Inside the layer the upper parcel leads by more

The upper parcel leads by the circulation took the most repeated wrong explanation of lift — that air parting at the nose must meet again at the tail — and turned its own quantity into a measurement. Two parcels released a whisker either side of the dividing streamline arrive at the trailing edge at different times; the time each spends creeping past the attachment point diverges, but identically for both, and the difference converges to the circulation over the stream speed squared at first order. The upper parcel leads, and by a definite amount. That calculation was for parcels just outside the boundary layer, and it ended by asking about parcels inside it, where the speeds are lower, the parcels nearest the wall barely move, and the layer is thicker on the decelerating upper surface.

The intuitive expectation is that the layer smears the difference out — that near the wall everything crawls and the lead shrinks. Computed, the lead inside the layer is larger, not smaller, and it grows the deeper the parcels go.

Parcels at a fixed fraction of the layer #

Inside a boundary layer a parcel cannot be released on a streamline and simply followed, because the streamlines inside a layer climb away from the wall as the layer thickens and the parcel’s distance from the wall in absolute units means nothing from one station to the next. The natural comparison is at a fixed fraction of the local thickness: a parcel held at a quarter of the layer’s thickness on each side, as a smoke filament injected at a quarter of the thickness would show. Such a parcel moves at the edge speed $U_e(s)$ times the profile’s value at that fraction, and its time to a station is the integral of $ds$ over that speed.

The edge speed is the potential flow’s, on the same Joukowski section as before. The layer is Thwaites’s: its momentum thickness from $\theta^2 = (0.45\nu/U_e^6)\int U_e^5\,ds$, its shape from $\lambda = \theta^2 U_e'/\nu$. The profile is Pohlhausen’s quartic, with its own shape parameter found from Thwaites’s $\lambda$. Both parcels start a small distance from the attachment point, one on each side, where the flow is Hiemenz’s stagnation-point layer and the same on both sides — so the time each takes to leave diverges identically and cancels, as it did outside the layer.

One consequence of this arrangement is worth stating before any number. Every speed is $U_e$ times a function of the fractional height and of $\lambda$, and $\lambda$ contains no Reynolds number. The lag at a fixed fraction of the layer’s thickness is the same at every laminar Reynolds number. The layer is thinner on a faster wing, but a parcel at a quarter of it sees the same history of speeds.

The lead grows with depth #

On a thin cambered section at zero incidence — thickness and camber parameters of 0.06 and 0.04, a lifting section with no angle — the lag just outside the layer grows along the chord and reaches 0.183 chord-over-speed units at mid-chord. At three-quarters of the layer’s thickness it is 0.195, barely more. At half the thickness it is 0.281, and at a quarter 0.695: nearly four times the inviscid value. The upper parcel is further ahead the deeper it is, and at every station along the chord, not only at mid-chord.

The curves stop at 62 per cent of the chord, where Thwaites’s method separates the upper laminar layer. Past that station the upper surface has no laminar parcels to time; what happens there depends on transition, a separation bubble or a turbulent layer, and none of them is in this calculation.

Why the slow fluid does not even things out #

The lead grows with depth because the two layers are not alike. The upper layer has been accelerated hard round the leading edge and is only gently decelerating by 30 per cent chord, with Thwaites’s $\lambda$ at −0.018; the lower layer has been decelerating more, at −0.047, and its profile is less full. At a quarter of the thickness the upper parcel moves at 0.45 of its edge speed and the lower at 0.42 — and the upper edge is itself 1.14 times faster than the lower. Both ratios favour the upper parcel, and their product is integrated along the whole surface.

Deeper still the effect is stronger, because near the wall a parcel’s speed is proportional to the wall shear times its height, and the wall shear is what an adverse gradient takes away first. The lower layer’s shear is smaller, so its slowest parcels are relatively slower than the upper layer’s, and a parcel at a tenth of the thickness on the lower surface falls far behind its partner. The slow fluid does not even things out; the slower of the two layers is slower in proportion to how close to the wall the comparison is made.

The layer remembers which side it came from #

The difference between the two layers at a given station is not set by the pressure gradient there alone. A layer that is an integral of everything upstream is the statement that a boundary layer’s state at a station carries its whole history: Thwaites’s integral weights the edge speed’s fifth power from the attachment point onwards, so the upper layer’s hard acceleration round the nose keeps it thin and full well after its own gradient has turned adverse, and the lower layer’s early deceleration keeps it thick long after. Two layers at the same chordwise station, under similar local gradients, can have quite different profiles because one was formed under a strong favourable gradient and the other was not.

That memory is why the factor grows along the chord in the first figure rather than settling: the upper layer’s advantage is laid down in its first fifth and spent slowly, and the parcels deep in the layer integrate it. It is also why the factor depends so strongly on incidence: at 2° the lower surface’s early history is more favourable, its layer less depleted near the wall, and the factor at a tenth of the thickness falls from 13 to 2.8.

What a parcel at a fixed fraction is, and is not #

A parcel held at a quarter of the layer’s thickness is a construction, not a piece of fluid. A real fluid parcel inside a growing layer is carried outward by the small velocity normal to the wall that the layer’s thickening implies, so a tracer released at a quarter of the thickness near the leading edge is at a smaller fraction later — the layer grows past it — and its speed falls relative to the edge’s. A fixed-fraction parcel is what a sequence of smoke injections at a fixed fraction would show, each read locally; a material parcel is what a single injection followed downstream shows. The line the dye actually draws is the same distinction between a streakline and a path.

For the comparison here the distinction moves numbers and not signs. A material parcel on each side sinks deeper into its layer as it goes, and the lower one, in the thicker and faster-growing layer, should sink further; both effects should push its arrival later still. The fixed-fraction lead is then likely a lower bound on the lead two material parcels released at the same fraction would show, and the conclusion — that depth exaggerates the upper parcel’s lead — is if anything understated. Following material parcels needs the velocity normal to the wall inside the layer, which an integral method does not supply, and is not done here.

In milliseconds #

A light aircraft’s wing of 1.5-metre chord at fifty metres a second has a chord-over-speed time of thirty milliseconds. On a section like the thin one here, the inviscid lead at mid-chord is then 5.5 milliseconds; at half the layer’s thickness it is 8.4, and at a quarter 21. The layer there, at a chord Reynolds number of five million, is a few millimetres thick, so a smoke filament a millimetre from the skin and one five millimetres from it would show leads of quite different sizes. And the same leads in milliseconds hold for a model of a tenth the chord at a tenth the speed: the chord-over-speed time is the same, and at a fixed fraction of the layer’s thickness the Reynolds number does not enter.

How deep a smoke filament can be and still read the circulation #

The answer to the earlier essay’s last question — at what height a smoke-pulse photograph reads the inviscid value — is in the outer quarter of the layer. On all three cases computed, the thin section at 0° and 2° and a thicker one at 0°, the lag at 30 per cent chord is within a few per cent of the outside value above three-quarters of the thickness and rises steeply below. At a tenth of the thickness it is 13 times the outside value on the thin section at 0°, 17 on the thicker one. The thin section at 2°, whose lower surface is more favourable, is less extreme: 2.8 times.

For a photograph that is a reassurance and a warning. Smoke released outside the layer — which is where smoke pulses in a wind tunnel are usually released — reads the circulation, as the earlier essay said. Smoke that strays into the layer, or a pulse released from a wall port, reads a lead that depends steeply on how deep it went, and on a lifting section always exaggerates the inviscid one.

A layer that runs out #

The comparison has a natural end, and it is earlier than the trailing edge on the side that matters. Along the upper surface Thwaites’s $\lambda$ is favourable for the first 16 per cent of the chord, then turns adverse and reaches the separation value of −0.09 at 62 per cent. Along the lower surface it is adverse almost throughout but never reaches −0.09 before the trailing edge. The upper parcels inside the layer are therefore timed to 62 per cent chord and no further.

The layer’s failure is the more ordinary fact here — the beginning of where the straight line stops as incidence rises — and it is the reason a laminar transit calculation cannot answer the question all the way to the trailing edge on any real lifting section: how much uphill a layer can take puts the limit at a modest adverse gradient, and the suction side of a lifting section asks for more than that over its rear half.

Incidence brings the end forward #

Raise the incidence and the suction peak sharpens, the recovery behind it steepens, and Thwaites’s separation moves forward: on the thin section from 72 per cent chord at −2° to 33 per cent at 4°; on the thicker one from 57 per cent to 30. At the incidences a wing cruises at, the laminar upper layer survives over half the chord on the thinner section and less on the thicker. That is the region in which a laminar-flow aerofoil is designed to keep its layer attached, by shaping the pressure distribution so that $\lambda$ stays favourable further aft, and it is also the region over which the parcels inside the layer can be compared at all.

Thickness and the factor between the inside and the outside #

The factor between the lag inside and outside the layer is a property of the two surfaces’ different histories, so it grows with whatever makes them more different. Camber at zero incidence makes them different in a particular way: the lower surface is the concave side and decelerates throughout, and that is where the large factors come from. A symmetric section at zero incidence has identical surfaces, no circulation and no lag anywhere, inside or outside the layer. A lifting symmetric section has its lower surface accelerating gently aft of the leading edge and its upper surface decelerating, which brings the two profiles’ shapes closer and the factor down; that is the thin section at 2°, whose factor at a tenth of the thickness is 2.8 against 13 at 0°. Thickness hides a third of the lag outside the layer; inside it, the same section shape decides how much depth exaggerates what is left.

The outer lag, Thwaites and Pohlhausen, against what they must reproduce #

The most important check is that the surface calculation, with the parcels held outside the layer and timed to the trailing edge along each surface, reproduces the earlier essay’s lag, computed by following parcels released upstream either side of the dividing streamline through the full potential flow. The two agree to a quarter of a per cent, 0.408 against 0.407, which is the step and the starting distance; their divergent parts cancel the same way. On a flat plate Thwaites’s momentum thickness is $0.671\sqrt{\nu x/U}$ against Blasius’s 0.664, the method’s known one per cent, and Pohlhausen’s shape parameter is zero at zero gradient. How thick is thin derived the Blasius layer itself.

The convention: lower minus upper, at a fixed fraction #

The lag is the lower parcel’s time minus the upper’s, positive when the upper parcel leads, in units of the chord over the stream speed. Heights are fractions of Pohlhausen’s thickness, at which the profile reaches the edge speed exactly; “outside the layer” is the edge speed itself. Stations are fractions of the chord from the leading edge. Thwaites’s separation is $\lambda = -0.09$. The start is a distance of a thousandth of the chord-scale from the attachment point on each side; halving it changes no quoted lag in its third figure, which is the cancellation of the two parcels’ identical crawl away from the stagnation point doing its work. The sections are Joukowski’s, with the thickness and camber parameters of the earlier essay’s mapping.

What the laminar picture cannot show #

Transition, which on a real wing — even one that is flat — usually happens before laminar separation on the upper surface, replaces the profile with a turbulent one that is fuller still, and would reduce the upper parcel’s lead near the wall. The separation bubble that often forms where a laminar layer separates and reattaches turbulent, inside which there is no meaningful transit at all. Pohlhausen’s profile, a one-parameter family that is crude near separation; the trend with depth is robust and the factors near the wall are not to better than tens of per cent. And the wake, behind the trailing edge, where the two layers merge and the parcels meet the question equal transit time was originally about: they do not meet.

Who timed the layer #

Thwaites’s integral method is from 1949 and Pohlhausen’s profile from 1921. Smoke-pulse photographs of the transit-time difference, with the upper pulse well ahead, are the visual centrepiece of Babinsky’s 2003 article on how wings work, taken outside the boundary layer; the earlier essay computed what they show. Timing parcels inside the layer at a fixed fraction of its thickness, and the observation that the result does not depend on the Reynolds number, follow from putting the two classical methods together.

Still open: the turbulent upper surface #

The comparison stops at the upper layer’s laminar separation. The next calculation continues it on a wing whose upper layer becomes turbulent before that point — a transition location chosen from a stated criterion, a turbulent integral method for the momentum thickness and the shape factor from there to the trailing edge, and Coles’s composite profile to place parcels at a fraction of the turbulent layer’s thickness — and asks whether the fuller turbulent profile pulls the upper parcels’ lead deep in the layer back towards the inviscid value, or whether the lower laminar layer, slower near its wall, keeps the exaggeration all the way to the trailing edge.