# Kato-Launder modification

### From CFD-Wiki

Line 1: | Line 1: | ||

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 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. | ||

+ | |||

+ | ==Basic equations== | ||

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

Line 64: | Line 66: | ||

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

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

+ | </math> | ||

+ | |||

+ | ==Ad-hoc production term modification== | ||

+ | |||

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

+ | |||

+ | Hence | ||

+ | |||

+ | :<math> | ||

+ | P = \mu_t S S | ||

+ | </math> | ||

+ | |||

+ | Where | ||

+ | |||

+ | :<math> | ||

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

+ | </math> | ||

+ | |||

+ | and | ||

+ | |||

+ | :<math> | ||

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

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

## Revision as of 15:41, 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.

## 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

## Ad-hoc production term modification

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

Hence

Where

and

## 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.