Mach number

Mach number

Discussion

Encyclopedia
Mach number ( or ) (generally icon, sometimes ˈ or ˈ) is the speed of an object moving through air, or any other fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

substance, divided by the speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

as it is in that substance for its particular physical conditions, including those of temperature and pressure. It is commonly used to represent the speed of an object when it is traveling close to or above the speed of sound.

where is the Mach number is the relative velocity of the source to the medium and is the speed of sound in the medium

The Mach number is named after Austria
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...

n physicist and philosopher Ernst Mach
Ernst Mach
Ernst Mach was an Austrian physicist and philosopher, noted for his contributions to physics such as the Mach number and the study of shock waves...

, a designation proposed by aeronautical engineer Jakob Ackeret
Jakob Ackeret
Jakob Ackeret was a Swiss aeronautical engineer. He is widely viewed as one of the foremost aeronautics experts of 20th century.- Birth and education :...

. Because the Mach number is often viewed as a dimensionless quantity
Dimensionless quantity
In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

rather than a unit 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
Fathom
A fathom is a unit of length in the imperial and the U.S. customary systems, used especially for measuring the depth of water.There are 2 yards in an imperial or U.S. fathom...

), which was also unit-first, and may have influenced the use of the term Mach. In the decade preceding faster-than-sound human flight, aeronautical engineers referred to the speed of sound as Mach's number, never "Mach 1."

In French, the Mach number is sometimes called the "nombre de Sarrau" ("Sarrau number") after Émile Sarrau
Émile Sarrau
Jacques Rose Ferdinand Émile Sarrau was a French chemist. He worked most of his career at the laboratory in the Dépôt Central des Poudres et Salpêtres . He did research on explosive shock waves, the effects of explosives and he developed new explosives...

, researching on explosions in the 1870s and 1880s.

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 nozzle
Nozzle
A nozzle is a device designed to control the direction or characteristics of a fluid flow as it exits an enclosed chamber or pipe via an orifice....

s, diffuser
Diffuser
Diffuser can refer to any device that diffuses in some manner such as:* In aerodynamics** Diffuser , a shaped section of a car's underbody which improves the car's aerodynamic properties** divergent part of the flute...

s or wind tunnel
Wind tunnel
A wind tunnel is a research tool used in aerodynamic research to study the effects of air moving past solid objects.-Theory of operation:Wind tunnels were first proposed as a means of studying vehicles in free flight...

s. As it is defined as a ratio of two speeds, it is a dimensionless number. At Standard Sea Level
International Standard Atmosphere
The International Standard Atmosphere is an atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes. It has been established to provide a common reference for temperature and pressure and consists of tables of...

conditions (corresponding to a temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

of 15 degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

), the speed of sound is 340.3 m/s (1225 km/h, or 761.2 mph, or 661.5 knots, or 1116 ft/s
Feet per second
The foot per second is a unit of both speed and velocity . It expresses the distance in feet traveled or displaced, divided by the time in seconds...

) in the Earth's atmosphere
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

. The speed represented by Mach 1 is not a constant; for example, it is mostly dependent on temperature and atmospheric composition and largely independent of pressure.
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 20°C or 68°F will experience shock waves in much the same manner as when it is traveling at Mach 1 at 11,000 m (36,000 ft) at -50°C or -58F, even though it is traveling at only 86% of its speed at higher temperature like 20°C or 68°F.

High-speed flow around objects

Flight can be roughly classified in six categories:
Regime Subsonic
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

Transonic
Transonic
Transonic speed is an aeronautics term referring to the condition of flight in which a range of velocities of airflow exist surrounding and flowing past an air vehicle or an airfoil that are concurrently below, at, and above the speed of sound in the range of Mach 0.8 to 1.2, i.e. 600–900 mph...

Sonic
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

Supersonic
Supersonic
Supersonic speed is a rate of travel of an object that exceeds the speed of sound . For objects traveling in dry air of a temperature of 20 °C this speed is approximately 343 m/s, 1,125 ft/s, 768 mph or 1,235 km/h. Speeds greater than five times the speed of sound are often...

Hypersonic
Hypersonic
In aerodynamics, a hypersonic speed is one that is highly supersonic. Since the 1970s, the term has generally been assumed to refer to speeds of Mach 5 and above...

High-hypersonic
Mach <1.0 0.8–1.2 1.0 1.2–5.0 5.0–10.0 >10.0

For comparison: the required speed for low Earth orbit
Low Earth orbit
A low Earth orbit is generally defined as an orbit within the locus extending from the Earth’s surface up to an altitude of 2,000 km...

is approximately 7.5 km/s = Mach 25.4 in air at high altitudes. The speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

in a vacuum corresponds to a Mach number of approximately 881,000 (relative to air at sea level).

At transonic speeds, the flow field around the object includes both sub- and supersonic parts. The transonic period begins when first zones of M>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 speed increases, the zone of M>1 flow increases towards both leading and trailing edges. As M=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; M<1 (a) and M>1 (b).

When an aircraft exceeds Mach 1 (i.e. the sound barrier
Sound barrier
The sound barrier, in aerodynamics, is the point at which an aircraft moves from transonic to supersonic speed. The term, which occasionally has other meanings, came into use during World War II, when a number of aircraft started to encounter the effects of compressibility, a collection of several...

) a large pressure difference is created just in front of the aircraft
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...

. This abrupt pressure difference, called a shock wave
Shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...

, 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
Sonic boom
A sonic boom is the sound associated with the shock waves created by an object traveling through the air faster than the speed of sound. Sonic booms generate enormous amounts of sound energy, sounding much like an explosion...

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 M=1 it is hardly a cone at all, but closer to a slightly concave plane.

At fully supersonic speed, 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
Shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...

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 speeds 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 becomes supersonic, one significant change takes place. The conservation of mass flow rate
Mass flow rate
Mass flow rate is the mass of substance which passes through a given surface per unit time. Its unit is mass divided by time, so kilogram per second in SI units, and slug per second or pound per second in US customary units...

leads 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 sonic speeds, and the diverging section continues the acceleration. Such nozzles are called de Laval nozzle
De Laval nozzle
A de Laval nozzle is a tube that is pinched in the middle, making a carefully balanced, asymmetric hourglass-shape...

s and in extreme cases they are able to reach incredible, hypersonic
Hypersonic
In aerodynamics, a hypersonic speed is one that is highly supersonic. Since the 1970s, the term has generally been assumed to refer to speeds of Mach 5 and above...

speeds (Mach 13 at 20°C).

An aircraft Machmeter
Machmeter
A Machmeter is an aircraft pitot-static system flight instrument thatshows the ratio of the true airspeed to the speed of sound,a dimensionless quantity called Mach number...

or electronic flight information system (EFIS) can display Mach number derived from stagnation pressure (pitot tube
Pitot tube
A pitot tube is a pressure measurement instrument used to measure fluid flow velocity. The pitot tube was invented by the French engineer Henri Pitot Ulo in the early 18th century and was modified to its modern form in the mid-19th century by French scientist Henry Darcy...

) and static pressure.

Calculation

Assuming air to be an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

, the formula to compute Mach number in a subsonic compressible flow is derived from Bernoulli's equation
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

for M<1:

where: is Mach number is impact pressure
Impact pressure
In compressible fluid dynamics, impact pressure is the difference between total pressure and static pressure. In aerodynamics notation, this quantity is denoted as q_c or Q_c....

and is static pressure
Static pressure
In fluid mechanics the term static pressure has several uses:* In the design and operation of aircraft, static pressure is the air pressure in the aircraft’s static pressure system....

is the ratio of specific heat
Heat capacity ratio
The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume . It is sometimes also known as the isentropic expansion factor and is denoted by \gamma or \kappa . The latter symbol kappa is...

of a gas at a constant pressure to heat at a constant volume (1.4 for air).

The formula to compute Mach number in a supersonic compressible flow is derived from the Rayleigh
Rayleigh number
In fluid mechanics, the Rayleigh number for a fluid is a dimensionless number associated with buoyancy driven flow...

Supersonic Pitot equation:

or for air, a simplfied formula:

where: is now impact pressure
Impact pressure
In compressible fluid dynamics, impact pressure is the difference between total pressure and static pressure. In aerodynamics notation, this quantity is denoted as q_c or Q_c....

measured behind a normal shock.

The Mach number at which an aircraft is flying at can be calculated by

where: is Mach number is velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

of the moving aircraft and is the speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

at the given altitude

Note that the dynamic pressure can be found as:

• Critical Mach number
Critical Mach number
In aerodynamics, the critical Mach number of an aircraft is the lowest Mach number at which the airflow over any part of the aircraft reaches the speed of sound....

• Machmeter
Machmeter
A Machmeter is an aircraft pitot-static system flight instrument thatshows the ratio of the true airspeed to the speed of sound,a dimensionless quantity called Mach number...

• Ramjet
Ramjet
A ramjet, sometimes referred to as a stovepipe jet, or an athodyd, is a form of airbreathing jet engine using the engine's forward motion to compress incoming air, without a rotary compressor. Ramjets cannot produce thrust at zero airspeed and thus cannot move an aircraft from a standstill...

• Scramjet
Scramjet
A scramjet is a variant of a ramjet airbreathing jet engine in which combustion takes place in supersonic airflow...

• Speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

• True airspeed
True airspeed
True airspeed of an aircraft is the speed of the aircraft relative to the airmass in which it is flying. True airspeed is important information for accurate navigation of an aircraft.-Performance:...