Visco-inertial effects

In 1987, Johnson Koplik and Dashen [JKD87] proposed a semi-phenomenological model to describe the complex density of an acoustical porous material with a motionless skeleton having arbitrary pore shapes. This expression is:

\[ \widetilde{\rho}(\omega) = \frac{ \alpha_{\infty} \rho_{0} }{ \phi } \Bigg[ 1 + \frac{ \sigma \phi }{ j \omega \rho_{0} \alpha_{\infty}} \sqrt{ 1 + j\frac{ 4 \alpha_{\infty}^{2} \eta \rho_{0} \omega }{ \sigma^{2} \Lambda^{2} \phi^{2}} }\Bigg] \]

4 parameters are invloved in the calculation of this dynamic density: the static air flow resistivity $\sigma$, the open porosity $\phi$, the high frequency limit of the tortuosity $\alpha_{\infty}$ and the viscous characteristic length $\Lambda$.

The low frequency limit of the real part of the dynamic density expression is not exact.

Thermal effects

In 1991, Champoux and Allard [CA91] introduced an expression for the dynamic bulk modulus for the same kind of porous material based on the previous work by Johnson et al.

\[ \widetilde{K}(\omega) = \displaystyle{\frac{\gamma P_{0}/\phi} {\gamma - (\gamma-1) \left[ 1-j\displaystyle{\frac{8\kappa} {\Lambda'^{2}C_{p}\rho_{0}\omega}} \sqrt{ 1+j \displaystyle{\frac{\Lambda'^{2}C_{p}\rho_{0}\omega} {16\kappa}} } \, \right]^{-1} }} \]

2 parameters are invloved in the calculation of this dynamic bulk modulus: the open porosity $\phi$ and the thermal characteristic length $\Lambda'$.

Limitations

Wrong dynamic mass density behavior at low frequencies

The expression given by Johnson, Koplik & Dashen does not describe the exact behavior of the dynamic mass density as $\omega$ tends to zero: the imaginary part of $\widetilde{\rho}$ is underestimated at low frequencies.

Pride, Morgan & Gangi [PMG93] have proposed a modified expression of the dynamic density. However, the expression given by Pride, Morgan & Gangi and further modified by D. Lafarge [Laf93] is rarely used as new required parameters cannot be characterized yet.

Changes to the JCA model by Pride et al. and D. Lafarge are detailed in the Johnson-Champoux-Allard-Pride-Lafarge model.

Anomalous interdependence between visco-inertial and thermal effects

Having a closer look at the dynamic bulk modulus expression given by Champoux and Allard, one can notice that visco-inertial and thermal parameters are linked together:

\[ \frac{s}{s'}=\frac{\Lambda'}{\Lambda} \]

This observation leads Lafarge et al. [LLAT97] to introduce a new parameter, the static thermal permeability $k'_{0}$, in order to break the dependence between visco-inertial and thermal parameters while keeping a "symmetry" between the two dissipative phenomena.

Changes to the JCA model by Lafarge et al. are detailed in the Johnson-Champoux-Allard-Lafarge model.