# Mach Number

## Introduction to the 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), lbf/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

## Related Topics

• Fluid Mechanics - The study of fluids - liquids and gases. Involves velocity, pressure, density and temperature as functions of space and time

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