CEE 598, GEOL 593

TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS

LECTURE 8

LAYER-AVERAGED GOVERNING EQUATIONS FOR TURBIDITY CURRENTS

z x q u ambient water c turbid water 1

SOME DEFINITIONS

x = boundary-attached (seafloor-attached) streamwise coordinate z = boundary-perpendicular upward normal coordinate u = 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.

g  (g , g g ) x y z  (gS, 0  g) z x q u ambient water c turbid water 2

SOME MORE DEFINITIONS

 a  f = density of ambient water = density of flowing water-sediment mixture in turbidity current R = submerged specific gravity of sediment in suspension    f a   a but    f a  f  Rc  Rc  1 z x q u ambient water c turbid water 3

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

Flow discharge per unit width = q f , volume suspended sediment discharge per unit width = q s , streamwise momentum discharge per unit width = q m :  q f   0 udz  UH q s   0  ucdz  UCH q m   a  0    a clear water U Thus layer-averaged quantities can be defined as UH   0  udz UCH    0   0  ucdz H u c C z 4

THE EQUATIONS: SUSPENDED SEDIMENT OF UNIFORM SIZE

 UH   t H  t   CH  t    x  UH  x    UCH  x    1 2 Rg v (E s s  CH 2   x r C) o   f 2 In the above equations, the bed shear stress  b is given by the relation    b a C U f 2 and v s e w = sediment fall velocity [L/T], = dimensionless rate of entrainment of ambient water into the turbidity current, r E s = dimensionless rate of entrainment of bed sediment into the turbidity current [1] o = c b /C > 1, where c b is a near-bed suspended sediment concentration 5

MOMENTUM BALANCE

 UH  t    x   1 Rg 2  CH 2  x   f a) b) c) d) e) 2 What the terms mean: a) time rate of change of depth-integrated momentum b) inertial force term c) pressure force term d) downstream gravitational force e) bed resistive force 6

What the terms mean:

FLUID MASS BALANCE

 H  t   UH  x  e U w a) b) c) a) time rate of change of depth-integrated mass in flow (multiply by  a ) b) streamwise change in flow discharge per unit width c) rate at which flow incorporates ambient fluid from above by mixing across interface ambient water z x u c q turbid water 7

SEDIMENT MASS BALANCE

 CH  t   UCH  x  v (E s s  r C) o a) b) c) d) 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 width c) rate at which sediment is eroded from the bed into suspension d) rate at which sediment settled out from the flow onto the bed z x q u ambient water c turbid water 8

A SAMPLE RELATION: ENTRAINMENT OF AMBIENT WATER

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.

1 e w  0.075

Ri

2.4

Ri

 RgCH U 2 

Fr

d  2 Note that subcritical turbidity currents tend to have less mixing of ambient water across their interface than supercritical turbidity currents.

0.1

0.01

0.001

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

0.0001

0.001

0.01

0.1

Ri

1 9 10

FROM PARKER ET AL. (1987)

e w  0.075

Ri

2.4

Ri

 RgCH U 2 

Fr

d  2 10

A SAMPLE RELATION: ENTRAINMENT OF SEDIMENT INTO SUSPENSION

E S  1  AZ 5 u A 0.3

Z 5 u

Re

p  RgD D  , Z u  u  v s

Re

0.6

p , u    b  a , 1.00E+00 1.00E-01 A  1.3x10

 7 1.00E-02 The relation is from Garcia and Parker (1991).

1.00E-03 1.00E-04 1.00E-05 1.00E-06 1.00E-07 0.1

1 10

Z

11 100

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

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

z H The volume downward flux of suspended sediment onto the bed is given as v s c b . In a layer averaged model, c b must be related to the layer-averaged value C by means of a dimensionless coefficient r o : c b  r C o   o 20 ?

b c C c b Sample relation: Parker (1982), estimated from the Rouse () profile for rivers: r o    u  v s     1.46

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TURBIDITY CURRENTS AND RIVERS RIVERS: flowing water under ambient air

 a = density of air <<  f  f = density of water    f   f   a  f  1

Fr

d 2 

Fr

2  U 2 gH

TURBIDITY CURRENTS: flowing dilute suspension of dirty water

under ambient clear water

 a = density of clear water  f = density of dirty water =  a R    s C    s   1.65

 

Fr

f d 2    a  U a 2 RCgH  

Ri

a   RC

Fr

d  2   1 RCgH U 2 C = volume concentration of sediment << 1 (dilute!) 13

COMPARISON OF 1D LAYER-INTEGRATED (APPROXIMATE) GOVERNING EQUATIONS FOR TURBIDITY CURRENTS AND RIVERS

Turbidity current: Compare with river:  UH   t H  t   CH  t    x  UH  x    UCH  x    1 Rg 2 s  CH 2  x    UH   t H  t   CH  t    x  UH  x    UCH  x 0    1 2 g  v (E s s H 2  x   r C) o  f 2  2 14

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.

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