3.2 Mach number

Mach number quantifies how fast an object moves relative to the speed of sound in the surrounding medium. It's the single most important parameter for classifying compressible flow behavior, and it drives nearly every design decision for high-speed aircraft, from airfoil shape to engine configuration.

Definition of Mach number

Mach number is named after Ernst Mach, the Austrian physicist who studied supersonic fluid dynamics in the late 19th century. It tells you whether a flow is "slow" or "fast" relative to how quickly pressure disturbances (sound waves) can travel through the fluid.

Ratio of flow velocity to local speed of sound

Mach number ($M$) is defined as:

$M = \frac{v}{a}$

where $v$ is the flow velocity and $a$ is the local speed of sound.

The local speed of sound depends on the thermodynamic state of the fluid:

$a = \sqrt{\gamma R T}$

• $\gamma$ = specific heat ratio (1.4 for air at standard conditions)
• $R$ = specific gas constant (287 J/(kg·K) for air)
• $T$ = absolute temperature in Kelvin

At sea level and standard conditions (15°C / 288.15 K), the speed of sound in air is about 340 m/s. An aircraft traveling at 680 m/s at those conditions would be flying at Mach 2.

Dimensionless quantity

Mach number has no units. This makes it useful for comparing flow behavior across different fluids, altitudes, and temperatures. Dimensionless parameters like this (another example is the Reynolds number) let engineers identify similar flow physics even when the physical scales are completely different.

Significance in aerodynamics

Mach number determines how a fluid responds to the presence of an object. It controls whether disturbances can propagate upstream, whether shock waves form, and how pressure distributes over surfaces. Getting the Mach number regime right is the starting point for any compressible flow analysis.

Characterization of flow regimes

Different Mach number ranges produce fundamentally different flow physics. The four main regimes are subsonic, transonic, supersonic, and hypersonic. Each regime brings its own set of aerodynamic challenges, and the design strategies that work in one regime can fail in another.

Subsonic vs supersonic flow

In subsonic flow ($M < 1$), pressure disturbances travel faster than the flow itself. This means the fluid "knows" an object is coming and can adjust smoothly. Changes in fluid properties (pressure, density, velocity) are gradual and continuous.

In supersonic flow ($M > 1$), the flow outruns its own pressure disturbances. The fluid has no warning of the object ahead, so it adjusts abruptly through shock waves. These are extremely thin regions where pressure, density, and temperature jump almost instantaneously.

This upstream-vs-downstream propagation difference is the core reason subsonic and supersonic flows behave so differently.

Transonic regime and critical Mach number

The transonic regime (roughly $M = 0.8$ to $1.2$) is where subsonic and supersonic flow coexist around the same object. Even if the freestream is subsonic, the flow can accelerate past Mach 1 locally, especially near the point of maximum thickness on an airfoil.

The critical Mach number is the freestream Mach number at which the flow first reaches Mach 1 somewhere on the surface. Once you exceed it:

• Local shock waves form on the surface
• Wave drag increases sharply
• Aerodynamic loads and stability can change dramatically

This is why the transonic regime demands careful design attention.

Calculation of Mach number

Velocity and speed of sound relationship

To find Mach number, you need two things: the flow velocity and the local speed of sound.

1. Determine the flow velocity $v$ (from measurement or given conditions)
2. Calculate the local speed of sound: $a = \sqrt{\gamma R T}$
3. Divide: $M = \frac{v}{a}$

Flow velocity can be measured using Pitot-static systems, laser Doppler velocimetry, or particle image velocimetry, depending on the application.

Variation with altitude and temperature

Because $a = \sqrt{\gamma R T}$, the speed of sound depends directly on temperature. As altitude increases in the troposphere, temperature drops, and so does the speed of sound.

• At sea level (15°C): $a \approx 340$ m/s
• At 11,000 m / 36,000 ft (about −56.5°C): $a \approx 295$ m/s

This means the same aircraft velocity corresponds to a higher Mach number at altitude than at sea level. An aircraft cruising at 250 m/s is at $M \approx 0.74$ at sea level but $M \approx 0.85$ at 11,000 m.

Mach number in different fluids

Mach number applies to any fluid, not just air. The speed of sound varies widely between fluids because $\gamma$ and $R$ (or more generally, the bulk modulus and density) differ.

• Speed of sound in water at room temperature: about 1,480 m/s
• Speed of sound in helium at room temperature: about 1,007 m/s

A flow at 340 m/s is Mach 1 in air but only about Mach 0.23 in water. Always use the correct fluid properties when calculating Mach number.

Compressibility effects

At low Mach numbers (below about 0.3), density changes in the flow are negligible and you can treat the fluid as incompressible. As Mach number increases beyond this, density variations become significant and compressibility must be accounted for.

Density changes at high Mach numbers

In subsonic flow, density changes are small enough to ignore for most engineering purposes. Once you enter the transonic and supersonic regimes, density can change substantially across the flow field. These density variations directly affect lift, drag, and pressure distributions, so incompressible assumptions break down and compressible flow equations become necessary.

Shock wave formation

Shock waves form when supersonic flow encounters an obstacle it cannot smoothly navigate around. The two main types are:

• Normal shocks: perpendicular to the flow direction. The flow downstream of a normal shock is always subsonic.
• Oblique shocks: inclined at an angle to the flow. The downstream flow can remain supersonic depending on the shock angle and Mach number.

Shock waves produce a sudden rise in pressure, temperature, and density, along with a drop in velocity. They are a major source of wave drag in transonic and supersonic flight.

Mach cone and Mach angle

When an object moves supersonically, the sound waves it generates pile up along a conical surface called the Mach cone. The half-angle of this cone is the Mach angle $\mu$:

$\sin \mu = \frac{1}{M}$

At Mach 2, the Mach angle is 30°. At Mach 5, it shrinks to about 11.5°. Higher Mach numbers produce narrower cones, meaning the region of influence becomes more tightly focused downstream.

Mach number regimes

Subsonic ($M < 0.8$)

Flow velocity stays below the local speed of sound everywhere around the object. No shock waves form, and fluid properties change smoothly. Most general aviation aircraft and low-speed wind tunnels operate here.

Transonic ($0.8 < M < 1.2$)

Mixed subsonic and supersonic regions coexist around the object. Local shock waves can form even when the freestream is subsonic. This regime brings wave drag, potential buffeting (flow-induced vibration), and rapid changes in aerodynamic coefficients. Most commercial airliners cruise in the lower transonic range (around Mach 0.78 to 0.85) to balance speed and fuel efficiency.

Supersonic ($1.2 < M < 5$)

The entire flow field around the object is supersonic (except possibly in small subsonic pockets behind strong shocks). Shock waves and Mach cones are prominent features. Fighter jets and missiles typically operate in this regime, requiring sharp-edged aerodynamic surfaces and specialized inlet designs.

Hypersonic ($M > 5$)

At these speeds, additional physics come into play beyond standard compressibility. The thin shock layer hugs the body surface, temperatures can reach thousands of degrees, and the gas may undergo chemical changes: molecular dissociation ($O_2$ and $N_2$ breaking apart) and even ionization. Spacecraft re-entry vehicles and hypersonic weapons operate here, requiring advanced thermal protection systems and heat-resistant materials.

Mach number effects on aerodynamics

Lift and drag coefficients

Both lift coefficient ($C_L$) and drag coefficient ($C_D$) vary with Mach number due to compressibility.

• In subsonic flow, $C_L$ generally increases with Mach number (the Prandtl-Glauert correction predicts this trend).
• Near the critical Mach number, shock formation can cause $C_L$ to drop and $C_D$ to spike. This sharp drag increase in the transonic regime is called drag divergence.
• In supersonic flow, wave drag becomes a dominant component of total drag.

Pressure distribution on airfoils

At low Mach numbers, the pressure distribution over an airfoil is smooth, with a suction peak near the leading edge on the upper surface. As Mach number increases into the transonic range, this suction peak intensifies until a shock wave terminates the supersonic pocket on the surface. The shock causes an abrupt pressure rise that can trigger boundary layer separation.

In fully supersonic flow, pressure changes occur primarily through oblique shocks and expansion fans, producing a more angular pressure distribution compared to subsonic flow.

Boundary layer behavior

The boundary layer is the thin layer of fluid near the surface where viscous effects dominate. Mach number influences it in several ways:

• Higher Mach numbers thin the boundary layer and steepen velocity gradients
• Shock waves create strong adverse pressure gradients that can cause the boundary layer to separate, increasing drag
• In hypersonic flow, shock-boundary layer interaction becomes a major concern, producing localized heating and complex separated flow regions
• Aerodynamic heating within the boundary layer increases with Mach number, making thermal management critical at supersonic and hypersonic speeds

Mach number in aircraft design

Airfoil and wing shape optimization

Airfoil and wing geometry are tailored to the intended Mach number range:

• Subsonic: Rounded leading edges, moderate thickness, and smooth curvature to keep flow attached and minimize pressure drag
• Transonic: Supercritical airfoil profiles with flattened upper surfaces to weaken shocks and delay drag divergence
• Supersonic: Thin, sharp-edged profiles (diamond or biconvex shapes) to minimize wave drag

Wing sweep is widely used on transonic and supersonic aircraft. Sweeping the wing back reduces the effective Mach number component perpendicular to the leading edge, delaying shock formation and lowering wave drag.

Engine inlet and nozzle design

The engine must receive air at conditions it can handle, regardless of flight Mach number.

• Subsonic inlets use smooth, gradually contracting ducts to decelerate air with minimal pressure loss.
• Supersonic inlets incorporate compression ramps, cones, or spikes to decelerate the supersonic flow through a series of oblique shocks before a final normal shock, maximizing pressure recovery.
• Nozzles expand exhaust gases to generate thrust. Subsonic exhaust uses a simple convergent nozzle. Supersonic exhaust requires a convergent-divergent (de Laval) nozzle to accelerate the flow past Mach 1.

Structural considerations for high Mach flight

Higher Mach numbers mean higher dynamic pressures and greater aerodynamic heating. Structural design must account for:

• Increased aerodynamic loads from shock-induced pressure distributions
• Thermal stresses from kinetic heating (surface temperatures on the SR-71 exceeded 300°C at Mach 3+)
• Aeroelastic effects like flutter (dangerous oscillations from the coupling of aerodynamic and structural forces) and divergence
• Material selection: titanium alloys, nickel superalloys, and ceramic thermal protection systems replace aluminum at high Mach numbers

Measurement techniques

Pitot-static system

The Pitot-static system measures Mach number by comparing total (stagnation) pressure to static pressure.

1. A Pitot tube facing into the flow captures the total pressure $p_0$
2. Static ports on the aircraft or probe surface measure the undisturbed static pressure $p$
3. The pressure ratio $p_0 / p$ is related to Mach number through isentropic flow relations (for subsonic flow) or the Rayleigh Pitot tube formula (for supersonic flow)

This system is simple and reliable, though it can be affected by flow angularity and probe placement.

Schlieren photography

Schlieren photography visualizes density gradients in a flow, making shock waves and compression regions visible. The setup works by passing collimated light through the test section and onto a knife edge that partially blocks the light. Regions of varying density bend the light differently, producing bright and dark patterns in the image.

From schlieren images, you can identify shock wave locations and measure shock angles. Combined with the oblique shock relations, these angles give you the local Mach number.

Computational fluid dynamics (CFD) simulations

CFD solves the governing equations of fluid motion (Navier-Stokes equations) numerically on a discretized mesh. For compressible flow problems, CFD can predict Mach number distributions, shock positions, pressure fields, and heat transfer rates across the entire flow domain.

CFD is especially valuable for exploring design variations before building physical models, though results must be validated against experimental data (wind tunnel tests, flight data) to ensure accuracy.

Historical milestones

Breaking the sound barrier

On October 14, 1947, Chuck Yeager flew the Bell X-1 rocket plane past Mach 1, becoming the first person to achieve controlled supersonic flight. This was a major engineering achievement because the transonic regime had caused severe control problems and structural failures in earlier attempts. The X-1's design, with its thin straight wings and all-moving horizontal tail, was specifically engineered to handle transonic aerodynamics.

High-speed aircraft development

Supersonic flight opened the door to a series of progressively faster aircraft:

• Lockheed F-104 Starfighter: Mach 2 fighter with extremely thin, short wings
• Concorde: Mach 2 commercial airliner that flew transatlantic routes from 1976 to 2003, featuring a slender delta wing and droop-nose design
• Lockheed SR-71 Blackbird: Mach 3+ reconnaissance aircraft built largely from titanium, with inlet cones that managed shock positioning across a wide speed range

Each of these aircraft required breakthroughs in aerodynamic shaping, propulsion, and materials to operate at their design Mach numbers.

Mach number in space exploration

Mach number is critical during both launch and atmospheric re-entry. During launch, rockets accelerate through the atmosphere and must pass through the transonic drag rise. Re-entry vehicles like the Space Shuttle experienced Mach numbers up to about 25, generating extreme heating (surface temperatures exceeding 1,600°C on the leading edges).

These conditions demand specialized thermal protection, from reinforced carbon-carbon on the hottest surfaces to silica tiles elsewhere. Mach number considerations also apply to missions at other worlds: Mars entry vehicles (like the Curiosity rover's heat shield) must be designed for the specific atmospheric composition and density of Mars, where the speed of sound differs from Earth.