Skip to main content
SpringerLink
Log in

Turbulence in Open-Channel Flows

  • Chapter
  • First Online:
Fluvial Hydrodynamics

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

Abstract

The turbulence in a fluid flow is characterized by irregular and chaotic motion of fluid particles. It is a complex phenomenon. In this chapter, the turbulence characteristics are discussed with reference to flow over a sediment bed. An application of Reynolds decomposition and time-averaging to the Navier–Stokes equations yields the Reynolds-averaged Navier–Stokes (RANS) equations, containing terms of Reynolds stresses. The RANS equations along with the time-averaged continuity equation are the main equations to analyze turbulent flow. The classical turbulence theories were proposed by Prandtl and von Kármán. Prandtl simulated the momentum exchange on a macro-scale to explain the mixing phenomenon in a turbulent flow establishing the mixing length theory, while von Kármán’s relationship for the mixing length is based on the similarity hypothesis. The velocity distribution in open-channel flow follows the linear law in viscous sublayer, the logarithmic law in turbulent wall shear layer, and the wake law in the outer layer. The determination of bed shear stress is always a challenging task. Different methods for the determination of bed shear stress are discussed. Flow in a narrow channel exhibits strong turbulence-induced secondary currents, and as a result, the maximum velocity appears below the free surface, known as dip phenomenon. Isotropic turbulence theory deals with the turbulent kinetic energy (TKE) transfer from the large-scale motions to smaller-scale motions until attaining an adequately small length scale so that the fluid molecular viscosity can dissipate the TKE into heat. Anisotropy in turbulence is analyzed by the anisotropic invariant mapping (AIM) and the anisotropy invariant function to quantify the degree of the departure from isotropy. Higher-order correlations are given by skewness and kurtosis of velocity fluctuations, TKE flux, and budget. This chapter also includes most of the modern development of turbulent phenomena, such as coherent structures and burst phenomena and double-averaging of heterogeneous flow over gravel beds.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Laminar flow changes over to turbulent flow, if the Reynolds number Re [=UD/υ for pipe flow and U(4R b)/υ for open-channel flow] is greater than 2000. Here, U is the average flow velocity, D is the pipe diameter, υ is the coefficient of kinematic viscosity, and R b is the hydraulic radius.

  2. 2.

    Eddies can be defined as swirls of fluid parcels with highly irregular shapes and wide range of sizes that are in a continual state of generation, evolution, and decay, as a cyclic process (Middleton and Southard 1984).

  3. 3.

    The velocity at level 1 is expressed as \( \bar{u}\left( {z + l} \right) \, = \bar{u}\left( z \right) \, + {\text{ d}}\bar{u} \). If \( \bar{u} \)(z + l) is expanded in a Taylor series up to the linear term only, then \( {\text d}\bar{u} = \bar{u}(z + l) - \bar{u}(z) = l{\text d}\bar{u}/dz \). Hence, the velocity at level 1 is given by ū + ldū/dz. However, velocity at level 2 is \( \bar{u} \).

  4. 4.

    The modified mixing length model of van Driest (1956) that incorporates the viscous effect is as follows:

    \( l = \kappa z\varGamma (\tilde{z}^{ + } )\quad\wedge\quad \varGamma (\tilde{z}^{ + } ) = 1 - \exp \left( {\frac{{\tilde{z}^{ + } }}{{B_{\text{d}} }}} \right)\quad\vee\quad{\tilde{z}}^{ + } = \frac{{zu_{ * } }}{\upsilon } \)

    where \( \varGamma \left( {\tilde{z}^{ + } } \right) \) is the van Driest damping function and B d is the damping factor.

  5. 5.

    More explicitly, the flow regimes are called (1) hydraulically smooth flow regime, (2) hydraulically rough flow regime, and (3) hydraulically transitional flow regime of turbulent flow.

  6. 6.

    The forms of the log-wake law for smooth and rough flows are as follows:

    \( \begin{aligned} {\text{Smooth flow:}}\,u^{ + } & = \left( {\frac{1}{\kappa }\ln \tilde{z}^{ + } + \left. B \right|_{\text{smooth}} } \right) + \frac{2\varPi }{\kappa }\sin^{2} \left( {\frac{\pi }{2}\tilde{z}} \right) \\ {\text{Rough flow:}}\,u^{ + } & = \left( {\frac{1}{\kappa }\ln \tilde{z}^{ + } + \left. B \right|_{\text{rough}} } \right) + \frac{2\varPi }{\kappa }\sin^{2} \left( {\frac{\pi }{2}\tilde{z}} \right). \\ \end{aligned} \)

  7. 7.

    The Colebrook–White equation is an implicit equation. An explicit form of the Colebrook–White equation was given by Haaland (1983). It is as follows:

    $$ \frac{1}{{\lambda_{\text{D}}^{0.5} }} = - 0.782\ln \left[ {\left( {\frac{{k_{\text{s}} P}}{14.8A}} \right)^{1.1} \,+\,\frac{6.9}{Re}} \right] .$$
  8. 8.

    The following transformation then applies:

    $$ S_{uu} (k_{\text{w}} ) = \frac{{\bar{u}}}{2\pi }F(f) \quad\wedge \quad k_{\text{w}} = 2\pi \frac{f}{{\bar{u}}}. $$
  9. 9.

    The TKE budget equation in tensor form is as follows:

    $$ \underbrace {{\bar{u}_{j} \frac{\partial k}{{\partial x_{j} }}}}_{\text{Advection}} = - \underbrace {{\frac{1}{\rho } \cdot \frac{{\partial (\overline{{p^{\prime}u^{\prime}_{i} }} )}}{{\partial x_{i} }}}}_{{p_{\text{D}} }} - \underbrace {{\frac{{\partial (\overline{{ku^{\prime}_{i} }} )}}{{\partial x_{j} }}}}_{{t_{\text{D}} }} + \underbrace {{\upsilon \frac{{\partial^{2} k}}{{\partial x_{j}^{2} }}}}_{{v_{\text{D}} }} - \underbrace {{\upsilon \overline{{\frac{{\partial u^{\prime}_{i} }}{{\partial x_{j} }} \cdot \frac{{\partial u^{\prime}_{i} }}{{\partial x_{j} }}}} }}_{\varepsilon } + \underbrace {{\left( { - \overline{{u^{\prime}_{i} u^{\prime}_{j} }} \frac{{\partial \bar{u}_{i} }}{{\partial x_{j} }}} \right)}}_{{t_{\text{P}} }} .$$

References

  • Absi R (2011) An ordinary differential equation for velocity distribution and dip-phenomenon in open channel flows. J Hydraul Res 49(1):82–89

    Google Scholar 

  • Andreopoulos J, Durst F, Zaric Z, Javonovic J (1984) Influence of Reynolds number on characteristics of turbulent wall boundary layers. Exp Fluids 2(1):7–16

    Google Scholar 

  • Antonia RA, Atkinson JD (1973) High-order moments of Reynolds shear stress fluctuations in a turbulent boundary layer. J Fluid Mech 58:581–593

    Google Scholar 

  • ASCE Task Force (1963) Friction factor in open channels. J Hydraul Div 89(2):97–143

    Google Scholar 

  • Bose SK (2009) Numeric computing in Fortran. Narosa, New Delhi

    Google Scholar 

  • Bose SK, Dey S (2007) Curvilinear flow profiles based on Reynolds averaging. J Hydraul Eng 133(9):1074–1079

    Google Scholar 

  • Bose SK, Dey S (2010) Universal probability distributions of turbulence in open channel flows. J Hydraul Res 48(3):388–394

    Google Scholar 

  • Bradshaw P (1967) Conditions for the existence of an inertial subrange in turbulent flow, Aerodynamics report number 1220. National Physical Laboratory, Teddington

    Google Scholar 

  • Camenen B, Bayram A, Larson M (2006) Equivalent roughness height for plane bed under steady flow. J Hydraul Eng 132(11):1146–1158

    Google Scholar 

  • Cardoso AH, Graf WH, Gust G (1989) Uniform flow in smooth open-channel. J Hydraul Res 27(5):603–616

    Google Scholar 

  • Choi KS, Lumley JL (2001) The return to isotropy of homogeneous turbulence. J Fluid Mech 436:59–84

    Google Scholar 

  • Chow VT (1959) Open channel hydraulics. McGraw-Hill Book Company, New York

    Google Scholar 

  • Clauser FH (1954) Turbulent boundary layers in adverse pressure gradients. J Aeronaut Sci 21(2):91–108

    Google Scholar 

  • Colebrook CF, White CM (1937) Experiments with fluid friction in roughened pipes. Proc R Soc London A 161(906):367–381

    Google Scholar 

  • Coleman NL (1981) Velocity profiles with suspended sediment. J Hydraul Res 19(3):211–227

    Google Scholar 

  • Coleman NL, Alonso CV (1983) Two-dimensional channel flows over rough surfaces. J Hydraul Eng 109(2):175–188

    Google Scholar 

  • Coles D (1956) The law of the wake in a turbulent boundary layer. J Fluid Mech 1:191–226

    Google Scholar 

  • Corino ER, Brodkey RS (1969) A visual investigation of the wall region in turbulent flow. J Fluid Mech 37:1–30

    Google Scholar 

  • Cramer H (1999) Mathematical methods of statistics. Princeton University Press, Princeton

    Google Scholar 

  • Dean RB (1976) A single formula for the complete velocity profile in a turbulent boundary layer. J Fluids Eng 98(4):723–726

    Google Scholar 

  • Detert M, Weitbrecht V, Jinka GH (2010) Laboratory measurements on turbulent pressure fluctuations in and above gravel beds. J Hydraul Eng 136(10):779–789

    Google Scholar 

  • Dey S (2003) Incipient motion of bivalve shells on sand beds under flowing water. J Eng Mech 129(2):232–240

    Google Scholar 

  • Dey S, Das R (2012) Gravel-bed hydrodynamics: a double-averaging approach. J Hydraul Eng 138(8):707–725

    Google Scholar 

  • Dey S, Das R, Gaudio R, Bose SK (2012) Turbulence in mobile-bed streams. Acta Geophys 60(6):1547–1588

    Google Scholar 

  • Dey S, Lambert MF (2005) Reynolds stress and bed shear in nonuniform unsteady open-channel flow. J Hydraul Eng 131(7):610–614

    Google Scholar 

  • Dey S, Raikar RV (2007) Characteristics of loose rough boundary streams at near-threshold. J Hydraul Eng 133(3):288–304

    Google Scholar 

  • Dey S, Sarkar S, Ballio F (2011) Double-averaging turbulence characteristics in seeping rough-bed streams. J Geophys Res 116(F03020), doi:10.1029/2010JF001832

  • Dong Z, Wang J, Chen C, Xia Z (1991) Turbulence characteristics of open-channel flows over rough beds. In: Proceedings of the twenty fourth meeting of International Association for Hydraulic Research, Delft, pp C33–C40

    Google Scholar 

  • Dyer KR (1986) Coastal and estuarine sediment dynamics. Wiley, Chichester

    Google Scholar 

  • Franca MJ, Lemmin U (2009) The simultaneous occurrence of logarithmic and S-shaped velocity profiles in gravel-bed river flows. Arch Hydro-Eng Environ Mech 56(1–2):29–41

    Google Scholar 

  • Frenkiel FN, Klebanoff PS (1967) Higher order correlations in a turbulent field. Phys Fluids 10(3):507–520

    Google Scholar 

  • Gad-el-Hak M, Bandyopadhyay PR (1994) Reynolds number effects in wall-bound turbulent flow. Appl Mech Rev 47(8):307–365

    Google Scholar 

  • Galletti B, Bottaro A (2004) Large-scale secondary structures in duct flow. J Fluid Mech 512:85–94

    Google Scholar 

  • Galperin B, Kantha LH, Hassid S, Rosati A (1988) A quasi-equilibrium turbulent energy-model for geophysical flows. J Atmos Sci 45(1):55–62

    Google Scholar 

  • García MH (2008) Sediment transport and morphodynamics. In: García MH (ed) Sedimentation engineering: processes, measurements, modeling, and practice, Manuals and reports on engineering practice number 110. American Society of Civil Engineers, Reston, pp 21–163

    Google Scholar 

  • Gessner FB (1973) The origin of secondary flow in turbulent flow along a corner. J Fluid Mech 58:1–25

    Google Scholar 

  • Gradshteyn IS, Ryzhik IM (1980) Table of integrals, series and products. Academic Press, New York

    Google Scholar 

  • Grass AJ (1971) Structural features of turbulent flow over smooth and rough boundaries. J Fluid Mech 50:233–255

    Google Scholar 

  • Grass AJ, Stuart RJ, Mansour-Tehrani M (1991) Vortical structures and coherent motion in turbulent flow over smooth and rough boundaries. Philos Trans R Soc London A 336(1640):35–65

    Google Scholar 

  • Gross TF, Nowell ARM (1985) Spectral scaling in a tidal boundary layer. J Phys Oceanogr 15(5):496–508

    Google Scholar 

  • Guo J, Julien PY, Meroney RN (2005) Modified log-wake law for zero-pressure-gradient turbulent boundary layers. J Hydraul Res 43(4):421–430

    Google Scholar 

  • Haaland SE (1983) Simple and explicit formulas for the friction factor in turbulent flow. J Fluids Eng 105(5):89–90

    Google Scholar 

  • Head MR, Bandyopadhyay P (1981) New aspects of turbulent boundary-layer structure. J Fluid Mech 107:297–338

    Google Scholar 

  • Hinze JO (1987) Turbulence. McGraw-Hill, New York

    Google Scholar 

  • Ikeda S (1981) Self forced straight channels in sandy beds. J Hydraul Div 107(4):389–406

    Google Scholar 

  • Irwin HPAH (1973) Measurements in a self-preserving plane wall jet in a positive pressure gradient. J Fluid Mech 61:33–63

    Google Scholar 

  • Jiménez J (2004) Turbulent flows over rough walls. Ann Rev Fluid Mech 36:173–196

    Google Scholar 

  • Kampé de Fériet J (1966) The Gram-Charlier approximation of the normal law and the statistical description of homogenous turbulent flow near statistical equilibrium, David Taylor model basin report 2013, Naval Ship Research and Development Center, Washington DC

    Google Scholar 

  • Keulegan GH (1938) Laws of turbulent flow in open channels. J Res National Bur Stand 21(6):707–741

    Google Scholar 

  • Kim S-C, Friedrichs CT, Maa JP-Y, Wright LD (2000) Estimating bottom stress in tidal boundary layer from acoustic Doppler velocimeter data. J Hydraul Eng 126(6):399–406

    Google Scholar 

  • Kirkgöz MS (1989) Turbulence velocity profiles for smooth and rough open-channel flow. J Hydraul Eng 115(11):1543–1561

    Google Scholar 

  • Kironoto BA, Graf WH (1994) Turbulence characteristics in rough uniform open-channel flow. Water Marit Energy Proc Inst Civ Eng (London) 106(4):333–344

    Google Scholar 

  • Kline SJ, Reynolds WC, Straub FA, Runstadler PW (1967) The structure of turbulent boundary layers. J Fluid Mech 30:741–773

    Google Scholar 

  • Kolmogorov AN (1941) The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers. Dokl Akad Nauk SSSR 30:299–303

    Google Scholar 

  • Krogstad PÅ, Antonia RA (1999) Surface roughness effects in turbulent boundary layers. Exp Fluids 27:450–460

    Google Scholar 

  • Lee MK, Eckelman LD, Hanratty TJ (1974) Identification of turbulent wall eddies through the phase relation of the components of the fluctuating velocity gradient. J Fluid Mech 66:17–33

    Google Scholar 

  • López F, García MH (1999) Wall similarity in turbulent open-channel flow. J Hydraul Eng 125(7):789–796

    Google Scholar 

  • Lu SS, Willmarth WW (1973) Measurements of the structures of the Reynolds stress in a turbulent boundary layer. J Fluid Mech 60:481–511

    Google Scholar 

  • Lumley JL (1978) Computational modeling of turbulent flows. Adv Appl Mech 18(4):123–176

    Google Scholar 

  • Lumley JL, Newman GR (1977) The return to isotropy of homogeneous turbulence. J Fluid Mech 82:161–178

    Google Scholar 

  • Manes C, Pokrajac D, McEwan I (2007) Double-averaged open-channel flows with small relative submergence. J Hydraul Eng 133(8):896–904

    Google Scholar 

  • Middleton GV, Southard JB (1984) Mechanics of sediment movement, Short course number 3, 2nd edn, Society of Economic Paleontologists and Mineralogists, Tulsa

    Google Scholar 

  • Nakagawa H, Nezu I (1977) Prediction of the contributions to the Reynolds stress from bursting events in open-channel flows. J Fluid Mech 80:99–128

    Google Scholar 

  • Nakagawa H, Nezu I, Tsujimoto T (1981) Turbulence structure with and without cellular secondary currents over various bed configurations. Ann, Disaster Prev Res Inst Kyoto Univ 24B:313–338 (in Japanese)

    Google Scholar 

  • Nezu I (1977) Turbulent structure in open channel flow. PhD thesis, Kyoto University, Kyoto

    Google Scholar 

  • Nezu I, Azuma R (2004) Turbulence characteristics and interaction between particles and fluid in particle-laden open channel flows. J Hydraul Eng 130(10):988–1001

    Google Scholar 

  • Nezu I, Nakagawa H (1993) Turbulence in open-channel flows. Balkema, Rotterdam

    Google Scholar 

  • Nezu I, Rodi W (1986) Open-channel flow measurements with a laser Doppler anemometer. J Hydraul Eng 112(5):335–355

    Google Scholar 

  • Nikora V, Goring D (1998) Spectral scaling for gravel-bed open-channel flows. In: Proceedings of the NATO advanced research workshop on stochastic models of hydrological processes and their applications to problems of environmental preservation. Water Problems Institute, Moscow, pp 239–245

    Google Scholar 

  • Nikora V, Goring D (2000) Flow turbulence over fixed and weakly mobile gravel beds. J Hydraul Eng 126(9):679–690

    Google Scholar 

  • Nikora V, Goring D, McEwan I, Griffiths G (2001) Spatially averaged open-channel flow over rough bed. J Hydraul Eng 127(2):123–133

    Google Scholar 

  • Nikora V, Koll K, McEwan I, McLean S, Dittrich A (2004) Velocity distribution in the roughness layer of rough-bed flows. J Hydraul Eng 130(10):1036–1042

    Google Scholar 

  • Nikora V, McEwan I, McLean S, Coleman S, Pokrajac D, Walters R (2007a) Double-averaging concept for rough-bed open-channel and overland flows: theoretical background. J Hydraul Eng 133(8):873–883

    Google Scholar 

  • Nikora V, McLean S, Coleman S, Pokrajac D, McEwan I, Campbell L, Aberle J, Clunie D, Koll K (2007b) Double-averaging concept for rough-bed open-channel and overland flows: applications. J Hydraul Eng 133(8):884–895

    Google Scholar 

  • Nikuradse J (1933) Strömungsgesetze in rauhen rohren. Verein Deutscher Ingenieure, Forschungsheft 361:1–22

    Google Scholar 

  • Nychas SG, Hershey HC, Brodkey RS (1973) A visual study of turbulent shear flow. J Fluid Mech 61:513–540

    Google Scholar 

  • Offen GR, Kline SJ (1973) Experiments on the velocity characteristics of ‘bursts’ and on the interactions between the inner and outer regions of a turbulent boundary layer. Report MD-31, Department of Mechanical Engineering, Stanford University, Stanford

    Google Scholar 

  • Offen GR, Kline SJ (1975) A proposed model of the bursting process in turbulent boundary layers. J Fluid Mech 70:209–228

    Google Scholar 

  • Perkins HJ (1970) The formation of streamwise vorticity in turbulent flow. J Fluid Mech 44:721–740

    Google Scholar 

  • Pope SB (2001) Turbulent flows. Cambridge University Press, Cambridge

    Google Scholar 

  • Prandtl L (1925) Bericht über untersuchungen zur ausgebildeten turbulenz. Zeitschrift für Angewandte Mathematik und Mechanik 5(2):136–139

    Google Scholar 

  • Prandtl L (1933) Neuere ergebnisse der turbulenzforschung. Zeitschrift des Vereines Deutscher Ingenieure 77(5):105–114

    Google Scholar 

  • Prandtl L (1952) Essentials of fluid dynamics. Blackie and Son, London

    Google Scholar 

  • Raupach MR (1981) Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers. J Fluid Mech 108:363–382

    Google Scholar 

  • Reynolds AJ (1974) Turbulent flows in engineering. Wiley, London

    Google Scholar 

  • Reynolds O (1883) An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels. Philos Trans R Soc London A 174:935–982

    Google Scholar 

  • Reynolds O (1895) On the dynamical theory of incompressible viscous fluids and the deformation of the criterion. Philos Trans R Soc London A 186:123–164

    Google Scholar 

  • Richardson LF (1922) Weather prediction by numerical process. Cambridge University Press, Cambridge

    Google Scholar 

  • Robinson SK (1991) Coherent motions in the turbulent boundary layers. Ann Rev Fluid Mech 23:601–639

    Google Scholar 

  • Rodríguez JF, García MH (2008) Laboratory measurements of 3-D flow patterns and turbulence in straight open channel with rough bed. J Hydraul Res 46(4):454–465

    Google Scholar 

  • Sarkar S (2010) Turbulence in loose boundary streams. PhD thesis, Department of Civil Engineering, Indian Institute of Technology, Kharagpur

    Google Scholar 

  • Sarkar S, Dey S (2010) Double-averaging turbulence characteristics in flows over a gravel-bed. J Hydraul Res 48(6):801–809

    Google Scholar 

  • Schlichting H (1979) Boundary layer theory. McGraw-Hill Book Company, New York

    Google Scholar 

  • Schlichting H, Gersten K (2000) Boundary layer theory. Springer, Berlin

    Google Scholar 

  • Simpson RL, Chew YT, Shivaprasad BG (1981) The structure of a separating turbulent boundary layer, part 2: higher-order turbulence results. J Fluid Mech 113:53–73

    Google Scholar 

  • Smith CR (1996) Coherent flow structures in smooth-wall turbulent boundary layers: facts, mechanisms and speculation. In: Ashworth P, Bennett S, Best JL, McLelland S (eds) Coherent flow structures in open channels. Wiley, Chichester, pp 804–813

    Google Scholar 

  • Smith CR, Metzler SP (1983) The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer. J Fluid Mech 129:27–54

    Google Scholar 

  • Smith JD, McLean SR (1977) Spatially averaged flow over a wavy surface. J Geophys Res 82(12):1735–1746

    Google Scholar 

  • Song T (1994) Velocity and turbulence distribution in nonuniform and unsteady open-channel flow. Doctoral dissertation, Ecole Polytechnique Federale de Lausanne, Lausanne

    Google Scholar 

  • Song T, Graf WH, Lemmin U (1994) Uniform flow in open channels with movable gravel bed. J Hydraul Res 32(6):861–876

    Google Scholar 

  • Soulsby RL, Dyer KR (1981) The form of the near-bed velocity profile in a tidally accelerating flow. J Geophys Res Oceans Atmos 86(NC9):8067–8074

    Google Scholar 

  • Spalding DB (1961) A single formula for the law of the wall. J Appl Mech 28(3):455–458

    Google Scholar 

  • Stapleton KR, Huntley DA (1995) Seabed stress determinations using the inertial dissipation method and the turbulent kinetic energy method. Earth Surf Proc Land 20(9):807–815

    Google Scholar 

  • Steffler PM, Rajaratnam N, Peterson AW (1985) LDA measurements in open channel. J Hydraul Eng 111(1):119–130

    Google Scholar 

  • Sumer BM, Kozakiewicz A, Fredsøe J, Deigaard R (1996) Velocity and concentration profiles in sheet-flow layer of movable bed. J Hydraul Eng 122(10):549–558

    Google Scholar 

  • Taylor GI (1935) Statistical theory of turbulence: part I–III. Proc R Soc London A 151(873):421–464

    Google Scholar 

  • Townsend AA (1976) The structure of turbulent shear flow. Cambridge University Press, Cambridge

    Google Scholar 

  • Tracy HJ, Lester CM (1961) Resistance coefficients and velocity distribution, smooth rectangular channel, Survey water-supply paper 1592-A. United States Geological, Washington, DC

    Google Scholar 

  • van Atta CW, Chen WY (1968) Correlation measurements in grid turbulence using digital harmonic analysis. J Fluid Mech 34:497–515

    Google Scholar 

  • van Driest ER (1956) On turbulent flow near a wall. J Aeronaut Sci 23(10):1007–1011

    Google Scholar 

  • van Rijn LC (1984) Sediment transport, part I: bed load transport. J Hydraul Eng 110(10):1431–1456

    Google Scholar 

  • van Rijn LC (1993) Principles of sediment transport in rivers, estuaries and coastal seas. Aqua Publications, The Netherlands

    Google Scholar 

  • Vanoni VA (1975) Sedimentation engineering, ASCE manual number 54. American Society of Civil Engineers, New York

    Google Scholar 

  • von Kármán T (1930) Mechanische ähnlichkeit und turbulenz. Nachrichten von der Gesellschaft der Wissenschaften Göttingen, Mathematisch-Physikalische Klasse 1:58–76

    Google Scholar 

  • Voulgaris B, Trowbridge JH (1998) Evaluation of the acoustic Doppler velocimeter (ADV) for turbulence measurements. J Atmos Oceanic Technol 15(1):272–289

    Google Scholar 

  • 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):211–217

    Google Scholar 

  • Wilson KC (1987) Analysis of bed-load motion at high shear stress. J Hydraul Eng 113(1):97–103

    Google Scholar 

  • Yalin MS (1977) Mechanics of sediment transport. Pergamon, Oxford

    Google Scholar 

  • Yalin MS (1992) River mechanics. Pergamon, Oxford

    Google Scholar 

  • Yang S-Q, Lim S-Y (1997) Mechanism of energy transportation and turbulent flow in a 3D channel. J Hydraul Eng 123(8):684–692

    Google Scholar 

  • Yang S-Q, Tan SK, Lim S-Y (2004) Velocity distribution and dip-phenomenon in smooth uniform open channel flows. J Hydraul Eng 130(12):1179–1186

    Google Scholar 

  • Zhou Y (1993) Interacting scales and energy transfer in isotropic turbulence. Phys Fluids A 5(10):2511–2524

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Indian Institute of Technology, Kharagpur, India

    Subhasish Dey

Authors
  1. Subhasish Dey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Subhasish Dey .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Dey, S. (2014). Turbulence in Open-Channel Flows. In: Fluvial Hydrodynamics. GeoPlanet: Earth and Planetary Sciences. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19062-9_3

Download citation

Publish with us

Policies and ethics

Search

Navigation