Four vortices bend faster, and bend the wrong one

A wake ends by bending, not by fading followed the two trailing vortices of a clean wing to their end. Each sits in the strain of the other, a bend of the right length grows in that strain, and in still air the pair pinches itself into a chain of rings: the fastest bend is 8.54 times the spacing long and grows by 0.827 e-folds each time the pair sinks by its own spacing, which for an airliner is every 27 seconds. The calculation had two vortices because a clean wing’s sheet rolls up into two.

A wing with its flaps down does not. The loading has a step at each flap’s outer edge, the sheet rolls up at each step as well as at each tip, and the wake leaves with four cores: a strong one at each tip and a weaker one at each flap edge, turning the same way as the tip vortex on its side. The earlier essay ended on that wake. Each vortex now has a close neighbour as well as a distant partner, the strains are larger, and the question is whether a wake like that — the wake of every airliner on approach, which is where the wake hazard matters — ends faster.

Four vortices, and a strain that turns #

The linearised theory is the same, extended from two vortices to any number. Every vortex is a straight line of circulation $\Gamma_i$ displaced by a small wave along its length, and the velocity at vortex $i$ has three parts: the strain of every other vortex, acting on $i$’s own displacement; the displacement of every other vortex, reaching $i$ through Crow’s kernels $\psi(kb)$ and $\chi(kb)$ in modified Bessel functions of the wavenumber times the distance; and $i$’s own bending, through the cut-off self-induction that stands in for its core. For two antiparallel vortices the matrix this builds is Crow’s, and its eigenvalues return his symmetric and antisymmetric growth rates to a part in $10^{10}$ at five wavelengths, which is the first check.

Four vortices add one thing two do not have. A pair of equal and opposite vortices sinks together at a steady speed, so its strain field is fixed and the stability matrix is constant in time. The tip and flap vortices on one side turn the same way, and two vortices turning the same way circle one another about their centroid of vorticity while the whole wake sinks. The strain each feels therefore rotates, and the linearised system is periodic in time. Its growth over one orbit is found the way any periodic system’s is: integrate the displacement equations together with the vortices’ motion over one period, starting from each of the eight independent displacements, and read the growth from the eigenvalues of the resulting map — the Floquet multipliers. The growth rate is the logarithm of the largest multiplier divided by the period.

Everything is measured in the units of the equivalent single pair: lengths in the spacing of the two sides’ centroids of vorticity, and time in the time the wake takes to sink by that spacing. A wake of given lift has a given total circulation per side and a given centroid spacing — that is what the pair records of the wing — so every four-vortex wake compared here carries the same lift as Crow’s pair. The cores are the earlier essay’s, a tenth of the spacing in radius.

Each side is a pair in orbit #

The first figure is the base flow. With a flap vortex of 0.3 of the tip’s strength, starting at 0.4 of the tip’s spanwise station, the pair circles its common centroid once in 0.91 of the time the wake takes to sink one spacing — for an airliner, about 25 seconds. The flap vortex travels a wide circle and the tip vortex a small one, in proportion to their strengths. With a stronger flap vortex, 0.5 of the tip’s, placed further inboard at 0.3, the orbit is wider and slower, 1.88 descent times.

These orbits are what flow visualisation behind flapped wings shows: the flap-edge vortex winds round the tip vortex, one or two turns, before the two merge into one. The merger is not in this calculation, which treats the cores as fixed, and the time it takes sets how long anything computed here can act. That limit is returned to below.

Twice as fast, on shorter waves #

The second figure is the result the question asked for. Crow’s pair has one band of unstable waves, peaking at 0.827 at 8.54 spacings. The flapped wake with a flap vortex of 0.3 at 0.4 has two. The first is Crow’s band, slightly weakened — 0.79 at 7.7 spacings. The second is new: a band of short waves, peaking at 1.57 on waves only 1.8 spacings long, growing almost twice as fast as anything in the clean wake. The wake with the stronger, further inboard flap vortex merges the two into one broad band peaking at 1.64 at 4.4 spacings.

So a flapped wing’s wake is more unstable than a clean one’s, by a factor close to two in growth rate, on waves between a fifth and a half of Crow’s. At an airliner’s scale that is an e-fold every 17 seconds instead of every 33. This is what the earlier essay anticipated from the strain alone — a close neighbour strains harder than a distant partner — and it is what experiments behind flapped wings have reported: a faster, shorter-wave distortion of the wake than a clean wing’s.

The narrow bands to the left of the new one, at shorter waves still, are resonances of the same kind at higher harmonics of the orbit, and they are weaker.

The two-dimensional motion underneath all of this is not complicated, and it is worth being clear that it is not the source of the new band. Four point vortices in mirror symmetry, once the common descent is removed, have three coordinates and two conserved quantities, their energy and their impulse, so the motion is confined to a closed curve and repeats, which is why it is periodic rather than chaotic, unlike the general motion of four vortices. A long wave, which displaces a whole vortex, merely moves the orbit to a neighbouring one, and nothing grows. The growth belongs to bends a few spacings long, short enough for the vortices’ self-induction and their neighbours’ kernels to differ along the bend, and long enough for the bend to be a bend rather than a wobble of the core.

The fast wave bends the flap vortex #

A growth rate says how fast a wave grows. It does not say what moves, and for a wake that matters more than anything, because the hazard to a following aircraft is the tip vortices — the strong ones — and the wake ends when the two tips are pulled together and link. The third figure reads the shape of the fastest wave at each wavelength from its Floquet eigenvector and plots the share of the motion the two tip vortices carry.

In Crow’s band, about eight spacings long, the tips carry half of the motion: tip and flap vortices on each side bend together, and the two sides bend as mirror images, which is the long bend that brings the tips together. In the new band near two spacings the tips carry 1.3 per cent. The wave is almost entirely the flap vortex, bent as it orbits the tip vortex, while the tip vortex stays straight. It is also not a mirror image across the wake: it is the antisymmetric shape, which does not bring the two sides together at all.

That is the result the growth rate alone would have hidden. The flapped wake is twice as unstable, and the extra instability is spent on the vortex that does not matter: it shortens the life of the flap vortex as a separate core, hastening its winding into the tip vortex, and leaves the tips to end by Crow’s instability at its old rate or a little slower. The strain argument was right about the rate and silent about the victim.

The same selectivity turns up wherever a weak vortex meets a strong one. A strained vortex holds its shape only until the strain reaches a fixed fraction of its own vorticity, so a weak vortex beside a strong one is the one that is pulled out of shape; and a vortex is a waveguide whose bending waves turn at a rate set by its own circulation — which suggests why the weak vortex responds: its bends turn slowly enough for the orbit’s strain to keep pace with them, while the strong vortex’s bends outrun it. The calculation shows the selectivity; this reading of it is an interpretation, not a separate result.

The reason it is the flap vortex is plain once stated. A weak vortex orbiting a strong one sits in a strain the strong one dominates, a strain that turns once per orbit, and a bend on it grows whenever the orbit’s timing lets the rotating strain stretch the bend repeatedly in the same sense. The strong vortex sits in a strain from the weak one, three times smaller, and barely responds. Equal vortices would share the motion; unequal ones hand it to the weaker.

Where the flap vortex sits decides the rate #

The fourth figure asks which flapped wakes are most unstable, the question the earlier essay posed as “which arrangements destroy a wake fastest”. It holds the lift — each side’s total circulation and centroid — and varies the flap vortex’s strength from a tenth of the tip’s to equal, at four stations from 0.3 to 0.6 of the tip’s.

Once the new band exists, its rate hardly depends on how strong the flap vortex is: at 0.4 of the tip’s station it is 1.53 to 1.58 for strengths from a fifth to equal. It depends on where the flap vortex sits: about 1.7 with the flap vortex far inboard at 0.3 of the tip’s station, 1.4 with it at 0.6, close to the tip. A weak flap vortex close to the tip — a tenth of the tip’s strength at half its station, or up to three-tenths at 0.6 — adds no new band at all, and the wake is Crow’s.

A flap schedule — the flaps’ extent and deflection — therefore moves the fast band’s rate by up to a quarter and its wavelength by more than a factor of ten, and it moves where the energy of the bend goes by nothing: in every co-rotating case the fastest wave is the flap vortex’s. No co-rotating arrangement at fixed lift makes the tips link faster than Crow’s pair does.

A counter-rotating inner vortex is another matter #

There is a four-vortex wake in which the tips do take part, and it is the one a flapped wing does not make. If the inner vortex on each side turns the other way — which happens when the loading inboard is pulled down, by a negatively loaded section or by a horizontal tail carrying a download close behind the wing — the fifth figure shows every wavelength growing, and fast. At the longest wave drawn, thirty spacings, the rate is 2.5, 4.3 and 6.6 for inner vortices of a tenth, a fifth and three-tenths of the tip’s strength: three to eight times Crow’s fastest. At shorter waves it grows faster still, up to the scale of the cores, where the cut-off model stops being trustworthy.

The growth at long waves is the tell. A bending instability vanishes as the wave gets longer, because a very long bend is just a displacement of the whole vortex and does nothing to its neighbours. Growth that survives at long waves means the two-dimensional motion itself is unstable: the symmetric dance of four counter-rotating vortices does not survive a sideways nudge, and the pairs on each side veer off it and are thrown about. This is the configuration that proposals for active wake alleviation reach for — wakes deliberately given a counter-rotating inner pair, or flap loads oscillated to excite the fast bands — and it is why they reach for it. It is not the wake of a wing that simply has its flaps down.

An airliner on approach #

The numbers become concrete at an airliner’s scale. The earlier essay’s heavy airliner shed a pair of circulation 508 square metres a second 47 metres apart, which sinks one spacing in 27 seconds. On approach, flaps down, a flap vortex of about three-tenths of the tip’s strength at 0.4 of its station orbits the tip vortex once every 25 seconds, and the fast band e-folds every 17 seconds on waves about 85 metres long — against Crow’s 33 seconds on waves 400 metres long.

The comparison that matters is between the fast band and the merger. If the flap vortex winds into the tip vortex within one or two orbits — a minute or less behind the aircraft — the fast band has one and a half to three e-folds to work with, a factor of four to twenty on whatever bend the atmosphere seeded. That is a large distortion of the flap vortex and a small one of the tips. The pair that remains after the merger then proceeds as Crow’s pair would, linking after the two to three minutes the earlier essay found, because the drag the wake keeps however it rolls up is the same drag and the pair’s circulation and spacing are the same lift.

What this predicts for the separation standards between landing aircraft is therefore nothing new: the flapped wake’s extra instability is spent before the wake is old enough to matter to the aircraft behind. It is the counter-rotating configuration, which a normal landing does not produce, that would change the answer.

What was checked #

The ledger holds five checks and three results. Two antiparallel vortices reproduce Crow’s rates to $10^{-10}$. The Bessel kernels, read from a table for speed, match the direct quadrature to $10^{-10}$. A flap vortex of a millionth of the tip’s strength reproduces Crow’s fastest rate to $2\times10^{-7}$, which tests the Floquet calculation against the constant- matrix answer through a genuinely orbiting base flow. The linearised system has no trace — the strain terms are symmetric and the self-induction is a rotation — so the determinant of the one-orbit map must be one, and it is to $3\times10^{-8}$. And halving the integration step moves the growth rate by $2\times10^{-7}$.

What a line-vortex wake leaves out #

Merger. Real tip and flap vortices merge after one or two orbits, when their cores grow to a sizeable fraction of their separation, as two rings that leapfrog end by merging rather than parting. The four-vortex phase then ends and the wake is Crow’s pair. The fast band can only amplify a bend for the time the orbit lasts; at 1.57 e-folds per descent time and an orbit of 0.91, that is about one and a half e-folds per orbit.

Fixed cores. The cut-off method assumes a thin core that keeps its size. A flap vortex bent strongly on a short wave approaches the core’s own scale, where the core’s internal waves and its elliptic instability take over.

Uneven sinking. The flapped wake sinks faster near the ground and slower with the wing’s downwash still acting close behind the aircraft; the calculation begins from a rolled-up wake in free, still air.

Linear amplitudes. Every rate here is for small bends. The linking of the tips, and the flap vortex’s winding into the tip vortex, are nonlinear events the rates only lead up to.

The convention: the equivalent pair’s units #

Lengths are in the spacing of the two sides’ centroids of vorticity; time in the time the wake sinks by that spacing, $2\pi b_0^2/\Gamma_0$ with $\Gamma_0$ each side’s total circulation; growth rates in e-folds per that time. The flap vortex’s strength is a fraction of the tip vortex’s on the same side, and its station a fraction of the tip vortex’s distance from the centre plane. The cut-off length is $0.642a$ for a core of radius $a$, as for Kelvin’s uniform core.

Who worked it out #

Crow’s analysis of the vortex pair is from 1970. Jimenez in 1975 showed that a co-rotating pair is stable to Crow’s long bend, which is why the tips here keep their old rate. Crouch in 1997 carried the long-wave analysis to two pairs of unequal vortices with Floquet theory and found fast instabilities behind flapped wings; Fabre, Jacquin and Loof extended it and examined counter-rotating inner vortices for wake alleviation in the early 2000s; Crouch, Miller and Spalart proposed oscillating the flap loads to excite the fast bands, and tested the idea in towing tanks.

Still open: whether the flap vortex’s bend reaches the tip before they merge #

The fast band bends the flap vortex and not the tip, in the linear theory. But the flap vortex is winding into the tip vortex, and when it merges it brings its bend with it. The next calculation follows the four filaments past the linear stage — thin cores advected by their own Biot–Savart velocity, the flap vortex’s bend seeded on the fast band — through the flap vortex’s merger, and asks how much of its bend the merged tip vortex inherits: whether a merger that happens while the flap vortex is strongly bent leaves the tip vortex with a seed on the fast band’s wavelength, which Crow’s instability would then grow from a larger start, or whether the merger averages the bend away.