On the Taylor-Prandtl controversy

In 1932, G. I. Taylor wrote a fascinating article suggesting that, in two dimensional turbulent flows at least, 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, and that therefore the conservation equations for the velocity u and the concentration of a passive scalar c must be different. Taylor's `vorticity transport' theory thus predicts that, in a 2D wake or a jet, the c-profile across the jet (scaled by its maximal value) is exactly twice as large as the axial u-profile (i. e. u(r)=c(r)²).

We reexamine critically this problem on hand of several experiments with plane (2D) and round (3D) turbulent jets seeded with high and low diffusing scalars, and conclude that the microscopic equations for u and c have no reasons to be different in general, but that the difference between the u and c-fields is a genuine mixing problem. Denoting Sc=ν/D the Schmidt number of the scalar, we find that u=c^1+1/√Sc is quantitatively consistent with all the observations we have on hand to date. Earlier measurements dating back two the 1930’s-40’s were all made for heat transport in air with Sc≈1, leading indeed to u=c², an agreement with Taylor’s vision which is only coincidental. The relation above is grounded in a coupling, by molecular diffusion, between the eddies (lamellae, sheets) and the bath in which they move. This observation is also strictly at odd with Reynolds' analogy, although essentially an adaptation of it to eddies transporting momentum and mass, but liable to exchange mass with a smooth reservoir as they move along their brownian path.