Mach Number: Formula, Flow Regimes & Calculator

Overview

The Mach number ($Ma$) is the ratio of flow velocity to the local speed of sound. It classifies compressible flow regimes and determines when compressibility effects become significant.

Variables

Symbol	Description	Unit
$Ma$	Mach number	—
$v$	Flow velocity	m/s
$c$	Speed of sound	m/s
$\gamma$	Ratio of specific heats	—
$R$	Specific gas constant	J/(kg·K)
$T$	Absolute temperature	K

Formula

$Ma = \frac{v}{c} = \frac{v}{\sqrt{\gamma R T}}$

Calculator

Notes

• Results are approximate and should be verified for critical applications
• Input values should be within reasonable engineering ranges

Mach Number Classification

The Mach number is used to classify flow regimes based on the ratio of flow speed to the speed of sound:

• Subsonic: $M < 1$
• Transonic: $M \approx 1$
• Supersonic: $M > 1$
• Hypersonic: $M \gg 1$ (typically $M > 5$)

Example Calculation

An aircraft flies at 500 mph at an altitude of 35,000 ft where the surrounding temperature is -60 °F.

Step 1: Calculate the local speed of sound. $c = \sqrt{kRT} = \sqrt{1.4 \times 1716 \, \frac{\text{ft·lb}}{\text{slug·°R}} \times ((-60 + 460) \, \text{°R})} \approx 980 \, \text{ft/s}$ where $k=1.4$ (ratio of specific heats for air) and $R=1716 \, \frac{\text{ft·lb}}{\text{slug·°R}}$.

Step 2: Convert the aircraft speed to consistent units. $v = 500 \, \frac{\text{miles}}{\text{hr}} \times \frac{5280 \, \text{ft}}{\text{mile}} \times \frac{1 \, \text{hr}}{3600 \, \text{s}} \approx 733 \, \text{ft/s}$

Step 3: Calculate the Mach number. $M = \frac{v}{c} = \frac{733 \, \text{ft/s}}{980 \, \text{ft/s}} \approx 0.75$ The aircraft is flying at subsonic speed.

References

• Original Source