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Mach number (M or Ma) (; ) is a dimensionless quantity in
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Moravian physicist and philosopher
Ernst Mach Ernst Waldfried Josef Wenzel Mach ( , ; 18 February 1838 – 19 February 1916) was a Moravian-born Austrian physicist and philosopher, who contributed to the physics of shock waves. The ratio of one's speed to that of sound is named the Mac ...
. : \mathrm = \frac, where: : is the local Mach number, : is the local flow velocity with respect to the boundaries (either internal, such as an object immersed in the flow, or external, like a channel), and : is the speed of sound in the medium, which in air varies with the square root of the thermodynamic temperature. By definition, at Mach1, the local flow velocity is equal to the speed of sound. At Mach0.65, is 65% of the speed of sound (subsonic), and, at Mach1.35, is 35% faster than the speed of sound (supersonic). Pilots of high-altitude
aerospace Aerospace is a term used to collectively refer to the atmosphere and outer space. Aerospace activity is very diverse, with a multitude of commercial, industrial and military applications. Aerospace engineering consists of aeronautics and astrona ...
vehicles use flight Mach number to express a vehicle's true
airspeed In aviation, airspeed is the speed of an aircraft relative to the air. Among the common conventions for qualifying airspeed are: * Indicated airspeed ("IAS"), what is read on an airspeed gauge connected to a Pitot-static system; * Calib ...
, but the flow field around a vehicle varies in three dimensions, with corresponding variations in local Mach number. 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 traveling 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 Wind tunnels are large tubes with air blowing through them which are used to replicate the interaction between air and an object flying through the air or moving along the ground. Researchers use wind tunnels to learn more about how an aircraft ...
channeling the medium. As the Mach number is defined as the ratio of two speeds, it is a dimensionless number. If  < 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.


Etymology

The Mach number is named after Moravian physicist and philosopher
Ernst Mach Ernst Waldfried Josef Wenzel Mach ( , ; 18 February 1838 – 19 February 1916) was a Moravian-born Austrian physicist and philosopher, who contributed to the physics of shock waves. The ratio of one's speed to that of sound is named the Mac ...
, and is a designation proposed by aeronautical engineer Jakob Ackeret in 1929. As the Mach number is a dimensionless quantity rather than a unit of measure, the number comes ''after'' the unit; the second Mach number is ''Mach2'' instead of ''2Mach'' (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 faster-than-sound human flight, aeronautical engineers referred to the speed of sound as ''Mach's number'', never ''Mach 1''.


Overview

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 There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ari ...
, standard temperature of , the speed of sound is . 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 , the speed of sound also decreases. For example, the standard atmosphere model lapses temperature to at altitude, with a corresponding speed of sound (Mach1) of , 86.7% of the sea level value.


Appearance in the continuity equation

As a measure of flow compressibility, the Mach number can be derived from an appropriate scaling of the continuity equation. The full continuity equation for a general fluid flow is: + \nabla\cdot(\rho ) = 0 \equiv - = \nabla \cdot where D/Dt is the material derivative, \rho is the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, and is the flow velocity. For isentropic pressure-induced density changes, dp = c^d\rho where c is the speed of sound. Then the continuity equation may be slightly modified to account for this relation:- = \nabla \cdot The next step is to nondimensionalize the variables as such:^ = /L, \quad t^ = Ut/L, \quad ^ = /U, \quad p^ = (p-p_)/\rho_U^, \quad \rho^ = \rho/\rho_where L is the characteristic length scale, U is the characteristic velocity scale, p_ is the reference pressure, and \rho_ is the reference density. Then the nondimensionalized form of the continuity equation may be written as:- = \nabla^ \cdot ^ \implies -\text^ = \nabla^ \cdot ^where the Mach number \text = U/c. In the limit that \text \rightarrow 0, the continuity equation reduces to \nabla \cdot = 0 - this is the standard requirement for incompressible flow.


Classification of Mach regimes

While the terms ''subsonic'' and ''supersonic'', in the purest sense, refer to speeds below and above the local speed of sound respectively, aerodynamicists often use the same terms to talk about particular ranges of Mach values. This occurs because of the presence of a ''transonic regime'' around flight (free stream) M = 1 where approximations of the Navier-Stokes equations used for subsonic design no longer apply; the simplest explanation is that the flow around an airframe locally begins to exceed M = 1 even though the free stream Mach number is below this value. Meanwhile, the ''supersonic regime'' is usually used to talk about the set of Mach numbers for which linearised theory may be used, where for example the ( air) flow is not chemically reacting, and where heat-transfer between air and vehicle may be reasonably neglected in calculations. In the following table, the ''regimes'' or ''ranges of Mach values'' are referred to, and not the ''pure'' meanings of the words ''subsonic'' and ''supersonic''. Generally,
NASA The National Aeronautics and Space Administration (NASA ) is an independent agency of the US federal government responsible for the civil space program, aeronautics research, and space research. NASA was established in 1958, succeedi ...
defines ''high'' hypersonic as any Mach number from 10 to 25, and re-entry speeds as anything greater than Mach 25. Aircraft operating in this regime include the
Space Shuttle The Space Shuttle is a retired, partially reusable low Earth orbital spacecraft system operated from 1981 to 2011 by the U.S. National Aeronautics and Space Administration (NASA) as part of the Space Shuttle program. Its official program na ...
and various space planes in development.


High-speed flow around objects

Flight can be roughly classified in six categories: For comparison: the required speed for
low Earth orbit A low Earth orbit (LEO) is an orbit around Earth with a period of 128 minutes or less (making at least 11.25 orbits per day) and an eccentricity less than 0.25. Most of the artificial objects in outer space are in LEO, with an altitude never m ...
is approximately 7.5 km/s = Mach 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 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) 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 The sound barrier or sonic barrier is the large increase in aerodynamic drag and other undesirable effects experienced by an aircraft or other object when it approaches the speed of sound. When aircraft first approached the speed of sound, ...
), a large pressure difference is created just in front of the
aircraft An aircraft is a vehicle that is able to fly by gaining support from the air. It 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. ...
. This abrupt pressure difference, called a
shock wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
, 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 A sonic boom is a sound associated with shock waves created when an object travels through the air faster than the speed of sound. Sonic booms generate enormous amounts of sound energy, sounding similar to an explosion or a thunderclap to t ...
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 In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
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 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 nozzles and in extreme cases they are able to reach hypersonic speeds ( at 20 °C). An aircraft Machmeter or electronic flight information system ( EFIS) can display Mach number derived from stagnation pressure ( pitot tube) and static pressure.


Calculation

When the speed of sound is known, the Mach number at which an aircraft is flying can be calculated by : \mathrm = \frac where: : M is the Mach number : ''u'' is
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of the moving aircraft and : ''c'' is the speed of sound at the given altitude (more properly temperature) and the speed of sound varies with the thermodynamic temperature as: :c = \sqrt, where: : \gamma\, is the ratio of specific heat of a gas at a constant pressure to heat at a constant volume (1.4 for air) : R_* is the specific gas constant for air. : T, is the static air temperature. If the speed of sound is not known, Mach number may be determined by measuring the various air pressures (static and dynamic) and using the following formula that is derived from Bernoulli's equation for Mach numbers less than 1.0. Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is:Olson, Wayne M. (2002). "AFFTC-TIH-99-02, ''Aircraft Performance Flight Testing''."
PDF
. Air Force Flight Test Center, Edwards AFB, CA, United States Air Force.
:\mathrm = \sqrt\, where: : ''qc'' is impact pressure (dynamic pressure) and : ''p'' is static pressure : \gamma\, is the ratio of specific heat of a gas at a constant pressure to heat at a constant volume (1.4 for air) : R_* is the specific gas constant for air. The formula to compute Mach number in a supersonic compressible flow is derived from the
Rayleigh Rayleigh may refer to: Science *Rayleigh scattering *Rayleigh–Jeans law *Rayleigh waves *Rayleigh (unit), a unit of photon flux named after the 4th Baron Rayleigh *Rayl, rayl or Rayleigh, two units of specific acoustic impedance and characte ...
supersonic pitot equation: : \frac = \left frac\mathrm^2\right\frac \cdot \left frac\right\frac


Calculating Mach number from pitot tube pressure

Mach number is a function of temperature and true airspeed. Aircraft flight instruments, however, operate using pressure differential to compute Mach number, not temperature. Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is found from Bernoulli's equation for (above): : \mathrm = \sqrt\, The formula to compute Mach number in a supersonic compressible flow can be found from the Rayleigh supersonic pitot equation (above) using parameters for air: : \mathrm \approx 0.88128485 \sqrt where: :''qc'' is the dynamic pressure measured behind a normal shock. As can be seen, M appears on both sides of the equation, and for practical purposes a root-finding algorithm must be used for a numerical solution (the equation's solution is a root of a 7th-order polynomial in M2 and, though some of these may be solved explicitly, the Abel–Ruffini theorem guarantees that there exists no general form for the roots of these polynomials). It is first determined whether M is indeed greater than 1.0 by calculating M from the subsonic equation. If M is greater than 1.0 at that point, then the value of M from the subsonic equation is used as the initial condition for fixed point iteration of the supersonic equation, which usually converges very rapidly. Alternatively, Newton's method can also be used.


See also

* * * * * * *


Notes


External links


Gas Dynamics Toolbox
Calculate Mach number and normal shock wave parameters for mixtures of perfect and imperfect gases.

Interactive calculator for Mach number.
NewByte standard atmosphere calculator and speed converter
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