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

Passive scalar — where it appears

A quantity carried by a flow without affecting it: a dye, a temperature difference too small to drive buoyancy, a trace gas. It obeys an advection-diffusion equation with the velocity given rather than solved for, which makes it the tractable half of a turbulent transport problem.

Named by 2 essays across 2 fields — 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 Four over the dimension, and three dimensions is where the three comes from. The constant in front of the exact law, against the number of dimensions the flow lives in. Both exact results — Yaglom's for a scalar and the velocity's law for the mixed third moment — have this same constant, because both come from the same statement: an isotropic radial flux in separation space whose divergence is a constant sink. Integrating that divergence gives 4Q r/d and nothing else. The four-thirds everybody quotes is four over three, and the three is the space rather than anything about turbulence. The dots are the quadrature, which agrees with the closed form to 2·10⁻⁹.

The fraction that is really four thirds

Turbulence has two exact results, not one. The second is about a scalar carried by the flow, its constant is four thirds rather than four fifths, and the difference between the two fractions has nothing in it about turbulence at all — it is the price of writing a three-component object in terms of one component.

turbulence · Structure function

Named alongside it

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

DimensionlessBatchelor scaleBlasiusBoundary conditionBoundary layerCascadeCorrelationDiffusionDissipationFour-fifths lawHeat transferInertial range

All concepts