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vortices.py
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vortices.py
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"""
Description
@author: Mattias Lazda
@collab: Jules Faucher, David Tucci
Feb 4, 2022
Simulation of leapfrog vortices.
"""
import numpy as np
import matplotlib.pyplot as pl
# Set number of steps and time step
dt = 1
Nsteps = 100
# Setting up initial conditions
y_v = np.array([5,5,-5,-5])
x_v = np.array([-25,-17,-25,-17])
k_v = np.array([1,1,-1,-1])
# Setting up the plot
pl.ion()
fig, ax = pl.subplots(1,1, figsize=(10,10))
# Mark the initial positions of the vortices
p, = ax.plot(x_v, y_v, 'k+', markersize=20, linewidth=5)
# Draw the initial velocity streamline
ngrid = 30
Y,X = np.mgrid[-ngrid//2:ngrid//2:360j, -ngrid:ngrid:360j]
vel_x = np.zeros(np.shape(X)) # this holds x-velocities
vel_y = np.zeros(np.shape(Y)) # this holds y-velocities
# Masking radius for better visualization of the vortex centres
r_mask = 1
for i in range(len(x_v)):
# Get x and y distance from source
x_diff = X - x_v[i]
y_diff = Y - y_v[i]
# Calculate r
r_sqr = np.power(x_diff,2) + np.power(y_diff, 2)
# Compute x and y velocities due to vortex located at x_i, y_i
vel_x_temp = k_v[i] * (-y_diff) / r_sqr
vel_y_temp = k_v[i] * (x_diff) / r_sqr
# Add to the total velocity field
vel_x += vel_x_temp
vel_y += vel_y_temp
# Mask out region around vortex
vel_x[r_sqr <= r_mask**2 ] = np.nan
vel_y[r_sqr <= r_mask**2 ] = np.nan
# Set boundaries of the simulation box
ax.set_xlim([-ngrid, ngrid])
ax.set_ylim([-ngrid//2, ngrid//2])
# Initial plot of the streamlines
ax.streamplot(X, Y, vel_x, vel_y, density = [1,1], color = 'firebrick')
pl.title('Starting Position')
fig.canvas.draw()
# Evolution
count = 0
while count < Nsteps:
# Re-initialize velocity field
vel_x = np.zeros(np.shape(X))
vel_y = np.zeros(np.shape(Y))
# Initialize advection velocities
adv_vel_x = np.zeros(len(x_v))
adv_vel_y = np.zeros(len(x_v))
indices = np.arange(len(x_v))
for i in indices:
# Remove current index to avoid calculating advection velocity due to itself
index_use = indices[indices!=i]
# Initialize advection velocities
adv_vel_x_i = 0
adv_vel_y_i = 0
for j in index_use:
x_diff_j = x_v[i] - x_v[j]
y_diff_j = y_v[i] - y_v[j]
r_sqr = np.power(x_diff_j,2) + np.power(y_diff_j, 2)
# Compute x and y velocities due to vortex located at x_i, y_i
adv_vel_x[i] += k_v[j] * (-y_diff_j) / r_sqr
adv_vel_y[i] += k_v[j] * (x_diff_j) / r_sqr
# Update positions of vortices
x_v = x_v + dt*adv_vel_x
y_v = y_v + dt*adv_vel_y
# Get updated velocity field
for i in range(len(x_v)):
# Get x and y distance from source
x_diff = X - x_v[i]
y_diff = Y - y_v[i]
# Calculate r
r_sqr = np.power(x_diff,2) + np.power(y_diff, 2)
# Compute x and y velocities due to vortex located at x_i, y_i
vel_x_temp = k_v[i] * (-y_diff) / r_sqr
vel_y_temp = k_v[i] * (x_diff) / r_sqr
# Add to the total velocity field
vel_x += vel_x_temp
vel_y += vel_y_temp
# Determine indices of range around the origin
vel_x[r_sqr <= r_mask**2 ] = np.nan
vel_y[r_sqr <= r_mask**2 ] = np.nan
# Update plot of the streamlines
# Clear previous streamlines
ax.collections = []
ax.patches = []
p.set_xdata(x_v)
p.set_ydata(y_v)
ax.streamplot(X,Y,vel_x,vel_y, density = [1,2], color = 'firebrick')
pl.title('Time from beginning, {0} s'.format(count*dt))
fig.canvas.draw()
pl.pause(0.0001)
count+=1