+ All Categories
Home > Documents > 1 LECTURE 13 TURBIDITY CURRENTS AND HYDRAULIC JUMPS CEE 598, GEOL 593 TURBIDITY CURRENTS:...

1 LECTURE 13 TURBIDITY CURRENTS AND HYDRAULIC JUMPS CEE 598, GEOL 593 TURBIDITY CURRENTS:...

Date post: 17-Dec-2015
Category:
Upload: kimberly-richard
View: 232 times
Download: 2 times
Share this document with a friend
37
1 LECTURE 13 TURBIDITY CURRENTS AND HYDRAULIC JUMPS CEE 598, GEOL 593 TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS Hydraulic jump of a turbidity current in the laboratory. Flow is from right to left.
Transcript

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


Recommended