A trailing wake hurries a pair together

Drag makes every free pair meet let two cylinders loose in an ideal stream, as a free pair turns, and forty-five degrees decides had, and gave each one a drag on its motion relative to the stream. The ideal pair swings like a pendulum towards side by side and either collides or parts depending on which side of forty-five degrees it is released. The damped pair does not swing: a drag coefficient of order one takes the swing’s energy, lets no pair get far from where it started, and brings every one round to meet near side by side.

That left an awkward fact. Real pairs of bodies do not all meet. Spheres falling one above the other draft, touch, tumble, and then separate and fall side by side at a spacing that depends on the Reynolds number; two cylinders towed side by side at moderate Reynolds numbers push apart. Something a real pair has and the ideal pair lacks must be repulsive, and the earlier essay named the obvious candidate: the wakes. A cylinder beside another’s wake sits in a stream that is slower on one side, and a body in a shear is pushed towards the faster fluid — away from the wake. The question it left was whether that push, added to the damped pair, finds a spacing where it balances the stream’s pull, and whether that spacing tracks the Reynolds number as measured separations do.

The push is there, and it is strong. But it does not hold the pair apart. It points the wrong way, and the reason is geometric: a wake is behind its body, and a pair side by side is not behind anything.

Two wakes that carry their drag #

The wake model is the simplest one that is honest about momentum. Behind each cylinder the stream has a deficit, a Gaussian across the stream whose width grows by diffusion with distance behind the cylinder, as Oseen’s far wake does, and whose integral across the stream is the cylinder’s drag. A wake is a momentum deficit, and a wake that keeps the drag and forgets the body found that far enough behind, the deficit’s integral is all a wake remembers of what made it; that integral is fixed here at the drag. The width starts at the cylinder’s radius and grows with the square root of distance, at a rate set by the Reynolds number on the diameter, so at a Reynolds number of 200 the wake is still only a radius and a third wide twenty radii back, and deep. Ahead of a cylinder and level with it there is no wake at all; behind it the deficit is switched on smoothly over the first radius.

A cylinder sitting in the other’s wake feels two forces. It meets slower water, so its drag is less than the leader’s, and the difference draws it forward relative to the leader — the drafting cyclists and racing cars use. And it sits in a stream faster on one side than the other, which pushes it towards the faster side. For a circular cylinder in a weak inviscid shear that push is the fluid’s density times the cylinder’s area times the stream’s speed times its gradient, with a coefficient of one, the cylinder’s added-mass coefficient, as Auton’s coefficient is a half for a sphere. The same coefficient is no accident: the force of getting going found that the added mass is the fluid’s inertia a body drags with it, and a body held in a shear is accelerating that borrowed fluid across the stream’s gradient. Each force acts on the cylinder in the wake, and the pair’s relative motion feels half the difference of what the two cylinders feel. Everything else — the ideal stream’s attraction from the images, the two added masses, the anisotropic drag on the relative motion — is the earlier essays’, unchanged, with the drag coefficient taken from the Reynolds number by White’s fit, about 1.3 at 200.

The follower is thrown out, not held off #

The first figure releases pairs six radii apart at small angles from tandem, one cylinder nearly behind the other, at a Reynolds number of 200, and plots the follower’s path relative to the leader. Without wakes, the faint paths, a near-tandem pair first drifts apart: the ideal stream pushes a tandem pair apart, and released at two degrees the pair reaches thirteen radii before the slow turn brings it round and the drag lets it close. With wakes the paths are completely different. The follower draws in towards the leader, then swings sharply out to the side, and from there turns towards side by side and closes. No pair with wakes gets beyond 6.3 radii, and every one touches, at between 84 and 85 degrees from the stream — close to side by side, where the pairs without wakes touch too.

The second figure follows the pair released at two degrees through time, and the motion has three stages. First drafting: the follower meets the leader’s wake head-on, its drag drops, and the spacing closes from six radii to 4.75 in about five radii of travel. Then the throw: a follower two degrees off the wake’s axis sits on the wake’s flank, where the stream speeds up across it, and the shear lift pushes it outward. That push grows as it moves out, because the flank is steepest a little off the axis, and by seven radii of travel the pair is twenty degrees from tandem. Then the close: out of the wake, the follower is back in the ideal stream’s hands, which turn the pair towards side by side and draw it together, and it touches at 84.5 degrees after 76 radii of travel. Without wakes the same release takes 2,006.

The sequence is the one falling spheres show — drafting, kissing, tumbling — with two differences. The cylinders turn before they touch rather than after, because the throw starts as soon as the follower is off the axis; and the fourth stage, the separation, is missing.

The meeting brought forward #

The third figure measures how much sooner. It plots the time to touch against the release angle at Reynolds numbers of 50 and 1,000. Without wakes the time climbs steeply towards tandem: a pair released nearly in line spends a long time drifting apart and coming back, 2,500 radii at two degrees and a Reynolds number of 50. With wakes the same release touches in 39. The wake brings the meeting forward by a factor of seven to fourteen at five degrees and of twenty-six to sixty-four at two, and the factor is larger at the lower Reynolds number, whose wakes are wider and whose drag coefficient is higher. At ten degrees it is still a factor of three or four.

Away from tandem the curves join. A follower released thirty degrees off the axis at a Reynolds number of 1,000, or forty at 50, starts outside the leader’s wake and stays outside it, and the wake changes nothing. So the wake is not a small correction spread over all releases. It is a large effect confined to a cone behind each cylinder, and inside that cone it is entirely attractive in its consequence: it makes the pair meet sooner.

Why the push is the wrong way #

The shear lift is a repulsion from the wake’s centre line. It is not a repulsion from the other cylinder. Those two directions agree only for a follower directly beside the wake — and a follower directly beside the wake, at the wake’s own station, is not in the wake at all, because the wake starts behind the leader. For every follower that does feel the wake, outward from the wake’s centre line is round towards side by side.

The fourth figure makes this exact. It plots the force that turns the pair — the torque on its line of centres — against the angle from the stream, at a spacing of six radii, for the ideal stream alone and with the wakes added. The ideal stream’s torque is positive everywhere between tandem and side by side: a free pair turns, and forty-five degrees decides found that the stream turns a pair to look as wide as possible across it, the pair’s version of a body with no lift and a moment anyway. The wake’s torque is positive too. It peaks at nine degrees, where it is 31 times the stream’s, adds six per cent at thirty degrees, and is gone by thirty-five. The two turn the pair the same way. A force that could hold a pair apart would have to push it back towards tandem against the stream’s couple, and nothing in a wake that trails behind its body can do that.

The one balance, in tandem #

There is one place the wake opposes the ideal stream, and it is the line of centres in tandem. There the stream pushes the pair apart — the earlier essays found that a tandem pair parts — and drafting pulls the follower in.

The fifth figure plots the force along the line of centres for a tandem pair against spacing, for the ideal stream alone and with drafting at five Reynolds numbers. The ideal push falls off as the inverse cube of the spacing. Drafting falls off much more slowly, since a narrow, deep laminar wake holds its deficit for tens of radii. Below a Reynolds number of about 109 drafting wins at every spacing: the follower is drawn in all the way to contact. Above it the two cross once, within a radius of contact, where the images’ push becomes strong — at 2.25 radii, a quarter-radius gap, at a Reynolds number of 200, and at 2.76 at 20,000.

The sixth figure follows that balance across Reynolds numbers, and measures what kind of balance it is. Along the line of centres it is stable: closer, the images push the follower back out; farther, drafting draws it back in. Across the line it is unstable: a follower nudged off the axis is thrown further off by the shear lift and turned further by the stream, with a stiffness of about three per radian at every Reynolds number. It is a saddle. A pair balanced there would hold its spacing and lose its alignment, and the alignment it loses goes, as the fourth figure showed, towards side by side, where there is no balance at all.

That answers the earlier essay’s question directly. The wakes do create a spacing at which the push balances the pull, and the spacing does depend on the Reynolds number through the wake’s width. But it is a tandem gap of a fraction of a radius, it is unstable to the smallest turn, and the side-by-side separation of falling or towed pairs — several diameters, and stable — is nowhere near it.

What the calculation was checked against #

With the wakes switched off the pair is the earlier essay’s to the last digit: released at fifty degrees with a drag coefficient of one, it touches at the same time and angle. The wake’s deficit integrates across the stream to the drag coefficient at three distances behind the cylinder to a part in a thousand trillion, after the switch-on factor is divided out. The two forces’ signs were checked at single points: the shear lift points away from the wake’s centre on both sides and vanishes on it; drafting reduces the follower’s drag; and a cylinder ahead of another feels nothing from it. The checks refuse a release inside contact, a Reynolds number of zero and a negative drag coefficient.

The integration is the earlier essays’ fourth-order scheme with the step shortened near contact. Its own checks — the energy budget of the damped pair, the overdamped limit against an independent first-order integration — carry over, since the wakes enter only as a force.

What the wake model leaves out #

The near wake. The deficit here is a far wake’s, Gaussian and steady. Within a few diameters of a cylinder at these Reynolds numbers the real wake is a recirculating bubble, and above about 47 it sheds vortices — the street that a point-vortex model of the Kármán wake stands in for. A follower in a leader’s vortex street feels an oscillating force, sideways as well as along the stream, which the time-averaged deficit replaces with its mean; whether the oscillation, rectified by the follower’s own response, adds a mean push the average misses is a question this model cannot ask. Drafting in a recirculating bubble is stronger than any far-wake deficit, so the tandem gap here is if anything too large.

Turbulent wakes. At the higher Reynolds numbers the wake is turbulent and spreads far faster than the laminar width used here, with a shallower deficit. That weakens drafting and moves the tandem balance out, but it cannot change the geometry: a turbulent wake is still behind its body.

The flow between the bodies. Side by side, the real repulsion at moderate Reynolds numbers comes from the gap between the cylinders: the boundary layers on the facing sides, the jet through the gap, and the way each cylinder’s shed vorticity deflects the other’s. Those are near-field and viscous, and they are what a model built from the ideal stream plus far wakes cannot hold. The earlier essay said “the wakes” and meant all of this; this calculation separates the far wake from the rest and finds the far wake attractive.

Rotation. Real pairs spin as they tumble, and a spinning cylinder in a stream carries a circulation and a lift. The ideal pair here does not rotate.

Three stages of four #

The ideal pair, damped and given wakes, reproduces three of the four stages falling pairs are known for. It drafts, because a wake is a deficit and a follower in it meets less drag. It kisses, or nearly does, and it tumbles, because the stream’s couple and the wake’s throw both turn it towards side by side. The fourth stage, the separation that leaves real pairs falling side by side at a spacing, it cannot reach, and now the reason is sharper than the earlier essay could make it. The separation needs a force that grows as a pair approaches side by side; every force in the ideal stream and in a trailing wake vanishes or attracts there.

That is no drag at all’s paradox at one more remove. The ideal theory gets the forces between bodies right where they are inviscid, as the paradox is for the pair, not each body showed, and its viscous corrections, added in the simplest forms that conserve momentum, extend it — drag damps the swing, the wake explains drafting. What it cannot supply is a viscous effect that acts across the stream between two bodies side by side, because in this model viscosity only ever acts behind a body. The Reynolds number decides the wake’s width and its drag; it does not, in a far-wake picture, decide anything about the space beside a body.

Who worked it out #

The drafting–kissing–tumbling sequence was named by Fortes, Joseph and Lundgren in 1987 from spheres falling in a tube, and Joseph’s group made it a standard test of particle-resolved simulation; side-by-side cylinder pairs and their gap flows have been measured since Zdravkovich’s reviews of the 1970s and 80s. The Gaussian far wake is the similarity solution of the linearised boundary-layer equations, Oseen’s and Goldstein’s; the inviscid shear lift is Auton’s of 1987, given for a sphere, with the cylinder’s coefficient following from its added mass. The ideal pair’s potential and its added masses are the method of images going back to Hicks in the 1880s.

Still open: the gap between two cylinders side by side #

The repulsion this model lacks is in the gap. Two cylinders side by side at a few diameters’ spacing squeeze the stream between them into a faster jet, and their boundary layers on the facing sides are thinned and their separation delayed, which tilts each one’s drag outward. The next calculation keeps the ideal pair but gives each cylinder a separation point set by the local stream on each side, from the images’ surface velocity, and lets the asymmetric separation give each a sideways force; it asks whether that force is repulsive side by side, at what spacing it balances the ideal stream’s attraction, and whether that spacing grows with the Reynolds number as the measured separations of falling pairs do.