MORPHODYNAMICS OF RECIRCULATING AND FEED FLUMES

1
CHAPTER 22 MORPHODYNAMICS OF RECIRCULATING AND
FEED FLUMES

• Laboratory flumes have proved to be valuable
  tools in the study of sediment transport and
  morphodynamics. Here the case of flumes with
  vertical, inerodible walls are considered. There
  are two basic types of such flumes
• Recirculating flumes and
• Feed flumes
• In addition, there are several variant types, one
  of which is discussed in a succeeding slide.

Recirculating flume in the Netherlands used by A.
Blom and M. Kleinhans to study vertical sediment
sorting by dunes. Image courtesy A. Blom.
2
THE MOBILE-BED EQUILIBRIUM STATE
The flume considered here is of the simplest
type it has a bed of erodible alluvium, constant
width B and vertical, inerodible walls. The bed
sediment is covered by water from wall to wall.
One useful feature of such a flume is that if it
is run long enough, it will eventually approach a
mobile-bed equilibrium, as discussed in Chapter
14. When this state is reached, all quantities
such as water discharge Q (or qw Q/B), total
volume bed material sediment transport rate Qt
(or qt Qt/B), bed slope S, flow depth H etc.
become spatially constant in space (except in
entrance and exit regions) and time. (The
parameters in question are averaged over bedforms
such as dunes and bars if they are present.)
3
THE EQUILIBRIUM STATE REVIEW OF MATERIAL FROM
CHAPTER 14
The hydraulics of the equilibrium state are those
of normal flow. Here the case of a plane bed (no
bedforms) is considered as an example. The bed
consists of uniform material with size D. The
governing equations are (Chapter 5)
Water conservation
Momentum conservation
Friction relations where kc is a composite
bed roughness which may include the effect of
bedforms (if present).
Generic transport relation of the form of
Meyer-Peter and Müller for total bed material
load where ?t and nt are dimensionless constants
4
THE EQUILIBRIUM STATE REVIEW contd.
In the case of the Chezy resistance relation, for
example, the equations governing the normal state
reduce to
In the case of the Manning-Stickler resistance
relation, the equations governing the normal
state reduce with to
Let D, kc and R be given. In either case above,
there are two equations for four parameters at
equilibrium water discharge per unit width qw,
volume sediment discharge per unit width qt, bed
slope S and flow depth H. If any two of the set
(qw, qt, S and H) are specified, the other two
can be computed.
5
THE RECIRCULATING FLUME
In a recirculating flume all the water and all
the sediment are recirculated through a pump.
The total amounts of water and sediment in the
system are conserved. In addition to the
sediment itself, the operator is free to specify
two parameters in operating the flume the water
discharge per unit width qw and the flow depth H.
The water discharge (and thus the discharge
per unit width qw) is set by the pump setting.
(More properly, what are specified are the
head-discharge relation of the pump and the
setting of the valve on the return line, but in
many recirculating systems flow discharge itself
can be set with relative ease and accuracy.)
The constant flow depth H reached at equilibrium
is set by the total amount of water in the
system, which is conserved. Increasing the total
amount of water in the system increases the depth
reached at final mobile-bed equilibrium.
Thus in a recirculating flume, equilibrium qw and
H are set by the flume operator, and total volume
sediment transport rate per unit width qt and bed
slope S evolve to equilibrium accordingly.
6
THE FEED FLUME
In a feed flume all the water and all the
sediment are fed in at the upstream and allowed
to wash out at the downstream end. Water is
introduced (usually pumped) into the channel at
the desired rate, and sediment is fed into the
channel using e.g. a screw-type feeder at the
desired rate. In addition to the sediment
itself, the operator is thus free to specify two
parameters in the operation of the flume the
water discharge per unit width qw and the total
volume sediment discharge per unit width qt
reached at final equilibrium, which must be equal
to the feed rate qtf.
Thus in a feed flume, equilibrium qw and qt are
set, by the flume operator, and equilibrium flow
depth H and bed slope S evolve accordingly.
7
A HYBRID TYPE THE SEDIMENT-RECIRCULATING,
WATER-FEED FLUME
In the sediment-recirculating, water-feed flume
the sediment and water are separated at the
downstream end. Nearly all the water overflows
from a collecting tank. The sediment settles to
the bottom of the collecting tank, and is
recirculated with a small amount of water as a
slurry. The water discharge per unit width qw
is thus set by the operator (up to the small
fraction of water discharge in the recirculation
line). The total amount of sediment in the flume
is conserved. In addition, a downstream weir
controls the downstream elevation of the bed.
The combination of these two conditions
constrains the bed slope S at mobile-bed
equilibrium. Adding more sediment to the flume
increases the equilibrium bed slope.
Thus in a sediment-recirculating, water-feed
flume, qw and S are set by the flume operator and
qt and H evolve toward equilibrium accordingly.

8
MORPHODYNAMICS OF APPROACH TO EQUILIBRIUM

• The final mobile-bed equilibrium state of a flume
  is usually not precisely known in advance. Flow
  is thus commenced from some arbitrary initial
  state and allowed to approach equilibrium. This
  motivates the following two questions
• How long should one wait in order to reach
  mobile-bed equilibrium?
• What is the path by which mobile-bed
  equilibrium is reached?
• It might be expected that the answer to these
  questions depends on the type of flume under
  consideration. Here two types of flumes are
  considered a) a pure feed flume and b) a pure
  recirculating flume.
• In performing the analysis, the following
  simplifying assumptions (which can easily be
  relaxed) are made
• The flow is always assumed to be subcritical in
  the sense that Fr lt 1.
• The channel is assumed to have a sufficiently
  large aspect ration B/H that
• sidewall effects can be neglected.
• Bed resistance is approximated in terms of a
  constant resistance coefficient
• Cf, so that the details of bedform
  mechanics are neglected.
• The sediment has uniform size D.
• The analysis presented here is based on Parker
  (2003).

9
THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING
FLUMES
In the world of sediment flumes, there is a
persistent legend concerning recirculating flumes
that is rarely documented in the literature.
That is, these flumes are said to develop
sediment lumps that recirculate round and
round, either without dissipating or with only
slow dissipation. The author of this e-book has
heard this story from T. Maddock, V. Vanoni and
N. Brooks. One of the authors graduate students
encountered these lumps in a recirculating,
meandering flume and showed them to the author
(Hills, 1987).
10
THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING
FLUMES
In the world of sediment flumes, there is a
persistent legend concerning recirculating flumes
that is rarely documented in the literature.
That is, these flumes are said to develop
sediment lumps that recirculate round and
round, either without dissipating or with only
slow dissipation. The author of this e-book has
heard this story from T. Maddock, V. Vanoni and
N. Brooks. One of the authors graduate students
encountered these lumps in a recirculating,
meandering flume and showed them to the author
(Hills, 1987).
11
THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING
FLUMES
In the world of sediment flumes, there is a
persistent legend concerning recirculating flumes
that is rarely documented in the literature.
That is, these flumes are said to develop
sediment lumps that recirculate round and
round, either without dissipating or with only
slow dissipation. The author of this e-book has
heard this story from T. Maddock, V. Vanoni and
N. Brooks. One of the authors graduate students
encountered these lumps in a recirculating,
meandering flume and showed them to the author
(Hills, 1987).
12
THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING
FLUMES
In the world of sediment flumes, there is a
persistent legend concerning recirculating flumes
that is rarely documented in the literature.
That is, these flumes are said to develop
sediment lumps that recirculate round and
round, either without dissipating or with only
slow dissipation. The author of this e-book has
heard this story from T. Maddock, V. Vanoni and
N. Brooks. One of the authors graduate students
encountered these lumps in a recirculating,
meandering flume and showed them to the author
(Hills, 1987).
13
THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING
FLUMES
In the world of sediment flumes, there is a
persistent legend concerning recirculating flumes
that is rarely documented in the literature.
That is, these flumes are said to develop
sediment lumps that recirculate round and
round, either without dissipating or with only
slow dissipation. The author of this e-book has
heard this story from T. Maddock, V. Vanoni and
N. Brooks. One of the authors graduate students
encountered these lumps in a recirculating,
meandering flume and showed them to the author
(Hills, 1987).
14
PARAMETERS

• x streamwise coordinate
• t time
• H H(x, t) flow depth
• U U(x, t) depth-averaged flow
• velocity
• ? ?(x, t) bed elevation
• S - ??/?x bed slope
• g gravitational acceleration
• qt volume bed material sediment transport
  rate per unit
• width
• qw UH water discharge per unit
• width
• ?b boundary shear stress at bed
• L flume length
• B flume width
• D sediment size
• ?p porosity of bed deposit of sediment

15
KEY APPROXIMATIONS AND ASSUMPTIONS

• Flume has constant width B.
• Sediment is of uniform size D.
• H/B ltlt 1 flume is wide and sidewall effects
  can be neglected.
• Flume is sufficiently long so that entrance
  and exit regions can be neglected.
• Flow in the flume is always Froude- subcritical
  Fr U/(gH)1/2 lt 1.
• qt/qw ltlt 1 volume transport rate of sediment
  is always much lower than that of water.
• Resistance coefficient Cf is approximated
• as constant.

16
GOVERNING EQUATIONS 1D FLOW
Flow mass balance
Flow momentum balance
Sediment mass balance
Closure relation for shear stress Cf
dimensionless bed friction coefficient
The condition qt/qw ltlt 1 allows the use of the
quasi-steady approximation introduced in Chapter
13, according to which the time-dependent terms
in the equations of flow mass and momentum
balance can be neglected.
17
REDUCTION TO BACKWATER FORM
The equations of flow mass and momentum balance
reduce to the standard backwater equation
introduced in Chapter 5.
or thus
where
18
SEDIMENT TRANSPORT RELATION
Sediment transport is characterized in terms of
the same generic sediment transport relation as
used in Chapter 20, except that the parameter ?s
is set equal to unity. Thus where
D grain size (uniform) ?s sediment
density ? water density R (?s/ ?) 1 ?
1.65
Einstein number
Shields number
19
CONSTRAINTS ON A RECIRCULATING FLUME
Water discharge qw is set by the pump.
The total amount of water in the flume is
conserved. With constant width, constant storage
in the return line and negligible storage in the
entrance and exit regions (L sufficiently large),
the constraint is (where C1 is a constant) At
final equilibrium, when H Ho, the constraint
reduces to HoL C1, according to which Ho is set
by the total amount of water.
The total amount of sediment in the flume is
conserved. Neglecting storage in the return line
and the head box, the constraint is (where C2 is
another constant)
Integrate the equation of sediment mass
conservation
to get
So a cyclic boundary condition is obtained
But from above
20
Three constraints
CONSTRAINTS ON A FEED FLUME
Water discharge qw is set by the pump.
The upstream sediment discharge is set by the
feeder. Where qtf is the sediment feed rate
Let ? ? H denote water surface elevation. The
downstream water surface elevation ?d is set by
the tailgate
The long-term equilibrium approached in a
recirculating flume (without lumps) should be
dynamically equivalent to that obtained in a
sediment-feed flume (e.g. Parker and Wilcock,
1993).
21
MOBILE-BED EQUILIBRIUM
The equation for water conservation reduces to
At mobile-equilibrium the equation of momentum
balance reduce to the relation for normal flow
introduced in Chapter 5
The sediment transport relation reduces to the
form which applies whether or not
mobile-bed equilibrium is reached.
Let R, g, D, ?t, nt, ?c and Cf be specified.
The equations in the boxes define three equations
in five parameters Uo, Ho, qto, qw and So at
mobile-bed equilibrium.
22
MOBILE-BED EQUILIBRIUM contd.
Recirculating flume Water discharge/width qw is
set by pump. Total amount of water Vw in system
is conserved. Assuming constant storage in
return line and neglecting entrance and exit
storage, Vw HLB ? depth Ho is set. Solve three
equations for qto, U0, So.
The subscript o denotes mobile-bed equilibrium
conditions.
Feed flume Water discharge/width qw is set by
pump. Sediment discharge/width qto is set by
feeder. Solve three equations for uo, Ho, So.
23
APPROACH TO EQUILIBRIUM NON-DIMENSIONAL
FORMULATION
Dimensionless parameters describing the approach
to equilibrium (denoted by the tilde or downward
cup) are formed using the values Ho, qto, So
corresponding to normal equilibrium.
Bed elevation ? is decomposed into into a
spatially averaged value and a deviation
from this ?d(x, t), so that by definition
The above two parameters are made dimensionless
as follows
From the above relations,
the dimensionless flume number
where
24
APPROACH TO EQUILIBRIUM contd.
The dimensionless relations governing the
approach to equilibrium are thus as follows
where Fro and denote the Froude and Shields
numbers at mobile-bed equilibrium, respectively,
The backwater relation is
The relation for sediment conservation decomposes
into two parts
The sediment transport relation is
where
25
APPROACH TO EQUILIBRIUM IN A FEED FLUME
In a feed flume, the boundary condition on the
sediment transport rate at the upstream end is
, or in dimensionless
variables, Thus the relations for sediment
conservation of the previous slide reduce to
The boundary condition on the backwater equation
is where ?d is a constant downstream elevation
set by a tailgate. This condition must hold at
all flows, including the final mobile-bed
equilibrium. Now the datum for elevation is set
(arbitrarily but conveniently) to be equal to the
bed elevation at the center of the flume (x 0.5
L) at equilibrium, so that ?ao 0 and Between
the above two relations and the
nondimensionalizations, it is found that
26
APPROACH TO EQUILIBRIUM IN A FEED FLUME SUMMARY
The equations
must be solved with the sediment transport
relation and boundary conditions
and a suitable initial condition, e.g. where SI
is an initial bed slope and ?aI is an initial
value for flume-averaged bed elevation, , ?a
?aI, ?d SIL0.5 (x/L), or in dimensionless
terms.
Note furthermore that ?aI ( ) and SI must be
chosen so as to yield subcritical flow in the
flume at t 0 ( ).
27
APPROACH TO EQUILIBRIUM IN A FEED FLUME FLOW OF
THE CALCULATION
In the case of a feed flume, the calculation
flows directly with no iteration. At any time
(e.g. ) the bed elevation profile, e,g,
the parameters and , are known. The
backwater equation is then solved for using
the standard step method of Chapter 5 upstream
from the downstream end, where the boundary
condition is Once is known everywhere,
is obtained everywhere from the sediment
transport relation of the previous slide. The
bed elevation one time step later is determined
from a discretized version of
where the second equation above is solved subject
to the boundary condition
28
APPROACH TO EQUILIBRIUM IN A RECIRCULATING FLUME
As shown at the bottom of Slide 19, the sediment
transport in a recirculating flume must satisfy a
cyclic boundary condition. In dimensionless
terms, this becomes As a result, the relations
for sediment conservation of Slide 24 reduce to
The total amount of water in the flume is
conserved. Evaluating the constant in the
equation at the top right of Slide 19 from the
final equilibrium, then, In dimensionless
form, this constraint becomes
29
APPROACH TO EQUILIBRIUM IN A RECIRCULATING FLUME
SUMMARY
The equation for can be dropped because
flume-averaged bed elevation cannot change in a
recirculating flume. The remaining backwater and
sediment conservation relations
must be solved with the sediment transport
relation and constraints
The initial condition is
where the initial slope SI must be chosen so as
to yield subcritical flow everywhere.
30
APPROACH TO EQUILIBRIUM IN A RECIRCULATING FLUME
FLOW OF THE CALCULATION
The method of solution is similar to that for the
feed flume with one crucial difference iteration
is required to solve the backwater equation over
a known bed subject to the integral
condition That is, for any guess
it is possible to solve the backwater equation
and test to see if the integral condition is
satisfied. The value of necessary to satisfy
the backwater equation can be found by trial and
error, or as shown below, by a more systematic
set of methods.
31
RESPONSE OF A RECIRCULATING FLUME TO AN INITIAL
BED SLOPE THAT IS BELOW THE EQUILIBRIUM VALUE
The low initial bed slope causes the depth to be
too high upstream. Total water mass in the flume
can be conserved only by constructing an M2 water
surface profile. The result is a shear stress,
and thus sediment transport rate at the
downstream end that is higher than the upstream
end. This sediment is immediately recirculated
upstream, where it cannot be carried, resulting
in bed aggration there.
32
RECIRCULATING FLUME ITERATIVE SOLUTION OF
BACKWATER EQUATION
The shooting method is combined with the
Newton-Raphson method to devise a scheme to solve
the backwater equation iteratively. Now for each
guess it is possible to solve the backwater
equation for , so that in general the
solution can be written as Now define the
function such that The correct value
of is the one for which the integral
constraint is satisfied, i.e.
33
ITERATIVE SOLUTION OF BACKWATER EQUATION contd.
Now according to the Newton Raphson method, if
is an estimate of the solution for , then
a better estimate (where w 1, 2, 3 is
an iteration index) is given as Reducing this
relation with the definition of ?, results in
the iteration scheme This iterative scheme
requires knowledge of the parameter
.
34
ITERATIVE SOLUTION OF BACKWATER EQUATION contd.
Now define the variational parameter Hv as
The governing equation and boundary condition of
the iterative scheme are
Taking the derivative of both equations with
respect to results in the variational
equation and boundary condition below
35
ITERATIVE SOLUTION OF BACKWATER EQUATION contd.
For any guess , then, the following two
equations and boundary conditions can be used to
find and for all values of .
The improved guess is then given as
The iteration scheme is continued until
convergence is obtained.
36
NUMERICAL IMPLEMENTATION
Numerical implementations of the previous
formulation for both recirculating and feed
flumes are given in RTe-bookRecircFeed.xls. The
GUI for the case of recirculation is given in
worksheet Recirc, and the GUI for the case of
feed is given in worksheet Feed. The
corresponding codes are in Module 1 and Module
2. Both these formulations use a) a
predictor-corrector method to compute backwater
curves, and b) pure upwinding to compute spatial
derivatives of qt in the various Exner equations
of sediment conservation. The discretization
given below is identical to that used in Chapter
20. The ghost node for sediment feed is
not used, however, in the implementation for the
recirculating flume.
37
NUMERICAL IMPLEMENTATION contd.
The backwater equation is solved using a
predictor-corrector scheme. The solution
proceeds upstream from the downstream node M1,
where for a feed flume and for a recirculating
flume is obtained from the iteration
scheme using the Newton-Raphson and shooting
techniques. The discretized backwater forms for
dimensionless depth are In the case of a
recirculating flume, the variational parameter Hv
must also be computed subject to the boundary
condition Hv,M1 1. The corresponding forms are
38
NUMERICAL IMPLEMENTATION contd.
In the case of a feed flume, the discretized
forms for sediment conservation are In
the case of a recirculating flume, the
corresponding form is
39
CRITERION FOR EQUILIBRIUM
It is assumed that equilibrium has been reached
when the bed slope is everywhere within a given
tolerance of the equilibrium slope So. A
normalized bed slope SN that everywhere equals
unity at equilibrium is defined as For the
purpose of testing for convergence, the
normalized bed slope SN,i at the ith node is
defined as At equilibrium, then, SN,i should
everywhere be equal to unity. The error ?i
between the bed slope and the equilibrium bed
slope at the ith node is Convergence is
realized when where ?t is a tolerance. In
RTe-bookRecircFeed.xls ?t has been set equal
to 0.01 (parameter epslope in a Const
statement).
40
PHASE PLANE INTERPRETATION
One way to interpret the results of the analysis
is in terms of a phase plane. Let Sup denote the
bed slope at the upstream end of the flume, and
Sdown denote it at the downstream end,
where Further define the normalized slope SN
as equal to S/So, so that The initial
normalized slope is denoted as SNI. At mobile-bed
equilibrium slope S is everywhere equal to the
equilibrium value So, so that SN 1 everywhere.
That is, one indicator of mobile-bed equilibrium
is the equality In a phase plane
interpretation, SN,up is plotted against SN, down
at every time step, and the approach toward (1,
1) is visualized. This equilibrium point (1, 1)
is called the fixed point of the phase problem.
41
PHASE PLANE INTERPRETATION contd.
The initial condition for the bed profile is S
SI (SN SNI) everywhere, where SI is not
necessarily equal to the mobile-bed equilibrium
value So (SNI is not necessarily equal to 1) .
For example, in the case SI 0.5 So, (SN,up,
SNdown) begin with the values (0.5, 0.5) and
gradually approach (1, 1). An approach in the
form of a spiral is usually indicative of damped
sediment waves, or lumps.
42
INPUT PARAMETERS USED IN THE CODES
Input for recirculating flume
SI initial bed slope So equilibrium bed slope
Input for feed flume
43
SAMPLE CALCULATION WITH RTe-bookRecircFeed.xls
RECIRCULATING FLUME
final equilibrium
This calculation requires a dimensionless time
of 11.94 in order to reach mobile-bed
equilibrium. The spiralling is indicative of a
damped sediment wave or lump. This is shown in
more detail in the next slide.
initial
44
BED EVOLUTION IN RECIRCULATING FLUME
45
SAMPLE CALCULATION WITH RTe-bookRecircFeed.xls
FEED FLUME
final equilibrium
The input parameters are comparable to that of
the previous case of recirculation. Mobile-bed
equilibrium is reached in a dimensionless time
of only 4.33. The phase diagram shows no
spiralling.
initial
46
BED EVOLUTION IN FEED FLUME
47
APPLICATION EXAMPLE
The dimensionless numbers of the previous two
calculations, i.e. Fro 0.4, nt 1.5, Fl 10,
3, SNI 0.5 and 0 (feed case
only) are now converted to dimensioned numbers
for a sample case, for which D 1 mm, R
1.65 ks 2.5 D Ho 0.2 m (equilibrium
depth) The sediment is assumed to move
exclusively as bedload. The assumed bedload
relation, given below, is from Chapter 6. The
assumed resistance relation, given below, is from
Chapter 5. From Ho, ks and the above resistance
relation, Cf 0.00354 From Ho and Fro
0.4, qw 0.560 m2/s From 3,
?o 0.148 From ?o, D, R and the load
relation, qto 1.57 x 10-5 m2/s From ?o
HoSo/(RD), Ho, R and D, So 0.00122 From Fl
Ho/(SoL), Ho and So, L 16.3 m
48
APPLICATION EXAMPLE contd.
In the example, then, the equilibrium parameters
are qw 0.560 m2/s Ho 0.2 m So
0.00122 qto 1.57 x 10-5 m2/s ( 1250
grams/minute for a flume width B 0.5 m and the
flume length is L 16.3 m The time to
equilibrium tequil is related to the
corresponding dimensionless parameter
as Assuming the value for bed porosity ?p of
0.4, the time to equilibrium for the
rercirculating case of Slide 43 and the feed case
of Slide 45 are then Recirculating
flume tequil 41.3 hours Feed flume tequil
15.0 hours
49
SOME CONCLUSIONS
The following tentative conclusions can be
reached concerning sediment lumps in flumes.

1. Cyclic lumps do occur in recirculating flumes.
2. These lumps do eventually dissipate.
3. Similar cyclic lumps are not manifested in
   sediment-feed flumes.
4. Feed flumes reach mobile-bed equilibrium faster
   than recirculating flumes.
5. Both flume types eventually reach the same
   mobile-bed equilibrium.

50
REFERENCES FOR CHAPTER 22
Hills, R., 1987, Sediment sorting in meandering
rivers, M.S. thesis, University of Minnesota, 73
p. figures. Parker, G., 2003, Persistence of
sediment lumps in approach to equilibrium in
sediment-recirculating flumes, Proceedings, XXX
Congress, International Association of Hydraulic
Research, Thessaloniki, Greece, August 24-29,
downloadable at http//cee.uiuc.edu/people/parkerg
/conference_reprints.htm . Parker, G. and
Wilcock, P., 1993, Sediment feed and
recirculating flumes a fundamental difference,
Journal of Hydraulic Engineering, 119(11),
1192-1204.