Dimensionless Quantities

Dimensionless Quantity	Definition	Description	Associated Regime
Reynolds	\( Re = \frac{\rho V L}{\mu} \)	Ratio between inertial and viscous forces	Stokes / steady or unsteady laminar / turbulent
Prandtl	\( Pr = \frac{\nu}{\alpha} = \frac{c_p \mu}{k} \)	Ratio between momentum diffusivity and thermal diffusivity	Momentum or thermal diffusivity driven flows
Rayleigh	\( Ra = \frac{\rho \beta \Delta T g L^3}{\mu \alpha} \)	Ratio between thermal diffusion and thermal convection time scales	Associated with laminar or turbulent free convection flows (buoyancy-driven); can also be relative to pure conduction regimes or mixed convection flows
Nusselt	\( Nu = \frac{h L}{k} \)	At a boundary, ratio between convective and conductive heat transfer	Pure conduction \( (<= 1) \), laminar or turbulent regime above
Atwood	\( A = \frac{\rho_{heavy} - \rho_{light}}{\rho_{heavy} + \rho_{light}} \)	Normalized relative density ratio, mostly used for studying the Rayleigh-Taylor instability	Same density \( = 0 \); massless light fluid \( = 1 \)

With:

\( L \): characteristic length ( \( m \))
\( V \): velocity ( \( m.s^{-1} \))
\( T \): temperature ( \( K \))
\( \beta \): thermal expansion coefficient ( \( K^{-1} \))
\( \rho \): density ( \( kg.m^{-3} \))
\( \mu \): dynamic viscosity ( \( Pa.s \equiv kg.m^{-1}.s^{-1} \))
\( \nu = \frac{\mu}{\rho} \): kinematic viscosity
\( c_p \): specific heat ( \( J.kg^{-1}.K^{-1} \))
\( k \): thermal conductivity ( \( W.m^{-1}.K^{-1} \))
\( \alpha = \frac{k}{\rho c_p} \): thermal diffusivity
\( h = \frac{q}{\Delta T} \): heat transfer coefficient ( \( W.m^{-2}.K^{-1} \)), with \( q \) the heat flux ( \( W.m^{-2} \))
\( g \): gravity constant ( \( m.s^{-2} \))