File:Lamb–Oseen vortex animation.gif

#!/usr/bin/env python3

import numpy as np
from scipy.special import expi
from matplotlib import pyplot as plt
from matplotlib import collections

twopi = 2*np.pi

# kinematic viscosity m^2/s
viscosity = 1.5e-5

# fixed circulation of the lamb-onseen vortex, m^2/s
circulation = 10e-3

# radial range in which to place particles, m
rmax = 0.2
rmark = 0.003 # radius of 'marker'
# number of particles
N = 600

# radial coordinate of particles, this does not change over time
# place them with sqrt of uniform 1D-distributed so they are uniform in 2D
part_r = rmax * np.sqrt(np.linspace(0, 1, N+1)[1:])
# precalculate some things
invr2 = 1/part_r**2
neg_tprime = -part_r**2 / (4 * viscosity) # negative of transition time for this radius
# angular coordinate of particles, set them randomly.
part_th = twopi*np.random.rand(N)
# rotation of particles - start them all out as horizontal
part_rot = np.zeros((N,))

dt = 0.01
i = 0

frames = []
frametimes = []
nextframetime = 0

fig = plt.figure()
fig.set_size_inches(4,4)
ax = plt.axes((0,0,1,1))
ax.set_xlim(-0.7*rmax, +0.7*rmax)
ax.set_ylim(-0.7*rmax, +0.7*rmax)
ax.set_aspect('equal')
ax.set_axis_off()

linesegs = np.empty((2*N, 2, 2))

while True:
    i += 1
    t = i*dt
    if t > 25:
        break

    # use finite difference to increment angular coordinate and rotation of particles.
    # we oould calculate these in closed form in terms of Ei(x) function, but this
    # is easy and good enough.
    part_th += (-dt * circulation / twopi) * np.expm1(neg_tprime/t) * invr2
    # note the rotation of particle is half of the vorticity.
    part_rot += (dt * circulation / (4*twopi*viscosity * t)) * np.exp(neg_tprime/t)

    # only show every tenth frame
    if i % 10 == 0:
        frame = f"frames/frame{i}.png"
        frames.append(frame)
        frametimes.append(t)

        [c.remove() for c in ax.lines]
        [c.remove() for c in ax.collections]
        [c.remove() for c in ax.texts]
        x = part_r*np.cos(part_th)
        y = part_r*np.sin(part_th)
        cr = rmark*np.cos(part_rot)
        sr = rmark*np.sin(part_rot)
        # first line segments of the crosses
        linesegs[0::2, 0, 0] = x + cr
        linesegs[0::2, 0, 1] = y + sr
        linesegs[0::2, 1, 0] = x - cr
        linesegs[0::2, 1, 1] = y - sr
        # second line segments of the crosses - perpendicular
        # note that having them distinct length or shade makes it visually MUCH easier to see
        # distinction between the rotational core and irrotational exterior, due to distinct texture.
        linesegs[1::2, 0, 0] = x - sr
        linesegs[1::2, 0, 1] = y + cr
        linesegs[1::2, 1, 0] = x + sr
        linesegs[1::2, 1, 1] = y - cr
        linecoll = collections.LineCollection(linesegs, linewidths=1, colors=((0., 0., 0.), (0.75, 0.75, 0.75)),
                               linestyle='solid')
        ax.add_collection(linecoll)

        [c.remove() for c in fig.texts]
        fig.text(0.01,0.01,f't = {t:.02f} s', fontsize='x-large', bbox=dict(facecolor='w', alpha=1))
        fig.savefig(frame)

# gif frame times are in centiseconds
delays = list(np.diff(np.round(np.array(frametimes) * 100)))
# show the last frame for a bit extra time
delays.append(delays[-1] + 200)

print(len(frames), len(delays))
assert(len(delays) == len(frames))

### Assemble animation using ImageMagick ###
calllist = 'convert'.split()
for delay,frame in zip(delays,frames):
    calllist += ['-delay',str(int(delay))]
    calllist += [frame]
calllist += '-loop 0 -layers Optimize _animation.gif'.split()
f = open('anim_command.txt','w')
f.write(' '.join(calllist)+'\n')
f.close()

print("composing into animated gif...", end='')
import sys, subprocess, os
sys.stdout.flush()
subprocess.call(calllist)
print("      done")
os.rename('_animation.gif','animation.gif')