The cheapest fork loses only its sideways speed, and pricing it closes the fork

The cheapest turbulent pipe redid Murray’s minimisation with turbulent friction. A vessel’s bill, as in the radius that costs least, is the power to push fluid along it plus an upkeep proportional to its volume, and with Blasius’s smooth-pipe friction the cheapest vessel carries a flow growing as its radius to the power 27/11 rather than 3. Its junctions follow. A symmetric fork has daughters $2^{-1/k}$ of the parent’s radius, 0.754, and a half-angle with $\cos\theta = 2^{2/k-1}$, 28.4° — narrower and thinner than Murray’s 37.5° and 0.794.

Every number in that essay priced the pipes and none priced the forks. In a turbulent network a junction has a loss of its own, a share of the parent’s dynamic pressure destroyed by the mixing at the split, and that essay ended by asking what pricing it would do: whether the cheapest fork narrows further, whether the daughters thicken to pay for a gentler split, and at what Reynolds number the junctions rather than the pipes decide a network’s shape.

The first answer is an identity, and it makes the other two straightforward. At the cheapest fork, each daughter already runs at the parent’s velocity component along itself. So the split destroys only the sideways part of the parent’s motion, and no resizing of the daughters can make it destroy less. What can is narrowing the fork, and because the pipes’ bill is flat at its best angle, a split that costs very little moves the angle a long way. The daughters do not thicken; they thin.

Borda’s loss with an angle in it #

A loss with no viscosity in it derived Borda’s result for a sudden widening: momentum and energy, balanced over the same control volume, disagree by exactly half the square of the velocity change, and that half is destroyed in the eddies of the mixing whatever the viscosity. The derivation needs only that the fluid’s velocity changes abruptly and that the change is undone by mixing rather than by a smooth passage.

A split is the same kind of event with a direction in it. Fluid arriving along the parent at speed $V_p$ leaves along a daughter at speed $V_d$, turned through the half-angle $\theta$. The natural extension of Borda’s argument charges the kinetic energy of the difference between those two velocities as vectors:

$$J = Q\,\tfrac{1}{2}\rho\,\lvert \mathbf{V}_p - \mathbf{V}_d \rvert^2 = Q\,\tfrac{1}{2}\rho\left(V_p^2 + V_d^2 - 2V_pV_d\cos\theta\right).$$

That is the loss model this essay uses, and it is a model. Measured loss coefficients for dividing junctions are larger than it gives, because the flow separates on the inside of the turn into a daughter and the separation adds a loss the abrupt-mixing argument does not count. So every calculation below is repeated with the loss doubled, as a stand-in for that separation, and where the doubling moves an answer the essay says so.

The rest of the bill is the earlier essay’s, unchanged: water, Blasius’s friction factor $0.3164\,\mathrm{Re}^{-1/4}$ for a smooth turbulent pipe, the pumping power per unit length $Q f\rho V^2/4r$, and an upkeep of fifty watts per cubic metre of pipe. A parent runs from a source to the junction and two daughters run on to two sinks, placed so that without the split’s loss the cheapest fork is the earlier essay’s, with daughters of a chosen length and a parent twice as long. The bill is then minimised over the junction’s position along the axis and over both radii at once.

The identity at the cheapest fork #

The first figure draws the split’s loss, as a multiple of the flow times the parent’s dynamic pressure, against the daughters’ radius, with the angle held at the cheapest value for each of three pipe laws: Murray’s laminar exponent 3, Blasius’s 27/11, and the fully rough 7/3. Each curve has a minimum, and each minimum sits exactly at that law’s own cheapest radius ratio — 0.794, 0.754, 0.743 — with the value $\sin^2\theta$ there.

That is not a coincidence of the three cases. The daughters each carry half the flow through a radius $r_d$, so their speed over the parent’s is $\tfrac12(r_p/r_d)^2$. The cheapest ratio is $2^{-1/k}$, which makes the speed ratio $2^{2/k-1}$. And the cheapest angle, from the triangle of forces the angle a junction chooses drew, has $\cos\theta = 2^{2/k-1}$. The same number, for every exponent:

$$\frac{V_d}{V_p} = \cos\theta.$$

Each daughter’s speed is exactly the component of the parent’s velocity along that daughter. The loss is then the kinetic energy of the component across it, $Q\,\tfrac12\rho V_p^2\sin^2\theta$: 0.37 of the dynamic pressure at Murray’s 37.5°, 0.226 at the smooth turbulent 28.4°, 0.18 at the rough 25.1°. And because $V_d = V_p\cos\theta$ is exactly where the vector difference is shortest for a given direction, the loss is stationary in the daughters’ radius at the cheapest fork. Resizing the daughters, up or down, can only increase what the split destroys.

The two halves of the identity come from different places. The radius ratio is a property of each pipe on its own — the cheapest radius for the flow it carries. The angle is a property of the force triangle, in which each pipe pulls on the junction with its own bill per unit length. Both give $2^{2/k-1}$ because, at the cheapest radius, a pipe’s bill per unit length is proportional to the square of its radius in every regime, the result the earlier essay used to carry the force triangle over from laminar flow. A force proportional to $r^2$ is a force proportional to the flow divided by the speed; the triangle is, in disguise, a balance of the flows’ momentum, and a momentum balance with no sideways force is what puts the daughters’ speed at the parent’s component along them. It is the same accounting that what a jet cannot push sideways uses to split a jet striking a plate, where the absence of a force along the surface fixes how the stream divides.

Why so small a loss moves the angle so far #

The second figure is the heart of the matter. A fork whose daughters are fifty parent diameters long has its radii held at the pipes-only values and its junction slid along the axis, which sweeps the half-angle from 10° to 34°. The pipes’ bill, dashed, is least at 28.4°, and it is flat there: a few degrees either side costs hundredths of a per cent, because a minimum found by a vanishing derivative rises only as the square of the error. The split’s loss is not flat at all. It is $\sin^2\theta$, which at 28.4° is rising steadily with the angle — at first order, not second.

Add a first-order slope to a second-order bowl and the bottom moves. Here it moves to 21.4°, seven degrees, while the split costs only 1.3 per cent of the bill there. The narrower fork saves more in the split than it spends in the longer daughters it needs to reach the same sinks, and it keeps doing so for seven degrees because over that range the pipes’ extra bill is still tiny.

That flatness was already the main lesson of the angle a junction chooses, which found five degrees off Murray’s angle costing a tenth of a per cent and warned that measured branching angles agreeing with the prediction confirmed very little. The same flatness has a second consequence it did not need then: any small cost that depends on the angle at first order decides the angle. The pipes fix only a broad bowl, and whatever else is priced at the junction sets where in it the fork sits. It is the situation the optimum that does not matter describes for a wing’s planform, seen from the other side: the flatness that makes an optimum forgiving also makes it easy to move.

Shorter daughters, narrower forks #

How far the angle moves depends on how much pipe there is to set against the split. The split’s loss is a fixed amount of power at the junction; the pipes’ bill grows with their length. The third figure varies the daughters’ length from ten to fifteen hundred parent diameters, for water at a hundredth of a cubic metre a second, a parent Reynolds number near 93,000.

Long daughters barely notice the junction: at five hundred diameters the cheapest angle is 27.2°, at fifteen hundred within half a degree of 28.4°. Short ones notice it a great deal. At fifty diameters the angle is 21.4°, at twenty 17.6°. Doubling the split’s loss, for the separation the bare form leaves out, moves the fifty-diameter fork to 18.6° — about the same as halving the daughters’ length. The answer to the earlier essay’s first question is therefore yes, the fork narrows, and by an amount fixed mainly by the length of pipe between junctions, measured in diameters.

The dial changes the flow by factors of ten. At a ten-thousandth of a cubic metre a second the parent’s Reynolds number is about 6,000 and the fifty-diameter fork sits at 23.8°; at a thousandth, about 24,000 and 22.7°. A faster flow forks narrower at the same length in diameters, and the reason is the friction factor: Blasius’s factor falls as the Reynolds number rises, so a pipe of a given length in diameters costs less in friction per unit of dynamic pressure, and the split’s loss, which does not fall, is the larger share.

The daughters thin #

The earlier essay guessed that the daughters might thicken to pay for a gentler split. The identity says they cannot, to first order: at the cheapest fork the split’s loss is stationary in their radius, so nothing in the split pulls them either way. What moves them is the angle. A narrower fork has a larger $\cos\theta$, and the split’s loss at that angle is least when the daughters run at $V_p\cos\theta$ — faster than before, so thinner.

The fourth figure shows the effect, and it is small. The ratio falls from 0.754 to 0.743 at fifty diameters and 0.732 at twenty. The daughters do not move all the way to the split’s own optimum, because their pipes’ bills pull back towards 0.754: at fifty diameters they run at 0.906 of the parent’s speed against the 0.931 the split alone would choose. The radii and the angle are adjusted together, and the angle does almost all the work.

A thinner daughter is a faster one, so a priced network carries its fluid down the tree with less slowing at each fork than the pipes alone would give it. At a pipes-only smooth turbulent fork the total cross-section grows by 13.7 per cent; at the fifty-diameter priced fork it grows by 10.4 per cent, the same as a fully rough pipe’s — a coincidence of the numbers, but a reminder that the area a network opens at each junction is not fixed by its pipe law alone.

The fork, drawn #

The fifth figure draws two forks between the same source and the same pair of sinks, with daughters thirty parent diameters long. The unpriced fork splits 120 radii from the source at 28.4°. The priced one splits at 91.5 radii, nearer the source, at 19.4°. Its daughters are longer and its parent shorter, and it is the cheaper network by 1.4 per cent with the split’s loss counted in both.

That is the geometric form of the result. Narrowing a fork between fixed points means moving the junction back towards the source, so that the daughters run more nearly along the parent’s line. In the limit of a split that cost a great deal, the cheapest network would put the junction as near the source as the geometry allows and run two nearly parallel pipes from it — the arrangement of a manifold, in which the junction is a header the daughters leave almost straight.

When the split decides #

The third question was at what Reynolds number the junctions rather than the pipes decide a network’s shape. The sixth figure gives the quantity that answers it: the daughters’ length, in parent diameters, at which the pipes-only fork’s split loss equals the pumping power of one daughter. Below that length the split is the larger loss.

It is short — 5.6 diameters at a Reynolds number of 6,000, 11 at 93,000 — and it grows with the Reynolds number for the reason the dial showed: friction per diameter falls and the split’s loss does not. So for any network whose segments are tens of diameters long the pipes still dominate the losses, and the junctions still decide the angle. The two questions have different answers. The split is a small part of the bill wherever the pipes are long enough to be called pipes, and it decides the angle anyway, because the angle is the one thing the pipes are indifferent to.

The right-hand panel says the same thing at a fixed length: for fifty-diameter daughters, the cheapest angle falls from 23.8° to 21.4° as the Reynolds number rises across the range Blasius’s law covers.

Where it would show #

The prediction is that a turbulent network built at least cost, whose segments are a few tens of diameters long, forks at half-angles near twenty degrees rather than the twenty-eight its pipe law alone would give, and that its daughters are a little thinner than the pipe law says. Engineered networks are not usually built that way — a water main’s branches are laid along streets, and a duct system’s along a building’s bays — so the geometry is chosen and the junction’s loss is minimised by its own shape instead, with radiused or swept fittings that reduce the separation. A swept Y-fitting is the practical form of a narrower fork.

Biological trees are a better test of the principle, and a harder one. The largest vessels and airways are turbulent at peak flow and their segments are only a few diameters long — exactly where the split matters most and where the model is least trustworthy, as the list of what it leaves out below says. A tree’s angles are also shaped by growth, by packing into a body and by pulsatile flow, all acting on the same few degrees the split’s price moves. The calculation predicts the direction a priced fork should lean, narrower than its pipe law; it does not show that any measured tree leans that way for this reason.

The convention: a symmetric fork priced in watts #

Every quantity is a power. The pipes’ pumping is the flow times the friction pressure drop along their length; the upkeep is fifty watts per cubic metre of pipe, the same stand-in for a vessel’s metabolic or an engineer’s capital cost the earlier essays used; the split’s loss is the flow times the kinetic energy per unit volume of the velocity difference. The fork is symmetric: equal daughters carrying equal halves at equal angles. Lengths are measured in parent diameters at the pipes-only optimum. The friction is Blasius’s, which is the smooth-pipe law between Reynolds numbers of about 4,000 and $10^5$ — the walls are smooth in the sense the roughness a wall cannot feel defines, their bumps buried in the viscous sublayer — and every flow here keeps the parent inside that range.

What the calculation does not price #

The split’s real loss. Borda’s form counts the abrupt change of velocity and nothing else. Real dividing junctions separate on the inside of each turn, and measured coefficients exceed the bare form; the doubled loss stands in for that, and it narrows the fork further. A radiused junction loses less than either.

Pipes too short to be pipes. Blasius’s factor describes fully developed flow. A daughter a few diameters long is still developing, its velocity profile still recovering from the split, and its friction is not the developed value; how far before a duct forgets puts a turbulent pipe’s development length at tens of diameters. The results below twenty diameters inherit that error, and the short-segment limit in which the split dominates is exactly where the model is weakest.

Asymmetric splits, and more than two daughters. An unequal split turns its two streams through different angles and the identity holds for each separately; that calculation was not made.

Laminar forks. Below a Reynolds number of a few hundred the split’s loss is not Borda’s — the flow turns smoothly and loses energy by viscosity in the turn, a loss that scales with the flow rather than its square. The laminar curve in the first figure shows what the identity says; the laminar fork’s priced angle is a different calculation.

Who priced the junction #

Murray’s minimisation of 1926 priced the pipes, and Zamir and others carried it to the angles of branching in the 1970s with the force triangle the earlier essays used; Borda’s loss is from 1766 and its extension to a change of direction is in every handbook of pipe-fitting losses, with measured coefficients for dividing tees and wyes collected by Idelchik and others. The identity drawn here — that at the cheapest fork each daughter carries the parent’s velocity component along it, so the bare split loses only the sideways speed — follows in two lines from the earlier results and is this calculation’s, stated without a search of the literature for it.

How the fork was checked #

The identity was checked at five exponents, from 2.2 to 3: the daughters’ speed ratio and the cosine of the cheapest angle agree to rounding error, the split’s loss equals the sideways form to the same precision, and its slope with the daughters’ radius is a few parts in a hundred million of its size. With the split left unpriced, the three-variable search returns the earlier essay’s fork — ratio 0.7540 and 28.41° — to nine figures. And the priced angle at fifty diameters does not move when the geometry is perturbed by a millionth.

Still open: the laminar fork’s own loss #

The identity holds for every exponent, Murray’s included, but the split’s loss in a laminar network is not Borda’s. At Reynolds numbers of a few hundred — a small artery, a capillary bed’s feeding arterioles, an airway past the eighth generation — the flow turns smoothly into each daughter and loses energy by viscosity in the turn, at a rate proportional to the flow rather than its square, and there is a Dean-type secondary flow in the bend. The next calculation would replace Borda’s loss with the laminar junction’s, computed or taken from Stokes-flow solutions for a branching tube, and ask whether a laminar fork priced for its split also narrows, by how much at the segment lengths a real tree has, and whether the first-order argument here survives a loss that is first order in the angle in a different way.