Dimensionless Numbers

Grashof Number

• Definition: $Gr={\frac {g\beta \Delta TV}{\nu ^{2}}}$, where $g$ is the gravitational acceleration, $\beta$ is the volumetric thermal expansion coefficient, $\Delta T$ is the temperature differrence, $V$ is the volume, and $\nu$ is the kinematic viscosity.
• Interpretation: The ratio of buoyant forces to viscous forces.
• Analysis: When $Gr>>1$, the viscous force is negligible compared to the buoyancy and inertial forces, and the flow starts a transition to the turbulent regime.

Mach Number

• Definition: $M=U/c$, where $U$ is the characteristic velocity scale, and $c$ is the speed of sound.
• Interpretation: The ratio of inertia force to compressibility force.
• Analysis: Compressibility effects can be neglected if $M<0.3$.

Prandtl Number

• Definition: $Pr=\nu /\kappa$, where $\nu$ is the viscosity, and $\kappa$ is the thermal diffusivity.
• Interpretation: The ratio of momentum diffusivity (i.e. kinematic viscosity) to heat diffusivity.
• Analysis: The typical Prandtl number for water is around 7 for water (At 20 degrees Celsius). See also Schmidt number.

Reynolds Number

• Definition: $Re={\frac {UL}{\nu }}$, where $U$ and $L$ are the characteristic velocity and length scales, and $\nu$ is the kinematic viscosity.
• Interpretation: The ratio of inertia force to viscous force.
• Analysis: Small Reynolds numbers are often associated with viscous flows, whereas large Reynolds numbers are typically found in turbulent flows.

Richardson Number (flux)

• Definition: $Rf={\frac {-g\alpha {\overline {wT'}}}{-{\overline {uw}}U_{z}}}$, where the overlines denote the ensemble averages of variables.
• Interpretation: The ratio of the buoyant destruction of turbulent kinetic energy to the shear production.
• Analysis: It is related to the gradient Richardson number by $Ri={\frac {\nu _{T}}{\kappa _{T}}}Rf$, where the ratio $\nu _{T}/\kappa _{T}$ is the turbulent Prandtl number.

Richardson Number (gradient)

• Definition: $Ri=N^{2}/U_{z}^{2}$, where $N$ is the buoyancy frequency, and $U$ 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, $Ri<0.25$ 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.

Rossby Number

• Definition: $Ro={\frac {U}{fL}}$, where $U$ is the characteristic velocity, and $f$ is the Coriolis frequency, and $L$ is the characteristic length scale.
• Interpretation: The ratio of the advective forces to the Coriolis pseudo-forces.
• Analysis: When $Ro\ll 1$, Coriolis effects dominate and the system approaches geostrophic balance.

Schmidt Number

• Definition: $Sc=\nu /\kappa$, where $\nu$ is the viscosity, and $\kappa$ is the mass diffusivity.
• Interpretation: The ratio of momentum diffusivity (i.e. kinematic viscosity) to mass diffusivity.
• Analysis: 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.