Stress

  • 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.