Rule of thumb · MechanicalNº 106 / 167

To go supersonic, the pipe has to get wider

Below the speed of sound, squeezing a duct speeds the gas up — the garden-hose intuition, and it is right. Above the speed of sound it is exactly backwards: squeezing slows the flow, and only a WIDENING duct keeps it accelerating.

Why it works

Mass cannot pile up, so density times area times speed is constant. Below Mach 1 the gas barely compresses, so a smaller area forces a higher speed. Above Mach 1 the density falls faster than the area does, so the area has to grow to keep the product fixed. The relation dV/V = −(dA/A)/(1−M²) simply changes sign at Mach 1 — and at Mach 1 itself it permits only dA = 0, which is why sonic flow sits exactly at the throat and nowhere else.

When it fails

It tells you nothing about whether the flow will stay attached to the wall. Expand a nozzle too far for the air outside and the atmosphere pushes back up the bell until the boundary layer lets go, tearing the flow off the wall and shaking the engine sideways. That is why a vacuum-optimised bell cannot be lit at sea level, and no amount of area ratio fixes it.

Do it exactly

Estimate with the rule, then check it against the calculator that models it properly.

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Why is a rocket nozzle shaped like an hourglass?

Below the speed of sound, squeezing a duct speeds the gas up — the garden-hose intuition, and it is right. Above the speed of sound it is exactly backwards: squeezing slows the flow, and only a WIDENING duct keeps it accelerating. Mass cannot pile up, so density times area times speed is constant.

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