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.

1. Introduction

The Herschel–Quincke (HQ) tube is composed of a waveguide with side branches. It is a common noise reduction device and uses the path difference of sound waves in different tubes to form interference and achieve noise reduction.

C. L. Morfey [1] theoretically processed a sound field in a rigid duct with flow. The propagation of axial waves in the duct was studied by using impedance boundary conditions at the end. The results indicate the influence of axial flow and swirling flow on the radiated sound power for different types of sound sources. P. Merkli [2] compared the theoretical and experimental values of resonance frequency in a T-tube by simplifying the actual geometric model and found that the excited resonance frequencies had good agreement with the analysis results. M. K. Myers [3] showed that the boundary condition at an impermeable surface is not equivalent to the continuity requirement of an acoustic particle displacement. Selamet et al. [4] derived the acoustic propagation loss characteristics of an HQ tube. The nonlinear one-dimensional difference model, combined with the state equation of the ideal gas, was adopted to calculate the balance equations of mass, momentum, and internal energy. W. Eversman and D. Okunbor [5] developed a finite element code to predict the radiated acoustic field of an aft fan duct based on the assumption that the flow is steady and irrotational. The calculation results indicate that the continuity of pressure is accurately enforced. Sjoerd W. Rienstra and Walter Eversman [6] proposed a numerical finite element method to solve the problem of sound propagation in varying ducts with mean flow and acoustic lining, and a test geometry was used to verify the effectiveness of numerical calculation methods. Eversman [7] restructured the Myers acoustic boundary condition at an admittance wall in a non-uniform duct; the 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. Willi and Mhring [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. Eversman [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.

Lori A. Brady [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. J.S. Anagnostopoulos 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. Poirier 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. Redonf et al. [13] considered the acoustic scattering problem of an infinite tube covered with sound-absorbing materials and assumed that the waveguide is filled with the uniform mean flow. They derived non-reflective boundary conditions on artificial boundaries based on Dirichlet-to-Neumann (DtN) mapping using a modal decomposition method and demonstrated the effectiveness of the method through numerical validation. Liu and Yin [14] studied noise reduction in an HQ tube and established a numerical model to calculate acoustic characteristics using GT-POWER. Both the tube without flow and with mean flow were considered, while revealing the impact of the angle and the changes of the diameters during flow. Liu et al. [15] proposed a multi-dimensional computational fluid dynamics (CFD) method to calculate the transfer matrix of the muffler. The absorptive material, heat transfer and mean flow effects were considered. Due to the simplicity of the simulation model and the low noise generated by the flow, the predicted noise was very consistent with the measured results. Alonso et al. studied the enhancement of HQ tube for adaptive ability. Due to the narrowband frequency associated with resonance in HQ waveguides, a variable cross-section was placed inside the waveguide to generate frequency shift in resonance. They mainly considered two devices: the first variable cross-section was a fixed-diameter ball that could move axially in the waveguide to obtain frequency shift by the ball position function; the second type of variable cross-section was composed of a fixed-position diaphragm, which obtained a variable cross-section through the deformation of the diaphragm, in which case, the frequency shift depended on the diaphragm deflection function. The internal acoustics and dynamics of these two devices were analyzed and experimentally studied, and the calculated and experimental results were compared and analyzed to verify the reliability of the model [16].

Recently, Mi et al. [17] investigated the sound pressure amplification mechanism of acoustic metamaterials for ultra-low frequency sound attenuation and constructed an HQ tube with q flexible side panel. A new peak in sound transmission loss (STL) in the system revealed the pressure amplification mechanism generated at the interface of the embedded panel. They also demonstrated through experiments the noise reduction effect of acoustic metamaterials on ultra-low frequency noise. Ahmadian et al. [18] studied the acoustic performance in an HQ tube, including sensitivity analysis and multi-objective optimization of design variables. A model utilizing the flow characteristics of HQ tubes was established, and a multi-objective genetic algorithm for the performance of HQ tubes was proposed. The variables were mainly designed to include frequency, temperature, area ratio, and the length of the tube. The final results indicated that the length of the bypass tube was a key factor in the sensitivity index of transmission loss, and also demonstrated the enormous potential of the proposed optimization algorithm in practical applications. Kessemtini et al. [19] processed sound propagation in the tube containing fluids and reduced computational time by reducing the size of finite elements in the model. For the tube with slowly changing cross-sections, local wave numbers are found through eigenvalue problems, and the amplitude changes between two points along the propagation axis are determined through energy conservation. Hong [20] established a nonlinear physical model based on three meshless sub-models to understand the physical process of vortex sound interaction. The feedback effect of sound waves was considered, and the velocity of sound was added to the potential flow of vortex shedding. Simultaneously, the interaction between the flow field, sound field and their unsteady state was obtained and the simulation results were compared and analyzed with the experimental data. Da Young Kim et al. [21] constructed a virtual HQ tube system using an acoustic metamaterial. This silencer system has high acoustic and geometric efficiency and achieved broadband and low-frequency attenuation by combining the effects of resonance, periodicity, phase difference and impedance. The predicted transmission loss (TL) has been experimentally validated effectively. Li researched a noise control device based on HQ pipes and electromechanical coupling. This device consists of a main pipeline section and a set of connected speakers, playing a side-by-side role in a traditional HQ tube. The sound waves applied to the upstream speaker can be immediately transmitted to another speaker through the connecting circuit, which represents a fast channel compared to the wave transmission through the fluid medium in the main channel. It is the noise reduction mechanism of the muffler. The performance of the muffler was studied using the transfer matrix method. A periodic muffler array was also developed to broaden the bandwidth and increase the amplitude of noise attenuation. The predicted results of plane wave theory are in good agreement with the results of three-dimensional finite element simulation [22].

This study investigates the resonance in the HQ tube containing differenct kinds of flow (such as mean flow and non-uniform flow) using the finite element method (FEM). The computational fluid dynamic (CFD) is applied to calculate the influence of the fluid field on the acoustic field. The Perfectly Matched Layer (PML) is used as the boundary condition to simulate the unbounded domain. The influence of changes in cross-section and the different types of flow on resonance is also considered.

2. Governing Equations and Solution Method

2.1. Governing Equations

2.1.1. Acoustic Formulation

With the continuous growth of computing resources, researchers are able to study the phenomenon of flow-induced noise or sound propagation effects related to any velocity field. To describe the physical process of the influence of any velocity field on sound propagation, Galrun employed an Euler–Lagrangian mixed framework to describe fluid dynamics disturbances. Although few people know about this method, the Galrun equation provided the original framework, and many researchers have since further developed this equation.

Aeroacoustics mainly involves the generation and propagation of sound in moving media and there are many publications about it. The key idea of this research method is to re-derive the Navier–Stokes equation to obtain the wave operator of a stationary fluid as an equivalent source on the right side of the equation. These sound sources generate an equivalent sound field in a stationary fluid. In a sense, this method describes the sound source and provides accurate analogies. The physical properties of sound sources in fluids have not been studied deeply [23].

The Galrun equation can be used to describe the evolution of Lagrangian displacement and study the acoustic with flow. It is a second-order partial differential equation derived from a first-order system of partial differential equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{d\rho }{dt}}+\rho \nabla ·\mathbf{v}=0\hfill \\ \rho {\displaystyle \frac{d\mathbf{v}}{dt}}+\nabla p=\mathbf{0}\hfill \\ {\displaystyle \frac{dp}{dt}}-c^2{\displaystyle \frac{d\rho }{dt}}=0,\hfill \end{array}\right.$$

which is called Euler’s equations. $\left\{\rho ,\mathbf{v},p\right\}$ is the density, particle velocity and pressure, respectively. $d/dt=\partial /\partial t+\mathbf{v}·\nabla$ is the substantial derivative. The two terms to the right of the equal sign are both Euler descriptions, where $\partial /\partial t$ is the local derivative and $\mathbf{v}·\nabla$ is the convective derivative. c is the speed of sound. In the case of perfect gas isentropy, the relationship is as follows:

$${\displaystyle \frac{p}{{\rho }^{\gamma }}}=\mathrm{const},c^2=\gamma {\displaystyle \frac{p}{\rho }}$$

(1)

where $\gamma$ is the heat capacity ratio.

The Euler perturbation of any physical quantity is represented by the exponent E and can be defined as the difference between the total disturbance $\mathrm{\Psi }$ and driving configuration ${\mathrm{\Psi }}_0$ at the geometric point ${\mathbf{x}}_{\mathbf{0}}$:

$$ϵ{\mathrm{\Psi }}^E({\mathbf{x}}_{\mathbf{0}},t)=\mathrm{\Psi }({\mathbf{x}}_{\mathbf{0}},t)-{\mathrm{\Psi }}_0({\mathbf{x}}_{\mathbf{0}},t)$$

(2)

where $ϵ$ is the dimensionless coefficient that defines the disturbance amplitude. In the case of a perfect flow isentropic, the linearized Euler equation (LEE) system takes the following form:

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mathrm{d}}_0{p}^E}{dt}}+{\mathbf{v}}^E·\nabla p_0+\gamma p_0\nabla ·{\mathbf{v}}^E+\gamma p^E\nabla ·{\mathbf{v}}_{\mathbf{0}}=0\hfill \\ {\displaystyle \frac{{\mathrm{d}}_0{\mathbf{v}}^E}{dt}}+{\mathbf{v}}^E·\nabla {\mathbf{v}}_{\mathbf{0}}+{\displaystyle \frac{1}{{\rho }_0}}\nabla p^E+{\displaystyle \frac{p^E}{{\rho }_0{c}_0^2}}{\displaystyle \frac{{\mathrm{d}}_0{\mathbf{v}}_{\mathbf{0}}}{dt}}=\mathbf{0}\hfill \end{array}\right.$$

(3)

where $p^E$, ${\mathbf{v}}^E$, ${\mathbf{v}}_{\mathbf{0}}$ and ${\rho }_0$ are the Euler perturbation of pressure, the average density, the flow speed and the average density, respectively.

The Galrun equation is derived from a mixed Euler–Lagrange perturbation of the Euler equation. It has the form of second-order differential equations in space and time, and is written as a Lagrange displacement:

$${\rho }_0{\displaystyle \frac{{\mathrm{d}}_0^2{\mathbf{w}}^L}{d{t}^2}}-\nabla ({\rho }_0{c}_0^2\nabla ·{\mathbf{w}}^L)-(\nabla ·{\mathbf{w}}^L){\rho }_0{\displaystyle \frac{{\mathrm{d}}_0{\mathbf{v}}_{\mathbf{0}}}{\mathrm{d}t}}{-}^T\nabla {\mathbf{w}}^L·\nabla p_0=0$$

(4)

where ${\mathrm{d}}_0/dt=\partial /\partial t+({\mathbf{v}}_0·\nabla )$ is the convective derivative, and ${\mathbf{w}}^L$ is the Lagrangian perturbation of the particle displacement. The Galrun equation can also be written in mixed form, as follows:

$$\left\{\begin{array}{c}{\rho }_0{\displaystyle \frac{{\mathrm{d}}_0^2{\mathbf{w}}^L}{d{t}^2}}+\nabla p^L-(\nabla ·{\mathbf{w}}^L){\rho }_0{\displaystyle \frac{{\mathrm{d}}_0{\mathbf{v}}_{\mathbf{0}}}{\mathrm{d}t}}{-}^T\nabla {\mathbf{w}}^L·\nabla p_0=0\hfill \\ p^L=-{\rho }_0{c}_0^2\nabla ·{\mathbf{w}}^L\hfill \end{array}\right.$$

(5)

where $p^L$ is the Lagrangian pressure disturbance.

2.1.2. PML for the Galbrun Equation

Sound waves can propagate in very large or even undefined areas. However, the area of numerical calculation is limited, so the calculation domain needs to be artificially truncated and ensure that sound waves can propagate normally. For example, Figure 1 shows that the infinite domain R is truncated by an artificial boundary $\mathrm{\Gamma }$, forming a bounded sound domain $\mathrm{\Omega }$. The boundary $\mathrm{\Gamma }$ must be transparent to prevent reflection from affecting the numerical solution in the $\mathrm{\Omega }$ domain [24].

The geometric transformation of a two-dimensional numerical model is

$$\widehat{x}=x\widehat{y}=y-c_0{M}_{y}t\widehat{t}=t$$

(6)

where M is the Mach number. The original time and space coordinates $(x,y,t)$ are restored:

$$\left\{\begin{array}{c}\partial /\partial \overline{x}\\ \partial /\partial \overline{y}\\ \partial /\partial \overline{t}\end{array}\right\}=\underset{Q^{-1}}{\underbrace{\left(\begin{array}{ccc}1& -{\alpha }_x{M}_y& -{\alpha }_x\\ 0& {\displaystyle \frac{1}{\sqrt{1-M_x^2}}}& 0\\ 0& M_y& 1\end{array}\right)}}\left\{\begin{array}{c}\partial /\partial x\\ \partial /\partial y\\ \partial /\partial t\end{array}\right\}$$

(7)

The final formula for the PML equation is as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{d_{12}^2{w}^L}{d{t}^2}}+{\nabla }_{12}{p}^L+\left({\nabla }_{12}.w^L\right){\nabla }_{12}{p}_0-{}^T\nabla _{12}·w^L{\nabla }_{12}{p}_0=0\hfill \\ p^L+{\rho }_0{c}_0^2{\nabla }_{12}·w^L=0\hfill \end{array}\right.$$

(8)

2.1.3. The Equation for Flow Field

Since the 1960s, computational fluid dynamics (CFD) has had a significant impact in many other applications, such as biomedical, mineral processing, aerodynamics, aeroacoustic [25,26], and the entire processing industry. This widespread application has led to a significant development of models and multi-physics concepts, which have been implemented on various simulation platforms [27]. CFD is widely used in the aerospace industry and has significant applications in industrial operations, including turbulence models, high-order numerical algorithms, output-based mesh adaptation, and geometric centrality in numerical design optimization [28].

CFD is a subject that uses numerical methods to solve mathematical equations by computer, revealing the physical laws of fluid motion, and studying the spatial physical characteristics of steady flow motion and the spatiotemporal physical characteristics of unsteady flow motion [29]. The basic idea can be summarized as follows: replace the fields of continuous physical quantities in the time and space domains, such as velocity and pressure fields, with a set of variable values at a finite number of discrete points; establish an algebraic equation system through certain principles and methods regarding the relationship between the field variables at these discrete points; and then solve the algebraic equation system to obtain an approximate value of the field variables.

The governing equations are the Navier–Stokes equations, including the continuity equation and the momentum equation, as written below:

$$\frac{\partial u_i}{\partial x_i}=0$$

(9)

$$\frac{\partial u_i{u}_j}{\partial x_i}=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\nu {\nabla }^2{u}_i$$

(10)

where $u_i$ is the velocity component in the i direction, $x_i$ is the coordinate component in the i direction, $\rho$ is the fluid density, $\nu$ is its dynamic viscosity coefficient and p is the pressure.

Turbulence is modeled with the Reynolds averaging method, and the velocity is decomposed into the time-averaged and fluctuating components. The Reynolds stress model is adopted to close the steady Reynolds-averaged Navier–Stokes (RANS) equations, and the Reynolds stress tensor is written as

$$\overline{u_i^{\prime }{u}_j^{\prime }}={\nu }_t(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})+\frac{2}{3}k{\delta }_{ij}$$

(11)

where $u_i^{\prime }$ is the fluctuation velocity component in the i direction, ${\nu }_t$ is the turbulent viscosity, k is the turbulent energy and ${\delta }_{ij}$ is the Kronecker delta.

2.2. Solution Method

This study investigates the characteristics of acoustic field in the HQ duct in a 2D coordinate system for computational aeroacoustics (CAA). The coupled calculation method is based on a mixed FEM and CFD method to solve the coupling calculation of sound field and flow field. The PML is placed at the end of the tube as an absorption boundary to simulate the unbounded waveguide.

Finite Element Method

To solve the Galbrun equation system, a weak form of FEM for discretizing variables is adopted. Two trial functions, $W^{*}$ and $P^{*}$, are multiplied by the acoustic equations. After applying the boundary conditions and after integration into the frequency domain, the following equation is obtained:

$$\begin{array}{cc}\hfill & {-\omega }^2\underset{\mathrm{\Omega }}{\int }{\rho }_0{W}^{*}·W^{L}d\mathrm{\Omega }-\mathrm{i}\omega \underset{\mathrm{\Omega }}{\int }{\rho }_0{W}^{*}·\left(v_0·\nabla W^L\right)d\mathrm{\Omega }+\mathrm{i}\omega \underset{\mathrm{\Omega }}{\int }{\rho }_0\left(v_0·\nabla W^{*}\right)·W^{L}d\mathrm{\Omega }\hfill \\ \hfill & -\underset{\mathrm{\Omega }}{\int }{\rho }_0\left(v_0·\nabla W^{*}\right)\left(v_0·\nabla W^L\right)d\mathrm{\Omega }+\underset{\mathrm{\Omega }}{\int }\left(W^{*}·\nabla P^L+\nabla P^{*}·W^L\right)d\mathrm{\Omega }-\underset{\mathrm{\Omega }}{\int }{\displaystyle \frac{1}{{\rho }_0{c}_0^2}}{P}^{*}{P}^{L}d\mathrm{\Omega }\hfill \\ \hfill & +\underset{\mathrm{\Omega }}{\int }{W}^{*}·\left[\left(\nabla ·W^L\nabla p_0\right)-\left({}^T\nabla W^L·\nabla p_0\right)\right]d\mathrm{\Omega }=\hfill \\ \hfill & \underset{S_2}{\int }{P}^{*}{W}_{n}dS+\underset{\mathrm{\Omega }}{\int }[w^{*}·F^L-{\displaystyle \frac{1}{{\rho }_0{c}_0^2}}{P}^{*}{G}^L]d\mathrm{\Omega }\hfill \\ \hfill & \forall (W^{*},P^{*})\mathrm{with}\left(W^{*}{|}_{S_1}=0,P^{*}{|}_{S_1}=0\right)\hfill \end{array}$$

(12)

where $\mathrm{\Omega }$ can be the region of interest or the areas where PML is located. In this paper, the interest domain is $A_1$, $A_2$, $A_3$ and $A_4$, while $A_5$ is the the PML. The continuous boundary conditions for all the physical quantities are used at the interface between $A_4$ and $A_5$ ($\mathrm{\Omega }=A_1\cup A_2\cup A_3\cup A_4\cup A_5$). The standard process of the finite element method is used to discretize the weak form (12). The right interpolation method is adopted for pressure and displacement and it must be ensured through the inf-sup conditions. It can be seen from $p^L=-{\rho }_0{c}_0^2\nabla ·w^L$ that there is a difference in the interpolation order of pressure and displacement.

The triangle finite element called “T4-3c” is selected to solve the numerical problems. After the assembly of matrices and the application of boundary conditions, the formula for global discrete variables can be written as follows:

$$\left[-{\omega }^2\left(\begin{array}{cc}\mathbf{M}& 0\\ 0& 0\end{array}\right)-i\omega \left(\begin{array}{cc}\mathbf{C}& {\mathbf{C}}_{\mathbf{12}}\\ {\mathbf{C}}_{\mathbf{21}}& 0\end{array}\right)+\left(\begin{array}{cc}{\mathbf{K}}_{\mathbf{11}}& {\mathbf{K}}_{\mathbf{12}}\\ {\mathbf{K}}_{\mathbf{21}}& {\mathbf{K}}_{\mathbf{22}}\end{array}\right)\right]\left\{\begin{array}{c}{w}^L\\ P^L\end{array}\right\}=\left\{\begin{array}{c}{F}_1^L\left(\omega \right)\\ S_1^L\left(\omega \right)\end{array}\right\}$$

(13)

Finally, the following equation can be obtained:

$$\left[K\left(\omega \right)\left(\omega \right)\right]\left\{U\left(\omega \right)\right\}=\left\{F\left(\omega \right)\right\}$$

(14)

3. Numerical Setup

In 1833, Herschel proposed the concept of destructive wave interference in one-dimensional waveguides, and the destructive interference of an acoustic wave was used as an example. Thirty years later, Quincke further developed this idea and confirmed this theory through experiments. The HQ tube is shown below.

In this paper, the normal mode is located at the entrance of the tube to excite the the acoustic field, while PML is placed at the end of the tube to simulate the infinite duct (as shown in Figure 2). If the influence of non-uniform flow field on acoustic field is considered, the Helmholtz equation is no longer applicable and LEE or Galbrun is adopted. For the flow field, all the parameters (${\rho }_0$, $p_0$, $c_0$, $v_0$) are obtained from the commercial software ANSYS.

For the numerical model, our key points focus on the dimensions of the main and bypass ducts, as well as the impact of different types of fluid on the sound field. The geometrical parameters of pressure, density, velocities and angular frequency were normalized with duct height H, density ${\rho }_0$, speed of sound $c_0$, ${\rho }_0{c}_0$, and $c_0/H$, respectively. The parameter of the tube is the ratio of the cross-sections of the main and bypass sections $\frac{A_1}{A_2}$. There are three kind of cases (without flow, mean flow and non-uniform flow) considered. For the numerical part, an in-house FEM code is used in Matlab to solve the Galbrun equation and the $k-ϵ$ model is adopted to solve the Navier–Stokes equation.

4. Results and Discussion

4.1. The Pressure Field without Flow

The first case concerns the infinite tube without flow. The Mach number is zero and the cross-section ratio between the main duct and bypass duct is $\frac{A_1}{A_2}$. The Perfectly Matched Layer is the absorption boundary. The acoustic resonance in the HQ tube and the influence of the cross-section is considered in this part. The distribution of the dynamic pressure in the tube is plotted for different cases (see Figure 3). In this paper, all the parameters are normalized.

Figure 3 shows the pressure distribution of the sound field with the frequency $f=0.4$, Mach number $M=0.0$ and the cross-section ratio $\frac{A_1}{A_2}=1.2$. At this time, the sound pressure in this area reaches its peak and most of the acoustic energy is concentrated in the main duct, and the mode is almost symmetrical.

To reveal the influences of cross-section ratio on the acoustic field,

Table 1. The resonance frequency for different cross-section ratios: $M=0.0$.

Cross-section	1.0	1.2	1.5	1.8
Resonance frequency	0.46	0.43	0.41	0.37

indicates the trend of resonance frequency for the HQ tube with different geometric sizes. The value of cross-section ratio is from 1.0 to 1.8 and we can see that the frequency decreases with the increase in cross-section ratio.

4.2. The Pressure Field with Uniform Mean Flow

The second case is the infinite tube with the uniform mean flow. The Mach number varies from 0.1 to 0.3 and the cross-section ratio between main duct and bypass duct $\frac{A_1}{A_2}$ is from 1.0 to 1.8. The acoustic resonance in the HQ tube is calculated in this part. Both the influence of cross-section and the Mach number on the acoustic resonance are considered in this section.

The distribution of the dynamic acoustic pressure in the tube is plotted. Figure 4a is the pressure field when $\frac{A_1}{A_2}=1.5$, $M=0.2$ at the frequency $f=0.39$. It can be seen that the resonance mode is concentrated on the main duct area. Figure 4b shows the acoustic resonance of $f=0.32$ when $\frac{A_1}{A_2}=1.2$, $M=0.3$. In this case, most of the energy is concentrated in the bypass tube. Both the resonances in Figure 4a,b are symmetric about y-axis. The maximum pressure amplitude of downstream and upstream interface is also calculated (see Figure 5).

Figure 5 shows the maximum pressure amplitude for different Mach numbers with the cross-section ratio $\frac{A_1}{A_2}=1.5$. The picture Figure 5a depicts that the maximum pressure appears at the frequency $f=0.41$ when the Mach number $M=0.0$. Figure 5b indicates the resonance frequency of $f=0.39$ and $M=0.1$. We can see from Figure 5c that the peak value of the resonance occurs when $f=0.39$ and $M=0.2$. Figure 5d reveals the resonance with $f=0.37$ and $M=0.3$. These four curb figures depict a phenomenon whereby the resonance frequency decreases with increasing Mach number.

Table 2. The resonance frequency for different Mach numbers: $\frac{A_1}{A_2}=1.5$.

Mach number	0.0	0.1	0.2	0.3
Acoustic pressure	2.2	2.0	3.7	3.1

shows the peak of acoustic pressure with the change in Mach number.

Furthermore, the change in resonance frequency with the variation in cross-section and Mach number is also investigated. As can be seen from Figure 6, the two curves show the fluctuation of resonance frequency. The horizontal axis is the cross-section and the vertical axis represents the resonance frequency. The black solid line and black dashed line represent the results where the Mach number is equal to 0.1 and 0.2, respectively. It can be seen that the resonance frequency decreases with the increase in cross-section ratio whether $M=0.1$ or $M=0.2$. However, for the definite cross-section ratio, there is no definite relationship between the resonance frequency and Mach number.

5. Conclusions

In this paper, the acoustic resonance in the HQ duct with a bypass tube is researched. The numerical model which contains the excitation acoustic source and PML is established. The finite element method is used to solve Galbrun equation, which can take the influence of flow field into consideration. The resonance frequency below the cut-off frequency is discussed under the normal mode incidence. The impact of cross-section ratio of the HQ tube and the Mach number of the flow field on the acoustic field is also discussed. The general rules of resonance frequency variation with cross-sectional ratio and Mach number is revealed. It is found that the acoustic resonances are all symmetric and there is a peak value when the resonance appears. The resonance frequency decreases with an increase in the cross-section ratio of the tube.