Non-Dimensional Quantities

The most common non-dimensional quantity used in turbulence modeling is the Reynolds number, which is the ratio of inertial over viscous forces. The following definitions are used throughout the theory of turbulence modeling:

Wall-distance Reynolds number:

where:

• $k$ is the turbulent kinetic energy.
• $d$ is the distance to the wall.
• $\nu$ is the kinematic viscosity.

Turbulent Reynolds number:

or:

or:

where:

• $\epsilon$ is the turbulent dissipation rate.
• $\omega$ is the specific dissipation rate.
• ${\mu }_t$ is the turbulent eddy viscosity.
• $\mu$ is the dynamic viscosity.

Kolmogorov Reynolds number:

Strain-rate Reynolds number:

where $S$ is given by Eqn. (1129).

Another useful number is the Rayleigh number, defined as:

where:

• $\Vert g_i\Vert$ is the norm of the gravity vector.
• $\beta$ is the thermal expansion coefficient.
• $\mathrm{\Delta }\text{T}$ is the temperature difference (typically between fluid and containing walls).
• $L_{ref}$ is the characteristic length.
• $\alpha$ is the thermal diffusivity.

$\alpha$ is usually defined as $\alpha =\frac{\nu }{\mathrm{Pr}}$, with $\mathrm{Pr}$ the Prandtl number defined as:

where:

• $C_p$ is the specific heat.
• $\kappa$ is the thermal conductivity.