Reynolds Stress Tensor and Turbulent Shear Stress

These stresses when applying a Time Average on the Navier stokes equations.

$$\overline{u_i^{\prime }{u}_j^{\prime }}=\left(\begin{array}{ccc}\overline{u^{\prime 2}}& \overline{u^{\prime }{v}^{\prime }}& \overline{u^{\prime }{w}^{\prime }}\\ \overline{u^{\prime }{v}^{\prime }}& \overline{v^{\prime 2}}& \overline{v^{\prime }{w}^{\prime }}\\ \overline{u^{\prime }{w}^{\prime }}& \overline{v^{\prime }{w}^{\prime }}& \overline{w^{\prime 2}}\end{array}\right)$$

They represent the average interactions between the fluctuating components.

Turbulent Shear Stress

$$\boxed{{\tau }_{ij}^{turb}=-\overline{u_i^{\prime }{u}_j^{\prime }}}$$