The other layer, and the one number that separates them

Everything on this site so far has been about one field: the velocity. A wall imposes a condition on it, a layer grows to satisfy that condition, and the thickness of the layer decides the drag.

A wall in a real flow imposes a second condition, on a second field. It has a temperature, or it is insulated and has none, and either way the fluid next to it has to be told something about heat. That second condition grows a second layer, in the same place, obeying nearly the same equation — and the two layers are not the same thickness.

The number that separates them is a property of the fluid and of nothing else.

The second equation is the first one, almost #

With the flow supplied by Blasius’ solution, the temperature satisfies

$$T'' + \tfrac{Pr}{2}\,f\,T' = 0,$$

where $f$ is the same stream function the velocity solve produced. The velocity satisfies $f''' + \tfrac{1}{2}f f'' = 0$.

Two things about that pairing decide everything below.

The temperature equation is linear. Once $f$ is known, $T$ satisfies a linear ordinary differential equation with no eigenvalue in it, so there is no shooting, no iteration, and exactly one solution. The velocity problem is nonlinear and needed a root-find; this one integrates in closed form up to two quadratures. That is not a computational convenience — it is the mathematical shadow of the temperature being a passive scalar: it is carried by the flow and exerts no force on it, so it cannot be unstable, cannot separate, and cannot have more than one answer.

And the only difference between them is $Pr$. Set the Prandtl number to one and the equation for $T$ is the equation for $f'$, with the same conditions at both ends. The two profiles are then the same function — not similar, identical.

This site computes them by routes that share only a grid: the velocity by fourth-order marching with a secant search on the wall slope, the temperature by two trapezoid quadratures over the result. At $Pr = 1$ they agree to $3\times10^{-8}$.

The thickness ratio, and where the famous exponent comes from #

The measured ratio $\delta_T/\delta$ is 1.141 for air against $Pr^{-1/3} = 1.121$; 0.499 for water against 0.523; 0.204 for oil against 0.215. The one-third power is a good fit and it is a fit — the exponent measured over the range from 0.6 to 10 is 0.343, not $0.3333$, and the difference is real rather than numerical.

Below $Pr \approx 0.1$ the fit stops being one at all. The exponent measured over the range of liquid metals is 0.4545; a decade lower it is 0.4744; a decade lower again, 0.4853. It is approaching a half, which is the exact answer in the limit where the velocity layer is so thin compared with the thermal one that the temperature sees a uniform stream — a much easier problem, whose answer is $Nu = 0.564\sqrt{Pe}$.

Mercury sits at $Pr = 0.025$. It is on the bend rather than in either limit, which means neither exponent is a fact about any fluid: both are the shape of one curve, and every real substance sits somewhere on it.

Two layers, three orders of magnitude #

The physical content of the Prandtl number is a ratio of two diffusivities: momentum’s $\nu$ against heat’s $\alpha$. Both spread outwards from the wall; whichever spreads faster reaches further.

• A liquid metal conducts heat far better than it diffuses momentum: $Pr \approx 0.01$, the thermal layer is nearly eight times the velocity layer, and the temperature field extends well beyond anything the wall has slowed.
• A gas has $Pr \approx 0.7$ for the good reason that both quantities are carried by the same molecules making the same journeys — the ratio cannot be far from one in a gas, and for air it is 0.71 at every temperature and pressure worth quoting.
• Water is 7 at room temperature and 1.8 near boiling.
• A heavy oil is in the hundreds, its thermal layer is a fifth of its velocity layer, and heat transfer to it is a wall phenomenon in a way that momentum transfer is not.

The gas case deserves the extra sentence, because the near-coincidence is not luck. In a gas both momentum and energy are carried by the same molecules travelling the same mean free path between the same collisions, so the two diffusivities are built from the same quantities and their ratio is a number of order one for kinetic-theory reasons rather than empirical ones. Every gas is between 0.65 and 0.75. That is why the Reynolds analogy — which is exact only at $Pr = 1$ — is nearly exact for every gas anybody has ever built a heat exchanger for, and why it is useless for the liquid metals that cool reactors.

The analogy, and where it stops #

At $Pr = 1$ the two profiles coincide, and the consequence is the most useful statement in convective heat transfer.

The wall shear is $\mu(\partial u/\partial y)_w$ and the heat flux is $k(\partial T/\partial y)_w$. If the two profiles are the same function, the two gradients stand in a fixed ratio, so a measurement of drag is a measurement of heat transfer. In dimensionless terms $St = c_f/2$ exactly, and this site’s two independent solves return that identity to $10^{-6}$.

The corrected version at the Prandtl number of air comes to 0.993 rather than one, which is why a century of heat-exchanger design has used it without much anxiety. At $Pr = 100$ it is 1.020, and at $Pr = 0.01$ it is 0.721 — a liquid-metal heat exchanger designed on the analogy would be nearly thirty per cent wrong, and that is exactly the application where the analogy is never used.

The engineering consequence of the analogy is worth stating in its blunt form, because it constrains design rather than merely describing it. Nothing that increases heat transfer at a wall leaves the drag alone. Roughening a surface, tripping the layer, adding turbulators: each of them improves the transfer of heat and pays for it in pressure drop, in a ratio the analogy fixes in advance. A heat-exchanger designer is not choosing between them; the choice is where on one curve to sit.

Why the ratio is a Péclet number in disguise #

The Prandtl number contains no flow. It is $\nu/\alpha$, a property of the substance, and it is constant for air whether the air is in a wind tunnel or a hurricane.

What the flow contributes is the Péclet number, $Pe = Re\,Pr = UL/\alpha$, which is the ratio of heat carried by the flow to heat carried by conduction — the thermal twin of the Reynolds number, and the number that says whether a temperature field has a boundary layer at all. Every result above is really a statement about $Pe$: $Nu \sim \sqrt{Pe}$ at small $Pr$, $Nu \sim Re^{1/2}Pr^{1/3}$ at moderate $Pr$, and both are the same statement that a layer’s thickness is set by a competition between carrying and spreading.

That competition is the same one Taylor dispersion is about, where a shear and a diffusivity multiply into an effective diffusivity a thousand times either. The difference is geometric rather than physical: there the two act along a tube, here across a layer.

One number from that figure is worth extracting. The Nusselt number — the ratio of actual heat transfer to what pure conduction across the same distance would give — comes out as $0.2942\sqrt{Re_x}$ for air, against the textbook $0.332\,Pr^{1/3}\sqrt{Re_x} = 0.2962\sqrt{Re_x}$. Seven tenths of a per cent apart, from a solve that used no correlation at any point. The agreement matters less than what it means: at a Reynolds number of a million, moving air past a wall carries about three hundred times as much heat as still air would conduct, and every bit of that factor is the layer being thin rather than the air being different.

The reverse reading of the analogy is the one that gets used in the wind tunnel. Since heat transfer and skin friction are the same measurement, a surface that can be persuaded to show where it is losing heat is showing where it is losing momentum — which is what an infrared camera on a model does, and why a thermal image of a wing in a tunnel is read as a map of transition. The bright line across it is not a temperature feature; it is the friction rising by a factor of three where the layer goes turbulent.

What the picture cannot show #

Every property is constant. Viscosity, conductivity and density are held fixed, which is exactly what makes the temperature a passive scalar. In a real flow with a hot wall the viscosity varies across the layer — for a gas it rises with temperature, for a liquid it falls sharply — and the velocity profile is no longer independent of the temperature. Heat transfer to an oil is the standard case where this matters, and correlations carry a viscosity-ratio factor for it that is measured rather than derived.

Buoyancy is absent. Nothing here lets the hot fluid rise. The whole calculation assumes forced convection dominates, which fails at low speeds and is a different subject — one this collection reaches from the stability end rather than the boundary-layer end.

And there is no dissipation. The right-hand side of the energy equation has been left at zero, which is legitimate at low speed and abandons the whole of high-speed heat transfer. Restoring it is what gives a wall told nothing about its temperature a temperature of its own, and that term is negligible until it is decisive.

And a third layer, on the same wall #

Nothing in the temperature equation used the fact that the scalar was a temperature. It used that the scalar is carried by the flow, spread by its own diffusivity, and exerts no force — so any passive scalar obeys the same equation with its own diffusivity in place of $\alpha$, and grows its own layer on the same wall.

The commonest such scalar is a concentration: a dissolving solid, a reacting species at a catalyst, an ion arriving at an electrode, water evaporating from a wet surface. Its ratio to the momentum diffusivity is the Schmidt number $Sc = \nu/D$, it plays exactly the part $Pr$ plays above, and every result in this essay transfers to it by substitution — including the Chilton–Colburn correction, which carries $Sc^{2/3}$ where it carried $Pr^{2/3}$.

What changes is the size of the number. Molecular diffusion of a solute in a liquid is enormously slower than either momentum or heat: a small ion in water has $D \approx 10^{-9}\ \mathrm{m^2/s}$ against a kinematic viscosity of $10^{-6}$, so $Sc \approx 1000$, and a protein is at $10^5$ or above. The concentration layer’s thickness is $Sc^{-1/3}$ of the velocity layer — a tenth for a small ion and a hundredth for a protein — which puts the whole of the transport into a film a few microns thick against a boundary layer of millimetres.

That single ratio is why so much of electrochemistry, corrosion and dissolution is a stirring problem rather than a chemistry problem. The reaction at the surface can be as fast as anybody likes and the rate is still set by how quickly the reactant crosses a few microns of nearly stationary liquid; a rotating-disc electrode exists to make that thickness calculable, and the classic Levich equation for its current is this essay’s $Nu \propto Re^{1/2}Sc^{1/3}$ with the constants worked out for a disc.

In a gas the three diffusivities converge again, and for the same kinetic-theory reason: momentum, energy and the molecules themselves all travel the same mean free path between the same collisions. Water vapour in air has $Sc \approx 0.6$ against $Pr = 0.71$, so their ratio — the Lewis number $Le = \alpha/D$ — is close to one, and the thermal and concentration layers on a wet surface in air are very nearly the same thickness.

That near-coincidence is doing more work in ordinary life than any other number in this essay. It is why a wet-bulb thermometer reads what it does: the surface cools until the heat arriving by conduction balances the heat leaving as latent heat of evaporation, and the balance depends on the ratio of the two transfer coefficients — which, because $Le \approx 1$, is one. The wet-bulb temperature therefore coincides with the adiabatic saturation temperature, psychrometric charts can be drawn without a separate mass-transfer coefficient, and evaporative cooling can be designed from a temperature measurement alone.

None of that would be true in a liquid, where $Le$ is in the hundreds, and none of it is true for a vapour much heavier than water. It is a fact about air and water vapour that has been absorbed into the instruments so thoroughly that it is rarely stated as a fact at all — which is the same thing this essay says about the Reynolds analogy holding for gases and failing for mercury.

Where the number comes from, physically #

The Prandtl number is a ratio of two diffusivities and both of them are the same kind of quantity — a length multiplied by a speed, divided by two or three depending on the derivation — so the fact that it is not always one takes explaining.

In a gas it is nearly one for the reason given above: momentum and energy ride on the same molecules. The small departure from unity is a real effect of how energy is stored, since a molecule carries rotational and vibrational energy that does not contribute to its momentum transfer, and the Eucken correction relates the two through the specific heat ratio. Every gas lands between 0.65 and 0.75 and the correction explains why.

In a liquid the mechanisms part company completely. Momentum is transferred by molecules jostling their neighbours in a cage, which is slow and strongly temperature-dependent; heat is carried by the same jostling and by vibrational modes propagating through the cage, which is much faster and barely temperature-dependent. So a liquid’s Prandtl number is large and falls steeply as it warms: water is 13 near freezing, 7 at room temperature and 1.8 near boiling, almost entirely because its viscosity falls by a factor of seven while its conductivity rises by a fifth.

In a liquid metal the electrons carry the heat, at a conductivity two orders of magnitude above anything the molecular motion could produce, while the viscosity remains that of an ordinary liquid. That is the whole of why mercury sits at 0.025, and it is why the thermal layer in a reactor coolant is thicker than the pipe’s boundary layer rather than a sliver inside it.

Who worked it out, and when #

Pohlhausen solved the temperature equation on Blasius’ profile in 1921 and produced the $Pr^{1/3}$ result; Reynolds had proposed the analogy in 1874, on physical grounds and without either equation, by arguing that the same eddies must carry both momentum and heat. Colburn’s correction is from 1933 and Chilton and Colburn’s version from 1934 — a fit, made by people who had a factory to run and needed something that worked between water and oil.

The surprising connection is that Reynolds’ argument was about turbulent flow, where the eddies carry both quantities in the same parcels, and the exact identity above is a laminar result, where molecular diffusion carries them and the two coefficients are equal only if $\nu = \alpha$. The analogy holds in both regimes for different reasons, which is why it survived so long before anybody was in a position to say what it was.

Where the ladder goes next #

The rung above is the same equation with dissipation restored, which is the high-speed case: a wall that heats itself, a recovery factor that is very nearly $\sqrt{Pr}$, and a heat flux driven from a temperature the flow invents rather than from the free stream’s.

The one beside it is 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 — which is why turbulent Prandtl numbers are near 0.9 for everything, and why liquid-metal heat exchangers behave so differently from everything else in the same pipe.