The cheapest turbulent pipe follows two and a half, not three

The radius that costs least found Murray’s law by minimising two bills: the power to push a flow through a vessel, which falls steeply as the vessel widens, and the cost of keeping the vessel’s volume, which rises with it. The minimum puts the flow proportional to the cube of the radius, and it comes with an invariant that makes the result more than tidy: the wall shear stress is the same in every vessel of the network, so endothelial cells that widen a vessel when the shear is above a set point and narrow it when below build the optimal network without anything computing a cost. The angle a junction chooses took the same cost to the junctions and found 37.467° for a symmetric split.

Every step of that used one formula, Poiseuille’s: the pressure drop per unit length of a laminar pipe flow, $8\mu Q/\pi r^4$. It holds while the Reynolds number is below about two thousand. A water main, a ventilation duct, a large artery at peak systole, an oil pipeline and the trachea at a sneeze are not laminar. This essay redoes the minimisation with the friction of a turbulent pipe and asks what is left of the law, the invariant and the rule.

The cost with turbulent friction #

Write the pumping power per unit length of a pipe of radius $r$ with the Darcy friction factor $f$,

$$\frac{Q\,\Delta p}{\ell} = Q\,\frac{f\rho V^2}{4r}, \qquad V = \frac{Q}{\pi r^2}, \qquad \mathrm{Re} = \frac{2\rho Q}{\pi\mu r},$$

and add the upkeep $b\pi r^2$. In laminar flow $f = 64/\mathrm{Re}$ and this is Poiseuille’s dissipation. Whenever $f$ is a power of the Reynolds number the pumping term is a power of the radius, $Q^q r^{-n}$, and the minimum of $Q^q r^{-n} + b\pi r^2$ sits where $n$ times the pumping bill equals twice the upkeep, with $r^{n+2} \propto Q^q$.

Three cases have closed forms. Laminar, $q = 2$ and $n = 4$: $r^6 \propto Q^2$, the cube law. Blasius’s smooth-pipe law $f = 0.316\,\mathrm{Re}^{-1/4}$, good from about 4,000 to $10^5$: the pumping goes as $Q^{11/4}r^{-19/4}$, so $r^{27/4} \propto Q^{11/4}$ and

$$Q \propto r^{27/11} = r^{2.4545}.$$

A fully rough pipe, whose friction factor no longer depends on the Reynolds number: pumping as $Q^3 r^{-5}$, $r^7 \propto Q^3$, and $Q \propto r^{7/3}$. The share of the bill that is pumping at the optimum is $2/(n+2)$: a third laminar, which is the laminar result’s “exactly a third whatever the constants”, then $8/27$ smooth and $2/7$ rough. Turbulence moves a little of the cost from the pump to the pipe.

What a real friction law gives #

The closed forms are limits, and a real pipe sits between them. The figure carries Colebrook’s equation for turbulent friction — the formula behind the Moody chart — with Poiseuille’s law below a Reynolds number of 2,300 and a stated blend across the transitional band to 4,000. The flow is water, the upkeep fifty watts per cubic metre, and the minimisation is done numerically for each flow rate; the exponent is read off by differentiating the optimum radius against the flow.

In the laminar range the exponent is 3 to six figures, as it must be. Past transition it drops at once to about 2.47 and then drifts slowly down: 2.45 at a Reynolds number of $2\cdot10^4$, 2.40 at $10^7$. That is Blasius’s 27/11 where Blasius’s fit holds, and a little below it at higher Reynolds numbers, where Colebrook’s friction falls more slowly than Blasius’s quarter power. Giving the wall a millimetre of roughness hardly moves the curve. The reason is the interesting part.

Seven thirds needs a roughness that grows with the vessel #

The 7/3 law is the one usually quoted for turbulent vessels, and it needs a friction factor that does not change with the vessel’s size. A fixed absolute roughness does not give that. The cheapest pipe for a large flow is a large pipe, and a millimetre of roughness in a pipe a metre across is a relative roughness of a thousandth, at which a wall can barely feel it until the Reynolds number is very high. The bigger the flow, the smoother the optimal pipe is in the only sense friction cares about, and the exponent stays near the smooth value.

Hold the roughness at a fixed fraction of the diameter instead — a wall whose texture is scaled with the vessel, as a geometrically similar network would have — and the friction factor does go to a constant, and the exponent goes to 7/3: 2.334 at the largest flows for $\varepsilon/D = 10^{-3}$, 2.333 for $10^{-2}$. So the two turbulent laws describe two different kinds of network. A family of pipes made of one material, with one surface finish at every size, follows 27/11. A family whose roughness scales with its size follows 7/3. Neither is Murray’s.

The invariant goes #

The wall shear stress is $\tau = (r/2)\,\Delta p/\ell$. In laminar flow at the optimum it does not depend on the radius — 0.224 pascals at this upkeep in every vessel from a millimetre to the transition — and that is the invariant. Past transition it grows: with Blasius’s law, $\tau \propto Q^{7/4}r^{-15/4}$, and along the optimum $Q \propto r^{27/11}$, so $\tau \propto r^{6/11}$. The figure’s slope over the turbulent range is about 0.56, close to that, and the shear reaches 7.3 pascals in the cheapest vessel of 1.9 metres radius. The constant-shear property is not a general feature of least-cost networks. It is a coincidence of Poiseuille’s exponents: $n = 4$ is the one value for which the shear at the optimum is a constant.

The rule a cell could follow stops finding the optimum #

The laminar argument for Murray’s law as biology was that the optimum could be reached by a local rule — keep the shear at a set point — with nobody computing a cost. In turbulent flow that rule still produces a definite law, and it is the wrong one. A shear held constant with Blasius’s friction needs $Q^{7/4}r^{-15/4}$ fixed, so $Q \propto r^{15/7}$, an exponent of 2.14, well below the optimum’s 2.45. Since the rule gives a vessel wider than the optimum at every turbulent flow, and the error grows with the flow, the figure’s warm lines rise: at ten times the flow of the last laminar vessel the shear-sized vessel is 1.18 times too wide; at a thousand times, 1.55, costing 1.73 times the least.

The other natural mistake is to carry the cube law on past transition, sizing a turbulent trunk as though it were a large laminar twig. That makes it too narrow, and narrowness costs more than width: at a thousand times the flow the cube-law vessel is 0.54 of the right radius and costs 5.8 times the least, because the pumping bill rises as the radius to the power $-4.75$ and the upkeep falls only as its square. The two laminar rules bracket the turbulent optimum, one on each side, and both fail by more the further the network extends above transition.

What that means for an artery is a question about flow rather than about vessels. The aorta’s mean Reynolds number is around a thousand and a half and its peak several thousand; its flow is turbulent for a fraction of each beat at most. Murray’s law is not the rule for a pulse found a different reason for the largest arteries to depart from the cube law — the reflection of the pressure wave at junctions — and its best exponents, 2.15 to 2.6, overlap the range found here. Two independent mechanisms push the largest vessels’ exponent below three in the same direction, and measured exponents in the largest arteries do sit below three; which mechanism the measurement is reading cannot be decided from the exponent alone.

Junctions narrow #

The junction follows from the exponent. A symmetric split with $Q \propto r^k$ gives daughters $2^{-1/k}$ of the parent’s radius: 0.7937 for Murray’s 3, 0.7540 for 27/11, 0.7430 for 7/3. The total cross-section therefore grows by 26 per cent at a laminar junction, 13.7 per cent at a smooth turbulent one and 10.4 per cent at a rough one, so the fluid in a turbulent network slows less on the way down.

The angle comes from the same force triangle the junction essay used, with each vessel pulling with its cost per unit length. At the optimum that cost is proportional to $r^2$ in every regime, because both bills scale together there, so the weights are the same squares of radii and only the radii have changed: $\cos\theta = 2^{2/k-1}$. The angle is 37.47° laminar, 28.41° smooth turbulent and 25.08° rough. A turbulent trunk forks at a narrower angle into thinner daughters, closer to a straight run, because each daughter is a larger share of the parent’s cost and pulls harder along the axis.

A tree with a turbulent trunk #

A network whose root is turbulent and whose leaves are not obeys both laws. The figure starts from one cubic metre a second of water — a city main — and halves the flow at each of thirty junctions. The first fourteen junctions are turbulent and obey 2.41 to 2.47, rising slowly as the Reynolds number falls. The two that straddle transition obey 2.38 and 1.90: the friction factor jumps across the transitional band, so the cheapest radius jumps too, and a junction that crosses the band can have almost any exponent the blend allows. That pair is the least trustworthy number in the essay, since the blend is a stated choice rather than a model of transition, and the number that is not a number is the reason no better choice exists. From the seventeenth junction on the flow is laminar and every junction is Murray’s to six figures.

So the “exponent of a network” is a property of where its flow is, not of the network. A measurement that pools junctions across transition and fits one exponent will get something between 2.4 and 3 that describes no junction in it.

What the upkeep decides, and what it does not #

The upkeep $b$ is the one number in the cost that has no measured value, and it is worth separating what it controls. It does not touch any exponent: every power law above came from the friction law alone, and a tenfold change in $b$ moves every curve in the first figure sideways in flow rate without changing its height. What it does set is where on the flow axis transition falls. The cheapest laminar vessel has $r \propto (\mu Q^2/b)^{1/6}$, so its Reynolds number grows as $Q^{2/3}b^{1/6}$, and a costlier wall — a larger $b$ — makes the optimal vessel narrower and faster and reaches transition at a smaller flow. At fifty watts per cubic metre the last laminar vessel carries 0.016 litres a second in a radius of 4.5 millimetres at 0.25 metres a second; at five hundred, 0.009 litres a second in 2.6 millimetres at 0.45; at five, 0.029 litres a second in 8.1 millimetres at 0.14. The transitional flow moves as $b^{-1/4}$.

At the other end of the tree the same arithmetic gives a recognisable pipe. A district water main carrying a tenth of a cubic metre a second has its cheapest diameter at 354 millimetres at this upkeep, a velocity of 1.0 metre a second and a Reynolds number of $3.6\cdot10^5$ — inside the one to two metres a second that design guides give for mains, though the agreement depends on a $b$ chosen for tissue rather than for steel. The exponent, the shares and the junction angle at that main do not depend on that choice at all, and they are the results here.

Engineers have been here #

Tissue costs its volume, so its upkeep is $r^2$. A steel pipe’s installed cost per metre does not go as its cross-section; it rises more slowly with size, roughly as the diameter to a power between one and one and a half. With an upkeep $\propto r^m$ the optimum radius goes as $Q^{q/(n+m)}$, and the figure plots that exponent. At $m = 1.5$ it is 0.44 for Blasius’s friction and 0.46 for constant friction.

Chemical engineers have sized pipes by an economic-diameter rule since the 1930s, and the rule’s standard form puts the cheapest diameter in turbulent flow proportional to the flow to the power 0.45. That correlation was fitted to pumping and pipe costs, not derived; its exponent is where the turbulent lines of this figure sit for a wall cost between $r^{1.1}$ and $r^{1.7}$. It is the turbulent Murray law with a cheaper wall, found by accountants. The corresponding laminar exponent would be 0.36 at $m = 1.5$, and nobody sizing a water main would use it.

How the optimum was checked #

The optimum is found by golden-section search on the logarithm of the radius to a relative tolerance of $10^{-13}$, and the exponents by differentiating it across a flow change of a tenth of a per cent. With pure laminar friction the optimisation returns 3.00002 against 3; with Blasius’s law, 2.45454 against 27/11; with constant friction, 2.33333 against 7/3. The laminar optimum matches Murray’s closed form $r^6 = 16\mu Q^2/b\pi^2$ to two parts in a thousand million, and the pumping shares match a third, 8/27 and 2/7 to one part in a hundred million. Colebrook’s equation, iterated to convergence, gives 0.01799 for a smooth pipe at a Reynolds number of $10^5$, the Moody-chart value, and at $\varepsilon/D = 10^{-3}$ and $\mathrm{Re} = 10^9$ it reaches von Kármán’s fully rough limit to two parts in a hundred thousand.

The convention: steady flow, straight pipes, a volume upkeep #

The cost is per unit length of straight pipe in steady flow. Junction losses are not in it: the energy lost at a fork is a local loss coefficient times the dynamic pressure, and in a turbulent network it is not small, which pushes the cheapest junction towards smaller angles still. The upkeep is proportional to volume except in the engineering figure. The Reynolds number is based on diameter. The transitional blend is linear in the logarithms of the Reynolds number and friction factor between the laminar value at 2,300 and Colebrook’s at 4,000, and is stated as a choice.

What the picture cannot show #

The figures cannot show transition itself. A real pipe near a Reynolds number of 2,000 to 4,000 is intermittently turbulent, with puffs whose share of the pipe depends on the inlet and the history, and the friction factor there is not a function of the Reynolds number at all. They also cannot show pulsation: the averaging of a turbulent friction over a cardiac cycle is not the friction of the mean flow, because the friction law is nonlinear. And they cannot say what an artery senses. The finding that a shear set point builds the wrong turbulent network is a statement about the rule as written; a vessel that sensed something else — the shear’s fluctuations, or its peak — would follow a different law, and which it does is a measurement this calculation cannot make.

Who found it, and when #

Murray derived the cube law in 1926 from exactly this cost with Poiseuille’s resistance. Uylings pointed out in 1977 that the same argument with a fully turbulent friction factor gives an exponent of 7/3, and that real arterial and bronchial trees might sit between the two; the intermediate Blasius value follows from the same algebra and appears in later work on optimal networks. The engineering economic diameter is older than both: it was being fitted to pipe costs in the 1930s, and textbooks of process design still print it with its exponent of 0.45. The observation that the constant-shear invariant is special to the laminar exponent follows directly from writing the shear at the optimum for a general friction law.

Still open: a junction that loses energy of its own #

Every number here prices the pipes and not the forks. In a turbulent network a junction has a loss coefficient of its own — a fraction of the dynamic pressure lost to the mixing and separation at the split — and that loss depends on the angle and on the daughters’ radius ratio. The next calculation adds a junction loss of the standard form to the cost and minimises the angle and the radii together, asking whether the optimal turbulent fork narrows further than the 28° found here, whether the daughters thicken to pay for a gentler split, and at what Reynolds number the junction losses rather than the pipe friction decide a network’s shape.