Modelling the transport equation of the scalar structure function

A model for the third-order mixed velocity-scalar structure function (represents the spatial increment of the quantity; is the velocity and is a scalar) is proposed for closing the transport equation of the second-order moment of the scalar increment. The closure model is based on a gradient-type hypothesis with an eddy-viscosity model which exploits the analogy between the turbulent kinetic energy and a passive scalar when the Prandtl number, is equal to 1. The solutions of the closed transport equation agree well with both measurements and numerical simulations, when is not too different from 1. However, the agreement deteriorates when differs significantly from 1. Nevertheless, the calculations capture the effect expected when varies. For example, as increases, a range of scales emerges where remains constant. This emerging scaling range should correspond to the spectral range in the three-dimensional scalar spectrum commonly denoted the viscous-convective range. Also when decreases below 1, the calculations reproduce the emergence of an expected inertial-diffusive range for scales larger than the Kolmogorov length scale.