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A second-order integral model for buoyant jets with background homogeneous and isotropic turbulence

  • Adrian C. H. Lai (a1), Adrian Wing-Keung Law (a1) (a2) and E. Eric Adams (a3)


Buoyant jets or forced plumes are discharged into a turbulent ambient in many natural and engineering applications. The background turbulence generally affects the mixing characteristics of the buoyant jet, and the extent of the influence depends on the characteristics of both the jet discharge and ambient. Previous studies focused on the experimental investigation of the problem (for pure jets or plumes), but the findings were difficult to generalize because suitable scales for normalization of results were not known. A model to predict the buoyant jet mixing in the presence of background turbulence, which is essential in many applications, is also hitherto not available even for a background of homogeneous and isotropic turbulence (HIT). We carried out experimental and theoretical investigations of a buoyant jet discharging into background HIT. Buoyant jets were designed to be in the range of $1<z/l_{M}<5$ , where $l_{M}=M_{o}^{3/4}/F_{o}^{1/2}$ is the momentum length scale, with $z/l_{M}<\sim 1$ and $z/l_{M}>\sim 6$ representing the asymptotic cases of pure jets and plumes, respectively. The background turbulence was generated using a random synthetic jet array, which produced a region of approximately isotropic and homogeneous field of turbulence to be used in the experiments. The velocity scale of the jet was initially much higher, and the length scale smaller, than that of the background turbulence, which is typical in most applications. Comprehensive measurements of the buoyant jet mixing characteristics were performed up to the distance where jet breakup occurred. Based on the experimental findings, a critical length scale $l_{c}$ was identified to be an appropriate normalizing scale. The momentum flux of the buoyant jet in background HIT was found to be conserved only if the second-order turbulence statistics of the jet were accounted for. A general integral jet model including the background HIT was then proposed based on the conservation of mass (using the entrainment assumption), total momentum and buoyancy fluxes, and the decay function of the jet mean momentum downstream. Predictions of jet mixing characteristics from the new model were compared with experimental observation, and found to be generally in agreement with each other.


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Cheng, N. S. & Law, A. W. K. 2001 Measurements of turbulence generated by oscillating grid. J. Hydraul. Engng 127 (3), 201208.
Ching, C. Y., Fernando, H. J. S. & Robles, A. 1995 Break down of line plumes in turbulent environments. J. Geophys. Res. 100 (C3), 47074713.
Craske, J. & van Reeuwijkl, M. 2015 Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets. J. Fluid Mech. 763, 500537.
Craske, J. & van Reeuwijkl, M. 2016 Generalised unsteady plume theory. J. Fluid Mech. 792, 10131052.
Craven, B. A. & Settles, G. S. 2006 A computational and experimental investigation of the human thermal plume. J. Fluids Engng 128 (6), 12511258.
Cuthbertson, A. J. S., Malcangio, D., Davies, P. A. & Mossa, M. 2006 The influence of a localized region on turbulence on the structural development of a turbulent, round, buoyant jet. Fluid Dyn. Res. 38, 683698.
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J. & Brooks, N. H. 1979 Mixing in Inland and Coastal Waters. Academic Press.
Guo, Y., Malcangio, D., Davies, P. A. & Fernando, H. J. S. 2005 A laboratory investigation into the influence of a localized region on turbulence on the evolution of a round turbulent jet. Fluid Dyn. Res. 36, 7889.
Hubner, J.2004 Buoyant plumes in a turbulent environment. PhD thesis, University of Cambridge.
Hunt, J. C. R. 1994 Atmospheric jets and plumes. In Recent Research Advances in the Fluid Mechanics of Turbulent Jets and Plumes (ed. Davies, P. A. & Valente Neves, M. I.), NATO ASI Series E, vol. 255, pp. 309334. Springer.
Hunt, J. C. R., Eames, I. & Westerweel, J. 2006 Mechanics of inhomogeneous turbulence and interfacial layers. J. Fluid Mech. 554, 499519.
Hussein, H. J., Capp, S. P. & George, W. K. 1994 Velocity measurements in a high-Reynolds-number, momentum-conserving, axisymmetric, turbulent jet. J. Fluid Mech. 258, 3175.
Khorsandi, B., Gaskin, S. & Mydlarski, L. 2013 Effect of background turbulence on an axisymmetric turbulent jet. J. Fluid Mech. 736, 250286.
Kiya, M., Ohyama, M. & Hunt, J. C. R. 1986 Vortex pairs and rings interacting with shear layer vortices. J. Fluid Mech. 172, 115.
Law, A. W. K., Cheng, N. S. & Davidson, M. J. 2001 Jet spreading in oscillating-grid turbulence. In Proceedings of the 3rd International Symposium on Environmental Hydraulics, vol. 36, pp. 16. International Association of Hydro-Environment Engineering and Research (IAHR).
Lee, J. H. W. & Chu, V. H. 2003 Turbulent Jets and Plumes: A Lagrangian Approach. Kluwer.
Linden, P. F. 1999 The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31, 201238.
Linden, P. F., Lane-Serff, G. F. & Smeed, D. A. 1990 Emptying filling boxes: the fluid mechanics of natural ventilation. J. Fluid Mech. 212, 309335.
Loomans, M.1998 The measurement and simulation of indoor airflow. PhD thesis, Technical University of Eindhoven.
Maxey, M. R. 1987 The velocity skewness measured in grid turbulence. Phys. Fluids 30, 935939.
Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 123.
Papanicolaou, P. N. & List, E. J. 1988 Investigations of round vertical turbulent buoyant jets. J. Fluid Mech. 195, 341391.
Perez-Alvarado, A.2016 Effect of background turbulence on the scalar field of a turbulent jet. PhD thesis, McGill University.
Petersen, J. E., Sanford, L. P. & Kemp, W. M. 1998 Coastal plankton responses to turbulent mixing in experimental ecosystems. Mar. Ecol. Prog. Ser. 171, 2341.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
van Reeuwijkl, M. & Craske, J. 2015 Energy-consistent entrainment relations for jets and plumes. J. Fluid Mech. 782, 333355.
Shabbir, A. & George, W. K. 1994 Experiments on a round turbulent buoyant plume. J. Fluid Mech. 275, 132.
Turner, J. S. & Campbell, I. H. 1987 A laboratory and theoretical study of the growth of ‘black smoker’ chimneys. Earth Planet. Sci. Lett. 82, 3648.
Variano, E. A. & Cowen, E. A. 2008 A random-jet-stirred turbulence tank. J. Fluid Mech. 604, 132.
Wang, H. & Law, A. W. K. 2002 Second-order integral model for a round turbulent buoyant jet. J. Fluid Mech. 459, 397428.
Wood, I. R., Bell, R. G. & Wilkinson, D. L. 1993 Ocean Disposal of Wastewater. World Scientific.
Wright, S. J. 1994 The effect of ambient turbulence on jet mixing. In Recent Research Advances in the Fluid Mechanics of Turbulent Jets and Plumes (ed. Davies, P. A. & Valente Neves, M. I.), NATO ASI Series E, vol. 255, pp. 1327. Springer.
Zhang, W., He, Z. & Jiang, H. 2017 Scaling for turbulent viscosity of buoyant plumes in stratified fluids: PIV measurement with implications for submarine hydrothermal plume turbulence. Deep-Sea Res. I 129, 8998.
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A second-order integral model for buoyant jets with background homogeneous and isotropic turbulence

  • Adrian C. H. Lai (a1), Adrian Wing-Keung Law (a1) (a2) and E. Eric Adams (a3)


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