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.

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INTERNAL HYDRAULIC JUMPS AND TURBIDITY CURRENTS

Stepped profile, Niger MarginFrom Prather et al. (2003)

Slope break: good place for a hydraulic jump

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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

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ADD MATERIAL ABOUT JUMP SIGNAL!AND CONTINUE WITH BORE!

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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

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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”.

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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

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

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THE ANALYSIS ALSO WORKS IN THE OTHER DIRECTION

Ud Uu

c

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

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

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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

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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

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SPEED OF INFINITESIMAL SHALLOW WATER WAVE

Froude number = flow velocity/shallow water wave speed

gHc

cUc

sw

sw

gH

UFr