Role of the Stress Tensor in RANS

By definition, RANS modelling averages or filters the temporal fluctuations in the flow. Thereby also filtering turbulent behaviour of the fluid.

Hence the average effect of the Turbulent behaviour is contributed to integral motion of the fluid through the Reynolds Stress Tensor in the momentum equation. It is represented as ${\tau }_{ij}^{turb}$.

The RANS Momentum Equation is given by

$$\boxed{\frac{\partial {\bar{u}}_i}{\partial t}+\frac{\partial {\bar{u}}_i{\bar{u}}_j}{\partial x_j}=-\frac{1}{\rho }\frac{\partial \bar{p}}{\partial x_i}+\frac{\partial }{\partial x_j}\left[v\frac{\partial {\bar{u}}_i}{\partial x_j}+{\tau }_{ij}^{turb}\right]}$$

where ${\tau }_{ij}^{turb}=-\overline{u_i^{\prime }{u}_j^{\prime }}$ represents the Reynolds Stresses