Concept

Vibrational excitation — where it appears

The filling of a molecule's vibrational mode as a gas is heated, which raises its heat capacity and lowers the ratio of specific heats. It enters only the energy equation, so behind a shock it moves the temperature and the density a great deal and the pressure hardly at all.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The density ratio and the nose pressure coefficient, against Mach number. Two quantities across a normal shock, on a logarithmic Mach axis. Both approach limits that depend on γ and on nothing else: six for the density ratio and 1.8394 for the pressure coefficient at the stagnation point. By Mach five the second is within three per cent of its limit and by Mach twenty within a tenth of a per cent. Above that the flow round a blunt body has stopped depending on how fast it is going and started depending on what the gas is.

A shock that lies on the body

Above about Mach eight the flow round a blunt body stops depending on how fast it is going. The density ratio, the nose pressure coefficient and the shock standoff all reach limits set by γ alone — and going from a perfect gas to a dissociating one halves the standoff.

compressible · Shock layer

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

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.

Model limit Shock structure Density Mach independence Measurement Normal shock Relaxation time Shock standoff Shock wave Acoustics Bulk viscosity Crocco theorem

All concepts