Isotropic vs Anisotropic Turbulence

Anisotropic Turbulence

Here the diagonal components of the Reynolds Stresses have different values. This means that there is a preference for the direction of fluctuations i.e. non-Diagonal components have non-Zero values.

Isotropic Turbulence

Here the Reynolds Normal Stresses are equal to each other. Therefore the fluctuations do not have a preferred directions and the non-Diagonal components are zero.

Turbulent Kinetic Energy

$$k=\frac{1}{2}(\overline{u^{\prime 2}}+\overline{v^{\prime 2}}+\overline{w^{\prime 2}})=\frac{1}{2}\overline{u_i^{\prime 2}}$$