Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T21:56:14.909Z Has data issue: false hasContentIssue false
  • English
  • Français

Buoyancy transfer in a two-layer system in steady state. Experiments in a Taylor–Couette cell

Published online by Cambridge University Press:  08 June 2020

Diana Petrolo Open the ORCID record for Diana Petrolo [Opens in a new window]  and
Sandro Longo Open the ORCID record for Sandro Longo [Opens in a new window]
Diana Petrolo
Affiliation:
Dipartimento di Ingegneria e Architettura (DIA), Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124Parma, Italy
Sandro Longo*
Affiliation:
Dipartimento di Ingegneria e Architettura (DIA), Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124Parma, Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Our experimental study focuses on the density and velocity field in two layers of fluid separated by a sharp density interface. Turbulence is generated by a non-invasive stirrer, a Taylor–Couette tank, and the interface is stabilized with a source of saline fluid and a source of fresh water at the bottom and top of the tank, respectively. The same volume fluxes are withdrawn by two sinks to maintain a constant volume of fluid in the tank. Our results confirm past experiments and show that a strong vertical exchange of fluid occurs close to the inner cylinder and across the interface, where the vertical turbulent length scales appear to be suppressed. For low values of kinetic energy supplied to the system, the interface may act as a rigid boundary for the turbulent eddies, with a reduction of the vertical length scales although it seems not to affect the horizontal length scales. The vertical buoyancy flux extracted at the top of the tank is fairly well reproduced by the measured correlation $\overline{\unicode[STIX]{x1D70C}^{\prime }w^{\prime }}$ between density and vertical velocity fluctuations across the interface. Quadrant analysis of the correlation terms reveals that the greatest contribution to salt flux is given by eddies that carry the lighter fluid from top to bottom across the interface. The mixing process is accompanied by a single wake-like disturbance, with a radial front advancing in the azimuthal direction across the interface, acting as a blade, and with a period that decreases with rotation rate. The wake favours the smoothing of the density step and, in a simplified model, we assume that the turbulent diffusion is active during a fraction of the cycle in the wake-mixing region, with diffusivity proportional to the transverse length scale and the speed of the wake. The mixing region is the domain between the nose of the wave-like perturbation and the section where the interface becomes ‘darker’ again after being mixed by the vortexes. The results of this model are in a fair agreement with the experiments. The potential energy of the interfacial perturbations is only a small part of the missing turbulent kinetic energy, defined as the difference in the turbulent kinetic energy between a well-mixed fluid and a two-layer fluid. Further analysis is needed to explain the mechanism of generating these perturbations and the factors that control their periodicity.

JFM classification

Geophysical and Geological Flows: Stratified flows Mixing: Turbulent mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 896 , 10 August 2020 , A27
Copyright
© The Author(s), 2020. Published by 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

Balmforth, N. J., Llewellyn Smith, S. G. & Young, W. R. 1998 Dynamics of interfaces and layers in a stratified turbulent fluid. J. Fluid Mech. 355, 329–358.CrossRefGoogle Scholar
Battjes, J. A. & Sakai, T. 1981 Velocity field in a steady breaker. J. Fluid Mech. 111, 421–437.CrossRefGoogle Scholar
Boubnov, B. M., Gledzer, E. B. & Hopfinger, E. J. 1995 Stratified circular Couette flow: instability and flow regimes. J. Fluid Mech. 292, 333–358.CrossRefGoogle Scholar
Boyer, D. L., Davies, P. A. & Guo, Y. 1997 Mixing of a two-layer stratified fluid by a rotating disk. Fluid Dyn. Res. 21 (5), 381–401.CrossRefGoogle Scholar
Briggs, D. A., Ferziger, J. H., Koseff, J. R. & Monismith, S. G. 1998 Turbulent mixing in a shear-free stably stratified two-layer fluid. J. Fluid Mech. 354, 175–208.CrossRefGoogle Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64 (4), 775–816.CrossRefGoogle Scholar
Brumley, B. H. & Jirka, G. H. 1987 Near-surface turbulence in a grid-stirred tank. J. Fluid Mech. 183, 235–263.CrossRefGoogle Scholar
Burin, M. J., Ji, H., Schartman, E., Cutler, R., Heitzenroeder, P., Liu, W., Morris, L. & Raftopolous, S. 2006 Reduction of Ekman circulation within Taylor–Couette flow. Exp. Fluids 40 (6), 962–966.CrossRefGoogle Scholar
Carminati, M. & Luzzatto-Fegiz, P. 2017 Conduino: affordable and high-resolution multichannel water conductivity sensor using micro USB connectors. Sensors Actuators B 251, 1034–1041.CrossRefGoogle Scholar
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23 (1), 455–493.CrossRefGoogle Scholar
Fernando, H. J. S. & Long, R. R. 1985 On the nature of the entrainment interface of a two-layer fluid subjected to zero-mean-shear turbulence. J. Fluid Mech. 151, 21–53.CrossRefGoogle Scholar
Fernando, H. J. S. & Long, R. R. 1988 Experiments on steady buoyancy transfer through turbulent fluid layers separated by density interfaces. Dyn. Atmos. Oceans 12 (3–4), 233–257.CrossRefGoogle Scholar
Goda, Y. 2010 Random Seas and Design of Maritime Structures. World Scientific.CrossRefGoogle Scholar
Guyez, E., Flor, J.-B. & Hopfinger, E. J. 2007 Turbulent mixing at a stable density interface: the variation of the buoyancy flux–gradient relation. J. Fluid Mech. 577, 127–136.CrossRefGoogle Scholar
Hannoun, I. A., Fernando, H. J. S. & List, E. J. 1988 Turbulence structure near a sharp density interface. J. Fluid Mech. 189, 189–209.CrossRefGoogle Scholar
Herlina & Jirka, G. H. 2008 Experiments on gas transfer at the air–water interface induced by oscillating grid turbulence. J. Fluid Mech. 594, 183–208.CrossRefGoogle Scholar
Komori, S., Murakami, Y. & Ueda, H. 1989 The relationship between surface-renewal and bursting motions in an open-channel flow. J. Fluid Mech. 203, 103–123.CrossRefGoogle Scholar
Leclercq, C., Partridge, J. L., Augier, P., Caulfield, C. P., Dalziel, S. B. & Linden, P. F.2016a Nonlinear waves in stratified Taylor–Couette flow. Part 1. Layer formation. arXiv:1609.02885.Google Scholar
Leclercq, C., Partridge, J. L., Augier, P., Caulfield, C. P., Dalziel, S. B. & Linden, P. F.2016b Nonlinear waves in stratified Taylor–Couette flow. Part 2. Buoyancy flux. arXiv:1609.02886.Google Scholar
Linden, P. F. 1980 Mixing across a density interface produced by grid turbulence. J. Fluid Mech. 100 (4), 691–703.CrossRefGoogle Scholar
Longo, S. 2010 Experiments on turbulence beneath a free surface in a stationary field generated by a Crump weir: free-surface characteristics and the relevant scales. Exp. Fluids 49 (6), 1325–1338.CrossRefGoogle Scholar
Longo, S. 2011 Experiments on turbulence beneath a free surface in a stationary field generated by a Crump weir: turbulence structure and correlation with the free surface. Exp. Fluids 50 (1), 201–215.CrossRefGoogle Scholar
Longo, S., Liang, D., Chiapponi, L. & Jiménez, L. A. 2012 Turbulent flow structure in experimental laboratory wind-generated gravity waves. Coast. Engng 64, 1–15.CrossRefGoogle Scholar
Longo, S., Ungarish, M., Di Federico, V., Chiapponi, L. & Addona, F. 2016 Gravity currents produced by constant and time varying inflow in a circular cross-section channel: experiments and theory. Adv. Water Resour. 90, 10–23.CrossRefGoogle Scholar
Mackenzie, K. V. 1981 Nine-term equation for sound speed in the oceans. J. Acoust. Soc. Am. 70 (3), 807–812.CrossRefGoogle Scholar
McKenna, S. P. & McGillis, W. R. 2004 Observations of flow repeatability and secondary circulation in an oscillating grid-stirred tank. Phys. Fluids 16 (9), 3499–3502.CrossRefGoogle Scholar
Mujica, N. & Lathrop, D. P. 2006 Hysteretic gravity-wave bifurcation in a highly turbulent swirling flow. J. Fluid Mech. 551, 49–62.CrossRefGoogle Scholar
Odier, P., Chen, J. & Ecke, R. E. 2012 Understanding and modeling turbulent fluxes and entrainment in a gravity current. Physica D 241 (3), 260–268.CrossRefGoogle Scholar
Oglethorpe, R. L. F.2014 Mixing in stably stratified turbulent Taylor–Couette flow. PhD thesis, University of Cambridge.CrossRefGoogle Scholar
Oglethorpe, R. L. F., Caulfield, C. P. & Woods, A. W. 2013 Spontaneous layering in stratified turbulent Taylor–Couette flow. J. Fluid Mech. 721, R3.CrossRefGoogle Scholar
Park, Y.-G., Whitehead, J. A. & Gnanadeskian, A. 1994 Turbulent mixing in stratified fluids: layer formation and energetics. J. Fluid Mech. 279, 279–311.CrossRefGoogle Scholar
Peregrine, D. H. & Svendsen, I. A. 1978 Spilling breakers, bores, and hydraulic jumps. In Coastal Engineering 1978, pp. 540–550. ASCE.CrossRefGoogle Scholar
Petrolo, D. & Woods, A. W. 2019 Measurements of buoyancy flux in a stratified turbulent flow. J. Fluid Mech. 861, R2.CrossRefGoogle Scholar
Pouquet, A., Sen, A., Rosenberg, D., Mininni, P. D. & Baerenzung, J. 2013 Inverse cascades in turbulence and the case of rotating flows. Phys. Scr. 2013 (T155), 014032.Google Scholar
Prandtl, L. 1942 Bemerkungen zur theorie der freien turbulenz. Z. Angew. Math. Mech. 22 (5), 241–243.CrossRefGoogle Scholar
Shravat, A., Cenedese, C. & Caulfield, C. P. 2012 Entrainment and mixing dynamics of surface-stress-driven stratified flow in a cylinder. J. Fluid Mech. 691, 498–517.CrossRefGoogle Scholar
Singh, K. N., Augier, P., Caulfield, C. P., Dalziel, S. B., Leclercq, C. & Partridge, J. L. 2018 Layering, instability, mixing, interfaces and turbulence in a stratified Taylor–Couette flow. In International Conference on Rayleigh-Bénard Turbulence, pp. 8–9 Enschede, The Netherlands, The University of Twente.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Thorpe, S. A. 1973 Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61 (4), 731–751.CrossRefGoogle Scholar
Thorpe, S. A. 2016 Layers and internal waves in uniformly stratified fluids stirred by vertical grids. J. Fluid Mech. 793, 380–413.CrossRefGoogle Scholar
Troy, C. D. & Koseff, J. R. 2005 The generation and quantitative visualization of breaking internal waves. Exp. Fluids 38 (5), 549–562.CrossRefGoogle Scholar
Turner, J. S. 1965 The coupled turbulent transports of salt and and heat across a sharp density interface. Intl J. Heat Mass Transfer 8 (5), 759–767.CrossRefGoogle Scholar
Variano, E. A. & Cowen, E. A. 2013 Turbulent transport of a high-Schmidt-number scalar near an air–water interface. J. Fluid Mech. 731, 259–287.CrossRefGoogle Scholar
Wessels, F. & Hutter, K. 1996 Interaction of internal waves with a topographic sill in a two-layered fluid. J. Phys. Oceanogr. 26 (1), 5–20.2.0.CO;2>CrossRefGoogle Scholar
Whitehead, J. A. & Stevenson, I. 2007 Turbulent mixing of two-layer stratified fluid. Phys. Fluids 19 (12), 125104.CrossRefGoogle Scholar
Wolanski, E. J. & Brush, L. M. Jr 1975 Turbulent entrainment across stable density step structures. Tellus 27 (3), 259–268.CrossRefGoogle Scholar
Woods, A. W., Caulfield, C. P., Landel, J. R. & Kuesters, A. 2010 Non-invasive turbulent mixing across a density interface in a turbulent Taylor–Couette flow. J. Fluid Mech. 663, 347–357.CrossRefGoogle Scholar

Petrolo and Longo supplementary movie 1

Exp 6, Ω = 2.00 rad/s. Top view of dye steaks at the interface.

Download Petrolo and Longo supplementary movie 1(Video)
Video 59.1 MB

Petrolo and Longo supplementary movie 2

Exp 1, Ω = 2.75 rad/s. Generation process of the interfacial perturbations. Two cameras recordings.

Download Petrolo and Longo supplementary movie 2(Video)
Video 37.9 MB