Total Bed-Material Discharge in Alluvial Channels

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Total Bed-Material Discharge in Alluvial Channels Total Bed-Material Discharge in Alluvial Channels GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-1 Total Bed-Material Discharge in Alluvial Channels By F. M. CHANG, D. B. SIMONS, and E. V. RICHARDSON STUDIES OF FLOW IN ALLUVIAL CHANNELS GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-1 UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1965 UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY William T. Pecora, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 - Price 20 cents (paper cover) CONTENTS Page Symbols_ _______________________________________________________ iv Abstract__ _____________________________________________________ I 1 Introduction._ ____________________________________________________ 1 Scope._______________________________________________________ 1 Acknowledgments. ____________________________________________ 3 Analysis of sediment size.__________________________________________ 3 Velocity distribution in alluvial channels____________________________ 5 Bed-material discharge._------__-______-_-_________________________ 8 Contact-bed-material discharge________________________________ 10 Suspended-bed-material discharge.______________________________ 13 Total bed-material discharge._________-_____-_-___________----__ 16 Evaluation.__________________________________________________ 18 Summary and conclusions__--_-__________-_-__-___-_______--_-___ 20 References_ _ ____________________________________________________ 22 ILLUSTRATIONS Page FIGURE 1. Bed-material size distribution of sands for flume data____ 12 2. Relation of sediment Reynolds number and von Karman's coefficient. ______________________________ _ ______ 7 3-6. Comparison of theoretical and measured velocity dis­ tribution for flume data 3. For de=0.19-mm sand ___ ___________________ 7 4. For de=0.33-, 0.35-mm sand. _________------_- 8 5. For de=0.50-, 0.52-mm sand___._._ _ ___._.__ 9 6. For rfe=0.93-mm sand ____ __._ ______ _-__ 9 7. Comparison of theoretical and measured velocity distri­ bution for canal data. _ __________________---_-_-- 10 8. Function Ii in terms of £ for various values of z ____ -___ 15 9. Function 72 in terms of £ for various values of z _____ __- 15 U Tg 10. KT versus _ relation for various sands from U* flume data_____________________________________-_ 17 11-13. Comparison of the estimated and measured total bed- material discharge for 11. Flumedata_ r _____________________---_----_ 19 12. River data. __________ _____ _ _____ _ _ 20 13. Both flume and river data __ __________-__-- 21 m SYMBOLS Symbol Definition Dimension AI Integration constant-_____-_____--_____--.__---_-__-----_ _______ A3 Volume constant of particle__.________---____--_________ _______ a Thickness of contact-bed-material layer____________________ L b e Apparent thickness of contact-bed-material layer (only static L force is considered)_---_-___-_--______--_____-_---_-_-_ L Ca Concentration at distance a from the bed.__________________ F/L3 Cm Mean concentration of contact-bed-material layer ___________ F/L3 c Concentration in dry weight per unit volume at distance y from the bed__________________________________________ F/L3 Cft Weight of contact bed material in a unit volume of the water- bed-material mixture__ _______________________________ F/L3 c, Weight of suspended bed material in a unit volume of the water- bed-material mixture_________________________________ F/L3 D Depth of flow_-----_-__---____---__-----------_-----_--_ L d Bed-material diameter. ___-_-___-____-__--_------__-----_ L de Equivalent diameter (definition in the text, p. I 14) _________ L d50 Median diameter______________________-___-__-----__-_-_ L e Porosity of bed material_______________-_---___-__-----_ _______ f Vector notation of force acting on particle__-_____---__--___ F g Gravitational acceleration. _____--______--____----____--__ L/T2 /i Integral value__-___-_-_________--___-__----_--____----__ _______ /2 Integral value_____-_-_________-_--_-----_---------_----_ _______ j Experimental coefficient, j=a/6c_-_____-_---_--_-_-----_--_ _______ #6 M-tf5e____________-_________-______.________ ___________ _______ Kbe Contact-bed-material coefficient--------------------------- _______ KT Total bed-material discharge coefficient__-___-___-_-__--_-_ _______ I Mixing length______________________----_--_-__-------- L M Coefficient M= Mr Af.___..__________________________ _______ MI Sediment-discharge distribution coefficient--.--------------- ------- M2 Ratio of concentrations between contact-bed-material layer and bed material in stationary bed- _________________________ _______ N Number of particles. ____________________________________ ------- n Number of moving layers in contact-bed-material layer ___ _______ p Percentage by weight of the bed material with diameter d_ ___ _______ pt Total percentage by weight of the bed material- _________ _______ Qb Contact-bed-material discharge per unit width of channel _ F/TL QD Contact-bed-material discharge per unit area._______________ Ff TL2 Q, Suspended-bed-material discharge per unit width of channel.__ F/TL QT Total bed-material discharge per unit width of channel_______ F/TL Rs Ratio of suspended-bed-to contact-bed-material discharge.__ _______ re Ratio of the velocity at £=£<, to the mean velocity of the fluid. _ _______ *S Slope of channel-_______________---_--_------_----_------ --__-__ U Mean velocity of flow.________-_--___-------_----.-------- L/T t/5 Mean velocity of contact-bed-material layer,_______________ L/T IV SYMBOLS Symbol Definition Dimension Um Velocity at water surface_____--_---_-_-----__------------ L/T U* Shear velocity__--_..-________-_.__________L___-.-_ L/T u Velocity at distance y from the bed________________________ L/T u' Fluctuating component of velocity in the x direction.________ L/T Ub Actual velocity of the contact bed material.________________ L/T us Actual velocity of the suspended Bed material.______________ L/T T7 Vector notation of the relative velocity between particle and fluid._-_-_-------_----_---------------_------------_- L/T v' Fluctuating component of velocity in the y direction. ________ L/T W Work done by the relative motion between particle and fluid. _ FL Wb Work done by bed material.______________________________ FL Wf Work done by fluid_._...-______-_-_--_--_-_----.___-_--- FL y Distance from the bed surface___________________________ L 2 Exponent for suspended-bed-material distribution ___________ _______ a Slope of bed (angle)--____________-__----__________---_-_- _______ 6 Ratio of the sediment transfer coefficient to the diffusion co­ efficient of fluid._____________---_-_-_-_____-------_-_- _______ 7 Specific weight of fluid._________-_-___---_____-_-_-__-_-_ F/L3 ys Specific weight of bed material.___________________________ F/L3 Ay Submerged specific weight of bed material._________________ _______ e Efficiency.-_ ____________________________________________ _______ e0 Diffusion coefficient of the fluid.__________________________ _______ f, Sediment transfer coefficient._____________________________ _______ « von Karman's coefficient--------------------------------- ____--_ M Dynamic viscosity of fluid.________--_---______-__-__----- FT/L2 v Kinematic viscosity of fluid.____________-__________--_---- L2/T STUDIES OF FLOW IN ALLUVIAL CHANNELS TOTAL BED-MATERIAL DISCHARGE IN ALLUVIAL CHANNELS By F. M. CHANG, D. B. SIMONS, and E. V. KICHAKDSON ABSTRACT This is a study of total bed-material discharge (wash load excluded) in alluvial channels, based partly on existing theories. The contact-bed-material discharge was obtained by utilizing the energy-work relation of the fluid and the bed material. An equation for velocity distribution was obtained by integrating the Reynolds equation through use of the Prandtl's hypothesis of mixing length. Through application of the basic equation for the distribution of suspended bed material by M. P. O'Brien, the suspended-bed-material discharge was in­ vestigated in terms of contact-bed-material discharge. Then, the total bed- material discharge was obtained simply by adding the contact-bed-material discharge and the suspended-bed-material discharge. The results were checked with available la boratory and field data and appeared to be mutually consistent and satisfactory. INTRODUCTION SCOPE No purely theoretical approach to the general principles of sediment discharge in alluvial channels seems possible at present because of insufficient quantitative knowledge of fluid turbulence and its effect on sediment within the fluid. Hence, the approach used in this report is semitheoretical and is based on existing theory of turbulent flow and on general ideas derived from observation and reasonable speculation. This study concerns itself principally with 1. The classification of the graded sands which comprise the bed material of natural alluvial channels. 2. The development of the velocity distribution, which includes consideration of the body force of the flow. 3. The subdivision of the total bed-material discharge into com­ ponents based upon significant forces to which the particles are subjected. 4. Relations for estimating suspended-bed-material discharge, contact- bed-material discharge, and hence, total bed-material discharge. I l 12 STUDIES OF FLOW IN ALLUVIAL CHANNELS 5. Evaluation of the formula on total bed-material discharge by use of laboratory and field data. The flume data (Simons, 1961b) used for the investigation were collected by the U.S. Geological
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