Vortex Dynamics Simulator
Vortex Dynamics Simulator models 2D fluid vortices past a barrier, visualizing motion and circulation.
Formulas Used in Vortex Dynamics Simulator
The simulator models the motion of 2D point vortices past a barrier:
Vortex Motion (Inviscid):
\\[ \frac{dx_i}{dt} = u_i = \frac{1}{2\pi} \sum_{j \neq i} \Gamma_j \frac{y_i – y_j}{(x_i – x_j)^2 + (y_i – y_j)^2} \\] \\[ \frac{dy_i}{dt} = v_i = -\frac{1}{2\pi} \sum_{j \neq i} \Gamma_j \frac{x_i – x_j}{(x_i – x_j)^2 + (y_i – y_j)^2} \\]Circulation Evolution (Viscous):
\\[ \Gamma_i(t) = \Gamma_i(0) e^{-\nu t / L^2} \\]Where:
- \\(x_i, y_i\\): Position of vortex \\(i\\) (m)
- \\(u_i, v_i\\): Velocity of vortex \\(i\\) (m/s)
- \\(\Gamma_i\\): Circulation of vortex \\(i\\) (m²/s)
- \\(\nu\\): Kinematic viscosity (m²/s)
- \\(L\\): Barrier length (m)
- \\(t\\): Time (s)
Example Calculation
Example: \\(u_0 = 1 \, \text{m/s}, \nu = 0.01 \, \text{m}^2/\text{s}, L = 0.1 \, \text{m}, N = 5\\)
Vortices initialized behind barrier with \\(\Gamma_i \approx 0.05 \, \text{m}^2/\text{s}\\). Motion computed via point vortex equations, with viscous decay.