By definition, the characteristic viscous length should be lower or equal to the thermal characteristic length:
\[ \Lambda \leq \Lambda' \]for fibrous or felts it is usually admitted that the static air-flow resistivity increases with mass density [Tar96]
The tortuosity for fibrous or felt materials can be approximated with (see e.g. Umnova et al. [UAL00])
\[ \alpha_{\infty} \simeq 1/\phi \]which is sometimes called Archie’s empirical law [Arc42]: $(1/\phi)^r$. From a different reasoning, Tournat et al. [TPLJ04] derived the expression $\alpha_\infty = 1+(1-\phi)$ for sound flow perpendicular to the axis of parallel identical cylinders for large porosity values. Obviously, this last expression is a Taylor expansion around $\phi = 1$ up to the order 1 of the expression $\alpha_{\infty} \simeq 1/\phi$. From a graphical observation, it appears these two expressions for $\alpha_{\infty}$ diverge for $\phi \lesssim$ 0.96
The tortuosity of a stack of identical solid spheres is calculated following J. G. Berryman [Ber80] as:
\[ \alpha_{\infty} = 1 + \displaystyle\frac{1-\phi}{2\phi} \]In addition, one can check the value of
- • the viscous pore shape factor $M$ for models based on Johnson Koplik & Dashen works.
- • the thermal pore shape factor $M'$ for models based on Champoux-Allard-Lafarge works.
The pore shape factors $M$ and $M'$ should have orders of magnitude of the order around 1 for usual acoustical porous materials. $M=M'=1$ for straight cylindrical pores.
\[ \begin{align} M &=& \displaystyle{\frac{8\alpha_{\infty}\eta}{\sigma\phi\Lambda^{2}}} \\ M' &=& \displaystyle{\frac{8k'_{0}}{\phi\Lambda'^{2}}} \end{align} \]