1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 10:
RELATIONS FOR THE ENTRAINMENT AND 1D TRANSPORT OF
SUSPENDED SEDIMENT
Dredging mine-derived of sand carried down predominantly by
suspension in the Ok Tedi, Papua New Guinea
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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,
qs
 H

 v s E  c b 
 0 c dz  
 x
t 
 E  cb
u
z
c
x
p

b
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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 cb  E there. Note that the
definition of b is peculiar to each researcher, but in general b/H << 1.
2. Determine the boundary shear stress b, or if bedforms are present the
component due to skin friction bs. Here we use the notation bs so as to always
admit the possibility of form drag.
3. If the sediment can be approximated as uniform with size D, compute s* =
bs/(RgD) and plot E versus s* to determine an entrainment rate.
4. If the sediment is to be treated as a mixture of sizes Di with fractions Fi in the
bed surface layer, from the measured entrainment rates Ei determine the
entrainment rates per unit content in the surface layer Eui = Ei/Fi, and plot Eui
versus si* = bs/(RgDi).
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ENTRAINMENT RELATIONS FOR UNIFORM MATERIAL
Garcia and Parker (1991) reviewed seven entrainment relations and
recommended three of these; Smith and McLean (1977), van Rijn (1984) and
(surprise surprise) Garcia and Parker (1991).
Smith and McLean (1977) offer the following entrainment relation.

 



E  0.65  o  bs  1 1   o  bs  1 ,  o  0.0024 , s  bs
RgD
 bc
 
 bc

The reference height is evaluated at what the authors describe as the top of the
bedload layer; where ks denotes the Nikuradse roughness height,
1 , bs  bc

b 
bs

1

26
.
3
, bs  bc
ks 
Rgk s

The authors give no guidance for the choice of bc. It is suggested here that it
might be computed as bc=RgDc*, where c* is given by the Brownlie (1981) fit
to the Shields relation:
c  0.22 Rep0.6  0.06  10
( 7.7 Rep 0.6 )
, Rep 
RgD D

4
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ENTRAINMENT RELATIONS FOR UNIFORM MATERIAL contd.
The entrainment relation of van Rijn (1984) takes the form
D50
E  0.015
b
1 .5


RgD D
  1 Rep0.2 , Rep 




s

c
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 ks or 0.01 H
when bedforms are absent or bedform height is not
known.
The critical Shields number can be evaluated with the Brownlie (1981) fit to
the Shields curve:

c
 0 .6
p
  0.22 Re
 0.06  10
( 7.7 Rep 0.6 )
, Rep 
RgD D

5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ENTRAINMENT RELATIONS FOR UNIFORM MATERIAL contd.
Garcia and Parker (1991) use a reference height b = 0.05 H;
AZu5
E
A 5
1
Zu
0.3
,
Zu 
us
Rep0.6 , us 
vs
bs
,

A  1.3 x10 7
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:
AZu5
E
A 5
1
Zu
0.3
,
us
Zu 
Rep0.6 S0.07 ,
vs
A  5.7x10 7
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ENTRAINMENT RELATIONS FOR SEDIMENT MIXTURES
Garcia and Parker (1991) generalized their relation to sediment mixtures. The
relation for mixtures takes the form
 D 
AZui5
u
Ei
Eui 

, Zui   m s Repi0.6  i 
Fi 1  A Z5
v si
 D50 
ui
0.3
 m  1  0.298 , A  1.3 x10 7
0.2
, Repi 
RgD i Di

where Fi denotes 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:
u

 D 
Zui   m  s Repi0.6 S0.08  i 
 v si

 D50 
0.2
A  7.8x107
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ENTRAINMENT RELATIONS FOR SEDIMENT MIXTURES contd.
McLean (1992; see also 1991) offers the following entrainment formulation for
sediment mixtures. Let ET denote the volume entrainment rate per unit bed area
summed over all grain sizes, pi denote the fractions in the ith grain size range in the
bedload transport and psbi = Ei/ET denote the fractions in the ith grain size range in
the sediment entrained from the bed. Then where p denotes bed porosity,


E T  1   p   o  bs  1
 bc

p sbi 
ipi
N
p
i1
i i
 a D

b  max  D 84  ,
 ao B (D84 ) 
,

 bs



1



1

o
 bc


1 for us / v si  1

u  u
i   s
, us 
c
for us / v si  1
 v si  uc


A 1 bs  1
 bc

B (D84 )  D84


1  A 2  bs  1
 bc

,
 o  0.004
bs

, uc 
bc

aD  0.12 , a0  0.056 ,
A1  0.68
A 2  0.0204(nD 84 )2  0.022(nD 84 )  0.0709
The critical boundary shear stress bc is evaluated using bed material D50;
again the Brownlie (1981) fit to the Shields curve is suggested here.
8
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
LOCAL EQUATION OF CONSERVATION OF SUSPENDED SEDIMENT
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

scxyz  suc  x yz  suc  x  x yz  s vc  y xz  s vc  y  y xz
t
 s w  v s )c  z xy  s w  v s )c  z  z xy
y+y
z+z
or thus
c uc vc ( w  v s )c



0
t
x
y
z
y
z
x
x+x
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
AVERAGING OVER TURBULENCE
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.
u
c  c  c , u  u  u ,
v  v  v ,
w  w  w
u
u
By definition, then,
c  u  v  w  0
t
The equation of conservation of suspended sediment mass is now averaged over
turbulence, using the following properties of ensemble averages: a) the average
of the sum = the sum of the average and b) the average of the derivative = the
derivative of the average, or
A B  A  B
,
A A

t
t
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
AVERAGING OVER TURBULENCE contd.
Recalling that vs is a constant, substituting the decompositions
c  c  c , u  u  u ,
v  v  v ,
w  w  w
into the equation of mass conservation of suspended sediment results in
c uc vc ( w  v s )c c uc vc ( w  v s )c








t
x
y
x
t
x
y
x
( c  c ) ( u  u)( c  c ) ( v  v )( c  c ) ( w  w   v s )( c  c )



0
t
x
y
z
Now for example
( u  u)( c  c)  ( u c  u c  cu  uc)  u c  u c  cu  uc  u c  uc
so that the final form of the averaged equation is
c  u c v c wc
c
uc  vc  w c



 vs



t
x
y
z
z
x
y
z
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
LOCAL STREAMWISE MOMENTUM CONSERVATION
The convective flux of any quantity is the quantity per unit volume times the velocity it
is being fluxed. So, for example, the convective flux of streamwise momentum in the
upward direction is wu = wu. The viscous shear stress acting in the x (streamwise)
direction on a face normal to the z (upward) direction is
u
 u w 
 zx   




z

x
z


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
y+y
z+z
y
z
x
x+x

uxyz  uu  x yz  uu  x  x yz  vu  y xz  vu  y  y xz
t
 wu  z xy  wu  z  z xy   zx z  z xy   zx z xy   
p x yz  p x  x yz  gxyzS
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
LOCAL STREAMWISE MOMENTUM CONSERVATION contd.
A reduction yields the relation
u u2 uv uw
1 p
 2u




  2    gS
t x
y
z
 x
z
Averaging over turbulence in the same way as before yields the result
u u 2 u v u w
1 p 1  zx Rzx 




 

    gS
t
x
y
z
 x   z
z 
where
zx  
u
z
,
Rzx  uw 
Here Rzx  uw  denotes the z-x component of the Reynolds stress generated by
the turbulence; the term uw  is known as the Reynolds flux of streamwise
momentum in the upward direction. For fully turbulent flow, the Reynolds stress 13
Rzx is usually far in excess of the viscous stress zx , which can be dropped.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
LOCAL STREAMWISE MOMENTUM CONSERVATION FOR NORMAL FLOW
The shear Reynolds stress Rzx is
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
u
z
c
x
u u
u v u w
1 p 1 





 gS
t
x
y
z
 x  z
2
or thus
p

b
d
 gS
dz
Integrating this equation under the condition of vanishing shear stress at the
water surface z = H yields the result
z

  b  1   ,
 H
b  gHS
14
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REYNOLDS FLUX OF SUSPENDED SEDIMENT
The terms uc , vc and w c denote convective Reynolds fluxes of 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).
c  0 , w   0  w c  0
z
c  0 , w   0  w c  0
15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
REYNOLDS FLUX OF STREAMWISE MOMENTUM
The shear stress   Rzx  uw  , or equivalently the Reynolds flux uw  of
streamwise (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 downward toward 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
to achieve momentum
balance. This downward
high streamwise
flux is maintained by
momentum u:
maintaining a streamwise
u
u'>0, v'<0
momentum profile that has
u
high velocity in the upper
low streamwise
part of the flow and low
momentum u:
velocity in the lower part of
u'<0, v'>0
the flow. This in turn
z
generates a negative value
of uw  and a positive
 uw   0
16
value of   uw  .
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REPRESENTATION OF REYNOLDS FLUX WITH AN EDDY DIFFUSIVITY
The concentration of any quantity in a flow is the quantity per unit volume. Thus
the 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:
c w    st
c
z
z
c ( z)
Reynolds flux of streamwise momentum in the z direction:
uw    t
u
z
dc
dc
 0  wc  st
0
dz
dz
In the above relations st is the kinematic eddy diffusivity of suspended sediment
[L2/T] and t is the kinematic eddy diffusivity (eddy viscosity) of momentum.
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
EDDY VISCOSITY FOR TURBULENT OPEN CHANNEL FLOW
The standard equilibrium velocity profile for hydraulically rough turbulent openchannel flow is the logarithmic profile;
u 1 z
1 
z



 n   8.5  n 30 
u   k s 
  ks 
where  = 0.4 and u* = (gHS)1/2. The eddy diffusivity of momentum can be backcalculated from this equation;
  uw    t
du
u
z

  t   u2 1  
dz
z
 H
Solving for t, a parabolic form is obtained;
z

 t  u z1  
 H

or
t
 1    ,
uH

t
uH
z
H
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM VERTICAL DISTRIBUTION OF SUSPENDED SEDIMENT
According to the Reynolds analogy, turbulence transfers any quantity, whether it
be 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:
z

 st   t  u z1  
 H
u
z
c
p
x

b
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
c  u c v c wc
c
uc  vc  w c



 vs



t
x
y
z
z
x
y
z
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM SUSPENSIONS contd.
The balance equation of suspended sediment thus becomes
dF
dc
 vs
0
dz
dz
, F  w c 
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
F  vsc  0
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
F  w c   st
dc
dz
,
z

 st  u z1  
 H
The balance equation thus reduces to:
z  dc

u z1  
 v sc  0
 H  dz
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOLUTION FOR THE ROUSE-VANONI PROFILE
The balance equation is:
dc

dz
vs
c 0
z


u z1  
 H
The boundary condition on this equation is a specified upward flux, or
entrainment rate of sediment into suspension at the bed:

z  d c

F z b   uz1  
 H  dz

 v sE
z b
Rouse (1939) solved this problem and obtained the following result,
vs
u
c  1    /  


cb  (1  b ) / b 
z
b
, c b  E ,   , b 
H
H
which is traditionally referred to as the Rouse-Vanoni profile.
21
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
u 1 z
1 
z
 n   8.5  n 30 
u   k s 
  ks 
yields a value for u 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 c cannot be
computed down to z = 0 either. The entrainment boundary condition must be
applied at z = b  ks/30.
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
AND NOW IT’S TIME FOR SPREADSHEET FUN!!
Go to RTe-bookRouseSpreadsheetFun.xls
Rouse-Vanoni Equilibrium Suspended Sediment Profile Calculator
b/H
vs
u
Input
0.05
3 cm/s
0.2 m/s
 1    /  
c


c b  (1  b ) / b 
6.6667
Sample Fall Velocities,
R = 1.65,  = 0.01 cm2/s
vs
D
cm/s
m
0.0000421
0.0002031
0.0010048
0.0048709
0.0356491
0.0816579
0.1798665
0.3256999
0.5117601
0.7484697
1.0785878
1.4376162
2.2156534
3.04447576
5.65004674
1.0
2.0
4.0
8.0
20.0
30.0
45.0
62.0
80.0
100.0
125.0
150.0
200.0
250
400

,
z
,
H
b 
b
,
H
cb  E
Rouse-Vanoni Profile of Suspended Sediment
Concentration
c/cb
u/vs
vs
u
1
0.756
0.635
0.557
0.5
0.455
0.418
0.386
0.357
0.331
0.307
0.285
0.263
0.241
0.22
0.197
0.173
0.145
0.11
0.077
0.046
0
0
1
z/H
ref
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
0.98
0.995
1
1
0.9
0.8
0.7
0.6
z
H
0.5
0.4
0.3
0.2
0.1
0
-0.2
0
0.2
0.4
0.6
c
cb
0.05
0.05
The above values were computed from the Dietrich (1982) fall velocity relation
0.8
1
This spreadsheet
allows calculation
of the suspended
sediment profile
from specified
values of b/H, vs
and u* using the
Rouse-Vanoni
profile.
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
1D SUSPENDED SEDIMENT TRANSPORT RATE FROM EQUILIBRIUM SOLUTION
The volume suspended sediment transport rate per unit width is qs computed as
H
H
0
b
qs   u cdz   u cdz
 1    /  
c  E

 (1  b ) / b 
vs
u
z
b
,   , b 
H
H
In order to perform the calculation, however, it is necessary to know the velocity
profile u( z) over a bed which may include bedforms. This velocity profile may be
specified as

u
z
u   n 30
  kc
1  z

  u  n

   kc


  8.5


where kc is a composite roughness height. If bedforms are absent, kc = ks =
nkDs90. If bedforms are present, the total friction coefficient Cf = Cfs + Cff may be
evaluated (using a resistance predictor for bedforms if necessary) and kc may be
back-calculated from the relation
Cz  Cf 1/ 2 
1  H
11 H
n11   k c  (  Cz)
  kc 
e
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
1D SUSPENDED SEDIMENT TRANSPORT RATE FROM EQUILIBRIUM SOLUTION
It follows that qs is given by the relations
u EH
qs  

 u H

 , , b 
 vs kc

 u H
 1  (1  ) /  
 , , b    
b (1   ) /  
v
k
b
b
 s c


vs
u

H
n 30
 kc

 d

The integral is evaluated easily enough using a spreadsheet. This is done in the
next chapter.
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
u
rigid bed
c
erodible bed
•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.
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE 1D PICKUP PROBLEM contd.
H
u
c
z
rigid bed
x
erodible bed
Governing equation
c
c
c
c
F
u
v
 w  v s 

t
x
y
z
z

c
c
  c 
u
 vs

 t

x
z z  z 
Boundary conditions
c 

 0,
 v sc   t

z  z H

Solution yields
the result that
c ( z, x )  c equil ( z)
 c 
  t
  v sE ,
 z  z b
as
x
A method for estimating Lsr is given in Chapter 21.
c x 0  0
Can be used to find
adaptation length Lsr for
suspended sediment
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
WHICH VERSION OF THE EXNER EQUATION OF BED SEDIMENT
CONTINUITY SHOULD BE USED FOR A MORPHODYNAMIC PROBLEM
CONTROLLED BY SUSPENDED SEDIMENT?
Should the formulation be
(1  p )

q
 - b  v s cb  E
t
x
with E computed based on local flow conditions, or
qb qs
qt

(1   p )
t
x x
x
with qs computed from the quasi-equilibrium relation

uEH  u H
 , , b 
  vs kc

applied to local flow conditions?
qs 
Selenga Delta, Lake Baikal,
Russia: image from NASA
https://zulu.ssc.nasa.gov/mrsid/mrsid.pl
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 Ls required for the flow to reach a quasi-equilibrium suspension. If L < Ls 28
the former formulation should be used. If L > Ls the latter formulation can be used.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SELF-STRATIFICATION OF THE FLOW DUE TO SUSPENDED SEDIMENT
A flow is stably stratified if heavier fluid lies below lighter fluid. The density
difference suppresses turbulent mixing.
The city of Phoenix, Arizona, USA
during an atmospheric inversion
Sediment-laden flows are self-stratifying
lighter up here
Well, somewhere down there
susp  (1  c )  sc  (1  Rc )
susp  

 e  Rc
c
heavier down here
Here susp = density of the suspension and e =
fractional excess density due to the presence of
suspended sediment.
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FLUX AND GRADIENT RICHARDSON NUMBERS
The damping of turbulence due to stable stratification is controlled by the flux
Richardson number Rif.
ge w 
Rgcw 
Rif 


du
du
 uw 
 uw 
dz
dz




[Rate of expenditure of turbulent kinetic
energy in holding the (heavy) sediment in
suspension]/[Rate of generation of
turbulent kinetic energy by the flow]
Turbulence is not suppressed at all for Rif = 0. Turbulence is killed completely
when Rif reaches a value near 0.2 (e.g. Mellor and Yamada, 1974)
Now let
Then
uw    t
dc
 Rg
dz  Ri
Ri 
f
2
d
u




dz


du
dc
, cw    t
dz
dz
where Ri denotes the gradient
Richardson Number
30
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SUSPENSION WITH SELF-STRATIFICATION:
SMITH-MCLEAN FORMULATION
Smith and McLean (1977), for example, propose the following relation for damping
of mixing due to self-stratification:

dc 


Rg
z


dz 
 t   to 1  4.7Ri  u z1   1  4.7
2
 H 
 du  


 
 dz  

The balance equations and boundary conditions take the forms:
dc
dc


 v sE
t
 v sc  0
t
dz zb
dz
t
du
z

 u2 1  
dz
 H
u zb
u

1 
b
ln 30 
  kc 
These relations may be solved iteratively for concentration and velocity profiles in
31
the presence of stratification.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SUSPENSION WITH SELF-STRATIFICATION:
GELFENBAUM-SMITH FORMULATION
The workbook RTe-bookSuspSedDensityStrat.xls implements the formulation for
stratification-mediated suppression of mixing due to Gelfenbaum and Smith (1986);
t 
 to
1  10 X
,
X
dc
dz
, Ri  
2
 du 


 dz 
Rg
1.35 Ri
1  1.35 Ri
It also uses the specification b = 0.05 H. The balance equations and boundary
conditions take the forms:
dc
dc
t
 v sc  0
 t
 v sE
dz
dz zb
t
du
z

 u2 1  
dz
 H
u zb
u

1 
b
ln 30 
  kc 
These relations may be solved iteratively for concentration and velocity profiles in
the presence of stratification. The workbook RTe-bookSuspSedDensityStrat.xls 32
provides a numerical implementation.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ITERATION SCHEME
The governing equations for flow velocity and suspended sediment concentration
can be integrated to give the forms
 z vs 
u2 
z
  
u  ub  
1

dz
,
c

c
exp
dz 


b
b 
b
H
t 
t 

z
where
ub 
u 
b
ln 30 
  kc 
, cb  E
Rg
Ri 
The relations of the previous slide can be rearranged to give
vs
c
t
 u2 
z 
1




H

 t
2
The iteration scheme is commenced with the logarithmic velocity profile
for velocity and the Rouse-Vanoni profile for suspended sediment:
u (0) 
 1    /  
u 
z
n 30  , c (0)  cb 

  kc 
 (1  b ) / b 
vs
u
, 
z
b
, b 
H
H
,
z

(t0 )  u z 1  
 H
33
where the superscript (0) denotes the
0th
iteration (base solution).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ITERATION SCHEME contd.
The iteration then proceeds as
Rg
(n1)
Ri

v s (n)
c
(tn)
u 
z 
1




H



u
2

(n)
t
(n1)
 ub  
z
b
2
,
(n1)
t

(t0)

1  10X
1.35 Ri(n1)
, X
1  1.35 Ri(n1)
 z vs

u2 
z
(n1)


1

dz
,
c

c
exp

dz

b
(n1) 
(n1)


b
t  H 
t


Iteration continues until u (n1) is tolerably close to u (n ) and c (n1) is tolerably close to c (n )
.
A dimensionless version of the above scheme is implemented in the workbook RtebookSuspSedDensityStrat.xls. More details about the formulation are provided in the
document Rte-bookSuspSedStrat.doc.
34
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INPUT VARIABLES FOR Rte-bookSuspSedDensityStrat.xls
The first step in using the workbook is to input the parameters R+1 (sediment
specific 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 kc is
equal to the grain roughness ks. In the presence of bedforms, kc is predicted from
one of the relations of Chapter 9 and the equations
11 H
k c  (  Cz) , Cz  Cf 1/ 2
e
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 cb can be specified by the user, or it can be calculated from the GarciaParker (1991) entrainment relation. In the former case, a value for cb must be
input. In the latter case, a value for the shear velocity due to skin friction us must
be input. It follows that in the latter case us can 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
35
profiles. Note: the iterative scheme may not always converge!
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION (a) with Garcia-Parker entrainment relation
Dimensionless Concentration Profiles versus Normalized
Depth
c
cb
cno (no stratification), cn
(stratification
1
0.9
0.8
0.7
qs with stratification = 0.72 x
qs without stratification
Stratification neglected
0.6
cno
cn
0.5
0.4
0.3
0.2
Dimensionless Velocity Profiles versus Normalized Depth
0.1
0
100
R
D
H
kc
0.4
z
H
1.65
Stratification
0.2 mm
5m
50 mm
u*
4 cm/s
u*s
2 cm/s

0.2
2
0.01 cm /s
0.6
0.8
1

included
cb  0.000115
u
u
uno (no stratification), un
(stratification
0
Stratification included
uno
un
10
Stratification neglected
1
0.01
0.1

1
z
H
36
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION (b) with Garcia-Parker entrainment relation
Dimensionless Concentration Profiles versus Normalized
Depth
c
cb
cno (no stratification), cn
(stratification
1
0.9
qs with stratification = 0.39 x
qs without stratification
0.8
0.7
Stratification neglected
0.6
cno
cn
0.5
0.4
0.3
0.2
Dimensionless Velocity Profiles versus Normalized Depth
0.1
100
0
0.2
0.4
0.6

R
D
H
kc
u*
u*s

z
H
0.8
Stratification included
1.65
0.2 mm
5m
50 mm
6 cm/s
4 cm/s
2
0.01 cm /s
cb  0.00363
1
u
u
uno (no stratification), un
(stratification
0
Stratification included
uno
un
10
Stratification neglected
1
0.01
0.1

1
z
H
37
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
García, M., and G. Parker, 1991, Entrainment of bed sediment into suspension, Journal of
Hydraulic Engineering, 117(4): 414-435.
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