A mathematical model on depth-averaged β-factor in open-channel turbulent flow

  • Punit Jain
  • Manotosh Kumbhakar
  • Koeli Ghoshal
Original Article


The well-known Rouse equation is the most widely used equation to determine the vertical distribution of suspended sediment concentration in an open-channel flow. The exponent of Rouse equation, known as Rouse number, contains the parameter β defined by the ratio of sediment diffusion coefficient to turbulent diffusion coefficient. As such to measure sediment concentration accurately, an appropriate expression for β is essentially needed. The present study, therefore, focuses on the derivation of depth-averaged β through modified expressions of sediment and turbulent diffusion coefficients. A regression analysis is done to establish the relation between β and normalized settling velocity, and the relation is used to determine suspension concentration.


Turbulent flow Sediment diffusion coefficient Momentum diffusion coefficient Depth-averaged β-factor 


  1. Absi R (2010) Concentration profiles for fine and coarse sediments suspended by waves over ripples: an analytical study with the 1-DV gradient diffusion model. Adv Water Resour 33(4):411–418CrossRefGoogle Scholar
  2. Absi R (2011) An ordinary differential equation for velocity distribution and dip-phenomenon in open channel flows. J Hydraul Eng 49(1):82–89CrossRefGoogle Scholar
  3. Cellino M, Graf W (1999) Sediment-laden flow in open-channels under noncapacity and capacity conditions. J Hydraul Eng 125(5):455–462CrossRefGoogle Scholar
  4. Cheng N-S (1997) Simplified settling velocity formula for sediment particle. J Hydraul Eng 123(2):149–152CrossRefGoogle Scholar
  5. Cheng P, Zhu H, Zhong B, Wang D (2015) Transport mechanisms of contaminants released from fine sediment in rivers. Acta Mech Sin 31(6):791–798CrossRefGoogle Scholar
  6. Chien N (1956) The present status of research on sediment transport. Trans Am Soc Civ Eng 121(1):833–868Google Scholar
  7. Coleman NL (1970) Flume studies of the sediment transfer coefficient. Water Resour Res 6(3):801–809CrossRefGoogle Scholar
  8. Coleman NL (1981) Velocity profiles with suspended sediment. J Hydraul Eng 19(3):211–229CrossRefGoogle Scholar
  9. Coleman NL (1986) Effects of suspended sediment on the open-channel velocity distribution. Water Resour Res 22(10):1377–1384CrossRefGoogle Scholar
  10. Dey S (2014) Fluvial hydrodynamics. Springer, BerlinCrossRefGoogle Scholar
  11. Fick A (1855) V. On liquid diffusion. Philos Mag Ser 4 10(63):30–39CrossRefGoogle Scholar
  12. Graf WH (1984) Hydraulics of sediment transport. Water Resources Publication, LittletonGoogle Scholar
  13. Graf W, Cellino M (2002) Suspension flows in open channels; experimental study. J Hydraul Eng 40(4):435–447CrossRefGoogle Scholar
  14. Hongwei F (1996) The study on three-dimensional mathematical model of river bed erosion for water-sediment two-phase flow. Acta Mech Sin 12(1):85–91CrossRefGoogle Scholar
  15. Hsu TW, Jan CD (1998) Calibration of Businger-Arya type of eddy viscosity model’s parameters. J Waterway Port Coastal Ocean Eng 124(5):281–284CrossRefGoogle Scholar
  16. Imamoto H (1988) Measurement of secondary flow in an open channel. In: Proceedings of 6th IAHR-APD Congress, Kyoto, JapanGoogle Scholar
  17. Kinoshita R (1967) An analysis of the movement of flood waters by aerial photography. J Jpn Soc Photogramm 6(1):1–17CrossRefGoogle Scholar
  18. Kirkgöz MS (1989) Turbulent velocity profiles for smooth and rough open channel flow. J Hydraul Eng 115(11):1543–1561CrossRefGoogle Scholar
  19. Kironoto B, Yulistiyanto B (2009) The validity of rouse equation for predicting suspended sediment concentration profiles in transversal direction of uniform open channel flow. In: International conference on sustainable development for water and waste water treatment, YogyakartaGoogle Scholar
  20. Kundu S, Ghoshal K (2012) An analytical model for velocity distribution and dip-phenomenon in uniform open channel flows. Int J Fluid Mech Res 39(5):13CrossRefGoogle Scholar
  21. Lyn D (1988) A similarity approach to turbulent sediment-laden flows in open channels. J Fluid Mech 193:1–26CrossRefGoogle Scholar
  22. Majumdar H, Carstens MR (1967) Diffusion of particles by turbulence: Effect of particle size. Water Resources Center, Georgia Inst Technol Rep WRC-0967, Dec 1967. 102 p, 12 fig, 12 tab, 19 ref. FWPCA Grant 5 R 01 WP 00912-02 ESEGoogle Scholar
  23. Mazumder B, Ghoshal K, Dalal D (2005) Influence of bed roughness on sediment suspension: experimental and theoretical studies. J Hydraul Eng 43(3):245–257CrossRefGoogle Scholar
  24. Montes V (1973) Interaction of two dimensional turbulent flow with suspended particles. Ph.D. thesis, Massachusetts Institute of Technology, CambridgeGoogle Scholar
  25. Nezu I (2002) Open-channel turbulence and its research prospect in the new century. In: Proc. 13th IAHRAPD Cong., pp 8Google Scholar
  26. Nezu I, Rodi W (1986) Open-channel flow measurements with a laser doppler anemometer. J Hydraul Eng 112(5):335–355CrossRefGoogle Scholar
  27. Nielsen P, Teakle IA (2004) Turbulent diffusion of momentum and suspended particles: a finite-mixing-length theory. Phys Fluids 16(7):2342–2348CrossRefGoogle Scholar
  28. Pal D, Ghoshal K (2014) Effect of bed roughness on grain-size distribution in an open channel flow. J Hydro Environ Res 8(4):441–451CrossRefGoogle Scholar
  29. Pal D, Ghoshal K (2016) Effect of particle concentration on sediment and turbulent diffusion coefficients in open-channel turbulent flow. Environ Earth Sci 18(75):1–11Google Scholar
  30. Prandtl L (1932) Recent results of turbulence research. Technical Memorandum 720, National Advisory Committee for AeronauticsGoogle Scholar
  31. Rouse H (1937) Modern conceptions of the mechanics of turbulence. Trans ASCE 102:463–543Google Scholar
  32. Tsai CT, Tsai CH, Weng CH, Bair JJ, Chen CN (2010) Calculation of bed load based on the measured data of suspended load. Paddy Water Environ 8(4):371–384CrossRefGoogle Scholar
  33. Van Rijn L (1984) Sediment transport, part II: suspend load transport. J Hydraul Eng ASCE 1104(11):16131641Google Scholar
  34. Vanoni VA (1946) Transportation of suspended sediment by water. Trans ASCE 111:67–102Google Scholar
  35. Wren D, Bennett S, Barkdoll B, Kuhnle R (2005) Distributions of velocity, turbulence, and suspended sediment over low-relief antidunes. J Hydraul Eng 43(1):3–11CrossRefGoogle Scholar
  36. Wu P, Jin Y (2010) Parameters used in modeling sediment-laden flow in open channels. In: Environmental Hydraulics, Two Volume Set: Proceedings of the 6th International Symposium on Environmental Hydraulics, Athens, Greece, 23–25 June 2010. CRC Press, London, p 265Google Scholar
  37. Xinyu L, Changzhi C, Zengnan D (1995) Turbulent flows in smooth-wall open channels with different slope. J Hydraul Eng 33(3):333–347CrossRefGoogle Scholar
  38. Yang SQ (2005) Interactions of boundary shear stress, secondary currents and velocity. Fluid Dyn Res 36(3):121–136CrossRefGoogle Scholar
  39. Yang SQ, Tan SK, Lim SY (2004) Velocity distribution and dip-phenomenon in smooth uniform open channel flows. J Hydraul Eng 130(12):1179–1186CrossRefGoogle Scholar
  40. Zhu HW, Cheng PD, Li W, Chen JH, Pang Y, Wang DZ (2017) Empirical model for estimating vertical concentration profiles of re-suspended, sediment-associated contaminants. Acta Mech Sin 33(5):846–854CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

Personalised recommendations