View source for Mach number

{{Short description|Ratio of speed of an object moving through fluid and local speed of sound}}
{{Use British English|date=January 2020}}

The '''Mach number''' ('''M''' or '''Ma'''), often only '''Mach''', ({{IPAc-en|m|ɑː|k}}; {{IPA|de|max|lang}}) is a [[dimensionless quantity]] in [[fluid dynamics]] representing the ratio of [[flow velocity]] past a [[Boundary (thermodynamic)|boundary]] to the local [[speed of sound]].<ref name="Young_et_al">{{cite book|last=Young|first=Donald F.|title=A Brief Introduction to Fluid Mechanics|date=2010-12-21|publisher=John Wiley & Sons|isbn=978-0-470-59679-1|edition=5th|first2=Bruce R. | last2 =  Munson |first3=Theodore H. | last3 = Okiishi | author-link3 = Theodore H. Okiishi |first4 = Wade W. | last4 = Huebsch  |page=95 | ol = OL24479108M | oclc = 667210577 | lccn = 2010038482 | df = dmy-all}}</ref><ref name="Graebel">{{cite book|last=Graebel|first=William P.|title=Engineering Fluid Mechanics|date = 2001-01-19 |publisher=[[CRC Press]] | edition = 1st |isbn=978-1-56032-733-2 |page=16|ol = OL9794889M | oclc = 1034989004 | df = dmy-all }}</ref>
It is named after the  [[Austria|Austrian]] physicist and philosopher [[Ernst Mach]].
: <math>\mathrm{M} = \frac{u}{c},</math>

where:
: {{serif|M}} is the local Mach number,
: {{mvar|u}} 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
: {{mvar|c}} is the speed of sound in the medium, which in air varies with the square root of the [[thermodynamic temperature]].

By definition, at Mach{{nbsp}}1, the local flow velocity {{mvar|u}} is equal to the speed of sound. At Mach{{nbsp}}0.65, {{mvar|u}} is 65% of the speed of sound (subsonic), and, at Mach{{nbsp}}1.35, {{mvar|u}} is 35% faster than the speed of sound (supersonic). Pilots of high-altitude [[aerospace]] vehicles use flight Mach number to express a vehicle's [[true airspeed|true]] [[airspeed]], but the flow field around a vehicle varies in three dimensions, with corresponding variations in local Mach number.

[[File:FA-18 Hornet breaking sound barrier (7 July 1999).jpg|right|thumb|An [[F/A-18 Hornet]] creating a [[vapor cone]] at [[transonic speed]] just before reaching the [[speed of sound]].]]

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 [[velocity|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]], [[jet engine#Air intake|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 {{serif|M}}&nbsp;<&nbsp;0.2–0.3 and the flow is [[steady flow|quasi-steady]] and [[isothermal flow|isothermal]], compressibility effects will be small and simplified incompressible flow equations can be used.<ref name="Young_et_al" /><ref name="Graebel" />

==Etymology==
The Mach number is named after the physicist and philosopher [[Ernst Mach]]<ref>{{cite encyclopedia|title=Ernst Mach|encyclopedia=[[Encyclopædia Britannica]]|url=https://www.britannica.com/biography/Ernst-Mach|year=2016|access-date=January 6, 2016}}</ref> according to a proposal by the aeronautical engineer [[Jakob Ackeret]] in 1929.<ref>Jakob Ackeret: Der Luftwiderstand bei sehr großen Geschwindigkeiten. Schweizerische Bauzeitung 94 (Oktober 1929), pp. 179–183. See also: N. Rott: Jakob Ackert and the History of the Mach Number. Annual Review of Fluid Mechanics 17 (1985), pp. 1–9.</ref> 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 [[International_System_of_Units#Unit_names|unit of measure]], the number comes ''after'' the word Mach; the second Mach number is ''Mach{{nbsp}}2'' instead of ''2{{nbsp}}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 [[sound barrier#Breaking the sound barrier|faster-than-sound human flight]], aeronautical engineers referred to the speed of sound as ''Mach's number'', never ''Mach 1''.<ref>Bodie, Warren M., ''The Lockheed P-38 Lightning'', Widewing Publications {{ISBN|0-9629359-0-5}}.</ref>

== Overview ==
[[File:Comparison US standard atmosphere 1962.svg|thumb|The speed of sound (blue) depends only on the temperature variation at altitude (red) and can be calculated from it since isolated density and pressure effects on the speed of sound cancel each other. The speed of sound increases with height in two regions of the stratosphere and thermosphere, due to heating effects in these regions.]]
Mach number is a measure of the [[compressible flow|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.<ref name=NASA>{{cite web |url=http://www.grc.nasa.gov/WWW/k-12/airplane/mach.html |work=[[NASA]] |title=Mach Number |editor=Nancy Hall}}</ref> As modeled in the [[International Standard Atmosphere]], dry air at [[mean sea level]], standard temperature of {{convert|15|C|F}}, the speed of sound is {{convert|340.3|m/s|ft/s mph km/h kn|sp=us|sigfig=5}}.<ref>Clancy, L.J. (1975), Aerodynamics, Table 1, Pitman Publishing London, {{ISBN|0-273-01120-0}}</ref> 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 {{convert|11000|m|ft|sp=us|sigfig=5}}, the speed of sound also decreases. For example, the standard atmosphere model lapses temperature to {{convert|-56.5|C|F}} at {{convert|11000|m|ft|sp=us|sigfig=5}} altitude, with a corresponding speed of sound (Mach{{nbsp}}1) of {{convert|295.0|m/s|ft/s mph km/h kn|sp=us|sigfig=4}}, 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#Fluid dynamics|continuity equation]].<ref>{{Cite book |last1=Kundu |first1=P.J. |title=Fluid Mechanics |last2=Cohen |first2=I.M. |last3=Dowling |first3=D.R. |publisher=Academic Press |year=2012 |isbn=978-0-12-382100-3 |edition=5th |pages=148–149}}</ref> The full continuity equation for a general fluid flow is:<math display="block">{\partial \rho\over{\partial t}} + \nabla\cdot(\rho {\bf u}) = 0
\equiv -{1\over{\rho}}{D\rho\over{Dt}} = \nabla \cdot {\bf u}</math>where <math>D/Dt</math> is the [[material derivative]], <math>\rho</math> is the [[density]], and <math>{\bf u}</math> is the [[flow velocity]].  For [[Isentropic process|isentropic]] pressure-induced density changes, <math>dp = c^{2}d\rho</math> where <math>c</math> is the speed of sound.  Then the continuity equation may be slightly modified to account for this relation:<math display="block">-{1\over{\rho c^{2}}}{Dp\over{Dt}} = \nabla \cdot {\bf u}</math>The next step is to [[Non-dimensionalization and scaling of the Navier–Stokes equations|nondimensionalize]] the variables as such:<math display="block">{\bf x}^{*} = {\bf x}/L, \quad t^{*} = Ut/L, \quad {\bf u}^{*} = {\bf u}/U, \quad p^{*} = (p-p_{\infty})/\rho_{0}U^{2}, \quad \rho^{*} = \rho/\rho_{0}</math>where <math>L</math> is the characteristic length scale, <math>U</math> is the characteristic velocity scale, <math>p_{\infty}</math> is the reference pressure, and <math>\rho_{0}</math> is the reference density.  Then the nondimensionalized form of the continuity equation may be written as:<math display="block">-{U^{2}\over{c^{2}}} {1\over{\rho^{*}}} {Dp^{*}\over{Dt^{*}}} = \nabla^{*} \cdot {\bf u}^{*} \implies -\text{M}^{2}{1\over{\rho^{*}}} {Dp^{*}\over{Dt^{*}}} = \nabla^{*} \cdot {\bf u}^{*}</math>where the Mach number <math>\text{M} = U/c</math>. In the limit that <math>\text{M} \rightarrow 0</math>, the continuity equation reduces to <math>\nabla \cdot {\bf u} = 0</math> &mdash; this is the standard requirement for [[incompressible flow]].

== Classification of Mach regimes ==
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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]] 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]] and various space planes in development.

{| class="wikitable"
|-
! rowspan=2 | Regime
! colspan=5 | Flight speed
! rowspan=2 | General plane characteristics
|-
! (Mach)
! (knots)
! (mph)
! (km/h)
! (m/s)
|-
! style="background-color: #FFFFFF;" | [[Subsonic aircraft|Subsonic]]
| <0.8
| <530
| <609
| <980
| <273
| Most often propeller-driven and commercial [[turbofan]] aircraft with high aspect-ratio (slender) wings, and rounded features like the nose and leading edges.
The subsonic speed range is that range of speeds within which, all of the airflow over an aircraft is less than Mach 1.  The critical Mach number (Mcrit) is lowest free stream Mach number at which airflow over any part of the aircraft first reaches Mach 1.  So the subsonic speed range includes all speeds that are less than Mcrit.
|-
! style="background-color: #00ff00;" | [[Transonic]]
| 0.8–1.2
| 530–794
| 609–914
| 980–1,470
| 273–409
| Transonic aircraft nearly always have [[swept wing]]s, causing the delay of drag-divergence, and often feature a design that adheres to the principles of the Whitcomb [[Area rule]].
The transonic speed range is that range of speeds within which the airflow over different parts of an aircraft is between subsonic and supersonic. So the regime of flight from Mcrit up to Mach 1.3 is called the transonic range.
|-
! style="background-color: #FF8181;" | [[Supersonic]]
| 1.2–5.0
| 794–3,308
| 915–3,806
| 1,470–6,126
| 410–1,702
| The supersonic speed range is that range of speeds within which all of the airflow over an aircraft is supersonic (more than Mach 1).  But airflow meeting the leading edges is initially decelerated, so the free stream speed must be slightly greater than Mach 1 to ensure that all of the flow over the aircraft is supersonic.  It is commonly accepted that the supersonic speed range starts at a free stream speed greater than Mach 1.3.
Aircraft designed to fly at supersonic speeds show large differences in their aerodynamic design because of the radical differences in the behavior of flows above Mach 1. Sharp edges, thin [[aerofoil]]-sections, and all-moving [[tailplane]]/[[canard (aeronautics)|canards]] are common. Modern [[combat aircraft]] must compromise in order to maintain low-speed handling; "true" supersonic designs include the [[F-104 Starfighter]], [[Mikoyan MiG-31|MiG-31]], [[North American XB-70 Valkyrie]], [[SR-71 Blackbird]], and BAC/Aérospatiale [[Concorde]].
|-
! style="background-color: #FF4242;" | [[Hypersonic]]
| 5.0–10.0
| 3,308–6,615
| 3,806–7,680
| 6,126–12,251
| 1,702–3,403
| The [[North American X-15|X-15]], at Mach 6.72, is one of the fastest manned aircraft. Also, cooled [[nickel]]-[[titanium]] skin; highly integrated (due to domination of interference effects: non-linear behaviour means that [[Superposition principle|superposition]] of results for separate components is invalid), small wings, such as those on the Mach 5 [[Boeing X-51|X-51A Waverider]].
|-
! style="background-color: #FF0303; color: #FFFFFF;" | High-hypersonic
| 10.0–25.0
| 6,615–16,537
| 7,680–19,031
| 12,251–30,626
| 3,403–8,508
| The [[NASA X-43]], at Mach 9.6, is one of the fastest aircraft. Thermal control becomes a dominant design consideration. Structure must either be designed to operate hot, or be protected by special silicate tiles or similar. Chemically reacting flow can also cause corrosion of the vehicle's skin, with free-atomic [[oxygen]] featuring in very high-speed flows. Hypersonic designs are often forced into [[Atmospheric reentry#Blunt body entry vehicles|blunt configurations]] because of the aerodynamic heating rising with a reduced [[Radius of curvature (mathematics)|radius of curvature]].
|-
! style="background-color: #C00000; color: #FFFFFF;" | [[Re-entry]] speeds
| >25.0
| >16,537
| >19,031
| >30,626
| >8,508
| [[Ablative heat shield]]; small or no wings; blunt shape. Russia's [[Avangard (hypersonic glide vehicle)]] is claimed to reach up to Mach 27.
|}

== High-speed flow around objects ==
Flight can be roughly classified in six categories:
{| class="wikitable" width=60%
! Regime
! [[Subsonic aircraft|Subsonic]]
! [[Transonic]]
! [[Speed of sound]]
! [[Supersonic]]
! [[Hypersonic]]
! [[Hypervelocity]]
|-
! Mach
|align=center| <0.8
|align=center| 0.8–1.2
|align=center| 1.0
|align=center| 1.2–5.0
|align=center| 5.0–10.0
|align=center| >8.8
|}
For comparison: the required speed for [[low Earth orbit]] is approximately 7.5&nbsp;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)

{| border="0"
| [[File:Transsonic flow over airfoil 1.svg|300px]]
| [[File:Transsonic flow over airfoil 2.svg|300px]]
|-
| (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]]), 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 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]] 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 travelling 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 nozzle]]s and in extreme cases they are able to reach [[hypersonic]] speeds ({{convert|13|Mach}} at 20&nbsp;°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
:<math>
\mathrm{M} = \frac{u}{c}
</math>

where:
: M is the Mach number
: ''u'' is [[velocity]] 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:

:<math>c = \sqrt{\gamma \cdot R_* \cdot T},</math>

where:
: <math>\gamma\,</math> is the [[Heat capacity ratio|ratio of specific heat]] of a gas at a constant pressure to heat at a constant volume (1.4 for air)
: <math> R_*</math> is the [[specific gas constant]] for air.
: <math> T, </math> 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 principle|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:<ref name="Olson">Olson, Wayne M. (2002). "AFFTC-TIH-99-02, ''Aircraft Performance Flight Testing''." ([http://www.aviation.org.uk/pdf/Aircraft_Performance_Flight_Testing.pdf PDF]). Air Force Flight Test Center, Edwards AFB, CA, United States Air Force. {{webarchive |url=https://web.archive.org/web/20110904190008/http://www.aviation.org.uk/pdf/Aircraft_Performance_Flight_Testing.pdf |date=September 4, 2011 }}</ref>
:<math>\mathrm{M} = \sqrt{\frac{2}{\gamma- 1 }\left[\left(\frac{q_c}{p} + 1\right)^\frac{\gamma - 1}{\gamma} - 1\right]}\,</math>

where:
: ''q<sub>c</sub>'' is [[impact pressure]] (dynamic pressure) and
: ''p'' is [[static pressure]]
: <math>\gamma\,</math> is the [[Heat capacity ratio|ratio of specific heat]] 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 number|Rayleigh]] supersonic pitot equation:
: <math>\frac{p_t}{p} =
  \left[\frac{\gamma + 1}{2}\mathrm{M}^2\right]^\frac{\gamma}{\gamma-1} \cdot
  \left[\frac{\gamma + 1}{1 - \gamma + 2\gamma\, \mathrm{M}^2}\right]^\frac{1}{\gamma - 1}
</math>

===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 {{nowrap|M < 1}} (above):<ref name="Olson" />
: <math>\mathrm{M} = \sqrt{5\left[\left(\frac{q_c}{p} + 1\right)^\frac{2}{7} - 1\right]}\,</math>


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:
: <math>\mathrm{M} \approx 0.88128485 \sqrt{\left(\frac{q_c}{p} + 1\right)\left(1 - \frac{1}{7\,\mathrm{M}^2}\right)^{2.5}}</math>

where:
:''q<sub>c</sub>'' 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 is a [[septic equation]] in M<sup>2</sup> 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.<ref name=Olson/> Alternatively, [[Newton's method]] can also be used.

== See also ==
*{{annotated link|Critical Mach number}}
*{{annotated link|Machmeter}}
*{{annotated link|Ramjet}}
*{{annotated link|Scramjet}}
*{{annotated link|Speed of sound}}
*{{annotated link|True airspeed}}
*{{annotated link|Orders of magnitude (speed)}}

== Notes ==
{{Reflist|30em}}

== External links ==
*[http://web.ics.purdue.edu/~alexeenk/GDT/index.html Gas Dynamics Toolbox] Calculate Mach number and normal shock wave parameters for mixtures of perfect and imperfect gases.
*[http://www.grc.nasa.gov/WWW/K-12/airplane/mach.html NASA's page on Mach Number] Interactive calculator for Mach number.
*[http://www.newbyte.co.il/calculator/index.php NewByte standard atmosphere calculator and speed converter]

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[[Category:Ernst Mach]]
[[Category:Aerodynamics]]
[[Category:Airspeed]]
[[Category:Dimensionless numbers of fluid mechanics]]
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