CEE 598, GEOL 593

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CEE 598, GEOL 593 TURBIDITY CURRENTS
MORPHODYNAMICS AND DEPOSITS
LECTURE 8 LAYER-AVERAGED GOVERNING EQUATIONS FOR
TURBIDITY CURRENTS
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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 ? ? tan(?) S ltlt
1. Thus x is almost horizontal and z is
almost vertical.
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SOME MORE DEFINITIONS
?a density of ambient water ?f density of
flowing water-sediment mixture in turbidity
current R submerged specific gravity of
sediment in suspension
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LETS JUMP TO THE 1D APPROXIMATE LAYER-INTEGRATED
EQUATIONS GOVERNING A TURBIDITY CURRENT
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
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THE EQUATIONS SUSPENDED SEDIMENT OF UNIFORM SIZE
In the above equations, the bed shear stress ?b
is given by the relation and vs 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 gt 1, where cb is a near-bed
suspended sediment concentration
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MOMENTUM BALANCE
a) b) c)
d) e)

• What the terms mean
• time rate of change of depth-integrated momentum
• inertial force term
• pressure force term
• downstream gravitational force
• bed resistive force

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FLUID MASS BALANCE
a) b) c)

• What the terms mean
• time rate of change of depth-integrated mass in
  flow (multiply by ?a)
• streamwise change in flow discharge per unit
  width
• rate at which flow incorporates ambient fluid
  from above by mixing across interface

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SEDIMENT MASS BALANCE
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
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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. 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).
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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).
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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 ltlt 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
Sample relation Parker (1982), estimated from
the Rouse () profile for rivers
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TURBIDITY CURRENTS AND RIVERS
RIVERS flowing water under ambient air
?a density of air ltlt ?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 ltlt 1 (dilute!)
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COMPARISON OF 1D LAYER-INTEGRATED (APPROXIMATE)
GOVERNING EQUATIONS FOR TURBIDITY CURRENTS AND
RIVERS
Turbidity current
Compare with river
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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.