Momentum, vorticity and scalar transport in turbulence: the Taylor–Prandtl controversy

The ‘vorticity transport’ theory by G. I. Taylor states that, in two-dimensional (2-D) turbulent flows, it is not the momentum of the eddies which is conserved from one step of their random walk to the other (the so-called Reynolds–Prandtl analogy), but their vorticity, implying that the conservation laws for the time-averaged profiles for the velocity u and concentration of a passive scalar c must be different. This theory predicts that, across a 2-D wake or a jet, both fields (scaled by their maximal value) are exactly related to each other by u=c². We reexamine critically this problem on hand of several experiments with plane and round turbulent jets seeded with high and low diffusing scalars, and conclude that the microscopic equations for u and c are identical, but that the differences between the u - and c -fields is a genuine mixing problem, sensitive to the dimensionality of the flow and to the intrinsic diffusivity of the scalar D , through the Schmidt number ( Sc=ν /D ) dependence of the flow coarsening scale. We observe that u=cβ with β =2 in plane jets irrespective of $Sc$ , β =3/2 in round jets at $Sc=1$ and β =1 in round jets for Sc→ ∞ . We explain why, because measurements dating back to the 1930s–40s were all made for heat transport in air ( Sc≈ 1 ), agreement with Taylor‘s vision was only coincidental. The experiments and the new representation proposed here are strictly at odds with Reynolds’ analogy, although essentially an adaptation of it to eddies transporting momentum and mass, but liable to exchange mass with a smooth reservoir along their Brownian path.