Environmental Fluid Mechanics

, Volume 15, Issue 6, pp 1163–1179 | Cite as

Entropic approach to estimate the mean flow velocity: experimental investigation in laboratory flumes

Original Article


The paper deals with the linear entropic relationship between the maximum velocity, u max , and the mean flow velocity, u m , through a dimensionless parameter Φ(M), in open-channel flow. The analysis is conducted with the aid of experimental data collected in straight laboratory flumes under different bed and side-walls roughness conditions. In particular, rough/vegetated beds and smooth/rough side-walls conditions have been investigated. The results show that, in the investigated conditions (with exception of low-submergence vegetated bed—h/k v  < 2), Φ(M) can be assumed equal to a value that is very close to that found in natural channels. This demonstrates that Φ(M) is able to implicitly reflect the different hydraulic behavior which is determined in rough and submerged vegetated beds. Thus, the entropy-based Manning’s roughness formula has been validated and the sensitivity analysis of Manning’s coefficient with the values of y o (location of the zero-velocity plane) has been also performed. It is found that this formula is quite robust to represent the observed flow resistance also in the presence of vegetation.


Flow discharge Entropy Flow velocity profile Experimental data 


  1. 1.
    Ammari A, Remini B (2009) Estimation of Algerian rivers discharges based one Chiu’s equation. Arab J Geosc. doi: 10.1007/s12517-009-0056-y Google Scholar
  2. 2.
    Carollo G, Ferro V, Termini D (2002) Flow velocity measurements in vegetated channels. J Hydraul Eng 128(7):664–673CrossRefGoogle Scholar
  3. 3.
    Carollo G, Ferro V, Termini D (2005) Flow resistance law in channels with flexible submerged vegetation. J Hydraul Eng 131(7):554–564CrossRefGoogle Scholar
  4. 4.
    Carollo G, Ferro V, Termini D (2007) Analysing longitudinal turbulence intensity in vegetated channels. J Agric Eng 4:25–35CrossRefGoogle Scholar
  5. 5.
    Carollo G, Ferro V, Termini D (2008a) Determinazione del profilo di velocità e di intensità della turbolenza in canali vegetati, 31° Convegno Nazionale di Idraulica e Costruzioni Idrauliche, 9-12 settembre Perugia, Italia, ISBN/EAN: 978-88-6074-220-9 (in Italian)Google Scholar
  6. 6.
    Carollo G, Ferro V, Termini D (2008b) Flow velocity profile and turbulence characteristics in a vegetated straight flume. In: International congress riverflow 2008—Cesnme, Izmir, 3–5 SeptGoogle Scholar
  7. 7.
    Chiu CL (1988) Entropy and 2-D velocity distribution in open channels. J Hydraul Eng 114(7):738–756CrossRefGoogle Scholar
  8. 8.
    Chiu CL (1991) Application of entropy concept in open-channel flow study. J Hydraul Eng 117(5):615–628CrossRefGoogle Scholar
  9. 9.
    Chiu CL, Murray DW (1992) Variation of velocity distribution along non-uniform open-channel flow. J Hydraul Eng 118(7):989–1001CrossRefGoogle Scholar
  10. 10.
    Chiu CL, Said CAA (1995) Maximum and mean velocitites and entropy in open-channel flow. J Hydraul Eng 121(1):26–35CrossRefGoogle Scholar
  11. 11.
    Domeneghetti A, Castellarin A, Brath A (2012) Assessing rating-curve uncertainty and its effects on hydraulic model calibration. Hydrol Earth Syst Sci 16:1191–1202CrossRefGoogle Scholar
  12. 12.
    Engelund F, Hansen E (1967) A monograph on sediment transport in alluvial streams. Teknik Vorlag, CopenhagenGoogle Scholar
  13. 13.
    Ferro V (1999) Friction factor for gravel bed channel with high boulder concentration. Proc ASCE J Hydraul Eng 125(7):771–778CrossRefGoogle Scholar
  14. 14.
    Finelli CM (2000) Velocity and concentration distributions in turbulent odor plumes in the presence of vegetation mimics: a flume study. Mar Ecol Prog Ser 207:297–309CrossRefGoogle Scholar
  15. 15.
    Folkard AM (2011) Vegetated flows in their environmental context: a review. Eng Comput Mech 164(EM1):3–24Google Scholar
  16. 16.
    Freeman GE, Rahmeyer WH, Copeland RR (2000) Determination of resistance due to shrubs and woody vegetation. ERDC/CHL TR-00-25, U.S.ACEGoogle Scholar
  17. 17.
    Ghisalberti M, Nepf HM (2002) Mixing layer and coherent structures in vegetated acquatic flows. J Geophys Res 107(2):1–11Google Scholar
  18. 18.
    Greco M, Mirauda D (2014) Entropy parameter estimation in large-scale roughness open channel. J Hydrol Eng. doi: 10.1061/(ASCE)HE.1943-5584.0001009 Google Scholar
  19. 19.
    Hart DD, Finelli CM (1999) Physical-biological coupling in streams: the pervasive effects of flow on benthic organisms. Annu Rev Ecol Syst 30:363–395CrossRefGoogle Scholar
  20. 20.
    Ikeda S, Kanazawa M (1996) Three dimensional organized vortices above flexible water plants. J Hydraul Eng 122(11):634–640CrossRefGoogle Scholar
  21. 21.
    Katul G, Wiberg P, Albertson J, Hornberger G (2002) A mixing layer theory for flow resistance in shallow streams. Water Resour Res 38(11):32-1/32-7Google Scholar
  22. 22.
    Kamphius JW (1974) Determination of Sand Roughness for Fixed Beds. J Hydraul Res 12(2):193–203CrossRefGoogle Scholar
  23. 23.
    Kowobary TS, Rice CE, Garton JE (1972) Effect of roughness elements on hydraulic resistance for overland flow. Trans Am Soc Agric Eng 15(5):979–984CrossRefGoogle Scholar
  24. 24.
    Kutija V, Hong HTM (1996) A numerical model for assessing the additional resistance to flow introduced by flexible vegetation. J Hydraul Res 34(1):99–114CrossRefGoogle Scholar
  25. 25.
    Lane SN, Hardy RJ, Elliott L, Ingham DB (2002) High resolution numerical modelling of three-dimensional flow over complex river bed topography. Hydrol Process 16(11):2261–2272CrossRefGoogle Scholar
  26. 26.
    Marini G, De Martino G, Fontana N, Fiorentino M, Singh VP (2011) Entropy approach for 2D velocity distribution in open-channel flow. J Hydraul Res 49(6):784–790. doi: 10.1080/00221686.2011.635889 CrossRefGoogle Scholar
  27. 27.
    Moramarco T, Singh VP (2010) Formulation of the entropy parameter based on hydraulic and geometric characteristics of river cross sections. J Hydrol Eng 15(10):852–858CrossRefGoogle Scholar
  28. 28.
    Moramarco T, Saltalippi C, Singh VP (2004) Estimation of mean velocity in natural channel based on Chiu’s velocity distribution equation. J Hydrol Eng 9(1):42–50CrossRefGoogle Scholar
  29. 29.
    Raudkivi AJ (1990) Loose Boundary Hydraulics, 3rd edn. Peregamon, New YorkGoogle Scholar
  30. 30.
    Ree WO, Palmer VJ (1949) Flow of water in channels protected by vegetative linings. Soil Conservation Service, United States Department of Agricolture, Washington DC. Technical Bullettin n. 967Google Scholar
  31. 31.
    Ribberink, J. and Chen, Z. 1993. Sediment transport of fine sand under asymmetric oscillatory flow, Delft Hydraulics, Report H840, part VII, The NetherlandsGoogle Scholar
  32. 32.
    Singh VP (2011) Derivation of power law and logarithmic velocity distributions using the Shannon entropy. J Hydrol Eng 16(5):478–483CrossRefGoogle Scholar
  33. 33.
    Smith RJ, Hacock NH, Ruffin JL (1990) Flood flow through tall vegetation. Agric Water Manag 18(4):317–332CrossRefGoogle Scholar
  34. 34.
    Statzner B (2008) How views about flow adaptations of benthic stream invertebrates changesd over the last centyury. Int Rev Hydrobiol 93(4–5):593–605CrossRefGoogle Scholar
  35. 35.
    Tanino Y, Nepf H (2008) Laboratory investigation on mean drag in a random array of Rigid, Emergent Cylinders. J Hydraul Eng 134(1):4–41CrossRefGoogle Scholar
  36. 36.
    Temple DM, Robinson KM, Ahring RM, Davis AG (1987) Stability design of grass-lined open channels. Agriculture handbook 667, Agricultural Research Service. United States Department of Agricolture, Washington DCGoogle Scholar
  37. 37.
    Termini D, Sammartano V (2007) Analysis of the role of turbulent structure in bed-forms formation in a rectilinear flume. In: 32nd IAHR congress—harmonizing the demands of arts and nature in Hydraulics, Venice, 1–6 JulyGoogle Scholar
  38. 38.
    Termini D, Sammartano V (2008) Experimental analysis of relation between coherent turbulent structures and bed-forms formation. Arch Hydroeng Environ Mech 55(3–4):29–47Google Scholar
  39. 39.
    Termini D, Sammartano V (2008) Evolution of coherent turbulent structures and relation with bed-forms formation in a straight flume. In: International congress riverflow 2008—Cesnme, Izmir, 3–5 SeptGoogle Scholar
  40. 40.
    Termini D, Sammartano V, Bonvissuto G (2008) Studio sperimentale dell’evoluzione di strutture turbolente in un canale rettilineo di laboratorio per due condizioni di scabrezza delle pareti, 31° Convegno Nazionale di Idraulica e Costruzioni Idrauliche, 9–12 Settembre 2008, Perugia (in Italian)Google Scholar
  41. 41.
    Termini D (2011) Experimental analysis of flow in a laboratory flume with flexible vegetation—EUROMECH Colloquium 523: Ecohydraulics: linkages between hydraulics, morphodynamics and ecological processes in rivers Clermont-Ferrand, France, 15–17 JuneGoogle Scholar
  42. 42.
    Termini D (2015) Flexible vegetation behaviour and effects on flow conveyance: experimental observations. J River Basin Manag. doi: 10.1080/15715124.2015.1012519 Google Scholar
  43. 43.
    Xia R (1997) Relation between mean and maximum velocities in a natural river. J Hydraul Eng 123(8):720–723CrossRefGoogle Scholar
  44. 44.
    Yalin MS (1992) River mechanics. Pregamon Press, LondonGoogle Scholar
  45. 45.
    Wilson CAME (2007) Flow resistance models for flexible submerged vegetation. J Hydrol 342(3–4):213–222CrossRefGoogle Scholar
  46. 46.
    Wu F, Shen HW, Chou Y (1999) Variation of roughness coefficients for unsubmerged and submerged conditions. J Hydraul Eng 125(9):934–942CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Research Institute for Geo-Hydrological Protection, CNRPerugiaItaly
  2. 2.Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, dei MaterialiUniversity of PalermoPalermoItaly

Personalised recommendations