Lab 17 — Fluid Dynamics

Bernoulli — Where It Runs Fast, the Pressure Drops

An incompressible fluid must carry the same volume through every cross-section (A₁v₁ = A₂v₂), so pinching the pipe forces the fluid to speed up. Energy is conserved, so whatever it gains in speed it must give up in static pressure. That is Bernoulli’s principle. Below, water flows through a horizontal 100 mm Venturi — so the ρgz term is constant. In the graph the gold line is total pressure, the green line is static pressure, and the gap between them is the dynamic pressure ½ρv². Squeeze the throat and the green line dives.

p+\tfrac12\rho v^2+\rho g z=\mathrm{const.}\qquad A_1v_1=A_2v_2\qquad Q=C_d\,A_2\sqrt{\frac{2\,\Delta p}{\rho\,(1-\beta^4)}}

Tighten the throat or raise the flow and the throat pressure falls below atmospheric, then all the way to water’s vapour pressure — where it boils at room temperature. That is cavitation.

Throat ratio β = d₂/d₁0.55

Flow rate Q9.0 L/s

Colour

Condition

Velocity—

Pressure—

As a flowmeter—

Bernoulli’s equation holds only for inviscid, incompressible, steady flow along a single streamline — and is enormously useful anyway. Reading flow rate from this Δp is exactly what an ISO 5167 Venturi meter does; a real one has a discharge coefficient of about 0.984 for β between 0.3 and 0.75. The “real (with losses)” mode here is a one-parameter stand-in — head loss taken proportional to the throat velocity head — whose coefficient was chosen so that C_d lands in that neighbourhood. It carries no Reynolds or geometry dependence (push β to 0.9 and it falls to 0.96), so read the agreement as order-of-magnitude, not as a match to the standard. The throat’s suction is what drives atomisers, carburettors and water aspirators; the difference between total and static pressure is what an aircraft’s Pitot tube measures. One warning: “air travels farther over the top of a wing, so it goes faster, so Bernoulli makes lift” is wrong — nothing requires the two flows to arrive together. Lift comes from circulation and from deflecting air downward; Bernoulli only relates the resulting speeds and pressures correctly.