On the estimation of swimming and flying forces from wake measurements

The transfer of momentum from an animal to fluid in its wake is fundamental to many swimming and flying modes of locomotion. Hence, properties of the wake are commonly studied in experiments to infer the magnitude and direction of locomotive forces. The determination of which wake properties are necessary and sufficient to empirically deduce swimming and flying forces is currently made ad hoc. This paper systematically addresses the question of the minimum number of wake properties whose combination is sufficient to determine swimming and flying forces from wake measurements. In particular, it is confirmed that the spatial velocity distribution (i.e. the velocity field) in the wake is by itself insufficient to determine swimming and flying forces, and must be combined with the fluid pressure distribution. Importantly, it is also shown that the spatial distribution of rotation and shear (i.e. the vorticity field) in the wake is by itself insufficient to determine swimming and flying forces, and must be combined with a parameter that is analogous to the fluid pressure. The measurement of this parameter in the wake is shown to be identical to a calculation of the added-mass contribution from fluid surrounding vortices in the wake, and proceeds identically to a measurement of the added-mass traditionally associated with fluid surrounding solid bodies. It is demonstrated that the velocity/pressure perspective is equivalent to the vorticity/vortex-added-mass approach in the equations of motion. A model is developed to approximate the contribution of wake vortex added-mass to locomotive forces, given a combination of velocity and vorticity field measurements in the wake. A dimensionless parameter, the wake vortex ratio (denoted Wa), is introduced to predict the types of wake flows for which the contribution of forces due to wake vortex added-mass will become non-negligible. Previous wake analyses are re-examined in light of this parameter to infer the existence and importance of wake vortex added-mass in those cases. In the process, it is demonstrated that the commonly used time-averaged force estimates based on wake measurements are not sufficient to prove that an animal is generating the locomotive forces necessary to sustain flight or maintain neutral buoyancy.