On algebraic second moment models

Equilibrium and bifurcation analysis is used to explore algebraic second mo ment models. It is shown that the three-dimensional, explicit algebraic str ess solution for the anisotropy tensor precludes rotational stabilization u nless two invariants of the mean velocity gradient vanish. If these vanish the irrotational part of the flow must be a plane strain: essentially the m odel 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 f low is steady. The full equilibrium analysis described herein provides a co nsistent picture of a model with equilibria that respond appropriately to r otation. However, if the algebraic stress approximation is used as a constitutive eq uation, 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 f ormulas that do not satisfy the bifurcation criterion preclude rotational s tabilization. The bifurcation criterion is a simple and powerful guidance t o turbulence model formulations.