An investigation of the dynamics of coherent structures in a turbulent channel flow with a vertical sidewall obstruction

Abstract

The physics of the flow around a vertical-wall obstruction attached to one of the sidewalls of a straight channel is numerically investigated using detached eddy simulation (DES) at a high channel Reynolds number, Re=5×105 (case HR). In particular, the study investigates the role played by the large-scale coherent structures in the sediment entrainment processes at the bed for conditions close to the initiation of scour (flat bed) in a loose-bed channel. Scale effects are investigated by comparing the results of the present DES simulation with results from a large eddy simulation performed at a much lower Reynolds number, Re=18 000 (case LR). Similar to laboratory flume studies of flow and scour around in-stream obstructions, the incoming flow in the simulations was fully turbulent and contained unsteady velocity fluctuations. The main necklace vortex of the horseshoe vortex (HV) system forming near the upstream base of the flow obstruction was subject to bimodal large-scale oscillations. The intensity of the bimodal oscillations peaked at vertical sections cutting through the tip of the obstruction. Present results show the size of the region of high turbulence amplification within the HV system decreases with the increase in the Reynolds number. Patches of vorticity were observed to detach from the leg of the main necklace vortex and to be convected at a small distance from the bed. Before dissipating, these patches could induce relatively large values of the bed shear stress beneath them. In case HR, the formation of these patches was primarily determined by the interaction of the main necklace vortex with the leg of the secondary necklace vortex rather than the interaction of the main necklace vortex with the tip of the obstruction, as was the case in the LR simulation. The degree of deformation of the cores of the vortex tubes shed in the upstream part of the separated shear layer (SSL) originating at the tip of the obstruction was much larger in the HR simulation. This provided an additional mechanism for the amplification of the horizontal vorticity in the near-bed region. Large-scale hairpinlike structures formed in the downstream part of the SSL. The legs of these vortices were oriented parallel to the interface between the high-speed outer flow and the recirculating flow past the obstruction. The lower leg of some of these hairpin vortices was situated, at times, at a small distance from the bed and was able to strongly amplify the local bed shear stress values. Consequently, the bed shear stress distribution in the instantaneous flow fields displayed a streaky structure over part of the SSL region. The changes in the relative position and size of the regions of high turbulence amplification inside the HV system and the SSL induced noticeable scale effects on the distributions of the shear stress and pressure root-mean-square (rms) fluctuations at the bed which control the sediment entrainment processes. In particular, due to the richer eddy content of the SSL, the intensity of the nondimensional pressure rms fluctuations beneath the SSL was found to be about two times larger in the HR simulation.

Full text

An investigation of the dynamics of coherent structures in a turbulent
channel flow with a vertical sidewall obstruction
Mete Koken
a兲
and George Constantinescu
b兲
Department of Civil and Environmental Engineering, IIHR-Hydroscience and Engineering,
Stanley Hydraulics Laboratory, The University of Iowa, Iowa City, Iowa 52242, USA
共Received 28 October 2008; accepted 1 July 2009; published online 26 August 2009兲
The physics of the flow around a vertical-wall obstruction attached to one of the sidewalls of a
straight channel is numerically investigated using detached eddy simulation 共DES兲 at a high channel
Reynolds number, Re=5 ⫻ 10
5
共case HR兲. In particular, the study investigates the role played by the
large-scale coherent structures in the sediment entrainment processes at the bed for conditions close
to the initiation of scour 共flat bed兲 in a loose-bed channel. Scale effects are investigated by
comparing the results of the present DES simulation with results from a large eddy simulation
performed at a much lower Reynolds number, Re=18 000 共case LR兲. Similar to laboratory flume
studies of flow and scour around in-stream obstructions, the incoming flow in the simulations was
fully turbulent and contained unsteady velocity fluctuations. The main necklace vortex of the
horseshoe vortex 共HV兲 system forming near the upstream base of the flow obstruction was subject
to bimodal large-scale oscillations. The intensity of the bimodal oscillations peaked at vertical
sections cutting through the tip of the obstruction. Present results show the size of the region of high
turbulence amplification within the HV system decreases with the increase in the Reynolds number.
Patches of vorticity were observed to detach from the leg of the main necklace vortex and to be
convected at a small distance from the bed. Before dissipating, these patches could induce relatively
large values of the bed shear stress beneath them. In case HR, the formation of these patches was
primarily determined by the interaction of the main necklace vortex with the leg of the secondary
necklace vortex rather than the interaction of the main necklace vortex with the tip of the
obstruction, as was the case in the LR simulation. The degree of deformation of the cores of the
vortex tubes shed in the upstream part of the separated shear layer 共SSL兲 originating at the tip of the
obstruction was much larger in the HR simulation. This provided an additional mechanism for the
amplification of the horizontal vorticity in the near-bed region. Large-scale hairpinlike structures
formed in the downstream part of the SSL. The legs of these vortices were oriented parallel to the
interface between the high-speed outer flow and the recirculating flow past the obstruction. The
lower leg of some of these hairpin vortices was situated, at times, at a small distance from the bed
and was able to strongly amplify the local bed shear stress values. Consequently, the bed shear stress
distribution in the instantaneous flow fields displayed a streaky structure over part of the SSL region.
The changes in the relative position and size of the regions of high turbulence amplification inside
the HV system and the SSL induced noticeable scale effects on the distributions of the shear stress
and pressure root-mean-square 共rms兲 fluctuations at the bed which control the sediment entrainment
processes. In particular, due to the richer eddy content of the SSL, the intensity of the
nondimensional pressure rms fluctuations beneath the SSL was found to be about two times larger
in the HR simulation. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3207859兴
I. INTRODUCTION
An isolated vertical-wall obstruction mounted on one of
the sidewalls of a channel with a turbulent incoming flow
causes a very complex, unsteady, three-dimensional 共3D兲
highly turbulent flow field 共see sketch in Fig. 1兲 to occur
within its proximity. The flow field is characterized by the
presence of turbulent structures over a wide range of scales
and by a pronounced downflow at the leading face of the
vertical-wall obstruction. As the flow approaches the ob-
struction, as a result of the adverse pressure gradients, the
incoming boundary layer separates. The main vortical sys-
tems in the flow are the horseshoe vortex 共HV兲 system near
the upstream base of the obstruction, the recirculating flow
region upstream of the obstruction, the separated shear layer
共SSL兲 originating at the tip of the obstruction, and the wake
flow past the obstruction. The unsteady necklace vortices
that form the HV system follow the junction line between the
bed and the upstream face of the obstruction. Then, past the
obstruction these vortices are stretched and bend in the di-
rection of the incoming flow. These necklace vortices con-
trol, to a great extent, the evolution of the scour process in
front and around the extremity 共tip兲 of the obstruction.
a兲
Present address: Civil Engineering Department, Middle East Technical
University, Ankara, Turkey.
b兲
Author to whom correspondence should be addressed. Telephone:
319 384-0630. Fax: 319 335-5238. Electronic mail:
sconstan@engineering.uiowa.edu.
PHYSICS OF FLUIDS 21, 085104 共2009兲
1070-6631/2009/21共8兲/085104/16/$25.00 © 2009 American Institute of Physics21, 085104-1
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Because of the presence and interaction among the large-
scale coherent structures, large shear stresses and pressure
fluctuations at the bed can be induced in some regions situ-
ated around the obstruction. Consequently, scour will de-
velop around the obstruction in loose-bed channels. If the
scour around the obstruction is excessive, the structural sta-
bility of the in-stream structure can be endangered. The prob-
lem is particularly relevant to river reaches containing river
training structures whose shapes are close to that of a
vertical-wall obstruction. They include spur dikes and
groynes used in river restoration and bridge abutments. Al-
though many experimental investigations were conducted to
study the scouring pattern and predict the maximum scour
depth for different hydraulic conditions and shapes of the
obstruction 共for a review see Refs. 1 and 2兲, less effort has
been made to understand the intricate flow physics behind
the local scouring process at these river training structures
and, in particular, the role played by the large-scale coherent
structures present around the flow obstruction. In river and
coastal engineering applications the incoming channel flow
is highly turbulent and the channel Reynolds numbers de-
fined with the channel depth, D, and the bulk channel veloc-
ity, U, are high 共ReⰇ10
5
兲.
Most of the previous work that focused on understanding
the physics of junction flows and, in particular, the dynamics
of the turbulent HV system has concentrated on flow past
obstructions situated away from the channel sidewalls. Such
typical examples relevant for applications in aerodynamics,
turbomachinery, and hydrodynamics are flow past surface
piercing and submerged cylinders of different shapes includ-
ing circular and square cylinders,
3–7
wing-shaped cylinders,
8
and cubes.
9,10
In these cases the dynamics of the necklace
vortices part of the HV system is mainly determined by the
interaction of the eddies advected with the incoming separat-
ing boundary layer and the outer flow with the upstream face
of the cylinder.
In the case of a vertical-wall obstruction mounted on one
of the channel sidewalls, Koken and Constantinescu
11
showed that the fluctuations in the coherence of the main
necklace vortex are significantly affected by the temporal
variations of the intensity of the main tornadolike corner
vortex. This vortex is part of the vortical system contained
within the recirculation region forming at the junction be-
tween the upstream face of the wall obstruction and the chan-
nel sidewall. Paik and Sotiropoulos
12
provided an in-depth
discussion of the dynamics, structure, and Lagrangian time
scales of the eddies inside the upstream recirculation region
and of the role of the shear layer forming at its interface with
the fast-moving outer flow. Both studies found that the main
role of the tornadolike vortices is to convect circulation and
momentum from the free surface region in which the flow is
recirculating into the core of the necklace vortices.
Koken and Constantinescu
11,13
provided the first in-
depth discussion of the structure and unsteady dynamics of
the HV system forming at the base of a vertical sidewall-
attached wall obstruction with flat and deformed scoured bed
based on results from a well-resolved 共no wall functions
were used兲 large eddy simulation 共LES兲 and dye visualiza-
tion experiments. Despite the fact that their numerical inves-
tigation was conducted at a channel Reynolds number of
only 18 000, the structure of the HV system was found to
resemble the one observed in previous investigations of junc-
tion flows conducted at much higher Reynolds numbers. In
particular, they found that the core of the main necklace vor-
tex is subject to bimodal aperiodical oscillations with char-
acteristics that are similar to those observed by Devenport
and Simpson
8
in their pioneering experimental study of the
HV system forming at the base of a wing-shaped cylinder in
a flat-bed channel at Re⬃1.15⫻10
5
共the length scale in the
definition of the Reynolds number is the maximum thickness
of the wing兲. This finding is consistent with the observation
made by Simpson
14
that the approach flow Reynolds number
is not a dominant factor for the existence of the bimodal
behavior. The presence of large-scale self-induced aperiodic
oscillations explained why the turbulence 关e.g., turbulent ki-
netic energy 共TKE兲, turbulence energy production, and Rey-
nolds stresses兴 levels within the HV system region were
about one order of magnitude higher compared to the levels
in the incoming turbulent boundary layer.
Experiments
15
showed that the low-frequency oscilla-
tions are due to injection of low-momentum high-vorticity
and high-momentum low-vorticity eddies in the HV region.
Most of these eddies are first recirculated with the downflow
as they reach the upstream face of the bluff body. Experi-
ments have also demonstrated the large statistical coherency
between the velocity and the pressure fluctuations at the
channel bed, at the dominant frequencies corresponding to
the large-scale bimodal oscillations of the necklace vortices.
The reader is referred to the review paper by Simpson
14
for a
detailed discussion of the dynamics of the turbulent HV sys-
tem. Several numerical investigations using eddy resolving
techniques have observed the presence of bimodal oscilla-
tions of the turbulent HV system forming at the base of a
bluff body mounted on a flat-bed channel.
9,16–19
Recently,
Koken and Constantinescu
13
found that the core of the main
necklace vortex continues to be subjected to bimodal aperi-
odic oscillations as the loose bed evolves between conditions
at the initiation of scour 共flat bed兲 and at the end of the scour
process 共equilibrium deformed bed兲. The effect of the devel-
FIG. 1. 共Color online兲 Sketch showing the main coherent structures and
related flow features around the vertical-wall obstruction.
085104-2 M. Koken and G. Constantinescu Phys. Fluids 21, 085104 共2009兲
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opment of the scour hole was to gradually reduce the ampli-
tude of the large-scale aperiodic oscillations.
Besides the HV system, the strongly accelerated flow
near the tip of the obstruction, the downflow, and the vortex
tubes shed inside the SSL can locally induce large bed shear
stresses and intensify the near-bed turbulence around the
flow obstruction.
11
Thus, these flow features are erosive flow
mechanisms of primary importance.
A first goal of the present detached eddy simulation
共DES兲 study is to explain the intricate flow physics around a
vertical-wall surface-piercing obstruction mounted in a chan-
nel with vertical sidewalls at a Reynolds number 共Re=5
⫻10
5
兲 that is representative of conditions present in applica-
tions of practical interest where these flow obstructions are
encountered 共e.g., in river reaches兲. A second goal is to elu-
cidate the roles played individually by the large-scale ener-
getic eddies as well as their interactions with each other and
with the flow obstruction, and to understand how they affect
the distributions of the shear stress,
␶
, and pressure root
mean square 共rms兲 fluctuations,
p
⬘
2
, at the bed. Sediment
erosion processes in loose-bed channels are controlled by the
levels of the bed shear stress and of the turbulent pressure
fluctuations in the near-bed region. Eddy-resolving tech-
niques such as LES and DES allow obtaining detailed quan-
titative and qualitative information on these flow quantities
that are very difficult to obtain from experiments. A third
goal is to study scale effects and thus to try to understand to
what extent the findings from relatively low-Reynolds num-
ber 共Re⬃10
4
兲 experimental or LES studies are relevant for
practical applications of junction flows in river engineering
and other disciplines in which the Reynolds numbers are
much higher 共Re⬎ 10
5
兲.
The paper is organized as follows. Following a short
description of the numerical method, DES model, and simu-
lation setup, the dynamics of the large-scale coherent struc-
tures and their role in controlling sediment entrainment phe-
nomena are discussed. This paper also analyzes the effect of
the presence of these coherent structures on the mean flow
and turbulence statistics. Scale effects are discussed based on
comparison of DES with results from a LES simulation at
Re=18 000.
11
In both simulations the incoming flow con-
tained unsteady velocity fluctuations and the ratio between
the width of the obstruction, W, and the channel depth, D ,
was the same 共W/ D = 1.5兲. Finally, the main findings are
summarized.
II. DISCUSSION OF SIMULATION APPROACH
The most popular numerical approach for predicting
complex engineering flows at realistic Reynolds numbers is
based on employing Reynolds averaged Navier–Stokes
共RANS兲 turbulence models. The role of the RANS model is
to account for the influence of the scales present in the flow
共eddies, vortices兲 on the mean flow. Past experience shows
that both steady and unsteady RANS models fail to predict
important aspects of massively separated flows dominated by
unsteady vortex shedding and large-scale vortex
interactions.
16,20,21
Previous RANS simulations of the flow in
channels containing a vertical-wall obstruction adjacent to a
sidewall are discussed in Ref. 11. The shortcoming of RANS
models of different degrees of complexity 共linear, nonlinear,
and second-order兲 to predict flow past surface-mounted cyl-
inders and cubes are discussed by Simpson
14
and Paik et al.
19
Also, the knowledge of the mean flow is not enough to un-
derstand the dynamics of the highly energetic coherent struc-
tures and their role in the entrainment of sediment from the
bed.
Eddy-resolving methods such as direct numerical simu-
lation 共DNS兲 and LES were shown to be much more success-
ful to predict massively separated flows and, in particular,
junction flows.
11,17,18,22,23
In LES all the dynamically impor-
tant scales are directly resolved and only the effect of the
smaller 共subgrid兲 scales on the resolved ones is modeled. To
maintain accuracy, the use of nondissipative numerical algo-
rithms is required in LES and DNS. Unfortunately, at very
large Reynolds numbers 共Re⬎10
5
兲, the mesh and time-step
requirements make the use of LES without wall modeling
共e.g., wall functions兲 not feasible to study junction flows.
Still, highly resolved low-Reynolds number investigations of
junction flows are relevant. For example, a DNS study of the
flow past a surface-mounted cube
22
was able to capture the
overall features of the HV system, including its location,
observed in an experiment conducted at a much higher Rey-
nolds number.
Hybrid RANS-LES techniques such as DES can resolve
the dynamically most important eddies in the flow and allow
studying the dynamics of the coherent structures at Reynolds
numbers that are comparable to those encountered in realistic
applications.
12,19,24,25
DES uses the same base turbulence
model in the RANS and LES regions. No special treatment is
required to match the solutions at the boundary between the
LES and RANS regions. The level of numerical dissipation
in the viscous solver has to be kept at a minimum to avoid
damping of the turbulent eddies in the resolved range which
can compromise the accuracy of the results obtained using
this type of methods. Additionally, more accurate results can
be obtained by using fine meshes that resolve the viscous
sublayer of the attached boundary layers 共the use of wall
functions is avoided兲.
An important problem for DES simulations of flow past
surface-mounted obstacles with an approaching turbulent
flow is that the numerical predictions are dependent on the
presence, or absence, of unsteady velocity fluctuations in the
incoming flow. All experimental investigations of local scour
at river training structures are conducted with an incoming
fully turbulent channel flow to mimic as close as possible the
conditions present in the field. Ideally, the inflow conditions
in channel flow simulations containing surface-mounted ob-
structions should contain the proper eddy content present at
the Reynolds number at which the simulation is conducted,
similar to the field or experimental conditions. This is needed
for the simulation to capture in a realistic way the nature of
the interactions between the incoming turbulent eddies in the
channel and the large-scale coherent structures forming in
massively separated flows. Chang et al.
26
studied the effect
of the inflow fluctuations in DES for the separated flow over
a bottom cavity in a channel flow. In the absence of inflow
velocity fluctuations the DES predictions of the coherent
085104-3 An investigation of the dynamics Phys. Fluids 21, 085104 共2009兲
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structures populating the shear layer on top of the cavity and
the distribution of the Reynolds stresses were found to differ
significantly from that of a well-resolved LES at the same
channel Reynolds number. With the exception of the recent
study of Kirkil and Constantinescu
33
for surface-mounted
high-aspect-ratio rectangular cylinders, all up-to-date DES
predictions of junction flows
12,19
used steady conditions.
Several studies
18,19
showed that eddy resolving simulations
conducted with steady inflow conditions predicted a HV sys-
tem that was located upstream of the measured location de-
spite successfully capturing the presence of bimodal oscilla-
tions and other relevant flow and turbulence quantities.
Moreover, the absence of small-scale unsteadiness in the
incoming flow allows the development of secondary insta-
bilities on the core of the large-scale vortices in a much
easier way. It is possible that some of the flow dynamics
共e.g., the formation of hairpinlike vortices wrapping around
the core of the main necklace vortex in the flow past a
surface-mounted wing-shaped cylinder
19
兲 present in the DES
simulations of junction flows with steady inflow conditions
will change if the inflow contains velocity fluctuations that
try to mimic, as close as possible, those present in a turbulent
channel flow. Such hairpinlike vortices did not form around
the core of the necklace vortices in a LES simulation
11
共Re
=18 000兲 of the flow past a vertical-wall obstruction attached
to a sidewall. Although we suspect this was mainly because
of the fact that realistic turbulence was fed through the in-
flow section in LES, another possibility to explain the ab-
sence of the hairpin vortices was the low value of the Rey-
nolds number in the LES simulation. One should also
mention that the LES study of Chang et al.
27
observed the
formation of similar large-scale hairpin vortices on the core
of the primary spanwise vortices shed in the SSL forming on
top of the cavity in the flow past a bottom cavity with steady
incoming flow. When resolved turbulence was added, the
hairpinlike vortices disappeared due to the strong jittering
effect of the large-scale eddies inside the SSL by the outer
flow turbulent eddies. These observations point toward the
importance of the inflow conditions in eddy resolving simu-
lations and the need to perform simulations with conditions
that are as close as possible to those present in applications
of engineering interest.
III. NUMERICAL MODEL AND COMPUTATIONAL
DETAILS
The present study uses the Spalart–Allmaras 共SA兲
RANS model as the base model in DES.
24
The SA RANS
model is based on a transport equation for the modified eddy
viscosity,
␯
˜
. The eddy 关subgrid scale 共SGS兲兴 viscosity
␯
t
is
related to
␯
˜
using
␯
t
=
␯
˜
f
v
1
, where f
v
1
is an empirical model
wall function. The SA version of DES is obtained by replac-
ing the turbulence length scale d 共distance to the nearest
wall兲 in the destruction term of the transport equation for
␯
˜
with a new length scale d
DES
=min共d,C
DES
⌬兲, where the
model parameter C
DES
is equal to 0.65 共Ref. 24兲 and ⌬ is a
measure of the local grid size. When the production and
destruction terms are balanced, the length scale in the LES
regions d
DES
=C
DES
⌬ becomes proportional to the local grid
size and yields an eddy viscosity proportional to the mean
rate of strain and ⌬
2
as in LES with a Smagorinsky model,
which allows the energy cascade down to grid size. The eddy
viscosity predicted by DES in the LES regions goes to zero if
the local grid size decreases to zero as in classical LES. For
rough wall boundaries, the formulation of Spalart
24
is used.
A general description of the code, governing equations,
and DES model used to perform the DES simulations is
given in Refs. 25 and 26. The 3D incompressible Navier–
Stokes equations are integrated using a fully implicit
fractional-step method. The governing equations are formu-
lated in generalized curvilinear coordinates on a nonstag-
gered grid. The convective terms in the momentum equations
are discretized using a blend of fifth-order accurate upwind
biased scheme and second-order central scheme.
28
This is
needed to minimize the level of numerical dissipation away
from solid boundaries. All other terms in the momentum and
pressure-Poisson equations are approximated using second-
order central differences. The discrete momentum 共predictor
step兲 and turbulence model equations are integrated in
pseudotime using the alternate direction implicit approxi-
mate factorization scheme. Time integration in the DES code
is done using a double time-stepping algorithm and local
time stepping is used to accelerate the convergence at each
physical time step. Source terms in the turbulence model
equations are treated implicitly. The time discretization is
second order accurate. The simulations were performed us-
ing a parallel 共
MPI兲 version of the code.
The geometry of the computational domain is depicted
in Fig. 2. The length scale is selected to be the flow depth, D.
The mean velocity in the channel, U, is used as the velocity
scale. The width of the vertical-wall obstruction mounted on
the right sidewall is 1.5D. The computational domain ex-
tends 8D upstream of the axis of the flow obstruction situ-
ated at x / D = 0 and 40D downstream of it. The domain width
is 5.6D. The nondimensional length, height, and width of the
obstruction are identical to those used in the study of Koken
and Constantinescu
11
who performed an LES simulation at
Re=18 000. Thus, the comparison of the present DES simu-
lation results at Re=5 ⫻ 10
5
with the LES simulation results
at Re=18 000 allows us to investigate scale effects. The code
used to perform the LES simulation employs a collocated
finite-volume nondissipative scheme to solve the filtered
FIG. 2. 共Color online兲 Computational domain in the HR simulation.
085104-4 M. Koken and G. Constantinescu Phys. Fluids 21, 085104 共2009兲
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Navier–Stokes equations with the dynamic Smagorinsky
model on unstructured meshes.
29,30
The fractional-step algo-
rithm is second order accurate in both space and time. All the
operators in the LES code including the convective terms are
discretized using central schemes.
The LES and DES simulations were performed on
meshes that were sufficiently fine in the wall-normal direc-
tion to avoid the use of wall functions. The computational
domain was meshed using close to 4 ⫻10
6
cells in LES and
7.4⫻10
6
cells 共480⫻192⫻80 grid points in the streamwise,
spanwise, and vertical directions兲 in the DES simulation at
Re=5⫻10
5
. The larger number of mesh cells in DES is re-
quired to resolve the attached boundary layers in the wall-
normal direction and the regions containing the dynamically
important eddies in the flow 共e.g., necklace vortices in the
HV region, eddies populating the SSL兲 as the size of these
eddies decreases with the Reynolds number.
The grid spacing in the wall-normal direction in DES
was 2⫻ 10
−5
D 共⬃0.5 wall units assuming a nondimensional
friction velocity u
␶
/ U = 0.04兲 on all the wall surfaces. The
maximum grid aspect ratio in the wall region was 1500. The
aspect ratio decreases as the upstream wall of the obstruction
is approached. The distance from the wall at which DES
switches from RANS mode to LES mode varied from 0.04D
away from the obstruction to about 0.017D close to the ob-
struction. By comparison, the average height of the core of
the main necklace vortex was around 0.2D.
In the upstream part of the SSL where vortex tubes are
shed, the mesh spacing in the horizontal directions was 30–
150 wall units. This is sufficient to resolve the flow in the
region where these vortex tubes are convected away from the
tip of the obstruction. The cell spacing was 150–600 wall
units inside the HV region. At most locations the core of the
main necklace vortex was resolved with at least 20 grid
points in the directions contained in a plane perpendicular to
the axis of the vortex. The grid spacing was significantly
finer in the critical region situated around the tip of the side-
wall obstruction. In the near wake, away from the obstruc-
tion and the other walls, the average mesh spacing was 500–
1500 wall units. The mesh spacing in the critical regions
expressed in nondimensional wall units was similar to the
one used in other DES investigations performed using the
same code.
25,26,31–33
Some of these investigations were per-
formed at comparable or higher Reynolds numbers. Grid de-
pendency studies and/or detailed comparison with experi-
ment were reported in these studies. Although no detailed
experimental data were available to validate the present DES
predictions of the flow past a sidewall obstruction, results
共e.g., the distributions of the nondimensional TKE and tur-
bulence energy production in the symmetry plane upstream
of the cylinder兲 from a DES investigation 共Re=10
6
兲 con-
ducted using the same code of the flow past a surface-
mounted cylinder in a flat-bed channel of depth equal to the
cylinder diameter were shown to be in good agreement with
experiments
8
conducted for the flow past a winged-shaped
cylinder with a thick incoming turbulent boundary layer. A
mesh containing about 8 ⫻ 10
6
cells was used and the dimen-
sions of the computational domain and nondimensional mesh
spacing in the HV region and SSLs were similar to those
considered in the present investigation. In particular, DES
successfully predicted the relative location of the core of
the main necklace vortex in the mean flow relative to the
cylinder.
The mean inflow velocity distribution was obtained from
a periodic channel RANS simulation conducted at Re=5
⫻10
5
. The velocity fluctuations were obtained from a pre-
liminary LES simulation of fully developed turbulent flow in
a periodic channel at a lower Reynolds number 共Re
=18 000兲. The mean velocity field predicted by LES was
subtracted from the instantaneous LES velocity fields to ob-
tain zero-mean fluctuations. In these periodic channel simu-
lations, the channel cross section was identical to the one in
the domain containing the vertical-wall obstruction. The ve-
locity fields 共mean component obtained from RANS and
fluctuating component obtained from LES兲 were then fed in
a time-accurate manner through the inflow section of the
computational domain containing the obstruction. As this
method does not allow generating a “true” fully developed
turbulent channel flow solution at the Reynolds number of
the simulation, the unsteady incoming flow has an artificial
character. However, the method allows obtaining an incom-
ing flow containing unsteady eddies that can jitter the large-
scale coherent structures induced by the presence of the flow
obstruction in a sufficiently realistic way. For the present
simulation in which the flow obstruction is the determining
factor in the formation of the energetically dominant coher-
ent structures in the flow 共e.g., eddies in the HV region, the
SSL, and the wake兲, the physical content of the unsteady
turbulent structures of the inflow is not a key issue. Whether
or not simpler methods to generate synthetic turbulence at
the inflow section can be successfully used instead of per-
forming preliminary LES or DES periodic channel flow
simulations was not investigated. Another option may be per-
forming a periodic channel DES simulation at the Reynolds
number of the simulation containing the flow obstruction. In
this case, the use of a backscatter model
34
is recommended to
reduce the problems associated with the development of an
unphysical DES buffer layer region and generation of “su-
perstreaks” in the channel. These flow phenomena are re-
sponsible for the observed inaccuracies in the prediction of
the mean velocity profile in the DES simulations of periodic
channels.
At the outflow, a convective boundary condition was
used. This boundary condition allows the coherent structures
to exit the computational domain in a time-accurate way and
without producing unphysical oscillations.
35
The free surface
is treated as a rigid, horizontal slip wall. Symmetry boundary
conditions are applied for all the flow variables, with the
exception of the vertical velocity component which is set
equal to zero. This is a reasonable approximation in cases in
which the Froude number 共Fr = U /
冑
gD, g is the gravitational
acceleration兲 is smaller than 0.5.
11,12
Moreover, to check the
validity of the assumption for a test case having the same
nondimensional flow and geometrical parameters as the one
used in DES, we run an experiment in a 20 m long and 3 m
wide flume containing a vertical-wall obstruction. In the ex-
periment, the flow depth was D = 0.53 m and the mean chan-
nel velocity was U =0.9 m / s 共Fr=0.39, Re⬵5 ⫻10
5
兲. The
085104-5 An investigation of the dynamics Phys. Fluids 21, 085104 共2009兲
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experiment confirmed the surface deformations around the
vertical-wall obstacle of width W = 0.8 m 共1.5D兲 were neg-
ligible. No-slip boundary conditions were applied at all the
wall surfaces.
This study considers the case in which the incoming
channel flow is fully turbulent. These conditions correspond
to the ones that are encountered in basically all applications
of interest in river and coastal engineering, which were the
primary motivation for the present study. The incoming
boundary layer thickness can strongly modify the structure
of the HV system when the ratio between the thickness of the
boundary layer at the position of the obstruction and the
obstacle size or channel depth is very small 共⬍0.2兲 . In these
cases the size of the HV region is significantly smaller than
the one forming in a much thicker incoming boundary layer.
These configurations are encountered much less often in
practical applications and are not studied here. As will be
discussed in the result sections, the width of the obstruction
共1.5D兲 was sufficiently large to induce the largest amplifica-
tion of the turbulence inside the HV region at vertical sec-
tions situated around the tip of the obstruction. Again, this
case is the more relevant one for river and coastal engineer-
ing applications, where obstacles of similar shapes 共e.g., spur
dikes and bridge abutments兲 with W / D ⬎ 1 are present.
Based also on our experimental observations, the overall co-
herence of the HV system is expected to slightly increase
with W / D, especially in the region situated around the tip of
the obstruction. However, the physics remains similar. For
the less relevant case for environmental fluid mechanics ap-
plications where W / D ⬍ 1, the interaction between the corner
vortex forming in the recirculation region upstream of the
obstruction and the HV system modifies the flow structure
around the obstacle so the findings of the present study are
not directly relevant. Finally, the ratio between the obstruc-
tion width and the channel width 共⬃0.26兲 was sufficiently
low such that the dynamics of the interactions between the
necklace vortices, the tip of the obstruction, and the eddies
convected in the SSLs were not significantly affected by the
flow blockage.
To investigate the capability of the DES code to accu-
rately capture the dynamics of the unsteady coherent eddies
and turbulence structure in the flow past a vertical-wall ob-
struction, a preliminary DES simulation at Re=18 000 was
performed. The DES simulation at Re= 18 000 used inflow
velocity data sets similar to the ones used in the LES of
Koken and Constantinescu,
11
such that the two solutions are
directly comparable. The wall-normal grid spacing was close
to 0.5 wall units. The distance from the wall at which DES
switches from RANS mode to LES mode was, on average,
20% larger than the one in the DES simulation at Re=5
⫻10
5
. Despite the different numerics and grid topologies
共unstructured versus structured兲 in the two codes, and the use
of a DES model instead of a dynamic Smagorinsky model,
the agreement between the two simulations was found to be
very good. This was true not only for the SSL and for the
wake region downstream of the obstruction, where DES is
expected to perform well, but also for the critical HV region
situated upstream of the obstruction which is tougher to pre-
dict using DES. For example, the levels of amplification of
the TKE inside the HV region in DES at representative sec-
tions were found to be, on average, about 20% lower com-
pared to the LES predictions. Same level of agreement was
found for the circulation of the main necklace vortex in the
mean flow. Moreover, the differences in the position of the
core of the main necklace vortex in the mean flow were
found to be insignificant between the LES and DES simula-
tions. In agreement with LES, DES predicted bimodal veloc-
ity histograms in vertical sections situated close to the tip of
the obstruction, where the turbulence intensity inside the HV
system was the highest. In sections cutting through the leg of
the main necklace vortex situated far from the tip of the
obstruction, the velocity histograms switched to a one-peak
shape 共no bimodal oscillations兲, consistent with the LES re-
sults. Still, because LES using a nondissipative code and a
dynamic Smagorinski model is expected to be able to capture
more correctly the eddy content of a low-Reynolds number
complex turbulent flow, in the present paper we discuss scale
effects based on the comparison between the DES solution at
Re=5⫻10
5
共HR solution兲 and the LES solution at Re
=18 000 共LR solution兲.
IV. HORSESHOE VORTEX SYSTEM
The mean flow coherent structures present in the up-
stream recirculation region are visualized in Fig. 3 using the
Q criterion.
36
In both LR and HR simulations, the role of the
main corner vortex CV1 is to convect fluid and momentum
from the upper levels of the upstream recirculation region
into the core of HV1. Path lines launched into the core of
CV1 close to the free surface 共not shown兲 follow a helicoidal
trajectory before being entrained into the core of HV1. This
happens in both the mean and instantaneous flow fields. As a
result, the shape of CV1 resembles that of a tornadolike vor-
FIG. 3. 共Color online兲 Vortical structure of the mean
flow in the vicinity of the obstruction. The eddies are
educed using the Q criterion. The SSL forming at the
tip of the obstruction was eliminated. 共a兲 Case HR; 共b兲
case LR.
085104-6 M. Koken and G. Constantinescu Phys. Fluids 21, 085104 共2009兲
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tex whose core size decreases with the distance from the free
surface.
At both Reynolds numbers, the cores of main 共HV1兲 and
secondary 共HV2兲 necklace vortices are initially parallel to
the upstream face of the obstruction and then follow the
shape of the SSL. In the HR simulation 关Fig. 3共a兲兴, the legs
of HV1 and HV2 merge at some distance downstream of the
obstruction. On the other hand, HV1 and HV2 remain sepa-
rated in the LR simulation 关Fig. 3共b兲兴. Also, in the HR simu-
lation the coherence of HV2 is larger and the circulation of
HV1 is more than twice that of HV2 in vertical sections that
are perpendicular to the axis of HV1 and are situated be-
tween the sidewall and the tip of the obstruction. However,
in sections situated past the tip of the obstruction, the circu-
lation of HV1 decreases rather sharply, while that of HV2
remains approximately constant. In sections situated close to
the merging region, the circulation of HV1 is only 20%
higher than that of HV2. By comparison, in the LR simula-
tion the circulation of HV2 is less than one-third that of HV1
in all vertical sections cutting through the axes of HV1 and
HV2.
In both simulations the circulation of HV1 peaks at lo-
cations situated around the tip of the obstruction. The ejec-
tion of patches of vorticity of opposite sign to the one inside
the core of HV1 from the channel-bottom region and the
merging with secondary necklace vortices convected from
the region where the boundary layer separates are the two
main mechanisms that modulate the coherence of HV1 in
time. Although the size and coherence of HV1 are subject to
large temporal variations, on average, the main necklace vor-
tex is more stable in the HR simulation.
Snapshots in time showing the distribution of the hori-
zontal vorticity magnitude
␻
h
共D / U兲 in a horizontal plane
situated at 0.05D from the bed 共Fig. 4兲 illustrate the temporal
variations of the structure of the HV system and of the co-
herence of the necklace vortices for case HR. Figure 4共a兲
shows the vorticity distribution at a time instant when HV1
and HV2 are strongly coherent. The relative position and
extent of the cores of HV1 and HV2 are very similar to those
observed in the mean flow fields in Fig. 3共a兲. The structure of
the instantaneous HV system is over about 50% of the simu-
lated time relatively close to the one observed in Fig. 4共a兲.
However, at times the coherence of the HV system can be
very small. This kind of event, illustrated in Fig. 4共b兲,is
generally triggered by the interaction of HV1 with the tip of
the obstruction, or with the vortex tubes shed in the upstream
part of the SSL, or with the small but strongly coherent junc-
tion vortex forming at the upstream base of the obstruction.
When these random interactions occur, the necklace vortices
are strongly jittered and HV1 loses most of its coherence for
a relatively short amount of time 共1–2D/ U兲.
Another interesting phenomenon illustrated in Fig. 5 is
the detachment of patches of vorticity from the downstream
part of the leg of HV1. Similar to case LR, these events
occur randomly in time and can be triggered by the interac-
tion of HV1 with an increasingly coherent HV2 vortex or
with a patch of vorticity convected with the incoming turbu-
lent flow in the near-bed region. HV1 is highly coherent in
Fig. 5共a兲. A short time later the downstream part of its leg
starts separating. The separation is complete 0.8D / U later
关Fig. 5共b兲兴. Before it starts dissipating this patch, containing
mostly horizontal vorticity, can strongly amplify the bed
shear stress beneath it, which can result in sediment entrain-
ment at the bed. After another 1.0D / U 关Fig. 5共c兲兴, the vor-
FIG. 4. 共Color online兲 Horizontal vorticity contours
␻
h
共D / U兲 in a horizon-
tal plane 共z / D=0.05兲 cutting through the cores of the necklace vortices 共case
HR兲. 共a兲 t =t
1
; 共b兲 t =t
2
. The regions of small magnitude of
␻
h
共D / U兲 were
blanked.
FIG. 5. 共Color online兲 Horizontal vorticity contours
␻
h
共D / U兲 in a horizon-
tal plane 共z / D= 0.05兲 cutting through the cores of the necklace vortices 共case
HR兲. 共a兲 t =t
3
; 共b兲 t =t
3
+0.8D / U; 共c兲 t = t
3
+1.8D / U. The regions of small
magnitude of
␻
h
共D / U兲 were blanked.
085104-7 An investigation of the dynamics Phys. Fluids 21, 085104 共2009兲
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ticity levels inside the first patch have decayed considerably
and a second patch starts detaching from the downstream
part of HV1, close to the location where a newly formed
secondary necklace vortex 共HV2兲 tries to merge with HV1.
These events illustrate one of the main mechanisms for the
growth of the scour hole in the lateral direction, away from
the tip of the obstruction, in the initial stages of the scour
process 共flat-bed channel兲. In the HR simulation these events
are not primarily triggered by the interaction of HV1 with
the tip of the obstruction, as was the case in the LR simula-
tion 共see the discussion based on LES results and dye visu-
alizations in Ref. 11兲, but rather by the interaction with the
incoming turbulent eddies and with the leg of HV2 at times
when the coherence of HV2 is relatively high.
Figure 6 shows the distributions of the resolved TKE and
pressure rms fluctuations,
p
⬘
2
, along with the two-
dimensional 共2D兲 mean velocity streamline patterns at sev-
eral vertical sections for case HR. Consistent with previous
experimental and numerical investigation of junction flows,
14
the pressure and velocity fluctuations are strongly amplified
inside the region where the core of HV1 undergoes large-
scale oscillations between two preferred states 共modes兲. The
levels of amplification of the TKE and
p
⬘
2
can be as much as
30 times higher than the levels corresponding to the back-
ground turbulence. By comparison, present results show that
the amplification of the turbulence inside the core of HV2
above the levels associated with the incoming fully turbulent
channel flow is not significant. This suggests that the core of
HV2 does not undergo large-scale oscillations.
Examination of the instantaneous flow fields show that
the flow in the HV region switches aperiodically between
two extreme states that are similar to the zero-flow and back-
flow modes described for the first time by Devenport and
Simpson.
8
Figures 7 shows the instantaneous velocity vec-
tors in a plane cutting through the tip of the obstruction at
two time instants when HV1 is in the zero-flow mode and the
back-flow mode for case HR. When a high-momentum irro-
tational patch of fluid 共i.e., coming from the free surface兲
reaches the upstream face of the vertical-wall obstruction, it
is transported toward the bottom with the downflow. As it
interacts with the bottom of the channel, the patch tries to
preserve its irrotationality and forms a strong near-wall jet in
the reverse flow direction. This wall jet pushes HV1 away
from the obstruction. As a result, the core of HV1 assumes a
more elliptical shape, leading to the back-flow mode 关Fig.
7共b兲兴. The other extreme situation 共zero-flow mode兲 occurs
when a patch of highly rotational and relatively low-
momentum fluid convected from the separating incoming
boundary layer reaches the obstruction. In this case separa-
tion occurs earlier, the core of HV1 has a smaller size and is
situated closer to the obstruction 关Fig. 7共a兲兴. Although some
differences are observed in terms of the occurrence and lo-
cation of the weak recirculation regions upstream of HV1
between cases HR 共Fig. 7兲 and LR 共Fig. 8兲 and the wall jets
in both modes are less strong in case HR, the fundamental
mechanism associated with the switching between the two
modes remains the same.
In case HR, the size of HV1 has more than doubled in
the back-flow mode compared to the zero-flow mode. How-
ever, the distance between the axes of HV1 in the two modes
is only about 0.1D. By comparison, the same distance is
close to 0.2D in case LR. Also, the wall jet is weaker and
more aligned with the bottom wall in case HR compared to
case LR. This results into a smaller average amplitude of the
large-scale oscillations of HV1 as the vortex switches be-
tween the two modes. The difference in the size of HV1
between the two modes is smaller in case LR compared to
case HR. Consequently, in case HR, the pressure rms fluc-
FIG. 6. 共Color online兲 2D mean flow streamline patterns, nondimensional
TKE, k/ U
2
, and pressure rms fluctuations, p
⬘
2
/ 共
␳
2
U
4
兲, in representative ver-
tical sections for case HR. The orientation of the sections is shown in the
inset in frame 共a兲. The vertical dashed line in frame 共c兲 marks the position of
the obstruction.
FIG. 7. 共Color online兲 Instantaneous velocity vectors in a plane cutting
through the tip of the obstruction 关see inset in frame 共c兲兴 showing HV1 in
the zero-flow mode 关frame 共a兲兴 and the back-flow mode 关frame 共b兲兴 for case
HR. Frame 共c兲 shows the distribution of the pressure rms fluctuations,
p
⬘
2
/ 共
␳
2
U
4
兲.
085104-8 M. Koken and G. Constantinescu Phys. Fluids 21, 085104 共2009兲
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tuations in the same plane 关Fig. 7共c兲兴 do not show the
double-peak distribution inside the patch of high p
⬘
2
value
observed in the LR simulation 关Fig. 8共c兲兴. The double-peak
distribution was observed in case LR at all sections where
the amplitude of the bimodal oscillations was sufficiently
high such that the regions of high
p
⬘
2
values associated with
each mode did not merge into a single patch of elliptical
shape.
The levels of amplification of the nondimensional values
of
p
⬘
2
inside the region where bimodal oscillations are
present are similar in the two simulations at equivalent sec-
tions 关e.g., compare Figs. 7共c兲 and 8共c兲兴. However, the patch
of high
p
⬘
2
values is situated closer to the channel bed in
case HR. In case LR 关Fig. 8共c兲兴 the values of
p
⬘
2
at the bed
in the HV region are less than 10% of the peak levels inside
the patch of high
p
⬘
2
values and are comparable to the levels
in the incoming turbulent flow, whereas in case HR the val-
ues of
p
⬘
2
at the bed are only 30%–40% smaller than the
maximum levels inside the same patch. This means that
based on the level of the turbulent pressure fluctuations at the
bed, the capacity of HV1 to entrain sediment particles from
the bed is significantly higher in case HR compared to case
LR, especially in the region situated around the tip of the
obstruction.
In sections cutting the axis of HV1 perpendicularly and
situated away from the tip of the obstruction, closer to the
sidewall containing the obstruction 关e.g., section 共a兲 in Fig.
6兴 or cutting through the leg of HV1 关e.g., section 共c兲 in Fig.
6兴, the shape of the patch of high
p
⬘
2
values becomes closer
to circular. This is the first indication that there is a sharp
decay of the amplitude of the bimodal oscillations away from
the region surrounding the tip of the obstruction. The vertical
band of large values of
p
⬘
2
situated to the right of the HV
region in Fig. 6 corresponds to the downflow 关section 共a兲兴 in
which eddies are convected toward the channel bottom close
to the upstream face of the obstruction or to the SSL 关sec-
tions 共b兲 and 共c兲兴 in which energetic vortex tubes are shed.
Observe the interaction between the regions of high turbu-
lence amplification associated with the HV system and the
SSL in the sections situated downstream of the obstruction.
The TKE distributions are qualitatively very similar in
the two simulations. As shown in Fig. 6 for case HR, the
TKE amplification inside the HV system peaks in sections
situated around the tip of the obstruction 关e.g., section 共b兲 in
Fig. 6兴. There are two regions of high TKE amplification
induced by the bimodal oscillations of HV1. The first one is
situated close to the bed and is due to the variations in the
extent and strength of the wall jet forming beneath HV1
关e.g., compare Figs. 7共a兲 and 7共b兲兴. The second one is asso-
ciated with the changes in the position of the core of HV1.
The presence of these two regions of high TKE amplification
in sections where the strength of the bimodal oscillations is
high was also observed in measurements
8
conducted in the
symmetry plane of the flow past a wing-shaped cylinder. In
the sections situated around the tip of the obstruction, the
region of high positive values of the turbulence production
term in the transport equation for the TKE, P
TKE
, has a
mushroomlike shape 关Fig. 9共a兲兴. A region of negative values
of P
TKE
is situated downstream of the patch of high positive
values. Both of these features of the distribution of P
TKE
in
the HV region were observed in experiments.
8
As shown in
Fig. 9, the distributions of the nondimensional values of
P
TKE
in the region where the bimodal oscillations are strong
are qualitatively similar in the LR and HR simulations. In the
HR simulation, the patch of high positive P
TKE
is situated
closer to the bed, which is consistent with the changes ob-
served in the mean position and the size of the core of HV1
because of the increase in the Reynolds number. Observe
also the presence of a secondary region containing relatively
high values of P
TKE
in the HR simulation. This region is
situated upstream of the main patch of positive values of
P
TKE
, around the location of HV2 in the mean flow. There is
no noticeable amplification of P
TKE
at the location of HV2 in
the LR simulation 关see Fig. 9共b兲兴.
In both simulations, as one approaches the sidewall to
which the flow obstruction is attached 关e.g., section 共a兲 in
FIG. 8. 共Color online兲 Instantaneous velocity vectors in a plane cutting
through the tip of the obstruction 关see the inset in frame 共c兲兴 showing HV1
in the zero-flow mode 关frame 共a兲兴 and the back-flow mode 关frame 共b兲兴 for
case LR. Frame 共c兲 shows the distribution of the pressure rms fluctuations,
p
⬘
2
/ 共
␳
2
U
4
兲.
FIG. 9. 共Color online兲 Distribution of the turbulence
production term in the transport equation for the TKE,
P
k
共D / U
3
兲, in section 共b兲共see the inset in Fig. 6兲. 共a兲
Case HR; 共b兲 case LR. The regions of low magnitude of
the turbulence production 共兩P
k
共D / U
3
兲兩⬍ 0.04兲 were
blanked.
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Fig. 6兴, the size of the patch of high TKE values increases
but the TKE levels are lower compared to those observed at
section 共b兲. At sections cutting through the leg of HV1 关e.g.,
section 共c兲兴, the size of the region of TKE amplification and
the TKE levels are decaying monotonically with the distance
from the tip of the obstruction. Same is true for the region of
high positive values of P
TKE
. In all the three sections ana-
lyzed in Fig. 6 for the HR simulation, the nondimensional
TKE levels in the HV region are by about 20%–40% lower
than the ones observed in the LR simulation 共see Fig. 8 in
Ref. 11兲, which is fully consistent with the results shown in
Fig. 9 for the distribution of P
TKE
in section 共b兲.
Thus, the overall effect of the increase in the Reynolds
number is to make the main necklace vortex more stable and
to decrease the amplitude of the bimodal oscillations in the
horizontal plane. The bimodal nature of the oscillations of
the core of HV1 in the HR simulation is proven by the shape
of the histograms of the probability density function of the
velocity in Fig. 10. Similar to case LR 共e.g., see Fig. 11 in
Ref. 11兲, the histograms inside the core of HV1 display a
one-peak shape at a distance of more than 1D from the tip of
the obstruction. This proves that the downstream part of the
leg of the main necklace vortex is not subject to bimodal
oscillations, which is consistent with the previous discussion
of the change in the shape of the region of high
p
⬘
2
values
from elliptical, in sections situated close to the tip of the
obstruction, to circular, in sections situated downstream of
the obstruction. The analysis of the velocity histograms con-
firmed the absence of bimodal oscillations in the region
where HV2 is situated in the mean flow.
Some of the velocity histograms in Fig. 10 show an in-
teresting feature, not present in case LR. The three peaks
present in the velocity histograms at sections 共a兲 and 共b兲 in
Fig. 10 suggest the presence of three modes. However, in-
spection of the time series shows that the large-scale oscilla-
tions remain bimodal. What happens is that very large-scale
temporal modulations 共⬎20D / U兲 are present in some of the
velocity time series. Such large-scale temporal modulations
were also observed in a high Reynolds number DES simula-
tion of flow past an infinitely long cylinder.
37
As a result of
these large-scale temporal modulations, the threshold value
associated with the mode switching should change in time to
remain consistent with the large-scale oscillations observed
in the velocity time series. Once that is done, the histograms
display a two-peak shape even at sections 共a兲 and 共b兲. The
horizontal bar in the histograms at sections 共a兲 and 共b兲 in Fig.
10 indicates which are the two peaks corresponding to the
vortex being in one mode. The third peak corresponds to the
other mode. The average nondimensional time corresponding
to successive mode switching is 2.8D / U and is about 20%
lower than the value estimated for case LR. Devenport and
Simpson
8
estimated a value of 4.14D / U for the flow past a
winged-shaped cylinder, where D is the maximum width of
the cylinder. In their experiment the thickness of the bound-
ary layer at the location of the body was about 0.5D.
The range of dominant nondimensional frequencies 共St
= fD/ U, f is the frequency兲 at which HV1 switches from one
mode to the other and then back is 0.1⬍St⬍0.3 for the
sidewall obstruction case. The mean value corresponding to
the observed average period HV1 spends in one mode is St
=0.18. The range of dominant frequencies associated with
the shedding of secondary necklace vortices from the region
where the incoming boundary layer separates is St
⬃0.3–0.5 共mean value St= 0.4兲 for the HR case. The mean
value was St=0.32 in the LR case. The ratio between the
mean Strouhal number associated with the bimodal oscilla-
tions and the Strouhal number associated with shedding of
necklace vortices in the region where the incoming boundary
layer separates is about 0.45 for the HR case, which is close
to the value 共0.5兲 determined experimentally.
8
Figure 13共a兲 visualizes the vortical structure of the in-
stantaneous flow around the obstruction for case HR. Al-
though the structure of the flow in the HV region is very
complex and the core of HV1 is locally disturbed by its
interaction with the surrounding turbulent eddies, no large-
scale hairpinlike vortices were observed to form around the
core of HV1 during the transition to the back-flow mode, as
observed in a DES simulation with steady inflow conditions
at a comparable Reynolds number.
19
V. SEPARATED SHEAR LAYER AND WAKE REGIONS
Figures 11 and 12 illustrate the main types of interac-
tions through which the vortex tubes in the SSL are jittered
by the eddies detaching from the boundary layer separating
on the sidewall, upstream of the obstruction. For example, in
Fig. 11共a兲 the vorticity patch 共VP1兲 present inside the de-
tached shear layer bordering the main corner vortex is con-
vected over the tip of the obstruction and starts interacting
with the last vortex tube 共VT1兲 shed in the SSL. The follow-
ing patch of vorticity advected in the detached shear layer
共VP2 in Fig. 11共b兲兲 is also convected past the tip of the
obstruction and past the newly formed vortex tube VT2 be-
FIG. 10. 共Color online兲 Velocity histograms in four vertical sections at the
location of the axis of HV1 in the mean flow 共case HR兲. The positions of the
sections are shown in the inset in Fig. 6.
085104-10 M. Koken and G. Constantinescu Phys. Fluids 21, 085104 共2009兲
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fore merging with one of the eddies convected inside the
upstream part of the SSL.
At other time instances, the vorticity patches that are
shed in the detached shear layer follow a trajectory which
intersects the upstream wall of the obstruction 关e.g., VP3 in
Figs. 12共a兲 and 12共b兲兴. Then, the patch of vorticity is con-
vected parallel to the face of the obstruction 关Fig. 12共b兲兴 and
merges with the vortex tube forming at the tip of the obstruc-
tion 关Fig. 12共c兲兴. The frequency of occurrence of events
similar to those illustrated in Fig. 12 is much higher com-
pared to that observed for case LR.
Finally, a third mechanism responsible for the jittering of
the SSL in the near-bed region is the interaction of the newly
shed eddies with the leg of the small junction vortex forming
at the base of the upstream face of the obstruction. At times,
the coherence of this vortex increases and the downstream
part 共leg兲 of the junction vortex extends some small distance
past the tip of the obstruction. The intrusion of the leg
strongly disturbs the shedding and interactions among the
SSL eddies in the near-bed region.
The vorticity distribution inside the SSL is more irregu-
lar in case HR, in particular, over the lower part of the chan-
nel 共z / D ⬍0.3兲 where most of the vortex tubes are strongly
deformed, starting in the formation region. This is clearly
observed by comparing Fig. 13共a兲 共case HR兲 and Fig. 13共b兲
共case LR兲. In the HR simulation the core of the second vor-
tex tube is strongly deformed in the vertical direction. As a
result, the vorticity vector inside the vortex tube reorients in
a direction that makes a relatively low angle with the bed.
Through this mechanism additional horizontal vorticity is in-
duced in the near-bed region. This has obvious consequences
on the sediment entrainment phenomena beneath the SSL. As
one moves downstream, the growth of secondary instabilities
is very rapid, such that at a distance of 1.5D from the ob-
struction large-scale eddies are connecting the distorted cores
of the vortex tubes. By comparison, the cores of the vortex
tubes remain close to vertical in case LR.
As shown in Figs. 13共c兲 and 13共d兲, large-scale hairpin-
like eddies are observed at several locations inside the SSL,
in the HR simulation. However, the two legs of the hairpins
are not contained in a plane parallel to the bed, as expected
to happen if these coherent structures would form as a result
of the interaction of the incoming turbulent flow with the
bed. Rather, the legs of the hairpins are contained in a plane
that is close to vertical. The large-scale hairpin structures
form as a result of the growth of secondary instabilities as-
sociated with the transition between the high-speed flow on
the outward side of the SSL and the low-speed flow inside
the main recirculation region in the wake. The hairpins are
contained mostly inside the SSL and the high-speed flow
side. The presence of large-scale hairpin structures in the
transition region 共shear layer兲 between a high-speed stream
of fluid and a low-speed region was also observed in flow
past cavities
27
and at the interface between a high Reynolds
number lock-exchange gravity current and the surrounding
flow. At some locations, the vorticity levels in the legs of the
hairpin structures are comparable to those in the vortex
tubes. In particular, examination of the instantaneous flow
fields show that the lower leg is situated, at times, at small
FIG. 11. Instantaneous-flow out-of-plane vorticity,
␻
n
共D / U兲, in a horizontal
plane 共z / D = 0.8兲 showing successive patches of vorticity 共VP1, VP2兲 being
convected over the tip of the obstruction and interacting with the vortex
tubes 共VT1, VT2兲 present in the upstream part of the SSL 共case HR兲. 共a兲 t
=t
4
; 共b兲 t =t
4
+0.8D / U.
FIG. 12. Instantaneous-flow out-of-plane vorticity,
␻
n
共D / U兲, in a horizontal
plane 共z/ D =0.8兲 showing a patch of vorticity 共VP3兲 being convected toward
the obstruction and then parallel to the upstream face of the obstruction
before disturbing the flow in the region where the vortex tubes are forming
共case HR兲. 共a兲 t =t
5
; 共b兲 t =t
5
+1.6D / U; 共c兲 t = t
5
+2.0D / U.
085104-11 An investigation of the dynamics Phys. Fluids 21, 085104 共2009兲
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distances from the bed and can induce large bed shear stress
values.
Figure 14 visualizes the main vortical eddies in the wake
region. A strongly coherent vortex VA is observed near the
junction line between the bed and the sidewall. The flow
inside the core of this vortex moves in the upstream direction
for x / D ⬍ 13 and in the opposite direction for x/ D ⬎13 共x is
measured from the center of the obstruction兲. This value is
very close to the one observed in the LR simulation. This
feature of the flow inside vortex VA was confirmed by dye
visualization experiments conducted at lower Reynolds
numbers.
11
The flow moving in the upstream direction is
re-entrained into the recirculation eddy VB present immedi-
ately downstream of the obstruction. In the case of a loose
bed, vortex VA plays an important role in the formation of
the elongated deposition hill of sediment close to the
sidewall.
13
In the mean flow, the sizes of the main recirculation
region downstream of the obstruction are very close in the
LR and HR simulations. The SSL reattaches at x/ D ⬃13.5 in
both simulations. However, in the HR simulation the up-
stream part of the SSL is oriented at a lower angle with the
streamwise direction. In fact, close to the bed 关e.g., z / D
=0.1 in Fig. 15共b兲兴 the SSL is nearly parallel to the stream-
wise direction. By comparison, the angle between the up-
stream part of the SSL and the streamwise direction was
close to 25° in the LR simulation at z / D =0.1–0.2 共see Fig.
16 in Ref. 11兲. The maximum width of the recirculation re-
gion in the mean flow is 20% larger in the LR simulation.
Another difference observed in the near-bed region 关e.g., at
z/ D =0.1 in Fig. 15共b兲兴 is the presence in the HR simulation
of an elongated region of relatively high-vorticity values up-
stream of the one corresponding to HV1. This region is in-
duced by the presence of a relatively stable secondary neck-
lace vortex HV2. No such region was observed in the LR
FIG. 13. 共Color online兲 Visualization
of vortical structure of the instanta-
neous flow around the obstruction us-
ing the Q criterion. 共a兲 General view
共case HR兲; 共b兲 general view 共case LR兲;
共c兲 top view 共case HR兲; 共d兲 detail view
showing the presence of hairpinlike
structures in the SSL 共case HR兲.
FIG. 14. 共Color online兲 Visualization of the bed-sidewall junction vortex VA
and the main vortex in the recirculation region downstream of the obstruc-
tion, VB, using 3D mean flow streamlines 共case HR兲. The arrows show the
direction of the flow.
FIG. 15. Distribution of the vorticity magnitude in the mean flow 共left兲 and
instantaneous flow 共right兲 for case HR. 共a兲 z/ D =1 共free surface兲; 共b兲 z / D
=0.1 共near-bed region兲.
085104-12 M. Koken and G. Constantinescu Phys. Fluids 21, 085104 共2009兲
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simulation 共see Fig. 16 in Ref. 11兲 in which the secondary
necklace vortices were very intermittent and, on average,
much less coherent.
Similar to case LR, a wide range of energetic scales is
observed in the instantaneous distributions of the vorticity
magnitude in the z / D =0.1 and z / D =0.9 sections which are
shown in Fig. 15. The nondimensional vorticity levels asso-
ciated with the eddies convected inside the downstream part
of the SSL are, at most times, larger than the ones associated
with the eddies present inside the main recirculation region
in the wake. This is observed over the whole depth of the
flow. By comparison, in case LR 共see Fig. 16 in Ref. 11兲 the
nondimensional vorticity levels in the downstream part of
the SSL and wake recirculation region were similar. This
shows that the SSL is more energetic in the HR simulation
共recall, this comparison is done for the nondimensional vari-
ables兲. This result is consistent with the richer range of ed-
dies and stronger nonlinear interactions among the eddies
populating the SSL, observed in the HR simulation 共see dis-
cussion of Fig. 13兲. Another consequence of the fact that the
SSL is more energetic in case HR is that the capacity of the
SSL eddies to entrain sediment from the loose bed during the
initial stages of the scour process will be higher. As the ed-
dies inside the SSL maintain their coherence in the near-bed
region over much longer distances 共about 50% longer for
same level of the nondimensional vorticity magnitude兲 com-
pared to case LR, the surface over which the SSL eddies can
entrain sediment is expected to be higher in case HR. As
observed from Fig. 15共b兲, the instantaneous values of the
vorticity magnitude inside these eddies are couple of times
higher than the mean values in the same region of the SSL.
The difference between the instantaneous and mean values of
the bed shear stress is expected to be of the same order.
VI. SEDIMENT ENTRAINMENT CAPACITY
AT THE CHANNEL BED
Knowledge of the spatial and temporal distributions of
the bed shear stress and of quantities characterizing the tur-
bulence intensity in the near-bed region is critical for deter-
mining the incipient motion of sediment particles in the case
where the channel bed is loose. Such information is very
difficult to obtain from experiments.
Figures 16共a兲 and 16共b兲 visualize the distribution of the
bed friction velocity, u
␶
/ U, for case HR at an arbitrary time
instant and in the mean flow, respectively. Based on the
Shields diagram, the threshold value for sediment entrain-
ment for a sediment size of d
50
=1.05 mm is u
␶
c0
/ U = 0.056.
This is the typical sediment size employed in local scour
experiments performed in the same flume that was used to
check the validity of the rigid lid approximation for case HR
共D= 0.53 m, U=0.9 m / s兲. The regions where sediment is
expected to be entrained due to large values of the bed fric-
tion velocity 共u
␶
⬎u
␶
c0
兲 are delimited in Figs. 16共a兲 and
16共b兲 using a solid line. The presence of relatively high pres-
sure fluctuations at the bed may produce sediment entrain-
ment even if u
␶
⬍u
␶
c0
.
The largest values of u
␶
共⬃0.07–0.1U兲 in the mean flow
field occur beneath the upstream part of the SSL and the
main necklace vortex. One should stress that an important
contribution to the magnitude of the friction velocity in the
region situated around the tip of the obstruction comes from
the amplification of the near-bed velocity due to the accel-
eration of the flow as it passes the obstruction, rather than to
the presence of HV1. Similar to the LR simulation, the bed
friction velocity beneath the upstream part of HV1 remains
lower than u
␶
c0
. A distinct patch of relatively high u
␶
values
共⬃0.065U兲 associated with HV2 can be observed in Fig.
16共b兲. A third region of relatively high bed friction velocity
共⬃0.06U兲 is present between the outward side of the SSL
and the sidewall that does not contain the obstruction. This
region is associated with the increase in the mean streamwise
velocity due to the decrease in the effective area over which
the flow is convected as it passes the obstruction 共contraction
scour effects兲.
The distribution of u
␶
in the instantaneous flow 关Fig.
16共a兲兴 is qualitatively similar to the one observed in the
mean flow 关Fig. 16共b兲兴. The main difference is the presence
of streaks of high bed friction velocity beneath the SSL re-
gion in the instantaneous flow. The primary reason for the
formation of these streaks is the presence of hairpinlike
structures in the vicinity of the bed 共see discussion of Fig.
13兲. The amplification of u
␶
inside these streaks can be im-
portant, up to two times the mean value of u
␶
at the same
location. The time series of u
␶
at such a point 共Fig. 17兲
clearly captures the effect of the passage of these highly
energetic coherent structures at a small distance from the
bed. In the upstream part of the SSL 共distances less than 1D
from the tip of the obstruction兲, it is the passage of the vortex
tubes that is mainly responsible for the local amplification of
u
␶
. Figure 16共c兲 shows the distribution of u
␶
in the instanta-
neous flow for case LR. Consistent with the eddy content of
FIG. 16. Distribution of bed friction velocity. 共a兲 Instantaneous flow, case
HR; 共b兲 mean flow, case HR; 共c兲 instantaneous flow 共case LR兲. The solid
contour line in frames 共a兲 and 共b兲 corresponds to u
␶
c0
/ U = 0.056.
085104-13 An investigation of the dynamics Phys. Fluids 21, 085104 共2009兲
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the SSL in the LR simulation 关no hairpinlike structures are
present in Fig. 13共b兲兴, no streaks of high u
␶
values are ob-
served beneath the downstream part of the SSL.
Figure 18 provides more details on the temporal varia-
tion of the instantaneous bed shear stress distribution,
␶
=
␳
u
␶
2
. At time instances at which the coherence of HV1 is
very high 关e.g., see Fig. 18共a兲兴, the amplification of the bed
shear stress beneath HV1 can be larger than the one typically
induced by the convection of the vortex tubes in the forma-
tion region. However, animations show that at most times,
␶
is the largest in the region where the vortex tubes are form-
ing 关e.g., see Fig. 18共b兲兴. Observe also the presence of a large
patch of high
␶
values 共see arrow兲 in the downstream part of
the SSL in Fig. 18共b兲. The analysis of the instantaneous flow
fields show that the patch formed due to the merging of a
vortex tube with a high-vorticity eddy convected toward the
outward part of the SSL in the near-bed region, similar to the
scenario discussed in Fig. 11. As it is convected downstream,
the eddy is stretched and eventually dissipates.
The other main factor that determines the capacity of the
flow to entrain sediment particles from the bed is the level of
the pressure rms fluctuations. The distribution of
p
⬘
2
/ 共
␳
2
U
4
兲
at the bed is plotted in Fig. 19. In both simulations,
p
⬘
2
is
large around the tip of the obstruction and beneath the SSL.
Observe also the relatively high
p
⬘
2
values around the region
where the flow reattaches to the sidewall 共x / D ⬃13兲 and, in
particular, the amplification of
p
⬘
2
beneath the core of vortex
VA 共Fig. 14兲. Although the distributions are qualitatively
similar, the nondimensional
p
⬘
2
levels are, on average, two
times larger in the HR simulation. This increase is explained
by the richer eddy content of the SSL, especially in the near-
bed region.
VII. SUMMARY AND CONCLUSIONS
The present numerical investigation of the flow past a
vertical-wall obstruction in a flat-bed channel provided a bet-
ter understanding of the flow physics and dynamics of the
coherent structures, in particular, of the large-scale eddies
and mechanisms responsible for sediment entrainment
around the flow obstruction. The relatively high Reynolds
number of the HR DES simulation 共Re=5⫻ 10
5
兲 and the
presence of unsteady velocity fluctuations in the incoming
flow allowed us to study these mechanisms at conditions that
are close to those encountered in engineering applications,
while using computational resources that are much smaller
than those required by sufficiently well-resolved LES with-
out wall functions. Comparison with results from the LR
simulation that was conducted at a lower Reynolds number
共Re= 18 000兲 allowed us to better understand scale effects.
The quantification of scale effects is of great interest as most
experimental studies of junction flows that are relevant for
environmental engineering and, in particular, of local scour
studies are conducted in laboratory flumes at Reynolds num-
bers that are much lower than those encountered in the field.
The structure and dynamics of the HV system in the two
simulations were found to be qualitatively similar. The core
of the main necklace vortex HV1 was shown to undergo
bimodal aperiodic oscillations, similar to the ones observed
in experimental investigations of other junction flows.
14
The
intensity of the bimodal oscillations was found to peak at
sections cutting through the tip of the obstruction, where the
overall amplification of the turbulence intensity 共e.g., TKE
and pressure rms fluctuations兲 was the highest. The bimodal
oscillations gradually disappeared in the leg of HV1. As a
result of the increase in the Reynolds number, the main neck-
lace vortex became more stable, the size of its core de-
creased, and the average amplitude of the aperiodic oscilla-
tions got lower. It is also significant to mention that the shape
of the main necklace vortex and turbulence structure in the
high Reynolds number DES simulations at Re⬎10
5
show
better agreement than predictions obtained from LES/DES
simulations at Re=O 共10
4
兲 when compared to the ones deter-
mined experimentally
8
at Re⬃1.15⫻ 10
5
. Additionally, the
secondary necklace vortex was more coherent, the nondi-
mensional pressure rms fluctuations at the bed were signifi-
cantly stronger beneath the HV region and the nondimen-
sional TKE values inside the HV system were 20%–40%
smaller in the HR simulation.
FIG. 17. Time series of bed friction velocity at point P1 关see Fig. 16共a兲兴.
The arrows show the relative maxima that are induced by the passage of
energetic SSL eddies 共e.g., lower legs of hairpinlike vortices兲 at a small
distance from the bed 共case HR兲.
FIG. 18. Distribution of the bed shear stress,
␶
/
␳
U
2
, in the instantaneous flow 共case HR兲: 共a兲 t =t
6
; 共b兲 t =t
7
.
085104-14 M. Koken and G. Constantinescu Phys. Fluids 21, 085104 共2009兲
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In both simulations, the presence of velocity fluctuations
in the incoming channel flow did not allow the formation of
large-scale hairpinlike eddies around the core of the necklace
vortices, similar to those observed in the study of Paik et
al.
19
Based also on the LR simulation results and the findings
of other studies,
26,27,33
we think that this large-scale instabil-
ity can develop only if the incoming flow contains velocity
fluctuations of very low intensity. Also, the present study
demonstrates that bimodal quasiperiodic oscillations can
form independent of the presence of this secondary instabil-
ity. Thus, in practical applications 共e.g., in river and coastal
engineering兲 in which the incoming flow is strongly turbu-
lent and contains turbulent eddies over a wide range of
scales, we do not expect this secondary instability to play a
major role.
Similar to the LR simulation, the coherence of HV1
along the direction defined by its axis in the mean flow was
found to be highly variable in time. Patches of vorticity were
observed to detach from the leg of HV1 and to be convected
at a small distance from the bed. Before dissipating, these
patches were observed to locally induce relatively large val-
ues of the bed shear stress. In contrast to the LR simulation,
the formation of these patches was primarily determined by
the interaction of HV1 with the leg of the secondary necklace
vortex rather than to the interaction of HV1 with the tip of
the obstruction.
The jittering of the SSL was mainly due to the convec-
tion of highly vortical eddies from the boundary layer sepa-
rating on the sidewall, upstream of the obstruction, toward
the extremity of the obstruction wall. Some of these eddies
were convected over the tip of the obstruction and interacted
directly with the vortex tubes in the upstream part of the
SSL, while others reached the upstream face of the obstruc-
tion wall, were then convected parallel to the face of the
obstruction and jittered the SSL in the region where vortex
tubes are forming.
A main finding of this study is that the eddy content of
the SSL changes considerably between the two Reynolds
numbers. The degree of deformation of the cores of the vor-
tex tubes in the upstream part of the SSL was much larger in
the HR simulation. This provided an additional mechanism
for the amplification of the horizontal vorticity in the near-
bed region. Additionally, the secondary instabilities acting on
the cores of the vortex tubes were much stronger in the HR
simulation. Large-scale hairpinlike eddies formed in the
downstream part of the SSL. The legs of these vortices were
oriented parallel to the interface between the high-speed
outer flow and the recirculating flow behind the obstruction.
The lower leg of some of these hairpin vortices was situated,
at times, at a small distance from the bed and was able to
strongly amplify the bed shear stress. Consequently, in the
HR simulation the bed shear stress distribution in the instan-
taneous flow fields displayed a streaky structure over part of
the SSL region. Similar to the LR simulation, the largest
values of the bed shear stress occurred in the strong accel-
eration region around the tip of the obstruction, and beneath
the main necklace vortex and the upstream part of the SSL.
The overall nondimensional levels of the bed shear stress in
the instantaneous and mean flow were smaller in the HR
simulation. This is consistent with the expected decrease of
the nondimensional bed shear stress with the increase in the
Reynolds number in fully developed turbulent channel flows.
As a result of the richer eddy content of the SSL in the
HR simulation, the values of the nondimensional pressure
rms fluctuations beneath the SSL were found to be around
two times larger compared to the LR simulation. This means
that flume experiments conducted at relatively low-Reynolds
numbers substantially underestimate the effect of the pres-
sure fluctuations on sediment entrainment.
ACKNOWLEDGMENTS
The authors would like to thank Taiwan’s National Cen-
ter for High Performance Computing 共NCHC兲 and the Trans-
portation Research and Analysis Computer Center 共TRACC兲
at the Argonne National Laboratory for providing the com-
putational resources needed to perform most of the simula-
tions.
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085104-15 An investigation of the dynamics Phys. Fluids 21, 085104 共2009兲
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