3.3.1. Flames with repetitive extinction and ignition

Figure 6 presents the typical OH^* chemiluminescence signal, flame position ( $x_f$ ) and propagation speed ( $S_f$ ) in a FREI cycle. Ignition is marked by simultaneous peaks in OH^* chemiluminescence and flame speed profiles and is followed by a decay in both signals as the flame propagates upstream. The flame eventually extinguishes after traversing a characteristic distance.

Figure 6. (a) Flame position is plotted alongside the corresponding mean flow temperature of the unburnt reactants in a typical FREI cycle. (b) The OH^* chemiluminescence signal is plotted alongsidethe flame propagation speed ( $S_{f}$ ). The plots correspond to $Re = 48$ at an equivalence ratio of $1.0$ . In the figure, $t_{ie}$ represents the ignition-to-extinction time scale, $t_{ei}$ denotes the flame re-ignition time scale and $T$ is the time period of FREI oscillations.

The observed flame dynamics can be understood using the analytical model for flame propagation in narrow channels developed by Daou & Matalon (Reference Daou and Matalon2002). It should be noted that while the model was developed for the asymptotic limit where $2l_f/d_{i} \gt \gt 1$ (where $l_f$ is the flame thickness), the qualitative trends– which are the primary focus of the current study – remain valid for $2l_f/d_{i} \geq 0.135$ (Daou & Matalon Reference Daou and Matalon2002). Based on our experimental data, the minimum value of $2l_f/d_{i}$ corresponding to the conditions in this study is 0.21. Therefore, the model is applicable to the experimental space explored in this work. Therefore, we have

(3.6) \begin{align} V^2\ln {\left (V\right )}=\ -\kappa. \end{align}

The model non-dimensionalises flame propagation speed as $V={2}/{3} ({1}/{\bar {u}}\left |{{\rm d}x_f}/{{\rm d}t}\right |+1 )$ , which can be rewritten in terms of $S_{f}$ as $V={2}/{3} ({S_f}/{\bar {u}} )$ . The heat losses at the combustor walls are accounted for using a non-dimensional parameter, $\kappa$ , defined as

(3.7) \begin{align} \kappa =\frac {Eq_w^{\prime \prime }l_f^2}{R_o^2T_a^2r_i}, \end{align}

where $E$ is the activation energy of the one-step reaction that describes the chemical activity, $q_{w}^{\prime \prime }$ is the heat flux at the inner walls of the combustor tube, $l_{f}$ describes the flame thickness, $R_{o}$ is the universal gas constant and $T_{a}$ is the adiabatic flame temperature. The wall heat flux, $q_{w}^{\prime \prime }$ , can be expressed as $\bar {h} (T_a-T_{w,i} ),$ where $\bar {h}$ is the effective heat transfer coefficient. The plots in figure 5(a) depict that the inner wall temperatures are close to the mean flow temperatures at a given axial location. We can thus approximate $ (T_a-T_{w,i} )$ as $ (T_a-T_m )$ in the expression for $q_{w}^{\prime \prime }$ . Here, $ (T_a-T_m )$ can be further scaled as $({Q_RY_F})/{C_p}$ based on energy balance between the reactants and products (Daou & Matalon Reference Daou and Matalon2002; Law Reference Law2006). Thus, $\kappa$ reduces to

(3.8) \begin{align} \kappa =E\bar {h}\left (\frac {Q_RY_F}{C_p}\right )\frac {l_f^2}{R_o^2T_a^2r_i}. \end{align}

As evident from the above equation, for a fixed value of Reynolds number and equivalence ratio, $\kappa$ scales inversely with $T_a^2$ :

(3.9) \begin{align} \kappa \propto \frac {1}{T_a^2}. \end{align}

As the FREI flame auto-ignites in the primary heating zone and propagates upstream, it encounters reactants with progressively decaying mean flow temperatures (see figure 6a). Since $T_a$ scales as $ (T_m+\ {(Q_RY_F)}/{C_p} )$ , $T_a$ drops during the propagation phase, increasing $\kappa$ . This reduces $V$ as per (3.6), causing $S_f$ to drop during the propagation phase, and this is evident in figure 6(b), which depicts the same trend experimentally. To plot in section S4 of the supplementary materials depicts the inverse dependence of $V$ on $\kappa$ . Given that the flame speed ( $S_f$ ) scales with the reaction rate (Law Reference Law2006), $\omega$ , a corresponding decay is expected in $\omega$ , and is corroborated by the OH^* chemiluminescence signal of the flame (figure 6b), which also scales with the reaction rate.

Following the analytical model outlined in (3.6) and the plot in section S4 of the supplementary materials, the non-dimensional heat loss parameter, $\kappa$ , increases and reaches a local maximum of 0.18 as the upstream traversing flame nears extinction (Daou & Matalon Reference Daou and Matalon2002). This is accompanied by a simultaneous decay in $V$ , which drops to a minimum of 2/3 close to extinction. At this point, ${{\rm d}x_f}/{{\rm d}t}$ will reduce to zero (as per the definition of $V$ ), causing the flame to cease propagation and extinguish.

Figure 7 presents the ignition, extinction, flame speed and OH^* chemiluminescence characteristics of the baseline FREI regime as a function of the premixture Reynolds number. As $Re$ increases at a given equivalence ratio, the reactant mass flow rate increases. Mathematically, this can be expressed as

(3.10) \begin{align} \dot {m}=\ \rho _u\left (\pi r_i^2\right )\bar {u}=\ \rho _u\left (\pi r_i^2\right )\left (Re\frac {\nu }{2r_i}\right )\ \sim \left (MW\right )_r\omega. \end{align}

Figure 7. (a,b) Ignition and extinction location are plotted alongside the corresponding mean flow temperature of the unburnt reactants across the space of $Re$ at which FREI is observed. Panel (c) shows $\textit{OH}^*_{ign}$ plotted alongside the flame propagation speed at ignition, $S_{f,ign}$ . (d) Flame travel distance and repetition frequency of FREI cycles are plotted for different values of $Re$ . In the figure, solid lines correspond to stoichiometric conditions and dotted lines correspond to $\Phi =1.2$ .

The last term in (3.10) is based on the mass consumption rate of the reactants (Law Reference Law2006). Here $(MW)_r$ is the molecular weight of the reactant mixture and $\omega$ is the global reaction rate. Equation (3.10) implies that the reaction rate increases with an increase in the Reynolds number ( $\omega \ \sim Re$ ) to supplement the higher consumption rate of the reactants at higher $Re$ . This trend is corroborated by the plots in figure 7(c), which show that the OH^* chemiluminescence signal of the flame at ignition increases with an increase in the Reynolds number. Here, $S_{f}$ at ignition exhibits a similar trend against $Re$ owing to the direct correspondence between $\omega$ and $S_f$ (figure 7c). It is interesting to note that the mean flow temperature of the reactants at ignition follows a similar trend, and increases with $Re$ (figure 7b). Given that $T_a$ scales as $ (T_m+ {Q_RY_F}/{C_p} )$ , an associated increase in $T_{a}$ is anticipated at higher Reynolds numbers. This increment in $T_a$ with increasing $Re$ positively contributes to the required increase in the reaction rate ( $\omega$ ) at higher Reynolds numbers (alongside the contribution due to increased concentrations of the reactants at higher $Re$ ), as indicated by a positive correlation between $T_a$ and $\omega$ , in the following equation:

(3.11) \begin{align} \omega \ \sim AT_a^b\exp {\left (-\frac {E_a}{R_oT_a}\right )}\left [Fuel\right ]^c\left [Air\right ]^d. \end{align}

Here $b$ , $c$ and $d$ are constants specific to the fuel–oxidiser type. However, as illustrated in § 3.2,which formulates the heat transfer from the walls into the premixed reactants, the mean flow temperature ( $T_m$ ) of the reactants at a given axial distance ( $x$ ) drops with increasing $Re$ (figure 5a). Thus, to attain higher mean flow temperatures at higher $Re$ , the reactant mixture has to travel further downstream into regions of higher wall temperatures. This trend is evident in figure 7(a), which demonstrates that the ignition location moves downstream with increasing $Re$ . Similar observations suggesting a downstream shift in the ignition locations at higher Reynolds numbers were reported earlier by Maruta et al. (Reference Maruta, Kataoka, Kim, Minaev and Fursenko2005) and Di Stazio et al. (Reference Di Stazio, Chauveau, Dayma and Dagaut2016a ).

Once auto-ignition is achieved, the flame propagates upstream with a decaying profile in $S_f$ and $V$ (figure 6b). Equation (3.6) can be used to estimate the decay rate of the upstream travelling flames (drop in $V$ against $x$ ), $ {{\rm d}V}/{{\rm d}x}$ , as

(3.12) \begin{align} \frac {{\rm d}V}{{\rm d}x}=\frac {1}{V\left (1+2\ln {\left (V\right )}\right )}\left (\frac {2c}{T_a^3}\right )\left (\frac {{\rm d}T_a}{{\rm d}x}\right ), \end{align}

where $c=E\bar {h} ( {Q_RY_F}/{C_p} ) ({l_f^2}/{R_o^2r_i})$ is a constant. In the above equation, $ ( {{\rm d}T_a}/{{\rm d}x} )$ can be substituted with $ ( {{\rm d}T_m}/{{\rm d}x} )$ since $T_a$ $= [T_m+ {Q_RY_F}/{C_p} ]$ , where $ {Q_RY_F}/{C_p}$ is a constant. Thus, differentiating this equation yields $ ( {{\rm d}T_a}/{{\rm d}x} ) = ( {{\rm d}T_m}/{{\rm d}x} )$ . Evaluating the variation of $ {{\rm d}V}/{{\rm d}x}$ with respect to $Re$ can help us understand how the flame travel distance (distance between the ignition and extinction location) varies with the Reynolds number. To simplify the analysis, we can estimate a scale for $ {{\rm d}V}/{{\rm d}x}$ by evaluating it at $x_{ign}$ (ignition location):

(3.13) \begin{align} p=\ \frac {{\rm d}V}{{\rm d}x} \sim \frac {c^\prime e^{g x_{ign}}}{V_{ign}(1+2\ln {(V_{ign})})}. \end{align}

In the above equation, $c^\prime ={2cg T_{m,o}}/{ ({QY_F}/{C_p} )^3}$ and $V_{ign}=V(x_{ign})$ . The formulation employs an exponential profile to approximate $T_m$ as $T_{m,o}\exp (g x)$ for $x\le x_{ign}$ . This approximation is depicted in figure 5(a). Differentiating (3.13) with respect to $Re$ yields

(3.14) \begin{align} & \frac {\partial p}{\partial (Re)}\nonumber\\& \sim \frac {\left [\left [V_{ign}\!\left (\!1\!+\!2\ln\! {\left (V_{ign}\right )}\!\right )c^\prime e^{g x_{ign}}\!\right ]\!\left ( {\partial x_{ign}}/{\partial \left (Re\right )}\right )\!-\!\left [c^\prime e^{g x_{ign}}\!\left (3\!+\!2\ln\! {\left (V_{ign}\right )}\!\right )\!\right ]\!\left ( {\partial V_{ign}}/{\partial \left (Re\right )}\right )\!\right ]}{[V_{ign}(1+2\ln {(V_{ign})}]^2}. \end{align}

Figure 7 shows that $ ({\partial x_{ign}}/{\partial (Re )} )\gt 0$ (explained earlier) and the plot in section S5 of the supplementary materials depicts that $ ({\partial V_{ign}}/{\partial (Re )} )\lt 0$ in the FREI regime. A physical reasoning based on the laminar flame propagation theory is presented in section S6 of the supplementary materials to justify the observed variation of $V$ with $Re$ . Substituting these inequalities in the above equation yields

(3.15) \begin{align} \frac {\partial p}{\partial (Re)}\gt 0. \end{align}

The above-obtained inequality can now be used to deduce the trends in the travel distance ( $d_{tr}$ ) upon varying the Reynolds numbers. Following the definition of $p$ , we can obtain the following scaling law for $d_{tr}$ :

(3.16) \begin{align} p=\frac {{\rm d}V}{{\rm d}x} \sim \frac {V_{ign}}{d_{tr}}\Longrightarrow d_{tr} \sim \frac {V_{ign}}{p}. \end{align}

Differentiating the above scaling law with respect to Re gives

(3.17) \begin{align} \frac {\partial \left (d_{tr}\right )}{\partial \left (Re\right )} \sim \frac {1}{p}\left (\frac {\partial V_{ign}}{\partial \left (Re\right )}\right )-\frac {d_{tr}}{p}\left (\frac {\partial p}{\partial \left (Re\right )}\right ). \end{align}

Since ${\partial p}/{\partial (Re)}\gt 0$ (3.15) and $ ({\partial V_{ign}}/{\partial (Re )} )\lt 0$ (see the plots in the section S5 of the supplementary figure), the above equation yields

(3.18) \begin{align} \frac {\partial \left (d_{tr}\right )}{\partial \left (Re\right )}\lt 0. \end{align}

This implies that the flame’s travel distance decreases with increasing $Re$ , and the trend is evident in figure 7(d). However, since, $d_{tr}=x_{ign}-x_{ext}$ , we can now estimate the expected trend of $x_{ext}$ against $Re$ :

(3.19) \begin{align} \frac {\partial \left (x_{ext}\right )}{\partial \left (Re\right )}=\ \frac {\partial \left (x_{ign}\right )}{\partial \left (Re\right )}-\frac {\partial \left (d_{tr}\right )}{\partial \left (Re\right )}. \end{align}

In the above equation, ${\partial (x_{ign} )}/{\partial (Re )}\gt 0$ (figure 7a) and ${\partial (d_{tr} )}/{\partial (Re )}\lt 0$ (3.18). This implies that

(3.20) \begin{align} \frac {\partial \left (x_{ext}\right )}{\partial \left (Re\right )}\gt 0. \end{align}

The inequality implies that the extinction location moves downstream as the Reynolds number increases. This is corroborated by the trends observed in figure 7(a). Equation (3.20) also implies that the flame extinguishes at higher mean flow temperatures at higher $Re$ (since ${{\rm d}T_m}/{{\rm d}x}\gt 0$ in the extinction region of the FREI regime) and is evident from the plots in figure 7(b).

The plots in figure 7(d) suggest that the frequency of the FREI cycles increases with increasing Reynolds numbers. To understand this trend, let us break down the time period associated within a FREI cycle ( $T$ ) into different time scales. A FREI cycle has three major events: ignition, extinction and re-ignition. Accordingly, there are two associated time scales (figure 6b): the time between ignition and extinction ( $t_{ie}$ ) andthe time between extinction and re-ignition ( $t_{ei}$ ). Therefore, we have

(3.21) \begin{align} T=t_{ie}+t_{ei} .\end{align}

The time scale, $t_{ie}$ , describes the time associated with the flame to propagate upstream, post-ignition, and traverse a distance of $d_{tr}$ . Likewise, $t_{ei}$ describes the time associated with the fresh reactant mixture to travel a distance of $d_{tr}$ and auto-ignite at the ignition location, initiating the next FREI cycle. We can thus scale $T$ as

(3.22) \begin{align} \left (\frac {1}{f}\right ) \sim T\ \sim \frac {d_{tr}}{\left ( {{\rm d}x_f}/{{\rm d}t}\right )_{ign}}+\frac {d_{tr}}{\bar {u}}. \end{align}

Differentiating the above equation with respect to $Re$ , we obtain

(3.23) \begin{align} \frac {\partial T}{\partial \left (Re\right )}\! \sim \left (\!\frac {1}{\left ( {{\rm d}x_f}/{{\rm d}t}\right )_{ign}}+\frac {1}{\bar {u}}\right )\left (\frac {\partial (d_{tr})}{\partial \left (Re\right )}\right )-\frac {d_{tr}}{\left (\!\left ( {{\rm d}x_f}/{{\rm d}t}\right )_{ign}\right )^2}\left (\frac {\partial \left ( ( {{\rm d}x_f}/{{\rm d}t}\right )_{ign} )}{\partial \left (Re\right )}\right )-\frac {d_i d_{tr}}{{\nu Re}^2}. \end{align}

However, the plots in the section S5 of the supplementary figure depict that $({\partial ( ({{\rm d}x_f}/{{\rm d}t} )_{ign} )})/{\partial (Re )}\gt 0$ , while (3.18) illustrates that ${\partial (d_{tr})}/{\partial (Re )}\lt 0$ . Therefore, the above equation simplifies to

(3.24) \begin{align} \frac {\partial T}{\partial \left (Re\right )}\lt 0. \end{align}

Equation (3.24) implies that the time period of the FREI oscillations decreases with increasing $Re$ , and thus, justifies the increasing trend of FREI frequencies with $Re$ , depicted in figure 7(d). The trends described above are variations with respect to the Reynolds number. Similar trends can be obtained against the equivalence ratio as well.

As the equivalence ratio increases from 1.0 to 1.2, the flame thickness ( $l_f$ ) increases and the adiabatic flame temperature ( $T_a$ ) reduces. Thus, as per (3.8), $\kappa$ increases, which in turn drops $V$ (see (3.6)). Thus, the flame propagation speed ( $S_f$ ) is expected to drop with increasing $\Phi$ , and is corroborated by the plots in figure 7(c). The plots illustrate that both $S_f$ and OH^* chemiluminescence signals at ignition drop with increasing $\Phi$ . Ignition locations are comparable at the equivalence ratios of 1.0 and 1.2 across the space of $Re$ (figure 7a). However, the extinction location is observed to shift downstream reducing $d_{tr}$ at $\Phi =1.2$ . This trend can be understood by estimating how ${{\rm d}V}/{{\rm d}x}$ changes with increasing $\Phi$ . Following (3.12), we see that

(3.25) \begin{align} p\left (=\frac {{\rm d}V}{{\rm d}x}\right )\ \propto \frac {l_f^2}{\left ( {QY_F}/{C_p}\right )^2}\left (\frac {1}{V_{ign}\left (1+2\ln {\left (V_{ign}\right )}\right )}\right ). \end{align}

As $ {QY_F}/{C_p}$ and $V_{ign}$ tend to decrease, while $l_f$ increases as the mixture shifts from stoichiometric ( $\Phi =1.0$ ) to fuel-rich conditions ( $\Phi =1.2$ ), it is expected that $p$ will be higher at $\Phi =1.2$ . Thus,

(3.26) \begin{align} \frac {\partial p}{\partial \Phi }\gt 0\ \quad \text {for } \Phi \gt 1. \end{align}

Following a similar process as depicted earlier, we can formulate ${\partial (d_{tr} )}/{\partial (\Phi )}$ by differentiating (3.16) with respect to $\Phi$ :

(3.27) \begin{align} \frac {\partial \left (d_{tr}\right )}{\partial \left (\Phi \right )}=\frac {1}{p}\left (\frac {\partial V_{ign}}{\partial \left (\Phi \right )}\right )-\frac {d_{tr}}{p}\left (\frac {\partial p}{\partial \left (\Phi \right )}\right ) .\end{align}

However, since, $ ({\partial V_{ign}}/{\partial (\Phi )} )\lt 0$ (figure 7c) and $ ({\partial p}/{\partial (\Phi )} )\gt 0$ (see (3.26)) in fuel-rich conditions, the travel distance ( $d_{tr}$ ) is expected to drop as $\Phi$ increases from 1.0 to 1.2. This causes the flame to extinguish at higher values of $x_{ext}$ (at $\Phi\,=\,1.2$ ), in comparison with the stoichiometric conditions. A drop in the travel distance is also accompanied by a drop in the time period of FREI oscillation, leading to an increase in the repetition frequency at $\Phi =1.2$ (figure 7d).

3.3.2. Propagating flames

The OH^* chemiluminescence signature of a typical PF, alongside its instantaneous flame location and flame propagation speed, is plotted in figure 8. Similar to that observed in the FREI regime, the flame auto-ignites in the high-temperature primary heating zone and starts to propagate upstream into regions of lower mean flow temperatures. Interestingly, unlike FREI, flames in this regime develop instabilities (reflected as fluctuation in the OH^* chemiluminescence signal) as they propagate upstream. These instabilities grow in magnitude, turn violent over the period (figure 8b) and become evident in flame imaging as back-and-forth motion of the flame front while still continuing to propagate upstream (figure 9a).

Figure 8. (a) Flame position is plotted alongside the flame propagation speed for a typical PF cycle. (b) Plot of the corresponding OH^* chemiluminescence signal and the pressure signal from the microphone. The profiles correspond to the Reynolds number of $48$ . In the figure, $t_{ins}$ denotes the time associated with the flame to traverse a distance of $x_{ins}$ and $t_{ei}$ is the time between extinction and re-ignition.

Figure 9. (a) Image sequence depicting the violent back-and-forth motion of a PF at $Re=48$ for the baseline case. (b) Similar back-and-forth motion of the flame front at the same Reynolds number for $d/d_{i} = 18$ .

These fluctuations are accompanied by a reduction in the mean flame propagation speeds and OH^* chemiluminescence signal (figure 8b). When decomposed in the frequency space, it becomes evident that these fluctuations in the OH^* chemiluminescence signature exhibit a distinct peak frequency close to the natural harmonic of the combustor tube (figure 10b). Interestingly, the microphone captures pressure fluctuations at the same frequency (figures 8b and 10b). A phase plot between heat release rate (OH^* chemiluminescence fluctuations) and pressure fluctuations reveals that these fluctuations are coupled (figure 10a,b) and that the heat release rate fluctuations tend to lag behind the pressure fluctuations. For the plots corresponding to figure 10(a,b), the heat release signal was found to lag behind the pressure signal by a phase angle of 30.8 $^{\circ}$ (estimated from the cross-power spectral density at the frequency corresponding to the peak power density, presented in figure 10b). It is important to note that in the plots presented in figure 10, $p^{\prime }$ and $q^{\prime }$ represent the fluctuating components of the pressure and PMT signals, respectively, which were obtained by subtracting the moving mean values of these signals from the original data.

Figure 10. (a) Phase plot between $q^{\prime }$ and $p^{\prime }$ during the thermoacoustic phase of a PF. (b) The FFT of the $p^{\prime }$ and $q^{\prime }$ signals is plotted alongside the cross-power spectral density of $p^{\prime }$ and $q^{\prime }$ . (c) Thermoacoustic frequency ( $f_{ins}$ ) and the mean growth rate of the thermoacoustic instability is plotted against the Reynolds number. (d) The root-mean-square value of the pressure fluctuations and the PMT fluctuations are plotted across the space of $Re$ . It should be noted that the pressure fluctuations presented in the figure have been corrected to account for the spatial location of the microphone with respect to the instantaneous location of the flame.

The plots in figure 10(a,b) make it clear that the observed instability is thermoacoustic in nature, in which the fluctuating heat release rate at the flame adds energy to the acoustic field, causing the pressure fluctuations to amplify, which, in turn, causes velocity fluctuations upstream of the flame and aggravate the heat release rate fluctuations. These fluctuations, however, dampen as the flame propagates to reach the upstream end of the combustor tube and encounters the meshed constriction, where it extinguishes. The process repeats again when the reactant mixture re-ignites with a characteristic time delay, generating a new PF. The authors hypothesise that the developed thermoacoustic instability is responsible for the PF (observed at $\Phi =0.8$ ) to sustain till the upstream end of the tube, while unsteady flames at $\Phi =1.0$ and $1.2$ extinguish after a characteristic propagation distance. The instability causes the flame front to move back and forth, which enhances the mixing between the upstream reactant mixture and the product gases (figure 9). The resulting preheating of the reactants might explain why the flame is able to sustain till the upstream end of the tube, wherein the mean flow temperature of the upstream gases is close to 300K (ambient levels).

The observed thermoacoustic coupling phase is similar to that of a flame propagating in a tube filled with a quiescent reactant mixture, as reported by Dubey et al. (Reference Dubey, Koyama, Hashimoto and Fujita2021), Castela et al. (Reference Castela, Correa, Alam, Jason and Lacoste2021) and Flores-Montoya et al. (Reference Flores-Montoya, Muntean, Pozo-Estivariz and Martínez-Ruiz2023). However, the primary distinction lies in the nature of the flame behaviour: our observations correspond to a periodically repeating flame, whereas in the referenced studies, the premixed flame is established only once, as the reactants are consumed during propagation and are not continuously replenished by a steady reactant flow, as in the present study.

Figure 10(c) plots the variation of the observed thermoacoustic coupling frequency ( $f_{ins}$ ) against $Re$ . Here, $f_{ins}$ is found to increase with increasing $Re$ . The figure also plots the variation in the growth rate ( $\alpha _g$ ) of the OH^* chemiluminescence fluctuations, which tends to attain a state of limit cycle, as depicted in figure 8(b). The growth rate $\alpha_g$ tends to decrease with increasing $Re$ . However, the decay rate of the fluctuations ( $\alpha _d$ ) was found to remain at a near-constant value close to −8.23 (with a standard deviation of 1.2) across the space of $Re$ . The constancy of the decay rate suggests that meshed constriction at the upstream end of the tube, which is a geometrical parameter independent of the Reynolds number, causes the instability to decay while extinguishing the flame. It should be noted that the growth and decay rates of the OH^* chemiluminescence signal were obtained by approximating the envelope of the fluctuation (obtained from Hilbert transform of the signal) using an exponential profile (= exp( $\alpha t$ )) . It is also interesting to note that alongside the growth rates, the root-mean-square (r.m.s.) value of the fluctuations also tends to drop with increasing $Re$ (figure 10d).

The variation of the flame characteristics with Reynolds number is similar to those observed in the FREI regime. As the Reynolds number increases, the flame tends to ignite at higher mean flow temperatures, causing the ignition location to shift downstream (figure 11a). Concurrently, the spatial location where the PF develops thermoacoustic instability ( $x_{ins}$ ) also shifts downstream with increasing Reynolds numbers (figure 11a). The mean propagation velocity ( $\bar {S}_{f,ins}$ ) at which the flame propagates beyond $x_{ins}$ ( $x\lt x_{ins}$ ) is found to remain relatively constant (till the flame extinguishes) with a weak increment with $Re$ .

Figure 11. (a) Ignition locations and thermoacoustic coupling locations are plotted alongside their corresponding mean flow temperatures across $Re$ for the baseline configuration. (b) The mean flame propagation speed, for $x \leq x_{ins}$ , is plotted against $Re$ , alongside the frequency of repetition of PF cycles.

A time-scale analysis, analogous to that presented for the FREI regime can be used to estimate the trends in the repetition frequency (number of PF cycles per second) associated with PFs with increasing $Re$ . The OH^* chemiluminescence signature of the PF presented in figure 8(b) clearly reveals two dominant time scales: one associated with the propagation of the flame from $x_{ins}$ to the upstream end of the combustor tube ( $t_{ins}$ ) and the other associated with the re-ignition of the fresh reactant mixture initiating the next PF cycle, post-extinction of the present cycle ( $t_{ei}$ ). It is to be noted that the time scale associated with the flame to propagate from $x_{ign}$ to $x_{ins}$ is negligible in comparison with $t_{ins}$ and $t_{ei}$ (figure 8b). Scaling the dominant time scales in terms of the associated length and velocity scales, we obtain

(3.28) \begin{align} T\ \sim t_{ins}+t_{ei\ } \sim \frac {x_{ins}}{\overline{\left({dx_f}/{dt}\right)}_{ins}}+\frac {x_{ign}}{\bar {u\ }} .\end{align}

In the above equation, $\overline{ ( {{\rm d}x_f}/{{\rm d}t} )}_{ins}=\bar {S}_{f,ins}-\bar {u}$ . Differentiating the equation with respect to $Re$ , we obtain

(3.29) \begin{align} \frac {{\rm d}T}{{\rm d}\left (Re\right )} &\sim \left [ \frac {1}{\overline{\left ( {{\rm d}x_f}/{{\rm d}t}\right )}_{ins}}\left (\frac {{\rm d}x_{ins}}{{\rm d}\left (Re\right )}\right )+\frac {1}{\bar {u}}\left (\frac {{\rm d}x_{ign}}{{\rm d}\left (Re\right )}\right ) \right ]\nonumber\\ & - \left [ \frac {x_{ins}}{\left (\overline {\left ( {{\rm d}x_f}/{{\rm d}t}\right )}_{ins}\right )^2} \frac {{\rm d} \left ( \overline{\left ( {{\rm d}x_f}/{{\rm d}t}\right )}_{ins} \right )}{{\rm d}(Re)} + \frac {d_i x_{ign}}{\nu Re^2}\right ]. \end{align}

It should be noted that, $ {d ( \overline{ ({{\rm d}x_f}/{{\rm d}t} )}_{ins} )}/{d(Re)}$ in the above equation can be expressed as

(3.30) \begin{align} \frac {{\rm d} \left( \overline { \left( {{\rm d}x_f}/{{\rm d}t} \right)}_{ins} \right)}{{\rm d}(Re)} = \frac {{\rm d} ( \bar {S}_{f,ins} )}{{\rm d}(Re)} - \frac {\nu }{d_i} .\end{align}

It is evident from figure 11(b) that ${d ( \bar {S}_{f,ins} )}/{d(Re)} \sim 0$ . Thus, $ {d ( \overline { ({{\rm d}x_f}/{{\rm d}t} )}_{ins} )}/ {d(Re)} \sim 0$ since the second term in (3.30) is negligible. Additionally, the term $ {d_i x_{ign}}/{\nu Re^2}$ in (3.29) is negligible in comparison with the first two terms (since the term scales inversely with the square of $Re$ ). Thus, the expression for $ {{\rm d}T}/{d (Re )}$ reduces to

(3.31) \begin{align} \frac {{\rm d}T}{{\rm d}\left (Re\right )} \sim \frac {1}{\overline{\left ( \frac{{\rm d}x_f}{{\rm d}t}\right )}_{ins}}\left (\frac {{\rm d}x_{ins}}{{\rm d}\left (Re\right )}\right )+\frac {1}{\bar {u}}\left (\frac {{\rm d}x_{ign}}{{\rm d}\left (Re\right )}\right ) .\end{align}

Since both $ ( {{\rm d}x_{ins}}/{d (Re )} )$ and $ ( {{\rm d}x_{ign}}/{d (Re )} )$ are greater than zero (figure 11a), $T$ tends to increase with $Re$ , decreasing the repetition frequency of the PF cycle (figure 11b).

3.3.3. Combined flames

At the Reynolds number of $32$ and equivalence ratio of $0.8$ , a flame exhibiting characteristics of both FREI and PFs is observed. The OH^* chemiluminescence signature of the flame is plotted in figure 12, alongside its corresponding instantaneous flame position and flame propagation speeds. The image sequence depicting the flame dynamics is presented in figure 2(e).

Figure 12. (a) The position of the flame is plotted alongside the corresponding mean flow temperature of the unburnt reactants for a combined flame (CF). (b) Plot of the corresponding pressure signal alongside its OH^* chemiluminescence signature.

It is evident from the figures that there is no periodicity in the observed flame dynamics. The convective time scale ( $t_c$ ) is hence used to non-dimensionalise time and represent time-series variation in figure 12. The flame tends to exhibit a series of finite travel FREI cycles followed by a PF cycle (wherein the flame travels to the upstream end of the combustor tube). The number of FREI cycles between consecutive PF cycles was stochastic and is found not to exhibit a characteristic repetition pattern. Not only was the flame behaviour qualitatively switching between FREI and PF, but FREI descriptors (extinction locations, flame travel distance) showed quantitative changes between consecutive FREI cycles.

It is to be noted that the flame regime was consistently observed at $Re=32$ and $\Phi =0.8$ , across three independent experimental trials, even after extended waiting periods in each trial. As the observation of this flame type was limited to a single data point ( $Re=32$ , $\Phi =0.8$ ), further data analysis was not possible to establish any trends with respect to $Re$ . This requires an independent study that deals with meso-scale flame dynamics at $Re\lt 32$ and is beyond the scope of the present work.

3.3.4. Comparison between the flame regimes

This section details a comparison between the flame regimes in terms of their heat release rates, repetition frequency and flame propagation speeds. To make an effective comparison in terms of heat release, it is essential to establish the correlation between the intensity signal from the PMT and the theoretical output power. The theoretical output power ( $q_{th}$ ) can be estimated based on the mass consumption rate of the fuel ( $\dot {m_{f}}$ ) and the enthalpy of combustion ( $Q_R$ ). It can then be related to the PMT intensity ( $q_{pmt}$ ) using a power law formulation, as reported by Lee & Santavicca (Reference Lee and Santavicca2003). Section S7 of the supplementary materials details the method used to formulate the power law ( $q_{pmt} = Aq_{th}^b$ ) and the correlations established based on our experimental data. It should, however, be noted that the formulation is developed based on the data from steady-state flames.

To enable a meaningful comparison between the different unsteady flame regimes, a reference standard is necessary. For this purpose, the developed correlations between $q_{th}$ and $q_{pmt}$ are used to estimate the PMT output intensity for an equivalent steady-state flame that could hypothetically exist under the operating conditions of the unsteady flames (FREI, PF and CF). This estimate is denoted as $Q_{pmt,SS}$ and is compared against the experimentally observed $Q_{pmt}$ for these unsteady flames. It should be noted that $Q_{pmt}$ represents the integrated PMT signal output over a reference period $\tau$ ( $Q_{pmt} = \int ^{\tau } q_{pmt} {\rm d}t$ ), and is therefore an indicator of the total energy release for the period of $\tau$ . Similarly, $Q_{pmt,SS} = \tau \cdot q_{pmt,SS}$ , where $q_{pmt,SS}$ denotes the power output of an equivalent steady-state flame established under the same operating conditions, as estimated using the developed correlation (see section S7of the supplementary materials).

The plots in figure 13(a) clearly illustrate that the $Q_{pmt}$ values corresponding to FREI and PFs are higher in comparison with $Q_{pmt,SS}$ . This observation suggests that the net heat release rates in unsteady flames of FREI and PFs exceed those of steady-state flames (if stabilised under the same conditions). This phenomenon is likely attributed to the preheating of the upstream combustor walls during the establishment of unsteady flames inside the combustor tube (Ju & Maruta Reference Ju and Maruta2011; Kaisare & Vlachos Reference Kaisare and Vlachos2012). Since unsteady flames undergo cycles of ignition, propagation and extinction, the wall regions upstream of the ignition zone experience preheating during the propagation phase of the unsteady flames (as the flame travels upstream following ignition). Post flame extinction, the fresh reactant mixture passes through this preheated zone, causing the temperature of the unburnt reactant mixture to rise. Consequently, the increased reactant temperatures enhance the reaction rates, leading to a rise in $Q_{pmt}$ . The plots in figure 13(a) also suggest that the ratio $X = Q_{pmt}/Q_{pmt,SS}$ is higher for FREI regimes in comparison with PF regimes, and the value of $X$ tends to decrease with increasing Reynolds number in both regimes. It should be noted that estimating the absolute efficiency of these flame regimes (comparison against the theoretical output power) requires exhaust gas analysis and is beyond the scope of the current work.

Figure 13. (a) Plot of $Q_{pmt}/Q_{pmt,SS}$ in the parametric space where FREI and PFs are observed. (b) Variation of the repetition frequency across the space of Re in the FREI ( $\Phi\,=\,1.0$ and 1.2) and PF ( $\Phi\,=\,0.8$ ) regimes. (c) Peak flame propagation speed plotted against Reynolds number.

Other interesting parameters for comparing the unsteady flame regimes of FREI and PFs are the repetition frequency and the peak flame propagation speeds. The frequency plots in figure 13(b) clearly differentiate the two flame regimes: PFs have their repetition frequency nearly an order of magnitude lower than that of FREI flames. However, their peak flame speeds are comparable, as shown in figure 13(c). It is important to note that the peak flame speed of PFs is distinct from $S_{f,ins}$ (the flame speed following thermoacoustic coupling, plotted in figure 11d) and is nearly an order of magnitude lower than $S_{f,peak}$ .

3.4.1. Flames with repetitive extinction and ignition

The introduction of the flat flame (secondary heater) divides the FREI regimes into two sub-regimes: one that retains the qualitative features of the baseline FREI and another termed the D-FREI regime. The D-FREI regime is characterised by distinct OH^* chemiluminescence and flame speed profiles that include a post-ignition peak deviating from the baseline behaviour (figure 14b). The D-FREI was observed in a characteristic range of Reynolds number ( $Re$ ), equivalence ratios ( $\Phi$ ) and separation distances ( $d$ ). To understand the presence of D-FREI and the reasons for the deviation in the OH^* chemiluminescence signal from its baseline behaviour, we first examine the ignition–extinction characteristics of the flame.

Figure 14. (a) The position of the flame is plotted alongside the corresponding mean flow temperature of the unburnt reactants for a typical D-FREI. (b) The flame propagation speed is plotted alongside its OH^* chemiluminescence signature. The plots correspond to $Re=48$ and an equivalence ratio of $1.0$ for $d/d_{i} = 15$ .

As the secondary heater is introduced at different separation distances ( $d$ ), ignition locations (figure 15, plotted in solid lines) did not appear to change significantly, as ignition remained confined to the primary heating zone, which was unaffected by the addition of the secondary heater. However, the variations in the extinction locations (figure 15(a), plotted in dotted lines) were prominent when the flame deviated from its baseline behaviour to exhibit D-FREI. It is to be noted that D-FREI was observed only in a characteristic range of $Re$ and $d$ at stoichiometric conditions (figure 15a,b). At $d/d_i=18$ , the regime was observed for $Re\geq 48$ , and for $d/d_i=15$ , D-FREI was observed across the space of $Re$ . In the rest of the parametric space, the characteristics were comparable to the baseline FREI.

Figure 15. (a) Ignition and extinction locations are plotted against Reynolds numbers for different wall heating conditions at the equivalence ratio of $1.0$ . (b) Here, $x_{ign}$ and $x_{ext}$ are traced out for different values of $Re$ and $d/d_{i}$ at the equivalence ratio of $1.2$ . (c) The parameter $R$ is plotted against $Re/({\rm d}\Phi )$ across the FREI regime for all values of $d/d_{i}$ . (d) The FREI repetition frequency is plotted against $Re$ for all values of $d/d_{i}$ . The solid lines in this plot correspond to $\Phi =1.0$ , while the dashed lines correspond to $\Phi =1.2$ .

The plots in figure 15(a) depict that, in the D-FREI regime, the flame travels beyond its baseline extinction location (extinction location at the corresponding $Re$ and $\Phi$ in the baseline configuration) and extinguishes further upstream beyond the secondary heating zone. An image sequence depicting the corresponding flame dynamics is presented in figure 2(c). To comprehend this, letus assess how the introduction of the secondary heater at different separation distances changes the mean flow temperatures and the corresponding reaction rates at the baseline extinction location ( $x_{ext,b}$ ). For this, let us define a parameter, $R$ as

(3.32) \begin{align} R= \left[\frac {\exp { \left(- \frac{E_a}{R_oT_{m,d} (x_{ext,b} )} \right)}}{\exp { \left(- \frac{E_a}{R_oT_{m,b} (x_{ext,b} )} \right)}} \right]_{Re,\Phi },\end{align}

where $T_{m,d}(x_{ext,b})$ is the mean flow temperature at the baseline extinction location when the secondary heater is at a separation distance of $d$ and $T_{m,b}(x_{ext,b})$ is the mean flow temperature at the extinction location in the baseline configuration; both evaluated at a fixed value of $Re$ and $\Phi$ . The variation of $R$ across the space of $Re$ , $\Phi$ and $d$ is plotted in figure 15(c). The parameter $R$ serves as an indicator to characterise the extent to which the ability of a premixed reactant mixture to sustain combustion at $x_{ext,b}$ has increased due to the introduction of the secondary heater at a separation distance of $d$ .

The figure clearly shows that, for cases where the flame behaviour remains consistent with the baseline behaviour, $R$ stays below approximately 1.4. In contrast, cases exhibiting diverging behaviour (D-FREI) have $R$ values ranging from 1.9–4.2. The map distinctly divides the space into two regions, indicating that if the value of $R$ exceeds a certain threshold range, the flame is likely to sustain combustion reactions and diverge from baseline behaviour to exhibit D-FREI. It is to be noted that a similar map can be created if $R$ is evaluated based on the adiabatic flame temperature corresponding to $T_{m,d}(x_{ext,b})$ and $T_{m,b}(x_{ext,b})$ .

When the flame exhibits D-FREI behaviour, it propagates beyond the baseline extinction location into the secondary heating zone. Here, the flame traverses two distinct regions: region $A$ , characterised by progressively increasing mean flow temperatures of the upstream reactant mixture (figure 14a,b), followed by region $D_s$ , wherein the mean flow temperatures progressively decrease (figure 14a,b). These are additional zones introduced by the secondary heater. In region $A$ , since the upstream mixture temperature increases along the propagation direction, we expect a proportional trend in the flame temperature, flame speeds and reaction rates, and this is reflected as a coupled spike in the OH^* chemiluminescence and the flame speed plots in figure 14(b). The flame attains its absolute peak in the OH^* chemiluminescence and the flame speed profiles close to the exit of region $A$ (figure 14b). This peak is greater in magnitude than the one attained post-ignition. As the flame continues propagating into region $D_{s}$ , the OH^* chemiluminescence signal and the flame propagation speeds start to decay (figure 14b) as the flame encounters reactants with progressively decaying mean flow temperatures similar to that observed in the propagation phase of the baseline case, and finally extinguishes after a characteristic travel distance.

Extending these observations in the ignition–extinction characteristics, D-FREI is bound to have a change in its characteristic time scales in comparison with its baseline counterpart. It is evident from the OH^* chemiluminescence profile (figure 14b) that flame divergence from its baseline behaviour introduces an additional time scale, $t_{ip}$ , which characterises the time between ignition and the instant the flame attains its global peak in OH^* chemiluminescence and $S_f$ profiles. However, a comparison of the different time scales involved reveals that the time between extinction and re-ignition ( $t_{ei}$ ) is still the most dominant (figure 14b), and thus, the frequency variation with respect to Reynolds number is expected to follow the same increasing trend as that observed in the previous section (baseline case). However, $t_{ip}$ introduces a shift in the time period of D-FREI, and this shift is reflected in the frequency plot in the form of a negative offset (figure 15d).

3.4.2. Propagating flames

Propagating flames, unlike FREI, extinguish only at the upstream end of the combustor tube. Hence, they pass through regions $A$ and $D_{s}$ , and thus, exhibit a flame acceleration phase post-ignition. This is reflected in the OH^* chemiluminescence and flame speed signals as a heightened peak (figure 16a,b).

Figure 16. (a) Flame position is plotted alongside the corresponding mean flow temperature of the unburnt reactants. (b) Plot of the corresponding OH^* chemiluminescence signal and the pressure signal from the microphone. The profiles correspond to a Reynolds number of $48$ and $d/d_{i}=18$ . (c) Ignition locations and thermoacoustic coupling locations are plotted alongside their corresponding mean flow temperatures across $Re$ for all values of $d/d_{i}$ . (d) The mean flame propagation speed for $x \leq x_{ins}$ is plotted against $Re$ , alongside the frequency of repetition of PF cycles, across the parametric space of $d/d_{i}$ .

Similar to our observations in the previous section, the ignition locations are not altered by the introduction of the secondary heater (figure 16c). However, $x_{ins}$ show significant deviation from the baseline observations. Thermoacoustic instabilities are found to affect flame propagation only after the flame crosses the secondary heating zone, implying that $x_{ins}$ shifts upstream due to the introduction of the secondary heater.

Alongside this, the upstream shift in $x_{ins}$ is found to increase with $Re$ , which contrasts with the observations in the baseline case (figure 16c). This reversal in the trend in $x_{ins}$ against $Re$ might explain the reversal in the trend of PF repetition frequency (in comparison with the baseline observation) following (3.31). In the presence of the secondary heater, the repetition frequency of the PF cycle ( $f$ ) is found to increase with increasing $Re$ (figure 16d). However, $S_{f,ins}$ follows a similar trend to that observed in the baseline case and tends to increase with increasing $Re$ (figure 16d).

The thermoacoustic coupling frequency ( $f_{ins}$ ) is found to increase with increasing Reynolds number (figure 17a), while the growth rates ( $\alpha _g$ ) and the r.m.s. value of fluctuations exhibits a decreasing trend (figure 17b), which is synonymous with the observation in the baseline configuration. It should be noted that the fluctuations in pressure and heat release rates drop to the order of the noise level as the Reynolds number increases to $80$ , and hence, the corresponding values of $f_{ins}$ , $p^{\prime }_{rms}$ and $q^{\prime }_{rms}$ at $Re=80$ are not reported here.

Figure 17. (a) Thermoacoustic frequency ( $f_{ins}$ ) and the mean growth rate of the thermoacoustic instability is plotted against $Re$ . (b) The r.m.s. value of the pressure fluctuations and the PMT fluctuations are plotted across the space of $Re$ . The plots correspond to all the values of $d/d_{i}$ explored in the current work.