The upper parcel leads by the circulation

The story about air meeting up again took apart the most widely taught account of lift — that air going over a wing’s longer upper surface must speed up to rejoin the air beneath it at the trailing edge. It tracked parcels through a solved flow and found the upper one arriving first, by a wide margin. Then it did something instructive: it gave up on transit time as a measurement. A parcel released exactly on the stagnation streamline never arrives, because it slows to a stop at the nose; one released a hair off it takes as long as one likes to get past; so the essay compared speed ratios instead, which depend on nothing arbitrary. The wing that is flat and flies then tested the shape premise directly.

The objection to transit time is correct for a single parcel. It is not correct for the difference between two, and that difference turns out to be one of the cleanest numbers in aerodynamics: on a thin section it is the circulation divided by the square of the speed. This essay computes it.

Two parcels, a whisker apart #

Release two parcels thirty chord-lengths’ worth of units upstream of a Joukowski section, at the same moment, a small stream-function increment $\varepsilon$ above and below the streamline that divides at the nose, and time each until it crosses a line the same distance downstream. Both drift towards the stagnation point at the nose and slow almost to rest there; then one is carried over the top and the other underneath. The upper one crosses the far line first, by 1.13 time units for this section — 0.28 of the time the stream takes to cover a chord. Behind the section both travel at the stream’s speed, side by side, and the gap between them, once open, stays open.

That is the claim the old story denies, made quantitative: the parcels do not meet, and the upper one is ahead by a definite amount.

Why the difference exists when the times do not #

Each transit time diverges as the parcels are released closer to the dividing streamline. Near the stagnation point the flow is a hyperbolic strain — speed proportional to distance from the point — and a parcel that arrives at a distance proportional to $\varepsilon$ spends a time proportional to $\ln(1/\varepsilon)$ getting past. The figure’s left panel shows it: both times grow by the same amount for every tenfold reduction in $\varepsilon$.

But both parcels pass the same stagnation point, one on each side, and the strain there is the same for both. So the divergent part of their crawl is the same, and it cancels in the difference. The right panel shows what is left: 1.1383 time units at $\varepsilon = 10^{-2}$, 1.1270 at $10^{-4}$ and 1.1268 at $10^{-5}$. At the trailing edge there is nothing to cancel, because the Kutta condition makes the flow leave the edge smoothly with no stagnation point there. The quantity the first essay could not define is well defined as soon as it is defined as a difference.

The circulation, as a time #

The linear estimate is short. Along a path at speed $U + u'$, the time to cover $dx$ is $dx/(U + u') \approx dx/U - u'\,dx/U^2$. Subtract the upper path’s time from the lower’s: the $dx/U$ terms cancel, and what remains is $(1/U^2)$ times the integral of $u'$ round the loop the two paths make together, which is the circulation $\Gamma$. So

$$\Delta t \approx \frac{\Gamma}{U^2},$$

and since a section’s lift per unit span is $\rho U\Gamma$, the lag is the lift divided by $\rho U^3$ — or, for a section of chord $c$ at lift coefficient $C_L$, the time the stream takes to cover half a chord multiplied by $C_L$. A wing at a lift coefficient of one makes the parcel over its top arrive half a chord-transit-time early. The circulation that Kutta–Joukowski turns into lift is the same number that the parcels turn into a lead.

In seconds #

The lag is short in absolute terms, which is part of why it went unmeasured. A light aircraft’s wing of 1.5-metre chord at fifty metres a second and a lift coefficient of 0.4 has $\Gamma/U^2$ of six milliseconds, and with its thickness taken into account about four and a half. An airliner’s five-metre chord at 230 metres a second and a lift coefficient of a half: about four milliseconds. A sailplane thermalling at twenty-five metres a second with a lift coefficient near one, over an 80-centimetre chord: about fourteen. What makes the lag visible is not its duration but the distance it opens — the stream’s speed times the lag, which is half the lift coefficient in chords and so a large fraction of the wing whatever the speed. Babinsky’s smoke pulses are photographed at a fixed instant, and what the photograph records is that distance.

Path length does not enter #

A flat plate has an upper and a lower surface of exactly the same length, and it flies. Its parcels have exactly the lag its circulation gives, to 0.4 per cent at small incidence. A thin circular arc at zero incidence, whose upper surface is longer than its lower, carries the same lag for the same circulation: 1.001 of $\Gamma/U^2$. So the lag depends on the circulation and not on any difference in path length, at first order. Path length enters only at second order, through the speed the thickness and camber give both surfaces, which is the shortfall the thick sections show — and there it reduces the lead, the opposite of what the story’s longer-path reasoning would have it do.

Where the estimate holds #

The estimate’s neglect is the square of the perturbation velocity, so it should hold on a thin plate at small incidence and fail in proportion to how large the perturbations are. It does. On a plate a thousandth of its chord thick the lag is $\Gamma/U^2$ to 0.4 per cent at half a degree, and falls to 0.94 of it at ten degrees — a second-order effect of the incidence, as the expansion says. A thin circular arc at zero incidence, cambered and lifting but with no thickness, gives 1.001.

The symmetric section 11.8 per cent thick is a different story. It keeps between 0.66 and 0.69 of the estimate at every incidence from half a degree to ten. Its shortfall has nothing to do with incidence and everything to do with its thickness.

Thickness hides a third of the lag #

Swept through thickness at a fixed four degrees, the lag falls steadily: 0.82 of the estimate at six per cent, 0.68 at twelve, 0.56 at seventeen. The reason is in the expansion’s neglected term. The circulation makes the upper surface faster and the lower slower by about the same amount; the thickness makes both surfaces faster, by a perturbation that does not change sign. A time is a distance divided by a speed, so a difference in times is a difference in speeds divided by the square of the speed, and on a thick section the speed in that denominator has been raised by the thickness on both sides. The lag is the circulation divided by the square of the speed the parcels actually travel at, near the surface, not the stream’s.

This is the mirror image of the first essay’s measurement. There, the speed ratio between the surfaces was the honest quantity and the transit time was hopeless; here the transit time is honest, and it carries the thickness’s effect on the speed that the ratio of speeds hid.

The gap that never closes #

Once both parcels are behind the section and moving at the stream’s speed, the upper one is ahead by $U\Delta t$, a distance that never closes because nothing downstream distinguishes the two. On a thin plate that distance is half the lift coefficient in chords — 0.28 chords at a lift coefficient of 0.58 — and on the cambered 11.8 per cent section about seven-tenths of that. A wing of five-metre chord cruising at a lift coefficient of a half leaves the parcels that straddled its nose about a metre apart, for ever.

That is the precise sense in which the equal-transit story is wrong, and it is stronger than “the upper parcel arrives first”. The story requires the gap to be zero, as a principle. The flow makes the gap proportional to the lift, so a wing that obeyed the principle would be a wing that did not lift.

A broken line of smoke #

The same lag has a picture that needs no timing at all. Release a vertical line of smoke upstream of the wing, every point at the same instant, and follow it. Far from the wing it is carried along unchanged. Near the dividing streamline the points just above and just below take different times to pass the section, so when they are behind it, they are not in a line: the line is broken at the wake, with the upper part ahead of the lower by $U\Delta t$. The break is the gap in the last figure, drawn by a whole column of parcels at once — a streakline’s cousin, a timeline, carrying the circulation as a step.

The step also tells the history of the wing. A wing started from rest takes its circulation over a few chords of travel, as the lift that arrives late computed, and sheds an equal and opposite starting vortex into the wake; a timeline laid down before the start and carried over the wing records the circulation the wing had while the line was passing it. A line that passes during the start is broken by less than one that passes later, and the growth of the step along a sequence of timelines is the growth of the circulation, read off the smoke.

Along a real wing’s span #

On a finite wing the circulation varies along the span, from its largest near the root to zero at the tips, and the lag follows it station by station: parcels straddling the leading edge near the root arrive with the largest lead, those near the tip with almost none. An elliptically loaded wing’s lag has the ellipse’s shape. The broken timeline of the last section is then broken by a step that varies along the span, largest in the middle, and the step’s spanwise change is the trailing vorticity that leaves the wing — the same bookkeeping the price of having ends does with circulation, written in parcels arriving at different times. A photograph of smoke pulses taken from above a finite wing would show the pulses that passed over the tips barely displaced from those that passed under, and the ones near the root far apart.

A parcel’s-eye account of the Kutta condition #

The lead has a reading that is easy to miss. With no circulation — a symmetric section at zero incidence — the two parcels arrive together, by symmetry, and the computed lag is exactly zero. It is the circulation, which the sharp edge decides, that makes one arrive first. So the equal-transit story is not merely wrong about the mechanism; it takes as given the one condition, equal arrival, that holds precisely when the wing makes no lift. A flow in which parcels did rejoin at the trailing edge would be the flow with the stagnation point moved off the edge to wherever the zero-lift circulation puts it, which is the flow the Kutta condition rules out.

How it was computed and checked #

The flow is the Joukowski map of a circle with a doublet and the Kutta circulation. Rather than mapping velocities into the aerofoil plane, where the trailing edge is a singular point of the map, each parcel is integrated in the circle plane at the velocity that moves its image correctly, the conjugate of the complex velocity divided by the squared modulus of the map’s derivative — finite at the trailing edge because the Kutta condition makes the circle plane’s velocity vanish there too, to $5\cdot10^{-17}$. A fourth-order Runge–Kutta step shortened near the circle resolves the stagnation crawl, and halving it changes the lag in the sixth figure. The start and finish lines are found by bisection on the stream function. A symmetric section at zero incidence gives a lag of exactly zero; the lag at $\varepsilon = 10^{-5}$ differs from that at $10^{-4}$ by two parts in ten thousand; and the thin arc’s 1.001 is the linear theory’s own value checked by the nonlinear calculation.

The convention: a lag measured between two straight lines #

Times are in units of a Joukowski map parameter divided by the stream speed, and quoted also in chord-transit times, the chord divided by the speed. The start and finish lines are thirty map units upstream and downstream, which is far enough that moving them changes the lag by less than the convergence figure’s resolution: the parcels are at the stream’s speed there, and any further distance adds the same time to both. The release offset is in stream function; the lag is lower parcel minus upper parcel, positive when the upper arrives first.

What the picture cannot show #

Real air at the nose of a wing is a boundary layer, and the parcels this calculation follows, an infinitesimal distance from the dividing streamline, are inside it — where they are slowed by friction and, near the trailing edge, carried into the wake. The lag computed here is that of the inviscid flow just outside the layer, which is the flow the lift is computed from. It is also a two-dimensional section; on a finite wing the trailing vortices add a spanwise drift and a downwash, and parcels over the top and underneath pass on different sides of the wake sheet as well as at different times. The sections are Joukowski’s, with cusped trailing edges; a real section’s finite trailing-edge angle puts a stagnation point there, and the parcels hugging the surface then crawl past the trailing edge too — symmetrically, since both arrive at the same stagnation point, so the difference still converges, but its value near the surface depends on how the edge is shaped. And the flow is incompressible: at high subsonic speed the upper surface’s supersonic pocket changes the speeds, and so the lag, by more than the thickness does here.

Who found it, and when #

The observation that the upper parcel arrives first is old and is in every careful account of lift; Babinsky’s smoke-pulse photographs of 2003 showed it directly in a wind tunnel, the upper pulses far ahead of the lower. The linear statement that the transit-time difference across a thin section is the circulation divided by the square of the speed follows in two lines from the definition of circulation, and has been used in teaching to replace the equal-transit argument with a quantitative one. The convergence of the difference, the effect of thickness, and the gap downstream are computed here.

Still open: the lag inside the boundary layer #

The parcels here are just outside the layer. The next calculation puts them inside it: a laminar boundary layer computed over the same Joukowski section with the outer velocity as given, parcels released at a stated fraction of the layer’s thickness on each side of the attachment point, and their times to the trailing edge compared. It asks whether the lag inside the layer — where the speeds are lower and the parcels nearest the wall barely move — is larger than the inviscid one, as slower speeds would suggest, or smaller, because the layer is thicker on the decelerating upper surface and the upper parcel is held back more; and at what height in the layer a smoke-pulse photograph like Babinsky’s reads the inviscid value.