In the case of Navier-Stokes flow with constant density the gravity do not influence the fluid motion and is commonly neglected from the balance momentum equation and the pressure is measured relative to its local hydrostatic value. Is this always true? Even for the case of a system which is not closed, i.e. there are inflow and outflow and the velocity is never zero?
If the density is constant, then div v = 0 and so the right hand side f can be neglected whenever its Leray projection Pf = f - \nabla\Delta^{-1}\nabla\cdot f can be defined and Pf=0. Since the gravity g is potential, obviously Pg=0 and hence it can be neglected whenever the boundary conditions are simple enough to define P. With outflows and inflows, this becomes a different story, I would say..
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