Dimensionless Numbers

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Burger Number
  • Definition:  S = \frac{c}{fr_0} , where  c = \sqrt{gH} is the nonrotational baroclinic phase speed,  f is the inertial frequency under the f-plane approximation, and  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 f \approx 10^{-4} \frac{1}{s}. If the lake has radius r_0 = 1000 m and a depth of H=10 m, then c = \sqrt{g H} = \sqrt{9.81 \cdot 10} \approx 9.9 \frac{m}{s} and so  S = \frac{c}{fr_0} = 99, which means a stratification dominated lake.
Froude Number
  • Definition:  Fr \propto \left[ \frac{\rho U^{2}/L}{\rho g}\right] ^{1/2} = \frac{U}{\sqrt{gL}} , where  g is the gravitational acceleration, and  U and  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.  Fr = 1 describes critical flow;  Fr > 1 describes supercritical flow;  Fr < 1 describes subcritical flow. The Froude number is analogous to the Mach number in compressible flow.
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 difference,  V is the volume, and  \nu is the kinematic viscosity.
  • Interpretation: The ratio of buoyant forces to viscous forces.
  • Analysis: When  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:  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 .
  • Example: The typical sound wave speed is  c \approx 340 m/s in air. For wind with a speed  U = 10 m/s, the Mach number is  M \approx 0.03 , and hence the compressibility is negligible.
Prandtl Number
  • Definition:  Pr = \nu/\kappa , where  \nu is the kinematic viscosity, and  \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:  Ra = \frac{g \alpha \Gamma d^4}{\kappa \nu} , where  g is the acceleration due to gravity,  \alpha is the coefficient of thermal expansion,  \Gamma is the vertical temperature gradient of the background state,  d is the depth of the fluid layer,  \kappa is the thermal diffusivity, and  \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:  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.
  • Example: Take water with a  \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  Re=\frac{UL}{\nu} = \frac{10^2}{10^{-6}} = 10^8, and we expect a turbulent flow.
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: The flux Richardson number 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: 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  \mathrm{day}^{-1} or  f \approx 10^{-4} \mathrm{s}^{-1} (for mid-latitudes). Assuming the characteristic speed of fluid flow is  U = 0.1 \mathrm{m}/\mathrm{s}, then a flow in a laboratory tank of length  L = 1 \mathrm{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 \mathrm{km} has a Rossby number  Ro \approx 0.1 and will be affected by Coriolis effects.
Schmidt 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.