Optimal mixing of buoyant jets and plumes in stratified fluids: theory and experiments

Abstract

The influence of ambient fluid stratification on buoyant miscible jets and plumes is studied theoretically and experimentally. Given a fixed set of jet/plume parameters, and an ambient fluid stratification sandwiched between top and bottom homogeneous densities, a theoretical criterion is identified to show how step-like density profiles constitute the most effective mixers within a broad class of stable density transitions. This is assessed both analytically and experimentally, respectively by establishing rigorous a priori estimates on generalized Morton–Taylor–Turner (MTT) models (Morton et al., Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23; Fischer et al., Mixing in Inland and Coastal Waters. Academic, 1979), and by studying a critical phenomenon determined by the distance between the jet/plume release height with respect to the depth of the ambient density transition. For fluid released sufficiently close to the background density transition, the buoyant jet fluid escapes and rises indefinitely. For fluid released at locations lower than a critical depth, the buoyant fluid stops rising and is trapped indefinitely. A mathematical formulation providing rigorous estimates on MTT models is developed along with nonlinear jump conditions and an exact critical-depth formula that is in good quantitative agreement with the experiments. Our mathematical analysis provides rigorous justification for the critical trapping/escaping criteria, first presented in Caulfield & Woods (J. Fluid Mech., vol. 360, 1998, pp. 229–248), within a class of algebraic density decay rates. Further, the step-like background stratification is shown to be the most efficient mixing profile amongst a broad family of stably stratified profiles sharing the same density transition within a fixed distance. Finally, the analysis uncovers surprising differences between the Gaussian and top-hat profile closures concerning initial mixing of the jet and ambient fluid.