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

Shock structure — where it appears

The internal profile of a shock, about which the jump conditions say nothing. It is set entirely by viscosity and conduction, and above about Mach three it is thinner than a mean free path — a structure the continuum equations that drew it cannot support.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

γ for air, against temperature. The ratio of specific heats for air as a mixture of nitrogen and oxygen, with the vibrational mode filling according to the Einstein function. It is 1.400 at room temperature, where only translation and rotation are available; 1.337 at a thousand kelvin; and 1.288 at six thousand. The usual compressible-flow tables use 1.4, and that is the value for a gas that is not hot — which, behind any shock worth drawing, it is not.

When gamma stops being a number

Every perfect-gas result uses γ = 1.4, which counts the ways a nitrogen molecule can hold energy at room temperature. Behind a Mach 10 shock the gas is at 3,800 kelvin and the count is different — and the pressure barely moves while the temperature falls by fifteen per cent.

compressible · Real gas The velocity through a shock at Mach two, as a function of position. The Becker profile, integrated outwards from its own inflection point. It runs from the upstream velocity to the downstream one — the two states the jump conditions give, which appear here as the equilibria of a first-order differential equation — and its steepest gradient matches the closed form to one part in ten thousand. The horizontal axis is in units of the thickness, which for this shock is 139 nanometres.

The discontinuity that has a thickness

The jump conditions do not contain the viscosity, which is why they are exact. The thickness is entirely viscosity — 289 nanometres at Mach 1.5, 35 at Mach 5, against a mean free path of 64. At Mach five the continuum equations have produced a structure thinner than the distance between collisions.

compressible · Shock The coefficient Stokes threw away is not a correction. What absorbs sound in dry air, split into the three transport effects that add to make the diffusivity of sound. Shear viscosity is the largest single contribution and it is not most of it; the bulk viscosity — which Stokes set to zero in 1845, saying in the same paper that he had no argument for doing so — is a quarter, and thermal conduction is a fifth. Setting ζ to zero does not make a small error in the absorption; it makes a 24 per cent one.

The viscosity nobody uses

A fluid has two viscosities. One resists a change of shape and appears in every viscous calculation; the other resists a change of volume, was set to zero by Stokes in 1845 in a paper that says he had no argument for doing so, and for carbon dioxide is fifteen hundred times larger than the one everybody quotes.

viscous · Bulk viscosity Seven shock structures, and the two states they all connect. The velocity through the shock for seven dissipation models — Prandtl numbers from a quarter to two, viscosities from constant to linear in temperature. Each curve is shifted so its midpoint sits at the origin. They start at the same speed, end at the same speed, and are nothing alike in between.

The jump does not ask what made it

Seven different dissipation mechanisms are made to smear the same shock. Their interiors are a factor of two and a third apart in thickness, their entropies overshoot the final value by between a quarter and a doubling, and the state they all arrive at agrees to seven parts in ten billion — because the end states are conservation and the interior is transport.

compressible · Shock Dry air makes a slow bang. The 10-to-90 per cent rise time of a steady shock against relative humidity, for jumps of 25, 50 and 90 pascals, all below the strength at which a discontinuity returns. Each falls roughly as the nitrogen relaxation time does, and doubles when the jump is halved. The rule marks the thermoviscous rise time of the 50-pascal shock, a few microseconds — two to three orders of magnitude below any of the curves.

Oxygen makes a boom's crack, nitrogen its rise time

The shock at the front of a sonic boom is not the viscous shock of a textbook, a few microseconds thick. It is spread over a large fraction of a millisecond by the vibrational relaxation of the air's molecules, and the two gases do different jobs. Oxygen, fast, removes the discontinuity for any boom weaker than about ninety pascals and decides how much of the front is left at the frequencies the ear weighs most; nitrogen, slow, sets the rise time that is measured. Water vapour speeds both, so a dry day makes a softer bang.

compressible · Sonic boom Smooth below one Mach number, a jump and a tail above it. The velocity change through steady shocks in carbon dioxide, as a fraction of the whole change, against distance in relaxation lengths — the equilibrium sound speed times the relaxation time, 0.737 mm at one atmosphere. Shocks slower than the frozen sound speed, M below 1.041 referred to the equilibrium speed, are smooth throughout: the whole rise is the vibration catching up. Faster ones jump first, at the frozen speed's own shock, and relax afterwards: at M = 1.3 the jump takes 83.9 per cent of the change at once.

A bulk viscosity holds a shock together until it splits

Carbon dioxide's bulk viscosity, fifteen hundred times its shear viscosity, is a vibrational relaxation seen from below its frequency, and a shock is where that description is tested hardest. Carried through a steady shock, the relaxation reproduces the coefficient exactly for the weakest shocks. At a pressure rise of nine and a half per cent the shock outruns the frozen sound speed and splits into a jump and a tail, and the coefficient then draws a shock that does not exist.

viscous · Bulk viscosity

Named alongside it

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

DissipationEntropyModel limitDiscontinuityNormal shockVibrational excitationBulk viscosityIrreversibilityPrandtl numberRankine–Hugoniot conditionsRelaxation timeSound absorption

All concepts