Vector field streamlines

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Streamlines of the flow around a vortex pair, coloured by speed.

A counter-rotating vortex pair of circulation $\pm\Gamma$ at $(\pm a, 0)$ induces the velocity

$$\mathbf{u}(\mathbf{x}) = \frac{\Gamma}{2\pi}\sum_{i=\pm} \pm\frac{\hat{\mathbf{z}} \times (\mathbf{x} - \mathbf{x}_i)}{|\mathbf{x} - \mathbf{x}_i|^2}.$$

This example uses matplotlib, so the page shows a PNG.

Source code

vector_field.py

import matplotlib.pyplot as plt
import numpy as np

gamma, a = 2.0 * np.pi, 1.0
x, y = np.meshgrid(np.linspace(-4.0, 4.0, 200), np.linspace(-3.0, 3.0, 150))

u = np.zeros_like(x)
v = np.zeros_like(x)
for sign, x0 in [(+1.0, -a), (-1.0, a)]:
    dx, dy = x - x0, y
    r2 = np.maximum(dx**2 + dy**2, 1e-3)
    u += sign * gamma / (2.0 * np.pi) * (-dy) / r2
    v += sign * gamma / (2.0 * np.pi) * dx / r2

speed = np.hypot(u, v)
fig, ax = plt.subplots(figsize=(8, 6))
lines = ax.streamplot(
    x, y, u, v, color=np.minimum(speed, 3.0), cmap="viridis", density=1.8, linewidth=1.2
)
fig.colorbar(lines.lines, ax=ax, label="speed")
ax.plot([-a, a], [0, 0], "o", color="crimson")
ax.set(xlabel="x", ylabel="y", title="Counter-rotating vortex pair", aspect="equal")
fig.tight_layout()
plt.show()