Mach number
Introduction
The Mach number (M) is a dimensionless quantity in fluid dynamics representing the ratio of the speed of an object moving through a fluid to the local speed of sound in that fluid. Named after the Austrian physicist and philosopher Ernst Mach, the Mach number is a critical parameter in aerodynamics, aeronautics, and astronautics, influencing the behavior of objects moving at high velocities.
Definition and Calculation
The Mach number is defined as:
\[ M = \frac{v}{a} \]
where:
- \( v \) is the velocity of the object relative to the fluid,
- \( a \) is the speed of sound in the fluid.
The speed of sound in a fluid is determined by the properties of the medium, including temperature and pressure. For an ideal gas, the speed of sound \( a \) can be calculated using the formula:
\[ a = \sqrt{\gamma \cdot R \cdot T} \]
where:
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( R \) is the specific gas constant,
- \( T \) is the absolute temperature of the gas.
Classification of Flow Regimes
The Mach number is used to classify different flow regimes:
- **Subsonic**: \( M < 1 \)
- **Transonic**: \( 0.8 < M < 1.2 \)
- **Supersonic**: \( 1 < M < 5 \)
- **Hypersonic**: \( M > 5 \)
Each regime has distinct characteristics and implications for the behavior of the fluid and the object moving through it.
Subsonic Flow
In subsonic flow, the Mach number is less than 1. The flow is characterized by smooth streamlines and relatively low compressibility effects. The aerodynamic forces and pressure distributions are primarily governed by the principles of incompressible flow.
Transonic Flow
Transonic flow occurs when the Mach number is close to 1, typically between 0.8 and 1.2. This regime is marked by the coexistence of both subsonic and supersonic flow regions. Shock waves and expansion fans can form, leading to complex flow patterns and significant changes in pressure and density.
Supersonic Flow
In supersonic flow, the Mach number exceeds 1. The flow is dominated by compressibility effects, and shock waves are a common feature. The behavior of the fluid and the aerodynamic forces are significantly influenced by the presence of these shock waves, which cause abrupt changes in pressure, temperature, and density.
Hypersonic Flow
Hypersonic flow occurs at Mach numbers greater than 5. This regime is characterized by extremely high temperatures and significant chemical reactions within the fluid. The aerodynamic heating and thermal stresses become critical factors in the design of vehicles operating in this regime, such as re-entry vehicles and high-speed aircraft.
Applications in Aerodynamics
The Mach number is a fundamental parameter in the design and analysis of aircraft, missiles, and spacecraft. It influences various aspects of aerodynamic performance, including lift, drag, stability, and control.
Aircraft Design
In aircraft design, the Mach number determines the aerodynamic characteristics and performance of the vehicle. For example, commercial airliners typically operate in the transonic regime to balance fuel efficiency and travel time. Supersonic and hypersonic aircraft, such as the Concorde and experimental vehicles like the X-15, are designed to withstand the extreme conditions associated with high Mach numbers.
Missile and Spacecraft Design
Missiles and spacecraft often operate in the supersonic and hypersonic regimes. The Mach number is crucial in designing these vehicles to ensure they can withstand the aerodynamic forces and thermal loads encountered during flight. For example, the design of re-entry vehicles must account for the intense heating and deceleration experienced at hypersonic speeds.
Shock Waves and Expansion Fans
Shock waves and expansion fans are phenomena associated with supersonic and hypersonic flows. Shock waves are characterized by sudden increases in pressure, temperature, and density, while expansion fans involve gradual decreases in these properties.
Normal and Oblique Shock Waves
Shock waves can be classified into normal and oblique types. Normal shock waves occur perpendicular to the flow direction and result in a significant reduction in Mach number, while oblique shock waves occur at an angle to the flow and cause a change in flow direction.
Expansion Fans
Expansion fans, also known as Prandtl-Meyer expansions, occur when a supersonic flow turns around a convex corner. These fans involve a series of expansion waves that decrease the pressure, temperature, and density of the flow while increasing the Mach number.
Mach Angle and Mach Cone
The Mach angle (\( \mu \)) is the angle between the direction of the flow and the shock wave generated by an object moving at supersonic speeds. It is given by:
\[ \sin(\mu) = \frac{1}{M} \]
The Mach cone is the conical region formed by the shock waves emanating from a supersonic object. The angle of the Mach cone is equal to the Mach angle, and it defines the region within which the effects of the shock wave are felt.
Effects of Compressibility
Compressibility effects become significant at high Mach numbers, leading to changes in the density and pressure of the fluid. These effects are particularly important in the transonic and supersonic regimes, where shock waves and expansion fans play a crucial role in the behavior of the flow.
Prandtl-Glauert Singularity
The Prandtl-Glauert singularity is a phenomenon that occurs in transonic flow, where the pressure coefficient becomes infinite at a specific Mach number. This singularity is associated with the formation of shock waves and rapid changes in the flow properties.
Area Rule
The area rule is a design principle used to minimize drag in transonic and supersonic aircraft. It states that the cross-sectional area of the aircraft should change smoothly along its length to reduce the formation of shock waves and associated drag. This principle was first applied in the design of the Convair F-102 Delta Dagger.
Mach Number in Different Media
The Mach number is not limited to air but can be applied to any fluid, including water and other gases. The speed of sound varies with the medium, affecting the Mach number and the associated flow characteristics.
Water and Other Liquids
In water, the speed of sound is much higher than in air, approximately 1500 m/s at room temperature. This results in different Mach numbers for objects moving through water compared to air. For example, a submarine traveling at 30 m/s has a Mach number of approximately 0.02 in water, indicating subsonic flow.
Other Gases
The Mach number can also be used in other gases, such as hydrogen or helium. The speed of sound in these gases varies with their molecular properties, affecting the Mach number and the behavior of the flow. For example, the speed of sound in helium is approximately 1000 m/s, leading to different flow characteristics compared to air.
Historical Development
The concept of the Mach number was developed in the early 20th century as scientists and engineers sought to understand the behavior of high-speed flows. Ernst Mach's work on shock waves and supersonic flow laid the foundation for the development of this important parameter.
Early Research
Early research on high-speed flows focused on understanding the behavior of projectiles and the effects of compressibility. The development of wind tunnels and other experimental techniques allowed scientists to study these phenomena in greater detail.
Supersonic and Hypersonic Flight
The advent of supersonic and hypersonic flight in the mid-20th century brought new challenges and opportunities for the application of the Mach number. The development of supersonic aircraft, such as the Bell X-1, and hypersonic vehicles, like the North American X-15, demonstrated the importance of understanding and controlling high-speed flows.