matlab 流动经典例子MIT

It is an example of a simple numerical method for solving the Navier-Stokes equations. It

contains fundamental components, such as discretization on a staggered grid, an implicit

viscosity step, a projection step, as well as the visualization of the solution over time. The

main priorities of the code are

The code does not provide:

2 Incompressible Navier-Stokes Equations

We consider the incompressible Navier-Stokes equations in two space dimensions

on a rectangular domain Ω = [0, l

conditions on each wall, i.e.

A derivation of the Navier-Stokes equations can be found in [2]. The momentum equations

(1) and (2) des cribe the time evolution of the velocity field (u, v) under inertial and viscous

forces. The pressure p is a Lagrange multiplier to satisfy the incompressibility condition

(3). Note that the momentum equations are already put into a numerics-friendly form. The

nonlinear terms on the right hand s ide equal

side, because it is closer to a conservation form.

tion. We incorporate this condition by using a projection approach [1]: Evolve the momen-

tum equations neglecting the pressure, then project onto the subspace of divergence-free

velocity fields.

3 Visualization

Every fixed number of time steps the current pressure and velocity field are visualized.

The pressure field is shown as a c olor plot with some contour lines. On top of this, the

normalized velocity field is shown as a quiver plot, i.e. little arrows indicate the direction of

the flow. Since in a lid driven cavity the flow rate varies significantly over different areas,

Applying the 2d-curl to this equation yields

The stream function exists since the compatibility condition ∇ · u = u

4 Numerical Solutio n Approach

The general approach of the code is described in Section 6.7 in the book Computational

(n + 1)