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A compressible flow is a situation in which the density of the flow cannot be assumed to be constant. In general, this is the case where the Mach number in part or all of the flow approaches or exceeds 0.3 (This is rather arbitrary but is a rule of thumb generally used). It can be proven that for gas flows with Mach number less than 0.3 the change in density is less than 5%.

While the above definition for compressible flow holds for most of the cases, it is possible to have flows with varying density and the flow is still incompressible. For example, consider a flow of water in oceans or air flow in atmosphere. The density varies as we move to different heights, but the flow can still be treated as incompressible. The factor that distinguishes a flow from being compressible or incompressible is the fact that in compressible flow the changes in the velocity of the flow can lead to changes the temperature which are not negligible. On the other hand in case of incompressible flow, the changes in the internal energy (i.e. temperature) are negligible even if the entire kinetic energy of the flow is converted to internal energy (i.e. the flow is brought to rest).

These definitions, though they seem to be inconsistent, are all saying one and the same thing: the Mach number of the flow is high enough so that the effects of compressibility can no longer be neglected.

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 Prandtl-Glauert 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 cannot propagate 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.

• Gas dynamics