Time scales of stratified turbulent flows and relations between second- order closure parameters and flow numbers

The description of turbulent mixing and chemical reactions by Lagrangian probability density function methods offers some significant advantages over other methods, mainly due to the simulation of mixing processes and the exact treatment of chemical transformations. A key problem of such methods is the information on the time scales of processes, because they determine the dynamics and intensity of mixing. This question is considered for stratified flow. Different models are presented for the development of these time scales in time and their stationary spatial patterns in dependence on shear and stratification. The model predictions are shown to be in agreement with large-eddy simulations of stratified homogeneous shear flow. Two further applications of these models are considered: the description of transitions between flow regimes characterized by different scaling quantities in the stationary atmospheric surface layer and, second, the simulation of buoyant plume rise. It is shown that the predictions of the stationary frequency model agree with measured data. The consideration of limit cases of this model leads to connections between second-order closure parameters and critical flow numbers that characterize these transitions. These relationships are shown to be very advantageous for the application of closure models. A new flow number that characterizes the transition to free convective flow under unstable stratification is introduced here in analogy to the critical gradient Richardson number, which characterizes the onset of turbulence in stably stratified flow. The second application provides a new theory for buoyant plume rise. Two parameters that describe the turbulent mixing in the entrainment and extrainment stages of plume rise are explained as ratios of the relevant time scales. The two-thirds power law of buoyant plume rise, which is observed for nonturbulent and neutrally stratified flow, is obtained without having to make ad hoc assumptions. For turbulent flow, the plume’s leveling-off is calculated in accord with measurements. © 1998 American Institute of Physics. S1070-6631 98 02504-5