Large amplitude fish swimming

A fish swims by stimulating its muscles and causing its body to "wiggle", which in turn generates
the thrust required for propulsion. The relationship between the forces generated by the fish
muscles and the observed pattern of movement is governed by the mechanics of the internal
structure ofthe fish, and the fluid mechanics of the surrounding water. The mathematical modell ing
of how fish swim involves coupling the external "biofluiddynamics" to the body's internal solid
mechanics.
The best-known theory for the hydrodynamics of fish swimming is Lighthill's elongated body
theory (Lighthill, 1975). In Lighthill's theory the curvature of the fish is assumed small and
the effect on the fish of the vortex wake is neglected. Cheng et al. (1991) did not make these
simplifications in developing their vortex lattice panel method, but the fish was assumed to be
infinitely thin and its undulations of small amplitude.
Lighthill's "recoil correction" is the addition of a solid-body motion to ensure that an imposed
"swimming description" satisfies the conservation of momentum and angular momentum. A
real fish is expected to minimize such sideways translation and rotation to avoid wasteful vortex
shedding. Cheng and Blickhan (1994) found that the panel method model required a smaller recoil
than did Lighthill's model. Our approach is to extend Cheng's model to large amplitude. Thus we
include the effect of the wake on the fish, and the self-induced deformation of the wake itself.
In studying the internal mechanics of the body we model the fish as an active bending beam.
Using the equations of motion of cross-sectional slices of the body we can form a set of coupled
differential equations for the bending moment distribution. At large amplitude the bending moment
equations involve the tangential forces acting on the body (which may be neglected in the small
amplitude version). Consequently we include the boundary layer along the fish in order to estimate
the viscous drag directly.
The panel method has been used successfully for the fluid mechanical calculations associated with
large-amplitude fish swimming. We are able to use its results as input to calculate the bending
moment distribution. The boundary layer calculations are based on a crude model; solutions to the
large amplitude bending moment equations should also be considered in this light.