Mach number

Mach number

Mach number [for E. Mach], ratio between the speed of an object and the speed of sound in the medium in which the object is traveling. An airplane that has the velocity of Mach 3.0 is traveling at three times the speed of sound as measured in the prevailing atmospheric conditions.

Mach number (mathrm{Ma} or M) (generally , sometimes /ˈmɑːx/ or /ˈmæk/) is the speed of an object moving through air, or any fluid substance, divided by the speed of sound as it is in that substance. It is commonly used to represent an object's (such as an aircraft or missile) speed, when it is travelling at (or at multiples of) the speed of sound.

M = frac
where
M is the Mach number
v_o is the velocity of the object relative to the medium and
v_s is the velocity of sound in the medium

The Mach number is named after Austrian physicist and philosopher Ernst Mach. Unlike most units of measure, with Mach, the number comes after the unit; the second Mach number is "Mach 2" instead of "2 Mach" (or Machs). This is somewhat reminiscent of the early modern ocean sounding unit "mark" (a synonym for fathom), which was also unit-first, and may have influenced the use of the term Mach. In the decade preceding man's flying faster than sound, aeronautical engineers referred to the speed of sound as Mach's number, never "Mach 1".

Overview

The Mach number is commonly used both with objects traveling at high speed in a fluid, and with high-speed fluid flows inside channels such as nozzles, diffusers or wind tunnels. As it is defined as a ratio of two speeds, it is a dimensionless number. At a temperature of 15 degrees Celsius and at sea level, the speed of sound is 340.3 m/s (1225 km/h, or 761.2 mph, or 1116 ft/s) in the Earth's atmosphere. The speed represented by Mach 1 is not a constant; for example, it is dependent on temperature and atmospheric composition. In the stratosphere it remains constant irrespective of altitude even though the air pressure varies with altitude.

Since the speed of sound increases as the temperature increases, the actual speed of an object traveling at Mach 1 will depend on the fluid temperature around it. Mach number is useful because the fluid behaves in a similar way at the same Mach number. So, an aircraft traveling at Mach 1 at sea level (340.3 m/s, 761.2 mph, 1,225 km/h) will experience shock waves in much the same manner as when it is traveling at Mach 1 at 11,000 m (36,000 ft), even though it is traveling at 295 m/s (654.6 mph, 1,062 km/h, 86% of its speed at sea level).

High-speed flow around objects

Flight can be roughly classified in five categories:

(For comparison: the required speed for low Earth orbit is ca. 7.5 km·s-1 = Ma 25.4 in air at high altitudes)

At transonic speeds, the flow field around the object includes both sub- and supersonic parts. The transonic period begins when first zones of Ma>1 flow appear around the object. In case of an airfoil (such as an aircraft's wing), this typically happens above the wing. Supersonic flow can decelerate back to subsonic only in a normal shock; this typically happens before the trailing edge. (Fig.1a)

As the velocity increases, the zone of Ma>1 flow increases towards both leading and trailing edges. As Ma=1 is reached and passed, the normal shock reaches the trailing edge and becomes a weak oblique shock: the flow decelerates over the shock, but remains supersonic. A normal shock is created ahead of the object, and the only subsonic zone in the flow field is a small area around the object's leading edge. (Fig.1b)

|- (a) (b)

Fig. 1. Mach number in transonic airflow around an airfoil; Ma<1 (a) and Ma>1 (b).

When an aircraft exceeds Mach 1 (i.e. the sound barrier) a large pressure difference is created just in front of the aircraft. This abrupt pressure difference, called a shock wave, spreads backward and outward from the aircraft in a cone shape (a so-called Mach cone). It is this shock wave that causes the sonic boom heard as a fast moving aircraft travels overhead. A person inside the aircraft will not hear this. The higher the speed, the more narrow the cone; at just over Ma=1 it is hardly a cone at all, but closer to a slightly concave plane.

At fully supersonic velocity the shock wave starts to take its cone shape, and flow is either completely supersonic, or (in case of a blunt object), only a very small subsonic flow area remains between the object's nose and the shock wave it creates ahead of itself. (In the case of a sharp object, there is no air between the nose and the shock wave: the shock wave starts from the nose.)

As the Mach number increases, so does the strength of the shock wave and the Mach cone becomes increasingly narrow. As the fluid flow crosses the shock wave, its speed is reduced and temperature, pressure, and density increase. The stronger the shock, the greater the changes. At high enough Mach numbers the temperature increases so much over the shock that ionization and dissociation of gas molecules behind the shock wave begin. Such flows are called hypersonic.

It is clear that any object traveling at hypersonic velocities will likewise be exposed to the same extreme temperatures as the gas behind the nose shock wave, and hence choice of heat-resistant materials becomes important.

High-speed flow in a channel

As a flow in a channel crosses M=1 becomes supersonic, one significant change takes place. Common sense would lead one to expect that contracting the flow channel would increase the flow speed (i.e. making the channel narrower results in faster air flow) and at subsonic speeds this holds true. However, once the flow becomes supersonic, the relationship of flow area and speed is reversed: expanding the channel actually increases the speed.

The obvious result is that in order to accelerate a flow to supersonic, one needs a convergent-divergent nozzle, where the converging section accelerates the flow to M=1, sonic speeds, and the diverging section continues the acceleration. Such nozzles are called de Laval nozzles and in extreme cases they are able to reach incredible, hypersonic velocities (Mach 13 at sea level).

An aircraft Machmeter or electronic flight information system (EFIS) can display Mach number derived from stagnation pressure (pitot tube) and static pressure.

Calculating Mach Number

Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is derived from Bernoulli's equation for M<1:

{M}=sqrt{frac{2}{gamma-1}left[left(frac{q_c}{P}+1right)^frac{gamma-1}{gamma}-1right]}

where:

M is Mach number
q_c is impact pressure and
P is static pressure
gamma is the ratio of specific heats

The formula to compute Mach number in a supersonic compressible flow is derived from the Rayleigh Supersonic Pitot equation:

{M}=0.88128485sqrt{left[left(frac{q_c}{P}+1right)left(1-frac{1}{[7M^2]}right)^{2.5}right]}
where:
q_c is now impact pressure measured behind a normal shock

See also

References

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