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12.7.3 Modeling Turbulent Diffusive Transport

$D_{T,ij}$ can be modeled by the generalized gradient-diffusion model of Daly and Harlow [ 75]:

 D_{T,ij} = C_s \; \frac{\partial}{\partial x_k} \left( \rh... ...partial \overline{u'_{i} u'_{j}}}{\partial x_{\ell}}\right) (12.7-2)

However, this equation can result in numerical instabilities, so it has been simplified in FLUENT to use a scalar turbulent diffusivity as follows [ 208]:
 D_{T,ij}= \frac{\partial}{\partial x_k} \left ( \frac{\mu_... ...c{\partial \overline{u'_{i} u'_{j}}}{\partial x_k}\right ) (12.7-3)

The turbulent viscosity, $\mu_t$, is computed using Equation  12.7-33.

Lien and Leschziner [ 208] derived a value of $\sigma_{k}= 0.82$ by applying the generalized gradient-diffusion model, Equation  12.7-2, to the case of a planar homogeneous shear flow. Note that this value of $\sigma_{k}$ is different from that in the standard and realizable $k$- $\epsilon$ models, in which $\sigma_{k} = 1.0$.

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