do not forget that you have a system of equations therefore you have to satisfy continuity and momentum equation simultaneously

Thank you FMDenaro.
What is the checking condition for the momentum?

The residuals of the two momentum equations

Do you mean the absolute change in u,v from one itetation to the next one?

I mean the residuals of the equations. The difference between two iterations is not the residual.

Thank you.
The check has to be done in the provisional velocities or to the corrected in SIMPLE?

Something else.
Do I have to use periodic boundary conditions mandatorily?

It would be better

Can I avoid them with other way?

The theoretical solution requires two plates estending infinitely in both directions.

Using fictious inlet/outlet BC.s requires that you prescribe the exact solution at inflow and a developed condition at outlet. I never tried such BC.s but I suppose they will drive to the exact solution at the steady state.

So at the inflow and outflow I can give linear distribution of u and p=0 at both ends. Is it right?

Thank you FMDenaro,

How can I implement periodic boundary conditions in this problem? Can you suggest me a book to study?

Thank you FMDenaro,

Because I have not dealt with periodic boundary conditions before can you suggest me a book to study about this matter?

You cannot find a complete book about the periodic boundary conditions... it is a numerical issue that is simply implemented in a code by using a link between boundary nodes.
I give you an example in 1D for a periodicity over a lenght L. You define the nodes 1,2...N in such a way that x(i)=(i-1)*h+Xs. This way x(N)=Xs+L.
Now, you have that nodes 1 and N are linked by periodicity. You compute the equation either in node 1 or N and prescribe the computed value at the node you do not compute.

Thank you,

Let me ask you:
What is Xs?
Do I have to alter all the momentum and pressure equations in SIMPLE at two ends or only the u,v,p at these ends?

Xs is only the general value of the position of the left point, it could be also 0. The periodicity involves all the variables.

I cann't understand if I write x(N)=Xs+L with Xs=0 so X(N)=L, how the code will understand that the start and finish node are the same?

Solve all the equations at i=1 and set the obtained values at i=N

So all the loops will not calculate anything at the i=N and set the calculated values at i=1 to i=N. Is it right this?

Thank you

yes, correct