# Kato-Launder modification

### From CFD-Wiki

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==Production term modification== | ==Production term modification== | ||

- | The proposal by Kato and Launder is to replace one of the strain-rates, <math>S</math>, in the turbulent production term with the vorticity, <math>\Omega</math>. | + | The proposal by Kato and Launder is to replace one of the strain-rates, <math>S</math>, in the turbulent production term with the vorticity, <math>\Omega</math>. The Kato-Launder modified production then becomes: |

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

- | P = \mu_t S | + | P = \mu_t S \Omega |

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

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\Omega \equiv \sqrt{\frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right)^2 } | \Omega \equiv \sqrt{\frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right)^2 } | ||

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

+ | |||

+ | ==Discussion== | ||

In pure shear-flows like boundary-layers and wakes the Kato-Launder modified production term will give exactly the same result as the unmodified production term. However, outside of boundary-layers and wakes the Kato-Launder modified production term will give very different results. Essentially what it does is to turn off the turbulent production outside of the boundary-layers and wakes. This has the good effect that | In pure shear-flows like boundary-layers and wakes the Kato-Launder modified production term will give exactly the same result as the unmodified production term. However, outside of boundary-layers and wakes the Kato-Launder modified production term will give very different results. Essentially what it does is to turn off the turbulent production outside of the boundary-layers and wakes. This has the good effect that |

## Revision as of 15:54, 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.

## Contents |

## Basic equations

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:

Hence

Where

## Production term modification

The proposal by Kato and Launder is to replace one of the strain-rates, , in the turbulent production term with the vorticity, . The Kato-Launder modified production then becomes:

Where

and

## Discussion

In pure shear-flows like boundary-layers and wakes the Kato-Launder modified production term will give exactly the same result as the unmodified production term. However, outside of boundary-layers and wakes the Kato-Launder modified production term will give very different results. Essentially what it does is to turn off the turbulent production outside of the boundary-layers and wakes. This has the good effect that

## 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.