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1
LECTURE 13 TURBIDITY CURRENTS AND HYDRAULIC JUMPS
CEE 598, GEOL 593TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS
Hydraulic jump of a turbidity current in the laboratory. Flow is from right to left.
2
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
3
THE CHARACTERISTICS OF HYDRAULIC JUMPS
subcritical
flow
supercritical
Hydraulic jumps in open-channel flow are characterized a drop in Froude number Fr, where
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.
gH
UFr
4
WHAT CAUSES HYDRAULIC JUMPS?
The conditions for a hydraulic jump can be met wherea) the upstream flow is supercritical, andb) slope suddenly or gradually decreases downstream, orc) the supercritical flow enters a confined basin.
Fr > 1
Fr > 1
Fr < 1
Fr < 1
5
INTERNAL HYDRAULIC JUMPS
Photo by Robert Symons, USAF, from the Sierra Wave Project in the 1950s.
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.
6
DENSIMETRIC FROUDE NUMBER
d
U
RCgHFr
U = flow velocityg = gravitational accelerationH = flow thicknessC = volume suspended sediment concentrationR = s/ - 1 1.65
Subcritical: Frd < 1 Supercritical: Frd > 1
Water surface
internal hydraulic jump
entrainment of ambient fluid
Internal hydraulic jumps are mediated by the densimetric Froude number Frd, which is defined as follows for a turbidity current.
7
INTERNAL HYDRAULIC JUMPS AND TURBIDITY CURRENTS
Stepped profile, Niger MarginFrom Prather et al. (2003)
Slope break: good place for a hydraulic jump
8
INTERNAL HYDRAULIC JUMPS AND TURBIDITY CURRENTSFrd > 1
Frd < 1Frd << 1
jump
coarser top/foreset
finer bottomset
ponded flow
Flow into a confined basin: good place for a hydraulic
jump
9
ANALYSIS OF THE INTERNAL HYDRAULIC JUMP
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
Definitions: “u” upstream and “d” downstreamU = flow velocityC = volume suspended sediment concentrationz = upward vertical coordinatep = pressurepref = pressure force at z = Hd (just above turbidity currentfp = pressure force per unit widthqmom = momentum discharge per unit widthFlow in the control volume is steady.USE TOPHAT ASSUMPTIONS FOR U AND C.
10
x
HU
1
In time t a fluid particle flows a distance Ut
The volume that crosses normal to the section in time t = UtH1The flow mass that crosses normal to the section in time t is density x volume crossed = (1+RC)UtH1 UtHThe sediment mass that crosses = sCUtH 1The momentum that cross normal to the section is mass x velocity =(1+RC)UtH1U U2tH
H = depthU = flow velocityChannel has a unit width 1
VOLUME, MASS, MOMENTUM DISCHARGE
UtH1Ut
11
x
HU
1
qf = UtH1/(t1) thus qf = UH
qmass = UtH1/ (t1) thus qmass = UH
qsedmass = sCUtH1/ (t1) thus qsedmass = sCUH
qmom = UtH1U /(t1) thus qmom = U2H
qf = volume discharge per unit width = volume crossed/width/timeqmass = flow mass discharge per unit width = mass crossed/width/timeqsedmass = sediment mass discharge per unit with = mass crossed/width/timeqmom = momentum discharge/width = momentum crossed/width/time
VOLUME, MASS, MOMENTUM DISCHARGE (contd.)
UtH1Ut
12
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
FLOW MASS BALANCE ON THE CONTROL VOLUME
/t(fluid mass in control volume) = net mass inflow rate
massu massd mass
u u d d mass f
f
0 q q q const
or
0 U H U H q cons tant q
where q UH flow discharge / width
13
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
FLOW MASS BALANCE ON THE CONTROL VOLUME contd/
Thus flow discharge
is constant across the hydraulic jump
fq UH
14
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
BALANCE OF SUSPENDED SEDIMENT MASS ON THE CONTROL VOLUME
/t(sediment mass in control volume) = net sediment mass inflow rate
sedmassu sedmassd sedmass
s u u u s d d d sedmass
0 q q q const
or
0 C U H C U H q cons tant
15
Thus if the volume sediment discharge/width is defined as
then qsedvol = qsedmass/s is constant across the jump.
But if
then C is constant across the jump!
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
BALANCE OF SUSPENDED SEDIMENT MASS ON THE CONTROL VOLUME contd
sedvolq CUH
f sedvolq UH const , q CUH const
16
PRESSURE FORCE/WIDTH ON DOWNSTREAM SIDE OF CONTROL VOLUME
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
drefz H
ref d
dpg(1 RC) , p p
dz
p p g(1 RC)(H z)
dH 2pd d0
1f pdz 1 g(1 RC)H
2
17
PRESSURE FORCE/WIDTH ON UPSTREAM SIDE OF CONTROL VOLUME
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
d
u drefz H
u
ref d u d
ref d u u u
g , H z Hdp, p p
g(1 RC) , 0 z Hdz
p g(H z) , H z Hp
p g(H H ) g(1 RC)(H z) ,0 z H
18
PRESSURE FORCE/WIDTH ON UPSTREAM SIDE OF CONTROL VOLUME contd.
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
ref d u d
ref d u u u
p g(H z) , H z Hp
p g(H H ) g(1 RC)(H z) ,0 z H
d u d
u
H H H
pu 0 0 H
2 2ref u d u u u ref d u d u
2 2ref d u d u u d u
f pdz 1 pdz 1 pdz 1
1 1p H g(H H )H (1 RC)H p (H H ) g(H H )
2 21 1
p H (1 RC)H g(H H )H g(H H )2 2
19
NET PRESSURE FORCE
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
2 2pnet pu pd ref d u d u u d u
2ref d d
2 2u d
1 1f f f p H (1 RC)H g(H H )H g(H H )
2 21
p H (1 RC)gH2
1RCg H H
2
20
STREAMWISE MOMENTUM BALANCE ON CONTROL VOLUME
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
/(momentum in control volume) = forces + net inflow rate of momentum
2 2 2 2u d u u d d
1 10 RCgH RCgH U H U H
2 2
21
REDUCTION
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
22 f
f f
2 2 2 2u d u u d d
2 22 2 f fu d
u d
qUH q U H Uq
Hthus
1 10 RCgH RCgH U H U H
2 2
q q1 10 RCgH RCgH
2 2 H H
22
REDUCTION (contd.)
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
2 22 2 w wu d
u d
q q1 10 RCgH RCgH
2 2 H H
Now define = Hd/Hu (we expect that 1). Also
u fdu 3 / 2
u u
U q
RCgH RCgH Fr
Thus 2 2du
12 1 1 0
Fr
23
REDUCTION (contd.)
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
But )1)(1(111
1 2
2du
( 1)2 ( 1)( 1) 0
Fr
2 2du- 2 0 Fr
24
RESULT
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
2ddu
u
H 11 8 1
H 2 Fr
This is known as the conjugate depth relation.
25
Conjugate Depth Relation
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3 3.5
Fru
Hd/H
uRESULT
2ddu
u
H 11 8 1
H 2 Fr
pp
qmomu qmomd
control volume
fpu
fpd
Hu
Ud
Hd
Uu
pref
z
Frdu
26
ADD MATERIAL ABOUT JUMP SIGNAL!AND CONTINUE WITH BORE!
27
SLUICE GATE TO FREE OVERFALL
Define the momentum function Fmom such that
H
qgH
2
1)H(F
22
mom
Then the jump occurs where
rightmomleftmom )H(F)H(F
sluice gate, Fr > 1
free overfall, Fr = 1
Fr > 1
Fr < 1
The fact that Hleft = Hu Hd = Hright at the jump defines a shock
28
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
gH
U~
gH21
HU~
2
2
22
Fr
Here “~” means “scales as”, not “equals”.
29
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
30
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
31
ANALYSIS FOR A BORE
Ud Uu
c
The bore migrates with speed c
The flow becomes steady relative to a coordinate system moving with speed c.
Ud - cUu - c
32
THE ANALYSIS ALSO WORKS IN THE OTHER DIRECTION
Ud Uu
c
The case c = 0 corresponds to a hydraulic jump
33
CONTROL VOLUME
q = (U-c)H qmass = (U-c)H qmom = (U-c)2H
pp
qmomu qmomd
control volume
fpufpd
HuHd
Ud - cUu - c
Mass balance
Momentum balance
dduu H)cU(H)cU(0
d2
du2
u2d
2u H)cU(H)cU(gH
2
1gH
2
10
34
EQUATION FOR BORE SPEED
d
uud H
H)cU()cU(
d
2u2
uu2
u2d
2u H
H)cU(H)cU(gH
2
1gH
2
10
)HH(g2
11
H
HH)cU( 2
d2u
d
uu
2u
1HH
H
)HH(g21
Uc
d
uu
2d
2u
u
35
LINEARIZED EQUATION FOR BORE SPEED
Let
H2
1HHH
2
1HH
U2
1UUU
2
1UU
)HH(2
1H)UU(
2
1U
du
du
dudu
Limit of small-amplitude bore:
1U
U1
H
H
36
LINEARIZED EQUATION FOR BORE SPEED (contd.)
Limit of small-amplitude boregHUc
HH
oHH
HH
21
1H
HH
2gH21
U
U
2
11Uc
1
HH
21
1
HH
21
1
HH
21
1H
HH
21
1HH
21
1Hg21
U
U
2
11Uc
2
2
222
37
SPEED OF INFINITESIMAL SHALLOW WATER WAVE
Froude number = flow velocity/shallow water wave speed
gHc
cUc
sw
sw
gH
UFr
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