## 1 Introduction  
Premixed combustion thermoacoustic oscillations commonly occur in gas turbines, aircraft engines, and gas boilers. These oscillations disrupt the stable operation of the combustion system and cause burner resonance, which may lead to structural damage [1]. Additionally, the thermoacoustic oscillations can cause deviations in the burner operations and lead to fluctuations in the flame equivalence ratio and changes in the flame structure, which significantly increase \(\mathbf{NO_{x}}\) emissions [2]. In recent years, with the widespread adoption of carbon neutrality, traditional power equipment, such as gas turbines, has shifted from being fueled by natural gas to co-firing with hydrogen fuel. This shift hasexacerbated issues such as thermoacoustic oscillations and NO\({}_{\rm x}\) emissions. However, the accurate prediction of thermoacoustic oscillations remains challenging [3]. Therefore, it is crucial to effectively suppress combustion thermoacoustic oscillations and reduce NO\({}_{\rm x}\) emissions. Combustion thermoacoustic oscillations exhibit complex multi-field coupling characteristics and can induce unstable phenomena, such as flashback and blowout, making their efficient control a major challenge [4]. Generally, control methods for thermoacoustic oscillations can be classified into active and passive control [5]. Active control methods utilize advanced measurements and execution mechanisms combined with advanced control techniques to achieve closed-loop control of combustion thermoacoustic oscillations [6]. However, these methods are still limited by the frequency and reliability of actuators and have not been widely adopted in actual gas turbine combustion systems [7]. Passive control methods typically utilize mufflers and other techniques to eliminate combustion chamber sound waves or disrupt the thermoacoustic coupling process, which alters the flame heat release rate through pilot fuel injection [8; 9]. Compared with active control methods, passive control methods provide higher reliability and a simpler design, but their suppression frequency band is narrower [10]. In complex and variable thermoacoustic oscillation modes, such as high-order, multi-mode, and time-varying thermoacoustic oscillations, the suppression effect of passive methods is less effective. With the decarbonization of existing gas turbine fuels, such as hydrogen (H\({}_{2}\)) and ammonia (NH\({}_{3}\))-blended natural gas, the application of these fuels will result in more complex thermoacoustic oscillation behaviors [11; 12]. These challenges hinder the effective suppression of thermoacoustic oscillations. Therefore, developing control methods capable of managing complex and variable thermoacoustic oscillations is crucial. Compared with traditional active or passive control methods, the use of jet-in-crossflow can effectively suppress combustion thermoacoustic oscillations and adjust the flame structure and chemical reaction processes [13; 14]. By changing the coupling between flame heat release rate and sound wave, jet-in-crossflow can suppress the problem of combustion instability. However, the jet-in-crossflow method is complex and involves a flow-flame-acoustic coupling process. Although numerous researchers have confirmed the control effect of microjets on combustion instabilities, the suppression process is influenced by several parameters, and the underlying suppression mechanism remains unclear. Researchers have explored the influence of various jet media and parameters on thermoacoustic oscillations, including hydrogen, air, inert gases, and oxygen-rich gases. [PERSON] and [PERSON] [15] investigated the effects of annular N\({}_{2}\)/O\({}_{2}\) and CO\({}_{2}\)/O\({}_{2}\) jets on combustion instabilities and NOx emissions in lean-premixed methane flames, achieving a synchronous suppression effect. [PERSON] and [PERSON] [16] controlled the combustion instability and NOx emissions using an adjustable swirl jet in lean-premixed flames. [PERSON] et al.