A variational approach for the dynamics of unsteady point vortices

A Lagrangian formulation for the dynamics of unsteady point vortices is introduced and implemented. The proposed Lagrangian is related to previously constructed Lagrangian of point vortices via a gauge-symmetry in the case of vortices of constant strengths; i.e., they yield the exact same dynamics. However, a different dynamics is obtained in the case of unsteady point vortices. The resulting Euler-Lagrange equation derived from the principle of least action exactly matches the Brown-Michael evolution equation for unsteady point vortices, which was derived from a completely different point of view; based on conservation of linear momentum. The proposed Lagrangian allows for applying Galerkin techniques to the weak formulation of the vortex dynamics. The resulting dynamic model of time-varying vortices is applied to the problem of an impulsively started flat plate as well as an accelerating and pitching flat plate. In each case, the resulting lift coefficient using the dynamics of the proposed Lagrangian is compared to that using previously constructed Lagrangian, other models in literature, and experimental data.