The statistical evolution of a two-dimensional polygonal, or ‘dry’, foam during diffusion of gas between bubbles lends itself to a very simple mathematical description by combining physical principles discovered by Young. Laplace, Plateau, and von Neumann over a period of a century and a half. Following a brief review of this ‘canonical’ theory, we report results of the largest numerical simulations of this system undertaken to date. In particular, we discuss the existence and properties of a scaling regime, conjectured on the basis of laboratory experiments on larger systems than ours by Glazier and coworkers, and corroborated in computations on smaller systems by Weaire and collaborators. While we find qualitative agreement with these earlier investigations, our results differ on important, quantitative details, and we find that the evolution of the foam, and the emergence of scaling, is very sensitive to correlations in the initial data. The largest computations we have performed follow the relaxation of a system with 1024 bubbles to one with O(10), and took about 30 hours of CPU time on a Cray-YMP supercomputer. The code used has been thoroughly tested, both by comparison with a set of essentially analytic results on the rheology of a monodisperse-hexagonal foam due to Kraynik & Hansen, and by verification of certain analytical solutions to the evolution equations that we found for a family of ‘fractal foams’.
It has been previously shown that relatively simple computational fluid dynamics (CFD) models can be used to calculate the transfer impedances, including the associated end corrections, of microperforated panels. The impedance is estimated by first calculating the pressure difference across a single hole when a transient input velocity is imposed, and then Fourier transforming the result to obtain the impedance as a function of frequency. Since the size of the hole and the dimensions of the inlet and outlet channels are very small compared to a wavelength, the flow through the hole can be modeled as incompressible. By using those procedures, Bolton and Kim extended Maa's classical theory to include a resistive end correction for sharp-edged cylindrical holes which differs from those previously proposed by the inclusion of a static component. Here it is shown that CFD models can also be used to compute end corrections for tapered holes. Since practical experimental characterization of perforated materials often involves measurement of the static flow resistance, a closed form empirical equation for that quantity has been developed. Finally, it is shown that configurations having equivalent static flow resistances can yield different acoustic absorptions.
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