Rte-bookAcronym1Notes.doc
Notes for
RTe-bookAcronym1.xls
The notes presented here serve as a companion to the Excel workbook program
Rte-bookAcronym1.xls for the computation of gravel bedload transport in rivers.
Introduction
The Acronym1 programs implement the Parker (1990a) surface-based bedload
transport relation in order to compute gravel bedload transport rates. Definitions
for the relation are given below.
The gravel is divided into N grain size ranges bounded by N+1 sizes Db,i, i = 1 to
N+1. The grain size distribution of the surface (active) layer of the bed is
specified in terms of the N+1 pairs (Db,i, Ff,i), i = 1..N+1, where Ff,i denotes the
percent finer in the surface layer. Here Db,1 must be the coarsest size, such that
Ff,1 = 100, and Db,N1 must be the finest size, such that Ff ,N1 = 0.
The finest size must equal or exceed 2 mm. That is, the sand must be removed
from the surface size distribution, and the fractions appropriately renormalized, in
determining the surface grain size distribution to be input into Acronym1.
The ith grain size range spans the size range (Db,i, Db,i+1) and has the
characteristic grain size Di and fraction in the surface layer Fi, where
Di  Db,iDb,i1
Fi  (Ff ,i  Ff ,i1 ) / 100
(1a,b)
for i = 1..N.
Grain sizes on the base-2 logarithmic  scale are computed as follows;
  n2 (D) 
og10 (D)
og10 (2)
(2a)
where D is specified in mm. Thus
i  n2 (Di ) 
og10 (Di )
og10 (2)
where Di is specified in mm.
material is then specified as
(2b)
The geometric mean size Dsg of the surface
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Rte-bookAcronym1Notes.doc
Dsg  2 s
N
s    iFi
(3a,b)
i1
The geometric and arithmetic standard standard deviations sg and ,
respectively of the surface material are given as
 sg  2
N
 2    i    Fi
2
(4a,b)
i1
The bedload transport relation
Parameters in the bedload transport relation are defined below.

s
R
g
b
u
qbi
=
=
=
=
=
=
=
qbT
=
density of water;
density of sediment;
(s/) – 1 = submerged specific density of sediment;
acceleration of gravity;
boundary shear stress on the bed;
b /  = shear velocity on the bed;
volume gravel bedload transport per unit width of grains in
the ith size range;
N
q
bi
= total volume gravel bedload transport rate per unit
i 1
pi
=
width summed over all sizes;
fraction of gravel bedload in the ith grain size range;
The transport relation can be expressed as
Wi 
Rgqbi
 0.00218G
Fiu3
(5a)
where
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Rte-bookAcronym1Notes.doc
D 
  sgo  i 
D 
 sg 
0.0951
,
sgo 
sg
ssrg
sg 
,
u2
,
RgD sg
4 .5

 0.853 
 for   1.59
54741 

 



G()  exp 14.2(  1)  9,28(  1)2 for 1    1.59

14.2 for   1



  1
O (sgo )  1
O (sgo )



ssrg  0.0386
(5b-g)

The functions O(sgo) and O(sgo) are specified in the tables on the worksheet
“Strain_Functions”.
In order to implement the above relation it is necessary to specify a) the surface
grain size distribution (Df,i, Ff,i) and b) the shear velocity u. This results in a
predicted values of qbi.
If the boundary shear stress at the bed includes a component of form drag, the
component must be removed before computing u.
Once the parameters qbi are known the total volume bedload transport rate per
unit width qbT and the fractions pi in the bedload can be calculated as
N
qbT   qbi
i1
pi 
qbi
qbT
(6a,b)
The results are presented in terms of qbT and the grain size distribution of the
bedload, which is computed from the values of p i. These same fractions pi are
used to compute the geometric mean and geometric standard deviation of the
bedload Dlg and lg, respectively, from the relations
Dlg  2
l
lg  2l
Np
l   ipi
i 1
Np
(7a,b)
l2   i  l  pi
2
(8a,b)
i 1
The percent finer in the bedload pf,i for the grain size Df,i is obtained from the
fractions pi as
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Rte-bookAcronym1Notes.doc
pf ,1  100
pf ,i  pf ,i1  100 pi1
(9a,b)
i  2..N  1
Let Dsx and Dlx denote sizes in the surface and bedload material, respectively,
such that x percent of the material is finer. For example, if x = 50 then D s50 and
Dl50 denote the median sizes of the surface and bedload material, respectively.
Once Ff,i is specified (pf,i is computed) the value Dsx (Dlx) can be computed by
interpolation. The interpolation should be done using a logarithmic scale for
grain size. For example, consider the computation of Dlx where pf,i  x  pf,i+1.
Then
Dlx  2lx
 lx   b,i1 
b,i  b,i1
( x  p f ,i1 )
p f ,i  p f ,i1
(10a,b)
where
b,i  n2 (Db,i )
(11)
Implementation
In order to carry out the above calculation it is necessary to specify specific
gravity of the sediment R + 1, the shear velocity of the flow u and the grain size
distribution of the material in excess of 2 mm in the surface layer (D b,i, Ff,i), i = 1
to N+1. The relation then predicts the total volume bedload transport rate per
unit width qbT of material in excess of 2 mm, as well as the grain size distribution
of this load (Db,i, pf,i).
The programs
“Acronym1” directly implements the above scheme. The Visual Basic code is
contained in Module 1 of this workbook. The code is implemented from the
worksheet “Acronym1”.
“Acronym1_R” combines the above scheme with a Manning-Strickler relation for
flow resistance. It first computes a value of u  from specified values of the water
discharge Qw, the channel width B and the bed slope H. It then implements the
same code as “Acronym1”. The code for “Acronym1_R” is contained in Module 2
of this workbook, and is implemented from the worksheet “Acronym1_R”. More
details about the resistance calculation are given in the worksheet
“IntroAcronym1_R”.
“Acronym1_D” combines the scheme of “Acronym1_R” with a flow duration
curve. The bedload transport rate and bedload grain size distribution are
computed for each flow of the curve, and then averaged to yield a mean bedload
transport rate and a mean bedload grain size distribution. The code for
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Rte-bookAcronym1Notes.doc
“Acronym1_D” is contained in Module 3 of this workbook, and is implemented
from the worksheet “Acronym1_D”. More details about the resistance calculation
are given in the worksheet “IntroAcronym1_D”.
Acronym1
The worksheet “Acronym1” is used for computing the volume bedload transport
rate per unit width and bedload grain size distribution from a specified surface
grain size distribution (with sand removed) (Db,i, Ff,i), i = 1..N+1, a bed shear
velocity u and a specific gravity of the sediment (here equal to R + 1). The
output includes the value of qbT, the Shields stress sg based on the surface
geometric mean size, where
sg 
u2
,
RgD sg
the bedload grain size distribution (Db,i, pf,i) and the values Dlg, lg, Dl90, Dl70, Dl50
and Dl30 for the load, as well as the corresponding values for the surface
material, Dsg, sg, Ds90, Ds70, Ds50 and Ds30.
The calculation is implemented from the worksheet “Acronym1”.
Basic code is contained in Module 1.
The Visual
Acronym1_R
The worksheet “Acronym1_R” is used for computing the volume bedload
transport rate per unit width and bedload grain size distribution from a specified
surface grain size distribution (with sand removed) (Db,i, Ff,i), i = 1..N+1, a specific
gravity of the sediment (here equal to R + 1), a water discharge Q, a channel
width B and a streamwise bed slope S. The output includes the value of q bT, the
Shields stress sg based on the surface geometric mean size, the flow depth H,
the shear velocity u, the bedload grain size distribution (Dd,i, pf,i) and the values
Dlg, lg, Dl90, Dl70, Dl50 and Dl30 for the bedload, as well as the corresponding
values for the surface ,material, Dsg, sg, Ds90, Ds70, Ds50 and Ds30.
The channel is assumed to be rectangular, with the vertical sidewalls having the
same roughness as the bed. The roughness height ks is computed as
k s  nkDs90
(1)
where nk is a user-specified dimensionless roughness factor. The author
suggests a value of 2 for nk. The bed and sidewall regions are partitioned in
accordance with the figure below, and the flow velocity in the sidewall regions is
approximated as identical to that in the bed region.
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Rte-bookAcronym1Notes.doc
Depth is computed according to the relation for momentum balance in the bed
region

bB  u2B  gS HB  H2

(2)
applicable to normal (steady, streamwise uniform) flow. Flow resistance on the
bed region is computed using a Manning-Strickler resistance relation,
1/ 6
H
U
  r  
u
 ks 
(3)
where r takes a value of 8.1 and
U
Q
BH
(4)
Reducing the above three relations, it is found that
 k 1 / 3Q 2 
H   2s 2 
  gB S 
 r

3 / 10
c1
 H
c1  1  
 B
 3 / 10
(5)
The above equation is solved iteratively for H in the code. Once H is known, the
shear velocity u is computed from (2), and the calculation proceeds using the
same algorithm as “Acronym1”. It is implicitly assumed in the calculation that all
of the boundary shear stress consists of skin friction, with form drag neglected.
The calculation is implemented from the worksheet “Acronym1_R”. The Visual
Basic code is contained in Module 2.
Acronym1_D
The worksheet “Acronym1_D” simply adds a flow duration curve to the algorithm
of “Acronym1_R” in order to compute the average volume gravel bedload
transport rate per unit width qbTa, as well as the average bedload grain size
distribution (Db,i, paf,i), i = 1..N+1. In addition, it computes the values Q a, Ha, ua
and ga corresponding to annual mean values of the water discharge, depth,
shear velocity and Shields stress based on surface geometric mean size. The
values Dalg, alg, Dal90, Dal70, Dal50 and Dal30 associated with the mean grain size
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Rte-bookAcronym1Notes.doc
distribution of the bedload are computed along with the corresponding values for
the surface material, Dsg, sg, Ds90, Ds70, Ds50 and Ds30. Finally, the program
computes the volume gravel bedload transport rate per unit width qbT, the water
discharge Qw, flow depth H, the shear velocity u and the Shields stress g
associated with each range in the flow duration curve, along with the fraction of
time pQ that the flow is in that range.
The flow duration curve is specified in terms of the pairs (Qwd,k, peQ,k), k = 1..M+1,
where Qwd,k denotes the kth discharge and peQ,k denotes the percentage of time
this flow is exceeded. Here k = 1 corresponds to the highest flow in the curve,
with an exceedance percentage peQ of zero, and k = M+1 corresponds to the
lowest flow in the curve, with an exceedance percentage p eQ of 100. The lowest
flow on the curve Q wd,M must exceed zero.
The characteristic flow Qwr,k in each range and fraction of time the flow is in that
range pQ,k ae computed as
Qwr ,k 
1
Qwd,k  Qwd,k 1 
2
pQ,k 
peQ,k 1  peQ,k
100
k  1..M
(1a,b,c)
Let Yk be any parameter defined for each of the flow ranges k = 1..M. The mean
value Ya averaged over the flow duration curve is then given as
M
Ya   YkpQ,k
(2)
k 1
For example, if the fractions in the bedload in each grain size range within flow
range k are given as pk,i then the average fractions of the bedload pai are given
as
M
pai   pk,ipQ,k
(3)
k 1
The calculation is implemented from the worksheet “Acronym1_D”. The Visual
Basic code is contained in Module 3.
Historical note
The programs given here are descendants of the Pascal program “Acronym1” of
Parker (1990b). The bedload transport relation remains that of Parker (1990a).
As before, “Acronym” stands for any convenient concoction of words that
possesses “Acronym” as its acronym. The author, however, leans toward the
following concoction: Algorithm causing the regurgitation of odious, numberyielding monstrosities.
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Rte-bookAcronym1Notes.doc
Caveat
Use these programs at your own risk.
References
Parker, G. 1990a Surface-based bedload transport relation for gravel rivers.
Journal of Hydraulic Research, 28(4), 417-436.
Parker, G. 1990b The "ACRONYM" series of Pascal programs for computing
bedload transport in gravel rivers. External Memorandum M-220, St. Anthony
Falls Hydraulic Laboratory, University of Minnesota.
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