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Bed-Material Load (Einstein's Method)

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Bed-Material Load (Einstein's Method) GOVERNMENT OF THE REPUBlIC OF INDONESIA MINISTRY OF PUBLIC WORKS • DIRECTORATE GENERAL OF WATER RESOURCES DEVELOPMENT PROGRAMME Of ASSISTANCE fOR THE IMPROVEMENT Of HYDROLOGIC DATA COLLECTION. PROCESSING AND EVALUATION IN INDONESIA BED- MATERIAL LDAD (EINSTEIN'S METHOOJ by M TRAVAGLIO po ~ SOCIETE CENTRALE ~~ POUR L" EQUIPEMENT DU TERRITOIRE INIERNATIONAt INTERNATIONAL BANDUNG MARCH 1981 f Bed-Materia1 Load (Einstein's Methed) by M. TRAVAGLIO Bandung, March 1981 Taole of Contents Page List of symbols • r Introduction 1 Einstein's Procedure 2 1. Hydraulic.Calculations 2 1.1 Test Reach • 2 1.2 Surface Drag and Bedform Drag (or Bar Resistance) 3 1.3 Mean velocity 4 a. Manning-Stricler's Equation 4 b. Logarithmic Type Formula ••• 5 1.4 step by Step Procedure for Hydraulic Calculations 6 2. Bed-Material Load Calculation . 8 2.1 Rouse Equation for Vertical Distribution of Suspended Matter 8 2. 2 Suspended Load Equation la 2.3 Einstein's Bed-Load Formula 11 2.4 Bed-Material Load Equation •• 13 3. Example of Bed-Material Load Calculation 21 Concluding Remarks 27 Annex 1 · 28 Annex 2 · 30 Annex 3 · . 31 Annex 4 · 33 Annex 5 35 References 37 l LIST OF SYMBOLS A cross-sectional area diameter of particle. In a mixture d d or median diameter d = 50 D depth of flow 2 g gravitational constant, mean value 9.81 rn/s gs bedload rate in weight per unit time and unit width gss suspended.load rate in weight per unit tirne and unit width gst bed-material load rate in weight per unit time and unit width G bedload rate in weight per unit time S G suspended load rate in weight per unit time SS bed-material loadrate in weight per unit time Gst n Manning roughness value p fraction of bed rnaterial in a given grain size p wetted perimeter 3 water discharge (m /s) A hydraulic radius ~ = p S channel slope u fluid velocity shear or friction velocity v settling velocity of particle 0 3 density of fluide For water at 20 C l = 1000 kg/m 3 density of particle.Usually taken as 2650 kg/m when the actual value is unknown 0 3 fluid specif~c weight. Water at 20 C = 1000 kgtm .; ... 3 partiele specifie weight. Taken usually as 2650 kgf/m when actual value is unknown o· kinernatic viscosity of fluide For water at 20 C. -2 2 = 10 cm /s shear stress or friction force per unit area exerted by the fluid at a depth y above the bed shear stress at the bottom 'Ï"'" = y RS or 1: = "(DS \0'0 H 0 other symbols are defined in due course in the following sections. l INTRODUCTION The bed-material load is made up of only those particles consisting of grain sizes represented in the bed. In theory the bed-material load can be predicted with the hydraulic knowledge of the streamJ that is, velocity bed composition and configuration shape of the measuring section water temperature concentration of fine sediment Therefore the problem at issue is to determine the relationship between the bed-material load and the prevailing hydraulic conditions such a problem has proved to be a difficult task and is not yet completely solved. 50 far comparisons of measured and calculated bed-material loads exhibit discrepancies which lead to think that first the problem o~ sediment transport is not fully understood and second great care must be taken in using bed-material load formulae. As pointed out by GRAF (see references at the end) "Einstein's method represents the most detailed and comprehensive treatment, from the point of fluid mechanics, that is presently available". This method is described in the following paragraphs. Nota We prefer the name "Bed-material load" to the name "Total load" since the so-called "washload" is not taken into account when one speaks of bed-material load. 2 EINSTEIN'S PROCEDURE Introduction The bed-materi~l load is divided in two parts according to the mode of transport. In the immediate vicinity of the bed in the so-called bed layer takes place the bedload whereas the suspended-load takes place above the bed layer where the particle's weight is supported by the surrounding fluid and thus the particles move with the flow at the same average velocity: Some researchers think the division of the bed-material load in two fractions is questionable. Actually such a division is rather artificial particularly when it comes to define a zone of demarcation between bed-load and suspended-load, nevertheless it is often convenient for the sake of clarity to distinguish these two modes of transport. Nota Figures number 2 ta number 9 are grouped fr.om page 15 to page 20. 1. HYDRAULIC C.l\LCULATIONS 1.1 Test Reach To calculate or measure the flow and the sediment transport in a stream, a test reach has to be selected first, the following requirements have to be fulfilled, the better they are the more reliable the results. It should be sufficiently long to determine rather accurately the slope of the channel It should have a fairly uniform and stable channel geometry with uniform flow conditions and bed material composition It should have a minimum of outside effects such as strong bends, islands, sills or excessive vegetation No tributaries should join the river within ~r immediatly above the test reach. It is worth noting that the foregoing requirements are those usually . sought-for to set up a gauging station. 3 1.2 Surface Drag and Bed-Form Drag (or Bar Resistance) To take into account the contribution the bedforms make to the channel roughness it was proposed that both the cross section area, denoted A, and the hydraulic radius, denoted ~, be di~ided into two parts: one related to the surface drag or grain roughness designated by A' and R' , the other related H to the bedform drag designated by A" and R~ respectively. In terms of hydraulic radii we have + R" = H It follows that both shear stress and friction velocity are in turn divided since: ~= = Y(RH ~ R")S and (1) = = (2) so we have: a. In terms of shear stresses 't" + 't;' (3) = o 0 b. in terms of friction velocities = (4) the "prime", 1 , used in the notation pertains to the surface àrag whereas the "double prime", fi , pertains to the bedform drag. Einstein and Barbarossa derived a curve fram data of river measurements which relates the "flow intensity" denoted y 35 and defined as 4 = is the bed sediment size forwhich (5 ) RES 35% of the material is finer) u to the ratio of the me an stream velocity, denoted u, to the friction u" * velocity due to the bar resistance denoted u;. This curve which has come to be known as "bar resistance curve" is shown in fig. 3. Nota: Different bedform shapes are sketched in Annex 1 1.3 Mean Velocity DePending on the surface roughness, Einstein and Barbarossa recommended use of either the Manning-Strickles equation or a"logarithmic type formula. a. Manning-Strickler's equation Is defined as u 7.66 (RH )i/6 (6 ) = d u' * 65 where d is the bed sediment size for which 65% of the bed material is 65 finer. The well-known Manning fonnula is defined as 1 1 2 u = R 3/2 5 / (7) n H 5 Let us assume firstly the velocity would be the same with a fIat bed and secondly the bedform would affect both the roughness coefficient n and the hydraulic radius i1i Be n' and ~ the values when no bedform exists. 50 we have l 3/2 u = R' si/2 {8} n' H By combin~ng {7} and {8} we get: n' ~ { }3/2 {9} n ~ and by combining· {6} and {8} we get d 1/6 n' 65 . {10} 24 Equations {9} and (lO) enable to ascertain whether there is a bedform drag or not and ta calculate ~ if need be. This is the case when direct measurement were made of the mean velocities for examp1e at a permanent gauging station. b. Logarithrnic Type Formula Einstein and Barbarossa chose the fo110wing equation which was derived from Nikuradse's experirnents by Keulegan. 12.27 ~ x U 2.3 log { } {11} u k d .*' 65 where k is the Prandtl - Von Karman coefficient equal to 0.4 for clear fluid and, x , is a correction factor for the transition from hydraulically (see AnneA 2 for a discussion about k) "rough to hydraulically srnooth surface, 6 d 65 the roughness being in turn related to the ratio T' where ~ is the thick- ness of the so-called laminar sublayer and is defined as 11.6 J) (J,I kinematic viscosity of the fluid) (12) u~ In figure 2, the factor x is given as a function of Use of Manning-Strickler's formula is recommended when the grain rough­ ness produces a hydraulically rough surface, i.e. when d65 is more than d 65 ~ about 5. Whereas use of a logarithmic formula when r ~s less than about 5 (see fig. 2). In case direct measurements of velocities are made, a trial and error procedure is used to determine R' and x. The chosen values have not only H to verify equation (11) but to verify both the 2 functions depicted by the curves given in figures 2 and 3. 1.4 Step by Step Procedure for Hydraulic Calculations Once· a test reach has been selected, the following informations are needed. l. Slope 2. Description of the cros~ section, that is, 2.1 Curve of ~ versus D A Cross section area 2.2 Curve of A versus DD Depth or stage 2.3 Curve of p versus D P Wetted perimeter 3.
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