Simple 2D incompressible Navier-Stokes solver on reguar Cartesian quad grid
###How to solve incompressible Navier-Stokes equations:
du/dt = -(u * nabla)u - (1/rho) * gradP + nu * laplacian(u)
div(u) = 0
We use splitting method based on the Helmoltz-Hodge decomposition: For any arbitrary vector field w:
w = u + grad(phi) div(u) = 0, rot(u) /= 0 ('/=' means 'not equal') div(grad(phi)) /= 0, rot(grad(phi)) === 0
So:
w = u + (dt/rho) * grad(p)
FD: u-uprev = dt*(nu*laplacian(uprev))
FD: u - uprev = dt*( -(u*nabla)u )
FD: - 4P(iy,ix) + P(iy+1,ix) + P(iy-1,ix) + P(iy,ix+1) + P(iy,ix-1) = (hh) * (rho/dt) * div(w) = S
P(iy,ix) = (1/4)*(P(iy+1,ix) + P(iy-1,ix) + P(iy,ix+1) + P(iy,ix-1) - S)
FD: u - uprev = dt*( - 1/rho*gradP );