Dimensionless Numbers

Reynolds Number

$Re={\frac {UL}{\nu }}$, where $U$ and $L$ are the characteristic velocity and length scales, and $\nu$ is the dynamic viscosity. It measures the ratio of the inertial force to the viscous force. Small Reynolds numbers are often associated with viscous flows, whereas large Reynolds numbers are typically found in turbulent flows.

Richardson Number

$Ri=N^{2}/U_{z}^{2}$, where $N$ is the buoyancy frequency and $U$ is the background horizontal velocity. It measures the ratio between the strength of the stratification and the velocity shear. It is a well known criterion for determining the linear stability of an inviscid stratified flow. A sufficient condition for the flow to be linearly stable is that the local Richardson number exceed 0.25 throughout the flow. However, $Ri<0.25$ 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 $Ri$ is not clear.

Schmidt Number

$Sc=\nu /\kappa$, where $\nu$ is the viscosity, and $\kappa$ is the diffusivity. It measures the ratio between the momentum diffusivity (i.e. viscosity) and the mass diffusivity. The typical Schmidt number for water is around 500, depending on the temperature and salinity. For direct numerical simulations, $Sc=1$ is commonly used in the literature. See also Prandtl number.