14th Step of the 12 steps to Navier-Stokes 😑

     After the tremendous success of 13th Step (thanks to the two people who read it, don't understand why they even bothered? 😂) The 14th step now exists! This case is called the case of flow around an obstacle! Like a box. ⬜ This is an unofficial continuation to this. If I get it approved by Dr. Barba, then it will be official. The original series is in Python but I coded this in MATLAB without using many MATLAB specific functions so the code can be translated to other programing languages 🖧 quite easily 😊.

     In terms of validation, Strouhal number is at 0.145 [1-4] for flow around a square cylinder at Re 100. This code gives St of 0.141. 🤓 Fig. 1 shows results from the code.

Fig. 1, Post processing

Code

%% clear and close

close all

clear

clc

beep off % annoying beep off :)

%% define spatial and temporal grids

l_square = 1; % length of square

h = l_square/10; % grid spacing

dt = 1; % time step

L = 40; % cavity length

D = 15; % cavity depth

Nx = round((L/h)+1); % grid points in x-axis

Ny = round((D/h)+1); % grid points in y-axis

nu = 0.000015111; % kinematic viscosity

Uinf = 0.0015111; % free stream velocity / inlet velocity  / lid velocity

cfl = dt*Uinf/h; % % cfl number

travel = 10; % times the disturbance travels entire length of computational domain

TT = travel*L/Uinf; % total time

ns = TT/dt; % number of time steps

Re = l_square*Uinf/nu; % Reynolds number

rho = 1.2047; % fluid density

%% initialize flowfield

u = Uinf*ones(Nx,Ny); % x-velocity

v = zeros(Nx,Ny); % y-velocity

p = zeros(Nx,Ny); % pressure

i = 2:Nx-1; % spatial interior nodes in x-axis

j = 2:Ny-1; % spatial interior nodes in y-axis

[X, Y] = meshgrid(0:h:L, 0:h:D); % spatial grid

maxNumCompThreads('automatic'); % select CPU cores

%% solve 2D Navier-Stokes equations

for nt = 1:ns

    pn = p;

    p(i, j) = (pn(i+1, j)+pn(i-1, j)+pn(i, j+1)+pn(i, j-1))/4 ...

        -h*rho/(8*dt)*(u(i+1, j)-u(i-1, j)+v(i, j+1)-v(i, j-1)); % pressure poisson

    p(1, :) = p(2, :); % dp/dx = 0 at x = 0

    p(Nx, :) = 0; % p = 0 at x = L

    p(:, 1) = p(:, 2); % dp/dy = 0 at y = 0

    p(:, Ny) = p(:, Ny-1); % dp/dy = 0 at y = D

    p(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % box geometry

    p(round(5*Nx/L), round(7*Ny/D:8*Ny/D)) = p(round(5*Nx/L)-1, round(7*Ny/D:8*Ny/D)); % left side

    p(round(6*Nx/L), round(7*Ny/D:8*Ny/D)) = p(round(6*Nx/L)+1, round(7*Ny/D:8*Ny/D)); % right side

    p(round(5*Nx/L:6*Nx/L), round(7*Ny/D)) = p(round(5*Nx/L:6*Nx/L), round(7*Ny/D)-1); % bottom side

    p(round(5*Nx/L:6*Nx/L), round(8*Ny/D)) = p(round(5*Nx/L:6*Nx/L), round(8*Ny/D)+1); % top side

    un = u;

    vn = v;

    u(i, j) = un(i, j)-dt/(2 * h)*(un(i, j).*(un(i+1, j)-un(i-1, j))+vn(i, j).*(un(i, j+1)-un(i, j-1))) ...

        -dt/(2*rho*h)*(p(i+1, j)-p(i-1, j)) ...

        +nu*dt/h^2*(un(i+1, j)+un(i-1, j)+un(i, j+1)+un(i, j-1)-4*un(i, j)); % x-momentum

    u(1, :) = Uinf; % u = Uinf at x = L

    u(Nx, :) = u(Nx-1, :); % du/dx = 0 at x = L

    u(:, 1) = 0; % u = 0 at y = 0

    u(:, Ny) = 0; % u = 0 at y = D

    u(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % box geometry

    u(round(5*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % left side

    u(round(6*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % right side

    u(round(5*Nx/L:6*Nx/L), round(7*Ny/D)) = 0; % bottom side

    u(round(5*Nx/L:6*Nx/L), round(8*Ny/D)) = 0; % top side

    v(i, j) = vn(i, j)-dt/(2*h)*(un(i, j).*(vn(i+1, j)-vn(i-1, j))+vn(i, j).*(vn(i, j+1)-vn(i, j-1))) ...

        -dt/(2*rho*h)*(p(i, j+1)-p(i, j-1)) ...

        + nu*dt/h^2*(vn(i+1, j)+vn(i-1, j)+vn(i, j+1)+vn(i, j-1)-4*vn(i, j)); % y-momentum

    v(1, :) = 0; % v = 0 at x = L

    v(Nx, :) = v(Nx-1, :); % dv/dx = 0 at x = L

    v(:, 1) = 0; % v = 0 at y = 0

    v(:, Ny) = 0; % v = 0 at y = D

    v(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % box geometry

    v(round(5*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % left side

    v(round(6*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % right side

    v(round(5*Nx/L:6*Nx/L), round(7*Ny/D)) = 0; % bottom side

    v(round(5*Nx/L:6*Nx/L), round(8*Ny/D)) = 0; % top side

end

%% post-processing

velocity_magnitude = sqrt(u.^2 + v.^2); % velocity magnitude

u1 = u; % u-velocity for plotting with box

v1 = v; % v-velocity for plotting with box

p1 = p; % p-velocity for plotting with box

u1(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = NaN; % step geometry

v1(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = NaN; % step geometry

p1(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = NaN; % step geometry

velocity_magnitude1 = sqrt(u1.^2 + v1.^2); % velocity magnitude with box

%% Visualize velocity vectors and pressure contours

hold on, axis off

contourf(X, Y, u1', 64, 'LineColor', 'none'); % contour plot

set(gca, 'FontSize', 20)

hh = streamslice(X, Y, u1', v1', 20); % streamlines

set(hh, 'Color', 'k','LineWidth', 01);

colorbar; % add color bar

colormap jet % set color map

axis equal % set true scale

xlim([0 L]); % set axis limits

ylim([2 13]);

xticks([0 L]) % set ticks

yticks([0 D]) % set ticks

xlabel('x [m]');

ylabel('y [m]');

Cite as:

Fahad Butt (2024). Flow Around Square Cylinder (https://fluiddynamicscomputer.blogspot.com/), Blogger. Retrieved Month Date, Year

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.

Python version:

     As the original series is in Python, here is the Python code for Step 14 of 12. Also, I removed mixed terms from pressure poisson equation, just because 😁.

# 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 / 10  # grid spacing

dt = 10  # 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 = 0.000015111  # kinematic viscosity

Uinf = 0.0015111  # free stream velocity / inlet velocity / lid velocity

cfl = dt * Uinf / h  # cfl number

travel = 10 # 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 = 1.2047  # 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

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 = L

    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 = L

    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

#%% 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()  # p-velocity 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

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 = '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()

References

[1] Khademinejad, Taha & Talebizadeh Sardari, Pouyan & Rahimzadeh, Hassan. (2015). Numerical Study of Unsteady Flow around a Square Cylinder in Compare with Circular Cylinder.

[2] Sohankar, A., Norbergb, C., Davidson, L., Numerical simulation of unsteady low-Reynolds number flow around rectangular cylinders at incidence, Journal of Wind Engineering and Industrial Aerodynamics, 69–71 (1997) 189-201.

[3] Cheng, M., Whyte, D. S., Lou, J., Numerical simulation of flow around a square cylinder in uniform-shear flow, Journal of Fluids and Structures, 23 (2007) 207–226.

[4] Lam, K., Lin, Y. F., L. Zou, Y. Liu, Numerical study of flow patterns and force characteristics for square and rectangular cylinders with wavy surfaces, Journal of Fluids and Structures, 28 (2012) 359–377.

13th Step of the 12 steps to Navier-Stokes 😑

     Indeed, the 13th step now exists! This case is called the case of the backward facing step (BFS)! ⬜ This is an unofficial continuation to this. If I get it approved by Dr. Barba, then it will be official. The original series is in Python but I coded this in MATLAB without using many MATLAB specific functions so the code can be translated to other programing languages 🖧 quite easily.

The Code

%% clear and close

close all

clear

clc

%% define spatial and temporal grids

l_square = 1; % length of square

h = l_square/50; % grid spacing

dt = 0.1; % time step

L = 21; % cavity length

D = 2; % cavity depth

Nx = round((L/h)+1); % grid points in x-axis

Ny = round((D/h)+1); % grid points in y-axis

nu = 0.000015111; % kinematic viscosity

Uinf = 0.0060444; % free stream velocity / inlet velocity / lid velocity

cfl = dt*Uinf/h; % % cfl number

travel = 4; % times the disturbance travels entire length of computational domain

TT = travel*L/Uinf; % total time

ns = TT/dt; % number of time steps

Re = l_square*Uinf/nu; % Reynolds number

rho = 1.2047; % fluid density

%% initialize flowfield

u = zeros(Nx,Ny); % x-velocity

v = zeros(Nx,Ny); % y-velocity

p = zeros(Nx,Ny); % pressure

i = 2:Nx-1; % spatial interior nodes in x-axis

j = 2:Ny-1; % spatial interior nodes in y-axis

[X, Y] = meshgrid(0:h:L, 0:h:D); % spatial grid

maxNumCompThreads('automatic'); % select CPU cores

%% solve 2D Navier-Stokes equations

for nt = 1:ns

pn = p;

p(i, j) = (pn(i+1, j)+pn(i-1, j)+pn(i, j+1)+pn(i, j-1))/4 ...

-h*rho/(8*dt)*(u(i+1, j)-u(i-1, j)+v(i, j+1)-v(i, j-1)); % pressure poisson

p(1, :) = p(2, :); % dp/dx = 0 at x = 0

p(Nx, :) = 0; % p = 0 at x = L

p(:, 1) = p(:, 2); % dp/dy = 0 at y = 0

p(:, Ny) = p(:, Ny-1); % dp/dy = 0 at y = D

p(round(1:1*Nx/L), round(1:1*Ny/D)) = 0; % step geometry

p(round(1*Nx/L), round(1:1*Ny/D)) = p(round(1*Nx/L)+1, round(1:1*Ny/D)); % dp/dx = 0 at x = 1 and y = 0 to 1

p(1:round(1*Nx/L), round(1*Ny/D)) = p(1:round(1*Nx/L), round(1*Ny/D)+1); % dp/dy = 0 at x = 0 to 1 and y = 1

p(1:round(1*Nx/L), 1) = p(1:round(1*Nx/L), 2); % dp/dy = 0 at x = 0 to 1 and y = 1

un = u;

vn = v;

u(i, j) = un(i, j)-dt/(2 * h)*(un(i, j).*(un(i+1, j)-un(i-1, j))+vn(i, j).*(un(i, j+1)-un(i, j-1))) ...

-dt/(2*rho*h)*(p(i+1, j)-p(i-1, j)) ...

+nu*dt/h^2*(un(i+1, j)+un(i-1, j)+un(i, j+1)+un(i, j-1)-4*un(i, j)); % x-momentum

u(round(1:1*Nx/L), round(1:1*Ny/D)) = 0; % step geometry

u(1, round(1:1*Ny/D)) = 0; % u = 0 at x = 0 and y = 0 to 1

u(1, round(1*Ny/D:2*Ny/D)) = Uinf; % u = Uinf at x = 0 and y = 1 to 2

u(round(1*Nx/L), round(1:1*Ny/D)) = 0; % u = 0 at x = 1 and y = 0 to 1

u(1:round(1*Nx/L), round(1*Ny/D)) = 0; % u = 0 at x = 0 to 1 and y = 1

u(1:round(1*Nx/L), 1) = 0; % u = 0 at x = 0 to 1 and y = 1

u(Nx, :) = u(Nx-1, :); % du/dx = 0 at x = L

u(:, 1) = 0; % u = 0 at y = 0

u(:, Ny) = 0; % u = 0 at y = D

v(i, j) = vn(i, j)-dt/(2*h)*(un(i, j).*(vn(i+1, j)-vn(i-1, j))+vn(i, j).*(vn(i, j+1)-vn(i, j-1))) ...

-dt/(2*rho*h)*(p(i, j+1)-p(i, j-1)) ...

+ nu*dt/h^2*(vn(i+1, j)+vn(i-1, j)+vn(i, j+1)+vn(i, j-1)-4*vn(i, j)); % y-momentum

v(round(1:1*Nx/L), round(1:1*Ny/D)) = 0; % step geometry

v(1, round(1:1*Ny/D)) = 0; % v = 0 at x = 0 and y = 0 to 1

v(1, round(1*Ny/D:2*Ny/D)) = 0; % v = Uinf at x = 0 and y = 1 to 2

v(round(1*Nx/L), round(1:1*Ny/D)) = 0; % v = 0 at x = 1 and y = 0 to 1

v(1:round(1*Nx/L), round(1*Ny/D)) = 0; % v = 0 at x = 0 to 1 and y = 1

v(1:round(1*Nx/L), 1) = 0; % v = 0 at x = 0 to 1 and y = 1

v(Nx, :) = v(Nx-1, :); % dv/dx = 0 at x = L

v(:, 1) = 0; % u = 0 at y = 0

v(:, Ny) = 0; % u = 0 at y = D

end

%% post-processing

velocity_magnitude = sqrt(u.^2 + v.^2); % velocity magnitude

u1 = u; % u-velocity for plotting with box

v1 = v; % v-velocity for plotting with box

p1 = p; % p-velocity for plotting with box

u1(1:round(1*Nx/L) , 1:round(1*Ny/D)) = NaN; % step geometry

v1(1:round(1*Nx/L) , 1:round(1*Ny/D)) = NaN; % step geometry

p1(1:round(1*Nx/L) , 1:round(1*Ny/D)) = NaN; % step geometry

velocity_magnitude1 = sqrt(u1.^2 + v1.^2); % velocity magnitude with box

%% Visualize velocity vectors and pressure contours

hold on

contourf(X, Y, u1', 64, 'LineColor', 'none'); % contour plot

set(gca, 'FontSize', 20)

% skip = 20;

% quiver(X(1:skip:end, 1:skip:end), Y(1:skip:end, 1:skip:end),...

% u1(1:skip:end, 1:skip:end)', v1(1:skip:end, 1:skip:end)', 1, 'k','LineWidth', 0.1); % Velocity vectors

% hh = streamslice(X, Y, u1', v1',2); % Streamlines

% set(hh, 'Color', 'k','LineWidth', 01);

colorbar; % Add color bar

colormap parula % Set color map

axis equal % Set true scale

xlim([0 L]); % Set axis limits

ylim([0 D]);

xticks([0 9 L]) % Set ticks

yticks([0 D]) % Set ticks

xt = [0 21]; % draw top wall

yt = [2 2];

xb = [1 21]; % draw bottom wall

yb = [0 0];

xbox = [0 1 1 0 0]; % draw box

ybox = [0 0 1 1 0];

plot(xbox, ybox, 'k', 'LineWidth', 2)

plot(xt, yt, 'k', 'LineWidth', 2)

plot(xb, yb, 'k', 'LineWidth', 2)

% clim([0 max(velocity_magnitude(:))]) % legend limits

% title('Velocity [m/s]');

xlabel('x [m]');

ylabel('y [m]');

     The results from this code at Re 400 are presented in Fig. 1. The re-attachment length is ~8 m from the trailing-edge of the box, which is same as previously published results, from example in [1].

Fig. 1, post-processes results

     Thank you for reading! If you want to hire me as your next PhD student, please do reach out!

References

 [1] Irisarri, D., Hauke, G. Stabilized virtual element methods for the unsteady incompressible Navier–Stokes equations. Calcolo 56, 38 (2019). https://doi.org/10.1007/s10092-019-0332-5

Open - Source coded 3D Navier–Stokes equations in C++

     Here are 3D Navier–Stokes equations configured for lid-driven cavity flow. The syntax is C++. The results from this code are shown in Fig. 1. From the pressure-Poisson equation, I removed mixed derivative terms to improve solution stability. 😆

p[i][j][k] = ((p[i+1][j][k] + p[i-1][j][k]) * h * h + (p[i][j+1][k] + p[i][j-1][k]) * h * h + (p[i][j][k+1] + p[i][j][k-1]) * h * h) / (2 * (h * h + h * h + h * h)) - h * h * h * h * h * h / (2 * (h * h + h * h + h * h)) * (rho * (1 / dt * ((u(i+1, j, k) - u(i-1, j, k)) / (2 * h) + (v(i, j+1, k) - v(i, j-1, k)) / (2 * h) + (w(i, j, k+1) - w(i, j, k-1)) / (2 * h))));

p[0][j][k] = p[1][j][k];

p[num_i - 1][j][k] = p[num_i - 2][j][k];

p[i][0][k] = p[i][1][k];

p[i][num_j - 1][k] = p[i][num_j - 2][k];

p[i][j][0] = p[i][j][1];

p[i][j][num_k - 1] = 0.0;

u[i][j][k] = u[i][j][k] - u[i][j][k] * dt / (2 * h) * (u[i+1][j][k] - u[i-1][j][k]) - v[i][j][k] * dt / (2 * h) * (u[i][j+1][k] - u[i][j-1][k]) - w[i][j][k] * dt / (2 * h) * (u[i][j][k+1] - u[i][j][k-1]) - dt / (2 * rho * h) * (p[i+1][j][k] - p[i-1][j][k]) + nu * (dt / (h * h) * (u[i+1][j][k] - 2 * u[i][j][k] + u[i-1][j][k]) + dt / (h * h) * (u[i][j+1] [k] - 2 * u[i][j][k] + u[i][j-1][k]) + dt / (h * h) * (u[i][j][k+1] - 2 * u[i][j][k] + u[i][j][k-1]));

u[0][j][k] = 0.0;

u[num_i - 1][j][k] = 0.0;

u[i][0][k] = 0.0;

u[i][num_j - 1][k] = 0.0;

u[i][j][0] = 0.0;

u[i][j][num_k - 1] = -Uinf;

v[i][j][k] = v[i][j][k] - u[i][j][k] * dt / (2 * h) * (v[i+1][j][k] - v[i-1][j][k]) - v[i][j][k] * dt / (2 * h) * (v[i][j+1][k] - v[i][j-1][k]) - w[i][j][k] * dt / (2 * h) * (v[i][j][k+1] - v[i][j][k-1]) - dt / (2 * rho * h) * (p[i][j+1][k] - p[i][j-1][k]) + nu * (dt / (h * h) * (v[i+1][j][k] - 2 * v[i][j][k] + v[i-1][j][k]) + dt / (h * h) * (v[i][j+1][k] - 2 * v[i][j][k] + v[i][j-1][k]) + dt / (h * h) * (v[i][j][k+1] - 2 * v[i][j][k] + v[i][j][k-1]));

v[0][j][k] = 0.0;

v[num_i - 1][j][k] = 0.0;

v[i][0][k] = 0.0;

v[i][num_j - 1][k] = 0.0;

v[i][j][0] = 0.0;

v[i][j][num_k - 1] = 0.0;

w[i][j][k] = w[i][j][k] - u[i][j][k] * dt / (2 * h) * (w[i+1][j][k] - w[i-1][j][k]) - v[i][j][k] * dt / (2 * h) * (w[i][j+1][k] - w[i][j-1][k]) - w[i][j][k] * dt / (2 * h) * (w[i][j][k+1] - w[i][j][k-1]) - dt / (2 * rho * h) * (p[i][j][k+1] - p[i][j][k-1]) + nu * (dt / (h * h) * (w[i+1][j][k] - 2 * w[i][j][k] + w[i-1][j][k]) + dt / (h * h) * (w[i][j+1][k] - 2 * w[i][j][k] + w[i][j-1][k]) + dt / (h * h) * (w[i][j][k+1] - 2 * w[i][j][k] + w[i][j][k-1]));

w[0][j][k] = 0.0;

w[num_i - 1][j][k] = 0.0;

w[i][0][k] = 0.0;

w[i][num_j - 1][k] = 0.0;

w[i][j][0] = 0.0;

w[i][j][num_k - 1] = 0.0;

Fig. 1, Velocity and pressure iso-surfaces

     Of course constants need to be defined, such as grid spacing in space and time, density, kinematic viscosity. These equations have been validated, as you might have read and here already! Happy coding!

     If you want to hire me as your PhD student in the research projects related to turbo-machinery, aerodynamics, renewable energy, please reach out. Thank you very much for reading.

1D Transient Diffusion (MATLAB code)

clear; clc;% clear the screen and memory

Nx=500; %number of space nodes

Nt=10000; %number of time nodes

Lx=0.3; %length of space (m)

Lt=10; %length of physical time (s)

dx=Lx/(Nx-1); %grid spacing

dt=Lt/(Nt-1); %time step

c=0.000097; %speed of wave (constant)

a=c*dt/dx.^2;

u=100*ones(Nx,1); %initialization of matrix for the property under investigation

x=zeros(Nx,1); %initialization of space

for i=1:Nx-1 %space loop

    x(i+1)=x(i)+dx;

end

u(1)=-20; %boundary condition

u(Nx)=50; %boundary condition

for t=0:dt:Lt %time loop

    un=u; %u(i,t)

    for i=2:Nx-1 %solution loop, backward in space forward in time

        u(i)=((1-2*a)*un(i))+a*(un(i+1)+un(i-1)); %discretized equation, u(i,t+1)

    end

    plot(x,u) %plotting

    title({t;'Time Elaplsed (s)'},'FontSize',38.5);

    pause(0.001)

end

 

 

1D non-Linear Convection (MATLAB code)

clear; clc;% clear the screen and memory

Nx=41; %number of space nodes

Nt=82; %number of time nodes

Lx=2; %length of space (m)

Lt=1; %length of time (s)

dx=Lx/(Nx-1); %grid spacing

dt=Lt/(Nt-1); %time step

a=dt/dx;

u=ones(Nx,1); %initialization of solution matrix

x=zeros(Nx,1); %initialization of space

hold on

for i=1:Nx-1 %space loop

    x(i+1)=x(i)+dx;

end

for i=0.5/dx:1/dx %initial conditions

    u(i)=2;

end

for t=1:Nt %time loop

    un=u; %u(i,t)

    for i=2:Nx %solution loop, backward in space forward in time

        u(i)=un(i)-u(i)*a*(un(i)-un(i-1)); %discretized equation, u(i,t+1)

    end

    plot(x,u,'k') %plotting

    pause(0.1)

end

1D Linear Convection (MATLAB code) (Updated with CFL condition)

clear; clc;% clear the screen and memory

Nx=41; %number of space nodes

Nt=82; %number of time nodes

Lx=2; %length of space (m)

Lt=1; %length of time (s)

dx=Lx/(Nx-1); %grid spacing

dt=Lt/(Nt-1); %time step

c=2; %speed of wave (constant)

a=c*dt/dx;

u=ones(Nx,1); %initialization of solution matrix

x=zeros(Nx,1); %initialization of space

hold on

for i=1:Nx-1 %space loop

    x(i+1)=x(i)+dx;

end

for i=0.5/dx:1/dx %initial conditions

    u(i)=2;

end

for t=1:Nt %time loop

    un=u; %u(i,t)

    for i=2:Nx %solution loop, backward in space forward in time

        u(i)=un(i)-a*(un(i)-un(i-1)); %discretized equation, u(i,t+1)

    end

    plot(x,u,'k') %plotting

    pause(0.1)

end



New Code, with the CFL condition satisfied.

clear; clc;% clear the screen and memory

Nx=13; %number of space nodes

Lx=2; %length of space (m)

Lt=1; %length of time (s)

dx=Lx/(Nx-1); %grid spacing

c=2; %speed of wave (constant)

dt=dx*0.1/c; %time step, Courant number = 0.1

Nt=(Lt/dt)+1; %number of time nodes

a=c*dt/dx;

u=ones(Nx,1); %initialization of solution matrix

x=zeros(Nx,1); %initialization of space

t=zeros(fix(Nt),1); %initialization of space

hold on

for i=1:Nt-1

    t(i+1)=t(i)+dt;

end

for i=1:Nx-1 %space loop

    x(i+1)=x(i)+dx;

end

for i=0.5/dx:1/dx %initial conditions

    u(i)=2;

end

for j=0:dt:Lt %time loop

    un=u; %u(i,t)

    for i=2:Nx %solution loop, backward in space forward in time

        u(i)=un(i)-a*(un(i)-un(i-1)); %discretized equation, u(i,t+1)

    end

    plot(x,u,'k') %plotting

    pause(0.1)

end