Modelling the transport equation of the scalar structure function

Abstract

A model for the third-order mixed velocity–scalar structure function $-\overline {(\delta u)( \delta \phi )^{2}}$ ( $\delta a$ represents the spatial increment of the quantity $a$ ; $u$ is the velocity and $\phi$ is a scalar) is proposed for closing the transport equation of the second-order moment of the scalar increment $\overline {( \delta \phi )^{2}}$ . 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, $Pr$ , is equal to 1. The solutions of the closed transport equation agree well with both measurements and numerical simulations, when $Pr$ is not too different from 1. However, the agreement deteriorates when $Pr$ differs significantly from 1. Nevertheless, the calculations capture the effect expected when $Pr$ varies. For example, as $Pr$ increases, a range of scales emerges where $\overline {( \delta \phi )^{2}}$ remains constant. This emerging scaling range should correspond to the $k^{-1}$ spectral range in the three-dimensional scalar spectrum commonly denoted the viscous–convective range. Also when $Pr$ decreases below 1, the calculations reproduce the emergence of an expected inertial–diffusive range for scales larger than the Kolmogorov length scale.