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

Momentum lift — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A control volume round an aerofoil. A rectangle drawn in the fluid around a lifting section. The arrows on the right-hand face show the downward velocity of the air leaving it, drawn to scale. Adding the momentum carried through all four faces to the pressure acting on them gives the force on whatever is inside, without the calculation ever going near the surface.

    Air must be pushed down, and the usual sum is wrong

    The momentum explanation of lift is the one physicists reach for, and it is right — a wing does hold itself up by throwing air downwards. The version usually given then does the accounting badly, and the face of the control volume it keeps turns out to carry the least of it.

    part 1 · misconceptions
  2. Three accounts of lift, one of them tuned to be exactly right at five degrees. Thin-aerofoil theory, which is the answer; Newtonian impact theory with its constant tuned so that it passes exactly through the truth at five degrees; and the equal-transit story, which has no free constant and sits along the bottom. At the tuning point the tuned model and the truth are indistinguishable, and no measurement there separates them.

    A right total from a wrong picture

    Newtonian impact theory has a free constant in it. Tuned at five degrees it reproduces a NACA 2412's lift exactly, and no measurement at five degrees can tell it from the truth. What separates them is the derivative — a lift-curve slope 2.8 times too steep — which is a second constraint rather than a better one.

    part 2 · misconceptions
  3. Two integrands, and the wrong one claims 9.58 per cent more drag. The two things that get integrated across a wake, each scaled to its own peak so the shapes can be compared. The momentum integrand u(U − u)/U² is the drag; the mass integrand (U − u)/U is the displacement thickness, and it is not a drag at all. They differ by a factor of u/U inside them, so the mass one is fatter wherever the deficit is deep — and its integral here is 1.1 times the momentum one's. That ratio is decided by how deep the wake is rather than by how wide: at a twentieth of this momentum thickness it falls to 1, and at twice it rises to 1.25. The error is smallest exactly where a survey is properly done, far downstream where the wake has spread and shallowed.

    Weighing what is missing

    A control volume drawn round a wing gets the lift out of it, on a flow that was solved exactly. The wake survey asks the same box for the drag on a flow nobody has solved, and it is how a real aerofoil's drag is known — with three assumptions, all of which are checkable, and one integral standing next to it that is wrong.

    part 3 · misconceptions
  4. The trailing sheet rolls up into two vortices, and nothing it carries is lost. The trailing vortex sheet behind an elliptically loaded wing, seen in a plane across the wake, at times 0, 0.05, 0.2, 0.6 in units of b²/Γ₀, represented by 160 point vortices with a smoothing length of 0.03 of the span. The tips curl up first and the sheet winds into two concentrated vortices while the whole system sinks under its own induced velocity; by t = 0.6 the pair's centroid has descended 0.122 of the span. Through all of it the crossflow energy — the induced drag — and the separation of the two halves' centroids, 0.7854 of the span, stay exactly what they were.

    The drag a wake keeps however it rolls up

    A plane drawn across the wake of a finite wing contains its induced drag as the kinetic energy of the swirling crossflow. The trailing sheet then rolls up into two vortices, and the energy does not change at all — roll-up moves the drag around the plane without spending any of it. What does spend it is viscosity, which turns crossflow energy into a total-pressure defect, so a plane farther back reads less induced drag, more profile drag, and the same total.

    part 4 · misconceptions

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