Environmental Fluid Mechanics

, Volume 4, Issue 1, pp 1–56 | Cite as

Integral Model for Turbulent Buoyant Jets in Unbounded Stratified Flows. Part I: Single Round Jet

  • Gerhard H. Jirka


The mechanics of buoyant jet flows issuing with a general three-dimensional geometry into an unbounded ambient environment with uniform density or stable density stratification and under stagnant or steady sheared current conditions is investigated. An integral model is formulated for the conservation of mass, momentum, buoyancy and scalar quantities in the turbulent jet flow. The model employs an entrainment closure approach that distinguishes between the separate contributions of transverse shear (leading to jet, plume, or wake internal flow dynamics) and of azimuthal shear mechanisms (leading to advected momentum puff or thermal flow dynamics), respectively. Furthermore, it contains a quadratic law turbulent drag force mechanism as suggested by a number of recent detailed experimental investigations on the dynamics of transverse jets into crossflow. The model is validated in several stages: First, comparison with basic experimental data for the five asymptotic, self-similar stages of buoyant jet flows, i.e., the pure jet, the pure plume, the pure wake, the advected line puff, and the advected line thermal, support the choice and magnitude of the turbulent closure coefficients contained in the entrainment formulation. Second, comparison with many types of non-equilibrium flows support the proposed transition function within the entrainment relationship, and also the role of the drag force in the jet deflection dynamics. Third, a number of spatial limits of applicability have been proposed beyond which the integral model necessarily becomes invalid due to its parabolic formulation. These conditions, often related to the breakdown of the boundary layer nature of the flow, describe features such as terminal layer formation in stratification, upstream penetration in jets opposing a current, or transition to passive diffusion in a turbulent ambient shear flow. Based on all these comparisons, that include parameters such as trajectories, centerline velocities, concentrations and dilutions, the model appears to provide an accurate and reliable representation of buoyant jet physics under highly general flow conditions.


Azimuthal Shear Advected Momentum Stable Density Stratification Pure Plume Entrainment Formulation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Zimm, W.: 1921, Ñber die Strömungsvorgänge im freien Luftstrahl, VDI-Forschungsheft 234.Google Scholar
  2. 2.
    Förthmann, E.: 1934, Ñber turbulente Strahlausbreitung, Ing.-Arch. 5, 42–54.Google Scholar
  3. 3.
    Tollmien, W.: 1926, Berechnung turbulenter Ausbreitungsvorgänge, ZAMM 6, 468–478.Google Scholar
  4. 4.
    Görtler, H.: 1942, Berechnung von Aufgaben der freien Turbulenz aus Grund eines neuen Näherungsansatzes, ZAMM 22, 244–254.Google Scholar
  5. 5.
    Reichardt, H.: 1942, Gesetzmäßigkeiten der freien Turbulenzen, VDI-Forschungsheft 414.Google Scholar
  6. 6.
    Schmidt, W.: 1941, Turbulente Ausbreitung eines Stromes erhitzter Luft, ZAMM 21, 265; 271.Google Scholar
  7. 7.
    Reichardt, H.: 1941, Ñber eine neue Theorie der freien Turbulenzen, ZAMM 21, 257–264.Google Scholar
  8. 8.
    Albertson, J.L., Dai, Y.B., Jensen, R.A. and Rouse, H.: 1950, Diffusion of submerged jets, Trans. ASCE 115, 639–664.Google Scholar
  9. 9.
    Rouse, H., Yih, C.S. and Humphreys, H.W.: 1952, Gravitational convection from a boundary source, Tellus 4.Google Scholar
  10. 10.
    Morton, B.R., Taylor, G.I. and Turner, J.S.: 1956, Turbulent gravitational convection from maintained and instantaneous sources, Proc. Roy. Soc. London A 234, 1–23.Google Scholar
  11. 11.
    Morton, B.R.: 1959, Forced plumes, J. Fluid Mech. 5, 151–163.Google Scholar
  12. 12.
    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
  13. 13.
    Jordinson, R.: 1956, Flow in a Jet Directed Normal to the Wind, R & M., No. 3974, British A.R.C.Google Scholar
  14. 14.
    Keffer, J.F. and Baines, W.D.: 1963, The round turbulent jet in a cross wind, J. Fluid Mech. 15, 481–496.Google Scholar
  15. 15.
    Bryant, L.W. and Cowdrey, C.F.: 1955, The effects of velocity and temperature of discharge on the shape of smoke plumes from a tunnel or chimney. Experiments in a wind tunnel. In: Proceedings of the Institute of Mechanical Engineering, London, 169, pp. 371-400.Google Scholar
  16. 16.
    Scorer, R.S.: 1958, Natural Aerodynamics, Pergamon Press, New York.Google Scholar
  17. 17.
    Csanady, G.T.: 1961, Some observations on smoke plumes, Int. J. Air Water Poll. 4, 47–51.Google Scholar
  18. 18.
    Turner, J.S.: 1960, A comparison between buoyant vortex rings and vortex pairs, J. Fluid Mech. 7, 419–432.Google Scholar
  19. 19.
    Richards, R.S.: 1963, Experiment on the motion of isolated cylindrical thermals through unstratified surroundings, Int. J. Air Water Pollut. 7, 17–34.Google Scholar
  20. 20.
    Abraham, G.: 1963, Jet Diffusion in Stagnant Ambient Fluid, Delft Hydraulics Lab., Publ. No. 29.Google Scholar
  21. 21.
    Fan, L.N.: 1967, Turbulent Buoyant Jets into Stratified or Flowing Ambient Fluids, Report No. KH-R-15, W.M. Keck Laboratory of Hydrology and Water Resources, California Institute of Technology, Pasadena, CA.Google Scholar
  22. 22.
    Wright, S.J.: 1977, Mean behavior of buoyant jets in a crossflow, J. Hydraul. Div., ASCE 103(HY5), 499–513; (5), 643-656.Google Scholar
  23. 23.
    Fischer, H.B., List, E.J., Koh, R.C.Y., Imberger, J. and Brooks, N.H: 1979, Mixing in Inland and Coastal Waters, Academic Press, New York, NY.Google Scholar
  24. 24.
    Frick,W.E.: 1984, Non-empirical closure of the plume equations, Atmos. Environ. 18, 653–662.Google Scholar
  25. 25.
    Lee, J.H.W. and Cheung, V.: 1990, Generalized Lagrangian model for buoyant jets in current, J. Environ. Engin. 116, 1085–1106.Google Scholar
  26. 26.
    Schatzmann, M.: 1978, The integral equations for round buoyant jets in stratified flows, J. Appl. Math. Phys. (ZAMP) 29, 608–630.Google Scholar
  27. 27.
    Wood, I.R., Bell, R.G. and Wilkinson, D.L.: 1993, Ocean Disposal of Wastewater, World Scientific Publishers, Singapore.Google Scholar
  28. 28.
    Jirka, G.H. and Fong, H.L.M.: 1981, Vortex dynamics and bifurcation of buoyant jets in crossflow, J. Engin. Mech. Div., ASCE 107, EM6.Google Scholar
  29. 29.
    Chu, P.C.K.: 1996, Mixing of Turbulent Advected Line Puffs, Ph.D. Thesis, University of Hong Kong.Google Scholar
  30. 30.
    Hanna, S.R., Briggs, G.A. and Hosker, Jr., R.P.: 1982, Handbook on Atmospheric Diffusion, Technical Information Center, U.S. Department of Energy, Oak Ridge, TN.Google Scholar
  31. 31.
    List, E.J.: 1982, Mechanics of turbulent buoyant jets and plumes, In: W. Rodi (ed.), Turbulent Jets and Buoyant Plumes, Pergamon Press.Google Scholar
  32. 32.
    Wang, H. and Law, A.W.K.: 2002, Second-order integral model for a round buoyant jet, J. Fluid Mech. 459, 397–428.Google Scholar
  33. 33.
    Fox, D.C.: 1970, Forced plume in a stratified fluid, J. Geophys. Res. 75(33), 6818–6835.Google Scholar
  34. 34.
    Fric, T.F. and Roshko, A.: 1994, Vortical structure in the wake of a transverse jet, J. FluidMech. 279, 1–47.Google Scholar
  35. 35.
    Smith, S.H. and Mungal, M.G.: 1998, Mixing, structure and scaling of the jet in crossflow, J. Fluid Mech. 357, 83–122.Google Scholar
  36. 36.
    Moussa, Z.M., Trischka, J.W. and Eskinazi, S.: 1977, The near field in the mixing of a round jet with a cross-stream, J. Fluid Mech. 80, 49–80.Google Scholar
  37. 37.
    Eiff, O.S. and Keffer, J.F.: 1997, On the structures in the near-wake region of an elevated turbulent jet in a crossflow, J. Fluid Mech. 333, 161–195.Google Scholar
  38. 38.
    Davidson, M.J. and Pun, K.L.: 1999, Weakly advected jets in cross-flow, J. Hydr. Engrg., ASCE 125, 47–58.Google Scholar
  39. 39.
    Chan, D.T.-L., Lin, J.-T. and Kennedy, J.F.: 1976, Entrainment and drag forces of deflected jets, J. Hydraulics Div., Proc. ASCE 102(HY5),615–635.Google Scholar
  40. 40.
    Margason, R.J.: 1993, Fifty years of jet in crossflow research, Computational and Experimental Assessment of Jets in Cross Flow, AGARD-CP-534, Winchester, U.K.Google Scholar
  41. 41.
    Morton, B.R. and Ibbetson, A.: 1996, Jets deflected in a crossflow, Exp. Therm. Fluid Sci. 12, 112–133.Google Scholar
  42. 42.
    Kelso, R.M. et al.: 1996, An experimental study of round jets in cross-flow, J. Fluid Mech. 306, 111–144.Google Scholar
  43. 43.
    Yuan, L.L., Street, R.L. and Ferziger, J.H.: 1998, Large-eddy simulations of a round jet in crossflow, J. Fluid Mech. 379, 71–104.Google Scholar
  44. 44.
    Abramovich, G.N.: 1963, The Theory of Turbulent Jets, The M.I.T. Press, Cambridge, MA.Google Scholar
  45. 45.
    Lee, J.H.W. and Jirka, G.H.: 1981, A vertical round buoyant jet in shallow water, J. Hydraul. Div., ASCE 107, HY 12.Google Scholar
  46. 46.
    Jirka, G.H. and Doneker, R.L.: 1991, Hydrodynamic classification of submerged single port discharges, J. Hydr. Engin. 117, 1095–1112.Google Scholar
  47. 47.
    Chu, P.C.K., Lee, H.H.W. and Chu, V.H.: 1999, Spreading of a turbulent round jet in coflow, J. Hydr Engrg., ASCE 125, 193–204.Google Scholar
  48. 48.
    Chen, C.J. and Rodi, W.: 1980), Vertical Buoyant Jets: A Review of Experimental Data, Pergamon Press, Oxford.Google Scholar
  49. 49.
    Jirka, G.H. and Harleman, D.R.F.: 1979, Stability and mixing of vertical plane buoyant jet in confined depth, J. Fluid Mech. 94, 275–304.Google Scholar
  50. 50.
    Nickels, T.B. and Perry, A.E.: 1996, The turbulent coflowing jet, J. Fluid Mech. 309, 157–182.Google Scholar
  51. 51.
    Wang, H.-J.: 2000, Jet Interaction in a Still or Co-Flowing Environment, Ph.D. Thesis, Hong Kong University of Science and Technology, Hong Kong.Google Scholar
  52. 52.
    Scorer, R.S.: 1978, Environmental Aerodynamics, Ellis Horwood, Chichester, UK.Google Scholar
  53. 53.
    Fai,W.C.: 1991, Advected Line Thermals and Puffs, M. Phil. Thesis, University of Hong Kong, Hong Kong.Google Scholar
  54. 54.
    Turner, J.S.: 1966, Jets and plumes with negative or reversing buoyancy, J. Fluid Mech. 26, 779–792.Google Scholar
  55. 55.
    Zhang, H. and Baddour, R.E.: 1998, Maximum penetration of vertical round dense jets at small and large Froude numbers, J. Hydr. Engin. 124, 550–553.Google Scholar
  56. 56.
    Abraham, G.: 1967, Jets with negative buoyancy in homogeneous fluid, J. Hydraulic Res. 5(4).Google Scholar
  57. 57.
    Roberts, P.J.W. and Toms, G.: 1987, Inclined dense jets in flowing current, J. Hydr. Engin. 113, 323–341.Google Scholar
  58. 58.
    Hutter, K. and Hofer, K.: 1978, Freistrahlen im homogenen und stratifizierten Medium-ihre Theorie und deren Vergleich mit dem Experiment, Mitteilungen der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie, ETH Zürich, Nr. 27.Google Scholar
  59. 59.
    Roberts, P.J.W., Maile, K. and Daviero, G.: 2001, Mixing in stratified jets, J. Hydr. Engin. 127, 194–200.Google Scholar
  60. 60.
    Roberts, P.J.W. and Matthews, P.R.: 1984, Dynamics of jets in two-layer stratified fluids, J. Hydr. Engin., ASCE 110, 1201–1217.Google Scholar
  61. 61.
    Akar, P.J. and Jirka, G.H.: 1994, Buoyant spreading processes in pollutant transport and mixing. Part I: Lateral spreading in strong ambient current, J. Hydraulic Res. 32, 815–831.Google Scholar
  62. 62.
    Jirka, G.H. and Arita, M.: 1987, Density currents or density wedges: Boundary layer influence and control methods, J. Fluid Mech. 177, 186–206.Google Scholar
  63. 63.
    Baines, P.G.: 1995, Topographic Effects in Stratified Flows, Cambridge Monographs on Mechanics, Cambridge University Press.Google Scholar
  64. 64.
    Wong, D.R.: 1984, Buoyant Jet Entrainment in Stratified Fluids, Ph.D. Thesis, Civil Engineering Department, The University of Michigan, Ann Arbor, MI.Google Scholar
  65. 65.
    Pratte, B.D. and Baines, W.D.: 1967, Profiles of the round turbulent jet in a crossflow, J. Hydr. Div., ASCE 93(HY6), 53–64.Google Scholar
  66. 66.
    Chu, V.H.: 1985, Oblique turbulent jets in a crossflow, J. Eng. Mech., ASCE 111, 1343–1360.Google Scholar
  67. 67.
    Margason, Ri.J.: 1968, The Path of a Jet Directed at Large Angles to a Subsonic Free Stream, NASA TN D-4919.Google Scholar
  68. 68.
    Chan, C.H.C. and Lam, K.M.: 1998, Centreline velocity decay of a circular jet in a counter-flowing stream, Phys. Fluids 10, 637–644.Google Scholar
  69. 69.
    Yoda, M. and Fiedler, H.E: 1996, The round jet in a uniform counterflow: Flow visualization and mean concentration measurements, Exp. Fluids 21, 427–436.Google Scholar
  70. 70.
    Cheung, V: 1991, Mixing a Round Buoyant Jet in a Current, Ph.D. Thesis, University of Hong Kong, Hong Kong.Google Scholar
  71. 71.
    Ayoub, G.M.: 1971, Dispersion of Buoyant Jets in a Flowing Ambient Fluid, Ph.D. Thesis, Imperial College, University of London.Google Scholar
  72. 72.
    Davidson, M.J., Gaskin, S. and Wood, I.R.: 2002, A study of a buoyant axisymmetric jet in a small co-flow, J. Hydr. Res. 40, 477–489.Google Scholar
  73. 73.
    Anderson, J.L., Parker, F.L. and Benedict, B.A.: 1973, Negatively Buoyant Jets in a Cross-Flow, Environmental Protection Technology Series, U.S. Environmental Protection Agency, Washington.Google Scholar
  74. 74.
    Chu, V.H.: 1975, Turbulent dense plumes in a laminar crossflow, J. Hydr. Res. 13, 263–279.Google Scholar
  75. 75.
    Nezu, I. and Nakagawa, H.: 1993, Turbulence in Open-Channel Flows, A.A. Balkema, Rotterdam.Google Scholar
  76. 76.
    Ayoub, G.M.: 1973, Test results on buoyant jets in injected horizontally in a cross flowing stream, Water, Air Soil Poll. 2, 409.Google Scholar
  77. 77.
    Wright, S.J.: 1984, Buoyant jets in density-stratified crossflow, J. Hydr. Engr. 110, (HY5), 643–656.Google Scholar
  78. 78.
    Hunter, G.C.: 1993, Experimental investigation of a buoyant jet in a stratified crossflow. In: S.D. Mobbs and J.C. King (eds.), Waves and Turbulence in Stably Stratified Flows, Clarendon Press, Oxford.Google Scholar
  79. 79.
    Huq, P.: 1997, Observations of jets in density stratified crossflows, Atmos. Environ. 31, 2011–2022.Google Scholar
  80. 80.
    Briggs, G.A.: 1969, Plume Rise, U.S. Atomic Energy Commission, Division of Technical Information Extension, Oak Ridge, TN.Google Scholar
  81. 81.
    Doneker, R.L. and Jirka, G.H.: 1991, Expert systems for design and mixing zone analysis of aqueous pollutant discharges, J. Water Resour. Plan. Manage. 117, 679–697.Google Scholar
  82. 82.
    Anwar, H.O.: 1972, Measurements on horizontal buoyant jets in calm ambient fluid, La Houille Blanche 27 (4).Google Scholar
  83. 83.
    Capp, S.P.: 1983, Experimental Investigation of the Buoyant Axisymmetric Jet, Ph.D. Thesis, University of Buffalo, State University of New York.Google Scholar
  84. 84.
    Cedervall, K.: 1963, The Initial Mixing of Jet Disposal into a Recipient, Tech. Reports 14 and 15, Div. of Hydraulics, Chalmers Institute of Technology, Goteborg, Sweden.Google Scholar
  85. 85.
    Corrsin, S. and Uberoi, M.S.: 1950, Further Experiments on the Flow and Heat Transfer in a Heated Turbulent Air Jet, NACA Report 998.Google Scholar
  86. 86.
    Crow, S.C. and Champagne, F.H.: 1971, Orderly structure in jet turbulence, J. Fluid Mech 48, 547–596.Google Scholar
  87. 87.
    Eiff, O.S. and Keffer, J.F.: 1999, Parametric investigation of the wake-vortex lock-in for the turbulent jet discharging from a stack, Exp. Thermal Fluid Sci. 19, 57–66.Google Scholar
  88. 88.
    Hansen, J. and Schroder, H.: 1968, Horizontal Jet Dilution Studies by Use of Radiocactive Isotopes, Acta Polytechnica Scandinavia, Civil Engineering and Building Construction Series No. 49, Copenhagen.Google Scholar
  89. 89.
    Hill, B.: 1972, Measurement of local entrainment rate in the initial region of axisymmetric turbulence air jets, J. Fluid Mech 51, 773–779.Google Scholar
  90. 90.
    Hussein, H.J., Capp, S.P. and George, W.K.: 1994, Velocity measurements in a high-Reynoldsnumber, momentum-conserving, axisymmetric, turbulent jet, J. Fluid Mech. 258, 31–75.Google Scholar
  91. 91.
    Labus, T.L. and Symons, E.P.: 1972, Experimental Investigation of an Axisymmetric Free Jet with an Initially Uniform Velocity Profile, NASA TN D-6783.Google Scholar
  92. 92.
    Papanicolaou, P.N. and List,W.J.: 1988, Measurement of round vertical axisymmetric buoyant jets, J. Fluid Mech. 195, 341–391.Google Scholar
  93. 93.
    Ricou, F.P. and Spalding, D.B.: 1961, Measurements of entrainment by axisymmetrical turbulent jets, J. Fluid Mech. 11, 21–32.Google Scholar
  94. 94.
    Rosler, R.S. and Bankoff, S.G.: 1963, Large scale turbulence characteristics of a submerged water jet, AIChE J. 9, 672–676.Google Scholar
  95. 95.
    Turner, J.S.: 1973, Buoyancy Effects in Fluids, Cambridge University Press.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Gerhard H. Jirka
    • 1
  1. 1.Institute for HydromechanicsUniversity of KarlsruheKarlsruheGermany

Personalised recommendations