Vortex dynamics: A variational approach using the principle of least action

The study of vortex dynamics using a variational formulation has an extensive history and a rich literature. The standard Hamiltonian function that describes the dynamics of interacting point vortices of constant strength is the Kirchhoff-Routh (KR) function. This function was not obtained from basic definitions of classical mechanics (i.e., in terms of kinetic and potential energies), but it was rather devised to match the already known differential equations of motion for constant-strength point vortices given by the Bio-Savart law. Instead, we develop a variational formulation for vortex dynamics based on the principle of least action. As an application, we consider two-dimensional massive vortices of constant strength. Interestingly, the obtained equations of motion are second-order differential equations defining vortex accelerations, not velocities. The resulting dynamics are more complex than those obtained from the KR formulation. For example, a pair of equal-strength, counter-rotating vortices could be initialized with different velocities, resulting in interesting patterns. Also, the developed model easily admits external body forces. When an electrodynamic force is considered, the interaction between it and the hydrodynamic vortex force leads to a rich, counterintuitive behavior that could not be handled by the KR formulation.

Figure 1

Schematic drawing for the problem formulation; $c_i$ is the boundary of the $i\mathrm{th}$ vortex in solid blue, ${\sigma }_i$ is the corresponding branch cut (barrier) in dashed red, and $R$ is the boundary of flow domain in black.

Figure 2

Simulation of proposed variational vortex dynamics model against the KR model for a pair of vortices. The initial velocities are set to five multiples of the Biot-Savart induced velocity in the same direction.

Figure 3

Simulation of the variational vortex dynamics model presenting the deviation of a counter-rotating vortex pair from the KR trajectory. Initial velocities for the vortices are $u_{1_0}=(10i,-0.5j)$ and $u_{2_0}=(0i,-0.5j)$, respectively.

Figure 4

Sensitivity study for different $a/b$ ratios. Trajectories for the right-hand side vortex are presented. The scale is adjusted to emphasize the differences between each of the cases.

Figure 5

Different simulations for counter-rotating pair of vortices with same charge placed in a constant-strength electric field. Simulations are initialized by the Biot-Savart induced velocity and electric field direction for each case is listed as (a) $\mathit{E}\downarrow$, (b) $\mathit{E}\uparrow$, and (c) $\mathit{E}\to$.

Figure 6

Different simulations for counter-rotating pair of vortices with opposite charge placed in a constant-strength electric field. Simulations are initialized by the Biot-Savart induced velocity; electric field direction and charge sign for each case are listed as (a) $\mathit{E}\uparrow$, $+q_1$, and $-q_2$, (b) $\mathit{E}\to$, $+q_1$, and $-q_2$, and (c) $\mathit{E}\to$, $-q_1$, and $+q_2$.

Figure 7

Comparison between the proposed variational vortex dynamics model and the model of Ragazzo et al. [33] when vortices are initialized with a velocity different from the Biot-Savart induced velocity. (a) Simulation of a counter-rotating pair initialized with five multiples of the Biot-Savart induced velocity, showing only one vortex. (b) Counter-rotating pair simulated as a single vortex in the half plane [33].

Figure 8

Comparison between the proposed variational vortex dynamics model and Richaud et al. [36] model. Vortices are initialized with a velocity different from the Biot-Savart induced velocity. (a) Simulation of co-rotating pair initialized with $u_{1_0}=(0i,5.83j)$ and $u_{2_0}=(0i,4.16j)$, showing only the first vortex on the right-hand side. (b) Single vortex simulated in a confined circular domain [36].

Figure 9

Comparison between the proposed variational vortex dynamics model and Richaud et al. [36] model when vortices are initialized with a velocity different from the Biot-Savart induced velocity. (a) Simulation of co-rotating pair initialized with $u_{1_0}=(-0.5i,0.83j)$ and $u_{2_0}=(0.5i,-0.83j)$. (b) Co-rotating vortices initially perturbed with a radial velocity in the inward direction [36].