HYDRAULICS OF MOUNTAIN RIVERS

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HYDRAULICS OF MOUNTAIN RIVERS Gary Parker,
University of Illinois
River in Taiwan Image courtesy C. Stark
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TOPICS COVERED

• This lecture is not intended to provide a full
  treatment of open channel flow. 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
• Approximations for the channel
• Shields number, Einstein number, generic
  bedload equation
• Boundary resistance in mountain streams Chezy
  and Manning-Strickler forms
• Skin friction and form drag in mountain rivers
• Backwater and the backwater length
• Normal (steady, uniform) flow
• Calculations of flow and sediment transport
  using the normal flow assumption

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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 the
lecture on hydraulic geometry, 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 reasonably 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 D
characteristic grain size (e.g. surface median
size Ds50) Recalling that R (?s/?) 1, the
Shields Number ? is defined as
It can be interpreted as scaling the ratio
impelling force of flow drag acting on a bed
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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THRESHOLD OF MOTION IN MOUNTAIN STREAMS
The threshold of motion is often expressed in
terms of a Shields curve (Shields, 1936). Let
denote the value of ? at the threshold of
motion, and Rep denote grain Reynolds number,
defined in the lecture on hydraulic geometry
as Based Neills (1968) work on coarse
sedimentg, Parker et al. (2003) amended the
Brownlie (1981) fit of the original Shields curve
to the form The asymptotic value of for
large Rep, i.e. coarse sediment, is 0.03.
Consider the case of quartz (R 1.65) in water
at 20?C (? 1x10-6 m2/s). The smallest value of
Ds50 for the data set introduced in the lecture
on hydraulic geometry is 27 mm, in which case Rep
17,850 and 0.0289. So an approximate
value of of 0.03 is appropriate for most
coarse-bedded mountain streams. In point of
fact, there is no sharply-defined threshold of
motion. The value of 0.03 should be interpreted
to be a value below which the bedload transport
rate is morphodynamically insignificant, not
precisely 0.
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SHIELDS NUMBER AT BANKFULL FLOW IN MOUNTAIN
STREAMS
It will be shown in Slide 26 that for the case of
mountain streams the Shields number at bankfull
flow can be estimated as where all parameters
were defined in the lecture on hydraulic
geometry. A plot of
versus is given to the right. The
data are those from the lecture on hydraulic
geometry. The average value of is
0.0486, i.e about 1.62 times the Shields number
at the threshold of motion. Most alluvial
gravel-bed streams move size Ds50 at bankfull
flow.
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THE EINSTEIN NUMBER A DIMENSIONLESS
QUANTIFICATION OF BEDLOAD TRANSPORT RATE
qb volume bedload transport rate per unit width
L2/T D characteristic grain size L R
(?s/?) 1 ? 1.65 for natural sediment 1
One standard approach to the quantification of
bedload transport is the specification of the
functional form
An example appropriate for the bedload transport
of gravel of uniform size is the modified form
of the bedload transport equation of Meyer-Peter
and Müller (1948) by Wong (2003) In field
gravel-bed rivers, however, a) the gravel is
rarely uniform and b) gravel is commonly
transported at Shields numbers below 0.0495 (see
previous slide).
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BOUNDARY RESISTANCE IN MOUNTAIN STREAMS
Let Q flow discharge L/T B water surface
width L H cross-sectionally averaged depth
L U Q/(BH) cross-sectionally averaged flow
velocity L u (?b/?)1/2 shear velocity
L/T Two dimensionless bed resistance
coefficients are defined here the Chezy
resistance coefficient Cz given as and the
standard bed resistance coefficient Cf ( f/8,
where f denotes the Darcy-Weisbach resistance
coefficient) Note that as the bed shear
stress increases, Cf increases and Cz decreases.
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RESISTANCE RELATIONS FOR HYDRAULICALLY ROUGH FLOW
Mountain streams are almost invariably in the
range of hydraulically rough flow, for which
resistance becomes independent of kinematic
viscosity ?. Keulegan (1938) offered the
following relation for hydraulically rough flow.
where ks a roughness height characterizing the
bumpiness of the bed L. A close approximation
is offered by the Manning-Strickler formulation
Parker (1991) suggested a value of ?r of 8.1 for
gravel-bed streams.
The roughness height over a flat bed of coarse
grains (no bedforms) is given as
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.
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TEST OF RESISTANCE RELATION AGAINST MOBILE-BED
DATA WITHOUT BEDFORMS FROM LABORATORY FLUMES
The data in question are from all the experiments
without bedforms used by Meyer-Peter and Müller
(1948) to develop their bedload relation.
Here Rb denotes the bed component of hydraulic
radius Rh rather than depth the flumes were too
narrow to allow the approximation Rh ? H.
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FORM DRAG AND SKIN FRICTION
The formulas of the previous slide hold only for
flat, coarse granular beds. River beds are
rarely flat. Lowland sand-bed streams typically
contain dunes. Dunes are not common in mountain
streams, but such bedforms as bars, pool-riffle
sequences and step-pool sequences are common.
All such features, as well as planform
irregularity, contributed added resistance, so Cz
is usually lower, and Cf is usually higher, than
predicted by the equation of the previous slide.
Step-pool pattern in the Hiyamizudani river,
Japan. Cour. K. Hasegawa
Bars in the Rhine River, Switzerland. Cour. M.
Jaeggi
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FORM DRAG AND SKIN FRICTION contd.
The component of the resistance coefficient Cfs
due to the flow acting on the grains themselves
is known as skin friction, and the extra
component Cff is denoted as form drag (due to
bedforms), to that the total resistance
coefficient Cf is given as
Cfs can be computed from the relations of Slides
10 and 11. In bar-dominated mountain streams,
form-drag is prominent at lower flows, but is
muted at flood flows (see images to the right).
In steeper streams with pool-riffle patterns, and
in particular step-pool patterns, form drag is
likely significant even at flood flows.
Elbow River, Alberta, Canada at low flow and
100-year flood. Cour. Alberta Research Council
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BACKWATER
If (most) alluvial streams are disturbed at a
point, the effect of that disturbance tends to
propagate upstream. For example, the effect of a
lake (slowing the flow down) or a waterfall
(speeding the flow up) is felt upstream, as
illustrated below.
Backwater from a lake
Backwater from a waterfall
Backwater effects are mediated by a dimensionless
number known as the Froude number Fr, where
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BACKWATER contd.
For most alluvial mountain streams, the Froude
number at bankfull flow Frbf satisfies the
condition As seen from the lecture on hydraulic
geometry. For the case Fr lt 1 backwater effects
propagate upstream, so that the effect of a
disturbance is felt upstream of it.
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BACKWATER contd.
Supercritical flow (Fr gt 1) does occur in very
steep mountain streams, and in particular streams
with step-pool patterns and bedrock streams. In
the case of a supercritical flow the effect of a
disturbance propagates downstream rather than
upstream.
Stream in the interior of British Columbia,
Canada. Cour. B. Eaton.
Sustained supercritical flow over an alluvial or
bedrock bed is unstable, and usually devolves
into a series of steps punctuated by hydraulic
jumps at formative flow.
Dry Meadow Creek, Calif., USA. Cour. M. Neumann
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THE BACKWATER LENGTH
The characteristic distance Lb upstream (in the
case of the more usual subcritical flow) or
downstream (in the case of supercritical flow) to
which the effect of a disturbance is felt is
known as the backwater length. Let Hd the flow
depth at the disturbance L S down-channel
slope of the river 1. The backwater length is
then given as Taking Hb to scale with Hbf,
some estimates of the backwater length are given
below. Estimates of the backwater length
obtained in this way average to 2.1 km, 1.1 km,
0.2 km, and 5.2 km for the data sets for Alberta,
Britain, Idaho and Colorado introduced in the
lecture on hydraulic geometry. That is,
backwater lengths tend to be very short in
mountain streams.
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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, in the absence of any perturbation due to
backwater. The resulting flow is constant in
time and in the downstream, or x direction. The
approximation of normal flow is often a very good
one in mountain streams.

• 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 the lecture on hydraulic
geometry, the bed slope S of most river, even
most mountain 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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CHEZY RESISTANCE COEFFICIENT AT BANKFULL FLOW
Using the normal flow approximation, it is found
between the relations
that Cz can be estimated as
Regression of all four data sets
This is how Czbf was estimated in the lecture on
hydraulic geometry, as shown to the right. If Cz
can be estimaged, the flow velocity U is then
given as This relation is known as Chezys law.
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MANNING-STRICKLER RESISTANCE RELATION FOR PLANE
BED
Using the normal flow approximation and the form
for Cz for a plane bed in the absence of bedforms
given in Slide 11, it is found that or solving
for U, This is known as a Manning-Strickler
resistance relation, where Mannings n (a
parameter that should be relegated to the dustbin
due to its perverse dimensions) is given
as (But you must remember to use MKS units
for n, whereas the equation for U works for any
consistent set of units).
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MANNING-STRICKLER RESISTANCE RELATION FOR
MOUNTAIN RIVERS AT BANKFULL FLOW
The regression of the data of Slide 21 for
mountain rivers at bankfull flow yields the
relation where Ubf Qbf/(HbfBbf), or
thus This represents a generalized
Manning-Strickler relation for mountain streams.
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FORM DRAG VERSUS SKIN FRICTION AT BANKFULL FLOW
In order to compare form drag versus skin
friction at bankfull flow, it is necessary to
estimate the roughness height ks. Here the
Kamphuis (1974) relation Is used in conjunction
with the reasonable estimate Then defining
Cf,bf, Cfs,bf and Cff,bf as the values of the
total resistance coefficient, the resistance
coefficient due to skin friction and the
resistance coefficient due to form drag,
respectively at bankfull flow, it is found
that The fraction of resistance that is form
drag Fform is thus given as
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FORM DRAG VERSUS SKIN FRICTION AT BANKFULL FLOW
contd.
The fraction of resistance that is form drag at
bankfull flow in mountain streams is less than
0.5. The fraction is 0.2 0.3 in relatively deep
mountain streams (Hbf/Ds50) gt 20, but can be
above 0.3 in relatively shallow mountain streams.
Deeper streams tend to have lower slopes, and
shallower streams tend to have higher slopes, as
shown in the next slide.
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SHIELDS NUMBER AT BANKFULL FLOW USING THE NORMAL
FLOW ASSUMPTION
Using the definition of the Shields number ? and
the estimate for bed shear stress ?b from the
normal flow approximation,
the following estimate is obtained for the
Shields number at bankfull flow This is the
origin of the estimate of Shields number used in
the chapter on hydraulic geometry and in Slide 7
of this lecture. A crude approximation of the
plot to the right yields so that S decreases as
Hbf/Ds50 increases.
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CALCULATING THE FLOW AT NORMAL EQUILIBRIUM
CHEZY FORMULATION
Between the relations
it can be shown that
Thus if the water discharge per unit width qw,
down-channel bed slope S, characteristic bed
grain size D and submerged specific gravity R are
known, and if the Chezy resistance coefficient Cz
can be estimated, the flow depth H, flow velocity
U, bed shear stress ?b and Shields number ? can
be computed as indicated above.
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CHEZY ? MANNING-STRICKLER
For the case of plane-bed rough flow, the
following formulation for resistance was given in
Slide 10
The corresponding relation based on data for
mountain rivers at bankfull flow is (Slide
20) Assuming that ks 2 Ds90 and Ds90 3
Ds50, the above relation can be cast into the form
Both relations can be cast in terms of a
generalized Manning-Strickler formulation, such
that
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CALCULATING THE FLOW AT NORMAL EQUILIBRIUM
MANNING-STRICKLER FORMULATION
Consider a generalized Manning-Strickler
resistance relation of the form where for
example ?g can be estimated as 5.92, ng can be
estimated as 0.210 and ks can be estimated as
2Ds90 for mountain gravel-bed streams at flood
flows (Slide 27). The relations for H, U, ?b and
? now become where
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CALCULATING BEDLOAD TRANSPORT AT NORMAL
EQUILIBRIUM
For no particularly good reason, most
formulations of bedload transport in gravel-bed
streams have ignored form drag. For the sake of
illustration, we do so here. Consider a
flume-like river with no form drag and
containing uniform gravel of size D, roughness
height ks ( 2D) and submerged specific gravity
R. The reach has bed slope S, and is conveying
water discharge per unit width qw. For this case
it is reasonable to assume ng 1/6 and ?g 8.1,
i.e. the relation of Slide 10. The Shields
number can be computed from the previous slide as
and the volume bedload transport rate per
unit width q can be estimated from Slide 7 as
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MANNING-STRICKLER STANDARD CASE OF ng 1/6
In the case of an exponent ng of 1/6 (the
standard Manning-Strickler exponent of Slide 10),
the relevant relations reduce to
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REFERENCES
Brownlie, W. R., 1981, Prediction of flow depth
and sediment discharge in open channels, Report
No. KH-R-43A, W. M. Keck Laboratory of Hydraulics
and Water Resources, California Institute of
Technology, Pasadena, California, USA, 232
p. 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. Neill, C. R., 1968, A reexamination of the
beginning of movement for coarse granular bed
materials, Report INT 68, Hydraulics Research
Station, Wallingford, England. Parker, G., 1991,
Selective sorting and abrasion of river gravel.
II Applications, Journal of Hydraulic
Engineering, 117(2) 150-171.
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REFERENCES
Parker, G., Toro-Escobar, C. M., Ramey, M. and S.
Beck, 2003, The effect of floodwater extraction
on the morphology of mountain streams, Journal of
Hydraulic Engineering, 129(11), 885-895. Shields,
I. A., 1936, Anwendung der ahnlichkeitmechanik
und der turbulenzforschung auf die
gescheibebewegung, Mitt. Preuss Ver.-Anst., 26,
Berlin, Germany. 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.
For more information see Gary Parkers e-book 1D
Morphodynamics of Rivers and Turbidity Currents
http//cee.uiuc.edu/people/parkerg/morphodynamics
_e-book.htm