On algebraic second moment models

Equilibrium and bifurcation analysis is used to explore algebraic second moment models. It is shown that the three-dimensional, explicit algebraic stress solution for the anisotropy tensor precludes rotational stabilization unless two invariants of the mean velocity gradient vanish. If these vanish the irrotational part of the flow must be a plane strain: essentially the model can only bifurcate and stabilize in two-dimensional mean flow. However, it is also shown that those same two invariants must vanish if the mean flow is steady. The full equilibrium analysis described herein provides a consistent picture of a model with equilibria that respond appropriately to rotation. However, if the algebraic stress approximation is used as a constitutive equation, without imposing full equilibrium, the bifurcation criterion [GRAPHICS] will not be met in three-dimensional flow. Hence the model cannot bifurcate to the stable solution branch. Similarly, ad hoc non-linear constitutive formulas that do not satisfy the bifurcation criterion preclude rotational stabilization. The bifurcation criterion is a simple and powerful guidance to turbulence model formulations.