Here are some sources would fit your problem,

Also, ANSYS  FLUENT  tutorial in the chapter of 'solvers'  has some information about simulation in PISO under different circumstances. 

For the boundaries, I think you can use forward difference for the inlet and backward difference for the outlet. For the upper parallel wall, you can use the backward difference and for the lower parallel wall, you can use forward difference. You can use center difference for the other grids. Good luck!

To inlet conditions, you have a half volume of control do X-direction grid, hence, the simplest condition is give to each volume from this boundary the velocity value U=U_inlet. 

The walls are like inlet condition, with of each half volume of control in Y-direction grid  has velocities equal to zero, V=0.

The out let you can  consider normal velocities gradient null and instead to prescribe with a interpolation, do a balance in these volume of control considering the half length. The null derivatives will vanish naturally.

The pressure need only one boundary with a Dirichilet condition, the inlet or outlet. You can set a inlet pressure or an outlet pressure. All others  boundaries, null gradient. Without a one Dirichilet condition, the problem is ill-posed, a Neumann problem...

As is a parallel plate, you can choose define inlet pressure and let free the outlet pressure as part of the solution. 

Technical Report (Driven Cavity Flow) Solução do Escoamento 2D Incompressível...

Hi,

Generally what has been said is correct, however there are lots of subtleties. For example it has been suggested to set V=0 for the walls, which is not very accurate. In a robust solver v=v_wall should not be used instead you need to set normal stresses to zero which also satisfies the continuity (see ferziger and peric p.204,205).

As for pressure you don't need to set the pressure on the whole boundary and infact you should not do that (At least not for SIMPLE algorithm), if inlet-outlet conditions assumed all boundaries including walls need a Neuman BC (zero gradient), you then only need a "reference" pressure to only be set at 'one grid point' not a whole boundary (see versteeg p.196). 

For outlets, you need to correct the fluxes to ensure global mass balance since extrapolation does not guarantee global mass balance...

Besides solving the problem on a collocated grid makes things much simpler and only a simple treatment for the pressure is required (rhie-chow interpolation e.g see ferziger p 248.)

I suggest you have a look at the codes from ftp site for the peric and ferziger book (ftp://ftp.springer.de/pub/technik/peric) and understand one of the simple 2d codes there before continuing ..

Good luck

Sina

@ Rezende @ Sina

If you see the attached figure, there are no half control volume of u along x direction. However when I am writing the equation near the wall,then the following equation:

a(p)*u(p)=ae*ue + aw*uw + an*un +as*uw +pressure term.

Now here un will be the wall term which I can set eaul to zero directly and an=0. The boundary condition enters the equation using source term S=Sc+Sp*un. Here, if we look closely, then Sc and Sp are same and equal to an calculates using dn =(mu/delta x) =dn/2.

Alternatively I read some where that I consider control volume outside the domain (like I have dine now) and take u (outside the domain)=-u(just near the wall) so it gives me u_wall=0.

However my main concern is the pressure correction equation that is being solved. I am solving pressure nodes from 2 to nx along x direction and 2 to ny along y out of total nx+! and ny+1 respectively. Now the concern is for node 2 along x direction when I write

ap*p' =ae*pe' +aw*pw' +an*pn' +as*ps' +mass_deficit

here as per boundary condition of inlet if I say pw'=0, and I can cut west node by setting aw=0. However when I need to calculate ap = ae+aw+an+as -Sp, then how do I calculate aw (or Sp. Mathematically the both are same)? Here coefficients are calculated using rho*Area^2/(ap) where ap is coefficient of velocity node, which is inlet here. 

But we have u at inlet, so we don't have any Sp here . As a result when I solve the linear system of equation, it gives me an error that matrix is almost singular and I solution (perror) of the order e^16, e^12 and so on. If I used some iterative solver it doesn't converge . I think there is some problem in formation of matrix of p_error due to boundary condition.

Can some one help in this regard. 

Thanks

(P.S: I have followed versetge in this regard)

 Hi Deewakar,  what Sina has written is correct and more robust, to consider  normal stress equal zero or  to  impose a reference node pressure is alternatives that works very well due physical and mathematical features. But, my first suggestions I consider simpler to coding. 

And really, there is no half VC in the figure, but they are there! This virtual displacement next to the boundaries will  do the VC's  will cross them. Attached there is a example. The first is the effective grid of U 2ns,  effective grid for V and the last, the real mesh used for P. 

The source terms are volumetric terms that generates or sinks the momentum you do not have any Sp. (e.g plasma accelerated by a magnetic field) Your BC's are not sources. the advective flux will be zero at the wall due V=0, but the diffusive part  dV/dY is not. So, An are not equal to zero, there is the viscous effect to consider which can be  coupled with the suggestion given by Sina.  You must to perform a balance and feed it with the BC that you choose to use. 

An=0 implicates in no viscosity at the wall. 

As you Ap coefficient depends on  U_inlet you can have a negative Ap. Have you monitoring the signal of them? 

Hi,

Are you talking about the attached equation. I didn't get how they have written Su in that way. Can you explain.? Thanks

In general source term is like some body force etc. which is zero in my case.

Hi,

Since my flow is incompressible, then div(u) will be zero in Smx as described in the attachment. So adding  those terms won't do any benefit.

Thanks

Dear Deewakar,

You don't need to include the extra terms from the stress tensor if you are doing incompressible simulations as you correctly mentioned yourself.

Back to your main question, what you need to do is to calculate the ap as usual

ap=ae+aw,an+as and aw is zero for first pressure cv and no extra contribution from the boundaries.

Then calculate mass dificit as (signs depend on how you define them)

Sp = fxw-fxe+fys-fyn, now for the first cv fxw is fixed by inlet velocity and can readily be calculated and by Sp I mean the source term at the rhs of linear equations.

Then you need to correct mass fluxed after solution to the pressure correction is found for nodes 2:nx-1, 2,ny-1 (assuming 1,nx,ny are boundaries). Note that the fluxes on the walls are zero (assuming impemeable wall) flux on inlet is fixed by the inlet velocity and I get back to the outlet flux.

Then correct the pressure and velocities.

Now for the outlet flux you need to extrapolate the velocity of nx-1 to nx then calculate a total outlet mass flux using these extrapolated velocities and use inflow mass flux to scale them such that the global mass conservation is gauranteed. (see versteeg)

Good luck

Sina

Hi. Regarding the inlet condition, you should specify u(in), v(in) (normally considered to be zero) and P_prime(in)=0.0 at the inlet port. If you have prescribed velocity inlet, you do not need to be worry about the pressure inlet boundary condition (which in normally the case). at the outlet boundary condition, gradient of all variables (except pressure) should be set to zero. Please read chapter 9 Versteeg titled "Implementation of boundary conditions". All you need is up there.

regards, Ali

Here are some sources would fit your problem,

Also, ANSYS  FLUENT  tutorial in the chapter of 'solvers'  has some information about simulation in PISO under different circumstances. 

Did you check Versteeg's book?