Extracting energy from unsteady flows through vortex control

Abstract

Vortex control is a new paradigm in fluid mechanics, with applications to propulsion and wake reduction. A heaving and pitching hydrofoil placed in a flow with an array of oncoming vortices can achieve a very high propulsive efficiency and reduced wake signature. The canonical example of flow with regular arrays of vortices is the Karman vortex street, and this is our model for the inflow to the foil. The problem of an oscillating foil placed within a Karman vortex street is investigated with a theoretical model and numerical simulation. The theoretical model is an adaptation of the classical linear theory for unsteady aerofoils. It combines the effects of nonuniform inflow and foil motion to predict the resulting thrust, lift, and moment. The numerical procedure allows for nonlinear interaction between the foil, performing large amplitude oscillations, and the oncoming vortex street. The method is based on two- dimensional potential flow and the theory of functions of a complex variable. Careful formulation of the velocity potential, and closed form expressions for force and moment on a Joukowski foil in the presence of point vortices, permits rapid evaluation of hydrodynamic performance. The theory and simulation results agree in their main conclusion: For optimum performance, the foil should try to intercept the vortices head on, while remaining inside the border of the oncoming vortex street. This mode is associated with a high degree of interaction between oppositely signed vorticity in the combined wake leading to reduced wake signature. The lowest efficiency is predicted when the foil avoids coming close to the vortices, here the combined wake consists of a row of very strong vortices of alternate sign. The theory also indicates that an oscillating foil can recover more of the energy contained in the vortex street than a stationary one, but this has not been confirmed in simulation. The interaction process in the wake is studied in more detail, using a much simplified model; the foil wake is modeled as a uniform shear layer of small but finite thickness, and an oppositely signed vortex is placed next to it to simulate the effect of one of the vortices in the Karman street. The subsequent interaction is simulated with the vortex method, assuming periodic boundary conditions. These simulations show that the shear layer rolls up and partially engulfs the vortex patch when two conditions are satisfied. The vortex must be close to the shear layer, and the circulation about the vortex and a representative segment of the shear layer must balance, such that neither one dominates the problem. In both of these simulations, a fast, O(N), vortex summation method based on multipole expansions is used, with special adaptations to account for the influence of image vortices.