AN ACOUSTIC ANALOGY MODEL FOR HELMHOLTZ RESONATORS WITH COOLING BIAS FLOW
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摘要: 亥姆霍兹共振器(HR)作为典型的被动消声装置, 常被安装于航空发动机和燃气轮机的燃烧室上用以吸收噪声进而抑制燃烧热声振荡. 在实际应用中, 为防止燃烧室内高温气体损坏HR, 常引入冷却气流从HR的背部空腔通过其颈部流入燃烧室, 以保护HR. 该冷却气流的温度一般显著低于燃烧室内的燃气温度. 将这样的HR安装到燃烧室上时, 该温差可能影响燃烧室中HR上、下游的声波与熵波关系, 并进而影响HR的吸声性能. 然而, 在以往研究HR对燃烧室内热声振荡的影响的模型中, 该温差的影响一般被忽略. 本文基于声类比的思想, 建立了一个可预测当带有冷却气流的HR被安装于一个一维声学管道中时的消声性能的理论模型. 该模型基于一维质量、动量和能量守恒方程, 在无黏性耗散作用, 忽略体积力、所有的外来热源和热扩散的假设下, 首次推导出了侧壁安装有带冷却气流的亥姆霍兹共振器的一维燃烧室中的带源项的波动方程. 该方程右侧的源项能体现出共振器对燃烧室内一维声场的影响, 通过该方程可以看到共振器所带来的声源/声耗散是由熵扰动与质量扰动项组成. 由此可以进一步看出由共振器温差所产生的熵扰动会以声源的形式进入燃烧室内的一维声波方程, 并显著地改变HR在其共振频率附近对燃烧室内声场的影响. 通过与已有的阶跃条件模型对比, 验证了该模型预测HR温差对管道内声场的影响的准确性.Abstract: Helmholtz resonators (HR), as a typical passive muffler device, are often installed in the combustor of aero engines and gas turbines to absorb noise and suppress combustion thermoacoustic oscillations. In practical applications, in order to prevent the high temperature gas in the combustion chamber from damaging the HR, a cooling airflow is often introduced from the back cavity of the HR into the combustor through its neck to protect the HR. The temperature of this cooling airflow is generally significantly lower than the temperature of the gas in the combustor. When such HR is installed on the combustor, the temperature difference may affect the relationship between the sound waves and entropy waves upstream and downstream of the HR in the combustor, and further affect the sound absorption performance of the HR. However, in previous models for studying the influence of HR on thermoacoustic oscillations in the combustor, the influence of this temperature difference is generally ignored. Based on the idea of acoustic analogy, we develop a theoretical model in this paper which can predict the acoustic performance of HR with cooling airflow. This HR is installed in a one-dimensional acoustic duct. The model is based on one-dimensional mass, momentum and energy conservation equations. Under the assumption of no viscous dissipation, ignoring volume forces, all external heat sources and thermal diffusion, we derived for the first time the wave equation with source term in a one-dimensional combustor with Helmholtz resonator with cooling airflow installed on the side wall.The source term on the right side of the equation reflects the effect of the resonator on the one-dimensional sound field in the combustor. It can be seen from this equation that the sound source/sink dissipation caused by the resonator is composed of entropy disturbance and mass disturbance terms. It can be further seen that the entropy disturbance generated by the temperature difference of the resonator will enter the one-dimensional sound wave equation in the combustor in the form of a sound source, significantly changing the effect of HR on the sound field in the combustor near its resonance frequency. By comparing with the existing jump condition model, we verified the accuracy of the model in predicting the effect of HR temperature difference on the sound field in the one-dimensional combustor.
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Key words:
- thermoacoustic oscillation /
- Helmholtz resonator /
- acoustic analogy /
- muffler /
- combustor
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3 HR温度1000 K时, 3个模型的压力扰动强度幅值、温度扰动和声吸收系数
3. The amplitude of pressure wave strength, temperature oscillationt and absorption coefficien of the three models with a HR temperature of 1000 K
4 3个模型的压力扰动强度幅值、熵波扰动强度幅值、温度扰动和声吸收系数
4. The amplitude of pressure wave strength, enthalpy wave strength, absorption coefficient and temperature oscillation of the three models
表 1 HR和燃烧室的几何与平均流参数
Table 1. Geometry and mean flow parameters of the HR and combustor
Parameter Value HR neck length L/m 0.005 neck sectional area $ {A_n}/{{\rm{m}}^2} $ 0.0001 mean cavity temperature $ {\bar T_n}/{\rm{K}} $ 500 ~ 1000 cavity volume $ V/{{\rm{m}}^3} $ 0.00025 neck mean Mach number $ {\bar M_n} $ 0.01 combustor upstream mean flow pressure $ {\bar p_1}/{\rm{MPa}} $ 2 pipe cross-sectional area $ {A_c}/{{\rm{m}}^2} $ 0.01π upstream mean flow temperature ${\bar T_1}/{\rm{K}}$ 1000 upstream mean Mach number $ {\bar M_1} $ 0.03 
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