# Prediction of the Absorption Characteristics of Non-Uniform Acoustic Absorbers with Grazing Flow
## 2 Full Navier-Stokes Equations (FNSE) Method  
Figure 1 shows a schematic diagram of a rectangular duct-acoustic absorber (RDAA) system with grazing flow (flow parallel to the facesheet). In Figure 1, all the walls in the RDAA system are assumed to be rigid, while a portion of the bottom wall is replaced with a sample absorber. In this paper, we mainly focus on the prediction method of the attenuation characteristics of the acoustically treated ducts with a grazing flow. The most straightforward way to accomplish this is to solve linearized Navier-Stokes equations using the FE technique, called the full Navier-Stokes equations (FNSE) method. The linearized Navier-Stokes equations are composed of continuity, momentum and energy equations [38], given by  
\[\mathrm{i}\omega\rho+\
abla\cdot(\rho_{0}\mathbf{u}+\rho\mathbf{u}_{0})=M_{0} \tag{1a}\] \[\rho_{0}[\mathrm{i}\omega\mathbf{u}+(\mathbf{u}\cdot\
abla) \mathbf{u}_{0}+(\mathbf{u}_{0}\cdot\
abla)\mathbf{u}]+\rho(\mathbf{u}_{0}\cdot \
abla)\mathbf{u}_{0}=\
abla\cdot\boldsymbol{\sigma}+\mathbf{F}-\mathbf{u}_{0} M_{0}\] (1b) \[\rho_{0}C_{p}(\mathrm{i}\omega T+\mathbf{u}\cdot\
abla T_{0}+ \mathbf{u}_{0}\cdot\
abla T)+\rho_{0}C_{p}\mathbf{u}_{0}\cdot\
abla T_{0}\] (1c) \[-\alpha_{p}T_{0}(\mathrm{i}\omega p+\mathbf{u}\cdot\
abla p_{0}+ \mathbf{u}_{0}\cdot\
abla p)-\alpha_{p}T\mathbf{u}_{0}\cdot\
abla p_{0}= \
abla\cdot(k\
abla T)+Q\]  
where \(p\), \(\mathbf{u}\), \(T\) and \(\rho\) are the perturbations of pressure, velocity, temperature and density, respectively. \(p_{0}\), \(\mathbf{u}_{0}\), \(T_{0}\) and \(\rho_{0}\) refer to the steady-state background pressure, velocity, temperature and density, respectively. \(\omega\) is the angular frequency of the incident wave, \(\mathbf{u}\) is  
Figure 1: Schematic diagram of the rectangular duct–acoustic absorber system under grazing flow. the vibration velocity of the particle, \(k\) is the wave number, \(\mathbf{\sigma}\) is the stress tensor, and \(C_{p}\) is the constant pressure heat capacity. The mass source term \(M_{0}\), force source term \(\mathbf{F}\) and heat source term \(Q\) on the right side of equations are zero by default. The stress tensor and the linearized state equation of an ideal gas [39] are expressed as  
\[\mathbf{\sigma}=-p\mathbf{I}+\mu[\
abla\mathbf{u}+(\
abla\mathbf{u})^{\mathrm{T}} ]+\bigg{(}\mu_{B}-\frac{2}{3}\mu\bigg{)}(\
abla\cdot\mathbf{u}_{t})\mathbf{I} \tag{2a}\] \[\rho=\rho_{0}(\beta_{T}p-\alpha_{p}T) \tag{2b}\]  
where \(\mu\) and \(\mu_{B}\) are the dynamic and bulk viscosity of the material, respectively. \(\mathbf{I}\) is the identity matrix. \(\alpha_{p}\) and \(\beta_{T}\) are the isobaric thermal expansion and isothermal compression coefficients, respectively, defined as  
\[\alpha_{p}=-\frac{1}{\rho_{0}}\bigg{(}\frac{\partial\rho_{0}}{\partial T_{0}} \bigg{)}_{p}=\frac{1}{c_{0}}\sqrt{\frac{C_{p}(\gamma-1)}{T_{0}}} \tag{3a}\] \[\beta_{T}=\frac{1}{\rho_{0}}\bigg{(}\frac{\partial\rho_{0}}{\partial\rho_{0}} \bigg{)}_{T}=\frac{1}{\rho_{0}}\frac{\gamma}{c_{0}^{2}}=\gamma\beta_{S} \tag{3b}\]  
where \(\gamma\) is the specific heat ratio, \(c_{0}\) is the speed of sound in air and \(\beta_{S}=1/\rho_{0}c_{0}^{2}\) is the compression coefficients in an adiabatic case. In an adiabatic case, the state equation given by Equation (2b) can be simplified as  
\[\rho=\rho_{0}\beta_{S}p=\rho_{0}\frac{\beta_{T}}{\gamma}p \tag{4}\]  
In order to simulate the influences of the background mean flow (grazing flow) on acoustic wave propagation, background mean flow temperature \(T_{0}\), absolute pressure \(p_{0}\) and velocity field \(\mathbf{u}_{0}\) should be defined. In this paper, \(T_{0}\) is taken as 293.15 K, while the absolute pressure and velocity fields are obtained from the turbulent physical field.