Mathematical solutions for constant potential vorticity critically controlled flow through ocean passages are complicated and not available in simple form. Therefore, to provide formulas for numerical circulation and ocean climate models, two simple formulas for volume flux are developed here. They are fitted to numerical values of the critical flux for constant potential vorticity flow over a flat bottom through a constriction. The two formulas of increasing complexity agree with the numerical values to better than 6% and 1.4%. These flux values are up to 24% less than the values of flux from zero potential vorticity formulas presently applied to ocean passages. The most precise new formula is used to predict flux magnitude through nine ocean passages that have current meter measurements. The size of the revisions compared to zero potential vorticity predictions is a few percent in the direction of better agreement. For further improvement between prediction and observation, other factors such as realistic bottom topography, friction, mixing, waves, and eddies must be included.