Comparison between Shannon and Tsallis entropies for prediction of shear stress distribution in open channels

  • Hossein BonakdariEmail author
  • Zohreh Sheikh
  • Mohtaram Tooshmalani
Original Paper


The concept of Tsallis entropy was applied to model the probability distribution functions for the shear stress magnitudes in circular channels (with filling ratios of 0.506, 0.666, 0.826), circular with flat bed (filling ratios of 0.333, 0.666), rectangular channel (1.34, 2, 3.94, 7.37 aspect ratios) and compound channel (with relative depths of 0.324, 0.46). The equation for the shear stress distribution was obtained according to the entropy maximization principle, and is able to estimate the shear stress distribution as much on the walls as the channel bed. The approach is also compared with the predictions obtained based on the Shannon entropy concept. By comparing the two prediction models, this study highlights the application of Tsallis entropy to estimate the shear stress distribution of open channels. Although the results of the two models are similar in the circular cross-section, the differences between them are more significant in circular with flat bed and rectangular channels. For a wide range of filling ratio values, experimental data are used to illustrate the accuracy and reliability of the proposed model.


Entropy Open channel Sediment Wall Bed 



The authors would like to express their appreciation to the anonymous reviewers for their helpful comments and to Ellen Vuosalo Tavakoli for the painstaking editing of the English text.


  1. Araujo JC, Chaudhry FH (1998) Experimental evaluation of 2-D entropy model for open channel flow. J Hydraul Eng 124(10):1064–1067CrossRefGoogle Scholar
  2. Berlamont JE, Trouw K, Luyckx G (2003) Shear stress distribution in partially filled pipes. J Hydraul Eng 129(9):697–705CrossRefGoogle Scholar
  3. Bonakdari H, Tooshmalani M (2010) Numerical study of the effect of roughness of solid walls on velocity fields and shear stress in rectangular open channel flow. In: 10th International symposium on stochastic hydraulics, water 2010 symposium, Quebec City, CanadaGoogle Scholar
  4. Bonakdari H, Larrarte F, Joannis C (2008) Study of shear stress in narrow channels: application to sewers. J Urban Water 5(1):15–20CrossRefGoogle Scholar
  5. Cao S (1995) Regime theory and geometric model for stable channels. PhD Thesis, School of Civil Engineering, University of BirminghamGoogle Scholar
  6. Cao S, Knight DW (1996) Shannon’s entropy-based bank profile equation of threshold channels. In: Tickle KS, Goulter JC, Xu C, Wasimi SA, Bouchart F (eds) Stochastic hydraulics’96, proceedings of the 7th IAHR international symposium on stochastic hydraulics, Central Queensland University, Australia, July. Balkema, Rotterdam, pp 169–175Google Scholar
  7. Chien N, Wan ZH (1999) Mechanics of sediment transport. ASCE Press, New YorkGoogle Scholar
  8. Chiu CL (1987) Entropy and probability concepts in hydraulics. J Hydraul Eng 114(7):738–756CrossRefGoogle Scholar
  9. Chiu CL (1991) Application of entropy concept in open channel flows. J Hydraul Eng 117(5):615–628CrossRefGoogle Scholar
  10. Cruff RW (1965) Cross channel transfer of linear momentum in smooth rectangular channels. Water Supply Paper 1592-B. USGS-Center, Washington, DC, pp 1–26Google Scholar
  11. Flintham TP, Carling PA (1988) The prediction of mean bed and wall boundary shear in uniform and compositely rough channels. In: Whith WR (ed) Proceedings of the 2nd international conference on river regime. Wiley, Chichester, pp 267–287Google Scholar
  12. Galip S, Neslihan S, Recep Y (2006) Boundary shear stress analysis in smooth rectangular channels. Can J Civ Eng 33(3):336–342CrossRefGoogle Scholar
  13. Ghosh SN, Roy N (1970) Boundary shear distribution in compound channel flow. J Hydraul Div ASCE 96(4):967–994Google Scholar
  14. Guo J, Julien PY (2005) Shear stress in smooth rectangular open-channel flow. J Hydraul Eng 131(1):30–37CrossRefGoogle Scholar
  15. Huai W, Chen G, Zeng Y (2013) Predicting apparent shear stress in prismatic compound open channels using artificial neural networks. J Hydroinform 15(1):138–146CrossRefGoogle Scholar
  16. Javid S, Mohammadi M (2012) Boundary shear stress in a trapezoidal channel. Int J Eng Trans A 25(4):323–332Google Scholar
  17. Julien PY (1995) Erosion and sedimentation. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  18. Kabiri-Samani A, Farshi F, Chamani MR (2013) Boundary shear stress in smooth trapezoidal open channel flows. J Hydraul Eng 139(2):205–212CrossRefGoogle Scholar
  19. Khodashenas SR, Paquier A (1999) A geometrical method for computing the distribution of boundary shear stress across irregular straight open channels. J Hydraul Res 37(3):381–388CrossRefGoogle Scholar
  20. Khodashenas SR, Paquier A (2002) River bed deformation calculated from boundary shear stress. J Hydraul Res 40(5):603–609CrossRefGoogle Scholar
  21. Knight DW (1981) Boundary shear in smooth and rough channels. J Hydraul Div ASCE 107(7):839–851Google Scholar
  22. Knight D, Sterling M (2000) Boundary shear in circular pipes partially full. J Hydraul Eng 126(4):263–275CrossRefGoogle Scholar
  23. Knight DW, Demetriou JD, Hamed ME (1984) Boundary shear stress in smooth rectangular channel. J Hydraul Eng 110(4):405–422CrossRefGoogle Scholar
  24. Knight DW, Yuen KWH, Al Hamid AAI (1994) Boundary shear stress distributions in open channel flow. In: Beven K, Chatwin P, Millbank J (eds) Physical mechanisms of mixing and transport in the environment. Wiley, New York, pp 51–87Google Scholar
  25. Lane EW (1953) Progress report on studies on the design of stable channels by the Bureau of Reclamation. Proc ASCE 79(280):1–30Google Scholar
  26. Lashkar-Ara B, Fathi-Moghadam M (2009) Wall and shear forces in open channel. Res J Phys 4(1):1–10Google Scholar
  27. Maszczyk T, Dush W (2008) Comparison of Shannon, Renyi and Tsallis entropy used in decision trees. Lect Notes Comput Sci 5097(2008):643–651CrossRefGoogle Scholar
  28. Myers WRC (1978) Momentum transfer in a compound channel. J Hydraul Res 16(2):139–150CrossRefGoogle Scholar
  29. Olivero M, Aguirre-Pey J, Moncada A (1999) Shear stress distributions in rectangular channels. In: 28th IAHR congress, Graz, AustraliaGoogle Scholar
  30. Rajaratnam N, Ahmadi R (1981) Hydraulics of channels with flood-plains. J Hydraul Res 19(1):43–60CrossRefGoogle Scholar
  31. Rhodes DG, Knight DW (1994) Distribution of shear force on the boundary of a smooth rectangular duct. J Hydraul Eng 120(7):787–807CrossRefGoogle Scholar
  32. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423, 623–656Google Scholar
  33. Singh VP, Luo H (2011) Entropy theory for distribution of one-dimensional velocity in open channels. J Hydrol Eng 16(9):725–735CrossRefGoogle Scholar
  34. Sterling M, Knight DW (2002) An attempt at using the entropy approach to predict the transverse distribution of boundary shear stress in open channel flow. Stoch Environ Res Risk Assess 16:127–142CrossRefGoogle Scholar
  35. Tominaga A, Nezu I, Ezaki K, Nakagawa H (1989) Three dimensional turbulent structure in straight open channel flows. J Hydraul Res 27:149–173CrossRefGoogle Scholar
  36. Tsallis C (1988) Possible generalization of Boltzmann–Gibbs statistics. J Stat Phys 52:479–487CrossRefGoogle Scholar
  37. Wilcock PR (1996) Estimating local bed shear stress from velocity observations. Water Resour Res 32(11):3361–3366CrossRefGoogle Scholar
  38. Yang SQ (2010) Depth-averaged shear stress and velocity in open-channel flows. J Hydraul Eng 136(11):952–958CrossRefGoogle Scholar
  39. Yang SQ, Lim SY (1997) Mechanism of energy transportation and turbulent flow in a 3D channel. J Hydraul Eng 123(8):684–692CrossRefGoogle Scholar
  40. Yuen KWH (1989) A study of boundary shear stress, flow resistance and momentum transfer in open channels with simple and trapezoidal cross section. PhD Thesis, University of BirminghamGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Hossein Bonakdari
    • 1
    • 2
    Email author
  • Zohreh Sheikh
    • 1
    • 2
  • Mohtaram Tooshmalani
    • 1
    • 2
  1. 1.Department of Civil Engineering, Faculty of EngineeringRazi UniversityKermanshahIran
  2. 2.Water and Wastewater Research CenterRazi UniversityKermanshahIran

Personalised recommendations