File:Stream function.png
Summary
| Description |
English: The w:stream function (below) can be used to make streamlines in velocity fields (red, above). |
| Date | |
| Source | Own work |
| Author | Nanite |
| PNG development |
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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Source code
Python source code.
#!/usr/bin/python3
"""
Make a stream function for a vector field.
We are going to cheat a bit for simplicity - make the stream function first
then derive vector field from it.
"""
import numpy as np
from matplotlib import pyplot as plt
# make a 1001 x 1001 grid
x = np.linspace(0,10,1001)
y = np.linspace(0,10,1001)[:,None]
dx = x[1]-x[0]
dy = y[1,0] - y[0,0]
extent = (x[0] - 0.5*dx, x[-1] + 0.5*dx, y[0,0] - 0.5*dy, y[-1,0] + 0.5*dy)
def vort(x0,y0,r):
return -np.log(np.sqrt((x - x0)**2 + (y - y0)**2 + r**2))
# construct some stream function with
# overall background flow in x direction
streamfunc = 0*x + 0.15*y
# put an irrotational vortex (singular) with negative vorticity
streamfunc -= 0.50 * vort(3.5,4.5,0)
# make a spread-out vortex with positive vorticity
streamfunc += 0.40 * vort(7,6,0.5)
# Create a hole in the domain by cutting off the singular vortex
streamfunc[(streamfunc < (+0.0)) & (y>2)] = np.nan
streammin = np.nanmin(streamfunc)
streammax = np.nanmax(streamfunc)
#calculate velocities; just forward difference for laziness.
ux = np.diff(streamfunc,axis=0) / dy
uy = np.diff(streamfunc,axis=1) / -dx
Nlevels = 9
contour_levels = (np.arange(0.5,(Nlevels+1))/(Nlevels+1)) * (streammax - streammin) + streammin
fig = plt.figure(figsize=(2,4), dpi=300)
# bottom axes: 3D plot
ax = fig.add_axes((0,0,1,0.5),projection = '3d', computed_zorder=False)
ax.view_init(30, -60, 0)
ax.set_xticks([])
ax.set_yticks([])
ax.set_zticks([])
ax.plot_surface(x, y, streamfunc, cmap='RdYlGn',
linewidth=0, antialiased=False,
)
ax.contour(x,y[:,0],streamfunc, levels=contour_levels, colors='k', linestyles='--', linewidths=0.5, zorder=1000)
ax = plt.axes((0,0.5,0.997,0.5))
ax.set_xticks([])
ax.set_yticks([])
# draw the hole in the domain as gray
plt.imshow(np.isnan(streamfunc), origin='lower', extent=extent, cmap='gray_r', vmin=0,vmax=4)
# plot every 67th velocity arrow
slicer = slice(33,None,67)
plt.quiver(x[slicer], y[slicer,:], ux[slicer,slicer], uy[slicer,slicer], scale=8., width=0.015, color='red')
plt.contour(x,y[:,0],streamfunc, levels=contour_levels, colors='k', linestyles='--', linewidths=0.75)
fig.savefig('stream function.png')
fig = plt.figure(figsize=(2,2), dpi=300)
ax = plt.axes((0,0,0.997,1))
plt.imshow(streamfunc, extent=extent, origin='lower')
plt.contour(x,y[:,0],streamfunc, levels=contour_levels, colors='k', linestyles='--', linewidths=0.75)
fig.savefig('stream function imshow.png')