Define Mach number.

Short Answer:

The Mach number is defined as the ratio of the velocity of a fluid (or object moving through a fluid) to the local speed of sound in that fluid. It is a dimensionless quantity that helps determine whether a flow is subsonic, transonic, supersonic, or hypersonic.

Mathematically, the Mach number is expressed as

where  is the velocity of the fluid or object, and  is the local speed of sound. The Mach number is widely used in aerodynamics and fluid mechanics to study compressible flow and high-speed motion.

Detailed Explanation:

Mach Number

The Mach number is a key parameter in compressible fluid flow, named after the Austrian physicist Ernst Mach, who studied the effects of high-speed motion on pressure waves. It expresses how fast an object or fluid is moving relative to the speed of sound in the same medium.

When a body moves through a fluid (like air), disturbances or pressure waves travel through the fluid at the speed of sound. The Mach number indicates how the speed of the moving body compares to this wave speed. Depending on its value, the flow behavior changes significantly, influencing pressure, temperature, and density around the moving object.

Definition and Formula

The Mach number (M) is defined as:

Where,

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

The speed of sound in a gas is given by:

Where,

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

From this, it is clear that the speed of sound depends on the temperature and nature of the gas. Therefore, the Mach number is influenced by both the flow velocity and the temperature of the surrounding fluid.

Physical Meaning of Mach Number

The Mach number represents the relative magnitude of the flow velocity compared to the speed of sound.

  • When : the object moves slower than sound waves (subsonic).
  • When : the object moves at exactly the speed of sound (sonic).
  • When : the object moves faster than sound waves (supersonic).

The Mach number determines whether pressure disturbances can move ahead of the object or are confined behind it. This understanding is essential in aerodynamics, nozzle design, and high-speed propulsion systems.

Classification of Flow Based on Mach Number

  1. Subsonic Flow (M < 1):
    • Flow velocity is less than the speed of sound.
    • Density changes are small, and the flow can often be treated as incompressible.
    • Example: Airflow around cars, ships, and low-speed aircraft.
  2. Sonic Flow (M = 1):
    • Flow velocity equals the speed of sound.
    • The flow experiences maximum disturbance, and the pressure waves cannot travel ahead of the moving object.
    • Example: Flow at the throat of a convergent-divergent nozzle.
  3. Transonic Flow (0.8 < M < 1.2):
    • Flow velocity is around the speed of sound.
    • Both subsonic and supersonic regions coexist, and shock waves begin to form.
    • Example: Flow over commercial jet aircraft wings.
  4. Supersonic Flow (1 < M < 5):
    • Flow velocity is greater than the speed of sound.
    • Shock waves form clearly, and strong changes in pressure, density, and temperature occur.
    • Example: Jet fighters and rockets.
  5. Hypersonic Flow (M > 5):
    • Flow velocity is much higher than the speed of sound.
    • Extremely high temperatures cause gas dissociation and ionization.
    • Example: Spacecraft reentry vehicles and missiles.

Importance of Mach Number

  1. Determines Flow Regime:
    • Helps classify flow as subsonic, transonic, or supersonic for analysis.
  2. Design of Aerodynamic Structures:
    • Used to design aircraft wings, nozzles, and diffusers suitable for specific speed ranges.
  3. Shock Wave Formation:
    • Predicts the formation and strength of shock waves in compressible flow.
  4. Performance of Compressible Devices:
    • Essential in designing compressors, turbines, and jet propulsion systems.
  5. Temperature and Pressure Relation:
    • At high Mach numbers, pressure and temperature variations become large and must be calculated for safety and performance.

Example Explanation

Consider an aircraft flying at a velocity of .
If the local speed of sound in air at that altitude is , then:

Hence, the aircraft is flying at Mach 1, i.e., at the speed of sound (sonic condition).

If the same aircraft flies at , then:

This means it is flying at Mach 2, which is supersonic speed.

Significance in Engineering Applications

  1. Aerodynamics:
    • The Mach number determines lift, drag, and stability of high-speed aircraft.
  2. Rocket and Jet Propulsion:
    • It affects exhaust velocity, nozzle design, and fuel efficiency.
  3. Compressible Flow Devices:
    • In nozzles and diffusers, Mach number governs pressure and velocity variations.
  4. Wind Tunnel Testing:
    • Wind tunnels are categorized by Mach number (subsonic, supersonic, or hypersonic).
  5. Shock and Expansion Waves:
    • Mach number helps predict the location and intensity of shocks or expansions in high-speed flow.
Conclusion

The Mach number is a fundamental parameter in fluid mechanics and aerodynamics that compares the flow velocity to the speed of sound in the medium. It helps classify flow regimes and predict the behavior of compressible fluids under different velocity conditions. The Mach number is critical in the design and analysis of high-speed systems such as aircraft, rockets, turbines, and nozzles. Understanding it enables engineers to ensure stable, efficient, and safe performance of machines and flow systems operating at various speeds.