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CEE 598, GEOL 593TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS

LECTURE 8LAYER-AVERAGED GOVERNING EQUATIONS FOR

TURBIDITY CURRENTS

x

zu

cq

turbid water

ambient water

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SOME DEFINITIONS

x = boundary-attached (seafloor-attached) streamwise coordinatez = boundary-perpendicular upward normal coordinateu = streamwise flow velocity (averaged over turbulence)c = volume suspended sediment concentration (averaged over turbulence)

Now we assume that q tan(q) = S << 1.Thus x is “almost” horizontal and z is “almost” vertical.

x

zu

cq

turbid water

ambient water

x y zg (g , g g ) (gS, 0 g)

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SOME MORE DEFINITIONS

a = density of ambient waterf = density of flowing water-sediment mixture in turbidity currentR = submerged specific gravity of sediment in suspension

x

zu

cq

turbid water

ambient water

f af a a

f

Rc(1 Rc) but Rc 1

1 Rc

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LETS’ JUMP TO THE 1D APPROXIMATE LAYER-INTEGRATED EQUATIONS GOVERNING A TURBIDITY

CURRENT

f 0

s 0

2 2m a a0

q udz UH

q ucdz UCH

q u dz U H

clear water

u

c

U

CH

z

Flow discharge per unit width = qf, volume suspended sediment discharge per unit width = qs, streamwise momentum discharge per unit width = qm:

Thus layer-averaged quantities can be defined as

0

0

2 2

0

UH udz

UCH ucdz

U H u dz

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THE EQUATIONS:SUSPENDED SEDIMENT OF UNIFORM SIZE

2 22

f

w

s s o

UH U H 1 CHRg RgCHS C U

t x 2 xH UH

e Ut xCH UCH

v (E r C)t x

In the above equations, the bed shear stress b is given by the relation

andvs = sediment fall velocity [L/T],ew = dimensionless rate of entrainment of ambient water into the turbidity

current,Es = dimensionless rate of entrainment of bed sediment into the turbidity current [1]ro = cb/C > 1, where cb is a near-bed suspended sediment concentration

2b a fC U

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MOMENTUM BALANCE

2 22

f

UH U H 1 CHRg RgCHS C U

t x 2 x

What the terms mean:

a) time rate of change of depth-integrated momentumb) inertial force termc) pressure force termd) downstream gravitational forcee) bed resistive force

a) b) c) d) e)

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FLUID MASS BALANCE

What the terms mean:

a) time rate of change of depth-integrated mass in flow (multiply by a)b) streamwise change in flow discharge per unit widthc) rate at which flow incorporates ambient fluid from above by mixing

across interface

a) b) c)

w

H UHe U

t x

x

zu

cq

turbid water

ambient water

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SEDIMENT MASS BALANCE

What the terms mean:a) time rate of change of depth-integrated sediment mass in flow (multiply

by s

b) streamwise change in sediment mass flow per unit widthc) rate at which sediment is eroded from the bed into suspensiond) rate at which sediment settled out from the flow onto the bed

a) b) c) d)

x

zu

cq

turbid water

ambient water

s s o

CH UCHv (E r C)

t x

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A SAMPLE RELATION:ENTRAINMENT OF AMBIENT WATER

w 2.4

2d2

0.075e

1 718

RgCH

U

Ri

Ri Fr

Here Ri denotes the bulk Richardson number. It is a ratio of the gravitational force resisting mixing to the inertial force of the flow. The larger is Ri, the less mixing there is across the interface between the turbidity current and the ambient flow.

Note that subcritical turbidity currents tend to have less mixing of ambient water across their interface than supercritical turbidity currents.

The relation is from Parker et al. (1987). 0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1 10

Ri

e w

10

w 2.4

2d2

0.075e

1 718

RgCH

U

Ri

Ri Fr

FROM PARKER ET AL. (1987)

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A SAMPLE RELATION:ENTRAINMENT OF SEDIMENT INTO SUSPENSION

The relation is from Garcia and Parker (1991).

50.6 7u b

u p5 s au

p

AZ u, Z , u , A 1.3x10

A v1 Z0.3

RgDD

Re

Re

SE

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.1 1 10 100

Z

Es

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RELATION OF NEAR-BED SUSPENDED SEDIMENT CONCENTRATION TO LAYER-AVERAGED VALUE

Let cb denote the suspended sediment concentration evaluated a distance z = b above the bed, where b/H << 1 (near-bed concentration).

The volume downward flux of suspended sediment onto the bed is given as vscb. In a layer-averaged model, cb must be related to the layer-averaged value C by means of a dimensionless coefficient ro:

b o

o

c r C

1 r 20?

z

c

cb

H

C

b

1.46

os

ur 1 31.5

v

Sample relation: Parker (1982), estimated from the Rouse () profile for rivers:

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RIVERS: flowing water under ambient air

gH

U1

222

df

af

f

FrFra = density of air << f

f = density of water

TURBIDITY CURRENTS: flowing dilute suspension of dirty water under ambient clear water

a = density of clear water

f = density of dirty water =

C = volume concentration of sediment << 1 (dilute!)

a a

f a

22 2d d 2

(1 RC)RC 1

(1 RC)

U RCgH

RCgH U

Fr Ri Fra s

s

(1 C) C (1 RC)

R 1.65

TURBIDITY CURRENTS AND RIVERS

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COMPARISON OF 1D LAYER-INTEGRATED (APPROXIMATE) GOVERNING EQUATIONS FOR

TURBIDITY CURRENTS AND RIVERS

Turbidity current:

2 22

f

w

s s o

UH U H 1 CHRg RgCHS C U

t x 2 xH UH

e Ut xCH UCH

v (E r C)t x

Compare with river:

2 22

f

s s o

UH U H 1 Hg gHS C U

t x 2 xH UH

0t xCH UCH

v (E r C)t x

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REFERENCES

Garcia, M, and G. Parker, 1991, Entrainment of bed sediment into suspension. Journal of Hydraulic Engineering, 117(4), 414‑435.

Parker, G., 1982, Conditions for the ignition of catastrophically erosive turbidity currents. Marine Geology, 46, pp. 307‑327, 1982.

Parker, G., M. H. Garcia, Y. Fukushima, and W. Yu, 1987, Experiments on turbidity currents over an erodible bed., Journal of Hydraulic Research, 25(1), 123‑147.