Hierarchical modeling of the dilute transport of suspended sediment in open channels

  • Fabián A. Bombardelli
  • Sanjeev K. Jha
Original Article


We propose, discuss and validate a theoretical and numerical framework for sediment-laden, open-channel flows which is based on the two-fluid-model (TFM) equations of motion. The framework models involve mass and momentum equations for both phases (sediment and water) including the interactive forces of drag, lift, virtual mass and turbulent dispersion. The developed framework is composed by the complete two-fluid model (CTFM), a partial two-fluid model (PTFM), and a standard sediment-transport model (SSTM). Within the umbrella of the Reynolds-Averaged Navier-Stokes (RANS) equations, we apply K–ε type closures (standard and extended) to account for the turbulence in the carrier phase (water). We present the results of numerical computations undertaken by integrating the differential equations over control volumes. We address several issues of the theoretical models, especially those related to coupling between the two phases, interaction forces, turbulence closure and turbulent diffusivities. We compare simulation results with various recent experimental datasets for mean flow variables of the carrier as well as, for the first time, mean flow of the disperse phase and turbulence statistics. We show that most models analyzed in this paper predict the velocity of the carrier phase and that of the disperse phase within 10% of error. We also show that the PTFM provides better predictions of the distribution of sediment in the wall-normal direction as opposed to the standard Rousean profile, and that the CTFM is by no means superior to the PTFM for dilute mixtures. We additionally report and discuss the values of the Schmidt number found to improve the agreement between predictions of the distribution of suspended sediment and the experimental data.


K–ε model Partial two-fluid model Reynolds-Averaged Navier-Stokes (RANS) equations Sediment transport Suspended sediments Turbulence modeling Two-fluid model (TFM) Two-phase flows 


  1. 1.
    Amoudry L, Hsu TJ, Liu PL (2005) Schmidt number and near-bed boundary condition effect on a two-phase dilute sediment transport. J Geophys Res 110(C09003). doi: 10.1029/2004JC002798
  2. 2.
    Axell L (2002) Wind-driven internal waves and Langmuir circulations in a numerical ocean model of the southern Baltic Sea. J Geophys Res. doi: 10.1029/2001JC000922
  3. 3.
    Best J, Bennett S, Bridge J, Leeder M (1997) Turbulence modulation and particle velocities over flat sand beds at low transport rates. J Hydrol Eng 123(12): 1118–1128 doi: 10.1061/(ASCE)0733-9429(1997)123:12(1118) CrossRefGoogle Scholar
  4. 4.
    Bombardelli FA (2003) Characterization of coherent structures from parallel, LES computations of wandering effects in bubble plumes. In: Proceedings of the 2003 world water & environmental resources congress, June 2003, Philadelphia, PA, EWRI/ASCEGoogle Scholar
  5. 5.
    Bombardelli FA (2004) Turbulence in multiphase models for aeration bubble plumes. PhD Thesis, Department of Civil and Environmental Engineering, University of Illinois at Urbana-ChampaignGoogle Scholar
  6. 6.
    Bombardelli FA, Buscaglia GC, García MH (2003) Parallel computations of the dynamic behavior of bubble plumes. In: Brust FW (ed) Proceedings of the pressure vessels and pipe division conference. Cleveland, vol PVP-464, Residual Stress, Fitness-for-Service, and Manufacturing Processes. ASME-PVP DivisionGoogle Scholar
  7. 7.
    Bombardelli FA, Buscaglia GC, Rehmann CR, Rincon LE, García MH (2007) Modeling and scaling of aeration bubble plumes: a two-phase flow analysis. J Hydraul Res 45(5): 617–630Google Scholar
  8. 8.
    Bombardelli FA, González AE, Niño YI (2008) Computation of the Basset force with a fractional-mathematics approach. J Hydr Eng, ASCE (in Press)Google Scholar
  9. 9.
    Brennen E (2005) Fundamentals of multiphase flow. Cambridge Press, New YorkGoogle Scholar
  10. 10.
    Buscaglia GC, Bombardelli FA, García MH (2002) Numerical modeling of large scale bubble plumes accounting for mass transfer effects. Int J Multiph Flow 28: 1763–1785 doi: 10.1016/S0301-9322(02)00075-7 CrossRefGoogle Scholar
  11. 11.
    Cao Z, Wei L, Xie J (1995) Sediment-laden flow in open channels from two-phase flow viewpoint. J Hydrol Eng 121(10): 725–735 doi: 10.1061/(ASCE)0733-9429(1995)121:10(725) CrossRefGoogle Scholar
  12. 12.
    Carrica P, Bonetto F, Drew D, Lahey R (1998) The interaction of background ocean air bubbles with a surface ship. Int J Numer Methods Fluids 28:571–600. doi:10.1002/(SICI)1097-0363(19980930)28:4<571::AID-FLD731>3.0.CO;2-EGoogle Scholar
  13. 13.
    Carrica P, Drew D, Bonetto F, Lahey R (1999) A polydisperse model for bubbly two-phase flow around a surface ship. Int J Multiph Flow 25: 257–305 doi: 10.1016/S0301-9322(98)00047-0 CrossRefGoogle Scholar
  14. 14.
    Cellino M, Graf WH (2002) Suspension flow in open channels; experimental study. J Hydraul Res 15: 435–447Google Scholar
  15. 15.
    Chen CP, Wood PE (1986) Turbulence closure modeling of the dilute gas-particle axisymmetric jet. AIChE J 32(1): 163–166 doi: 10.1002/aic.690320121 CrossRefGoogle Scholar
  16. 16.
    Chien N, Wan Z (1999) Mechanics of sediment transport. ASCE Press, USAGoogle Scholar
  17. 17.
    Coleman NL (1986) Effects of suspended sediment on the open-channel distribution. Water Resour Res 22(10): 1377–1384 doi: 10.1029/WR022i010p01377 CrossRefGoogle Scholar
  18. 18.
    Crowe CT, Sommerfeld M, Tsuji Y (1998) Multiphase flows with droplets and particles. CRC Press, FloridaGoogle Scholar
  19. 19.
    Czernuszenko W (1998) Drift velocity concept for sediment-laden flows. J Hydrol Eng 124(10): 1026–1033 doi: 10.1061/(ASCE)0733-9429(1998)124:10(1026) CrossRefGoogle Scholar
  20. 20.
    Dietrich WE (1982) Settling velocity of natural particles. Water Resour Res 18(6): 1626–1632 doi: 10.1029/WR018i006p01615 CrossRefGoogle Scholar
  21. 21.
    Dong P, Zhang K (1999) Two-phase flow modeling of sediment motions in oscillatory sheet flow. Coast Eng 36: 87–109 doi: 10.1016/S0378-3839(98)00052-0 CrossRefGoogle Scholar
  22. 22.
    Drew DA (1975) Turbulent sediment transport over a flat bottom using momentum balance. J Appl Mech 42: 38–44Google Scholar
  23. 23.
    Drew D, Passman S (1999) Theory of multicomponent fluids. Applied mathematical sciences, vol 135. Springer, New YorkGoogle Scholar
  24. 24.
    Einstein HA, Chien N (1955) Effects of heavy sediment concentration near the bed on velocity and sediment distribution. MRD Sediment Ser Rep No 8. Univ of California, Berkeley, US Army Corps of Engineers, Missouri DivGoogle Scholar
  25. 25.
    Elghobashi SE, Abou-Arab TW (1983) A two-equation turbulence model for two-phase flows. Phys Fluids 26(4): 931–938 doi: 10.1063/1.864243 CrossRefGoogle Scholar
  26. 26.
    Ferziger JH, Peric M (2002) Computational methods for fluid dynamics. Springer, New YorkGoogle Scholar
  27. 27.
    Graf WH, Cellino M (2002) Suspension flows in open channels: experimental study. J Hydraul Res 40(4): 435–447Google Scholar
  28. 28.
    Greimann BP, Muste M, Holly FM Jr (1999) Two-phase formulation of suspended sediment transport. J Hydraul Res 37: 479–500Google Scholar
  29. 29.
    Greimann BP, Holly FM Jr (2001) Two-phase flow analysis of concentration profiles. J Hydrol Eng 127(9): 753–762 doi: 10.1061/(ASCE)0733-9429(2001)127:9(753) CrossRefGoogle Scholar
  30. 30.
    Guo J, Julien PY (2001) Turbulent velocity profiles in sediment-laden flows. J Hydraul Res 39(1): 11–23Google Scholar
  31. 31.
    Hsu T, Jenkins JT, Liu PLF (2003a) On two-phase sediment transport: dilute flow. J Geophys Res 108(C3): 3057 doi: 10.1029/2001JC001276 CrossRefGoogle Scholar
  32. 32.
    Hsu T, Chang H, Hsieh C (2003b) A two-phase flow model of wave-induced sheet flow. J Hydraul Res 41(3): 299–310Google Scholar
  33. 33.
    Jiang J, Law AW, Cheng N-S (2004) Two-phase analysis of vertical sediment laden jets. J Eng Mech 131(3): 308–318 doi: 10.1061/(ASCE)0733-9399(2005)131:3(308) CrossRefGoogle Scholar
  34. 34.
    Kataoka I, Serizawa A (1989) Basic equations of turbulence in gas-liquid two-phase flow. Int J Multiph Flow 15(5): 843–855 doi: 10.1016/0301-9322(89)90045-1 CrossRefGoogle Scholar
  35. 35.
    Kobayashi N, Seo SN (1985) Fluid and sediment interaction over a plane bed. J Hydrol Eng 111(6): 903–919CrossRefGoogle Scholar
  36. 36.
    Lain S, Aliod R (2000) Study on the Eulerian dispersed phase equations in non-uniform turbulent two-phase flows: discussion and comparison with experiments. Int J Heat Fluid Flow 21: 374–380 doi: 10.1016/S0142-727X(00)00023-0 CrossRefGoogle Scholar
  37. 37.
    Landau L, Lifshitz E (2000) Fluid mechanics. Pergamon, OxfordGoogle Scholar
  38. 38.
    Launder BE, Spalding DB (1973) The numerical computation of turbulent flows. Comput Methods Appl Mech Eng 3: 269–289 doi: 10.1016/0045-7825(74)90029-2 CrossRefGoogle Scholar
  39. 39.
    Liu H, Sato S (2006) A two-phase flow model for asymmetric sheetflow conditions. Coast Eng 53: 825–843 doi: 10.1016/j.coastaleng.2006.04.002 CrossRefGoogle Scholar
  40. 40.
    Longo S (2005) Two-phase flow modeling of sediment motion in sheet-flows above plane beds. J Hydrol Eng 131(5): 366–379 doi: 10.1061/(ASCE)0733-9429(2005)131:5(366) CrossRefGoogle Scholar
  41. 41.
    Loth E (2007) Computational fluid dynamics of bubbles, drops and particles. Cambridge University Press, CambridgeGoogle Scholar
  42. 42.
    Lyn DA (1986) Turbulence and turbulent transport in sediment-laden open channel flows. PhD Thesis, Calif Inst of Technol, PasadenaGoogle Scholar
  43. 43.
    Lyn DA (1988) A similarity approach to turbulent sediment-laden flows in open channels. J Fluid Mech 193: 1–26 doi: 10.1017/S0022112088002034 CrossRefGoogle Scholar
  44. 44.
    Lyn DA (1991) Resistance in flat-bed sediment-laden flows. J Hydrol Eng 117(1): 94–114 doi: 10.1061/(ASCE)0733-9429(1991)117:1(94) CrossRefGoogle Scholar
  45. 45.
    Lyn DA (1992) Turbulence characteristics of sediment-laden flows in open channels. J Hydrol Eng 118(7): 971–987 doi: 10.1061/(ASCE)0733-9429(1992)118:7(971) CrossRefGoogle Scholar
  46. 46.
    Lyn DA (2008) Sedimentation engineering: theories, measurements, modeling and practice. In: García M (ed) Manual No 110, ASCEGoogle Scholar
  47. 47.
    McTigue DF (1981) Mixture theory for suspended sediment transport. J Hydraul Div 107(HY6): 659–673Google Scholar
  48. 48.
    Moraga FJ, Larreteguy AE, Drew DA, Lahey RT (2003) Assessment of turbulent dispersion models for bubbly flows in the low Stokes number limit. Int J Multiph Flow 29(4): 655–673 doi: 10.1016/S0301-9322(03)00018-1 CrossRefGoogle Scholar
  49. 49.
    Muste M, Patel VC (1997) Velocity profiles for particles and liquid in open-channel flow with suspended sediment. J Hydrol Eng 123(9): 742–751 doi: 10.1061/(ASCE)0733-9429(1997)123:9(742) CrossRefGoogle Scholar
  50. 50.
    Muste M, Fujita K, Yu I, Ettema R (2005) Two-phase versus mixed-flow perspective on suspended sediment transport in turbulent channel flows. Water Resour Res 41: W10402 doi: 10.1029/2004WR003595 CrossRefGoogle Scholar
  51. 51.
    Nezu I, Azuma R (2004) Turbulence characteristics and interaction between particles and fluid in particle-laden open channel flows. J Hydrol Eng 130: 988–1001 doi: 10.1061/(ASCE)0733-9429(2004)130:10(988) CrossRefGoogle Scholar
  52. 52.
    Nezu I, Nakagawa H (1993) Turbulence in open-channel flow. IAHR Monograph. A A Balkema Publishers, RotterdamGoogle Scholar
  53. 53.
    Nezu I, Rodi W (1986) Open-channel flow measurements with a laser doppler anemometer. J Hydrol Eng 112: 335–355CrossRefGoogle Scholar
  54. 54.
    Parker G (2004) 1D sediment transport morphodynamics with application to rivers and~turbidity currents. e-book downloadable at:
  55. 55.
    Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere, New YorkGoogle Scholar
  56. 56.
    Prosperetti A, Zhang DZ (1995) Finite-particle-size effects in disperse two-phase flows. Theor Comput Fluid Dyn 7: 429–440 doi: 10.1007/BF00418141 CrossRefGoogle Scholar
  57. 57.
    Rodi W (1984) Turbulence models and their application in hydraulics. International Association for Hydraulic Research, Delft, The NetherlandsGoogle Scholar
  58. 58.
    Rouse H (1937) Modern conception of the mechanics of turbulence. Trans ASCE 102: 463–543Google Scholar
  59. 59.
    Serizawa A, Kataoka I, Michiyosi I (1975) Turbulence structure of air-water flows: parts 1–3. Int J Multiph Flow 21(3): 221–259 doi: 10.1016/0301-9322(75)90011-7 CrossRefGoogle Scholar
  60. 60.
    Sokolichin A, Eigenberger G (1999) Applicability of the standard turbulence model to the dynamic simulation of bubble columns: part I. Detailed numerical simulations. Chem Eng Sci 54: 2273–2284 doi: 10.1016/S0009-2509(98)00420-5 Google Scholar
  61. 61.
    Sokolichin A, Eigenberger G, Lapin A (2004) Simulation of buoyancy driven bubbly flow: established simplifications and open questions. AIChE J 50: 24–45. doi: 10.1002/aic.10003 CrossRefGoogle Scholar
  62. 62.
    Sommerfeld M (1992) Modelling of particle-wall collisions in confined gas-particle flows. Int J Multiph Flow 18: 905–926 doi: 10.1016/0301-9322(92)90067-Q CrossRefGoogle Scholar
  63. 63.
    Svensson U (1998) Program for boundary layers in the environment—system description and manual. SMHI Reports. Oceanography 24:42 ppGoogle Scholar
  64. 64.
    Svensson U, Sahlberg J (1989) Formulae for pressure gradients in one-dimensional lake models. J Geophys Res 94: 4939–4946 doi: 10.1029/JC094iC04p04939 CrossRefGoogle Scholar
  65. 65.
    Svensson U, Axell L, Sahlberg J, Omstedt A (2002) Program for boundary layers in the environment—system description and manualGoogle Scholar
  66. 66.
    Taggart WC, Yermoli CA, Montes S, Ippen AT (1972) Effects of sediment size and gradation on concentration profiles for turbulent flow. MIT Report No 152Google Scholar
  67. 67.
    Tomiyama A, Shimada N (2001) A numerical method for bubbly flow simulation based on a multi-fluid model. Trans ASME J Press Vessel Technol 123(4): 510–520 doi: 10.1115/1.1388010 CrossRefGoogle Scholar
  68. 68.
    Troshko AA, Hassan YA (2001a) A two-equation turbulence model of turbulent bubbly flows. Int J Multiph Flow 27: 1965–2000 doi: 10.1016/S0301-9322(01)00043-X CrossRefGoogle Scholar
  69. 69.
    Troshko AA, Hassan YA (2001) Law of the wall for two-phase turbulent boundary layers. Int J Heat Mass Transfer 44: 871–875 doi: 10.1016/S0017-9310(00)00128-9 CrossRefGoogle Scholar
  70. 70.
    Vanoni VA (1946) Transportation of suspended sediment by water. Trans ASCE 111: 67–133Google Scholar
  71. 71.
    Vanoni VA (1975) Suspension of sediment. In: Vanoni VA(eds) Sedimentation engineering. ASCE, New YorkGoogle Scholar
  72. 72.
    Van Rijn LC (1984) Sediment transport. Part II: suspended load transport. J Hydrol Eng 110(11): 1613–1641CrossRefGoogle Scholar
  73. 73.
    Villaret C, Davies AG (1995) Modeling sediment-turbulent flow interactions. Appl Mech Rev 48(9): 601–609Google Scholar
  74. 74.
    Wang X, Qian N (1992) Velocity profiles of sediment laden flow. Int J Sediment Res 7(1): 27–58Google Scholar
  75. 75.
    Zhang D, Deen NG, Kuipers JAM (2006) Numerical simulation of the dynamic flow behavior in a bubble column: a study of closures for turbulence and interface forces. Chem Eng Sci 61: 7593–7608 doi: 10.1016/j.ces.2006.08.053 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of California, DavisDavisUSA

Personalised recommendations