A 3-D finite volume model for sediment transport in coastal waters

  • Weiming WuEmail author
  • Qianru Lin


A 3-D model has been developed to simulate sediment transport and bed change induced by currents and waves in coastal waters. The currents are calculated with the 3-D phase-averaged shallow water flow equations, the wave characteristics are determined with a horizontal 2-D wave spectral transformation model, and multiple-sized non-cohesive suspended load and bed load are simulated using a non-equilibrium transport model. The classical mixing length model is modified to determine the horizontal and vertical eddy viscosity considering the effects of current, waves, and wave breaking. A new approach is proposed in the present 3-D model framework to calculate the bed grain shear stress, which is applied to determine the equilibrium bed-load transport rate and near-bed suspended-load concentration. The flow and sediment transport equations are numerically solved using an implicit finite volume method on a hexahedral mesh constructed with horizontal telescoping (quadtree) rectangular cells and vertical sigma coordinate. The difference in the near-bed cells for flow and suspended load calculations is avoided by adding the bed load to the suspended load in the first flow cell above the bed. The developed model has been tested in four laboratory and field cases. It reasonably well reproduces the measured water levels, flow velocities, sediment concentrations, and bed changes.


Coastal morphodynamics Currents Waves Sediment transport Three-dimensional Multiple-sized 


  1. Andrews DG, McIntyre ME (1978) An exact theory of nonlinear waves on a Lagrangian-mean flow. J Fluid Mech 89:609–646. CrossRefGoogle Scholar
  2. Ardhuin F, Rascle N, Belibassakis KA (2008a) Explicit wave-averaged primitive equations using a generalized Lagrangian mean. Ocean Model 20:35–60. CrossRefGoogle Scholar
  3. Ardhuin F, Jenkins AD, Belibassakis KA (2008b) Comments on “The three-dimensional current and surface wave equations”. J Phys Oceanogr 38:1340–1350. CrossRefGoogle Scholar
  4. Battjes J (1975) Modeling of turbulence in the surf zone. Proc. Symp. Model Techniques, San Francisco, USA, pp 1050–1061Google Scholar
  5. Beck TM, Kraus NC (2010) Shark River Inlet, New Jersey, entrance shoaling: report 2, Analysis with coastal modeling system. Technical Report ERDC/CHL-TR-10-4. U.S. Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory, Vicksburg, MS.
  6. Bijvelds MDJP (2001) Numerical modelling of estuarine flow over steep topography. Doctoral Dissertation, Delft University of Technology, The NetherlandsGoogle Scholar
  7. Craik ADD, Leibovich S (1976) A rational model for Langmuir circulation. J Fluid Mech 73:401–426. CrossRefGoogle Scholar
  8. DHI (2008) Mike 3 flow model. Scientific Documentation, Danish Hydraulic InstituteGoogle Scholar
  9. Ding Y, Wang SSY, Jia Y (2006) Development and validation of a quasi-three dimensional coastal area morphological model. J Waterw Port Coast Ocean Eng 132(6):462–476CrossRefGoogle Scholar
  10. Fortunato, A.B., Oliviera, A. (2007). Improving the stability of a morphodynamic modeling system. J Coast Res Spec Issue 50, 486–490Google Scholar
  11. Gravens MB, Wang P (2007) Data report: laboratory testing of longshore sand transport by waves and currents; morphology change behind headland structures. Technical Report ERDC/CHL TR-07-8, U.S. Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory, Vicksburg, MississippiGoogle Scholar
  12. Groeneweg J (1999) Wave-current interactions in a generalized Lagrangian mean formulation. Ph.D. Thesis, Delft Univ. of Technology, Delft, NetherlandsGoogle Scholar
  13. Hanson H, Kraus NC (1989) GENESIS: generalized model for simulating shoreline change, report 1: technical reference. Technical Report CERC-89-19, U.S. Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, MississippiGoogle Scholar
  14. Hsu SA (1988) Coastal meteorology. In: Academic Press. San Diego, CaliforniaGoogle Scholar
  15. Johnson HK, Zyserman JA (2002) Controlling spatial oscillations in bed level update schemes. Coast Eng 46(2):109–126CrossRefGoogle Scholar
  16. Johnson BD, Kobayashi N, Gravens MB (2012) Cross-shore numerical model CSHORE for waves, currents, sediment transport, and beach profile evolution. Technical Report ERDC/CHL TR-12-22, U.S. Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory, Vicksburg, MississippiGoogle Scholar
  17. Kaihatu JM, Shi F, Kirby JT, Svendsen IA (2002) Incorporation of random wave effects into a quasi-3D nearshore circulation model. Naval Research laboratory, Oceanography Division, 1002 Balch Boulevard, Stennis Space Center, MS, USAGoogle Scholar
  18. Kraus NC, Larson M (1991) NMLONG—numerical model for simulating the longshore current, report 1: model development and tests. Technical Report DRP-91-1, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, USAGoogle Scholar
  19. Kubatko EJ, Westerlink JJ, Dawson C (2006) An unstructured grid morphodynamic model with a discontinuous Galerkin method for bed evolution. Ocean Model 15(1–2):71–89CrossRefGoogle Scholar
  20. Larson M, Kraus NC (1989) SBEACH: numerical model for simulating storm-induced beach change. Technical Report CERC-89-9, Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, MississippiGoogle Scholar
  21. Lesser G, Roelvink J, van Kester J, Stelling G (2004) Development and validation of a three-dimensional morphological model. Coast Eng 51(8–9):883–915CrossRefGoogle Scholar
  22. Lin L, Demirbilek Z, Mase H, Zheng J, Yamada F (2008) CMS-Wave: a nearshore spectral wave processes model for coastal inlets and navigation projects. Technical Report ERDC/CHL TR-08-13. U.S. Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory, Vicksburg, MississippiGoogle Scholar
  23. Longuet-Higgins MS, Stewart RW (1962) Radiation stress and mass transport in gravity waves with application to “surf beats”. J Fluid Mech 13:481–504. CrossRefGoogle Scholar
  24. Longuet-Higgins MS, Stewart RW (1964) Radiation stresses in water waves; a physical discussion with applications. Deep-Sea Res 11:529–562Google Scholar
  25. Marsooli R, Wu W (2014) Three-dimensional finite-volume model of dam-break flows with sediment transport over movable beds. J Hydraul Eng, ASCE, 04014066, 1–12,
  26. Mase H (2001) Multidirectional random wave transformation model based on energy balance equation. Coastal Eng Journal 43(4):317–337CrossRefGoogle Scholar
  27. Mase H, Oki K, Hedges TS, Li HJ (2005) Extended energy-balance-equation wave model for multidirectional random wave transformation. Ocean Eng 32(8–9):961–985CrossRefGoogle Scholar
  28. McWilliams JC, Restrepo JM, Lane EM (2004) An asymptotic theory for the interaction of waves and currents in coastal waters. J Fluid Mech 511:135–178. CrossRefGoogle Scholar
  29. Mellor GL (2003) The three-dimensional current and surface wave equations. J Phys Oceanogr 33:1978–1989CrossRefGoogle Scholar
  30. Mellor GL (2008) The depth-dependent current and wave interaction equations: a revision. J Phys Oceanogr 38:2587–2596CrossRefGoogle Scholar
  31. Nairn RB, Southgate HN (1993) Deterministic profile modelling of nearshore processes. Part 2. Sediment transport and beach profile development. Coast Eng 19(1–2):57–96CrossRefGoogle Scholar
  32. Newberger PA, Allen JS (2007) Forcing a three-dimensional, hydrostatic, primitive-equation model for application in the surf zone: I. Formulation. J Geophys Res 112:C08018. Google Scholar
  33. Nezu I (2005) Open-channel flow turbulence and its research prospect in the 21st century. J Hydraul Eng ASCE 131(4):229–246. CrossRefGoogle Scholar
  34. Pelnard-Considere R (1956) Essai de theorie de l’Evolution des forms de rivages en plage de sable et de galets, Fourth Journees de l’Hydrolique, les energies de la Mer, Question III, Rapport No. 1, 289–298Google Scholar
  35. Powell MD, Vickery PJ, Reinhold TA (2003) Reduced drag coefficient for high wind speeds in tropical cyclones. Nature 422:279–283CrossRefGoogle Scholar
  36. Prandtl L (1925) Über die ausgebildete Turbulenz. ZAMM 5:136–139CrossRefGoogle Scholar
  37. Raudkivi AJ (1998) Loose boundary hydraulics. A.A. Balkema, Rotterdam 496pGoogle Scholar
  38. Rhie TM, Chow A (1983) Numerical study of the turbulent flow past an isolated airfoil with trailing-edge separation. AIAA J 21:1525–1532CrossRefGoogle Scholar
  39. Rodi W (1993) Turbulence models and their applications in hydraulics, 3rd edn. IAHR Monograph, Rotterdam, p 104Google Scholar
  40. Roelvink D, Reniers A (2012) A guide to modeling coastal morphology. Advances in coastal and ocean engineering, World Scientific, Vol 12, p 274Google Scholar
  41. Roland A, Cucco A, Ferrarin C, Hsu T, Liau J, Ou S, Umgiesser G, Zanke U (2009) On the development and verification of a 2-D coupled wave-current model on unstructured meshes. J Mar Syst 78:S244–S254. CrossRefGoogle Scholar
  42. Ruessink BG, Miles JR, Feddersen F, Guza RT, Elgar S (2001) Modeling the alongshore current on barred beaches. J Geophys Res 106(C10):22451–22464CrossRefGoogle Scholar
  43. Saad Y (1993) A flexible inner-outer preconditioned GMRES algorithm. SIAM J Sci Comput 14:461–469CrossRefGoogle Scholar
  44. Saad Y (1994) ILUT: a dual threshold incomplete ILU factorization. Numerical Linear Algebra Appl 1:387–402CrossRefGoogle Scholar
  45. Sanchez A, Wu W, Beck TM (2016) A depth-averaged 2-D model of flow and sediment transport in coastal waters. Ocean Dyn 66:1475–1495. CrossRefGoogle Scholar
  46. Satkevich (Саткевич), А.А. (1934) Theoretical foundations of hydro-aerodynamics (Теоретические основи гидроазродинамики), Т. 2, Dynamics of Liquid Bodies (Динамика жидких тел)Google Scholar
  47. Sheng YP, Liu T (2011) Three-dimensional simulation of wave-induced circulation: comparison of three radiation stress formulations. J Geophys Res 116:C05021. CrossRefGoogle Scholar
  48. Soulsby RL (1997) Dynamics of marine sands, a manual for practical applications. H.R. Wallingford, Thomas TelfordGoogle Scholar
  49. Soulsby R, Whitehouse R (2005) Prediction of ripple properties in shelf seas: mark 1, predictor. Technical Report TR 150, HR Wallingford, UKGoogle Scholar
  50. Spalding DB (1972) A novel finite-difference formulation for differential expressions involving both first and second derivatives. Int J Numer Methods Eng 4:551–559. CrossRefGoogle Scholar
  51. Stelling GS, Van Kester JATM (1994) On the approximation of horizontal gradients in sigma coordinates for bathymetry with steep bottom slopes. Int J Numer Methods Fluids 18(10):915–935CrossRefGoogle Scholar
  52. Stive MJF, de Vriend HJ (1994) Shear stresses and mean flow in shoaling and breaking waves. Proc. the 24th Coastal Engineering International Conference, Kobe, Japan. American Society of Civil Engineers, New York, pp 594–608Google Scholar
  53. Svendsen IA (2006) Introduction to nearshore hydrodynamics. World Scientific, p 722Google Scholar
  54. Uchiyama Y, McWilliams JC, Shchepetkin AF (2010) Wave-current interaction in an oceanic circulation model with a vortex-force formalism: application to the surf zone. Ocean Model 34:16–35. CrossRefGoogle Scholar
  55. van der Salm GLS (2013) Coastline modelling with UNIBEST: areas close to structures. M.S. Thesis. Delft University of Technology, The NetherlandsGoogle Scholar
  56. van Doormal JP, Raithby GD (1984) Enhancements of the SIMPLE method for predicting incompressible fluid flows. Num Heat Transfer 7:147–163Google Scholar
  57. van Rijn LC (1986) Sedimentation of dredged channels by currents and waves. J Waterw Port Coast Ocean Eng 112(5):541–559CrossRefGoogle Scholar
  58. van Rijn LC (1993) Principles of sediment transport in rivers, estuaries and coastal seas. Aqua Publications, The NetherlandsGoogle Scholar
  59. van Rijn LC, Havinga FJ (1995) Transport of fine sands by currents and waves. J Waterw Port Coast Ocean Eng 121(2):123–133CrossRefGoogle Scholar
  60. Warner JC, Sherwood CR, Signell RP, Harris CK, Arango HG (2008) Development of a three-dimensional, regional, coupled wave, current, and sediment-transport model. Comput Geosci 34:1284–1306CrossRefGoogle Scholar
  61. Wu W (2004) Depth-averaged 2-D numerical modeling of unsteady flow and nonuniform sediment transport in open channels. J Hydraul Eng ASCE 130(10):1013–1024CrossRefGoogle Scholar
  62. Wu W (2007) Computational river dynamics. Taylor & Francis, Abingdon, p 494CrossRefGoogle Scholar
  63. Wu W (2014) A 3-D phase-averaged model for shallow water flow with waves in vegetated water. Ocean Dyn 64(7):1061–1071. CrossRefGoogle Scholar
  64. Wu W (2015) 3-D numerical modeling of undertow current and sediment transport in surf zone. Proc. 2015 International Conference on Coastal Sediments, San Diego, CAGoogle Scholar
  65. Wu W, Lin Q (2014) Nonuniform sediment transport under non-breaking waves and currents. Coast Eng 90:1–14.
  66. Wu W, Lin Q (2015) An implicit 3-D finite-volume model of shallow water flows. Adv Water Resour 83:263–276. CrossRefGoogle Scholar
  67. Wu W, Rodi W, Wenka T (2000a) 3-D numerical modeling of water flow and sediment transport in open channels. J Hydraul Eng ASCE 126(1):4–15CrossRefGoogle Scholar
  68. Wu W, Wang SSY (2006) Formulas for sediment porosity and settling velocity. J Hydraul Eng ASCE 132(8):858–862CrossRefGoogle Scholar
  69. Wu W, Wang SSY, Jia Y (2000b) Nonuniform sediment transport in alluvial rivers. J Hydraul Res IAHR 38(6):427–434CrossRefGoogle Scholar
  70. Xia H, Xia Z, Zhu L (2004) Vertical variation in radiation stress and wave-induced current. Coast Eng 51:309–321. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringClarkson UniversityPotsdamUSA
  2. 2.MicrosoftRedmondUSA

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