Longitudinal dispersion by bodies fixed in a potential flow

Abstract

We examine tracer dispersion by the potential flow through a random array of rigid bodies fixed relative to a mean flow. Both Darcy flow through permeable bodies and inviscid irrotational flow past impermeable bodies are treated within one theoretical framework. The variation of the longitudinal dispersivity with body shape and permeability κ is examined for the case of high Peclet number, Pe . In the absence of diffusive effects, the longitudinal dispersivity D xx (where the mean flow is parallel to the x –axis) is calculated by tracing the evolution of a material surface advected by the mean flow and distorted by the array of bodies. For a random array of identical bodies of volume V and low volume fraction α, D xx = α | D f | UL / V . The drift volume, D f , is defined as the volume between the final and initial position of a material surface distorted by a single body moving in an unbounded flow, and L is the length–scale characterizing the associated longitudinal displacement of the surface. The variation of D xx with permeability is illustrated by considering permeable cylinders and spheres, and the effect of body shape on dispersion is illustrated by considering impermeable spheroids. The longitudinal dispersivity arising from the flow past impermeable bodies is D xx = α C xx UL , where C xx is the added–mass coefficient characterizing the mean flow around the body. This indicates that bluff bodies enhance longitudinal dispersion by promoting the longitudinal stretching of fluid elements. For Darcy flow through bodies of low permeability, the longitudinal dispersivity is Caligraphic D xx α( C xx + 1) UL . The length–scale L , and thus D xx, is singular as κ→ 0, owing to the long retention time of fluid within the bodies. For highly permeable two–dimensional bodies, Caligraphic D xx = α( C yy + 1) UL , where C yy is the added–mass coefficient characterizing the flow around an impermeable body moving parallel to the y –axis. Consequently, dispersion by highly permeable bodies is enhanced when the bodies are slender, in contrast to the low–permeability limit. The influence of finite tracer diffusivity on longitudinal dispersion is demonstrated to make a negligible contribution when κ< 0 provided Pe ≫ max (1, 1/κ) and for impermeable bodies provided Pe ≫ 1. When Pe ≪ 1/κ, the longitudinal dispersion is dominated by diffusive effects and D xx = O (α U 2 a 2 / D 1).