Vorticity in Three-Dimensionally Random Porous Media

Abstract

The flow gradient tensor controls the rate of dissipation of concentration fluctuations due to local dispersion, and therefore determines the rate of dilution of solute in a spatially random flow field. Off-diagonal terms of the flow gradient tensor quantify the rotational characteristics of the flow. A leading order description of the vorticity $(\omega = (\omega _1 ,\omega _2 ,\omega _3 ))$, of flow in a three-dimensionally random spatially correlated hydraulic conductivity field $(K({\text{x}}))$ is made by relating the vorticity spectrum to the spectrum of ln $K({\text{x}})$. Distinct components of the vorticity are found to be linearly uncorrelated $(\overline {\omega _i \omega _j } = 0,i \ne j)$. The characteristic vorticity component in the bulk flow direction is zero $(\sigma _{\omega _1 } = 0)$, and the characteristic vorticity in the transverse directions $(\sigma _{\omega _2 } ,\sigma _{\omega _3 } )$ are inversely proportional to the hydraulic conductivity microscales in the other transverse direction, as exhibited in a numerical calculation of the vorticity.