Dimensionless Numbers

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 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 } .

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. 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: 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.

Richardson 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 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. Also called the gradient Richardson number.
• 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. 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.