4.1. SETAI Behavior under Asymmetric Supply Structure
Previous investigations have shown that
LP and
φ collectively influence the interaction between sound pressure oscillation and heat release oscillation, determining whether SETAI can be encouraged [
9,
25]. Specifically, when
LP/λ
P complies with Equation (5), the mode shape within the premixed chamber corresponds to a quarter to a half wave. In this scenario, a pressure node will exist upstream of the flame stabilizer, as the far end (inlet) of the premixed chamber acts as a pressure antinode due to its near-closed acoustic boundary condition. It should be noted that there is a potential error regarding the far end of the premixed duct not completely adhering to a closed boundary condition. Thus, when the instantaneous pressure at the flame stabilizer exceeds its average value, the pressure will increase in the flow direction. This pressure difference between the two sides of the flame stabilizer will reduce the mixture flow rate, causing a flow rate oscillation that is phase-led by 90° relative to the pressure oscillation. In contrast, the flow rate oscillation will be phase-lagged by 90° relative to the pressure oscillation when Equation (6) is satisfied. For combustion with
φ = 0.9,
τ satisfies Equation (7) (where
TM is the oscillation period). This indicates that the phase lag of heat release oscillation is less than 180° relative to the flow rate oscillation. In this case, the phase difference between sound pressure oscillation and heat release oscillation would be less than 90°, promoting the oscillation as long as Equation (5) is also satisfied. When Equations (6) and (7) are simultaneously satisfied, however, the sound pressure oscillation and heat release oscillation are out of phase, resulting in a negative Rayleigh criterion that suppresses oscillation. In the combustion of
φ = 0.7, however, Equation (8) is satisfied, and therefore the condition of
LP/
λP satisfying Equation (6) can result in an oscillation promotion, while Equation (5) results in oscillation suppression. In a symmetric supply structure, both sides have the same effect of either promoting or suppressing oscillation. In an asymmetric supply system, a different
LP value may lead to the interaction between heat release oscillation and sound pressure oscillation promoting each other on one side but suppressing each other on the other side.
The ratio of
LP to
λP was calculated on each side to analyze the mechanism of encouraging or suppressing oscillation under an asymmetric supply structure, combining the results with those obtained from measuring
PRMS, as shown in
Table 2,
Table 3 and
Table 4. In asymmetric supply systems, the heat release oscillation is probably in a different phase between the two sides, generating two sources (OS
1 and OS
2, respectively) with different effects on pressure oscillation. Taking the results in
Table 2 as an example, in the symmetric supply system with
L1 =
L2 = 2.06 m,
LP/
λP is 0.439. Under this condition, both OS
1 and OS
2 promote oscillation, resulting in a strong oscillation with
PRMS as high as 4108 Pa. However, when
L2 is changed, OS
2 may suppress the oscillation, as observed in cases 29, 30, 31, and 38, resulting in an overall weak oscillation. Furthermore, it is notable that although OS
1 still has a strong promotion effect in cases 28, 36, and 37, these conditions only exhibit combustion noise without obvious oscillation.
The results in
Table 3 and
Table 4 also demonstrate the same regularity, where most cases show that the whole system can only be pushed into a strong oscillation when OS
2 has a promotion effect. An exceptional circumstance occurs in cases 58 and 59, where both OS
1 and OS
2 have a promotion effect, but the whole system does not exhibit oscillation. This may be because the sound pressure antinode is too close to the stabilizer [
24], resulting in a relatively weak flow rate oscillation and muting the thermo-acoustic interaction. These observations indicate that, for an asymmetric supply structure, a necessary but not sufficient condition for strong oscillation is that both
L1 and
L2 satisfy oscillation promotion.
Table 5 lists the phase of sound pressure oscillation at seven measurement points under various cases. These data clearly demonstrate that the sound pressure differs within the two premixed chambers in an asymmetric supply structure. In particular, the pressure oscillation phase at the close end (P
2 and P
3) is very close in the symmetric supply structure but differs significantly in some asymmetric supply structure cases. This difference in pressure oscillation phase between P
2 and P
3 implies the asynchronized heat release oscillation between two inlets of the combustion chamber. In the symmetric supply system, the flow rate oscillation at the combustion chamber inlet is almost synchronous between the two sides, resulting in synchronous heat release oscillation. Therefore, the coupling between pressure and heat release oscillation is also the same between the two sides, and OS
1 and OS
2 have the same effect on the system’s SETAI.
For asymmetric supply systems, it is important to note that although OS
1 and OS
2 still experience the same pressure oscillation within the combustion chamber, their heat release oscillation may not be synchronized. Specifically, during certain periods within an oscillation cycle, the heat release oscillation may increase sound pressure on one side but have a reducing effect on the other side. In this case, there is a possibility that either OS
1 or OS
2 may individually encourage SETAI, but their promotional effects may partly cancel each other out. Consequently, the oscillation in an asymmetric supply system is generally weaker than in the two corresponding symmetric systems, as reflected in
Table 2,
Table 3 and
Table 4 and
Figure 4,
Figure 5 and
Figure 6. The non-synchronous sound pressure oscillation between the two premixed chambers is the reason for the reduced oscillation under the asymmetric supply structure, and thus the system SETAI could potentially be controlled simply by altering the length of one premixed chamber.
Figure 8 compares the sound pressure oscillation in case 64 with the two corresponding symmetric systems. The oscillation condition in case 64 (
L1= 1.65 m and
L2 = 3.87 m) is almost the same as in the two symmetric supply structures, in terms of both frequency and amplitude. This is unlike the phenomenon observed in
Figure 2 or
Figure 3. Case 64 was specially designed such that one premixed chamber is longer than the other by half
λP. This implies that the pressure mode shape at the upstream position of the flame stabilizer is the same between the two sides, despite having different
LP. Under the same combustion conditions, a synchronous heat release oscillation was obtained between OS1 and OS2, resulting in pressure oscillation behavior that is as strong as that in the corresponding symmetric system. The same results were observed under the condition of
φ = 0.7, as shown in
Figure 9.
The oscillation mechanism of these two cases was further investigated by analyzing the corresponding
LP/
λP and the sound pressure oscillation phase at each measurement point, as shown in
Table 6 and
Table 7, respectively. In each case, both OS
1 and OS
2 satisfy the requirements for prompting oscillation. Furthermore, what is significant is that the sound pressure oscillation is almost identical between P
2 and P
3, indicating an in-phase waveform of sound pressure that drives a synchronous flow rate oscillation between the two sides. With this and the same
φ, the pressure oscillation promotion from OS
1 and OS
2 is synchronous, and hence their combined result is the same as that of the symmetric supply system. This further confirms that the system oscillation characteristic is determined by the effects on each side and their cooperation. The opposite phase between P
1 and P
4 is attributed to an additional half wave formed within the longer premixed chamber [
18].
4.2. SETAI Behavior under Asymmetric Combustion Condition
Table 8 presents the ratio of
LP to
λP on each side and the measured
PRMS for two different asymmetric combustion conditions, namely cases 66 and 67. In case 67, both OS
1 and OS
2 promote oscillation, but neither satisfies the requirement for promoting oscillation in case 66. This observation aligns with the findings illustrated in
Figure 7, and it supports the notion that the occurrence of oscillation in an asymmetric condition is not determined by the overall
φ but is dependent on the specific condition (combination of
LP/
λP and
φ) on each side. Under an asymmetric combustion condition where
φ varies between the two sides, the value of
τ relies on the local
φ, rather than the global
φ. This is logical since the heat release at each flame should be controlled by the local
φ, rather than being influenced by the total
φ [
24,
26]. Consequently, the heat release oscillation differs between the two sides, even when there is an identical flow rate oscillation. This discrepancy leads to an unsynchronized effect on the pressure experienced on both sides.
It can be hypothesized that altering
φ on each side, even with a consistent total
φ, may impact the oscillation characteristic of the system, similar to adopting an asymmetric structure for air/fuel supply. To validate this hypothesis, experiments were conducted under symmetric supply structures (
L1 =
L2 = 1.85 m) but with asymmetric combustion conditions, as detailed in
Table 1e.
Figure 10 illustrates the observed oscillation characteristic, while
Table 9 analyzes the respective effects of OS
1 and OS
2. Previous experiments have demonstrated that a
φ range of 0.7–0.8 and 0.9–1.4, respectively, needs to be combined with an
Lp/
λP range of (2
n/4, (2
n + 1)/4) and (
n/4, 2
n/4) in order to generate a promotion effect. Among cases 68–73, where
LP is 1.85 m,
φ values of 0.9, 1.1, and 1.3 satisfy this requirement. Consequently, both sides have a promotion effect in cases 72 and 73, which is consistent with the observations in
Figure 10 and
Table 7, where strong oscillation occurs under these two cases. In cases where OS
2 promotes oscillation but OS
1 suppresses it, weak oscillation was observed in case 69, while non-oscillation occurs in cases 70 and 71.
In the case of an asymmetric supply structure with a symmetric combustion condition, τ on two sides is identical, but the sound pressure waveform is different, resulting in distinct effects on system oscillation. In comparison, the situation for an asymmetric combustion condition differs in approach but yields similar results. Consequently, SETAI can also be controlled by generating different effects through the asymmetric φ distribution between the two sides. For instance, in case 72, where the global φ is 0.91, a strong oscillation is observed under the symmetric combustion condition (φ1 = φ2 = 0.9) with the same supply system. In case 71, however, the asymmetric combustion condition of φ1 = 0.7 and φ2 = 1.3 suppresses the oscillation, without modifying the total heat release or structure. This finding holds significant importance for controlling SETAI in actual combustion systems.