6.2 Mach number

Mach number is the ratio of a flow's velocity to the local speed of sound. It's the single most important parameter for determining when compressibility effects matter and how a flow will behave at high speeds.

This guide covers the definition and formula, flow regime classification, factors that change Mach number, shock wave behavior, applications in aerodynamics and gas dynamics, and measurement techniques.

Definition of Mach number

Mach number is a dimensionless quantity that compares how fast a fluid (or an object through a fluid) is moving relative to the speed at which sound travels in that same fluid. In compressible flow analysis, it tells you whether density changes are negligible or dominant.

Ratio of flow velocity to local speed of sound

The definition is straightforward:

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

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

The word "local" matters here. The speed of sound isn't a fixed number; it depends on the fluid's temperature and composition, so it can vary from point to point within a flow field. When $M$ approaches 1, the flow velocity is near the local speed of sound, and compressibility effects become significant.

Symbol and formula for Mach number

The standard symbol is $M$ (sometimes written $Ma$). For an ideal gas, you can expand the definition by substituting the expression for the speed of sound:

$M = \frac{v}{\sqrt{\gamma R T}}$

• $v$ = flow velocity
• $\gamma$ = specific heat ratio (ratio of $c_p$ to $c_v$)
• $R$ = specific gas constant
• $T$ = absolute temperature

For air at standard conditions: $\gamma \approx 1.4$ and $R = 287 \text{ J/(kg·K)}$. At sea-level standard temperature (288.15 K), this gives a speed of sound of about 340 m/s.

Significance of Mach number

Mach number determines which physical effects dominate a flow and which mathematical models you need to use. A flow at $M = 0.1$ and a flow at $M = 3$ behave in fundamentally different ways, and Mach number is how you distinguish between them.

Mach number as indicator of compressibility effects

• Below about $M = 0.3$, density changes are small enough to ignore. You can treat the flow as incompressible, which simplifies the governing equations considerably.
• Between $M = 0.3$ and $M = 1$, compressibility effects grow progressively. Density variations, temperature changes, and pressure-velocity coupling all become important.
• Above $M = 1$, the flow is supersonic. Shock waves can form, and the flow physics change qualitatively: disturbances can no longer propagate upstream.

Critical Mach number and flow regimes

The critical Mach number ($M_{cr}$) is the freestream Mach number at which the flow first reaches sonic speed ($M = 1$) somewhere on the body's surface. For typical airfoils, $M_{cr}$ falls between about 0.6 and 0.8, depending on the shape and angle of attack. Beyond $M_{cr}$, localized supersonic regions and shock waves appear on the surface even though the freestream is still subsonic.

Flow regimes classified by Mach number:

• Subsonic: $M < 1$
• Transonic: $M \approx 0.8 \text{ to } 1.2$ (mixed subsonic and supersonic regions)
• Supersonic: $1 < M < 5$ (approximately)
• Hypersonic: $M > 5$

Subsonic vs supersonic flow

In subsonic flow, pressure disturbances travel in all directions (upstream and downstream). The flow adjusts smoothly and continuously to obstacles, and streamlines curve gradually.

In supersonic flow, the fluid moves faster than its own pressure signals can travel. Disturbances can only propagate downstream, so the flow "doesn't know" about an obstacle until it's right on top of it. This is why shock waves form: the flow must adjust abruptly rather than gradually.

The sonic condition ($M = 1$) is a mathematical singularity in the governing equations of compressible flow, which is why transonic flows are particularly challenging to analyze.

Factors influencing Mach number

Since $M = v/a$, anything that changes either the flow velocity or the local speed of sound will change the Mach number.

Relationship between Mach number and flow velocity

This one is direct. If you increase $v$ while holding $a$ constant, $M$ goes up proportionally. Doubling the flow speed doubles the Mach number (assuming the speed of sound doesn't change).

Effect of fluid properties on Mach number

The speed of sound in an ideal gas is $a = \sqrt{\gamma R T}$. Two things to note:

• Temperature: Higher temperature means higher speed of sound, which lowers the Mach number for a given flow velocity. A hot gas flow at 500 m/s has a lower Mach number than a cold gas flow at the same speed.
• Gas composition: Different gases have different $\gamma$ and $R$ values. For example, helium ($\gamma = 1.67$, $R = 2077 \text{ J/(kg·K)}$) has a much higher speed of sound than air at the same temperature, so the same flow velocity corresponds to a lower Mach number in helium.

Variation of Mach number with altitude

In Earth's atmosphere, temperature varies with altitude, and that changes the speed of sound:

• In the troposphere (sea level to ~11 km), temperature decreases with altitude. The speed of sound drops, so an aircraft flying at constant velocity sees its Mach number increase as it climbs.
• In the lower stratosphere (~11 to ~20 km), temperature is roughly constant at about 217 K, so the speed of sound stays nearly constant at about 295 m/s.
• Higher in the stratosphere (above ~20 km), temperature begins to increase again, raising the speed of sound.

This is why an aircraft's indicated Mach number can change even at constant airspeed.

Mach number in compressible flow

Shock waves are one of the most distinctive features of compressible flow, and Mach number governs when they form, how strong they are, and what happens across them.

Role of Mach number in shock wave formation

Shock waves form when the flow velocity exceeds the local speed of sound ($M > 1$). The upstream Mach number directly controls the shock's strength:

• A weak shock at $M = 1.1$ produces small jumps in pressure and temperature.
• A strong shock at $M = 5$ produces enormous jumps. For example, a normal shock at $M = 5$ in air produces a static pressure ratio of about 29:1 across the shock.

As the upstream Mach number increases, the shock becomes stronger and the downstream changes in pressure, temperature, and density all grow.

Normal vs oblique shock waves

Normal shocks are perpendicular to the flow direction. They typically occur inside ducts, at nozzle exits, or ahead of blunt bodies. The entire velocity component passes through the shock, so normal shocks produce the largest property changes for a given Mach number.

Oblique shocks are inclined at an angle to the flow. They form when supersonic flow encounters a wedge, cone, or other deflection. Only the velocity component normal to the shock front determines the shock strength. The upstream Mach number and the flow deflection angle together determine the shock angle and downstream conditions.

Mach number across shock waves

Across a normal shock, the Mach number always drops from supersonic to subsonic. This is a fundamental result: if $M_1 > 1$ upstream, then $M_2 < 1$ downstream.

The normal shock relation for downstream Mach number is:

$M_2^2 = \frac{M_1^2 + \frac{2}{\gamma - 1}}{\frac{2\gamma}{\gamma - 1} M_1^2 - 1}$

For oblique shocks, the downstream Mach number can still be supersonic if the shock is weak enough (i.e., the deflection angle is small). Only the Mach number component normal to the shock must drop below 1.

Applications of Mach number

Mach number in aerodynamic design

Aircraft design is heavily shaped by the intended Mach number range:

• Subsonic transports ($M < 0.8$) use swept wings and supercritical airfoils to delay the onset of shock waves and push $M_{cr}$ as high as possible.
• Transonic and supersonic fighters ($M \approx 0.8 \text{ to } 2.5$) use thin, highly swept or delta wings to manage shock waves and minimize wave drag.
• Hypersonic vehicles ($M > 5$) face extreme aerodynamic heating and require blunt nose shapes to spread the thermal load, along with specialized thermal protection systems.

The critical Mach number is a key design target because exceeding it causes shock-induced drag rise and potential buffeting.

Mach number effects on aircraft performance

• Drag: At high subsonic speeds ($M \approx 0.8$ to 1), shock waves form on the wing surface and cause wave drag, a sharp drag increase sometimes called the "drag divergence." Supersonic flight adds additional pressure drag and increased skin friction.
• Lift: Shock waves on the upper wing surface can cause boundary layer separation, reducing lift and changing the pitching moment.
• Propulsion: Different engine types are suited to different Mach regimes. Turbofans work well up to about $M = 0.9$, turbojets to about $M = 3$, ramjets from roughly $M = 2$ to $M = 5$, and scramjets above $M = 5$.

Mach number considerations in wind tunnel testing

Wind tunnels are classified by the Mach number range they can achieve:

• Low-speed (subsonic): $M < 0.3$
• Transonic: $M \approx 0.8$ to $1.2$
• Supersonic: $M \approx 1.2$ to $5$
• Hypersonic: $M > 5$

The test section Mach number must be carefully controlled to match the intended flight conditions. Beyond Mach number, other similarity parameters (especially Reynolds number) must also be matched for the results to scale accurately to full-size vehicles. Transonic tunnels are particularly challenging to operate because small changes in conditions can shift the flow between subsonic and supersonic.

Mach number in gas dynamics

Mach number appears throughout the analysis of internal compressible flows in ducts, nozzles, and diffusers.

Mach number in isentropic flow

Isentropic flow (no friction, no heat transfer, constant entropy) provides the baseline relations connecting Mach number to flow properties. The key isentropic relations are:

$\frac{T_0}{T} = 1 + \frac{\gamma - 1}{2} M^2$

$\frac{p_0}{p} = \left(1 + \frac{\gamma - 1}{2} M^2\right)^{\frac{\gamma}{\gamma - 1}}$

$\frac{\rho_0}{\rho} = \left(1 + \frac{\gamma - 1}{2} M^2\right)^{\frac{1}{\gamma - 1}}$

Here, the subscript 0 denotes stagnation (total) conditions. Given the Mach number at any point and the stagnation conditions, you can find the local temperature, pressure, and density.

Mach number in Fanno and Rayleigh flow

These are two canonical duct flow models:

• Fanno flow: Adiabatic (no heat transfer) with wall friction in a constant-area duct. Friction drives the Mach number toward 1 regardless of whether the flow starts subsonic or supersonic. Subsonic flow accelerates; supersonic flow decelerates. Maximum duct length for a given inlet Mach number is set by the choking condition ($M = 1$ at the exit).
• Rayleigh flow: Frictionless with heat addition or removal in a constant-area duct. Adding heat drives the Mach number toward 1 for both subsonic and supersonic inlet conditions. Removing heat drives it away from 1. Again, choking occurs when $M = 1$.

In both cases, Mach number is the primary variable that determines all other flow properties through the respective relations.

Mach number in nozzle and diffuser design

Converging-diverging (C-D) nozzles are the standard device for accelerating flow to supersonic speeds:

1. The flow enters subsonically through the converging section, accelerating as the area decreases.
2. At the throat (minimum area), the flow reaches $M = 1$. This is a necessary condition for supersonic flow downstream.
3. In the diverging section, the flow continues to accelerate supersonically as the area increases.

The area-Mach number relation governs this process:

$\frac{A}{A^*} = \frac{1}{M}\left[\frac{2}{\gamma + 1}\left(1 + \frac{\gamma - 1}{2}M^2\right)\right]^{\frac{\gamma + 1}{2(\gamma - 1)}}$

where $A^*$ is the throat area (where $M = 1$).

Diffusers do the reverse: they decelerate the flow. In supersonic inlets, the goal is to slow the flow to subsonic speeds with minimal total pressure loss, often using a combination of oblique shocks and a final normal shock before a subsonic diffuser section.

Measurement of Mach number

Techniques for measuring Mach number

Several methods exist, each with different strengths:

• Pitot-static tube: Measures stagnation and static pressures. The pressure ratio gives the Mach number through isentropic (subsonic) or normal shock (supersonic) relations. Simple, reliable, and widely used.
• Schlieren and shadowgraph imaging: Optical methods that visualize density gradients. They reveal shock wave locations and flow structure but give qualitative rather than directly quantitative Mach number data without calibration.
• Pressure-sensitive paint (PSP): A coating whose luminescence changes with local static pressure, providing a full surface pressure map from which Mach number distribution can be inferred.
• Laser Doppler velocimetry (LDV) and particle image velocimetry (PIV): Measure local flow velocity directly using laser light scattered by seed particles. Mach number is then calculated from the velocity and the local speed of sound.

Pitot-static tube and Mach meter

The Pitot-static tube has two ports:

1. A stagnation (total) pressure port facing directly into the flow.
2. A static pressure port perpendicular to the flow (or on the body surface away from the stagnation point).

For subsonic flow, the Mach number is calculated from the ratio $p_0/p$ using the isentropic relation. For supersonic flow, a bow shock forms ahead of the Pitot tube, so the Rayleigh Pitot tube formula (which accounts for the normal shock loss) is used instead. A Mach meter in an aircraft cockpit performs this calculation automatically from the measured pressures.

Limitations and uncertainties in Mach number measurement

• Pitot-static tubes lose accuracy in the transonic range ($M \approx 0.8$ to 1.2) because shock waves interact with the probe and the flow is unsteady. Probe alignment errors also introduce uncertainty.
• Optical techniques (schlieren, shadowgraph) are excellent for flow visualization but require careful calibration and reference data to extract quantitative Mach numbers.
• PSP has spatial resolution limits and can be affected by temperature variations (since luminescence depends on both pressure and temperature).
• LDV and PIV depend on seed particles faithfully following the flow, which can be an issue across shock waves where particles lag behind rapid velocity changes.

Proper calibration, alignment, and uncertainty analysis are necessary for any Mach number measurement to be reliable.