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<br /> <br />1688 <br /> <br />OCTOBER 1973 <br /> <br />HY10 <br /> <br />a function of U.djv according to Eq. 17. The data can be fairly well represented <br />by <br /> <br />v <br />~ = 2.05; <br />w <br /> <br />U.d <br />70<- <br />v <br /> <br />. . . . . . . . , . . , . . . . . . . . . . . . . . (19) <br /> <br />The scattering in Fig. 3 is mainly due to the inconsistant definitions of incipient <br />motion used by different investigators and the large range of variation of relative <br />roughness. <br />The basic forms of Eqs. 18 and 19 are derived fr~m concepts generally accepted <br />in fluid mechanics and boundary layer theory. n,.e coefficients in these equations <br />are determined from reliablc data indcpcndcntly collccted by cight different <br />investigators. No complicated calculation is involved in determining the dimen. <br />sionless critical velocity by using Eqs. 18 and 19. These two equations should <br />provide engineers with a useful criterion for incipient motion. <br /> <br />EaUIUBRlUM AND MAXIMUM CoNCENTRATlON <br /> <br />(,. <br /> <br />At very low water discharge, the rate of sediment transport in an alluvial <br />channel is zero. When water discharge and unit stream power arc increased <br />graduaUy. at some point and time particles on the bed will begin to move. <br />As water discharge and unit stream power arc further incrcased to and above <br />certain values, the flow condition of this water and sediment mixture is fully <br />developed, and reaches a dynamic equilibrium between total sediment concentra- <br />tion and effective unit stream power. This balance between total sediment <br />concentration and effective unit stream power in accordance with Eq. 1 was <br />weU demonstrated (40) despitc the existence of different bed forms. The high <br />correlations (40) between the total sediment concentration and effective unit <br />stream power can exist only if the condition that sediment particles are always <br />transported at the maximum or equilibrium condition for the given effective <br />unit stream power and constraints is satisfied. <br />Schumm and Khan (19) measured the total sediment concentration during <br />the development of channel patterns, with entrance placed at an angle, from <br />straight to meandering thalweg, and finally, to braided channels. A zero value <br />of critical unit stream power was assumed in Fig. 4, since no significant <br />improvement of accuracy can be made by using other values of critical unit <br />stream power. Fig. 4 indicates that the balance between total sediment concentra- <br />tion and effective unit stream power is maintained even if a freely developed <br />channel changes its pattern. <br />All the mentioned datu (19,40) were collcctcd under conditions which were <br />considered by thc original investigators as being in equilibrium. Thus, if the ' <br />value of VS i:; predetcrmined as in some laboratory experiments, the equilibrium <br />condition of an alluvial flow can be considered as the condition that the sediment <br />particles are transported at the maximum or equilibrium concent:ation for the <br />given effcctive unit stream power and constraints. If we conSider the total <br />sediment concentration and watcr discharge as independcnt variables of a natural <br />stream, then the natural stream will adjust its unit stream power through the <br />process of scour and deposition so that the (VS - Y cr S) / C!l ratio can reach <br />a minimum value which is determined. by the constramts applied to the stream. <br />A similar conclusion was made by Maddock (I5). <br /> <br />^ <br />t <br />, <br />r,. <br />l' <br /> <br />1, <br /> <br />'.i" <br /> <br />INCIPIENT MOTION <br /> <br />. <br /> <br />DIMENSIONLESS UNIT STREAM POWER EaUATlON <br /> <br />The mechanisms through which fractions of unit stream power are used in <br />transporting sediment by sliding, rolling, jumping, and suspending are very <br />complex. For example, when a sediment particle is sliding along the bed, the <br />power is used mainly to overcome the friction. While in suspension. the power <br />is consumed mainly in supporting the particle from falling. Whether a sediment <br />particle is transported by sliding, rolling, jumping, or suspending depends on <br />the instantaneous flow condition, and in most cases, it is random (41). It is <br />extremely difficult to associate a fraction of unit stream power with a particular <br />mode of transport. The fact that at equilibrium condition the total sediment <br />concentration is always at its maximum, and that a definite amount of effective <br />unit stream power is used in transporting this maximum or cqujJjbrium total <br />sediment concentration for a given sediment and fluid character, enables us <br />to consider the problem of sediment transport as a whole regardless of the <br />difference between bed load and suspended load. <br />Dimensional analysis is a powerful tool in dealing with a complex problem, <br />if it is properly applied. The outcome of a dimensional analysis depends on <br />the selection of variables. A meaningful and useful result can be expected only <br />if each variable selected for the analysis has a physical significance pertinent <br />to the problem involved. The variables involved in the determination of total <br />sediment concentration can be described by <br /> <br />He" VS, u., v, w. d) = 0 <br /> <br />. . . . . . , . . . . . . . . . (20) <br /> <br />The VS product, defined as the unit stream power, is considered as a single <br />variable because of its importance to the study of sediment transport. The <br />significance of other variables to the total sediment concentration is well known. <br />By the use of Buckingham's 1T theorem, Cf in Eq. 20, can be expressed by <br /> <br />C, = <(>' (VS ,~, W d) . . . . . . . . . . . . . . . . . . . . . . . . . . , (21) <br />W w v <br /> <br />Eq. 21 only suggests that C, is related to the dimensionless unit stream power, <br />VS/w, the ratio of shear velocity and fall velocity U./w, and the fall velocity <br />Reynolds number, Old/v. Because a critical unit stream power, VcrS, is required <br />to start the sediment movement, Eq. 21 is modified to <br /> <br />C,=<I> (VS _ VaS ,~, Wd) <br />Ol Ol W v <br /> <br />. (22) <br /> <br />The final form of Eq. 22 has to be detcrmincd from the analysis of actual <br />data. <br />The primary data used herein to test the validity of the concept of unit stream <br />power and the determination of the final form of Eq. 22 are summarized in <br />Table 2. Collectively, 1,093 sets of laboratory flume data with concentration <br />greater than 10 ppm werc collectcd by Gilbcrt (6), Nomicos (17), Vanoni and <br />Brooks (32), Kennedy (12), Stcin (26), Guy, Simons, and Richardson (8), and <br />Williams (36). Sixty-five sets of field data were collected by Colby and Hembree <br />(4), Hubbell and Matejka (9), and Jordan (10). From the analysis of these data, <br />it was found that the equation <br /> <br /> <br />.' <br /> <br />, ~ <br /> <br /> <br />i <br /> <br />