Reynolds-number power-law scaling of differential molecular diffusion in turbulent nonpremixed combustion

A full understanding of differential molecular diffusion (DMD) in turbulent combustion has its theoretical significance for improving models of turbulent combustion. The scaling of the effect of DMD with respect to the Reynolds number in turbulent combustion is of particular interest for developing physically consistent modeling approaches for DMD. Such a scaling has so far been mostly studied in simple nonreacting flow problems, and a simple power-law scaling has been reported before. The applicability of the power-law scaling to turbulent combustion problems where the chemical reaction is expected to strongly couple with DMD has not been thoroughly studied. In this work, we aim to examine such a scaling by developing a statistical analysis of the dependence of DMD on the Reynolds number in turbulent nonpremixed combustion. Three Sandia temporally evolving planar jet nonpremixed ${\mathrm{CO}/\mathrm{H}}_2$ direct numerical simulation flames [E. R. Hawkes et al., Proc. Combust. Inst. 31, 1633 (2007)] are chosen as the target flames for the study. The Reynolds-number scaling based on a statistical analysis is reported, which is found to be statistically consistent with previous theoretical results in nonreacting problems. The results provide supportive evidence to the existence of a universal power-law scaling of the effect of DMD with respect to the Reynolds number in turbulent nonreacting and reacting flow problems. The results are also important for constraining the development of Reynolds-number-scaling consistent physical models for treating DMD in the modeling and simulations of multicomponent turbulent diffusion systems.

Figure 1

Sketch of the Sandia temporally evolving jet ${\mathrm{CO}/\mathrm{H}}_2$ DNS flames [35].

Figure 2

Profiles of the turbulent kinetic energy $k$, the turbulence dissipation rate $\varepsilon$, the turbulence integral length scale $l$, the turbulence integral velocity scale $u$, the molecular viscosity $\nu$, and the turbulent Reynolds number ${\text{Re}}_t$ in the three Sandia ${\mathrm{CO}/\mathrm{H}}_2$ DNS flames [35] at the different times $t/t_0=10$, 15, 20, and 30 against $y/y_{1/2}$, where $y_{1/2}$ is the half-width of the mixing layer based on the profiles of ${\tilde{\xi }}_{\text{C}}$.

Figure 3

Profiles of mean mixture fraction ${\tilde{\xi }}_{\text{C}}$, rms of mixture fraction ${\xi }_{\text{C,}\text{rms}}$, mean ${\tilde{z}}_{\text{HC}}$, and rms $z_{\text{HC,rms}}$ against $y/y_{1/2}$ in the three Sandia ${\mathrm{CO}/\mathrm{H}}_2$ DNS flames at the different times $t/t_0=10$, 15, 20, and 30.

Figure 4

Conditional average ${\tilde{z}}_{\text{HC}}{|}_{\mathrm{C}}$ against ${\text{Re}}_t$ in the three ${\mathrm{CO}/\mathrm{H}}_2$ DNS flames: case L, red circles; case M, green squares; and case H, blue triangles. The dashed lines are reference lines with slope $-1$ in the log-log plot. The condition $\mathrm{C}(c_m,c_r,c_{\chi })$ for computing the conditional average ${\tilde{z}}_{\text{HC}}{|}_{\mathrm{C}}$ is given above each plot.

Figure 5

Conditional average of $z_{\text{HC,rms}}{|}_{\mathrm{C}}$ against ${\text{Re}}_t$ in the three ${\mathrm{CO}/\mathrm{H}}_2$ DNS flames: case L, red circles; case M, green squares; and case H, blue triangles. The dashed lines are reference lines with slope $-0.25$ in the log-log plot. The condition $\mathrm{C}(c_m,c_r,c_{\chi })$ for computing the conditional average $z_{\text{HC,rms}}{|}_{\mathrm{C}}$ is given above each plot.

Figure 6

The PDFs of scaling exponents ${\kappa }_m$ (left) and ${\kappa }_r$ (right). The solid lines are the Gaussian PDF with the mean and variance calculated from the data samples. The error bars are the estimated 95% confidence intervals.