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LECTURE 6HYDRAULICS AND SEDIMENT TRANSPORT:

RIVERS AND TURBIDITY CURRENTS

CEE 598, GEOL 593TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS

From PhD thesis of M. H. Garcia

Head of a turbidity current in the laboratory

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STREAMWISE VELOCITY AND CONCENTRATION PROFILES: RIVER AND TURBIDITY CURRENT

river

air

clear water

turbidity current

u

u

c

c

u = local streamwise flow velocity averaged over turbulencec = local streamwise volume suspended sediment concentration averaged

over turbulencez = upward normal direction (nearly vertical in most cases of interest)

z

z

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VELOCITY AND CONCENTRATION PROFILES BEFORE AND AFTER A HYDRAULIC JUMP

The jump is caused by a break in slope

Garcia and Parker (1989)

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x ut

A

u

The flux of any quantity is the rate at which it crosses a section per unit time per unit area.So flux = discharge/area

The fluid volume that crosses the section in time t is AutThe suspended sediment volume that crosses is cAutThe streamwise momentum that crosses is wuAut

The fluid volume flux = uThe suspended sediment volume flux = ucThe streamwise momentum flux = wu2

utA

VOLUME FLUX OF FLOWING FLUID AND SUSPENDED SEDIMENT

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LAYER-AVERAGED QUANTITIES: RIVER

In the case of a river, layer = depthH = flow depthU = layer-averaged flow velocityC = layer-averaged volume suspended sediment concentration

(based on flux) Now letqw = fluid volume discharge per unit width (normal to flow)qs = suspended sediment discharge per unit width (normal to flow)

discharge/width = integral of flux in upward normal direction

H

0s

H

0w

ucdzq

udzq

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FOR A RIVER:

H

0

H

0

ucdzUH

1C

udzH

1U

air

u

c

UC

H

0

H

0w

ss

H

0

H

0

ww

ucdzUH

1

q

qCorUCHucdzq

udzH

1

H

qUorUHudzq

Or thus

Flux-based average values U and C

z

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LAYER-AVERAGED QUANTITIES: TURBIDITY CURRENT

clear water

turbidity current

u

c

The upper interface is diffuse!

So how do we define U, C, H?

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USE THREE INTEGRALS, NOT TWO

clear water

turbidity current

u

c

Letqw = fluid volume discharge per unit widthqs = suspended sediment discharge per unit widthqm = forward momentum discharge per unit width

Integrate in z to “infinity.”

0

2wm

0s

0w

dzuq

ucdzq

udzq

z

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FOR A TURBIDITY CURRENT

HUdzuq

UCHucdzq

UHudzq

2w0

2wm

0s

0w

Three equations determine three unknowns U, C, H, which can be computed from u(z) and c(z).

clear water

u

c

U

CH

z

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BED SHEAR STRESS AND SHEAR VELOCITY

Consider a river or turbidity current channel that is wide and can be approximated as rectangular.

The bed shear stress b is the force per unit area with which the flow pulls the bed downstream (bed pulls the flow upstream) [ML-1T-2]

The bed shear stress is related to the flow velocity through a dimensionless bed resistance coefficient (bed friction coefficient) Cf, where

2w

bf U

C

The bed shear velocity u [L/T] is defined as

w

bu

Between the above two equations,

2/1fCCz

u

U

where Cz = dimensionless Chezy resistance coefficient

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SOME DIMENSIONLESS PARAMETERS

Flow Reynolds number ~ (inertial force)/viscous force): must be >~ 500 for turbulent flow

Froude number ~ (inertial force)/(gravitational force)

D = grain size [L] = kinematic viscosity of water [L2/T], ~ 1x10-6 m2/sg = gravitational acceleration [L/T2]R = submerged specific gravity of sediment [1]

)currturb(RCgH

U)river(

gH

Ud FrFr

UHRe

Particle Reynolds number ~ (dimensionless particle size)3/2

DRgDpRe

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SOME DIMENSIONLESS PARAMETERS contd.

Shields number ~ (impelling force on bed particle/ resistive force on bed particle): characterizes sediment mobility

RgD

u

RgD

UC

RgD

22fb

Now let c denote the “critical” Shields number at the threshold of motion of

a particle of size D and submerged specific gravity R. Modified Shields relation:

]1006.022.0[5.0 )7.7(6.0pc

6.0p ReRe

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SHIELDS DIAGRAM

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1 10 100 1000 10000 100000 1000000

Rep

c*

sandsilt gravel

The silt-sand and sand-gravel borders correspond to the values of Rep computed with R = 1.65, = 0.01 cm2/s and D = 0.0625 mm and 2 mm, respectively.

no motion

motion

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CRITERION FOR SIGNIFICANT SUSPENSION

wheregD

vsf

RR

But recall

DRgDpRe

Thus the condition

)( pff ReRR

defining the threshold for significant suspension.

1~v

u

s

fss v

RgD

RgD

u

v

u

Re

and

and the relation of Dietrich (1982):

specifies a unique curve

1v

u

s

)(function psus Re

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0.01

0.1

1

10

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Rep

suspension

motion

no motion

bedload transport

negligible suspension

bedload and suspended load transport

sand gravelsilt

50bf

svu

SHIELDS DIAGRAM WITH CRITERION FOR SIGNIFICANT SUSPENSION

Suspension is significant when u/vs >~ 1

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NORMAL OPEN-CHANNEL FLOW IN A WIDE CHANNEL

Normal flow is an equilibrium state defined by a perfect balance between the downstream gravitational impelling force and resistive bed force. The resulting flow is constant in time and in the downstream, or x direction.

Parameters:

x = downstream coordinate [L]H = flow depth [L]U = flow velocity [L/T]qw = water discharge per unit width [L2T-1]B = width [L]Qw = qwB = water discharge [L3/T]g = acceleration of gravity [L/T2] = bed angle [1]b = bed boundary shear stress [M/L/T2]S = tan = streamwise bed slope [1]

(cos 1; sin tan S)

w = water density [M/L3]

The bed slope angle of the great majority of alluvial rivers is sufficiently small to allow the approximations

1cos,Stansin

xB

x

gHxBS

bBx

H

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THE DEPTH-SLOPE RELATION FOR NORMAL OPEN-CHANNEL FLOW

UHBBqQUHq www

Conservation of downstream momentum:Impelling force (downstream component of weight of water) = resistive force

xBxSgHBsinxgHB bww

gHSwb

Reduce to obtain depth-slope product rule for normal flow:

Conservation of water mass (= conservation of water volume as water can be treated as incompressible):

xB

x

gHxBS

bBx

H

gHSu

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THE CONCEPT OF BANKFULL DISCHARGE IN RIVERS

QQbf

Let denote river stage (water surface elevation) [L] and Q denote volume water discharge [L3/T]. In the case of rivers with floodplains, tends to increase rapidly with increasing Q when all the flow is confined to the channel, but much less rapidly when the flow spills significantly onto the floodplain. The rollover in the curve defines bankfull discharge Qbf.

Bankfull flow ~ channel-forming flow???

Minnesota River and floodplain, USA, during the

record flood of 1965

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PARAMETERS USED TO CHARACTERIZE BANKFULL CHANNEL GEOMETRY OF RIVERS

In addition to a bankfull discharge, a reach of an alluvial river with a floodplain also has a characteristic average bankfull channel width and average bankfull channel depth. The following parameters are used to characterize this geometry.

Definitions:

Qbf = bankfull discharge [L3/T]Bbf = bankfull width [L]Hbf = bankfull depth [L]S = bed slope [1]Ds50 = median surface grain size [L]= kinematic viscosity of water [L2/T]R = (s/ – 1) = sediment submerged specific gravity (~ 1.65 for natural

sediment) [1]g = gravitational acceleration [L/T2]

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SETS OF DATA USED TO CHARACTERIZE RIVERS

Sand-bed rivers D 0.5 mmSand-bed rivers D > 0.5 mmLarge tropical sand-bed riversGravel-bed riversRivers from Japan (gravel and sand)

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SHIELDS DIAGRAM AT BANKFULL FLOW

0.001

0.01

0.1

1

10

100

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Rep

*

Sand-bed D < 0.5 mm

Sand bed D > 0.5 mm

Gravel-bed

motion threshold

suspension threshold

0.0625 mm

2 mm

16 mm

0.5 mm

Japan

Large Tropical Sand

sand-bed gravel-bed

Compared to rivers, turbidity currents have to be biased toward this region to be suspension-driven!

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FROUDE NUMBER AT BANKFULL FLOW

0.01

0.1

1

10

0.00001 0.0001 0.001 0.01 0.1

S

Fr b

f

Sand-bed D < 0.5 mmSand bed D > 0.5 mmGravel-bedLarge Tropical Sand

Turbidity currents?

)currturb(RCgH

U)river(

gH

Ud FrFr

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DIMENSIONLESS CHEZY RESISTANCE COEFFICIENT AT BANKFULL FLOW

1

10

100

0.00001 0.0001 0.001 0.01 0.1

S

Cz b

f

Sand-bed D < 0.5 mmSand bed D > 0.5 mmGravel-bedLarge Tropical Sand

Turbidity currents?

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DIMENSIONLESS WIDTH-DEPTH RATIO AT BANKFULL FLOW

1

10

100

1000

0.00001 0.0001 0.001 0.01 0.1

S

Bb

f/Hb

f Sand-bed D < 0.5 mmSand bed D > 0.5 mmGravel-bedLarge Tropical Sand

Turbidity currents?

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THE DEPTH-SLOPE RELATION FOR BED SHEAR STRESS DOES NOT NECESSARILY WORK FOR TURBIDITY

CURRENTS!

river

air

u

c

b

i

In a river, there is frictional resistance not only at the bed, but also at the water-air interface. Thus if I denotes the interfacial shear stress, the normal flow relation generalizes to:

gHSwib

But in a wide variety of cases of interest, I at an air-water interface is so small compared to b that it can be neglected.

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A TURBIDITY CURRENT CAN HAVE SIGNIFICANT FRICTION ASSOCIATED WITH ITS INTERFACE

If a turbidity current were to attain normal flow conditions,

gHSwib

whereclear water

turbidity current

u

c

b

i

2fiwi

2fwb

UC

UC

and Cf denotes a bed friction coefficient and Cfi denotes an interfacial frictional coefficient.

But turbidity currents do not easily attain normal flow conditions!

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REFERENCES

Garcia and Parker (1989)