Introduction and definition of 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)
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)
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)
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 Aircrafts Mach Number
An aircraft flies at speed 500 mph at an altitude of 35.000 ft. The surrounding temperature is -60 oF.
The sound speed at this altitude with this temperature can be calculated as
c = [k R T]1/2
= [ 1.4 (1716 ft lb/slug oR) ((-60 oF) + (460 oR)) ]1/2
= 980 ft/s
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