On an adaptation of the Reynolds number, applicable to body-caudal-fin aquatic locomotion

The Reynolds number, which describes the relative importance of viscous and inertial contributions is commonly used to analyze forces on fish and other aquatic animals. However, this number is based on steady, time-independent conditions, while all swimming motions have a periodic component. Here we apply periodic flow conditions to define a new non-dimensional group, which we name the “Periodic Swimming Number, P”, which rectifies this lacuna. This new non-dimensional number embodies the periodic motion and eliminates the arbitrariness of choosing a length scale in the Reynolds number for Body –Caudal-Fin (BCF) swimming. We show that the new number has the advantage of compressing known data on fish swimming to two orders of magnitude, vs. over six required when using the existing Reynolds number and can point to a new comparison of swimming effectiveness for swimming modes.