RTe-bookCh5Hydraulics.ppt

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
1
CHAPTER 5:
REVIEW OF 1D OPEN CHANNEL HYDRAULICS
Dam at Hiram Falls on the Saco River near Hiram, Maine, USA

2
TOPICS REVIEWED
This e-book is not intended to include a full treatment of open channel flow. It is
assumed that the reader has had a course in open channel flow, or has access
to texts that cover the field. Nearly all undergraduate texts in fluid mechanics for
civil engineers have sections on open channel flow (e.g. Crowe et al., 2001).
Three texts that specifically focus on open channel flow are those by Henderson
(1966), Chaudhry (1993) and Jain (2000).
Topics treated here include:
• Relations for boundary resistance
• Normal (steady, uniform) flow
• St. Venant shallow water equations
• Gradually varied flow
• Froude number: subcritical, critical and supercritical flow
• Classification of backwater curves
• Numerical calculation of backwater curves

3
SIMPLIFICATION OF CHANNEL CROSS-SECTIONAL SHAPE
River channel cross sections have complicated shapes. In a 1D analysis, it is
appropriate to approximate the shape as a rectangle, so that B denotes channel
width and H denotes channel depth (reflecting the cross-sectionally averaged depth
of the actual cross-section). As was seen in Chapter 3, natural channels are
generally wide in the sense that Hbf/Bbf << 1, where the subscript “bf” denotes
“bankfull”. As a result the hydraulic radius Rh is usually approximated accurately by
the average depth. In terms of a rectangular channel,
H
B
channel floodplain
floodplain
H
B
H
2
1
H
H
2
B
HB
Rh 











4
THE SHIELDS NUMBER:
A KEY DIMENSIONLESS PARAMETER QUANTIFYING SEDIMENT MOBILITY
gD
R
b




3
c
2
b
2
g
3
4
~










D
R
D
b = boundary shear stress at the bed (= bed drag force acting on the flow per unit
bed area) [M/L/T2]
c = Coulomb coefficient of resistance of a granule on a granular bed [1]
Recalling that R = (s/) – 1, the Shields Number * is defined as
It can be interpreted as a ratio scaling the ratio impelling force of flow drag acting on
a particle to the Coulomb force resisting motion acting on the same particle, so that
The characterization of bed mobility thus requires a quantification of boundary shear
stress at the bed.

5
QUANTIFICATION OF BOUNDARY SHEAR STRESS AT THE BED
2
/
1
f
C
u
U
Cz 



U = cross-sectionally averaged flow velocity ( depth-averaged
flow velocity in the wide channels studied here) [L/T]
u* = shear velocity [L/T]
Cf = dimensionless bed resistance coefficient [1]
Cz = dimensionless Chezy resistance coefficient [1]
2
b
f
U
C



BH
Q
U 




b
u

6
RESISTANCE RELATIONS FOR HYDRAULICALLY ROUGH FLOW
6
/
1
s
r
2
/
1
f
k
H
C
u
U
Cz 










 

Keulegan (1938) formulation:
90
s
k
s D
n
k 











 
 s
2
/
1
f
k
H
11
n
1
C
u
U
Cz 
where  = 0.4 denotes the dimensionless Karman constant and ks = a roughness
height characterizing the bumpiness of the bed [L].
Manning-Strickler formulation:
where r is a dimensionless constant between 8 and 9. Parker (1991) suggested
a value of r of 8.1 for gravel-bed streams.
Roughness height over a flat bed (no bedforms):
where Ds90 denotes the surface sediment size such that 90 percent of the
surface material is finer, and nk is a dimensionless number between 1.5 and 3.
For example, Kamphuis (1974) evaluated nk as equal to 2.

7
COMPARISION OF KEULEGAN AND MANNING-STRICKLER RELATIONS
r = 8.1
Note that Cz does not
vary strongly with depth.
It is often approximated
as a constant in broad-
brush calculations.
1
10
100
1 10 100 1000
H/ks
Cz
Keulegan
Parker Version of Manning-
Strickler
6
/
1
s
k
H
1
.
8
Cz 









8
BED RESISTANCE RELATION FOR MOBILE-BED FLUME EXPERIMENTS
Sediment transport relations for rivers have traditionally been
determined using a simplified analog: a straight, rectangular
flume with smooth, vertical sidewalls. Meyer-Peter and Müller
(1948) used two famous early data sets of flume data on
sediment transport to determine their famous sediment transport
relation (introduced later). These are a) a subset of the data of
Gilbert (1914) collected at Berkeley, California (D50 = 3.17 mm,
4.94 mm and 7.01 mm) and the set due to Meyer-Peter et al.
(1934) collected at E.T.H., Zurich, Switzerland (D50 = 5.21 mm
and 28.65 mm).
Bedforms such as dunes were present in many of the
experiments in these two sets. In the case of 116 experiments
of Gilbert and 52 experiments of Meyer-Peter et al., it was
reported that no bedforms were present and that sediment was
transported under flat-bed conditions. Wong (2003) used this
data set to study bed resistance over a mobile bed without
bedforms.
Flume at Tsukuba
University, Japan
(flow turned off).
Image courtesy H.
Ikeda. Note that
bedforms known
as linguoid bars
cover the bed.

9
BED RESISTANCE RELATION FOR MOBILE-BED FLUME EXPERIMENTS
contd.
Most laboratory flumes are not wide enough to prevent sidewall effects. Vanoni
(1975), however, reports a method by which sidewall effects can be removed from
the data. As a result, depth H is replaced by the hydraulic radius of the bed region
Rb. (Not to worry, Rb  H as H/B  0). Wong (2003) used this procedure to
remove sidewall effects from the previously-mentioned data of Gilbert (1914) and
Meyer-Peter et al. (1934).
The material used in all the experiments in question was quite well-sorted. Wong
(2003) estimated a value of D90 from the experiments using the given values of
median size D50 and geometric standard deviation g, and the following relation for
a log-normal grain size distribution;
Wong then estimated ks as equal to 2D90 in accordance with the result of
Kamphuis (1974), and s in the Manning-Strickler resistance relation as 8.1 in
accordance with Parker (1991). The excellent agreement with the data is
shown on the next page.
28
.
1
g
50
90 D
D 


10
1.00
10.00
100.00
1.00 10.00 100.00
Rb/ks
Cz
ETH 52
Gilbert 116
Parker Version of Manning-
Strickler
6
/
1
s
b
k
R
1
.
8
Cz 








TEST OF RESISTANCE RELATION AGAINST MOBILE-BED DATA WITHOUT
BEDFORMS FROM LABORATORY FLUMES

11
NORMAL FLOW
Normal flow is an equilibrium state defined by a perfect balance between the
downstream gravitational impelling force and resistive bed force. The
resulting flow is constant in time and in the downstream, or x direction.
Parameters:
x = downstream coordinate [L]
H = flow depth [L]
U = flow velocity [L/T]
qw = water discharge per unit width [L2T-1]
B = width [L]
Qw = qwB = water discharge [L3/T]
g = acceleration of gravity [L/T2]
 = bed angle [1]
b = bed boundary shear stress [M/L/T2]
S = tan = streamwise bed slope [1]
(cos   1; sin   tan   S)
 = water density [M/L3]
As can be seen from Chapter 3, the
bed slope angle  of the great
majority of alluvial rivers is sufficiently
small to allow the approximations
1
cos
,
S
tan
sin 






x
B
x
gHxBS
bBx
H

12
UHB
B
q
Q
UH
q w
w
w 


Conservation of downstream momentum:
Impelling force (downstream component of weight of water) = resistive force
x
B
xS
gHB
sin
x
gHB b 








gHS
b 


Reduce to obtain depth-slope
product rule for normal flow:
NORMAL FLOW contd.
Conservation of water mass (= conservation of water volume as water can be
treated as incompressible):

x
B
x
gHxBS
bBx
H
gHS
u 


13
ESTIMATED CHEZY RESISTANCE COEFFICIENTS FOR BANKFULL FLOW
BASED ON NORMAL FLOW ASSUMPTION FOR u*
The plot below is from Chapter 3
1
10
100
1 10 100 1000 10000 100000
Grav Brit
Grav Alta
Grav Ida
Sand Mult
Sand Sing
bf
Cz
Ĥ
50
bf
bf
bf
bf
bf
bankfull
bf
D
H
Ĥ
,
S
gH
H
B
Q
u
U
Cz 












14
Relation for Shields stress  at normal equilibrium:
(for sediment mobility calculations)
gHS
U
C 2
f 


RELATION BETWEEN qw, S and H AT NORMAL EQUILIBRIUM
D
R
HS
gD
R
b





D
R
S
g
q
C 3
/
2
3
/
1
2
w
f










3
/
1
2
w
f
gS
q
C
H 








2
/
1
2
/
1
z
2
/
1
2
/
1
f
S
H
g
C
S
H
C
g
U 

or
Reduce the relation for momentum conservation b = gHS with the resistance form
b = CfU2:
Generalized Chezy
velocity relation
Further eliminating U with the relation for water mass conservation qw = UH and
solving for flow depth:

15
ESTIMATED SHIELDS NUMBERS FOR BANKFULL FLOW
BASED ON NORMAL FLOW ASSUMPTION FOR b
The plot below is from Chapter 3
2
50
50
bf
50
bf
50
b
50
bf
D
gD
Q
Q̂
,
D
R
S
H
gD
R






1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 1.E+12 1.E+14
Grav Brit
Grav Alta
Sand Mult
Sand Sing
Grav Ida
Q̂

 50
bf

16
RELATIONS AT NORMAL EQUILIBRIUM WITH MANNING-STRICKLER
RESISTANCE FORMULATION
6
/
1
s
r
2
/
1
f
3
/
1
2
w
f
k
H
C
gS
q
C
H 

















 
RD
S
g
q
k 10
/
7
10
/
3
2
r
2
w
3
/
1
s











Relation for Shields stress  at normal equilibrium:
(for sediment mobility calculations)
10
/
3
2
r
2
w
3
/
1
s
gS
q
k
H 









2
/
1
3
/
2
6
/
1
s
r
2
/
1
2
/
1
f
S
H
k
g
S
H
C
g
U 

 6
/
1
s
r
2
/
1
3
/
2
k
g
n
1
,
S
H
n
1
U 


Manning-Strickler velocity relation
(n = Manning’s “n”)
Solve for H
to find
Solve for U
to find

17
BUT NOT ALL OPEN-CHANNEL FLOWS ARE AT OR CLOSE TO EQUILIBRIUM!
Flow into standing water (lake or
reservoir) usually takes the form
of an M1 curve.
Flow over a free overfall
(waterfall) usually takes the form
of an M2 curve.
A key dimensionless parameter describing the way
in which open-channel flow can deviate from
normal equilibrium is the Froude number Fr: gH
U

Fr
And therefore the calculation of bed shear stress as b = gHS is not always
accurate. In such cases it is necessary to compute the disquilibrium (e.g.
gradually varied) flow and calculate the bed shear stress from the relation
2
f
b U
C




18
NON-STEADY, NON-UNIFORM 1D OPEN CHANNEL FLOWS:
St. Venant Shallow Water Equations
0
x
UH
t
H






Relation for water mass conservation
(continuity):
Relation for momentum conservation:
2
f
2
2
U
C
x
gH
x
H
g
2
1
x
H
U
t
UH














x = boundary (bed) attached nearly horizontal coordinate [L]
y = upward normal coordinate [L]
 = bed elevation [L]
S = tan  - /x [1]
H = normal (nearly vertical) flow depth [L]
Here “normal” means “perpendicular to the bed” and has
nothing to do with normal flow in the sense of equilibrium.
x
y

H
Bed and water surface slopes
exaggerated below for clarity.

19
DERIVATION: EQUATION OF CONSERVATION OF OF WATER MASS
Q = UHB = volume water discharge [L3/T]
Q = Mass water discharge = UHB [M/T]
/t(Mass in control volume) = Net mass inflow rate
x
x
UH
B
UHB
UHB
t
x
HB
x
x
x
















0
x
UH
t
H






Reducing under assumption of
constant B:
H
B
x
Q
Q

20
STREAMWISE MOMENTUM DISCHARGE
Momentum flows!
Qm = U2HB = streamwise discharge of streamwise momentum [ML/T2]. The
derivation follows below.
Momentum crossing left face in time t = (HBU2t) = mass x velocity
Qm = momentum crossing per unit time, = (Momentum crossing in t)/ t = U2HB
HB
U
Q 2
m 

Ut
(HBUt)(U)
U
Note that the streamwise momentum discharge has the same units as force,
and is often referred to as the streamwise inertial force.

21
The flow is assumed to be gradually varying, i.e. the spatial scale Lx of variation in
the streamwise direction satisfies the condition H/Lx << 1. Under this assumption the
pressure p can be approximated as hydrostatic. Where z = an upward normal
coordinate from the bed,
STREAMWISE PRESSURE FORCE
 


H
0
2
p BgH
2
1
pdz
B
F
Fp = pressure force [ML/T2]
g
z
p





Integrate and evaluate the constant of integration under the condition of zero (gage)
pressure at the water surface, where y = H, to get:
 
z
H
g
p 


p = pressure (normal stress) [M/L/T2]
Integrate the above relation over the cross-sectional area
to find the streamwise pressure force Fp:
p
H
Fp
B

22
DERIVATION: EQUATION OF CONSERVATION OF STREAMWISE MOMENTUM
/t(Momentum in control volume) = net momentum inflow rate + sum of forces
Sum of forces = downstream gravitational force – resistive force + pressure force
at x – pressure force at x + x
xB
U
C
x
x
gHB
B
gH
2
1
B
gH
2
1
HB
U
HB
U
t
xU
HB 2
f
x
x
2
x
2
x
x
2
x
2

























2
f
2
U
C
x
gH
x
H
gH
x
H
U
t
HU














x
Qm
B
H
Qm
Fp
Fp
gHBxS
bBx
or reducing,

23
CASE OF STEADY, GRADUALLY VARIED FLOW
0
x
UH
t
H






2
f
2
2
U
C
x
gH
x
H
g
2
1
x
H
U
t
UH
















 w
q
UH
H
U
C
dx
d
g
dx
dH
g
dx
dU
U
2
f





Reduce equation of water mass
conservation and integrate:
dx
dH
H
q
dx
dU
H
q
U 2
w
w




Thus:
Reduce equation of streamwise
momentum conservation:
But with water conservation:
dx
dU
UH
dx
dUH
U
dx
H
dU2


So that momentum
conservation reduces to:
constant

24
THE BACKWATER EQUATION
to get the backwater equation:
2
f
f
2
3
2
w
2
C
S
,
gH
U
gH
q
,
x
S Fr
Fr 







2
f
1
S
S
dx
dH
Fr



H
U
C
x
g
x
H
g
dx
dU
U
2
f









Reduce
with
dx
dH
H
q
dx
dU
,
H
q
U 2
w
w



where
Here Fr denotes the Froude number of the flow and Sf denotes the friction slope.
For steady flow over a fixed bed, bed slope S (which can be a function of x) and
constant water discharge per unit width qw are specified, so that the backwater
equation specified a first-order differential equation in H, requiring a specified
value of H at some point as a boundary condition.

25
2
f
1
S
S
dx
dH
Fr



3
/
1
2
w
f
n
3
n
2
w
f
f
gS
q
C
H
gH
q
C
S
S 











Consider the case of constant bed slope S. Setting the numerator of the right-hand
side backwater equation = zero, so that S = Sf (friction slope equals bed slope)
recovers the condition of normal equilibrium, at which normal depth Hn prevails:
Setting the denominator of the right-hand side of the backwater equation = zero yields
the condition of Froude-critical flow, at which Fr = 1 and depth = the critical value Hc:
3
/
1
2
w
c
3
c
2
w
2
g
q
H
gH
q
1 










 Fr
NORMAL AND CRITICAL DEPTH
At any given point in a gradually varied flow the depth H may differ from both Hn and
Hc. If Fr = qw/(gH3)1/2 < 1 the flow slow and deep and is termed subcritical; if on the
other hand Fr > 1 the flow is swift and shallow and is termed supercritical. The
great majority of flows in alluvial rivers are subcritical, but supercritical flows
do occur. Supercritical flows are common during floods in steep bedrock rivers.

26
COMPUTATION OF BACKWATER CURVES
The case of constant bed slope S is considered as an example.
Let water discharge qw and bed slope S be given.
In the case of constant bed friction coefficient Cf, let Cf be given.
In the case of Cf specified by the Manning-Strickler relation, let r and ks be given.
Compute Hc:
Compute Hn
:
If Hn > Hc then (Fr)n < 1: normal flow is subcritical, defining a “mild” bed slope.
If Hn < Hc then (Fr)n > 1: normal flow is supercritical, defining a “steep” bed slope.
,
)
H
(
1
)
H
(
S
S
dx
dH
2
f
Fr



Requires 1 b.c. for
unique solution:
1
x
H
H
1

3
/
1
2
w
c
g
q
H 








3
/
1
2
w
f
n
gS
q
C
H 








10
/
3
2
r
2
w
3
/
1
s
n
gS
q
k
H 









or
3
2
w
3
/
1
s
2
r
f
3
2
w
f
f
3
2
w
2
gH
q
k
H
)
H
(
S
or
gH
q
C
)
H
(
S
,
gH
q
)
H
(














Fr
where x1 is a starting point. Integrate upstream
if the flow at the starting point is subcritical,
and integrate downstream if it is supercritical.

27
COMPUTATION OF BACKWATER CURVES contd.
Flow at a point relative to critical flow: note that
It follows that 1 – Fr2(H) < 0 if H < Hc, and 1 – Fr2 > 0 if H > Hc.
3
c
2
w
3
2
w
2
gH
q
1
,
gH
q
)
H
( 

Fr
Flow at a point relative to normal flow: note that for the case of constant Cf
3
n
2
w
f
3
2
w
f
f
gH
q
C
S
,
gH
q
C
)
H
(
S 

3
n
2
w
3
/
1
s
n
2
r
3
2
w
3
/
1
s
2
r
f
gH
q
k
H
S
,
gH
q
k
H
)
H
(
S
























and for the case of the Manning-Strickler relation
It follows in either case that S – Sf(H) < 0 if H < Hn, and S – Sf(H) > 0
if H > Hn.

28
MILD BACKWATER CURVES M1, M2 AND M3
M1: H1 > Hn > Hc
Again the case of constant bed slope S is considered. Recall that
M2: Hn > H1 > Hc
M3: Hn > Hc > H1






)
H
(
1
)
H
(
S
S
dx
dH
1
2
1
f
x1
Fr
Depth increases downstream,
decreases upstream
Depth increases downstream,
decreases upstream
Depth decreases downstream,
increases upstream
A bed slope is considered mild if Hn > Hc. This is the most common case in
alluvial rivers. There are three possible cases.
3
2
w
3
/
1
s
2
r
f
3
2
w
f
f
3
2
w
2
gH
q
k
H
)
H
(
S
or
gH
q
C
)
H
(
S
,
gH
q
)
H
(














Fr






)
H
(
1
)
H
(
S
S
dx
dH
1
2
1
f
x1
Fr






)
H
(
1
)
H
(
S
S
dx
dH
1
2
1
f
x1
Fr

29
M1 CURVE
M1: H1 > Hn > Hc






)
H
(
1
)
H
(
S
S
dx
dH
2
f
Fr
3
2
w
2
3
2
w
f
f
gH
q
,
gH
q
C
S 
 Fr
Water surface elevation  =  + H (remember H is measured normal to the bed, but
is nearly vertical as long as S << 1). Note that Fr < 1 at x1: integrate upstream.
Starting and normal (equilibrium) flows are subcritical.
As H increases downstream, both Sf and Fr decrease toward 0.
Far downstream, dH/dx = S  d/dx = d/dx(H + ) = constant: ponded water
As H decreases upstream, Sf approaches S while Fr remains < 1.
Far upstream, normal flow is approached.
Bed slope has
been exaggerated
for clarity.
H
Hc Hn
H1


The M1 curve
describes subcritical
flow into ponded water.

30
M2 CURVE
M1: Hn > H1 > Hc






)
H
(
1
)
H
(
S
S
dx
dH
2
f
Fr
3
2
w
2
3
2
w
f
f
gH
q
,
gH
q
C
S 
 Fr
Note that Fr < 1 at x1; integrate upstream. Starting and normal (equilibrium) flows
are subcritical.
As H decreases downstream, both Sf and Fr increase, and Fr increases toward 1.
At some point downstream, Fr = 1 and dH/dx = - : free overfall (waterfall).
As H increases upstream, Sf approaches S while Fr remains < 1.
Far upstream, normal flow is approached.
Bed slope has
been exaggerated
for clarity.
The M2 curve describes
subcritical flow over a free
overfall.
Hc Hn



31
M3 CURVE
M1: Hn > Hc > H1






)
H
(
1
)
H
(
S
S
dx
dH
2
f
Fr
3
2
w
2
3
2
w
f
f
gH
q
,
gH
q
C
S 
 Fr
Note that Fr > 1 at x1; integrate downstream. The starting flow is supercritical, but
the equilibrium (normal) flow is subcritical, requiring an intervening hydraulic jump.
As H increases downstream, both Sf and Fr decrease, and Fr decreases toward 1.
At the point where Fr would equal 1, dH/dx would equal . Before this state is
reached, however, the flow must undergo a hydraulic jump to subcritical flow.
Subcritical flow can make the transition to supercritical flow without a hydraulic
jump; supercritical flow cannot make the transition to subcritical flow without one.
Hydraulic jumps are discussed in more detail in Chapter 23.
Bed slope has
been exaggerated
for clarity.
The M3 curve describes
supercritical flow from a
sluice gate.
H1
Hc Hn


Hydraulic jump
M3 curve

32
HYDRAULIC JUMP
subcritical
flow
supercritical
In addition to M1, M2, and M3 curves, there is also the family of steep S1, S2 and S3
curves corresponding to the case for which Hc > Hn (normal flow is supercritical).
These curves tend to be very short, and are not covered in detail here.

33
CALCULATION OF BACKWATER CURVES
Here the case of subcritical flow is considered, so that the direction of integration is
upstream. Let x1 be the starting point where H1 is given, and let x denote the step
length, so that xn+1 = xn - x. (Note that xn+1 is upstream of xn.) Furthermore, denote
the function [S-Sf(H)]/(1 – Fr2(H)] as F(H). In an Euler step scheme,
or thus
)
H
(
x
H
H
dx
dH
n
1
n
n
F



 
x
)
H
(
H
H n
n
1
n 


 F
A better scheme is a predictor-corrector scheme, according to which
x
)
H
(
H
H n
n
1
n
,
p 


 F
  x
)
H
(
)
H
(
2
1
H
H 1
n
,
p
n
n
1
n 


 
 F
F
A predictor-corrector scheme is used in the spreadsheet
RTe-bookBackwater.xls. This spreadsheet is used in the calculations of the
next few slides.

34
BACKWATER MEDIATES THE UPSTREAM EFFECT OF BASE LEVEL
(ELEVATION OF STANDING WATER)
A WORKED EXAMPLE (constant Cz):
S = 0.00025
Cz = 22
qw = 5.7 m2/s
D = 0.6 mm
R = 1.65
H1 = 30 m H1 > Hn > Hc
so M1 curve
m
01
.
3
gS
q
C
H
3
/
1
2
w
f
n 









m
49
.
1
g
q
H
3
/
1
2
w
c 









H
Hc Hn
H1


Example: calculate the variation in H and b = CfU2 in x upstream of x1
(here set equal to 0) until H is within 1 percent of Hn

35
0
5
10
15
20
25
30
35
-140000 -120000 -100000 -80000 -60000 -40000 -20000 0
x, m
H
(m),

b
(N/m
2
),
U
(m/s)
H
U
tb
RESULTS OF CALCULATION: PROFILES OF DEPTH H, BED SHEAR STRESS b
AND FLOW VELOCITY U
b
U
H

36
RESULTS OF CALCULATION: PROFILES OF BED ELEVATION  AND WATER
SURFACE ELEVATION 
0
5
10
15
20
25
30
35
40
-140000 -120000 -100000 -80000 -60000 -40000 -20000 0
x m
bed
(

)
and
water
surface
(

)
elevations
m
h
x



37
REFERENCES FOR CHAPTER 5
Chaudhry, M. H., 1993, Open-Channel Flow, Prentice-Hall, Englewood Cliffs, 483 p.
Crowe, C. T., Elger, D. F. and Robertson, J. A., 2001, Engineering Fluid Mechanics, John Wiley
and sons, New York, 7th Edition, 714 p.
Gilbert, G.K., 1914, Transportation of Debris by Running Water, Professional Paper 86, U.S.
Geological Survey.
Jain, S. C., 2000, Open-Channel Flow, John Wiley and Sons, New York, 344 p.
Kamphuis, J. W., 1974, Determination of sand roughness for fixed beds, Journal of Hydraulic
Research, 12(2): 193-202.
Keulegan, G. H., 1938, Laws of turbulent flow in open channels, National Bureau of Standards
Research Paper RP 1151, USA.
Henderson, F. M., 1966, Open Channel Flow, Macmillan, New York, 522 p.
Meyer-Peter, E., Favre, H. and Einstein, H.A., 1934, Neuere Versuchsresultate über den
Geschiebetrieb, Schweizerische Bauzeitung, E.T.H., 103(13), Zurich, Switzerland.
Meyer-Peter, E. and Müller, R., 1948, Formulas for Bed-Load Transport, Proceedings, 2nd
Congress, International Association of Hydraulic Research, Stockholm: 39-64.
Parker, G., 1991, Selective sorting and abrasion of river gravel. II: Applications, Journal of
Hydraulic Engineering, 117(2): 150-171.
Vanoni, V.A., 1975, Sedimentation Engineering, ASCE Manuals and Reports on Engineering
Practice No. 54, American Society of Civil Engineers (ASCE), New York.
Wong, M., 2003, Does the bedload equation of Meyer-Peter and Müller fit its own data?,
Proceedings, 30th Congress, International Association of Hydraulic Research, Thessaloniki,
J.F.K. Competition Volume: 73-80.

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