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LECTURE 6HYDRAULICS AND SEDIMENT TRANSPORT:
RIVERS AND TURBIDITY CURRENTS
CEE 598, GEOL 593TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS
From PhD thesis of M. H. Garcia
Head of a turbidity current in the laboratory
2
STREAMWISE VELOCITY AND CONCENTRATION PROFILES: RIVER AND TURBIDITY CURRENT
river
air
clear water
turbidity current
u
u
c
c
u = local streamwise flow velocity averaged over turbulencec = local streamwise volume suspended sediment concentration averaged
over turbulencez = upward normal direction (nearly vertical in most cases of interest)
z
z
3
VELOCITY AND CONCENTRATION PROFILES BEFORE AND AFTER A HYDRAULIC JUMP
The jump is caused by a break in slope
Garcia and Parker (1989)
4
x ut
A
u
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 AutThe suspended sediment volume that crosses is cAutThe streamwise momentum that crosses is wuAut
The fluid volume flux = uThe suspended sediment volume flux = ucThe streamwise momentum flux = wu2
utA
VOLUME FLUX OF FLOWING FLUID AND SUSPENDED SEDIMENT
5
LAYER-AVERAGED QUANTITIES: RIVER
In the case of a river, layer = depthH = flow depthU = layer-averaged flow velocityC = layer-averaged volume suspended sediment concentration
(based on flux) Now letqw = 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
H
0s
H
0w
ucdzq
udzq
6
FOR A RIVER:
H
0
H
0
ucdzUH
1C
udzH
1U
air
u
c
UC
H
0
H
0w
ss
H
0
H
0
ww
ucdzUH
1
q
qCorUCHucdzq
udzH
1
H
qUorUHudzq
Or thus
Flux-based average values U and C
z
7
LAYER-AVERAGED QUANTITIES: TURBIDITY CURRENT
clear water
turbidity current
u
c
The upper interface is diffuse!
So how do we define U, C, H?
8
USE THREE INTEGRALS, NOT TWO
clear water
turbidity current
u
c
Letqw = fluid volume discharge per unit widthqs = suspended sediment discharge per unit widthqm = forward momentum discharge per unit width
Integrate in z to “infinity.”
0
2wm
0s
0w
dzuq
ucdzq
udzq
z
9
FOR A TURBIDITY CURRENT
HUdzuq
UCHucdzq
UHudzq
2w0
2wm
0s
0w
Three equations determine three unknowns U, C, H, which can be computed from u(z) and c(z).
clear water
u
c
U
CH
z
10
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
2w
bf U
C
The bed shear velocity u [L/T] is defined as
w
bu
Between the above two equations,
2/1fCCz
u
U
where Cz = dimensionless Chezy resistance coefficient
11
SOME DIMENSIONLESS PARAMETERS
Flow Reynolds number ~ (inertial force)/viscous force): must be >~ 500 for turbulent flow
Froude number ~ (inertial force)/(gravitational force)
D = grain size [L] = kinematic viscosity of water [L2/T], ~ 1x10-6 m2/sg = gravitational acceleration [L/T2]R = submerged specific gravity of sediment [1]
)currturb(RCgH
U)river(
gH
Ud FrFr
UHRe
Particle Reynolds number ~ (dimensionless particle size)3/2
DRgDpRe
12
SOME DIMENSIONLESS PARAMETERS contd.
Shields number ~ (impelling force on bed particle/ resistive force on bed particle): characterizes sediment mobility
RgD
u
RgD
UC
RgD
22fb
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:
]1006.022.0[5.0 )7.7(6.0pc
6.0p ReRe
13
SHIELDS DIAGRAM
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
1 10 100 1000 10000 100000 1000000
Rep
c*
sandsilt gravel
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.
no motion
motion
14
CRITERION FOR SIGNIFICANT SUSPENSION
wheregD
vsf
RR
But recall
DRgDpRe
Thus the condition
)( pff ReRR
defining the threshold for significant suspension.
1~v
u
s
fss v
RgD
RgD
u
v
u
Re
and
and the relation of Dietrich (1982):
specifies a unique curve
1v
u
s
)(function psus Re
15
0.01
0.1
1
10
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06
Rep
suspension
motion
no motion
bedload transport
negligible suspension
bedload and suspended load transport
sand gravelsilt
50bf
svu
SHIELDS DIAGRAM WITH CRITERION FOR SIGNIFICANT SUSPENSION
Suspension is significant when u/vs >~ 1
16
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]b = 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
1cos,Stansin
xB
x
gHxBS
bBx
H
17
THE DEPTH-SLOPE RELATION FOR NORMAL OPEN-CHANNEL FLOW
UHBBqQUHq www
Conservation of downstream momentum:Impelling force (downstream component of weight of water) = resistive force
xBxSgHBsinxgHB bww
gHSwb
Reduce to obtain depth-slope product rule for normal flow:
Conservation of water mass (= conservation of water volume as water can be treated as incompressible):
xB
x
gHxBS
bBx
H
gHSu
18
THE CONCEPT OF BANKFULL DISCHARGE IN RIVERS
QQbf
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
19
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]= kinematic viscosity of water [L2/T]R = (s/ – 1) = sediment submerged specific gravity (~ 1.65 for natural
sediment) [1]g = gravitational acceleration [L/T2]
20
SETS OF DATA USED TO CHARACTERIZE RIVERS
Sand-bed rivers D 0.5 mmSand-bed rivers D > 0.5 mmLarge tropical sand-bed riversGravel-bed riversRivers from Japan (gravel and sand)
21
SHIELDS DIAGRAM AT BANKFULL FLOW
0.001
0.01
0.1
1
10
100
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06
Rep
*
Sand-bed D < 0.5 mm
Sand bed D > 0.5 mm
Gravel-bed
motion threshold
suspension threshold
0.0625 mm
2 mm
16 mm
0.5 mm
Japan
Large Tropical Sand
sand-bed gravel-bed
Compared to rivers, turbidity currents have to be biased toward this region to be suspension-driven!
22
FROUDE NUMBER AT BANKFULL FLOW
0.01
0.1
1
10
0.00001 0.0001 0.001 0.01 0.1
S
Fr b
f
Sand-bed D < 0.5 mmSand bed D > 0.5 mmGravel-bedLarge Tropical Sand
Turbidity currents?
)currturb(RCgH
U)river(
gH
Ud FrFr
23
DIMENSIONLESS CHEZY RESISTANCE COEFFICIENT AT BANKFULL FLOW
1
10
100
0.00001 0.0001 0.001 0.01 0.1
S
Cz b
f
Sand-bed D < 0.5 mmSand bed D > 0.5 mmGravel-bedLarge Tropical Sand
Turbidity currents?
24
DIMENSIONLESS WIDTH-DEPTH RATIO AT BANKFULL FLOW
1
10
100
1000
0.00001 0.0001 0.001 0.01 0.1
S
Bb
f/Hb
f Sand-bed D < 0.5 mmSand bed D > 0.5 mmGravel-bedLarge Tropical Sand
Turbidity currents?
25
THE DEPTH-SLOPE RELATION FOR BED SHEAR STRESS DOES NOT NECESSARILY WORK FOR TURBIDITY
CURRENTS!
river
air
u
c
b
i
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:
gHSwib
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.
26
A TURBIDITY CURRENT CAN HAVE SIGNIFICANT FRICTION ASSOCIATED WITH ITS INTERFACE
If a turbidity current were to attain normal flow conditions,
gHSwib
whereclear water
turbidity current
u
c
b
i
2fiwi
2fwb
UC
UC
and Cf denotes a bed friction coefficient and Cfi denotes an interfacial frictional coefficient.
But turbidity currents do not easily attain normal flow conditions!
27
REFERENCES
Garcia and Parker (1989)