Rte-bookSuspSedStrat.doc
STRATIFICATION OF SUSPENDED SEDIMENT IN OPEN CHANNEL FLOW
Gary Parker
This document is a companion to the Excel workbook Rte-bookSuspSedDensityStrat.xls.
Definitions
Consider steady, equilibrium open channel flow over an erodible bed in a wide channel
with upward normal profiles of streamwise velocity u  u( z) and volume suspended
sediment concentration c  c ( z) , where z denotes a coordinate upward normal from
bed. The flow has depth H, shear velocity u and shear velocity due to skin friction
us  u . The bed has composite roughness kc, where kc is given as
11 H
e(  Cz)
kc 
, Cz  Cf 1/ 2 
U
u
where U denotes depth-averaged flow velocity. In the absence of bedforms k c becomes
equal to the grain roughness ks . In the above relations,. The water has density  and
kinematic viscosity .
The sediment is assumed to be uniform with size D, density s and fall velocity vs. The
submerged specific gravity of the sediment is given as
R
s
1

and the explicit particle Reynolds number Rep is given as
Rep 
RgD D

where g denotes gravitational acceleration. Particle fall velocity vs is specified by a
dimensionless relation of the form
R f  f (Rep )
where
Rf 
vs
RgD
and the functional relationship itself is given by Dietrich (1982).
Basic forms
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Rte-bookSuspSedStrat.doc
The equation of momentum conservation for the flow takes the form
t
du
z
 u2 (1  )
dz
H
(1)
where t denotes a kinematic eddy viscosity. The corresponding form for conservation
of suspended sediment is
v sc  t
dc
0
dz
(2)
Eddy viscosity t is specified in the following form:
 t  uH F1( ) F2 (Ri)
(3)
where  denotes the Karman constant, F1 and F2 are specified functions,

z
H
and Ri denotes the gradient Richardson number, given by
dc
Ri  Rg dz 2
 du 


 dz 
(4)
The bottom boundary condition for velocity is determined by a match to the rough
logarithmic law at normalized reference bed elevation r:
u
r
u

1
H
n(30r )

kc
(5)
where  denotes the Karman constant (0.4). The bottom boundary condition for
concentration of suspended sediment is likewise given in terms of a specification of
reference bed concentration cr at r. In the implementation of the workbook r is set
equal to 0.05.
Dimensionless forms
Define dimensionless velocity and concentration as follows:
u
u
u
c
c
cr
2
Rte-bookSuspSedStrat.doc
Equations (1), (2), (4) and (5) can then be reduced to
du
(1   )

d F1 ( )F2 (Ri)
(6)
dc
1
1

c
d
u r F1 ()F2 (Ri)
(7)
Ri  Ri
dc
d
 du 


 d 
(8)
2
u 
1
H
n(30r )

kc
(9)
ur 
u
vs
(10a,b)
r
where
Ri 
RgH cr
u2
In addition, the near-bed condition for normalized concentration c becomes
c  1
(11)
r
Forms for the functions F1 and F2
The standard form for the function F1 is the parabolic one
F1  (1 )
(12)
Smith and McLean (1977) offer the alternative form
F1 

  1.32892 2  16.86321 3  25.22663 4  r    0.3

2
3
0.3    1
0.160552  0.075605  0.1305618  0.1055945
Gelfenbaum and Smith (1986) offer the alternative form
F1   exp(   3.2 2 
2
3.2 2 )
3
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Rte-bookSuspSedStrat.doc
Smith and McLean (1977) provide the following form for F2:
F2  1 4.7 Ri
Gelfenbaum and Smith (1986) offer the alternative below:
F2 
1
1  10.0 X
X
1.35 Ri
1  1.35 Ri
(13)
Relations (12) and (13) are used in the implementation of the workbook.
Form for near-bed concentration
The workbook allows two options for near-bed concentration. Either it can be specified
by the user, or it can be computed using the relation of Garcia and Parker (1991). In the
latter case, the specification for reference concentration is as follows;
cr 
AX5e
A 5
1
Xe
0.3
Xe 
us
Rep0.6
vs
where A = 1.3x10-7. Note that if the entrainment relation of Garcia and Parker is used, it
is necessary to know both u and us , where u*s denotes the shear velocity due to skin
friction only. This requires the use of a resistance predictor that includes the effect of
bedforms when they are present.
Solution
The solution used in the spreadsheet uses the parabolic form (12) for F1, the form (13)
for F2 due to Gelfenbaum and Smith (1986) and the form for cr due to Garcia and
Parker (1991). The solution is implemented iteratively. In the zeroth-order solution F2 is
set equal to 1, corresponding to vanishing stratification (Ri = 0). This yields the solution
1
u
1
H
n(30 )

ks
 (1  ) /   ur
c

 (1   r ) r 
The above solution is then used to compute Ri, which is found after some manipulation
to take the form
Ri  Ri
 F2
c
ur (1  )
This in turn allows for an update of F2, and thus the solution for u and c. The solution is
iterated until convergence is obtained.
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Rte-bookSuspSedStrat.doc
The iterative scheme is implemented as follows.
iterations for u and c, respectively, and let
Ri(n 1)  Ri
F2(n1) 
Let u(n) and c(n) denote the nth
 F1F2(n) (n)
c
ur (1  )2
1
1  10.0 X
X
1.35 Ri(n1)
1  1.35 Ri(n1)
Equations (6) and (7) subject to (9) and (11) yield the following forms for the (n+1)th
iteration.
u (n 1) 

1
H
(1  )
n(r )  
d
 r F ()F( n  1)

ks
1
2
  1

1
c (n 1)  exp  
d
 r u F ()F(n 1)
r 1
2


The calculation is commenced with the logarithmic law for flow velocity and the RouseVanoni solution for suspended sediment concentration;
1
u (1) 
1
H
n(30 ) , c (1)

ks
 (1  ) /   ur


 (1  r )r 
, F2(1)  1
The iteration is continued until u(n+1) is acceptably close to u(n) and c(n+1) is acceptably
close to c(n).
Note: the iterative scheme is not always guaranteed to converge!
References
Dietrich, W. E. 1982 Settling velocity of natural particles. Water Resources Research,
18(6), 1615-1626.
Garcia, M. and Parker, G. 1991 Entrainment of bed sediment into suspension. J.
Hydraul. Engrg., ASCE, 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.
Smith, J. D. and McLean, S. R. 1977 Spatially averaged flow over a wavy surface. J.
Geophys. Res., 82(2), 1735-1746.
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