# Kato-Launder modification

### From CFD-Wiki

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</math> | </math> | ||

- | In incompressible flows, where <math>\frac{\partial u_i}{\partial x_i} = 0</math> the production term <math>P</math> can be rewritten as: | + | In incompressible flows, where <math>\frac{\partial u_i}{\partial x_i} = 0</math>, the production term <math>P</math> can be rewritten as: |

:<math> | :<math> | ||

\begin{matrix} | \begin{matrix} | ||

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

- | \ & = & \left[ 2 \mu_t S_{ij}^* - \frac{2}{3} \rho k \delta_{ij} | + | \ & = & \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} + | \ & = & \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) | + | \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} |

+ | \right] \frac{\partial u_i}{\partial x_j} \\ | ||

+ | \ & \approx & \mbox{ (assuming incompressible flow)} \\ | ||

+ | \ & \approx & \mu_t \left(\frac{\partial u_i}{\partial x_j} + | ||

+ | \frac{\partial u_j}{\partial x_i} \right) \frac{\partial u_i}{\partial x_j} \\ | ||

+ | \ & = & | ||

\end{matrix} | \end{matrix} | ||

</math> | </math> |

## Revision as of 15:21, 8 December 2005

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, , used in most two-equation models can be written as:

Where is the turbulent production normally given by:

is the turbulent shear stress tensor given by the Boussinesq assumption:

Where is the eddy-viscosity given by the turbluence model and is the trace-less viscous strain-rate defined by:

In incompressible flows, where , the production term can be rewritten as:

## References

**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.