Unresolved stress tensor modeling in turbulent premixed V-flames using iterative deconvolution: An a priori assessment

The unresolved stress tensor in large-eddy simulations is an important unclosed term which affects the evolution of the resolved kinetic energy and consequently the evolution of large-scale motions. The literature on modeling this term for nonreacting flows is substantial but very scarce for turbulent flames and reacting flows with nonuniform density in general. The modeling assumptions in classic models are often strong even for nonreacting flows, which limits their range of applicability. In this study, a method using an iterative reconstruction algorithm is used in order to model the unresolved stress tensor for a highly demanding flow configuration which involves reaction, modeled using detailed chemistry, and mean shear. The evaluation of the method is conducted a priori using direct numerical simulation data, which are filtered and then sampled onto a coarser mesh. The performance of the method is compared against eight different classic models in the literature which include both static and dynamic formulations.

Figure 1

Progress variable isosurface, $c=0.1$, for (a) case V97, (b) case V60H, and (c) case V60.

Figure 2

Averaged (in homogeneous $y$ direction) instantaneous progress variable, $\langle c\rangle$, for all three cases.

Figure 3

Averaged (in homogeneous $y$ direction) normal stress component ${\tau }_{11}^{+}$, for case V60H and ${\mathrm{\Delta }}^{+}=3$: averaging locations are shown in gray-dashed lines at $x_1^{+}=-1.5{\delta }_L$, $x_2^{+}=0$, $x_3^{+}=1.5{\delta }_L$, $x_4^{+}=3{\delta }_L$, and $x_5^{+}=5{\delta }_L$ relative to the rod location for all three DNS cases.

Figure 4

Orientation of the stresses in the two nonhomogeneous directions for this coordinate system.

Figure 5

Normalized histograms of instantaneously averaged values of (a) numerator $\langle L_{kk}\rangle$ and (b) denominator $\langle P\rangle$, used to calculate dynamic parameter $C_I$ in Eq. (11) (case V97, ${\mathrm{\Delta }}^{+}=3$).

Figure 6

Pearson correlation coefficients averaged across all cases and all filter widths, for individual stress-tensor components.

Figure 7

Pearson correlation coefficients averaged across all stress-tensor components and all cases, for individual filter widths.

Figure 8

Pearson coefficient averaged over all six stress-tensor components for different number of iterations using the IDEF method, for case V97 and ${\mathrm{\Delta }}^{+}=3$.

Figure 9

Average shear stresses (a) $\langle {\tau }_{11}^{+}\rangle$ and (b) $\langle {\tau }_{13}^{+}\rangle$, for different $x$ locations, case V97, ${\mathrm{\Delta }}^{+}=1.0$: $•$, DNS; $＿$ S-SMAG, $﹍$ D-SMAG $＿$ SIMB, $＿$SIMET, $＿$GRAD, $＿$S-CLARK, $﹍$D-CLARK-D, $＿$WALE, $＿$IDEF.

Figure 10

Average shear stresses (a) $\langle {\tau }_{11}^{+}\rangle$ and (b) $\langle {\tau }_{13}^{+}\rangle$, for different $x$ locations, case V97, ${\mathrm{\Delta }}^{+}=3.0$. Lines as in Fig. 9.

Figure 11

Instantaneously averaged actual shear stress $\langle {\tau }_{13}\rangle$, modeled stress, rate of strain tensor $\langle {\tilde{S}}_{13}\rangle$, and dynamic Smagorinsky parameter $\langle C_D\rangle$: case V97, ${\mathrm{\Delta }}^{+}=3$, at $x_3^{+}$.

Figure 12

Average shear stress $\langle {\tau }_{13}^{+}\rangle$ for cases V60, V60H, and V97 at streamwise position $x_3^{+}$. Lines as in Fig. 9.

Figure 13

Average dominant component of the divergence of the stress tensor $\langle f_1^{+}\rangle =\langle {\delta }_L\partial {\tau }_{1j}^{+}/\partial x_j\rangle$ for case V97: (a) ${\mathrm{\Delta }}^{+}=1$, (b) ${\mathrm{\Delta }}^{+}=3$; $•$, DNS, $＿$IDEF.

Figure 14

Alignment angle between modeled ${\tau }_{j1}^m$ and actual ${\tau }_{j1}$, for ${\mathrm{\Delta }}^{+}=3$, at $x_3^{+}$. Lines as in Fig. 9.

Figure 15

Normalized wall-clock times for modeling the unresolved stress tensor for all models: case V97, ${\mathrm{\Delta }}^{+}=3$.

Figure 16

Normalized wall-clock times for modeling the unresolved stress tensor against iteration count for IDEF: case V97, ${\mathrm{\Delta }}^{+}=3$.