In this section, we reiterate our study's focus by revisiting Searby's observations [6] and clearly defining our research subject. We have performed experimental observations of premixed flames propagating downward in an open-ended tube, adhering to Rayleigh's criterion [5]. Typically, a quarter-wavelength resonator exhibits opposing trends in velocity and pressure fields. Our laboratory observations, as depicted in Figure 2, have captured the process of thermoacoustic instability within such a resonator. Post-ignition, at the apex of the resonator, curved flames manifest hydrodynamic instability (or D-L, instability) with no detectable acoustic pressure. Midway down the tube, the emergence of a flat flame, resonating at a frequency marginally above the fundamental, is accompanied by the onset of nominal acoustic pressure. This frequency elevation is attributable to the accelerated sound speed within the resonator, a consequence of the hot gases produced during combustion. Following this phase, acoustic pressure stabilizes, sustaining the vibrating flat flame, indicative of primary acoustic instability. This is succeeded by a marked surge in acoustic pressure, coinciding with the formation of a corrugated flame surface—a hallmark of secondary instability. Such a phenomenon is exclusive to high laminar burning velocities and bears resemblance to Faraday instability. A pronounced vibration amplitude persists over 5-10 cycles before the flame transitions into a turbulent state, whereupon acoustic pressure sharply declines. This pattern consistently ensues post-secondary acoustic instability. It is pivotal to note that our study concentrates on elucidating the transition from secondary acoustic instability to turbulence.

Figure 2: 
Acoustic pressure at the bottom of the tube with sequential images for a propagating flame at Mix. 4

Figure 3 illustrates the time-dependent progression of the flame tip position. The flame advances at a uniform velocity until it reaches a specific juncture (roughly 30 to 50 cm from the resonator's apex), at which a distinct acceleration is observable. Complementary to Figure 3, Figure 4 displays a chronological array of flame images for various gas mixtures. For Mix. 1, which exhibits the lowest S_L and βM values, the flame's displacement speed (S_d) in the non-acoustic field emanating from the resonator's top is calculated to be 13.51, which is marginally higher than the actual S_L (8.5 cm/s). This increment is likely due to an increased flame surface area, a consequence of the D-L instability. An augmentation in flame surface area is evident at 1.30 seconds for Mix. 1 as depicted in Figure 4. A similar pattern is noted for other mixtures, with the S_d for Mix. 2 through 4 recorded at 17.14, 16.57, and 21.98 cm/s, respectively. These values represent a greater than 50% enhancement in S_d relative to S_L, a phenomenon previously attributed to the gas expansion effect stemming from density disparities between unburned and burned gases [11][12]. There is no measured acoustic frequency because the flame does not vibrate in this region.

Figure 3: 
Comparative analysis of flame tip position versus time for different gas mixtures Indicating primary and secondary acoustic instability regions

Figure 4: 
Sequential flame images demonstrating the propagation distance over time for various gas mixtures, corresponding to temporal flame tip positions in Figure 3

Moving on to the primary acoustic field, it can be observed for Mix. 1 between approximately 1.55 and 1.90 seconds, where the vibrating flat flame is a dominant feature (refer to Figure 4 at t = 1.62 s for Mix. 1). During this phase, S_d for Mix. 1 is measured at 8.73 cm/s, slightly lower than in the non-acoustic field, and approximates the S_L. This trend is consistent across the other mixtures, with S_d for Mix. 2 through 4 sequentially measured at 10.03, 14.60, and 15.42 cm/s, indicating values close to the S_L.

This suggests that the inherent instability of the flame, namely the D-L instability, is suppressed in the primary acoustic region, thus excluding the effect of flame curvature. Consequently, the values obtained are very similar to the theoretical flame propagation velocity. To elucidate this point further, Figure 5 presents the variation of S_d with S_L in the primary acoustic field.

As depicted, a linear relationship can be discerned, and the slope (= 1.01) of the fitting function is noted to be nearly 1, with a correlation coefficient of 0.84.

Figure 5: 
Displacement flame velocity (S_d) as a function of laminar burning velocity (S_L) in the primary acoustic field

In analyzing the acoustic frequency trends within the primary acoustic field, the frequencies for Mix. 1 through 4 were determined to be 91.95, 85.79, 85.98, and 85.84 Hz, respectively. These values are found to be very similar to the fundamental frequency, which is assumed to be 85 Hz for the speed of sound in air. Thus, it is evident that the acoustic frequencies in the primary acoustic field do not deviate from the fundamental frequency.

In examining the secondary acoustic field, the case of Mix. 1 can be observed between approximately 2.60 and 3.00 seconds, with a prominent characteristic being the formation of a corrugated structure on the flame surface. During this interval, S_d of Mix. 1 is noted to be 6.85 cm/s, which is the lowest when compared to both the non-acoustic and primary acoustic fields. A similar pattern is evident in other mixtures as well, where Mix. 2 through 4 exhibit S_d values of 5.65, 10.19, and 14.37 cm/s, respectively. These values represent the slowest flame velocities encountered during the flame's propagation within the resonator. It is crucial to understand that in this context, S_d refers to the average value of displacement flame velocity, not the instantaneous velocity during vibrations, and this average value of displacement flame velocity indicates a non-zero average displacement. Within the secondary acoustic field, the flame experiences 5-10 cycles of high amplitude vibrations before it transitions seamlessly into a turbulent flame.

In our examination of the secondary acoustic field, we focused on evaluating the growth rate of acoustic pressure. Figure 6 displays the time-dependent acoustic pressure for each of the gas compositions we selected.

The figure clearly shows that the acoustic pressure in each case undergoes exponential and explosive amplification over time. To conduct a quantitative analysis of the growth rate in this phase of amplification, we applied curve fitting using an exponential function to the crest of the acoustic fluctuations. The equation used for the curve fitting is as follows:

Figure 6: 
Grow rate of acoustic pressure in secondary acoustic instability with various gas mixtures

Here, the parameter ω, which denotes the slope of the growth rate is listed in the accompanying table for each figure. Y₀ and A are disregarded since they do not influence the grow rate but only impact the acoustic value itself. Notably, an increase in S_L or βM correlates with an increase in ω, suggesting that, as with the primary acoustic field found in previous studies [2], pressure coupling plays a crucial role in the growth rate of pressure waves during secondary acoustic instability.

Further analysis of acoustic frequency trends within the secondary acoustic field revealed values of 100.53, 110.53, 91.60, and 94.24 Hz for Mix. 1 through 4, respectively. These frequencies are 5-10 Hz higher compared to those in the primary acoustic field, which can be attributed to the expanded volume occupied by the burned gases within the resonator. This expansion leads to an increase in the speed of sound in the fluid, impacting the acoustic frequency.

Turning our attention to turbulent flames, let's start with Mix 1. As evidenced in Figure 3, the onset of turbulence is marked by a notable surge in S_d. Complementary to this, Figure 4 illustrates a rapid, explosive expansion in the flame's surface area, accompanied by the formation of characteristic eddies indicative of turbulent flames. In the case of Mix 1, S_d peaks at 151.07 cm/s, the highest velocity observed across all acoustic development stages. This high velocity trend is consistent across other mixtures as well, with Mix 2 to 4 showing S_d values of 139.95, 210.66, and 190.39 cm/s, respectively. These values represent the highest flame speeds during propagation within the resonator. Post the turbulent phase, the flame exhausts all the premixed gas in the resonator, leading to its natural extinction.

Furthermore, our study evaluates the acoustic decay rate in the damping zone of the turbulent flame. Figure 7 depicts the acoustic pressure over time for each gas mixture, revealing a uniform pattern of exponential and rapid decay in acoustic pressure across all mixtures.

Figure 7: 
Decay rate of acoustic pressure in turbulization with various gas mixtures

To quantify the decay rate in this damping phase, we employed exponential curve fitting on the crest of acoustic fluctuations. The calculated decay slope, represented by ω, is included in the table accompanying each figure. Intriguingly, an increase in S_L or βM results in a decrease in the absolute value of ω. This observation aligns with the shift in the relative position of secondary instability occurrence within the resonator, as shown in Figure 3. Specifically, a higher S_L or βM correlates with an upward shift in the onset of secondary instability. This shift suggests a shorter residence time for the turbulent flame. Further analysis of acoustic frequency trends within the turbulent revealed values of 129.0, 110.2, 125.5, and 114.2 Hz for Mix. 1 through 4, respectively Consequently, it is inferred that an increase in S_L or βM, which leads to a thinner flame thickness, precipitates an earlier occurrence of secondary instability, resulting in a diminished absolute value of the acoustic decay rate in the damping zone of the turbulent flame.

Figure 8 depicts the relationship of frequency in amplification zone (f_a) to acoustic frequency in damping zone (f_d) for various gas mixtures. It is observed that the acoustic frequency increases relatively in the damping zone of the flame. This can be attributed to the increase in heat release as the flame surface and velocity increase, acting as a scattering factor for the frequency.

Figure 8: 
Acoustic frequency in damping zone (f_d) as a function of Acoustic frequency in amplification zone (f_a) with various gas mixtures

Now, our focus shifts to analyzing the correlation between ω_a and ω_d in the amplification and damping zones, as derived from our experimental results. Figure 9 depicts the relationship of ω_a to ω_d for various gas mixtures we have utilized.

Figure 9: 
Decay rate (ω_d) in damping zone as a function of grow rate (ω_a) in amplification zone with various gas mixtures

To quantitatively analyze this relationship, we employed logistic regression analysis, one of the predictive models that also gives information about saturation, and fitted the data to the following equation:

As seen in Figure 9, in the amplification zone, an increase in ω_a leads to a decrease in the absolute value of ω_d, eventually reaching saturation at a specific value.

Additionally, an increase in the absolute value of ω_d in the damping zone correlates with a saturation of ω_a in the amplification zone. Utilizing extrapolation methods, the saturation values were determined as ω_a = 20.22 and ω_d = -14.85. It's important to reiterate that a high ω_a in the amplification zone physically signifies that the coupling strength between the flame and the acoustic field exceeds a critical threshold, leading to a rapid onset of nonlinear instability. This implies an earlier occurrence of secondary acoustic fields, which, in turn, increases the residence time of turbulent flames within the resonator. Consequently, we can infer that the increased residence time of turbulent flames in a confined space leads to a decrease in ω_d.

Lastly, to confirm the close relationship between the growth mechanism of secondary acoustic instability and pressure coupling, Figure 10 illustrates the correlation between ω_a and ω_d with the pressure coupling constant, βM.. As evident in the figure, both ω_a and ω_d exhibit a linear relationship with βM.. Below are the fitting functions and their respective correlation coefficients:

Figure 10: 
Dependence of growth rate (ω) in amplification and damping zone on βM values with various gas mixtures

In essence, as βM increases, the flame thickness diminishes, making the flame front more susceptible to pressure and temperature fluctuations. Subsequently, when the coupling strength between the flame and the acoustic field surpasses a critical threshold, it triggers rapid growth in secondary acoustic instability. Consequently, ω_a of the acoustic field increases, and due to this heightened ω_a, ω_d (in absolute terms) of the turbulent flame decreases.