RELATIONS FOR THE ENTRAINMENT AND 1D TRANSPORT OF SUSPENDED SEDIMENT

1
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
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,

• 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.

3
THE STRATEGY contd.

• For equilibrium open-channel suspensions,
• 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
  ltlt 1.
• 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.
• 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).

4
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. The reference height is
evaluated at what the authors describe as the top
of the bedload layer where ks denotes 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?RgD?c, where ?c is given by the Brownlie
(1981) fit to the Shields relation
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 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
6
ENTRAINMENT RELATIONS FOR UNIFORM MATERIAL contd.
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
7
ENTRAINMENT RELATIONS FOR SEDIMENT MIXTURES
Garcia and Parker (1991) generalized their
relation to sediment mixtures. The relation for
mixtures takes the form 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
8
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,
The critical boundary shear stress ?bc is
evaluated using bed material D50 again the
Brownlie (1981) fit to the Shields curve is
suggested here.
9
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 or thus
10
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. By definition, then,
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
11
AVERAGING OVER TURBULENCE contd.
Recalling that vs is a constant, substituting the
decompositions into the equation of mass
conservation of suspended sediment results
in Now for example so that the final
form of the averaged equation is
12
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
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
13
LOCAL STREAMWISE MOMENTUM CONSERVATION contd.
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
stress generated by the turbulence the term
is known as the Reynolds flux of
streamwise momentum in the upward direction. For
fully turbulent flow, the Reynolds stress ?Rzx
is usually far in excess of the viscous stress
, which can be dropped.
14
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
or thus Integrating this equation under the
condition of vanishing shear stress at the water
surface z H yields the result
Depth-slope product!
Linear distribution of shear stress!
15
REYNOLDS FLUX OF SUSPENDED SEDIMENT
The terms 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).
16
REYNOLDS FLUX OF STREAMWISE MOMENTUM
The shear stress , or
equivalently the Reynolds flux 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 flux
is maintained by maintaining a streamwise
momentum profile that has high velocity in the
upper part of the flow and low velocity in the
lower part of the flow. This in turn generates
a negative value of and a positive
value of .
17
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 Reynolds flux of
streamwise momentum in the z direction 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.
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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
19
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 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
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
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.
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.
23
AND NOW ITS TIME FOR SPREADSHEET FUN!! Go to
RTe-bookRouseSpreadsheetFun.xls
This spreadsheet allows calculation of the
suspended sediment profile from specified values
of b/H, vs and u using the Rouse-Vanoni profile.
24
1D SUSPENDED SEDIMENT TRANSPORT RATE FROM
EQUILIBRIUM SOLUTION
The volume suspended sediment transport rate per
unit width is qs computed as
In order to perform the calculation, however, it
is necessary to know the velocity profile
over a bed which may include bedforms. This
velocity profile may be specified as
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
25
1D SUSPENDED SEDIMENT TRANSPORT RATE FROM
EQUILIBRIUM SOLUTION
It follows that qs is given by the relations
The integral is evaluated easily enough using a
spreadsheet. This is done in the next chapter.
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.

• 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.

27
THE 1D PICKUP PROBLEM contd.
Governing equation
Boundary conditions
Can be used to find adaptation length Lsr for
suspended sediment
Solution yields the result that
A method for estimating Lsr is given in Chapter
21.
28
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 with E computed
based on local flow conditions, or with qs
computed 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 Ls required for the flow to reach a
quasi-equilibrium suspension. If L lt Ls the
former formulation should be used. If L gt Ls the
latter formulation can be used.
Selenga Delta, Lake Baikal, Russia image from
NASA https//zulu.ssc.nasa.gov/mrsid/mrsid.pl
29
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
Well, somewhere down there
Here ?susp density of the suspension and ?e
fractional excess density due to the presence of
suspended sediment.
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
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
where Ri denotes the gradient Richardson Number
Then
31
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
The balance equations and boundary conditions
take the forms
These relations may be solved iteratively for
concentration and velocity profiles in the
presence of stratification.
32
SUSPENSION WITH SELF-STRATIFICATION GELFENBAUM-SM
ITH FORMULATION
The workbook RTe-bookSuspSedDensityStrat.xls
implements the formulation for stratification-medi
ated suppression of mixing due to Gelfenbaum and
Smith (1986)
It also uses the specification b 0.05 H. The
balance equations and boundary conditions take
the forms
These relations may be solved iteratively for
concentration and velocity profiles in the
presence of stratification. The workbook
RTe-bookSuspSedDensityStrat.xls provides a
numerical implementation.
33
ITERATION SCHEME
The governing equations for flow velocity and
suspended sediment concentration can be
integrated to give the forms
where
The relations of the previous slide can be
rearranged to give
The iteration scheme is commenced with the
logarithmic velocity profile for velocity and the
Rouse-Vanoni profile for suspended sediment
where the superscript (0) denotes the 0th
iteration (base solution).
34
ITERATION SCHEME contd.
The iteration then proceeds as
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. More details
about the formulation are provided in the
document Rte-bookSuspSedStrat.doc.
35
INPUT VARIABLES FOR Rte-bookSuspSedDensityStrat.xl
s
The first step in using the workbook is to input
the parameters R1 (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 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 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 profiles. Note the iterative scheme
may not always converge!
36
SAMPLE CALCULATION (a) with Garcia-Parker
entrainment relation
qs with stratification 0.72 x qs without
stratification
Stratification neglected
Stratification included
Stratification included
Stratification neglected
37
SAMPLE CALCULATION (b) with Garcia-Parker
entrainment relation
qs with stratification 0.39 x qs without
stratification
Stratification neglected
Stratification included
Stratification included
Stratification neglected
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. García, 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 Hydraulic Engineering, 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. Wright, S. and G. Parker, 2004, Flow
resistance and suspended load in sand-bed
rivers simplified stratification model, Journal
of Hydraulic Engineering, 130(8), 796-805.