Compressible flow

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A flow is considered to be a compressible flow if the change in density of the flow with respect to pressure is non-zero along a streamline. In general, this is the case where the Mach number in part or all of the flow exceeds 0.3. The Mach .3 value is rather arbitrary, but it is used because gas flows with a Mach number below that value demonstrate changes in density with respect to the change in pressure of less than 5%. Furthermore, that maximum 5% density change occurs at the stagnation point of an object immersed in the gas flow and the density changes around the rest of the object will be significantly lower.

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:

\frac{a_c}{a_i} \sim \frac{1}{\sqrt{1-M^2}}

(ac is compressible lift curve slope, ai is the incompressible lift curve slope, and M is the Mach number). Note that this correction only yields acceptable results over a range of approximately 0.3<M<0.7.

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), ie. when the Mach number is greater than 1. 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. This effect is utilized in De Laval nozzles.

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