Dual scaling and the $2/3$ power law for the second-order scalar structure function in homogeneous turbulence with a mean scalar gradient

A dual scaling of the second-order scalar structure function $\overline {{(\delta \theta )}^2}$ , i.e. a scaling based on the Batchelor–Kolmogorov scales $\theta _B$ , $\eta$ and another based on $\theta '$ , $L$ , representative of the large-scale motion, is examined in the context of the transport equation for $\overline {{(\delta \theta )}^2}$ . Direct numerical simulation data over a relatively wide range of the Taylor microscale Reynolds number $Re_\lambda$ and a Schmidt number of order 1 in statistically stationary homogeneous isotropic turbulence with a uniform mean scalar gradient are used. It is observed that as $Re_\lambda$ increases, a dual scaling appears to emerge, where the scaling based on $\theta '$ , $L$ extends to increasingly smaller values of $r/L$ , where $r$ is the separation associated with the increment $ {{\delta \theta }}$ , while the scaling based on $\theta _B$ , $\eta$ extends to increasingly larger values of $r/\eta$ . This suggests that both scalings should eventually overlap over a range of scales as $Re_\lambda$ continues to increase. Further, it is shown that such a dual scaling leads to the power-law relation $\overline {{(\delta \theta )}^2} \sim r^{\zeta _2}$ , where $\zeta _2=2/3$ in the overlap region. The use of an empirical model for the local slope of $\overline {{(\delta \theta )}^2}$ (i.e. $\zeta _2$ ) shows that a value of $Re_\lambda$ of order $10^4$ is required for the slope to first reach the value $2/3$ . Clearly, values larger than $10^4$ will be required before a $r^{2/3}$ inertial range is established.