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fluid_sim.py
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#!/usr/bin/env python3
# LatticeBoltzmannDemo.py: a two-dimensional lattice-Boltzmann "wind tunnel"
# simulation
# Uses numpy to speed up all array handling.
# Uses matplotlib to plot and animate the curl of the macroscopic velocity
# field.
# Copyright 2013, Daniel V. Schroeder (Weber State University) 2013
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of
# this software and associated data and documentation (the "Software"), to
# deal in
# the Software without restriction, including without limitation the rights to
# use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies
# of the Software, and to permit persons to whom the Software is furnished to
# do
# so, subject to the following conditions:
# The above copyright notice and this permission notice shall be included in
# all
# copies or substantial portions of the Software.
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED,
# INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A
# PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHOR BE
# LIABLE FOR
# ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
# TORT OR
# OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE
# USE OR
# OTHER DEALINGS IN THE SOFTWARE.
# Except as contained in this notice, the name of the author shall not be used
# in
# advertising or otherwise to promote the sale, use or other dealings in this
# Software without prior written authorization.
# Credits:
# The "wind tunnel" entry/exit conditions are inspired by Graham Pullan's code
# (http://www.many-core.group.cam.ac.uk/projects/LBdemo.shtml). Additional
# inspiration from
# Thomas Pohl's applet (http://thomas-pohl.info/work/lba.html). Other
# portions of code are based
# on Wagner
# (http://www.ndsu.edu/physics/people/faculty/wagner/lattice_boltzmann_codes/)
# and Gonsalves
# (http://www.physics.buffalo.edu/phy411-506-2004/index.html; code adapted
# from Succi,
# http://global.oup.com/academic/product/the-lattice-boltzmann-equation-9780199679249).
# For related materials see http://physics.weber.edu/schroeder/fluids
import time
import matplotlib.animation
import matplotlib.pyplot
import numpy
# Define constants:
height = 80 # lattice dimensions
width = 200
viscosity = 0.02 # fluid viscosity
omega = 1 / (3 * viscosity + 0.5) # "relaxation" parameter
u0 = 0.1 # initial and in-flow speed
four9ths = 4.0 / 9.0 # abbreviations for lattice-Boltzmann weight factors
one9th = 1.0 / 9.0
one36th = 1.0 / 36.0
performanceData = False # set to True if performance data is desired
# Initialize all the arrays to steady rightward flow:
# particle densities along 9 directions
n0 = four9ths * (numpy.ones((height, width)) - 1.5 * u0**2)
nN = one9th * (numpy.ones((height, width)) - 1.5 * u0**2)
nS = one9th * (numpy.ones((height, width)) - 1.5 * u0**2)
nE = one9th * (numpy.ones((height, width)) + 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
nW = one9th * (numpy.ones((height, width)) - 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
nNE = one36th * (numpy.ones((height, width)) + 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
nSE = one36th * (numpy.ones((height, width)) + 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
nNW = one36th * (numpy.ones((height, width)) - 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
nSW = one36th * (numpy.ones((height, width)) - 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
rho = n0 + nN + nS + nE + nW + nNE + nSE + nNW + nSW # macroscopic density
ux = (nE + nNE + nSE - nW - nNW - nSW) / rho # macroscopic x velocity
uy = (nN + nNE + nNW - nS - nSE - nSW) / rho # macroscopic y velocity
# Initialize barriers:
# True wherever there's a barrier
barrier = numpy.zeros((height, width), bool)
barrier[int(height / 3) - 8 : int(height / 3) + 8, int(height / 3)] = (
True # simple linear barrier
)
barrierN = numpy.roll(barrier, 1, axis=0) # sites just north of barriers
barrierS = numpy.roll(barrier, -1, axis=0) # sites just south of barriers
barrierE = numpy.roll(barrier, 1, axis=1) # etc.
barrierW = numpy.roll(barrier, -1, axis=1)
barrierNE = numpy.roll(barrierN, 1, axis=1)
barrierNW = numpy.roll(barrierN, -1, axis=1)
barrierSE = numpy.roll(barrierS, 1, axis=1)
barrierSW = numpy.roll(barrierS, -1, axis=1)
# Move all particles by one step along their directions of motion (pbc):
def stream():
global nN, nS, nE, nW, nNE, nNW, nSE, nSW
# axis 0 is north-south; + direction is north
nN = numpy.roll(nN, 1, axis=0)
nNE = numpy.roll(nNE, 1, axis=0)
nNW = numpy.roll(nNW, 1, axis=0)
nS = numpy.roll(nS, -1, axis=0)
nSE = numpy.roll(nSE, -1, axis=0)
nSW = numpy.roll(nSW, -1, axis=0)
# axis 1 is east-west; + direction is east
nE = numpy.roll(nE, 1, axis=1)
nNE = numpy.roll(nNE, 1, axis=1)
nSE = numpy.roll(nSE, 1, axis=1)
nW = numpy.roll(nW, -1, axis=1)
nNW = numpy.roll(nNW, -1, axis=1)
nSW = numpy.roll(nSW, -1, axis=1)
# Use tricky boolean arrays to handle barrier collisions (bounce-back):
nN[barrierN] = nS[barrier]
nS[barrierS] = nN[barrier]
nE[barrierE] = nW[barrier]
nW[barrierW] = nE[barrier]
nNE[barrierNE] = nSW[barrier]
nNW[barrierNW] = nSE[barrier]
nSE[barrierSE] = nNW[barrier]
nSW[barrierSW] = nNE[barrier]
# Collide particles within each cell to redistribute velocities (could be
# optimized a little more):
def collide():
global rho, ux, uy, n0, nN, nS, nE, nW, nNE, nNW, nSE, nSW
rho = n0 + nN + nS + nE + nW + nNE + nSE + nNW + nSW
ux = (nE + nNE + nSE - nW - nNW - nSW) / rho
uy = (nN + nNE + nNW - nS - nSE - nSW) / rho
ux2 = ux * ux # pre-compute terms used repeatedly...
uy2 = uy * uy
u2 = ux2 + uy2
omu215 = 1 - 1.5 * u2 # "one minus u2 times 1.5"
uxuy = ux * uy
n0 = (1 - omega) * n0 + omega * four9ths * rho * omu215
nN = (1 - omega) * nN + omega * one9th * rho * (omu215 + 3 * uy + 4.5 * uy2)
nS = (1 - omega) * nS + omega * one9th * rho * (omu215 - 3 * uy + 4.5 * uy2)
nE = (1 - omega) * nE + omega * one9th * rho * (omu215 + 3 * ux + 4.5 * ux2)
nW = (1 - omega) * nW + omega * one9th * rho * (omu215 - 3 * ux + 4.5 * ux2)
nNE = (1 - omega) * nNE + omega * one36th * rho * (
omu215 + 3 * (ux + uy) + 4.5 * (u2 + 2 * uxuy)
)
nNW = (1 - omega) * nNW + omega * one36th * rho * (
omu215 + 3 * (-ux + uy) + 4.5 * (u2 - 2 * uxuy)
)
nSE = (1 - omega) * nSE + omega * one36th * rho * (
omu215 + 3 * (ux - uy) + 4.5 * (u2 - 2 * uxuy)
)
nSW = (1 - omega) * nSW + omega * one36th * rho * (
omu215 + 3 * (-ux - uy) + 4.5 * (u2 + 2 * uxuy)
)
# Force steady rightward flow at ends (no need to set 0, N, and S
# components):
nE[:, 0] = one9th * (1 + 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
nW[:, 0] = one9th * (1 - 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
nNE[:, 0] = one36th * (1 + 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
nSE[:, 0] = one36th * (1 + 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
nNW[:, 0] = one36th * (1 - 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
nSW[:, 0] = one36th * (1 - 3 * u0 + 4.5 * u0**2 - 1.5 * u0**2)
# Compute curl of the macroscopic velocity field:
def curl(ux, uy):
return (
numpy.roll(uy, -1, axis=1)
- numpy.roll(uy, 1, axis=1)
- numpy.roll(ux, -1, axis=0)
+ numpy.roll(ux, 1, axis=0)
)
# Here comes the graphics and animation...
theFig = matplotlib.pyplot.figure(figsize=(8, 3))
fluidImage = matplotlib.pyplot.imshow(
curl(ux, uy),
origin="lower",
norm=matplotlib.pyplot.Normalize(-0.1, 0.1),
cmap=matplotlib.pyplot.get_cmap("jet"),
interpolation="none",
)
# See http://www.loria.fr/~rougier/teaching/matplotlib/#colormaps for
# other cmap options
bImageArray = numpy.zeros((height, width, 4), numpy.uint8) # an RGBA image
bImageArray[barrier, 3] = 255 # set alpha=255 only at barrier sites
barrierImage = matplotlib.pyplot.imshow(
bImageArray, origin="lower", interpolation="none"
)
# Function called for each successive animation frame:
# startTime = time.
# frameList = open('frameList.txt','w') # file containing list
# of images (to make movie)
def nextFrame(arg): # (arg is the frame number, which we don't need)
# global startTime
# if performanceData and (arg % 100 == 0) and (arg > 0):
# endTime = time.clock()
# print("%1.1f" % (100 / (endTime - startTime)), 'frames per second')
# startTime = endTime
# frameName = "frame%04d.png" % arg
# matplotlib.pyplot.savefig(frameName)
# frameList.write(frameName + '\n')
for step in range(20): # adjust number of steps for smooth animation
stream()
collide()
fluidImage.set_array(curl(ux, uy))
return (fluidImage, barrierImage) # return the figure elements to redraw
animate = matplotlib.animation.FuncAnimation(theFig, nextFrame, interval=1, blit=True)
matplotlib.pyplot.show()