HIGHLIGHTS OF OPEN CHANNEL HYDRAULICS AND SEDIMENT TRANSPORT

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HIGHLIGHTS OF OPEN CHANNEL HYDRAULICS AND
SEDIMENT TRANSPORT
Dam at Hiram Falls on the Saco River near Hiram,
Maine, USA
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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.
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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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HYDRAULIC JUMP
Supercritical (Fr gt1) to subcritical (Fr lt 1)
flow.
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ILLUSTRATION OF BEDLOAD TRANSPORT
Double-click on the image to see a video clip of
bedload transport of 7 mm gravel in a flume
(model river) at St. Anthony Falls Laboratory,
University of Minnesota. (Wait a bit for the
channel to fill with water.) Video clip from the
experiments of Miguel Wong.
rte-bookbedload.mpg to run without relinking,
download to same folder as PowerPoint
presentations.
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ILLUSTRATION OF MIXED TRANSPORT OF SUSPENDED LOAD
AND BEDLOAD
Double-click on the image to see the transport of
sand and pea gravel by a turbidity current
(sediment underflow driven by suspended sediment)
in a tank at St. Anthony Falls Laboratory.
Suspended load is dominant, but bedload transport
can also be seen. Video clip from experiments of
Alessandro Cantelli and Bin Yu.
rte-bookturbcurr.mpg to run without relinking,
download to same folder as PowerPoint
presentations.
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PARAMETERS CHARACTERIZING SEDIMENT TRANSPORT
qb Volume bedload transport rate per unit width
L2/T qs Volume suspended load transport rate
per unit width L2/T qt qb qs volume total
bed material transport rate per unit width
L2/T qw Volume wash load transport rate
per unit width L2/T ? water density
M/L3 ?s sediment density M/L3 R (?s/?)
1 sediment submerged specific gravity
1 D characteristic sediment size (e.g. Ds50)
L ? dimensionless Shields number,
(HS)/(RD) for normal flow 1 Dimensionless
Einstein number for bedload transport Dimension
less Einstein number for total bed material
transport
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SOME GENERIC RELATIONS FOR SEDIMENT TRANSPORT
BEDLOAD TRANSPORT RELATIONS (e.g. gravel-bed
stream) Wongs modified version of the relation
of Meyer-Peter and Müller (1948) Parkers
(1979) approximation of the Einstein (1950)
relation TOTAL BED MATERIAL LOAD TRANSPORT
RELATION (e.g. sand-bed stream) Engelund-Hansen
relation (1967)
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REFERENCES
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.