Marcelo H. García
6.4 THRESHOLD CONDITION FOR SEDIMENT MOVEMENT
When a granular bed is subjected to a turbulent flow, virtually no motion of the grains is observed at some flows, but the bed is mobilized noticeably at other flows. Factors that affect the mobility of grains subjected to a flow are summarized below:
randomness
grain placement turbulence
forces on grain
fluid
lift mean & turbulent drag
gravity
In the presence of turbulent flow, random fluctuations typically prevent the clear defini-tion of a critical, or threshold condidefini-tion for modefini-tion: The probability for the movement of a grain is never precisely zero (Lavelle and Mofjeld, 1987). Nevertheless, it is possible to define a condition below which movement can be neglected for many practical purposes.
6.4.1 Granular Sediment on a Stream Bed
Figure 6.6 is a diagram showing the forces acting on a grain in a bed of other grains. When critical conditions exist and the grain is on the verge of moving, the moment caused by the critical shear stress τcabout the point of support is just equal to that of the weight of the grain. Equating these moments gives (Vanoni, 1975):
FIGURE 6.6 Forces acting on a sediment particle on an inclined bed
τc c in which γs specific weight of sediment grains, γ specific weight of water, D diam-eter of grains, is the slope angle of the stream, the angle of repose of the sediment, c1
and c2are dimensionless constants, and a1and a2are lengths shown in Fig. 6.6. Any consis-tent set of units can be used in Eq. (6.39). For a horizontal bed, Eq. (6.39) reduces to
τc c a1, and a2 are not known. Therefore, the relation between the pertinent quantities is expressed by dimensional analysis, and the actual relation is determined from experimen-tal data. Figure 6.7 is such a relation, first presented by Shields (1936) and carries his name. The curve is expressed by dimensionless combinations of critical shear stress τc, sediment and water specific weights γsandγ, sediment size D, critical shear velocity u*c
τ/ρc and kinematic viscosity of water ν.
These quantities can be expressed in any consistent set of units. Dimensional analysis yields,
The Shields values of τc*are commonly used to denote conditions under which bed sediments are stable but on the verge of being entrained. Not all workers agree with the results given by the Shields curve. For example, some workers give τc* 0.047 for the dimensionless critical shear stress for values of R* u*D/ν in excess of 500 instead of 0.06, as shown in Fig. 6.7. Taylor and Vanoni (1972) reported that small but finite amounts of sediment were transported in flows with values of τc*given by the Shields curve.
The value of τcto be used in design depends on the particular case at hand. If the sit-uation is such that grains that are moved can be replaced by others moving from upstream, some motion can be tolerated, and the Shields values can be used. On the other hand, if grains removed cannot be replaced, as on a stream bank, the Shields value of τcare too large and should be reduced.
The Shields diagram is not especially useful in the form of Fig. 6.7 because to find τc, one must know u* τ/ρc. The relation can be cast in explicit form by plotting τc* ver-sus Rep, noting the internal relation
u*
is the submerged specific gravity of the sediment. A useful fit is given by Brownlie (1981a):
τ*c 0.22Rep0.6 0.06 exp(17.77Rep0.06) (6.44)
RgD D
FIGURE 6.7Shields diagram for initiation of motion. Source Vanoni (1975)
FIGURE 6.8 Angle of repose of granular material. (Lane, 1955)
With this relation, the value of τc*can be computed readily when the properties of the water and the sediment are given.
The value of bed-shear stress τbfor a wide rectangular channel is given by τb γHS, as shown earlier. The average bed-shear stress for any channel is given by τb γRhS, in which Rh the hydraulic radius of the channel cross section.
6.4.2 Granular Sediment on a Bank
A sediment grain on a bank is less stable than one on the bed because the gravity force tends to move it downward (Ikeda, 1982). The ratio of the critical shear stress τwcfor a par-ticle on a bank to that for the same parpar-ticle on the bed τcis (Lane, 1955)
τ τ
w c
= cos φc 1
1
ttaa
nn
φφ1
2
(6.45)
whereφ1is the slope of the bank and θ is the angle of repose for the sediment. Values of θ are
given in Fig. 6.8 after Lane (1955) and also can be found in Simons and Senturk (1976).
6.4.3 Granular Sediment on a Sloping Bed
Equation (6.39) shows that τcdiminishes as the slope angle φ increases. For extremely smallφ’s, τcis given by Eq. (6.40). Taking the ratio between Eqs. (6.39) and (6.40) yields
ττ crit-ical shear stress for a bed with an extremely small slope. The value of τcocan be found from the Shields diagram or with Eq. (6.44). Equation (6.46) is for positive φ, which is positive for downward sloping beds. For beds with adverse slope,φ is negative and the term tan φ/tan θ in Eq. (6.46) is positive.
6.4.4 Sediment Mixtures
Several authors have offered empirical or quasi-theoretical extensions of the above rela-tions to the case of mixtures (e.g., Wilcock, 1988). Let Didenote the characteristic grain size of the ith size range in a mixture. Furthermore, let Dsgdenote the geometric mean size of the surface (exchange, active) layer. Most of the generalizations can be written in the following form (Parker, 1990):
whereτbciandτbcsgdenote the values of the dimensioned critical shear stress required to move sediment of sizes Diand Dsgin the mixture, respectively, and β is an exponent tak-ing a value given below;
β 0.9 (6.49)
Figure 6.9 shows the similarity between four different published expressions having the general form given by Eq. (6.47), which is of interest because it includes the effect of hiding. For uniform material, the critical Shields stress is defined by Eq. (6.44). Consider two flumes, one with uniform size Daand the other with uniform size Db. For sufficient-ly coarse material (u*D/ν » 1 or Rep» 1), the critical Shields stress must be the same for both sizes (Fig. 6.7). It follows from Eq. (6.42) that where τbcaandτbcbdenote the dimen-sioned boundary shear stresses for cases a and b respectively,
τbcb τbca