SINOPSIS

The paper is an experimental and analytical investigation of the analogy between the variance of the passive scalar field, θ² and the dynamical field, in particular the total kinetic energy q² (the sum of variances of the three velocity components). The analogy between the transport equation of θ² and q² is most likely to be valid under the constraint of statistical homogeneity (negligible effect of pressure) and for a Schmidt (or Prandlt) number equal to 1 (molecular or viscous diffusion predominates in the same range of scales).
Experimental data were inferred from simultaneous hot- and cold-wire measurements in a slightly heated axisymmetric shear layer, for which similar initial and boundary conditions were imposed for both θ and q (i.e., the mean temperature and velocity gradient are both present).
We show clear experimental evidence that:
i) at large scales, for which the production is dominant (either shear or mean temperature gradient), the analogy between the kinetic energy and the temperature variance is satisfactory.
ii) the smallest scales, especially for locations where the magnitude of the mean temperature and velocity gradient decreases, become shear-independent, and the analogy is not tenable.
We provide an analytical explanation for this behaviour, based on a simple model which is reasonably wellvalidated against experimental data. It is shown that:
i) local isotropy is not necessary for the similarity θ−q to be valid;
ii) the main factor which allows the similarity to hold is the production term in the one-point kinetic energy budget equation. When only production is present, a simple closure of this term based on a Prandtl-type model leads to simple, analytical solutions and the similarity can be explained.
iii) when other effects (e.g., decay) are present, departures from similarity can occur.