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

Equal transit time is the popular explanation of lift that says the air over the top of a wing must meet the air under it at the trailing edge, and so must go faster. The claim is false in fact: the upper parcel arrives first, and the upper parcel leads by the circulation showed by how much — just outside the boundary layer, the difference in arrival time is the circulation divided by the speed squared, the quantity the lift is made of. Inside the layer the upper parcel leads by more then went inside the boundary layers on the two surfaces, holding each parcel at a fixed fraction of the local thickness, and found that the slow fluid there does not even things out. The lead grows with depth: at a quarter of the thickness it was nearly four times the outside value at mid-chord, and at a tenth of it thirteen to seventeen times, because the suction side’s laminar layer has a fuller profile and a faster edge than the pressure side’s.

That essay stopped where the upper laminar layer separated, at 62 per cent of the chord at zero incidence and 33 per cent at four degrees, and it noted the obvious objection. No wing at a useful Reynolds number keeps a laminar layer that far. The upper layer, decelerating under its adverse pressure gradient, becomes turbulent first; the lower, accelerated, may stay laminar much further. A turbulent layer is full almost to the wall, and the question was whether that fullness pulls the deep lead back towards the outside value, or whether the lower laminar layer, slow near its wall, keeps the exaggeration all the way to the trailing edge.

The answer is that it depends on which side turns first, and the side that turns first is set by the incidence. That makes the result sharper than either expectation: the deep lead can be exaggerated, close to the outside value, or reversed, on the same section, at the same Reynolds number, a few degrees apart.

Laminar to transition, turbulent after #

The section is the earlier essay’s thin cambered Joukowski aerofoil, and its laminar layers are computed exactly as before: Thwaites’s method from the attachment point along each surface, with Pohlhausen’s quartic profile inside. What is new is that each layer is now allowed to turn turbulent. Michel’s criterion puts transition at the first station where the momentum-thickness Reynolds number exceeds an empirical function of the distance Reynolds number; it is the simplest criterion that responds to the pressure gradient through the layer’s own growth, and it is calibrated on wind-tunnel data for wings. If the laminar layer separates first, transition is taken at the separation point, a short bubble treated as instant. From transition the layer is turbulent and computed by Head’s entrainment method, the standard integral method of its kind, with Ludwieg and Tillmann’s skin friction, starting at a shape factor of 1.4.

Inside a turbulent layer the profile is Coles’s composite law — a logarithmic layer near the wall and a wake towards the edge — fitted at each station to the shape factor Head’s method gives and to the friction law Coles’s profile itself implies. An adverse gradient grows the wake used the same profile to show how a pressure gradient fattens the outer part of a turbulent layer, and that is what happens here on the upper surface as the gradient turns adverse.

A parcel held at a fixed fraction of the layer’s thickness moves at the edge speed times the profile’s value at that fraction. The thickness jumps at transition, and a parcel at a fixed fraction rises with it: the convention the earlier essay chose, carried over, and the one a smoke filament seeded at a fixed fraction of the layer would show.

Where each side turns #

The first figure is the transition station on each surface against incidence, at a chord Reynolds number of a million.

At four degrees the upper layer turns turbulent early, at 11 per cent of the chord, behind the suction peak at the leading edge; the lower, accelerated for most of its length, stays laminar to the trailing edge. That is the picture the earlier essay anticipated. At zero incidence it reverses. There the attachment point sits close to the nose, the lower surface’s speed peaks at 5 per cent of the chord, 1.11 times the stream’s, and falls from there all the way to the trailing edge, while the upper surface’s keeps rising to a fifth of the chord before it starts to fall. The lower layer, decelerating from almost the start, turns turbulent at 35 per cent of the chord; the upper, accelerated for longer, holds on to 51. The two cross between half a degree and one degree.

Nothing in that is exotic. A cambered wing at a low lift coefficient, as at a fast cruise or in a dive, flies near the incidence at which its lower surface’s pressure starts to resemble its upper’s, and lower-surface transition near the nose is a familiar feature of wing pressure distributions there. The number that is not a number made the general point that transition is not a fixed Reynolds number but an outcome of the disturbances a layer meets and the gradient it runs in; here the gradient is decided by the incidence.

The deep lead, through the turbulent layers #

The second figure is the lag at 95 per cent of the chord, lower parcel minus upper, as a multiple of the lag just outside the layer, against the parcels’ depth in the layer. Four cases: natural transition at zero and four degrees, and both layers tripped near the leading edge at the same two incidences.

With natural transition at four degrees the earlier essay’s exaggeration survives to the trailing edge. A tenth of the way into the layer the lag is 12.5 times the outside value, a quarter of the way in 3.5 and half way 1.2. The upper parcel has been in a full turbulent profile since 11 per cent of the chord; the lower has been in a laminar one the whole way, moving at less than half its edge speed a quarter of the way in. Every station of that laminar stretch adds to the upper parcel’s lead.

At zero incidence the exaggeration not only shrinks; deep in the layer it reverses. A quarter of the way in, the lag at 95 per cent of the chord is 0.61 of the outside value; a tenth of the way in, it is −1.95. The lower parcel, carried in a turbulent profile from 35 per cent of the chord while the upper was still laminar to 51, has made up the whole of the inviscid lead and more. At that depth the lower parcel arrives first. Equal transit time is not merely false there; it is false in the other direction.

The tripped cases are the control. With both layers turbulent from the nose the deep lag is close to the outside value: 1.5 and 1.0 a quarter of the way in, 1.8 and 1.3 a tenth of the way in, at zero and four degrees. The turbulent profiles on the two surfaces are nearly alike, and the small excess that remains is the earlier essay’s edge-speed asymmetry acting on two full profiles. So the turbulent layer’s fullness does pull the deep lag back towards the outside value, as the refuted claim supposed — but only if the two layers become turbulent together, which on a real wing they do not.

Along the chord #

The third figure follows the quarter-depth lag along the chord, with the transitions marked. At four degrees the lag climbs quickly behind the upper surface’s early transition, is 2.9 times the outside value at mid-chord and keeps rising, to 3.5 at 95 per cent, because the lower layer never turns and every station adds to the account. At zero incidence the lag falls behind the lower surface’s transition at 35 per cent, crosses below zero to −0.12 times the outside value at mid-chord, and recovers only partly once the upper layer turns at 51, to 0.61 at 95 per cent.

The picture is of a running account. Each parcel’s arrival time is the integral of the inverse of its speed along its whole path, and a layer that is an integral of everything upstream made the same point about the layer’s thickness: what a station shows depends on the history that reached it. The deep lag at the trailing edge is a record of where on each surface the layer turned.

The Reynolds number is back #

The earlier essay found one clean result: in a wholly laminar layer the lag at a fixed fraction of the thickness does not depend on the Reynolds number at all, because every speed in the layer is the edge speed times a function of the depth fraction and of a pressure-gradient parameter that does not contain it. A turbulent layer breaks that, twice. Transition moves forward as the Reynolds number rises, and the turbulent profile itself depends on the skin friction, which depends on the momentum-thickness Reynolds number.

The fourth figure plots the deep lags at 95 per cent of the chord against the chord Reynolds number. At four degrees the quarter-depth lag is 2.9 times the outside value at 3·10⁵, rises to 3.7 near a million and falls to 2.6 at 10⁷; the tenth-depth lag runs from 9.6 to 7.8 with a peak between. Two transitions move with the Reynolds number, and they pull in opposite directions. Rising from 3·10⁵, the upper layer turns earlier — 31 per cent of the chord, then 11 — and the upper parcel spends more of its run in a full profile, so the lag grows. Above a million the lower layer begins to turn as well, at 69 per cent of the chord at 3·10⁶ and 50 at 10⁷, and the laminar stretch that made the exaggeration shortens. The dial moves the incidence. At zero, the lowest Reynolds number gives back the laminar essay’s picture — both layers turn near 62 per cent of the chord and the deep lag is exaggerated, 7.9 times the outside value a tenth of the way in — but from a million upward the lower layer turns first, the tenth-depth lag stays negative, and the quarter-depth lag sits between a half and two thirds of the outside value. The same section, at the same incidence, changes the sign of its deep lag between a model aeroplane’s Reynolds number and a light aircraft’s.

Two profiles side by side #

The fifth figure shows why the side that turns first matters so much. At 60 per cent of the chord at zero incidence both layers are turbulent, and a quarter of the way in both parcels move at about 0.72 of their edge speeds. At four degrees the upper layer is turbulent and the lower still laminar, and a quarter of the way in the upper parcel moves at 0.68 of its edge speed and the lower at 0.46. A turbulent profile is nearly flat across most of the layer, with the whole of the velocity drop squeezed against the wall; a laminar profile drops steadily across the whole of it. A parcel held a quarter of the way into a laminar layer is in slow fluid; the same parcel in a turbulent layer is not.

The line the dye actually draws warned that a visualisation does not show what a parcel does unless it is seeded on the parcel. A smoke wire upstream of a wing, its filaments drawn into the layer, would show exactly these lags, and the photograph would depend on the incidence and the Reynolds number in a way that the inviscid lead does not.

What it means for the explanation #

The equal-transit-time explanation has always been defended, when it is defended at all, by an appeal to the air close to the wing: surely the slow, sticky air on the surface is what has to keep pace. These two essays answer that defence completely. Close to the wing there is no single answer to keep pace with. The lag deep in the layers is a property of where each layer turned turbulent, which is a property of the incidence, the Reynolds number and the state of the surface and the free stream, and it can be ten times the inviscid lead or the opposite of it. The one place the lag is fixed by the lift is outside the layers, and there the upper parcel leads by the circulation over the speed squared.

That is also the useful reading of the third figure for anyone who measures. A smoke-wire photograph taken a quarter of the way into the layer on a wing at zero incidence would show the parcels closer together at the trailing edge than the lift accounts for, and a photograph of the same wing at four degrees would show the upper parcel three and a half times further ahead. Neither photograph measures the circulation, and how thick is thin gives the reason in one number: at a chord Reynolds number of a million the layers are between a few thousandths and a hundredth of the chord thick, and a filament is either outside them, where the answer is clean, or inside, where it is not.

What was checked #

Coles’s fitted profile was checked to integrate back to the shape factor it was fitted to, within two parts in a hundred thousand, and its own skin friction was compared with Ludwieg and Tillmann’s: the two agree within fourteen per cent across the cases tried, which is the spread between two fits of one law. Head’s method on a flat plate grows the momentum thickness to within one and a half per cent of the one-seventh-power law at a Reynolds number of ten million, with a shape factor of 1.34. Parcels held outside the layer reproduce the outside lag exactly through the turbulent calculation, which checks that nothing in the new timing touches the inviscid answer. The layers refuse a Reynolds number too low for a turbulent layer and a negative thickness.

What the calculation leaves out #

Transition is a criterion, not a calculation. Michel’s correlation puts transition where wind-tunnel wings of moderate turbulence put it. A quiet free stream or a polished surface delays it, a rough one or a bug strike brings it forward, and how much uphill a layer can take set out why the gradient matters as much as the Reynolds number. The figures’ transition points are the model’s, and the sign of the deep lag at zero incidence depends on the order of the two.

Transition is a point. A real transition region runs over a length comparable with the laminar layer’s, and the profile changes across it; the parcel’s speed would change smoothly rather than in a step.

The parcel rises with the layer. At transition the layer thickens quickly, and a parcel held at a fixed fraction of it is carried outward. A parcel following the flow would not be — it would stay at its own height and find itself deeper in the layer. That is a different experiment, and it is the one the earlier essay’s convention was chosen to avoid, so that the two essays’ numbers could be compared.

Two dimensions, and no separation. The section is thin and the incidences moderate; the turbulent layers stay attached to the trailing edge in every case shown.

Who timed the layer #

Head’s entrainment method is of 1958, Ludwieg and Tillmann’s skin-friction law of 1949, Coles’s law of the wake of 1956, and Michel’s transition criterion of 1951. The equal-transit-time explanation has been criticised in print since at least the 1980s, and its inside-the-layer versions are this collection’s. Smoke-wire photographs of the boundary layer on a wing, which would test them, have been made since Mueller’s work at Notre Dame in the 1980s, though not, as far as the published record shows, timed parcel against parcel.

Still open: a parcel that is not held #

Every parcel in this essay and the last was held at a fixed fraction of the layer’s thickness, which is a convention rather than a motion. A real filament of smoke follows the flow: it starts at a height, is carried by the velocity there, and drifts outward only with the layer’s own slow normal velocity. The next calculation would follow true fluid particles through both layers, seeded at fixed heights upstream of the attachment point, through transition and the turbulent layer’s growth, and ask whether a particle that starts a few thicknesses above the wall stays deep enough in the layer for the transition order to decide its arrival, or whether the layer’s growth leaves it near the edge where the lag is the inviscid one.