Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Jet — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The jet is 61.1% of the hole. Flow out of a slot in a plane wall, solved by Kirchhoff's free-streamline method. The outer curve is not a wall and not a guess: it is the streamline on which the pressure is ambient, and where it goes is part of the solution. It leaves the edge of the slot travelling straight down the wall and turns through ninety degrees, settling to a jet whose width is π/(π+2) = 0.6110 of the opening. Every streamline drawn is a level set of the streamfunction the conformal map supplies.

    The hole that halves the flow

    A jet leaving a sharp-edged hole is narrower than the hole, and by an amount that is not measured but computed. One geometry gives exactly one half from momentum alone; another gives exactly π/(π+2) from a conformal map in which the shape of the free surface is part of the answer.

    part 1 · applied
  2. The jet divides 75% to 25%. A jet striking a plate at 60 degrees. Both sheets leave at the jet's own speed, because their surfaces are at ambient pressure and Bernoulli allows nothing else, and the plate can exert no force along itself because the fluid has no viscosity. Momentum along the plate then fixes the split at (1 + cos β)/2 = 0.7500, and the normal force at ṁV sin β = 0.8660. Nothing about the plate's material, size or roughness enters either.

    What a jet cannot push sideways

    A jet striking a plate divides in two, and how it divides is fixed by a single sentence — an inviscid fluid exerts no force along a surface. That one statement, plus mass, gives the split exactly — and the same sentence turns a flat plate into a bucket worth twice as much.

    part 2 · applied
  3. Most power at exactly half the jet speed, found by search at 0.5000. The power a bucket takes from a jet, against how fast the bucket runs, for four deflection angles. Every curve is a parabola with roots at zero — where the force is greatest and the bucket is not moving — and at the jet speed, where the bucket is running away and there is no force at all. The peak is halfway between, at U = V/2, and it is there for every angle and every flow rate. A golden-section search that knows none of the algebra puts it at 0.500000.

    Half the jet speed takes everything

    A bucket standing still feels the largest force and does no work; a bucket running with the jet does no work either. Between them the power peaks at exactly half the jet speed, for every bucket shape and every flow rate — and at that speed a perfect bucket leaves the water motionless.

    part 3 · applied

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