## 2 Methods
### Background Theory  
Computational fluid dynamics (CFD) is based upon the numerical solution of the underlying governing equations for a fluid flow, under specified boundary conditions. The propagation of an acoustic wave in a flow tube may be represented as an unsteady three-dimensional compressible flow, and the corresponding governing equations may be expressed in three-dimensional Cartesian coordinates as [33]  
Mass conservation equation:  
\[\frac{\partial\rho}{\partial t}+\text{div}\Big{(}\rho\overset{\rightarrow}{ \mathbf{u}}\Big{)}=0 \tag{1}\]  
Momentum conservation:  
\[\frac{\partial(\rho u)}{\partial t}+\text{div}\Big{(}\rho u\overset{ \rightarrow}{\mathbf{u}}\Big{)}=-\frac{\partial p}{\partial x}+\text{div} \big{(}\tau_{ix}\big{)}+S_{Mx} \tag{2}\]  
\[\frac{\partial(\rho v)}{\partial t}+\text{div}\Big{(}\rho v\overset{ \rightarrow}{\mathbf{u}}\Big{)}=-\frac{\partial p}{\partial y}+\text{div} \big{(}\tau_{iy}\big{)}+S_{My} \tag{3}\]\[\frac{\partial(\rho w)}{\partial t}+\text{div}\left(\rho w\overset{\rightarrow}{ \mathbf{u}}\right)=-\frac{\partial p}{\partial z}+\text{div}(\tau_{iz})+S_{Mz} \tag{4}\]  
where \(u,\ v,\ w\) are the components of the velocity vector \(\overset{\rightarrow}{\mathbf{u}}\) in \(x,\ y,\ z\) directions, respectively; \(\tau_{ij}\) represents a viscous stress in the \(j\) direction exerted on a surface perpendicular to the \(i\) direction; \(S_{M}\) is the momentum source term to include other sources or sinks of the momentum such as body forces; \(\rho,p\) are, respectively, the density and pressure of the fluid; and \(t\) is the time. Energy equation:  
\[\frac{\partial(\rho h)}{\partial t}+\text{div}\left(\rho h\overset{\rightarrow }{\mathbf{u}}\right)=\text{div}(k\text{ grad }T)+\frac{\partial p}{\partial t}+\text{div}\left(\tau_{ij} \overset{\rightarrow}{\mathbf{u}}\right)+S_{h} \tag{5}\]  
where \(h\) represents the specific total enthalpy, \(k\) is thermal conductivity, \(S_{h}\) is the enthalpy source term and \(T\) is the temperature. The ideal gas equation of state is applied to determine the density of the fluid. The PVPM model was originally developed for the representation of uniform and highly porous materials, with a rigid frame. The model introduced a porosity term, \(\sigma\), into the governing equations [33]. For example, when considering only the \(x\) direction within a porous zone, the continuity and momentum equations may be written as  
\[\frac{\partial(\sigma\rho)}{\partial t}+\text{div}\left(\sigma\rho\overset{ \rightarrow}{\mathbf{u}}\right)=0 \tag{6}\]  
\[\frac{\partial(\rho\sigma u)}{\partial t}+\text{div}\left(\rho\sigma u\overset {\rightarrow}{\mathbf{u}}\right)=-\sigma\frac{\partial p}{\partial x}+\text{ div}(\sigma\tau_{ix})+\sigma S_{Mx} \tag{7}\]  
Porosity \(\sigma\) is defined as the ratio of the void space to the overall bounding volume for the plate. \(u\) is the x direction velocity. The magnitude of the velocity vector \(\overset{\rightarrow}{\mathbf{u}}\) within the porous region has to be amplified by a factor \(1/\sigma\) for the transverse flow to satisfy the mass continuity. This resultant physical velocity is equivalent to the average real velocity within the porous material region. Therefore, mass flow rate conservation and the correct representation of the overall velocity changes across the porous region are simultaneously achieved by applying the physical velocity porous media model, which makes the acquisition of acoustic properties possible. \(S_{Mx}\) is the pressure source term for the PVPM model and \(S_{Mx}=\frac{\Delta p}{|\alpha|}=\frac{h^{2}}{\alpha}u\), where \(\ I_{\text{eff}}\) is the effective thickness of the porous media zone which will be described in Section 2.2.2, \(\mu\) is the dynamic viscosity of the medium in which the sound signal propagates and \(\alpha\) is the permeability of the porous media region.