# Mathematical expression

The mathematical definition of the high frequency limit of the tortuosity given by Johnson, Koplik and Dashen is:

APMR

Acoustical Porous Material Recipes

The mathematical definition of the high frequency limit of the tortuosity given by Johnson, Koplik and Dashen is:

\[
\alpha_{\infty} =
\frac{\displaystyle{\frac{1}{V}\int_{V}v^{2}dV}}
{\left( \displaystyle{\frac{1}{V}\int_{V}\vec{v}dV} \right)^{2}}
\]

where $V$ is the homogeneization volume and $\vec{v}$ is the velocity of the fluid particle at high frequencies, when the viscous boundary layer is much smaller than the characteristic size of the pores.

A physical description of the dynamic tortuosity $\alpha_{\infty}$ is given by Johnson, Koplik and Dashen:

We shall consider the value of $\alpha_{\infty}$ to be a measure of the disorder in the system [material].

For acoustical materials, the range of values of the tortuosity is approximately [1.00 3.00].
The theoritical lower value for $\alpha_{\infty}$ is 1.00 as the average of square quantities is greater than or equal to the square value of their average. This lower value corresponds to parallel streamlines of the velocity field (cf. fig. below).

A value of $\alpha_{\infty}$ greater than 3.00 is usually the result of an erroneous characterization (e.g. characterization of a multi-scale material using a single porosity model).

The high frequency limit of the dynamic tortuosity can be directly measured or estimated from ultrasound or impedance tube measurements for example. More information are reported on the $\alpha_{\infty}$ characterization page.

A web application, created by Vivek T. Ramamoorthy, to get more familiar with the tortuosity is available at https://vivektramamoorthy.github.io/TortuosityDemo/.