13.1 Speed of Sound and Mach Number

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

Mach number, the ratio of flow velocity to local sound speed, helps categorize flow regimes. It's crucial in aerodynamics, determining whether a flow is subsonic, transonic, supersonic, or hypersonic. 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: $c = \sqrt{\gamma R T}$
  • $\gamma$: ratio of specific heats (heat capacity ratio) represents the fluid's compressibility (air at standard conditions: 1.4)
  • $R$: gas-specific constant depends on the gas composition (air: 287 J/kg·K)
  • $T$: 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: $M = \frac{V}{c}$
  • $V$: flow velocity in the medium (aircraft speed, wind tunnel velocity)
  • $c$: local speed of sound depends on fluid properties and temperature
• Calculation steps:
  1. Determine flow velocity ($V$) from given information or flow equations (pitot tube measurement, numerical simulation)
  2. Calculate local speed of sound ($c$) based on fluid properties and temperature (ideal gas equation, 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: $M < 0.8$
  • Flow velocity below local sound speed minimal compressibility effects
  • Density variations small often treated as incompressible (low-speed wind tunnels, propeller aircraft)
• Transonic flow: $0.8 \leq M \leq 1.2$
  • Flow velocity near sound speed significant compressibility effects
  • Shock waves may form leading to abrupt changes in flow properties (transonic aircraft, high-speed wind tunnels)
• Supersonic flow: $1.2 < M < 5$
  • Flow velocity exceeds sound speed compressibility effects dominate
  • Shock waves present leading to discontinuities in flow properties (supersonic aircraft, rocket nozzles)
• Hypersonic flow: $M \geq 5$
  • 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 compressible flow)
• 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)