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**Mach number** (**M** or **Ma**) ( /mɑːk/ ; German: [max] ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound.^{ [1] }^{ [2] }

- Etymology
- Overview
- Classification of Mach regimes
- High-speed flow around objects
- High-speed flow in a channel
- Calculation
- Calculating Mach number from pitot tube pressure
- See also
- Notes
- External links

where:

- M is the local Mach number,
- 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
- 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 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). Pilots of high-altitude aerospace vehicles use flight Mach number to express a vehicle's true airspeed, 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 channeling the medium. As the Mach number is defined as the ratio of two speeds, it is a dimensionless number. 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.^{ [1] }^{ [2] }

The Mach number is named after Austrian physicist and philosopher Ernst Mach,^{ [3] } and is a designation proposed by aeronautical engineer Jakob Ackeret in 1929.^{ [4] } 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 *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 faster-than-sound human flight, aeronautical engineers referred to the speed of sound as *Mach's number*, never *Mach 1*.^{ [5] }

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.^{ [6] } 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; 661.49 kn).^{ [7] } 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; 573.4 kn), 86.7% of the sea level value.

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.

Regime | Flight speed | General plane characteristics | ||||
---|---|---|---|---|---|---|

(Mach) | (knots) | (mph) | (km/h) | (m/s) | ||

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. |

Transonic | 0.8–1.2 | 530–794 | 609–914 | 980–1,470 | 273–409 | Transonic aircraft nearly always have swept wings, 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. |

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/canards are common. Modern combat aircraft must compromise in order to maintain low-speed handling; "true" supersonic designs include the F-104 Starfighter, North American XB-70 Valkyrie, SR-71 Blackbird, and BAC/Aérospatiale Concorde. |

Hypersonic | 5.0–10.0 | 3,308–6,615 | 3,806–7,680 | 6,126–12,251 | 1,702–3,403 | The 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 of results for separate components is invalid), small wings, such as those on the Mach 5 X-51A Waverider. |

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 blunt configurations because of the aerodynamic heating rising with a reduced radius of curvature. |

Re-entry speeds | >25.0 | >16,537 | >19,031 | >30,626 | >8,508 | Ablative heat shield; small or no wings; blunt shape. |

Flight can be roughly classified in six categories:

Regime | Subsonic | Transonic | Speed of sound | Supersonic | Hypersonic | Hypervelocity |
---|---|---|---|---|---|---|

Mach | <0.8 | 0.8–1.2 | 1.0 | 1.2–5.0 | 5.0–10.0 | >8.8 |

For comparison: the required speed for low Earth orbit 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)

(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 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.

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 (Mach 13 (15,926 km/h; 9,896 mph) 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.

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

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)

Note that the dynamic pressure can be found as:

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:^{ [8] }

and the speed of sound varies with the thermodynamic temperature as:

where:

*q*is impact pressure (dynamic pressure) and_{c}*p*is static pressure- is the ratio of specific heat of a gas at a constant pressure to heat at a constant volume (1.4 for air)
- is the specific gas constant for air.

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

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 M < 1 (above):^{ [8] }

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:

where:

*q*is the dynamic pressure measured behind a normal shock._{c}

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 M^{2} 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.^{ [8] } Alternatively, Newton's method can also be used.

- Critical Mach number
- Machmeter – Flight instrument
- Ramjet – Jet engine designed to operate at supersonic speeds
- Scramjet – Jet engine where combustion takes place in supersonic airflow
- Speed of sound – Distance travelled during a unit of time by a sound wave propagating through an elastic medium
- True airspeed
- Orders of magnitude (speed)

- 1 2 Young, Donald F.; Bruce R. Munson; Theodore H. Okiishi; Wade W. Huebsch (2010).
*A Brief Introduction to Fluid Mechanics*(5 ed.). John Wiley & Sons. p. 95. ISBN 978-0-470-59679-1. - 1 2 Graebel, W.P. (2001).
*Engineering Fluid Mechanics*. Taylor & Francis. p. 16. ISBN 978-1-56032-733-2. - ↑ "Ernst Mach".
*Encyclopædia Britannica*. 2016. Retrieved January 6, 2016. - ↑ 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.
- ↑ Bodie, Warren M.,
*The Lockheed P-38 Lightning*, Widewing Publications ISBN 0-9629359-0-5. - ↑ Nancy Hall (ed.). "Mach Number".
*NASA*. - ↑ Clancy, L.J. (1975), Aerodynamics, Table 1, Pitman Publishing London, ISBN 0-273-01120-0
- 1 2 3 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. Archived September 4, 2011, at the Wayback Machine

- Gas Dynamics Toolbox Calculate Mach number and normal shock wave parameters for mixtures of perfect and imperfect gases.
- NASA's page on Mach Number Interactive calculator for Mach number.
- NewByte standard atmosphere calculator and speed converter

**Aerodynamics**, from Greek ἀήρ *aero* (air) + δυναμική (dynamics), is the study of motion of air, particularly when affected by a solid object, such as an airplane wing. It is a sub-field of fluid dynamics and gas dynamics, and many aspects of aerodynamics theory are common to these fields. The term *aerodynamics* is often used synonymously with gas dynamics, the difference being that "gas dynamics" applies to the study of the motion of all gases, and is not limited to air. The formal study of aerodynamics began in the modern sense in the eighteenth century, although observations of fundamental concepts such as aerodynamic drag were recorded much earlier. Most of the early efforts in aerodynamics were directed toward achieving heavier-than-air flight, which was first demonstrated by Otto Lilienthal in 1891. Since then, the use of aerodynamics through mathematical analysis, empirical approximations, wind tunnel experimentation, and computer simulations has formed a rational basis for the development of heavier-than-air flight and a number of other technologies. Recent work in aerodynamics has focused on issues related to compressible flow, turbulence, and boundary layers and has become increasingly computational in nature.

The **speed of sound** is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At 20 °C (68 °F), the speed of sound in air is about 343 metres per second, or a kilometre in 2.9 s or a mile in 4.7 s. It depends strongly on temperature as well as the medium through which a sound wave is propagating. At 0°C/32°F, the speed-of-sound is 1192 km/h, 741 mph.

In thermodynamics and fluid mechanics, the **compressibility** is a measure of the relative volume change of a fluid or solid as a response to a pressure change. In its simple form, the compressibility may be expressed as

**Compressible flow** is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number is smaller than 0.3. The study of compressible flow is relevant to high-speed aircraft, jet engines, rocket motors, high-speed entry into a planetary atmosphere, gas pipelines, commercial applications such as abrasive blasting, and many other fields.

The **Prandtl–Glauert singularity** is a theoretical construct in flow physics, often incorrectly used to explain vapor cones in transonic flows. It is the prediction by the Prandtl–Glauert transformation that infinite pressures would be experienced by an aircraft as it approaches the speed of sound. Because it is invalid to apply the transformation at these speeds, the predicted singularity does not emerge. The incorrect association is related to the early-20th-century misconception of the impenetrability of the sound barrier.

**Transonic** flow is air flowing around an object at a speed that generates regions of both subsonic and supersonic airflow around that object. The exact range of speeds depends on the object's critical Mach number, but transonic flow is seen at flight speeds close to the speed of sound, typically between Mach 0.8 and 1.2.

A **de Laval nozzle** is a tube that is pinched in the middle, making a carefully balanced, asymmetric hourglass shape. It is used to accelerate a hot, pressurized gas passing through it to a higher supersonic speed in the axial (thrust) direction, by converting the heat energy of the flow into kinetic energy. Because of this, the nozzle is widely used in some types of steam turbines and rocket engine nozzles. It also sees use in supersonic jet engines.

**Indicated airspeed** (**IAS**) is the airspeed read directly from the airspeed indicator (ASI) on an aircraft, driven by the pitot-static system. It uses the difference between total pressure and static pressure, provided by the system, to either mechanically or electronically measure dynamic pressure. The dynamic pressure includes terms for both density and airspeed. Since the airspeed indicator cannot know the density, it is by design calibrated to assume the sea level standard atmospheric density when calculating airspeed. Since the actual density will vary considerably from this assumed value as the aircraft changes altitude, IAS varies considerably from true airspeed (TAS), the relative velocity between the aircraft and the surrounding air mass. Calibrated airspeed (CAS) is the IAS corrected for instrument and position error.

An **oblique shock** wave is a shock wave that, unlike a normal shock, is inclined with respect to the incident upstream flow direction. It will occur when a supersonic flow encounters a corner that effectively turns the flow into itself and compresses. The upstream streamlines are uniformly deflected after the shock wave. The most common way to produce an oblique shock wave is to place a wedge into supersonic, compressible flow. Similar to a normal shock wave, the oblique shock wave consists of a very thin region across which nearly discontinuous changes in the thermodynamic properties of a gas occur. While the upstream and downstream flow directions are unchanged across a normal shock, they are different for flow across an oblique shock wave.

**Rayleigh flow** refers to frictionless, non-adiabatic flow through a constant area duct where the effect of heat addition or rejection is considered. Compressibility effects often come into consideration, although the Rayleigh flow model certainly also applies to incompressible flow. For this model, the duct area remains constant and no mass is added within the duct. Therefore, unlike Fanno flow, the stagnation temperature is a variable. The heat addition causes a decrease in stagnation pressure, which is known as the Rayleigh effect and is critical in the design of combustion systems. Heat addition will cause both supersonic and subsonic Mach numbers to approach Mach 1, resulting in choked flow. Conversely, heat rejection decreases a subsonic Mach number and increases a supersonic Mach number along the duct. It can be shown that for calorically perfect flows the maximum entropy occurs at M = 1. Rayleigh flow is named after John Strutt, 3rd Baron Rayleigh.

**Choked flow** is a compressible flow effect. The parameter that becomes "choked" or "limited" is the fluid velocity.

In aviation, stagnation temperature is known as **total air temperature** and is measured by a temperature probe mounted on the surface of the aircraft. The probe is designed to bring the air to rest relative to the aircraft. As the air is brought to rest, kinetic energy is converted to internal energy. The air is compressed and experiences an adiabatic increase in temperature. Therefore, total air temperature is higher than the static air temperature.

A supersonic expansion fan, technically known as **Prandtl–Meyer expansion fan**, a two-dimensional simple wave, is a centered expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of an infinite number of Mach waves, diverging from a sharp corner. When a flow turns around a smooth and circular corner, these waves can be extended backwards to meet at a point.

In fluid dynamics, a **moving shock** is a shock wave that is travelling through a fluid medium with a velocity relative to the velocity of the fluid already making up the medium. As such, the normal shock relations require modification to calculate the properties before and after the moving shock. A knowledge of moving shocks is important for studying the phenomena surrounding detonation, among other applications.

**Fanno flow** is the adiabatic flow through a constant area duct where the effect of friction is considered. Compressibility effects often come into consideration, although the Fanno flow model certainly also applies to incompressible flow. For this model, the duct area remains constant, the flow is assumed to be steady and one-dimensional, and no mass is added within the duct. The Fanno flow model is considered an irreversible process due to viscous effects. The viscous friction causes the flow properties to change along the duct. The frictional effect is modeled as a shear stress at the wall acting on the fluid with uniform properties over any cross section of the duct.

**Shock** is an abrupt discontinuity in the flow field and It occurs in flows when the local flow speed exceeds the local sound speed. More specifically, it is a flow whose Mach number exceeds 1.

The **Cauchy number** (**Ca**) is a dimensionless number in continuum mechanics used in the study of compressible flows. It is named after the French mathematician Augustin Louis Cauchy. When the compressibility is important the elastic forces must be considered along with inertial forces for dynamic similarity. Thus, the Cauchy Number is defined as the ratio between inertial and the compressibility force in a flow and can be expressed as

**Isentropic nozzle flow **describes the movement of a gas or fluid through a narrowing opening without an increase or decrease in entropy.

In gas dynamics, the ** Kantrowitz limit** refers to a theoretical concept describing choked flow at supersonic or near-supersonic velocities. When an initially subsonic fluid flow experiences a reduction in cross-section area, the flow speeds up in order to maintain the same mass-flow rate, per the continuity equation. If a near supersonic flow experiences an area contraction, the velocity of the flow will increase until it reaches the local speed of sound, and the flow will be choked. This is the principle behind the Kantrowitz limit: it is the maximum amount of contraction a flow can experience before the flow chokes, and the flow speed can no longer be increased above this limit, independent of changes in upstream or downstream pressure.

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