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10
th
International Conference on Hydroinformatics
HIC 2012, Hamburg,
GERMANY
APPICABILITY OF ENTR
AINMENT
FORMULAS
TO
MODELLING OF
GEOMORPHIC
OUTBURST
FLOWS
M
.
GUAN
(1)
, N.G. WRIGHT
(1)
, P.A. SLEIGH
(1)
(1):
School of Civil Engineering, University
of
Leeds, Leeds,
LS2
9JT,
UK
This paper presents a
non

capacity coupled model of dam

break induced geomorphic flows
which couples hydrodynamic model, morphodynamic model and bed evolution.
A
n upwind
Godunov

type scheme with a modified Ro
e
’s Riemann Solver
is
implemented
to solve the
hyperbolic system.
As the key closure equation for non

equilibrium sediment transport
model,
a
wide range of
previous
empirical, semi

empirical or theoretical
entrainment
functions are summarized and
their performance
is
investigated
by
appl
ication
to three
dam

break flow
cases
with
a range of
different
S
hields
’
parameter
s
.
The sensitivity
of
the
parameters
in
the
entrainment formulas
is
also evaluated
.
M
any of
the exist
ing
entrainment
formulas have rarely been verified thoroughly due to
diff
icult
y of
v
alidation
in the situation
of high

magnitude
outburst
flow
s
.
For non

capacity model
s
of dam

break transient flows,
an appropriate and applicable entrainment relationship should be selected and applied
thoroughly according to different hydraulic
conditions.
INTRODUCTION
Dam

break flows have
long
been the subject of scientific and technical research for many
hydraulic scientists and engineers.
In
the natural state,
such
outburst flows
can
usually
induce severe erosion and deposition
which interact with
the
water
flow
.
I
n
an attempt to
understand th
e
behaviour of such flows
,
several
models
of
dam

break
flow
over movable
beds
have been established
[
1

7
]
.
For
a
non

equilibrium model, i
t is important to determine
the
net flux
well, irrespective of whether the flow is
1D or 2D
.
F
or
example
, in
the
models
of
[
1
,
2
,
6
,
8
]
, the entrainment function is key for closing
the
governing
equations.
However, the understanding
of
sedime
nt transport is still limited such
that
the
entrainment
rate
determination
relies
on empirical relationships
in most circumstances.
In
previous
work
entrainment relations
are usually expressed by multiplying near

bed equilibrium
concentration by sediment sett
l
ing velocity.
Therefore,
the near

bed
equilibrium
concentration is widely
utilised
in
determining
the entrainment rate
. A plethora of studies
have
extensively investigated
this rate
either
theoretically
or
empirically
[
9

16
]
. Such
relations
hips
mainly
account for
the average effects of turbulent shear flow on
the
bed
surface
and involve some empirical parameters which
give
them
potentially
high
uncertainty
.
Other authors propose
empirically or semi

empirically
coupling equations
between
flow and sediment transport
[
5
,
8
]
.
Other
s have
theoretically derived the
entrainment function
with consideration of
the effects of turbulent burst
[
17
]
or based
on
kinetic theory
[
18
]
.
To our knowledge, no study has been
undertaken
to investigate the performance of
entrainment functions
in
outburst geomorphic flows.
Therefore the aim of the
paper
is
to
investigate the suitability and applicability of exist
ing
entrainment function
s
to
modelling
outburst
geomorphic
flows.
In
practice
, it is difficult to distinguish suspended and bed load
where they co

exist; therefore,
suspended dominant total load
is accounted for
via a
morphological model
. A
1D shallow water
model
which couples water
flow,
sediment
transport
and bed evolution
is
presented
and
solved by
an upwind
Godunov

type scheme
with a
R
oe
’s Riemann Solver
.
Lee & Wright’s source term treatment
[
3
]
is adapted to deal
with
irregular
bed geometry.
FORMULATION OF COMPUTATION MODEL
The
hydrodynamic
governing equations are the
one

dimensional Shallow Water Equatins
,
in which
the mass and momentum exchange between flow phase and sediment phase
are
accounted for.
They can be expressed
as follows.
t
Z
x
Q
t
A
b
)
(
(1)
t
Z
A
Q
t
C
b
gA
S
S
gh
b
gA
A
Q
x
t
Q
b
m
m
m
w
s
f
b
0
2
2
2
)
(
2
/
2
(2)
where
ρ
w
=
clear water density
;
ρ
s
=
sediment
particle
density;
ρ
m
=
density of water
and
sediment mixture
,
ρ
m
=(1

C) ρ
w
+Cρ
s
;
ρ
0
is the density of saturated bed material
ρ
0
=(1

p)ρ
s
+pρ
w;
p
is the sediment porosity;
C
is the volumetric sediment concentration;
Z
b
is bed
elevation;
S
b
is
bed slope term and
S
f
is
friction slope term
.
The
morphodynamic governing
equation
and bed
elevation equation can be
written
as follows
.
)
(
)
(
D
E
b
x
QC
t
AC
(3)
0
)
(
x
QC
t
w
ith
AC
Z
p
b
)
1
(
(
4
)
w
here
b
is the channel width;
E
and
D
are
the entrainment flux
and
deposition flux
of
sediment particles
respectively
.
To close the system, some parameters
need to
be
determined.
Firstly, the friction slope is determined by
the
Manning
equation
;
3
/
4
2
/


h
u
u
n
S
f
. T
he relation
proposed by
Cao (2004)
is used for
the
deposition flux:
m
C
C
D
1
0
; w
here
the
empirical coefficient
α=min (2, (1

p)/C);
m=2;
ω
0
is setting
velocity of
single
sediment particle
which can be
determined
by the equation
of
van Rijn
(1984)
.
For
the
irregular
bed slope
source
term, t
he homogeneous

form method of Lee &
Wright (2010
) is
ado
pted.
The
compact form
of Eq
s
. (1,2,3,4)
can be
written
as
ad
S
x
H
t
U
where
R
F
H
w
here
R represents the flux vector relative to the source terms, which can drive or
impede the flow of water.
The
modified
Shallow Water Equations
can be numerically
discretised
by using the FVM
.
k
i
ad
i
i
k
i
k
i
tS
H
H
x
t
U
U
,
*
2
/
1
*
2
/
1
1
I
n order to
maintain
stability of
the
numerical model, the well

known Courant

Friedrichs

Lewy (
0<CFL
≤
1
) stability condition is applied
in this study.
REVIEW OF SELECTED ENTRAINMENT FUNCTIONS
In this
paper, the typical or classical and high
ly
cited entrainment formul
as are selected;
they include:
Engelund
&
Fredsoe (1976)
(
EF
)
; Smith
&
McLean (1977)
(
SM
)
; Van Rijn
(1984)
(
vR
)
; Garcia
&
Parker (1991)
(
GP
)
; Zyserman
&
Fredsoe (1994)
(
ZF
)
; Cao (1999);
Sun
&
Parker (2005)
(
SP
)
; Zhong
&
Wang (2011)
(
ZW
)
; Li
&
Duffy (2011)
(
LD
)
.
Some
formulas are expressed by multiplying near

bed equilibrium concentration by sediment
setting velocity as
E=w
0
c
ae
; which involve
EF,
SM,
vR,
GP
and Z
F
.
T
hey are summarized
and catalogued in
Table 1
.
In
the
formul
as
,
c
ae
= the near

bed equilibrium concentration;
R
=
(ρ
s
/ρ
w

1)
;
u
*
’
=bed shear velocity related to grain friction;
u*
c
=critical bed

shear
velocity for initial motion of sediment;
v
=kinematic viscosity of water;
R
p
= sediment
particle Reynolds number;
d
= sediment particle diameter;
IF= impact factors.
Table 1. The
catalogue and
formulas of selected entrainment functions
F
unction
s
Entrainment formulas
Specification
EF
3
1
)
1
/(
65
.
0
b
ac
c
;
5
.
0
)
1
(
027
.
0
6
/
06
.
0
R
p
b
β=1;
IF
: only
θ.
SM
0
0
0
0
1
65
.
0
T
T
c
ac
c
c
T
0
γ
0
= 2.4 X 10

3
;
IF
: only
θ.
vR
3
.
0
*
5
.
1
015
.
0
d
T
a
d
c
ac
;
2
*,
2
*,
2
'
*
)
(
)
(
)
(
cr
cr
u
u
u
T
;
3
/
1
2
*
v
gR
d
d
a
=max(2d,0.005h);
IF
:
u
*
’
.
GP
)
3
.
0
/
1
/(
5
5
u
u
ac
AZ
AZ
c
;
s
n
p
u
v
R
u
Z
/
'
*
A=1.3*10

7; n=0.6;
IF
:
u
*
’
;
R
p
;
ZF
46
.
0
/
)
045
.
0
(
331
.
0
1
)
045
.
0
(
331
.
0
75
.
1
75
.
1
ac
c
IF
: only
θ
;
C
b,max
=0.54
when
θ>2
Cao
h
dU
p
R
E
c
c
)
(
)
1
(
160
8
.
0
U
=7u/6;
IF
:
θ
;
d
;
u
;
h
.
ZW
/
)
/
1
/
1
)(
4
/(
3
*
8
/
9
1
2
L
s
w
e
C
u
p
c
E
s
w
D
m
c
D
C
p
t
3
2
2
p = 0.94; c
µ
=0.09;
c
m
=0.6;
IF
:
θ;
R
p
;
C
L
.
SP
n
t
c
t
u
u
E
1
)
(
2
;
2
/
1
f
c
t
C
Rgd
u
1.5< n< 2.0;
IF
:
θ;
a
t
.
LD
2
2
)
(
v
u
h
E
c
in 2D model
IF
:
θ;
a
;
u
;
h
.
NUMERICAL MODEL PERFORMANCE
A
test is performed for assessing the accuracy of numerical model
of the hydrodynamic
equations only
.
T
he
experiment
of
dam

break flow over a triangular hump
(
EU CADAM
project
)
is reproduced
.
T
he triangular hump
is
symmetric
with
0.4 m
heig
ht
.
The
h
initial
=
0.75
m
before dam; h=0m at the
downstream
with a free open outlet
.
6
monitoring gauges are set
up located at 2 m
, 4 m, 8 m, 10 m, 11 m and 13 m
downstream of the dam.
The
Manning
coefficient
n
is set equal to
0
.
0125 throughout the domain. Simulation
is
run for 90 s
and
the comparisons
between
simulated
results
and
experiment
al
data
are
depicted in Fig
ure
.1
,
which
show the predictions of arriv
al
time and water depth
agree
with measurement data in
gauge
2, 4,
6
.
This test
indicates
the
good
stability and efficiency of proposed numerical
model to
simulate outburst
flow over irregu
lar bed and
wetting/drying case.
Figure
1
.
Dam

break
flow
over a hump:
comparison between simulated and measure
water
depth
against
time histories of
at
gauge
2, 4 and 6
RESULTS AND COMPARISONS
Initial simulation conditions
T
he entrainment equations
are close
ly
related to
Shields parameter
which
can be given by
gd
w
s
b
)
/(
.
S
everal ranges of Shields parameters are accounted for by setting up
different initial conditions and so enabling the analys
is
of the applicability of the
entrainment functions as a result of high shear stress triggered by outburst
geomorphic
flows. In this paper, three cases are
produced
by:
i)
h
initial
=
1m
before dam
;
h
down
=0m;
the
approximate range of
θ
is around 1.
ii)
h
initial
=2
m
before dam
;
h
down
=0m;
t
he approximate
range of
θ
is 1

2.
iii
) The last
belongs to
a high
bed shear stress case; h
initial
=
1
0
m
;
h
down
=0m;
t
he maximum
θ
can reach about 20. The performance of ea
ch formula is investigated in
terms of bed profile and water surface as well as the arrival time of the wave front.
Comparisons and analysis
Case 1
:
The
simulated
results
of this case
are displayed in Fig
.
2
which
shows that the
comparison goes well for the classical empirical formulas of EF, SM, GP, ZF and SP. The
location of bores and wave fronts is very close
for these methods
. However, for van Rijn’s
function, simulation results seems to have different features
, more specifically, there are
two scour holes that appear in the prediction of bed scour and two bores in water surface,
especially in longer time scales. There is a similar tendency with simulation results of
Cao’s formulas.
F
or Z
hong and
W
ang
’s
function, the bed scour is predicted to be very
small compared with that of the others so that water front runs faster with lower sediment
concentration. It is clear that Cao’s function prediction of the bed erosion is the most severe
and two scour holes o
ccur in bed profile, particularly in long term simulation
[
1
]
, w
hile
Z
hong and
W
ang
’s kinetic theory

based function has the opposite pre
diction. The
comparison indicates that selection of entrainment function in modelling
can have
a
significant impact on simulation results.
Fig
ure
2
.
a
for the results of EF, SM, GP, ZF, SP and vR’s formulas at t=7s;
b
for water
surface and bed profile of vR, GP, Cao and ZW’s formulas;
c
for sediment concentration
Case 2:
According to the experimental
investigation
[
19
]
,
an
increas
e
of
θ
will
increase
sediment particles
’
suspension intensity.
Compared with case 1,
the entrainment of
sediment particles
is
more severe
such
that suspension account
s
for a significant role
in
the
co

existent
flow of bed load and suspension
as
in this case
.
Here
t
hey do not have a very
good agreement
between
each oth
er
anymore
.
Several simulation results have fast
wave
front
s and relative
ly
small
bed erosion. By virtue of performance of such formulas, t
wo
groups are separated here
(Figure 4. a)
: group 1 including water and bed profiles of SM, vR
and SP
’s formulas
; group 2 including the results of EF, GP and ZF. Group 1 and 2 both
show
a
good agreement
in water front and bed scour
, but
an
increas
e in
θ makes the
bed
erosion
of group1
more severe
than that of group 2.
In Figure 4(b)
the classical entrainment functions
are compared
with new functions
considering turbulent burst effects.
It can clearly be seen
that significant uncertainty still
exists in the application of entrainment functions.
V
an
R
ijn
’s formula predicts
the
same
wate
r front but smaller bed scour compared with Cao’s f
ormula
. The bed erosion
of
Cao’s

0.3

0.1
0.1
0.3
0.5
0.7
0.9
30
40
50
60
70
80
elevation (m)
x (m)
EF
SM
vR
GP
ZF
SP

0.3

0.1
0.1
0.3
0.5
0.7
0.9
30
40
50
60
70
80
elevation (m)
x (m)
vR
GP
Cao
ZW
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
30
40
50
60
70
80
Sediment concentration
x (m)
vR
GP
Cao
ZW
a
b
c
formula is still the most severe and the predictions of Z
hong and
W
ang
’s novel entrainment
function are the smallest in bed erosion and fastest in water front.
Fig
ure
4
.
a
)
for Simulate results
of EF, SM, GP, ZF, and SP’s formulas at t=7s
;
b
)
are t
he
prediction results of
vR, GP, Cao and ZW
’s formulas
Case 3:
Usually, for
high water head outburst flows
,
S
hields
’
parameter is much higher
and
the turbulen
ce
is very intense
.
Most of s
ediment
particles are entrained into suspension
so that sediment
transport is dominated by suspended or wash load. In
such hydraulic
condition
s
, some entrainment functions, especially the empirical formulas
which are
only
related
to Shields parameter
,
are
possibly
not
applicable
anymore
resulting in the
predicted
bed profiles
that
are unreali
stic.
The predicted
results are shown in Fig
.
5
; therein, Fig.
5
(a)
gives
the water surface and bed profile predicted by classical
empirical formulas, while
Fig.
5
(b) compares the results of v
an
R
ijn
, G
arcia
&
P
arker
, Cao and Z
hong
&
W
an
g
’s
formulas.
Fig
ure
5
.
W
ater
surface
and bed profile at t=20s
for different formulas
Uncertainty
analysis of pa
rameter
in
entrainment function
s
1)
Threshold
of initial
sediment
motion
An
exact value of the critical Shields
’
parameter remains very difficult to define with great
accuracy.
Three
common
values are
ad
o
pted
here:
0.03, 0.047 and 0.06.
To
verify
the
uncertainty
of
critical
θ
,
these
three
values
are applied
to model.
The results show that

0.5
0
0.5
1
1.5
30
40
50
60
70
80
elevation (m)
x (m)
EF
SM
vR
GP
ZF
SP

0.6

0.1
0.4
0.9
1.4
30
40
50
60
70
80
elevation (m)
x (m)
vR
GP
Cao
ZW

4

2
0
2
4
6
8
10
100
200
300
400
elevation (m)
x (m)
EF
SM
vR
GP
ZF
SP

3

1
1
3
5
7
9
100
200
300
400
elevation (m)
x (m)
vR
GP
Cao
ZW
a
b
a
b
critical
θ
has no significant influence on bed erosion for
high shear stress case
.
For low
shear stress case, the critical
θ
is a
more
sensitive
factor for numerical predictions.
2
)
Shields parameter in
the
entrainment function of ZF, SM and EF
These
three entrainment functions have a common feature

the only factor influencing
entrainment is
S
hields parameter.
In other word
s
,
they are
so sensitive to
S
hields parameter
that the predic
tion will yield unrealistic
bed profile
s
when high
θ
occur
s
–
as in
case 3.
3
)
Sediment particle diameter in
entrainment function
s
Sediment particle size
d
can influence sediment movement
.
The formulas of
Engelund &
Fredsoe
,
Smith &
McLean,
Zyserman &
Fredsoe
,
Sun &
Parker
and
Li &
Duffy
account for
effects
of
d
in
S
hields parameter
;
while
v
an
R
ijn
’s formula
incorporate
grain

friction

related bed shear stress
, except that, G
arcia
&
P
arker
takes sediment particle Reynolds
number into consideration as well
;
Z
hong
&
W
ang
considers
the effects through
S
hields
parameter and
sediment particle Reynolds number
.
Such considerations are
based on
sediment mechanism, and the simulation can express the effects of sediment size.
For Cao’
s
formula, d

0.2
appears
in
the
entrainment equation
which
makes
the equation more sens
itive
to sediment particle size. When bed material is very fine, the bed erosion
will be
quite
severe and probably over
predicted
as
seen
in
all
three cases
.
Cao (2004) only analysed the
simulation
performance in coarse
sediment
particle.
Similar discussion
s
about its over
p
rediction appear
in the literature
[
6
]
.
DISCUSSIONS AND
C
ONCLUSIONS
This paper presents a non

capacity coupled model of
outburst
geomorphic flows and
summarize
s
a
wide range of exist
ing
entrainment formulas
.
T
he
ir
performance is
evaluated
by appl
ication
to three cases of
outburst geomorphic flows
with
different range of
θ
.
Further
t
he
ir
applicability
and
parameter
sensitivity are
also
analysed.
For low
er
S
hields
parameter (θ
<
2)
case
, the classical empirical functions of
Engelund &
Fredsoe
,
Smith &
McLean
,
Garcia &
Parker
and
Zyserman &
Fredsoe
can be applicable and they can
achieve
similar results
with
no big discrepancy
.
However,
with increasing
of
θ
,
their performance
become
s
unreliable
, so they
cannot
be
recommended
for high shear stress case
s
. The
uncertainty of empirical coefficient
s
in the
Sun &
Parker
and
Li &
Duffy
’s formulas makes
them too
uncertain
to be applied
successfully so it
is
can
not recommended to
use
th
ese
in
dam

break induced geomorphic flows.
Van
R
ijn
’s formula is based on bed shear stress
related to grain skin friction that is the combination of be
d grain and form shear stresses,
m
ore impor
tantly, it has fewer uncertain
empirical parameters
and the performance is stable
in
both
low and high shear stress. Cao’s formula
might
over
predict
bed scour according to
the
comparison in this paper.
This paper analyses the performance of entrainment functions on outburst geomorphic
flows qualitatively
.
In terms of
quantitative evaluation
,
uncertainties still exist
in the
simulation results.
Q
uantitative
prediction
to such cases
will be
the future research work.
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