5.1. Statistical analysis of kinetic energy transfer

Figure 8 shows the average of energy transfer terms for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. The energy transfer terms are normalized by the total dissipation $\epsilon _{T}=-\langle p \theta \rangle +\epsilon _{0}$. The average of the pressure–dilatation term $\langle \varPhi _{l}\rangle / \epsilon _{T}$ is plotted in figure 8(a). It is shown that for isothermal reactions ($Da=2$) at $M_{t}=0.2$ and 0.8, the magnitude of $\langle \varPhi _{l}\rangle / \epsilon _{T}$ is negligibly small at all $l / \eta$ scales. The results are consistent with the previous observations by Wang et al. (Reference Wang, Wan, Chen and Chen2018a) in simulating non-reacting compressible turbulence. For exothermic reaction ($Da=200$) at $M_{t}=0.2$, the magnitude of $\langle \varPhi _{l}\rangle / \epsilon _{T}$ is evidently larger than that for isothermal reactions at the same turbulent Mach number for all $l / \eta$ scales. At $l / \eta >20$, the magnitude of $\langle \varPhi _{l}\rangle / \epsilon _{T}$ gradually decreases with the increase of $l / \eta$. For exothermic reaction at $M_{t}=0.8$, heat release results in an increase of the magnitude of $\langle \varPhi _{l}\rangle / \epsilon _{T}$ at $10 \leqslant l / \eta \leqslant 100$ compared with values for isothermal reactions at the same turbulent Mach number. The results indicate that heat release through exothermic reactions can enhance the net contribution of pressure–dilatation to the kinetic energy transfer in a wide range of scales, especially at low turbulent Mach number. Figure 8(b) plots the average of SGS flux $\langle \varPi _{l}\rangle / \epsilon _{T}$. It is found that $\langle \varPi _{l}\rangle / \epsilon _{T}$ nearly collapse with each other for all cases. At $2 < l / \eta < 30$, $\langle \varPi _{l}\rangle / \epsilon _{T}$ grows rapidly and at scales $30 \leqslant l / \eta \leqslant 100$, $\langle \varPi _{l}\rangle / \epsilon _{T} \approx 1.0$. For exothermic reaction at $M_{t}=0.2$, $\langle \varPi _{l}\rangle / \epsilon _{T}$ is slightly larger than values for the other three cases over the entire $l / \eta$ scales, indicating that heat release can enhance the relative contribution of SGS flux to kinetic energy transfer. The average of viscous dissipations $\langle D_{l}\rangle / \epsilon _{T}$ are plotted in figure 8(c). It is found that $\langle D_{l}\rangle / \epsilon _{T}>0.5$ at $l / \eta \leqslant 10$, showing that viscous dissipations play an important role in kinetic energy transfer within this scale range. For all simulation cases, $\langle D_{l}\rangle / \epsilon _{T}$ decreases with the increase of filter width $l / \eta$ at $l / \eta >10$. This observation is consistent with previous theoretical analysis (Aluie Reference Aluie2011, Reference Aluie2013) and numerical results (Wang et al. Reference Wang, Wan, Chen and Chen2018a) in non-reacting compressible turbulence. For exothermic reaction ($Da=200$) at $M_{t}=0.2$, $\langle D_{l}\rangle / \epsilon _{T}$ is slightly larger than values for the other three cases at all $l / \eta$ scales, suggesting that strong heat release can lead to an increase of viscous dissipation. Figure 8(d) shows the overall energy transfer $\langle \varPhi _{l}+\varPi _{l}+D_{l}\rangle / \epsilon _{T}$. It can be found that $\langle \varPhi _{l}+\varPi _{l}+D_{l}\rangle / \epsilon _{T} \approx 1.0$ at scales $l / \eta \leqslant 64$ for all simulation cases and $\langle \varPhi _{l}+\varPi _{l}+D_{l}\rangle / \epsilon _{T}$ is insensitive to the change of turbulent Mach number and reaction heat release. The results indicate that the effects of large-scale external forcing in current simulations are localized to large scales. In the theoretical analysis of kinetic energy transfer in compressible turbulence, Aluie (Reference Aluie2013) proved that by using the Favre filtering for scale decomposition, the kinetic energy injection can be localized to large scales by proper stirring, such as a large-scale external acceleration field used in the current study. Results in current simulations show that heat release through chemical reactions will not essentially change the behaviour of large-scale external forcing applied to the numerical system.

Figure 8.

Average of energy transfer terms for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $\langle \varPhi _{l}\rangle / \epsilon _{T}$; (b) $\langle \varPi _{l}\rangle / \epsilon _{T}$; (c) $\langle D_{l} \rangle / \epsilon _{T}$; (d) $\langle \varPi _{l}+D_{l}+\varPhi _{l}\rangle /\epsilon _{T}$.

The normalized r.m.s. values of pressure–dilatation $\varPhi _{l}^{\prime } / \epsilon _{T}$ and SGS flux $\varPi _{l}^{\prime } / \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8 are shown in figure 9. Here, $\varPhi _{l}^{\prime }=\sqrt {\langle (\varPhi _{l}-\langle \varPhi _{l} \rangle )^{2} \rangle }$ and $\varPi _{l}^{\prime }=\sqrt {\langle (\varPi _{l}-\langle \varPi _{l} \rangle )^{2} \rangle }$. As shown in figure 9(a), $\varPhi _{l}^{\prime } / \epsilon _{T}$ decreases gradually with the increases of filter width $l / \eta$. For all simulation cases, $\varPhi _{l}^{\prime } / \epsilon _{T}>1$ is found at all $l / \eta$ scales. Particularly, for exothermic reaction ($Da=200$) at $M_{t}=0.2$, it is found that $\varPhi _{l}^{\prime } / \epsilon _{T}>100$ at $l / \eta \leqslant 200$, and $\varPhi _{l}^{\prime } / \epsilon _{T}$ is much larger than those for the other three cases at all $l / \eta$ scales. Thus, the pressure–dilatation plays an important role in the local energy transfer between kinetic energy and internal energy for exothermic reactions. Heat release through exothermic reaction can significantly enhance the local transfer between kinetic energy and internal energy through pressure work, which is consistent with previous studies on chemically reacting turbulence (Jaberi & Madnia Reference Jaberi and Madnia1998; Livescu et al. Reference Livescu, Jaberi and Madnia2002). The normalized r.m.s. value of SGS flux is shown in figure 9(b). It can be found that $\varPi _{l}^{\prime } / \epsilon _{T}$ increases with the increase of filter width at $l / \eta \leqslant 20$ for all simulation cases. At a given turbulent Mach number, $\varPi _{l}^{\prime } / \epsilon _{T}$ for exothermic reaction ($Da=200$) is larger than that for isothermal reaction ($Da=2$), which indicates that heat release for exothermic reactions can notably enhance the local kinetic energy transfer by SGS flux.

Figure 9.

The r.m.s. values of pressure–dilatation and SGS flux for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $\varPhi _{l}^{\prime } / \epsilon _{T}$; (b) $\varPi _{l}^{\prime } / \epsilon _{T}$.

Figure 10 depicts the p.d.f. of the normalized pressure–dilatation $\varPhi _{l} / \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. For isothermal reaction ($Da=2$) at $M_{t}=0.2$ as shown in figure 10(a), the p.d.f. curves of $\varPhi _{l} / \epsilon _{T}$ are nearly symmetric with respect to the line $\varPhi _{l} / \epsilon _{T}=0$, indicating that the compression and expansion are in a balanced state. For exothermic reaction ($Da=200$) at $M_{t}=0.2$ as shown in figure 10(b), both left- and right-hand tails of the p.d.f. become significantly longer than those for isothermal at the same turbulent Mach number. The observation shows that heat release through exothermic reaction significantly enhances both flow compression and expansion motions. For isothermal reaction ($Da=2$) at $M_{t}=0.8$ shown in figure 10(c), the right-hand tail of the p.d.f. is longer than its left-hand tail, suggesting that local compression can be stronger than local expansion at high turbulent Mach number due to the increase of flow compressibility. The result is consistent with observations by Wang et al. (Reference Wang, Wan, Chen and Chen2018a) in simulating non-reacting compressible turbulence at high turbulent Mach numbers. For exothermic reaction ($Da=200$) at $M_{t}=0.8$ as shown in figure 10(d), both sides of the p.d.f. become longer. For exothermic reactions at $M_{t}=0.2$ and 0.8, the right-hand tails of p.d.f. at $l/\eta =16$ are longer than their left-hand tails, suggesting that local compression motions can be stronger than local expansion motions at small scales.

Figure 10.

The p.d.f. of the normalized pressure–dilatation $\varPhi _{l} / \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (b) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.8$, $Da=200$.

Figure 11 shows the p.d.f. of the normalized SGS flux $\varPi _{l} / \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. The p.d.f. of SGS flux is skewed to the positive side, indicating that the kinetic energy transferred by SGS flux has a tendency to be from large scales to small scales. For all simulation cases, both positive and negative tails of p.d.f. become longer as the filter width $l/ \eta$ decreases. For isothermal reactions ($Da=2$), with the increase of turbulent Mach number from 0.2 to 0.8 as shown in figures 11(a) and 11(c), respectively, the right-hand tails of p.d.f. of SGS flux become longer especially at small scales $l/\eta =16$ and 32, due to the increase of flow compressibility. These observations are consistent with findings for non-reacting compressible isotropic turbulence (Wang et al. Reference Wang, Wan, Chen and Chen2018a). For exothermic reactions ($Da=200$) at $M_{t}=0.2$ and 0.8, the two tails of the p.d.f. of SGS flux become longer compared with the p.d.f. for isothermal reactions ($Da=2$) at the same turbulent Mach number. The results reveal that heat release through exothermic reactions significantly enhances local SGS energy transfer at all scales, especially at small scales. The right-hand tails of p.d.f. of SGS flux for exothermic reactions at $M_{t}=0.2$ and 0.8 are longer than their left-hand tails, suggesting that the positive SGS flux is predominant over the negative SGS flux.

Figure 11.

The p.d.f. of the normalized SGS flux $\varPi _{l} / \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (b) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.8$, $Da=200$.

The pressure–dilatation $\varPhi _{l}$ is decomposed into a positive component $\varPhi _{l}^{+}$ and a negative component $\varPhi _{l}^{-}$ (O'Brien et al. Reference O'Brien, Urzay, Ihme, Moin and Saghafian2014): $\varPhi _{l}=\varPhi _{l}^{+}+\varPhi _{l}^{-}$; here $\varPhi _{l}^{+}=\frac {1}{2} (\varPhi _{l}+|\varPhi _{l}|)$ and $\varPhi _{l}^{-}=\frac {1}{2} (\varPhi _{l}-|\varPhi _{l}|)$. The positive pressure–dilatation indicates kinetic energy destruction by flow compression while the negative pressure–dilatation implies kinetic energy production by flow expansion (O'Brien et al. Reference O'Brien, Urzay, Ihme, Moin and Saghafian2014). Figure 12 shows the normalized average of the positive and negative components of pressure–dilatation for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. As shown in figure 12(a), the positive component of pressure–dilatation for exothermic reaction ($Da=200$) at $M_{t}=0.2$ decreases gradually with the increase of filter width $l/\eta$. The magnitude of the negative component of pressure–dilatation is quite close to the positive component for all cases. For exothermic reaction ($Da=200$) at $M_{t}=0.2$, the magnitude of both positive and negative components of pressure–dilatation are significantly larger than the values for other cases at all $l/\eta$ scales. The results reveal that heat release through exothermic reaction at low turbulent Mach number can significantly increase both positive and negative components of pressure–dilatation.

Figure 12.

Normalized average of the positive and negative components of pressure–dilatation for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8.

Similarly, the SGS flux $\varPi _{l}$ is decomposed into a positive component $\varPi _{l}^{+}$ and a negative component $\varPi _{l}^{-}$: $\varPi _{l}=\varPi _{l}^{+}+\varPi _{l}^{-}$; here $\varPi _{l}^{+}=\frac {1}{2} (\varPi _{l}+|\varPi _{l}|)$ and $\varPi _{l}^{-}=\frac {1}{2} (\varPi _{l}-|\varPi _{l}|)$. Figure 13 shows the normalized average of the positive and negative components of SGS flux for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. It can be found that for isothermal reactions ($Da=2$) at $M_{t}=0.2$ and 0.8, the magnitudes of $\langle \varPi _{l}^{+}\rangle / \epsilon _{T}$ and $\langle \varPi _{l}^{-}\rangle / \epsilon _{T}$ become larger with the increase of turbulent Mach number. At a given turbulent Mach number, the magnitudes of $\langle \varPi _{l}^{+}\rangle / \epsilon _{T}$ and $\langle \varPi _{l}^{-}\rangle / \epsilon _{T}$ for exothermic reactions ($Da=200$) are larger than those for isothermal reactions ($Da=2$). Particularly, for exothermic reaction ($Da=200$) at $M_{t}=0.2$, the magnitudes of $\langle \varPi _{l}^{+}\rangle / \epsilon _{T}$ and $\langle \varPi _{l}^{-}\rangle / \epsilon _{T}$ at $l /\eta \geqslant 20$ are significantly larger than the values for the other three cases. The magnitudes of the positive component of SGS flux are comparably larger than their negative counterparts for all cases. These observations suggest that heat release through exothermic reactions can significantly enhance both forward-scatter and backscatter of kinetic energy through SGS flux.

Figure 13.

Normalized average of the positive and negative components of SGS flux for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8.

5.2. Influence of velocity divergence on SGS flux of kinetic energy

The influences of velocity divergence on SGS flux of kinetic energy are studied in this section. Figure 14 shows the joint p.d.f. of the normalized SGS kinetic energy flux $\varPi _{l} /\epsilon _{T}$ and the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ with the filter width $l/\eta =16$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8 for different normalized reaction times. Here, $\theta _{l}=\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {\tilde {u}}$ is the filtered velocity divergence. Inertial-range flow quantities can exhibit some universal statistical properties and are of great importance in the study of interscale energy transfer in turbulence. The filter width performance in capturing spectral information as shown in figure 7 suggests that the filter width $l/\eta =16$ can resolve flow quantities in the inertial-range. For isothermal reaction ($Da=2$) at turbulent Mach number $M_{t}=0.2$ as shown in figure 14(a1)–(a4), the shape of the joint p.d.f. is nearly symmetric with respect to the line $\theta _{l} / \theta _{l}^{\prime }=0$ at $t/\tau =0.9\sim 37$, indicating that the contributions of compression regions and expansion regions to the SGS flux are nearly equal to each other. At $t/\tau =33$ and 37 approaching the end of chemical reaction, the positive of SGS flux is slightly enhanced corresponding to the increase of fraction of the joint p.d.f. at the positive part of $\varPi _{l}$ in the first and second quadrants. For exothermic reaction ($Da=200$) at $M_{t}=0.2$ as shown in figure 14(b1)–(b4), the shape of the joint p.d.f. is nearly symmetric with respect to the line $\theta _{l} / \theta _{l}^{\prime }=0$ at the initial stage of the chemical reaction ($t/\tau =0.9$). At $t/\tau =3.7$ corresponding to the fast chemical reaction period, the fractions of the first and second quadrants of the joint p.d.f. become larger, suggesting that heat release enhances the positive SGS flux and the contributions of compression and expansion motions to the SGS flux are nearly equal to each other. At $t/\tau =33$ and 37, the fractions of the second and fourth quadrants of the joint p.d.f. become significantly larger than those in the first and third quadrants. The joint p.d.f. for this case indicates that compression regions have a major contribution to the direct SGS flux of kinetic energy, while expansion regions have a major contribution to the reverse SGS flux of kinetic energy. The fraction of the second quadrants of the joint p.d.f. is larger than that in the forth quadrants, indicating that the occurrence of strong compression largely enhances the direct SGS flux of kinetic energy from large scales to small scales. It is worth noting that stronger compressibility is observed at $t/\tau =33$, 37 than that at $t/\tau =3.7$, even though heat release by the reaction is more significant at $t/\tau =3.7$. The observation suggests that the effect of heat release on the flow compressibility is quite complicated: the internal energy increased by reaction heat can be transferred to the dilatational component of kinetic energy through pressure work, and then leads to generations of shocklets. The time scales of the overall flow dynamic processes are larger than the time scale of reaction. Consequently, the enhancement of compressibility by heat release becomes more significant after the stage of peak reaction rate. Thus, the temporal average of flow statistics based on flow fields at $t/\tau =25\sim 37$ can properly demonstrate the effect of heat release of reaction on the flow dynamics. For isothermal reaction ($Da=2$) and exothermic reaction ($Da=200$) at $M_{t}=0.8$ as shown in figures 14(c1)–(c4) and 14(d1)–(d4), respectively, the joint p.d.f. exhibits similar distributions in which the p.d.f. fractions in the second and fourth quadrants are larger than those in the first and the third quadrants at $t/\tau =0.9$. At $t/\tau =3.7$, the fractions at second and fourth quadrants are further increased for exothermic reaction ($Da=200$). At the end of reaction ($t/\tau =37$), the joint p.d.f. for isothermal and exothermic reactions at $M_{t}=0.8$ are similar to their initial values ($t/\tau =0.9$). The results are consistent with observations by Wang et al. (Reference Wang, Wan, Chen and Chen2018a) through simulating non-reacting compressible turbulence: at $M_{t}>0.6$, the second quadrant of the joint p.d.f. predominates over other three quadrants, indicating that the strong compression associated with the shocklet structures greatly enhances the direct SGS flux of kinetic energy from large scales to small scales. The influence of heat release on the joint p.d.f. through exothermic reactions is relatively less obvious at high turbulent Mach number. It should be noted that with the increase of filter width (the joint p.d.f. for $l/\eta =32$, 64 and 128 are not plotted in the paper), the joint p.d.f. of $\varPi _{l} /\epsilon _{T}$ and $\theta _{l} / \theta _{l}^{\prime }$ becomes smaller. The qualitative statistical properties of the joint p.d.f. are nearly unaffected by the filter width.

Figure 14.

Joint p.d.f. of $\varPi _{l} /\epsilon _{T}$ and $\theta _{l} / \theta _{l}^{\prime }$ for the filter width $l/\eta =16$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8 at different normalized reaction time: (a1–a4) $M_t=0.2$, $Da=2$; (b1–b4) $M_t=0.2$, $Da=200$; (c1–c4) $M_t=0.8$, $Da=2$; (d1–d4) $M_t=0.8$, $Da=200$.

Figure 15 plots the average of normalized SGS flux $\varPi _{l} / \epsilon _{T}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ with different filter widths $l/\eta =16$, 24, 32, 48, 64 for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. It is shown that for isothermal reaction ($Da=2$) at $M_{t}=0.2$ as shown in figure 15(a), data are quite scattered at $\theta _{l} / \theta _{l}^{\prime }\geqslant 3$ due to the lack of samples. Now $\langle \varPi _{l} / \epsilon _{T} | \theta _{l} / \theta _{l}^{\prime } \rangle \approx 1$ is found at $-3 \leqslant \theta _{l} / \theta _{l}^{\prime }\leqslant 3$, which indicates that the effect of local compression and expansion motions on the overall direct SGS flux of kinetic energy is negligibly small. For exothermic reaction ($Da=200$) at $M_{t}=0.2$ as shown in figure 15(b), data are scattered at $\theta _{l} / \theta _{l}^{\prime }\leqslant -5$. The strong compression leads to strong direct SGS flux of kinetic energy, especially for small scales $l/\eta =16$, 32. For isothermal reaction ($Da=2$) and exothermic reaction ($Da=200$) at $M_{t}=0.8$ as shown in figures 15(c) and 15(d), respectively, the strong compression also leads to strong direct SGS flux of kinetic energy. Wang et al. (Reference Wang, Wan, Chen and Chen2018a) derived a heuristic model for the conditional average of SGS flux in the compression region where $\theta _{l} \leqslant 0$ in non-reacting compressible turbulence and observed a good agreement of the derived model with simulated data at turbulent Mach numbers $M_{t}=0.6$, 0.8 and 1.0. It is reported that SGS flux of kinetic energy increases linearly with the square of filtered velocity divergence in strong compression regions. However, the derived model (Wang et al. Reference Wang, Wan, Chen and Chen2018a) is no longer suitable for predicting data for reacting turbulence, especially for exothermic reactions with strong heat release. Teng et al. (Reference Teng, Wang, Li and Chen2020) studied the Mach scaling of kinetic energy and kinetic energy dissipation in chemically reacting compressible turbulence and found that for exothermic reactions with strong heat release, the normalized kinetic energy and kinetic energy dissipation are nearly independent of turbulent Mach number. Based on the model provided by Wang et al. (Reference Wang, Wan, Chen and Chen2018a), a modified relation without consideration of the influence of turbulent Mach number is proposed for the conditional average of SGS flux in compression regions of chemically reacting compressible isotropic turbulence,

(5.1)\begin{equation} \langle \varPi_{l}/\epsilon_{T}|\theta_{l} / \theta_{l}^{\prime} \rangle=1+\beta_{0}(\theta_{l} / \theta_{l}^{\prime})^{2}, \end{equation}

where, $\beta _{0}=0.0034$ and 0.19 for isothermal reactions ($Da=2$) at $M_{t}=0.2$ and 0.8, respectively, and $\beta _{0}=0.54$ and 0.25 for exothermic reactions ($Da=200$) at $M_{t}=0.2$ and 0.8, respectively. It can be found that $\beta _{0}$ for exothermic reactions is larger than that for isothermal reactions at a given turbulent Mach number, revealing the fact that heat release enhances direct SGS flux of kinetic energy at both low and high turbulent Mach numbers.

Figure 15.

Average of normalized SGS flux $\varPi _{l} / \epsilon _{T}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ for different filter widths $l/\eta =16$, 24, 32, 48, 64 for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (b) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.8$, $Da=200$.

Figure 16 shows the average of normalized SGS flux $\varPi _{l} / \epsilon _{T}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ with different filter widths $l/\eta =16$, 24, 32, 48, 64 in expansion regions for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. Based on the heuristic model proposed by Wang et al. (Reference Wang, Wan, Chen and Chen2018a), a modified relation in reacting compressible turbulence in expansion regions ($\theta _{l} \geqslant 0$) is suggested as follows:

(5.2)\begin{equation} \langle \varPi_{l}/\epsilon_{T}|\theta_{l} / \theta_{l}^{\prime} \rangle=1-\beta_{1}(\theta_{l} / \theta_{l}^{\prime})^{1.2}. \end{equation}

As shown in figure 16, a good agreement is observed at $0 \leqslant \theta _{l} / \theta _{l}^{\prime } \leqslant 5$ for all simulated cases, where, $\beta _{1}=0.2$ and 0.7 for isothermal reactions ($Da=2$) at $M_{t}=0.2$ and 0.8, respectively, and $\beta _{1}=1.50$ and 1.01 for exothermic reactions ($Da=200$) at $M_{t}=0.2$ and 0.8, respectively. It can be noticed that heat release through exothermic reactions also intensify expansion motions which result in an increase of reverse SGS flux of kinetic energy. It can also be noticed that the magnitude of conditional average of normalized SGS flux $\varPi _{l} / \epsilon _{T}$ in strong compression regions is much higher than that in strong expansion regions, which is consistent with a general physical insight that the flow statistics in strong compression regions are more intermittent than that in strong expansion regions for highly compressible turbulence (Wang et al. Reference Wang, Shi, Wang, Xiao, He and Chen2012a, Reference Wang, Gotoh and Watanabe2017).

Figure 16.

Average of normalized SGS flux $\varPi _{l} / \epsilon _{T}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ for different filter widths $l/\eta =16$, 24, 32, 48, 64 in expansion regions for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (b) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.0$, $Da=200$.

Figure 17 shows instantaneous isosurfaces of SGS flux coloured by normalized filtered velocity $\theta _{l}/ \theta _{l}^{\prime }$ at $\varPi _{l} / \varPi _{l}^{\prime }=2$ with the filter width $l/\eta =16$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8 at $t/\tau =35$. At $M_{t}=0.2$ for isothermal reaction ($Da=2$) as shown in figure 17(a), the isosurfaces of $\varPi _{l} / \varPi _{l}^{\prime }=2$ can be identified as blob-like structures which are nearly uniformly distributed over the entire flow domain. Meanwhile, the fraction of compression regions and expansion regions on the isosurfaces are nearly equal to each other. For exothermic reaction ($Da=200$) at $M_{t}=0.2$ as shown in figure 17(b), the fraction of expansion regions are exceeded by the fraction of compression regions on the isosurfaces of $\varPi _{l} / \varPi _{l}^{\prime }=2$, which can be identified by sheet-like structures. This observation suggests that direct SGS flux of kinetic energy increases substantially due to the increase of flow compression induced by heat release at low turbulent Mach number. At $M_{t}=0.8$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions as shown in figures 17(c) and 17(d), respectively, with the increase of flow compressibility, the fraction of sheet-like structures of $\varPi _{l} / \varPi _{l}^{\prime }=2$ increases due to the generation of shocklets, associated with strong flow compression. Heat release through exothermic reaction ($Da=200$) promotes the formation of a larger patch of sheet-like structures of $\varPi _{l} / \varPi _{l}^{\prime }=2$ compared with the structures for the isothermal reaction situation at $M_{t}=0.8$, and consequently increases the direct SGS flux of kinetic energy.

Figure 17.

Isosurfaces of SGS flux coloured by normalized filtered velocity $\theta _{l}/ \theta _{l}^{\prime }$ at $\varPi _{l} / \varPi _{l}^{\prime }=2$ for the filter width $l/\eta =16$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions, at turbulent Mach number $M_{t}=0.2$ and 0.8 at $t/\tau =35$: (a) $M_t=0.2$, $Da=2$, $\varPi _{l} / \varPi _{l}^{\prime }=2$; (b) $M_t=0.2$, $Da=200$, $\varPi _{l} / \varPi _{l}^{\prime }=2$; (c) $M_t=0.8$, $Da=2$, $\varPi _{l} / \varPi _{l}^{\prime }=2$; (d) $M_t=0.8$, $Da=200$, $\varPi _{l} / \varPi _{l}^{\prime }=2$.

Figure 18 shows isosurfaces of SGS flux coloured by normalized filtered velocity $\theta _{l}/ \theta _{l}^{\prime }$ at $\varPi _{l} / \varPi _{l}^{\prime }=-0.5$ with the filter width $l/\eta =16$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8 at $t/\tau =35$. At $M_{t}=0.2$ for isothermal reaction ($Da=2$) as shown in figure 18(a), the isosurfaces of $\varPi _{l} / \varPi _{l}^{\prime }=-0.5$ can be identified as blob-like structures. However, more sparse distributions are observed compared with isosurfaces of $\varPi _{l} / \varPi _{l}^{\prime }=2$ at the same flow parameters. For exothermic reaction ($Da=200$) at $M_{t}=0.2$ as shown in figure 18(b), more blob-like structures associated with strong expansion regions are observed compared with isothermal reaction cases at $M_{t}=0.2$. Thus, heat release induced expansion motions substantially enhance the reverse SGS flux of kinetic energy for exothermic reaction at low turbulent Mach number $M_{t}=0.2$. For isothermal ($Da=2$) and exothermic ($Da=200$) reactions at $M_{t}=0.8$ as shown in figures 18(c) and 18(d), respectively, the field is largely occupied by blob-like structures associated with strong flow expansion. Furthermore, heat release leads to minor changes of negative SGS flux structures in expansion regions at high turbulent Mach number $M_{t}=0.8$.

Figure 18.

Isosurfaces of SGS flux coloured by normalized filtered velocity $\theta _{l}/ \theta _{l}^{\prime }$ at $\varPi _{l} / \varPi _{l}^{\prime }=-0.5$ for the filter width $l/\eta =16$ for isothermal ($Da=2$) and exothermic (Da = 200) reactions, at turbulent Mach number $M_{t}=0.2$ and 0.8 at $t/\tau =35$: (a) $M_t=0.2$, $Da=2$, $\varPi _{l} / \varPi _{l}^{\prime }=-0.5$; (b) $M_t=0.2$, $Da=200$, $\varPi _{l} / \varPi _{l}^{\prime }=-0.5$; (c) $M_t=0.8$, $Da=2$, $\varPi _{l} / \varPi _{l}^{\prime }=-0.5$; (d) $M_t=0.8$, $Da=200$, $\varPi _{l} / \varPi _{l}^{\prime }=-0.5$.

Wang et al. (Reference Wang, Wan, Chen and Chen2018a) defined a variable $\beta _{l}$ to investigate the relation between the large-scale strain and the SGS stress,

(5.3)\begin{equation} \beta_{l} = \frac{\tilde{\tau}_{i j} \tilde{{\mathsf{S}}}_{{\mathsf{ij}}}}{|\tilde{\tau}||\tilde{\boldsymbol{\mathsf{S}}}|}, \end{equation}

where $|\tilde {\tau }|=\sqrt {\tilde {\tau }_{{ij}} \tilde {\tau }_{{ij}}}$ and $|\tilde {\boldsymbol{\mathsf{S}}}|=\sqrt {\tilde {{\mathsf{S}}}_{{\mathsf{ij}}} \tilde {{\mathsf{S}}}_{{\mathsf{ij}}}}$. Based on this definition, $\beta _{l}$ is further decomposed into a positive component and a negative component: $\beta _{l}=\beta _{l}^{+}+\beta _{l}^{-}$ (O'Brien et al. Reference O'Brien, Urzay, Ihme, Moin and Saghafian2014), where the positive component $\beta _{l}^{+}=\frac {1}{2}(\beta _{l}+|\beta _{l}|)$ and the negative component $\beta _{l}^{-}=\frac {1}{2}(\beta _{l}-|\beta _{l}|)$. The positive component $\beta _{l}^{+}$ indicates interscale kinetic energy backscatter and the negative component $\beta _{l}^{-}$ corresponds to kinetic energy forward-scatter (Germano et al. Reference Germano, Piomelli, Moin and Cabot1991; Piomelli et al. Reference Piomelli, Cabot, Moin and Lee1991). Figures 19 and 20 show the average of two components of $\beta _{l}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ with different filter widths $l/\eta =16$, 24, 32, 48, 64 for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. For isothermal reaction ($Da=2$) at $M_{t}=0.2$ as shown in figures 19(a) and 20(a), the positive component $\beta _{l}^{+}$ and the negative component $\beta _{l}^{-}$ are primarily found at $-5\leqslant \theta _{l} / \theta _{l}^{\prime } \leqslant 5$, suggesting that both flow compression and expansion contribute to the kinetic energy forward-scatter and backscatter. The magnitude for negative component $\beta _{l}^{-}$ is much larger than that for the positive component $\beta _{l}^{+}$, indicating that the forward-scatter of kinetic energy is stronger than the energy backscatter. For exothermic reaction ($Da=200$) at $M_{t}=0.2$, the positive component $\beta _{l}^{+}$ is found at $0\leqslant \theta _{l} / \theta _{l}^{\prime } \leqslant 5$ (figure 19b) and the negative component $\beta _{l}^{-}$ is found at $-10\leqslant \theta _{l} / \theta _{l}^{\prime } \leqslant 0$ (figure 20b). The observations reveal that flow expansion has a major contribution to the kinetic energy backscatter and the flow compression has a major contribution to the kinetic energy forward-scatter. For isothermal and exothermic reactions at $M_{t}=0.8$, as shown in figures 20(c) and 20(d), respectively, the negative components $\langle \beta _{l}^{-}| \theta _{l} / \theta _{l}^{\prime } \rangle$, which are close to –1, are found at $-10\leqslant \theta _{l} / \theta _{l}^{\prime } \leqslant -8$, suggesting the antiparallel alignment between the large-scale strain and the SGS stress in strong compression regions. Heat release through exothermic reactions at $M_{t}=0.2$ (figure 20b) and 0.8 (figure 20d) increase the antiparallel alignment between the large-scale strain and the SGS stress.

Figure 19.

Average of positive $\beta _{l}^{+}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ for different filter widths $l/\eta =16$, 24, 32, 48, 64 for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (b) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.8$, $Da=200$.

Figure 20.

Average of negative $\beta _{l}^{-}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ for different filter widths $l/\eta =16$, 24, 32, 48, 64 for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (a) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.8$, $Da=200$.

5.3. Helmholtz decomposition on SGS flux of kinetic energy

Helmholtz decomposition can be applied to decompose a compressible vector field into a solenoidal part and a dilatational component. For compressible flow, the Helmholtz decomposition is applied to a density-weighted variable $\sqrt {\rho } \boldsymbol {u}$ to enforce the positive-definiteness of the spectra of the solenoidal and dilatational kinetic energies (Kida & Orszag Reference Kida and Orszag1990, Reference Kida and Orszag1992; Miura & Kida Reference Miura and Kida1995; Pierre & Claude Reference Pierre and Claude2008; Praturi & Girimaji Reference Praturi and Girimaji2019; Donzis & John Reference Donzis and John2020). Taking the filtered density-weighted variable $\tilde {\boldsymbol {w}}$ as $\tilde {\boldsymbol {w}}=\sqrt {\rho } \tilde {\boldsymbol {u}}$, the filtered kinetic energy can be expressed as $\tilde {\boldsymbol {w}}^{2} / 2$. Thus, the filtered equation for $\tilde {\boldsymbol {w}}$ is expressed as (Wang et al. Reference Wang, Wan, Chen and Chen2018a)

(5.4)\begin{equation} \frac{\partial \tilde{w}_{i}}{\partial t}+\tilde{u}_{j} \frac{\partial \tilde{w}_{i}}{\partial x_{j}}+\frac{\tilde{w}_{i}}{2} \frac{\partial \tilde{u}_{j}}{\partial x_{j}}={-}\frac{1}{\sqrt{\bar{\rho}}} \frac{\partial \bar{p}}{\partial x_{i}}-\frac{1}{\sqrt{\bar{\rho}}} \frac{\partial \bar{\rho} \tilde{\tau}_{i j}}{\partial x_{j}}+\frac{1}{\sqrt{\bar{\rho}} R e} \frac{\partial \bar{\sigma}_{i j}}{\partial x_{j}}. \end{equation}

Applying Helmholtz decomposition on $\tilde {\boldsymbol {w}}$ yields $\tilde {\boldsymbol {w}}=\tilde {\boldsymbol {w}}^{s}+\tilde {\boldsymbol {w}}^{d}$, where $\tilde {\boldsymbol {w}}^{s}$ is the solenoidal component and $\tilde {\boldsymbol {w}}^{d}$ is the dilatational component. The two components $\tilde {\boldsymbol {w}}^{s}$ and $\tilde {\boldsymbol {w}}^{d}$ satisfy $\boldsymbol {\nabla } \boldsymbol {\cdot } \tilde {\boldsymbol {w}}^{s}=0$ and $\boldsymbol {\nabla } \times \tilde {\boldsymbol {w}}^{d}=0$, respectively. Multiplying (5.4) by $\tilde {w}_{i}^{X}$ yields (Wang et al. Reference Wang, Wan, Chen and Chen2018a)

(5.5)\begin{equation} \tilde{w}_{i}^{X} \frac{\partial \tilde{w}_{i}}{\partial t}+\tilde{w}_{i}^{X}\left(\tilde{u}_{j} \frac{\partial \tilde{w}_{i}}{\partial x_{j}}+\frac{\tilde{w}_{i}}{2} \frac{\partial \tilde{u}_{j}}{\partial x_{j}}\right)={-}\frac{\tilde{w}_{i}^{X}}{\sqrt{\bar{\rho}}} \frac{\partial \bar{p}}{\partial x_{i}}-\frac{\tilde{w}_{i}^{X}}{\sqrt{\bar{\rho}}} \frac{\partial \bar{\rho} \tilde{\tau}_{i j}}{\partial x_{j}}+\frac{\tilde{w}_{i}^{X}}{\sqrt{\bar{\rho}} R e} \frac{\partial \bar{\sigma}_{i j}}{\partial x_{j}}, \end{equation}

where, $X=s,d$ denotes the solenoidal component and dilatational component, respectively. The filtered equation for the average of the solenoidal (or dilatational) component of the large-scale kinetic energy is expressed as (Wang et al. Reference Wang, Wan, Chen and Chen2018a)

(5.6)\begin{equation} \frac{\partial}{\partial t}\left\langle\frac{1}{2}\left(\tilde{w}_{i}^{X}\right)^{2}\right\rangle=\left\langle A_{l}^{X}\right\rangle-\left\langle\varPhi_{l}^{X}\right\rangle-\left\langle\varPi_{l}^{X}\right\rangle-\left\langle D_{l}^{X}\right\rangle. \end{equation}

Here, $A_{l}^{X}$ is the nonlinear advection term; $\varPhi _{l}^{X}$ is the large-scale pressure–dilatation term; $\varPi _{l}^{X}$ is the SGS kinetic energy flux; $D_{l}^{X}$ is the viscous dissipation term. These terms are defined as (Wang et al. Reference Wang, Wan, Chen and Chen2018a)

(5.7)\begin{gather} A_{l}^{X} \equiv{-}\tilde{w}_{i}^{X}\left(\tilde{u}_{j} \frac{\partial \tilde{w}_{i}}{\partial x_{j}}+\frac{\tilde{w}_{i}}{2} \frac{\partial \tilde{u}_{j}}{\partial x_{j}}\right), \end{gather}

(5.8)\begin{gather}\varPhi_{l}^{X} \equiv{-}\bar{p} \frac{\partial}{\partial x_{i}}\left(\frac{\tilde{w}_{i}^{X}}{\sqrt{\bar{\rho}}}\right), \end{gather}

(5.9)\begin{gather}\varPi_{l}^{X} \equiv{-}\bar{\rho} \tilde{\tau}_{i j} \frac{\partial}{\partial x_{j}}\left(\frac{\tilde{w}_{i}^{X}}{\sqrt{\bar{\rho}}}\right), \end{gather}

(5.10)\begin{gather}D_{l}^{X} \equiv \frac{\bar{\sigma}_{i j}}{Re} \frac{\partial}{\partial x_{j}}\left(\frac{\tilde{w}_{i}^{X}}{\sqrt{\bar{\rho}}}\right). \end{gather}

It is straightforward to derive the following relations: $\langle A_{l}^{s}+A_{l}^{d}\rangle = 0$, $\varPhi _{l}^{s}+\varPhi _{l}^{d}=\varPhi _{l}$, $\varPi _{l}^{s}+\varPi _{l}^{d}=\varPi _{l}$ and $D_{l}^{s}+D_{l}^{d}=D_{l}$.

Figure 21 shows the average of solenoidal and dilatational components of the nonlinear advection: $-\langle A_{l}^{s} \rangle / \epsilon _{T}$ and $\langle A_{l}^{d} \rangle / \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. For isothermal ($Da=2$) reaction at turbulent Mach number $M_{t}=0.2$, $-\langle A_{l}^{s} \rangle / \epsilon _{T}$ and $\langle A_{l}^{d} \rangle / \epsilon _{T}$ are close to 0. For exothermic ($Da=200$) reaction at turbulent Mach number $M_{t}=0.2$, $-\langle A_{l}^{s} \rangle / \epsilon _{T}$ and $\langle A_{l}^{d} \rangle / \epsilon _{T}$ are non-negative, however, the values are also small. The results show that the kinetic energy transfer from the solenoidal mode to the dilatational mode by nonlinear advection is relatively weak for isothermal and exothermic reactions at $M_{t}=0.2$. At $M_{t}=0.8$ for isothermal and exothermic reactions, both $-\langle A_{l}^{s} \rangle / \epsilon _{T}$ and $\langle A_{l}^{d} \rangle / \epsilon _{T}$ decrease gradually with the increase of filter width at $10 \leqslant l/\eta \leqslant 256$, suggesting that the dilatational mode absorbs kinetic energy from the solenoidal mode at different length scales. Heat release through exothermic reaction at low and high turbulent Mach numbers exerts a small influence on the two components of the nonlinear advection.

Figure 21.

Average of solenoidal and dilatational components of the nonlinear advection: (a) $-\langle A_{l}^{s} \rangle / \epsilon _{T}$ and (b) $\langle A_{l}^{d} \rangle / \epsilon _{T}$.

Figure 22 shows the average of solenoidal and dilatational components of the pressure–dilatation: $\langle \varPhi _{l}^{s} \rangle / \epsilon _{T}$ and $\langle \varPhi _{l}^{d} \rangle / \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. As shown in figure 22(a), the solenoidal component of the pressure–dilatation, $\langle \varPhi _{l}^{s} \rangle / \epsilon _{T}$, decreases with the increase of filter width $l/\eta$ at $l/\eta \leqslant 60$ for isothermal and exothermic reactions at $M_{t}=0.8$. The average values $\langle \varPhi _{l}^{s} \rangle / \epsilon _{T}$ are always very small for isothermal and exothermic reactions at $M_{t}=0.2$ and 0.8. The average of the dilatational component of the pressure–dilatation $\langle \varPhi _{l}^{d} \rangle / \epsilon _{T}$ is shown in figure 22(b). It can be found that for isothermal reactions at $M_{t}=0.2$ and 0.8, the magnitudes of dilatational component of the pressure–dilatation $\langle \varPhi _{l}^{d} \rangle / \epsilon _{T}$ are negligibly small compared with values for exothermic reaction cases. At $M_{t}=0.2$ for exothermic reaction, heat release leads to an increase of the dilatational component of the pressure–dilatation. Particularly, the magnitude of $\langle \varPhi _{l}^{d} \rangle / \epsilon _{T}$ for exothermic reaction at $M_{t}=0.2$ is significantly larger than that for exothermic reaction at $M_{t}=0.8$ at all $l/\eta$ scales. The magnitudes of the solenoidal component of the pressure–dilatation $\langle \varPhi _{l}^{s} \rangle / \epsilon _{T}$ for all cases are much smaller than those of the dilatational component of the pressure–dilatation $\langle \varPhi _{l}^{d} \rangle / \epsilon _{T}$, revealing that the dilatational component of the pressure–dilatation is much more important than the solenoidal component in kinetic energy transfer.

Figure 22.

Average of solenoidal and dilatational components of the pressure–dilatation: (a) $\langle \varPhi _{l}^{s} \rangle / \epsilon _{T}$ and (b) $\langle \varPhi _{l}^{d} \rangle / \epsilon _{T}$.

Figure 23 shows the average of solenoidal and dilatational components of the SGS flux: $\langle \varPi _{l}^{s} \rangle / \epsilon _{T}$ and $\langle \varPi _{l}^{d} \rangle / \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. As shown in figure 23(a), the average solenoidal SGS flux $\langle \varPi _{l}^{s} \rangle / \epsilon _{T}$ is nearly independent of turbulent Mach number and heat release in current simulations. At $l/\eta < 30$, $\langle \varPi _{l}^{s} \rangle / \epsilon _{T}$ grows rapidly with the increase of filter width for all simulating cases and in the range of $30\leqslant l/\eta \leqslant 80$, $\langle \varPi _{l}^{s} \rangle / \epsilon _{T}\approx 1.0$ which suggests that the solenoidal component of SGS flux has a major contribution to the net transfer of kinetic energy from large scales to small scales in the range of $30\leqslant l/\eta \leqslant 80$. In contrast, the average dilatational SGS flux $\langle \varPi _{l}^{d} \rangle / \epsilon _{T}$ increases evidently with the increase of turbulent Mach number for isothermal reactions ($Da=2$) as shown in figure 23(b). The average dilatational SGS flux $\langle \varPi _{l}^{d} \rangle / \epsilon _{T}$ for exothermic reactions ($Da=200$) are apparently larger than that for isothermal reactions ($Da=2$) at $M_{t}=0.2$ and 0.8, indicating that heat release can significantly enhance the interscale transfer of kinetic energy through the dilatational component of SGS flux. The average dilatational SGS flux $\langle \varPi _{l}^{d} \rangle / \epsilon _{T}$ for all cases are smaller than 0.09, implying that the contribution of dilatational component to the net flux of kinetic energy is quite small as compared with its solenoidal counterpart.

Figure 23.

Average of solenoidal and dilatational components of the SGS flux: (a) $\langle \varPi _{l}^{s} \rangle / \epsilon _{T}$ and (b) $\langle \varPi _{l}^{d} \rangle / \epsilon _{T}$.

The p.d.f.s of the normalized dilatational pressure–dilatation $\varPhi _{l}^{d}/ \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8 are plotted in figure 24. It is observed that for isothermal reaction ($Da=2$) at $M_{t}=0.2$ as shown in figure 24(a), the p.d.f.s for $16 \leqslant l/\eta \leqslant 128$ are nearly symmetric with respect to the line $\varPhi _{l}^{d} / \epsilon _{T}=0$, indicating that the positive part of dilatational pressure–dilatation is almost cancelled by the negative part, leading to a negligible net contribution of pressure–dilatation to the dilatational kinetic energy transfer. For isothermal reaction ($Da=2$) at $M_{t}=0.8$ as shown in figure 24(c), the p.d.f. at $l/\eta =16$ is skewed toward the positive side, suggesting that the flow compression is more important than expansion in the dilatational mode of pressure–dilatation at small scales. For exothermic reactions ($Da=200$) at $M_{t}=0.2$ and 0.8, as shown in figures 24(b) and 24(d), respectively, the skewness of the p.d.f.s for both sides are increased and in addition, the right-hand tails are longer than their left-hand tails, suggesting that heat release induced flow compression plays a more important role than flow expansion in transferring dilatational kinetic energy through pressure work, especially at small scales.

Figure 24.

The p.d.f. of the normalized dilatational pressure–dilatation $\varPhi _{l}^{d}/ \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (b) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.8$, $Da=200$.

Figure 25 shows that p.d.f.s of the normalized dilatational SGS flux $\varPi _{l}^{d} / \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. For isothermal reaction ($Da=2$) at $M_{t}=0.2$ as shown in figure 25(a), the p.d.f.s for $16 \leqslant l/\eta \leqslant 128$ are nearly symmetric along $\varPi _{l}^{d} / \epsilon _{T}=0$, indicating that the positive part of dilatational SGS flux is almost cancelled by the negative part, leading to a negligible net SGS flux of dilatational kinetic energy. At $M_{t}=0.8$ for isothermal reaction ($Da=2$) shown in figure 25(c), the p.d.f. is skewed toward the positive side, suggesting that the direct SGS flux of dilatational kinetic energy is larger than the reverse flux of dilatational kinetic energy. For exothermic reactions ($Da=200$) at $M_{t}=0.2$ and 0.8 shown in figures 25(b) and 25(d), respectively, the p.d.f. is skewed toward the positive side, suggesting that the direct SGS flux of dilatational kinetic energy is significantly larger than the reverse flux of dilatational kinetic energy due to the effect of heat release.

Figure 25.

The p.d.f. of the normalized dilatational SGS flux $\varPi _{l}^{d}/ \epsilon _{T}$ for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (b) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.8$, $Da=200$.

Figure 26 shows the spatial–temporal average of energy transfer terms in the filtered equation of dilatational component of kinetic energy for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8 averaged at $t/\tau =25\sim 37$. For isothermal reactions ($Da=2$) at $M_{t}=0.2$ and 0.8 as shown in figures 26(a) and 26(c), respectively, the average values of pressure–dilatation $-\langle \varPhi _{l}^{d} \rangle / \epsilon _{T}$ are very small. For isothermal reaction at $M_{t}=0.8$ as shown in figure 26(c), with the increase of flow compressibility, the nonlinear advection term $\langle A_{l}^{d} \rangle / \epsilon _{T}$ and the average dissipation term $-\langle D_{l}^{d} \rangle / \epsilon _{T}$ become more important at scales $l/\eta \leqslant 10$. The magnitudes of $\langle A_{l}^{d} \rangle / \epsilon _{T}$ and $-\langle D_{l}^{d} \rangle / \epsilon _{T}$ decay with the increase of filter width at $l/\eta > 10$. For exothermic reactions at $M_{t}=0.2$ and 0.8 as shown in figures 26(b) and 26(d), respectively, the average pressure–dilatation terms $-\langle \varPhi _{l}^{d} \rangle / \epsilon _{T}$ become important, indicating that flow compression dominates the pressure–dilatation and leads to an increase of pressure work. Heat release evidently increases the dilatational SGS flux at $10 \leqslant /\eta \leqslant 100$ and the nonlinear advection term $\langle A_{l}^{d} \rangle / \epsilon _{T}$ at scales $l/\eta \leqslant 10$. The overall average of dilatational components of kinetic energy transfer terms $\langle A_{l}^{d}-\varPhi _{l}^{d}-\varPi _{l}^{d}-D_{l}^{d} \rangle / \epsilon _{T}$ for all cases are close to 0.

Figure 26.

Average of energy transfer terms in the filtered equation of dilatational component of kinetic energy for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (b) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.8$, $Da=200$.

The average of normalized dilatational SGS flux $\varPi _{l}^{d} / \epsilon _{T}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ with different filter widths $l/\eta =16$, 24, 32, 48, 64 for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8 are shown in figure 27. The conditional average of dilatational SGS flux is nearly independent of filter width $l /\eta$, for $16 \leqslant l /\eta \leqslant 64$ at $-5 \leqslant \theta _{l} / \theta _{l}^{\prime } \leqslant 5$ for all simulating cases. The conditional average $\langle \varPi _{l}^{d}/\epsilon _{T}|\theta _{l} / \theta _{l}^{\prime } \rangle$ is positive in the compression region and is negative in the expansion region. For exothermic reactions ($Da=200$) at $M_{t}=0.2$ and 0.8 as shown in figures 27(b) and 27(d), respectively, the positive of $\langle \varPi _{l}^{d}/\epsilon _{T}|\theta _{l} / \theta _{l}^{\prime } \rangle$ dominates over their negative counterparts, indicating that heat release enhances the direct SGS flux of dilatational kinetic energy. For non-reacting dilatational turbulence, Wang et al. (Reference Wang, Wan, Chen and Chen2018a) derived a simple model for the conditional average of dilatational SGS flux in the compression region $\theta _{l}\leqslant 0$. However, the model is no longer applicable for exothermic reactions in chemically reacting turbulence. A modified relation is proposed based on the derived model by Wang et al. (Reference Wang, Wan, Chen and Chen2018a),

(5.11)\begin{equation} \langle \varPi_{l}^{d}/\epsilon_{T}|\theta_{l} / \theta_{l}^{\prime} \rangle=\beta_{0}^{d}(\theta_{l} / \theta_{l}^{\prime})^{2}, \end{equation}

where, $\beta _{0}^{d}=0.0018$ and 0.11 for isothermal reactions ($Da=2$) at $M_{t}=0.2$ and 0.8 as shown in figures 27(a) and 27(c), respectively, and $\beta _{0}^{d}=0.39$ and 0.15 for exothermic reactions ($Da=200$) at $M_{t}=0.2$ and 0.8 as shown in figures 27(b) and 27(d), respectively. It can be found that $\beta _{0}^{c}$ for exothermic reaction is larger than that for isothermal reaction at a given turbulent Mach number. Based on this relation, the average normalized solenoidal SGS flux in compression region can be derived as

(5.12)\begin{equation} \langle \varPi_{l}^{s}/\epsilon_{T}|\theta_{l} / \theta_{l}^{\prime} \rangle=1+\beta_{0}^{s}(\theta_{l} / \theta_{l}^{\prime})^{2}, \end{equation}

where, $\beta _{0}^{s}=\beta _{0}-\beta _{0}^{c}$. Consequently, $\beta _{0}^{s}=0.0016$ and 0.08 for isothermal reactions ($Da=2$) at $M_{t}=0.2$ and 0.8, respectively, and $\beta _{0}^{s}=0.15$ and 0.10 for exothermic reactions ($Da=200$) at $M_{t}=0.2$ and 0.8, respectively. The coefficient $\beta _{0}^{s}$ for exothermic reactions are also larger than that for isothermal reactions. The results indicate that heat release enhances both dilatational SGS flux and solenoidal SGS flux of kinetic energy in compression regions.

Figure 27.

Average of normalized dilatational SGS flux $\varPi _{l}^{d} / \epsilon _{T}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ for different filter widths $l/\eta =16$, 24, 32, 48, 64 for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (b) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.8$, $Da=200$.

Figure 28 shows the average of dilatational SGS flux $\varPi _{l}^{d} / \epsilon _{T}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ with different filter widths $l/\eta =16$, 24, 32, 48, 64 in expansion regions for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8. A modified relation based on the heuristic model derived by Wang et al. (Reference Wang, Wan, Chen and Chen2018a) is given for the SGS flux in expansion region of chemically reacting turbulence as

(5.13)\begin{equation} \langle \varPi_{l}^{d}/\epsilon_{T}|\theta_{l} / \theta_{l}^{\prime} \rangle={-}\beta_{1}^{d}(\theta_{l} / \theta_{l}^{\prime})^{1.2}, \end{equation}

where, $\beta _{1}^{d}=0.05$ and 0.30 for isothermal reactions ($Da=2$) at $M_{t}=0.2$ and 0.8 as shown in figures 28(a) and 28(c), respectively, and $\beta _{1}^{c}=1.19$ and 0.49 for exothermic reactions ($Da=200$) at $M_{t}=0.2$ and 0.8 as shown in figures 28(b) and 28(d), respectively. Correspondingly, the average normalized solenoidal SGS flux in the expansion region can be derived as

(5.14)\begin{equation} \langle \varPi_{l}^{s}/\epsilon_{T}|\theta_{l} / \theta_{l}^{\prime} \rangle=1-\beta_{1}^{s}(\theta_{l} / \theta_{l}^{\prime})^{1.2}, \end{equation}

where, $\beta _{1}^{s}=\beta _{1}-\beta _{1}^{d}$. Accordingly, $\beta _{1}^{s}=0.15$ and 0.40 for isothermal reactions ($Da=2$) at $M_{t}=0.2$ and 0.8, respectively, and $\beta _{1}^{s}=0.31$ and 0.52 for exothermic reactions ($Da=200$) at $M_{t}=0.2$ and 0.8, respectively, are obtained.

Figure 28.

Average of normalized dilatational SGS flux $\varPi _{l}^{d} / \epsilon _{T}$ conditioned on the normalized filtered velocity divergence $\theta _{l} / \theta _{l}^{\prime }$ for different filter widths $l/\eta =16$, 24, 32, 48, 64 in expansion regions for isothermal ($Da=2$) and exothermic ($Da=200$) reactions at turbulent Mach number $M_{t}=0.2$ and 0.8: (a) $M_t=0.2$, $Da=2$; (b) $M_t=0.2$, $Da=200$; (c) $M_t=0.8$, $Da=2$; (d) $M_t=0.8$, $Da=200$.