Transcript Slide 1

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
1
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)
air
river
clear
water
u
z
c
u
z
c
turbidity
current
2
VELOCITY AND CONCENTRATION PROFILES BEFORE AND
AFTER A HYDRAULIC JUMP
The jump is caused by a break in slope
Garcia and Parker (1989)
3
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
A
u
x
ut
utA
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
4
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
H
qw   udz
0
H
qs   ucdz
0
5
FOR A RIVER:
Flux-based average values U and C
qw 1 H
qw   udz  UH or U 
  udz
0
H H0
H
qs
1 H
qs   ucdz  UCH or C 

ucdz

0
qw UH 0
H
air
U
Or thus
1 H
U   udz
H 0
1 H
C
ucdz

0
UH
C
z
u
c
6
LAYER-AVERAGED QUANTITIES: TURBIDITY CURRENT
The upper interface is diffuse!
So how do we define U, C, H?
air
clear
water
c
u
c
turbidity
current
7
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.”

qw   udz
0
clear
water

qs   ucdz
0

qm   w  u dz
z
2
0
u
c
turbidity
current
8
FOR A TURBIDITY CURRENT

qw   udz  UH
0
Three equations determine three
unknowns U, C, H, which can be
computed from u(z) and c(z).

qs   ucdz  UCH
0

qm   w  u2dz   wU2H
0
clear
water
U
C
H
u
c
z
9
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
Cf 
b
 wU2
The bed shear velocity u [L/T] is defined as
u 
b
w
Between the above two equations,
U
 Cz  C f 1/ 2
u
where Cz = dimensionless Chezy
resistance coefficient
10
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)
Fr 
U
gH
(river) Frd 
U
RCgH
( turb curr)
Flow Reynolds number ~ (inertial force)/viscous force): must be >~ 500 for
turbulent flow
Re 
UH

Particle Reynolds number ~ (dimensionless particle size)3/2
Re p 
RgD D

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SOME DIMENSIONLESS PARAMETERS contd.
Shields number ~ (impelling force on bed particle/ resistive force on bed
particle): characterizes sediment mobility
b
Cf U2
u2
 


RgD RgD RgD

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:

c
  0.5 [0.22 Re
0.6
p
 0.06  10
( 7.7 Re p0.6 )
]
12
SHIELDS DIAGRAM
0.1
0.09
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.
0.08
0.07
 c*
0.06
0.05
motion
0.04
0.03
0.02
no motion
silt
0.01
sand
gravel
0
1
10
100
1000
Rep
10000
100000
1000000
13
CRITERION FOR SIGNIFICANT SUSPENSION
u
~ 1
vs
But recall
u
u

vs
RgD
where
vs
Rf 
RgD
Thus the condition
and
Re p 
RgD D

u
1
vs
and the relation of Dietrich (1982):
specifies a unique curve
RgD


vs
Re f
R f  R f (Re p )
sus  function(Re p )
defining the threshold for significant suspension.
14
SHIELDS DIAGRAM WITH CRITERION FOR SIGNIFICANT
SUSPENSION
Suspension is significant when u/vs >~ 1
10
bedload and suspended load transport
u  v s
negligible suspension
1
suspension


 bf 50
motion
0.1
bedload transport
no motion
silt
0.01
1.E+00
sand
1.E+01
gravel
1.E+02
1.E+03
Rep
1.E+04
1.E+05
1.E+06
15
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]
x
bBx
x

H
B
gHxBS
The bed slope angle  of the great
majority of alluvial rivers is sufficiently
small to allow the approximations
sin   tan   S , cos  1
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THE DEPTH-SLOPE RELATION FOR NORMAL OPENCHANNEL FLOW
Conservation of water mass (= conservation of water volume as water
can be treated as incompressible):
qw  UH
Qw  qwB  UHB
Conservation of downstream momentum:
Impelling force (downstream component of weight of water) =
resistive force
w gHBx sin  w gHBxS  bBx
Reduce to obtain depth-slope
product rule for normal flow:
b  w gHS
u  gHS
x
bBx
x

H
B
gHxBS
17
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???

Qbf
Minnesota River and
floodplain, USA, during the
record flood of 1965
Q
18
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]
19
SETS OF DATA USED TO CHARACTERIZE RIVERS
Sand-bed rivers D  0.5 mm
Sand-bed rivers D > 0.5 mm
Large tropical sand-bed rivers
Gravel-bed rivers
Rivers from Japan (gravel and sand)
20
SHIELDS DIAGRAM AT BANKFULL FLOW
100
sand-bed
gravel-bed
10
Compared to rivers, turbidity
currents have to be biased
toward this region to be
suspension-driven!
Sand-bed D < 0.5 mm
Sand bed D > 0.5 mm
Gravel-bed
1
motion threshold
*
suspension threshold
0.0625 mm
2 mm
0.1
16 mm
0.5 mm
Japan
0.01
Large Tropical Sand
0.001
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06
Rep
21
FROUDE NUMBER AT BANKFULL FLOW
10
Turbidity currents?
1
Frbf
Sand-bed D < 0.5 mm
Sand bed D > 0.5 mm
Gravel-bed
Large Tropical Sand
0.1
Fr 
0.01
0.00001
0.0001
0.001
S
U
gH
(river) Frd 
0.01
U
RCgH
( turb curr)
0.1
22
DIMENSIONLESS CHEZY RESISTANCE COEFFICIENT AT
BANKFULL FLOW
100
Czbf
Turbidity currents?
Sand-bed D < 0.5 mm
Sand bed D > 0.5 mm
Gravel-bed
Large Tropical Sand
10
1
0.00001
0.0001
0.001
S
0.01
0.1
23
DIMENSIONLESS WIDTH-DEPTH RATIO AT BANKFULL
FLOW
1000
Turbidity currents?
100
Bbf/Hbf
Sand-bed D < 0.5 mm
Sand bed D > 0.5 mm
Gravel-bed
Large Tropical Sand
10
1
0.00001
0.0001
0.001
S
0.01
0.1
24
THE DEPTH-SLOPE RELATION FOR BED SHEAR STRESS
DOES NOT NECESSARILY WORK FOR TURBIDITY
CURRENTS!
i
river
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:
air
clear
b  i  wwater
gHS
u
i
c
b
But in a wide variety of cases of
interest, I at an air-water interface
is so small compared to b that it
u
can be neglected.
c
turbi
25
A TURBIDITY CURRENT CAN HAVE SIGNIFICANT FRICTION
ASSOCIATED WITH ITS INTERFACE
If a turbidity current were to attain
normal flow conditions,
b  i  w gHS
clear
water
where
b   w C f U2
i
i   w C fiU2
u
c
b
turbidity
current
and Cf denotes a bed friction
coefficient and Cfi denotes an
interfacial frictional coefficient.
But turbidity currents do not easily
attain normal flow conditions! 26
REFERENCES
Garcia and Parker (1989)
27