# Research on the Impact of a Fluid Field on an Acoustic Field in Herschel-Quincke Tube
###### Abstract  
A study concerning the influence of flow on the Herschel-Quincke duct is presented here, which includes the numerical model, the acoustic source and the absorption condition called the Perfectly Matched Layer. For the excitation of a sound field, a normal mode wave is placed at the inlet of the tube. The function of PML is to simulate the infinite tube and avoid the reflection of acoustic wave. To investigate the influence of flow field on sound field, a coupled calculation method combining the finite element method and computational fluid dynamics is used to solve the linearized Euler equation, named the Galbrun equation. Firstly, the influence of the cross-section of the tube on the acoustic field is considered. Secondly, the effects of flow on the acoustic field is also investigated. Lastly, a comparative analysis of the simulation results reveals the influence of flow and other parameters of the tube on sound propagation. Both the Mach number and the cross-section ratio have an influence on the acoustic resonance, and the resonance frequency decreases with the increase in the cross-section ratio. Herschel-Quincke tube; finite element method; linearized Euler equation; computational fluid dynamics +
Footnote †: journal: _Sciencedence_  
0000-0001-6999-2227-1800][PERSON]. [PERSON], [PERSON], [PERSON], H.  
suitable form for finite element prediction is obtained by simplified formulas. After the simplification, the normal component of the average flow velocity is completely eliminated and the local values of density, tangential flow velocity and admittance are required. [PERSON] and [PERSON] [8] studied the sound wave propagation in an inhomogeneous duct composed of two semi-infinite uniform ducts with a steady flow. The close relation between energy conservation, time-reversal invariance and reciprocity with respect to the interchange between the source and observer, for sound waves propagating in an inhomogeneous duct with rigid or soft walls, was found. [PERSON] [9] found the reciprocal relationship of scaled reflection matrices in direct and reverse flow when a source is located at the end of the duct. He also obtained the equivalent one-dimensional reverse flow theorem, which can be well approximated to three-dimensional models at low frequencies, and this theorem relates acoustic power propagated at the two ends of the duct. This article also numerically verifies the interrelationships between multimodal propagation and low-frequency propagation in axisymmetric (circular and annular) ducts. [PERSON] [10] developed an analytical technique to research the control potential of the HQ tube for high-order modes. The acoustic field inside the HQ tube is modeled as a plane wave, while the sound field inside the main tube is modeled by expanding the sound field based on higher-order modes. The reduction performance of the variation of geometric parameters of HQ tubes (such as the axial position, length, distance between interfaces, and cross-sectional area) and the influence of the modal content of disturbance has been studied. [PERSON] et al. [11] studied the flow field in a square tube T-joint with time-dependent periodic inlet velocity and equal branch velocity based on previous experiments. They also found that the flow can endure a higher adverse pressure gradient before separation, compared with stable inlet conditions. [PERSON] et al. [12] used an HQ tube array in the model and combined it with a series connection of local liners to obtain the scattering matrix results for the combination installation of HQ resonators and acoustic pads. The main purpose of this method is to control pitch and broadband noise, which leads to the state of standing waves and improves linear efficiency. They compared the scattering matrix results obtained in the model with numerical and experimental data and obtained consistent results. [PERSON] et al.