The effect of a non-zero Lagrangian time scale on bounded shear dispersion

Abstract

AbstractPrevious studies of shear dispersion in bounded velocity fields have assumed random velocities with zero Lagrangian time scale (i.e. velocities are$\delta $-function correlated in time). However, many turbulent (geophysical and engineering) flows with mean velocity shear exist where the Lagrangian time scale is non-zero. Here, the longitudinal (along-flow) shear-induced diffusivity in a two-dimensional bounded velocity field is derived for random velocities with non-zero Lagrangian time scale${\tau }_{L} $. A non-zero${\tau }_{L} $results in two-time transverse (across-flow) displacements that are correlated even for large (relative to the diffusive time scale${\tau }_{D} $) times. The longitudinal (along-flow) shear-induced diffusivity${D}_{S} $is derived, accurate for all${\tau }_{L} $, using a Lagrangian method where the velocity field is periodically extended to infinity so that unbounded transverse particle spreading statistics can be used to determine${D}_{S} $. The non-dimensionalized${D}_{S} $depends on time and two parameters: the ratio of Lagrangian to diffusive time scales${\tau }_{L} / {\tau }_{D} $and the release location. Using a parabolic velocity profile, these dependencies are explored numerically and through asymptotic analysis. The large-time${D}_{S} $is enhanced relative to the classic Taylor diffusivity, and this enhancement increases with$ \sqrt{{\tau }_{L} } $. At moderate${\tau }_{L} / {\tau }_{D} = 0. 1$this enhancement is approximately a factor of 3. For classic shear dispersion with${\tau }_{L} = 0$, the diffusive time scale${\tau }_{D} $determines the time dependence and large-time limit of the shear-induced diffusivity. In contrast, for sufficiently large${\tau }_{L} $, a shear time scale${\tau }_{S} = \mathop{ ({\tau }_{L} {\tau }_{D} )}\nolimits ^{1/ 2} $, anticipated by a simple analysis of the particle’s domain-crossing time, determines both the${D}_{S} $time dependence and the large-time limit. In addition, the scalings for turbulent shear dispersion are recovered from the large-time${D}_{S} $using properties of wall-bounded turbulence.