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

The story about air meeting up again is that two parcels separated at a wing’s leading edge must rejoin at its trailing edge, so the upper one, with further to go, must go faster. They do not rejoin: the upper parcel leads by the circulation, arriving first by an amount set by the lift. Inside the layer the upper parcel leads by more followed parcels into the laminar boundary layers and found the lead growing with depth. And the side that turns turbulent first decides who arrives first near the wall gave the layers their transition: at a wing’s ordinary incidence the upper layer turns turbulent first, a turbulent layer is full almost to the wall, and the upper parcel’s lead near the wall became more than ten times the lead outside.

Every one of those parcels was held at a fixed fraction of its layer’s thickness, from the attachment point to the trailing edge. That is a convention, and the last essay said so: a real filament of smoke follows the flow, starting at a height and carried by the velocity there, drifting outward only with the layer’s slow normal velocity. It asked whether a filament that starts a little above the wall stays deep enough 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.

Neither, and the answer reverses the held parcels’ verdict. A filament starts outside the layer, not inside it: at the attachment point the layer is thinner than any filament’s height. The layer grows up round it, and the upper layer, turbulent and in an adverse gradient, grows faster and swallows its filament deeper. Followed this way, the parcels near the wall at 4° arrive nearly together.

A particle is a streamline #

The section and its layers are the last essay’s: a thin cambered Joukowski section, 6 per cent thickness parameter and 4 per cent camber, at a chord Reynolds number of a million. Each layer is laminar from the attachment point by Thwaites’s method, with Pohlhausen’s quartic profile, until Michel’s criterion marks its transition; it is turbulent after by Head’s entrainment method, with Coles’s composite profile. At 4° the upper layer turns turbulent at 10.6 per cent chord, and the lower stays laminar to the trailing edge.

What changes is the parcel. The flow is steady, so a fluid particle moves along a streamline, and inside a boundary layer a streamline is the line under which a fixed mass flux runs. At each station the stream function at a height is the edge speed times the integral of the profile up to that height; the particle on a given streamline is at the height where that integral matches its flux, and it moves at the local speed there. Above the layer the profile is uniform and the stream function grows with height at the edge speed. Each particle’s height is therefore read off the local profile at every station, its speed with it, and its time to each station follows.

The upper and lower particles start together just off the attachment point, on streamlines carrying the same flux either side of the dividing streamline that runs into the attachment point. The flux is given as a share of what the upper layer carries at the trailing edge: a streamline at a twentieth is deep in the upper layer by the end, one at one is at its edge, one beyond one passes outside it. Along its path the flux beneath each particle stays at its value to two parts in a thousand, which is the check that the particle really is on its streamline.

The answer, by depth #

The first figure is the result at 95 per cent chord, as a multiple of the lag outside both layers — the circulation’s lag, the one the earlier essays used as their unit. Far out, at a streamline beyond the upper layer’s trailing-edge flux, both curves are at one: outside the layers the followed particles are the inviscid ones.

At 4° the curve falls as the streamline goes deeper. At a fifth of the upper layer’s flux the lag is 0.47 of the outside one; at a tenth, almost zero; at a twentieth, −0.16 — the lower particle arrives first, by a sixth of the outside lag. At the deepest streamline drawn, a fiftieth, it is 0.46. The held parcels of the last essay gave the opposite shape at this incidence: a lead growing with depth to more than ten times the outside one.

At 0° the followed curve rises instead: deep particles lag by 1.5 to 1.7 times the outside lag, the upper leading by more than outside. The held parcels at 0° had the lower particle first near the wall. Following the flow reverses the depth dependence at both incidences.

Where the filaments enter #

The second figure shows why. It follows the two particles of the streamline a twentieth of the way into the upper layer’s trailing-edge flux and plots each one’s height as a fraction of its own layer’s local thickness. Both start above one — outside their layers — because at the attachment point the layer is only Hiemenz’s thin stagnation-point layer, and a streamline carrying any finite flux runs above it. Both then fall below one as the layers grow up round them: the upper particle at 3.6 per cent chord, the lower at 15.

From there the upper particle sinks deeper and faster. Its layer turns turbulent at 10.6 per cent and, in the adverse gradient behind the suction peak, thickens rapidly; a turbulent layer entrains fluid from above, so a particle on a fixed streamline finds ever more of the layer above it. By 95 per cent chord it sits at 0.11 of its layer. The lower particle’s layer stays laminar and thin, accelerated by a favourable gradient, and the lower particle ends at 0.35 of it — three times higher in its own layer, where the profile is faster.

The dial moves the streamline. A fiftieth of the flux enters both layers earlier and ends deeper; a fifth enters the upper layer late and ends at a third of it; one, the upper layer’s edge flux, stays outside the lower layer altogether. Every filament, however close to the wall it is aimed, begins in the inviscid race.

Held at a depth, and followed to it #

The third figure puts the two conventions side by side, against the upper parcel’s depth at 95 per cent chord. Held at a tenth of the layer, the parcels lag by 10.4 times the outside lag: the upper parcel, in a full turbulent profile, races; the lower, at the same tenth of a laminar profile, crawls. Followed particles that end at a tenth of the upper layer lag by −0.16.

The difference is history and partner. A held parcel spends the whole chord at its fraction — slow, deep in the layer, from the very start — and the held lower parcel is at the same fraction of its own layer. A followed particle spends the first part of its journey outside the layer, in the inviscid race, near the suction peak where the two surfaces’ speeds differ most; it reaches its deep position only near the end. And the lower particle on the same streamline is not at the same fraction of its layer but three times higher. The held convention compares two parcels at equal depth; smoke compares two parcels on equal mass flux, which the two layers spread over very different depths.

Along the chord #

The fourth figure follows the same streamline’s lag along the chord. At 4° it starts near the outside value, 0.92 at 15 per cent chord, where both particles have only just entered their layers; it falls as the upper particle sinks into its thickening turbulent layer, to 0.37 at mid-chord and −0.16 at 95 per cent. At 0° the opposite: 1.1 at 15 per cent, 1.6 at mid-chord, 1.7 near the trailing edge. At 0° the lower layer turns turbulent first, as the last essay found, at 35 per cent chord against the upper’s 51. Both particles enter their layers near 4.5 per cent, but from mid-chord it is the lower layer that thickens faster and buries its particle deeper, and the lower particle that loses time there — the reverse of the 4° case, through the same mechanism.

The pattern is the one the transition order sets, read through the layer’s thickness rather than its fullness. The held parcels said the side that turns turbulent first wins near the wall, because its profile is fuller. The followed particles say the side that turns turbulent first loses near the wall, because its layer is thicker and swallows its particle sooner and deeper. Both are true of their own parcels; only the second is true of smoke.

Across the Reynolds numbers of light wings #

The fifth figure varies the chord Reynolds number. At $3\cdot10^5$, a model aircraft or a small drone, the upper particle still leads at every depth — by 0.39 of the outside lag at a twentieth of the flux, and by more than the outside lag at the deepest streamline — because the laminar stretch before transition is longer in proportion and the turbulent layer has less chord in which to grow. At $10^6$ and $3\cdot10^6$ the near-tie appears: −0.16 and −0.17 at a twentieth, and at $3\cdot10^6$ the deepest streamline drawn is also slightly negative.

A higher Reynolds number thins every layer, so a streamline at a given share of the trailing-edge flux sits at about the same depth, but the transition moves forward and the turbulent part of the upper layer gets longer. The deep tie is a property of a wing with most of its upper surface turbulent, which is what full-size wings have.

Why the held lead was so large #

The held parcels’ lead of ten outside lags was not made at the start. Held at a tenth of their layers at 4°, the two parcels are only 0.4 time units apart by 10 per cent chord and 14 apart by 95 per cent: the gap opens steadily over the whole chord, because the lower held parcel, a tenth of the way into a laminar layer, moves at a fifth of the edge speed from end to end while the upper one, a tenth of the way into a full turbulent profile, moves at more than half. The held convention made the lower parcel crawl along the bottom of its layer for the whole journey.

A followed lower particle does not crawl. On the same streamline as an upper particle a tenth of the way into the thick turbulent layer, it sits at a third of the thin laminar layer beneath — three times higher in its own layer, in faster fluid — and that is most of the difference between the two conventions. The entry delay matters too: each followed particle spends its first few per cent of chord outside its layer at the edge speed, where held parcels were already deep, but the gap the held parcels opened was opened mostly later. The equal depth that the held convention imposed was the assumption doing the work, and how thick is thin is the reminder that two layers’ thicknesses are measured on different scales, so that equal fractions of them are not equal anything else.

It is the same lesson a layer that is an integral of everything upstream teaches about the layer itself: what is inside it at the trailing edge was put there along the way, and which fluid sits where depends on how much each layer has entrained.

What it says about the experiments #

Smoke and dye are the only way anyone has seen the transit lag, and the experiments that show the upper filament arriving first are conducted with filaments released upstream of the leading edge, often well above the dividing streamline, where the lag is the circulation’s. The line the dye actually draws is a streakline, and in a steady flow a streakline is a streamline, which is what makes the particles here the right model of it. What the calculation adds is that the near-wall filaments, if any experiment could release and follow them, would not show the dramatic lead the held parcels predicted; at a full-size wing’s Reynolds number and incidence they would arrive nearly together, and a few would be overtaken.

That leaves the equal-transit story where the earlier essays left it — refuted already by a flat plate, whose paths above and below are equal and which still flies — and makes one of its subtler defences fail. A defender might say that the air close to the wing, which actually makes the lift through the layer’s vorticity, might arrive together even though the outer air does not. The followed particles say that this is nearly true at 4° deep in the layer — and that it is a coincidence of two layers’ different thicknesses, true at one incidence and false at another, and false at model scale. Nothing about the lift depends on it.

The convention: streamlines and their flux #

A streamline is labelled by the mass flux between it and the dividing streamline, per unit span, as a share of the flux the upper layer carries at the trailing edge. Heights are fractions of each layer’s local thickness, the Pohlhausen thickness where laminar and Coles’s where turbulent, so they jump where a layer’s thickness does, at transition, as the last essay’s did. The lag is the lower particle’s arrival time minus the upper’s at a station, over the same difference for particles far outside both layers. Times start when both particles are a thousandth of a unit of arc length from the attachment point.

What the calculation leaves out #

The layer’s normal velocity, and the wall’s curvature. The particle’s height is found from the stream function, which includes the layer’s displacement of the streamlines; but the arc length along the surface is used as the particle’s path length, which ignores the small extra distance a particle at a height travels round a curved surface.

The transition region. Michel’s criterion puts transition at a point and Head’s layer starts there; a real transition takes a few per cent of the chord, and a particle crossing it sees a profile that is neither laminar nor turbulent. Particles that are swallowed near the transition point carry that uncertainty.

Unsteadiness. A turbulent layer’s particles do not follow the mean streamlines: they wander by the size of the large eddies, and a real smoke filament in a turbulent layer is dispersed across a large part of the layer within a few thicknesses. The followed particles here are mean-flow particles, which is what a time-averaged smoke picture shows and not what an individual filament does.

Who timed the layer #

The transit lag outside the layers is Kutta and Joukowski’s circulation, timed by many authors since the equal-transit story was first debunked; Babinsky’s smoke photographs of 2003 are the most reproduced demonstration. The boundary-layer methods are Thwaites’s, Michel’s, Head’s and Coles’s. The followed-particle calculation, and the finding that it reverses the held parcels’ depth dependence at both incidences, is this calculation’s, for the section and layers stated.

How it was checked #

Each row’s profile, integrated by quadrature, gives the momentum thickness over thickness its method states, to a part in a thousand. The flux beneath a followed particle stays at its streamline’s value along the whole path, to two parts in a thousand, which is the definition of a particle on a streamline. And particles forty times the upper layer’s trailing-edge flux away from the dividing streamline reproduce the inviscid lag exactly.

Still open: the filament a turbulent layer disperses #

The particles here follow the mean flow, and inside a turbulent layer no real particle does: eddies the size of the layer carry it up and down across a large part of the thickness within a few thicknesses of travel. A filament of smoke entering the upper turbulent layer is therefore spread into a cloud whose particles sample the whole profile, fast and slow. The next calculation would give the turbulent particles a random vertical motion with the eddy viscosity’s diffusivity, release clouds rather than particles, and ask how wide the spread of arrival times becomes at the trailing edge — whether the lag between the clouds’ centres is the followed particles’ near-tie, and whether the spread within each cloud is larger than the lag between them, which would make the question of who arrives first a question about distributions.