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.

Airflow Simulation in Empty / Occupied Rooms and Environments

     One fine morning, I decided to code the Navier–Stokes equations [read if you are bored 🤣] . This post has the results of  flow simulation inside close environments of various aspect ratios. As is customary with all my CFD work using commercial and 🏡made CFD codes, this too is inspired by the free lectures of Dr. Lorena Barba.

The first case has an aspect ratio of 1:1 while for the second case, the aspect ratio is at 3:1. The airflow is at 0.4555 m/s for both cases. Both cases are isothermal at 293 K. There is no turbulence 🌬model [free code ] 🤑.

The smallest resolved scale (~4x smallest mesh size) for 1:1 case is ~ 0.0045 m and for the 3:1 case is at 0.0112 m. Time scales (~4x time-step size) are at 0.0004 s and 0.0004 s, respectively. Fig. 1 shows results for 1:1 aspect ratio while the results for 3:1 case are shown in Fig. 2. For both cases, flow enters from top-left and exists at bottom-right of the rooms. The boundary conditions are taken from [1]. I compared the results with Fluent simulations I ran at same boundary conditions and stopped my simulations when eye-balling didn't revealed any difference 😆. What you expect? This is a blog not a journal 😝.

Fig. 1, 1:1 aspect ratio

Fig. 2, 3:1 aspect ratio

 

Fig. 3, 3:1 aspect ratio with partition

Fig. 4, Flow inside ducts

     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.

References

     [1] Horikiri, Kana & Yao, Yufeng. Validation Study of Convective Airflow in an Empty Room, "Recent Researches in Energy, Environment, Devices, Systems, Communications and Computers", ISBN: 978-960-474-284-4

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

Saithe Fish UDF (ANSYS Fluent)

     This post is about Fish Simulation in ANSYS Fluent using a User Defined Function (UDF). The UDF is mentioned below. The flow conditions are taken from [1]. This goes with the videos shown in Fig. 1-2. The CAD files for t=0 are available here (for UDF 01).

Fig. 1, Animation of motion achieved through the UDF 01.

 

 Fig. 2, Animation of motion achieved through the UDF 02 (Validated).

     The results of present simulations are compared with [1]. The results are in excellent agreement as the C_{l, max} from [1] is at 1.57 while the maximum C_{l, max} from present simulation is at 1.6. The drag coefficient [1], C_{d, max} in [1] is at 0.164; while from the present simulations I got, 0.151. The C_{d, avg} comes out to be 0.072 [1] form the present simulations. I got a value of 0.064 from he present simulation. These would gradually become more accurate with mesh refinement, which I will certainly do if I send some ideas I have for peer review.

UDF 01:

#include "udf.h"

#include "unsteady.h"

#include "dynamesh_tools.h"

#include "math.h"

DEFINE_GRID_MOTION(dynamic,domain,dt,time,dtime)

{

 Thread *tf = DT_THREAD(dt);

 face_t f;

 Node *v;

 int n;

 double x, y, y_ref_previous, y_ref;

 SET_DEFORMING_THREAD_FLAG(THREAD_T0(tf));  

 begin_f_loop(f,tf) {

  f_node_loop(f,tf,n) {

   v = F_NODE(f,tf,n);

   if (NODE_POS_NEED_UPDATE(v)) {

    NODE_POS_UPDATED(v);

    x = NODE_X(v);

    real amplitude = 0.02 + 0.01*x + 0.1*x*x;

    y_ref_previous = amplitude * cos(2*M_PI*x + 2*M_PI*0.8*(PREVIOUS_TIME));

    y_ref = amplitude * cos(2*M_PI*x + 2*M_PI*0.8*(CURRENT_TIME));

     

if (NODE_Y(v) > y_ref_previous){

     NODE_Y(v) = y_ref+fabs(NODE_Y(v)-y_ref_previous);

    }

    else 

     if (NODE_Y(v) < y_ref_previous){

      NODE_Y(v) = y_ref-fabs(NODE_Y(v)-y_ref_previous);

     }

     else {

      NODE_Y(v) = y_ref;

     }

    }

   }

  }

 }

 end_f_loop(f,tf);

UDF 02 (Validated):

#include "udf.h"

DEFINE_GRID_MOTION(dynamic,domain,dt,time,dtime)

{

 Thread *tf = DT_THREAD(dt);

 face_t f;

 Node *v;

 int n;

 double x, y_ref_previous, y_ref, amplitude, fr;

 SET_DEFORMING_THREAD_FLAG(THREAD_T0(tf));  

 begin_f_loop(f,tf) {

  f_node_loop(f,tf,n) {

   v = F_NODE(f,tf,n);

   if (NODE_POS_NEED_UPDATE(v)) {

    NODE_POS_UPDATED(v);

    x = fabs(NODE_X(v));

    amplitude = 0.02 - 0.0825 * x + 0.1625 * x * x;

    fr = 2;

    y_ref_previous = amplitude * cos(2 * M_PI * x - 2 * M_PI * fr * (PREVIOUS_TIME));

    y_ref = amplitude * cos(2 * M_PI * x - 2 * M_PI * fr * (CURRENT_TIME));

     

if (NODE_Y(v) > y_ref_previous){

     NODE_Y(v) = y_ref + fabs(NODE_Y(v) - y_ref_previous);

    }

    else 

     if (NODE_Y(v) < y_ref_previous){

      NODE_Y(v) = y_ref - fabs(NODE_Y(v) - y_ref_previous);

     }

     else {

      NODE_Y(v) = y_ref;

     }

    }

   }

  }

 }

 end_f_loop(f,tf);

If you want to hire me as you next shining PhD/Master student or collaborate in research, please reach out! Thank you for reading!

References

[1] Gen-Jin Dong, Xi-Yun Lu; Characteristics of flow over traveling wavy foils in a side-by-side arrangement. Physics of Fluids 1 May 2007; 19 (5): 057107. https://doi.org/10.1063/1.2736083

Lid-Driven Cavity MATLAB Code

     MATLAB code for 2D Lid-Driven Cavity. Includes labeled commands, plotting and is less than 100 lines of code. Resume possible, to resume comment close all, clear, clc, u, v, p and then run the code. Results are available here:

%% clear and close

close all

clear

clc

%% define spatial and temporal grids

h = 1/10; % grid spacing

cfl = h; % cfl number

L = 1; % cavity length

D = 1; % 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 / lid velocity

dt = h * cfl / Uinf; % time step

travel = 2; % times the disturbance travels entire length of computational domain

TT = travel * L / Uinf; % total time

ns = TT / dt; % number of time steps

l_square = 1; % length of square

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)) * h^2 + (pn(i, j+1) + pn(i, j-1)) * h^2) ./ (2 * (h^2 + h^2)) ...

        - h^2 * h^2 / (2 * (h^2 + h^2)) * (rho * (1 / dt * ((u(i+1, j) - u(i-1, j)) / (2 * h) + (v(i, j+1) - v(i, j-1)) / (2 * h)))); % pressure poisson

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

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

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

    p(:, Ny) = 0; % p = 0 at y = D 

    un = u;

    vn = v;

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

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

    u(1, :) = 0; % u = 0 at x = 0

    u(Nx, :) = 0; % u = 0 at x = L

    u(:, 1) = 0; % u = 0 at y = 0

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

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

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

    v(1, :) = 0; % v = 0 at x = 0

    v(Nx, :) = 0; % v = 0 at x = L

    v(:, 1) = 0; % v = 0 at y = 0

    v(:, Ny) = 0; % v = 0 at y = D

end

%% post-processing

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

% Visualize velocity vectors and pressure contours

hold on

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

set(gca,'FontSize',40)

% skip = 20;

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

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

hh = streamslice(X, Y, u', v', 5); % Streamlines

set(hh, 'Color', 'k','LineWidth', 0.1);

colorbar; % Add color bar

colormap hsv % Set color map

axis equal % Set true scale

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

ylim([0 D]);

xticks([0 L]) % Set ticks

yticks([0 D]) % Set ticks

clim([0 0.95*max(velocity_magnitude(:))]) % Legend limits

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

xlabel('x [m]');

ylabel('y [m]');

Cite as: Fahad Butt (2023). Lid-Driven Cavity (https://fluiddynamicscomputer.blogspot.com/2023/09/lid-driven-cavity-matlab-code.html), Blogger. Retrieved Month Date, Year

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.

Heaving Flat Plate Computational Fluid Dynamics (CFD) Simulation (In-House CFD Code)

     The adventure 🏕 🚵 I started a while ago to make my own CFD 🌬 code / software 💻 for my digital CV and to make a shiny new turbulence mode (one-day perhaps) is going along nicely. This post is about a 2D 10% flat plate undergoing forced heaving motion. Heaving motion is achieved by Eqn. 1.

h_y = H_o*sin(2*π*f_h*t)                                              Eqn. 1

     w.r.t. Eqn. 1 reduced frequency is defined as (f_h*H_o/U∞)), f_h is the frequency of oscillations, while t is the instantaneous time. H_o is the heaving amplitude and U∞ is the free stream velocity. h_y is the position of the flat plate. The animation is shown in Fig. 1.

     The Strouhal number is 0.228 and Reynolds number is at 500. As we can see, the in-house CFD code works very well for this complex CFD simulation. Validation of this work will never be completed 😆. As soon, I will move on to the next project without completing this one. Anyway, discretized Navier-Stokes equations are available here, in both C++ and MATLAB formats if you want to validate this non sense yourself! Good Luck!

The animation from in-house CFD simulation

     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.

Home-Made Computational Fluid Dynamics (CFD), (Includes CFD code)

     One fine morning, I decided to code the Navier–Stokes equations using the finite-difference method. This post has the results of this adventure 🏞️ (so-far). As is customary with all my CFD work using commercial and home-made CFD codes, this too is inspired by the free lectures of Dr. Lorena Barba.

Internal / External Fluid Dynamics - Lid-Driven Cavity

     I started using Lid-Driven Cavity example. Just because everyone else uses it to validate the code they write. The lid-driven cavity case is giving correct results up to ~Re 1,000 without any turbulence models or wall functions.

     The results shown in Fig. 1 correspond to at Re 1,000. It can be seen that the present code which is just vanilla Navier–Stokes; captures the vortices at both bottom edges well as compared to published data. But why would you want to use a vanilla CFD code which solved only ~Re 1,000?, anyways... Some validation... u-velocity at (0.5, 0.1719) is at -0.3869 m/s as compared to -0.38289 m/s [1]. Meanwhile, v-velocity at (0.2266, 0.5) is 0.3252 as compared to 0.30203. Furthermore, v-velocity at (0.8594, 0.5) is at -0.4128 m/s as compared to -0.44993 m/s. All in all, a good agreement with published data. An animation is uploaded here.  Fig. 1a shows a different variation of lid driven cavity.

Fig. 1, Lid driven cavity post-processing

Fig. 1a, Heated lid driven cavity post-processing

Internal / External Aerodynamics - Backward-Facing Step (BFS)

     The next example is the case of Backward Facing Step (BFS). This case is analogues to flow around a building or a pipe with different diameter or anything with sharp edge along the flow. Let's be honest, flow around a building 🏢 is at much higher Re than the measly ~1,000 this code solves. But for validation, here we are.

 

     The cases are self validating as the re-attachment length of the vortex just after the step is very well documented [4]. Fig. 2 compares the results from the vanilla code with published data. It can be seen that the results are in good agreement.

Fig. 2, Backward Facing Step (BFS) post processing at Re 200

Internal Fluid Dynamics - Aero-Thermal Pipe Flow

     This is the first case I ran as it is self validating. The product of area and velocity should be same at  the outlet as given at the inlet as a boundary condition, which it is! Furthermore, density calculated should be proportional to the temperature calculated, which it is! The results at Re ~3300 are shown in Fig. 3. This case is relevant to bio medical field as Reynolds number in human blood 🩸flow is around same flow regime. Moreover, flow through heated pipes etc. is also relevant.

Fig. 3, Flow through heated pipe / between flat plates

Clot Flow

     Fig. 4 shows an example of flow where a clot appears in a blood 🩸 vessel. Done on this same 🏡made CFD 🌬️ code used throughout this blog. Or in a pipe where oil 🛢️ flows. It can be seen how smooth /less chaotic the flow is without clot (less work for heart ❤️ ). Eat 🦐 🥑 🥙 🥕 🥒 🍲 healthy! Mechanical properties of fluid are taken as follows. 5 ltr/min flowrate. Blood vessel diameter 40 mm. Fluid density at 1,070 Kg/m³ and dynamic viscosity at 4.5 cP.

Fig. 4, Flow inside a blood vessel with and without clot

External Aerodynamics - Flow around a Square Cylinder

     Why not circular cylinder you ask? I am very lazy 😂. It is hard work to draw a circle using code. I am used to CAD. 😇. Again, low Reynolds number are very rare in practical applications but for the sake of completeness, I added this case as well. Fig. 5 shows flow around the cylinder at Re 100. Vorticity is shown in Fig. 5. The results are compared with experimental data. The Strouhal number from the home-made CFD code is at 0.158 as compared to a value of 0.148. The results are within 7%  of published literature [2-3].

Fig. 5, Post-processing

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

Code:

     I present to you the code-able discretized equations for solving fluid flow problems. At first, C++ version is presented followed by the MATLAB version. Equation 1-2 are Poisson equations for pressure. Equation 3 and 4 are x-momentum equations (without source). Equations 5 and 6 are y-momentum equations.

double p_ij = ((pn(i + 1, j) + pn(i - 1, j)) * dy * dy + (pn(i, j + 1) + pn(i, j - 1)) * dx * dx) / (2 * (dx * dx + dy * dy)) - dx * dx * dy * dy / (2 * (dx * dx + dy * dy)) * (rho * (1 / dt * ((u(i + 1, j) - u(i - 1, j)) / (2 * dx) + (v(i, j + 1) - v(i, j - 1)) / (2 * dy)) - ((u(i + 1, j) - u(i - 1, j)) / (2 * dx)) * ((u(i + 1, j) - u(i - 1, j)) / (2 * dx)) - 2 * ((u(i, j + 1) - u(i, j - 1)) / (2 * dy) * (v(i + 1, j) - v(i - 1, j)) / (2 * dx)) - ((v(i, j + 1) - v(i, j - 1)) / (2 * dy)) * ((v(i, j + 1) - v(i, j - 1)) / (2 * dy))));                (1)

p(i, j) = ((pn(i+1, j) + pn(i-1, j)) * dy^2 + (pn(i, j+1) + pn(i, j-1)) * dx^2) ./ (2 * (dx^2 + dy^2)) - dx^2 * dy^2 / (2 * (dx^2 + dy^2)) * (rho * (1/dt * ((u(i+1, j) - u(i-1, j)) / (2 * dx) + (v(i, j+1) - v(i, j-1)) / (2 * dy)) - ((u(i+1, j) - u(i-1, j)) / (2 * dx)).^2 - 2 * ((u(i, j+1) - u(i, j-1)) / (2 * dy) .* (v(i+1, j) - v(i-1, j)) / (2 * dx)) - ((v(i, j+1) - v(i, j-1)) / (2 * dy)).^2));                (2)

double u_ij = un(i, j) - un(i, j) * dt / dx * (un(i, j) - un(i - 1, j)) - vn(i, j) * dt / dy * (un(i, j) - un(i, j - 1)) - dt / (2 * rho * dx) * (p(i + 1, j) - p(i - 1, j)) + nu * (dt / (dx * dx) * (un(i + 1, j) - 2 * un(i, j) + un(i - 1, j)) + dt / (dy * dy) * (un(i, j + 1) - 2 * un(i, j) + un(i, j - 1)));                (3)

u(i, j) = un(i, j) - un(i, j) * dt/dx .* (un(i, j) - un(i-1, j)) - vn(i, j) * dt/dy .* (un(i, j) - un(i, j-1)) - dt / (2 * rho * dx) * (p(i+1, j) - p(i-1, j)) + nu * (dt/dx^2 * (un(i+1, j) - 2 * un(i, j) + un(i-1, j)) + (dt/dy^2 * (un(i, j+1) - 2 * un(i, j) + un(i, j-1))));                (4)

double v_ij = vn(i, j) - un(i, j) * dt / dx * (vn(i, j) - vn(i - 1, j)) - vn(i, j) * dt / dy * (vn(i, j) - vn(i, j - 1)) - dt / (2 * rho * dy) * (p(i, j + 1) - p(i, j - 1)) + nu * (dt / (dx * dx) * (vn(i + 1, j) - 2 * vn(i, j) + vn(i - 1, j)) + dt / (dy * dy) * (vn(i, j + 1) - 2 * vn(i, j) + vn(i, j - 1)));                (5)

v(i, j) = vn(i, j) - un(i, j) * dt/dx .* (vn(i, j) - vn(i-1, j)) - vn(i, j) * dt/dy .* (vn(i, j) - vn(i, j-1)) - dt / (2 * rho * dy) * (p(i, j+1) - p(i, j-1)) + nu * (dt/dx^2 * (vn(i+1, j) - 2 * vn(i, j) + vn(i-1, j)) + (dt/dy^2 * (vn(i, j+1) - 2 * vn(i, j) + vn(i, j-1))));                (6)
    

     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 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.

References

[1] U Ghia, K.N Ghia, C.T Shin, "High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method", Journal of Computational Physics, Volume 48, Issue 3, 1982, Pages 387-411, ISSN 0021-9991, https://doi.org/10.1016/0021-9991(82)90058-4
[2] Khademinejad, Taha & Talebizadeh Sardari, Pouyan & Rahimzadeh, Hassan. (2015). Numerical Study of Unsteady Flow around a Square Cylinder in Compare with Circular Cylinder
[3] Ávila, Ítalo & Santos, Gabriel & Ribeiro Neto, Hélio & Neto, Aristeu. (2019). Physical Mathematical and Computational Modeling of the Two-Dimensional Flow Over a Heated Porous Square Cylinder. 10.26678/ABCM.COBEM2019.COB2019-0854
[4] Irisarri, Diego & Hauke, Guillermo. (2019). Stabilized virtual element methods for the unsteady incompressible Navier–Stokes equations. Calcolo. 56. 10.1007/s10092-019-0332-5. 

CFD Basics: Code Vectorization

     This post is about comparing 2 codes to solve the 2D Laplace equation using finite difference method. A sample code mentioned under "Code 01" uses nested loops. We must, wherever possible avoid nested loops. The solution to the code is shown in Fig. 1. The second code uses vectorization instead of the nested loops. The vectorized version is mentioned under  "Code 02".

Code 01

clear

clc

close all

%% Parameters

Lx = 1; % Length of the domain in the x-direction

Ly = 1; % Length of the domain in the y-direction

Nx = 201; % Number of grid points in the x-direction

Ny = 201; % Number of grid points in the y-direction

dx = Lx / (Nx - 1); % Grid spacing in the x-direction

dy = Ly / (Ny - 1); % Grid spacing in the y-direction

%% Initialize temperature matrix

T = zeros(Nx, Ny);

T(:, 1) = 100;

T(:, Nx) = 0;

T(1, :) = 25;

T(Ny, :) = 50;

%% Gauss-Seidel iteration

max_iter = 50000; % Maximum number of iterations

tolerance = 1e-10; % Convergence tolerance

error = inf; % Initialize error

iter = 0; % Iteration counter

while error > tolerance && iter < max_iter

T_old = T;

% Solve Laplace equation using Gauss-Seidel iterations

for i = 2:Nx-1

for j = 2:Ny-1

T(i, j) = ((T(i+1, j) + T(i-1, j))*dy^2 + (T(i, j+1) + T(i, j-1))*dx^2) / (2*(dx^2 + dy^2));

end

end

% Compute error

error = max(abs(T(:) - T_old(:)));

% Increment iteration counter

iter = iter + 1;

end

%% plotting

[X, Y] = meshgrid(0:dx:Lx, 0:dy:Ly);

contourf(X, Y, T')

axis equal

colormap jet

colorbar

clim([0 100])

title('Temperature Distribution Non Vec')

xlabel('x')

ylabel('y')

zlabel('Temperature (T)')

colorbar

Code 02

clear

clc

close all

%% Parameters

Lx = 1; % Length of the domain in the x-direction

Ly = 1; % Length of the domain in the y-direction

Nx = 201; % Number of grid points in the x-direction

Ny = 201; % Number of grid points in the y-direction

dx = Lx / (Nx - 1); % Grid spacing in the x-direction

dy = Ly / (Ny - 1); % Grid spacing in the y-direction

i = 2:Nx-1;

j = 2:Ny-1;

%% Initialize temperature matrix

T = zeros(Nx, Ny);

T(:, 1) = 100;

T(:, Nx) = 0;

T(1, :) = 25;

T(Ny, :) = 50;

%% Gauss-Seidel iteration

max_iter = 50000; % Maximum number of iterations

tolerance = 1e-10; % Convergence tolerance

error = inf; % Initialize error

iter = 0; % Iteration counter

while error > tolerance && iter < max_iter

T_old = T;

% Solve Laplace equation using Gauss-Seidel iterations

T(i, j) = ((T(i+1, j) + T(i-1, j))*dy^2 + (T(i, j+1) + T(i, j-1))*dx^2) / (2*(dx^2 + dy^2));

% Compute error

error = max(abs(T(:) - T_old(:)));

% Increment iteration counter

iter = iter + 1;

end

%% plotting

[X, Y] = meshgrid(0:dx:Lx, 0:dy:Ly);

contourf(X, Y, T')

axis equal

colormap jet

colorbar

clim([0 100])

title('Temperature Distribution Vec')

xlabel('x')

ylabel('y')

zlabel('Temperature (T)')

colorbar

Result

Results say that vectorized code is ~1.5x faster than nested looped code for 200x200 matrix. Simulation results are now presented.

Fig. 1, Vectorized VS Nested Loops

Thank you for reading! I hope you learned something new! If you like this blog and want to hire me as your PhD student, please get in touch!