Short Answer:

The Mach number is a dimensionless number that represents the ratio of the velocity of a fluid (or object) to the speed of sound in that fluid. It is denoted by M and is given by the formula:

where   is the velocity of the object or fluid and   is the speed of sound in that medium.

In simple words, the Mach number shows whether a flow is subsonic, sonic, or supersonic. If the flow velocity is less than the speed of sound, it is subsonic; if equal, it is sonic; and if greater, it is supersonic. This concept is widely used in aerodynamics and gas dynamics.

Detailed Explanation:

Mach Number

The Mach number (M) is one of the most important dimensionless quantities in fluid dynamics and aerodynamics. It helps in understanding how compressibility effects influence the behavior of fluids, especially gases, when they move at high speeds. The Mach number defines the relationship between the velocity of an object or flow and the velocity of sound in the same medium.

It was named after the Austrian physicist and philosopher Ernst Mach, who studied the effects of motion at high speeds and the behavior of shock waves in gases. The Mach number is primarily used in the study of compressible flow, aerodynamics, jet propulsion, and supersonic flight.

1. Definition of Mach Number

The Mach number is defined as:

“The ratio of the speed of a fluid or an object to the speed of sound in the same medium.”

Mathematically,

where,

•  = Mach number (dimensionless)
•  = velocity of the object or fluid (m/s)
•  = speed of sound in the medium (m/s)

Since both   and   have the same units, the Mach number has no dimension and is simply a ratio.

2. Speed of Sound in a Medium

The speed of sound (a) in a gas depends on the elastic properties and density of the gas and is given by:

where,

•  = ratio of specific heats (Cp/Cv)
•  = specific gas constant (J/kg·K)
•  = absolute temperature of the gas (K)

From this formula, it is clear that the speed of sound depends on the temperature of the medium — higher temperature leads to higher speed of sound.

For example, the speed of sound in air at 20°C is approximately 343 m/s.

3. Classification of Flow Based on Mach Number

The Mach number helps classify the flow regimes depending on the speed of the fluid relative to the speed of sound:

1. Subsonic Flow (M < 1):
   • Flow velocity is less than the speed of sound.
   • Compressibility effects are negligible.
   • Example: Airflow around cars or low-speed aircraft.
2. Sonic Flow (M = 1):
   • Flow velocity equals the speed of sound.
   • The flow is said to be at Mach 1.
   • Example: Tip of a propeller blade or jet nozzle throat at sonic conditions.
3. Supersonic Flow (1 < M < 5):
   • Flow velocity is greater than the speed of sound.
   • Compressibility effects are significant, and shock waves form.
   • Example: Jet aircraft like fighter planes.
4. Hypersonic Flow (M > 5):
   • Flow velocity is much greater than the speed of sound.
   • Extreme compressibility and high-temperature effects occur.
   • Example: Spacecraft re-entry or missiles.

Thus, the Mach number indicates how compressibility and pressure changes affect fluid motion.

4. Physical Meaning of Mach Number

The Mach number expresses how fast a body moves through a fluid compared to the speed at which pressure disturbances (sound waves) move through that same fluid.

• When M < 1 (Subsonic): The fluid can adjust smoothly to the motion of the object, as pressure disturbances move faster than the object.
• When M = 1 (Sonic): The object moves at the same speed as sound waves, creating strong interactions and possible shock waves.
• When M > 1 (Supersonic): The object moves faster than pressure waves, causing sudden changes in pressure and temperature — known as shock waves.

This ratio determines whether the flow can be treated as incompressible or compressible in engineering calculations.

5. Importance of Mach Number

The Mach number is highly significant in the study of gas and aerodynamic flows for several reasons:

1. Determines Flow Regime:
   It helps classify flows into subsonic, transonic, supersonic, and hypersonic regions, which influence design parameters in aircraft and rocket systems.
2. Compressibility Effects:
   For low Mach numbers (below 0.3), compressibility effects are negligible. At higher Mach numbers, compressibility and density changes must be considered.
3. Design of Nozzles and Diffusers:
   In nozzles and diffusers, flow area changes depend on whether the flow is subsonic or supersonic, and Mach number determines how the pressure and velocity vary.
4. Shock Wave Formation:
   At supersonic speeds, Mach number helps predict the location and strength of shock waves formed around the object.
5. Similarity in Model Testing:
   When performing aerodynamic tests using scale models, maintaining the same Mach number ensures dynamic similarity between the model and the prototype.

6. Example Calculation

Suppose an aircraft is flying at a velocity of 680 m/s at a height where the speed of sound is 340 m/s.

Hence, the Mach number is 2, which means the aircraft is flying at supersonic speed (twice the speed of sound).

7. Applications of Mach Number

• Aerospace Engineering:
  Used to classify and design aircraft, missiles, and spacecraft according to their speed ranges.
• Gas Dynamics:
  Helps analyze compressible flow in nozzles, diffusers, turbines, and compressors.
• Wind Tunnel Testing:
  Mach number is maintained constant between model and prototype to ensure similarity.
• Jet Propulsion:
  Determines performance and flow conditions in jet engines.
• Shock Wave Studies:
  Used to predict behavior of shock and expansion waves in supersonic and hypersonic flows.

8. Relation to Other Dimensionless Numbers

The Mach number is related to other important dimensionless numbers such as:

• Reynolds Number: Ratio of inertial to viscous forces.
• Froude Number: Ratio of inertial to gravitational forces.
• Euler Number: Ratio of pressure to inertial forces.

Together, these numbers describe different physical effects acting in a fluid flow.

Conclusion

In conclusion, the Mach number is a dimensionless parameter that represents the ratio of a fluid’s velocity to the speed of sound in that medium. It helps classify flow regimes as subsonic, sonic, supersonic, or hypersonic and plays a crucial role in understanding compressible flows, shock waves, and aerodynamic design. The Mach number is essential in aerospace, gas dynamics, and propulsion studies because it defines the behavior of fluids when velocity becomes comparable to or greater than the speed of sound.