Compressible flow

A compressible flow is a situation in which the compressibility of the fluid
must be taken into account. In general, this is the case where the
Mach number in part or all of the flow approaches or exceeds 1. Under these circumstances, it is usual to neglect viscosity and the Euler equations are used (rather than the Navier-Stokes equations).

For subsonic compressible flows, it is sometimes possible to model
the flow by applying a correction factor to the answers derived from
incompressible calculations or modelling - for example, the Glauert-Prandtl
rule

(a_c is compressible lift curve slope, a_i is the
incompressible lift curve slope, and M is the Mach number).

For many other flows, their nature is qualitatively different to subsonic
flows. A flow where the local Mach number reaches or exceeds 1
will usually contain shock waves. A shock is an abrupt change
in the velocity, pressure and temperature in a flow; the thickness of a
shock scales with the molecular mean free path in the fluid (typically
a few micrometers).

Shocks form because
information about conditions downstream of a point of sonic or supersonic
flow can not propgate back upstream past the sonic point.

The behaviour of a fluid changes radically as it starts to move above
the speed of sound (in that fluid). For example, in subsonic flow, a
stream tube in an accelerating flow contracts. But in a supersonic
flow, a stream tube in an accelerating flow expands. To interpret this
in another way, consider steady flow in a tube that has a sudden expansion:
the tube's cross section suddenly widens, so the cross-sectional area
increases.

In subsonic flow, the fluid speed drops after the expansion (as expected).
In supersonic flow, the fluid speed increases. This sounds like a
contradiction, but it isn't: the mass flux is conserved but because
supersonic flow allows the density to change, the volume flux is not
constant.