Drag makes every free pair meet

A free pair turns, and forty-five degrees decides released two cylinders from rest in an ideal stream and followed them in both coordinates of their relative motion: the spacing between their centres and the angle of their line of centres to the stream. The stream’s pressure turns the pair towards side by side, the arrangement in which it looks biggest to the stream, and pulls a side-by-side pair together. The result was a pendulum that closes as it swings. Pairs released more than 45° from tandem collide, meeting at whatever angle the swing has reached; pairs released nearer tandem part for good. Far apart the dividing line is 45° exactly, because the pair’s potential is homogeneous of degree minus two and the Lagrange–Jacobi identity ties the curvature of the squared spacing to the sign of the energy.

Nothing in that calculation removes energy, which is what an ideal fluid means, and it ended by adding the simplest thing that does: a drag on each cylinder’s motion relative to the stream. The questions it posed were how strong the drag must be for a pair to settle side by side before it collides, whether pairs released beyond 45° from tandem still collide or are caught at a finite spacing, and whether that spacing depends on the Reynolds number the way the side-by-side spacing of falling spheres is measured to. The answers turn out to be simpler, and stranger, than the questions expected.

The drag a pair feels #

The stream in the pair’s frame is the speed at which the pair moves through the fluid — two particles settling under gravity, or two bodies towed. Each cylinder’s drag is the ordinary one, $\tfrac12\rho C_D (2a)|\mathbf U + \mathbf v|(\mathbf U + \mathbf v)$, where $\mathbf v$ is its velocity relative to the pair’s centre. When that relative motion is slow compared with the stream, the change in drag it causes is linear in it and not the same in both directions. A cylinder moving along the stream changes both the size and the direction of the flow it meets and feels twice the drag change of one moving across it, which only tilts the flow. In the units the earlier essays used — the cylinder’s radius, the stream’s speed and the mass of fluid a cylinder displaces — the relative motion gains a damping of $2C_D/\pi$ along the stream and $C_D/\pi$ across it.

This is a model, and it says what it is. It keeps the ideal stream’s pressure forces between the cylinders — the doublet images of a pair set free in a stream, the two added masses of closing and of sliding — and adds a drag with no wake behind it. Each cylinder feels the free stream, not its neighbour’s wake. What it buys is the right order of magnitude for the damping at the Reynolds numbers where both effects are present: a circular cylinder’s drag coefficient is 1.46 at a Reynolds number of 100 and a little over one from a thousand to a hundred thousand, so the damping rates are a few tenths in the pair’s units. The energy the pair keeps plus the energy the drag has dissipated equals the energy of release to about two parts in a million over each run, and with no drag the pair is the ideal pair to the last bit.

The swing, and what stops it #

The figure follows the earlier essay’s example, a pair released six radii apart at 50° from the stream. Ideally it turns to side by side over the first twenty radii of stream travel, swings through, comes back, and meets at 77.3° after 34.8 radii. A drag coefficient of 0.02 or 0.2 still lets it swing through side by side once, and it meets on the way back, at 69° and 75°. At a drag coefficient of one there is no swing left. The pair turns towards side by side without ever passing it, creeps in, and meets at 86.3° after 49.4 radii — later than the ideal pair, because it is moving slowly the whole way, and much closer to side by side, because it never overshot.

So the first question has an answer in two parts. A drag coefficient of order one is enough to stop the swing, and a pair then arrives near side by side rather than wherever its swing had got to. But it never settles there before colliding, because nothing about side by side holds the pair apart. Side by side is where the stream’s torque vanishes; it is not where the stream’s pull vanishes. The ideal potential pulls a side-by-side pair together at every spacing, and a damped pair that has turned to side by side simply keeps closing, more slowly.

Every pair comes round #

The second question has the surprising answer. Released six radii apart at 10° and 20° from tandem, the ideal pair parts for good: its energy is positive, and far apart the Lagrange–Jacobi identity makes its squared spacing curve upward for ever. Under a drag coefficient of one the same pairs start the same way, drifting apart while turning. But the drag takes their speed, and without speed a pair cannot outrun the torque that turns it. Once the angle passes 45° the potential pulls instead of pushes, and the pair comes back. The farthest any of them gets is 9.6 radii, and every one of them meets, near side by side.

The reason can be stated without integrating anything. Far apart the pair’s potential is $2 - 2\cos 2\varphi / D^2$, and with the drag dominating inertia the pair moves down the gradient of that potential at a speed set by the drag rather than accelerating. With a drag the same in both directions that motion conserves the ratio $\sin 2\varphi / D^2$, so a pair released at a small angle drifts out only until $\sin 2\varphi$ reaches one — 45° — and then comes back in along a path that turns it all the way to side by side. The real drag, twice as strong along the stream as across it, bends the path differently, and the integrations in the figure are what show that it ends the same way. Exactly in tandem, and only there, the pair parts. The Lagrange–Jacobi identity divided released pairs by the sign of a conserved energy, and a drag, however small, means there is no conserved energy to divide them by.

The earlier essay’s 45° is therefore a statement about an ideal fluid in the most literal sense. It is exact there, and the smallest dissipation removes it — not by moving it to a different angle but by removing the other side of it.

How far a pair gets #

How small is “however small”? The figure measures how far pairs released near tandem get before the drag turns them back. Under a drag coefficient of one a pair released 5° from tandem reaches 11.1 radii; under 0.1 it reaches 28.3. Under the smallest drags drawn, pairs released near tandem coast out past sixty radii, where the calculation stops following them. The far-field argument says they come back too, but at sixty radii the interaction between the two is so weak that anything else in a real flow — turbulence, a wall, a third particle — would decide their fate first.

So the practical statement is this: at drag coefficients of order one, which is every Reynolds number from a hundred to a hundred thousand, a pair released within a few diameters of each other stays within a few diameters, and meets. Parting is a feature of weak damping, and at weak damping it is not the stream’s forces between the pair that decide how far apart it ends, but whatever else is in the flow.

Where the pairs meet #

The earlier essay’s fourth figure found ideal pairs from four radii meeting anywhere between 48.8° and 89.5°, and pairs released below about 41° not meeting at all. Under a drag coefficient of 0.1 the meeting angles scatter less and the lowest releases begin to meet. Under a drag coefficient of one every release meets, and all between 85.9° and 89.4°: a pair released nearly in tandem and a pair released nearly side by side end the same way.

They end close to side by side and never exactly on it, because the pair is still turning when the gap closes. The torque’s strength and the pull’s strength both grow as the gap shrinks, the pull faster, so the last stretch of closing outruns the last few degrees of turning. A contact angle of 86° to 89° is the damped pair’s signature: it is what the drafting, kissing and tumbling of falling particles would look like if tumbling were all there were.

When the swing is overdamped #

Whether a pair swings at all depends on its spacing, and the dependence is steep. The torque that swings it about side by side comes from the difference between the two arrangements’ doublet sums, which falls as the inverse square of the spacing, and the pair’s moment of inertia about that swing grows as the square — so the swing’s stiffness falls as the inverse fourth power. The drag does not fall at all. The drag coefficient that critically damps the swing is therefore 2.0 at three radii between centres, 1.11 at four, 0.49 at six and 0.18 at ten.

Set against the drag coefficient a cylinder actually has, from 1.46 at a Reynolds number of 100 to 1.00 at a hundred thousand, the curve crosses at 3.5 to 4.2 radii. Beyond that spacing — beyond about one clear diameter between the cylinders — a real pair at these Reynolds numbers cannot swing; it can only turn monotonically towards side by side. The pendulum of the ideal calculation exists only for pairs already nearly touching, and there the collision comes first.

How long it takes #

The drag does not prevent the meeting, but it does delay it. For a pair released near side by side the figure plots the time to contact against the drag coefficient. With little drag the time is the ideal pair’s, set by inertia: the pair accelerates together under its mutual pull, most of its inertia borrowed from the fluid — the same force of getting going that makes any body accelerating in a fluid heavier than it is. With heavy drag the pair creeps at the speed at which the drag balances the pull, and the time grows nearly in proportion to the drag coefficient — the local slope from six radii is 0.88 at the heavy end and still rising towards one. It is the same crossover as a body falling in air and in syrup: inertia sets the time until the drag is strong enough to hold the speed at its terminal value, and after that the drag does.

What was checked #

Three checks bear on the damping itself. With no drag the damped integrator must reproduce the ideal pair exactly, and does, to the last bit. With drag the energy the pair keeps plus the energy the drag has taken must equal the energy of release, and does to two parts in a million at drag coefficients of 0.05, 0.5 and 2. And with heavy, equal drag in both directions the pair must follow the gradient path of its potential, which obeys a first-order equation, $dD/d\varphi = D^2 S_D / S_\varphi$, sharing nothing with the damped integrator but the potential; integrated on its own from ten radii at 15°, it agrees with the damped pair to about one part in a hundred thousand along the whole path, out to 14.0 radii and back to contact. The tests also refuse a negative drag coefficient, a Reynolds number outside the drag fit, and a critical damping asked for inside contact.

What the picture cannot show #

A wake. The drag here is the drag of a cylinder alone in the stream. A cylinder behind another sits in its wake and feels less drag — the drafting that brings falling particles into contact from tandem, which an ideal stream reverses — and a cylinder beside another feels the other’s wake as a sideways push. The vortex street a steady calculation cannot draw is the reason no steady calculation here can supply either.

Unsteady drag. A drag coefficient is a time average. The pair moves over tens of radii of stream travel, and a cylinder’s wake sheds vortices every five radii or so, so each cylinder feels a fluctuating side force at least as large as the interaction forces between them. The damped pair is the average behaviour and nothing more.

Spheres. As before, three dimensions are kinder: every interaction is weaker. The swing’s stiffness then falls as the inverse fifth power of the spacing and the crossover to overdamping comes even closer in.

No rest point, and what that says about falling spheres #

The third question can now be answered with a no. Falling spheres that tumble end side by side and separate, and the separation is measured to depend on the Reynolds number. The damped ideal pair has no rest spacing to depend on anything: side by side, its potential attracts at every spacing, and the drag only decides how slowly it closes. A real pair’s side-by-side separation must therefore come from something this model leaves out. At low and moderate Reynolds numbers that is the wakes: two bodies side by side each divert the other’s wake outward, which is a repulsion, and viscous forces between them are repulsive where the ideal forces are attractive. The measured separation is where those win.

That puts the ideal calculation in its place precisely. It explains the turning — the stream’s couple, which is a body with no lift and a moment anyway acting on a pair — and, with drag added, why a pair arrives near side by side rather than anywhere. It cannot explain the parting, and adding drag to it makes that clearer rather than less: it turns the ideal attraction into an unopposed creep. No drag at all was the paradox the argument started from, and the paradox is for the pair, not each body showed that the ideal forces between two bodies are real even when the drag on the pair is zero. What the free pair adds is that those forces are always attractive side by side, so that anything that lets a real pair settle apart must be something the ideal theory does not contain.

The convention: a drag coefficient of the stream #

Lengths are in cylinder radii and time in radii of stream travel, as in the earlier essays. The drag coefficient is the single cylinder’s, $C_D$, based on diameter, evaluated at the Reynolds number of the stream past the pair, $2aU/\nu$. The damping it gives is the linearisation of that drag about the stream, so it assumes the relative motion is slow compared with the stream. Under a drag coefficient of one each cylinder moves at up to a ninth of the stream’s speed more than four radii from its partner, a fifth between three and four, and up to half in the last half-radius before contact. The linearisation is good where the turning and the reach are decided and stretched where the pair closes; there it understates the drag, and the meeting would be a little slower still. A cylinder’s drag coefficient against Reynolds number is taken from White’s fit, $1 + 10\,\mathrm{Re}^{-2/3}$, good to about ten per cent from 1 to 2·10⁵, and the Reynolds number is the one number that fit is a function of.

Who worked it out #

The linearised drag of a body in a stream, anisotropic by a factor of two between along and across, follows from the drag law itself and is used in the stability of towed and falling bodies. The gradient-flow limit of a damped mechanical system — Aristotelian mechanics, force proportional to velocity — is standard in the theory of overdamped motion. The drafting, kissing and tumbling of falling spheres were named by Fortes, Joseph and Lundgren in 1987, and the side-by-side repulsion of two bodies at moderate Reynolds number has been computed and measured since; the ideal forces between bodies go back to Bjerknes and Kelvin.

Still open: the wake that holds a pair apart #

The model’s missing repulsion has a simplest form that can be added in the same frame. Each cylinder’s wake, far enough behind it, is a deficit in the stream’s speed spreading as the square root of distance, and a cylinder beside another’s wake feels a sideways force towards the faster fluid — away from the wake’s centre. The next calculation gives each cylinder an Oseen-like wake of the width and depth its drag coefficient sets, lets each feel the other’s wake as a change in the stream it meets, and asks whether the damped pair then finds a side-by-side spacing where the wake’s push balances the ideal pull — and whether that spacing, as a function of the Reynolds number through the wake’s width, falls where tumbling spheres are measured to separate.