Vorticity with velocity vectors

A Taylor-Green vortex array, the standard closed-form test case for incompressible flow solvers. The mesh carries scalar vorticity; a thinned quiver overlays the velocity field that produced it, so direction and rotation are readable together.

Two details make the pairing work. The color limits are set symmetrically about zero (vmin=-lim, vmax=+lim) so the diverging colormap puts irrotational flow at its neutral midpoint rather than wherever the data happens to straddle. And the arrows are subsampled by slicing the grid – one arrow per 14 cells – because a vector per cell would bury the field it is meant to annotate.

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import numpy as np
import polars as pl
import plotpress

g = np.linspace(0.0, 2.0 * np.pi, 260)
X, Y = np.meshgrid(g, g)

U = np.sin(X) * np.cos(Y)                 # velocity components
V = -np.cos(X) * np.sin(Y)
vorticity = 2.0 * np.sin(X) * np.sin(Y)   # curl of (U, V)

# One row per grid node -- the shape a flow solver's own field export is in,
# before it is gridded for the mesh and quiver.
field = pl.DataFrame({
    "x": X.ravel(), "y": Y.ravel(),
    "u": U.ravel(), "v": V.ravel(), "vorticity": vorticity.ravel(),
}).sort(["y", "x"])
g = field["x"].unique().sort().to_numpy()
shape = (g.size, g.size)
U = field["u"].to_numpy().reshape(shape)
V = field["v"].to_numpy().reshape(shape)
vorticity = field["vorticity"].to_numpy().reshape(shape)
X, Y = np.meshgrid(g, g)

lim = float(np.abs(vorticity).max())
fig, ax = plotpress.subplots(figsize=(7.0, 6.0))
mesh = ax.pcolormesh(g, g, vorticity, cmap="coolwarm", vmin=-lim, vmax=lim)

thin = slice(None, None, 14)
ax.quiver(X[thin, thin], Y[thin, thin], U[thin, thin], V[thin, thin],
          color="#222222")

bar = fig.colorbar(mesh, ax=ax)
bar.set_title("omega")
ax.set_aspect("equal")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_title("Taylor-Green vorticity with velocity vectors")
fig.tight_layout()