CHAPTER 4: RELATIONS FOR THE CONSERVATION OF BED SEDIMENT. This chapter is devoted to the derivation of equations describing the conservation of bed sediment. Definitions of some relevant parameters are given below. . q b = volume bedload transport rate per unit width [L 2 T -1 ] - PowerPoint PPT Presentation
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
CHAPTER 4:RELATIONS FOR THE CONSERVATION OF BED SEDIMENT
This chapter is devoted to the derivation of equations describing the conservation of bed sediment. Definitions of some relevant parameters are given below.
qb = volume bedload transport rate per unit width [L2T-1]
qs = volume suspended load transport rate per unit width [L2T-1]
qt = qb + qs = volume bed material transport rate per unit width [L2T-1]
gb = sqb = mass bedload transport rate per unit width [ML-1T-1]
(corresponding definitions for gs, gt) = bed elevation [L]p = porosity of sediment in bed deposit [1]
(volume fraction of bed sample that is holes rather than sediment: 0.25 ~ 0.55 for noncohesive material)
g = acceleration of gravity [L/T2]x = boundary-attached streamwise coordinate [L]y = boundary-attached transverse coordinate [L]z = boundary-attached upward normal (quasi-vertical) coordinate [L]t = time [T]
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
ILLUSTRATION OF BEDLOAD TRANSPORTDouble-click on the image to see a video clip of bedload transport of 7 mm gravel in a flume (model river) at St. Anthony Falls Laboratory, University of Minnesota. (Wait a bit for the channel to fill with water.) Video clip from the experiments of Miguel Wong.
rte-bookbedload.mpg: to run without relinking, download to same folder as PowerPoint presentations.
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
ILLUSTRATION OF MIXED TRANSPORT OF SUSPENDED LOAD AND BEDLOADDouble-click on the image to see the transport of sand and pea gravel by a turbidity current
(sediment underflow driven by suspended sediment) in a tank at St. Anthony Falls Laboratory. Suspended load is dominant, but bedload transport can also be seen. Video clip from
experiments of Alessandro Cantelli and Bin Yu.
rte-bookturbcurr.mpg: to run without relinking, download to same folder as PowerPoint presentations.
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
Let denote the volume concentration of sediment c in suspension at (x, z, t), averaged over turbulence. Here c = (sediment volume)/(water volume + sediment volume).
In the case of a dilute suspension of non-cohesive material,
Ecvxq
t)1( bs
bp
-
bss cvD
where cb denotes the near-bed value of c .
Similarly, a dimensionless entrainment rate E can be defined such that
EvE ss
Thus
bc
)t,z,x(c
zbss cvD
bcc
c
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
Definitions:z = upward normal coordinate from the bed [L] = local streamwise flow velocity averaged over turbulence [L/T] = local volume sediment concentration averaged over turbulence [1]H = flow depth [L]qs = volume transport rate of suspended sediment per unit width [L2/T]U = vertically averaged streamwise flow velocity [L/T]C = vertically flux-averaged volume concentration of sediment in suspension [1]
H
0s dzcuq
cu
u
y c
u
c
u
s
H
0
H
0
qdzcuUCH
dzuUH
z
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
1D EQUATION OF CONSERVATION OF SEDIMENT IN SUSPENSION
(mass of sediment in control volume)/t =net mass inflow rate of suspended sediment+ mass rate of entrainment of sediment into suspension– mass rate of deposition onto the bed
u
x 1
H
sCUH
sCUH
sEs
x sDs
x
xCUHCUHdzcxt sssxxsxs
H
0s
DE
or reducing with the relation qs = UCH and previous evaluations for Es and Ds,
bcE
s
sH
0v
xqdzc
t
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
The active, exchange or surface layer approximation (Hirano, 1972):
Sediment grains in active layer extending from - La < z’ < have a constant, finite probability per unit time of being entrained into bedload.Sediment grains below the active layer have zero probability of entrainment.
x
z'
qbi qbi La
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
REDUCTION contd.The total bedload transport rate summed over all grain sizes qbT and the fraction pbi of bedload in the ith grain size range can be defined as
bT
bibi
N
1ibibT q
qp,qq
The conservation relation can thus also be written as
Summing over all grain sizes, the following equation describing the evolution of bed elevation is obtained:
xpqLF
t)L(
tf)1( bibT
aiaIip
xq
t)1( bT
p
Between the above two relations, the following equation describing the evolution of the grain size distribution of the active layer is obtained:
x
qfxpq
tLfF
tFL)1( bT
IibibTa
Iiii
ap
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
where 0 1 (Hoey and Ferguson, 1994; Toro-Escobar et al., 1996). In the above relations Fi, pbi and fi denote fractions in the surface layer, bedload and substrate, respectively.
That is:The substrate is mined as the bed degrades.A mixture of surface and bedload material is transferred to the substrate as the
bed aggrades, making stratigraphy.Stratigraphy (vertical variation of the grain size distribution of the substrate)
needs to be stored in memory as bed aggrades in order to compute subsequent degradation.
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
1D GENERALIZATIONS: TECTONICS, SUSPENSION, TOTAL BED MATERIAL LOAD
To include tectonics, make the transformation - base in the above derivation (or integrate from z’’ = 0 to z’’ = - base, where z’’ = z’ - base) to obtain:
xpqLF
t)L(
tf)1( bibT
aiaIip
Repeating steps outlined previously for uniform sediment, if qtT denotes the total bed material load summed over all sizes and pti denotes the fraction of the bed material load in the ith grain size range,
To include suspended sediment, let vsi = fall velocity, Ei = dimensionless entrainment rate, and denote the near-bed volume concentration of sediment, all for the ith grain size range, so that the relation generalizes to:
ibisibibT
aiaIip EcvxpqLF
t)L(
tf)1(
bic
xpqLF
t)L(
tf)1( titT
aiaIip
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
Rivers often sort their sediment. An example is downstream fining: many rivers show a tendency for sediment to become finer in the downstream direction.
Long profiles showing downstream fining and gravel-sand
transition in the Kinu River, Japan (Yatsu,
1955)
elevation
bed slope
median bed material grain size
WHY THE CONCERN WITH SEDIMENT MIXTURES?
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
In order to make further progress, it is necessary to
•Develop a means for computing the bedload transport rate qb (qbi) as a function of the flow;•Develop a means for computing the dimensionless entrainment rate E (Ei) into suspension as a function of the flow;•Develop a model for tracking the concentration of sediment in suspension, so that can be computed.•Specify the thickness of the active layer La.
The key flow parameter turns out to be boundary shear stress .b
ibisibibT
aiaIip EcvxpqLF
t)L(
tf)1(
Ecvxq
t)1( bs
bp
-
)c(c i)c(c bib
Sediment approximated as uniform in size
Sediment mixtures
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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to
Hirano, M., 1971, On riverbed variation with armoring, Proceedings, Japan Society of Civil Engineering, 195: 55-65 (in Japanese).
Hoey, T. B., and R. I. Ferguson, 1994, Numerical simulation of downstream fining by selective transport in gravel bed rivers: Model development and illustration, Water Resources Research, 30, 2251-2260.
Paola, C., P. L. Heller and C. L. Angevine, 1992, The large-scale dynamics of grain-size variation in alluvial basins. I: Theory, Basin Research, 4, 73-90.
Parker, G., 1991, Selective sorting and abrasion of river gravel. I: Theory, Journal of Hydraulic Engineering, 117(2): 131-149.
Toro-Escobar, C. M., G. Parker and C. Paola, 1996, Transfer function for the deposition of poorly sorted gravel in response to streambed aggradation, Journal of Hydraulic Research, 34(1): 35-53.
Toro-Escobar, C. M., C. Paola, G. Parker, P. R. Wilcock, and J. B. Southard, 2000, Experiments on downstream fining of gravel. II: Wide and sandy runs, Journal of Hydraulic Engineering, 126(3): 198-208.
Yatsu, E., 1955, On the longitudinal profile of the graded river, Transactions, American Geophysical Union, 36: 655-663.