Standard applications: Venturi, Pitot and Torricelli

The Venturi tube

The Venturi tube is a duct featuring a local constriction. The conservation of flow rate (S*v = constant, the continuity equation for an incompressible fluid) requires an increase in velocity at the constriction. Applying Bernoulli’s principle at constant altitude, this results in a drop in pressure at this point:

P1 + (1/2)ρv1² = P2 + (1/2)ρv2²

By measuring the pressure difference P1 – P2 using a differential pressure gauge, and knowing the cross-sectional area ratio S1/S2, the volumetric flow rate Q = S1v1 = S2v2 can be calculated.

The Pitot tube

Used to measure the velocity of a flow (aircraft, wind tunnel, air flow), it compares the total pressure, measured at a stagnation point where the velocity is zero, with the static pressure of the free flow:

P_total = P_static + (1/2)ρv² -> v = √(2*(P_total - P_static)/ρ)

Torricelli’s formula

This describes the emptying of a reservoir through an orifice situated at a depth h below the free surface. By applying Bernoulli’s equation between the free surface (velocity virtually zero, atmospheric pressure) and the orifice (also atmospheric pressure):

v = √(2gh)

This velocity is identical to that of a free fall from a height h: a remarkable result which illustrates the conversion of potential energy into kinetic energy.

Pitfall to avoid

In Torricelli’s formula, it is implicitly assumed that the cross-sectional area of the reservoir is much larger than that of the orifice, which allows the downward velocity of the free surface to be neglected. Without this assumption, a corrective term must be added.