It's just my curiosity what could be the possible internal settings for presure B.C. (without given velocity) in CFD solvers? Since I got a lot precious comments from previous question, I become hungry for more feeding. And thank you!

From what I am using, a F.V. segregated solution code, pressure B.C. is not necessarily required unless a reference value or somehow approximated to form P gradient for V equations. No pressure eq. exists to apply the P B.C. The singular, consistent linear system of Poisson-continuity P or P' eq. can have solution only when velocity B.C. fulfill continuity as prerequisite. P boundary does help in many situations when velocity B.C is unknown. Since P B.c. is kind of nominal B.C., the actual internal settings will decide the result flow patterns.

Andy gave me a great point that assuming a periodic V at in/out bondaries while maintain continuity and shooting for balanced P drop and mass rate by trial and error. This is good, although a little twisting the reality, for straight in-out problems.

How about in a situation when in or out is uncertain or mixed at the boundary. This kind should be avoided, but sometimes in our application there is no choice otherwise. In this situations, results were usually unreasonable or unsteady. The total mass rate at the boundary could be zero, but in/out volume is out of control. Therefore, I wonder what could be the internal settings? and what could be done to fix the problem.

You are not the only person among the confused group of people. My suggestion is " try to keep the problem simple". In this way, you will be able to keep your thinking straight. You can try to define the following three sandard problems: 1) the square cavity flow problem with a moving lid. In this case , there's no inflow or out flow. 2). flow through a finite length pipe. Here, the inflow must be identical to the outflow. 3). flow over a cylinder. a steady-state low Reynolds number flow. All of the above cases are laminar, low Reynolds number flows ( Re=100). You can approach these problems from two sides, 1) the flow is incompressible, and 2) the flow is compressible ( low Mach number). Write down the equations, the formulations, the boundary conditions, then ask yourself, whether you can obtain the solution or not. ( step-by-step). There is one more standard case you can try, that is, a round jet into a stationary environment, where the the velocity distribution at the jet is given.

The best way for the understanding of the pressure boundary question for the N.S.not-compressible unsteady model is based,on my opinion, on the knowledge of the so-called Inverse Theorem of Vector Calculus.The ITVC has various forms and can be found in good Fluidynamic text books(e.g.Batcheloor) and ensures (in its simpler form) how a smooth field results to be uniquely fixed when its div ,rot,and boundary normal components (plus suitable circulations values for non linearly connected field regions) are given.Such is the case of the initial values of velocity time derivatives appearing in the N-S. eqns. when the initial velocity values are given together with the normal (inflow) boundary velocity component. Of course less simple situations may require generalized forms of the ITVC and the question appear even more complex for the weak solutions;on the other side the NS.closure problems are not yet fully solved!Works of French and Russian mathematicians are a good ref. on this point.