## 2 Materials and Methods  
### Flow Resistivity  
Characterization of porous materials can be performed based on properties such as the nominal pore size \(d_{p,0}\) of the microscopic structures or porosity \(\varphi\). The latter is defined as the ratio of the fluid volume \(V_{f}\) inside the pores divided by the total volume \(V_{t}\) of the porous material and only depends on geometrical properties. In this study, a permeable material consisting of an open-cell, porous metal foam with homogeneously distributed pores is used. The NiCrAl metal foam with a nominal pore size of \(800\,\mathrm{\SIUnitSymbolMicro m}\) is manufactured by the company Alantum. Figure 1 shows a microscopic image of the foam which is referred to as P800 in the following. A macroscopic relation between the constant seepage velocity \(v_{s}\) through the material and the related pressure drop \(\Delta p\) along the thickness \(t\) of the material exists in the form of the Hazen-Dupuit-Darcy relation [20]  
\[\frac{\Delta p}{t}=\frac{\mu}{K}v_{s}+C\rho v_{s}^{2}\,. \tag{1}\]  
The seepage velocity \(v_{s}=\dot{Q}/A\) is calculated from the volumetric flow rate \(\dot{Q}\) passing through a sample with cross section \(A\). The viscous term depends on the dynamic viscosity \(\mu\) and the flow permeability \(K\). The additional quadratic term represents effects of fluid expansion/contraction and changes in flow direction within porous media. It is referred to as the inertia term and depends on the fluid density \(\rho\) and the form coefficient \(C\). For low seepage velocities, the quadratic term can be neglected and the ratio \(\mu/K\) can directly be related to the pressure drop. A limit for dominant linear behavior can be defined based on the ratio between form forces and viscous forces [20]. The requirement \(F_{f}/F_{visc}<0.1\) is used here which translates to seepage velocities \(v_{s}<0.25\,\mathrm{m}\,\mathrm{s}^{-1}\) for the given porous material. At \(0^{\circ}\) incidence, the expected pressure difference between suction and pressure side solely depends on the magnitude of turbulent boundary layer pressure fluctuations \(p_{rms}\approx 0.1q\) with \(q\) being the dynamic inflow pressure [21]. From Equation (1) it follows that the seepage velocity which can be expected for the permeable materials used in this study is low enough to neglect the inertia term. In a previous campaign [16], the physical properties of the metal foam were specified as given in Table 1.  
From the definition of flow resistivity \(R=\mu/K\) in the viscous flow regime, it follows that pressure communication can actively be influenced by changing either the material properties (represented by the permeability) or the fluid conditions. The approximate relation between viscosity or air flow resistivity and air temperature can be expressed in the form of the power law  
\[\frac{\mu}{\mu_{0}}=\frac{R}{R_{0}}=\left(\frac{T}{T_{0}}\right)^{n}\,, \tag{2}\]  
where the exponent \(n\) depends on the fluid and reference temperature. For air, \(n=2/3\) is commonly chosen [22]. Combining this relation with Eq. 1, it is apparent that the noise reduction capability of permeable materials is a function of fluid temperature. When non-linear effects are taken into account, the pressure drop \(\Delta p\) is additionally affected by the fluid density (Equation (1)) which can be related to the temperature via the ideal gas law, according to which \(\rho\propto 1/T\). An experimental characterization of the porous metal foam properties for varying temperatures is performed by using the setup depicted in Figure 1. Pressurized air passes through an electrical heating unit consisting of a coiled heating wire. A desired air temperature up to 90 \({}^{\circ}\)C can be controlled manually via adjusting the electric current supply of the heating system.