Conformal Mapping Model of Vortex Interaction Forces on Foils of Realistic Shape

An analytical modeling approach is developed to compute the unsteady force on a foil, with any specified chordwise thickness profile, due to a passing point vortex. The model uses a generalized Joukowsky mapping technique to map the flow about a circular cylinder to a foil of any specified shape. An expression based on the unsteady Blasius force equation is developed to compute the force response of the foil due to individual, frozen vortices that drift by the foil along mean flow streamlines. Calculations of lift force spectra for NACA 0007-61 foil shapes show 6 and 10 decibels of reduction at reduced frequencies, based on half-chord and freestream speed, of 30 and 50, respectively, relative to a NACA 0007-65 foil shape, thus highlighting the importance of foil leading-edge shape in high-frequency lift response to incident vortices. The lift response for reduced frequencies above 10 is then approximated by the vortex inertia force, and normalization of this high-frequency force term by the thin foil response shows a frequency-dependent exponential relationship that can be merged with thin foil lift response theory to compute the lift spectrum over the broad reduced frequency range between 0.1 and 50.</p>