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

Equal transit — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two parcels released together do not arrive together. The most repeated explanation of lift says that air parting at the leading edge must meet again at the trailing edge, so the longer upper path forces a higher speed. Released into the solved field, the upper parcel arrives long before the lower one — the premise is simply false, and the real speed difference is larger than it would require.

    The story about air meeting up again

    The most repeated explanation of lift says that air parting at the nose must rejoin at the tail, so the longer upper path forces a higher speed. The premise is false, and the speed it predicts is wrong by a factor of twenty.

    part 1 · misconceptions
  2. Four sections, and what the shape story says about each. A flat plate, a symmetric section twelve per cent thick, a cambered one, and a section with a wavy upper skin. The first two have upper and lower surfaces of exactly equal length; the last has an upper surface two and a half per cent longer than its lower one, which is twice the cambered section's excess.

    The wing that is flat, and flies

    Refuting the equal-transit story by computing the parcels leaves its premise standing, and the premise is the part most readers believe: that the shape is what makes the lift. It is a claim about shapes, so it is tested with shapes — a flat plate, a symmetric section, and a cambered one flown upside down.

    part 2 · misconceptions
  3. The parcel over the top arrives first, and never waits for the other. Two parcels released together far upstream, a whisker above and below the streamline that divides at the nose of a Joukowski section 11.8% thick at 4°. Both crawl past the nose; then the upper one is carried over the top faster and reaches the far line 1.13 time units ahead — 0.279 chord-transit times — against the 1.59 that the circulation divided by the speed squared predicts. Behind the section they travel on side by side at the stream's speed and the gap between them never closes.

    The upper parcel leads by the circulation

    Transit time looked like a quantity that could not be measured: a parcel released near the dividing streamline crawls past the nose for as long as one likes. The difference between two parcels' transit times, one either side of that streamline, does not crawl. It converges, and on a thin section it is the circulation divided by the square of the speed — the lift, measured in seconds. The upper parcel arrives first by that much, and behind the wing the two never close the gap.

    part 3 · misconceptions
  4. Inside the layer the upper parcel's lead grows with depth. The transit lag, lower parcel minus upper, of parcels held at a quarter, a half and three-quarters of the boundary layer's thickness and just outside it, timed from either side of the attachment point to each chordwise station, on a thin cambered section at zero incidence, up to the upper layer's laminar separation at 62% chord. At mid-chord the lag outside the layer is 0.183 chord-over-speed units; at three-quarters of the thickness 0.195, at half 0.281 and at a quarter 0.695.

    Inside the layer the upper parcel leads by more

    Two parcels parting at a wing's attachment point arrive at different times, and outside the boundary layer the difference is the circulation over the speed squared. Inside the layer one might expect the slow, sticky fluid to wash the difference out. It does the opposite: deep in the layer the upper parcel's lead is several times the inviscid one, because the suction side's layer is fuller than the pressure side's. Only the layer's outer quarter reads the inviscid lead, and on the suction side the laminar layer gives out before the trailing edge.

    part 4 · misconceptions
  5. A quarter of the way into the layer, the lag follows the transitions along the chord. The transit lag a quarter of the way into the layer, as a multiple of the lag outside it, against chordwise station, at 0° and 4° on the thin cambered section at a chord Reynolds number of a million, with each surface's transition marked. At 0° the upper layer turns at 51% and the lower at 35%; the lag is −0.122 times the outside value at mid-chord and 0.61 at 95%. At 4° the upper layer turns at 11% and the lower not at all before the trailing edge; the lag is 2.91 times the outside value at mid-chord and 3.51 at 95%.

    The side that turns turbulent first decides who arrives first near the wall

    Inside a laminar layer the upper parcel's lead over the lower grows with depth. Real wings are not laminar to the trailing edge: one surface's layer turns turbulent first, and a turbulent layer is full almost to the wall. At the incidences a wing flies at, the upper side goes first and the exaggerated lead survives to the trailing edge, smaller than the laminar one and shrinking with the Reynolds number. Below about one degree the lower side goes first, and a tenth of the way into the layer the lower parcel arrives first.

    part 5 · misconceptions
  6. Followed along their streamlines, deep parcels at 4° arrive nearly together, and at 0° the upper leads by more than outside. The arrival lag at 95% chord, lower parcel minus upper, as a multiple of the lag outside both layers, against the streamline both parcels follow, as a share of the upper layer's mass flux at the trailing edge. At 4° it is 0.46 for the deepest streamline drawn, −0.16 at a twentieth, 0.47 at a fifth and 1 at the layer's edge; at 0°, 1.47, 1.68, 1.33 and 1.

    Followed rather than held, the parcels near a wing nearly tie

    Parcels held at a fixed fraction of a boundary layer's thickness made the side that turns turbulent first decide which arrives first near the wall, and at a wing's ordinary incidence they gave the upper parcel a lead ten times the one outside the layer. Real smoke is not held at a depth: it follows a streamline. Near the attachment point the layer is thinner than any filament's height, so every filament starts outside it and is swallowed later, and on the upper surface, whose turbulent layer grows faster, it is swallowed deeper. Followed this way, the deep parcels at 4° arrive nearly together, and at a Reynolds number of a million the lower one can come first.

    part 6 · misconceptions

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