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

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Abstract

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.

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References

  1. Zimm, W.: 1921, Ñber die Strömungsvorgänge im freien Luftstrahl, VDI-Forschungsheft 234.

  2. Förthmann, E.: 1934, Ñber turbulente Strahlausbreitung, Ing.-Arch. 5, 42–54.

    Google Scholar 

  3. Tollmien, W.: 1926, Berechnung turbulenter Ausbreitungsvorgänge, ZAMM 6, 468–478.

    Google Scholar 

  4. Görtler, H.: 1942, Berechnung von Aufgaben der freien Turbulenz aus Grund eines neuen Näherungsansatzes, ZAMM 22, 244–254.

    Google Scholar 

  5. Reichardt, H.: 1942, Gesetzmäßigkeiten der freien Turbulenzen, VDI-Forschungsheft 414.

  6. Schmidt, W.: 1941, Turbulente Ausbreitung eines Stromes erhitzter Luft, ZAMM 21, 265; 271.

    Google Scholar 

  7. Reichardt, H.: 1941, Ñber eine neue Theorie der freien Turbulenzen, ZAMM 21, 257–264.

    Google Scholar 

  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. Rouse, H., Yih, C.S. and Humphreys, H.W.: 1952, Gravitational convection from a boundary source, Tellus 4.

  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. Morton, B.R.: 1959, Forced plumes, J. Fluid Mech. 5, 151–163.

    Google Scholar 

  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. Jordinson, R.: 1956, Flow in a Jet Directed Normal to the Wind, R & M., No. 3974, British A.R.C.

  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. 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.

  16. Scorer, R.S.: 1958, Natural Aerodynamics, Pergamon Press, New York.

    Google Scholar 

  17. Csanady, G.T.: 1961, Some observations on smoke plumes, Int. J. Air Water Poll. 4, 47–51.

    Google Scholar 

  18. Turner, J.S.: 1960, A comparison between buoyant vortex rings and vortex pairs, J. Fluid Mech. 7, 419–432.

    Google Scholar 

  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. Abraham, G.: 1963, Jet Diffusion in Stagnant Ambient Fluid, Delft Hydraulics Lab., Publ. No. 29.

  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. 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. 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. Frick,W.E.: 1984, Non-empirical closure of the plume equations, Atmos. Environ. 18, 653–662.

    Google Scholar 

  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. Schatzmann, M.: 1978, The integral equations for round buoyant jets in stratified flows, J. Appl. Math. Phys. (ZAMP) 29, 608–630.

    Google Scholar 

  27. Wood, I.R., Bell, R.G. and Wilkinson, D.L.: 1993, Ocean Disposal of Wastewater, World Scientific Publishers, Singapore.

    Google Scholar 

  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. Chu, P.C.K.: 1996, Mixing of Turbulent Advected Line Puffs, Ph.D. Thesis, University of Hong Kong.

  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. List, E.J.: 1982, Mechanics of turbulent buoyant jets and plumes, In: W. Rodi (ed.), Turbulent Jets and Buoyant Plumes, Pergamon Press.

  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. Fox, D.C.: 1970, Forced plume in a stratified fluid, J. Geophys. Res. 75(33), 6818–6835.

    Google Scholar 

  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. 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. 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. 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. Davidson, M.J. and Pun, K.L.: 1999, Weakly advected jets in cross-flow, J. Hydr. Engrg., ASCE 125, 47–58.

    Google Scholar 

  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. 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.

  41. Morton, B.R. and Ibbetson, A.: 1996, Jets deflected in a crossflow, Exp. Therm. Fluid Sci. 12, 112–133.

    Google Scholar 

  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. 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. Abramovich, G.N.: 1963, The Theory of Turbulent Jets, The M.I.T. Press, Cambridge, MA.

    Google Scholar 

  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. Jirka, G.H. and Doneker, R.L.: 1991, Hydrodynamic classification of submerged single port discharges, J. Hydr. Engin. 117, 1095–1112.

    Google Scholar 

  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. Chen, C.J. and Rodi, W.: 1980), Vertical Buoyant Jets: A Review of Experimental Data, Pergamon Press, Oxford.

    Google Scholar 

  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. Nickels, T.B. and Perry, A.E.: 1996, The turbulent coflowing jet, J. Fluid Mech. 309, 157–182.

    Google Scholar 

  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. Scorer, R.S.: 1978, Environmental Aerodynamics, Ellis Horwood, Chichester, UK.

    Google Scholar 

  53. Fai,W.C.: 1991, Advected Line Thermals and Puffs, M. Phil. Thesis, University of Hong Kong, Hong Kong.

    Google Scholar 

  54. Turner, J.S.: 1966, Jets and plumes with negative or reversing buoyancy, J. Fluid Mech. 26, 779–792.

    Google Scholar 

  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. Abraham, G.: 1967, Jets with negative buoyancy in homogeneous fluid, J. Hydraulic Res. 5(4).

  57. Roberts, P.J.W. and Toms, G.: 1987, Inclined dense jets in flowing current, J. Hydr. Engin. 113, 323–341.

    Google Scholar 

  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.

  59. Roberts, P.J.W., Maile, K. and Daviero, G.: 2001, Mixing in stratified jets, J. Hydr. Engin. 127, 194–200.

    Google Scholar 

  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. 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. 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. Baines, P.G.: 1995, Topographic Effects in Stratified Flows, Cambridge Monographs on Mechanics, Cambridge University Press.

  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. 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. Chu, V.H.: 1985, Oblique turbulent jets in a crossflow, J. Eng. Mech., ASCE 111, 1343–1360.

    Google Scholar 

  67. Margason, Ri.J.: 1968, The Path of a Jet Directed at Large Angles to a Subsonic Free Stream, NASA TN D-4919.

  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. 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. Cheung, V: 1991, Mixing a Round Buoyant Jet in a Current, Ph.D. Thesis, University of Hong Kong, Hong Kong.

    Google Scholar 

  71. Ayoub, G.M.: 1971, Dispersion of Buoyant Jets in a Flowing Ambient Fluid, Ph.D. Thesis, Imperial College, University of London.

  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. 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. Chu, V.H.: 1975, Turbulent dense plumes in a laminar crossflow, J. Hydr. Res. 13, 263–279.

    Google Scholar 

  75. Nezu, I. and Nakagawa, H.: 1993, Turbulence in Open-Channel Flows, A.A. Balkema, Rotterdam.

    Google Scholar 

  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. Wright, S.J.: 1984, Buoyant jets in density-stratified crossflow, J. Hydr. Engr. 110, (HY5), 643–656.

    Google Scholar 

  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. Huq, P.: 1997, Observations of jets in density stratified crossflows, Atmos. Environ. 31, 2011–2022.

    Google Scholar 

  80. Briggs, G.A.: 1969, Plume Rise, U.S. Atomic Energy Commission, Division of Technical Information Extension, Oak Ridge, TN.

    Google Scholar 

  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. Anwar, H.O.: 1972, Measurements on horizontal buoyant jets in calm ambient fluid, La Houille Blanche 27 (4).

  83. Capp, S.P.: 1983, Experimental Investigation of the Buoyant Axisymmetric Jet, Ph.D. Thesis, University of Buffalo, State University of New York.

  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. Corrsin, S. and Uberoi, M.S.: 1950, Further Experiments on the Flow and Heat Transfer in a Heated Turbulent Air Jet, NACA Report 998.

  86. Crow, S.C. and Champagne, F.H.: 1971, Orderly structure in jet turbulence, J. Fluid Mech 48, 547–596.

    Google Scholar 

  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. 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.

  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. 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. 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.

  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. Ricou, F.P. and Spalding, D.B.: 1961, Measurements of entrainment by axisymmetrical turbulent jets, J. Fluid Mech. 11, 21–32.

    Google Scholar 

  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. Turner, J.S.: 1973, Buoyancy Effects in Fluids, Cambridge University Press.

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  1. Institute for Hydromechanics, University of Karlsruhe, Karlsruhe, Germany

    Gerhard H. Jirka

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Jirka, G.H. Integral Model for Turbulent Buoyant Jets in Unbounded Stratified Flows. Part I: Single Round Jet. Environmental Fluid Mechanics 4, 1–56 (2004). https://doi.org/10.1023/A:1025583110842

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