8/12/2019 Ch 10 Suspended Sediment
1/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
1
CHAPTER 10:
RELATIONS FOR THE ENTRAINMENT AND 1D TRANSPORT OFSUSPENDED SEDIMENT
Dredging mine-derived of sand carried down predominantly bysuspension in the Ok Tedi, Papua New Guinea
8/12/2019 Ch 10 Suspended Sediment
2/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
2
THE STRATEGY
Consider the case of an equilibrium suspension in an equilibrium (normal) 1D open
channel flow. Returning to the equation of conservation of suspended sediment
from Chapter 4,
x
z
b
c
u
p
bcE
ss
H
0v
xqdzc
t
bcE
Under equilibrium conditions the dimensionless entrainment rate E is equal to
the near-bed average concentration of suspended sediment! We can:
Obtain empirical relation for E versus boundary shear stress for equilibrium
conditions.
With luck, the relation can be applied to conditions that are not too strongly
disequilibrium.
8/12/2019 Ch 10 Suspended Sediment
3/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
3
THE STRATEGY contd.
For equilibrium open-channel suspensions,
1. Determine a position z = b near the bed and measure the volume concentration
of suspended sediment averaged over turbulence there. Note that the
definition of b is peculiar to each researcher, but in general b/H
8/12/2019 Ch 10 Suspended Sediment
4/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
4
Smith and McLean (1977) offer the following entrainment relation.
The reference height is evaluated at what the authors describe as the top of the
bedload layer; where ksdenotes the Nikuradse roughness height,
The authors give no guidance for the choice of bc. It is suggested here that it
might be computed as bc=RgDc*, where c* is given by the Brownlie (1981) fit
to the Shields relation:
ENTRAINMENT RELATIONS FOR UNIFORM MATERIAL
Garcia and Parker (1991) reviewed seven entrainment relations andrecommended three of these; Smith and McLean (1977), van Rijn (1984) and
(surprise surprise) Garcia and Parker (1991).
RgD
,0024.0,11165.0 bssobc
bso
bc
bso
E
bcbs
s
bs
bcbs
s
,
Rgk
3.261
,1
k
b
DgD
,1006.022.0 p)7.7(6.0
pc
6.0p
R
ReRe
Re
8/12/2019 Ch 10 Suspended Sediment
5/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
5
ENTRAINMENT RELATIONS FOR UNIFORM MATERIAL contd.
The entrainment relation of van Rijn (1984) takes the form
The reference level b is set as follows:
b = 0.5 b, where b= average bedform height, when known;
b = the larger of the Nikuradse roughness height ksor 0.01 H
when bedforms are absent or bedform height is notknown.
The critical Shields number can be evaluated with the Brownlie (1981) fit to
the Shields curve:
DRgD
,1b
D015.0 p
2.0
p
5.1
c
s50 ReReE
DgD
,1006.022.0 p)7.7(
6.0pc
6.0
p
RReRe
Re
8/12/2019 Ch 10 Suspended Sediment
6/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
6
Garcia and Parker (1991) use a reference height b = 0.05 H;
Wright and Parker (2004) found that the relation of Garcia and Parker
(1991) performs well for laboratory flumes and small to medium sand-bed
streams, but does not perform well for large, low-slope streams. Wright
and Parker (2004) have thus amended the relationship to cover this latter
range as well Again the reference height b = 0.05 H. This corrects Garcia
and Parker to cover large, low-slope streams:
7bss
6.0
p
s
su
5
u
5
u 10x3.1A,u,v
uZ,
Z
3.0
A1
AZ
ReE
707.06.0
p
s
su
5
u
5
u 10x7.5A,Sv
uZ,
Z
3.0
A1
AZ
ReE
ENTRAINMENT RELATIONS FOR UNIFORM MATERIAL contd.
8/12/2019 Ch 10 Suspended Sediment
7/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
7
Garcia and Parker (1991) generalized their relation to sediment mixtures. Therelation for mixtures takes the form
where Fidenotes the fractions in the surface layer and denotes the arithmetic
standard deviation of the bed sediment on the scale. The reference height b is
again equal to 0.05 H.
Wright and Parker (2004) amended the above relation so as to apply to large,
low-slope sand bed rivers as well as the types previously considered by Garcia
and Parker (1991). The relation is the same as that of Garcia and Parker (1991)
except for the following amendments:
ENTRAINMENT RELATIONS FOR SEDIMENT MIXTURES
7m
ii
pi
2.0
50
i6.0pi
si
smui
5
ui
5
ui
i
iui
10x3.1A,298.01
DRgD,
D
D
v
uZ,
Z3.0
A1
AZ
F
EE
ReRe
2.0
50
i08.06.0
pisi
s
mui D
D
Sv
u
Z
Re
7
10x8.7A
8/12/2019 Ch 10 Suspended Sediment
8/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
8
McLean (1992; see also 1991) offers the following entrainment formulation forsediment mixtures. Let ETdenote the volume entrainment rate per unit bed area
summed over all grain sizes, pidenote the fractions in the ith grain size range in the
bedload transport and psbi= Ei/ETdenote the fractions in the ith grain size range in
the sediment entrained from the bed. Then where pdenotes bed porosity,
ENTRAINMENT RELATIONS FOR SEDIMENT MIXTURES contd.
004.0,1111E obc
bso
bc
bsopT
bc
cbs
ssis
csi
cs
sis
iN
1i
ii
iisbi u,u,1v/ufor
uv
uu
1v/ufor1
,
p
pp
1A1
1A
D)D(,)D(a
Damaxb
bc
bs2
bc
bs1
8484B
84Bo
84D0709.0)nD(022.0)nD(0204.0A
68.0A,056.0a,12.0a
84
2
842
10D
The critical boundary shear stress bcis evaluated using bed material D50;
again the Brownlie (1981) fit to the Shields curve is suggested here.
8/12/2019 Ch 10 Suspended Sediment
9/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
9
yxc)vwyxc)vw
zxvczxvczyuczyuczyxct
zzsszss
yysysxxsxss
LOCAL EQUATION OF CONSERVATION OF SUSPENDED SEDIMENT
0z
c)vw(
y
vc
x
uc
t
c s
x x+x
y+yy
z+z
z
Once entrained, suspended sediment can be carried about by the turbulent flow.
Let c denote the instantaneous concentration of suspended sediment, and (u, v,
w) denote the instantaneous flow velocity vector. The instantaneous velocity
vector of suspended particles is assumed to be simply (u, v, w - vs) where vs
denotes the terminal fall velocity of the particles in still water. Mass balance of
suspended sediment in the illustrated control volume can be stated as
or thus
8/12/2019 Ch 10 Suspended Sediment
10/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
10
AVERAGING OVER TURBULENCE
www,vvv,uuu,ccc
0wvuc
t
A
t
A,BABA
In a turbulent flow, u, v, w and c all show fluctuations in
time and space. To represent this, they are decomposed
into average values (which may vary in time and space at
scales larger than those characteristic of the turbulence)
and fluctuations about these average values.
By definition, then,
t
u
u
u
The equation of conservation of suspended sediment mass is now averaged over
turbulence, using the following properties of ensemble averages: a) the averageof the sum = the sum of the average and b) the average of the derivative = the
derivative of the average, or
8/12/2019 Ch 10 Suspended Sediment
11/38
8/12/2019 Ch 10 Suspended Sediment
12/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
12
The convective flux of any quantity is the quantity per unit volume times the velocity itis being fluxed. So, for example, the convective flux of streamwise momentum in the
upward direction is wu = wu. The viscous shear stress acting in the x (streamwise)
direction on a face normal to the z (upward) direction is
LOCAL STREAMWISE MOMENTUM CONSERVATION
z
u
x
w
z
uzx
zSyxgzypzyp
yxyxyxwuyxwu
zxvuzxvuzyuuzyuuzyxut
xxx
zzxzzzxzzz
yyyxxx
The balance of streamwise momentum in the control
volume requires that:
(streamwise momentum)/t = net convective inflow
of momentum + net shear force + net pressure force
+ downslope force of gravity
x x+x
y+yyz+z
z
8/12/2019 Ch 10 Suspended Sediment
13/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
13
A reduction yields the relation
Averaging over turbulence in the same way as before yields the result
where
Here denotes the z-x component of the Reynolds stressgenerated by
the turbulence; the term is known as the Reynolds fluxof streamwise
momentum in the upward direction. For fully turbulent flow, the Reynolds stressRzxis usually far in excess of the viscous stress , which can be dropped.
LOCAL STREAMWISE MOMENTUM CONSERVATION contd.
gSz
u
x
p1
z
uw
y
uv
x
u
t
u2
22
gSzz
1
x
p1
z
wu
y
vu
x
u
t
u Rzxzx2
wu,zu
Rzxzx
wuRzx wu
zx
8/12/2019 Ch 10 Suspended Sediment
14/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
14
The shear Reynolds stress Rzxis
abbreviated as ; its value at the bed is
b.. When the flow is steady and uniform
in the x and y directions, streamwise
momentum balance becomes
LOCAL STREAMWISE MOMENTUM CONSERVATION FOR NORMAL FLOW
gSz
1
x
p1
z
wu
y
vu
x
u
t
u 2
x
z
b
c
u
p
or thus
Integrating this equation under the condition of vanishing shear stress at the
water surface z = H yields the result
gSdz
d
gHS,H
z
1 bb
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
15/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
15
REYNOLDS FLUX OF SUSPENDED SEDIMENT
The terms denote convective Reynolds fluxesof suspended
sediment. They characterize the tendency of turbulence to mix suspended
sediment from zones of high concentration to zones of low concentration, i.e.
down the gradient of mean concentration. In the case illustrated below
concentration declines in the positive z direction; turbulence acts to mix the
sediment from the zone of high concentration (low z) to the zone of low
concentration (high z).
cwandcv,cu
0cw0w,0c
0cw0w,0c
z
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
16/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
16
REYNOLDS FLUX OF STREAMWISE MOMENTUM
The shear stress , or equivalently the Reynolds flux ofstreamwise (x) momentum in the upward (z) direction characterizes the tendency of
turbulence to transport streamwise momentum from high concentration to low. In the
case of open channel flow, the source for streamwise momentum is the downstream
gravity force term gS. This momentum must be fluxed downwardtoward the bed
and exited from the system (where the loss of momentum is manifested as a
resistive force balancing the downstream pull of gravity) in order
wuRzx wu
low streamwise
momentum u:u'0, v'
8/12/2019 Ch 10 Suspended Sediment
17/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
17
REPRESENTATION OF REYNOLDS FLUX WITH AN EDDY DIFFUSIVITY
The concentration of any quantity in a flow is the quantity per unit volume. Thusthe concentration of streamwise momentum in the flow is u and the volume
concentration of suspended sediment is c. The tendency for turbulence to mix
any quantity down its concentration gradient (from high concentration to low
concentration) can be represented in terms of a kinematic eddy diffusivity:
Reynolds flux of suspended sediment in the z direction:
Reynolds flux of streamwise momentum in the z direction:
In the above relations stis the kinematic eddy diffusivity of suspended sediment
[L2/T] and tis the kinematic eddy diffusivity (eddy viscosity) of momentum.
z
cwc st
zuwu t
z)z(c
0dzcdcw0
dzcd
st
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
18/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
18
EDDY VISCOSITY FOR TURBULENT OPEN CHANNEL FLOW
The standard equilibrium velocity profile for hydraulically rough turbulent open-channel flow is the logarithmic profile;
where = 0.4 and u*= (gHS)1/2. The eddy diffusivity of momentum can be back-
calculated from this equation;
Solving for t
, a parabolic form is obtained;
or
ss k
z30n
15.8
k
zn
1
u
u
H
z1u
z
u
dz
udwu 2tt
H
z1zut
H
z,1
Hu
t
Hut
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
19/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
19
EQUILIBRIUM VERTICAL DISTRIBUTION OF SUSPENDED SEDIMENT
According to the Reynolds analogy, turbulence transfers any quantity, whether itbe momentum, heat, energy, sediment mass, etc. in the same fundamental way.
While it is an approximation, it is a good one over a relatively wide range of
conditions. As a result, the following estimate is made for the eddy diffusivity of
sediment:
For steady flows that are uniform in the x and z directions maintaining a
suspension that is similarly steady and uniform, the equation of conservation of
suspended sediment reduces to
Hz1zutst
x
z
b
c
u
p
z
cw
y
cv
x
cu
z
cv
z
cw
y
cv
x
cu
t
cs
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
20/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
20
EQUILIBRIUM SUSPENSIONS contd.
The balance equation of suspended sediment thus becomes
This equation can be integrated under the condition of vanishing net sediment
flux in the z direction at the water surface to yield the result
i.e. the upward flux of suspended driven by turbulence from high concentration
(near the bed) to low concentration (near the water surface) is perfectly balanced
by the downward flux of suspended sediment under its own fall velocity. The
Reynolds flux F can be related to the gradient of the mean concentration as
The balance equation thus reduces to:
H
z1zu,
dz
cdcwF stst
cwF,0dz
cdv
dz
dFs
0cvF s
0cvdz
cd
H
z
1zu s
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
21/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
21
SOLUTION FOR THE ROUSE-VANONI PROFILE
The balance equation is:
The boundary condition on this equation is a specified upward flux, or
entrainment rate of sediment into suspension at the bed:
Rouse (1939) solved this problem and obtained the following result,
which is traditionally referred to as the Rouse-Vanoni profile.
Evdz
cd
H
z1zuF s
bz
bz
0c
H
z1zu
v
dz
cd s
H
b,
H
z,Ec,
/)1(
/1
c
cbb
u
v
bbb
s
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
22/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
22
REFERENCE LEVEL
The reference level cannot be taken as zero. This is because turbulence cannot
persist all the way down to a solid wall (or sediment bed). No matter whether the
boundary is hydraulically rough or smooth, essentially laminar effects must
dominate right near the wall (bed).
It is for this reason that the logarithmic velocity law
yields a value for of - at z = 0. The point of vanishing velocity is reached at z
= ks/30. Since the eddy diffusivity from which the profile of suspended sediment
is computed was obtained from the logarithmic profile, it follows that cannot be
computed down to z = 0 either. The entrainment boundary condition must be
applied at z = b ks/30.
ss k
z30n
15.8
k
zn
1
u
u
u
c
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
23/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
23
AND NOW ITS TIME FOR SPREADSHEET FUN!!
Go toRTe-bookRouseSpreadsheetFun.xlsRouse-Vanoni Equilibrium Suspended Sediment Profile Calculator
Input
b/H 0.05
vs 3 cm/s
u 0.2 m/s
c/cb z/H ref u/vs 6.6667 1 0.05
0.756 0.1
0.635 0.15
0.557 0.2
Sample Fall Velocities, 0.5 0.25
R = 1.65, = 0.01 cm2/s 0.455 0.30.418 0.35
vs D 0.386 0.4
cm/s m 0.357 0.450.0000421 1.0 0.331 0.5
0.0002031 2.0 0.307 0.55
0.0010048 4.0 0.285 0.60.0048709 8.0 0.263 0.65
0.0356491 20.0 0.241 0.7
0.0816579 30.0 0.22 0.75
0.1798665 45.0 0.197 0.8
0.3256999 62.0 0.173 0.85
0.5117601 80.0 0.145 0.9
0.7484697 100.0 0.11 0.95
1.0785878 125.0 0.077 0.98
1.4376162 150.0 0.046 0.995
2.2156534 200.0 0 1
3.04447576 250 0 0.05
5.65004674 400 1 0.05
The above values were computed from the Dietrich (1982) fall velocity relation
Rouse-Vanoni Profile of Suspended Sediment
Concentration
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.2 0 0.2 0.4 0.6 0.8 1
bc
c
H
z
Ec,
H
b,
H
z,
/)1(
/1
c
cbb
u
v
bbb
s
This spreadsheetallows calculation
of the suspended
sediment profile
from specified
values of b/H, vs
and u*using theRouse-Vanoni
profile.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
24/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
24
1D SUSPENDED SEDIMENT TRANSPORT RATE FROM EQUILIBRIUM SOLUTION
H
b
H
0s dzcudzcuq
H
b,
H
z,
/)1(
/1Ec b
u
v
bb
s
5.8k
zn
1u
k
z30n
uu
cc
In order to perform the calculation, however, it is necessary to know the velocityprofile over a bed which may include bedforms. This velocity profile may be
specified as
)Cz(c
c
2/1f
e
H11k
k
H11n
1CCz
The volume suspended sediment transport rate per unit width is qscomputed as
)z(u
where kcis a composite roughness height. If bedforms are absent, k
c= k
s=
nkDs90. If bedforms are present, the total friction coefficient Cf= Cfs+ Cffmay be
evaluated (using a resistance predictor for bedforms if necessary) and kcmay be
back-calculated from the relation
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
25/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
25
b
cs
s ,
k
H,
v
uEHuq
1D SUSPENDED SEDIMENT TRANSPORT RATE FROM EQUILIBRIUM SOLUTION
It follows that qsis given by the relations
1
c
u
v
bbb
cs b
s
dk
H
30n/)1(
/)1(
,k
H
,v
u
The integral is evaluated easily enough using a spreadsheet. This is done in the
next chapter.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
26/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
26
CLASSICAL CASE OF DISEQUILIBRIUM SUSPENSION: THE 1D PICKUP
PROBLEM
Consider a case where sediment-free equilibrium open-channel flow over a rough,
non-erodible bed impinges on an erodible bed offering the same roughness.
H
ri id bed erodible bed
cu
The flow can be considered quasi-steady over time spans shorter than that by
which significant bed degradation occurs.
The flow but not the suspended sediment profile can be considered to be at
equilibrium.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
27/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
27
THE 1D PICKUP PROBLEM contd.
H
rigid bed erodible bed
cu
z
F
z
cvw
y
cv
x
cu
t
cs
z
c
zz
cv
x
cu ts
x
z
0c,Evz
c,0
z
ccv
0xs
bz
t
Hz
ts
xas)z(c)x,z(c equilSolution yields
the result that
Can be used to find
adaptation length Lsrfor
suspended sediment
Governing equation
Boundary conditions
A method for estimating Lsris given in Chapter 21.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
28/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
28
Should the formulation be
with E computed based on local flow conditions, or
with qscomputed from the quasi-equilibrium relation
applied to local flow conditions?
The answer depends on the characteristic length L of the phenomenon of interest
(one meander wavelength, length of alluvial fan etc.) compared to the adaptation
length Lsrequired for the flow to reach a quasi-equilibrium suspension. If L < Lsthe former formulation should be used. If L > Lsthe latter formulation can be used.
WHICH VERSION OF THE EXNER EQUATION OF BED SEDIMENT
CONTINUITY SHOULD BE USED FOR A MORPHODYNAMIC PROBLEM
CONTROLLED BY SUSPENDED SEDIMENT?
Ecvxt
)1( bsp
bq
-
bcs
s ,k
H
,v
uEHu
q
xxxt)1( p
tsb qqq
---
Selenga Delta, Lake Baikal,Russia: image from NASAhttps://zulu.ssc.nasa.gov/mrsid/mrsid.pl
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
29/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
29
SELF-STRATIFICATION OF THE FLOW DUE TO SUSPENDED SEDIMENT
A flow is stably stratifiedif heavier fluid lies below lighter fluid. The densitydifference suppresses turbulent mixing.
The city of Phoenix, Arizona, USA
during an atmospheric inversion
Well, somewhere down there
Sediment-laden flows are self-stratifying
Rc
)Rc1(c)c1(
e
susp
ssusp
c
lighter up here
heavier down here
Here susp= density of the suspension and e=
fractional excess density due to the presence ofsuspended sediment.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
30/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
30
FLUX AND GRADIENT RICHARDSON NUMBERS
The damping of turbulence due to stable stratification is controlled by the flux
Richardson number Rif.
[Rate of expenditure of turbulent kinetic
energy in holding the (heavy) sediment in
suspension]/[Rate of generation of
turbulent kinetic energy by the flow]
dz
udwu
wcRg
dz
udwu
wg ef
Turbulence is not suppressedat all for Rif= 0. Turbulence is killed completely
when Rifreaches a value near 0.2 (e.g. Mellor and Yamada, 1974)
dz
cd
wc,dz
ud
wu tt Now let
Thenf2
dz
ud
dz
cdRg
where Ridenotes the gradient
Richardson Number
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
31/38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
31
SUSPENSION WITH SELF-STRATIFICATION:
SMITH-MCLEAN FORMULATION
These relations may be solved iteratively for concentration and velocity profiles in
the presence of stratification.
2
tot
dz
ud
dz
cdRg
7.41H
z1zu7.41
0cv
dz
cdst
H
z1u
dz
ud 2t
Ev
dz
cds
bz
t
c
bz
k
b30ln
1
u
u
The balance equations and boundary conditions take the forms:
Smith and McLean (1977), for example, propose the following relation for damping
of mixing due to self-stratification:
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
32/38
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
32
SUSPENSION WITH SELF-STRATIFICATION:
GELFENBAUM-SMITH FORMULATION
These relations may be solved iteratively for concentration and velocity profiles in
the presence of stratification. The workbook RTe-bookSuspSedDensityStrat.xlsprovides a numerical implementation.
2
tot
dzud
dz
cdRg
,1.351
35.1X,
X101
RiRi
Ri
0cv
dz
cdst
H
z1u
dz
ud 2t
Ev
dz
cds
bz
t
c
bz
k
b30ln
1
u
u
It also uses the specification b = 0.05 H. The balance equations and boundary
conditions take the forms:
The workbook RTe-bookSuspSedDensityStrat.xls implements the formulation for
stratification-mediated suppression of mixing due to Gelfenbaum and Smith (1986);
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
33/38
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
33
ITERATION SCHEME
The governing equations for flow velocity and suspended sediment concentration
can be integrated to give the forms
z
bt
sb
z
bt
2
b dzv
expcc,dzH
z1
uuu
where
Ec,kb30lnuu b
cb
The relations of the previous slide can be rearranged to give 2
t
2
t
s
H
z1
u
cv
Rg
Ri
The iteration scheme is commenced with the logarithmic velocity profilefor velocity and the Rouse-Vanoni profile for suspended sediment:
,
H
b,
H
z,
/)1(
/1cc,
k
z30n
uu b
u
v
bb
b
)0(
c
)0(
s
H
z1zu)0(t
where the superscript (0) denotes the 0thiteration (base solution).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
34/38
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
34
ITERATION SCHEME contd.
The iteration then proceeds as
,
H
z1
u
cv
Rg
2
)n(t
2
)n(
)n(t
s
1)(nRi
z
b )1n(t
sb
)1n(z
b )1n(t
2
b
)1n( dzv
expcc,dzH
z1
uuu
1)(n
1)(n)0(
t)1n(
t1.351
35.1X,
X101
Ri
Ri
)n(u )1n(c Iteration continues until is tolerably close to and is tolerably close to.
A dimensionless version of the above scheme is implemented in the workbook Rte-
bookSuspSedDensityStrat.xls. Moredetails about the formulation are provided in the
document Rte-bookSuspSedStrat.doc.
)1n(u )n(c
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
35/38
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
35
INPUT VARIABLES FOR Rte-bookSuspSedDensityStrat.xls
The first step in using the workbook is to input the parameters R+1 (sedimentspecific gravity), D (grain size), H (flow depth), kc(composite roughness height
including effect of bedforms, if any), u(shear velocity) and (kinematic viscosity
of water). When bedforms are absent, the composite roughness height kcis
equal to the grain roughness ks. In the presence of bedforms, kcis predicted from
one of the relations of Chapter 9 and the equations
The user must then click a button to clear any old output. After this step, the user
is presented with a choice. Either the near-bed concentration of suspended
sediment can be specified by the user, or it can be calculated from the Garcia-
Parker (1991) entrainment relation. In the former case, a value for must be
input. In the latter case, a value for the shear velocity due to skin friction usmustbe input. It follows that in the latter case uscan be predicted using one of the
relations of Chapter 9.
Once either of these options are selected and the appropriate data input, a click
of a button performs the iterative calculation for concentration and velocity
profiles. Note: the iterative scheme may not always converge!
bc
2/1
f)Cz(c CCz,
eH11k
bc
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
36/38
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
36
Dimensionless Velocity Profiles versus Normalized Depth
1
10
100
0.01 0.1 1
uno(
nos
tratification),un
(stratification
uno
un
Dimensionless Concentration Profiles versus NormalizedDepth
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
cno(
nos
t
ratification),cn
(stra
tification
cno
cn
SAMPLE CALCULATION (a) with Garcia-Parker entrainment relation
bc
c
u
u
H
z
Hz
Stratification included
qswith stratification = 0.72 x
qswithout stratificationStratification neglected
Stratification included
Stratification neglected000115.0cb
R 1.65D 0.2 mm
H 5 m
kc 50 mm
u* 4 cm/s
u*s 2 cm/s
0.01cm2/s
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
37/38
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
37
Dimensionless Velocity Profiles versus Normalized Depth
1
10
100
0.01 0.1 1
uno(
nos
tratif
ication),un
(stratific
ation
unoun
Dimensionless Concentration Profiles versus NormalizedDepth
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
cno(
nos
tratification),cn
(stratification
cno
cn
SAMPLE CALCULATION (b) with Garcia-Parker entrainment relation
bc
c
uu
H
z
H
z
Stratification included
Stratification neglected
Stratification neglected
Stratification included
qswith stratification = 0.39 x
qswithout stratification
00363.0cb
R 1.65D 0.2 mm
H 5 m
kc 50 mm
u* 6 cm/s
u*s 4 cm/s
0.01cm2/s
1D SEDIMENT TRANSPORT MORPHODYNAMICS
8/12/2019 Ch 10 Suspended Sediment
38/38
with applications to
RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004
38
REFERENCES FOR CHAPTER 10
Brownlie, W. R., 1981, Prediction of flow depth and sediment discharge in open channels, Report
No. KH-R-43A, W. M. Keck Laboratory of Hydraulics and Water Resources, California
Institute of Technology, Pasadena, California, USA, 232 p.
Garca, M., and G. Parker, 1991, Entrainment of bed sediment into suspension, Journal of
Hydraulic Engineering, 117(4): 414-435.
Gelfenbaum, G. and Smith, J. D., 1986, Experimental evaluation of a generalized suspended-
sediment transport theory, in Shelf and Sandstones, Canadian Society of Petroleum
Geologists Memoir II, Knight, R. J. and McLean, J. R., eds., 133
144.McLean, S. R., 1991, Depth-integrated suspended-load calculations, Journal of Hydraulic
Engineering, 117(11): 1440-1458.
McLean, S. R., 1992, On the calculation of suspended load for non-cohesive sediments, 1992,
Journal of Geophysical Research, 97(C4), 1-14.
Mellor, G. and Yamada, T., 1974, A hierarchy of turbulence closure models for planetary
boundary layers: Journal of Atmospheric Science, v.31, 1791-1806.
van Rijn, L. C., 1984, Sediment transport. II: Suspended load transport Journal of HydraulicEngineering, 110(11), 1431-1456.
Rouse, H., 1939, Experiments on the mechanics of sediment suspension, Proceedings 5th
International Congress on Applied Mechanics, Cambridge, Mass,, 550-554.
Smith, J. D. and S. R. McLean, 1977, Spatially averaged flow over a wavy surface, Journal of
Geophysical Research, 82(12): 1735-1746.
W i ht S d G P k 2004 Fl i t d d d l d i d b d