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}

## Check consistency of your parameter values

 Static air flow resistivity ($\sigma$) N.s.m$^{-4}$ Open porosity ($\phi$) HF limit of the tortuosity ($\alpha_{\infty}$) Viscous characteristic length ($\Lambda$) m Thermal characteristic length ($\Lambda'$) m Static thermal permeability ($k'_0$) m$^2$ Viscous pore shape factor ($M$) Thermal pore shape factor ($M'$) Comments: Click "Check parameters" to see if your input parameters are valid.