HYDRAULICS AND SEDIMENT TRANSPORT:

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CEE 598, GEOL 593 TURBIDITY CURRENTS
MORPHODYNAMICS AND DEPOSITS
LECTURE 6 HYDRAULICS AND SEDIMENT
TRANSPORT RIVERS AND TURBIDITY CURRENTS
Head of a turbidity current in the laboratory
From PhD thesis of M. H. Garcia
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STREAMWISE VELOCITY AND CONCENTRATION PROFILES
RIVER AND TURBIDITY CURRENT
u local streamwise flow velocity averaged over
turbulence c local streamwise volume suspended
sediment concentration averaged over
turbulence z upward normal direction (nearly
vertical in most cases of interest)
z
z
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VELOCITY AND CONCENTRATION PROFILES BEFORE AND
AFTER A HYDRAULIC JUMP
The jump is caused by a break in slope
Garcia and Parker (1989)
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VOLUME FLUX OF FLOWING FLUID AND SUSPENDED
SEDIMENT
The flux of any quantity is the rate at which it
crosses a section per unit time per unit area. So
flux discharge/area
The fluid volume that crosses the section in time
?t is ?Au?t The suspended sediment volume that
crosses is c?Au?t The streamwise momentum that
crosses is ?wu?Au?t The fluid volume flux
u The suspended sediment volume flux uc The
streamwise momentum flux ?wu2
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LAYER-AVERAGED QUANTITIES RIVER
In the case of a river, layer depth H flow
depth U layer-averaged flow velocity C
layer-averaged volume suspended sediment
concentration (based on flux) Now let qw
fluid volume discharge per unit width (normal to
flow) qs suspended sediment discharge per unit
width (normal to flow) discharge/width
integral of flux in upward normal direction
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FOR A RIVER
Flux-based average values U and C
Or thus
z
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LAYER-AVERAGED QUANTITIES TURBIDITY CURRENT
The upper interface is diffuse! So how do we
define U, C, H?
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USE THREE INTEGRALS, NOT TWO
Let qw fluid volume discharge per unit width qs
suspended sediment discharge per unit width qm
forward momentum discharge per unit
width Integrate in z to infinity.
z
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FOR A TURBIDITY CURRENT
Three equations determine three unknowns U, C, H,
which can be computed from u(z) and c(z).
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BED SHEAR STRESS AND SHEAR VELOCITY
Consider a river or turbidity current channel
that is wide and can be approximated as
rectangular. The bed shear stress ?b is the
force per unit area with which the flow pulls the
bed downstream (bed pulls the flow upstream)
ML-1T-2 The bed shear stress is related to the
flow velocity through a dimensionless bed
resistance coefficient (bed friction coefficient)
Cf, where
The bed shear velocity u? L/T is defined as
Between the above two equations,
where Cz dimensionless Chezy resistance
coefficient
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SOME DIMENSIONLESS PARAMETERS
D grain size L ? kinematic viscosity of
water L2/T, 1x10-6 m2/s g gravitational
acceleration L/T2 R submerged specific
gravity of sediment 1
Froude number (inertial force)/(gravitational
force)
Flow Reynolds number (inertial force)/viscous
force) must be gt 500 for turbulent flow
Particle Reynolds number (dimensionless
particle size)3/2
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SOME DIMENSIONLESS PARAMETERS contd.
Shields number (impelling force on bed
particle/ resistive force on bed particle)
characterizes sediment mobility
Now let ?c? denote the critical Shields number
at the threshold of motion of a particle of size
D and submerged specific gravity R. Modified
Shields relation
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SHIELDS DIAGRAM
The silt-sand and sand-gravel borders correspond
to the values of Rep computed with R 1.65, ?
0.01 cm2/s and D 0.0625 mm and 2 mm,
respectively.
motion
no motion
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CRITERION FOR SIGNIFICANT SUSPENSION
But recall
where
and
Thus the condition
and the relation of Dietrich (1982) specifies a
unique curve
defining the threshold for significant suspension.
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SHIELDS DIAGRAM WITH CRITERION FOR SIGNIFICANT
SUSPENSION
Suspension is significant when u?/vs gt 1
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NORMAL OPEN-CHANNEL FLOW IN A WIDE CHANNEL
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)
• w water density M/L3

The bed slope angle ? of the great majority of
alluvial rivers is sufficiently small to allow
the approximations
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THE DEPTH-SLOPE RELATION FOR NORMAL OPEN-CHANNEL
FLOW
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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THE CONCEPT OF BANKFULL DISCHARGE IN RIVERS
Let ? denote river stage (water surface
elevation) L and Q denote volume water
discharge L3/T. In the case of rivers with
floodplains, ? tends to increase rapidly with
increasing Q when all the flow is confined to the
channel, but much less rapidly when the flow
spills significantly onto the floodplain. The
rollover in the curve defines bankfull discharge
Qbf. Bankfull flow channel-forming flow???
Minnesota River and floodplain, USA, during the
record flood of 1965
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PARAMETERS USED TO CHARACTERIZE BANKFULL CHANNEL
GEOMETRY OF RIVERS
In addition to a bankfull discharge, a reach of
an alluvial river with a floodplain also has a
characteristic average bankfull channel width and
average bankfull channel depth. The following
parameters are used to characterize this
geometry. Definitions Qbf bankfull discharge
L3/T Bbf bankfull width L Hbf bankfull
depth L S bed slope 1 Ds50 median surface
grain size L n kinematic viscosity of water
L2/T R (rs/r 1) sediment submerged
specific gravity ( 1.65 for natural sediment)
1 g gravitational acceleration L/T2
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SETS OF DATA USED TO CHARACTERIZE RIVERS
Sand-bed rivers D ? 0.5 mm Sand-bed rivers D gt
0.5 mm Large tropical sand-bed rivers Gravel-bed
rivers Rivers from Japan (gravel and sand)
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SHIELDS DIAGRAM AT BANKFULL FLOW
Compared to rivers, turbidity currents have to be
biased toward this region to be suspension-driven!
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FROUDE NUMBER AT BANKFULL FLOW
Turbidity currents?
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DIMENSIONLESS CHEZY RESISTANCE COEFFICIENT AT
BANKFULL FLOW
Turbidity currents?
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DIMENSIONLESS WIDTH-DEPTH RATIO AT BANKFULL FLOW
Turbidity currents?
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THE DEPTH-SLOPE RELATION FOR BED SHEAR STRESS
DOES NOT NECESSARILY WORK FOR TURBIDITY CURRENTS!
In a river, there is frictional resistance not
only at the bed, but also at the water-air
interface. Thus if ?I denotes the interfacial
shear stress, the normal flow relation
generalizes to
But in a wide variety of cases of interest, ?I at
an air-water interface is so small compared to ?b
that it can be neglected.
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A TURBIDITY CURRENT CAN HAVE SIGNIFICANT FRICTION
ASSOCIATED WITH ITS INTERFACE
If a turbidity current were to attain normal flow
conditions,
where
and Cf denotes a bed friction coefficient and Cfi
denotes an interfacial frictional
coefficient. But turbidity currents do not
easily attain normal flow conditions!
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REFERENCES
Garcia and Parker (1989)