# Kato-Launder modification

### From CFD-Wiki

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 many 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 the over-production of turbulent energy in stagnation regions and regions with very strong acceleration is eliminated.

## Applicability

The Kato-Launder modification can be used together with most two-equation models that have a production term formulated as above. The modification was originally developed for transient simulations of vortex-shedding begind square cylinders, where the normal k-epsilon model tends to produce too much turbulent energy in stagnation regions and in the small regions with strong acceleration and decelleration around the square corners. This over-production create too much turbulent viscosity which in turns affects the vortex-shedding and the development of the vortex-street downstream of the square cylinder. With the modified production term Kato and Launder was able to produce much better results.

The Kato-Launder modification has also been popular in the turbomachinery field. Where the stagnation-point-problem of two-euqation models can lead to significant errors. In turbomachinery application it is also common with regions with very high accelerations and decellerations (shocks, suction-side peaks, ...) where the Kato-Launder modification can help a two-equation model which predicts too much turbulence.

## 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.