Glossary

13.1 Speed of Sound and Mach Number

3 min read•july 19, 2024

Sound travels through fluids at different speeds, depending on the medium's properties. The is about 343 m/s, while in water it's much faster at 1,480 m/s. Temperature, , and all affect sound speed.

, the ratio of flow velocity to local sound speed, helps categorize flow regimes. It's crucial in aerodynamics, determining whether a flow is , , , or . This impacts vehicle design and performance in high-speed applications.

Speed of Sound and Mach Number

Speed of sound in fluids

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  • Speed at which small pressure disturbances travel through a fluid medium
  • For ideal gases, calculated using the equation: c=γRTc = \sqrt{\gamma R T}
    • γ\gamma: ratio of specific heats (heat capacity ratio) represents the fluid's compressibility (air at standard conditions: 1.4)
    • RR: gas-specific constant depends on the gas composition (air: 287 J/kg·K)
    • TT: absolute temperature of the gas affects molecular motion and energy (room temperature: 293 K)
  • Depends on fluid compressibility more compressible fluids have lower sound speeds (water: 1,480 m/s, air: 343 m/s)
  • Increases with temperature due to higher molecular motion and energy transfer (air at 20℃: 343 m/s, air at 100℃: 386 m/s)
  • Decreases with molecular weight heavier molecules have slower sound propagation (helium: 1,007 m/s, carbon dioxide: 259 m/s)

Calculation of Mach number

  • Dimensionless ratio of flow velocity to local speed of sound represents compressibility effects
  • Defined as: M=VcM = \frac{V}{c}
    • VV: flow velocity in the medium (aircraft speed, wind tunnel velocity)
    • cc: local speed of sound depends on fluid properties and temperature
  • Calculation steps:
    1. Determine flow velocity (VV) from given information or flow equations (pitot tube measurement, numerical simulation)
    2. Calculate local speed of sound (cc) based on fluid properties and temperature (, experimental data)
    3. Divide flow velocity by local speed of sound to obtain Mach number (subsonic: < 0.8, supersonic: > 1.2)

Flow regimes vs Mach number

  • Subsonic flow: M<0.8M < 0.8
    • Flow velocity below local sound speed minimal compressibility effects
    • small often treated as incompressible (low-speed wind tunnels, propeller aircraft)
  • Transonic flow: 0.8≤M≤1.20.8 \leq M \leq 1.2
    • Flow velocity near sound speed significant compressibility effects
    • may form leading to abrupt changes in flow properties (transonic aircraft, high-speed wind tunnels)
  • Supersonic flow: 1.2<M<51.2 < M < 5
    • Flow velocity exceeds sound speed compressibility effects dominate
    • Shock waves present leading to discontinuities in flow properties (supersonic aircraft, rocket nozzles)
  • Hypersonic flow: M≥5M \geq 5
    • Flow velocity greatly exceeds sound speed extreme compressibility and high-temperature effects
    • Strong shock waves and complex gas dynamics observed (hypersonic vehicles, reentry vehicles)

Significance of Mach number

  • Determines flow regime and applicable equations/assumptions (incompressible vs )
  • Indicates importance of compressibility effects (negligible at low Mach, significant at high Mach)
  • Affects shock wave formation and behavior (location, strength, and impact on flow properties)
  • Influences design of vehicles and devices in compressible flow (aerodynamics, propulsion, high-speed vehicles)
    • Subsonic aircraft design focuses on lift generation and drag reduction (wing shape, streamlining)
    • Supersonic aircraft design considers shock wave management and heat transfer (swept wings, thermal protection)
    • Rocket nozzle design optimizes expansion and thrust generation based on Mach number (converging-diverging nozzle)

Key Terms to Review (22)

Bulk Modulus: Bulk modulus is a measure of a material's resistance to uniform compression, defined as the ratio of volumetric stress to the change in volume strain. It reflects how much pressure is needed to compress a given volume of material and is crucial for understanding how sound propagates through different media and influences Mach number calculations.
Compressibility: Compressibility is the measure of a fluid's ability to change its volume in response to a change in pressure. This property is crucial because it affects how fluids behave under varying pressure conditions, impacting their density and flow characteristics. Understanding compressibility helps in analyzing phenomena such as sound propagation and shock waves, particularly in gases where the changes in density due to pressure variations are more pronounced.
Compressible flow: Compressible flow refers to the behavior of fluids when their density changes significantly due to variations in pressure and temperature. This phenomenon typically occurs in high-speed flows, such as those encountered in aerodynamics and gas dynamics, where the effects of compressibility cannot be ignored. Understanding compressible flow is essential for analyzing the performance of aircraft, rockets, and other systems operating at high velocities.
Density: Density is the mass per unit volume of a substance, typically expressed in units like kg/m³. It plays a crucial role in determining how fluids behave under various conditions, influencing buoyancy, pressure distribution, and flow characteristics.
Density variations: Density variations refer to changes in the mass per unit volume of a substance, which can significantly affect the behavior of fluids under different conditions. In fluid mechanics, these variations play a crucial role in understanding phenomena like buoyancy, stability, and the propagation of sound waves. Changes in density can be influenced by temperature, pressure, and phase changes, which are essential factors when analyzing fluid flow and related concepts such as the speed of sound and Mach number.
Humidity effect: The humidity effect refers to the influence of moisture content in the air on the speed of sound and the Mach number. Increased humidity lowers the density of air, which leads to an increase in the speed of sound. This is significant because as humidity changes, it can affect various aerodynamic calculations and the performance of aircraft, particularly when flying at high speeds.
Hypersonic: Hypersonic refers to speeds that exceed Mach 5, which is five times the speed of sound in air at sea level. At these speeds, the dynamics of fluid flow change dramatically, leading to significant aerodynamic heating and pressure effects that influence vehicle design and stability. Understanding hypersonic flight is essential for advancements in aerospace engineering and defense applications.
Ideal Gas Equation: The ideal gas equation is a fundamental relation in thermodynamics that describes the behavior of an ideal gas. It combines several gas laws into one equation: PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is the absolute temperature. Understanding this equation is crucial for analyzing the speed of sound in gases and determining the Mach number, as both concepts rely on the behavior of gas under varying conditions.
Incompressible Flow: Incompressible flow refers to a fluid flow where the fluid density remains essentially constant throughout the motion. This concept is crucial in fluid mechanics as it simplifies many calculations, especially when analyzing the behavior of liquids, which typically exhibit little to no compressibility under standard conditions. Recognizing a flow as incompressible allows for the application of specific equations and principles, streamlining the analysis of fluid dynamics.
Isaac Newton: Isaac Newton was a groundbreaking physicist and mathematician whose work laid the foundation for classical mechanics, especially through his laws of motion and universal gravitation. His insights into the behavior of objects in motion significantly advanced the understanding of fluid dynamics, particularly regarding the behavior of fluids in motion and the forces acting on them.
John William Strutt (Lord Rayleigh): John William Strutt, known as Lord Rayleigh, was a prominent British physicist and engineer who made significant contributions to the study of sound and acoustics, among other areas. His work in the late 19th century laid the foundation for understanding the speed of sound in various media and the concept of the Mach number, which describes the relationship between an object's speed and the speed of sound in the surrounding medium.
Mach Number: Mach number is a dimensionless quantity representing the ratio of the speed of an object to the speed of sound in the surrounding medium. This term is crucial in understanding fluid dynamics, particularly when analyzing compressible flows, shock waves, and various flow regimes that occur in different fluid mechanics applications.
Mach Number Formula: The Mach number formula is a dimensionless quantity that represents the ratio of the speed of an object moving through a fluid to the speed of sound in that same fluid. It is a crucial concept in fluid mechanics and aerodynamics, helping to categorize the flow regime of an object as subsonic, transonic, supersonic, or hypersonic based on its velocity relative to the local speed of sound.
Molecular weight: Molecular weight is the mass of a single molecule of a substance, typically measured in atomic mass units (amu) or grams per mole (g/mol). It plays a crucial role in determining the behavior of gases, including their speed of sound and the Mach number, as these properties are influenced by how heavy or light the molecules are. Understanding molecular weight helps explain why different gases can travel at varying speeds through a medium.
Shock Waves: Shock waves are abrupt changes in pressure, temperature, and density that occur when an object moves through a medium at a speed greater than the speed of sound in that medium. This phenomenon creates a wavefront that travels outward from the source, characterized by a steep gradient and leading to significant changes in local properties. Understanding shock waves is essential in fields like aerodynamics, where they play a crucial role in high-speed flows and interactions with surrounding media.
Sonic boom: A sonic boom is a loud explosive sound caused by the shock waves created when an object travels through the air at a speed greater than the speed of sound. This phenomenon occurs when an object, such as an aircraft, breaks the sound barrier, producing pressure waves that coalesce and create a powerful noise that can be heard on the ground.
Speed of sound in air: The speed of sound in air is the distance traveled by a sound wave in a unit of time, typically measured in meters per second (m/s). This speed is influenced by factors such as temperature, humidity, and atmospheric pressure, and it serves as a critical parameter in fluid mechanics, particularly in the study of compressible flows and shock waves.
Speed of sound in water: The speed of sound in water refers to the rate at which sound waves travel through water, which is influenced by various factors including temperature, salinity, and pressure. Sound travels faster in water than in air due to the higher density and incompressibility of water, allowing sound waves to transmit energy more efficiently. This concept is crucial for understanding underwater acoustics and has implications in fields such as marine biology, sonar technology, and fluid dynamics.
Subsonic: Subsonic refers to speeds that are less than the speed of sound in a given medium, typically air at sea level, which is about 343 meters per second (or 1,125 feet per second). This speed range is significant in fluid mechanics as it relates to the behavior of fluids when objects move through them at lower velocities, leading to different aerodynamic characteristics compared to supersonic or hypersonic flows.
Supersonic: Supersonic refers to speeds that exceed the speed of sound in a given medium, typically air, which is approximately 343 meters per second (1,125 feet per second) at sea level under standard conditions. When an object travels at supersonic speeds, it creates shock waves, resulting in phenomena such as sonic booms. This term is important for understanding various applications in aerodynamics, such as the design of aircraft and missiles that can travel faster than sound.
Temperature dependence: Temperature dependence refers to how certain properties or behaviors of materials change with variations in temperature. In the context of fluid dynamics, particularly concerning the speed of sound and Mach number, this term highlights how the speed of sound in a fluid is influenced by temperature changes, affecting various applications like aerodynamics and thermodynamics.
Transonic: Transonic refers to the range of speeds that are around the speed of sound, typically defined as speeds between Mach 0.8 and Mach 1.2. In this regime, the flow characteristics change significantly due to the presence of both subsonic and supersonic flow patterns. This unique condition can lead to phenomena such as shock waves and changes in lift and drag on objects moving through the fluid.
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