Eddy Viscosity or Boussinesq Hypothesis

$${\tau }_{ij}^{\text{turb }}=2v_t{\bar{S}}_{ij}-\frac{2}{3}k{\delta }_{ij}=v_t\left(\frac{\partial {\bar{u}}_i}{\partial x_j}+\frac{\partial {\bar{u}}_j}{\partial x_i}\right)-\frac{2}{3}k{\delta }_{ij}$$

For constant viscosity

$${\tau }_{ij}^{\text{turb }}=v_t\left(\frac{\partial {\bar{u}}_i}{\partial x_j}\right)-\frac{2}{3}k{\delta }_{ij}$$

Where,

• ${\tau }_{ij}^{\text{turb }}\to$ Reynolds Stress Tensor
• $v_t\to$ Eddy Viscosity or Turbulent Viscosity
• $S_{ij}\to$ Mean Strain Rate Tensor
• $k\to$ Turbulent Kinetic Energy
• $\delta \to$ Kronecker Delta

Note: $k$ and $v_t$ are left are left as unknowns. Separate models are needed for them.

See: 240814 Viscous Stress Tensor for a Newtonian Fluid