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.
