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Large-eddy simulation of the turbulent structure in compound open-
channel flows
Zhihua Xie, School of Engineering, Cardiff University, Cardiff, CF24 3AA, UK.
Email: [email protected]
Binliang Lin, State Key Laboratory of Hydroscience and Engineering, Tsinghua
University, Beijing, 10084, China; and
School of Engineering, Cardiff University, Cardiff, CF24 3AA, UK.
Email: [email protected]
Tel: +86(0)10-62796544; +44(0)29-20874696 (author for correspondence)
Roger A. Falconer, School of Engineering, Cardiff University, Cardiff, CF24 3AA,
UK.
Email: [email protected]
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ABSTRACT
A large-eddy simulation study has been undertaken to investigate the turbulent
structure of open-channel flow in an asymmetric compound channel. The dynamic
sub-grid scale model has been employed in the model, with the partial cell treatment
being implemented using a Cartesian grid structure to deal with the floodplain. The
numerical model was used to predict the: primary velocity and secondary currents,
boundary shear stress, turbulence intensities, turbulent kinetic energy, and Reynolds
stresses. These parameters were compared with experimental measurements published
in the literature, with relatively close agreement being obtained between both sets of
results. Furthermore, instantaneous flow fields and large-scale vortical structures were
predicted and are presented herein. These vorticial structures were found to be
responsible for the significant lateral exchange of mass and momentum in compound
channels.
Keywords: large-eddy simulation; compound channels; turbulence; hydraulics; river
engineering; secondary currents.
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1. Introduction
Compound open-channel flows appear in most natural rivers and are of vital
importance in river engineering and flood risk management. The shear layer generated
between the higher speed main-channel flow and lower speed flood-plain flow,
together with secondary currents, cause a significant lateral exchange of mass and
momentum. Therefore, it is important to investigate and be able to model accurately
the turbulent structure within such compound open-channel flows.
Many experimental studies have been carried out to investigate the mean velocity
and boundary shear stress in compound open-channel flows using a Pitot static tube
and a Preston tube (for example [1-5]). It has been found that the distributions of
velocity and shear stress are affected by the relative depths and widths of the
floodplain in compound channels. More detailed measurements of turbulent structure
and secondary currents in compound open-channel flows have been conducted by
making use of hot-film anemometers [6, 7] and two-component laser Doppler
anemometers [8-12]. Strong inclined secondary currents have been observed in the
vicinity of the junction between the main channel and the floodplain. These
experimental measurements, having provided much insight into the complex turbulent
flow structure, have not only helped to better understand the complicated interaction
between flows in the main channel and the floodplain, but have also assisted in
developing turbulence models in numerical model simulations.
There are several numerical model studies of turbulent flow in compound open
channels reported in the literature, which provide more detailed space-time resolution
of the flow field when compared to experimental investigations, acting as a
complementary approach to gain further insight into the turbulent flow dynamics.
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Most of the numerical model simulations were based on the Reynolds-averaged
Navier-Stokes (RANS) equations, in which all of the unsteadiness has been averaged
out and considered as a part of the turbulence, and simulated using an algebraic stress
model [13], a non-linear k model [14], a non-linear k model [15], and a more
sophisticated Reynolds stress model [16-18]. Such RANS models are able to provide
satisfactory mean flow fields, but cannot provide detailed instantaneous flow
dynamics and coherent turbulent structure predictions. In addition, many previous
RANS model studies were focused primarily on the distribution of the mean velocity
and boundary shear stress, where the turbulence statistics were neglected. To
overcome these limitations some highly resolved computations, based on the large-
eddy simulation (LES) approach with the Smagorinsky sub-grid scale (SGS) model,
have been performed for turbulent flow in compound open channels [19, 20]. In these
studies the mean velocity field, turbulence intensity, and Reynolds shear stresses were
predicted, with the results being generally found to be in good agreement with the
corresponding experimental results. However, most previous studies of compound
channel flows have focused mainly on the mean flow field, whereas little attention has
been paid to the instantaneous flow field.
The objective of the study presented in this paper is to report on the use of the
LES approach with a dynamic SGS model to investigate the turbulent structure in
compound open-channel flows, including large-scale vortical structures and
instantaneous secondary flows, which play an important part in the flow resistance
and hence the sediment transport fluxes. Model predicted mean velocity and boundary
shear stress distributions, secondary currents, and turbulence statistics are presented
and compared with the available detailed experimental measurements of Tominaga
and Nezu [12]. Details are given in the remainder of the paper of: (i) the mathematical
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model and numerical solution method, (ii) the computational model setup, (iii) the
numerical model results and discussions, and finally, (iv) some conclusions.
2. Mathematical model and numerical method
The large-eddy simulation approach is adopted in this study and the governing
equations used are based on the filtered Navier-Stokes equations, as given for mass
and momentum conservation by:
ui
xi
0 , (1)
(ui )
t(uiu j )
x j
p
xi
(2Sij )
xj
gi ij
x j
, (2)
where the overbar denotes the spatial filtering over the grid, xi represents the
Cartesian coordinates (i = 1, 2, 3 corresponding to x , y and z , meaning the
streamwise, vertical and spanwise directions, respectively), ui represents the resolved
velocity components (i = 1, 2, 3 corresponding to u , v and w , respectively), t is
time, p is the filtered pressure, Sij 1
2ui x j uj xi is the filtered strain-rate
tensor, gi gsin( ), gcos( ), 0 is the gravitational acceleration component in the
xi direction, g is the gravitational acceleration, is the angle of the channel to the
horizontal, is density and is the dynamic viscosity of the fluid.
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The term ij uiuj uiuj is the SGS stress tensor and the anisotropic part of the
SGS term is modelled by an eddy-viscosity model of the form:
ij 1
3 ij kk 2tSij , (3)
where
t Cd2 S , S 2SijSij 1 2
, (4)
with the cut-off length scale xyz 1 3 and a model coefficient Cd . In this
study, the dynamic sub-grid model [21, 22] is used to determine the model coefficient
Cd , as given by:
Cd 1
2
Lij Mij
Mij Mij
, (5)
where Lij uiu j uiu j and Mij
2 S Sij 2 S Sij . In these equations, the hat
represents spatial filtering over the test filter. The symbol for spatial filtering is
dropped hereinafter for simplicity.
In this study, the governing equations were discretised using the finite volume
method on a staggered Cartesian grid. The advection terms were discretised by a high-
resolution scheme [23], which combined the high order accuracy with monotonicity,
whereas the gradients in pressure and diffusion terms were obtained by central
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difference schemes. In order to deal with complex geometries in Cartesian grids, a
partial cell treatment [24] was utilised in the finite volume discretisation, in which the
advective and diffusive fluxes at cell faces, as well as cell volume, were modified in
cut cells. The SIMPLE algorithm [25] was employed for the pressure-velocity
coupling and Gear’s second-order method was used for the time derivative, which led
to an implicit scheme for the governing equations. The code was parallelised using
MPI (Message Passing Interface) and a domain decomposition technique.
3. Model setup
The computational model was set up to replicate the laboratory model study
undertaken by Tominaga and Nezu [12], in which detailed measurements were made
for the turbulent structure in compound open-channel flows at a test section 7.5 m
downstream from the inlet of the flume. In the present study case S-2, as referred to in
[12], in which the ratio of the flood-plain depth to the main-channel depth, i.e.
h / H 0.5 , and the smooth channel bed, was considered. A schematic view of a
compound open-channel flow is shown in Fig. 1, where the origin of the coordinate
system is located at the bottom of the sidewall in the main channel. The width of the
flume was B 0.4 m and the width of the main channel was b 0.2 m. The main-
channel depth H and the flood-plain depth h were 0.08 m and 0.04 m, respectively.
The length of the open channel was set to be L 10H in the simulations. The
corresponding Reynolds and Froude numbers, based on the mean bulk flow velocity
Ubulk 0.349 m/s and the hydraulic radius R 15H / 28 , were Re 67,000 and
Fr 0.39, respectively.
Detailed velocity measurements were acquired using a fibre-optic laser Doppler
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anemometer in the experiments. These data, shown as contour plots in [12], were used
to test the numerical model. The computational domain of 10H H 5H was
discretised using a grid of 192 96 384 cells in the streamwise, vertical, and
spanwise directions, respectively (shown in Fig. 1). Periodic boundary conditions
were used in the streamwise direction. The near-wall modelling approach [26] was
employed for the bed and sidewalls of the open channel, in which the instantaneous
shear stress was related to the velocity adjacent to the solid boundary, by using a time-
averaged wall law. The free surface was modelled as a rigid lid (i.e. a free-slip
boundary condition), which has been successfully used in previous LES studies of
turbulent open-channel flow [19, 27]. The flow was driven by the gravity force, i.e.
gsin( ) , in the streamwise direction. The calculations were started with initial
conditions that consisted of the mean bulk flow velocity, with random perturbations
superimposed in all three directions. After a statistically steady state was reached, the
simulations were continued for about 22 flow-through times (i.e. L /Ubulk ) for
acquiring turbulence statistics.
4. Results and discussion
In this section, the LES results of the mean flow field and turbulence statistics of the
compound open-channel flow are shown and discussed. Moreover, the instantaneous
flow fields, including the secondary flow, are presented and analysed. In the
following, the angular bracket, , represents averaging over time, and the resolved
variable is decomposed into a mean value and a resolved fluctuation as:
, (6)
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where the prime denotes fluctuation with respect to the mean resolved quantity. The
turbulence intensity is obtained using:
rms 2. (7)
It is worth noting that for comparisons to be made with the experimental data all of
the predicted mean flow and turbulence statistics were further averaged along the
streamwise direction.
4.1 Mean velocity distribution
One main feature of compound open-channel flows is that the fast moving flow in the
main channel is retarded by the slow moving flow on the floodplain. The primary
velocity distribution is usually linked to the discharge characteristics and the flow
resistance of the compound open-channel flow structure. Fig. 2 shows the predicted
mean streamwise velocity u distribution, where the predicted results are scaled by
the maximum velocity Umax . The experimental measurements [12] and the modelled
results obtained previously using a LES model, with the Smagorinsky SGS algorithm
[20] and a non-linear k model [14] are also shown in Fig. 2 for comparison
purposes.
The mean bulk flow velocity Ubulk and the maximum velocity Umax , obtained
from the present simulation, are 0.343 m/s and 0.404 m/s, respectively, which are in
reasonably close agreement with the measured values of 0.349 m/s and 0.389 m/s,
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respectively. The main flow pattern of compound open-channel flows is satisfactorily
reproduced by the present LES model, especially in locations near the main-
channel/flood-plain (M/F) junction edge and the main-channel bottom. The contour
lines bulge towards the free surface in the vicinity of the M/F junction edge, with flow
deceleration due to the low-speed flow being transported by the secondary currents
(which will be shown later in Fig. 3) away from the wall. In contrast, the contour lines
bulge towards the wall on both sides of the M/F junction edge, with flow acceleration
due to the high-speed flow being transported by the secondary currents towards the
wall, which is consistent with the experimental measurements [12]. The flow
deceleration can also be observed near the free surface in the sidewall region and near
the bottom of the main channel, which is well predicted in the present study.
4.2 Secondary currents
There are mainly two types of secondary currents based on the mechanism for
streamwise vorticity generation in three-dimensional turbulent flows [28]: one is
Prandtl’s first kind of secondary flow, as observed in curved channels and meandering
rivers, which is caused by the non-uniformity of the flow in the streamwise direction;
the other is Prandtl’s second kind of secondary flow, as observed in straight
compound channels, which is generated by Reynolds stresses. Fig. 3 shows the
predicted mean secondary currents ( v , w ), where the velocities are normalised by
the maximum velocity Umax and only a few selected vectors are plotted for clarity.
The predicted maximum magnitude of the secondary currents is about 4% of Umax ,
which agree well with the experimental measurements. A strong inclined upward flow
is observed in the vicinity of the M/F junction edge, with a pair of counter-rotating
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streamwise vortices occurring on both sides, namely: the main-channel vortex and the
flood-plain vortex [12]. Two additional vortices, namely: the free-surface vortex and
bottom vortex, appear in the sidewall region, which are in good agreement with the
experimental observations.
In the vicinity of the M/F junction edge, better results are obtained by the non-
linear k model [14] in terms of the inclined angle of the secondary currents. In the
present model, there is a slight discrepancy for the inclined angle when compared to
the experimental measurements, this is thought to be caused mainly by the rigid lid
treatment for the free surface and the corresponding limitations in predicting the
vertical velocities. In the sidewall region, the bottom vortex is not captured by the
non-linear k model, but it is predicted by both LES models. Similar secondary
circulation patterns are obtained for most part of the compound channel in the LES
model predictions. Due to the rigid lid treatment mentioned above, near the free
surface there is a discrepancy between the predicted secondary currents and the
experimental results on the main channel side of the junction, whereas better
agreement is obtained in [20] using a height function for the free surface. However,
their model is unable to predict the flow deceleration zone on the left side near the bed
(see Figs 2a and 2c), which is thought to be due to the weak bottom vortex predicted
in [20] (shown in Fig. 3c).
In general, the predicted secondary currents closely resemble the experimental
measurements, which affect the structure of the mean primary velocity (shown in Fig.
2) and the velocity-dip phenomenon in open-channel flows [28], and play an essential
role in the spanwise transport of mass, momentum, and heat in compound channels.
4.3 Bed shear stress distribution
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The bed shear stress distribution plays an important role in bank protection and
sediment transport [29]. Fig. 4 shows the predicted bed shear stress (calculated using
the log-law as used by Tominaga and Nezu [12]) distribution, where the bed shear
stress is normalised by the averaged value in the spanwise direction. The
corresponding experimental measurements and the previous predictions using a k
with algebraic stress model [13], a non-linear k model [14], and two LES models
with the Smagorinsky SGS algorithm [19, 20] are also included in Fig. 4 for
comparisons.
The main features of the bed shear stress in the compound channel are reasonably
well predicted in the present study. The shear stress peak is well predicted in the main
channel, whereas the value is under-predicted in the floodplain. In the main channel,
there is an inflection point for the distribution of the bed shear stress near the sidewall
region ( z / H 0.5 ) in the experimental data, which is reasonably well predicted by
the present LES model. In the floodplain, the bed shear stress is over-predicted by
approximately 8%, indicating that the amount of spanwise momentum transport from
the main channel to the floodplain is slightly over-predicted in the present simulation.
4.4 Turbulence statistics
The mean flow fields have been presented in the previous sections, whereas the
turbulence statistics are reported here to study the turbulent structure in compound
open-channel flows. Fig. 5 shows the predicted turbulence statistics (right column)
along with the corresponding experimental measurements (left column), where the
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turbulence intensities are scaled by the friction velocity u , and the turbulent kinetic
energy and Reynolds stresses are normalised by u2 .
All three components of the turbulence intensity, i.e. urms, vrms, and wrms (Figs 5
(a-c)), have higher values near the solid boundaries parallel to their own components
and lower values near the solid boundaries normal to their own components. In the
vicinity of the M/F junction edge, all turbulence intensities increase with different
trends. Higher values are observed for urms on both sides of the junction edge,
whereas vrms is higher on the main-channel side of the edge and wrms is higher on the
flood-plain side of the edge, respectively. It is worth noting that the minimum values
of urms and wrms occur below the free surface in the main channel, which correspond
to the velocity-dip phenomenon, seen in Fig. 2, and the decrease of vrms and the
increase of wrms are observed near the free surface, demonstrating the redistribution
effect of the free surface on turbulence intensity.
The distribution of turbulent kinetic energy k (Fig. 5(d)) is similar to the pattern
observed for the mean streamwise velocity shown in Fig. 2, with the contour lines
bulging in an inclined angle towards the free surface from the M/F junction edge. In
contrast to the mean streamwise velocity, higher values of k occur in the vicinity of
the sidewalls and bottom bed and lower values occur below the free surface in the
compound channel.
The distributions of the Reynolds stresses u v (Fig. 5(e)), and u w (Fig.
5(f)), are generally correlated to gradients of the mean streamwise velocity, which are
consistent with the experimental measurements [12]. Near the free surface negative
values of u v occur in the sidewall regions, whereas near the sidewalls u w
become positive in the main channel and negative in the floodplain, respectively. In
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the vicinity of the M/F junction edge, negative peak values of u v and u w
are found below the inclined upflow in the main-channel side, whereas positive peak
values of u v and u w are found above the inclined upflow in the flood-plain
side. However, the distributions of the two Reynolds stresses differ near the junction
edge, as u v increase in the vertical direction and u w increase in the
spanwise direction. All the signs of u v and u w correspond well to those for
u / y and u / z . It is worth noting that one important feature of compound
open-channel flows can be recognised from the distribution of u w near the
junction edge, indicating a spanwise transport of momentum from the main channel
towards the floodplain ( u w / z 0).
In general, the predicted distributions of turbulence statistics agree closely with
the experimental measurements, which provide a useful tool for investigating more
detailed turbulent structures in compound open-channel flows.
4.5 Anisotropy of turbulence
It can be seen from Figs 5 (b-c) that the spanwise turbulence intensity wrms and
vertical turbulence intensity vrms have different distributions in compound open-
channel flows. The difference between them, i.e., ( w 2 v 2 ) , denoting the
anisotropy of turbulence, is a driving force for the generation of secondary currents
observed in Fig. 3. Fig. 6 shows the predicted value of ( w 2 v 2 ), normalised by
u2 , along with the measured values. The predicted distribution of anisotropy of
turbulence generally follows the pattern in the experimental measurements, although
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the predicted negative region of ( w 2 v 2 ) is smaller and lower in the vicinity of
the M/F junction edge when compared to the measured values. This discrepancy is
thought to be due to the rigid lid approximation affecting the distribution of the
vertical turbulence intensity, and causing the difference between the predicted and
measured secondary currents near the junction edge, as shown in Fig. 3. The value of
( w 2 v 2 ) decreases in the vertical direction from the bottom bed, and increases
towards the free surface after attaining the negative peak value in the upper flow
region of the channel. At the M/F junction edge, the value of ( w 2 v 2 ) changes
from negative in the main channel to positive on the floodplain. The anisotropy of
turbulence in the compound open-channel flow is responsible for the generation of the
streamwise vortices, which is consistent with the experimental observations [12].
4.6 Instantaneous flow field
In order to better understand the vortical structure in compound open-channel flows,
the 2 method [30] was used in the present study to identify vortex cores, based on
the second invariant of the velocity gradient tensor. Fig. 7 shows a snapshot of the
instantaneous vortical structure, plotted as iso-surfaces of 2 20 and coloured by
the vertical distance y / H . Streamwise vortices are observed across the compound
channel, with most of the vortices being located near the bed and sidewalls where the
rate of strain is high. As the secondary currents are only 4% of the streamwise
velocity, most of the vortices are elongated and advected downstream. Near the
sidewalls, the streamwise vortices are apparent near the free surface and the bed,
corresponding to the free-surface vortex and bottom vortex observed in the mean
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secondary currents. In the vicinity of the M/F junction edge, streamwise vortices are
developed, resulting in secondary flows being formed, which are responsible for the
momentum transfer between the main channel and the floodplain. The location of the
vortex cores varies in the streamwise direction, corresponding to the chaotic flow
field at the cross-sections.
In order to study the temporal and spatial development of the secondary currents,
a detailed analysis has been undertaken for the streamwise vortices using the
instantaneous velocities predicted by the present LES model. Fig. 8 shows snapshots
of the distributions of instantaneous streamwise velocity u and secondary currents
(v,w) at two cross-sections: x / H = 5 (Fig. 8(a-c)) and x / H = 6 (Fig. 8(d-f)), for a
constant time interval of 0.1L /Ubulk . Both the instantaneous distributions of the
streamwise velocity and the secondary currents are different from their mean values,
as shown in Figs 2 and 3 respectively, demonstrating the temporal variations of the
turbulent flow field. In the near-wall region, the transport of low-speed flow towards
the wall and high-speed flow away from the wall can be recognised. It can be
observed that the flow near the bed or sidewalls is more turbulent than the flow in the
interior region, which is consistent with the high values for turbulence statistics in the
near-wall region. At x / H = 5 (from Fig. 8(a) to Fig. 8(c)), the development of a
strong clockwise vortex on the floodplain near the M/F junction edge is apparent,
corresponding to the streamwise vortices moving downstream towards the floodplain.
By comparing the instantaneous streamwise velocity and secondary flows between
x / H = 5 (Fig. 8(a-c)) and x / H = 6 (Fig. 8(d-f)) at a same time, it is shown that their
flow patterns are different, demonstrating spatial variations of the three-dimensional
turbulent flow field. However, it is interesting to note that similar flow patterns are
observed between Fig. 8(a) and Fig. 8 (e) and between Fig. 8(b) and Fig. 8(f). As the
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time interval between each figure from the top to the bottom is 0.1L /Ubulk ,
corresponding to a streamwise distance x / H = 1, it is suggested that the secondary
flows are affected by the main velocity, transported downstream with the streamwise
velocity. Most of the flow patterns observed at x / H = 5 can be seen at x / H = 6 at
the next time interval, with small differences occurring near the solid boundaries.
Fig. 9 shows an example of snapshots for the distribution of the instantaneous
streamwise velocity u and velocity vectors (u,w) for the shear layer at a horizontal
plane y / H 0.75 , for a constant time interval of 0.1L /Ubulk . As the streamwise
velocity is one order larger than the spanwise velocity, in Fig. 9 the values of the
spanwise velocity w are increased by a factor of 10 for the sake of clarity. It can be
seen that the flow in the main channel is generally moving faster than that on the
floodplain. In the vicinity of the M/F junction edge, a large shear region can be
observed, causing mass and momentum transfer between the main channel and the
floodplain. In Fig. 9a, a strong flow region on the floodplain at x / H 0.5 is moving
towards the main channel. This flow region is advected with the primary velocity and
moves to x / H 1.5 and 2.5 in Figs 9b and 9c, respectively, confirming the distance
of x / H 1 for each time interval, as found in the instantaneous secondary currents.
It is shown from Figs 9(a-c) that a similar turbulent flow structure can be observed at
different cross sections and at different times, indicating that the main flow pattern is
affected by the main velocity.
It is worth mentioning that although the instantaneous primary and secondary
flows vary in space and time, demonstrating the complex three-dimensional turbulent
flow structures, there is little variation for the time-averaged flow fields at different
cross sections and the main feature of compound open-channel flows can be
recognised from the mean primary velocity (Fig. 2) and secondary currents (Fig. 3).
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5. Conclusions
In this paper a large-eddy simulation model of the turbulent structure in compound
open-channel flows has been presented and the predictions were compared with
available experimental data. The principal flow features measured in the experiments
were generally well reproduced, including: the distributions of the mean velocity and
secondary currents, boundary shear stress, turbulence intensities, turbulent kinetic
energy, and Reynolds stresses. The anisotropy of the turbulence was also investigated
in this study, with this being a driving force for the generation of secondary currents.
Furthermore, the instantaneous flow fields and large-scale vortical structures were
presented, with a stronger turbulent flow in the near-wall region and a significant
lateral transport of momentum being shown. It was found that although there are
significant temporal and spatial variations for the three-dimensional turbulent flow
field, their time-averaged results are similar at different cross sections.
The study demonstrates the capability of the present LES model in providing
reliable detailed mean velocity field characteristics and turbulence statistics for
compound open-channel flows, which can act as a complementary approach to
experimental investigations to gain further insight into the turbulent flow dynamics.
The main future extension of this study would be to include a solute transport module
and roughness effect to investigate mass exchange between the main channel and the
floodplain.
Acknowledgements
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The research was supported by the UK Engineering and Physical Sciences Research
Council (EP/G014264/1). This work was performed using the computational facilities
of the Advanced Research Computing @ Cardiff (ARCCA) Division, Cardiff
University.
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Association for Hydraulic Research monograph series. Rotterdam: A.A. Balkema;
1993.
22
[29] Shiono K, Knight DW. Turbulent open-channel flows with variable depth across
the channel. J Fluid Mech 1991; 222: 617–646.
[30] Jeong J, Hussain F. On the identification of a vortex. J Fluid Mech 1995; 285:
69–94.
23
Figure captions
Fig. 1. Schematic of computational domain and numerical grid for turbulent open-
channel flow in a compound channel (only every sixth grid line plotted for clarity).
Fig. 2. Distribution of primary mean velocity u /Umax .
Fig. 3. Secondary current vectors.
Fig. 4. Distribution of bed shear stress.
Fig. 5. Comparison between predicted turbulence statistics (right column) and
experimental measurements [12] (left column): (a) streamwise turbulence intensity;
(b) vertical turbulence intensity; (c) spanwise turbulence intensity; (d) turbulent
kinetic energy; (f) Reynolds stress u v ; (g) Reynolds stress u w .
Fig. 6. Distribution of ( w 2 v 2 ).
Fig. 7. Snapshot of instantaneous vortical structure plotted as iso-surfaces of
2 20 and coloured by the vertical distance y / H in compound open-channel
flows.
24
Fig. 8. Snapshots of the distribution of instantaneous streamwise velocity u and
secondary velocity vectors (v,w) at two cross-sections: x / H = 5 and 6, for a constant
time interval of 0.1L /Ubulk .
Fig. 9. Snapshots for the distribution of instantaneous streamwise velocity u and
velocity vectors (u,w) for the shear layer at a horizontal plane y / H 0.75 , for a
constant time interval of 0.1L /Ubulk . The values of w are increased by a factor of 10
for the sake of clarity.
26
Figure 2
(a) experiment [12]
0.9750.95 0.9 0.85
0.8 0.75
0.70.6
z/H
y/H
<u>/ Umax
0 1 2 3 4 50
0.5
1
(b) present LES result
(c) LES model [20]
(d) non-linear k model [14]
27
Figure 3
(a) experiment [12]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
z/H
y/H
0.02Umax
(b) present LES result
(c) LES model [20]
(d) non-linear k model [14]
29
Figure 5
z/H
y/H
0.81.2
1.6
22.8
1.6
2.4
(a) Streamwise Turbulence Intensity urms
/u
0 1 2 3 4 50
0.5
1
z/H
y/H
(b) Vertical Turbulence Intensity vrms
/u
0.6
0.4
0.80.6
0.6 0.5
0.50.6
0.4
0 1 2 3 4 50
0.5
1
z/H
y/H
(c) Spanwise Turbulence Intensity wrms
/u
0.5
0.9
0.7 0.7
0.9
0.9
0.6
0 1 2 3 4 50
0.5
1
z/H
y/H
12 3 3.5
3.54
(d) Turbulent Kinetic Energy k/u2
0 1 2 3 4 50
0.5
1
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