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.
1
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
2
THE CHARACTERISTICS OF HYDRAULIC JUMPS
Hydraulic jumps in open-channel flow are characterized a drop in Froude
number Fr, where
U
Fr 
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
3
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
4
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.
5
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
7
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
8
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
9
control volume
VOLUME, MASS, MOMENTUM DISCHARGE
H = depth
U = flow velocity
Channel has a unit width 1
1
U
H
x
Ut
UtH1
In time t a fluid particle flows a distance Ut
The volume that crosses normal to the section in time t = UtH1
The flow mass that crosses normal to the section in time t is density x
volume crossed = (1+RC)UtH1 UtH
The sediment mass that crosses = sCUtH 1
The momentum that cross normal to the section is mass x velocity =
(1+RC)UtH1U U2tH
10
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
Ut
UtH1
qf = UtH1/(t1)
thus
qf = UH
qmass = UtH1/ (t1)
thus
qmass = UH
qsedmass = sCUtH1/ (t1) thus
qsedmass = sCUH
qmom = UtH1U /(t1)
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
where
 qmass  cons tan t  qf
qf  UH  flow discharge / width
pref
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
control volume
Hd
p
qmomd
12
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
13
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
14
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
15
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
16
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
17
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
18
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
19
STREAMWISE MOMENTUM BALANCE ON CONTROL
VOLUME
/(momentum in control volume) = forces + net inflow rate of momentum
1
1
2
0  RCgHu  RCgHd2  Uu2Hu  Ud2Hd
2
2
pref
f pd
f pu
Ud
qmomu
Hu
Uu
z
p
control volume
Hd
p
qmomd
20
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
21
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
RCg Hu3 / 2
 1
2Fr  1    1  2  0
 
2
du
22
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
23
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.
24
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
25
ADD MATERIAL ABOUT JUMP SIGNAL!
AND CONTINUE WITH BORE!
26
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) lef t  Fmom (H) right
27
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
gH2 gH
2
Here “~” means “scales as”, not “equals”.
28
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
29
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
30
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
31
THE ANALYSIS ALSO WORKS IN THE OTHER DIRECTION
c
Uu
Ud
The case c = 0 corresponds to a hydraulic jump
32
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
33
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

34
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
35
LINEARIZED EQUATION FOR BORE SPEED (contd.)
 1 U 
c  U1 

 2 U 
2
2


1  2  1 H   1 H   
gH 1 
  1 
 
2   2 H   2 H   


  1 H  
1 
 

 1 H    2 H  
H1 
1

 2 H   1  1 H  
  2 H  

1 2  H 
gH  2

2
H 
 1 U 

c  U1 

2

 2 U 
 1 H  H
 H  
H1 
 o

 
 2 H   H
 H  
 c  U  gH
Limit of small-amplitude bore
36
SPEED OF INFINITESIMAL SHALLOW WATER WAVE
c  U  c sw
c sw  gH
Froude number = flow velocity/shallow water wave speed
U
Fr 
gH
37