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Unconfined turbulent entrainment across density interfaces

  • Ajay B. Shrinivas (a1) and Gary R. Hunt (a2)


We present theoretical models describing the quasi-steady downward transport of buoyant fluid across a gravitationally stable density interface separating two unbounded quiescent fluid masses. The primary transport mechanism is turbulent entrainment resulting from the localised impingement of a vertically forced high-Reynolds-number axisymmetric jet with steady source conditions. The entrainment across the interface is examined in the large-time asymptotic state, wherein the interfacial gravity current, formed by the fluid entrained from the upper layer and the jet, becomes infinitesimally thin and a two-layer stratification persists. Characterising flows with small interfacial Froude numbers $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}({{\mathrm{Fr}}}_i)$ as an axisymmetric semi-ellipsoidal impingement dome, we combine conservation equations with a mechanistic model of entrainment and reveal that, in this regime, the dimensionless entrainment flux $E_i$ across the interface follows the power law $E_i = 0.24{{\mathrm{Fr}}}_i^2$ . For large- ${{\mathrm{Fr}}}_i$ impingements, modelled as a fully penetrating turbulent fountain, we show that $E_i$ no longer scales with ${{\mathrm{Fr}}}_i^2$ , but linearly on ${{\mathrm{Fr}}}_i$ , following $E_i = 0.42{{\mathrm{Fr}}}_i$ . We establish the intermediate range of ${{\mathrm{Fr}}}_i$ over which there is a transition between these quadratic and linear power laws, thus enabling us to classify the dynamics of entrainment across the interface into three distinct regimes. Finally, the close agreement of our solutions with existing experimental results is illustrated.


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Ansong, J. K., Kyba, P. K. & Sutherland, B. R. 2008 Fountains impinging on a density interface. J. Fluid Mech. 595, 115139.
Baines, W. D. 1975 Entrainment by a plume or jet at a density interface. J. Fluid Mech. 68 (2), 309320.
Baines, W. D., Corriveau, A. F. & Reedman, T. J. 1993 Turbulent fountains in a closed chamber. J. Fluid Mech. 255, 621646.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bloomfield, L. J. & Kerr, R. C. 2000 A theoretical model of a turbulent fountain. J. Fluid Mech. 424, 197216.
Cardoso, S. S. S. & Woods, A. W. 1993 Mixing by a turbulent plume in a confined stratified region. J. Fluid Mech. 250, 277305.
Ching, C. Y., Fernando, H. J. S. & Noh, Y. 1993 Interaction of a negatively buoyant line plume with a density interface. Dyn. Atmos. Oceans 19, 367388.
Coffey, C. J. & Hunt, G. R. 2010 The unidirectional emptying box. J. Fluid Mech. 660, 456474.
Cotel, A. J. & Breidenthal, R. E. 1997 A model of stratified entrainment using vortex persistence. Appl. Sci. Res. 57, 349366.
Cotel, A. J., Gjestvang, J. A., Ramkhelawan, N. N. & Breidenthal, R. E. 1997 Laboratory experiments of a jet impinging on a stratified interface. Exp. Fluids 23, 155160.
Cotel, A. J. & Kudo, Y. 2008 Impingement of buoyancy-driven flows at a stratified interface. Exp. Fluids 45, 131139.
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455493.
Fernando, H. J. S. & Long, R. R. 1983 The growth of a grid-generated turbulent mixed layer in a two-fluid system. J. Fluid Mech. 133, 377395.
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J. & Brooks, N. H. 1979 Mixing in Inland and Coastal Waters. Academic Press.
Hunt, G. R. & Coffey, C. J. 2010 Emptying boxes – classifying transient natural ventilation flows. J. Fluid Mech. 646, 137168.
Kaye, N. B., Flynn, M. R., Cook, M. J. & Ji, Y. 2010 The role of diffusion on the interface thickness in a ventilated filling box. J. Fluid Mech. 652, 195205.
Kaye, N. B. & Hunt, G. R. 2006 Weak fountains. J. Fluid Mech. 558, 319328.
Kumagai, M. 1984 Turbulent buoyant convection from a source in a confined two-layered region. J. Fluid Mech. 147, 105131.
Lin, Y. J. P. & Linden, P. F. 2005 The entrainment due to a turbulent fountain at a density interface. J. Fluid Mech. 542, 2552.
Linden, P. F. 1973 The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment. J. Fluid Mech. 60 (3), 467480.
Linden, P. F. 1975 The deepening of a mixed layer in a stratified fluid. J. Fluid Mech. 71 (2), 385405.
McDougall, T. J.1978 Some aspects of geophysical turbulence. PhD thesis, University of Cambridge.
Mizushina, T., Ogino, F., Takeuchi, H. & Ikawa, H. 1982 An experimental study of vertical turbulent jet with negative buoyancy. Wärme-und Stoffübertragung. 16, 1521.
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 123.
Rouse, H. & Dodu, J. 1955 Turbulent diffusion across a density discontinuity. La Houille Blanche 10, 522532.
Shapiro, M. A. 1980 Turbulent mixing within tropopause folds as a mechanism for the exchange of chemical constituents between the stratosphere and troposphere. J. Atmos. Sci. 37, 9941004.
Shy, S. S. 1995 Mixing dynamics of jet interaction with a sharp density interface. Exp. Therm. Fluid Sci. 10, 355369.
Turner, J. S. 1966 Jets and plumes with negative or reversing buoyancy. J. Fluid Mech. 26 (4), 779792.
Turner, J. S. 1968 The influence of molecular diffusivity on turbulent entrainment across a density interface. J. Fluid Mech. 33 (4), 639656.
Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431471.
Williamson, N., Armfield, S. W. & Lin, W. 2011 Forced turbulent fountain flow behaviour. J. Fluid Mech. 671, 535558.
Zhang, Y., Bellingham, J. G., Godin, M. A. & Ryan, J. P. 2012 Using an autonomous underwater vehicle to track the thermocline based on peak-gradient detection. IEEE J. Ocean. Eng. 37 (3), 544553.
Zilitinkevich, S. S. 1991 Turbulent Penetrative Convection. Avebury Technical.
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