FOANSS, i.e. Fadoobaba's Open Advanced Navier-Stokes 🌬 Solver is now capable to simulate flow with species with / without obstacles of course 🤓. In this example, the species is temperature. Gravity is not considered so the momentum equations are unchanged from Step 14. The non dimensional version of energy equation has been used. Mach number term is removed as Reynolds number is very low 😱. These blog posts are written for educational purposes, hence the over simplifications! A remainder that you are reading a blog, not a Q1 journal 😃.
The simulated case is called the case of flow around an obstacle. Like a heat sink fin ⬜. The fin / box is heated in the example shown in Fig. 1. For validation, read here. This code and blog post is not endorsed or approved by Dr. Barba, I just continue the open-source work of her. The 13th and 15th Steps are available for reading along with code.
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| Fig. 1, Temperature contours with streamlines |
Code
# Copyright <2024> <FAHAD BUTT>
# Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (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 AUTHORS OR COPYRIGHT HOLDERS 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.
#%% import libraries
import numpy as np
import matplotlib.pyplot as plt
#%% define spatial and temporal grids
l_square = 1 # length of square
h = l_square / 50 # grid spacing
dt = 0.002 # time step
L = 35 # domain length
D = 21 # 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 = 2 # 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
Pr = 0.7 # Prandtl number
rho = 1 # fluid density
#%% initialize flowfield
u = Uinf * np.ones((Nx, Ny)) # x-velocity
v = np.zeros((Nx, Ny)) # y-velocity
p = np.zeros((Nx, Ny)) # pressure
T = np.ones((Nx, Ny)) # temperature
X, Y = np.meshgrid(np.linspace(0, L, Nx), np.linspace(0, D, Ny)) # spatial grid
#%% solve 2D Navier-Stokes equations
for nt in range(ns):
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 * rho / (8 * dt) * (u[2:, 1:-1] - u[:-2, 1:-1] + v[1:-1, 2:] - v[1:-1, :-2]) # pressure
# boundary conditions for pressure
p[0, :] = p[1, :] # dp/dx = 0 at x = 0
p[-1, :] = 0 # p = 0 at x = L
p[:, 0] = p[:, 1] # dp/dy = 0 at y = 0
p[:, -1] = p[:, -2] # dp/dy = 0 at y = D
p[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box geometry
p[round(10 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = p[round(10 * Nx / L) - 1, round(10 * Ny / D):round(11 * Ny / D)] # box left
p[round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = p[round(11 * Nx / L) + 1, round(10 * Ny / D):round(11 * Ny / D)] # box right
p[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D)] = p[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D) - 1] # box bottom
p[round(10 * Nx / L):round(11 * Nx / L), round(11 * Ny / D)] = p[round(10 * Nx / L):round(11 * Nx / L), round(11 * Ny / D) + 1] # box top
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 * rho * h) * (p[2:, 1:-1] - p[:-2, 1:-1]) + nu * 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
# boundary conditions for x-velocity
u[0, :] = Uinf # u = Uinf at x = 0
u[-1, :] = u[-2, :] # du/dx = 0 at x = L
u[:, 0] = Uinf # u = 0 at y = 0
u[:, -1] = Uinf # u = 0 at y = D
u[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box geometry
u[round(10 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box left
u[round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box right
u[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D)] = 0 # box bottom
u[round(11 * Nx / L):round(11 * Nx / L), round(11 * Ny / D)] = 0 # box top
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 * rho * h) * (p[1:-1, 2:] - p[1:-1, :-2]) + nu * 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
# boundary conditions for y-velocity
v[0, :] = 0 # v = 0 at x = 0
v[-1, :] = v[-2, :] # dv/dx = 0 at x = L
v[:, 0] = 0 # v = 0 at y = 0
v[:, -1] = 0 # v = 0 at y = D
v[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box geometry
v[round(10 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box left
v[round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box right
v[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D)] = 0 # box bottom
v[round(11 * Nx / L):round(11 * Nx / L), round(11 * Ny / D)] = 0 # box top
Tn = T.copy()
T[1:-1, 1:-1] = Tn[1:-1, 1:-1] - (dt / (2 * h)) * (u[1:-1, 1:-1] * (Tn[2:, 1:-1] - Tn[:-2, 1:-1]) + v[1:-1, 1:-1] * (Tn[1:-1, 2:] - Tn[1:-1, :-2])) + (dt / (h**2 * Re * Pr)) * ((Tn[2:, 1:-1] - 2 * Tn[1:-1, 1:-1] + Tn[:-2, 1:-1]) + (Tn[1:-1, 2:] - 2 * Tn[1:-1, 1:-1] + Tn[1:-1, :-2])) # energy
T[0, :] = 1 # T = 0 at x = 0
T[-1, :] = T[-2, :] # dT/dx = 0 at x = L
T[:, 0] = 1 # T = 0 at y = 0
T[:, -1] = 1 # T = 0 at y = D
T[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box geometry
T[round(10 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box left
T[round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box right
T[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D)] = 0 # box bottom
T[round(11 * Nx / L):round(11 * Nx / L), round(11 * Ny / D)] = 0 # box top
#%% post-processing
velocity_magnitude = np.sqrt(u**2 + v**2) # velocity magnitude
u1 = u.copy() # u-velocity for plotting with box
v1 = v.copy() # v-velocity for plotting with box
p1 = p.copy() # pressure for plotting with box
T1 = T.copy() # temperature for plotting with box
# box geometry for plotting
u1[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = np.nan
v1[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = np.nan
p1[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = np.nan
T1[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = np.nan
velocity_magnitude1 = np.sqrt(u1**2 + v1**2) # velocity magnitude with box
# visualize velocity vectors and pressure contours
plt.figure(dpi = 500)
plt.contourf(X, Y, u1.T, 128, cmap = 'hsv')
# plt.colorbar()
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()
plt.figure(dpi = 500)
plt.contourf(X, Y, v1.T, 128, cmap = 'turbo')
# plt.colorbar()
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()
plt.figure(dpi = 500)
plt.contourf(X, Y, p1.T, 128, cmap = 'gist_rainbow')
# plt.colorbar()
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()
plt.figure(dpi = 500)
plt.contourf(X, Y, T1.T, 128, cmap = 'jet')
# plt.colorbar()
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()
plt.figure(dpi = 500)
plt.streamplot(X, Y, u1.T, v1.T, color = 'black', cmap = 'jet', density = 2, linewidth = 0.5,\
arrowstyle='->', arrowsize = 1) # plot streamlines
plt.gca().set_aspect('equal', adjustable = 'box')
plt.axis('off')
plt.show()
Thank you very much for reading! If you want to hire me as your new shinning post-doc or want to collaborate on the research, please reach out!
