In abundant spare time ⏳, yours truly has implemented the non-conservative and non-dimensional form of the discretized Navier-Stokes 🍃 equations. The code 🖳 in it's simplest form is less than 50 lines including importing libraries and plotting! 😲 For validation, refer here. More examples and free code is available here, here and here. Happy codding!
Code
# Copyright <2024> <FAHAD BUTT>
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import numpy as np
import matplotlib.pyplot as plt
l_square = 1 # length of square
h = l_square / 50 # grid spacing
dt = 0.001 # time step
L = 1 # domain length
D = 1 # domain depth
Nx = round(L / h) + 1 # grid points in x-axis
Ny = round(D / h) + 1 # grid points in y-axis
nu = 1 / 100 # kinematic viscosity
Uinf = 1 # free stream velocity / inlet velocity / lid velocity
cfl = dt * Uinf / h # cfl number
travel = 20 # times the disturbance travels entire length of computational domain
TT = travel * L / Uinf # total time
ns = int(TT / dt) # number of time steps
Re = round(l_square * Uinf / nu) # Reynolds number
u = np.zeros((Nx, Ny)) # x-velocity
v = np.zeros((Nx, Ny)) # y-velocity
p = np.zeros((Nx, Ny)) # pressure
for nt in range(ns): # solve 2D Navier-Stokes equations
pn = p.copy()
p[1:-1, 1:-1] = (pn[2:, 1:-1] + pn[:-2, 1:-1] + pn[1:-1, 2:] + pn[1:-1, :-2]) / 4 - h / (8 * dt) * (u[2:, 1:-1] - u[:-2, 1:-1] + v[1:-1, 2:] - v[1:-1, :-2]) # pressure
p[0, :] = p[1, :] # dp/dx = 0 at x = 0
p[-1, :] = p[-2, :] # dp/dx = 0 at x = L
p[:, 0] = p[:, 1] # dp/dy = 0 at y = 0
p[:, -1] = 0 # p = 0 at y = D
un = u.copy()
vn = v.copy()
u[1:-1, 1:-1] = (un[1:-1, 1:-1] - dt / (2 * h) * (un[1:-1, 1:-1] * (un[2:, 1:-1] - un[:-2, 1:-1]) + vn[1:-1, 1:-1] * (un[1:-1, 2:] - un[1:-1, :-2])) - dt / (2 * h) * (p[2:, 1:-1] - p[:-2, 1:-1]) + (1 / Re) * dt / h**2 * (un[2:, 1:-1] + un[:-2, 1:-1] + un[1:-1, 2:] + un[1:-1, :-2] - 4 * un[1:-1, 1:-1])) # x momentum
u[0, :] = 0 # u = 0 at x = 0
u[-1, :] = 0 # u = 0 at x = L
u[:, 0] = 0 # u = 0 at y = 0
u[:, -1] = Uinf # u = Uinf at y = D
v[1:-1, 1:-1] = (vn[1:-1, 1:-1] - dt / (2 * h) * (un[1:-1, 1:-1] * (vn[2:, 1:-1] - vn[:-2, 1:-1]) + vn[1:-1, 1:-1] * (vn[1:-1, 2:] - vn[1:-1, :-2])) - dt / (2 * h) * (p[1:-1, 2:] - p[1:-1, :-2]) + (1 / Re) * dt / h**2 * (vn[2:, 1:-1] + vn[:-2, 1:-1] + vn[1:-1, 2:] + vn[1:-1, :-2] - 4 * vn[1:-1, 1:-1])) # y momentum
v[0, :] = 0 # v = 0 at x = 0
v[-1, :] = 0 # v = 0 at x = L
v[:, 0] = 0 # v = 0 at y = 0
v[:, -1] = 0 # v = 0 at y = D
X, Y = np.meshgrid(np.linspace(0, L, Nx), np.linspace(0, D, Ny)) # spatial grid
plt.figure(dpi = 200)
plt.contourf(X, Y, v.T, 128, cmap = 'jet') # plot contours
plt.colorbar()
plt.streamplot(X, Y, u.T, v.T, color = 'black', cmap = 'jet', density = 2, linewidth = 0.5, arrowstyle='->', arrowsize = 1) # plot streamlines
plt.gca().set_aspect('equal', adjustable='box')
plt.xticks([0, L])
plt.yticks([0, D])
plt.xlabel('x [m]')
plt.ylabel('y [m]')
plt.show()
Thank you for reading! If you want to hire me as your next shinning post-doc, do let reach out!
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