Evolution of passive scalar statistics in a spatially developing turbulence

We investigate the evolution of passive scalar statistics in a spatially developing turbulence using direct numerical simulation. Turbulence is generated by a square grid element, which is heated continuously, and the passive scalar is temperature. The square element is the fundamental building block for both regular and fractal grids. We trace the dominant mechanisms responsible for the dynamical evolution of scalar-variance and its dissipation along the bar and grid-element centerlines. The scalar-variance is generated predominantly by the action of the mean scalar gradient behind the bar and is transported laterally by turbulent fluctuations to the grid-element centerline. The scalar-variance dissipation (proportional to the scalar-gradient variance) is produced primarily by the compression of the fluctuating scalar-gradient vector by the turbulent strain rate, while the contribution of mean velocity and scalar fields is negligible. Close to the grid element the scalar spectrum exhibits a well-defined $-5/3$ power-law, even though the basic premises of the Kolmogorov-Obukhov-Corrsin theory are not satisfied (the fluctuating scalar field is highly intermittent, inhomogeneous, and anisotropic, and the local Corrsin-microscale-Péclet number is small). At this location, the PDF of scalar gradient production is only slightly skewed towards positive, and the fluctuating scalar-gradient vector aligns only with the compressive strain-rate eigenvector. The scalar-gradient vector is stretched or compressed stronger than the vorticity vector by turbulent strain rate throughout the grid-element centerline. However, the alignment of the former changes much earlier in space than that of the latter, resulting in scalar-variance dissipation to decay earlier along the grid-element centerline compared to the turbulent kinetic energy dissipation. The universal alignment behavior of the scalar-gradient vector is found far downstream, although the local Reynolds and Péclet numbers (based on the Taylor and Corrsin length scales, respectively) are low.

Figure 1

Sketch of the computational domain: (a) front view, (b) side view at plane $\mathrm{z}/t_0=0$ (sketches not to scale).

Figure 2

Contours of (a) instantaneous temperature and (b) mean temperature fields. The isocontours range from 1 to 1.$5T_{\infty }$. The white dotted line in (b) is the isoline of normalized mean vorticity vector magnitude $(\langle |{\mathbf{\omega }}^{*}|\rangle t_0/U_{\infty })$ of 0.18. The black dotted-dashed line is the centerline of the grid element, and the bar centerline is represented by a pink dotted-dashed line.

Figure 3

(a) Evolution of scalar-variance and turbulent kinetic energy along the bar centerline; (b) evolution of budget terms of scalar-variance along the bar centerline. All terms are normalized by $U_{\infty }{T}_{\infty }^2/t_0$.

Figure 4

(a) Evolution of scalar-variance and turbulent kinetic energy along the grid-element centerline; (b) evolution of budget terms of scalar-variance along the grid-element centerline. All terms are normalized by $U_{\infty }{T}_{\infty }^2/t_0$. The transport due to fluctuations of the turbulent kinetic energy is also shown.

Figure 5

(a) Evolution of scalar-gradient variance and enstrophy along the bar centerline; (b) evolution of budget terms of scalar-gradient variance along the bar centerline. All terms are normalized by $U_{\infty }{T}_{\infty }^2/t_0^3$.

Figure 6

(a) Evolution of scalar-gradient variance and enstrophy along the grid-element centerline; (b) evolution of budget terms of scalar-gradient variance along the grid-element centerline. All the terms are normalized by $U_{\infty }{T}_{\infty }^2/t_0^3$.

Figure 7

(a) Spectra of the fluctuating scalar at different locations along the grid-element centerline; (b) compensated spectra plotted in linear-logarithmic axes at the same locations. The dotted lines in (a) indicate the $-5/3$ slope, and in (b) they represent a constant value of $E_{TT}{(f{t}_0/U_{\infty })}^{5/3}$.

Figure 8

PDF of scalar gradient production at different locations along the grid-element centerline: (a) $x/x^{*}=0.25$, (b) $x/x^{*}=0.5$, (c) $x/x^{*}=0.95$.

Figure 9

JPDF of eigenvalues of strain-rate tensor against the scalar gradient production: (a)–(c) at $x/x^{*}=0.25$ (near-grid-element inhomogeneous region), (d)–(f) at $x/x^{*}=0.95$ (decay region). The isocontours range from ${10}^1$ to ${10}^{-1}$.

Figure 10

PDF of cosine of the angle between scalar-gradient vector and strain-rate eigenvectors at different locations along the grid-element centerline: (a) $x/x^{*}=0.25$, (b) $x/x^{*}=0.35$, (c) $x/x^{*}=0.5$. The dotted lines in (c) are the results from Ref. [21].

Figure 11

(a) PDF of cosine of the angle between fluctuating scalar gradient and vorticity vectors at different locations along the grid-element centerline; (b), (c) comparison between alignments of instantaneous versus fluctuating scalar gradient and vorticity vectors at (b) $x/x^{*}=0.25$, (c) $x/x^{*}=0.5$.

Figure 12

PDF of eigencontribution to scalar gradient production at different locations along the grid-element centerline: (a) $x/x^{*}=0.25$, (b) $x/x^{*}=0.5$, (c) $x/x^{*}=0.95$.

Figure 13

Evolution of turbulent kinetic energy and scalar-variance dissipations along (a) the bar centerline and (b) the grid-element centerline. (c) Evolution of the time-averaged quantities $\chi$ and $\mathrm{{\rm Y}}$ (defined in the text) along the grid-element centerline.

Figure 14

PDF of scalar gradient production rate and enstrophy production rate at different locations along the grid-element centerline: (a) $x/x^{*}=0.25$, (b) $x/x^{*}=0.5$, (c) $x/x^{*}=0.95$.