Post on 05-Mar-2023
Proceeding of 2013 IAHR World Congress
ABSTRACT: Riverbed is likely to be damaged in the natural conditions or due to human activities,
which results in river morphology changed, and even the presence of extreme cases, such as crack or
geological fault due to earthquake. Thus, riverbed evolution is different from natural evolution owing to
flow structure around the damaged zone has changed. In order to facilitate research in this paper, the post-
damaged riverbed is simplified as a triangular pit which has a steep slope appeared on the riverbed. This
paper presents the investigation on riverbed evolution behavior in the vicinity of the pit in different flow
conditions by experiments and numerical simulations. This study mainly focuses on explaining the
mechanism of pit migration by measuring the shapes of the pit in different time during its development
and numerical simulating the flow field in corresponding moment. Two kinds of flow conditions are
included in experiments, clear-water and live-bed conditions. The experimental result under the former
condition shows that the migration of pit is not obvious; the phenomenon in the latter condition shows
that the upstream side slope of the pit moves forward and the downstream side slope is eroded, and the
retrogressive erosion to upstream is not found. A vertical Two-dimensional standard k-ε turbulent model
is used and its validation is achieved against experimental results of velocity field measured by an
acoustic Doppler velocimeter (ADV). The results of the numerical simulation show that the longitudinal
vortex in the pit hole develops to the bottom with migration of the pit and the velocity is small (<0.1m/s)
in the pit hole. The maximum value of bed-shear stress amplification appears at the upstream and
downstream ends of the pit. This paper only analyzes one post-damaged riverbed form, for other riverbed
forms in different flow and sediment conditions, this problem needs further studying.
KEY WORDS: Post-damaged riverbed, Evolution, Steep slope, Two-dimensional model, Shear stress.
1 INTRODUCTION
The post-damaged riverbed can be divided into two cases according to the causes, one is that
riverbed is damaged in the natural conditions, such as trench, crack or geological fault which usually has
a sharp longitudinal slope due to earthquake; the other case is man-made damage, such as sand and gravel
mining from riverbed for construction projects. Researches about this problem mainly focus on the latter
case, Lee et al. (1993) investigated the migration behavior of several rectangular pits of different sizes
composed of uniform bed material by a series of experiments, they distinguished two different periods in
the process of the pit migration, namely convection period and diffusion period, and empirical equations
for maximum scour depth and migration speed of the pits were obtained. Neyshabouri, Farhadzadeh &
Amini (2002) also reported an experimental study on the migration of rectangular mining pits and
variation of longitudinal profile in channel bed composed of uniform sediments, they concentrated on the
Experimental and Numerical Investigation of Riverbed Evolution in Post-Damaged Conditions Jinzhao Li
Ph.D. Student, Beijing Jiaotong University, Beijing 100044, China. Email: 12115290@bjtu.edu.cn
Meilan Qi
Professor, Beijing Jiaotong University, Beijing 100044, China. Email: mlqi@bjtu.edu.cn
Yakun Jin
Graduate Student, Beijing Jiaotong University, Beijing 100044, China. Email: 11121260@bjtu.edu.cn
2
influence of length/width ratio of the pit to the migration speed. The results showed the effect of width is
more important than the effect of length. Gill (1994) presented an approximately theoretical solution for
the behavior of the mining pits of regular geometrical shape using the one-dimensional St. Venant’s
equations of motion, but these theoretical solutions were only applicable to equilibrium condition of
sediment flow. Ribberink, Roos & Hulscher (2005) provided analytical formulas for the migration
velocity and infill time of trenches in the marine environment, using a harmonic analysis of simplified
one-dimensional basic equations for the morphodynamics system, and the results showed that damping
dominates the shortest trench, while migration and damping occur simultaneously for the long trench. Wu
& Wang (2008) simulated the flow through an initially dry mining pit and the associated morphology
changes using the generalized one-dimensional shallow water equations and nonequilibrum sediment
transport equations, the results showed the most significant erosions occur at the upstream and
downstream ends of the pit due to headcut and tailcut respectively. Azar, Namaee & Rostami (2012)
simulated the variation of bed profile due to mining pit using one-dimensional hydraulic engineering
software (HEC-RAS). Chen & Liu (2009) compared two commonly used hydrodynamic and sediment
transport model, HEC-RAS and CCHE2D (a depth-averaged two-dimensional model developed at the
University of Mississippi), and the results showed that the latter model was more robust in simulating
flood zone coverage, non-uniform sediment sorting, and channel geomorphologic changes.
In this paper, the former case, a steep longitudinal slope and small length/depth ratio of the pit, was
investigated. The post-damaged riverbed was simplified as a triangular pit appeared on the riverbed for
facilitating study. The following aims to illustrate the relationship between the hydrodynamic parameters
of the flow around the pit and the shape of the pit during its development. Two kinds of flow conditions
are considered, clear-water and live-bed condition. The riverbed evolution behaviors around the pit in
different flow conditions are presented by measuring riverbed elevation in different period. The flow field
in corresponding moment is calculated by numerical simulating (a vertical 2-D standard k-ε turbulent
model) which has been validated against experimental results of velocity field measured by an acoustic
Doppler velocimeter (ADV).
2 FLUME EXPERIMENTS
2.1 Experimental Apparatus
The experiments in this study were conducted in a circulating flume in which the slope is adjustable,
6m long, 25cm wide and 25cm deep. A false floor was installed in the flume leaving a
40cm×25cm×10cm recess in which sediments were placed, ensuring the enough thickness of sediment
used in live-bed experiment. A PVC grille at the inlet section was used to stabilize flow and an adjustable
weir at the downstream end of the flume was used to control the depth of flow. The flow discharge was
monitored using an electromagnetic meter and the accuracy is ±0.01m3/h. The water depth and bed
morphology were monitored by a non-contract ultrasonic elevation meter, the accuracy is ±0.01mm. An
acoustic Doppler velocimeter (ADV) was used which has a sampling rate and volume of 200Hz and
0.09cm3 respectively.
2.2 Calibration Test
The purpose of this test is to examine the accuracy of the experimental instruments, especially the
acoustic Doppler velocimeter (ADV). The test is a rigid-bed experiment which means there’s no layer of
sand on the bottom of the flume (smooth-bed).
The water depth was maintained at 10cm and the approach flow velocity at V=25.7cm/s which was
obtained from integration of velocity profile. The longitudinal slope of channel was fixed as 0.0002. The
test was conducted under the sub-critical (Fr=0.26) and uniform flow. The velocity measurements were
made by ADV in the longitudinal symmetrical line of the flume.
Fig. 1 shows the measured, undisturbed mean velocity profile at the measurement sections. In the
figure, u is the shear velocity,
u
(1)
3
and y
is the normalized distance from the bed,
*yuy
(2)
herein,
is the undisturbed bed shear stress, is the fluid density and is the kinematic
viscosity.
Fig. 1 illustrates the mean velocity profile can be approximated by the logarithmic law,
*
2.5ln( ) 4.5u
yu
(3)
which means there’s a good agreement between the measurement data and the theoretical result.
Figure 1 Undisturbed velocity profile for the smooth bed, *u =1.04cm/s
2.3 Migration Experiments
The purpose of these experiments are to investigate the migration of pit with sharp slope (exceeding
the angle of repose) in two flow conditions, one is clear-water condition which means the flow approach
velocity is smaller than critical velocity for the initiation of sediment motion; the other is in live-bed
regime when the shear stress induced by the water flow exceeds the critical shear stress of the bed
material.
The experimental flow conditions in this test are summarized in table 1, where i is bed longitudinal
slope; Q is the inflow discharge; u is mean velocity obtained from integration of velocity profile; cu
is the incipient velocity of sediment; Re is the Reynolds number; Fr is the Froude number; D and L are
the depth and length (along the flow direction) of the pit respectively. The bed material consisted of a
cohesionless, uniform sand with a median particle size of D50=0.6mm and a specific gravity of 2.65. The
geometric standard deviation of the bed material was σ=1.483. In this experiment, flow condition in run 1
is in clear-water regime and run 2 is in live-bed regime.
Table 1 Flow conditions for the experiments
Run i Q (m3/h) h(m) u (m/s) c
u (m/s) Re Fr D(cm) L(cm)
1 1/5000 11.27 0.06 0.21 0.24 52500 0.274 4 4
2 1/2000 13.65 0.05 0.303 0.235 15150 0.786 4.3 5
For creating the pit, a cubic metal mold with desired dimensions was used and was put into the
desired location. For prevention of moving bed materials, water with low speed entered into the flume.
Afterwards, the flow depth increased gently until to the required depth, then removed the metal mould.
The pit was located about 3m away from the flume entrance. During the experiments, the discharge was
kept constant and the channel bed elevation around the pit was measured in different moment until the
shape of the pit reached an equilibrium state.
0.01
0.1
1
10
0 10 20 30 40
y(c
m)
u(cm/s)
*/ 2.5ln( ) 4.5u u y
4
2.3.1 Clear-water experiment
Fig. 2 shows the migration of pit in clear-water condition. The result reveals the most significant
erosions occur at the upstream and downstream ends of the pit at the beginning period (0~5min) when the
transient flow through a dry pit.
After the pit hole is filled up with water (5min~55min), the migration of the pit is not obvious in
clear-water regime and the retrogressive erosion to upstream is not found. Moreover, the morphological
time scale of the evolution is small and reaches an equilibrium state soon. Another phenomenon is the
slope of the upstream side (22o) is greater than the downstream (16
o), which means the tailcut (7cm from
the pit to downstream) is greater than headcut (5cm from the pit to upstream)
Figure 2 Bed longitudinal profile and pit variation with time in clear-water condition.
2.3.2 Live-bed experiment
Live-bed experiment means that the sand on the riverbed is mobile due to the flow approach velocity
is larger than critical velocity for the initiation of sediment motion. Since no sediment was supplied from
upstream, the experiment was conducted in a nonequilibrium condition.
Fig. 3 shows that the temporal variation of the bed elevation around the pit is obvious in live-bed
conditions. The upstream slope of the pit moves forward and the downstream slope is eroded, a steep
upstream slope and a more gentle downstream slope slowly develops. This is because most of the
incoming sediment deposits in the pit hole and the water entering downstream channel is almost clear,
thus, erosion to downstream occurs and gradually extends downstream. The slope of the pit hole in the
upstream side is approximately equal to the angle of repose 32 . It can be also observed that the
lowest point of the pit does not move downstream and the behavior of infilling dominates the pit’s
development.
Figure 3 Bed longitudinal profile and pit variation with time in live-bed condition
Fig. 4 shows that the variation of bed elevation at the lowest point of the pit (x=0) with time. It can
be observed that the infilling phenomenon is significant, infilling velocity is high, at the beginning period
for the transient flow through a dry pit (0~1min). After the pit hole is filled up with water (1min~100min),
the infilling velocity is approximately constant for the relation between the bed elevation y and time t is
approximately linear.
0
1
2
3
4
5
6
7
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Bed
lev
el(c
m)
x(cm)
Orig.pit
5min
15min
35min
55min
0
1
2
3
4
5
6
-7 -5 -3 -1 1 3 5 7 9 11 13
Bed
lev
el(c
m)
x(cm)
Orig.pit
1min
5min
15min
30min
5
Figure 4 Infilling of the pit hole with time in live-bed conditions
It should be noted that retrogressive erosion to upstream was not found in this experiment. The shape
of the pit at 100 minute didn’t reach equilibrium state due to the irregular ripples were formed and the
morphology presented three-dimensional features after 100 minute.
3 NUMERICAL SIMULATIONS
3.1 Numerical Model Description
3.1.1 Governing equations
The CFD solver FLUENT used in the present study solves the vertical two-dimensional Reynolds-
averaged Navier-Stokes equations for incompressible flow,
( ) ( ) 0i j
j
u ut x
(4)
( ) ( )ij
i i j i
j i j
pu u u g
t x x x
(5)
in which t time; i
u represents velocity in i
x direction; p total pressure; i
g is gravitational
acceleration in the i direction; is the fluid density andij is stress tensor defined as
2
( )( ) ( )3
ji iij t t ij
j i i
uu uk
x x x
(6)
in which and t
, respectively represents kinematic molecular and turbulent viscosity; k turbulent
kinetic energy; and ij
is Kronecker delta.
Turbulent stresses in Reynolds-averaged equations can be closed using standard k-ε model which is
the simplest and most widely used two-equation turbulence model.
In the present study, the SIMPLE algorithm (Patankar 1980) was used to calculate the steady-state-
flow. In this algorithm, the pressure field is calculated and the velocity field is corrected so that the
continuity equation is satisfied in an iterative manner.
3.1.2 Boundary conditions
The boundaries of the computational domain (fig. 5) are inlet, outlet, symmetry boundaries and walls.
Inlet and outlet boundary conditions
(i) At the inlet, zero vertical velocity, v , was specified. The inlet profile for u , k and ε were based on
the equilibrium profiles obtained from uniform-channel flow calculations with similar flow settings.
(ii) At the outlet, the outflow which means an overall mass balance correction and zero diffusion flux
(zero normal gradients) for flow variables are defined.
Symmetry boundary
At the top surface of the computational domain, a symmetry boundary which means zero-gradient
conditions (Neumann conditions) for velocity and k and ε was applied,
0
1
2
3
4
0 20 40 60 80 100 120
Bed
ele
vat
ion
y(
cm)
Time t(min)
0.013 1.847y t
6
0, 0u k
v py y y
(7)
Wall (bed) (i) No-slip boundary condition is specified to set the velocity to be zero at the solid boundaries.
(ii) The law-of-the-wall (a series of semiempirical equations) is utilized to calculate the bottom shear
stressw
.
Figure 5 Boundary conditions
3.1.3 Computational mesh
In order to accurately calculate the bottom shear stress of the bed, the size of the cells adjacent to the
wall boundary is chosen to satisfy the limits of the wall units distance 11.225< y
<30. (Salaheldin. 2004)
The value of y
should greater than 11.225 to ensure that the nodes in the first layer of the mesh are not
located in the laminar sublayer, and then the values in the laminar sub layer are calculated by
semiemprical methods (law-of-wall) instead numerically calculating them. The value of y
should be
smaller than 30 to ensure that the law of the wall is applicable. The wall unit distance y
has been
defined in equation (2).
The grid in computational domain is unstructured and generated by a tool of GAMBIT (a grid
generation software). Most of the cells are set rectangular (2mm2mm) except the cells in vicinity of pit
are generated by the method of Y-type to the triangle shape.
3.2 Validation of Numerical Model
The numerical model has been validated against the present experiments. Three flow conditions are
considered and summarized in Table 2, where i is bed longitudinal slope; Q is the inflow discharge; u
is mean velocity obtained from integration of velocity profile; Re is the Reynolds number; Fr is the
Froude number.
Table 2 Flow Conditions for validating numerical model
Run i Q (m3/h) u(m/s) h(m) Re Fr L (cm) D (cm)
1 1/5000 11.27 0.21 0.06 52500 0.272 5 4.3
2 1/5000 23.16 0.257 0.1 64250 0.260 5 4.3
3 1/10000 7.96 0.147 0.06 36750 0.192 5 4.3
3.2.1 Qualitative validation
`Fig. 6 shows the measured velocity field around the pit by acoustic Doppler velocimeter (ADV)
compared with that obtained from the solution of numerical model. There is a good agreement between
the numerical simulations and the measurements qualitatively for the presence of vortex inside the pit
hole. However the comparisons also indicate that the center of vortex measured experimentally is located
at the downstream of the center vertical line of the pit obviously, whereas the vortex numerical simulated
is approximately located at the center of the pit hole.
7
(a) Run 1
(b) Run 2
(c) Run 3
Figure 6 Experimental (left) and numerical (right) results of velocity field in different flow conditions. The level of
velocity magnitude in numerical results is displayed in different color
3.2.1 Quantitative validation
In order to validate the results of numerical simulation quantitatively, fig. 7-9 compare the vertical
distribution of streamwise velocity Ux in different position and flow conditions between experimental
data and the numerical result. The y axis represents the distance from the measuring points to the bottom
of the pit. In each figure, (a) represents the center vertical line (x=0); (b) represents the upstream end of
the pit (x=-2.5cm); (c) represents the downstream end of the pit (x=2.5cm). The velocity at the bottom of
the pit cannot be measured due to the limitation of the ADV. It can be noted that the negative velocity
exists inside the pit which shows that the existence of vortex. The fig.7-9 ((b), (c)) show that the
numerical and experimental results agree well at the knickpoints (upstream and downstream ends) of pit.
The agreement between the present calculations and the measurements is also reasonable inside the pit
hole in run 3 conditions (fig. 9 (a)). However, in fig. 7 (a) and fig. 8 (a), for 0.02m<y<0.035m, the
velocity Ux is underpredicted by numerical model; for 0.035m<y<0.05m, the Ux is overpredicted by
numerical model, the maximum difference between the model and the experiment being more than 30%.
8
No clear explanation has been found for the observed discrepancy between the numerical model and the
experiments.
Figure 7 Vertical distribution of velocity Ux in run 1 condition.
Figure 8 Vertical distribution of velocity Ux in run 2 condition.
Figure 9 Vertical distribution of velocity Ux in run 3 condition.
In order to explain the phenomenon of riverbed evolution around the pit, numerical calculations in
different times during the pit’s migration have been conducted. The bed longitudinal profile and pit
variation with time have been measured in the flume experiment, thus the shapes of pit in different times
during its migration can be created in the numerical simulation. As the complication and irregularity of
the pit shape, the professional drawing software, AutoCAD, has been used, and then the model is
imported into the Gambit to generate grid. The numerical flow condition is the same as the experiment
condition which can be reviewed in table 1. The results and discussions are as follows.
4 RESULTS AND DISCUSSIONS
9
4.1 Clear-water Condition
Flow fields
Fig. 10 shows that the velocity fields around the pit in different times (the initial moment and the
equilibrium moment) in clear-water condition.
From the numerical calculation results, above mentioned phenomenon of riverbed evolution around
the pit can be explained. Fig. 10 shows that the presence of the vortex in the pit hole and the vortex
develops to the bottom of the hole with the pit’s migration. The velocity in the vortex region is very small
(0~0.1m/s) which can be displayed by the level color of velocity magnitude (blue), this calculation result
gives an explanation that the migration is not obvious in clear-water condition.
Figure 10 also shows that the vortex doom almost occupies the pit hole, in which the flow field is
different from some mining pit which has gentle slope and high length/depth ratio (>>1).
Figure 10 Velocity fields around the pit in clear-water condition, (a) displays the initial moment; (b) displays the
equilibrium moment.
Vorticity contours Fig. 11 exhibits the vorticity contours at the pit hole in different moment during migration. The
vorticity contours are computed from the velocity components. It is
v u
x y
(9)
where u and v represent the velocity magnitude in x axis and y axis direction respectively. In fig.
11, a right hand convection that is positive in the clockwise direction is adopted to define the vorticity.
The region of high vorticity concentrates near the upstream end of the pit during the initial stage of
migration and grows to the bottom of the pit with migration of pit. In fact, the point of the highest
vorticity is located at the point of the separated flow, which can be revealed from velocity vectors (fig.
10). Also, the magnitude of the largest vorticity decreases, from 30s-1
to 20s-1
, with the development of
pit hole. In general, the influence of the vortex on the flow field at the pit hole is weakened with the
development of the pit hole.
Figure 11 Contours of vorticity magnitude (in s-1) around the pit hole in different moment in clear-water condition. (a)
represents initial moment; (b) represents equilibrium moment.
10
Bed-shear stress
The incipient motion of sediment is mainly due to the action of flow drag force in present
experimental condition. Fig. 12 displays the bed shear stress amplification around the pit in clear-water
condition. Here the amplification of the bed-shear stress is defined by
w
(10)
in which w
is the bed-shear stress and
is the undisturbed bed-shear stress. The x axis is defined by a
dimensionless variable, x/L; here L is the pit’s length along the direction of flow.
Fig. 12 shows that the amplification of bed-shear stress is largest at the knickpoints (the upstream
and downstream ends) of the pit which explains the above phenomenon that occurs in the experiments
(headcut and tailcut erosion). This large amplification is caused by the combined effect of the contraction
of streamlines and the vortex. Fig. 12 also indicates that the amplification of the bed-shear stress is very
small inside the pit which also explains the above phenomenon in the experiment under clear-water
condition. The bed-shear stress amplification at the downstream end of the pit is greater than the upstream
end of the pit, this result reveals that the downstream end of the pit is more easily eroded in this condition.
Table 3 presents the range of 1 and 1 at initial and equilibrium moment respectively.
Figure 12 Bed-shear stress amplification along the x direction around the pit in different moment under clear-water
condition
Table 3 Amplification of bed-shear stress in different moment under clear-water condition
Initial moment Equilibrium moment
1 -0.65<x/L<-0.55, 0.55<x/L<0.65 -1.85<x/L<-0.8, 0.92<x/L<2.24
1 -0.55<x/L<0.55 -0.8<x/L<0.92
4.2 Live-bed Condition
Flow fields
Fig. 13 shows the velocity fields around the pit in four periods, the initial moment, 1minute,
30minute and 100minute, in live-bed condition. The size of the vortex becomes smaller and develops to
the bottom of the pit hole with migration of the pit; this result indicates the influence of the pit on the flow
field gradually becomes weaken. The velocity magnitude in the pit hole is small (0~0.15m/s) which
explains the phenomenon that the incoming sediment deposits in the pit hole, thus the water entering
downstream channel is almost clear.
Bed-shear stress
Fig. 14 illustrates that the largest point of bed shear stress changes as the riverbed evolution around
the pit, especially, the largest point at downstream side moves forward due to the migration of the pit. It
may be also noted that the amplification of shear stress at downstream end of the pit becomes smaller
which reveals that the erosion downstream.
Fig. 15 presents variation of the maximum amplification upstream and downstream respectively.
It can be found that the maximum value upstream remains constant approximately and are greater than
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-5 -3 -1 1 3 5 x/L
Initial
moment
Equilibrium
moment
11
1. However, the maximum value downstream decrease with time and are less than 1. Table 4 presents
the range of 1 and 1 in different moment during the pit migration.
Figure 13 Velocity fields around the pit in live-bed condition, (a) displays the initial moment; (b) displays the 1
minute moment; (c) displays the 30 minute moment; (d) displays the 100 minute moment.
Figure 14 Bed shear stress amplification along the x direction around the pit in different moment under live-bed
condition
Figure 15 The maximum amplification of bed-shear stress upstream and downstream respectively variation with time.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-2 -1 0 1 2 3 4 x/L
Initial moment
1min
30min
100min
0
0.4
0.8
1.2
1.6
2
0 20 40 60 80 100 120
Time t(min)
Upstream
Downstream
12
Table 4 Amplification of bed-shear stress in different moment under live-bed condition
Initial moment 1 minute 30 minute 100 minute
1 -0.7<x/L<-0.55, 0.5<x/L<0.65 -1.66<x/L<-0.75 -1.45<x/L<-0.62 -1.15<x/L<-0.35
1 -0.55<x/L<0.5 -0.75<x/L<4 -0.62<x/L<4 -0.35<x/L<4
5 CONCLUSIONS
In the present study, an investigation of riverbed evolution behavior around a triangular pit which
has a steep slope in two flow conditions (clear-water and live-bed conditions) was performed through
laboratory experiments and numerical simulations
(i) The experimental results show that the most significant erosions occur at the upstream and
downstream ends of the pit at the beginning period when the transient flow through a dry pit.
(ii) In clear-water condition, the migration of the pit is not obvious. The slope of the upstream side
( 22o) is greater than the downstream (16o ) when the pit reaches equilibrium state.
(iii) In live-bed condition, the upstream slope of the pit moves forward and equal to the angle of
repose 32 ; the downstream slope is eroded and retrogressive erosion to upstream is not found; the
infilling velocity of the pit hole is approximately constant.
(iv) The numerical results indicate that the presence of the vortex in the pit hole and the influence of
vortex on the flow field in the pit hole is weakened with the pit’s migration. The largest amplification of
bed-shear stress is located at the upstream and downstream ends of the pit.
(v) In clear-water condition, the velocity in the vortex region is very small (0~0.1m/s) and vortex
doom almost occupies the pit hole.
(vi) In live-bed condition, the maximum value downstream (<1) decrease with time.
However, this paper only analyzes one post-damaged riverbed form, for other riverbed forms in
different flow and sediment conditions, the riverbed evolution needs further studying.
ACKNOWLEDGEMENT
This study is supported by the National Natural Science Foundation of China (No.50778018.51078021),
the Doctoral Program of the Ministry of Education (No.20100009110016) and Basic Expense of
Scientific Research of Beijing Jiaotong University (C13JB00230).
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