Equivalent Fluid MPP Model

The equivalent fluid microperforate model estimates the input parameters to the JCA model based on the perforate diameter \((d)\), center-to-center distance \((b)\), and thickness \((t)\).

The internal _calc_dynamics method is then used to determine the dynamic mass density \((\tilde{\rho})\) and dynamic bulk modulus \((\widetilde{K})\) using the JCA model.

Estimation of JCA parameters

\[ \phi = \frac{\pi}{4}\Bigg(\frac{d}{b}\Bigg)^2 \]

\[ \sigma = \frac{32\eta}{\phi d^2} \]

\[ \Lambda = \frac{d}{2} \]

\[ \Lambda^{\prime} = \frac{d}{2} \]

\[ \tau = 1+\frac{2*fok}{t} \]

where \(fok\) is:

\[ fok = \frac{4d}{3\pi}(1-1.13eps-0.09eps^2+0.27eps^3) \]

and \(eps\) is:

\[ eps = 2\sqrt{\frac{\phi}{\pi}} \]

The Add_MPP_EF_Layer method then converts the modified dynamic mass density and bulk modulus to the characteristic impedence \((Z_{c})\) and wavenumber \((k_{c})\) for use in the layer transfer matrix.

\[ Z_{c} = \sqrt{\tilde{\rho}\widetilde{K}} \]

\[ k_{c} = {\omega}\sqrt{\frac{\tilde{\rho}}{\widetilde{K}}} \]

Model Parameters:

Using the following nomenclature --- Symbol = [Units] (name)

Model Specific

\[ d = \Bigg[m\Bigg]\tag{perforate diameter} \]

\[ b = \Bigg[m\Bigg]\tag{center-to-center distance} \]

\[ t = \Bigg[m\Bigg]\tag{layer thickness} \]

JCA Parameters

\[ \sigma = \Bigg[\frac{Pa*s}{m^2}\Bigg]\tag{static airflow resistivity} \]

\[ \phi = \Bigg[unitless\Bigg]\tag{porosity} \]

\[ \tau = \Bigg[unitless\Bigg]\tag{tortuosity} \]

\[ \Lambda = \Bigg[{\mu}m\Bigg]\tag{viscous characteristic length} \]

\[ \Lambda^{\prime} = \Bigg[{\mu}m\Bigg]\tag{thermal characteristic length} \]

Defining Other Symbols:

Using the following nomenclature --- Symbol = [Units] (name)

\[ \eta = \Bigg[Pa*s\Bigg]\tag{viscosity of air} \]