Mach Number

The Mach Number is a dimensionless value useful for analyzing fluid flow dynamics problems where compressibility is a significant factor.

The Mach Number can be expressed as

M = v / c                                  (1)

where

M = Mach number

v = fluid flow speed (m/s, ft/s)

c = speed of sound (m/s, ft/s)

Alternatively the Mach Number can be expressed with the density and the bulk modulus for elasticity as

M = v (ρ / E)^1/2                            (2)

where

ρ = density of fluid (kg/m³, lb/ft³)

E = bulk modulus elasticity (N/m²(Pa), lb_f/in² (psi))

The bulk modulus elasticity has the dimension pressure and is commonly used to characterize the fluid compressibility.

The square of the Mach number is the Cauchy Number.

M² = C                                (3)

where

C = Cauchy Number

Subsonic and Supersonic speed

• If the mach number is < 1, the flow speed is lower than the speed of sound - and the speed is subsonic.
• If the mach number is ~ 1, the flow speed is approximately like the speed of sound - and the speed is transonic.
• If the mach number is > 1, the flow speed is higher than the speed of sound - and the speed is supersonic.
• If the mach number is >> 1, the flow speed is much higher than the speed of sound - and the speed is hypersonic.

Example - Calculating an Aircraft Mach Number

An aircraft flies at speed 500 mph at an altitude of 35000 ft. The surrounding temperature is -60 ^oF.

The speed of sound at this altitude and temperature can be calculated

c = (k R T)^1/2

    = (1.4 (1716 ft lb/slug ^oR) ((-60 ^oF) + (460 ^oR)))^1/2

    = 980 ft/s

where

k = 1.4

R = 1716 (ft lb/slug ^oR)

The speed of the aircraft can be calculated as

v = (500 miles/hr) (5280 ft/miles) / (3600 sec/hr)

    = 733 ft/sec

The Mach Number can be calculated as

M = (733 ft/s) / (980 ft/s)

= 0.75 - the aircraft is flying at subsonic speed