REVIEW OF 1D OPEN CHANNEL HYDRAULICS

1
CHAPTER 5 REVIEW OF 1D OPEN CHANNEL HYDRAULICS
Dam at Hiram Falls on the Saco River near Hiram,
Maine, USA
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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

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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 ltlt 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,
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THE SHIELDS NUMBER A KEY DIMENSIONLESS PARAMETER
QUANTIFYING SEDIMENT MOBILITY
?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.
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QUANTIFICATION OF BOUNDARY SHEAR STRESS AT THE BED
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
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RESISTANCE RELATIONS FOR HYDRAULICALLY ROUGH FLOW
Keulegan (1938) formulation
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.
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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.
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.
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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.
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TEST OF RESISTANCE RELATION AGAINST MOBILE-BED
DATA WITHOUT BEDFORMS FROM LABORATORY FLUMES
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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
• tb 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
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NORMAL FLOW contd.
Conservation of water mass ( conservation of
water volume as water can be treated as
incompressible)
Conservation of downstream momentum Impelling
force (downstream component of weight of water)
resistive force
Reduce to obtain depth-slope product rule for
normal flow
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ESTIMATED CHEZY RESISTANCE COEFFICIENTS FOR
BANKFULL FLOW BASED ON NORMAL FLOW ASSUMPTION FOR
u The plot below is from Chapter 3
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RELATION BETWEEN qw, S and H AT NORMAL EQUILIBRIUM
Reduce the relation for momentum conservation ?b
?gHS with the resistance form ?b ?CfU2
Generalized Chezy velocity relation
or
Further eliminating U with the relation for water
mass conservation qw UH and solving for flow
depth
Relation for Shields stress ?? at normal
equilibrium (for sediment mobility calculations)
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ESTIMATED SHIELDS NUMBERS FOR BANKFULL FLOW BASED
ON NORMAL FLOW ASSUMPTION FOR ?b The plot
below is from Chapter 3
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RELATIONS AT NORMAL EQUILIBRIUM WITH
MANNING-STRICKLER RESISTANCE FORMULATION
Solve for H to find
Solve for U to find
Manning-Strickler velocity relation (n
Mannings n)
Relation for Shields stress ?? at normal
equilibrium (for sediment mobility calculations)
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BUT NOT ALL OPEN-CHANNEL FLOWS ARE AT OR CLOSE TO
EQUILIBRIUM!
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
Flow over a free overfall (waterfall) usually
takes the form of an M2 curve.
Flow into standing water (lake or reservoir)
usually takes the form of an M1 curve.
A key dimensionless parameter describing the way
in which open-channel flow can deviate from
normal equilibrium is the Froude number Fr
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NON-STEADY, NON-UNIFORM 1D OPEN CHANNEL
FLOWS St. Venant Shallow Water Equations

• 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.

Bed and water surface slopes exaggerated below
for clarity.
Relation for water mass conservation (continuity)
Relation for momentum conservation
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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
Reducing under assumption of constant B
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STREAMWISE MOMENTUM DISCHARGE Momentum flows!
Qm ?U2HB streamwise discharge of streamwise
momentum ML/T2. The derivation follows below.
Momentum crossing left face in time Dt
(?HBU2Dt) mass x velocity Qm momentum
crossing per unit time, (Momentum crossing in
Dt)/ Dt ?U2HB
Note that the streamwise momentum discharge has
the same units as force, and is often referred to
as the streamwise inertial force.
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STREAMWISE PRESSURE FORCE
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
ltlt 1. Under this assumption the pressure p can
be approximated as hydrostatic. Where z an
upward normal coordinate from the bed,
p pressure (normal stress) M/L/T2
Integrate and evaluate the constant of
integration under the condition of zero (gage)
pressure at the water surface, where y H, to
get
Integrate the above relation over the
cross-sectional area to find the streamwise
pressure force Fp
Fp pressure force ML/T2
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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
or reducing,
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CASE OF STEADY, GRADUALLY VARIED FLOW
Reduce equation of water mass conservation and
integrate
constant
Thus
Reduce equation of streamwise momentum
conservation
But with water conservation
So that momentum conservation reduces to
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THE BACKWATER EQUATION
Reduce
with
to get the backwater equation
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.
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NORMAL AND CRITICAL DEPTH
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
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 lt 1 the flow slow and deep and is
termed subcritical if on the other hand Fr gt 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.
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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 gt Hc then (Fr)n lt 1 normal flow is
subcritical, defining a mild bed slope. If Hn lt
Hc then (Fr)n gt 1 normal flow is supercritical,
defining a steep bed slope.
or
where x1 is a starting point. Integrate upstream
if the flow at the starting point is
subcritical, and integrate downstream if it is
supercritical.
Requires 1 b.c. for unique solution
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COMPUTATION OF BACKWATER CURVES contd.
Flow at a point relative to critical flow note
that
It follows that 1 Fr2(H) lt 0 if H lt Hc, and 1
Fr2 gt 0 if H gt Hc.
Flow at a point relative to normal flow note
that for the case of constant Cf
and for the case of the Manning-Strickler
relation
It follows in either case that S Sf(H) lt 0 if H
lt Hn, and S Sf(H) gt 0 if H gt Hn.
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MILD BACKWATER CURVES M1, M2 AND M3
Again the case of constant bed slope S is
considered. Recall that
A bed slope is considered mild if Hn gt Hc. This
is the most common case in alluvial rivers.
There are three possible cases.
Depth increases downstream, decreases upstream
M1 H1 gt Hn gt Hc
Depth decreases downstream, increases upstream
M2 Hn gt H1 gt Hc
Depth increases downstream, decreases upstream
M3 Hn gt Hc gt H1
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M1 CURVE
M1 H1 gt Hn gt Hc
Water surface elevation ? ? H (remember H is
measured normal to the bed, but is nearly
vertical as long as S ltlt 1). Note that Fr lt 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 lt 1. Far upstream, normal flow is
approached.
The M1 curve describes subcritical flow into
ponded water.
Bed slope has been exaggerated for clarity.
30
M2 CURVE
M1 Hn gt H1 gt Hc
Note that Fr lt 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 lt
1. Far upstream, normal flow is approached.
The M2 curve describes subcritical flow over a
free overfall.
Bed slope has been exaggerated for clarity.
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M3 CURVE
M1 Hn gt Hc gt H1
Note that Fr gt 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.
The M3 curve describes supercritical flow from a
sluice gate.
Bed slope has been exaggerated for clarity.
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HYDRAULIC JUMP
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 gt Hn
(normal flow is supercritical). These curves
tend to be very short, and are not covered in
detail here.
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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 xn1
xn - ?x. (Note that xn1 is upstream of xn.)
Furthermore, denote the function S-Sf(H)/(1
Fr2(H) as F(H). In an Euler step scheme,
or thus
A better scheme is a predictor-corrector scheme,
according to which
A predictor-corrector scheme is used in the
spreadsheet RTe-bookBackwater.xls. This
spreadsheet is used in the calculations of
the next few slides.
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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
gt Hn gt Hc so M1 curve
Example calculate the variation in H and tb
?CfU2 in x upstream of x1 (here set equal to 0)
until H is within 1 percent of Hn
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RESULTS OF CALCULATION PROFILES OF DEPTH H, BED
SHEAR STRESS ?b AND FLOW VELOCITY U
H
tb
U
36
RESULTS OF CALCULATION PROFILES OF BED ELEVATION
h AND WATER SURFACE ELEVATION x
x
h
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.