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

Abstract

Previous 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.

The Generation of Surfzone Eddies in a Strong Alongshore Current

“…These horizontal eddies (hereafter termed eddies) have rotational (as opposed to irrotational) velocities associated with vertical vorticity (hereafter termed vorticity). When alongshore current shear was strong, drifter-derived alongshore diffusivities were well predicted by a shear dispersion theory that includes nonzero Lagrangian time scale (Spydell and Feddersen 2012a). Drifterderived time-dependent absolute cross-shore diffusivities were consistent with stirring due to surfzone eddies with Lagrangian (not Eulerian) time scales of O(100 s) (Spydell and Feddersen 2012b).…”

“…The magnitude of these eddy velocities was also generally related to V (Noyes et al 2004). For example, on a monotonic, alongshore uniform beach with V 5 0 m s 21 , the presence of a scale-dependent relative diffusivity indicated the presence of an energetic eddy field with scales varying from 10 to 50 m (Spydell et al 2007;Spydell and Feddersen 2009). For example, on a monotonic, alongshore uniform beach with V 5 0 m s 21 , the presence of a scale-dependent relative diffusivity indicated the presence of an energetic eddy field with scales varying from 10 to 50 m (Spydell et al 2007;Spydell and Feddersen 2009).…”

Time persistence of floating-particle clusters in free-surface turbulence

“…Surface compressibility may play an important role in determining the motion of passive tracers like pollutants and nutrients but also the spreading rate of active ocean surfactants, such as phytoplankton [2]. Our findings provide useful indications to parameterize the relevant timescales characterizing dispersion of such species and, therefore, can assist in developing reliable predictive models [20]. …”

Anisotropy in pair dispersion of inertial particles in turbulent channel flow

“…In this case, the shear profile is a nonlinear function of the wall distance and its effects on particles are complicated by the interactions with a flow structure which is anisotropic along the wall-normal direction. This has an influence on the current framework used to model particle dispersion in sheared flows, since the model scales, as for instance the Lagrangian time scale of the fluid, 36 are strongly non-uniform across the channel. 37 Phenomena involving the effects of both shear and turbulent dispersion are currently modelled with approximated or ad hoc chosen scales.…”

“…37 Phenomena involving the effects of both shear and turbulent dispersion are currently modelled with approximated or ad hoc chosen scales. 36 Such modelling choices could be improved on the basis of statistical characterization of particle pairs coming from detailed numerical simulations.…”