They are nonlinear, partial differential equations which are about the worst kinds of equations to solve. About the only things that can make them worse are changes in state (e.g. steam condensing) or non-Newtonian fluids (e.g. mudflows). Numerous researchers have shown that it is impossible to arrive at a closed-form, analytical solution in the form v(r, φ, z) =… When people work with these equations these days, they almost always use numerical solutions. I heard a story once that Albert Einstein started his research in solving the Navier-Stokes equations but gave up because it was too difficult. He moved on to easy topics like quantum physics and relativity. I haven’t been able to verify this, but it makes for a good story.

One of the most important inputs to transient analysis is the speed of the pressure wave (called celerity) moving through the pipes. The speed depends on the fluid properties and pipe properties and has a significant impact on the magnitude of the transient pressure wave and the speed at which it moves through the pipes.

In order to do any kind of hydraulic modeling, whether it is for water distribution, sewage or stormwater, it is necessary to determine the head loss over a range of flow rates. Much of the early days of hydraulic engineering were consumed with engineers trying to come up with the right equation to do these calculations.