Continuity and Bernoulli

Two equations do most of the work in hydraulics. The continuity equation expresses conservation of mass: for an incompressible fluid in steady flow, the discharge Q = A V is the same at every section of a pipe or channel. Where the pipe narrows, velocity rises; where it widens, velocity falls.

Bernoulli's equation expresses conservation of energy along a streamline for steady, frictionless, incompressible flow: pressure head plus velocity head plus elevation head is constant. Each term is an energy per unit weight measured in metres of fluid. Real flows lose energy to friction and turbulence, so practical forms add a head loss term between two sections, and pumps add head while turbines remove it. Plotting the energy grade line and hydraulic grade line along a pipeline shows at a glance where pressure is high or low and where it might drop below atmospheric.

Applications are everywhere. A venturimeter or orifice meter measures flow from the pressure difference between a wide and a narrow section. A Pitot tube measures velocity from the difference between stagnation and static pressure. Flow from an orifice in a tank follows Torricelli's result, V = √(2gh), reduced by a discharge coefficient. Siphons, spillways, nozzles and pump suction lines are all checked with Bernoulli, keeping pressure above vapour pressure to avoid cavitation.