Mach number

The Mach number is a dimensionless quantity used in fluid dynamics to express the ratio between the velocity of a flow and the local speed of sound. Named after the Austrian physicist Ernst Mach, it provides a critical measure for understanding compressibility effects in gases and is central to the study of aerodynamics, propulsion systems, and high-speed flight.

Definition and Fundamental Concept

The Mach number MMM is defined as:
M=ucM = \frac{u}{c}M=cu​
where uuu is the velocity of an object or fluid relative to its surroundings, and ccc is the speed of sound in the medium. In air, the speed of sound varies with the square root of the absolute temperature, meaning that Mach values depend strongly on atmospheric conditions rather than on altitude alone.
By definition:

• At Mach 1, the flow velocity equals the speed of sound.
• At Mach 0.65, the flow travels at 65 % of the speed of sound, within the subsonic regime.
• At Mach 1.35, the flow is 35 % faster than sound, within the supersonic regime.

The Mach number is crucial in determining whether compressibility effects must be included in modelling. For flows with M<0.2 M < 0.2M<0.2–0.30.3 0.3, air behaves as if it were incompressible and simpler equations can be used.

Etymology and Conventions

The term “Mach number” honours Ernst Mach, following a proposal by the aeronautical engineer Jakob Ackeret in 1929. “Mach” is always capitalised because it derives from a proper name, and the unit is written as “Mach x”, not “Mx”, since it is dimensionless.

Speed of Sound and Atmospheric Influence

The speed of sound in a gas depends on:
c=γRTc = \sqrt{\gamma R T}c=γRT​
where:

• γ\gammaγ is the heat-capacity ratio (approximately 1.4 for dry air),
• RRR is the specific gas constant for air,
• TTT is the static temperature.

As temperature normally decreases with altitude in the lower atmosphere, the speed of sound also falls, meaning that an aircraft may reach Mach 1 at a lower true airspeed at high altitude.

Classification of Mach Regimes

Aerodynamic behaviour changes dramatically depending on Mach number, and flows are classified into several regimes.
Subsonic (below critical Mach number)In this regime, all airflow around a body is slower than Mach 1. The critical Mach number is the lowest free-stream Mach value at which airflow over any surface first reaches Mach 1.
Transonic (approximately Mach 0.8 to 1.3)Transonic flows contain a mixture of subsonic and supersonic regions. Shock waves begin to appear locally, especially above wings. As speed approaches Mach 1, a vapour cone may form around aircraft due to condensation effects in low-pressure zones. Flow equations designed for subsonic regimes no longer apply reliably.
Supersonic (Mach 1.3 to 5)Shock waves form ahead of moving bodies. Aerodynamic design must accommodate abrupt pressure changes, using thin aerofoils, sharp leading edges, and all-moving tailplanes.
Hypersonic (Mach 5 to 10)Shock waves are extremely strong and temperatures high enough to cause dissociation of air molecules. Heat-resistant materials and specialised thermal protection become necessary.
High hypersonic (Mach 10 to 25)Flow conditions include ionisation and plasma formation. Vehicles such as spaceplanes or re-entry bodies operate briefly in this regime.
Re-entry speeds (above Mach 25)Thermal loads become extreme, requiring advanced ablative or ceramic shielding.

High-Speed Flow Around Objects

At transonic speeds, supersonic patches first appear over an aircraft wing. These terminate in normal shocks, which rapidly decelerate flow back to subsonic speeds. As the Mach number increases:

• The supersonic region expands toward the leading and trailing edges.
• At Mach 1, the normal shock reaches the trailing edge.
• Beyond Mach 1, a weak oblique shock replaces the normal shock as flow remains mostly supersonic.
• A Mach cone forms, its angle narrowing with increasing speed. This cone produces the sonic boom heard on the ground.

At hypersonic speeds, temperatures behind shocks become so high that gas dissociation and ionisation occur, strongly influencing drag and heat transfer.

High-Speed Flow in Channels

In channels or nozzles, compressible flow displays reversed behaviour once sonic speed is reached:

• Subsonic flow: decreasing area increases velocity.
• Supersonic flow: increasing area increases velocity.

This leads to the convergent–divergent (de Laval) nozzle, which accelerates gas to sonic speed at the throat and to supersonic or hypersonic speeds in the diverging section. Such nozzles are essential in rocket engines and high-speed wind tunnels.

Calculation of Mach Number

If the speed of sound is known, the Mach number is calculated directly using:
M=ucM = \frac{u}{c}M=cu​
When ccc is not known, it is obtained from temperature:
c=γRTc = \sqrt{\gamma R T}c=γRT​
For subsonic compressible flow, Mach number may also be derived from measured static and dynamic pressures using formulas based on Bernoulli-type relations, assuming ideal-gas behaviour.

Aerodynamic Relevance and Applications

The Mach number underpins the modelling of compressible flows in aviation, aerospace propulsion, and high-speed fluid systems. It determines when shocks form, when compressibility corrections must be applied, and how air behaves around lifting surfaces and engine inlets. Its classification regimes guide the design of aircraft, missiles, rockets, and hypersonic vehicles and inform the choice of materials capable of withstanding high-temperature gas dynamics.