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 <2025> <FAHAD BUTT>
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import numpy as np
import matplotlib.pyplot as plt
l_square = 1 # length of square
h = l_square / 500 # grid spacing
dt = 0.00002 # 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 = 200 # 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()
Lid-Driven Cavity
The case of lid-driven cavity in the turbulent flow regime can now be solved in reasonable amount of time. The results are shown in Fig. 1. I stopped the code while the flow is still developing as you are reading a blog and not a Q1 journal. 😆 Within Fig. 1, streamlines, v and u component of velocity and pressure are shown going from left to right and top to bottom. At the center of Fig. 1, the velocity magnitude is superimposed. As this is DNS, the smallest spatial scale resolved is ~8e-3 m [8 mm]. While, the smallest time-scale ⌛ resolved is ~8e-4 s [0.8 ms].
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| Fig. 1, The results at Reynolds number of 10,000 |
Free-Jet
The case of free jets in the turbulent flow regime can now be solved in reasonable amount of time. The results are shown in Fig. 2. I stopped the code while the flow is still developing. Once again, I remind you that you are reading a blog and not a Q1 journal. 😆 Within Fig. 2, streamlines and species are shown. As this is DNS, the smallest spatial scale resolved is ~0.02 m [2 cm]. While, the smallest time-scale ⌛ resolved is ~4e-4 s [0.4 ms]. The code for implementing species, in this case temperature using the energy equation is available on the previous post.
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| Fig. 2, Free jet at Reynolds number 10000 |
Heated Room
The benchmark case of mixed convection in an open room in the turbulent flow regime can now be solved in reasonable amount of time as well. The results are shown in Fig. 3. I stopped the code while the flow field stopped showing any changes. 😆 As this is DNS, the smallest spatial scale resolved is ~0.0144 m [1.44 cm]. While, the smallest time-scale ⌛ resolved is ~1e-4 s [1 ms]. The code for implementing species, in this case temperature using the energy equation is available on the previous post. In the previous post, the momentum equation has no changes as the gravity vector is at 0 m/s2. For this example, Boussinesq assumption is used.
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| Fig. 3, Flow inside a heated room at Reynolds number of 5000 |
Thank you for reading! If you want to hire me as your next shinning post-doc, do let reach out!
