# Scalar dissipation

Scalar dissipation is a very important quantity in non-premixed combustion modelling. It often provides the connection between the mixing field and the combustion modelling. It is specially important in flamelet and RANS models.

In a laminar flow the scalar dissipation rate is defined (units are 1/s) as $\chi \equiv 2 D \left( \frac{\partial Z}{\partial x_j} \right)^2$

where $D$ is the diffusion coefficient of the scalar.

In turbulent flows, the scalar dissipation is seen as a scalar energy dissipation} and its role is to destroy (dissipate) scalar variance (scalar energy) analogous to the dissipation of the turbulent energy $\epsilon$. This term is known as the turbulent scalar dissipation and is written as $\chi_t \equiv 2 D \left( \frac{\partial \widetilde{Z''}}{\partial x_j} \right) ^2$

Opposite to the kinetic energy dissipation, most of the scalar dissipation occur at the finest scales.

In Conditional Moment Closure (CMC) and Flamelet based on conserved scalar models, the quantity of interest is the " main scalar dissipation rate", $\widetilde{\chi}$. From Favre Averaging the laminar dissipation rate $\widetilde{\chi} = 2D \widetilde{\left(\frac{\partial Z}{\partial x_j}\right) ^2} \approx 2 D \left( \frac{\partial \tilde{Z}}{\partial x_j} \right) ^2 + \chi_t$

Under RANS assumptions gradient of the scalar fluctuations are much larger than gradients of the means, and therefore the mean scalar dissipation rate is approximately the turbulent dissipation rate $\widetilde{\chi} \approx \chi_t$