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3 An Approach to the Sediment Transport Problem 2 OH 02 From General Physics <x> tGO GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-1 GO H W OH feq c^ cb O OS CLH og OH CO sz; « GO E-i O o -«1 o o I?p An Approach to the Sediment Transport Problem From General Physics By R. A. BAGNOLD PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-1 From considerations oj energy balance and oj mechanical equilibrium, a mathematical expression is derived relating the rates of sediment transport as bedload and as suspended load to the expenditure ofpower by a statistically steady flow of water UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1966 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 35 cents (paper cover) CONTENTS Page Page Introduction _______________________________________ II The final transport rate relationship_.--_---_________ 118 General considerations_____ ---.._ .__._______ 2 Existing flume data_________--___------_-----___--_. 19 Essential features of granular flow._______________ 2 Limitations and uncertainties..____--_---______-- 19 Implication: the existence of an upward supporting Wall drag and flow depth._____________________ 20 stress, ________----__---_-_____-____________ 2 Estimation of appropriate fall velocity V. _________ 21 Restrictions of conditions to be considered. ________ 3 Comparison of the theory with the experimental flume Bedload and suspended load and the work rates of their data.-...-.________._---.-_._._.____ 21 transport- _ __________________________________ 3 Gilbert 5.0-mm pea gravel-_______-----_-_______- 21 Definitions- _. _ _______________________________ 3 Gilbert 0.787-mm sand__ . __ __________ 21 Transport work rates..________________________ 4 Gilbert 0.507-mm sand____-___-____._______ 21 Principle of solid friction and bedload work rate____ 4 Simons, Richardson, and Albertson 0.45-mm sand_ 26 Suspended-load work rate.--.-________________ 5 Gilbert 0.376-mm sand__________-_-_______ 26 Transport work rates and available power; the general Gilbert 0.307-mm sand________________ 26 sediment-transport relationship_______________ 5 Barton and Lin 0.18-mm sand--_---_-_----___--_- 26 The general power equation.___________________ 5 Laursen 0.11-mm silt._________________________ 26 Available stream power_______________________ 5 General comments.______-_____-_-_-_-_____._--- 26 The general transport relationship in outline form___~ G Experiments by Vanoni, Brooks, and Nomicos__ __ 27 Bedload transport efficiency et> at high transport stages.- 6 Relation.between i and co over the lower, transitional Concept of a moving flow boundary.____________ 6 stages-----_----_____-_--_------------_____-- 28 Critical stage beyond which the bedload efficiency Transport of fine silts in experimental flumes...____ 28 66 should be predictable.______________________ 8 The available river data, nature and uncertainties. _____ 29 Effect of inadequate flow depth on the values of b~ 9 Energy-slope estimation...... .__.__. ... 29 Variation of the dynamic bedload friction coefficient Bedload and suspended load._____--______--___-_ 29 tan a. _______________________________________ 10 Deficiency of sediment supply___.__.___._________ 29 The suspended transport efficiency es of shear turbulence. 12 Values of the stage criterion 6. ___________________ 30 Critical flow stages for suspension_________________ 15 Comparison of predicted with measured river transport The effective fall velocity V for suspension of rates____________________________________________ . 30 heterogeneous sediments.____-_-_-______-____ 17 Comparison of river transport data with data for wind- The effective mean grain size D for heterogeneous transported sand--_--__-________________________ 35 bedloads.___- _--___--____----_____--____- 18 Conclusion _ __ ____________________________________ 35 Washload._____ _ ___-____---_____--___________ 18 References-_______^________________________________ 37 Pago FIGURE 1. Diagram of the relation of normal force or stress to tangential force or stress between solids in moving contact- . 14 2-5. Graphs: 2. Flow relative to moving boundaries..__.__-___._-._-__._._-__.________.____, 3. Values of theoretical bedload efficiency factors, in terms of mean flow velocity, for quartz-density solids in an adequate depth of fully turbulent water_____-_-_---___--_______-___-_-_-__----_______- 4. Values of the solid-friction coefficient tan a in terms of the bed-stress criterion 6=Tl(<r p)gD for quartz- density solids of various sizes in water, derived from figure 5, and the predicted critical values of 6 beyond which the theory should be applicable------___---_--___-__---___-_-----_---_-____-_- 5. Values of the solid-friction coefficient tan a in terms of the Reynolds-number criterion G for granular shear. _____________.___________________________________________________ 11 6. Diagram of characteristic fluid motion close to the boundary.._________________________________________ 13 7. Schematic diagram showing postulated asymmetry in the normal eddy velocity components of shear turbulence. 13 8. Graph showing theoretical values of the suspension criterion 8a =V2/gD together with Shields' values of 0 at the threshold of bed movement________________________________________________________________ 16 in IV CONTENTS FIGUBES 9-13. Comparative plots of theoretical and experimental transport rates (flume experiments): Page 9. Gilbert 5.0-mm gravel and 0.787-mm sand__,__-____-____ _________________ _._ .._ 122 10. Gilbert 0.507-mm sand and Simons, Richardson, and Albertson 0.45-mm sand._____________________ 23 11. Gilbert 0.376-mm and 0.307-mm sand________.________ _.__-.__ _ __ __ 24 12. Barton and Lin 0.18-mm sand and Laursen 0.11-mm sand______________________________________ 25 13. Vanoni and Brooks 0.14-mm sand, Nomicos 0.172-mm sand, and Laursen 0.04-mm silt __ 27 14. Graph of transport-rate discrepancies (predicted/measured), from table 1, grouped according to river and station______-____-____-- ________________________________________________________ 34 15. Graphs showing examples of anomalous size distributions of assumedly suspended loads, associated with anomalous transport rates predicted from them (suggesting that the sampling had included a proportion of coarser bedload material)______-____.,____-__----_..----_---------_----_--------- 36 TABLE Page TABLE 1. Data from 146 individual river measurements comparing measured with predicted transport rates. 130 SYMBOLS Where Symbol meanings preceded by f refer to quantities involving intro­ the gravity acceleration g. These quantities have ambiguous Symbol duced Meaning dimensions and values depending on the ambiguous definition of I 14 fTransport rate of solids by immersed weight. In relations between such quantities, g cancels out and weight per whole width of flow therefore has been omitted, leaving the quantities expressed in practical units. However, g is necessarily retained in relations Transport rate of solids by dry mass between such quantities and, for example, velocities. The per unit width. shear stress T, for instance, is sometimes expressed as p gdS and { Transport rate of solids by dry mass sometimes as p dS, according to the context. There is no real per whole width of flow. ambiguity, because all relations are dimensionally correct. fMass of transported solids over unit Meanings of dimensionless symbols are preceded by {. bed area. Where 4 flmmersed weight is m'g= mg. intro­ IT Symbol duced Meaning 4 Stress transmitted from solid to solid b I 14 {Ratio F/w*. normal to the plane of shear. 19 {Transport concentration by immersed 17 {Proportion of unity, by weight, of a weight. given size grade in a sample of 9 {Static bed concentration by volume heterogeneous granular material. = (1 porosity). 5 Discharge. c 14 {Flow coefficient u/u*. 7 {Conventional Reynolds number for D 7 Grain size. fluid flow. d 5 Flow depth. 5 {Energy slope. eb 6 {Bedload transport efficiency 4 fStress transmitted from solid to solid _ bedload work rate. tangential to the plane of shear. ~ stream power 4 Mean transport velocity of solids. ec, et 7 {Factors of e_. Uf 7 Transport velocity, relative to ground, e. 6 {Suspension efficiency of a notional nongranular moving suspension work rate flow boundary. ~~ steam power 6 Mean flow velocity 14 {Residual upward momentum flux, an Q internal fluid stress. cross-sectional area IV Gravity acceleration. 13 Root-mean-square velocity com­ G 10 {Special Reynolds number for solids ponents of turbulent fluctuations. sheared in fluid= »/^ u* 14 Shear velocity. ij V X 13 Root-mean-square value of the down­ 4 fTransport rate of solids by immersed ward part of t'. weight per unit width =- - gj. 13 Root-mean-square value of the up­ ward part of "'. SYMBOLS Where Where intro­ intro­ Symbol duced Meaning Symbd duced Meaning V 15 Fall velocity of suspended solids. co 15 fStream power per unit boundary Effective fall velocity of hetero­ area=rtt. geneous solids =pVj,. 6 Subscript pertaining to bedload. tan a 4 J Coefficient of solid friction = T/P. c Subscript pertaining to a moving 10 Absolute viscosity of fluid. boundary. Subscript pertaining to conditions 9 JDimensionless shear stress = (*-p)gD within stationary bed. 10 tLinear concentration 17 Subscript pertaining to the grain size mean D of solids present in proportion p. mean free distance between
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