How body torque and Strouhal number change with swimming speed and developmental stage in larval zebrafish

Abstract

Small undulatory swimmers such as larval zebrafish experience both inertial and viscous forces, the relative importance of which is indicated by the Reynolds number (Re). Re is proportional to swimming speed (vswim) and body length; faster swimming reduces the relative effect of viscous forces. Compared with adults, larval fish experience relatively high (mainly viscous) drag during cyclic swimming. To enhance thrust to an equally high level, they must employ a high product of tail-beat frequency and (peak-to-peak) amplitude fAtail, resulting in a relatively high fAtail/vswim ratio (Strouhal number, St), and implying relatively high lateral momentum shedding and low propulsive efficiency. Using kinematic and inverse-dynamics analyses, we studied cyclic swimming of larval zebrafish aged 2-5 days post-fertilization (dpf). Larvae at 4-5 dpf reach higher f (95 Hz) and Atail (2.4 mm) than at 2 dpf (80 Hz, 1.8 mm), increasing swimming speed and Re, indicating increasing muscle powers. As Re increases (60 → 1400), St (2.5 → 0.72) decreases nonlinearly towards values of large swimmers (0.2-0.6), indicating increased propulsive efficiency with vswim and age. Swimming at high St is associated with high-amplitude body torques and rotations. Low propulsive efficiencies and large yawing amplitudes are unavoidable physical constraints for small undulatory swimmers.

Keywords: biomechanics; body torque; development; larval zebrafish; swimming.

Figure 1.

A 5 dpf fish larva swimming at approximately 50 ℓ_body s^–1. (a) Sequence of body midlines at 0.5 ms intervals (blue); path of snout tip (green), tail tip (red) and CoM (black). (b) Body shape at 3 ms intervals. CoM (red sphere) periodically falls outside the body. (c) Translational speed of CoM and (d) force on CoM in the direction of (continuous curve) and perpendicular to (dotted curve) the instantaneous velocity vector. The blue and red ‘+’ signs in panel (c) and (d) correspond to the minima and maxima in α_body (figure 2a). The time between two ‘+’ signs of the same colour represents a full tail-beat cycle.

Figure 2.

Body angle, moment of inertia and power for episode in figure 1. (a) Instantaneous body angle α_body (black; blue and red ‘+’ signs: minima and maxima) and head angle α_head (green); (b) body angular velocity ω_body; (c) body angular acceleration (black: by differentiation of ω_body; red: according to electronic supplementary material, equation (S3.18)); (d) moment of inertia about the CoM J_body; dashed blue line shows (i.e. J_body for the straight fish), (e) rate of change of the moment of inertia ; (f) total specific body power based on the rate of change of kinetic energy.

Figure 3.

Body torque and angle-of-attack of the tail at 0.96 ℓ_body for the episode in figure 1. (a) Instantaneous inertial body torque τ_body (circles: selected extrema). (b) Angle-of-attack of the tail with respect to the velocity vector of the tail (β_tail). The blue and red ‘+’ signs in panel (a) and (b) correspond to the minima and maxima in α_body (figure 2a). The time between two ‘+’ signs of the same colour represents a full tail-beat cycle. (c) Distribution of the contribution to τ_body by the 51 body segments expressed as torque per unit length . Each blue curve represents the torque distribution at a particular instant (at 1 ms intervals). The heavy black curves show the envelope. The narrowest location along of the ‘neck’ of the envelope corresponds to the location of the straight-body CoM.

Figure 4.

Estimated torques. (a) Contour plot of inertial torque per unit length with respect to the CoM against normalized position along the body () and time. (b) Inertial body torque τ_body against time (black); torque on the body due to dynamic pressure, τ_dynp (green). (c) Contour plot of torque on the body per unit length due to dynamic pressure against and time. (d) Contour plot of torque on the body per unit length due to shear-force distribution against and time. (e) Inertial body torque τ_body against time (black); torque on the body torque due to shear-force distribution, τ_fric (blue) and acceleration reaction forces (τ_acc) × 0.2 (red). (f) Contour plot of torque per unit length on the body due to acceleration reaction forces against and time.

Figure 5.

Specific kinetic energy for (a) a near-cyclic swimming event at approximately 50 ℓ_body s^–1 (age 5 dpf) (same as figure 1) and (b) approximately 8 ℓ_body s^–1 (age 3 dpf). Total kinetic energy of the body (; black), kinetic energy due to speed of the CoM (; red), and sum of kinetic energy due to rotation of the segmental masses around CoM and rotation of the segments about their central vertical axis (; blue). .

Figure 6.

Swimming kinematics during (near-)cyclic swimming for 9–11 swimming events per age group (2–5 dpf). (a) Mean swimming speed along the path of the CoM (○) and along a straight line approximation of the path of motion (+) against cycle frequency f. Both speeds differ little, indicating small sideslip. 2, 3 dpf larvae reach lower speeds for a given f than 4, 5 dpf larvae. (b) Specific swimming speed () against f (same dataset as (a)). (c) Peak-to-peak tail-beat amplitude A_tail against f. Data in (c) and (d) were fitted by total least squares. In (c), the black curve shows the fit for the total dataset. The combined datasets for 2 and 3 dpf (blue-cyan line) and 4 and 5 dpf (red-green line) were fitted also separately. The 4–5 dpf age group tended to use higher A_tail than the 2–3 dpf group for f > 50 Hz, and they vary A_tail more over the frequency range. Panel (d) shows same data as (c) for dimensionless tail-beat amplitude (). Parameter values for each curve fit are given in electronic supplementary material, table S2.

Figure 7.

Dynamics of (near-)cyclic swimming for same dataset as figure 6. (a) Amplitude of torque peaks (mean per event, colour code: age 2–5 dpf) over mean swimming speed (). (b) As (a), but non-dimensionalized. (c) Amplitude of torque peaks against Reynolds number (Re). (d) As (c) but non-dimensionalized. (e) Peak-to-peak amplitude of body angle (mean per event) against . (f) Idem, but against dimensionless swimming speed. Second-order polynomial curve fits are shown for (a)–(f). (g) St against Re, with a fit of a (negative) power function plus a constant. (h) Re against swimming number Sw (logarithmic scales). The curve fit follows from the fit of St against Re, by using St = Sw/Re. Parameter values for each curve fit are given in electronic supplementary material, table S3.

Figure 8.

Specific kinetic energy due to translation E*_tr,CoM and rotation E*_rot,body during (near-)cyclic swimming for the same dataset as figure 6. (a,b) Maximum (○) and mean (+) values against mean swimming speed Second-order polynomial curve fits for maxima (continuous curve) and mean (dashed curve) are shown in (a,b). (c,d) Ratio of the maxima (c) and the means (d) of E_tr,CoM and E_rot,body over . The horizontal dashed line indicates a ratio of 1. Parameter values for each curve fit in (a) and (b) are given in electronic supplementary material, table S4.

Figure 9.

Maximum specific power computed as rate of change of kinetic energy of the body per body mass during (near-)cyclic swimming for same dataset as figure 6. (a) Total (○), translational (×) and rotational (+) power over mean swimming speed. Second-order polynomial curve fits for maximum total (continuous curve), and translational, and rotational (dashed curves) powers are shown. (b) Ratio of maximum translational to rotational power. The horizontal dashed line indicates a ratio of 1. Parameter values of the curve fits in (a) are given in electronic supplementary material, table S4.