Effects of Biogas Composition on the Edge Flame Propagation in Igniting Turbulent Mixing Layers

To understand the effects of both the turbulence intensity and biogas composition on the overall ignition performance, the temporal evolutions of the maximum non-dimensional temperature \(T_{\max }^+\) (where \(T^+ = ( \hat{T} - T_0 )/(T_\text {ad}^0 - T_0 )\)) and the volume defined by \(c \geqslant 0.9\) normalised by the energy deposition volume \(V_\text {sp} = ( 4/3 ) \pi R_\text {sp}^3\) are shown in Fig. 3.

The behaviour observed here agrees with earlier computational studies (Turquand d’Auzay et al. 2019b; Chakraborty and Mastorakos 2006; Chakraborty et al. 2007), with a rapid increase of the maximum temperature value during the energy deposition until it first reaches a value larger than the adiabatic temperature of the stoichiometric \(\hbox {CH}_4\)/air mixture (\(T^+ = 1\)) and then a maximum at \(t^+ = 1\). At \(1 \leqslant t^+ \leqslant 4\), the large thermal gradient present at the centre of the energy deposition region induces a large heat transfer rate to the cold reactants and leads to a rapid decrease of the maximum temperature. In case of successful propagation, \(T^+_{\max }\) stabilises to a value at most equal to the adiabatic flame temperature of the diluted stoichiometric mixture (\(T_\text {ad}^{0,\psi }\)). In the other cases, \(T^+_{\max }\) drops to a value too low to sustain combustion and the kernel eventually quenches (e.g. D05u08).

The temporal evolution of the burned gas volume is similar to what has been previously observed (Chakraborty et al. 2007), it decreases monotonically as \(u'/s_l^0\) increases. This is explained by an increase of the turbulent diffusivity (\(D_t \sim u' l_t\)) which alters the competition between the heat release and the thermal diffusion by increasing the latter. However, as the present work considers decaying turbulence, both the turbulent diffusivity and turbulent heat transfer rate decrease with time which results in an increasing growth rate. Consistent with the decrease of the laminar burning velocity shown in Fig. 1, the burned gas volume also decreases monotonically as \(\psi\) increases. Note that the difference is very small at early times as the growth is mainly driven by the very large temperature values (and thus very large flame speeds) found in the energy deposition region. The temporal evolution of \(V_{c \geqslant 0.9}/V_\text {sp}\) depends on the competition between the heat release due to combustion and the heat transfer from the hot gas kernel to the surrounding cold mixture, and the growth indicates that the former overcomes the latter. In the D05u04 and D00u08 cases, the heat transfer from the kernel supersedes the combustion heat release once the energy deposition effects have subsided (e.g. \(T^+_\text {max}\) decreases and converges towards the adiabatic flame temperature of the stoichiometric mixture). Consequently, a decrease in the burned gas volume can be observed, but as turbulence decays with time, so does the heat transfer to the surrounding mixture. The heat release rate from combustion is thus able to overcome again the heat transfer rate such that the kernel can resume its growth. In the most extreme cases, the shrinking of the kernel can also lead to a global quenching, which is indicated by the burned gas volume decreasing to zero (e.g. D05u08 and D025u08, whose evolution is not shown here for conciseness but follows that of D05u08).

Thus, both the turbulence and mixture composition strongly affect the flame kernel resulting from a thermal runaway. This is explained by the \(\hbox {CO}_2\) influence on the burning velocity and the heat release rate per unit flame area, while the turbulence affects both the flame surface area and the turbulent diffusivity. Hence the competition between the overall heat release (combined effect of the flame surface area and consumption rate per unit surface area) and the heat transfer rates is governed by both \(\psi\) and \(u'/s_l^0\) and ultimately decides on the success of the energy deposition.

Before analysing the flame structure in more details, it is worthwhile to visualise its development in three-dimensions as to understand better the interplay between the mixture fraction and fuel mass fraction gradients (i.e. the mixture fraction and fuel mass fractions iso-surface curvatures and the scalar dissipation rate), as well as the topology of the edge flame. Figure 4 illustrates the temporal evolution of the kernel development by showing both the stoichiometric mixture fraction and \(T^+ = 0.6\) iso-surfaces for different turbulence intensities.

A triple flame is found along the intersection between these two iso-surface, that visibly drives the kernel growth under laminar conditions. For the case D00u00, the \(\xi = \xi _\text {st}\) iso-surface remains unchanged in time whereas the volume defined by the temperature iso-surface increases in time. The growth rate is particularly pronounced along the \(\xi = \xi _\text {st}\) iso-surface compared with the relatively limited expansion of the kernel in the direction of the mixture fraction gradient, where mixtures outside the flammability range are encountered. This highlights the key role played by the triple flame propagation in the kernel development.

Under turbulent conditions (i.e. cases D00u04 and D00u08), the \(\xi = \xi _\text {st}\) iso-surface gets significantly deformed and wrinkled by the fluid motion which visibly impedes the edge flame propagation and decreases the kernel growth rate. This is mostly due to the combined action of large mixture fraction and fuel mass fraction gradients and iso-surface curvature. The kernel shape is also significantly affected by the surrounding fluid motion and its shape increasingly departs from the laminar one as the value of \(u'/s_l^0\) increases. Similarly to the laminar conditions, the kernel develops more readily in slightly rich mixtures as evidenced by the absence of hot regions in the lean part of the domain at late times (e.g. D00u08 at \(t^+ = 21\)). Furthermore, the absence of intersection between the temperature and mixture fraction iso-surfaces indicates that the lean branch of the triple flame may have collapsed on the stoichiometric mixture fraction as reported by Buckmaster (1996).

The dimensionless temperature (\(T^+\)), normalised fuel mass fraction (\(Y_{\text {CH}_4}^+ = Y_{\text {CH}_4} / ( 2 \xi _\text {st} Y_{\text {CH}_4}^\infty )\)) and normalised fuel reaction rate magnitude (\(\dot{\omega }_{\text {CH}_4}^+ = | \dot{\omega }_{\text {CH}_4} | / \max { | \dot{\omega }_{\text {CH}_4} |}\)) contours are shown in Fig. 5 for the laminar cases D00u00 and D05u00 at \(t^+ = 17.5\) in the xy mid-plane.

A triple flame structure at the stoichiometric mixture fraction iso-surface is visible for both D00u00 and D05u00 cases on Fig. 5. The fuel reaction rate magnitude appears larger around the stoichiometric mixture fraction iso-surface than towards the lean and rich sides. The peak is observed for slightly rich mixtures, which is consistent with the earlier findings of Fig. 1, while two premixed flames are visible on both the rich and lean sides of the diffusion flame stabilised on the \(\xi _\text {st}\) iso-surface. This finding is in good agreement with the observations previously reported by Ray et al. (2001) where using detailed chemistry simulations, the peak reaction rate magnitude was also observed for fuel-rich mixtures. The lean flame starts to extinguish along the symmetry axis, suggesting that mixtures whose mixture fraction is locally below the lean flammability limit are encountered.

Neither the shape nor the flame structure appear affected by the mixture composition, although as shown in Fig. 3, the growth rate is, which is also consistent with Fig. 1. Similarly, in the case where \(\hbox {CO}_2\) is present, the burned gas temperature has been found to be consistently lower than in the undiluted case, and this trend strengthens with increasing dilution.

Figure 6 presents the contours of \(T^+\), \(Y_{\text {CH}_4}^+\) and \(\dot{\omega }_{\text {CH}_4}^+\) for the turbulent cases D00u04 and D05u04 at \(t^+ = 12.25\). The previous observations qualitatively hold, but as turbulent motions wrinkle the flame front, the kernel shape is significantly affected and deviates considerably from the laminar shape. Note that as the flames studied here are nominally in the thickened flame regime, instances of local thickening may be observed due to the penetration of energetic eddies into the preheat zone.

The amount of \(\hbox {CO}_2\) dilution does not qualitatively change these observations, although, similarly to the laminar cases, the kernel growth rate is smaller and the burned gas temperature decreases when \(\hbox {CO}_2\) is added to the mixture.

As shown in Figs. 4, 5 and 6, the presence of a triple-flame is evident following successful ignition. As the kernel growth rate is governed directly by the propagation of the triple point, it is instructive to analyse the effects of the \(\hbox {CO}_2\) dilution and turbulence intensity on the edge flame propagation behaviour. Following Ruetsch et al. (1995) and accounting for the presence of \(\hbox {CO}_2\) in the mixture, the theoretical laminar edge flame speed can be expressed as \(s_e^\psi = s_l^{0,\psi } \sqrt{ \rho _0 / \rho _b^{0,\psi }}\), where \(\rho _b^{0,\psi }\) is the burned gas density for the stoichiometric diluted mixture. By using the fact that \(\rho _b^{0,\psi } \sim \rho _0 / ( 1 + \tau ^\psi )\) in unity Lewis number adiabatic stoichiometric combustion (\(\tau ^\psi = ( T_\text {ad}^{0,\psi } - T_0 ) / T_0\)), one gets : \(s_e^\psi = s_l^{0,\psi } \sqrt{ 1 + \tau ^\psi }\).

The Probability Density Functions (PDFs) of \(s_e / s_e^\psi\) are shown in Fig. 7a for \(u'/s_l^0 = 4\) at different time instants, while the cases with \(u'/s_l^0 = 8\) behave qualitatively similarly. At early times, the most probable values of the edge flame speeds of both diluted and undiluted mixtures are greater than their respective theoretical limit due to the lasting effects of the energy deposition. As time progresses, the PDFs broaden and the probability of locally finding a negative value of the edge flame speed becomes non zero and increases with increasing \(\psi\). A peak near \(s_e^\psi / s_e \approx 1\) can still be observed for both dilution values which suggests that the edge flame propagates at a speed that is consistent with the theoretical estimates in the regions in which the conditions allow for its propagation.

The flame kernel shrinks and it fails to exhibit self-sustained propagation when the probability of finding a negative value of \(s_e\) becomes larger than the one of finding a positive one, i.e. when the mean edge flame speed \(\langle s_e \rangle\) becomes negative. Figure 7b presents the temporal evolution of \(\langle s_e \rangle\) for different cases. As seen previously, the mean value is initially very large but decays relatively fast to a value smaller than the theoretical limit at \(t^+ \approx 3\) at which point different behaviours can be observed. For low values of \(u'/s_l^0\), the mean value stabilises at a value slightly smaller than the theoretical limit, and this value decreases with \(\psi\). This is explained by the small size of the kernel in the diluted mixtures compared to that in undiluted ones, which increases the flame area to volume ratio which in turns acts to increase the heat transfer rate from the hot gas kernel compared to the undiluted mixture. As the dilution increases, \(\langle s_e \rangle\) also decreases and may assume negative values suggesting that the edge flame mostly retreats along the \(\xi _\text {st}\) surface (e.g. cases D05u04 and D05u08). As the turbulence level increases, \(\langle s_e \rangle / s_e^\psi\) appears consistently lower for the diluted cases than in the undiluted ones.

Figure 8 presents the decomposition of the edge flame speed into its different components, i.e. \(s_z^+\) and \(s_d^+\) for different cases. For laminar conditions (see Fig. 8a), the mixture fraction field remains almost unchanged which implies \(s_z \approx 0\), and in turn implies that \(s_e \approx s_d\). In the turbulent cases, the most probable value of the contribution arising from \(s_z^+\) remains close to 0.0. On the other hand, the most probable value of the \(s_d^+\) contribution corresponds to the theoretical limit of the edge flame speed (i.e. \(s_l^{0,\psi } \sqrt{ 1 + \tau ^\psi }\)), indicating that this surface propagates locally against the local flow field. Overall, the displacement of the \(\xi = \xi _\text {st}\) surface contributes only moderately but negatively to the edge flame speed as observed from the most probable value of the edge flame speed being smaller than the most probable value of the contribution arising from \(s_d^+\). The addition of \(\hbox {CO}_2\) does not qualitatively change these conclusions but its main effect appears to be the increase of the probability of finding negative values of \(s_d^+\), and to skew the PDFs of \(s_z^+\) towards the negative values. It can be noted that in cases where the mixture fraction surface has a velocity distribution with a non-zero mean along the edge flame, these conclusions might not hold entirely, and this is beyond the scope of the current investigation and thus will not be addressed further.

Finally, it is worthwhile to consider the influence of the biogas composition on the influence of strain rate, curvature and scalar gradients on the local behaviour of the edge flame speed \(s_e\). These statistics are presented in Figs. 9, 10 and 11 at \(t^+ = 1.75, \, 5.25\) and 8.75 to illustrate the time evolution at different stages after thermal runaway, i.e. shortly after the end of the energy deposition, in a transition state where the energy deposition effects may still be observable and after these effects have completely subsided, respectively.

As shown in Fig. 9a and similarly to previous studies on localised ignition of mixing layers (Chakraborty and Mastorakos 2006), the edge flame speed appears weakly correlated with \(|\nabla Y_\text {fu} |\) at early times, but the correlation strengthens with time. For large values of the gradient, i.e. \(|\nabla Y_\text {fu} |\times \delta _z^\psi / Y_{\text {fu}}^\infty \approx 0.35\), this correlation sign changes to negative while for both moderate and small values of \(|\nabla Y_\text {fu} |\) it remains positive. At \(t^+ = 5.25\), negative values of the edge flame speed are visible at low values of \(|\nabla Y_\text {fu} |\), but this tendency disappears later. This coincides with the time at which the triple flame structure is fully established following the decay of the thermal runaway effects induced by the localised ignition. It can indeed be observed from Fig. 3 that in the cases where a successful self-sustained kernel growth is obtained, for \(t^+ \geqslant 5\), \(T^+_\text {max}\) stabilises to a time independent value corresponding the stoichiometric adiabatic temperature of the diluted mixture. Small values of \(|\nabla Y_\text {fu} |\) in the progress variable range \(0.55 \leqslant c \leqslant 0.95\) at early stages of the flame kernel evolution imply a depletion of fuel due to high temperature, which in turn leads to negative values of \(s_e\) due to the contributions of \(s_z\) and the molecular diffusion component of \(s_d\). Once the edge flame structure is established beyond the period of influence of external energy deposition, the mean edge flame speed assumes positive values for all values of \(|\nabla Y_\text {fu} |\times \delta _z^\psi / Y_\text {fu}^\infty\). The addition of \(\hbox {CO}_2\) does not qualitatively affect these conclusions, especially at early times where \(\langle s_e \rangle _{|Y_\text {fu} |}\) is similar regardless of the value of \(\psi\). However, a difference arises in the value of \(|\nabla Y_\text {fu} |\times \delta _z^\psi / Y_\text {fu}^\infty\) at which the peak edge flame speed is found, which increases with \(\psi\). Furthermore, across most of the range of \(|\nabla Y_\text {fu} |\), the normalised mean edge flame speed decreases as \(\psi\) increases, which indicates potential long lasting effects of the stretch rate history. Indeed, without these long lasting effects caused by the increased curvature of the fuel mass fraction iso-surface as \(\psi\) increases, the normalised edge flame speed curves should collapse.

Figure 9b presents the mean value of \(s_e\) conditioned on \(|\nabla \xi |\). Following previous studies (Chakraborty and Mastorakos 2006), a similar non monotonic \(|\nabla \xi |\) dependence of \(s_e\) has been observed here. For the non-premixed mode of combustion, an increase of \(|\nabla Y_\text {fu} |\) acts to increase \(|\nabla \xi |\), whereas in the premixed mode, an increase of \(|\nabla Y_\text {fu} |\) may be associated with small values of \(|\nabla \xi |\). At early times, irrespective of the biogas composition, a weak negative correlation has been found between \(|\nabla \xi |\) and \(|\nabla Y_\text {fu} |\) for the cases with \(u'/s_l^0 = 4.0\), as visible in Fig. 10. Thus, the combination of positive \(s_e\) and \(|\nabla Y_\text {fu} |\) correlation and negative correlation between \(|\nabla \xi |\) and \(|\nabla Y_\text {fu} |\) leads to the negatively correlated branch of \(s_e\) and \(|\nabla \xi |\). However, at later times (i.e. \(t^+ \geqslant 8.0\)) and regardless of the dilution, the interdependence between \(|\nabla \xi |\) and \(|\nabla Y_\text {fu} |\) becomes non monotonic and shows a positively correlated branch (see Fig. 10). This eventually leads to a predominantly positive correlation between \(s_e\) and \(|\nabla \xi |\). Increasing the amount of \(\hbox {CO}_2\) in the fuel stream does not qualitatively change these correlations, however, for large values of \(|\nabla \xi |\times \delta _z^\psi / \xi _\text {st}\), the value \(\langle s_e \rangle _{|\nabla \xi |} / s_e^\psi\) remains identical irrespective of the \(\hbox {CO}_2\) dilution.

Finally, the edge flame speed dependence on the tangential strain rate and on the fuel mass fraction iso-surface curvature are shown in Fig. 11 for the cases D00u04 and D05u04. It is evident from Fig. 11a that \(s_e\) is mostly negatively correlated with the curvature of the fuel mass fraction iso-surface, and the strength of this correlation increases with time. At early times, the negative correlation is almost linear, albeit weak, while at a later time a positively correlated branch appears for moderately positive values of the curvature, which originates from the non-linear curvature \(\kappa _m\) dependence of the sum of the normal diffusion and reaction components of the fuel mass fraction displacement speed with \(\kappa _m\) as reported in previous studies (Chakraborty et al. 2010; Hesse et al. 2009, 2012). It can be noted that the mixture composition seems to have only a relatively minor effect on the mean value of \(s_e\) conditioned upon \(\kappa _m\) as all curves almost collapse onto a single one and this irrespective of the time.

Figure 11b presents the edge flame speed response to the strain rate tangential to the fuel mass fraction iso-surface, that exhibits a moderately positive correlation that becomes strongly non-monotonic for intermediate times before weakening considerably at later times, while the magnitude of \(a_T\) decreases monotonously. This decrease in magnitude of \(a_T\) is due to the decaying nature of the turbulent field in which the kernel develops. Note that this might be different in presence of a turbulence generating mechanism, e.g. mean shear, but this is beyond the scope of the current investigation. The strong non-linear \(a_T\) dependence of \(s_e\) is consistent with previous findings (Chakraborty et al. 2010; Hesse et al. 2009, 2012) in which this was attributed to the fuel mass fraction displacement speed components (reaction, tangential and normal diffusion) correlation trends with the tangential strain rate, whose relative strengths lead to the complex overall behaviour observed here. Long after the effects of the energy deposition have subsided (e.g. \(t^+ \geqslant 5\)), a strong effect of the mixture composition can be noted on the mean value of \(s_e\) conditioned upon \(a_T\), much larger than observed from the response of \(s_e\) to \(|\nabla \xi |\), \(|\nabla Y_\text {fu} |\) or \(\kappa _m\). This is particularly visible when the turbulence has not yet significantly decayed (\(t^+ \approx 5.25\)), where for small positive values of \(a_T\), the mean value of \(s_e\) decreases strongly as \(\psi\) increases. On the contrary, for large values of \(a_T\), larger mean edge flame speed are obtained as \(\psi\) increases (within the considered range). This stronger influence of the mixture composition appears directly linked to the larger curvature found for the diluted mixture. Indeed, a negative correlation between the tangential strain rate and the fuel mass fraction curvature has been observed previously both experimentally (Renou et al. 1998) and numerically (Chakraborty and Cant 2004; Chakraborty 2007; Klein et al. 2006). This negative correlation is compounded here by the strong heat sink effect of added \(\hbox {CO}_2\) due to the chemical heat release. Thus, the edge flame develops in regions with high positive values of \(\kappa _m\) leading to small values of the tangential strain rate as \(\psi\) increases. This in turn leads to a smaller edge flame speed due to the positive correlation between \(s_e\) and \(a_T\) existing at all times for small values of \(a_T\) as shown in Fig. 11b. This difference in strain rate history during the kernel development directly leads to a decreasing mean edge flame speed with dilution at late times, even in the laminar case as observed in Fig. 7b where the mean edge flame speed was smaller than its theoretical estimate.

The findings regarding the edge flame speed statistics gathered from Figs. 7, 8, 9, 10 and 11 indicate that as the qualitative nature of the edge flame propagation does not change with the addition of \(\hbox {CO}_2\), the combustion models developed for conventional fuels (e.g. \(\hbox {CH}_4\)/air mixtures) are likely to hold in the case of partially-premixed biogas/air turbulent combustion.