Statistics of incremental averages of passive scalar fluctuations

Whereas statistical moments of differences of turbulent quantities measured over a given separation (viz., structure functions) have been extensively studied, statistics of incremental sums (or equivalently averages, e.g., $\mathrm{\Sigma }\theta \equiv \left[\theta \right(x+r)+\theta (x\left)\right]/2$) of the same quantities have only been the subject of recent research. The present work investigates incremental averages of a turbulent passive scalar (temperature), measured in nearly homogeneous, and isotropic (passive and active), grid-generated turbulence, for turbulent Reynolds numbers in the range $94\le R_{\lambda }(\equiv u_{\mathrm{rms}}\lambda /\nu )\le 582$. The scalar field is generated by the action of the turbulent velocity field against an imposed mean temperature gradient. Following the approach of Mouri and Hori [Phys. Fluids 22, 115110 (2010)] for the velocity field, we examine statistics of incremental averages of the passive scalar field as a function of separation (viz., incremental average structure functions) for different Reynolds numbers, comparing them with both the results of Mouri and Hori, as well as the corresponding incremental average structure functions for the velocity field for the flows studied herein. While the statistics of $\mathrm{\Sigma }\theta$ are primarily large-scale quantities, and would therefore be expected to be flow dependent, they exhibit certain similarities to the statistics of incremental averages of velocity ($\mathrm{\Sigma }{u}_{\alpha }$), measured both in the flow under consideration as well as the different classes of flows studied by Mouri and Hori. Finally, we derive a scale-dependent evolution equation for the incremental average of the scalar field fluctuations, $\mathrm{\Sigma }\theta$. We discuss its relationship to Yaglom's four-thirds law for differences in passive scalar fluctuations and compare the results with the experimental data.

Figure 1

Scale-by-scale evolution of the terms in Eq. (9) at $R_{\lambda }=222$. (a) The terms are nondimensionalized by $\frac{4}{3}\langle {\epsilon }_{\theta }\rangle \eta$, which is a constant. (b) The terms are nondimensionalized by $\frac{4}{3}〈{\epsilon }_{\theta }r〉$, which is a nondimensionalization that is both a function of scale and analogous with Yaglom's equation.

Figure 2

Normalized (a) second- and (b) fourth-order incremental averages of $u$, $v$, and $\theta$. $R_{\lambda }=582$.

Figure 3

Reynolds number dependence of the normalized (a) second- and (b) fourth-order incremental average structure functions of temperature.

Figure 4

Kurtosis structure functions of incremental averages. (a) $\langle {\left(\mathrm{\Sigma }u\right)}^4\rangle /{\langle {\left(\mathrm{\Sigma }u\right)}^2\rangle }^2$, $\langle {\left(\mathrm{\Sigma }v\right)}^4\rangle /{\langle {\left(\mathrm{\Sigma }v\right)}^2\rangle }^2$, and $\langle {\left(\mathrm{\Sigma }\theta \right)}^4\rangle /{\langle {\left(\mathrm{\Sigma }\theta \right)}^2\rangle }^2$ for $R_{\lambda }=582$. (b) The Reynolds number dependence of $\langle {\left(\mathrm{\Sigma }v\right)}^4\rangle /{\langle {\left(\mathrm{\Sigma }v\right)}^2\rangle }^2$. (c) The Reynolds number dependence of $\langle {\left(\mathrm{\Sigma }\theta \right)}^4\rangle /{\langle {\left(\mathrm{\Sigma }\theta \right)}^2\rangle }^2$.

Figure 5

Normalized, mixed velocity-temperature second-order incremental average structure function $\left(\mathrm{\Sigma }v\mathrm{\Sigma }\theta \right)$.

Figure 6

Probability density function for the first-order structure function of (a) the incremental difference and (b) the incremental average at $R_{\lambda }=582$. Circles, $r=10\eta$; squares, $r=100\eta$; triangles, $r=1000\eta$; crosses, (a) $\mathrm{PDF}\left(\frac{\partial \theta }{\partial x}/〈{\left(\frac{\partial \theta }{\partial x}\right)}^2〉\right)$ and (b) $\mathrm{PDF}(\theta /{\theta }_{\mathrm{rms}})$. The solid lines are Gaussian fits to the (a) incremental difference at $r/\eta =1000$ (mean, 0; standard deviation, 1.1) and (b) incremental average at $r/\eta =1000$ (mean, $-0.01$; standard deviation, $-1.2$).

Figure 7

Joint probability density functions of (a) $u$ and $v$, (b) $u$ and $\theta$, (c) $v$ and $\theta$, (d) $\mathrm{\Sigma }u$ and $\mathrm{\Sigma }v$, (e) $\mathrm{\Sigma }u$ and $\mathrm{\Sigma }\theta$, and (f) $\mathrm{\Sigma }v$ and $\mathrm{\Sigma }\theta$. $R_{\lambda }=582$. $r/\eta =100$.

Figure 8

Scale dependence of the (nondimensionalized) expectations of (a) $\langle {\left(\mathrm{\Sigma }{u}_{\alpha }\right)}^2|\mathrm{\Delta }{u}_{\alpha }\rangle$, (b) $\langle {\left(\mathrm{\Sigma }\theta \right)}^2|\mathrm{\Delta }\theta \rangle$, and (c) $\langle {\left(\mathrm{\Sigma }\theta \right)}^2|\mathrm{\Delta }{u}_{\alpha }\rangle$. In (a) and (c), the solid line represents data for which $u_{\alpha }=u$ (the longitudinal velocity fluctuation), and the dashed line represents $u_{\alpha }=v$ (the transverse velocity fluctuation). $r=10\eta$ (bottom), $r=100\eta$ (middle, offset by 0.5), and $r=1000\eta$ (top, offset by 1.0). $R_{\lambda }=582$.

Figure 9

Reynolds-number dependence of the nondimensionalized expectations of (a) $\langle {\left(\mathrm{\Sigma }{u}_{\alpha }\right)}^2|\mathrm{\Delta }{u}_{\alpha }\rangle$, (b) $\langle {\left(\mathrm{\Sigma }\theta \right)}^2|\mathrm{\Delta }\theta \rangle$, and (c) $\langle {\left(\mathrm{\Sigma }\theta \right)}^2|\mathrm{\Delta }{u}_{\alpha }\rangle$. In (a) and (c), the solid line represents data for which $u_{\alpha }=u$ (the longitudinal velocity fluctuation), and the dashed line represents $u_{\alpha }=v$ (the transverse velocity fluctuation). $R_{\lambda }=94$ (bottom), $R_{\lambda }=306$ (middle, offset by 0.5), and $R_{\lambda }=582$ (top, offset by 1.0). $r/\eta =10$.

Figure 10

Scale-by-scale evolution of the terms in Eq. (26). $R_{\lambda }=222$.