Mach number

The Mach number (M or Ma), often only Mach, (/mɑːx/; German: [max]) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after Austrian physicist and philosopher Ernst Mach.

$${\displaystyle \mathrm {M} ={\frac {u}{c}},}$$

where:

By definition, at Mach 1, the local flow velocity u is equal to the speed of sound. At Mach 0.65, u is 65% of the speed of sound (subsonic), and, at Mach 1.35, u is 35% faster than the speed of sound (supersonic).

The local speed of sound, and hence the Mach number, depends on the temperature of the surrounding gas. The Mach number is primarily used to determine the approximation with which a flow can be treated as an incompressible flow. The medium can be a gas or a liquid. The boundary can be travelling in the medium, or it can be stationary while the medium flows along it, or they can both be moving, with different velocities: what matters is their relative velocity with respect to each other. The boundary can be the boundary of an object immersed in the medium, or of a channel such as a nozzle, diffuser or wind tunnel channelling the medium. As the Mach number is defined as the ratio of two speeds, it is a dimensionless quantity. If M < 0.2–0.3 and the flow is quasi-steady and isothermal, compressibility effects will be small and simplified incompressible flow equations can be used.

The Mach number is named after the physicist and philosopher Ernst Mach, in honour of his achievements, according to a proposal by the aeronautical engineer Jakob Ackeret in 1929. The word Mach is always capitalized since it derives from a proper name and since the Mach number is a dimensionless quantity rather than a unit of measure. This is also why the number comes after the word Mach. It was also known as Mach's number by Lockheed when reporting the effects of compressibility on the P-38 aircraft in 1942.

Mach number is a measure of the compressibility characteristics of fluid flow: the fluid (air) behaves under the influence of compressibility in a similar manner at a given Mach number, regardless of other variables. As modeled in the International Standard Atmosphere, dry air at mean sea level, standard temperature of 15 °C (59 °F), the speed of sound is 340.3 meters per second (1,116.5 ft/s; 761.23 mph; 1,225.1 km/h; 661.49 kn). The speed of sound is not a constant; in a gas, it increases proportionally to the square root of the absolute temperature, and since atmospheric temperature generally decreases with increasing altitude between sea level and 11,000 meters (36,089 ft), the speed of sound also decreases. For example, the standard atmosphere model lapses temperature to −56.5 °C (−69.7 °F) at 11,000 meters (36,089 ft) altitude, with a corresponding speed of sound (Mach 1) of 295.0 meters per second (967.8 ft/s; 659.9 mph; 1,062 km/h; 573.4 kn), 86.7% of the sea level value.

The Mach number arises naturally when the continuity equation is nondimensionalized for compressible flows. If density variations are related to pressure through the isentropic relation ${\displaystyle dp=c^{2}\,d\rho }$, the nondimensionalized continuity equation contains a prefactor ${\displaystyle (u/c)^{2}=M^{2}}$. This shows that the Mach number directly measures the importance of compressibility effects in a flow. In the limit ${\displaystyle M\to 0}$, the equation reduces to the incompressibility condition ${\displaystyle \nabla \cdot \mathbf {u} =0}$.