A kinematic model of Kármán gaiting in rainbow trout

A mechanistic understanding of how fishes swim in unsteady flows is challenging despite its prevalence in nature. Previous kinematic studies of fish Kármán gaiting in a vortex street behind a cylinder only report time-averaged measurements, precluding our ability to formally describe motions on a cycle-by-cycle basis. Here we present the first analytical model that describes the swimming kinematics of Kármán gaiting trout with 70-90% accuracy. We found that body bending kinematics can be modelled with a travelling wave equation, which has also been shown to accurately model free-stream swimming kinematics. However, free-stream swimming and Kármán gaiting are separated in the parameter space; the amplitude, wavelength and frequency values of the traveling wave equation are substantially different for each behavior. During Kármán gaiting, the wave is initiated at the body center, which is 0.2L (where L is total body length) further down the body compared with the initiation point in free-stream swimming. The wave travels with a constant speed, which is higher than the nominal flow speed just as in free-stream swimming. In addition to undulation, we observed that Kármán gaiting fish also exhibit substantial lateral translations and body rotations, which can constitute up to 75% of the behavior. These motions are periodic and their frequencies also match the vortex shedding frequency. There is an inverse correlation between head angle and body angle: when the body rotates in one direction, the head of the fish turns into the opposite direction. Our kinematic model mathematically describes how fish swim in vortical flows in real time and provides a platform to better understand the effects of flow variations as well as the contribution of muscle activity during corrective motions.