- Viscous and turbulent stress tensors are symmetric.
- This means their eigenvectors are orthogonal.
- So, you can rotate the coordinate system to make the stress tensor diagonal
- The trace (sum of diagonal elements) is the sum of the eigenvalues, and is independent of the coordinate system orientation.
- The mean stress is
$$\frac{1}{3}σ_{i,i} =\frac{1}{3}(σ_{1,1} + σ_{2,2} + σ_{3,3}).$$
- Let $Π = \frac{1}{3}σ_{k,k}δ_{i,j}$.
- Then (σ - Π) is called the deviatoric stress with trace = 0.
- This effectively subtracts off the mean stress, (or the mean normal stress).
- The mean $Π$ is associated with dilatation.
- $σ-Π$ is associated with deformation.
- The $Π$ part is like a pressure and is absorbed into the pressure gradient term.
- The $σ-Π$ part is responsible for the transport of momentum.
- This is what is modeled by the Smagorinski model.

- See Turbulent Flows by S.B. Pope Appendix B, and page. 88, 581.