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

Total enthalpy — where it appears

The enthalpy of a moving gas plus its kinetic energy per unit mass, which is conserved along a streamline in steady adiabatic flow with no work done. In a boundary layer it is what the energy equation carries, and at unit Prandtl number on a flat plate it is linear in the velocity.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

What the skin settles at, before anything is done to it. The adiabatic wall temperature against Mach number, in air at 216.7 K, with the stagnation temperature above it. The gap between the two is the recovery factor, which is 0.8417 here and stays there at every Mach number — it is a property of the Prandtl number and not of the speed. At Mach 2 the skin sits at 363 K, at Mach 3 at 545 K, and at Mach 5 at 1128 K, which is past what aluminium will do. Nothing has been burnt and nothing has been rubbed: the air was brought to rest, and this is where its kinetic energy went.

The wall that heats itself

A surface told nothing about its temperature does not settle at the air's. It settles most of the way to the stagnation temperature, and the heat flux is driven from that invented temperature rather than from the free stream's — so a wall hotter than the air can be being heated by it.

compressible · Recovery A cooled wall at β = 1: the linear relation is 158 K out inside the layer. The static temperature across a laminar layer at Mach 5 with an edge temperature of 220 K, over a wall whose total enthalpy is 0.5 of the edge's, at a pressure gradient β = 1: exact (thick) and from the Crocco–Busemann linear relation (thin), at a Prandtl number of one with constant properties. The wall is at 660 K in both. The exact profile peaks at 684 K and the linear one at 759 K; the largest difference, −158.3 K, is at η = 0.81.

The gradient the heat never hears

On a flat plate at a Prandtl number of one, a boundary layer's total enthalpy is a straight-line function of its velocity, whatever the wall's temperature. Put the same layer in a pressure gradient and the straight line fails everywhere except on an insulated wall, because the gradient enters the velocity's equation and not the enthalpy's — and a favourable gradient can leave a band of gas colder than the free stream above a wall three times hotter than it.

compressible · Recovery A cold wall holds its layer on longer; a hot one lets it go sooner. The pressure-gradient parameter β at which the wall shear vanishes, against the wall's total enthalpy over the edge's. With the velocity deaf to the temperature it is −0.1988 for every wall. Coupled, it is -0.3264 for a wall at absolute zero, -0.2623 for a wall at half the edge's total enthalpy, -0.1988 at one, -0.1573 at one and a half and -0.1295 at two. Cooling delays separation by more than the same heating hastens it.

A cold wall holds its layer on longer

A compressible boundary layer's pressure gradient pushes on its density, and its density is its temperature. Solve the velocity and the enthalpy together and the wall's temperature reaches the velocity: a cold wall holds its layer on to an adverse gradient two-thirds steeper, a hot one lets it go a third sooner, the stagnation point's heat-to-friction ratio more than doubles between them, and the cold band an uncoupled layer put above a cooled nose disappears.

compressible · Recovery

Named alongside it

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

Aerodynamic heatingBoundary layerHeat transferAdverse pressure gradientAnalogyEnergy equationFalkner–SkanMach numberPrandtl numberSimilarity solutionSkin frictionAdiabatic wall

All concepts