Realizability of Reynolds-Stress Turbulence Models

Abstract

It is shown that certain existing phenomenological models of turbulence in terms of differential equations for the Reynolds stresses R_α_β=(ul_α ul_β) do not guarantee realizable solutions. The known realizability conditions are R_α_β ≥ 0 for α=β and R²_α_β ≥ R_α_αR_β_β for α≠β. A stronger requirement is that the matrix R_α_β be positive semi-definite. This implies three conditions like non-negative eigenvalues or non-negative principal invariants. Conditions are given which must be satisfied by the model itself in order to guarantee realizable solutions for any realizable initial and boundary conditions. Some means are proposed which can be used to change the existing models into realizable ones.