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Kato-Launder modification

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The Kato-Launder modification is an ad-hoc modification of the turbulent production term in the k equation. The main purpose of the modification is to reduce the tendency that two-equation models have to over-predict the turbulent production in regions with large normal strain, i.e. regions with strong acceleration or decelleration.

The transport equation for the turbulent energy, k, used in most two-equation models can be written as:

\frac{\partial}{\partial t} \left( \rho k \right) +
\frac{\partial}{\partial x_j} 
 \rho k u_j - \left( \mu + \frac{\mu_t}{\sigma_k} \right) 
 \frac{\partial k}{\partial x_j}
P - \rho \epsilon - \rho D

Where P is the turbulent production normally given by:

P = \tau_{ij}^{turb} \frac{\partial u_i}{\partial x_j}

\tau_{ij}^{turb} is the turbulent shear stress tensor given by the Boussinesq assumption:

\tau_{ij}^{turb} \equiv 
- \overline{\rho u''_i u''_j} \approx
2 \mu_t S_{ij}^* - \frac{2}{3} \rho k \delta_{ij}

Where \mu_t is the eddy-viscosity given by the turbluence model and S_{ij}^* is the trace-less viscous strain-rate defined by:

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}

In incompressible flows, where \frac{\partial u_i}{\partial x_i} = 0 the production term P can be rewritten as:

P & = & \tau_{ij}^{turb} \frac{\partial u_i}{\partial x_j} \\
\ & = & \left[ 2 \mu_t S_{ij}^* - \frac{2}{3} \rho k \delta_{ij}] \right] \frac{\partial u_i}{\partial x_j} \\
\ & = & \left[ 2 \mu_t \left( \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} \right)
 \right] \frac{\partial u_i}{\partial x_j} - \frac{2}{3} \rho k \delta_{ij}


Kato, M. and Launder, B. E. (1993), "The Modeling of Turbulent Flow Around Stationary and Vibrating Square Cylinders", Proc. 9th Symposium on Turbulent Shear Flows, Kyoto, August 1993, pp. 10.4.1-10.4.6.

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