Semi-infinite vortex trails, and their relation to oscillating airfoils

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Abstract
Semi-infinite double rows of vortices are used to study the periodic wake of both oscillating and stationary two-dimensional bodies immersed in a uniform incompressible stream. Analytical expressions for the induced Velocities on the body, for trails with constant spacing, which are valid for small values of the oscillation amplitude are presented while, for the general case of vortex shedding, an iterative procedure for the representation of trails of variable spacing is developed and used. Vortex streets due to oscillating bodies are obtained as a function of three non-dimensional parameters: the Strouhal number (initial spacing ratio), a non-dimensional vortex strength and the downstream spacing ratio. Criteria establishing when such trails are expected to widen, become narrow or stay of constant width are presented, as well as expressions for the induced velocities.The trails and their induced velocities enable the calculation of the vortex strength from measurable quantities. Thus they can serve as a method for estimating the hydrodynamic forces on the airfoil due to large amplitude oscillations, such as those observed in the propulsive movements of fish and cetaceans, as well as the small amplitude oscillations due to hydroelastic interactions.

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