Convergence criteria for pressure correction method solving Unsteady N-S equtions
 

I am using a pressure correction method for solving Unsteady Navier-Stokes equations, i want to know what is the convergence criteria i have to use for every time step.

Convergence must imply that you fulfill the divergence-free constraint (at some tolerance).

So, for example, the maximum cell value in the Divergence matrix must be less than 1e-6 for a time step!!!
what is the criteria for this value i.e 1e-6 ??
what if the maximum cell value in the Divergence matrix is not satisfying this condition ??
which equations i have to iterate in the same time step??

Any reply?

What you ask is not very clear, at least to me. I will refer you again to Ferziger and Peric's book "computational methods for fluid dynamics", pages 124-127.

I am not talking about the convergence criteria of poisson Eq. solver.
I am talking about the convergence criteria for one unsteady time step.
I think the answer of Filippo Maria Denaro is correct.
however, what i want to know if i am using a staggered grid, then:
(for example)
U matrix will be 100x99 cells
V matrix will be 99x100 cells
the Divergence matrix will be 100x100

So what i want to know:

1- Is the cell with with the maximum value in comparison with all other cells in the divergence matrix must be less than the tolerance we assume?
2- how can we decide the value of the tolerance?
3- Do I have to iterate all the equations i attach in the 1st post to reach to the required tolerance?
4- Isn't these iterations means that i am going forward in time although i am doing iteration for the same time step ?

If you wish to iterate out the non-linearity in the momentum equation I would suggest using a different scheme which includes an estimate of the u^n+1 value in the differencing of the convection terms. Including the diffusion terms implicitly in the momentum equation is normally done primarily for reasons of stability not accuracy. If the diffusion number remains stable it would be more efficient to use a fully explicit momentum equation. In practice a tight tolerance on the diffusion terms which are by definition small in high Reynolds number flows is unlikely to be worth the computational cost so long as the linear solver is one that gives reasonable values with a few steps.

The solution of the pressure related quantity is going to take most of the computational time and so one tends to use a larger tolerance during any settling down phase and tighten it during the accurate phase. Some have put forward the view that the residual errors from not fully solving the Poisson equation only need to be significantly less than the truncation error. Although one can see where this is coming from, small violations in continuity have a long history in destabilising flows and causing problems at boundaries (boundary conditions for schemes using pressure related quantities like your phi also need care). Perhaps the simplest thing to do is to perform a representative simulation with the Poisson equation solved into roundoff and then repeat with looser tolerances and look at what changes.