The paradox is for the pair, not for each body

No drag at all proves d’Alembert’s paradox: a body held in a steady, irrotational stream of an inviscid fluid, with nothing else in the flow, feels no force. The proof is a control volume. Draw a large surface round the body; the momentum crossing it far away is exactly what it would be with no body there, because the disturbance a closed body makes dies away fast enough; so the momentum balance leaves nothing for the body to push against. The force of getting going finds the first way round it, an unsteady stream, and the drag that is made of waves the second, a free surface that carries energy away.

The proof has a feature that is easy to read past. The surface it draws goes round the whole flow’s bodies. With two bodies in the stream the argument goes through unchanged and proves that the pair, together, feels nothing. It proves nothing at all about each body separately. This essay computes the force on each of two cylinders in an ideal stream and finds that it is not zero, that it can be large, and that it points in directions nobody would guess from the paradox.

Building the flow round two cylinders #

The flow past one cylinder is the stream plus a doublet at its centre — the image of the stream in the circle, by the circle theorem. With two cylinders, each one’s doublet disturbs the flow at the other, which must be closed again by reflecting that doublet in the second circle; the reflected doublet disturbs the first, and so on. The images march inwards towards two limit points, one inside each cylinder, as they do for two cylinders carrying circulation, and their strengths fall off geometrically, faster the further apart the cylinders are. For two cylinders a fifth of a radius apart, 38 rounds of reflection bring the last doublet’s strength below $10^{-15}$.

The construction is checked by what it has to achieve. The velocity normal to each surface should vanish, and it does, to three parts in $10^{15}$ at three spacings and two arrangements. The force on each cylinder then follows from Blasius’ theorem, the integral of the square of the complex velocity round its own surface, evaluated by the trapezoid rule, which is spectrally accurate for a periodic integrand.

The first figure is the flow past two cylinders side by side, across the stream, with a gap of a fifth of a radius between them. The streamlines crowd through the gap: it is narrow, and the flow that would have gone round the inner side of each cylinder has to share it. The flow round the outer sides is barely changed. The fore-and-aft symmetry of the single cylinder survives — the pattern is its own mirror image front to back — and that is why neither cylinder feels any drag. The up-and-down symmetry of each cylinder’s own flow does not survive, and that is where the force comes from.

A pressure the neighbour takes away #

The second figure reads the pressure off the lower cylinder’s surface. A lone cylinder’s pressure coefficient is $1 - 4\sin^2\theta$, with a minimum of −3 at its shoulders. Facing the gap, the neighbour drives it to −21.4: the flow through the gap is fast enough that its dynamic pressure is more than seven times the lone cylinder’s lowest. On the far side the pressure falls too, to −4.2, because the stream displaced by the pair is squeezed round the outside as well. The stagnation points, where the pressure coefficient is one, move off the axis, round towards the far side.

Integrated round the surface, the fore-and-aft symmetric part of that distribution gives no drag, as it must. The top-to-bottom asymmetry gives a force across the stream: a suction on the side facing the gap, much stronger than the suction on the far side. Each cylinder is pulled towards the other with a force of 3.60 ρU²a, per unit length — in coefficient form, 3.6 times the dynamic pressure acting on a diameter. That is nearly four times the drag a real cylinder feels at the Reynolds numbers where it is largest, and it is there in a theory that is famous for predicting no force.

Why the proof cannot see each body’s force #

The control-volume proof fails for one body of a pair for a reason that can be stated exactly. Draw the surface round one cylinder only and the flow crossing it is not a uniform stream plus a disturbance that dies away; it is the stream plus the neighbour’s disturbance, which is still there however large the surface is drawn, because the neighbour is inside the region where the surface has to pass. The momentum flux through the surface does not reduce to the undisturbed value, and the balance leaves a force.

The cleaner statement is Lagally’s theorem. A body in an ideal flow can be replaced by the singularities inside it — here the doublets — and the force on it is the sum, over those singularities, of their strength times the velocity induced at them by everything else. A single cylinder in a uniform stream has one doublet at its centre, and the velocity at its centre from everything else is the uniform stream, which exerts no force on a doublet: a doublet feels a force only in a velocity gradient. Two cylinders each sit in the gradient of the other’s field, and that is the whole of the interaction. The far-field law below is Lagally’s theorem with one doublet on each side; the full calculation is the same theorem with the whole image sequence.

That also shows why the forces are equal and opposite. Each doublet’s force is the product of its strength with the gradient of the other’s field, and the two products are the same quantity seen from the two ends, as the forces between two electric dipoles are. The analogy is not loose: the potential of a doublet is the potential of a line dipole, and the method of successive images used here is the one used for the capacitance of two parallel wires.

One behind the other #

Put the cylinders one behind the other and the force turns through a right angle and changes sign. The third figure shows why. Between two cylinders in line the stream has nowhere to go: the fluid there is nearly stagnant, its pressure is close to the full stagnation pressure, and it pushes the two cylinders apart. The rear cylinder is pushed downstream, which looks like drag. The front cylinder is pushed upstream, into the stream flowing past it. Half a radius apart the push is 0.50 ρU²a on each; a fifth of a radius apart, 0.61.

The front cylinder’s upstream force is the result the paradox makes most surprising and least suspicious. The pair’s total is zero, so what the rear cylinder feels as a drag, the front one feels as a thrust. In ideal flow a body can be pulled forward against a stream simply by having another body behind it.

From contact to far apart #

The fourth figure follows both forces from the point where the cylinders nearly touch to twenty radii apart. Far apart they agree with a simple law. A cylinder sitting in the velocity field of a distant neighbour’s doublet is in a slightly non-uniform stream, and a body in a non-uniform ideal stream is pushed towards faster flow with a force equal to the mass of fluid it displaces plus the mass it has to borrow, times the gradient of half the square of the speed. For a circular cylinder the two masses are equal, the gradient is set by the neighbour’s doublet, and the force comes out as $4\pi\rho U^2 a^4/D^3$ at a centre spacing $D$ — towards the neighbour side by side, where the neighbour speeds up the flow, and away from it in tandem, where the neighbour slows it down. At twenty radii the computed forces agree with it to half a per cent.

Close together the two arrangements part company. Side by side, the force grows without limit as the gap closes — 3.60 at a gap of a fifth of a radius, 6.23 at a tenth — because the flow through a vanishing gap has to go ever faster. In tandem the push levels off near two-thirds of ρU²a, because the stagnant fluid between the cylinders can push no harder than the stagnation pressure, however close they come.

What the far field sees #

From far away the pair is one body, and what the far field remembers of a body in an ideal stream is its total doublet strength: the sum of every image’s moment, which sets how much the pair displaces the stream around it and how much fluid it drags with it when it accelerates. Two isolated cylinders have a total of two. Side by side a fifth of a radius apart, the sum of the image sequence is 2.63: the pair blocks the stream as though it were a third larger than two separate cylinders, because the gap between them passes far less than its share of the flow and the stream is pushed round the outside instead. In tandem at the same gap it is 1.69, less than two, because each cylinder sits partly in the other’s sheltered flow and the pair presents a slimmer outline.

Those two numbers are the far-field face of the forces. A side-by-side pair that attracts is a pair whose combined displacement grows as its members approach, and a tandem pair that repels is one whose displacement shrinks. Nothing in the pair’s far field reveals the internal forces directly — the paradox guarantees it cannot — but the pair’s effective size does, and it changes with spacing in the direction the forces push. It is also the number that sets the pair’s added mass, so two cylinders side by side are harder to accelerate along the stream’s direction than two apart, which is the start of the unsteady interaction that the steady calculation leaves out.

The force turns as the pair turns #

The fifth figure turns the pair. At three radii apart, the upstream cylinder of a pair in line is pushed upstream with 0.35 ρU²a; side by side it is pulled across towards its neighbour with 0.65. In between, the streamwise force changes sign at a stagger of 25 degrees and the cross-stream force at 55: over the whole band between, the upstream cylinder is pushed downstream and away from its neighbour, and at 58 degrees the downstream push reaches 0.56 — larger than the tandem push the other way. The downstream cylinder always feels exactly the opposite. None of this is a drag in the usual sense, since the pair as a whole feels nothing; it is a mutual force, which in a real flow would move the cylinders if they were free to move.

A spring with the wrong sign #

The side-by-side force has a consequence that matters well beyond two cylinders. It grows as the gap closes: at a centre spacing of 2.5 radii it rises by 3.2 ρU²a for every radius the cylinders come closer, at 2.2 radii by 15. For two tubes that are free to flex, that is a spring with the wrong sign — a negative stiffness. Displace them towards each other and the fluid pulls them further in; apart, and it lets go. A tube held by its own elastic stiffness is stable only while that stiffness exceeds the fluid’s negative one, and the fluid’s grows with the square of the flow speed.

Past that speed the pair buckles towards each other, a static divergence rather than a vibration. It is one of the mechanisms behind the fluid-elastic instability of tube bundles in heat exchangers and steam generators, where hundreds of flexible tubes sit a fraction of a diameter apart in a cross-flow. There the flow separates and the full picture needs the wake’s delay as well, but the displacement-dependent force that starts it is of this kind, and its steep rise at close spacing is why tube arrays are packed with care. The far-field law makes the same point in a formula: a force falling as $1/D^3$ has a stiffness falling as $1/D^4$, and a gap that halves quadruples the stiffness faster still.

The ships that were drawn together #

The side-by-side flow is the flow round two ships steaming in parallel at the same speed, seen from the ships, and the attraction is real. In September 1911 the liner Olympic, leaving Southampton, and the cruiser HMS Hawke, running alongside in the Solent, came together and the cruiser’s bow was driven into the liner’s side. The Admiralty’s case, which the court accepted, was that the liner’s displacement flow had drawn the smaller ship in, beyond the reach of its rudder. The phenomenon is now treated routinely in ship handling: two ships passing close are drawn together and their bows and sterns are pushed round, and a ship close to a bank is drawn towards it. In a channel the “bank effect” is the same image system with a wall for the mirror — a wall made by reflection, which is also why a wing near the ground feels its own image.

Real ships are not ideal cylinders, and the forces are modified by viscosity, the free surface and the finite length of the hulls. But the sign and the steep rise as the gap closes are the ideal flow’s, and they are the reason the interaction is dangerous: it is weak at a comfortable distance and grows faster than the helmsman’s instinct as the distance shrinks.

What was checked #

The sixth figure is the ledger. Both surfaces are streamlines, which is what the image construction is for. The pair’s total force — the sum of two Blasius integrals each taken round one cylinder alone — vanishes to six parts in $10^{15}$ in twelve arrangements of spacing and stagger; nothing in the construction imposes that, and it is d’Alembert’s theorem for the pair, recovered as a check rather than assumed. And far apart each force approaches the one a cylinder feels in its neighbour’s velocity gradient, which is derived independently of the images.

What the picture cannot show #

Separation. A real cylinder in a stream separates and sheds a wake, and the wake of the upstream cylinder of a tandem pair engulfs the downstream one; the forces on real cylinders in tandem at ordinary Reynolds numbers are dominated by that, not by the ideal interaction. Side-by-side cylinders in a real flow have a biased gap flow that switches from side to side. The ideal forces are what an attached flow would give, and they are the right answer only where the flow stays attached — at the start of the motion, or round streamlined bodies.

Steady flow. If the cylinders move relative to each other, the added-mass terms of the force of getting going add to these, and an unsteady pair can feel a net force.

Two dimensions. Spheres interact in the same way with a force falling faster with distance, as $1/D^4$.

The convention the numbers depend on #

Forces are per unit length of cylinder, in units of $\rho U^2 a$ with $a$ the radius; a force coefficient on a diameter would be the same number. $D$ is the distance between the centres in radii, so a gap of a fifth of a radius is $D = 2.2$. The stagger angle is between the stream and the line of centres. Pressure coefficients are referred to the free stream.

Who found it, and when #

The interaction of bodies in an ideal fluid was worked out in the nineteenth century: Kelvin and Tait, and Bjerknes, studied the apparent attractions and repulsions between pulsating and moving spheres, which Bjerknes offered as a mechanical model of electrostatics. The method of successive images for two circles goes back to Kirchhoff and to the theory of two charged conducting cylinders, and Lagally in 1922 gave the general theorem expressing the force on a body in terms of the singularities inside it. The Olympic–Hawke collision of 1911 made ship interaction a legal and practical question.

Still open: the pair that is free to move #

The forces here are for cylinders held in place. A pair free to move in the stream — two bubbles rising side by side, two spheres settling — would respond to them, and the ideal theory then predicts the motion, not just the force. Two bubbles rising side by side are drawn together by exactly this attraction, and measured bubble pairs at high Reynolds number do approach each other; the next calculation lets the two cylinders move under their mutual force and their added masses, and asks whether a side-by-side pair collides, how long it takes, and whether the tandem pair, pushed apart, settles at a spacing.