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

Momentum lift — the series

7 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
  5. The face behind a finite wing carries the share its own shape gives it. The share of the lift leaving through the face of the box 500 spans behind the wing as downward momentum, and the share carried as pressure on the faces above and below, against the box's height over its width, for boxes of thirty-six square spans, with the far-wake share (2/π)·arctan(H/W). A box sixteen times as tall as it is wide sends 0.96 of the weight through the face behind, a square one 0.501, and one sixteen times as wide 0.04.

    Behind a finite wing the wake face keeps its share

    Draw a box round a two-dimensional section and the face behind it carries almost none of the lift, so "the air is thrown down" keeps the wrong term. A finite wing is different. Its trailing vortices carry downward momentum through the face behind it however far back the face is. The share is set by the face's own shape — all of the weight through a tall face, half through a square one, none through a wide one — and the only things a box cannot change are the total and the induced drag.

    part 5 · misconceptions
  6. In stable air a wide-body's wake sinks a few hundred metres and stops tens of kilometres behind it. The depth the trailing vortex pair of a wide-body airliner has sunk, against distance behind the aeroplane, for buoyancy frequencies of 0.005, 0.01 and 0.02 per second and in neutral air: weight 3 MN, span 60 m, 250 m/s, air density 0.4 kg/m³, so the pair starts down at 2.15 m/s. It stops 638, 319 and 160 m down, 117, 58.3 and 29.2 km behind; in neutral air it would go on sinking.

    Stable air takes back the momentum a wing pushes down

    Behind a wing in neutral air the trailing vortices carry the downward momentum the wing gave the air for ever, and a face across the wake keeps its share of the weight however far back it is drawn. In stably stratified air the column the pair carries down is lighter than the air around it, and buoyancy pushes back. The pair stops a few hundred metres down, its momentum returned, after a time that is the same for every aeroplane — 2.33 divided by the buoyancy frequency, about four minutes — and from there on the whole weight crosses the box's upper and lower faces as pressure.

    part 6 · misconceptions
  7. An oval that sheds its fluid is braked less, so the pair sinks further and stops later, until it sheds too fast to stop at all. The depth of a trailing vortex pair below its starting level, in spacings, against time in spacings over its initial speed, in air of N = 0.22: Saffman's oval, which keeps its fluid, and one detraining with α = 0.05. Saffman's stops at depth 6.75 after 10.6; this one at 6.93 after 12.4.

    A wake that sheds its fluid sinks further, and past a point does not stop

    In stable air the pair of trailing vortices behind a wing carries a column of lighter air down with it, and buoyancy pushes the column back until the pair has given up its momentum and stops. That picture keeps every parcel of the column. A real pair sheds the light air as it sinks, and shedding does two things at once: it takes away part of the momentum the buoyancy had to return, and it takes away the buoyancy that was returning it. The second wins. A pair that sheds a little stops later and deeper; one that sheds faster than a threshold loses its column before buoyancy can stop it. The shed air keeps a quarter of the momentum swinging up and down behind the aeroplane.

    part 7 · misconceptions

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