<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 />
|