Study of subgrid-scale velocity models for reacting and nonreacting flows

A study is conducted to identify advantages and limitations of existing large-eddy simulation (LES) closures for the subgrid-scale (SGS) kinetic energy using a database of direct numerical simulations (DNS). The analysis is conducted for both reacting and nonreacting flows, different turbulence conditions, and various filter sizes. A model, based on dissipation and diffusion of momentum (LD-D model), is proposed in this paper based on the observed behavior of four existing models. Our model shows the best overall agreements with DNS statistics. Two main investigations are conducted for both reacting and nonreacting flows: (i) an investigation on the robustness of the model constants, showing that commonly used constants lead to a severe underestimation of the SGS kinetic energy and enlightening their dependence on Reynolds number and filter size; and (ii) an investigation on the statistical behavior of the SGS closures, which suggests that the dissipation of momentum is the key parameter to be considered in such closures and that dilatation effect is important and must be captured correctly in reacting flows. Additional properties of SGS kinetic energy modeling are identified and discussed.

Figure 1

(a) Isosurface of enstrophy (green) for case NR2 threshold at a value $\mu +2\sigma$ and (b) isosurface of heat release (red) (threshold at 0.8 of the maximum) for case R2 with isosurface of enstrophy (green).

Figure 2

PDFs of ${log}_{10}{k}^{+}$ (in standardized form) from DNS $(–\cdot –)$ are compared with standard Gaussian PDF $(—)$ for nonreacting (a) and reacting (b) cases and different filter widths.

Figure 3

Scatter plots of SGS kinetic energy from DNS vs modeled SGS kinetic energy for case NR1 of Table 1. The scatter plots are shown for the four models of Sec. 1 and two filter sizes. The Pearson linear correlation coefficient, $r$, is indicated for each scatter plot.

Figure 4

Scatter plots of SGS kinetic energy from DNS vs modeled SGS kinetic energy for case NR2 of Table 1. The scatter plots are shown for the four models of Sec. 1 and two filter sizes. The Pearson linear correlation coefficient, $r$, is indicated for each scatter plot.

Figure 5

The $q\text{−}q$ plots (in log scale) of normalized $k$ from DNS vs modeled $k$ are shown for cases NR1 to NR3 of Table 1. Five different models are presented for $k$: SRV model, Eq. (4) $(•••)$; Bardina's model, Eq. (5) $(—)$; Lilly's model, Eq. (6) $(—)$; Colin's model, Eq. (7) $(—)$; and the LD-D model, Eq. (9) $(––)$. Results obtained using two different filter sizes are shown for each case.

Figure 6

PDFs of normalized SGS kinetic energy, $k^{+}$, obtained from DNS $(–\cdot –)$ using two different filter sizes, are compared (in semilog plot) with those obtained using five different models: SRV model, Eq. (4) $(•••)$; Bardina's model, Eq. (5) $(—)$; Lilly's model, Eq. (6) $(—)$; Colin's model, Eq. (7) $(—)$; and the LD-D model, Eq. (9) $(––)$. The PDFs are multiplied by $k$ so that equal areas correspond to equal probabilities in the semilog plot.

Figure 7

Scatter plots of SGS kinetic energy from DNS vs modeled SGS kinetic energy for case R1 of Table 2. The scatter plots are shown for the four models of Sec. 1 and the LD-D model of Eq. (9), and results are presented for two different filter sizes. The quantities $r$ and $r_f$, indicated in the plots, represent the Pearson linear correlation coefficients obtained respectively using the entire domain and the flame region only, indicated here as the region where $0.5\le \tilde{c}\le 0.8$. Colors represent the progress variable field, $\tilde{c}$.

Figure 8

Scatter plots of SGS kinetic energy from DNS vs modeled SGS kinetic energy for case R2 of Table 2. The scatter plots are shown for the four models of Sec. 1 and the LD-D model of Eq. (9), and results are presented for two different filter sizes. The quantities $r$ and $r_f$, indicated in the plots, represent the Pearson linear correlation coefficients obtained respectively using the entire domain and the flame region only, indicated here as the region where $0.5\le \tilde{c}\le 0.8$. Colors represent the progress variable field, $\tilde{c}$.

Figure 9

The $q\text{−}q$ plots (in log scale) of normalized $k$ from DNS vs modeled $k$ are shown for cases R1 and R2 of Table 2. Five different models are presented for $k$: SRV model, Eq. (4) $(•••)$; Bardina's model, Eq. (5) $(—)$; Lilly's model, Eq. (6) $(—)$; Colin's model, Eq. (7) $(—)$; and LD-D model, Eq. (9) $(––)$. Results obtained using two different filter sizes are shown for each case.

Figure 10

PDFs of normalized SGS kinetic energy, $k^{+}$, obtained from DNS $(–\cdot –)$ using two different filter sizes, are compared (in semilog scale) for cases R1 and R2 of Table 2 with those obtained using five different models: SRV model, Eq. (4) $(•••)$; Bardina's model, Eq. (5) $(—)$; Lilly's model, Eq. (6) $(—)$; Colin's model, Eq. (7) $(—)$; and LD-D model, Eq. (9) $(––)$. The PDFs are multiplied by $k$ so that equal areas correspond to equal probabilities in the semilog plot.

Figure 11

Conditional averages of normalized SGS kinetic energy, $k^{+}$, obtained from DNS $(–\cdot –)$ using two different filter sizes, are compared for cases R1 and R2 of Table 2 with those obtained using five different models: SRV model, Eq. (4) $(•••)$; Bardina's model, Eq. (5) $(—)$; Lilly's model, Eq. (6) $(—)$; Colin's model, Eq. (7) $(—)$; and LD-D model, Eq. (9) $(––)$.