| The vertical dispersion of fluid particles in stably stratified homogeneous turbulence with mean shear is investigated. An analysis framework which describes the associated flow energetics in the Lagrangian frame is developed. This provides a more clear and consistent interpretation of the behavior of the mean square vertical displacement which can be related to the total potential energy (TPE) of a given set of fluid particles. The analysis considers TPE in terms of the available potential energy (APE), associated with the nonequilibrium displacement, and the reference potential energy (RPE), associated with the change in particle equilibrium height, i.e., the equilibrium displacement. The corresponding evolution equations describe the key sequence of processes. As fluid particles move away from their equilibrium height, vertical kinetic energy is converted (reversibly) to APE. This establishes nonequilibrium displacement and increases TPE. Without molecular diffusion, gravity will reduce the vertical velocity and the particle will eventually return to its original equilibrium height; APE is converted back to KE in this reversible process. With molecular diffusion, fluid particles will change their density, and therefore their equilibrium height, such to reduce density fluctuation; i.e., some of the APE will be dissipated and converted to RPE. Thus, molecular diffusion acts to preserve displacements and reduce the reconversion of PE to KE. In this manner, fluid particles can move further away from their original equilibrium level and the mean square vertical displacement can grow without limit.;The evolution equations are integrated in time and give a relation for mean square vertical displacement. At long time, the RPE will dominate the TPE; mean square vertical displacement is then a measure of the total APE dissipated by the flow. The significance of this with respect to the total energy dissipated is given by the cumulative mixing efficiency, O c, which depends on the strength of stratification. In the case of decaying turbulence, mean square vertical displacement evolves to a constant value, proportional to Oc. In the case of stationary turbulence, the (constant) rate of growth of mean square vertical displacement is proportional to Oc. The analysis is demonstrated using direct numerical simulations of homogeneous shear flows with decaying, stationary, and growing turbulence. Results for the latter case show mean square vertical displacement to continually increase, and to have a reduced dependence on and reduced values of Oc as the strength of stratification decreases. In general, simulation results are in agreement with the analysis and confirm that, in homogeneous stratified flows with mean shear, an effective time scale for vertical dispersion at long time is that of the turbulence decay time. |
