# Turbulence dissipation rate

(Difference between revisions)
 Revision as of 10:21, 13 June 2011 (view source)Peter (Talk | contribs)m (moved Turbulence dissipation to Turbulence dissipation rate: more complete name)← Older edit Latest revision as of 01:00, 11 April 2015 (view source) (Abbreviations for units no longer italicized.) (One intermediate revision not shown) Line 1: Line 1: - Turbulence dissipation, $\epsilon$ is the rate at which [[turbulence kinetic energy]] is converted into thermal internal energy. The SI unit of $\epsilon$ is $J / kg s = m^2 / s^3$. + Turbulence dissipation, $\epsilon$ is the rate at which [[turbulence kinetic energy]] is converted into thermal internal energy. The SI unit of $\epsilon$ is $\mathrm{J} / (\mathrm{kg} \cdot \mathrm{s}) = \mathrm{m}^2 / \mathrm{s}^3$. $\epsilon \, \equiv \, \nu \overline{\frac{\partial u_i'}{\partial x_k}\frac{\partial u_i'}{\partial x_k}}$ $\epsilon \, \equiv \, \nu \overline{\frac{\partial u_i'}{\partial x_k}\frac{\partial u_i'}{\partial x_k}}$ + + + For compressible flows the definition is most often slightly different: + + $\epsilon \, \equiv \, \frac{1}{\overline{\rho}} \overline{\tau_{ij} \frac{\partial u_i''}{\partial x_j}}$ + + Where the viscous stress, $\tau_{ij}$, using Stokes law for mono-atomic gases, is given by: + + $\tau_{ij} = 2 \mu S^*_{ij}$ + + where + + $S^*_{ij} \equiv \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}$

## Latest revision as of 01:00, 11 April 2015

Turbulence dissipation, $\epsilon$ is the rate at which turbulence kinetic energy is converted into thermal internal energy. The SI unit of $\epsilon$ is $\mathrm{J} / (\mathrm{kg} \cdot \mathrm{s}) = \mathrm{m}^2 / \mathrm{s}^3$.

$\epsilon \, \equiv \, \nu \overline{\frac{\partial u_i'}{\partial x_k}\frac{\partial u_i'}{\partial x_k}}$

For compressible flows the definition is most often slightly different:

$\epsilon \, \equiv \, \frac{1}{\overline{\rho}} \overline{\tau_{ij} \frac{\partial u_i''}{\partial x_j}}$

Where the viscous stress, $\tau_{ij}$, using Stokes law for mono-atomic gases, is given by:

$\tau_{ij} = 2 \mu S^*_{ij}$

where

$S^*_{ij} \equiv \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}$