Knudsen Number - Relationship To Mach and Reynolds Numbers in Gases

Relationship To Mach and Reynolds Numbers in Gases

The Knudsen number can be related to the Mach number and the Reynolds number:

Noting the following:

Dynamic viscosity,

Average molecule speed (from Maxwell-Boltzmann distribution),

thus the mean free path,

dividing through by L (some characteristic length) the Knudsen number is obtained:

where

  • is the average molecular speed from the Maxwell–Boltzmann distribution,
  • T is the thermodynamic temperature,
  • μ is the dynamic viscosity,
  • m is the molecular mass,
  • kB is the Boltzmann constant,
  • ρ is the density, .

The dimensionless Mach number can be written:

where the speed of sound is given by

where

  • U is the freestream speed,
  • R is the Universal gas constant, (in SI, 8.314 47215 J K−1 mol−1),
  • M is the molar mass,
  • is the ratio of specific heats, and is dimensionless.

The dimensionless Reynolds number can be written:

Dividing the Mach number by the Reynolds number,

and by multiplying by ,

yields the Knudsen number.


The Mach, Reynolds and Knudsen numbers are therefore related by:

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