The high frequency limit of the
dynamic tortuosity,
term abusively reduced to
"tortuosity", characterizes, partly, the
visco-inertial effects at high frequencies (when the viscous boundary layer
is small compared to the characteritic size of the pores).
It is usually identified by
the symbol
$\alpha_{\infty}$
and is a dimensionless quantity.
We shall consider the value of $\alpha_{\infty}$ to be a measure of the disorder in the sytem [material]. [JKD87]
The mathematical definition of the high frequency limit of the tortuosity is given by:
| $ \alpha_{\infty} = \frac{\displaystyle{\frac{1}{V}\int_{V}v^{2}dV}} {\left( \displaystyle{\frac{1}{V}\int_{V}\vec{v}dV} \right)^{2}}$ | (1) |
where $V$
is the homogeneization volume and
$\vec{v}$
is the velocity of the fluid particule
at high frequencies, when the viscous boundary layer is
much smaller than the characteristic size of the pores.
$\alpha_{\infty}$
is thus linked to the dispersion of the microscopic fluid particule velocity
around their macroscopic value.
Range of values
The theoritical low bound value for
$\alpha_{\infty}$
is 1.00 . This last value corresponds to parallel streamlines of the velocity field.
For acoustical materials, the range of values of the tortuosity is approximately [1.00
3.00]. 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 simple porosity model).
Schematic representations of three porous media with different open porosities ($\phi$), static air flow resistivities ($\sigma$) and high frequency limit of the dynamic tortuosities ($\alpha_{\infty}$).
The high frequency limit of the dynamic tortuosity can be directly measured [] indirectly characterized [ACHL94,OPT02].
References
[ACHL94] Allard J.-F., Castagnède B., Henri M. and
Lauriks W., Evaluation of the tortuosity in acoustic porous
materials saturated by air, Rev. Sci. Instrum. 65, 1994
[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
[OPT02]
Olny X., Panneton P. and Tran-Van J.,
An indirect acoustical method for determining intrinsic parameters of porous materials,
proceedings of the 2nd Biot's conference (Grenoble France), 2002
.
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