Modelling the transport equation of the scalar structure function

A model for the third-order mixed velocity-scalar structure function -<((delta u)(delta phi)(2))over bar> (delta alpha represents the spatial increment of the quantity alpha; 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 <((delta phi)(2))over bar> . 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 differs significantly from 1. Nevertheless, the calculations capture the effect expected when varies. For example, as increases, a range of scales emerges where <((delta phi)(2))over bar> 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.

Automated summary

A model for closing the transport equation of the second-order moment of the scalar increment is proposed in this study. The model is based on a gradient-type hypothesis and an eddy-viscosity model that exploits the analogy between turbulent kinetic energy and a passive scalar. The findings show that the closure model agrees well with measurements and numerical simulations when the Prandtl number, pr, is not too different from 1, but the agreement deteriorates when the difference is significant. However, the calculations capture the expected effect when pr varies.