.
Fig 9.
Measured propulsive force from anguilliform and carangiform sheet simulations.
Axial propulsive force vs. specific wavelength for additional small–sheet simulations with prescribed anguilliform (▲) and carangiform (⚫) amplitude profiles. Data are available in S2 Data.
Fig 10.
Measured swimming speed and force from realistic eel and mackerel simulations.
Axial swimming speed and propulsive force computed from free–swimming (green) and translation–locked (black) simulations of undulating (a) eel bodies, and (b) mackerel bodies, plotted against the specific wavelength. In free–swimming simulations, the forward swimming speed of the undulating body was an output parameter of the simulation. These simulations were carried out at Relat = 1.12 × 102, with corresponding swimming–speed Reynolds number range 1.378 × 103 < Re < 5.95 × 103. Data are available in S2 Data.
Fig 11.
Vortical structures shed from free–swimming eel and mackerel.
Three-dimensional vortical structures visualized for free–swimming simulations of an eel (a–c) and mackerel (d–f) at three different SW values. The wakes are visualized using isosurfaces of q–criterion, where , where A and S are the antisymmetric and symmetric parts of the fluid velocity gradient tensor ∇u, respectively.
Table 1.
Mean specific wavelength, lateral Reynolds number, and aspect ratio for the organisms studied in this work.
*Anguilliform swimmers. †Swimmer has a distinct caudal fin. All Clupea harengus and Danio rerio specimens were larval. Aspect ratio data are measured from side–view images or schematics of swimmer and not related to the listed Ni value.
Fig 12.
Wake and body visualizations from free–swimming simulations.
Top figures show the midline kinematics (black) over time for the three different types of undulating bodies considered in the present numerical study. Dashed red line denote the amplitude function ±a(x). Middle and bottom figures show contours of vorticity magnitude for the three bodies. Middle figures show the top–view with undulations present in the lateral direction, while bottom figures show the side–view of each body. (a), (b), & (c) An undulating flat plate with SW = 10 and Relat = 4.49; the low Reynolds number causes the wake to remain large and mostly attached. (d), (e) & (f) An undulating eel body with SW = 10 and Relat = 1.12 × 102. (g), (h), & (i) An undulating mackerel body with SW = 10 and Relat = 1.12 × 102.
Fig 13.
Computational setup and grid refinement validation case for an undulating sheet.
(a) Computational setup for a translation–locked simulation of an undulating sheet with L = 20 cm, h = 2 cm, f = 3 Hz, a = 1 cm and λ = 20 cm, which corresponds to Relat = 3.37 × 102. Three adaptive mesh levels are shown and the immersed structure is always placed on the finest mesh level. (b) Time evolution of axial force generated by the sheet for two different grid spacing values: ΔX0 = (3.125, 3.125, 3.125) × 10−3 L (orange) and ΔX1 = (1.5625, 1.5625, 1.5625) × 10−3 L (blue).