INITIATION OF MOTION AND BED LOAD TRANSPORT OF NONUNIFORM SEDIMENTS

Abstract

Knowledge of the entrainment and transport of sediments is necessary in solving various practical problems like soil erosion and conservation, stable channel design and computation of bed level changes in streams. The Critical Tractive Stress (CTS) and Bed Load transport of uniform sediments is reasonably well understood at present. However, the entrainment and transport characteristics of the different size fractions in a nonuniform sediment bed are different from those of uniform sediment bed, on account of exposure and sheltering effects in nonuniform sediment beds. In general, sediment nonuniformity increases with increase in the average size of the sediment and hence its effect is expected to be more important in initiation and bed load transport of coarse sediments. Egiazaroff (1965), Ashida-Michiue (1971) and Hayashi et al. (1980) presented relationships, which reveal that the CTS of a particular size fraction, dj, is dependent on the ratio of this size to the arithmetic mean size, da, as well as the CTS of the arithmetic mean size. There is no satisfactory method at present of calculating the CTS of the arithmetic mean size. Some investigators like Parker et al. (1982) and Wilcock and Southard (1988), presented relationships based on the median size (d50) and advocated the notion of equal-mobility, viz. that all particles on a sediment bed start moving at the same bed shear stress and it could be taken as the CTS of the median size. Further, Wilcock and Southard (1988) recommended that the CTS of median size could be estimated as per modified Shields' criterion, but these values are less than those recommended by Parker et al. (1982). Nakagawa et al. (1982) considered the geometric standard deviation, o-e, as an additional parameter influencing the CTS of nonuniform sediments. u) Studies by Wiberg-Smith (1987), Bridge and Bennett (1992) and Wilcock (1993) give results at variance from those mentioned earlier. Einstein (1950) proposed the first semi-theoretical model for fractionwise calculation of bed load transport rates. Misri et al. (1984) pointed out the limitation of Einstein's method, viz. nonconsideration of exposure effects of coarse fractions and presented the concept of an exposure-cum-sheltering parameter for the calculation of bed load transport rates - a concept also used by Proffit et al. (1983) and Samaga et al. (1986). Bridge and Bennett (1992) presented a semi-theoretical model for bed load transport of nonuniform sediments, but it involves estimation of some model parameters like angle of repose, <p, and dragpartition about which there are some uncertainties. Mittal et al. (1990) checked many of the available relationships including that of Ashida-Michiue (1971), for their accuracy and proposed a modified form of Samaga et al. (1986) relationship. This is based on data with M* 0.18 and may thus need modification for lower values of M. Here Mis Kramer's uniformity coefficient. The critical review on the subject thus suggested the need for an in-depth study related to the CTS and bed load transport of nonuniform sediments and the same was taken up during this investigation. An extensive set of experiments was conducted in the Hydraulics Laboratory of the University of Roorkee, Roorkee, India, in a tilting, recirculating flume having 12.0 m working length, 0.4 m width and 0.52 mdepth. Five different sediments were used for the present study. The arithmetic mean size, da, and geometric standard deviation, <rg, of sediment mixtures were varied from 3.16 mm to 6.10 mm and 1.73 to 2.90 respectively. The median - sized fractions of the first Ui) two mixtures (3.35 mm - 4.00 mm) were painted with white colour, while red colour was used to paint the median sized fractions (2.00 mm - 2.80 mm) of the remaining three mixtures, to enable visual observation of their incipient motion conditions. Apart from visual observations of the incipient motion conditions for the median sizes, bed load transport rates were measured for different fractions under different conditions. The data from previous investigations were also used in the analysis and they covered a range of <rg from 1.41 to 7.43 and da from 1.81 mm to 41.65 mm. The data were first used for verifying the existing relationships for CTS of nonuniform sediments. None of the methods gave satisfactory results over a wide range of conditions. Hence it was felt that there is need to develop a new relationship considering all the relevant parameters influencing the CTS of nonuniform sediments. A relationship based on the representative sizes CI35, d9o and Kramer's uniformity coefficient, M, was proposed. The reason for selecting d35 and doo as the characteristic sediment sizes is that they have been used, respectively, in several resistance and bed load transport studies (Garde and Ranga Raju, 1985). After a number of trials it was found that T^/x^gQ was showing a systematic variation with d,/d35 and M. The proposed relationship based on these parameters is given by :*ci . . dj *c90 ' d35 where t*cj = dimensionless CTS of a particular size fraction, dj, and computed as xci/(A*s.dj); t*cqo = dimensionless CTS of d9Q, and computed as xc9o/(A*s.d9o); -.986 1.552 [ J7L_ { 1.908 M - 0.073} J (1) i iii) Tci and Tc90 are tne CTS of a particular size fraction, dj, and 6gQ respectively, A?s = ys - yp ys, yf are unit weights of sediments and water respectively. Interestingly, t^q was found to be constant and equal to 0.018. Thus Eq. (1) reduces to d: -0.986 x*ci = 0.0279 I —i- { 1.908 M- 0.073} J (2) d35 Hence using Eq. (2), the CTS of a particular size fraction, dj, can be estimated. Check on several existing relationships for bed load transport revealed that none of the methods gave good results over a wide range of variables. As such, the data were analysed afresh to get a reasonably good transport relationship. The bed load transport law, for uniform sediments, proposed by Misri et al. (1984) and subsequently extended by Samaga et al. (1986), was taken as the basis for studying the effect of nonuniformity. The bed load transport rate of a particular size fraction, dj, can be estimated, if its exposure and sheltering Teff correction factor, £B, is known. Here, £b = —T_. where xeff is the grain shear To stress, which would cause the same transport rate of a uniform size fraction, dj, as caused by t0 in a nonuniform sediment bed of that particular fraction, dj. Semi-analytical and empirical approaches were both used for the development of a relationship for ^g. Uv) Following Misri et al. (1984), a semi-analytical approach was adopted for developing a predictor for £q. The following were the important premises made in this approach. (i) Particles finer than arithmetic mean size, da, are entrained due to lift force only, while drag and lift are both responsible for entraining the particles coarser than the arithmetic mean size. (ii) Arithmetic mean size, da, is not influenced by exposure and I sheltering effects. The expressions for £3 were obtained on the above premises as : ?b =kl [i +4-{^^ •t >^tci •t H. for di<da 0) J x0 dj ^L da di ? and, CB = kD 10.719 {log {30.2 (-1 - 0.5)})z\, for di>da (4) da Here Kl is a parameter, which takes into account the reduction in lift force, as the particle is lifted up from the bed and Kp is a parameter accounting for the velocity distribution being not truly logarithmic. While Kl was found to be related to M, dj/da and t0/(a?s da), Krj was found to be only a function of T6/(A^s.da)- Despite some theoretical support for the expression of £b. the eventual errors in the bed load transport rates computed using this model were rather large for some data. (v) Keeping in mind the inadequacies of the existing relationships and the aforesaid lift and drag model, an empirical bed load transport relationship has .!• been proposed. Following the earlier studies at Roorkee, one could write ^B = f [M, t^/toc , T0/(Ays<di) 1 (5) Here toc = CTS of the arithmetic mean size as per Shields. Detailed analysis of the data led to the equation : To -0.75144 Cm . CB = 0.0713 [ Cs. j^—jl (6) where Cm = 0.7092 flog (M) 1 + 1.293, for 0.05 < M < 0.38 (7a) Cm = 1.0, for M * 0.38 (7b) and log (Cs) = -0.1957 -0.9571 [log (—)]- Toc 0.19485 [log (—)l2+0.0644 [log (—)13 (8) Toc Toc Using Eqs. (6), (7) and (8) exposure-cum-sheltering correction factor £g lor a particular size fraction can be estimated and hence bed load transport rate of that fraction by substituting CB-T6^A^s-di) instead of x0/(Ars.d) in bed load transport law for uniform sediments.