Leading-edge vortex dynamics and impulse-based lift force analysis of oscillating airfoils

The vortex dynamics and lift force generated by a sinusoidally heaving and pitching airfoil during dynamic stall are experimentally investigated for reduced frequencies of $k=fc/U_{\infty }=0.06-0.16$, pitching amplitude of ${\theta }_0={75}^{\circ }$ and heaving amplitude of $h_0/c=0.6$. The lift force is calculated from the velocity fields using the finite-domain impulse theory. The concept of moment-arm dilemma associated with the impulse equation is revisited to shed light on its physical impact on the calculated forces. It is shown that by selecting an objectively defined origin of the moment-arm, the impulse force equation can be greatly simplified to two terms that have a clear physical meaning: (1) the time rate of change of impulse of vortical structures within the control volume and (2) Lamb vector that indirectly captures the contribution of vortical structures outside of the control volume. The results show that the trend of the lift force is dependent on the formation of the leading-edge vortex, as well as its time rate of change of circulation and chord-wise advection relative to the airfoil. Additionally, the trailing-edge vortex, which is observed to only form for $k\le 0.10$, is shown to have lift-diminishing effects that intensify with increasing reduced frequency. Lastly, the concept of optimal vortex formation is investigated. The leading-edge vortex is shown to attain the optimal formation number of approximately 4 for $k\le 0.1$, when the scaling is based on the leading-edge shear velocity. For larger values of k the vortex growth is delayed to later in the cycle and does not reach its optimal value. The result is that the peak lift force occurs later in the cycle. This has consequences on power production which relies on correlation of the relative timing of lift force and heaving velocity.