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Friction factor plays a fundamental role in hydraulic analysis and design. In coarse-bed rivers friction factor depends on grain size and bed forms. Considering boundary-layer characteristics, this study first determines the total friction factor in gravel-bed rivers with bed forms. Then it determines the grain friction factor by the Keulegan and the Shields parameter methods and compares these methods. Finally it determines the form friction factor by subtracting the grain friction factor from the total friction factor. Field observations are employed to test the methods.
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TECHNICAL NOTES
Determination of Form Friction Factor
Hossein Afzalimehr1; Vijay P. Singh, F.ASCE2; and Elham Fazel Najafabadi3
Abstract: Friction factor plays a fundamental role in hydraulic analysis and design. In coarse-bed rivers friction factor depends on grain
size and bed forms. Considering boundary-layer characteristics, this study first determines the total friction factor in gravel-bed rivers with
bed forms. Then it determines the grain friction factor by the Keulegan and the Shields parameter methods and compares these methods.
Finally it determines the form friction factor by subtracting the grain friction factor from the total friction factor. Field observations are
employed to test the methods.
DOI: 10.1061/ASCEHE.1943-5584.0000175
CE Database subject headings: Friction; Bedforms; Boundary layers; Parameters; River beds.
Author keywords: Friction factor; Bed forms; Boundary-layers characteristics; Shields parameter.
Introduction
Prediction of friction factor is fundamental for river hydraulic
analysis, including design of stable channels, estimation of flow
velocities for determination of design floods, flow and sediment
routing, and determination of channel flood capacity. Meyer-Peter
and Muller 1948, Petit 1989, Robert 1997, among others,
have emphasized that evaluation of incipient motion and sediment
transport processes requires determination of friction factor. In
coarse-bed rivers, the value of friction factor depends not only on
grain size but also on bed forms. Bed forms can have a significant
effect on friction factor and transport of sediment in alluvial chan-
nel Garcia 2008. Further, form friction is the major source of
boundary resistance and is vital in determining flow depth
Brownlie 1983. Ecologically, bed forms are important features,
for they affect flow characteristics and influence fish habitat.
Existing friction factor relations underestimate flood discharge
by as much as 100% for coarse-bed rivers Jarrett 1991. This is
because estimation of friction factor is a complex problem, not
easy to quantify only by the grain friction factor. Further, the bed
configuration varies with variations of flow conditions and makes
it considerably difficult to obtain the friction due to these bed
forms by estimating only the grain friction factor.
The friction factor can be divided into two parts Yalin 1972;
Yen 1991
f=f+f1
in which primed fis the grain friction factor and the double prime
is the form friction factor. The grain friction factor fis due to the
bed shear stress applied on the grains; the form friction factor f
is due to the local energy loss on the downstream side of bed
forms. Yen 2002surveyed investigations on grain and form fric-
tion factors. Fedele and Garcia 2001showed that the friction
factor in alluvial channels in the presence of bed forms was a
nonlinear and nonunique function of the Shields parameter, grain
size, and the relative flow depth.
Review of literature on the division of friction factor shows
that the grain friction factor fis normally computed using the
Reynolds number and the relative roughness d50 /R, where d50 is
the median grain size and Ris the hydraulic radius, as indepen-
dent variables e.g., Vanoni and Hwang 1967, Alam and
Kennedy 1969, Acaroglu 1972, and van Rijn 1982, 1984兲兴.
van Rijn 1984expressed the grain friction factor fas
1/f0.5 = 2.03 log12.2h/d90兲共2
in which his the flow depth and d90 is the grain size for which
90% of grains are finer.
In gravel-bed rivers, pools and riffles are dominant bed forms.
Prestegaard 1983and Griffiths 1989estimated friction factor
along the riffle-pool sequence and attributed part of the total fric-
tion factor directly to bed undulations. Investigations on the in-
fluence of pools and riffles on form friction factor have applied
average flow conditions at the reach scale rather than measure-
ments of velocity profiles along bed forms.
The ASCE Task Force 1963in Hydromechanics Committee
stated that any progress in friction factor problem depended on
the concept of boundary layer. This means more sophisticated
approaches based on measurement of the velocity distribution are
required along and across a river to predict friction factor. This
technical note seeks to ameliorate the accuracy of form friction
factor for coarse-bed rivers by using velocity profiles and
boundary-layer characteristics, including displacement and mo-
1Associate Professor, Dept. of Water Engineering, Isfahan Univ. of
Technology, Isfahan 84156, Iran. E-mail: hafzali@cc.iut.ac.ir
2Caroline and William N. Lehrer Distinguished Chair in Water Engi-
neering, Professor of Biological and Agricultural Engineering and Profes-
sor of Civil and Environmental Engineering, Dept. of Biological and
Agricultural Engineering, Texas A&M Univ., College Station, TX 77802-
2117. E-mail: vsingh@tamu.edu
3Graduate Student, Dept. of Water Engineering, Isfahan Univ. of
Technology, Isfahan 84156, Iran corresponding author. E-mail:
e.fazel@ag.iut.ac.ir
Note. This manuscript was submitted on March 3, 2009; approved on
August 19, 2009; published online on August 24, 2009. Discussion period
open until August 1, 2010; separate discussions must be submitted for
individual papers. This technical note is part of the Journal of Hydro-
logic Engineering, Vol. 15, No. 3, March 1, 2010. ©ASCE, ISSN 1084-
0699/2010/3-237–243/$25.00.
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mentum thicknesses. Accordingly, the objective of this study
therefore is to 1apply boundary-layer characteristics displace-
ment thickness and momentum thickness to evaluate the
total friction factor in gravel-bed rivers in the presence of bed
forms; 2identify the regions of the highest velocities and shear
stresses in a gravel reach with bed forms; and 3determine and
compare form friction factors obtained by using the Keulegan and
the Shields parameter methods.
Determination of Form Friction Factor
When grains of sediment begin to move, random patterns of sedi-
mentation and erosion generate small perturbations in the bed
surface elevation. These perturbations develop various bed forms
over the bed surface. The friction factor depends on the configu-
ration of these bed forms.
Determination of Total Friction Factor
Friction factor is normally estimated by the dimensionless Darcy-
Weisbach equation, which assumes uniform flow. In mountain
rivers, due to bed and flow nonuniformities, it is difficult to justify
this assumption and this equation may therefore not hold. To take
into account the influence of flow nonuniformity in the Darcy-
Weisbach equation, it is necessary to estimate shear velocity in
this equation using the velocity distribution and boundary-layer
parameters. Accordingly, the total friction factor fis determined
Yalin 1972as
f=8
u
2
um
2
3
where um=weighted value of velocity in a cross section and the
shear velocity uis calculated using the boundary-layer charac-
teristics method BLCMas follows Afzalimehr and Anctil
2001:
u=umax
C
4
where C=4.4 Afzalimehr and Anctil 2001;=displacement
thickness; =momentum thickness; and umax= maximum velocity
observed over a velocity profile; these thicknesses are defined
Schlichting and Gersten 2000as
=
0
h
1− u
umax
dy 5
=
0
hu
umax
1− u
umax
dy 6
Eq. 4not only uses the boundary-layer characteristics but also
contains the effect of flow nonuniformity via velocity profile. The
displacement thickness reveals that the thickness of the water
should be augmented so that the fictious uniform nonviscous flow
has the same mass flow rate properties as actual viscous flow
Munson et al. 1994. Also, the momentum thickness is an
index that is proportional to the decrement in momentum flow
due to the nonuniformity of velocity profile. Therefore, presents
a height proportional to the missing momentum flow at the free
stream condition. Furthermore, it should be stressed that the esti-
mated at a given section is proportional to the grain friction
factor of that section. Ludwieg and Tillmann see Young 1989兲兴
were the pioneers in the application of and for predicting
100
Section 4 - Pool
Section
5
Riffle
10
n
er
Section
5
-
Riffle
1
1
10
100
%F
i
n
1
10
100
Grain size
(
mm
)
Fig. 1. Grain size distribution in the selected reach
1568.2
1568.4
8
1568
1567.6
.
8
Y
(
m
)
1567.2
1567.4
Bed river
1567
0 102030405060708090100110
Distance from upstream (m)
Fig. 2. Velocity profiles along the selected reach
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friction factor, and their methodology has found considerable sup-
port in aeronautical engineering.
Determination of Grain Friction Factor
Considering the division of the total friction factor into two parts,
grain part fand form part f, it should be noted that the value
of grain friction factor cannot be measured in the field and can
only be estimated by assuming that part of the roughness is due to
surface shear stress applied directly to the grains as if the river
bed would have a plane rough surface Julien et al. 2002.
Keulegan Method
To calculate the grain friction, the equation of Keulegan 1938
was applied
f=
2.03 log
12.2h
ks
−2
7
in which ks=Nikuradse equivalent roughness size. Yen 1991
presented a list of ksvalues reported in the literature, with ks
1567.4
1567.6
1567.8
1568
1568.2
1
5
68
.4
Y(m)
Bed river
1567
1567.2
0 5 10 15 20 25 30 35 40
Distance from left bank
(
m
)
Fig. 3. Velocity profiles across the selected reach
Fig. 4. aTopographic map of the selected reach; bschematic of
the three-dimensional bed form in the selected reach. Flow over runs
from top right to bottom left.
Fig. 5. aVelocity distribution map for the selected reach; bshear
stress distribution map for the selected reach
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varying from 1.23d35 Ackers and White 1973to 3d90 van Rijn
1982or 6.6d50 Hammond et al. 1984. However, it was found
that application of ks=d50 without any coefficient was most com-
patible with the other method that entailed the critical Shields
parameter.
Shields Parameter Method
The critical Shields parameter cris defined Buffington and
Montgomery 1997as
cr =
SG −1d50
8
Considering the shear stress as =hSfChow 1959; Keshavarzy
and Ball 1997, the median grain size can be determined by Eq.
8as
d50 =hSf
SG −1cr
9
in which Sf=friction slope; SG=s/= 2.65; sand = specific
gravities of sediment and water, respectively; and the value of cr
is determined from the equation of Lamb et al. 2008as
cr = 0.15S0.25 10
where S=bed slope.
Using a large set of experimental field data for studies on
incipient motion where the grain Reynolds number was larger
than 100, Lamb et al. 2008found that much of the data were not
within 0.03⬍␶cr 0.06; however they found a relationship be-
tween cr and bed slope Ssuch as Eq. 10. Graf and Suszka
1987found that for the grain Reynolds number greater than 500,
the Shields parameter depended on slope. In the grain Reynolds
Table 1. Measured and Calculated Hydraulic Parameters
Reach name Section number SfD50m
h
ma
um
m/sb
uBL
m/sf
f
Kulega
n with
d50
f
Kulega
n with
d50
f
Kulega n with sheilds
fKulegan
with sheildsRe103
Zayanderud 1 0.0031 0.010 0.85 0.65 0.07 0.089 0.027 0.062 0.044 0.045 0.70
2 0.0083 0.009 0.80 0.76 0.08 0.092 0.026 0.066 0.058 0.034 0.72
3 0.0015 0.009 0.71 0.74 0.08 0.102 0.028 0.075 0.036 0.066 0.791
4 0.0026 0.012 0.65 0.86 0.09 0.093 0.030 0.062 0.042 0.051 1.10
5 0.0025 0.013 0.64 0.84 0.09 0.083 0.032 0.052 0.041 0.042 1.11
Kaj 1 0.0109 0.014 0.28 0.83 0.09 0.091 0.043 0.048 0.064 0.027 1.28
2 0.0028 0.012 0.22 0.63 0.06 0.080 0.044 0.036 0.042 0.037 0.73
3 0.0028 0.012 0.24 0.61 0.06 0.069 0.042 0.027 0.042 0.027 0.66
4 0.0028 0.012 0.32 0.45 0.04 0.056 0.038 0.018 0.042 0.014 0.44
5 0.0009 0.008 0.32 0.54 0.04 0.054 0.034 0.020 0.032 0.023 0.38
6 0.002 0.008 0.25 0.78 0.08 0.075 0.036 0.039 0.039 0.036 0.57
7 0.0017 0.008 0.27 0.63 0.06 0.061 0.035 0.026 0.037 0.024 0.42
8 0.0023 0.008 0.26 0.64 0.06 0.060 0.035 0.024 0.040 0.019 0.42
Gamasyab 1 0.0037 0.018 0.28 1.10 0.10 0.064 0.047 0.017 0.046 0.018 1.76
2 0.004 0.018 0.28 1.08 0.10 0.066 0.047 0.019 0.047 0.019 1.76
3 0.00396 0.018 0.32 1.14 0.11 0.069 0.044 0.025 0.047 0.023 1.91
4 0.0052 0.021 0.25 0.90 0.10 0.107 0.052 0.056 0.051 0.057 2.18
5 0.0057 0.020 0.28 1.18 0.12 0.081 0.048 0.032 0.052 0.029 2.38
6 0.00457 0.022 0.27 1.08 0.12 0.096 0.051 0.044 0.048 0.047 2.60
7 0.00537 0.020 0.45 1.62 0.16 0.081 0.041 0.040 0.051 0.030 3.26
8 0.0027 0.021 0.18 0.62 0.06 0.086 0.059 0.027 0.042 0.044 1.34
9 0.0026 0.017 0.26 0.88 0.08 0.064 0.047 0.017 0.042 0.022 1.34
10 0.0026 0.022 0.38 1.20 0.11 0.062 0.045 0.017 0.041 0.021 2.33
11 0.0030 0.018 0.37 1.28 0.12 0.064 0.042 0.022 0.043 0.021 2.07
12 0.0028 0.014 0.30 0.98 0.08 0.058 0.041 0.017 0.042 0.016 1.18
13 0.0033 0.019 0.40 1.37 0.12 0.065 0.042 0.024 0.044 0.021 2.36
14 0.0019 0.016 0.34 1.05 0.09 0.058 0.042 0.016 0.038 0.020 1.42
15 0.0034 0.016 0.33 1.08 0.09 0.061 0.042 0.019 0.045 0.017 1.52
16 0.0024 0.019 0.33 1.03 0.09 0.063 0.046 0.017 0.040 0.023 1.75
17 0.0028 0.023 0.23 0.80 0.08 0.070 0.055 0.015 0.042 0.028 1.72
18 0.0029 0.023 0.21 0.78 0.08 0.076 0.058 0.018 0.043 0.033 1.75
19 0.003 0.017 0.22 0.87 0.08 0.061 0.050 0.011 0.043 0.018 1.29
20 0.0028 0.018 0.26 0.87 0.08 0.066 0.048 0.019 0.042 0.024 1.42
21 0.0027 0.018 0.25 0.80 0.08 0.072 0.049 0.023 0.042 0.030 1.37
ahis the weighted value of depth in a cross section.
bumis the weighted value of velocity in a cross section.
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number Re=ud50 /in which is the kinematic viscosityin-
dependent region of the Shields diagram, cr is considerably
lower when dealing with fully exposed spherical grains. Coleman
1967and Fenton and Abbott 1977obtained values of cr close
to 0.01 for this region. Afzalimehr et al. 2007showed that under
decelerating flow cr is lower than the values reported in the
literature. Keshavarzy and Ball 1997reasoned that sediment
particles commence to move at flow depths lower than those es-
timated from Shields’ diagram. This is because the instantaneous
shear stress applied to sediment particles is higher than the mean
shear stress Keshavarzy and Ball 1999.
Eq. 10does not represent the effect of flow nonuniformity
due to the application of bed slope S. On the other hand, the use
of friction slope Sfin Eq. 10which is different in nonuniform
flow from bed slope Sallows us to take into account the influ-
ence of flow nonuniformity on the estimation of cr. If one as-
sumes that the coefficient and the exponent of Eq. 10reflect the
morphological characteristics of rivers, then application of Sfin-
stead of Sin Eq. 3leads to
cr = 0.15Sf
0.25 11
The friction slope Sfcan be defined by the St. Venant equation as
Sf=Sdh
dx 1−Fr
2兲共12
where dh/dx= water surface variation; Fr=Froude number de-
fined as Fr=um/gh0.5;h= weighted value of depth in a cross
section; and g=gravitational acceleration.
Substituting Eq. 9into Eq. 7gives
f=2.03 log20.13cr/Sf兲兴−2 13
Putting cr from Eq. 11into Eq. 13and using the logarithm
rules, fcan be written as
f=0.9742 – 1.5225 logSf兲兴−2 14
This equation shows that fcan be predicted from the friction
slope and hence Eq. 14includes the effect of flow nonunifor-
mity. Employing flow data from a wide range of gauging sites,
Golubtsov 1969found a positive dependency of friction factor
on slope for slopes in the range of 0.4–20%. At lower slopes no
specific dependency was observed. A physical explanation for a
slope dependency in the friction factor can be due to the change
in the river morphology which typically occurs on moving up-
stream along river profiles Bathurst 2002.
Determination of Form Friction
Now returning to Eq. 1, the form friction factor fcan be
estimated as
f=ff15
Field Experimentation and Testing
A 100-m-long reach of Zayandehrood River in central Iran was
selected for this study. The width of reach varies from 37 to 42 m.
The selected reach was almost straight and was formed of gravel
grains. The grain size distribution was calculated using the Wol-
man method Fig. 1. Surveying along this reach and reading the
water surface and bed levels, the profiles of water and bed sur-
faces were constructed. Five sections in the selected reach, spaced
20–29 m from each other, were chosen to measure velocity pro-
files Fig. 2. These sections were selected based on bed form
variations. The velocity was measured in such a way that one
profile was obtained in the center and two profiles were taken on
the both sides of the river Fig. 3. The point velocities were
measured in the vertical direction from the bed to the water sur-
face by using a butterfly current meter with horizontal axes. The
distance between measurement points was between 1 to 2 cm in
the 20% depth near the bed and between 3 to 5 cm in the upper
80% depth. In this manner smaller velocity gradient in the upper
80% depth in comparison to 20% flow depth near the bed was
adequately accommodated. The time step of each point velocity
measurement was taken as 50 s with three repetitions. Further-
more, for constructing a topographic map of the reach Fig. 4and
distributions of velocity and shear stress Fig. 5along the reach,
62 points were randomly measured. At these points, flow veloci-
ties were measured at 0.2hand 0.8hfrom the water surface. Table
1 presents hydraulic characteristics of the selected reach. Further-
more, using the same method data were obtained for other reaches
of two gravel-bed rivers in Iran. In total 34 sections were used in
this study.
Computation and Discussion of Results
For a reasonable estimate of form friction factor, one needs to
determine the total and the grain friction factors. The flow char-
acteristics, such as the longitudinal nonuniformity of flow charac-
terized by deceleration or acceleration, can change the velocity
profile and hence the determination of and in Eq. 4兲共Afza-
limehr and Rennie 2009. Accordingly, using Eqs. 5and 6, the
value of uwas calculated for three velocity profiles in each sec-
tion and then its weighted value along with the weighted velocity
at each section were used in Eq. 3for estimating the total fric-
tion factor f. Also, the grain friction factor was calculated using
Eqs. 7and 14. However, Eq. 7needs only the weighted flow
depth and median grain size at each section. On the other hand,
for estimation of fby Eq. 14, one needs to calculate friction
slope from Eq. 12. Therefore, application of Eq. 14to estimat-
ing fdepends on flow conditions at the upstream via S,dh/dx,
and Fr. Using the calculated ffrom Eq. 3兲兴 and ffrom Eqs. 7
or 14兲兴 the value of fis determined by Eq. 15. The results
showed that the calculated form friction factors using grain fric-
tion of Eqs. 7and 14were in agreement. It should be stressed
that both methods Eqs. 7and 14兲兴 take into account flow non-
uniformity; however, Eq. 14uses it directly by applying the
friction slope and Eq. 7uses it indirectly by applying and in
ffor the determination of f.
The BLCM including and Afzalimehr and Rennie 2009
can be reasonably applied for the estimation of shear velocity in
the presence of bed forms by using Eq. 4. The results not pre-
sented here inrevealed that BLCM had an acceptable agreement
with the log law method. However, since the later method uses
the near bed data, it is sensitive to the reference level Hinze
1975, while the former applies all of velocity profile data and is
not affected by the reference level and illustrates the effect of bed
or flow nonuniformity by the velocity shapes.
Fig. 4ashows a topographic map of the selected reach. At the
upstream end of the reach, bed level was 1,567.6 m and along the
reach it became 1,567.5 m and at the end of the reach it became
again 1,567.5 m. This shows that the central part of the reach
formed a pool. However, on the left margin of the reach topo-
graphic variation was more pronounced than the central part. Fig.
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4bexhibits a three-dimensional view of the bed level in the
reach.
Plan form maps of flow velocity and bed shear stress for the
reach are shown in Fig. 5. The highest velocities and shear
stresses occurred over the central part of the reach. Fig. 5 also
shows lower shear stresses along the channel margins, especially
on the left bank and in the deeper part of the reach.
Fig. 6 compares the form friction factor values obtained using
the two methods discussed above. It is observed that 82% of data
are included in the limits of 35%, illustrating a good agreement
between the simple method of Keulegan which involves only
depth and grain size to calculate grain friction factor with the
other method which involves the friction slope. The latter method
takes into account the water surface variation and flow nonunifor-
mity for the estimation of fand is better from a theoretical point
of view. However, the former method is more practical from an
engineering point of view. Part of the scatter in the data must
therefore be a function of the unknown effects of, for example,
bed material size distribution, shape, and orientation.
Conclusions
The following conclusions can be drawn from this study:
1. Application of the boundary-layer parameters and ,
based on the ASCE Task Force recommendation, to the esti-
mation of the total friction factor fshows that the major
part of friction factor in gravel-bed rivers is due to the form
friction factor;
2. Application of d50 for the Nikuradse equivalent roughness
size kswithout any coefficient in the Keulegan equation
displays the best agreement with the method of based on the
use of the critical Shields parameter for calculation of the
grain friction factor;
3. Using the friction slope in the equation of Lamb et al. is
more realistic than the application of bed slope because the
former parameter takes into account the flow structure in the
estimation of the critical Shields parameter; and
4. There is a reasonable agreement between the methods of
Keulegan and that which applies the Shields parameter to
estimate the form friction factor.
Notation
The following symbols are used in this paper:
d50 median diameter of sediment particles;
ffriction factor;
fgrain friction factor;
fbed form friction factor;
Fr Froude number defined by um/gh0.5;
ggravitational acceleration;
hflow depth;
h/d50 relative flow depth;
ksNikuradse’s equivalent roughness;
Rhydraulic radius;
Re Reynolds number that is umh/;
Reparticle Reynolds number ud50 /;
Sbed slope;
Sffriction slope;
ummean flow velocity;
umax maximum velocity observed over a velocity
profile;
ushear velocity;
vvertical component of flow velocity;
Wwidth of river;
wlateral component of flow velocity;
xdeviation point of the inner layer from the
outer layer;
ydistance from the reference level of channel
bed;
␥⫽specific gravity of water;
sspecific gravity of sediment;
boundary-layer displacement;
␪⫽momentum thickness;
␬⫽von Karman’s constant;
␯⫽kinematic viscosity;
␶⫽bed shear stress; and
cr critical shear stress.
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0
.
08
%
35
+%35
0
.
04
0.05
0.06
0.07
h
us
i
ng SP
)
-
%
35
0.01
0.02
0.03
0
.
04
f
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(k
u
l
egan w
i
t
h
0
0 0.02 0.04 0.06 0.0
8
f ''(kule
g
an with usin
g
d50)
Fig. 6. Comparison of form friction fvalues estimated using the
Keulegan and the Shields parameter application methods
242 / JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MARCH 2010
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... The friction attributable to a specific source can be calculated by subtracting the other sources of friction from the total friction. This method has been used extensively for Darcy-Weisbach friction factor (Afzalimehr et al. 2010a;Schoneboom et al. 2010;Afzalimehr et al. 2017) and Manning's coefficient (Green 2005;Shields et al. 2017). Equation 1 is a linear superposition of grain friction (f′), vegetative friction (f′′), and friction due to bedforms (f′′): ...
... The friction due to bedforms can be estimated by subtracting the grain and vegetative friction components from the total friction (f). Manning's equation is a conventional method for the calculation of channel resistance and has been used extensively in the field for vegetated channels (Green 2005;Verschoren et al. 2017;Errico et al. 2018Errico et al. , 2019, however, it has been reported that the friction calculated by the Boundary Layer Characteristics Method developed by (Afzalimehr and Anctil 2000) is a better method to calculate the shear velocity and then the friction as it incorporates nonuniformities of flow (Afzalimehr et al. 2010a(Afzalimehr et al. , 2019c. The objectives of this study therefore, were: (1) to assess the applicability of the Boundary Layer Characteristics Method to the calculation of friction factor and bed shear stress and estimate the bedform friction using a linear superposition of friction factors, and (2) to determine the CONTACT Sadegh Derakhshan sadeghderakhshan@yahoo.com distribution of velocity and bed shear stress around vegetation patches and bedforms in the field. ...
... The BLCM is better suited for the calculation of shear velocity in comparison to conventional methods in this study, because of the following reasons: 1-Darcy-Weisbach's friction factor f is developed for uniform flow (Equation 2) and BLCM accounts for the nonuniformities of flow (caused by the presence of vegetation and possibly bedforms) given that BLCM uses the whole velocity profile and the non-uniformities of flow are incorporated into the velocity profiles (Afzalimehr et al. 2010a(Afzalimehr et al. , 2019c. ...
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Aquatic vegetation and morphology are integral components of fluvial systems, study of which is important due to ecological and engineering projects. Often, studies in fluvial systems are carried out in the laboratory as collecting data in the field is expensive and difficult, and even risky. A field study was carried out to measure the hydraulic and geometric parameters of the bed and the banks of a gravel bed river with instream emergent vegetation in northern Iran. The aim of this study was to gain insight into the effect of bed forms and vegetation patches on flow resistance and distribution of velocity and bed shear stress. The boundary layer characteristics method was employed to calculate the shear velocity and friction factor, the resulted friction values are compared with that of Manning’s equation. The friction of bed forms was estimated using a linear superposition of friction. Results showed that the boundary layer characteristics method was suitable for calculating the shear velocity under various conditions, moreover, the distribution of velocity and bed shear stress were discussed. The findings of this study will be useful in river management and restoration.
... Amongst the well-known friction and flow resistance indices, the Darcy-Weisbach friction factor is one of the most applied ones in hydraulic calculations. The total friction factor (f ) can be written as the following [13,14]: ...
... This equation is valid for both uniform and non-uniform flow if presents the true gradient of the energy loss. On the other hand, τ*cr can be written in association with as the following [14,26]: * 0.15 . By considering Einstein's Equation, the Darcy-Weisbach friction factor can be calculated as: ...
... This equation is valid for both uniform and non-uniform flow if S f presents the true gradient of the energy loss. On the other hand, τ* cr can be written in association with S f as the following [14,26]: Assuming k s = d 50 in Equation (5), and by comparing Equations (6) and (7), f can be calculated via logarithm rules. The final equation can be written as [14]: ...
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The motion of sediment and development of bed forms in natural rivers still constitute intriguing phenomena that always challenge engineers and researchers involved in a variety of problems including river hydraulics and other geophysical and environmental flows. Resistance to flow in alluvial streams has been widely investigated. Since the pioneer work of Exner [12] and, later, Einstein and Barbarossa [8], the profound effects of movable sediments on the flow structure have been recognized as relevant in order to determine the nature of flow resistance (Kennedy [21]; Yalin [40]; Vanoni [36]; García and Parker [14]). This paper presents a simple methodology, based on energy balance, to compute the components of the total shear stress (grain and form-drag) acting on a uniform, two-dimensional flow over fully developed dunes. The method considers mainly an analysis of spatially-averaged (over several dune wavelengths) shear stress distributions. Then, boundary-layer theory is used in order to find an equivalent turbulent-wall-shear flow for the actual spatially-averaged flow over dunes and to derive a characteristic roughness length for the logarithmic velocity distribution above the dunes. The corresponding roughness function that also appears in the logarithmic velocity profile is also investigated and related to dimensionless sediment transport parameters. Alluvial roughness is found to be a function of a single parameter which includes the total Shields stress, a particle Reynolds number, and the ratio between flow depth and sediment diameter.
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Data compiled from eight decades of incipient motion studies were used to calculate dimensionless critical shear stress values of the median grain size, τ* (c)50. Calculated τ* (c)50 values were stratified by initial motion definition, median grain size type (surface, subsurface, or laboratory mixture), relative roughness, and flow regime. A traditional Shields plot constructed from data that represent initial motion of the bed surface material reveals systematic methodological biases of incipient motion definition; τ* (c)50 values determined from reference bed load transport rates and from visual observation of grain motion define subparallel Shields curves, with the latter generally underlying the former: values derived from competence functions define a separate but poorly developed field, while theoretical values predict a wide range of generally higher stresses that likely represent instantaneous, rather than time-averaged, critical shear stresses. The available data indicate that for high critical boundary Reynolds numbers and low relative roughnesses typical of gravel-bedded rivers, reference-based and visually based studies have τ*(c)50 ranges of 0.052-0.086 and 0.030-0.073, respectively. The apparent lack of a universal τ*(c)50 for gravel-bedded rivers warrants great care in choosing defendable τ*