Turbulent diffusion of chemically reacting flows: Theory and numerical simulations

The theory of turbulent diffusion of chemically reacting gaseous admixtures developed previously [T. Elperin et al., Phys. Rev. E 90, 053001 (2014)] is generalized for large yet finite Reynolds numbers and the dependence of turbulent diffusion coefficient on two parameters, the Reynolds number and Damköhler number (which characterizes a ratio of turbulent and reaction time scales), is obtained. Three-dimensional direct numerical simulations (DNSs) of a finite-thickness reaction wave for the first-order chemical reactions propagating in forced, homogeneous, isotropic, and incompressible turbulence are performed to validate the theoretically predicted effect of chemical reactions on turbulent diffusion. It is shown that the obtained DNS results are in good agreement with the developed theory.

Figure 1

Spectrum of turbulent kinetic energy.

Figure 2

Instantaneous $c$ field obtained in the DNS at $t/{\tau }_f=5.4$ (top row), 6.2 (middle row), and 7 (bottom row) for $\text{Re}=200$, with $S_L/u_0=0.1$ (left column) and 2.0 (right column). Gray isosurfaces are associated with the peak reaction rate, i.e., $c(x,y,z,t)=c^{*}$, where $W\left(c^{*}\right)$ is the maximum value. Top-sliced planes are colored from blue to red in order to represent an increase in the local $c$ from zero to unity. Yellow vortex tubes visualize turbulence field by showing regions characterized by large values $Q>Q^{*}=2u_0^2/{\tau }_f\nu$ of the $Q$ criterion.

Figure 3

Dependences of the normalized turbulent scalar flux $\langle u^{\prime }{c}^{\prime }\rangle /u_0$ on the mean reaction progress variable $\langle c\rangle$, computed at different ratios of $S_L/u_0$ specified in the legend for (a) $\mathrm{Re}=50$, (b) $\mathrm{Re}=100$, and (c) $\mathrm{Re}=200$.

Figure 4

Dependences of the normalized mean turbulent wave thickness ${\delta }_{\mathrm{t}}/{\ell }_{11}$ on the normalized wave speed $S_L/u_0$ computed at three different turbulent Reynolds numbers $\mathrm{Re}$ specified in legend.

Figure 5

Dependences of the normalized turbulent scalar diffusivity $D_T/u_0{\ell }_{11}$ on the mean reaction progress variable $\langle c\rangle$, computed for (a) $\mathrm{Re}=50$, (b) $\mathrm{Re}=100$, and (c) $\mathrm{Re}=200$. Values of $S_L/u_0$ are specified near the curves.

Figure 6

Reaction-rate ratio $\theta =\max \{\langle W\rangle (\langle c\rangle \left)\right\}/\max \left\{W\right(c\left)\right\}$ vs $S_L/u_0$. Symbols show DNS data, with the Reynolds numbers Re being specified in the legend.

Figure 7

Theoretical dependence ${\tilde{D}}_T\equiv D_T/D_0^T$ on Damköhler number $\mathrm{Da}$ determined by Eq. (12) for different values of the Reynolds number $\mathrm{Re}=50$ (blue), 100 (black), and 200 (red) at $\mathrm{Pr}=1$. The DNS data on $\widehat{D_T}$ normalized using $u_0{l}_{11}$ are shown in blue triangles ($\mathrm{Re}=50$), black squares ($\mathrm{Re}=100$), and red circles ($\mathrm{Re}=200$).