# Kato-Launder modification

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

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

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:

$\begin{matrix} P & = & \tau_{ij}^{turb} \frac{\partial u_i}{\partial x_j} \\ \ & = & \left[ 2 \mu_t S_{ij}^* - \frac{2}{3} \rho k \delta_{ij} \right] \frac{\partial u_i}{\partial x_j} \\ \ & = & \left[ 2 \mu_t \left( \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij} \right] \frac{\partial u_i}{\partial x_j} \\ \ & \approx & \mu_t \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \frac{\partial u_i}{\partial x_j} \\ \ & = & \mu_t \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \\ \end{matrix}$

Hence

$P = \mu_t S S$

Where

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

## Production term modification

The proposal by Kato and Launder is to replace one of the strain-rates, $S$, in the turbulent production term, $P$, above with the vorticity, $\Omega$.

The Kato-Launder modified production term then becomes:

$P = \mu_t S S$

Where

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

and

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

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 because in these flows the strain-rate is equal to the vorticity.

## 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.