D3 Wake Analysis

Topic

The wake left in a fish's path is the central element of this model: it represents the structure of the disturbed fluid left behind as the fish swims, containing all the information regarding the forces the fish exerted on the fluid during the process. According to Newton's third law, the momentum the fish imparts to the wake is equal and opposite to the momentum the fluid exerts on the fish itself—specifically, the thrust driving it forward and the drag opposing its advance. Thanks to this relationship, analyzing the wake makes it possible to calculate the fish's net propulsive force, propulsive efficiency, and cost of transport without the need to instrument the fish directly; this makes the analysis a particularly valuable tool for studying fish swimming in their natural environment.

Velocity data within the wake are obtained primarily using the same particle image velocimetry technique described in the previous model, or a digital variant thereof. In a fish that relies on its body and caudal fin as the primary source of propulsion, the wake displays a succession of ring-shaped vortices; these are arranged with their axes perpendicular to the plane in which the caudal fin beats from side to side. Each half-stroke of the caudal fin generates one of these vortex rings, and the ring's hydraulic impulse—which directly determines the thrust generated during that half-stroke—depends on three factors: fluid density, the circulation of the ring itself, and the area enclosed by the ring. The circulation of each vortex ring is, in turn, measured by integrating the fluid's vorticity over the area occupied by the vortex within the same measurement plane illuminated by the laser. While the fish's thrust is derived from the vortex rings left in its wake, total drag is determined differently—based on what is known as the momentum deficit in the wake. By integrating the fluid velocity profile downstream of the fish across a plane transverse to its path, the difference between the momentum flux measured at that plane and the momentum flux of the free stream (in the absence of the fish) yields the drag force directly, following the same principle of momentum conservation applicable to any control volume.

Comparing this thrust—derived from the impulse of the vortex rings—with the drag—derived from the momentum deficit—allows for the verification of the fish's complete energy balance. It also enables the detection of discrepancies that reveal force contributions not captured by the two-dimensional measurement plane. For a fish swimming at a constant speed, this balance leads to an expected, verifiable result: the ratio of thrust (measured via vortices) to drag (measured via momentum deficit) should be close to unity. This confirms that the wake itself—as described at the start of this model—contains all the information needed to account for the forces acting on the fish, without ever having touched or instrumented its body.

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