Statistics of incremental averages of passive scalar fluctuations

Whereas statistics of differences in turbulent quantities measured over a given separation have been extensively studied, statistics of the incremental averages of the same quantities (e.g., $\Sigma \theta \equiv [\theta(x+r) + \theta(x)]/2$) have only been the focus of recent research. The present work studies incremental averages of a fluctuating passive scalar (temperature) in nearly homogeneous, isotropic turbulence, generated by an active grid, with an imposed mean scalar gradient. Following the arguments of Mouri and Hori ({\it Phys. Fluids}, 2010) for the velocity field, we derive a scale-dependent budget for the incremental average of the scalar field fluctuations, $\Sigma \theta$. We discuss its relationship to Yaglom's four-thirds law (which pertains to differences of passive scalar fluctuations) and compare the results with the experimental data. Furthermore, the statistics of $\Sigma \theta$ are compared with those of the incremental averages of the velocity fluctuations ($\Sigma u_\alpha$).