Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming

Abstract

Animal movements result from a complex balance of many different forces. Muscles produce force to move the body; the body has inertial, elastic, and damping properties that may aid or oppose the muscle force; and the environment produces reaction forces back on the body. The actual motion is an emergent property of these interactions. To examine the roles of body stiffness, muscle activation, and fluid environment for swimming animals, a computational model of a lamprey was developed. The model uses an immersed boundary framework that fully couples the Navier-Stokes equations of fluid dynamics with an actuated, elastic body model. This is the first model at a Reynolds number appropriate for a swimming fish that captures the complete fluid-structure interaction, in which the body deforms according to both internal muscular forces and external fluid forces. Results indicate that identical muscle activation patterns can produce different kinematics depending on body stiffness, and the optimal value of stiffness for maximum acceleration is different from that for maximum steady swimming speed. Additionally, negative muscle work, observed in many fishes, emerges at higher tail beat frequencies without sensory input and may contribute to energy efficiency. Swimming fishes that can tune their body stiffness by appropriately timed muscle contractions may therefore be able to optimize the passive dynamics of their bodies to maximize peak acceleration or swimming speed.

Fig. 1.

Example flow patterns around the reference swimmer (shown in gray). Arrows indicate flow velocity; background color shows vorticity. The green line indicates the path of the center of mass.

Fig. 2.

The swimmer reaches a steady speed and produces net positive axial impulse along most of its body. (A) Swimming speed (solid line, left axis) and force per unit height in the axial direction (gray regions, right axis). Arrow indicates the time of Fig. 1. (B) Axial impulse per unit height produced over a cycle period during steady swimming. Impulse values due to normal and tangential stresses are in open and filled gray bars, respectively. Net axial impulse per unit height is shown with a thick line.

Fig. 3.

Comparison of fluid forces as calculated from analytical models and the CFD model. The plot shows time course of lateral forces from the Navier–Stokes solution (left axis, open symbols), Lighthill’s large amplitude elongated body theory (right axis, “reactive,” dashed line) (22), and Taylor’s resistive model (right axis, “resistive,” dotted line) (11, 23). Arrows indicate two peaks due to vortex shedding.

Fig. 4.

Identical swimmers in fluids of different viscosity have different kinematics. (A) Outlines of swimmers in three viscosities (gray, 0.5×; black, 1×; and white, 10×water) at the same time. (B) Body wavelength in fluid of different viscosities. Activation wavelength is identical.

Fig. 5.

For a given muscle activation pattern, there are different optimal stiffness values for maximum acceleration or steady swimming speed. The plots show four swimmers with increasing stiffness: tan dotted line, simulation 4, Table 1; green long dashes, simulation 5; black, reference simulation; and cyan short dashes, simulation 6. (A) Swimming speed vs. time. (B) Body outlines for each swimmer at the time indicated by the arrow on panel A. (C) Mean acceleration during the first tail beat. (D) Mean steady swimming speed. (E) Muscle power coefficient C_P,mus.

Fig. 6.

A phase lag between muscle activation and curvature corresponds to negative muscle work near the tail and is produced when muscle forces are low compared to fluid forces. (A) Phase of curvature and muscle activation along the body. Gray bar shows the activated muscle region, while lines show the position of zero curvature (black solid line, reference; dark blue dots, 1.5 Hz; purple short dashes, floppy body with weak muscles; red long dashes, high viscosity). (B) Negative muscle work produced in the last 20% of the body as a fraction of the total muscle work, plotted against rms muscle force divided by rms fluid forces in the lateral direction. Numbers indicate simulation number, Table 1.