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.

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!