Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-25T06:38:50.326Z Has data issue: false hasContentIssue false
  • English
  • Français

Confined turbulent entrainment across density interfaces

Published online by Cambridge University Press:  14 August 2015

Ajay B. Shrinivas  and
Gary R. Hunt
Ajay B. Shrinivas
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
Gary R. Hunt*
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
*
†Email address for correspondence: gary.hunt@eng.cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

In pursuit of a universal law for the rate of entrainment across a density interface driven by the impingement of a localised turbulent flow, the role of the confinement, wherein the environment is within the confines of a box, has to date been overlooked. Seeking to unravel the effects of confinement, we develop a phenomenological model describing the quasi-steady rate at which buoyant fluid is turbulently entrained across a density interface separating two uniform layers within the confines of a box. The upper layer is maintained by a turbulent plume, and the localised impingement of a turbulent fountain with the interface drives entrainment of fluid from the upper layer into the lower layer. The plume and fountain rise from sources at the base of the box and are non-interacting. Guided by previous observations, our model characterises the dynamics of fountain–interface interaction and the steady secondary flow in the environment that is induced by the perpetual cycle of vertical excursions of the interface. We reveal that the dimensionless entrainment flux across the interface $E_{i}$ is governed not only by an interfacial Froude number $\mathit{Fr}_{i}$ but also by a ‘confinement’ parameter ${\it\lambda}_{i}$, which characterises the length scale of interfacial turbulence relative to the depth of the upper layer. By deducing the range of ${\it\lambda}_{i}$ that may be regarded as ‘small’ and ‘large’, we shed new light on the effects of confinement on interfacial entrainment. We establish that for small ${\it\lambda}_{i}$, a weak secondary flow has little influence on $E_{i}$, which follows a quadratic power law $E_{i}\propto \mathit{Fr}_{i}^{2}$. For large ${\it\lambda}_{i}$, a strong secondary flow significantly influences $E_{i}$, which then follows a cubic power law $E_{i}\propto \mathit{Fr}_{i}^{3}$. Drawing on these results, and showing that for previous experimental studies ${\it\lambda}_{i}$ exhibits wide variation, we highlight underlying physical reasons for the significant scatter in the existing measurements of the rate of interfacial entrainment. Finally, we explore the implications of our results for guiding appropriate choices of box geometry for experimentally and numerically examining interfacial entrainment.

JFM classification

Convection: Convection Convection: Plumes/thermals Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 779 , 25 September 2015 , pp. 116 - 143
Check for updates
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baines, W. D. 1975 Entrainment by a plume or jet at a density interface. J. Fluid Mech. 68 (2), 309–320.Google Scholar
Baines, W. D., Corriveau, A. F. & Reedman, T. J. 1993 Turbulent fountains in a closed chamber. J. Fluid Mech. 255, 621–646.CrossRefGoogle Scholar
Cardoso, S. S. S. & Woods, A. W. 1993 Mixing by a turbulent plume in a confined stratified region. J. Fluid Mech. 250, 277–305.Google Scholar
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, 367–388.CrossRefGoogle Scholar
Coffey, C. J. & Hunt, G. R. 2010 The unidirectional emptying box. J. Fluid Mech. 660, 456–474.CrossRefGoogle Scholar
Cotel, A. J., Gjestvang, J. A., Ramkhelawan, N. N. & Breidenthal, R. E. 1997 Impingement of buoyancy-driven flows at a stratified interface. Exp. Fluids 23, 155–160.CrossRefGoogle Scholar
Devenish, B. J., Rooney, G. G., Webster, H. N. & Thomson, D. J. 2010 The entrainment rate for buoyant plumes in a crossflow. Boundary-Layer Meteorol. 134, 411–439.Google Scholar
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455–493.Google Scholar
Hoult, D. P. & Weil, J. C. 1972 Turbulent plume in a laminar cross flow. Atmos. Environ. 6, 513–531.CrossRefGoogle Scholar
Hunt, G. R. & Burridge, H. C. 2015 Fountains in industry and nature. Annu. Rev. Fluid Mech. 47, 195–220.Google Scholar
Hunt, G. R. & Coffey, C. J. 2010 Emptying boxes: classifying transient natural ventilation flows. J. Fluid Mech. 646, 137–168.Google Scholar
Jirka, G. H. & Harleman, D. R. F. 1979 Stability and mixing of a vertical plane buoyant jet in a confined depth. J. Fluid Mech. 94, 275–304.Google Scholar
Kaye, N. B. & Hunt, G. R. 2006 Weak fountains. J. Fluid Mech. 558, 319–328.CrossRefGoogle Scholar
Kaye, N. B. & Linden, P. F. 2004 Coalescing axisymmetric turbulent plumes. J. Fluid Mech. 502, 41–63.Google Scholar
Kumagai, M. 1984 Turbulent buoyant convection from a source in a confined two-layered region. J. Fluid Mech. 147, 105–131.Google Scholar
Lee, J. H. W. & Chu, V. H. 1979 Turbulent Jets and Plumes: A Lagrangian Approach. Kluwer Academic.Google Scholar
Lin, Y. J. P. & Linden, P. F. 2005 The entrainment due to a turbulent fountain at a density interface. J. Fluid Mech. 542, 25–52.Google Scholar
Linden, P. F. 1974 The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment. J. Fluid Mech. 60 (3), 467–480.Google Scholar
Mizushina, T., Ogino, F., Takeuchi, H. & Ikawa, H. 1982 An experimental study of vertical turbulent jet with negative buoyancy. Wärme-Stoffübertrag. 16, 15–21.Google Scholar
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Shrinivas, A. B. & Hunt, G. R. 2014 Unconfined turbulent entrainment across density interfaces. J. Fluid Mech. 757, 573–598.CrossRefGoogle Scholar
Shy, S. S. 1995 Mixing dynamics of jet interaction with a sharp density interface. Exp. Therm. Fluid Sci. 10, 355–369.CrossRefGoogle Scholar
Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431–471.Google Scholar