Dimensionless Numbers

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Reynolds Number
  • Definition: , where and are the characteristic velocity and length scales, and is the dynamic viscosity.
  • Interpretation: The ratio of inertial force to viscous force.
  • Analysis: Small Reynolds numbers are often associated with viscous flows, whereas large Reynolds numbers are typically found in turbulent flows.
Gradient Richardson Number
  • Definition: , where is the buoyancy frequency and is the background horizontal velocity.
  • Interpretation: The ratio between the strength of stratification and velocity shear.
  • Analysis: A sufficient condition for the flow to be linearly stable is that the local Richardson number exceed 0.25 throughout the flow. However, does not mean the flow is necessarily unstable (the criterion is not sufficient). When the flow is not a parallel shear flow, the meaning of is not clear.
Schmidt Number
  • Definition: , where is the viscosity, and is the diffusivity.
  • Interpretation: The ratio between the momentum diffusivity (i.e. viscosity) and mass diffusivity.
  • Analysis: The typical Schmidt number for water is around 500, depending on the temperature and salinity. For direct numerical simulations, is commonly used in the literature. See also Prandtl number.