Scaling laws for propulsive performance of pitching foils with non-uniform flexibility*

Aquatic animals swim efficiently in different environments, using a combination of flexible tails and appendages with non-uniform flexibility. Using a simplified pure pitching foil with non-uniform flexibility, we develop a model based on the scaling of the added mass and circulatory forces. The classic linear theory is augmented by additional nonlinearities and modified for non-uniform flexibility by considering the effective flexibility of the foil.  The scaling relation then is verified using inviscid numerical simulations and experiments over a wide range of variables, including the dimensionless amplitude, dimensionless flexion ratio, reduced frequency, and Strouhal number. The developed relations were found to be in excellent agreement with the numerical and experimental data. These scaling laws are used to identify physical mechanisms that influence the thrust generation and efficiency of a non-uniformly flexible propulsor and can be used as a guide for improving performance. These results are relevant in the development of bio-inspired underwater vehicles.