Forcing of globally unstable jets and flames
In the analysis of thermoacoustic systems, a flame is usually characterised by the way its heat release responds to acoustic forcing. This response depends on the hydrodynamic stability of the flame. Some flames, such as a premixed bunsen flame, are hydrodynamically globally stable. They respond only at the forcing frequency. Other flames, such as a jet diffusion flame, are hydrodynamically globally unstable. They oscillate at their own natural frequencies and are often assumed to be insensitive to low-amplitude forcing at other frequencies. If a hydrodynamically globally unstable flame really is insensitive to forcing at other frequencies, then it should be possible to weaken thermoacoustic oscillations by detuning the frequency of the natural hydrodynamic mode from that of the natural acoustic modes. This would be very beneficial for industrial combustors. In this thesis, that assumption of insensitivity to forcing is tested experimentally. This is done by acoustically forcing two different selfexcited flows: a non-reacting jet and a reacting jet. Both jets have regions of absolute instability at their base and this causes them to exhibit varicose oscillations at discrete natural frequencies. The forcing is applied around these frequencies, at varying amplitudes, and the response examined over a range of frequencies (not just at the forcing frequency). The overall system is then modelled as a forced van der Pol oscillator. The results show that, contrary to some expectations, a hydrodynamically self-excited jet oscillating at one frequency is sensitive to forcing at other frequencies. When forced at low amplitudes, the jet responds at both frequencies as well as at several nearby frequencies, and there is beating, indicating quasiperiodicity. When forced at high amplitudes, however, it locks into the forcing. The critical forcing amplitude required for lock-in increases with the deviation of the forcing frequency from the natural frequency. This increase is linear, indicating a Hopf bifurcation to a global mode. The lock-in curve has a characteristic ∨ shape, but with two subtle asymmetries about the natural frequency. The first asymmetry concerns the forcing amplitude required for lock-in. In the non-reacting jet, higher amplitudes are required when the forcing frequency is above the natural frequency. In the reacting jet, lower amplitudes are required when the forcing frequency is above the natural frequency. The second asymmetry concerns the broadband response at lock-in. In the non-reacting jet, this response is always weaker than the unforced response, regardless of whether the forcing frequency is above or below the natural frequency. In the reacting jet, that response is weaker than the unforced response when the forcing frequency is above the natural frequency, but is stronger than it when the forcing frequency is below the natural frequency. In the reacting jet, weakening the global instability – by adding coflow or by diluting the fuel mixture – causes the flame to lock in at lower forcing amplitudes. This finding, however, cannot be detected in the flame describing function. That is because the flame describing function captures the response at only the forcing frequency and ignores all other frequencies, most notably those arising from the natural mode and from its interactions with the forcing. Nevertheless, the flame describing function does show a rise in gain below the natural frequency and a drop above it, consistent with the broadband response. Many of these features can be predicted by the forced van der Pol oscillator. They include (i) the coexistence of the natural and forcing frequencies before lock-in; (ii) the presence of multiple spectral peaks around these competing frequencies, indicating quasiperiodicity; (iii) the occurrence of lock-in above a critical forcing amplitude; (iv) the ∨-shaped lock-in curve; and (v) the reduced broadband response at lock-in. There are, however, some features that cannot be predicted. They include (i) the asymmetry of the forcing amplitude required for lock-in, found in both jets; (ii) the asymmetry of the response at lock-in, found in the reacting jet; and (iii) the interactions between the fundamental and harmonics of both the natural and forcing frequencies, found in both jets.