CHAPTER 7: RELATIONS FOR 1D BEDLOAD TRANSPORT. Let q b denote the volume bedload transport rate per unit width (sliding, rolling, saltating). It is reasonable to assume that q b increases with a measure of flow strength, such as depth-averaged flow velocity U or boundary shear stress b .
RELATIONS FOR 1D BEDLOAD TRANSPORT
Let qb denote the volume bedload transport rate per unit width (sliding, rolling, saltating). It is reasonable to assume that qb increases with a measure of flow strength, such as depth-averaged flow velocity U or boundary shear stress b.
A dimensionless Einstein bedload number q* can be defined as follows:
A common and useful approach to the quantification of bedload transport is to empirically relate qb* with either the Shields stress * or the excess of the Shields stress * above some appropriately defined “critical” Shields stress c*. As pointed out in the last chapter, c* can be defined appropriately so as to a) fit the data and b) provide a useful demarcation of a range below which the bedload transport rate is too low to be of interest.
The functional relation sought is thus of the form
All the bedload relations in this chapter pertain to a flow condition known as “plane-bed” transport, i.e. transport in the absence of significant bedforms. The influence of bedforms on bedload transport rate will be considered in a later chapter.
The “mother of all modern bedload transport relations” is that due to Meyer-Peter and Müller (1948) (MPM). It takes the form
The relation was derived using flume data pertaining to well-sorted sediment in the gravel sizes.
Recently Wong (2003) and Wong and Parker (submitted) found an error in the analysis of MPM. A re-analysis of the all the data pertaining to plane-bed transport used by MPM resulted in the corrected relation
If the exponent of 1.5 is retained, the best-fit relation is
The “critical Shields stress” c* of either 0.047 or 0.0495 in either the original or corrected MPM relation(s) must be considered as only a matter of convenience for correlating the data. This can be demonstrated as follows.
Consider bankfull flow in a river. The bed shear stress at bankfull flow bbf can be estimated from the depth-slope product rule of normal flow:
The corresponding Shields stress bf50* at bankfull flow is then estimated as
where Ds50 denotes a surface median size. For the gravel-bed rivers presented in Chapter 3, however, the average value of bf50* was found to be about 0.05 (next page).
According to MPM, then, these rivers can barely move sediment of the surface median size Ds50 at bankfull flow. Yet most such streams do move this size at bankfull flow, and often in significant quantities.
There is nothing intrinsically “wrong” with MPM. In a dimensionless sense, however, the flume data used to define it correspond to the very high end of the transport events that normally occur during floods in alluvial gravel-bed streams. While the relation is important in a historical sense, it is not the best relation to use with gravel-bed streams.
Some commonly-quoted bedload transport relations with good data bases are given below.
Ashida & Michiue (1972)
Engelund & Fredsoe (1976)
Fernandez Luque & van Beek (1976)
Parker (1979) fit to
E = Einstein
AM = Ashida-Michiue
EF = Engelund-Fredsoe
P approx E = Parker approx of Einstein
FLBSand = Fernandez Luque-van Beek, tc* = 0.038
FLBGrav = Fernandez Luque-van Beek, tc* = 0.0455
The bedload relation of Einstein (1950) contains no critical Shields number. This reflects his probabilistic philosophy.
All of the relations except that of Einstein correspond to a relation of the form
In the limit of high Shields number. In dimensioned form this becomes
where K is a constant; for example in the case of Ashida-Michiue, K = 17. Note
that in this limit the bedload transport rate becomes independent of grain size!!
Some of the scatter between the relations is due to the face that c* should be a
function of Rep. This is reflected in the discussion of the Fernandez Luque-van
Beek relation in the next slide. (Recall that .)
plotted well outside of the data used to derive them. For example, in data used
to derive Fernandez Luque-van Beek, * never exceeded 0.11, whereas
the plot extends to * = 1.
In the experiments on which the relation is based;
a) Streamwise bed slope angle varied from near 0 to 22.
The material tested included five grain sizes and three specific gravities, as
given below; also listed are the values of Rep and the critical Shields number co* determined empirically at near vanishing bed slope angle.
c) It is thus possible to check the effect of Rep and on the transport relation of Fernandez Luque and van Beek (FLvB).
The experimental values of co* generally track the modified Shields relation of Chapter 6, but are high by a factor ~ 2. This reflects the fact that they correspond to a condition of “very small” transport determined in a consistent way (see original reference).
The ratio c*/co* decreases with streamwise angle as predicted by the relation of Chapter 6, but to obtain good agreement r must be set to the rather high value of 47 (c = 1.07).
Double-click on the image to run the video.
D = 0.116 mm
S = 0.035
U = 1.05 m/s
Fr = 1.85
sheet* = 0.51
rte-booksheetpeng.mpg: to run without relinking, download to same folder as PowerPoint presentations.
Video clip courtesy P. Gao and A. Abrahams; Gao (2003)
A number of mechanistic derivations of bedload transport relations are available. These are basically of two types, based on two forms for bedload continuity. Let bl = the volume of sediment in bedload transport per unit area, ubl = the mean velocity of bedload particles, Ebl = the volume rate of entrainment of bed particles into bedload movement (rolling, sliding or saltation, not suspension) and Lsbl = the average step length of a bedload particle (from entrainment to deposition, usually including many saltations). The following continuity relations then hold:
In a Bagnoldean approach, separate predictors are developed for ubl and bl, the latter determined from the Bagnold constraint (Bagnold, 1956). Models of this type include the macroscopic models of Ashida and Michiue (1972) and Engelund and Fredsoe (1976), and the saltation models of Wiberg and Smith (1985, 1989), Sekine and Kikkawa (1992), and Nino and Garcia (1994a,b). Recently, however, Seminara et al. (2003) have shown that the Bagnold constraint is not generally correct.
In the Einsteinean approach the goal is to develop predictors for Ebl and Lsbl. It is the former relation that is particularly difficult to achieve. Models of this type include
the Einstein (1950), Tsujimoto (e.g. 1991) and Parker et al. (2003).
Again, the case under consideration is plane-bed bedload transport (no bedforms).
As a preliminary, define a dimensionless sediment transport rate W* as
Now all previously presented bedload transport rates for uniform sediment can be rewritten in terms of W* as a function of *:
Ashida & Michiue (1972)
Engelund & Fredsoe (1976)
Fernandez Luque & van Beek (1976)
Parker (1979) fit to
Consider the bedload transport of a mixture of sizes. The thickness La of the active (surface) layer of the bed with which bedload particles exchange is given by as
where Ds90 is the size in the surface (active) layer such that 90 percent of the material is finer, and na is an order-one dimensionless constant (in the range 1 ~ 2). Divide the bed material into N grain size ranges, each with characteristic size Di, and let Fi denote the fraction of material in the surface (active) layer in the ith size range. The volume bedload transport rate per unit width of sediment in the ith grain size range is denoted as qbi. The total volume bedload transport rate per unit width is denoted as qbT, and the fraction of bedload in the ith grain size range is pbi, where
Now in analogy to *, q* and W*, define the dimensionless grain size specific Shields number i*, grain size specific Einstein number qi* and dimensionless grain size specific bedload transport rate Wi* as
It is now assumed that a functional relation exists between qi* (Wi*) and i*, so that
The bedload transport rate of sediment in the ith grain size range is thus given as
According to this formulation, if the grain size range is not represented in the surface (active) layer, it will not be represented in the bedload transport.
Basic transport relation
Effective critical Shields stress for surface geometric mean size
Note This relation has been modified slightly from the original formulation. Here the relation specifically uses surface fractions Fi, and surface geometric mean size Dsg is specified in preference to the original arithmetic mean size Dsm = DiFi.
Modified version of Egiazaroff (1965) hiding function
This relation is appropriate only for the computation of gravel bedload transport rates in gravel-bed streams. In computing Wi*, Fi must be renormalized so that the sand is removed, and the remaining gravel fractions sum to unity, Fi = 1. The method is based on surface geometric size Dsg and surface arithmetic standard deviation s on the scale, both computed from the renormalized fractions Fi.
In the above O and O are set functions of sgospecified in the next slide.
It is not necessary to use the above chart. The calculations can be performed using the Visual Basic programs in RTe-bookAcronym1.xls
An example of the renormalization to remove sand is given below.
The Microsoft Excel workbook RTe-bookAcronym1.xls is an example of a workbook in this e-book that uses code written in Visual Basic for Applications (VBA). VBA is built directly into Excel, so that anyone who has Excel (versions 2000 or later) can execute the code directly from the relevant worksheet in the workbook RTe-bookAcronym1.xls. The relevant code is contained in three modules, Module1, Module2 and Module3 of the workbook, which may be accessed from the Visual Basic Editor.
All the code in this e-book is written in VBA in Excel modules. A self-teaching tutorial in VBA is contained in the workbook RTe-bookIntroVBA.xls. Going though the tutorial will not only help the reader understand the material in this e-book, but also allow the reader to write, execute and distribute similar code.
To take the tutorial, open RTe-bookIntroVBA.xls and follow the instructions.
The workbook RTe-bookAcronym1.xls provides interfaces for three different implementations.
In the implementation of “Acronym1” the user inputs the specific gravity of the sediment R+1, the shear velocity of the flow and the grain size distribution of the bed material. The code computes the magnitude and size distribution of the bedload transport.
In the implementation of “Acronym1_R” the user inputs the specific gravity of the sediment R+1, the flow discharge Q, the bed slope S, the channel width B, the parameter nk relating the roughness height ks to the surface size Ds90 and the grain size distribution of the bed material. The code computes the shear velocity using a Manning-Strickler resistance formulation and the assumption of normal flow, and then computes the magnitude and size distribution of the bedload transport.
The implementation of “Acronym1_D” uses the same formulation as “Acronym1_R”, but allows specification of a flow duration curve so that average annual gravel transport rate and grain size distribution can be computed.
EXAMPLE: INTERFACE FOR “Acronym1” IN Rte-bookAcronym1.xls
The sand is not excluded in the fractions Fi used to compute Wi*. The method is based on the surface geometric mean size Dsg and fraction sand in the surface layer Fs.
EFFECT OF SAND CONTENT IN THE SURFACE LAYER ON GRAVEL MOBILITY IN A GRAVEL-BED STREAM
Wilcock and Crowe (2003) have shown that increasing sand content in the bed surface layer of a gravel-bed stream renders the surface gravel more mobile. This effect is captured in their relationship between the reference Shields number for the surface geometric mean size (a surrogate for a critical Shields number) and the fraction sand Fs in the surface layer:
Note how decreases as Fs increases. The surface layers of gravel-bed streams rarely contain more than 30% sand; beyond this point the gravel tends to be buried under pure sand.
BEDLOAD RELATION FOR MIXTURES DUE TO POWELL, REID AND LARONNE (2001)
The sand is excluded in the fractions Fi used to compute Wi*. The method is based on the surface median size Ds50, computed after excluding sand..
More information about bedload transport relations for mixtures can be
found in Parker (in press; downloadable from http://www.ce.umn.edu/~parker/).
Assumed grain size distribution of the bed surface: Dsg = surface geometric mean, sg = surface geometric standard deviation, Fs = fraction sand in surface layer.
Dsg = 40.7 mm, sg = 2.36
Dsg = 22.3 mm, sg = 4.93, Fs = 0.16
Other input parameters:
R = 1.65
u* = 0.15 to 0.40 m/s
A-M = Ashida and Michiue (1972), sand not excluded
P = Parker (1990), sand excluded
P-R-L = Powell et al. (2001), sand excluded
W-C = Wilcock and Crowe (2003), sand not excluded
qG = volume gravel bedload transport rate per unit width
DGg = geometric mean size of the gravel portion of the transport
Gg = geometric standard deviation of the gravel portion of the bedload
pG = fraction gravel in the bedload transport (only for A-M and W-C):
fraction sand = 1 - pG
It is necessary to have a flow duration curve to perform the calculation. The flow duration curve specifies the fraction of time a given water discharge is exceeded, as a function of water discharge.
This curve is divided into M bins k = 1 to M, such that pQk specifies the fraction of time the flow is in range k with characteristic discharge Qk. The value u*k must be computed for each range. For example, in the case of normal flow with constant width B, the Manning-Strickler resistance relation from Chapter 5 yields
The grain size distribution of the bed material must be specified; ks can be computed as 2 Ds90 (for a plane bed). Once u*,k is computed, either qb,k (material approximated as uniform) or qbi,k (mixtures) is computed for each flow range, and the annual sediment yield qba or total yield qbTa and grain size fractions of the yield pai are given as
Expect the flood flows to contribute disproportionately to the annual sediment yield.
An implementation is given in “Acronym1_D” of Rte-bookAcronym1.xls .
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