Validation and implications of an energy-based bedload transport equation

Transcription

1 Sedimentology (2012) 59, doi: /j x Validation and implications of an energy-based bedload transport equation PENG GAO Department of Geograpy, Syracuse University, Syracuse, NY 13244, USA ( maxwell.syr.edu) ABSTRACT A recently developed bedload equation (Abraams & Gao, 2006) as te form i b = xg 3 4, were i b is te immersed bedload transport rate, x is te stream power per unit area, G =1) c /, is te dimensionless sear stress and c is te associated tresold value for te incipient motion of bed grains. Tis equation as a parsimonious form and provides good predictions of transport rate in bot te saltation and seetflow regimes (i.e. flows wit low and ig values, respectively). In tis study, te equation was validated using data independent of tose used for developing it. Te data represent bedload of identical sizes transported in various steady, uniform, fully roug and turbulent flows over plane, mobile beds. Te equation predicted i b quite well over five orders of magnitude. Tis equation was furter compared wit six classic bedload equations and sowed te best performance. Its teoretical significance was subsequently examined in two ways. First, based on collision teory, te parameter G was related to te ratio of grain-to-grain collisions to te total collisions including bot grain-to-grain and grain-to-bed collisions, P g by P g = G 2, suggesting tat G caracterizes te dynamic processes of bedload transport from te perspective of granular flow, wic partly accounts for te good performance of te equation. Moreover, examining te ability of two common equations to predict bedload in gravel-bed rivers revealed tat G can also be used to simplify equations for predicting transport capacities in suc rivers. Second, a simple dimensionless form of te equation was created by introducing B = i b /x. Te teoretical nature of te term B was subsequently revealed by comparing tis equation wit bot te Bagnold model and two commonly used parameters representing dimensionless bedload transport rates. Keywords Bedload transport, stream power, granular flow, gravel-bed rivers. INTRODUCTION Intrinsic properties of natural rivers, suc as eterogeneous sediment transport, te interaction between sediment supply and bed surface adjustment, and te ydrodynamics of bedform (for example, sand bars) evolution, make te relationsip between bedload transport rates and ydraulic variables extremely complex. Altoug scientists and engineers ave gained profound insigt into te mecanics of bedload transport ever since te development of te DuBoys equation (DuBoys, 1879), te first pysically based bedload transport equation, an apparently simple question still cannot be answered: For given ydraulic and sedimentary caracteristics, wat is te rate of bedload transport in an alluvial cannel? In oter words, tere is no single bedload equation tat can be applied universally to all rivers (Gomez & Curc, 1989; Gomez, 1991; Simons & Senturk, 1992; Yang & Huang, 2001; Almedeij & Diplas, 2003). Tis lack of universal caracterization of bedload transport is caused partly by practical limitations, suc as errors in measurement of sear stress and sediment sampling (Gomez & Curc, 1989), and is partly due to complexities introduced by eterogeneity of sediment sizes (Parker et al., 1982; Wilcock, 2001) and non-uniform and unsteady flows in natural rivers. Te latter includes signif Ó 2012 Te Autor. Journal compilation Ó 2012 International Association of Sedimentologists

2 An energy-based bedload transport equation 1927 icant canges of water surface slope during floods (Meirovic et al., 1998; Powell et al., 2006), cross-cannel variations of bed sear stress, flow velocity and transport rates (Powell et al., 1999, 2006; Ferguson, 2003) or bedload pulsing due to te pase difference between sediment and water discarges (Reid & Frostick, 1987). Witout tese limitations and complexities, can bedload transport be universally caracterized by a single equation? A recently developed bedload transport equation (Abraams & Gao, 2006) for steady, uniform, fully roug and turbulent flows, transporting a bedload of omogeneous grains over plane, mobile beds (ereafter referred to as ideal flow) as provided an affirmative answer. Tis equation was developed based on te Bagnold energy model (Bagnold, 1973): i b tan a ¼ e b x ð1þ were tan a is te dynamic friction coefficient, e b is te transport efficiency, i b = q b g(q s ) q) is te immersed bedload transport rate (J s )1 m )2 ), q b is te volumetric bedload transport rate (m 2 s )1 ), g is te acceleration of gravity (m s )2 ), q s is te density of bedload grains (kg m )3 ), q is te flow density (kg m )3 ), x = su is te stream power per unit area (or unit stream power) (W m )2 ), s = qgs is te bed sear stress (N m )2 ), is te mean flow dept (m), S is te energy slope and u is te mean flow velocity (m s )1 ). Te dynamic friction coefficient, tan a is defined as te ratio of te tangential sear stress due to grain collisions, T g to te associated normal dispersive stress, P, wic is equivalent to te submerged weigt of bedload grains, W. In general, bedload may be transported in eiter te saltation or seetflow regimes, wic are ydraulically distinguised by 0Æ5 (Gao, 2008), were = qs/((q s q)d 50 ) is te dimensionless sear stress and D 50 is te median size of bedload grains (m). In te saltation regime, grains move along te bed by sliding, rolling or saltating wit values less tan 0 5. Te total sear stress available for transporting bedload, T, wic is equivalent to t ) t c, were s c is te tresold value of s for te incipient motion of grains, contains not only T g but also a component tat is transferred to te bed due to friction or turbulence to maintain te equilibrium of te fluid (Graf, 1998), T f. Tus, T = t ) t c = T g + T f. In te seetflow regime were > 0 5 and grains mainly travel in loosely defined granular layers witin a zone tat is muc ticker tan D 50, te proportion of T f becomes smaller and smaller as increases. At very ig values, T T g. Terefore, Eq. 1 is teoretically confined to flows wit ig values in te seetflow regime. By replacing tan a in Eq. 1 wit te stress coefficient s b = T/W, Abraams & Gao (2006) modified te original energy model as: i b s b ¼ e b x and developed equations for bot s b and e b : s b ¼ 06G 2 e b ¼ 06G 14 ð2þ ð3aþ ð3bþ A new bedload equation was subsequently establised by combining Eq. 2 wit Eqs 3a and 3b: were: i b ¼ xg 34 G ¼ 1 c ¼ 1 u2 c u 2 ð4þ ð5þ u * =(gs) 0 5 is te sear velocity (m s )1 ), and c and u *c are te tresold values of and u * for te incipient motion of grains on te bed. Equation 4 applies for bot te saltation and seetflow regimes and serves as a universal equation to predict bedload transport rates under any ydraulic and sedimentary conditions in te ideal flows. Altoug of Eq. 4 is only valid in te ideal flows, its simplicity merits furter investigation. Tis article presents suc an investigation. It begins by validating Eq. 4 using an independent set of bedload data, wic is followed by comparing te predictive power of Eq. 4 wit tat of several classic bedload equations. Ten, te teoretical implications of Eq. 4 are revealed by demonstrating te meaning of G in Eq. 4 and illustrating te advantage of a new dimensionless form of Eq. 4, respectively. Te article ends wit a discussion of te two dimensionless variables in a new dimensionless form. VALIDATING THE PREDICTABILITY OF EQUATION 4 Altoug Eq. 4 was tested in Abraams & Gao (2006), furter validating it not only assures its robustness and generality but also justifies te subsequent analysis in tis article.

3 1928 P. Gao Data compilation A data set tat is for bedload of omogeneous grains and entirely independent of tat used by Abraams & Gao (2006) was compiled to validate Eq. 4. Using tis data set avoids te possible errors raised from validating an equation by employing part of te data used to develop it (Gomez & Curc, 1989). Te validation data set consists of 264 flume or closed-conduit experiments from Recking (2006) and Nnadi & Wilson (1992), and tose compiled by Jonson (1943), Smart & Jäggi (1983) and Gomez & Curc (1988). To ensure te compatibility of data from different experiments, te sidewall-drag effect was corrected using te metod of Williams (1970) except for te data from Nnadi & Wilson (1992) wo corrected te original data using teir own metod. Te effect of slope on te bed sear stress was corrected using te same metod employed by Abraams & Gao (2006). No teoretical metod exists to accurately determine values of c (Buffington & Montgomery, 1997). Terefore, for eac series of experiments, te value of c was determined by fitting lines troug plots of i b against, extending tese lines to te axis and recording te value of, were i b = 0. For experiments were te smallest values of i b were significantly greater tan 0, suc as tose from Smart & Jäggi (1983) and Nnadi & Wilson (1992), te extrapolation metod failed to provide a reasonable estimate of c and tus te value of c was set to equal to Te selected data were filtered to only keep tose representing turbulent and fully roug flows, wic means tat te flow Reynolds number R =4u/m > 8000 and te rougness Reynolds number R ks = k s u * /m > 70 were m is te kinematic viscosity (m 2 s )1 ) and k s was determined in te same way as in Abraams & Gao (2006). Wen a grain is entrained in a flow, weter it moves as bedload or goes into suspension depends on te tresold value of te dimensionless settling velocity W = w/u *, were te settling velocity, w, was determined using te equation developed by Ceng (1997a). In te saltation regime, suspension occurs wen W < 1Æ5, wile in te seetflow regime, te initiation of suspension occurs wen W <0Æ8 (Abraams & Gao, 2006). However, in te seetflow regime, bedload concentration is significantly iger tan tat in te saltation regime. Given tat w is positively correlated wit concentration (Ceng, 1997b), te true value of w sould be greater tan tat calculated using te Ceng (1997a) equation. Inasmuc as suspended concentration begins to increase significantly wen W =0Æ15 (Wilson, 2005), te tresold value of W in tis study was set as 0Æ55, between 0Æ8 and 0Æ15, to take sediment concentration into consideration. Tose measurements wit calculated values of W less tan 0Æ55 in te seetflow regime ave significant suspended sediment and were excluded from te validation data set. Validation Data from 186 experiments (Table 1) performed in te ideal flows were terefore selected for validating Eq. 4. Tese data ad values ranging from Table 1. Ranges of relevant ydraulic and sedimentary variables of te compiled independent experimental data in te ideal flows. Sources Nnadi & Wilson (1992) Recking (2006) Jonson (1943) Smart & Jäggi (1983) Gomez & Curc (1988) Total Number of experiments Re ( 10 6 ) 0Æ135 0Æ256 0Æ014 0Æ128 0Æ039 0Æ303 0Æ014 5Æ04 0Æ347 0Æ574 0Æ014 5Æ04 Re * Fr* 2Æ53 3Æ22 1Æ12 2Æ58 0Æ48 1Æ85 0Æ42 1Æ67 0Æ80 0Æ95 0Æ45 3Æ22 D 50 ( 10 )3 m) 0Æ7 2Æ3 9Æ0 1Æ7 7Æ1 5Æ2 28Æ7 6Æ5 0Æ7 28Æ7 0Æ964 1Æ850 0Æ091 0Æ312 0Æ049 0Æ260 0Æ05 0Æ144 0Æ034 0Æ074 0Æ034 1Æ850 c 0Æ04 0Æ04 0Æ03 0Æ053 0Æ04 0Æ029 0Æ029 0Æ053 B 0Æ858 1Æ168 0Æ088 0Æ627 2Æ9E-04 0Æ481 1Æ7E-3 0Æ190 0Æ0005 0Æ0534 2Æ9E-04 1Æ168 *Froude number Fr = u/(g) 0Æ5. Tese data contain several sets of flume experiments, eac of wic as a c value.

4 An energy-based bedload transport equation Æ034 to 1Æ85 and ence covered bedload transport in bot te saltation and seetflow regimes. Bot subcritical and supercritical flows were included and values of D 50 in tese data ranged from sand (0Æ0007 m) to gravel (0Æ0287 m). Terefore, te compiled data covered a wide range of ydraulic and sedimentary conditions. Comparison of predicted wit measured i b (Fig. 1) sowed tat data points were generally located symmetrically around te line of perfect agreement over five orders of magnitude. Only less tan 9% of te total points were plotted outside of (but still close to) te zone between te discrepancy ratios (van Rijn, 1984; Almedeij & Diplas, 2005; Camenen & Larson, 2005) of 0Æ5 and 2, wic furter indicates te general agreement between measured and predicted values of i b. Given tat te selected data are independent of tose used in Abraams & Gao (2006), te good agreement between measured and predicted i b confirms te general success of Eq. 4 for predicting bedload transport rates of omogeneous sediments in various steady, uniform, fully roug and turbulent flows over plane, mobile beds. By converting i b into te well-known dimensionless form / = q b /(g(q s -q)/qd 50 ) 0 5 D 50, first introduced by Einstein (1950), Eq. 4 may be made dimensionless (Abraams & Gao, 2006): / ¼ 15 G 34 u u ð6þ Te predictive ability of Eq. 4 was subsequently compared wit tat of six classic and well-known bedload equations developed in terms of excess dimensionless sear stress, ) c (see te first six equations in Table 2). All of tese equations can be presented in dimensionless forms similar to Eq. 6. Te last two equations in Table 2 were developed for fractional bedload transport over beds of eterogeneous grains and were selected for illustrating a property of G tat is elaborated below. Using te independent data described previously, i b values predicted by eac of te first six equations were compared wit te measured transport rates (Fig. 2A to F). Te equations of Meyer-Peter and Müller, Sields, Bagnold and Fernandez Luque & van Beek can only predict a small proportion of te data reasonably well (i.e. tose falling in te zone between te two discrepancy lines). Te equations of bot Smart and Yalin can predict medium and ig bedload transport rates reasonably well, but generate significantly larger errors in predictions for low bedload transport rates. None of tese equations predict bedload transport rates in bot te saltation and seetflow regimes as well as Eq. 4 does. Te results sown in Figs 1 and 2 verify tat bedload transport rates for omogeneous sediments in ideal flows can generally be predicted by Eq. 4. Matematically, te first six equations in Table 2 ave a coefficient wit different values, suggesting tat tey can only perform well for flows tat ave similar ydraulic and sediment conditions to tose of te data used to develop tem. By contrast, te dimensionless form of Eq. 4 (i.e. Eq. 6) as no coefficient, implying tat Eq. 4 reflects te functional relationsip between bedload transport rates and te ydraulic variables involved. Some insigt may be gained by investigating te teoretical meaning of te variables in Eq. 4. THEORETICAL SIGNIFICANCE OF EQUATION 4 Fig. 1. Predicted i b using Eq. 4 versus measured i b. Te two dased lines represent te discrepancy ratios of 0 5 and 2, respectively. Te boundary i b value between te saltation and seetflow regimes is ca 0Æ65 kg m )1 s )1. Te parameter G Te pysical meaning of te dimensionless parameter G in Eq. 4 can be uncovered using te collision teory (Leeder, 1979). In tis teory, Leeder (1979) assumed tat bedload grains moving in saltation are controlled by grain-to-bed (GB) and grain-to-grain (GG) collisions. Leeder (1979) ten derived an expression for te mean

5 1930 P. Gao Table 2. Eigt selected bedload equations. Te first six are based on excess dimensionless sear stress c, wile te last two are based on te transport stage / c. Sources Meyer-Peter & Müller (1948) Fernandez Luque & van Beek (1976) Bagnold [obtained from Yalin (1977)] Sields [obtained from Simons & Senturk (1992)] Smart (1984) Bedload equations / =8( c ) 1Æ5 =8 1Æ5 G 1Æ5 / ¼ 57ð c Þ 15 ¼ G 15 / =4Æ25 0Æ5 ( ) c )=4Æ25 1Æ5 G / ¼ 10ð cþðu=sþ ½gDðs 1ÞŠ ¼ 102 Gðu=sÞ ½gDðs 1ÞŠ ; s ¼ qs q 1 / ¼ 4S 06 u u 05 ð c Þ¼4S 06 u u 15 G Yalin (1977) / ¼ A 05 ð c Þ¼A 15 G; A ¼ 0635 lnð1 þ asþ 1 c as ; as ¼ S04 c 1 c " Parker et al. (1982) / ¼ exp # 2 1 c c / ¼ = c c Wilcock & Crowe (2003) / ¼ <135 c c! / ¼ ð= c Þ c c 165 free pat lengt of a saltating grain, k, in terms of te gaseous kinetic teory and compared it wit te measured mean lengt of a saltation trajectory, L, to caracterize te nature of GG and GB collisions. Tis expression led to te following outcomes: for u * /u *c <2,k > L meaning tat te transport of bedload grains is dominated by GB collisions; for u * /u *c >2,k < L suggesting tat GG collisions dominate. At u * /u *c 2, k = L signifying tat te probability of GB and GG collisions is te same. Based on te concept of GG and GB collisions, a new variable, P g (Abraams & Gao, 2006), was defined as te relative frequency of GG collisions wit regard to te total (GG and GB) collisions during bedload transport at a given transport stage, wic is commonly defined using u * /u *c (Leeder, 1979), s/s c (or / c ) (Wiberg & Smit, 1989). Tis variable sould be negatively related to k/l but te specific function cannot be derived directly. However, by definition, wen k = L, P g =0Æ5; tis provides a reference point for constructing a simple function tat satisfies te negative trend between P g and k/l: P g ¼ 1 ð7þ n þ 1 were n ¼ k=l. Several pairs of u * /u *c and n were reproduced from te data sown in figure 4B of Leeder (1979). Values of n were subsequently used to calculate P g (Table 3). Examining te relationsip between P g and a variety of ydraulic variables sowed tat te data in Table 3 may be best modelled by (Fig. 3): P g ¼ G 2 ð8þ Equation 8 signifies tat G is directly related to te relative frequency of GG collisions during bedload transport in bot te saltation and seetflow regimes. As increases, more grains can be transported as bedload and ence te probability of GG collisions increases, wic is quantitatively associated wit te increase of G.

6 An energy-based bedload transport equation 1931 Fig. 2. Comparison of predicted bedload transport rates using te selected eigt equations (Table 2) wit te measured ones: MPM Meyer-Peter and Müller; LvB Fernandez Luque and van Beek; PKM Parker, Klingeman and Mclean; WC Wilcock and Crowe.

7 1932 P. Gao Table 3. Values of u * /u *c and P g obtained from te data in figure 4B of Leeder (1979). u * /u *c P g Æ Æ Æ Æ Æ Æ85 Fig. 3. P g versus G. Te open circles are te data reproduced from te data in figure 4B of Leeder (1979), te solid curve is Eq. 8, and te two solid dots are two extreme points tat satisfy Eq. 8 (i.e. te boundary conditions). Terefore, G is a critical variable tat captures te dynamic processes of bedload transport for omogeneous grains over mobile beds wen bedload transport is viewed from te perspective of granular flow (i.e. bedload transport is te process of grain movement). It follows tat tis parameter alone can account for te canges of bedload transport rates wit values and lead to a simple bedload equation witout any coefficient (i.e. Eq. 4 or Eq. 6) tat performs well in bot te saltation and seetflow regimes. Altoug G is also included in te six classic equations (Table 2), it is essentially derived from te commonly used dimensionless excess sear stress, ) c and is not an independent variable as in Eq. 6. Natural rivers do not generally transport omogeneous sediments and flows are more complex tan te ideal flows in te experiments used to define and validate Eq. 4. Tis model cannot, terefore, be used to predict bedload for eterogeneous sediments in gravel-bed rivers. However, te parameter G still provides insigt into te associated transport mecanics. For instance, two well-known bedload equations for bedload of eterogeneous sediments (te last two equations in Table 2) involve two different expressions for estimating bedload transport rates in two different ranges of / c values. Tis suggests tat bedload of finer sediments for low / c values follows different ydraulic rules from tat of coarse sediments for ig / c values. However, wen te two equations were applied to te independent data set for predicting bedload of identical sediments, te trends of te predicted i b values are approximately parallel to te line of equality (see Fig. 2G and H), altoug neiter of tem predicts bedload transport rates as well as Eq. 4 does. Tis implies tat use of G for bedload of eterogeneous grains migt enable collapse of te two different expressions for te two different ranges of / c values. Following from tis, a bedload equation tat predicts bedload transport capacity of eterogeneous sediments in gravel-bed rivers as been developed recently using only one independent variable, G (Gao, 2011). Te utility of G for caracterizing te mecanics of bedload transport provides furter evidence to support te emerging assertion tat bedload transport sares many common properties wit granular flows, suc tat more insigt may be gained if bedload transport is treated as te process of granular interaction/collision (Frey & Curc, 2011). A different dimensionless form of Equation 4 Equation 4 indicates tat as sear stress increases, submerged bedload transport rate, i b increases faster tan te unit stream power, x. At very ig sear stress, te bedload transport rate, i b numerically approaces x. Tis finding is at odds wit te Bagnold (1973) model. Rearranging Eq. 1 gives: i b ¼ e b tan a x ¼ 167e bx ð9þ were tan a 0Æ6. Bagnold (1973) asserted tat wen te transport efficiency reaces its maximum, e b = 100% and i b /x =1Æ67. In oter words, i b can be numerically greater tan x. Te source of tis errant assertion is tat te transport efficiency can never reac 100%. Rater, its maximum value is 60% (see Eq. 3b and Abraams & Gao, 2006). Terefore, te classic representation of a range of e b values (up to 100%) in te plot of i b versus x as a series of curves (Emmett, 1976; Leopold & Emmett, 1976; Reid & Laronne, 1995; Knigton, 1998) sould be modified.

8 An energy-based bedload transport equation 1933 It follows tat x not only represents te total energy a flow as (i.e. te ability of doing work) but also serves as a numerical scale tat can normalize i b tat is, i b in open-cannel flows cannot be greater tan x. Tus, a different dimensionless form tan Eq. 6 may be easily generated as: B ¼ i b x ¼ G34 ¼ 1 34 c ð10þ In addition to its simplicity, Eq. 10 also as a teoretical advantage tat may be revealed by comparing B wit te two widely used parameters representing dimensionless bedload transport rates. Te first is / and te second is W*, wic is defined as W* =(q s /q ) 1)gq b /u 3 * (Parker & Klingeman, 1982). Teir matematical relationsips are: B ¼ i b x ¼ W u u ¼ / u 15 ð11þ u qffiffi Given tat u u ¼ f 8 were f is Darcy Weisbac friction factor, Eq. 10 suggests tat B incorporates te independent impact of resistance to flow on bedload transport rates into bot / and W *. Because f may vary greatly wit G in bot te saltation and seetflow regimes (Fig. 4), use of B circumvents te need for additional variables or constants to account for suc variation and furter explains te generality of Eq. 10. Te implication of Equation 10 Te ydraulic implication of Eq. 10 can be revealed by te following reasoning, based on te assumption tat bedload grains always ave te same size and, ence, c remains invariable. Wit tis assumption, te magnitude of i b could be controlled by, x or bot. Wen two flows ave te same value, meaning te same product of flow dept () and energy slope (S), one flow may ave iger velocity and unit discarge, wic leads to a iger stream power tan te oter. Tis effect can be acieved by aving lower flow resistance in te first flow. Tus te flow wit iger x will transport more bedload (i.e. iger i b ). Wen two flows ave te same x value, te product of unit discarge (Q) and energy slope (S) is te same in te two flows. If te first flow as a ig enoug Q and relatively gentle bed slope wit relatively narrow cannel widt, ten it is possible tat tis flow as a iger sear stress, altoug te associated flow velocity may be low and flow resistance may be ig. Terefore, te iger value will give rise to a iger i b. Tese two cases indicate tat eiter or x may vary, as one of tem remains constant and, consequently, causes a cange of bedload transport rates; tis suggests tat neiter x nor controls bedload transport alone. Pysically, Eq. 10 reveals tat te proportion of bedload transport rate to stream power (i.e. B) is correlated to te relative frequency of GG collisions wit respect to te total collisions, wic can be determined by G. Numerically, bot B and G vary from 0 to 1. Tus, Eq. 10, wic is simple and general, is determined by bot G and B. It suggests tat under simplified, ideal flow conditions, te mecanics of bedload transport is remarkably simple: any combination of x and i b leads to a unique B tat can be determined simply using te associated G. In natural gravel-bed rivers were bedload sediments and tose from bot te bed surface and substrate are eterogeneous, bedload transport is more complex tan described by Eq. 10. Tus, Eq. 10 cannot be used directly to estimate bedload transport rates in natural rivers. However, B and G can still be used to provide some quantitative bencmarks for more complex investigation of bedload transport, as demonstrated in te equation for bedload transport capacities of gravel-bed rivers establised in Gao (2011). CONCLUSIONS Fig. 4. Variations of resistance to flow, f, wit G in bot te saltation and seetflow regimes. A data set was carefully compiled to validate, in two ways, a previously publised equation for predicting bedload under ideal conditions (steady, uniform, fully roug and turbulent flows transporting omogeneous grains over plane, mobile beds) (Equation 4; Abraams & Gao,

9 1934 P. Gao 2006). First, predicted i b values were compared wit measured values and sowed good agreement. Second, six classic bedload equations were selected and compared wit Eq. 4 using te compiled data set. Te results sowed tat Eq. 4 as te best predictive power. Tus, Eq. 4 is a general equation tat can predict bedload transport rates under a wide range of uniform, fully roug and turbulent flows transporting bedload of omogeneous grains over plane, mobile beds. Its form as some significant implications regarding te mecanics of bedload transport. Te parameter G is found to be related to te proportion of grain-to-grain collisions to te total [te sum of grain-to-grain (GG) and grain-to-bed (GB)] collisions, P g by P g = G 2. Tis relation suggests tat if bedload is viewed from te perspective of granular movement, canges in bedload transport rates wit ydraulics are essentially controlled by te cange of te relative frequency of GG and GB collisions, wic is caracterized by G. Altoug flows in gravel-bed rivers are more complex tan te ideal flows, G enables a collapse of two different expressions associated wit te two different ranges of / c in two bedload equations for gravel-bed rivers (Fig. 2G and H) and can be used to determine bedload transport capacities of eterogeneous grains in gravel-bed rivers (Gao, 2011). Terefore, te parameter G is a fundamental ydraulic variable tat captures te processes of bedload transport. Moving x in Eq. 4 to te left gives rise to a muc simpler dimensionless form (i.e. Eq. 10). Comparing te term B wit te two commonly used parameters for dimensionless bedload transport rate, / and W*, sowed tat B accounts for te effect of te cange of resistance to flow on bedload transport and tus as te ydraulic advantage of merging data into a single relation. Te dimensionless bedload transport rate, B in Eq. 10 is fundamentally different from i b /x used by Bagnold (1973) in tat te latter can be greater tan 1 for ig values. Numerically, B ranges from 0 to 1, suggesting tat x can normalize i b suc tat te cange of B is consistent wit te cange of te relative frequency of GG and GB collisions, and ence B can be simply determined using G. REFERENCES Abraams, A.D. and Gao, P. (2006) A bed-load transport model for roug turbulent open-cannel flows on plane beds. Eart Surf. Processes and Landforms, 31, Almedeij, J.H. and Diplas, P. (2003) Bedload transport in gravel-bed streams wit unimodal sediment. J. Hydraul. Eng., 129, Almedeij, J. and Diplas, P. (2005) Bed load sediment transport in epemeral and perennial gravel bed streams. EOS Trans. Am. Geopys. Union, 86, 429. Bagnold, R.A. (1973) Te nature of saltation and of bed-load transport in water. Proc. R. Soc. London 332, Buffington, J.M. and Montgomery, D.R. (1997) A systematic analysis of eigt decades of incipient motion studies, wit spatial reference to gravel-bedded rivers. Water Resour. Res., 33, Camenen, B. and Larson, M. (2005) A general formula for noncoesive bed load sediment transport. Estuarine Coastal Self Sci., 63, Ceng, N. (1997a) Effect of concentration on settling velocity of sediment particles. J. Hydraul. Eng., 123, Ceng, N. (1997b) Simplified settling velocity formula for sediment particle. J. Hydraul. Eng., 123, DuBoys, M.P. (1879) Etudes du regime et l action exercee par les eaux sur un lit a fond de graviers indefiniment affouilabie. Ann. Ponts Causses 5(18), Einstein, H.A. (1950) Te Bed Load Function for Sediment Transportation in Open Cannel Flows. United States Department of Agriculture, Soil Conservation Service, Wasington, D.C. Emmett, W.W. (1976) Bedload transport in two large, gravelbed rivers, Idao and Wasington. Proceedings, Tird Federal Inter-agency Sedimentation Conference, Marc 1976 Vol. 4, , Denver, Colorado. Ferguson, R.I. (2003) Te missing dimension: effects of lateral variation on 1-D calculations of fluvial bedload transport. 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