# Turbulence dissipation rate

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 Revision as of 19:34, 12 June 2011 (view source)Peter (Talk | contribs) (Finally fixed the \nu and \equiv problems in the LaTeX rendering)← Older edit Revision as of 10:56, 13 June 2011 (view source)Peter (Talk | contribs) (Added compressible definition using shear stress)Newer edit → (One intermediate revision not shown) Line 2: Line 2: $\epsilon \, \equiv \, \nu \overline{\frac{\partial u_i'}{\partial x_k}\frac{\partial u_i'}{\partial x_k}}$ $\epsilon \, \equiv \, \nu \overline{\frac{\partial u_i'}{\partial x_k}\frac{\partial u_i'}{\partial x_k}}$ + + + 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 + + $S^*_{ij} \equiv \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}$

## Revision as of 10:56, 13 June 2011

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

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

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

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