Spectrum of density fluctuations in a particle-fluid system— i. monodisperse spheres

A method for calculating the density autocorrelation ′(x)′(x + r) for a homogeneous particle-fluid system in both physical and Fourier transform space has been developed. The density autocorrelation was related to two quantities, the Overlap function which is defined as the volume of intersection of two spheres as a function of the separation distance and the radial distribution function (RDF) of the particles. In dimensionless co-ordinates, the parameter that characterizes the density autocorrelation is the volume fraction of particles, ₁, , or equivalently the dimensionless mean separation distance (normalized by the particle diameter), . For an isotropic randomly distributed system of particles, the density autocorrelation was observed to oscillate with the correlation distance r, with a wavelength that was proportional to λ. The Fourier transform of the autocorrelation likewise oscillated with the wavenumber k, however the effect of changes in the particle volume fraction was limited to the first peak only. Subsequent peaks were more closely associated with the Overlap function.

The results for the density autocorrelation were extended to a particle-fluid system which experienced an asymptotically large pressure gradient. This initially produced a uniform relative motion between the two fields. In this limit, other higher-order moments such as the Reynolds stress can be related to the density autocorrelation in a straightforward manner. Moreover the spectral shapes of all moments collapse onto the density autocorrelation spectrum in this limit. It was pointed out that the uniform relative motion will eventually become unstable because of hydrodynamic forces on the particles induced by the relative motion. This effect was estimated by introducing a mildly attractive force into the RDF. The results demonstrated that the induced hydrodynamic force promoted a shift in the density spectrum toward small k (large scale) indicating an alternative mechanism for growth in the integral length scale.