Specified static pressure outlet boundary condition
 

Hello everyone,
I have been studying the derivation of specified static pressure outlet boundary condition in finite volume method for incompressible flows from Textbook "The Finite Volume Method in Computational Fluid Dynamics:An Advanced Introduction with OpenFOAM and Matlab "by F. Moukalled , L. Mangani and M. Darwish.
I have a question regarding the derivation (attached). Equ(15.141) simplifies to
Vb = VC
is that true? and if so, what is the benefit from Equ(15.140) then?

Thanks in advance

141 is derived from 140. If you're studying derivations, it's helpful to remember from where they are actually derived. It's equivalent to asking what is the usefulness of knowing that even numbers are divisible by 2 if you already know that 4 is an even number? Well how did you arrive at that conclusion in the first place? Assuming the premise is a big logical no-no in derivations and proofs. And looking at the answer in the back of the book also is not a proof that you know how the answer was obtained, actually it's an indirect confirmation that you have no clue how it was obtained.

Also, 140 is a general flux extrapolation scheme and that formula can be applied in a general sense, such as non-zero fluxes at the boundary and new boundary conditions. To reuse the same example. How do you test if 6 is odd or even? Knowing that 4 is even will not help you but knowing that even numbers are divisible by 2 certain will. If you are only interested in how to implement a pressure outlet BC with a zero flux condition then 141 is the answer you seek. But if you want pressure outlets for other flux conditions, as well as other non-presssure BCs then suddenly 141 is useless and 140 is invaluable.

Thanks LuckyTran for your reply. I agree with you regarding derivations and proofs. However, how is Equ(140) general and can be used for non-zero flux conditions although it's equivalent to (if you dot product both sides with eb) :
grad(Vb).eb = 0

You are assuming that the boundary flux is 0. Again, assuming the premise. You are tunnel visioning on the outcome that you expect.


How about... not assuming anything. Consider for example that the exit boundary face is not fully developed, is not constant area, or even that you might want a specified flux as a boundary condition.