Electric liquid saturation

The first method used to measure $\alpha_{\infty}$ is the one described by Brown in 1980 [Bro80]. This method is based on the measurement of the electric resistance of a porous sample with a pore network saturated with an electric liquid. The higher the tortuosity, the higher the electrical resistance. This method provides a direct measurement of the tortuosity however it requires that the skeleton of the material is an electrical insulator and it requires the value of the porosity. Finally, one difficulty is to obtain the saturation of the pore network.

To overcome this last issue, Johnson and colleagues [JPSPK82] have proposed a similar method with Helium 2, a super fluid without measurable viscosity. This method allows better signal to noise ratios than Brown's method. The main drawback of the method is that the set-up is not accessible to usual acousticians.

Estimation at ultrasound frequencies

The first method used to estimate the high frequency limit of the dynamic tortuosity in air is the one described by Allard et al. [ACHL94]. This work (probably one of the fastest written article in Acoustics: outlined during a PhD defense by J.-F. Allard), is based on the increase of the time of flight of the air-borne compressional wave between two ultrasound transducers when there is no material and then a material placed in between them.

While developing a method used to estimate the viscous characteristic length and the thermal characteristic length, Leclaire et al. [LKLGT96], then [LKLMBC96], present a method to estimate $\alpha_{\infty}$. In both papers, the estimation of $\alpha_{\infty}$ is not the main purpose and is only briefly presented (and not even used for the first paper). The general drawback to measurements at ultrasonic frequencies, highlighted in [LKLMBC96], is the effect of the multiple scattering of waves by the skeleton structure. To overcome this issue, measurements in different fluids (air and helium for [LKLMBC96]) is required.

In 2003, Fellah et al. [FBLDAC03] developed a method to estimate $\alpha_{\infty}$ (and the open porosity) of a material sample from the measurements of its reflection coefficients at two oblique incidences (and always at ultrasonic frequencies). This technique gives estimations of the parameters at the surface of the material sample. They might differ from the bulk parameters due to the foaming process or due to the cutting of the sample itself. A special attention should thus be given to the surface of the material samples before any measurements. However, it can be noticed that this method could provide the opportunity to measure the "surface" open porosity and the "surface" high frequency limit of the dynamic tortuosity of a material designed with a skin effect without separating the skin from the core.


Inspired by the work of Panneton & Olny (described in the next section), Groby et al. [GORSL10] have introduced a method to estimate, from two measurements (allowing one to retrieve the transmitted and reflected coefficients of a material at ultrasonic frequencies), $\alpha_{\infty}$, the the viscous characteristic length and the thermal characteristic length as well as the the open porosity. Obtaining a good signal to noise ratio for both reflection and transmission, at ultrasonic frequencies, for the same material sample is the main difficulty when using this technique.

Works based on these methods have been published since. They usually try overcome one issue with more or less success.

Estimation at acoustic frequencies

An estimation of the high frequency limit of the dynamic tortuosity can be obtained from the measurements of the dynamic density of a porous medium, at audible frequencies with prior knowledge of the static air-flow resistivity and the open porosity.
The method, introduced by Olny, Panneton & Tran-van [OPT02] and further described by Panneton & Only [P006] (and Olny & Panneton [0P08]) also allows estimations of the viscous characteristic length, the thermal characteristic length and the static thermal permeability from the same data.
Here, audible frequencies ensure the acoustic wavelength is much larger than the size of the Representative Elementary Volume (REV).
The method is based on an analytical inversion of the formula for the dynamic mass density $\widetilde{\rho}(\omega)$ following Johnson, Koplik and Dashen model (see e.g. "Visco-inertial effects" section for Johnson-Champoux-Allard model).

Note that :
• [PO06] also introduces an estimation of the static air flow resistivity which is now part of the appendix of [ISO 9053-1:2018].
• Jaouen, Gourdon & Glé [JGG20] present estimations of the open porosity from the low and high frequency asymptotes of the dynamic bulk modulus making it possible to characterize all parameters of the Johnson-Champoux-Allard-Lafarge model from an impedance tube.

Finally, to estimate the high frequency limit of the dynamic tortuosity of acoustical facings as modeled by Atalla & Sgard modelling [AS07] (i.e. $\alpha_{\infty}$ also accounts for the flow distortion around the perforations) a method has been developed by Jaouen & Bécot [JB11]. This method relies on the measurement of the acoustical impedance for plane wave and normal incidence of the facing backed by an air gap of known thickness.