Dimensionless Numbers

Burger Number

• Definition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = \frac{c}{fr_0} } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = \sqrt{gH} } is the nonrotational baroclinic phase speed, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } is the inertial frequency under the f-plane approximation, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_0 } is the radius of basin. The Burger number is sometimes denotes as Bu.
• Interpretation: The ratio of stratification to rotation of a circular lake.
• Analysis: Small Burger numbers correspond to shallow water, wide lakes, or rotation dominated dynamics. Large Burger numbers correspond to deep water, narrow lakes, or stratification dominated dynamics.
• Example: Take a lake in the mid latitudes, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \approx 10^{-4} \frac{1}{s}} . If the lake has radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_0 = 1000 m} and a depth of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=10 m} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = \sqrt{g H} = \sqrt{9.81 \cdot 10} \approx 9.9 \frac{m}{s} } and so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = \frac{c}{fr_0} = 99} , which means a stratification dominated lake.

Froude Number

• Definition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Fr \propto \left[ \frac{\rho U^{2}/L}{\rho g}\right] ^{1/2} = \frac{U}{\sqrt{gL}} } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g } is the gravitational acceleration, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L } are the characteristic velocity and length scales.
• Interpretation: The square root of the ratio of inertia force to gravity force.
• Analysis: The Froude number describes different regimes of open channel flow, including hydraulic jumps. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Fr = 1 } describes critical flow; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Fr > 1 } describes supercritical flow; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Fr < 1 } describes subcritical flow. The Froude number is analogous to the Mach number in compressible flow.

Grashof Number

• Definition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Gr = \frac{g\beta\Delta TV}{\nu^2} } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g } is the gravitational acceleration, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta } is the volumetric thermal expansion coefficient, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta T } is the temperature difference, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } is the volume, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } is the kinematic viscosity.
• Interpretation: The ratio of buoyant forces to viscous forces.
• Analysis: When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Gr \gg 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = U/c } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U } is the characteristic velocity scale, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c } is the speed of sound.
• Interpretation: The ratio of inertia force to compressibility force.
• Analysis: Compressibility effects can be neglected if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M < 0.3 } .
• Example: The typical sound wave speed is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \approx 340 } m/s in air. For wind with a speed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = 10 } m/s, the Mach number is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \approx 0.03 } , and hence the compressibility is negligible.

Prandtl Number

• Definition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Pr = \nu/\kappa } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } is the viscosity, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa } is the thermal diffusivity.
• Interpretation: The ratio of momentum diffusivity (i.e. kinematic viscosity) to heat diffusivity.
• Example: The typical Prandtl number for water is around 7 for water (At 20 degrees Celsius). See also Schmidt number.

Rayleigh Number

• Definition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ra = \frac{g \alpha \Gamma d^4}{\kappa \nu} } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g } is the acceleration due to gravity, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha } is the coefficient of thermal expansion, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma } is the vertical temperature gradient of the background state, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d } is the depth of the fluid layer, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa } is the thermal diffusivity, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } is the kinematic viscosity.
• Interpretation: The ratio of the destabilizing effect of buoyancy force to the stabilizing effect of viscous force.

Reynolds Number

• Definition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Re = \frac{UL}{\nu} } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L } are the characteristic velocity and length scales, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.
• Example: Take water with a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu = 10^{-6} \frac{m^2}{s}} , and imagine a cube with side length 10 meters submerged in this water. The Reynolds number in this case is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Re=\frac{UL}{\nu} = \frac{10^2}{10^{-6}} = 10^8} , and we expect a turbulent flow.

Richardson Number (flux)

• Definition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: The flux Richardson number is related to the gradient Richardson number by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ri = \frac{\nu_T}{\kappa_T}Rf } , where the ratio Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_T/\kappa_T } is the turbulent Prandtl number.

Richardson Number (gradient)

• Definition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: The gradient Richardson number is used 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.

Rossby Number

• Definition: $Ro={\frac {U}{fL}}={\frac {1/f}{L/U}}$, 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, or the ratio of the planetary time scale to the time scale of fluid motion.
• Analysis: When $Ro\ll 1$, Coriolis effects dominate and the system approaches geostrophic balance.
• Example: The Coriolis frequency of Earth is $f=1$ day$^{-1}$ or $f\approx 10^{-4}$ s$^{-1}$ (for mid-latitudes). Assuming the characteristic speed of fluid flow is $U=0.1$ m/s, then the flow in a laboratory tank of length $L=1$ m has a Rossby number $Ro\approx 10^{3}$ and is not affected by the Coriolis force, whereas a flow on the ocean scale with $L=10$ km has a Rossby number $Ro\approx 0.1$ and will be affected by Coriolis effects.

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.
• Example: 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.

Wedderburn Number

• Definition: $W={\frac {gH^{2}}{Fr_{0}}}$, where $g$ is the gravitational acceleration, $H$ is the depth, $F$ is the wind forcing, and $r_{0}$ is the radius of basin.
• Interpretation: The ratio of stratification to wind forcing of a circular lake.
• Analysis: Small Wedderburn numbers correspond to large wind stress, wide lakes, and shallow water. Large Wedderburn numbers correspond to low wind stress, narrow lakes, and deep water.