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Bedload resistance in supercritical flow
International Journal of Sediment Research 24 (2009) 400-409
Bedload resistance in supercritical flow Alireza HABIBZADEH1 and Mohammad Hossein OMID2
Abstract The movement of bedload in subcritical flow produces additional roughness as compared to flow in a rigid bed. The magnitude of this bed load roughness is proportional to the thickness of the sediment layer moving along the bed, the particle size and the sediment concentration. In a supercritical flow, however, further resistance is expected due to the momentum absorption by the high flow velocity. In this study the effect of sediment movement on the flow resistance in supercritical flow was experimentally investigated. The experiments included flows over smooth and rough beds carrying sediment of mean diameters D50=2.80, 5.42 and 7.06 mm in a rigid rectangular channel. The results show that the sediment transport may increase the friction factor by up to 90% and 60% in smooth and rough beds, respectively. Bedload extracts its momentum from the flow, which causes a reduction of near bed flow velocity and steeper velocity gradient near the bed resulting in an increase in shear velocity as well as in roughness height. The increase in friction factor is directly related to bedload concentration and particle size. Key Words: Hydraulic resistance, Friction factor, Bedload transport, Supercritical flow
1 Introduction Sediment transport along rigid channels is of special importance to the design and operation of lined canals in irrigation networks and in channels in hydroelectric plants. The interaction between the moving sediment particles and the flow is not fully understood. It is generally believed that particles moving as bedload extract momentum from the flow, causing a reduction in flow velocity and an apparent increase in bed roughness. The increase in roughness scale is caused by the wake of each individual moving particle. This bedload roughness is proportional to bedload layer thickness (Wiberg and Rubin 1989). Arora et al. (1986) investigated the process of resistance to flow in rigid boundary channels carrying sediment-laden flow. Their results show that the friction factor may increase or decrease as compared to the clear water value depending on factors such as sediment concentration, particle fall velocity, flow mean velocity and energy slope. Stonestreet et al. (1994) studied the resistance of supercritical flow in the presence of sediment load. Their results showed that the bedload movement can increase the Manning friction coefficient from 0.014 to 0.0153 m1/3/s. Abrahams and Li (1998) conducted a series of experiments on the resistance due to saltation load for supercritical sheet flow. They observed that sediment load was mostly as saltation load and this load accounted for 20.8% of friction factor and 89% of bed roughness. Song et al. (1998) studied the effect of bedload on the flow resistance of near-critically flow in pipes and rectangular open channels. The sediment injected to flow was uniform gravel of 12.3 mm diameter. They found that bedload transport may increase the friction factor up to 50%, causing an increase in flow 1 Postgraduate student, Dept. of Irrigation and Reclamation Eng., University of Tehran, Karaj, Iran; Tel./Fax: +98-261-2241119, E-mail: [email protected] 2 Assoc. Prof., Dept. of Irrigation and Reclamation Eng., University of Tehran, Karaj, Iran; E-mail: [email protected] Note: The original manuscript of this paper was received in Sept. 2008. The revised version was received in Nov. 2008. Discussion open until Dec. 2010. - 400 -
International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
depth by 14%. Omid et al. (2003) studied the effect of bedload transport on the flow resistance in a rigid boundary channel. They found that friction factor in sub-critical flow increases with the bedload concentration. Bergeron and Carbonneau (1999) had the same results for bedload resistance. Gao and Abrahams (2004) focused on the resistance effect due to bedload transport of supercritical flow in a rough-bed open channel. The experiments were in the range of fully turbulent rough flow and relative submergence (depth to roughness height ratio) varying from 1 to 20. Sediment particles had diameters between 5 and 7.8 mm. They introduced a bedload friction factor fb and developed an equation relating fb to parameters such as the bedload concentration, the dimensionless particle diameter and the relative submergence. Hu and Abrahams (2004) studied the resistance effects of turbulent overland flow. They found that the friction factor due to bedload movement was 22.06% of total resistance. This friction component is controlled by factors such as concentration, particle diameter, relative submergence, bed slope and Froude number. Hu and Abrahams (2005) compared bedload friction factor in mobile and fixed beds and found that bedload friction factor in mobile beds is more than that in fixed beds. The reason was reported to be the mechanism of momentum absorption from the flow. They related the friction factor due to bedload transport in mobile beds to total friction factor and dimensionless particle diameter. The hydraulic flow resistance is usually considered with the Darcy-Weisbach friction factor f defined as ⎛u ⎞ f = 8⎜ * ⎟ ⎝U ⎠
2
(1)
where u* = τ 0 / ρ is the shear velocity (m/s) in which τ 0 is the bed shear stress, ρ is the water density and U is the mean flow velocity. In sediment-laden flow over alluvial beds, the friction factor may be divided into four components, namely grain resistance, form resistance, sediment transport resistance (bedload and suspended load) and wave resistance (Hu and Abrahams, 2006). In a steady uniform flow the friction factor is a function of the Reynolds number Re = ρUR μ (R is the hydraulic radius and μ is the dynamic viscosity of water), Froude number Fr = U gD (g is the gravitational acceleration and D is the hydraulic depth of flow) and relative roughness ks/R (the ratio of roughness height ks to hydraulic radius R). The Moody (1944) diagram is based on such flow conditions, provided that the Froude number effect is negligible (Yen, 2002). Rouse (1965) observed that if Fr is well in excess of unity, it influences f. He concluded that the gravitational influence on open channel resistance was related to free surface instability involved in the formation of roll waves. When Fr exceeds a certain value Fs (Fr at which instability of free surface occurs) the flow would be non-uniform and unsteady. Under these conditions, if the channel is long enough, roll waves develop (Sarma et al., 1991). The friction factor may be easily calculated if the shear velocity is known. The shear velocity may be predicted using the logarithmic velocity profile and the Saint-Venant method (Rowinski et al., 2005). Graf and Song (1995) studied the bed shear stress in non-uniform and unsteady open channel flows. They concluded that the bed shear stress can be reasonably predicted with one of the methods of logarithmic velocity profile and Saint-Venant method. In the logarithmic velocity profile the bed shear stress is calculated using the velocity profile slope near the bed, where the logarithmic profile is dominant. In the Saint-Venant method the friction slope is related to the water surface slope and calculated by solving gradually varied flow equation. Velocity distribution near the bed can be described with a logarithmic law called “the law of the wall”. This equation is dominant in the inner region of the boundary layer, which contains up to 15-20% of the boundary layer thickness. In this region, the velocity distribution is mostly affected by the bed characteristics. In the outer region flow properties are affected by other parameters, resulting in a reduction in flow velocity and deviation from the logarithmic law. Velocity distribution in the outer region is expressed by “velocity defect law”. The law of the wall for a hydraulically rough bed may be presented as follows u 1 ⎛ y⎞ (2) = ln⎜ ⎟ + B u* κ ⎜⎝ k s ⎟⎠ where u is velocity at a distance y above the smooth bed, κ is Von Karman’s constant, ks is equivalent roughness and B is the integration constant. The value of ks is usually taken as a factor of the median size of the particles, D50 (the diameter for which 50 percent of the particles are finer). International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
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Tominaga and Nezu (1992) found that the log-law distribution of velocity is valid in the case of supercritical flow on steep slopes with κ = 0.41 , but the intercept is somehow different. The friction factor calculated from the velocity profile is a local factor but the one estimated using water surface slope (Saint Venant method) is friction factor related to a reach. Knowing this, the logarithmic profile may be used to find the friction factor and to compare the variation in different conditions. Values of u* then were found from each velocity profile from the slope and intercept of the regression equation obtained by fitting a straight line to the lower points (inner region of the boundary layer) of the profile on a semi-logarithmic scale. In this study the effect of sediment transport on the flow resistance of supercritical flow in smooth and rough beds was experimentally investigated (to extend the results of the previous study by Omid and Habibzadeh, 2007). 2 Experimental setup and procedure The experiments consisted of creating supercritical flow conditions carrying sediment by injecting fine gravels (D50 = 2.8, 5.4, 7.1 mm) into the flow in a rigid rectangular channel. Experiments were performed in a horizontal channel with glass sides and smooth bed of rectangular cross-section 0.50 m wide and 9 m long. The upstream end of the channel was a stilling tank provided with a rectangular sharp-crested weir and an adjustable submerged gate to obtain various flow depths and Froude numbers. A collecting basket was provided at the downstream end of the channel. The weight of sediment collected in the basket was measured to calculate the sediment injection rate. In each test clear water was supplied at a steady rate from the upstream tank. A single layer of uniform sediment with D50 = 2.8, 5.4 and 7.1 mm was glued onto the bed for the rough bed experiments and the same size of sediment was used for sediment injection. Flow velocities were measured using a 6 mm diameter static Pitot tube. The experiments were carried out with clear water over either smooth or rough beds as well as sediment-laden flow over either of the beds. In the first series, a number of preliminary experiments were carried out to determine the friction factor of smooth and rough rigid beds. The flow depth was measured using a series of piezometers installed on the bed of the channel. For the experiments with sediment-laden flow, sediment was injected into the flow at different rates using a roller belt system installed upstream of the sluice gate. The injection rates of sediment were so adjusted for each prescribed flow discharge to be lower than the rate at which initiation of sediment deposition occurs on the bed. For each injection rate I, expressed in terms of mass rate, the corresponding volumetric sediment concentration C is: I (3) C= Qρ s where Q is water discharge and ρ s is sediment density (≈ 2,650 kg/m3). The experimental ranges are: Sediment diameter 2.8, 5.4, 7.1 mm Sediment concentration 0.03% to 0.25% Discharge 26.5 to 46.5 l/s Froude number 1.40 to 8.54
Velocity profiles were measured in three cross sections located at 1, 2 and 3 m from the gate along the centerline of the channel. Starting from 3 mm above the bed, as dedicated by Pitot tube, velocities were measured at vertical intervals of 2 mm up to the water surface. The depth-averaged velocity U in each section was obtained by integrating the velocity profile over the whole depth of flow. The shear velocity u* and the friction factor f were then calculated using Figs. (2) and (1), respectively. The friction factor associated with the bed was calculated using Vanoni-Brooks sidewall effect elimination method. 3 Results Two examples of velocity profiles over smooth and rough beds are shown in Fig. 1. These show the velocity profiles for zero transport (clear water) and for sediment laden flow for two different sediment concentrations. These profiles represent the response of the flow velocity to the sediment movement particularly in the near-bed layers. The average velocity of the particles is smaller than that of the flow - 402 -
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farther from the wall while it is larger than the flow velocity very near the wall. The flow turbulence is also increased by the presence of the particles in the near wall regions and this plays an important role in particle-fluid interactions. The vertical component of the particles turbulence intensity is dominant to the streamwise one, in the near wall region (Nezu and Azuma, 2004). The sediment movement causes a steeper velocity gradient near the bed than clear water resulting in an increase in shear velocity and roughness coefficient. The results also suggest that the reduction in mean flow velocity is accompanied by an increase in flow depth.
Fig. 1 Velocity profiles over a (a) smooth and (b) rough bed
The friction factor f as a function of Re for clear water over a smooth bed is presented in Fig. 2. The theoretical line represents the values of f, calculated from Colebrook-White formula. As is seen, agreement between the two sets of data is good. Based on the experimental data, B=3.2 was found in Eq. (2). The deviation of this intercept from the other equations found in the literature is attributed to the parameters chosen as the roughness height and reference level. Kironoto and Graf (Afzalimehr and Anctil, 1999) suggested that the reference level lies 0.2D50 below the top of the roughness elements; whereas Tu and Graf (Afzalimehr and Anctil, 1999) used 0.25D50. In this study the median sediment diameter, D50 was used as the roughness parameter and the reference level was assumed to be at the top of the elements. The value of 3.2 is comparable to that proposed by Gao and Abrahams (2004) who found B=2.8.
Fig. 2
Friction factor f versus Reynolds number Re in smooth bed channel with clear water
Using the experimental data and combining Figs. (1) and (2), the friction factor due to grain roughness in rough beds for clear water conditions was fitted as (Fig. 3) International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
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⎛ h ⎞ 8 ⎟⎟ + 3.2 = 2.5 ln⎜⎜ f ⎝ d 50 ⎠
(4)
The measured friction factor as a function of relative roughness is shown in Fig. 3. In this figure is also shown friction factor calculated from Eq. (4).
Fig. 3 Friction factor f against relative submergence h/ks for clear water experiments
The variation of the friction factor against the Reynolds number for all the experiments is plotted in Fig. 4. The friction factor of sediment-laden flow on the smooth bed is seen to be less than that due to clear water flow over rough beds. This is different from the results of bedload movement in sub-critical flow reported by Omid et al. (2003). For supercritical flow, small depth of flow associated with the high flow velocity makes the particle size comparable to the flow depth. It means that for small relative submergences, the contribution of the grain roughness to the total friction factor is more than that due to sediment transport.
Fig. 4 Variation of friction factor versus Reynolds number - 404 -
International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
The variation of the friction factor in terms of the Froude number is shown in Fig. 5. As an average, the injection of sediment to the supercritical flow caused 18% reduction in Froude number for the specified discharge (Fig. 6).
Fig. 5 Variation of friction factor versus Froude number for different flow conditions
Friction factor as a function of relative submergence for all experiments is shown in Fig. 7. The figure includes all the data collected for various sediment diameters, sediment concentration and flow rates considered in the experiments. In this figure are also shown calculated values of friction factor from Eq. (4) for clear water condition. It can be seen from the graph that injection of sediment caused a dramatic increase in friction factor. In this figure is also shown the important role of grain roughness in small relative submergence where the friction factor of clear-water flow over rough bed is more than that of sediment-laden flow over smooth bed.
Fig. 6 Reduction of Froude number due to bedload movement International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
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Fig. 7
Variation of friction factor against relative submergence
The effect of sediment concentration on friction factor for smooth and rough beds are demonstrated in Figs. 8 and 9, respectively. The results show that in the range of non-deposit injection rates, the friction factor is increased up to 60% and 90% on the smooth and rough beds, respectively. The results also show that the rate of increase in friction factor due to sediment concentration is nearly the same for smooth and rough rigid beds.
Fig. 8 Effect of sediment concentration on friction factor of smooth bed
The variation of f/fc (fc is clear water friction factor) against bedload concentration is shown in Fig. 10. It can be seen from the figure that f / fc increases with increase in sediment concentration. - 406 -
International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
Fig. 9
Effect of sediment concentration on friction factor of rough bed
Fig. 10
Variation of f / fc versus sediment concentration
The ratio of f f c as a function of C√D* is plotted in Fig. 11 to take into account both the sediment concentration and grain size. In this figure D* is the dimensionless diameter of the particles defined as 13 D* = ⎡⎣( S − 1) g ν 2 ⎤⎦ D50 . This figure also showns the calculated values of f from the equations proposed by Song et al. (1998) for sub-critical flow and Gao and Abrahams (2004) for supercritical flows. 4 Conclusion The effect of the bedload transport on the friction factor in supercritical flow was experimentally investigated in terms of the sediment concentration and the grain size. The study is restricted to steady flow and to bedload of nearly uniform sediment size. The followings are the conclusions: 1) The measured friction factors of supercritical flow on the smooth and rough bed channels fit the Colebrook-White equation, 2) the effects of bedload transport on the flow resistance of supercritical flow are similar to the results of Song et al. (1998), Bergeron and Carbonneau (1999) and Omid et al. (2003) on the effects of bedload movement on sub-critical flow resistance, showing an considerable increase of friction factor due to the movement of sediment particles over the bed, International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
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Fig. 11
Variation of f / fc versus non-dimensional sediment concentration and grain size
3) bedloads in supercritical flow in the range of our experiments show an increase in the friction factor up to 90% and 60% for smooth and rough beds, respectively, causing a reduction in Froude number by 18%, 4) the increase in sediment concentration and/or particle size make a significant increase in the friction factor of supercritical flow in a smooth or rough rigid bed channel, 5) the results can be used to find the characteristics of hydraulic jump with incoming flow of different sediment concentrations. The studies are expected to be useful in the design of stilling basin downstream of sediment sluice gates. Acknowledgements The authors wish to express their gratitude to the University of Tehran for the financial support granted during the whole course of this research. References Abrahams A. D. and Li G. 1998, Effect of saltating sediment on flow resistance and bed roughness in overland flow. Earth Surface Processes and Landforms, Vol. 23, No. 10, pp. 953–960. Afzalimehr H. and Anctil F. 1999, Velocity distribution and shear velocity behaviour of decelerating flows over gravel bed. Canadian Journal of Civil Engineering, Vol. 26, No. 4, pp. 468–475. Arora A. K., Ranga Raju K. G., and Garde R. J. 1986, Resistance to flow and velocity distribution in rigid boundary channels carrying sediment-laden flow. Water Resources Research, Vol. 22, No. 6, pp. 943–951. Bergeron N. E. and Carbonneau P. 1999, The effect of sediment concentration on bedload roughness. Hydrological Processes, Vol. 13, No. 16, pp. 2583–2589. Chow V. T. 1956, Open Channel Hydraulics. McGraw-Hill Kogakusha Ltd., International Student Edition. Gao P. and Abrahams A. D. 2004, Bedload transport resistance in rough open channel flows. Earth Surface Processes and Landforms, 29, pp. 423–435. Graf W.H. and Song T. 1995, Bed-shear stress in non-uniform and unsteady open-channel flows. Journal of Hydraulic Research, Vol. 33, No. 5, pp. 699–704. Hu S. and Abrahams A. D. 2006, Partitioning resistance to overland flow on rough mobile beds. Earth Surface Processes and Landforms, Vol. 31, No. 10, pp. 1280–1291. Hu S. and Abrahams A. D. 2005, The effect of bed mobility on resistance to overland flow. Earth Surface Processes and Landforms, Vol. 30, No. 11, pp. 1461–1470. Hu S. and Abrahams A. D. 2004, Resistance to overland flow due to bed load transport on plane mobile beds. Earth Surface Processes and Landforms, Vol. 29, No. 13, pp. 1691–1701. Moody L. F. 1944, Friction factors for pipe flow. Transactions of the ASME, Vol. 66.
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Nezu, I. and Azuma, R. 2004, Turbulence characteristics and interaction between particles and fluid in particle-laden open channel flows. Journal of Hydraulic Engineering, ASCE, Vol. 130, No. 10, pp. 988–1001. Omid M. H. and Habibzadeh A. 2007, Effect of bedload movement on supercritical flow resistance in rigid boundary channels. Proceeding CD, 32nd Congress of IAHR, Venice, Italy, July 1-6. Omid M. H., Mahdavi A., and Narayanan R. 2003, Effect of bedload transport on flow resistance in rigid boundary channels. Proceeding, 30th IAHR Congress, Thessaloniki, Greece, Aug. 24-29. Rouse H. 1965, Critical analysis of open-channel resistance. Journal of the Hydraulic Division, ASCE, Vol. 91, No. HY4, pp. 1–25. Rowinski P. M., Aberle J., and Mazurczyk A. 2005, Shear velocity estimation in hydraulic research. Acta Geophysica Polonica, Vol. 53, No. 4, pp. 567–583. Sarma K. V. N. and Syamala. 1991, Supercritical flow in smooth open channels. Journal of Hydraulic Engineering, ASCE, Vol. 117, No. 1, pp. 54–63. Song T., Chiew Y. M., and Chin, C. O. 1998, Effect of bed-load movement on flow friction factor. Journal of Hydraulic Engineering, ASCE, Vol. 124, No. 2, pp. 165–175. Stonestreet S. E., Copeland R. R., and Mcvan D. C. 1994, Bed load roughness in supercritical flow. Proceeding ASCE Hydraulic Engineering Conference, pp. 747–751. Tominaga A. and Nezu I. 1992, Velocity profiles in steep open channel flows. Journal of Hydraulic Engineering, ASCE, Vol. 118, No. 1, pp. 73–90. Wiberg P. L. and Rubin D. M. 1989, Bed roughness produced by saltating sediment. Journal of Geophysical Research, 94, pp. 5011–5016. Yen B.C. 2002, Open channel flow resistance. Journal of Hydraulic Engineering, ASCE, Vol. 128, No. 1, pp. 20–39.
International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
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Bedload resistance in supercritical flow Alireza HABIBZADEH1 and Mohammad Hossein OMID2
Abstract The movement of bedload in subcritical flow produces additional roughness as compared to flow in a rigid bed. The magnitude of this bed load roughness is proportional to the thickness of the sediment layer moving along the bed, the particle size and the sediment concentration. In a supercritical flow, however, further resistance is expected due to the momentum absorption by the high flow velocity. In this study the effect of sediment movement on the flow resistance in supercritical flow was experimentally investigated. The experiments included flows over smooth and rough beds carrying sediment of mean diameters D50=2.80, 5.42 and 7.06 mm in a rigid rectangular channel. The results show that the sediment transport may increase the friction factor by up to 90% and 60% in smooth and rough beds, respectively. Bedload extracts its momentum from the flow, which causes a reduction of near bed flow velocity and steeper velocity gradient near the bed resulting in an increase in shear velocity as well as in roughness height. The increase in friction factor is directly related to bedload concentration and particle size. Key Words: Hydraulic resistance, Friction factor, Bedload transport, Supercritical flow
1 Introduction Sediment transport along rigid channels is of special importance to the design and operation of lined canals in irrigation networks and in channels in hydroelectric plants. The interaction between the moving sediment particles and the flow is not fully understood. It is generally believed that particles moving as bedload extract momentum from the flow, causing a reduction in flow velocity and an apparent increase in bed roughness. The increase in roughness scale is caused by the wake of each individual moving particle. This bedload roughness is proportional to bedload layer thickness (Wiberg and Rubin 1989). Arora et al. (1986) investigated the process of resistance to flow in rigid boundary channels carrying sediment-laden flow. Their results show that the friction factor may increase or decrease as compared to the clear water value depending on factors such as sediment concentration, particle fall velocity, flow mean velocity and energy slope. Stonestreet et al. (1994) studied the resistance of supercritical flow in the presence of sediment load. Their results showed that the bedload movement can increase the Manning friction coefficient from 0.014 to 0.0153 m1/3/s. Abrahams and Li (1998) conducted a series of experiments on the resistance due to saltation load for supercritical sheet flow. They observed that sediment load was mostly as saltation load and this load accounted for 20.8% of friction factor and 89% of bed roughness. Song et al. (1998) studied the effect of bedload on the flow resistance of near-critically flow in pipes and rectangular open channels. The sediment injected to flow was uniform gravel of 12.3 mm diameter. They found that bedload transport may increase the friction factor up to 50%, causing an increase in flow 1 Postgraduate student, Dept. of Irrigation and Reclamation Eng., University of Tehran, Karaj, Iran; Tel./Fax: +98-261-2241119, E-mail: [email protected] 2 Assoc. Prof., Dept. of Irrigation and Reclamation Eng., University of Tehran, Karaj, Iran; E-mail: [email protected] Note: The original manuscript of this paper was received in Sept. 2008. The revised version was received in Nov. 2008. Discussion open until Dec. 2010. - 400 -
International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
depth by 14%. Omid et al. (2003) studied the effect of bedload transport on the flow resistance in a rigid boundary channel. They found that friction factor in sub-critical flow increases with the bedload concentration. Bergeron and Carbonneau (1999) had the same results for bedload resistance. Gao and Abrahams (2004) focused on the resistance effect due to bedload transport of supercritical flow in a rough-bed open channel. The experiments were in the range of fully turbulent rough flow and relative submergence (depth to roughness height ratio) varying from 1 to 20. Sediment particles had diameters between 5 and 7.8 mm. They introduced a bedload friction factor fb and developed an equation relating fb to parameters such as the bedload concentration, the dimensionless particle diameter and the relative submergence. Hu and Abrahams (2004) studied the resistance effects of turbulent overland flow. They found that the friction factor due to bedload movement was 22.06% of total resistance. This friction component is controlled by factors such as concentration, particle diameter, relative submergence, bed slope and Froude number. Hu and Abrahams (2005) compared bedload friction factor in mobile and fixed beds and found that bedload friction factor in mobile beds is more than that in fixed beds. The reason was reported to be the mechanism of momentum absorption from the flow. They related the friction factor due to bedload transport in mobile beds to total friction factor and dimensionless particle diameter. The hydraulic flow resistance is usually considered with the Darcy-Weisbach friction factor f defined as ⎛u ⎞ f = 8⎜ * ⎟ ⎝U ⎠
2
(1)
where u* = τ 0 / ρ is the shear velocity (m/s) in which τ 0 is the bed shear stress, ρ is the water density and U is the mean flow velocity. In sediment-laden flow over alluvial beds, the friction factor may be divided into four components, namely grain resistance, form resistance, sediment transport resistance (bedload and suspended load) and wave resistance (Hu and Abrahams, 2006). In a steady uniform flow the friction factor is a function of the Reynolds number Re = ρUR μ (R is the hydraulic radius and μ is the dynamic viscosity of water), Froude number Fr = U gD (g is the gravitational acceleration and D is the hydraulic depth of flow) and relative roughness ks/R (the ratio of roughness height ks to hydraulic radius R). The Moody (1944) diagram is based on such flow conditions, provided that the Froude number effect is negligible (Yen, 2002). Rouse (1965) observed that if Fr is well in excess of unity, it influences f. He concluded that the gravitational influence on open channel resistance was related to free surface instability involved in the formation of roll waves. When Fr exceeds a certain value Fs (Fr at which instability of free surface occurs) the flow would be non-uniform and unsteady. Under these conditions, if the channel is long enough, roll waves develop (Sarma et al., 1991). The friction factor may be easily calculated if the shear velocity is known. The shear velocity may be predicted using the logarithmic velocity profile and the Saint-Venant method (Rowinski et al., 2005). Graf and Song (1995) studied the bed shear stress in non-uniform and unsteady open channel flows. They concluded that the bed shear stress can be reasonably predicted with one of the methods of logarithmic velocity profile and Saint-Venant method. In the logarithmic velocity profile the bed shear stress is calculated using the velocity profile slope near the bed, where the logarithmic profile is dominant. In the Saint-Venant method the friction slope is related to the water surface slope and calculated by solving gradually varied flow equation. Velocity distribution near the bed can be described with a logarithmic law called “the law of the wall”. This equation is dominant in the inner region of the boundary layer, which contains up to 15-20% of the boundary layer thickness. In this region, the velocity distribution is mostly affected by the bed characteristics. In the outer region flow properties are affected by other parameters, resulting in a reduction in flow velocity and deviation from the logarithmic law. Velocity distribution in the outer region is expressed by “velocity defect law”. The law of the wall for a hydraulically rough bed may be presented as follows u 1 ⎛ y⎞ (2) = ln⎜ ⎟ + B u* κ ⎜⎝ k s ⎟⎠ where u is velocity at a distance y above the smooth bed, κ is Von Karman’s constant, ks is equivalent roughness and B is the integration constant. The value of ks is usually taken as a factor of the median size of the particles, D50 (the diameter for which 50 percent of the particles are finer). International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
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Tominaga and Nezu (1992) found that the log-law distribution of velocity is valid in the case of supercritical flow on steep slopes with κ = 0.41 , but the intercept is somehow different. The friction factor calculated from the velocity profile is a local factor but the one estimated using water surface slope (Saint Venant method) is friction factor related to a reach. Knowing this, the logarithmic profile may be used to find the friction factor and to compare the variation in different conditions. Values of u* then were found from each velocity profile from the slope and intercept of the regression equation obtained by fitting a straight line to the lower points (inner region of the boundary layer) of the profile on a semi-logarithmic scale. In this study the effect of sediment transport on the flow resistance of supercritical flow in smooth and rough beds was experimentally investigated (to extend the results of the previous study by Omid and Habibzadeh, 2007). 2 Experimental setup and procedure The experiments consisted of creating supercritical flow conditions carrying sediment by injecting fine gravels (D50 = 2.8, 5.4, 7.1 mm) into the flow in a rigid rectangular channel. Experiments were performed in a horizontal channel with glass sides and smooth bed of rectangular cross-section 0.50 m wide and 9 m long. The upstream end of the channel was a stilling tank provided with a rectangular sharp-crested weir and an adjustable submerged gate to obtain various flow depths and Froude numbers. A collecting basket was provided at the downstream end of the channel. The weight of sediment collected in the basket was measured to calculate the sediment injection rate. In each test clear water was supplied at a steady rate from the upstream tank. A single layer of uniform sediment with D50 = 2.8, 5.4 and 7.1 mm was glued onto the bed for the rough bed experiments and the same size of sediment was used for sediment injection. Flow velocities were measured using a 6 mm diameter static Pitot tube. The experiments were carried out with clear water over either smooth or rough beds as well as sediment-laden flow over either of the beds. In the first series, a number of preliminary experiments were carried out to determine the friction factor of smooth and rough rigid beds. The flow depth was measured using a series of piezometers installed on the bed of the channel. For the experiments with sediment-laden flow, sediment was injected into the flow at different rates using a roller belt system installed upstream of the sluice gate. The injection rates of sediment were so adjusted for each prescribed flow discharge to be lower than the rate at which initiation of sediment deposition occurs on the bed. For each injection rate I, expressed in terms of mass rate, the corresponding volumetric sediment concentration C is: I (3) C= Qρ s where Q is water discharge and ρ s is sediment density (≈ 2,650 kg/m3). The experimental ranges are: Sediment diameter 2.8, 5.4, 7.1 mm Sediment concentration 0.03% to 0.25% Discharge 26.5 to 46.5 l/s Froude number 1.40 to 8.54
Velocity profiles were measured in three cross sections located at 1, 2 and 3 m from the gate along the centerline of the channel. Starting from 3 mm above the bed, as dedicated by Pitot tube, velocities were measured at vertical intervals of 2 mm up to the water surface. The depth-averaged velocity U in each section was obtained by integrating the velocity profile over the whole depth of flow. The shear velocity u* and the friction factor f were then calculated using Figs. (2) and (1), respectively. The friction factor associated with the bed was calculated using Vanoni-Brooks sidewall effect elimination method. 3 Results Two examples of velocity profiles over smooth and rough beds are shown in Fig. 1. These show the velocity profiles for zero transport (clear water) and for sediment laden flow for two different sediment concentrations. These profiles represent the response of the flow velocity to the sediment movement particularly in the near-bed layers. The average velocity of the particles is smaller than that of the flow - 402 -
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farther from the wall while it is larger than the flow velocity very near the wall. The flow turbulence is also increased by the presence of the particles in the near wall regions and this plays an important role in particle-fluid interactions. The vertical component of the particles turbulence intensity is dominant to the streamwise one, in the near wall region (Nezu and Azuma, 2004). The sediment movement causes a steeper velocity gradient near the bed than clear water resulting in an increase in shear velocity and roughness coefficient. The results also suggest that the reduction in mean flow velocity is accompanied by an increase in flow depth.
Fig. 1 Velocity profiles over a (a) smooth and (b) rough bed
The friction factor f as a function of Re for clear water over a smooth bed is presented in Fig. 2. The theoretical line represents the values of f, calculated from Colebrook-White formula. As is seen, agreement between the two sets of data is good. Based on the experimental data, B=3.2 was found in Eq. (2). The deviation of this intercept from the other equations found in the literature is attributed to the parameters chosen as the roughness height and reference level. Kironoto and Graf (Afzalimehr and Anctil, 1999) suggested that the reference level lies 0.2D50 below the top of the roughness elements; whereas Tu and Graf (Afzalimehr and Anctil, 1999) used 0.25D50. In this study the median sediment diameter, D50 was used as the roughness parameter and the reference level was assumed to be at the top of the elements. The value of 3.2 is comparable to that proposed by Gao and Abrahams (2004) who found B=2.8.
Fig. 2
Friction factor f versus Reynolds number Re in smooth bed channel with clear water
Using the experimental data and combining Figs. (1) and (2), the friction factor due to grain roughness in rough beds for clear water conditions was fitted as (Fig. 3) International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
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⎛ h ⎞ 8 ⎟⎟ + 3.2 = 2.5 ln⎜⎜ f ⎝ d 50 ⎠
(4)
The measured friction factor as a function of relative roughness is shown in Fig. 3. In this figure is also shown friction factor calculated from Eq. (4).
Fig. 3 Friction factor f against relative submergence h/ks for clear water experiments
The variation of the friction factor against the Reynolds number for all the experiments is plotted in Fig. 4. The friction factor of sediment-laden flow on the smooth bed is seen to be less than that due to clear water flow over rough beds. This is different from the results of bedload movement in sub-critical flow reported by Omid et al. (2003). For supercritical flow, small depth of flow associated with the high flow velocity makes the particle size comparable to the flow depth. It means that for small relative submergences, the contribution of the grain roughness to the total friction factor is more than that due to sediment transport.
Fig. 4 Variation of friction factor versus Reynolds number - 404 -
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The variation of the friction factor in terms of the Froude number is shown in Fig. 5. As an average, the injection of sediment to the supercritical flow caused 18% reduction in Froude number for the specified discharge (Fig. 6).
Fig. 5 Variation of friction factor versus Froude number for different flow conditions
Friction factor as a function of relative submergence for all experiments is shown in Fig. 7. The figure includes all the data collected for various sediment diameters, sediment concentration and flow rates considered in the experiments. In this figure are also shown calculated values of friction factor from Eq. (4) for clear water condition. It can be seen from the graph that injection of sediment caused a dramatic increase in friction factor. In this figure is also shown the important role of grain roughness in small relative submergence where the friction factor of clear-water flow over rough bed is more than that of sediment-laden flow over smooth bed.
Fig. 6 Reduction of Froude number due to bedload movement International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
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Fig. 7
Variation of friction factor against relative submergence
The effect of sediment concentration on friction factor for smooth and rough beds are demonstrated in Figs. 8 and 9, respectively. The results show that in the range of non-deposit injection rates, the friction factor is increased up to 60% and 90% on the smooth and rough beds, respectively. The results also show that the rate of increase in friction factor due to sediment concentration is nearly the same for smooth and rough rigid beds.
Fig. 8 Effect of sediment concentration on friction factor of smooth bed
The variation of f/fc (fc is clear water friction factor) against bedload concentration is shown in Fig. 10. It can be seen from the figure that f / fc increases with increase in sediment concentration. - 406 -
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Fig. 9
Effect of sediment concentration on friction factor of rough bed
Fig. 10
Variation of f / fc versus sediment concentration
The ratio of f f c as a function of C√D* is plotted in Fig. 11 to take into account both the sediment concentration and grain size. In this figure D* is the dimensionless diameter of the particles defined as 13 D* = ⎡⎣( S − 1) g ν 2 ⎤⎦ D50 . This figure also showns the calculated values of f from the equations proposed by Song et al. (1998) for sub-critical flow and Gao and Abrahams (2004) for supercritical flows. 4 Conclusion The effect of the bedload transport on the friction factor in supercritical flow was experimentally investigated in terms of the sediment concentration and the grain size. The study is restricted to steady flow and to bedload of nearly uniform sediment size. The followings are the conclusions: 1) The measured friction factors of supercritical flow on the smooth and rough bed channels fit the Colebrook-White equation, 2) the effects of bedload transport on the flow resistance of supercritical flow are similar to the results of Song et al. (1998), Bergeron and Carbonneau (1999) and Omid et al. (2003) on the effects of bedload movement on sub-critical flow resistance, showing an considerable increase of friction factor due to the movement of sediment particles over the bed, International Journal of Sediment Research, Vol. 24, No. 4, 2009, pp. 400–409
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Fig. 11
Variation of f / fc versus non-dimensional sediment concentration and grain size
3) bedloads in supercritical flow in the range of our experiments show an increase in the friction factor up to 90% and 60% for smooth and rough beds, respectively, causing a reduction in Froude number by 18%, 4) the increase in sediment concentration and/or particle size make a significant increase in the friction factor of supercritical flow in a smooth or rough rigid bed channel, 5) the results can be used to find the characteristics of hydraulic jump with incoming flow of different sediment concentrations. The studies are expected to be useful in the design of stilling basin downstream of sediment sluice gates. Acknowledgements The authors wish to express their gratitude to the University of Tehran for the financial support granted during the whole course of this research. References Abrahams A. D. and Li G. 1998, Effect of saltating sediment on flow resistance and bed roughness in overland flow. Earth Surface Processes and Landforms, Vol. 23, No. 10, pp. 953–960. Afzalimehr H. and Anctil F. 1999, Velocity distribution and shear velocity behaviour of decelerating flows over gravel bed. Canadian Journal of Civil Engineering, Vol. 26, No. 4, pp. 468–475. Arora A. K., Ranga Raju K. G., and Garde R. J. 1986, Resistance to flow and velocity distribution in rigid boundary channels carrying sediment-laden flow. Water Resources Research, Vol. 22, No. 6, pp. 943–951. Bergeron N. E. and Carbonneau P. 1999, The effect of sediment concentration on bedload roughness. Hydrological Processes, Vol. 13, No. 16, pp. 2583–2589. Chow V. T. 1956, Open Channel Hydraulics. McGraw-Hill Kogakusha Ltd., International Student Edition. Gao P. and Abrahams A. D. 2004, Bedload transport resistance in rough open channel flows. Earth Surface Processes and Landforms, 29, pp. 423–435. Graf W.H. and Song T. 1995, Bed-shear stress in non-uniform and unsteady open-channel flows. Journal of Hydraulic Research, Vol. 33, No. 5, pp. 699–704. Hu S. and Abrahams A. D. 2006, Partitioning resistance to overland flow on rough mobile beds. Earth Surface Processes and Landforms, Vol. 31, No. 10, pp. 1280–1291. Hu S. and Abrahams A. D. 2005, The effect of bed mobility on resistance to overland flow. Earth Surface Processes and Landforms, Vol. 30, No. 11, pp. 1461–1470. Hu S. and Abrahams A. D. 2004, Resistance to overland flow due to bed load transport on plane mobile beds. Earth Surface Processes and Landforms, Vol. 29, No. 13, pp. 1691–1701. Moody L. F. 1944, Friction factors for pipe flow. Transactions of the ASME, Vol. 66.
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