Fluctuations and correlations of reactive scalars near chemical equilibrium in incompressible turbulence

The statistical properties of species undergoing chemical reactions in a turbulent environment are studied. We focus on the case of reversible multicomponent reactions of second and higher orders, in a condition close to chemical equilibrium sustained by random large-scale reactant sources, while the turbulent flow is highly developed. In such a state a competition exists between the chemical reaction that tends to dump reactant concentration fluctuations and enhance their correlation intensity and the turbulent mixing that on the contrary increases fluctuations and removes relative correlations. We show that a unique control parameter, the Damköhler number (${\text{Da}}_{\theta }$), which can be constructed from the scalar Taylor microscale, the reactant diffusivity, and the reaction rate, characterizes the functional dependence of fluctuations and correlations in a variety of conditions, i.e., at changing the reaction order, and the Reynolds and Schmidt numbers. The larger is such a Damköhler number the more depleted are the scalar fluctuations as compared to the fluctuations of a passive scalar field in the same conditions, and vice versa the more intense are the correlations. A saturation in this behavior is observed beyond ${\text{Da}}_{\theta }\simeq O\left(10\right)$. We provide an analytical prediction for this phenomenon which is in excellent agreement with direct numerical simulation results.

Figure 1

(a) Evolution of the root mean square of scalar fluctuations and mean values for the case of $n=2$, $\text{Da}=0.1$, $\text{Sc}=1$, and ${\text{Re}}_{\lambda }=150$. Time is normalized by the integral time $k/\varepsilon$ with $k=3u^{\prime 2}/2$. The dashed vertical line marks the initial time for the computation of statistical quantities in this study. (b) PDF of scalar fields in dimensionless units (main panel) and normalized with respect to their standard deviations (inset, the black lines are Gaussian curves). For clarity of visualization, these curves are artificially shifted.

Figure 2

Correlation coefficients between $R_1$ and $R_2$ ($r(R_1,R_2)$), $R_1$ and $P$ ($r(R_1,P)$), $R_2$ and $P$ ($r(R_2,P)$) as functions of ${\text{Da}}_{\theta }$, under the conditions of (a) $n=1$ (b) $n=2$, and (c) $n=3$ and the Schmidt number $\text{Sc}=1$. Theoretical predictions are shown as black lines.

Figure 3

Correlation coefficients between $R_1$ and $R_2$ ($r(R_1,R_2)$), $R_1$ and $P$ ($r(R_1,P)$), $R_2$ and $P$ ($r(R_2,P)$) as functions of ${\text{Da}}_{\theta }$, under the conditions of (a) ${\text{Re}}_{\lambda }=20$, (b) ${\text{Re}}_{\lambda }=40$, and (c) ${\text{Re}}_{\lambda }=80$. The order of $R_2$ ($n$) is 1. Theoretical predictions are shown as black lines.

Figure 4

The fluctuations of the reactive scalars normalized by the fluctuation of the passive scalar ($T$) as functions of ${\text{Da}}_{\theta }$, under the conditions of (a) $n=1$, (b) $n=2$, and (c) $n=3$, and the Schmidt number $\text{Sc}=1$. Theoretical predictions are shown as black lines.

Figure 5

(a) Taylor microscales of scalars (computed on $T$) as functions of Sc. The red line is $0.3/\text{St}$. A power law fit of the form $a{\text{Sc}}^b$ on the ${\text{Re}}_{\lambda }=150$ data set gives $a=0.29\pm 0.1$ and $b=-0.93\pm 0.02$. (b) Taylor microscales of the reactive scalars with respect to the passive scalar with different orders of reaction, as functions of Da for all the simulations at ${\text{Re}}_{\lambda }=150$ and $\text{Sc}=1$.

Figure 6

(a) Correlation coefficients of scalar evolving in a coarse-grained turbulent flow field, compared with theoretical predictions, and (b) Taylor microscales of the scalars convected by filtered velocity (${\tilde{\lambda }}_{\theta }$) as functions of the maximum wave number of filtered velocity ($K$), at $\text{Sc}=1$, ${\text{Re}}_{\lambda }=80$, and $\text{Da}=0.05$ (${\text{Da}}_{\theta }=1.42$).

Figure 7

Energy spectra of reactive and passive scalar fields, i.e., $E_{\theta }\left(k\right)$ with $\theta =R_1$, $R_2$, $P$, and $T$, and velocity field $E_u\left(k\right)$ (solid light blue line) in the condition ${\text{Re}}_{\lambda }=150$, $\text{Sc}=1$, $n=1$, and $\text{Da}=10$, 1, 0.1, 0.01, and 0.001 (from top to bottom). Each spectra is compensated with the KOC scaling ${\left(k\eta \right)}^{5/3}$ and normalized by the global energy $\langle {\theta }^{\prime 2}\rangle$. For clarity, the energy spectra of scalars are shifted vertically by a multiplicative factor of 0.1.

Figure 8

Coherency spectra of the reactive scalars, under the conditions of ${\text{Re}}_{\lambda }=80$, $\text{Sc}=1$, and $n=1$. The horizontal axis is normalized in terms of the Kolmogorov scale $\eta$. The dashed vertical line marks the maximum wave number at which the scalar source terms act.

Figure 9

The global correlation coefficients of the reactive scalars and their gradients along the $x$ direction, under the condition of ${\text{Re}}_{\lambda }=150$, $\text{Sc}=1$, and $n=1$.