# Incompressible 2D pressure-driven flow: for Navier-Stokes equations in nondimensional form, how should I express pressure boundary conditions?

*I have updated this question.

I'm using two codes for modeling incompressible, two-dimensional flow out of a tank. My fluid dynamics experience is pretty basic; I'm using the codes to understand a chemical process. The codes haven't been used much, so I'm running a benchmark/test (running both codes on same flow problem) to make sure I understand them. Unfortunately, I can't post the codes.

Problem: In the test, I want a pressure gradient to drive flow (a constant pressure at the bottom of the tank and 0 at the tank outlet). So far, velocities calculated from one code are much higher than velocities from the other.

One code uses the Navier-Stokes equations in dimensionless form solved by Galerkin finite elements with a perturbation method (penalty formulation). This works by eliminating pressure from the equations. I managed to track down the developer and he ran an example for me. Our results are identical, so I'm not making a mistake with the inputs. The other code uses a volume of fluid method and pressure is not eliminated.

I'm now wondering if the dimensionless code (where pressure is eliminated) can handle pressure boundary conditions correctly. If anyone has any ideas about this, I'd love to hear them. Thanks a lot.

Edit for Dan

Here is the tank geometry I'm dealing with, plus the dimensionless groups that (according to the documentation) are used in the code that is based on dimensionless Navier-Stokes. The dimensionless parameters are the ones with the bars over the letters. $\rho$ is fluid density; $\eta$ is fluid viscosity. I know it is possible to nondimensionalize pressure by using velocity in the nondimensionalization, but I don't know the velocity at the inlet so don't want to do that.

There is nothing in the documentation about how to deal with a pressure boundary condition. I have used the pressure dimensionless term, but with $A$, not $a$, since I want to define an inlet boundary condition. This gives me a pressure that is too high because it leads to velocities that are greater than in the other code. I hope this is clear.

• To help with 1, it would be really helpful to have the full set of non-dimensional parameters. They all interact. I'm particularly interested in the use of the inlet length. Is that the length scale that you're using in the rest of the dimensionless values for the problem? – Dan Apr 25 '16 at 4:53
• Thanks for your interest, Dan. I edited my question. I hope I have managed to be coherent! – Ant Apr 25 '16 at 10:15
• @Ant one thing I would do is pick test values that make some sense for a typical fluid. So for example, if the code was used for a liquid previously, use typical values for water. If instead it was used for a gas, use typical values for an ideal gas at room temperature and pressure. That way you can make simple calculations about what the velocity should be. Right now you are using a viscosity typical of the most viscous liquid and a density typical of a gas. – Salomon Turgman May 27 '16 at 15:26
• @Ant it is difficult to know without seeing the code. For example, for the non-dimensional based code, are you sure it expects non-dimensional input? Maybe it expects SI units or something else and it converts them internally? – Salomon Turgman May 27 '16 at 15:28
• Thanks, sturgman. Regarding your second comment, I checked in with the person who wrote the non-dimensional code. He said that for my test inputs, the non-dimensional values are equivalent to the dimensional ones. – Ant May 27 '16 at 15:33