15th Step of the 12 steps to Navier-Stokes 😑

     FOANSS, i.e. Fadoobaba's Open Advanced Navier-Stokes 🌬 Solver is now capable to simulate flow around curved geometries. In this example, flow around a circular cylinder ⭕ is presented. The main challenge in handling the curved boundaries 🗻 in discrete world is application of the Neumann boundary condition for pressure. The method employed in FOANSS is to use polar ⭗ coordinate system for normal derivative calculation. The idea came while yours's truly was developing FOAMNE i.e. Fadoobaba's Open Advanced 🔨 Mechanics 🧠 Neural Engine. More details can be read here.

     In the CFD code, ∂p/dn is replaced by ∂p/dx * cos(θ) + ∂p/dy * sin(θ). After some trickery, the pressure on the curved surface can be represented by equation 1. Within equation 1, p is the pressure at current time-step. i and j are nodes in the x and y axis, respectively. cos_theta and sin_theta represent radial angles of the polar coordinate system (r θ).

((p[i + 1, j] + p[i - 1, j]) * cos_theta + (p[i, j + 1] + p[i, j - 1] * sin_theta)) / (2 * (cos_theta + sin_theta)) (1)

     At first, flow around a circular cylinder at Reynolds number 100 is simulated. The accuracy of the results is measured using the Strouhal number. For Reynolds number 100, Strouhal number for flow around a circular cylinder is 0.16. The Strouhal number obtained from the code on a coarse mesh, i.e. only 50 cells per cylinder diameter, is 0.17. This is within 8% of the published data. The Strouhal number will never reach 0.16 as the mesh is cartesian and the cylinder surface is represented as a stairstep. Still, very good for teaching purposes! The results are shown in Fig. 1. It can be seen that the vortex shedding phenomenon is calculated relatively accurately. A remainder that you are reading a blog, not a Q1 journal 😃. The vorticity plot is shown in Fig. 2.

Fig. 1, The results from the code

Code

#%% import libraries

import numpy as np

import matplotlib.pyplot as plt

#%% define spatial and temporal grids

l_square = 1 # length of square or diamater of circle

h = l_square / 50 # grid spacing

dt = 0.0001 # time step

L = 15 # domain length

D = 10 # 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

rho = 1 # fluid density

#%% initialize flowfield

u = np.ones((Nx, Ny)) # x-velocity

v = np.zeros((Nx, Ny)) # y-velocity

p = np.zeros((Nx, Ny)) # pressure

x = np.linspace(0, L, Nx) # x-axis vector

y = np.linspace(0, D, Ny) # y-axis vector

XX, YY = np.meshgrid(x, y, indexing="ij") # grid for circle boundary

#%% create circle

x_c = 5 # circle center y

y_c = 5 # circle center y

radius = 0.5 # circle radius

mask = (XX - x_c)**2 + (YY - y_c)**2 <= radius**2 # circular region

tol = 0.125 * h # tolerance for boundary points

circle_boundary = np.abs(np.sqrt((XX - x_c)**2 + (YY - y_c)**2) - radius) <= tol # boundary mask

boundary_indices = np.argwhere(circle_boundary) # circle boundary

# compute radial distances and angles (polar coordinates)

r_values = []

cos_theta_values = []

sin_theta_values = []

for idx in boundary_indices:

    i, j = idx

    r = np.sqrt((XX[i, j] - x_c)**2 + (YY[i, j] - y_c)**2)

    cos_theta = (XX[i, j] - x_c) / r

    sin_theta = (YY[i, j] - y_c) / r

    r_values.append(r)

    cos_theta_values.append(cos_theta)

    sin_theta_values.append(sin_theta)

# define quadrant masks (circle is divided in 4 parts) and corner points

quadrant1 = (XX > x_c) & (YY > y_c) & circle_boundary

quadrant2 = (XX > x_c) & (YY < y_c) & circle_boundary

quadrant3 = (XX < x_c) & (YY < y_c) & circle_boundary

quadrant4 = (XX < x_c) & (YY > y_c) & circle_boundary

top_pt= (XX == x_c) & (YY > y_c) & circle_boundary

bottom_pt = (XX == x_c) & (YY < y_c) & circle_boundary

left_pt = (XX < x_c) & (YY == y_c) & circle_boundary

right_pt = (XX > x_c) & (YY == y_c) & circle_boundary

#%% 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[mask] = 0 # circle

    # apply Neumann boundary condition for pressure in polar coordinates

    p[quadrant1] = (((p[i + 1, j] + p[i - 1, j]) * cos_theta) + ((p[i, j + 1] + p[i, j - 1]) * sin_theta)) / (2 * (cos_theta + sin_theta))

    p[quadrant2] = (((p[i + 1, j] + p[i - 1, j]) * cos_theta) + ((p[i, j + 1] + p[i, j - 1]) * sin_theta)) / (2 * (cos_theta + sin_theta))

    p[quadrant3] = (((p[i + 1, j] + p[i - 1, j]) * cos_theta) + ((p[i, j + 1] + p[i, j - 1]) * sin_theta)) / (2 * (cos_theta + sin_theta))

    p[quadrant4] = (((p[i + 1, j] + p[i - 1, j]) * cos_theta) + ((p[i, j + 1] + p[i, j - 1]) * sin_theta)) / (2 * (cos_theta + sin_theta))

    p[top_pt] = p[i, j + 1]

    p[bottom_pt] = p[i, j - 1]

    p[left_pt] = p[i - 1, j]

    p[right_pt] = p[i + 1, j]

    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[mask] = 0 # circle

    u[circle_boundary] = 0 # circle

    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[mask] = 0 # circle

    v[circle_boundary] = 0 # circle

#%% post-processing

velocity_magnitude = np.sqrt(u**2 + v**2) # velocity magnitude

X, Y = np.meshgrid(np.linspace(0, L, Nx), np.linspace(0, D, Ny)) # spatial grid

# visualize velocity vectors and pressure contours

plt.figure(dpi = 500)

plt.contourf(X, Y, u.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.colorbar(orientation='vertical')

plt.show()

plt.figure(dpi = 500)

plt.contourf(X, Y, v.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.colorbar(orientation='vertical')

plt.show()

plt.figure(dpi = 500)

plt.contourf(X, Y, p.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.colorbar(orientation='vertical')

plt.show()

plt.figure(dpi = 500)  # make a nice crisp image :)

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.axis('off')

plt.show()

Fig. 2, Vorticity with streamlines

     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! The code might be available soon!

2D Heat Equation Code (Finite Difference Method)

     After 🎉 successfully teaching 👨‍🏫 the neural network about the 🔥 heat / diffusion 💉 equation, yours truly thought it is the time ⏱️ to write a simple code 🖥️ for discretized domain as well. A code yours truly created in abundant spare time is mentioned in this blog 📖.

     A complex (sinusoidal) 🌊 boundary condition is implemented. Dirichlet and Neumann boundary conditions are also implemented. The code is vectorized ↗ so there is only one loop 😁.  CFL condition is implemented in the time-step calculation so time-step ⏳ is adjusted based on the mesh size automatically 😇.

     For simple geometries, traditional numerical methods are is still better than PINNs. 🧠

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 numpy as np

import matplotlib.pyplot as plt

# mesh parameters

L = 1 # length of plate

D = np.pi # width of plate

h = 1 / 50 # grid size

Nx = int((L / h) + 1) # grid points in x-axis

Ny = int((D / h) + 1) # grid points in y-axis

alpha = 1 # thermal diffusivity

dt = h**2 / (4 * alpha) # time step (based on CFL condition)

total_time = 1 # total time

nt = int(total_time / dt) # total time steps

# initialization

T = np.zeros((Nx, Ny)) # initial condition

T_new = np.zeros_like(T)

x = np.linspace(0, L, Nx)

y = np.linspace(0, D, Ny)

# solve 2D-transient heat equation

for n in range(nt):

    T_new[1:-1, 1:-1] = T[1:-1, 1:-1] + ((alpha * dt) / h**2) * (T[2:, 1:-1] + T[:-2, 1:-1] + T[1:-1, 2:] + T[1:-1, :-2] - 4 * T[1:-1, 1:-1])

    T[:, :] = T_new

    # apply boundary conditions

    T[0, :] = np.sin(2 * np.pi * y / D) # T = sin(y) at x = 0

    T[-1, :] = T[-2, :] # dT/dx = 0 at x = L

    T[:, 0] = 0 # T = 0 at y = 0

    T[:, -1] = 0 # T = 0 at y = D

# plotting

X, Y = np.meshgrid(x, y, indexing="ij")

plt.figure(dpi = 500)

plt.contourf(X, Y, T, levels = 64, cmap = "jet")

plt.gca().set_aspect('equal', adjustable = 'box')

plt.colorbar(label = "Temperature")

plt.title("Temperature Distribution")

plt.xlabel("x")

plt.ylabel("y")

plt.show()

     The code creates the output as shown in Fig. 1.

Fig. 1, Temperature distribution

     Thank you for reading! If you want to hire me as a post-doc researcher in the fields of thermo-fluids and / or fracture mechanics, do reach out!