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Turbulence dissipation rate

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(Finally fixed the \nu and \equiv problems in the LaTeX rendering)
(Added compressible definition using shear stress)
 
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<math>\epsilon \, \equiv \, \nu \overline{\frac{\partial u_i'}{\partial x_k}\frac{\partial u_i'}{\partial x_k}}</math>
<math>\epsilon \, \equiv \, \nu \overline{\frac{\partial u_i'}{\partial x_k}\frac{\partial u_i'}{\partial x_k}}</math>
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For compressible flows the definition is most often slightly different:
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<math>\epsilon \, \equiv \, \frac{1}{\overline{\rho}} \overline{\tau_{ij} \frac{\partial u_i''}{\partial x_j}}</math>
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Where the viscous stress, <math>\tau_{ij}</math>, using Stokes law for mono-atomic gases, is given by:
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<math>\tau_{ij} = 2 \mu S^*_{ij}</math>
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where
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<math>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}</math>

Latest 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}

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