Scalar flux modeling in turbulent flames using iterative deconvolution

In the context of large eddy simulations, deconvolution is an attractive alternative for modeling the unclosed terms appearing in the filtered governing equations. Such methods have been used in a number of studies for non-reacting and incompressible flows; however, their application in reacting flows is limited in comparison. Deconvolution methods originate from clearly defined operations, and in theory they can be used in order to model any unclosed term in the filtered equations including the scalar flux. In this study, an iterative deconvolution algorithm is used in order to provide a closure for the scalar flux term in a turbulent premixed flame by explicitly filtering the deconvoluted fields. The assessment of the method is conducted a priori using a three-dimensional direct numerical simulation database of a turbulent freely propagating premixed flame in a canonical configuration. In contrast to most classical a priori studies, the assessment is more stringent as it is performed on a much coarser mesh which is constructed using the filtered fields as obtained from the direct simulations. For the conditions tested in this study, deconvolution is found to provide good estimates both of the scalar flux and of its divergence.

Figure 1

The deconvolution and explicit-filtering process applied to $\overline{\varphi }=\overline{\rho uc}$. Going from left to right are the original DNS field (a) $\rho uc$, the filtered field $\overline{\rho uc}$, the deconvoluted field ${\left\{\rho uc\right\}}^{*}$, and the filtered-deconvoluted field $\overline{{\left\{\rho uc\right\}}^{*}}$ for each filter in (b)–(j). The data correspond to an instantaneous snapshot in the $z$ direction for case C, and are normalized using the maximum instantaneous DNS value. Reactants flow from the bottom and hot products exit from the top. Note that deconvolution is conducted in three dimensions; the figures are simply two-dimensional contours.

Figure 2

Sample output from successive iterations $n$ using the deconvolution algorithm. Starting from the left is the filtered field (a) $\overline{\varphi }=\overline{\rho uc}$ for ${\mathrm{\Delta }}^{+}=2.0$, and moving to the right (b)–(d) are the deconvoluted fields on the LES mesh. Notice the gradual sharpening of the image as smaller-scale structures are recovered, particularly in the reactants side (blue side).

Figure 3

Case A instantaneous scatterplot of the $x$ component of the scalar-flux term, $f_1$, against the DNS result for ${\mathrm{\Delta }}^{+}=1.0$: (a) using IDEF and (b) using Clark's model.

Figure 4

Case A instantaneous scatterplot of the $x$ component of the scalar-flux term, $f_1$, against the DNS result for ${\mathrm{\Delta }}^{+}=3.0$: (a) using IDEF and (b) using Clark's model.

Figure 5

Case C instantaneous scatterplot of the $x$ component of the scalar-flux term, $f_1$, against the DNS result for ${\mathrm{\Delta }}^{+}=1.0$: (a) using IDEF and (b) using Clark's model.

Figure 6

Case C instantaneous scatterplot of the $x$ component of the scalar flux term, $f_1$, against the DNS result for ${\mathrm{\Delta }}^{+}=3.0$: (a) using IDEF and (b) using Clark's model.

Figure 7

Averaged (in homogeneous directions $y$ and $z$) flux term $f_{1x}$ for case A: (a) ${\mathrm{\Delta }}^{+}=1$, (b) ${\mathrm{\Delta }}^{+}=2$, and (c) ${\mathrm{\Delta }}^{+}=3$.

Figure 8

Averaged (in homogeneous directions $y$ and $z$) flux term $f_{1x}$ for case B: (a) ${\mathrm{\Delta }}^{+}=1$, (b) ${\mathrm{\Delta }}^{+}=2$, and (c) ${\mathrm{\Delta }}^{+}=3$.

Figure 9

Averaged (in homogeneous directions $y$ and $z$) flux term $f_{1x}$ for case C: (a) ${\mathrm{\Delta }}^{+}=1$, (b) ${\mathrm{\Delta }}^{+}=2$, and (c) ${\mathrm{\Delta }}^{+}=3$.

Figure 10

Averaged (in homogeneous directions $y$ and $z$) flux term $F_x$ for case A: (a) ${\mathrm{\Delta }}^{+}=1$, (b) ${\mathrm{\Delta }}^{+}=2$, and (c) ${\mathrm{\Delta }}^{+}=3$.

Figure 11

Averaged (in homogeneous directions $y$ and $z$) flux term $F_x$ for case B: (a) ${\mathrm{\Delta }}^{+}=1$, (b) ${\mathrm{\Delta }}^{+}=2$, and (c) ${\mathrm{\Delta }}^{+}=3$.

Figure 12

Averaged (in homogeneous directions $y$ and $z$) flux term $F_x$ for case C: (a) ${\mathrm{\Delta }}^{+}=1$, (b) ${\mathrm{\Delta }}^{+}=2$, and (c) ${\mathrm{\Delta }}^{+}=3$.