Transcript Slide 1
CEE 598, GEOL 593
TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS
LECTURE 13
TURBIDITY CURRENTS AND HYDRAULIC JUMPS
Hydraulic jump of a turbidity current in the laboratory. Flow is
from right to left.
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WHAT IS A HYDRAULIC JUMP?
A hydraulic jump is a type of shock, where the flow undergoes a sudden
transition from swift, thin (shallow) flow to tranquil, thick (deep) flow.
Hydraulic jumps are most familiar in the context of open-channel flows.
The image shows a hydraulic jump in a laboratory flume.
flow
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THE CHARACTERISTICS OF HYDRAULIC JUMPS
Hydraulic jumps in open-channel flow are characterized a drop in Froude
number Fr, where
Fr
U
gH
from supercritical (Fr > 1) to subcritical (Fr < 1) conditions. The result is a
step increase in depth H and a step decrease in flow velocity U passing
through the jump.
flow
supercritical
subcritical
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WHAT CAUSES HYDRAULIC JUMPS?
The conditions for a hydraulic jump can be met where
a) the upstream flow is supercritical, and
b) slope suddenly or gradually decreases downstream, or
c) the supercritical flow enters a confined basin.
Fr < 1
Fr > 1
Fr < 1
Fr > 1
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INTERNAL HYDRAULIC JUMPS
Hydraulic jumps in rivers are associated with an extreme example of flow
stratification: flowing water under ambient air.
Internal hydraulic jumps form when a denser, fluid flows under a lighter
ambient fluid. The photo shows a hydraulic jump as relatively dense air
flows east across the Sierra Nevada Mountains, California.
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Photo by Robert Symons, USAF, from the Sierra Wave Project in the 1950s.
DENSIMETRIC FROUDE NUMBER
Internal hydraulic jumps are mediated by the densimetric Froude number
Frd, which is defined as follows for a turbidity current.
Frd
U
RCgH
U = flow velocity
g = gravitational acceleration
H = flow thickness
C = volume suspended sediment concentration
R = s/ - 1 1.65
Subcritical: Frd < 1
Supercritical: Frd > 1
Water surface
internal hydraulic jump
entrainment of ambient fluid6
INTERNAL HYDRAULIC JUMPS AND TURBIDITY CURRENTS
Slope break: good place
for a hydraulic jump
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Stepped profile, Niger Margin
From Prather et al. (2003)
INTERNAL HYDRAULIC JUMPS AND TURBIDITY CURRENTS
Frd > 1
jump
Frd < 1
Frd << 1
coarser top/foreset
ponded flow
Flow into a confined basin:
good place for a hydraulic
jump
finer bottomset
"ultimate" profile
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ANALYSIS OF THE INTERNAL HYDRAULIC JUMP
Definitions: “u” upstream and “d” downstream
U = flow velocity
C = volume suspended sediment concentration
z = upward vertical coordinate
p = pressure
pref = pressure force at z = Hd (just above turbidity current
fp = pressure force per unit width
qmom = momentum discharge per unit width
Flow in the control volume is steady.
USE TOPHAT ASSUMPTIONS FOR U AND C.
pref
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
Hd
p
qmomd
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control volume
VOLUME, MASS, MOMENTUM DISCHARGE
H = depth
U = flow velocity
Channel has a unit width 1
1
U
H
x
Ut
UtH1
In time t a fluid particle flows a distance Ut
The volume that crosses normal to the section in time t = UtH1
The flow mass that crosses normal to the section in time t is density x
volume crossed = (1+RC)UtH1 UtH
The sediment mass that crosses = sCUtH 1
The momentum that cross normal to the section is mass x velocity =
(1+RC)UtH1U U2tH
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VOLUME, MASS, MOMENTUM DISCHARGE (contd.)
qf = volume discharge per unit width = volume crossed/width/time
qmass = flow mass discharge per unit width = mass crossed/width/time
qsedmass = sediment mass discharge per unit with = mass crossed/width/time
qmom = momentum discharge/width = momentum crossed/width/time
1
U
H
x
Ut
UtH1
qf = UtH1/(t1)
thus
qf = UH
qmass = UtH1/ (t1)
thus
qmass = UH
qsedmass = sCUtH1/ (t1) thus
qsedmass = sCUH
qmom = UtH1U /(t1)
qmom = U2H 11
thus
FLOW MASS BALANCE ON THE CONTROL VOLUME
/t(fluid mass in control volume) = net mass inflow rate
0 qmassu qmassd qmass const
or
0 UuHu UdHd qmass cons tan t qf
where
qf UH flow discharge / width
pref
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
control volume
Hd
p
qmomd
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FLOW MASS BALANCE ON THE CONTROL VOLUME contd/
Thus flow discharge
qf UH
is constant across the hydraulic jump
pref
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
control volume
Hd
p
qmomd
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BALANCE OF SUSPENDED SEDIMENT MASS ON THE
CONTROL VOLUME
/t(sediment mass in control volume) = net sediment mass
inflow rate
0 qsedmassu qsedmassd qsedmass const
or
0 sCuUuHu sCdUdHd qsedmass cons tan t
pref
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
Hd
p
qmomd
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control volume
BALANCE OF SUSPENDED SEDIMENT MASS ON THE
CONTROL VOLUME contd
Thus if the volume sediment discharge/width is defined as
qsedvol CUH
then qsedvol = qsedmass/s is constant across the jump.
But if
qf UH const , qsedvol CUH const
then C is constant across the jump!
pref
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
Hd
p
qmomd
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control volume
PRESSURE FORCE/WIDTH ON DOWNSTREAM SIDE OF
CONTROL VOLUME
dp
g(1 RC) , p z H pref
d
dz
p pref g(1 RC)(Hd z)
fpd
Hd
0
pref
1
pdz 1 g(1 RC)Hd2
2
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
control volume
Hd
p
qmomd
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PRESSURE FORCE/WIDTH ON UPSTREAM SIDE OF
CONTROL VOLUME
dp g , Hu z Hd
dz g(1 RC) , 0 z Hu
, p z H pref
d
pref g(Hd z) , Hu z Hd
p
pref g(Hd Hu ) g(1 RC)(Hu z) ,0 z Hu
pref
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
control volume
Hd
p
qmomd
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PRESSURE FORCE/WIDTH ON UPSTREAM SIDE OF
CONTROL VOLUME contd.
pref g(Hd z) , Hu z Hd
p
pref g(Hd Hu ) g(1 RC)(Hu z) ,0 z Hu
Hd
Hu
Hd
0
0
Hu
fpu pdz 1 pdz 1 pdz 1
1
1
pref Hu g(Hd Hu )Hu (1 RC)Hu2 pref (Hd Hu ) g(Hd Hu )2
2
2
1
1
pref Hd (1 RC)Hu2 g(Hd Hu )Hu g(Hd Hu )2
pref
2
2
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
Hd
p
qmomd
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control volume
NET PRESSURE FORCE
1
1
2
fpnet fpu fpd pref Hd (1 RC)Hu g(Hd Hu )Hu g(Hd Hu )2
2
2
1
pref Hd (1 RC)gHd2
2
1
RCg Hu2 H2d
2
pref
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
control volume
Hd
p
qmomd
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STREAMWISE MOMENTUM BALANCE ON CONTROL
VOLUME
/(momentum in control volume) = forces + net inflow rate of momentum
1
1
2
0 RCgHu RCgH2d Uu2Hu U2dHd
2
2
pref
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
control volume
Hd
p
qmomd
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REDUCTION
f pd
f pu
Ud
qmomu
Hu
Uu
Hd
p
qmomd
p
control volume
UH qf
2
q
U2H Uqf f
H
thus
1
1
2
0 RCgHu RCgHd2 Uu2Hu Ud2Hd
2
2
2
2
q
q
1
1
0 RCgHu2 RCgH2d f f
2
2
Hu Hd
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REDUCTION (contd.)
f pd
f pu
Ud
qmomu
Hu
Hd
Uu
p
qmomd
p
control volume
2
2
q
q
1
1
0 RCgHu2 RCgH2d w w
2
2
Hu Hd
Now define = Hd/Hu (we expect that 1). Also
Frdu
Thus
Uu
RCgHu
qf
RCgHu3 / 2
1
2Fr 1 1 2 0
2
du
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REDUCTION (contd.)
f pd
f pu
Ud
qmomu
Hu
Hd
Uu
p
qmomd
p
control volume
But
1 1
1
1 2 ( 1)( 1)
( 1)
2Fr
( 1)( 1) 0
2
du
2
2 - 2Frdu
0
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RESULT
f pd
f pu
Ud
qmomu
Hu
Uu
Hd
p
qmomd
p
control volume
Hd 1
2
1 8Frdu 1
Hu 2
This is known as the conjugate depth relation.
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pref
RESULT
f pd
f pu
Ud
qmomu
Hu
Hd
Uu
z
p
p
qmomd
control volume
Conjugate Depth Relation
4
3.5
3
Hd/Hu
2.5
2
1.5
Hd 1
2
1 8Frdu
1
Hu 2
1
0.5
0
0
0.5
1
1.5
2
Fr
Frdu
u
2.5
3
3.5
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ADD MATERIAL ABOUT JUMP SIGNAL!
AND CONTINUE WITH BORE!
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SLUICE GATE TO FREE OVERFALL
sluice gate, Fr > 1
Fr < 1
free overfall, Fr = 1
Fr > 1
Define the momentum function Fmom such that
1 2 q2
Fmom (H) gH
2
H
Then the jump occurs where
Fmom (H) left Fmom (H) right
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The fact that Hleft = Hu Hd = Hright at the jump defines a shock
SQUARE OF FROUDE NUMBER AS A RATIO OF FORCES
Fr2 ~ (inertial force)/(gravitational force)
inertial force/width ~ momentum discharge/width ~ U2H
gravitational force/width ~ (1/2)gH2
2
2
U
H
U
Fr 2 ~
~
1
gH
2
gH
2
Here “~” means “scales as”, not “equals”.
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MIGRATING BORES AND THE SHALLOW WATER WAVE
SPEED
A hydraulic jump is a bore that has stabilized and no longer
migrates.
Tidal bore, Bay of Fundy,
Moncton, Canada
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MIGRATING BORES AND THE SHALLOW WATER WAVE
SPEED
Bore of the Qiantang River,
China
Pororoca Bore, Amazon River
http://www.youtube.com/watch?v=
2VMI8EVdQBo
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ANALYSIS FOR A BORE
The bore migrates with speed c
c
Uu
Ud
The flow becomes steady relative to a coordinate system moving with speed
c.
Uu - c
Ud - c
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THE ANALYSIS ALSO WORKS IN THE OTHER DIRECTION
c
Uu
Ud
The case c = 0 corresponds to a hydraulic jump
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CONTROL VOLUME
q = (U-c)H
qmass = (U-c)H
qmom = (U-c)2H
f pu
Ud - c
qmomu
Hu Uu - c
Hd
p
f pd
qmomd
p
control volume
Mass balance
Momentum
balance
0 (Uu c )Hu (Ud c )Hd
1
1
2
0 gHu gH2d (Uu c )2 Hu (Ud c )2 Hd
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2
2
EQUATION FOR BORE SPEED
Hu
(Ud c ) (Uu c )
Hd
2
H
1
1
0 gHu2 gH2d (Uu c )2 Hu (Uu c )2 u
2
2
Hd
Hu
1
(Uu c ) Hu
1 g(Hu2 H2d )
Hd
2
2
c Uu
1
g(Hu2 H2d )
2
Hu
Hu
1
Hd
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LINEARIZED EQUATION FOR BORE SPEED
Let
1
U (Uu Ud )
2
1
Uu U U
2
1
Hu H H
2
1
H (Hu Hd )
2
1
Ud U U
2
1
Hd H H
2
Limit of small-amplitude bore:
H
1
H
U
1
U
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LINEARIZED EQUATION FOR BORE SPEED (contd.)
1 U
c U1
2 U
2
2
1 2 1 H 1 H
gH 1
1
2 2 H 2 H
1 H
1
1 H 2 H
H1
1
2 H 1 1 H
2 H
1 2 H
gH 2
2
H
1 U
c U1
2
2 U
1 H H
H
H1
o
2 H H
H
c U gH
Limit of small-amplitude bore
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SPEED OF INFINITESIMAL SHALLOW WATER WAVE
c U c sw
c sw gH
Froude number = flow velocity/shallow water wave speed
U
Fr
gH
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