The heat the eddies do not carry

The other layer put a heated plate in a laminar stream and found two boundary layers on one wall: one for velocity, one for temperature, with thicknesses in the ratio $Pr^{1/3}$. Across ordinary fluids that is a factor of twenty, from a liquid metal whose thermal layer is five times the velocity layer to an oil whose thermal layer is a fifth of it. At exactly one Prandtl number the two profiles coincide.

That essay’s closing section pointed at the turbulent version, where “the two layers are set by eddies rather than by molecules and the ratio between them stops being a fluid property”. This is that version. It turns out to be half true in an instructive way: the ratio stops being a fluid property for every fluid an engineer normally meets, and becomes one again, with a vengeance, for the fluids used to cool nuclear reactors.

Why the eddies should carry heat like momentum #

In a turbulent pipe almost all the transport across the flow is done by eddies. A parcel of fluid thrown from near the wall towards the axis carries its low momentum and its high temperature with it, and the rates at which those two quantities are carried depend on the same motion. So the eddy viscosity $\nu_t$ and the eddy heat diffusivity $\alpha_t$ should be nearly equal, and their ratio — the turbulent Prandtl number, measured at around 0.85 in air and water — should not depend much on the fluid.

That is Reynolds’ analogy of 1874, and it is why heat-transfer engineering can take a friction factor, which is easy to measure, and turn it into a heat transfer coefficient, which is not. But it has a condition hidden in it. The eddies carry heat like momentum only where they carry more heat than the molecules do. Close to the wall the eddies are damped, and there both transports are molecular: momentum by viscosity, heat by conduction. The question is how far from the wall the molecules keep the upper hand, and for heat that depends on the fluid.

The calculation #

The pipe is fully developed, with a friction Reynolds number $Re_\tau = u_\tau R/\nu$. The shear stress falls linearly from wall to axis, and it is carried by molecular viscosity plus an eddy viscosity from a mixing length: Nikuradse’s distribution for a pipe, damped near the wall by van Driest’s factor. The wall is heated at a uniform flux, and the heat is carried inwards by molecular conduction plus an eddy diffusivity $\nu_t/Pr_t$ with $Pr_t = 0.85$:

$$\Big(\frac{1}{Pr} + \frac{\nu_t/\nu}{Pr_t}\Big)\frac{d\theta^+}{dy^+} = \frac{q}{q_w}.$$

The heat flux itself falls towards the axis in proportion to how much flow lies inside each radius, because that flow absorbs the heat that has not yet reached the axis. The Nusselt number follows from the mixing-cup temperature.

The mixing length, van Driest’s constant and the turbulent Prandtl number are a closure and are borrowed, and so every Nusselt number here is the closure’s. Two things are not. Setting the eddy viscosity to zero must return the laminar values exactly — a Nusselt number of 48/11 at every Prandtl number and $f\cdot Re = 64$ — and it does, to $2\times10^{-8}$. And the closure’s friction factor must be a pipe’s: it matches Prandtl’s friction law to 1.2 per cent from a Reynolds number of 17,000 to 500,000.

Air, water and a liquid metal in one pipe #

The picture is the answer, and the numbers make it precise. Within thirty wall units of the wall — the viscous sublayer and the buffer layer — the velocity completes 54 per cent of its rise from wall to axis. Air’s temperature completes 49 per cent in the same distance. Water’s completes 81 per cent, because at a Prandtl number of seven the thermal sublayer is thinner than the viscous one. And liquid sodium’s completes 4 per cent: its temperature is still close to the wall’s where the velocity is halfway to the axis’s.

So for air and water the laminar factor of twenty has collapsed. Both profiles are set in the same thin layer against the wall, and the turbulent core above it is nearly uniform in both velocity and temperature. For liquid sodium the laminar picture has survived intact: a thermal layer that fills the pipe, sitting on a velocity layer that does not.

Where the eddies win #

The diffusivity ratio explains the profiles directly. It is $Pr\,(\nu_t/\nu)/Pr_t$, and the eddy viscosity is the same in every fluid at the same Reynolds number, so the Prandtl number simply slides the curve up and down. For air it crosses one about eleven wall units from the wall and reaches 135 in the core: beyond the buffer layer the eddies carry nearly all the heat, as they carry nearly all the momentum. For water the crossing is earlier and the peak is 1,350.

For liquid sodium, at a Prandtl number of 0.005, the ratio peaks at 0.58. There is no point in the pipe where the eddies carry as much heat as conduction does. The turbulence is exactly as vigorous as in air at the same Reynolds number — the velocity profile is identical — and for heat it is largely irrelevant. Sodium conducts so well, through its electrons, that an eddy moving a hot parcel across the pipe is overtaken by conduction before it gets there.

A threshold with almost no Reynolds number in it #

The condition for the eddies ever to win is that the peak of that curve exceeds one:

$$Pr\cdot\Big(\frac{\nu_t}{\nu}\Big)_\text{max} > Pr_t.$$

The peak eddy viscosity in a pipe grows in proportion to the Reynolds number. In this closure it is 0.082 of the pipe radius in wall units at every Reynolds number from 6,000 to a million, which is a statement about the mixing length’s outer shape: the largest eddies scale with the pipe. So the threshold Prandtl number falls as one over the Reynolds number, and the product that matters is a Péclet number.

The threshold Péclet number is between 310 and 550 across the whole range, drifting slowly because the bulk velocity grows logarithmically relative to the friction velocity. Below a Péclet number of about four hundred, heat in a turbulent pipe travels as it would in a laminar one; above it, the eddies start to take over, first in the middle of the pipe where they are largest.

That is why the Péclet number, rather than the Reynolds number, is the number liquid-metal heat transfer is correlated on. For ordinary fluids the threshold is irrelevant — air’s Péclet number is always above it once the flow is turbulent at all, since its Prandtl number is near one. For a liquid metal at a Prandtl number of 0.01, a turbulent pipe at a Reynolds number of 40,000 has a Péclet number of 400 and sits exactly on it.

The Nusselt number, and which correlation it obeys #

The computed Nusselt numbers fall into the two families engineers have long used. For air they follow Dittus and Boelter’s correlation, $0.023\,Re^{0.8}Pr^{0.4}$, to within a few per cent: 88.5 at a Reynolds number of 38,000 against 92, and 234 at 131,000 against 248. For a liquid metal at $Pr = 0.01$ they follow Lyon’s correlation of 1951, $Nu = 7 + 0.025\,Pe^{0.8}$: 10.0 against 9.9 at a Péclet number of 380, 16.2 against 14.8 at 1,300, 33 against 30 at 5,000.

The air-and-water correlation applied to the liquid metal would have been badly wrong. At a Reynolds number of 131,000 it gives 45.3 where the pipe gives 16.2 — a factor of 2.8. The error is not in the exponent on the Prandtl number, which a larger fit could adjust. It is structural: the correlation’s form assumes the heat goes where the momentum goes, and its constant term is zero because a turbulent air flow has no laminar floor worth keeping. Lyon’s form keeps the floor, a constant near seven, because below a Péclet number of a few hundred the floor is most of the answer.

The analogy, and where it breaks #

The same breakdown shows in the analogy that turns friction into heat transfer. Colburn’s form of it, $St\,Pr^{2/3} = f/8$, holds in this pipe to within 4 per cent for air — 0.956 — and stays between three-quarters and one and a half across the Prandtl numbers from 0.3 to 100, the band that includes every gas, water and most oils. Below a Prandtl number of 0.1 it collapses, and for a liquid metal at 0.01 it is 0.320: the friction factor overstates the heat transfer threefold, because the momentum is carried by eddies and the heat is not.

The surprising connection is that the laminar layer and the liquid metal are the same physics met from two directions. In a laminar layer there are no eddies, so heat and momentum each diffuse molecularly at their own rates and the profiles differ by the Prandtl number. In a liquid metal there are eddies, but the molecular conduction is so fast that the eddies do not matter for heat, so the temperature diffuses as if the flow were laminar while the velocity does not. A turbulent liquid metal is thermally laminar and dynamically turbulent at once, and every correlation for it has to be built around that split.

What turbulence does to a liquid metal’s Prandtl number #

There is a way to state the split as a single number, and it is worth doing because it shows the eddies are not doing nothing to the heat.

Where the eddies are strongest, halfway to the axis, the fluid behaves as though it had an effective Prandtl number: the total diffusivity of momentum, molecular and turbulent, over the total diffusivity of heat. For air at $Re_\tau = 2000$ that is 0.85 — the turbulent Prandtl number, because the eddies swamp both molecular terms. For mercury it is 0.71. For liquid sodium it is 0.42, eighty times its molecular value of 0.005.

So the eddies do move a liquid metal’s heat, and by a great deal compared with nothing. What they cannot do is move it faster than the electrons already do. The effective Prandtl number is lifted towards the turbulent value and stops well short of it, and the shortfall is exactly the conduction that the eddies never overtake. At a lower Reynolds number, where the eddies are weaker, the lift is much smaller: at $Re_\tau = 200$ sodium’s core behaves as a fluid with a Prandtl number of 0.076.

This is also the cleanest way to see why a liquid metal is thermally laminar and not merely poorly turbulent. In air the effective Prandtl number of the core is the turbulent one and has forgotten the molecules. In sodium it is a mixture weighted towards the molecules, and the wall’s temperature profile inherits that weighting all the way to the axis. The buffer layers that shape the velocity near the wall shape the temperature only where the eddies are the stronger carrier, and for sodium that is nowhere.

Why this is a reactor engineer’s problem #

Liquid metals are not an academic curiosity. They carry heat out of fast-neutron reactor cores — sodium in most designs, lead or lead–bismuth in others — because they conduct well, stay liquid over a wide range of temperature at low pressure, and do not slow neutrons the way water does. The first reactor to use one, Clementine at Los Alamos in 1946, was cooled by mercury.

In those cores the coolant runs through narrow gaps between fuel pins at Reynolds numbers of tens of thousands, and with a Prandtl number of 0.005 the Péclet number is a few hundred: squarely on the threshold. The heat-transfer coefficient there is neither the laminar one nor the one a water-cooled design would use, and a design that borrowed a water correlation would predict fuel temperatures that were wrong by a large factor in the unsafe direction — a heat-transfer coefficient two or three times too high means a cladding temperature estimate that is too low. The existence of a separate family of liquid-metal correlations, all of them in the Péclet number, is the engineering record of this essay’s threshold.

The checks #

The laminar limit is the check that matters, because it tests the heat-flux distribution and the mixing-cup average without any help from the closure. A mistake in the way the flux is shared out across the radius, or in the weighting of the mixing-cup temperature, would show there as a Nusselt number other than 48/11, and at a different wrong value for each Prandtl number — which is why the check is run at three of them rather than one. With the eddy viscosity switched off, the same integrations return a Nusselt number of 4.36364 at Prandtl numbers of 0.01, 1 and 100, against 48/11 = 4.363636, and a friction factor times Reynolds number of 64.0000.

What the mixing length cannot show #

The turbulent Prandtl number in a liquid metal. It was held at 0.85 here, the value measured in air and water. In liquid metals it is measured to be larger, rising to one or two at low Péclet numbers, because conduction smears the temperature fluctuations an eddy carries before it can deliver them. That would move the threshold to a somewhat higher Péclet number and lower the Nusselt numbers near it; it would not remove the threshold.

Near-wall eddies. The van Driest damping is an empirical fit to velocity data. The thermal sublayer of a high-Prandtl-number fluid sits deep inside it, where the fit is least certain, which is why the computed Colburn factor rises to 1.5 for oils where measurements are closer to one.

Buoyancy and property variation. Every property is constant. A real heated liquid metal, or a real oil, changes viscosity and conductivity with temperature across the pipe.

Magnetic fields. A liquid metal conducts electricity as well as heat, and in a strong magnetic field — the case in the blankets proposed for fusion reactors — the eddies drive currents that damp them. The eddy viscosity then falls below its value at the same Reynolds number without a field, and the threshold Péclet number rises with it, sometimes far enough that the heat transfer is laminar at Reynolds numbers where the flow would otherwise be fully turbulent. None of that is in a mixing length calibrated on air and water.

Entrance effects. The pipe is fully developed. A liquid metal’s thermal entrance length is long — at low Péclet number the temperature keeps developing for many diameters — and most real heated lengths are partly in it.

Who worked it out #

Osborne Reynolds proposed the analogy between heat and momentum transport in 1874. Allan Colburn gave it the $Pr^{2/3}$ form in 1933, and Frank Dittus and Llewellyn Boelter’s correlation is from 1930. Robert Martinelli analysed heat transfer to molten metals in 1947 and showed that the molecular conduction term could not be dropped from the turbulent analysis; Richard Lyon’s correlation followed in 1951, from the reactor programmes that needed it. Edward van Driest’s damping function is from 1956.

Still open: the combined entrance a liquid metal lives in #

A liquid metal’s thermal layer takes many diameters to reach the pipe’s axis, because conduction rather than eddies carries the heat across it, while the velocity profile develops in the few diameters turbulence needs. So a heated liquid-metal pipe spends much of its length with a developed velocity profile and a developing temperature one — the case the thermal entrance essay solved for laminar flow.

The next calculation carries that entrance problem into the turbulent pipe with the same closure, and asks how the entrance length scales: whether, for a liquid metal below the threshold Péclet number, it follows the laminar scaling in the Péclet number — as the thermally laminar picture here predicts — and at what Péclet number it switches over to the few diameters of an ordinary turbulent fluid.