Velocity Head (h = v²/2g)

Velocity head is Torricelli's law read backwards. Instead of asking how fast a fluid moves after falling a height $h$, it asks how high a moving stream could climb if all its speed were traded back for elevation. The answer is the same expression rearranged: $h_v = v^2/2g$. It is kinetic energy per unit weight of fluid, and expressing it as a length is what lets it be added directly to elevation and pressure head in one budget.

Water at 3 m/s carries $3^2/19.613 = 0.459$ m of velocity head. That is a useful benchmark, because it shows how small this term usually is. At the 1.5 m/s a hydronic loop typically runs, velocity head is 0.115 m — about a tenth of a metre against a pump head of perhaps 15 m, so under 1% of the energy budget. In ordinary piping, velocity head is nearly always negligible; in nozzles, orifices and jets it is the entire story. Knowing which situation you are in saves a great deal of unnecessary arithmetic.

The three heads together are Bernoulli's equation: elevation head plus pressure head plus velocity head, constant along a streamline. Where a pipe narrows, continuity forces the velocity up, velocity head rises as $v^2$, and the pressure head falls to pay for it — which is the venturi effect and the reason a pitot tube works, since the difference between a stagnation tap and a static tap is exactly $h_v$. Multiply velocity head by ρg and you have the dynamic pressure $q$ from the neighbouring page; they are the same quantity in different currency.

Here the honest correction matters, because textbooks state Bernoulli far too broadly. It is a statement of energy conservation along a single streamline in steady, inviscid flow — not a general law of fluids. It does not hold across a pump, which adds energy; it does not hold across a turbine, which removes it; and it does not hold through any region where friction is significant, because friction converts head irreversibly into heat. Writing "total head is constant" across 100 m of pipe is simply false — that is precisely where the head goes. The usable form carries explicit terms for pump head added and friction head lost, and the pure three-term version applies only over short, smooth, unobstructed runs.

Two smaller points. The $v$ here is the average velocity, but kinetic energy depends on $v^2$, and squaring an average is not the same as averaging the squares. The true kinetic energy of a real velocity profile is always higher, which engineers handle with a correction factor α — about 1.05 in turbulent flow, where the profile is fairly flat, but exactly 2.0 in laminar flow, where the parabolic profile means the simple formula understates the kinetic energy by half. And velocity head, like pressure head, is a length and not a pressure until it is multiplied by ρg, so it can only be added to other heads of the same fluid.