Evidence of Non-Universality of von Kármán’s κ

Chapter
Part of the GeoPlanet: Earth and Planetary Sciences book series (GEPS)

Abstract

A notable universal feature of wall-bounded turbulent flows is the universal logarithmic law of the wall deduced by Theodore von Kármán. This law of the wall describes how time-averaged streamwise velocity changes with distance from the wall. Despite the law of the wall having a universal von Kármán constant κ = 0.41 that governs the slope of the log-law velocity profile, as commonly known over a period of about 80 years, in fluvial streams there are a number of instances of the non-universality of κ. To be specific, it behaves as a variable in flows with low relative submergence, or where there is bed-load and/or suspended-load sediment transport. This article focuses on the aspect of non-universality of κ by inviting various open questions relating to future research directions.

Keywords

Suspended Sediment Acoustic Doppler Velocimeter Roughness Layer Relative Submergence Streamwise Spacing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The Taylor and Francis Group is kindly acknowledged for having granted free permission of reuse of the content of the chapter Gaudio et al. (2010) (www.tandfonline.com).

References

  1. Bayazit M (1976) Free surface flow in a channel of large relative roughness. J Hydraul Res 14(2):115–126CrossRefGoogle Scholar
  2. Bennett SJ, Bridge JS (1995) The geometry and dynamics of low-relief bed forms in heterogeneous sediment in a laboratory channel, and their relationship to water flow and sediment transport. J Sediment Res A65:29–39Google Scholar
  3. Best J, Bennett S, Bridge J, Leeder M (1997) Turbulence modulation and particle velocities over flat sand beds at low transport rates. J Hydraul Eng 123(12):1118–1129CrossRefGoogle Scholar
  4. Bohlen WF (1969) Hotwire anemometer study of turbulence in open-channel flows transporting neutrally buoyant particles. Report number 69-1, Experimental sedimentology laboratory, Department of earth and planetary sciences, Massachusetts institute of technology, Cambridge, Mass, USAGoogle Scholar
  5. Cellino M, Graf WH (1999) Sediment-laden flow in open-channels under noncapacity and capacity conditions. J Hydraul Eng 125(5):455–462CrossRefGoogle Scholar
  6. Cioffi F, Gallerano F (1991) Velocity and concentration profiles of solid particles in a channel with movable and erodible bed. J Hydraul Res 29(3):387–401CrossRefGoogle Scholar
  7. Coleman NL (1981) Velocity profiles with suspended sediment. J Hydraul Res 19(3):211–227CrossRefGoogle Scholar
  8. Coleman NL (1986) Effects of suspended sediment on the open-channel velocity distribution. Water Resour Res 22(10):1377–1384CrossRefGoogle Scholar
  9. Cooper JR (2006) Spatially-induced momentum transfer over water-worked gravel beds. PhD thesis, The University of Sheffield, Sheffield, UKGoogle Scholar
  10. Dey S, Raikar RV (2007) Characteristics of loose rough boundary streams at near-threshold. J Hydraul Eng 133(3):288–304CrossRefGoogle Scholar
  11. Dey S, Das R, Gaudio R, Bose SK (2012) Turbulence in mobile-bed streams. Acta Geophysica (in press)Google Scholar
  12. Dittrich A, Hammann de Salazar K (1993) Bed instability caused by clear water flow. Final report of project Eroslope (EV5 V-CT92-0179), Institute for hydraulic engineering, Braunschweig technical university, GermanyGoogle Scholar
  13. Dittrich A, Koll K (1997) Velocity field and resistance of flow over rough surface with large and small relative submergence. Int J Sedim Res 12(3):21–33Google Scholar
  14. Einstein HA, Chien N (1955) Effects of heavy sediment concentration near the bed on velocity and sediment distribution. MRD sediment series report number 8, University of California, Berkeley, US Army Corps of Engineers, Missouri Division, St. Louis, MO, USAGoogle Scholar
  15. Elata C, Ippen AT (1961) The dynamics of open channel flow with suspensions of neutrally buoyant particles. Technical report number 45, Massachusetts institute of technology, Boston, MA, USAGoogle Scholar
  16. Gallagher M, McEwan I, Nikora V (1999) The changing structure of turbulence over a self-stabilising sediment bed. Internal report number 21, Department of Engineering, University of Aberdeen, Aberdeen, UKGoogle Scholar
  17. Gaudio R, Miglio A, Calomino F (2011) Friction factor and von Kármán’s κ in open channels with bed-load. J Hydraul Res 49(2):239–247CrossRefGoogle Scholar
  18. Gaudio R, Miglio A, Dey S (2010) Non-universality of von Kármán’s κ in fluvial streams. J Hydraul Res 48(5):658–663CrossRefGoogle Scholar
  19. Guo J, Julien PY (2001) Turbulent velocity profiles in sediment-laden flows. J Hydraul Res 39(1):11–23CrossRefGoogle Scholar
  20. Gust G, Southard JB (1983) Effects of weak bed load on the universal law of the wall. J Geophys Res 88(C10):5939–5952Google Scholar
  21. Gyr A, Schmid A (1997) Turbulent flows over smooth erodible sand beds in flumes. J Hydraul Res 35(4):525–544CrossRefGoogle Scholar
  22. Hetsroni G, Zakin JL, Mosyak A (1997) Low-speed streaks in drag-reduced turbulent flow. Phys Fluids 9(8):2397–2404CrossRefGoogle Scholar
  23. Hino M (1963) Turbulent flow with suspended particles. J Hydraul Div 89(HY4):161–185Google Scholar
  24. Hughes RL (2007) A mathematical determination of von Kármán’s constant, κ. J Hydraul Res 45(4):563–566CrossRefGoogle Scholar
  25. Ippen AT (1971) A new look at sedimentation in turbulent streams. J Boston Soc Civil Eng 58(3):131–163Google Scholar
  26. Kirkbride A (1993) Observations of the influence of bed roughness on turbulence structure in depth limited flows over gravel beds. In: Clifford NJ, French JR, Hardisty J (eds.) Turbulence: perspectives on flow and sediment transport, Wiley, Chichester, pp 185–196Google Scholar
  27. Koll K (2002) Feststofftransport und Geschwindigkeitsverteilung in Raugerinnen. Karlsruhe University, Fak. f. Bauingenieur- und Vermessungswesen, Diss. v. 12.07.2002, http://www.ubka.uni-karlsruhe.de/cgibin/
  28. Koll K (2006) Parameterisation of the vertical velocity profile in the wall region over rough surfaces. In: Ferreira RML, Alves ECTL, Leal JGAB, Cardoso AH (eds.) River flow 2006, Taylor & Francis, London, Proceedings of international conference of fluvial hydraulics, Lisbon, Portugal, pp 163–171Google Scholar
  29. Lo TS, L’vov VS, Pomyalov A, Procaccia I (2005) Estimating von Kármán’s constant from homogeneous turbulence. Europhys Lett 72(6):943–949CrossRefGoogle Scholar
  30. Long CE, Wiberg PL, Nowell ARM (1993) Evaluation of von Kármán’s constant from integral flow parameters. J Hydraul Eng 119(10):1182–1190CrossRefGoogle Scholar
  31. Lyn DA (1986) Turbulence and turbulent transport in sediment-laden open-channel flows. Report number KH-R-49, In: Keck WM (ed.) Laboratory of hydraulic and water resources, California Institute of Technology, Pasadena, CA, USAGoogle Scholar
  32. Muste M (2002) Sources of bias errors in flume experiments on suspended-sediment transport. J Hydraul Res 40(6):695–708CrossRefGoogle Scholar
  33. Nezu I, Azuma R (2004) Turbulence characteristics and interaction between particles and fluid in particle-laden open channel flows. J Hydraul Eng 130(10):988–1001CrossRefGoogle Scholar
  34. Nikora VI, Goring DG (1999) Effects of bed mobility on turbulence structure. NIWA Internal report number 48, NIWA, Christchurch, New ZealandGoogle Scholar
  35. Nikora V, Goring D (2000) Flow turbulence over fixed and weakly mobile gravel beds. J Hydraul Eng 126(9):679–690CrossRefGoogle Scholar
  36. Nikora V, Goring D, McEwan I, Griffiths G (2000) Spatially averaged open-channel flow over rough bed. J Hydraul Eng 127(2):123–133CrossRefGoogle Scholar
  37. Nouh M (1989) The von-Kármán coefficient in sediment laden flow. J Hydraul Res 27(4):477–499CrossRefGoogle Scholar
  38. Owen PR (1964) Saltation of uniform grains in air. J Fluid Mech 20:225–242CrossRefGoogle Scholar
  39. Packman AI, Salehin M, Zaramella M (2004) Hyporheic exchange with gravel beds: Basic hydrodynamic interactions and bedform-induced advective flows. J Hydraul Eng 130(7):647–656CrossRefGoogle Scholar
  40. Paintal AS, Garde RJ (1964) Discussion of ‘suspended transportation mechanics: suspension of sediment’. J Hydraul Div 90(HY4):257–265Google Scholar
  41. Pokrajac D, Finnigan JJ, Manes C, McEwan I, Nikora V (2006) On the definition of the shear velocity in rough bed open channel flows. In: Ferreira RML, Alves ECTL, Leal JGAB, Cardoso AH (eds.) River flow 2006, Taylor & Francis, London, UK, Proceedings of international conference of fluvial hydraulics, Lisbon, Portugal pp 88–96Google Scholar
  42. Rand W (1953) Discussion of ‘some effects of suspended sediment on flow characteristics’. In: Proceedings of fifth hydraulics conference, Bulletin 34, State University of Iowa, Iowa City, Iowa, USA pp 156–158Google Scholar
  43. Sirovich L, Karlsson S (1997) Turbulent drag reduction by passive mechanisms. Nature 388:753–755CrossRefGoogle Scholar
  44. Smith JD, McLean SR (1977) Spatially averaged flow over a wavy surface. J Geophys Res 82(12):1735–1746CrossRefGoogle Scholar
  45. Tiederman WG, Luchik TS, Bogard DG (1985) Wall layer structure and drag reduction. J Fluid Mech 156:419–437CrossRefGoogle Scholar
  46. Vanoni VA (1946) Transportation of suspended sediment by water. Trans Am Soc Civil Eng 111:67–133Google Scholar
  47. Vanoni VA, Nomicos GN (1960) Resistance properties of sediment laden stream. Trans Am Soc Civil Eng 125:1140–1175Google Scholar
  48. van Rijn LC (1993) Principles of sediment transport in rivers, estuaries and coastal seas. Aqua Publications, The NetherlandsGoogle Scholar
  49. von Kármán T (1930) Mechanische ähnlichkeit and turbulenz. Nachrichten der Adademie der Wissenchaften Göttingen, Mathematisch-Physikali-Sche Klasse 58–76Google Scholar
  50. Wang X, Qian N (1992) Velocity profiles of sediment-laden flow. Int J Sedim Res 7(1):27–58Google Scholar
  51. Wang X, Wang ZY, Yu M, Li D (2001) Velocity profile of sediment suspensions and comparison of log-law and wake-law. J Hydraul Res 39(2):211–217CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Dipartimento di Difesa del Suolo “V. Marone”Università della CalabriaRende (CS)Italy
  2. 2.Department of Civil EngineeringIndian Institute of TechnologyKharagpurIndia

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