Nobody derived Navier-Stokes equations to reach the formulas that show the erosion of a river bank, or the discharge calculation of a weir. Doing so would be nearly impossible given the extremely high level of complexity.
So instead of trying to start with the fundamental equations and build "up", you start by analyzing the behavior and modeling it.
It isn't any different than what Kepler, Newton, Galileo, etc. did: you look at the real world and build models that describe the world with enough precision for what you need.
Sure, you might say that hydraulics isn't a science per-se, but an applied field and so on. In that case, so is economics.