Hello, FIR
Here's a basic workflow that you can follow:
- Discretize the equation.
- Set up initial and boundary conditions.
- Let user define ( v, w, dt ) (and others as needed).
- Iterate over time using numerical integration (Euler, Runge-Kutta, etc.).
- Visualize or output the results.
You can try out something like this
% Parameters
Nx = 100; % Number of spatial points
L = 1; % Length of domain
dx = L/(Nx-1); % Spatial step
dt = 0.001; % Time step (user-defined)
Nt = 200; % Number of time steps
u = zeros(Nx,1); % Velocity u
p = zeros(Nx,1); % Pressure
rho = 1; % Density
r = 0.01; % Viscosity
% Initial conditions
u(:) = 0; % or user-defined
p(:) = 0; % or user-defined
% Main time-stepping loop
for n = 1:Nt
u_old = u;
p_old = p;
for i = 2:Nx-1
du_dx = (u_old(i+1) - u_old(i-1))/(2*dx);
d2u_dx2 = (u_old(i+1) - 2*u_old(i) + u_old(i-1))/(dx^2);
dp_dx = (p_old(i+1) - p_old(i-1))/(2*dx);
% specify v, w, du/dt, dv/dt, dw/dt as needed
v = 0; % or user-defined
w = 0; % or user-defined
% Time derivative (explicit Euler)
u(i) = u_old(i) + dt * ( ...
- u_old(i) * du_dx ...
- (1/rho) * dp_dx ...
+ r * d2u_dx2 ...
);
end
% Boundary conditions
u(1) = 0; u(end) = 0; % Example BCs
end
plot(linspace(0,L,Nx), u)
xlabel('x'); ylabel('u');
title('Velocity profile');
