Added mass effect on flapping foil

Unsteady effects caused by accelerating bodies in water play a very important role in biological propulsion. However, such propulsion mechanisms are challenging for simulation as both body geometry and locomotion patterns are quite complex and a natural first step is to model the motion of a standard man-made airfoil. The work presents simulations of harmonic oscillations of a NACA 0012 foil in water and the hydrodynamic forces generated were obtained by a Boundary Element Method (panel method) code. The focus is placed on one of the most important unsteady effects, the added mass effect, which has not been sufficiently addressed in the literature. The corresponding unsteady forces were obtained through appropriately devised two-dimensional added mass tensor. The computational results were compared to existing analytical ones and a maximum error of 10−6 was obtained for the added mass coefficients of the circle of unit diameter. The development of a dedicated numerical approach for the calculation of the added mass tensor is necessitated by the lack of analytical solution for a variety of wing shapes such as NACA foils. The simulations showed that for the range of investigated parameters the inertia thrust and lift generated by the flapping foil increase sharply when the added mass contribution is considered. For example, if the Strouhal number is set to 0.3 and the ratio between the wing and fluid densities to 0.3, the time average of the inertia thrust increases by 23 times and the maximum of the inertia lift is ca. 37 times larger when the added mass effect is considered. Generally, a densities' ratio of order 1 results in an increase of the time-average inertia thrust of order 10. It was confirmed that, as the densities' ratio becomes larger, the contribution of the added mass to the generated inertia forces decreased. As the Strouhal number increases, the added mass effect was found to be more dominant due to the imposed motion kinematics, i.e. the pitch amplitude. The obtained results show clearly that for the specific case of flapping flight in dense fluids the unsteady effects caused by the object acceleration are of prime importance for two reasons: (i) accurate estimate of the generated thrust and (ii) realistic assessment of the resulting structural loads.