Relationship between the intrinsic radial distribution function for an isotropic field of particles and lower-dimensional measurements

Abstract

In this paper, we present relationships between the intrinsic radial distribution function (RDF) for a three-dimensional, isotropic system of particles and the lower-dimensional RDFs obtained experimentally from either two-dimensional or one-dimensional sampling of the data. The lower-dimensional RDFs are shown to be equivalent to integrals of the three-dimensional function, and as such contain less information than their three-dimensional counterpart. An important consequence is that the lower-dimensional RDFs are attenuated at separation distances below the characteristic length scale of the measurement. In addition, the inverse problem (calculating the three-dimensional RDF from the lower-dimensional measurements) is not well posed. However, recent results from direct numerical simulations (Reade & Collins 2000) showed that the three-dimensional RDF for aerosol particles in a turbulent flow field obeys a power-law dependence on r for r [Lt ] η, where η is the Kolmogorov scale of the turbulence. In this case, the inverse problem is well posed and it is possible to obtain the prefactor and exponent of the power law from one- or two-dimensional measurements. A procedure for inverting the data is given. All of the relationships derived in this paper have been validated by data derived from direct numerical simulations.