Johnson-Champoux-Allard model
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) = \displaystyle{\frac{\alpha_{\infty}\rho_{0}}{\phi}} \left[ 1 + \displaystyle{\frac{\sigma\phi} {j\omega\rho_{0}\alpha_{\infty}}} \displaystyle{\sqrt{ 1 + j\displaystyle{\frac{4\alpha_{\infty}^{2}\eta\rho_{0}\omega} {\sigma^{2}\Lambda^{2}\phi^{2}}} } \,\, \right] $ | (1) |
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 $\Lambda$.
Only the low frequency limit of the real part of this dynamic density expression is not exact.
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) |
2 parameters are invloved in the calculation of this dynamic bulk modulus: the open porosity $\phi$, and the thermal characteristic length $\Lambda'$.
References
[JKD87] Johnson D. L., Koplik J. and Dashen R.,
Theory of dynamic permeability and tortuosity in fluid-saturated porous media,
J. Fluid Mech., 176, 1987, pp. 379-402
[CA91] Champoux Y. and Allard J.-F.,
Dynamic tortuosity and bulk modulus in air-saturated porous media,
J. Appl. Phys., 70, 1991, pp. 1975-1979
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