Surface-averaged quantities in turbulent reacting flows and relevant evolution equations

While quantities conditioned to an isosurface of reaction progress variable $c$, which characterizes fluid state in a turbulent reacting flow, have been attracting rapidly growing interest in the recent literature, a mathematical and physical framework required for research into such quantities has not yet been elaborated properly. This paper aims at filling two fundamental gaps in this area, i.e., (i) ambiguities associated with a definition of a surface-averaged quantity and (ii) the lack of rigorous equations that describe evolutions of such quantities. In the first (theoretical) part of the paper, (a) analytical relations between differently defined (area-weighted and unweighted) surface-averaged quantities are obtained and differences between them (quantities) are discussed, (b) a unified method for deriving an evolution equation for bulk area-weighted surface-averaged value of a local characteristic $\varphi$ of a turbulent reacting flow is developed, and (c) the method is applied for deriving evolution equations for the bulk area-weighted surface-averaged reaction-surface density $\left|\mathbf{\nabla }c\right|$, local reaction-wave thickness $1/\left|\mathbf{\nabla }c\right|$, and local displacement speed $S_d$, i.e., the speed of an isosurface of the $c(\mathit{x},t)$ field with respect to the local flow. In the second (numerical) part of the paper, direct numerical simulation data obtained recently from a highly turbulent reaction wave are analyzed in order to (1) highlight substantial differences between area-weighted and unweighted surface-averaged quantities and (2) show that various terms in the derived evolution equations are amenable to accurate numerical evaluation in spite of appearance of the so-called zero-gradient points [C. H. Gibson, Phys. Fluids 11, 2305 (1968)] in a highly turbulent medium. Finally, the obtained analytical and numerical results are used to shed light on the paradox of local flame thinning and broadening which is widely discussed in the turbulent combustion literature.

Figure 1

A 2D snapshot of the fully developed $c^s$ field in case C. Four (solid and dotted) black lines show isosurfaces of $c=0.2$, 0.4, 0.8, and 0.95 from left to right.

Figure 2

Snapshots of the $c$ field in case A at two transient instants of (a) $t=0.8{\tau }_F$ and (b) $1.6{\tau }_F$ and (c) at the fully developed state of $t_{\infty }$. Five black lines show isosurfaces of $c=0.05$, 0.2, 0.4, 0.8, and 0.95 from left to right.

Figure 3

Snapshots of the $c$, vorticity, and $S_d$ fields in case B at a single representative instant. Only a part of the computational domain, which contains the most disturbed reaction wave is plotted. Four dashed and solid lines show the isosurfaces of $c=0.02$, 0.1, 0,3 and 0.98, local velocity vectors are plotted with line arrows.

Figure 4

Comparison of the lhs and the rhs of Eq. (27), as well as residuals $\mathcal{R}=\sum \text{rhs}-\text{lhs}$. Tweaked residuals ${\mathcal{R}}^{*}$ are computed by substituting all the area-weighted surface-averaged terms ${〈·〉}_{{}_s}{|}_{\widehat{c},t}$ on the rhs of Eq. (27) with the counterpart unweighted surface-averaged terms ${〈·〉}_{{}_v}{|}_{\widehat{c},t}$. Fine-grained results obtained for $\widehat{c}=0.1$ and $\widehat{c}=0.88$ are plotted in the left and right columns, respectively. Results computed in cases A, B, and C are reported in the up, middle, and bottom rows, respectively. All terms are normalized with $S_L/{\delta }_F^2$, time is normalized with ${\tau }_F^{*}={\tau }_F$ in cases A and B, or ${\tau }_F^{*}={\tau }_F/22.5$ in case C.

Figure 5

Dependencies of the lhs, the rhs, and various terms on the rhs of Eq. (27) on the value of $\widehat{c}$ that results are conditioned to. Black stars show residuals $\mathcal{R}=\sum \text{rhs}-\text{lhs}$. Tweaked residuals ${\mathcal{R}}^{*}$ are computed by substituting all the area-weighted surface-averaged terms ${〈·〉}_{{}_s}{|}_{\widehat{c},t}$ on the rhs of Eq. (27) with the counterpart unweighted surface-averaged terms ${〈·〉}_{{}_v}{|}_{\widehat{c},t}$. Results computed at $t=0.045{\tau }_F^{*}$ and $t=0.32{\tau }_F^{*}$ are plotted in the left and middle columns, respectively, with ${\tau }_F^{*}={\tau }_F$ in cases A and B or ${\tau }_F^{*}={\tau }_F/22.5$ in case C. Results obtained from the fully developed waves are reported in the right column. Results computed in cases A, B, and C are shown in the up, middle, and bottom rows, respectively. All terms are normalized with $S_L/{\delta }_F^2$.

Figure 6

Evolution of normalized terms in Eq. (29) (see caption to Fig. 4 for further details).

Figure 7

$\widehat{c}$ profiles of normalized terms in Eq. (29) (see caption to Fig. 5 for further details).

Figure 8

Evolution of normalized terms in Eq. (30) (see caption to Fig. 4 for further details).

Figure 9

$\widehat{c}$ profiles of terms in Eq. (30), appropriately normalized based on $S_L$ and ${\delta }_F$ and plotted similarity as in Fig. 5. $\mathcal{C}=1$ in cases A and B (up and middle row) or $\mathcal{C}=100$ in case C (bottom row). All 10 terms listed in legends are shown in the right column, but 5 terms are difficult to distinguish because their magnitudes are low.