Conceptual problem: how to define pressure boundary conditions for stratified fluid in units of pressure/density?

Question

This may be a trivial problem, but I'm having a tough time with conceptualization.

I'm using a code for modeling flow out of a tank. I'm not experienced in fluid dynamics; I'm more experienced with chemical aspects of the problem. I want to use pressure boundary conditions (a constant pressure at the bottom of the tank and atmospheric pressure at the tank outlet). The code is for incompressible flow and I must input the pressure values in terms of pressure/density (units of m2 s-2).

The problem is I don't understand which densities to use for each boundary condition. At the bottom of the tank, I'm guessing that I should use the tank fluid density (or some weighted average if there is more than one kind of fluid layer in the tank). But what should I use at the top (where the boundary condition is the pressure of the atmosphere)? When I divide atmospheric pressure by the density of air, I get a value that is higher than the bottom boundary of the tank and flow goes the wrong way (down, not up to the outlet). When I divide atmospheric pressure by the same density that I use for the bottom boundary (the density of the fluid in the tank), my output shows pressure values that are lower than atmospheric. It seems like I should get this, but I don't. Any help would be appreciated, but unfortunately, I'm not allowed to post the code.

Answer 1

The reason you're dividing the pressure's by the density is that in the naiver stokes equations that describe fluid flow have three types of terms: inertia terms that are multiplied by the fluid density, pressure gradient terms, and viscous terms that are multiplied by the fluid viscosity. If you divide the whole equation by the fluid density then you can combine your absolute viscosity and your fluid density to get the kinematic density. Then you only have one fluid property to worry about, which makes solving equations easier, but you have to divide your pressure by your fluid density.

This means that the density you want to divide by is the density of the fluid you're simulating flow for. If you're simulating multiple fluid densities in the same model you cannot use this simplification. However, it seems like you aren't simulating the air, so you can safely just use the fluid density.

One other thing to note is that it's only pressure gradients that matter in incompressible flow, so often times people will use gauge pressure (absolute pressure minus one atmosphere) as it doesn't make a difference in the flow results and remembering to add the one atmosphere everywhere is annoying. So if the pressure at your inlet is in gauge pressure than the pressure at the top of your tank should be 0 not one 1 atm.