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 δ-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 τL. A non-zero τL results in two-time transverse (across-flow) displacements that are correlated even for large (relative to the diffusive time scale τD) times. The longitudinal (along-flow) shear-induced diffusivity DS is derived, accurate for all τL, 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 DS. The non-dimensionalized DS depends on time and two parameters: the ratio of Lagrangian to diffusive time scales τL/τD and the release location. Using a parabolic velocity profile, these dependencies are explored numerically and through asymptotic analysis. The large-time DS is enhanced relative to the classic Taylor diffusivity, and this enhancement increases with √τL. At moderate τL/τD = 0. 1 this enhancement is approximately a factor of 3. For classic shear dispersion with τL = 0, the diffusive time scale τD determines the time dependence and large-time limit of the shear-induced diffusivity. In contrast, for sufficiently large τL, a shear time scale τS = (τLτD)1/2, anticipated by a simple analysis of the particle's domain-crossing time, determines both the DS time dependence and the large-time limit. In addition, the scalings for turbulent shear dispersion are recovered from the large-time DS using properties of wall-bounded turbulence. © 2011 Cambridge University Press.