Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Peclet — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. One layer, and the four others inside and outside it. The velocity profile over a flat plate, and the temperature profile in the same layer at four Prandtl numbers: a liquid metal at 0.01, air at 0.71, water at 7 and a heavy oil at 100. The equations differ by one number and the profiles differ by a factor of twenty in thickness. At Pr = 1 the two are the same function — not similar, identical, to eight decimal places — because the equations and the conditions are then the same, which is what every statement called a Reynolds analogy rests on.

    The other layer, and the one number that separates them

    A wall in a stream carries two boundary conditions and grows two layers. Their thicknesses differ by a factor of twenty across ordinary fluids, and at exactly one Prandtl number the two profiles are not similar but identical.

    part 1 · regimes
  2. The number a duct settles at is an eigenvalue. The local Nusselt number against x⁺ = x/(D·Re·Pr), from a Crank–Nicolson march that contains no eigenvalue anywhere. It settles at 3.6568, which is λ₀²/2 for the Graetz eigenvalue problem — a completely separate calculation. The mark at x⁺ = 0.05 is the entry length every textbook quotes: it delivers a Nusselt number 1.45 per cent above the developed value, which is a perfectly reasonable tolerance and is never the one stated.

    How far before the heat arrives

    A duct's thermal entry length is quoted everywhere as x/(D·Re·Pr) = 0.05, with no tolerance attached. Working out what it delivers gives a Nusselt number 1.45 per cent above the developed value — and the developed value itself is not a term ratio at all but an eigenvalue, 3.6568, which is also the rate at which the duct forgets its inlet.

    part 2 · regimes
  3. Where the eddies outconduct the molecules. The ratio of turbulent to molecular heat diffusivity across the pipe at Reτ = 2000, for five fluids, on logarithmic axes. Wherever it is above one the eddies carry more heat than conduction does. For air it passes one inside the buffer layer and reaches 135; for water, earlier and higher. For liquid sodium it never reaches one anywhere: at its peak, halfway to the axis, the eddies carry just over half what conduction carries, and the temperature profile is set by conduction across the whole pipe.

    The heat the eddies do not carry

    In a laminar layer the temperature and the velocity have different thicknesses in every fluid but one. In a turbulent pipe the eddies carry both, and the difference nearly vanishes — for air, water and oil alike. It does not vanish for a liquid metal, whose molecules conduct heat faster than the eddies can, and the boundary between the two behaviours is a Péclet number of about four hundred at every Reynolds number.

    part 3 · regimes
  4. The entrance grows with the Péclet number, peaks, and then shrinks. The distance from a wall-temperature step at which the local Nusselt number has come within 5 per cent of its developed value, in diameters, against the Péclet number, for turbulent pipes at Reτ = 500, 1000, 2000 and 5000, with the slug's laminar line L/D = 0.029·Pe. The liquid metals sit on the line; the entrance peaks at 11.6 diameters at Pe ≈ 4700 for Reτ = 1000 and falls to about three and a half diameters for water.

    The longest thermal entrance belongs to neither metal nor gas

    A liquid metal carries its heat across a turbulent pipe by conduction, as a laminar flow would, and so its thermal entrance should be long. It is long only when its Péclet number is large. Below a few hundred the entrance grows in proportion to the Péclet number with the constant of a fluid moving as a solid block, and it leaves that scaling close to the threshold at which the eddies first match conduction. The longest entrance in a turbulent pipe belongs to the fluids between the metals and the gases, and how long it is depends on a number the measurements have never pinned down.

    part 4 · regimes
  5. A resistive wall lengthens a slug's thermal entrance by half, and no more. The 5 per cent thermal entrance length of a slug flow, as a multiple of its length with the wall held at a fixed temperature, against the wall Biot number hw·D/kf. A wall that conducts well changes nothing; as the wall's resistance grows the entrance lengthens to a peak of 1.501 times at a Biot number of 2.29, and then shortens slightly to 1.457 times, the fixed-flux value, for a wall that conducts hardly at all.

    A steel wall lengthens a sodium entrance by half

    A liquid metal's thermal entrance is long because its heat crosses the pipe by conduction, and a steel tube conducts a tenth as well as sodium, so the wall is part of the path. Put it in series with the fluid and the entrance does grow — but by half at the very most, at a wall Biot number near two, which is where sodium in a steel tube happens to sit. A wall that resists more changes the boundary condition from a fixed temperature to a fixed flux, not the length.

    part 5 · regimes

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