Vortex Blinking Simulation in Hamiltonian Chaos

Formulation

1. Stream Function (Ψ): In potential flow theory, the stream function is a scalar function whose contours represent the flow lines of the fluid. For a vortex, the stream function Ψ at a point (x, y) in the plane is given by: $$ Ψ(x, y) = \frac{Γ}{2π} \ln\sqrt{(x - x_0)^2 + (y - y_0)^2} $$ Here, $Γ$ is the circulation of the vortex, and $(x_0, y_0)$ is the position of the vortex center. The circulation $Γ$ determines the strength and the rotational direction of the vortex.

2. Velocity Field: The velocity components (u, v) at any point (x, y) in the flow field can be derived from the stream function. They are given by the partial derivatives of Ψ: $$ u(x, y) = \frac{\partial Ψ}{\partial y}, \quad v(x, y) = -\frac{\partial Ψ}{\partial x} $$ Substituting the expression for Ψ and simplifying, we get: $$ u(x, y) = -\frac{Γ}{2π} \frac{y - y_0}{(x - x_0)^2 + (y - y_0)^2}, \quad v(x, y) = \frac{Γ}{2π} \frac{x - x_0}{(x - x_0)^2 + (y - y_0)^2} $$ These expressions describe the velocity field around the vortex.

3. Regularization with Delta (δ): In numerical simulations, a small parameter δ is often added to the denominator to avoid the singularity at the vortex center: $$ u(x, y) = -\frac{Γ}{2π} \frac{y - y_0}{(x - x_0)^2 + (y - y_0)^2 + δ^2}, \quad v(x, y) = \frac{Γ}{2π} \frac{x - x_0}{(x - x_0)^2 + (y - y_0)^2 + δ^2} $$ This δ term helps to avoid numerical issues when the point (x, y) is very close to the vortex center (x_0, y_0).

Parameter Overview

• delta: A small numerical value to avoid singularities
• vortex_frequency: Controls how frequently the active vortex switches
• orientation: Sets the initial orientation of vortex interaction