PPE in Projection Method
 

Hello! I am solving the incompressible flows (vortex spin-off or lid-driven cavity in square geometry) using the prejection method. I cannot project out the divergence generating component of velocity field. (Just in the first time step) I am guessing that the problem is coming from the PPE. Laplacian(P) = Div(u)/dt For the vortex spin-off problem, the initial velocity field is given as u=0.25(y-0.5), v=0.25(0.5-x) and u=v=0 at the four boundaries (x=0,x=1,y=0,y=1). From these initial conditions, Div(u) is not zero near the boundaries but total sum of Div(u) over the entire domain is zero. I enforced the homogeneous Neumann BC for PPE at all boundaries but I cannot get a pressure field which has zero gradient at the boundary. Non-zero pressure gradient generate non-zero velocity component at the boundaries. Of course, I can reset these non-zero velocities by applying zero velocities at the boundaries. Does this non-zero pressure gradient arise from the source terms Div(u) at the boundaries or am I missing something?

How can I get rid of this problem? If you have some experience about this kind problem, please help me.

Regards,

Woo Tae

ps) Collocated variables arrangement is used in my code.

Re: PPE in Projection Method
 

Collocated variables arrangement is a bad,yet ? choice for projection method. The Poission eq. generally comes from discrete DIV & Grad. In staggered grid, there is no need for preesure BC. In Colocated grids, the resulting Possion eq. has multiple solutions, or even-odd point decoupling. One must specially treat this. If the Possion eq. comes from applying differential Div to momentum eq, this will need compactbility condition. You pressure Possion as well as its BC must consider this. There are lots of papers dealing with colocated grid for projection method.

Re: PPE in Projection Method
 

Thank you for your kind comments. Could you reference some papers dealing with colocated grid for projection method?