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

Reynolds analogy — where it appears

The idea that turbulent eddies carry heat and momentum across a flow in the same way, so a wall's heat transfer can be read from its friction. It holds for gases and ordinary liquids and fails for liquid metals, whose molecules conduct heat faster than eddies carry it.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

regimes · Peclet 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.

regimes · Peclet

Named alongside it

The objects these essays reach for when they reach for this one.

Heat transferPeclet numberPrandtl numberBlasiusBoundary conditionBoundary layerCorrelationDiffusionDimensionlessDimensionless numberEddy viscosityMixing length

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