A pair set free in a stream collides or parts

The paradox is for the pair, not for each body put two circular cylinders in a steady ideal stream, held them in place, and computed the force on each. D’Alembert’s theorem held for the pair, whose total force was zero to fifteen figures, and failed for each body: side by side they were pulled together, with a force that grows without limit as the gap closes, and in tandem they were pushed apart, the front one upstream against the stream flowing past it. The calculation ended on the obvious next question. The cylinders were held. Two bubbles rising side by side are not, nor are two bodies drifting in a current, and a force on a body that is free to move is the beginning of a motion.

This essay lets the pair go. The cylinders are released from rest in the stream and allowed to move along the line joining their centres — towards each other or apart — under the stream’s force and their own inertia, which in a fluid includes the fluid they have to push. The questions are the ones the held pair raised: whether a side-by-side pair collides, how long it takes, and whether a tandem pair, pushed apart, comes to rest at some spacing.

The stream’s force is a potential #

The first step turns out to simplify everything that follows. From far away, the pair in the stream looks like a single doublet — the far field of any closed body in an ideal stream is a doublet, as what the far field remembers showed — and its strength, the sum of every image in the successive-image construction, measures how large the pair looks: two for two isolated cylinders, 2.63 for a pair side by side a fifth of a radius apart, 1.69 for a pair in tandem at the same gap. Call that sum $S(D)$, as a function of the distance between centres.

The first figure puts two numbers side by side at every spacing. One is the force on each cylinder along the line of centres, from Blasius’ integral of the pressure round its own surface, exactly as the essay before computed it. The other is $-\pi\,dS/dD$, the slope of the pair’s doublet sum, in units of $\rho U^2 a$. They agree to nine figures, side by side and in tandem, at every spacing tried.

That is not a coincidence of the numbers but a theorem, a form of the one Kelvin used for bodies in a moving fluid: in an ideal stream the interaction energy of a set of bodies is proportional to their total doublet strength, and the forces between them are its gradient. Stated as a rule for motion, a pair free to move rearranges itself to look bigger to the stream. Side by side, the doublet sum grows as the gap closes — the gap passes less than its share of the stream and the pair blocks more of it — so the pair is drawn together. In tandem the sum shrinks as the gap closes, since each cylinder shelters in the other’s flow, so the pair is pushed apart.

A force that is the gradient of a potential gives an energy integral, and the motion can be written down without integrating an equation of motion step by step.

Why looking bigger is the rule #

The rule has an exact twin in electrostatics, and the twin makes it less surprising. The potential of a doublet is the potential of a line dipole, and the successive images used here are the ones used for two charged cylinders; the stream plays the part of a uniform applied field, and the doublet sum plays the part of the pair’s polarisability. Two conductors held at a fixed voltage by a battery move so as to increase their capacitance, because the battery does work on them as they do: the force is half the square of the voltage times the rate at which the capacitance grows. Two cylinders held in a fixed stream move so as to increase their doublet sum, because the stream does work on them as they do, and the force is the stream’s dynamic pressure times the rate at which the sum grows.

The twin also settles where the energy comes from. As the pair closes, the cylinders speed up and the disturbance they make in the stream grows stronger, so both the bodies and the fluid round them gain energy together. Neither is paid for by the other: both are drawn from the stream far away, which is held at its speed whatever the pair does, just as a battery holds its voltage and pays for both the field’s energy and the work the conductors do as they move. What is held fixed changes that bookkeeping and not the force, which is why the same rule — towards the larger doublet — would hold for a pair coasting through still fluid with its momentum fixed.

The inertia of a pair that is closing #

The other half of the motion is inertia. A cylinder accelerated through a fluid has to accelerate some of the fluid too — the force of getting going — and the mass a body has to borrow for a circular cylinder alone is exactly the mass of fluid it displaces. Two cylinders moving towards each other borrow more, because the fluid between them has to be pushed out of the gap sideways, and the amount depends on the gap.

By symmetry, two cylinders approaching each other at equal speeds are the same problem as one cylinder approaching a plane wall halfway between them: the plane of symmetry is a streamline, and a wall is nothing else. The added mass follows from a second set of successive images — each moving cylinder’s own doublet, reflected in the other as though the other were fixed, and back again — with the fluid’s kinetic energy computed round both surfaces. The second figure plots it against the gap, with a panel method of the kind used for a body near a wall at three gaps as an independent check; the two agree to four parts in a hundred thousand, and at a gap of a fiftieth of a radius, checked separately, to one in twenty thousand.

The curve’s surprise is how slowly it grows. Far apart each cylinder borrows its own displaced mass. At a gap of four-tenths of a radius it borrows 1.47 displaced masses, at a tenth 1.78, at a fiftieth 2.03, at a five-hundredth 2.20: a logarithm, not a divergence. Squeezing an ideal fluid out of a closing gap costs very little, because the fluid in the gap can escape as fast as it likes; the gap is narrow for only a short stretch, and the fast flow there carries little kinetic energy.

That is exactly the opposite of what a real fluid does. The last of the oil and the squeeze-film essays found viscous resistance to squeezing growing without limit as a gap closes — a resistance that a torn film keeps even when it cavitates — so that two surfaces pressed together in oil never quite touch. In an ideal fluid the cushioning is inertial, and inertia does not cushion much.

Released side by side, the pair collides #

The third figure is the motion. With the stream’s force a potential and the inertia a function of the gap, each cylinder’s energy is conserved in the form

$$\tfrac12 \left(m + m_c(h)\right)\dot h^2 = \tfrac12\left[S(2h) - S(D_0)\right],$$

with $h$ the half-spacing, $m$ the cylinder’s own mass in displaced masses — one for a neutrally buoyant cylinder, zero for a bubble — and $D_0$ the spacing it was released at. The time to reach any spacing is an integral, and it was checked against a step-by-step integration of the equation of motion, with the velocity-dependent force that a changing added mass brings, to one part in ten million.

Side by side, the pair closes and collides. Neutrally buoyant cylinders released four radii apart — two radii of clear water between them — collide after 6.5 radii of stream have passed, which for two ten-centimetre pipes in a one-metre-per-second current is a third of a second. Released three radii apart, after 2.9; bubbles, with nothing of their own to accelerate, from four radii after 4.8. They speed up all the way in, the pull growing faster than the inertia, and arrive at about the stream’s speed: a neutral pair from four radii is closing at 0.97 of it a tenth of a radius from contact, having peaked at 1.09. The logarithmic added mass slows the last approach by a tenth and no more.

How long a collision takes #

The fourth figure is the collision time against the starting spacing. Far apart, the force is the far-field law of the essay before, $4\pi\rho U^2 a^4/D^3$, and the inertia is one displaced mass each for the fluid plus the cylinder’s own. The motion is then a particle falling in an inverse-cube potential, which integrates in closed form: the time from rest at $D_0$ to contact is $D_0^2\sqrt{1 - 4/D_0^2}\,/\,2$ radii of stream for neutral cylinders, and $1/\sqrt2$ of that for bubbles.

The exact calculation follows the law within a per cent beyond ten radii and falls below it closer in: at a starting spacing of 2.5 radii the collision takes 1.43 radii of stream where the far-field law says 1.88, because close together the true pull is much stronger than the far-field cube law allows, as the held pair’s force showed. The dominant fact is the square: double the starting spacing and the collision takes four times as long, since the pull falls as the cube of the distance and the pair has further to go.

Bubbles rising side by side #

The side-by-side pair in a stream is, in the pair’s own frame, two bubbles rising side by side through still liquid, and the calculation says such bubbles are drawn together at high Reynolds number. That is what is measured. Bubble pairs rising side by side in clean water at Reynolds numbers in the hundreds approach each other, and the approach is predicted well by potential-flow interaction corrected for the thin boundary layers on the bubbles. At low Reynolds numbers the same pairs drift apart instead: the vorticity the bubbles shed, which ideal flow does not have, produces a repulsion that wins when the boundary layers are thick. The crossover sits at a Reynolds number of a few tens.

What the measured bubbles do not do is collide at the stream’s speed. As they close, the liquid film between them has to drain, viscosity takes over from inertia in the last fraction of a radius, and the pair either bounces apart or coalesces depending on how fast the film drains against how hard the bubbles are pressed. The ideal-flow calculation gets them to the edge of that film and is silent about what happens there.

Where a free pair in a current matters #

Two cylinders free to move across a current are not an idealisation invented for the purpose. The vertical pipes that carry oil and gas from the sea bed to a floating platform — risers — hang in clusters a few diameters apart, tensioned at the top and free to swing, in currents of a metre a second. They do collide, and riser clashing is a design case for every deep-water field. The forces that bring them together in practice are mostly the wake of the upstream riser, which shelters the downstream one and lets it drift, but the side-by-side attraction computed here acts on every pair whatever the wake does, and it is strongest at exactly the small spacings where a clash begins. The same arithmetic, with a wall for the mirror, is why ships passing close are drawn together, which the essay on the held pair followed to the collision of 1911 between a liner and a cruiser.

A tandem pair parts for ever #

The fifth figure is the tandem pair, and here the energy integral gives the whole answer at once. Pushed apart, the pair moves towards spacings where it looks larger, and it keeps doing so for ever, since the doublet sum rises monotonically towards two. Far apart each cylinder borrows one displaced mass, so the final relative speed is $2\sqrt{(2 - S_0)/(m+1)}$ times the stream speed. Released a fifth of a radius apart, a neutral pair parts at 0.79 of the stream speed and a pair of bubbles at 1.12; released four radii apart, at 0.48 and 0.68. There is no spacing at which a tandem pair settles, because nothing in an ideal stream pulls it back.

A real tandem pair does the opposite. The rear body sits in the front body’s wake, where the fluid is moving with the front body and the drag on the rear one is much lower than on the front, so the rear one catches up — drafting, as cyclists and geese do. Two spheres falling one behind the other in a liquid close up, touch, and then tumble to fly side by side, a sequence observed often enough in the 1980s to be named “drafting, kissing and tumbling”. The ideal calculation predicts the first stage backwards, because the wake is exactly what it does not have; but it predicts the last stage correctly, since side by side is where the doublet sum is largest.

What was checked #

The sixth figure is the ledger. The stream’s force from the doublet sum agrees with Blasius’ integral round each surface to three parts in $10^9$ in five arrangements, which is the theorem the whole motion rests on, verified rather than assumed. The moving pair’s added mass by images agrees with the panel method’s circle approaching a wall at three gaps to four parts in $10^5$, and sixty radii apart it is one displaced mass to four figures. And the closing time from four radii by the energy integral, 6.50 radii of stream, agrees with a fourth-order Runge–Kutta integration of the equation of motion, including the force from the changing added mass, to one part in $10^7$.

What the ideal pair leaves out #

Wakes. Every real pair at a Reynolds number where ideal flow is tempting has a wake, and a wake reverses the tandem pair’s behaviour and changes the side-by-side pair’s strength.

The last film. The collision is followed to a gap of a hundredth of a radius. Below that a real fluid’s viscosity takes over in the gap and decides between bouncing and touching.

Motion across the line of centres. The cylinders move only towards or away from each other. A pair free to move in any direction also turns, since the stream’s force has a component across the line of centres whenever the pair is staggered, as the essay before found.

Two dimensions. Spheres interact in the same way with faster-falling forces — the far-field pull falls as $1/D^4$ — and their collision time grows faster with the starting spacing.

The convention the numbers depend on #

Lengths are in cylinder radii, spacings between centres; times are in radii over the stream speed; forces per unit length in $\rho U^2 a$; masses in units of the fluid a cylinder displaces, $\rho\pi a^2$ per unit length. A neutrally buoyant cylinder has a mass of one in those units and a bubble zero. The doublet sum $S$ is in units of $a^2U$ and is two for two isolated cylinders.

Who found it, and when #

Kelvin and Bjerknes worked out in the nineteenth century that bodies in an ideal fluid move as though under forces derived from the fluid’s kinetic energy, and Bjerknes built pulsating spheres that attracted and repelled as charges do. Lagally’s theorem of 1922 gave the force on a body in terms of the singularities inside it. The interaction of rising bubble pairs was worked out in potential flow by Biesheuvel and van Wijngaarden, Kok and others, and Legendre, Magnaudet and Mougin in 2003 computed side-by-side pairs at finite Reynolds number and found the attraction turning to repulsion as the Reynolds number falls. Drafting, kissing and tumbling were named by Fortes, Joseph and Lundgren in 1987.

Still open: a pair that can turn #

The rule the calculation found — the pair moves to look bigger to the stream — has a consequence the line-of-centres motion cannot show. At a fixed spacing the doublet sum is largest side by side and smallest in tandem, so a pair free to move in the plane should turn as well as close: a staggered pair would swing round towards side by side while the two close. The next calculation frees both coordinates of each cylinder, and asks whether every released pair ends side by side before it collides, how long the turning takes against the closing, and whether the tumbling that falling spheres show is, in its last stage, nothing but this.