The Von-Kármán coefficient in sediment laden flow

This article was downloaded by: [The University of Manchester Library] On: 04 December 2014, At: 06:38 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Hydraulic Research Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tjhr20 The Von-Kármán coefficient in sediment laden flow M. Nouh a a College of Engineering , King Saud University , P.O. Box 70178, Riyadh-Diriyah, 11567, Kingdom of Saudi Arabia Published online: 19 Jan 2010. To cite this article: M. Nouh (1989) The Von-Kármán coefficient in sediment laden flow, Journal of Hydraulic Research, 27:4, 477-499, DOI: 10.1080/00221688909499125 To link to this article: http://dx.doi.org/10.1080/00221688909499125 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. 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Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/ terms-and-conditions http://www.tandfonline.com/loi/tjhr20 http://www.tandfonline.com/action/showCitFormats?doi=10.1080/00221688909499125 http://dx.doi.org/10.1080/00221688909499125 http://www.tandfonline.com/page/terms-and-conditions http://www.tandfonline.com/page/terms-and-conditions The Von-Karman coefficient in sediment laden flow Constante de Von-Karman pour les écoulements chargés en sediments SUMMARY The paper presents the results of an experimental investigation on the behaviour of Von-Karman coefficient in sediment laden flow in straight and in curved reaches of smooth rectangular rigid boundary open channels. The investigation shows the effect of varying the Reynolds number of sediment laden flow, the suspended sediment concentration in the flow, the radius-breadth ratio of the channel, and the central angle of the bend. RÉSUMÉ L'article présente les résultats d'une recherche expérimentale sur Ie comportement de la "constante de Von- Karman" pour des écoulements chargés en sediments dans Ie cas d'écoulements a surface libre dans des canaux rectilignes ou en courbe, a paroi lisses non mobiles. L'étude montre l'influence de la variation des paramètres suivants: nombre de Reynolds de l'écoulement chargé de sediments, concentration de l'écoule- ment en matières en suspension, rapport rayon hydraulique sur largeur du canal et angle de rotation de la fibre moyenne du canal en courbe. KEY WORDS Von-Karman coefficient; Sediment laden flow; Sediment transport; Open channel flow; Curved channels; Spiral motion; Velocity distribution; Concentration of suspended sediment; Reynolds number. Introduction The Von-Karman coefficient in sediment laden flow is of importance in the field of river engi­ neering. It is mostly used in Prandtl's logarithmic law to evaluate the vertical velocity profile of the flow. Results of measurements in laboratory straight flumes and in river straight reaches reported by Vanoni [1], Einstein and Chien [3], Graf [4], Hino [6], and Zagustin et al. [11] indicate that the Von-Karman coefficient decreases with the increase of suspended sediment concentra­ tion in an open channel flow. Other investigations made by Imamoto et al. [7] show that this coefficient increases with the increase of suspended sediment concentration in the flow. On the other hand, such variation of the Von-Karman coefficient with suspended sediment concentra­ tion has not been found by Gust [5]. He concluded that the coefficient is independent on the amount of suspended sediment in an open channel flow. This conclusion is supported by Coleman [2] and Vetter [10] who explained the coefficient variation with suspended sediment concentration, found in earlier studies, due to errors induced by ignoring the existence of the wake flow region. Revision received December 5, 1988, Open for discussion till March 31, 1990. JOURNAL OF HYDRAULIC RESEARCH, VOL. 27. 1989. NO. 4 477 M. NOUH Prof, of Civil Engineering, College of Engineering, King Saud University, P.O. Box 70178, Riyadh-Diriyah 11567, Kingdom of Saudi Arabia Present address: Dept. of Civil Engineering, College of Engineering, Sultan Qaboos University, P.O. Box 32483, Al-khod, Muscat, Sultanate of Oman D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 While these studies, among others, contribute significantly towards better understanding of the mechanics and dynamics of open channel flow with sediment transport it must be emphasized that they are only applicable to low velocity flows with low concentrations of suspended sedi­ ments in straight open channels. They do not extend to include high velocity flows that contain a lot of sediments in suspension, or to include flows in curved channels where spiral motion represents a significant flow feature. This study is an initial trial to fill such gap of knowledge. Its main objective is to present the results of some experiments designed to investigate: 1. the effect of varying the Reynolds number of sediment laden flow in straight and in curved open channels on the value of Von-Karman coefficient in the flow; 2. the effect of varying the amount of suspended sediments in a straight and in a curved open channel flow on the value of the Von-Karman coefficient in the flow; 3. the variation of the Von-Karman coefficient in a sediment laden flow in a curved channel with variation of the radius-breadth ratio and with variation of central bend angle of the channel. The investigations are based on measurements of suspended sediment and flow velocity profiles in laboratory flume channels designed for this purpose. Test facility and experimental procedures The laboratory investigations were performed on a rigid 17.0 m long recirculating plexiglass flume having a rectangular cross-section 0.40 m wide by 0.40 m deep. These particular dimen­ sions of the flume were found to be best suited the space available in the laboratory. Because the intend was to determine the Von-Karman coefficient from measurements of flow velocities, depth of flow in the flume was selected to be as large as possible (in the range between 0.275 m and 0.295 m) to suit the measurements at several points along the depth. The flume was supported 1.0 m above the laboratory floor on an adjustable-sloping frame so that the flume slope could be adjusted to maintain uniform flow. It consisted of straight and curved sections that could be assembled to give a desired alignment. Four curved sections having mean radius-breadth ratios, rcjb, of 0.5,1.0,2.0 and 3.0 with a fixed central angle of 90° were used in four series of experimental runs. Each series of the experimental runs was performed on one of the curved sections that was preceded and followed by straight tangents 12.0 m and 3.0 m long, respectively; there were no other bends in the system. The 12.0 m long straight channel reach, upstream from the bend section, was sufficiently long to promote good entrance conditions to the latter. Flow entered the head tank via two circular orifices located in the base of the tank and ap­ proached the entrance to the channel through a smooth transition curved section. An adjustable- height overflow pipe, located on the axis of the tank, provided additional control over the head generated therein. A gated control section was provided at the downstream end of the channel to regulate the depth of flow in the channel. Electronic point gauges located at approximately 4.0 m intervals along the flume centerline provided a mean to check the depth of flow and water surface slope during the experimental runs. The flow was driven by a propeller pump, and discharge was measured by a calibrated venturi meter in the flume return pipeline. The measurements of suspended sediment and flow velocity profiles were taken at fixed locations in the four experimental series of runs. These locations were at a cross-section 6.0 m upstream from the bend; i.e. in the straight section preceding the bend, and at cross-sections of central angle 0°, 5°, 10°, 15°, 20°, 30°, 40°, 50°, 60°, 70°, 80°, and 90° from the bend entrance. At 478 JOURNAL DE RECHERCHES HYDRAULIQUES. VOL. 27. 1989. NO. 4 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 each of these cross-sections the measuring points were selected at 5, 10,15,20, 25, 30,35, 40,45, 50, 60, 75, 100, 125, 150, 175, 200, 220,240, 255, and 265 mm above the flume bottom centerline. A Preston tube of 6.35 mm along with a static pressure probe placed above it were used to measure bed shear stresses at the selected locations on the flume centerline. The Preston tube was calibrated at every cross-section selected for measurements. Calibration curves were developed to relate bed shear stress at the flume centerline to the pressure difference between Preston tube and pressure probe for mean concentration of suspended sediment, C, median diameter of suspended sediment, Z)50, radius-breadth ratio, rcjb, and central angle of the bend as parameters. A Pitot sphere was used to measure velocity components in the three planes. It is 7-mm diameter, and contains five 1.8 mm diameter stainless steel tubes, connected to pressure manometers at the top ends and to a 10 mm diameter sphere at the bottom ends. Detailed description of this Pitot sphere is given by Shukry [9]. The differential pressure across the Pitot sphere was read by means of differential manometers inclined to an angle of 15 degrees from the horizontal to magnify the reading. The Pitot sphere was calibrated. Calibration curves were developed to relate the angle of inclination of a point velocity vector of flow, with suspended sediment of a specified mean concentration and a specified size, with the dynamic coefficients and with the inclination factor (defined as a function between any angle of inclination and its corresponding manometer heads in three directions) at the point. These curves provided an easy mean to determine the angle of inclination of a velocity vector from the measurements of the corresponding manometric heads. The procedure utilised for constructing these curves is exactly the same as that previously reported by Shukry [9]. In the calibration process, the Pitot sphere was allowed to travel with uniform speeds, ranging between 0.10 m/s and 3.0 m/s, in a stillwater 120 m long channel and the dynamic heads were recorded during the motion. Angles of inclination ranging between ± 60°, mean concentrations of suspended sediment ranging between 0 and 60 gpl, and size distributions of suspended sediment having median diameter ranging between 0.03 mm and 0.24 mm were considered. A system consisting of a pump and regulating valves was added to the Pitot sphere in order to collect samples of suspended sediments at measuring points at velocities equal to observed local velocities; an idea reported by Coleman [2], among others. Provision was made for installing the Preston tube temporarily in place of the Pitot sphere for measuring the bed shear stress at the flume centerline. A thermometer was placed 2.0 m from the flume entrance to measure water temperatures. For each series of the experimental runs, the flume was set to a desired slope and clear water at a predetermined rate was allowed into the flume. Uniformity of flow approaching the center- line of measuring cross-sections was ensured by adjusting the depth of flow constant near the entrance and exit reaches of the flume. Flow velocity profiles and bed shear stresses at the standard locations were then measured. After the clear water experiment, a 12.0 kg increment of sand of a predetermined size distribution was injected very slowly to the flow ensuring that the sand was completely in suspension and no deposition was allowed. The depth of flow and water surface slope were checked using the electronic point gauges and the flume slope and discharge were adjusted as needed to maintain uniform flow. Suspended sediment and flow velocity profiles as well as bed shear stresses were then measured at the standard locations. Another flow rate was selected and the above procedure was repeated. Then, the above whole sequence of experimentation was repeated with another increment of sand being added each time. The experimental series was terminated with a experiment in which mean suspended sediment JOURNAL OF HYDRAULIC' RESEARCH, VOL. 27. 1989. NO. 4 479 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 concentration was about 60 gpl. In such concentration no stationary sand was observed anywhere along the flume. Then, the same experimentation was repeated using sand of other size distribu­ tion. Four sands of different size distributions were used in the experiments. Table 1 shows the main characteristics of the sand. In addition, the range of data collected during the experiments is given in Table 2. Table 1. Main characteristics of sands used in the experiments sand SI S2 S3 S4 Table 2. name of experi­ mental series of runs EXR1 EXR2 EXR3 EXR4 relative density 2.65 2.65 2.65 2.65 Range of data collected during the radius- breadth of bend rjb 0.50 1.0 2.0 3.0 sand SI S2 S3 S4 SI S2 S3 S4 SI S2 S3 S4 SI S2 S3 S4 num­ ber of runs 49 42 42 38 41 43 38 36 37 36 36 29 38 35 36 36 discharge x l0~ 2 Q (m3/s) 5.06-27.03 4.86-28.11 4.94-29.05 5.11-28.17 4.94-29.56 5.06-27.93 5.11-28.41 5.06-28.20 4.88-28.46 4.93-29.38 4.88-28.47 4.98-29.14 4.89-29.48 4.92-28.17 5.21-29.63 5.07-29.05 diameter median £>50 (mm) 0.030 0.060 0.120 0.240 experiments total depth of flow at channel center- line yj (m) 0.280-0.295 0.275-0.280 0.280-0.288 0.280-0.291 0.280-0.288 0.275-0.281 0.280-0.293 0.280-0.295 0.280-0.283 0.275-0.276 0.280-0.283 0.275-0.281 0.280-0.286 0.280-0.287 0.280-0.287 0.275-0.278 maximum Dm IX (mm) 0.040 0.080 0.160 0.320 average velocity v% (m/s) 0.53-2.49 0.49-2.53 0.51-2.42 0.49-2.51 0.53-2.49 0.49-2.42 0.48-2.38 0.49-2.49 0.48-2.63 0.49-2.57 0.48-2.61 0.49-2.68 0.48-2.41 0.49-2.51 0.53-2.64 0.49-2.61 energy slope x l0~ 3 sE 1.70-41.6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 80-42.7 80-42.3 70-41.9 80-42.1 70-40.4 80-43.6 80-42.1 70-45.2 80-41.9 80-42.0 80-40.0 70-41.9 80-42.6 80-45.0 70-41.9 mean sediment concen­ tration C (gpl) 0.00-66.2 0.00-64.8 0.00-62.8 0.00-58.3 0.00-69.6 0.00-66.8 0.00-60.2 0.00-57.4 0.00-63.6 0.00-63.0 0.00-60.2 0.00-57.3 0.00-71.5 0.00-68.1 0.00-63.9 0.00-56.8 minimum *^min (mm) 0.015 0.030 0.060 0.120 temper­ ature T (°C) 24.6-25.4 24.5-25.2 24.1-24.9 24.0-24.9 24.1-24.8 24.1-24.6 24.0-25.2 23.9-25.1 24.3-24.9 24.0-24.8 23.7-24.6 .4-25.3 23.5-24.7 23.6-24.6 24.4-25.1 24.0-24.9 Method of analysis As it has been mentioned earlier, the behaviour of the Von-Karman coefficient, x, with variations of the Reynolds number, R, and suspended sediment concentrations in sediment laden flow was achieved through measurements of suspended sediment and flow velocity profiles in laboratory flumes of different alignment (see Table 2). The flow velocity measurements near the flume bottom; i.e. within the zone of yjZ flume bottom centerline, and Z is the boundary layer thickness (defined as the depth of flow from the flume bottom centerline to the point of maximum velocity) of the respective flow in the flume, were used to evaluate x from the transformed logarithmic velocity law as X = 2.30/7, (lOg [ j ^ D / k y , " «ry2) Ü) where «ryi and i/ry2 are the mean longitudinal velocities rectified to be in the direction of bed shear stress at points of depth yx and y2, respectively, and U, is the shear velocity of the flow. The shear velocity at the flume centerline was evaluated from the following relationship U, = [gyj(SE-Sw)]05 (2) where g is the gravitational acceleration, yT is the total depth of flow, SE is the energy gradient of flow, and 5W is the energy loss due to the flume channel walls and the channel bed away from the centerline. For each cross-section selected for measurements, curves were developed to relate Sw toyT for Q, SE, and rc/b as parameters; an idea introduced by Coleman [2]. In these curves, the values of 5W were determined using Preston tube measurements of true centerline bed shear stress rc as s w = sn - Tc/eg>-T (3) where Q is the density of flow; defined at a depth y above the channel bottom centerline as ey = ew + (QS-QJCY (4) where QW and gs and the water and sediment densities, respectively, and Cy is the volumetric suspended sediment concentration at the point. The curves provided as easy way to evaluate Sw during the experiments. To evaluate z o n a flow from measurements of the velocity profile at the center of a specified cross-section in the flumes, the following procedure was utilised. The value of rc\b and the measurements of Q and SE were used to identify the appropriate curve from which 5W was estimated for the measured value ofyT. Then, equation (2) was used to compute the value of U, in the flow. The percentage error, defined as the percentage ratio of the absolute difference between U, estimated from the curves and U, determined from direct Preston tube measurements to the U, determined from the Preston tube measurements, was in the order of 2% in the straight section and 5% in the bend sections. The resultant velocities measured at the standard depths were projected on the rc-plane; the direction of the resultant velocity as close as possible to the boundary and was determined during the Preston tube measurements. The projected velocities are called the rectified longitudinal velocities at the cross-section. Then, a semilogarithmic graph of the rectified velocity profile (i.e. log y/yT against «ry; the rectified velocity at depth y) was constructed and its best fitted slope within the zone y\Z sediment concentration Cz at the top of the boundary layer was determined by reading the plot at y = Z, and the bed concentration Q was determined by extrapolating the plot to y = 0. The Reynolds number, R, within the boundary layer was computed as R = UrZlv (6) where Ur is the rectified mean longitudinal velocity within the boundary layer, and v is the mean kinematic viscosity of sediment laden flow within the boundary layer. The rectified mean longitudinal velocity was computed as 1 ] Ut = -\undy (7) ^ o where wry is the rectified longitudinal flow velocity on the rc-plane at depth y from the channel bottom centerline. The kinematic viscosity of sediment laden flow, vy, at a depths from the channel bottom center- line was computed from the following formula /AvQ + 2.50Cy + 6.25Cy2 + 15.62Cy3) vy = ~i v ^ ' ° ) where ,uw is the dynamic viscosity of clear water. The mean kinematic viscosity of flow, v, within the boundary layer was evaluated as v=-)vydy (9) ^ o The integration of equations (7) and (9) was done with the same method utilised in evaluating the integration of equation (5). The variation of x with C and R in the curved segments was also related to the strength of spiral motion, 5, generated due to bend curvature. There are many ways of interpreting the strength of spiral motion. In this study, the strength of such motion at any cross-section is taken as the percentage ratio of the mean kinetic energy of the lateral motion to the total kinetic energy of flow at the cross-section. Its value at any cross-section was determined from the contour lines of the resultant velocity vector, u, and the longitudinal velocity component (i.e. tangent to the flume centerline), ux, using a planimeter. The utilised procedure was as follow. Each cross-section was subdivided into eight cross-section compartments to form with the standard elevations of meas­ urements a grid. The velocity components in three directions were measured at the intersections of the grid using the calibrated Pitot sphere. Contour lines of w and wx were then constructed, and the following parameters were evaluated. The mean kinetic energy of the forward motion (i.e. in the direction tangent to the flume centerline): / U} \ i t Wx in which U, is the mean forward velocity; defined as U, = QjA, A is the area of the flow cross- section, and u% is the forward (longitudinal) velocity of an elemental area. The mean total kinetic energy: IU2\ 1 t «3 T =JTASTU^A ( 1 1 ) \ 2g }m UXA o 2g 482 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 27, 1989, NO. 4 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 The difference between the value determined by equation (11) and that determined by equation (10) at any cross-section gives the mean kinetic energy of the lateral motion, M, at the cross- section. Thus \2g)B (12) Equations (10) and (11) for any cross-section were solved graphically by a planimeter from u- contours and wx-contours. The strength of spiral motion, S, was then computed as S = 0.01 0.00 0.30 0.60 0.90 1.20 1.50 1.80 2.10 2.40 Fig. la. VELOCITY.Ury.in m/s Selected velocity profile of sediment laden flows in straight and curved flumes. Choix de profils de vitesse d'écoulements chargés de sediments dans des canaux droits et en courbe. JOURNAL OF HYDRAULIC RESEARCH, VOL. 27, 1989, NO. 4 483 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 1.00 >. Ll_ o co LU D _ l < > 0.10 o ST001101 o CR901101 9 r c / b ie 0.5 0.415 90 0.5 0.461 CR901101 1 1 1 L ST001 101 — 0.0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 VELOCITY,Ury, in m/s Fig. lb. Selected velocity profile of sediment laden flows in straight and curved flumes. Choix de profils de vitesse d'écoulements chargés de sediments dans des canaux droits et en courbe. to a computer program developed to determine at each selected location of each experiment the values of R, C, and x using the previously described procedures. Selected velocity profiles of some experiments are presented in Figs. 1 to 3 to show the effect of R, C, and channel alignment on the value of x. The main characteristics of these experiments, among others, are given in Table 3. Referring to this table, the letters ST and CR refer to the straight and curved sections, respectively. The first two digits, after ST or CR, refer to angle of bend (in degrees). The middle two digits refer to the experimental series of runs and to the suspended sand used, respectively. The last two digits gives the number of the experiment in the experimental series of runs. For example, CR301110 refers to profiles measured in the curved section under the following conditions: angle of bend = 30 degrees; experimental series of runs is EXR1, i.e. rz\b = 0.50; sand used is SI; and number of experiment in the series is 10. Fig. la shows the effect of R on flow velocity profiles in straight and in curved flumes. These profiles are almost of the same mean concentration of suspended sediment (see Table 3). Comparing ST001110 profile with ST001117 profile (in straight flume), and CR901110 profile 484 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 27. 1989, NO. 4 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 Table 3. Main characteristics of selected experiments case of experiment ST001101 ST001110 ST001113 ST001117 CR001101 CR001110 CR001113 CR001117 CR301101 CR301110 CR301113 CR301117 CR601101 CR601110 CR601113 CR601117 CR901101 CR901110 CR901113 CR901117 CR001107 CR002128 CR003117 CR004129 CR301107 CR302128 CR303117 CR304129 CR601107 CR602128 CR603117 CR604129 CR901107 CR902128 CR903117 CR904129 flume section straight straight straight straight curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved curved rjb - - - - 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 1.00 2.00 3.00 0.50 1.00 2.00 3.00 0.50 1.00 2.00 3.00 0.50 1.00 2.00 3.00 angle of bend - - - - 00° 00° 00° 00° 30° 30° 30° 30° 60° 60° 60° 60° 90° 90° 90° 90° 00° 00° 00° 00° 30° 30° 30° 30° 60° 60° 60° 60° 90° 90° 90° 90° Q x l O - 2 (m3/s) 5.96 5.96 5.96 15.17 5.96 5.96 5.96 15.17 5.96 5.96 5.96 15.17 5.96 5.96 5.96 15.17 5.96 5.96 5.96 15.17 16.48 17.60 17.86 17.90 16.48 17.60 17.86 17.90 16.48 17.60 17.86 17.90 16.48 17.60 17.86 17.90 T (°C) 24.2 25.1 24.6 24.6 24.2 25.1 24.6 24.6 24.2 25.1 25.0 24.6 24.2 25.1 25.0 24.6 24.2 25.1 25.0 24.6 24.0 24.6 24.3 24.5 24.6 24.6 24.3 24.5 24.6 24.6 24.3 24.5 24.6 24.6 24.3 24.5 y-x (m) 0.282 0.281 0.282 0.282 0.282 0.281 0.282 0.282 0.280 0.281 0.282 0.282 0.282 0.282 0.282 0.285 0.283 0.283 0.284 0.284 0.285 0.284 0.285 0.285 0.285 0.285 0.283 0.285 0.288 0.285 0.285 0.286 0.280 0.283 0.280 0.280 u, (m/s) 0.534 0.530 0.521 1.334 0.538 0.541 0.525 1.349 0.541 0.547 0.528 1.358 0.554 0.550 0.528 1.435 0.564 0.575 0.527 1.439 1.435 1.426 1.572 1.640 1.445 1.506 1.575 1.643 1.536 1.519 1.589 1.648 1.545 1.520 1.595 1.650 R xlO 5 3.89 4.04 5.03 12.81 3.91 4.15 5.05 12.97 3.94 4.22 5.06 13.21 4.04 4.17 5.07 13.06 4.04 5.22 5.07 13.25 11.03 11.52 12.33 12.97 11.70 12.30 12.92 13.64 14.46 14.30 14.89 15.83 17.07 16.97 17.62 18.88 C (gpl) 00.00 25.68 25.28 26.11 00.00 25.11 25.25 25.80 00.00 24.93 25.23 25.14 00.00 24.81 26.17 25.28 00.00 24.16 24.26 25.33 9.35 9.14 9.25 9.31 12.17 12.32 12.25 12.37 25.14 24.55 25.92 25.50 33.16 33.85 33.50 33.08 X 0.415 0.315 0.350 0.465 0.418 0.327 0.362 0.470 0.430 0.349 0.382 0.483 0.454 0.354 0.406 0.493 0.461 0.354 0.414 0.502 0.422 0.420 0.418 0.418 0.447 0.442 0.438 0.430 0.517 0.488 0.481 0.477 0.551 0.533 0.528 0.508 with CR901117 profile (in curved flume), one can see that the increase of R in a sediment laden flow (within the range of these particular experiments) in straight flumes or in curved flumes increases the velocity gradient close to the boundary, which results in an increase in the Von- Karman coefficient in the flow. Such variation of/? with x is more noticable in curved flumes than in straight flumes. This indicates that both curvature of channel and R affect the value of x in a sediment laden flow. Fig. lb shows the effect of channel curvature on velocity gradient of flow close to the boundary. The profiles ST001101 and CR901101 are for flows free of sediment and having amost the same value of R. It is apparent that there is a considerable effect of channel curvature on the velocity JOURNAL OF HYDRAULIC RESEARCH, VOL. 27, 1989. NO. 4 485 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 1.00 •v. >. LL O C/Ï LU = > _ l < > 0.10 0.01 CR301101 CR001101 CR901101 CR601101 CR001101 CR301101 CR601101 CR901101 9 00 30 60 90 rc/b 0.5 0.5 0.5 0.5 0.418 0.430 0.454 0.461 0.45 0.60 0.75 VELOCITY.Ury, in m/s 0.90 1.05 Fig. 2. Selected velocity profile of clear water flows in curved flumes. Choix des profils de vitesse d'écoulements d'eau claire dans des canaux courbes. gradient of flow. In addition, the effect of C on the velocity gradient of flow in straight flumes maybe seen by comparing ST001101 profile (Fig. lb), case of clear water, with ST001110 profile (Fig. la). In curved flumes, similar effect can be seen by comparing CR901101 (Fig. lb), case of clear water, with CR901110 profile (Fig. la). Both angle of bend and radius-breadth ratio of curved flumes were found to have effects on the velocity gradient of flow. Typical examples for such effects are shown in Figs. 2 and 3. These findings support the previous results reported by Rozovskii [8], among others. Generally, as it can be seen from Figs. 1 to 3, channel curvature, represented by angle of bend and radius-breadth ratio, and properties of sediment laden flow, represented by R and C, affect the velocity gradient of flow close to the boundary. The velocity gradient increases as angle of bend, R, and C increase, and as the radius-breadth ratio decreases. The result of the above variation of velocity gradient is changes in the value of the Von-Karman coefficient of the flow. Typical examples for such changes are given in Table 3. The suspended sand used in the flow of the experiments mentioned in this table is SI (see Table 1). The upper part of the table shows the effect ofR on the Von-Karman coefficient in sediment laden flow in 486 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 27, 1989. NO. 4 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 1.00 LL o co 0.10 LU < > ~i 1 1 r CR001117 CR301117 CR601117 CR901117 0.0 0.60 °CR001117 •CR301117 ■CR601117 °CR901117 e 00 30 60 90 rc/b < 0.5 0.470 0.5 0.483 0.5 0.493 0.5 0.502 0.90 1.20 VELOCITY ,Ury,in m/s 1.50 1.80 Fig. 3a. Selected velocity profile of sediment laden flows in curved flumes. Choix de profils de vitesse d'écoulements chargés en sediments dans des canaux courbes. straight and in curved flumes, whereas the lower part of the table shows the effect of flume align­ ment on the behaviour of the Von-Karman coefficient in sediment laden flow in curved flumes. It can be seen that the behaviour of the Von-Karman coefficient in sediment laden flow follows that of velocity gradient close to the boundary (see Figs. 1 to 3). Both channel alignment and flow properties affects the value of Von-Karman coefficient in a sediment laden flow. In the following, an evaluation of the Von-Karman coefficient in sediment laden flow in straight and in curved flumes is made. Straight flumes The suspended sediment and flow velocity profiles measured in the straight portion of each investigated flume alignment, at the center of a cross-section 6.0 m upstream from the bend enterance, were used to evaluate the Von-Karman coefficient and the corresponding C in flows of different values of R, using the previously described methods of calculation. Interpolation technique was utilised to determine the value of the Von-Karman coefficient that corresponds to a specified value of R and a specified value of C. In this regard, values of R(x 105) = 4, 4.5, 5, JOURNAL OF HYDRAULIC RESEARCH, VOL. 27. 198^. NO. 4 487 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 1.00 O 0.10 co LU r>< > T 1 1 1 1 r / u '' /o • o /" l'/fm° ml lO/a • CR903117 CR904129 CR901137 CR902128 // ■CR901107 °CR902128 °CR903117 • CR904129 e 90 90 90 90 rc/b K 0.5 0.5511 1.0 0.533 2.0 0.528 3.0 0.508 0.0' ' ' ' ' u—* *—>->— 0 .00 0.30 0.60 0.90 1.20 1.50 1.80 2.10 2.40 VELOCITY,Ury,in m/s Fig. 3b. Selected velocity profile of sediment laden flows in curved flumes. Choix de profils de vitesse d'écoulements chargés en sediments dans des canaux courbes. 6,7,10,15, and 20, and values of C = 0,10,20,30,40,50, and 60 gpl were considered. The variation of x with those of C and R were found almost the same for all size distribution of suspended sediment used in this study (see Table 1) and for all alignments of flume downstream. Thus, it was decided to average the values of x that were interpolated for a specified value of C and a specified value of/?, and that resulted from different size distributions and channel alignments, and to consider this average as the value of x corresponding to the specified values of C and R in the straight flume flow. The variations of the averaged values of x with C and R are shown in Fig. 4. Inspection of the figure reveals that the Von-Karman coefficient decreases with the increase in mean suspended sediment concentrations in sediment laden flows of Reynolds number less that 7.0 x 105, whereas it increases with the increase in mean suspended sediment concentrations in sediment laden flows of Reynolds number more than 7.0 x 105. In sediment laden flows of Reynolds number approximately equal to 7.0 x 105 (called, in this study, the critical Reynolds number), the Von-Karman coefficient seems to be independent on the amount of suspended sediments in the flows. 488 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 27. 1989. NO. 4 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 0.6 i - I 0.5 h. U O u z o > 0,4 0.3 0.2 • R=20xlfj5 0 R=15xl05 ■ R=10xl0 5 D R=7 xlO5 + R=6 xlO5 * R=5 xlO5 « R=4.5x l0 5 m R = 4 x i o 5 Sand SI ^ ^ ^ - " 1 ( i ' n • _»—•" ■ P r 5 R;20x10 R : 1 5 x 1 0 S " R = 10x10 5 j 1 R =7 x10 5 U h R = 6x 10 È _ R = 5 x 105 ' ^___R_=Ax 10 s ' ( J [ ] 10 20 30 40 50 60 MEAN SUSPENDED SEDIMENT CONCENTRATION, in grams per liter Fig. 4. Variation of Von-Karman coefficient with mean suspended sediment concentration for different values of Reynolds number in straight flume. Variation de la constante de Von-Karman avec la concentration moyenne de sediment en suspen­ sion pour différentes valeurs du nombre de Reynolds dans un canal rectiligne. It is believed, while it has not been tested by the author, that the level of turbulence close to the boundary decreases in flows of Reynolds number less than the critical Reynolds number, and increases in flows of Reynolds number lager than the critical one, as C increases. Such variations in the level of turbulence are believed to be the main reason for the variations of the Von-Karman coefficient with C and R. It may be hypothesized that the level of turbulence close to the boundary in sediment laden flow increases as C decreases [1] and as R increases, and that the rate of such increase of turbulence level depends on the values of C and R. In flows of Reynolds number less than 7.0 x 105, the effect on the level of turbulence of C is greater than that of R, resulting in an overall decrease in the turbulence level and associated decrease in the value of x. On the other hand, the effect on the turbulence level of/? is greater than that of C in flows of Reynolds number larger than 7.0 x 105. This results in an increase of the turbulence level and associated value of x. The effect on the turbulence level ofR is almost the same as that of C in flows of Reynolds number approximately equal to 7.0 x 105. In this case, there is no significant effect of suspended sediment concentra­ tions on the value of x. To show the effect of size distribution of suspended sediment on the variation of x with R in sediment laden flow, Fig. 5 is presented. The figure indicates that the rate of variation of x with R is larger in flows of high suspended sediment concentrations than that in flows of low suspended sediment concentrations, and is larger in fine suspended sediments than that in relatively coarse suspended sediments. It can also be seen that as the concentration of suspended sediments of a given size distribution decreases the curve describing the variation of x with R tends to rotate around a point at which R is approximately equal to the critical value of R (= 7.0 x 105), and the rate of such variation decreases. The value of x at the rotating point depends on the size distribution of suspended sediment. It has a value in flow with fine suspended sediment larger than that in flow with relatively coarse suspended sediment. In all cases, the JOURNAL OF HYDRAULIC RESEARCH, VOL. 27. 1989, NO. 4 489 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 U 6 8 10 12 H 16 18 20 REYNOLDS NUMBER ( x ifj5) Fig. 5. Variation of Von-Karman coefficient with Reynolds number for different grain size distributions of suspended sands in straight flume. Variation de la constante de Von-Karman en fonction du nombre de Reynolds pour différentes granulométries de sable en suspension dans un canal rectiligne. value of x in sediment laden flow of Reynolds number approximately equal to 7.0 x 105 (i.e. at the rotating point) is less than the value of x in clear water flow (=0.415). In flows with negligiable amount of suspended sediment (i.e. clear water flows, the curve describing the variation of x with R (Fig. 5) becomes horizontal, meaning that the Von-Karman coefficient in clear water flow is independent on the Reynolds number of the flow. From the above results, if one accepts the hypothesis that the change of % in a given flow is the result of variation in the turbulence level close to the boundary, it may be concluded that the reduction in turbulence level produced by coarse suspended sediment of a given concentration is larger than that produced by fine suspended sediment of the same concentration. This is apparent from Fig. 5 in which the value of the Von-Karman coefficient in flow with coarse suspended sediment is always less than that in flow with fine suspended sediment. Curved flumes The effect of stream curvature in the behaviour of the Von-Karman coefficient in sediment laden flows in curved flumes was evaluated and is presented hereafter. Measurements of suspended sediment concentrations and flow velocities at each of the standard cross-sections were used to compute S, x, R, and C under various experimental conditions at the cross-section. Using these computed values, an interpolation technique was utilised to determine for a given values of R and C the corresponding values of x and S at the cross-section. In this regard, values of R(x 105) = 4,6, 8,10,12,14,16,18, and 20, and values of C = 0,10,20,30,40,50, and 60 gpl were considered. To study the effect of stream curvature on the behaviour of the Von-Karman coeffi­ cient in sediment laden flows in curved flumes, the relative Von-Karman coefficient, xr, was evaluated and then related to various parameters identifying the flow and channel alignment. The relative Von-Karman coefficient is taken for a certain flow in a given flume alignment as the 490 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 27, 1989. NO. 4 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 1.20 1.15 Row \LX ▼90 105 1.00 1.20 1.15 7 x 1 0 * s O C= 60gpl • C= 50gpl + C - 40gpl n c =3ogPi ■ C =20gpl OC =10gpl C = 0 4.0 s 3-0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0/90 Fig. 6a. Variation of relative Von-Karman coefficient, xr, with angle of bend for different mean concentra­ tion of suspended sand (upper) rc/b = 2.0, (lower) rc/6 = 3.0. Variation relative de la constante de Von-Karman, xr, en fonction de Tangle du coude pour différentes concentrations moyennes de sable en suspension, rcjb = 2fi (supérieur), rjb = 3,0 (inférieur). ratio of the Von-Karman coefficient evaluated at the center of a cross-section in the curved portion of the flume to the Von-Karman coefficient evaluated at the center of the cross-section in the straight portion preceding the bend of the flume alignment. Fig. 6 shows the variation of x, with angle of bend for different concentrations of suspended sediment in the investigated flume alignments, and the relation of such variation to the strength of spiral motion generated in the bends. The suspended sand used in the experiments of Fig. 6 was sand SI (see Table 1). Typical results were obtained from flows with the other sands considered in this study. The presented JOURNAL OF HYDRAULIC- RLSEARCH. VOL. 27, 1989, NO. 4 491 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 0.0 0.1 u E L L U O u 1.20 1.15 1.10 z o > u >< 1.05 1.00 0.0 0.1 0/90 Fig. 6b. Variation of relative Von-Karman coefficient, x„ with angle of bend for different mean concentra­ tion of suspended sand (upper) rjb = 0.50, (lower) rc/6 = 1.0. Variation relative de la constante de Von-Karman, xn en fonction de Tangle du coude pour différentes concentrations moyennes de sable en suspension, rjb = 0,50 (supérieur), rjb = l,0 (inférieur). results in this figure are for flows of a Reynolds number equal to 7.0 x 105. This particular value of Reynolds number was selected because in its flows the Von-Karman coefficient is independent on the concentration of suspended sediment, so the presented variation of xr with angle of bend for a certain concentration of suspended sediments in the flow is only due to flume curvature (alignment). Inspection of Fig. 6 reveals that the Von-Karman coefficient in the curved sections is larger than that in the straight sections of a given flume alignment, and this is valid in either clear water or 492 JOURNAL DE RECHERCHES HYDRAUL1QUES, VOL. 27. 1989. NO. 4 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 1.15 R (xiO ) Fig. 7a. Variation of relative Von-Karman coefficient, x„ and strength of spiral motion, S, with Reynolds number, R, for different values of mean concentration of suspended sand (rjb = 0.50). Variation relative de la constante de Von-Karman, x„ et intensité du mouvement hélicoïdal, S, en fonction du nombre de Reynolds, R, pour différentes valeurs de la concentration moyenne du sable en suspension (/•,.//> = 0,50). JOURNAL OF HYDRAULIC' RESEARCH, VOL. 27, 1989. NO. 4 493 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 * S ■ C=20gpl O C=60gpl O C=10gpl • C=50gpl X C=0 =40gpl =30gpl 9 : 6 0 0.0 s 3-0 0.0 10 12 R WO5) K 16 18 20 Fig. 7b. Variation of relative Von-Karman coefficient, x„ and strength of spiral motion, 5, with Reynolds number, R, for different values of mean concentration of suspended sand (rjb = 1.0). Variation relative de la constante de Von-Karman, x„ et intensité du mouvement hélicoïdal, S, en fonction du nombre de Reynolds, R, pour différentes valeurs de la concentration moyenne su sable en suspension (/•{./ö = 1,50). sediment laden flows. In addition, the relative value of Von-Karman coefficient increases as angle of bend increases, as mean-breadth ratio of bend decreases, as strength of spiral motion increases, and as mean concentration of suspended sediment increases. 494 JOURNAL DL RECHERCHES HYDRAULIQUES, VOL. 27, 1989, NO. 4 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 R ( x 1(T) Fig. 7c. Variation of relative Von-Karman coefficient, x„ and strength of spiral motion, S, with Reynolds number, R, for different values of mean concentration of suspended sand (rc/6 = 2.0). Variation relative de la constante de Von-Karman, x„ et intensité du mouvement hélicoïdal, S, en fonction du nombre de Reynolds, /?, pour différentes valeurs de la concentration moyenne su sable en suspension (rjb = 2,0). On the other hand, typical variation of xr with R for different concentrations of suspended sediment is shown in Fig. 7. The sand used in the experiments of this figure was SI (see Table 1). Fig. 7 combines the effects of both R and channel alignment on the vaule of the Von-Karman JOURNAL OF HYDRAULIC' RESEARCH, VOL. 27. 1989. NO. 4 495 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 1.10 1.15 1.10 C=60gpl C=50gpl C=40gpl C=20gpl C=0 r c / b ; 3- 0 9 ; 60° »C,gpt O 90 30 2.0 1.0 0.0 s 3-0 1.15 1.10 1.05 1.00 R (xl(T) Fig. 7d. Variation of relative Von-Karman coefficient, xn and strength of spiral motion, S, with Reynolds number, R, for different values of mean concentration of suspended sand (rjb = 3.0). Variation relative de la constante de Von-Karman, xr, et intensité du mouvement hélicoïdal, 5, en fonction du nombre de Reynolds, R, pour différentes valeurs de la concentration moyenne su sable en suspension (>c/6 = 3,0). coefficient in sediment laden flows. It can be seen that xr decreases as R increases. Comparing this figure (Fig. 7) with Fig. 5, in which x increases as R increases, it may be concluded that the effect on x in a sediment laden flow of channel alignment is less than that of R. 496 JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 27, 1989, NO. 4 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 Referring to Fig. 7, it can be seen that xr varies with R up to a limit after which xr becomes almost independent on R. This limit depends on the amount of suspended sediment in the flow as well as on the characteristics of the curved section (i.e. angle of bend and mean radius-breadth ratio). It is smaller in flows of low concentrations of suspended sediments than in flows of high con­ centrations of suspended sediments, in cross-sections of small angle of bend than in those of large angle of bend, and in cross-sections of large mean radius-breadth ratio than those of small mean radius-breadth ratio. Generally, the variational behaviour of xT follows that of 5 in curved flumes. Conclusions Experimental investigations are made on the behaviour of the Von-Karman coefficient in sediment laden flow in straight and in curved flumes. From the results obtained the following conclusions can be drawn: 1. The variation of Von-Karman coefficient with mean concentration of suspended sediments in an open channel flow depends on the value of the Reynolds number of the flow. With the increase in mean concentration of suspended sediment, the Von-Karman coefficient decreases in flows having Reynolds number less than 7.0 x 105 but increases in flows having Reynolds number larger than 7.0 x 105. The variation of the Von-Karman coefficient with mean concentration of suspended sediment is insignificant in flows having Reynolds number approximately equal to 7.0 x 105. 2. The Von-Karman coefficient increases as Reynolds number increases in straight streams. Such increase of Von-Karman coefficient in flows with fine sizes of suspended sediments is larger than that in flows with relatively coarse sizes of suspended sediments, and is larger in flows of high suspended sediments concentrations than that in flows of low concentrations of suspended sedi­ ments. In any case, the rate of such increase in steeper in flows having Reynolds number les than 9.0 x 105 than that in flows having Reynolds number larger than 9.0 x 105. 3. The value of the Von-Karman coefficient in curved streams is larger than that in straight streams, and this is valid either in clear water flows or in sediment laden flows. 4. The relative value of Von-Karman coefficient (defined as the ratio of the Von-Karman coeffi­ cient in a flow in a curved channel cross-section to that in the flow in a straight channel cross- section) increases as the amount of suspended sediment in the flow increases, as the Reynolds number of the flow decreases, as the angle of bend increases, as the mean radius-breadth ratio of the curved section decreases, and as the strength of spiral motion (defined as the percentage ratio of the mean kinetic energy of the lateral motion to the total kinetic energy of flow) at a curved channel cross-section increases. 5. The strength of spiral motion at a curved channel cross-section increases as angle of bend of the channel increases, as mean radius-breadth ratio of the channel decreases, and as the Reynolds number of the channel flow decreases. The variation of the relative Von-Karman coefficient with mean concentration of suspended sediment and with Reynolds number of the flow at cross-sections of high strength of spiral motion is more significant than that at cross- sections of low strength of spiral motion. 6. In a clearwater flow in straight channels the Von-Karman coefficient is independent on the Reynolds number of the flow. 7. In a sediment laden flow in a curved channel the relative Von-Karman coefficient depends on the Reynolds number of the flow up to a limit affected by characteristics of the curved channel and by the concentration of suspended sediment in the flow. JOURNAL 01- HYDRAULIC RESEARCH, VOL. 27, 1989. NO. 4 497 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 Acknowledgements The work described in this paper was partly supported by the Research Center of the College of Engineering, King Saud University. The help recieved from Professor A. Shukry, from Alexandria University, Eng. M. Badia and Eng. M. Abaza, from the Research Center of King Saud University, are very much appreciated. Notations C mean concentration of suspended sediment Cy volumetric suspended sediment concentration at depth y from channel bottom centerline g gravitational acceleration R Reynolds number within the boundary layer S strength of spiral motion at a cross-section of channel SE energy gradient Sw energy loss per unit channel length due to periphery other than at channel centerline U mean velocity vector of flow at a cross-section of channel U, rectified mean longitudinal velocity within the boundary layer Ux mean forward velocity of flow at a cross-section of channel U, shear velocity at flume centerline u velocity vector of an elemental area «ry rectified longitudinal flow velocity at depth y from channel bottom centerline ux forward velocity component of an elemental area y depth of flow yT total depth of flow Z thickness of boundary layer x Von-Karman coefficient xT relative Von-Karman coefficient Q density of sediment laden flow QS density of sediment ew density of clear water rc shear stress at channel centerline v mean kinematic viscosity of sediment laden flow within the boundary layer vy kinematic viscosity of sediment laden flow at d e p t h s from channel bottom centerline Hv dynamic viscosity of clear water 498 JOURNAL DE RECHERCHES HYDRAULIQUES. VOL. 27, 1989, NO. 4 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14 References / Bibliographie 1. American Society of Civil Engineers, Sedimentation Engineering: ASCE Manual and Reports on Engi­ neering Practice, No. 54, V. A. Vanoni (editor), 1975, p. 745. 2. COLEMAN, N. L., Velocity Profiles With Suspended Sediment, Journal of Hydraulic Research, IAHR, Vol. 19, No. 3, 1981, pp. 211-229. 3. EINSTEIN, H. A. and CHIEN, N., Effects of Heavy Sediment Concentration Near the Bed on Velocity and Sediment Distributions, Missouri River Division, Sediment Series No. 8, Fluid Mechanics Laboratory, University of California, Berkeley, California, U.S.A., 1955, p. 47. 4. GRAF, E. H., Hydraulics of Sediment Transport, McGraw-Hill Book Co., Inc., New York, U.S.A., 1971, p. 513. 5. GUST, G., Observations on Turbulent Drag Reduction in a Dilute Suspension of Clay in Sea-Water, Journal of Fluid Mechanics, Vol. 75, Part 1, 1976, pp. 29-47. 6. HINO, M., Turbulent Flow With Suspended Particles, Journal of the Hydraulics Division, ASCE, Vol. 89, No. HY4, April 1963, pp. 161-185. 7. IMAMOTO, H., ASANO, T. and ISHIGAKI, T., Experimental Investigation of a Free Surface Shear Flow With Suspended Sand Grains, Proceedings of the 17th Congress of the IAHR, 1977, pp. 105-112. 8. ROZOVSKII, I. L., Flow of Water in Bends of Open Channels, Academy of Sciences of Ukranian SSR, Kiev, U.S.S.R., 1957, (Translated by Y. Prushansky, The Israel Program for Scientific Translations, No. OTS 60-51133, National Technical Information Service, United States Department of Commerce, 1961). 9. SHUKRY, A., Flow Around Bends in Open Flume, Transactions of the American Society of Civil Engineers, Vol. 115, 1950, pp. 751-779. 10. VETTER, M., Velocity Distribution and Von-Karman Constant in Open Channel Flows With Sediment Transport, Proceedings of the 3rd International Conference on River Sedimentation, The University of Mississippi, University, Mississippi, U.S.A., 1986, pp. 814-823. 11. ZAGUSTIN, A. and ZAGUSTIN, K, Mechanics of Turbulent Flow in Sediment Laden Streams, Proceedings of the 13th Congress of the IAHR, 1969, pp. 317-324. JOURNAL OF HYDRAULIC RESEARCH, VOL. 27, 1989, NO. 4 499 D ow nl oa de d by [ T he U ni ve rs ity o f M an ch es te r L ib ra ry ] at 0 6: 38 0 4 D ec em be r 20 14