Shear-free turbulent diffusion—classical and new scaling laws

We consider the problem of turbulence generation at a vibrating grid in the x₂–x₃ plane. Turbulence diffuses in the x₁-direction. Analyzing the multi-point correlation equation using Lie-group analysis, we find three different invariant solutions (scaling laws): classical diffusion-like solution (heat equation like), decelerating diffusion-wave solution and finite domain diffusion due to rotation. All solutions have been obtained using Lie-group (symmetry) methods. It is shown that if only one spatial dimension is considered, models based on Reynolds averaging are only capable to model either the diffusion-like solution or the decelerating diffusion-wave solution. The latter solution is only admitted under certain algebraic constraints on the model constants; e.g. in case of the K– model the model constants need to obey the relation c₂σ/σ_K = 2. Turbulent diffusion on a finite domain induced by rotation is not admitted by any of the classical models. Finally, in the appendix it is shown that Lele's transformation (Phys. Fluids 28(1) (1985) 64) leads to a complete analytic solution of the steady diffusion problem modelled by the K– equation.