Skip to main content Accessibility help

Turbulence, entrainment and low-order description of a transitional variable-density jet

  • B. Viggiano (a1), T. Dib (a1), N. Ali (a1), L. G. Mastin (a2), R. B. Cal (a1) and S. A. Solovitz (a3)...


Geophysical flows occur over a large range of scales, with Reynolds numbers and Richardson numbers varying over several orders of magnitude. For this study, jets of different densities were ejected vertically into a large ambient region, considering conditions relevant to some geophysical phenomena. Using particle image velocimetry, the velocity fields were measured for three different gases exhausting into air – specifically helium, air and argon. Measurements focused on both the jet core and the entrained ambient. Experiments considered relatively low Reynolds numbers from approximately 1500 to 10 000 with Richardson numbers near 0.001 in magnitude. These included a variety of flow responses, notably a nearly laminar jet, turbulent jets and a transitioning jet in between. Several features were studied, including the jet development, the local entrainment ratio, the turbulent Reynolds stresses and the eddy strength. Compared to a fully turbulent jet, the transitioning jet showed up to 50 % higher local entrainment and more significant turbulent fluctuations. For this condition, the eddies were non-axisymmetric and larger than the exit radius. For turbulent jets, the eddies were initially smaller and axisymmetric while growing with the shear layer. At lower turbulent Reynolds number, the turbulent stresses were more than 50 % higher than at higher turbulent Reynolds number. In either case, the low-density jet developed faster than a comparable non-buoyant jet. Quadrant analysis and proper orthogonal decomposition were also utilized for insight into the entrainment of the jet, as well as to assess the energy distribution with respect to the number of eigenmodes. Reynolds shear stresses were dominant in Q1 and Q3 and exhibited negligible contributions from the remaining two quadrants. Both analysis techniques showed that the development of stresses downstream was dependent on the Reynolds number while the spanwise location of the stresses depended on the Richardson number.


Corresponding author

Email address for correspondence:


Hide All
Adrian, R. J., Christensen, K. T. & Liu, Z.-C. 2000 Analysis and interpretation of instantaneous turbulent velocity fields. Exp. Fluids 29 (3), 275290.
Agrawal, A. & Prasad, A. K. 2003 Integral solution for the mean flow profiles of turbulent jets, plumes, and wakes. Trans. ASME J. Fluids Engng 125 (5), 813822.
Amielh, M., Djeridane, T., Anselmet, F. & Fulachier, L. 1996 Velocity near-field of variable density turbulent jets. Intl J. Heat Mass Transfer 39 (10), 21492164.
Arndt, R. E. A., Long, D. F. & Glauser, M. N. 1997 The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet. J. Fluid Mech. 340, 133.
Berkooz, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech. 25 (1), 539575.
Caraballo, E., Samimy, M., Scott, J., Narayanan, S. & Debonis, J. 2003 Application of proper orthogonal decomposition to a supersonic axisymmetric jet. AIAA J. 41 (5), 866877.
Chen, C. J. & Rodi, W.1980 Vertical turbulent buoyant jets: a review of experimental data. NASA STI/Recon Tech. Rep. A 80.
Chojnicki, K. N., Clarke, A. B., Adrian, R. J. & Phillips, J. C. 2014 The flow structure of jets from transient sources and implications for modeling short-duration explosive volcanic eruptions. Geochem. Geophys. Geosyst. 15 (12), 48314845.
Costa, A., Suzuki, Y. J., Cerminara, M., Devenish, B. J., Esposti Ongaro, T., Herzog, M., Van Eaton, A. R., Denby, L. C., Bursik, M., de’Michieli Vitturi, M. et al. 2016 Results of the eruptive column model inter-comparison study. J. Volcanol. Geotherm. Res. 326, 225.
Dimotakis, P. E. 2000 The mixing transition in turbulent flows. J. Fluid Mech. 409, 6998.
Djeridane, T., Amielh, M., Anselmet, F. & Fulachier, L. 1996 Velocity turbulence properties in the near-field region of axisymmetric variable density jets. Phys. Fluids 8 (6), 16141630.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press.
Dunn, J. R. 1953 The origin of the deposits of tufa in Mono Lake. J. Sedim. Res. 23 (1), 1823.
Falcone, A. M. & Cataldo, J. C. 2003 Entrainment velocity in an axisymmetric turbulent jet. Trans. ASME J. Fluids Engng 125 (4), 620627.
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J. & Brooks, N. H. 1979 Mixing in Inland and Coastal Waters. Academic.
Gerashchenko, S. & Prestridge, K. 2015 Density and velocity statistics in variable density turbulent mixing. J. Turbul. 16 (11), 10111035.
Gharbi, A., Amielh, M. & Anselmet, F. 1995 Experimental investigation of turbulence properties in the interface region of variable density jets. Phys. Fluids 7 (10), 24442454.
Ginster, U., Mottl, M. J. & Von Herzen, R. P. 1994 Heat flux from black smokers on the endeavour and cleft segments, Juan de Fuca ridge. J. Geophys. Res. 99 (B3), 49374950.
Hamilton, N., Kang, H. S., Meneveau, C. & Cal, R. B. 2012 Statistical analysis of kinetic energy entrainment in a model wind turbine array boundary layer. J. Renew. Sustainable Energy 4 (6), 063105.
Han, D.2001 Study of turbulent nonpremixed jet flames using simultaneous measurements of velocity and CH distribution. PhD thesis, Stanford University.
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.
Kaminski, E., Tait, S. & Carazzo, G. 2005 Turbulent entrainment in jets with arbitrary buoyancy. J. Fluid Mech. 526, 361376.
Katul, G., Poggi, D., Cava, D. & Finnigan, J. 2006 The relative importance of ejections and sweeps to momentum transfer in the atmospheric boundary layer. Boundary-Layer Meteorol. 120 (3), 367375.
Kays, W. M. & Crawford, M. E. 1993 Convective Heat and Mass Transfer, 2nd edn. McGraw-Hill.
Kotsovinos, N. E. 1976 A note on the spreading rate and virtual origin of a plane turbulent jet. J. Fluid Mech. 77 (2), 305311.
Krug, D., Chung, D., Philip, J. & Marusic, I. 2017 Global and local aspects of entrainment in temporal plumes. J. Fluid Mech. 812, 222250.
Kwon, S. J. & Seo, I. W. 2005 Reynolds number effects on the behavior of a non-buoyant round jet. Exp. Fluids 38 (6), 801812.
Kyle, D. M. & Sreenivasan, K. R. 1993 The instability and breakdown of a round variable-density jet. J. Fluid Mech. 249, 619664.
Lumley, J. L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation, pp. 166178. Nauka.
Mastin, L. G 2007 A user-friendly one-dimensional model for wet volcanic plumes. Geochem. Geophys. Geosyst. 8 (3), Q03014.
Mi, J. B., Kalt, P., Nathan, G. J. & Wong, C. Y. 2007 PIV measurements of a turbulent jet issuing from round sharp-edged plate. Exp. Fluids 42 (4), 625637.
Monkewitz, P. A., Lehmann, B., Barsikow, B. & Bechert, D. W. 1989 The spreading of self-excited hot jets by side jets. Phys. Fluids A 1 (3), 446448.
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234 (1196), 123.
Namer, I. & Ötügen, M. V. 1988 Velocity measurements in a plane turbulent air jet at moderate Reynolds numbers. Exp. Fluids 6 (6), 387399.
Nolan, K. P., Walsh, E. J. & McEligot, D. M. 2010 Quadrant analysis of a transitional boundary layer subject to free-stream turbulence. J. Fluid Mech. 658, 310335.
O’Neill, P., Soria, J. & Honnery, D. 2004 The stability of low Reynolds number round jets. Exp. Fluids 36 (3), 473483.
Örlü, R. & Alfredsson, P. H. 2008 An experimental study of the near-field mixing characteristics of a swirling jet. Flow Turbul. Combust. 80 (3), 323350.
Paillat, S. & Kaminski, E. 2014 Second-order model of entrainment in planar turbulent jets at low Reynolds number. Phys. Fluids 26 (4), 045110.
Panchapakesan, N. R. & Lumley, J. L. 1993 Turbulence measurements in axisymmetric jets of air and helium. Part 2. Helium jet. J. Fluid Mech. 246, 225247.
Papanicolaou, P. N. & List, E. J. 1988 Investigations of round vertical turbulent buoyant jets. J. Fluid Mech. 195, 341391.
Patrick, M. 2007 The gas content and buoyancy of strombolian ash plumes. J. Volcanol. Geotherm. Res. 166 (1), 16.
Patte-Rouland, B., Lalizel, G., Moreau, J. & Rouland, E. 2001 Flow analysis of an annular jet by particle image velocimetry and proper orthogonal decomposition. Meas. Sci. Technol. 12 (9), 1404.
Poggi, D. & Katul, G. G. 2007 The ejection-sweep cycle over bare and forested gentle hills: a laboratory experiment. Boundary-Layer Meteor. 122 (3), 493515.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Raffel, M., Willert, C. E. & Kompenhans, J. 2013 Particle Image Velocimetry: a Practical Guide. Springer.
Raupach, M. R. 1981 Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers. J. Fluid Mech. 108, 363382.
Recker, E., Wagemakers, R., Janssens, B. & Gilson, B.2012 PIV study of variable-density round jets. In 9th National Congress on theoretical and applied mechanics, Brussels, 9-10-11 May 2012.
Ricou, F. P. & Spalding, D. B. 1961 Measurements of entrainment by axisymmetrical turbulent jets. J. Fluid Mech. 11 (01), 2132.
Rona, P. A., Klinkhammer, G., Nelsen, T. A., Trefry, J. H. & Elderfield, H. 1986 Black smokers, massive sulphides and vent biota at the Mid-Atlantic ridge. Nature 321 (6065), 3337.
Saffaraval, F., Solovitz, S. A., Ogden, D. E. & Mastin, L. G. 2012 Impact of reduced near-field entrainment of overpressured volcanic jets on plume development. J. Geophys. Res. 117 (B05209).
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Part I: Coherent structures. Q. Appl. Maths 45 (3), 561571.
Solovitz, S. A., Mastin, L. G. & Saffaraval, F. 2011 Experimental study of near-field entrainment of moderately overpressured jets. Trans. ASME J. Fluids Engng 133 (5), 051304.
Sorey, M. L., Kennedy, B. M., Evans, W. C., Farrar, C. D. & Suemnicht, G. A. 1993 Helium isotope and gas discharge variations associated with crustal unrest in Long Valley Caldera, California, 1989–1992. J. Geophys. Res. 98 (B9), 1587115889.
Sparks, R. S. J. 1986 The dimensions and dynamics of volcanic eruption columns. Bull. Volcanol. 48 (1), 315.
Sparks, R. S. J., Bursik, M. I., Carey, S. N., Gilbert, J. S., Glaze, L. S., Sigurdsson, H. & Woods, A. W. 1997 Volcanic Plumes. Wiley.
Sreenivasan, K. R. & Antonia, R. A. 1978 Joint probability densities and quadrant contributions in a heated turbulent round jet. AIAA J. 16, 867868.
Stevenson, D. S. 1993 Physical models of fumarolic flow. J. Volcanol. Geotherm. Res. 57 (3–4), 139156.
Suresh, P. R., Srinivasan, K., Sundararajan, T. & Das, S. K. 2008 Reynolds number dependence of plane jet development in the transitional regime. Phys. Fluids 20 (4), 044105.
Suzuki, Y. J., Costa, A., Cerminara, M., Ongaro, T. E., Herzog, M., Van Eaton, A. R. & Denby, L. C. 2016 Inter-comparison of three-dimensional models of volcanic plumes. J. Volcanol. Geotherm. Res. 326, 2642.
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.
Thring, M. W. & Newby, M. P. 1953 Combustion length of enclosed turbulent jet flames. In Symposium (International) on Combustion, vol. 4, pp. 789796. Elsevier.
Tinney, C. E., Glauser, M. N. & Ukeiley, L. S. 2008 Low-dimensional characteristics of a transonic jet. Part 1. Proper orthogonal decomposition. J. Fluid Mech. 612, 107141.
Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431471.
Wallace, J. M. 2016 Quadrant analysis in turbulence research: history and evolution. Annu. Rev. Fluid Mech. 48, 131158.
Wang, H. & Law, A. W.-K. 2002 Second-order integral model for a round turbulent buoyant jet. J. Fluid Mech. 459, 397428.
Wang, P., Fröhlich, J., Michelassi, V. & Rodi, W. 2008 Large-eddy simulation of variable-density turbulent axisymmetric jets. Intl J. Heat Fluid Flow 29 (3), 654664.
White, F. M. 1991 Viscous Fluid Flow, 2nd edn. McGraw-Hill.
Woods, A. W. 1993 Moist convection and the injection of volcanic ash into the atmosphere. J. Geophys. Res. 98 (B10), 1762717636.
Yoon, J.-H. & Lee, S.-J. 2003 Investigation of the near-field structure of an elliptic jet using stereoscopic particle image velocimetry. Meas. Sci. Technol. 14 (12), 2034.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed