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The validity of 2D numerical simulations of vortical structures around a bridge deck
MATHEMATICAL COMPUTER MODELLING Mathematical
PERGAMON
and Computer
Modelling
37 (2003) 795-828 www.elsevier.com/locate/mcm
The Validity of 2D Numerical Simulations of Vertical Structures Around a Bridge Deck L. BRUNO* Department of Structural Engineering Politecnico di Torino Viale Mattioli 39, 10126 Torino, Italy [email protected]
S. KHRIS OptiFlow, Consulting Company in Numerical Fluid Mechanics IMT-TechnopBle de Chateau-Gombert, Cedex 20, 13451 Marseille, France khrisQoptiflow.fr (Received February 2002; accepted April 2002) Abstract-This paper deals with a computational study for evaluating the capability of 2D numerical simulation for predicting the vertical structure around a quasibluff bridge deck. The laminar form, a number of BANS equation models, and the LES approach are evaluated. The study was applied to the deck section of the Great Belt East Bridge. The results are compared with wind-tunnel data and previously conducted computational simulations. Sensitivity of the results with regard to the computational approaches applied for each model is discussed. Finally, the study confirms the importance of safety-barrier modelling in the analysis of bridge aerodynamics. @ 2003 Elsevier Science Ltd. All rights reserved.
Keywords-Computational wind engineering, Bridge aerodynamics, methods, Turbulence modelling.
Vortex shedding, Grid-based
1. INTRODUCTION The study of aerodynamic characteristics plays a prominent role in the design of long-span bridges. In particular, the vortex-induced forces acting on the bridge deck are of great interest from the practical point of view. In fact, this phenomenon occurs at a relatively low range of wind speeds and can markedly affect the durability and serviceability of the structure. Traditionally, these studies are experimentally carried out by means of wind-tunnel tests, which are economically
burdensome and in some cases not realistic, owing to the lack of aerodynamic
similitude of the model.
In order to respect the conditions of aerodynamic similarity without.
using cumbersome models, wind-tunnel tests often employ section models in the preliminary bridge design. In fact, rigidly mounted models provide a good description of the complex flows that develop around bluff-shaped cylinders at high Reynolds numbers. The first author wishes to express his appreciation to the Fluent Italia and the Hewlett Packard Italia Corporations *Author to whom all correspondence should be addressed. 0895-7177/03/$ - see front matter @ 2003 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(03)00087-S
Typeset
by d,@‘IEX
L. BRUNO AND S. KHRIS
796
A number of fundamental studies, critically reviewed by Buresti [l], have investigated the unsteady
flow around
basic
bluff sections,
such as circular,
results obtained from such studies represent a useful of industrial fields, such as bridge aerodynamics. Generally
speaking,
ticity and by massive
flows of this kind are characterized separation
which involves
and in its wake.
important content.
role in the production of aerodynamic Three-dimensional structures ultimately
spite the nominally
Both of these phenomena,
two-dimensional
geometry
or rectangular
ones.
applications
motion
of the vertical
which are strictly
The
in a number
by shear layers presenting
unsteady
the body
square,
basis for further
marked
structures
vor-
around
interconnected,
play an
forces in terms of mean value and frequency appear in the process of wake formation de-
of the body,
as a result
of the vortex
lines being
distorted. The case study of the rectangular section best applies to the shape of long-span bridge decks. In regard
to these
cylindrical
bodies,
it is widely
recognized
that
their
width-depth
B/D
ratio
strongly affects the characteristics of the separated shear layer that is generated at the leading edge [2]. From this point of view, such sections are classified into two categories, namely, the “separated-type”
sections
cross-sections) latter
(compact
[3]. The former
present
more complex
cross-sections)
are generally
phenomena,
and B/D = 6 or the double
mode
and the ‘?eattached-type”
prone to Karman-type
such as discontinuities
in the lift fluctuations
sections
vortex
shedding,
in Strouhal
the
at B/D = 2.8
number
due to unsteady
(elongated whereas
reattachment
of the
separated shear layer on the side surfaces at 2.0 < B/D < 2.8 [4]. Likewise, most of the bridge decks are quite elongated (B/D z 7) so that occurrence of a reattached shear layer calls for a proper evaluation of other characteristics of the flow, such as the extent of the separation the convection of the vortices along the surfaces of the deck, and the interaction between shed from the windward During
the last
plementary efficiency tunnel
and leeward
decade,
computational (accuracy-cost
tests.
ratio)
To achieve
is ,required
to the potential
flow (see, for example,
Reynolds-averaged A number accuracy time,
imposed
mensions, markedly
The latter
Navier-Stokes
(RANS)
boundary
and grid size. inflates
(NSEs).
numerical
memory
of numerical
equation
parameters
in grid-based conditions,
procedures
simplified
introduced decks.
that
a com-
An increased
as against
have been proposed.
methods
(panel involve
may be further
(DNS),
large-eddy
windThese
methods)
applied
numerical
solution
classified
simulation
as regards (LES),
and
models.
have to be taken methods:
extension
On the other
category
simulation
of bridge
to be attractive
[6]), as well as approaches
numerical
simulation
has gradually
analysis
approach
categories:
into direct
of other
of the
into two main
equations
modelling
computers
for this
this goal, a number
classified
turbulence
of powerful
to the aerodynamic
can be broadly
of the Navier-Stokes
edges [5].
the advent approach
bubble, vortices
into consideration
discretization
of the computational
schemes
to increase
in both
domain
space
in two or three
the and di-
hand,
3D analysis and refinement of spatial resolution Murakami [i’] reports that usage and processing time. For instance,
LES computation of vortex-shedding flow past a square section requires only 60 hours for the 2D model but more than 42 days for the 3D model on a convex ~240 machine. As may be readily understood, even when possible the extension of 3D simulations to industrial applications characterized by complex geometries and high Reynolds numbers (as in the case of long-span bridge decks) is likely to prove extremely burdensome and excessively time-consuming as compared to wind-tunnel tests. In order to reduce the computational costs, many authors have examined the applicability of the above-mentioned approaches to the 2D simulations of vortex-shedding flow past a square cylinder. According to the authors of the present paper, three main studies obtained the most representative results. Tamura et al. [8] discussed the reliability of two-dimensional direct numerical simulation for They emphasized the importance of the mix unsteady flows around cylinder-type structures. between grid size and discretization an excessive amount of numerical
schemes diffusion.
of the nonlinear convection term According to the above authors,
in order to avoid two-dimensional
797
2D Numerical Simulations calculations
cannot
in the process
limitations
properly
simulate
of 2D direct
More recently, square cylinder
the three-dimensional
Consequently,
of wake formation. numerical
simulation
50% with respect
with the Smagorinsky
order to compensate
for the limited
LES computations
with the experimental
results,
discrepancies.
out a wide spectrum
square
According
other
mechanism,
out a critical
sections
enables
hand,
and extended
two-dimensional
an incompleteness
present
by 2D computations
review
the spanwise model
number
analysis,
related
this
through-
R.ANS simula-
diffusion
sections.
process
They
by an eddy
region.
On the
in the underesti-
cross sections. further
to such flows is very high
respecting
using
rectangular
was detected
study of unsteady flows around bridge decks introduces both to physical and to computational aspects. Reynolds
cert,ain
it is fundamentally
even in the high Reynolds-number
The related
the
present
mechanism
performed
lift fluctuations
First,
as well as with those
a close correspondence
because
to elongated
mated
in computational
of 2D and 3D
a 3D phenomenon.
of the tests
reattached-type
b?
(Cs = O.lOj in
of the 2D LES computations
this approach
analyses
was increased
The results
using RANS models,
of the ensemble-averaged
for stationary
direction.
which is essentially
and
class of flows.
et al., the energy-transfer
to Murakami be reproduced
appear
flow past a two-dimensional Smagorinsky-type subgrid
for 2D simulations
of the 3D computations the results
ultimately
used in 3D simulations
in the spanwise
that the k - E model, which incorporates
demonstrated viscosity,
whereas
et al. [3] carried
Shimada tions around
generally
with those obtained
range cannot
due to the vortex-stretching
constant
constant
The results
which
be paid to the reliability
for the above-mentioned
diffusion
were compared
from experiments.
significant
must
Murakami et al. [9] predicted the vortex-shedding using large-eddy simulation (LES). The standard
model was used, and the value of the Smagorinsky
obtained
structures
attention
number
involves
important
difficulties
(1.e + 5 < Re < 1.e + 08):
particular
requirements
in spatial
discretization. Second, acterized
the most widely
used deck sections
by intermediate
currently
levels of “bluffness”
adopted
(elongated
sections
and leeward). Consequently, the flow around them is generally bubbles and shedding of vertical structures that are extremely Third,
the actual
finished
and crash barriers.
bridge
According
be disregarded.
Using
itatively
showed
the influence
response
of bridge
deck carries
to various
flow visualization
decks.
of partial
Adopting
several
authors,
for long-span
bridges
and faces inclined
windward
characterized by small separation variable in frequency and size.
items of equipment,
such as side railings
the effects of such section-model
techniques
in wind-tunnel
streamlining
and traffic
the same approach,
are char-
Scanlan
tests, barriers
details
Bienkiewicz
cannot
[lo] qual-
on the vortex-induced
et al. [ll] pointed
out the criti-
cal dependence of bridge-flutter derivatives upon even minor details, such as deck railings They likewise identified a critical component of the experimental approach in terms of accuracy and similitude
requirements
ers has recently the case of steady To provide present
paper.
in the modelling
been confirmed flow around
an example The
Great
of the section
by computational a streamlined
Belt East
The noticeable performed
influence
of barri-
using RANS models
in
deck [12].
of such peculiarities, Bridge
significant benchmark from the standpoint peculiarities mentioned above fully apply
details.
simulations
we shall introduce (GBEB)
the study
deck is here considered
case adopted
in the
as a particularly
of vortex shedding for three main reasons. First, the to this deck. Second, large vortex-induced oscillat,ions
were measured on the main span just after the deck’s completion. Third, the deck has been the subject of various experimental measurements [13] and computational studies, and thus, provides a significant data base to be used for assessing new simulations. Figure 1 is a schematic illustration of the flow pattern around the deck at Re = UoB/u = 1.5e+5 (where B is the deck width) and o = 0” using the computed instantaneous streamlines. The instants tr and tz correspond to a local maximum value and a local minimum value of the lift force, respectively. The three main characteristics may be easily recognized:
L.
798
BRUNO AND S. KHRIS
vg vortex street behind
the barriers
v3 vortex shedding v, advected boundary separation
vortices
v2 lower vortex
at the lower side
layer point
in the deck wake
interaction reattachment
points
vi-2 coalescence Figure
(a) the strong
1.
interaction
vertical
between
vortices
structures
around
deck.
bubble
Independently,
downstream another
of interaction.
of the lower sharp recirculation
The barriers First,
is accelerated. edge and prevent
perturbations
the flow passing
between
The side barriers
reattach
a larger
vi e
v2
corner,
travel
separation
bubble
and in the near
across
region
from forming.
on account
layer downstream
Second,
the vortex
of the close to
corner,
of two main
of the deck and the lateral
shear
from the
the lower surface
vi passes the lower leeward
surface
the separated
and
vl, which develop
in the near-wake
in the flow pattern
the upper
the vortices;
at its lower surface
zone (~2) is present
seem to enhance
between
The vortices
the lower slope. When the advected procession of vortices coalesces with the vortex 212and is shed in the wake. nomena.
of vortices
vi, ~2, and ~4, ~5;
the deck are located
wake. We shall focus on their mechanism separation
v4 e 3
Flow pattern around the deck: instantaneous streamlines.
(b) the differences in size, intensity, and shedding frequency (c) the flow perturbations induced by the safety barriers. The main
between
due to the barriers
it
phe-
barriers
of the upwind street
us behind
the trailing barriers seems to interact with the vortex v4 in the deck wake. Table 1 summarizes the wind-tunnel setup and the conditions of computation of the most relevant studies. The results obtained are expressed in terms of the following integral parameters: the mean values of the drag coefficient CD = D/(1/2 pUsB) and lift coefficient C:L = L/(1/2 pUaB), and the Strouhal number St = fc,D/U,,where D is the deck depth. Early section-model tests were conducted by Reinhold et al. [14] on a number of candidate cross-sectional shapes. The 1:80 modei selected, designated as H9.1 and characterized by a width-depth ratio Bf D x 7.2, was tested using equipment fully described in [14]. Steady-state wind-load coefficients were subsequently measured on the same cross section using a 1:300 tautstrip model 1151. The overall agreement between the results obtained from models of different size indicates that Reynolds-number influence on the global aerodynamic
effects, even though they are present, do not have a marked behaviour of the deck. The most important discrepancy regards the value of the lift coefficient. In the opinion of the authors of the present study, this
79!l
2D Numerical Simulations Table 1. Outline of published studies-zero Author Reinhold
[141
[151
Larose
Frandsen
Model
Method
[161
Larsen [17,18]
angle-of-attack
(cx = 0’).
Geom.
Re
CD
CL
St
1.e + 5
0.08
+ 0.01
0.109 - 0.158
0.10
- 0.08
0.11
-
_
0.08 - 0.15
0.06
+ 0.06
0.100 - 0.168
Exp.
Section
det
Exp.
Taut-strip
det
7.e + 4
Exp.
Full scale
det
1.7e+7
DVM
pot. + w
2Dba.s
1.e + 5
Taylor [191
DVM
pot. + w
2Dbas
1.e + 5
0.05
+ 0.07
0.16 - 0.18
Frandsen [20]
DVM
pot. + w
2Dbss
1.6e + 7
0.08
+ 0.06
0.09
FEM
NSE lam
2Dbas
1.6e + 7
0.06
- 0.09
0.25
FDM
NSE lam
2Dba.s
3.e + 5
0.07
- 0.19
0.101 - 0.168
Selvam [22,23]
FEM
NSE les
2Dba.s
l.e+5
0.06
- 0.34
0.168
Jenssen [24]
FVM
NSE les
3Ddet
4.5e + 4
0.06
+ 0.04
0.16
FEM
NSE les
PDdet
7.e + 4
0.07
+ 0.08
0.17
Kuroda
[21]
Bnevoldsen
difference model.
[25]
may be related
to the unachieved
In fact, as demonstrated
the higher reduced.
flow velocity The various
and the stronger
vortex-shedding
the spread frequency content The full-scale measurements of the flow. The results lower bound
similitude
suction
of the barriers
along the lower surface.
mechanisms
in the 1:300
blockage effect of the railings
highlighted
in Figure
involves
The lift force is thus 1 probably
account
for
of the lift force that was found experimentally by Reinhold et al. [14]. obtained by Frandsen [16] substantially confirm this characteristic
obtained
of the range
aerodynamic
in [12], the most important
by Larose
[15] seem to indicate
has the most important
that the Strouhal
effects on the structural
number
response
at the
of the deck.
Most of the computational simulations assume the same Reynolds number used by Reinhold et al. [14]; exceptions are represented by [26,27]. On the other hand, only Enevoldsen et al. [25,28] state
that
account
they
include
the deck equipment
the inflow turbulent
Three
general
conclusions
may be drawn
the first place, the underestimated the simulations.
in the computational
level experimentally
from an overall
prediction
This error is certainly
model.
No author
takes into
set at It M 7.5%. analysis
of the drag coefficient
due to the absence
of barriers
of the results is a common
obtained.
feature
and to the incoming
In
in all of smooth
flow. Second, most of the simulations fail to highlight the spread frequency content of the lift force, indicating only one Strouhal number. This difficulty mainly concerns the approaches that solve the Navier-Stokes equations using grid-based methods. Finally, almost all the simulations are conducted
in the 2D computational
of the fact that excessively A number
3D simulations
burdensome
domain.
remain
for industrial
of applications
This common
scarcely
applicable
approach
in extended
provides
confirmation
parametric
studies
and
applications.
are based on the so-called
discrete
vortex
method
(DVM),
which is
comprehensively reviewed in [29]. This approach is widely adopted in bridge aerodynamics (see. for example, [6]) for two main reasons. First, the Lagrangian nature of the method markedly reduces
the difficulties
encountered
in grid-based
methods
(mesh
quality,
numerical
diffusion,
modelling of small details of the deck section). Second, the description of the flow field is obtained by an essentially potential-flow method, so that the computational effort is significantly reduced as compared
to the methods
involving
solution
of the Navier-Stokes
equations.
On the other hand, the above advantages are obtained thanks to a fundamental assumption regarding the physical feature of the flow, namely that “for high Reynolds numbers, viscous effects may not be essential in the development of the wake, as the vorticity that is introduced in the wake from the separation
point
remains
confined
to narrow
regions
and its dynamics
is
mainly dominated by convection, with viscous diffusion only playing a secondary role” [11. From the numerical point of view, this assumption requires that the amount of vorticity w shed from the separation point (which must be known a priori) be introduced by means of particles of finite core size in each time step. This approach has been applied to the study case by Larsen
L. BRUNO AND S. KHRIS
800
et al. [17,18], who obtained
an impressive
agreement
between
data, even if compared with more complex approaches. all applications of vortex models to unsteady separated have shown
that
the
circulation
of the vortices
their
results
and the experimental
According to Sarpkaya [29], “practically flows past two-dimensional bluff bodies
should
be reduced
as a function
of time
and
space in ad hoc manner in order to bring the calculated forces, pressure and circulations in close agreement with those measured”. As a result, the evaluation of the performance of this approach and of its suitability of the aforesaid
for application
numerical
the drag coefficient
by addition
“care must
be taken
Other
approach, of a flat plate
when
analysing
effect is largely
authors
have predicted using
of the barriers.
treated these
element
finite volume
as [ .
results
] porosity
the Great
to these
are
authors.
effects are neglected”; the nature
e.g., finite
difference
i.e.,
of the flow
Belt East Bridge by solving
methods,
value of
the side railings
According
in such a way as to change
of discretization
In the framework
the underestimated
Consequently,
as a solid geometry.
the flow around
a number
very problematical.
and Vezza [19] attribute
overestimated
Stokes equations method,
Taylor
to the lack of modelling
introduced the blockage completely.
to other shapes remains
the Navier-
method,
finite
method.
Among the various works that neglect turbulence modelling (laminar tained by Kuroda [21] are in good agreement with the wind-tunnel data.
form), the results obIt seems that the error
in approximating
balances
turbulence between
the convection
damping. numerical
order to justify
diffusion,
More recently,
obtained
paid to boundary-layer
damping studies
and the scheme
that
are reported
adopted
the absence
of
on the relationship
(fifth-order
upwinding)
in
in a 2D simulation.
on the mesh-density modelling.
a numerical
no parametric
et al. [27] applied
Frandsen study
hand,
grid resolution,
the accuracy
case. A parametric
term provides
On the other
the FEM-without-turbulence sensitivity
As a result,
was initiated
shedding-frequency
model
to the study
with particular predictions
attention
were generally
inaccurate even for the most refined grid so that the authors of the paper concluded that “further studies are needed to establish the model requirements”. Subsequently, Frandsen [20] tested the DVM in order to compare other studies. To the knowledge study.
Only
the two different
of the present
authors,
Lee et al. [30] adopted
loading
on a fully bluff bridge
without
law of the wall.
approaches
deck.
RANS
a RANS
with one another
models
approach
have never
for predicting
The renormalization
The quadratic
upwind
group
interpolation
scheme was employed to approximate the convection term. indirectly validated by performing dynamic structural analysis at the bridge any conclusions
midspan
with the full-scale
to be drawn
In the last few years,
measurements.
on the reliability
the LES approach
been
applied
to the case
the vortex-induced
(RNG)
of
k - E model
wind was used
for convective
kinetics
The unsteady and comparing
wind loading was the displacements
This approach
of the statistical
has been applied
and with the results
clearly
(QUICK)
does not permit
approach. to the case study.
Selvam
[23] dis-
cussed the reliability of the 2D simulation, considering the deck section of the Great Belt East Bridge approach span. Both 2D and 3D computational models were used, neglecting the barriers. No parametric studies were performed. The author assumed the value of the Smagorinsky constant without
proposed by Murakami any further investigations.
drag coefficient
with reasonable
[9] for the square cylinder (C, = 0.15 in 2D, C, = 0.10 in 3D), According to Selvam, only 3D models are able to capture the accuracy.
In a subsequent
paper,
the same author
[22] employed
two different levels of grid refinement and two near-wall treatments (i.e., low Reynolds number and law of the wall) in the 2D model of the Great Belt East Bridge main span. The results of the study indicate an impressive sensitivity of the mean flow to the parameters mentioned no (ACD M 50%, ACL M 400%, ASt M 20%). B ecause of the limited number of computations, conclusions were drawn regarding this sensitivit,y. In the case of nonperiodic flows and spread frequency contents of the aerodynamic forces, the integral parameters only provide a partial physical insight into the flow. Figure 2 compares the
2D Numerical
upper surface
I
I
I
,, Ic
X01
Simulations
I
I
<
0.4
0
0.2
..
0.4
0.6
0.7
0.6
0.8
_
. +
turbulent flow: taut ship model [ 1fl turbulent flow: full-scale [16]
x/B 1
Kuroda(fdm) [21] ~ Jenssen (fvm) [24] -----Frandsen (fern) [ 203 - - - - Frandsen (dvm) [To] .-..-. ..... u”
-1.0 -
I
I
1
,
I
I
lower surface Figure
distributions a number
of the mean of simulations.
the computational First,
2. Published
pressure
minor
thus
imperfections
estimated
both
on the deck
on the deck obtained
discrepancies
clearly
0 I x/B 5 0.2 the experimental of the leading
overestimating
may be related
Second,
coefficient
Two main
lower surface just downstream They
Cp distribution
mean
appear
(a = 0’)
in wind-tunnel between
tests
and in
the experimental
and
distributions.
in the range
phenomenon,
studies:
to the high anisotropy of the leading
the lengths is restricted
simulations.
bubbles
suction
simulations
in the forebody
model cannot tests
on the
ignore this are not clear.
region
[31], but
be ruled out a priori.
at the lower and upper
In the wind-tunnel
the range
a moderate
for this discrepancy
of the turbulence
edge of the geometrical
within
reveal
All the computational The reasons
of the separation
in the computational
the lower surface
point.
the pressure.
data
surfaces
are over-
the recirculation
zone at
0.2 < z/B 5 0.35, and recovery
of the pressure
rapidly occurs within the range 0.35 5 x/B 5 0.45. Except for the DVM approach, all the other numerical methods predict a pressure recovery in the range 0.6 5 x/B < 0.7 and underestimate the suction
peak.
It is to be pointed
out that
the phenomena
occurring
in this region
play a
major role in the vortex-formation process, as emerges from Figure 1. Consequently, the lack of accuracy in this region deeply affects the overall reliability of the simulations. The reason for these errors is not clear. However, if it is taken into account that the characteristics of the incoming flow are rather similar, the differences between computations different computational procedures adopted in the studies. The overestimated
length
of the bubble
at the upper
surface
seem to be related
seems to be directly
to the
related
t,o
the interference effects of the windward side barrier. These effects have a minor impact on the dynamics of the flow but markedly affect the mean value of the lift force. Jenssen [24] has modelled such details, obtaining the local reattachment of the shear layer. Nevertheless, the
802
L. BRUNO AND S. KHRIS
constant value of the pressure zone that is not experimentally From the present primarily
address
critical
downstream detected.
of the railings
review of the studies
the question
of validation
suggests
published,
the presence
it follows that
of the codes employed.
fail to provide any closer insight into the various physical thermore, the effects of various computational parameters
of a recirculation
most of the researches
In many
cases, the results
mechanisms involved in the flow. Furon the solutions is not systematically
clarified. The purpose
of the present
to verify the suitability To do this,
particular
parameters
to be taken
LES approach regards
attention
so enabling
not only
effects of the details
The incompressible,
unsteady,
study.
in the model
models.
In particular, but
The suitability
both to complete
the
also as
of each of
model.
Finally,
the experimental
measurements,
and deck.
computational
parameters
model.
of the aerodynamic
setup,
and to point
up the
field.
EQUATIONS
two-dimensional
The nondimensional
suitable
on the basis of the optimized
2. GOVERNING in the present
turbulence
to be made with wind-tunnel
on the characteristics
the most
subgrid
studies
flow past a bluff bridge
the above-mentioned
is discussed
is included
comparisons
of the previous
unsteady
in the various
as regards
of the Smagorinsky
to turbulence
of the section
correct
the incompleteness
for predicting
has been paid to determining
parameter
approaches
the equipment
is to reduce
into consideration
is optimized
the free input
the various
paper
of 2D simulations
Navier-Stokes
instantaneous
equations
continuity
of motion
and momentum
are solved equations
are div u = 0 and dU
dt where
u, p, and t are, respectively,
by the reference The Reynolds
velocity number
lence-modelling closure
Re results
vector,
pressure,
and
turbulence
are applied.
modelling)
In the framework
a second-order
models
are the standard
(STD)
model.
The Reynolds-averaged
closure model [32] and th e renormalization
incompressible
E$+U.!L!&-L 3 axj
axi
approach
According
to the standard by means
nondimensionalized viscosity
k - E model
u.
$+&!p2
equations
(p_+g1+2& P
turbu-
both first-
are expressed
as
C”+ vt) Sij,
3
[32], the turbulent
axj
approach,
are adopted. The first-order closure group (RNG) (331 forms of the Ic - E
Navier-Stokes
of their transportation
LAM and various
of the statistical
where ut is the isotropic eddy viscosity given as vt = pC,k2/E, and SQ is the mean flow strain rate expressed as follows:
rate E are computed
and time
p, and the kinematic
from these variables.
(i.e., without
approaches models
the velocity
+ & Au,
Us, the deck chord B, the air density
Both the laminar-state order
+ u. grad u = -gradp
k is the turbulent
kinetic
energy
kinetic
energy,
k and its dissipation
equations
[(‘+z)&]+Pk-E
and (5)
803
2D Numerical Simulations where Pk is the production
constant
term of the turbulent
are the conventional
kinetic
energy,
and the values
of the empirical
ones [32].
In order to cope with the well-known
excessive
production
of turbulent
kinetic
energy
model in nonequilibrium situations, the k - E form suggested by the renormalization is also applied. This model introduces a further production term into the transport the dissipation
of this
group [33] equation of
rate E in the form c/J13 (1 - rllrlo) E2 lfP$ X-’
Ii-=
(6)
where
and the constants In order
70, ,0, uE, ok, C,,,
to account
better
CEz, C, are given in [33].
for important
Reynolds stresses, diffusive transport
the energy transfer mechanisms-F’ranke
to the calculation
of vortex
Abandoning solving
the isotropic
transport
such equations turbulent
past
a square
transport
and modelling term
of the flow-such
as the anisotropy
and qbtained
the RSM closes the
Reynolds assumptions
interesting RANS
Rij = q.
stresses
are required.
and the pressure-strain
of the
flow and the mean flow, or the the Reynolds stress model (RSM)
cylinder
hypothesis,
for the individual
are unknown,
diffusive
shedding eddy-viscosity
equations
aspects
between the turbulent and Rodi [31] applied
term,
equations
Several
In the present
are respectively,
results. by
terms
in
study,
the
expressed
by
the models proposed by Lien and Leschziner [34] and by Gibson and Launder [35]. Model equations for the large eddy simulation (LES) are expressed in the form
and a?!&
aii,
--
dt+fijFdz
where the overline
indicates
scale model
subgrid-scale
(8)
dXi
3
subgrid
d
the filtered
value of the variable.
[36]: ksGs indicates expressed by
is employed
eddy viscosity
USGS =
The subgrid-scale tations
mixing
length
PLSGS
Ls~s is related
component
Smagorinsky-type of k, and USGS the
2/2SijSij.
19)
to the Smagorinsky
constant
Cs in 2D compu-
by LSGS = min
where
The standard
the subgrid
n = 0.42, A is the cell surface,
(
ted, C,A1/2
and d the distance
>
,
between
(10) the cell and the nearest
wall
boundary. 2.1.
Near- Wall Turbulence Modelling
Near-wall modelling wall-bounded flows. A suitable for application However, Franke et al.
has a significant impact upon the reliability of numerical simulations of number of works that use various turbulence models render these models throughout the boundary layer by means of the wall-function approach. [31] have demonstrated that “the assumptions of a logarithmic velocity
distribution and of local equilibrium of turbulence are violated in separated flows, especially near separation and reattachment regions” [31]. Bearing in mind the purpose of the present study, the low Reynolds number one-equation model (“two-layer model”) has been adopted in the viscosity-affected region (Re, = p&y/v > 200) in
804
L. BRUNO AND S. KHRIS
conjunction with the statistical approach. This approach was successfully applied by Franke and Rodi [31] in the study case of the square cylinder and by Shimada and Ishihara (31 in predicting vortex
shedding
past rectangular
are here calculated
cylinders.
using the turbulent
ut = The length
The turbulent
kinetic
energy
viscosity
rate E
k from the relations k3/2
pc, JiFl,,
scales I, and 1, are expressed
ut and the dissipation
(11)
by the following
relations:
(12) where the values of the constant When strain
the LES model
in the formulas
is applied,
are taken
from the laminar
stress-
relationship ii
-_=%
where u, = (~~/p)‘.~
is the friction
PWY
Computations method.
were carried
In the grid-based
on fundamental
coupling
unstructured
In particular,
out
method
using
PROCEDURES
the FLUENT
simulation
importance. grid type
~5.4 code,
of flow around
In this study,
and structured
the mapped
LJ'
velocity.
3. NUMERICAL
takes
from Chen and Pate1 [37].
the wall shear stress ru is obtained
mesh types
based
complex
the computational by means
simulations
computations are shown
and complies
using turbulent in Figure
with the mesh requirements
models.
The computational
3. The size of the domain
parametric study reported in [12]. The pressure-velocity coupling is achieved with splitting equation,
of operators
whilst
enforcing
the continuity
of the total
of a parametric
and the boundary
consistently
by means of the pressure-implicit approach
study
approach
SIMPLE
fine for
in the
conditions
with the results
to advance
by grid.
in order to achieve
of the two-layer
domain
is adopted
[38], using a predictor-corrector
are generated
of substructuring
is used in the near solid boundary
volume
mesh generation
domains
resolution of the flow structure in this region. The cell thickness y,,, adjacent to the solid wall forms the subject laminar
on the finite
geometries,
of the
algorithm
the momentum
equation.
-.-.-.-.-.-.-.-.-.-.-.~.~.~.~.~.~.-.~.~.~ periodic (LES) I U=l; v=o; I =o I
U=l;
v=o;I =o
periodic (LEb) ,.,.,.,.-.-.-.-.-.-.-.~.~.~.~,~..m.~.~......a 15B Figure 3. Analytical
I 30B
domain and boundary
conditions.
805
2D Numerical Simulations
For time discretization, mensional
the fully implicit
time step in the laminar
At* < 2.e- 2 with respect
vergence criteria. initial conditions
to the expected
derivative
cept for the nonlinear (second-order
terpolation formula
Euler
and statistical frequency
scheme
approach
content
is adopted.
The nondi-
varies in the range 5.e - 4 <
of the phenomenon
and to the con-
The time advancement in LES undergoes optimization during the study. The are set equal to the inlet boundary conditions (impulsively started simulation).
All the spatial schemes
second-order
approach
terms
are discretized
by the second-order
convection
terms.
The convection
terms
central
difference,
second-order
upwind
kinetics-QUICK
[40]) that
for convective
central
are discretized
differencing, according
[39], and quadratic
are generally
expressed
ex-
to threcl
upwind
according
in-
to the
an d d:,f The weights
= -~e.fe
for the discretization
schemes
adopted
Table 2. Coefficients I
As reviewed
of a signal-without
diffusive schemes Notwithstanding assertions
extent
diffusion.
have a diffusion the signal
the relative
of different
physical
problems
and to numerical
effects (markedly
dispersion
effects
they
On the other
hand,
it is not
and according
affecting
stabilizes
appropriate
Scheme
to cause phase
Second-Order
Upwind
QUICK
universal may vary
time-marching
schemes,
structures errors
of convective schemes.
I---Central
to make
performance
flow seems to be significantly
Leading Term
Second-Order
the computation.
[8]), and computational-domain
frequencies). Table 3. Accuracy
terms
of the above-mentioned
since their
the small vertical
(i.e., it appears
(as in content
even leading-order
to different
(see, for example,
terms
alter the frequency
The accuracy
of these schemes
(i.e., 2D or 3D). The physics of the investigated diffusion
IEe
effect, which generally
grid density
I
I KP
[41]), the odd leading-order
in amplitude.
performance
(see [7,8,42]),
ku
effects-i.e.,
is summarized in Table 3. the above general considerations,
models
to numerical shedding
numerical
but reducing
regarding
in the presence turbulent
scheme)
wiggles
have dispersive
schemes. I
I
&Jw
(see, for example,
schemes)
involving
(as in the QUICK removing
I
authors
are given in Table 2.
of convective
Scheme
by various
the case of second-order
Kwfw- Kwwfww
- ~p.fp -
Effect
sensitive
around
between
both
the deck)
the different
806
L. BRUNO AND S. KHRIS
However, even though a number of studies have been devoted to the evaluation of these schemes, in the present paper a parametric approach.
study is performed using both the laminar form and the LES
4. APPLICATION 4.1.
Simulations
Without
Turbulence
AND
RESULTS
Modelling
The simulations assume the experimental conditions adopted by Reinhold et al. [14] during the section-model tests (see Table l), except for incoming turbulence intensity It = 0. The angle of incidence of the flow is zero. 4.4.1.
Effects
of grid spacing
and discretization
schemes
on the flow field
The effects of the near-wall cell thickness yw in 2D simulations without turbulence are discussed in this section. characterized
To do this, three grid systems are taken into account.
by a nondimensional
grid spacing adjacent
modelling These are
to the solid boundary equal to yw =
1.3e - 2, yu, = 2.0e - 3, and yw = 2.2e - 4 (width B = 1) and will be designated in what follows respectively by lam-l, lam-2, and lam-3. The performance of these three systems is first evaluated in conjunction
with a second-order upwind scheme for convection terms. cellnumber
1.3~2
2.Oe-3
2.2e-4
45512
28767 25865 0.01
0.001
Y$
o.ooo1
(aI
@I
Figure 4. Near-wall cell thickness versus total number of cells (a); grid system near the deck (b).
The reduction of the first cell width involves an increased number of control volumes in the neighbourhood of the wall. The outer part of the computational domain retains the same grid. Figure 4a shows the evolution of the total number. of cells of the model versus near-wall cell thickness. The close-up view of the grid system near the deck is also shown in the case of the most refined mesh (lam-3 Figure 4b). In order to establish a relationship
between the grid spacing and the simulated instantaneous
flow pattern, Figure 5 shows the time history of the lift force in the range 25.6 < At < 27.6, where the nondimensional time is expressed as t = TUo/B. A number of instants are selected on the time track, and the instantaneous streamline patterns are plotted for each model. The coarsest mesh provides a steady solution, i.e., without fluctuations of the lift force. The flow remains completely attached to the deck on the side surfaces, as shown by the streamline pattern just below the diagram. The other grid systems enable observation of major fluctuations of the lift force. The time history of the lift coefficient CL related to the medium level of refinement appears very regular, suggesting that just one mechanism plays a dominant role in the vortex-shedding process. The flow-pattern visualizations in the left-hand side column confirm the above hypothesis. The separation bubble at the lower surface is very large, but the level of vorticity in the recirculation region seems too low to involve vortex shedding. Consequently, the vortex does not interact with the vortex developing in the near-wake region. Local maxima in the lift track (t = 26.0
807
2D Numerical Simulations I I I I,
0 I I I r 1 ’ I * 1 r
yw=l .3e-2 y,=2.0e3 ---.--y,12&4 ------
CL
+0.055
-
I m
:
cL lmed
yw = 1.3e - 2
yw = 2.Oe - 3
Yw = 2.2e - 4 0
A
t = 26.0
t = 26.2
t = 26.2
t = 26.4
t = 26.4
t = 26.7
t = 26.7 Figure 5. Time history of the lift coefficient and instantaneous
and t = 26.4)
are regularly
observed
(t = 26.2 and t = 26.7) are related
streamlines.
when this vortex is shed in the wake. Conversely, to the maximum
development
of the vortex
the minima
close to the wall.
The grid spacing lam-3 enables detection of a fundamental feature of the flow, namely the interaction and merging between the vortices travelling along the lower surface and the vortices (t = 25.9) and minimum (t = 26.2) emerging at the downstream edges. The first maximum substantially
match
those simulated
by means
of the grid lam-2.
But subsequently
the lift force
in the lam-2 model performs one period of oscillation between the instants t = 26.2 and t = 26.7: whereas the CL~ just covers a half period (local maximum at t = 26.7). The streamlines clearly show the merging bubble
of the vortices
downstream
at that moment.
surface
and the clockwise
in the upper area of the deck wake. However, the role played shedding process does not seem to be of primary importance.
by these structures
The pressure in Figure
of the windward
The most refined grid also detects
distribution
6. These
results
edge at the upper
on the lower surface confirm
the differences
at the above-mentioned between
the models
the separation circulating
instants lam-2
flow
in the vort,exis presented
and lam-3.
In the
the pressure fluctuations are always separated by the sharp corner at x/B = 0.8 When the finest grid is used, the pressure fluctuations move backward, without interaction. reducing the extent of the separation bubble. Owing to the larger amount of vorticity produced at the separation point, the pressure at x/B = 0.8 does not remain constant, but reaches a deep trough when the vortices at the lower surface pass the edge. It should be noted that as former
model
L. BRUNO AND S. KHRIS
808 1
1
CP
CP
0.5
0.5
0 -0.5 -1
4
-1.5
-1
‘.
-1.5
J
0
0.2
0.6
0.4
0.8
?IB
1
t , 0
0.2
Figure 6. Instantaneous surface.
approximation with sharp
is enlarged,
case of semibluff
distributions
major
characterized
sections,
of the pressure coefficient
the model just
does not involve corners
simulates
problems
by massive
XII3
1
process,
thus
in the simulation
of flow around
and large-sized
conceals
leading
Cp on the lower
one large eddy on the side surface.
separation
a coarse mesh completely
basis of the vortex-formation
0.8
(b) yw = 2.2e - 4.
(a) yU = 2.0e - 3.
the grid spacing
0.6
0.4
vortices. error.
bluff cylinders Instead,
in the
structures
at the
A number
of grid
the small vertical
to a fundamental
This
points is required both in order to ensure a cell size smaller than the smallest vortex length and to prevent an excessive amount of numerical diffusion related to the discretization scheme adopted for the diffusive
terms.
In order to extend
the discussion
treatment
of the results
the model
lam-3.
is called for. Figure
As may be readily
of this sort, as has already very important extent, Figure
to the integral
for the extraction
in time and space, a statistical
7a shows the CD and CL time traces
appreciated,
been pointed
flow parameters the signals
out by Taylor
of meaningful
are completely
[19], the extent
statistical
simulated
random.
of the sampling
parameters.
To optimize
with
In a case window
is
the sampling
the computations run as long as the lower Strouhal number Sti remains constant (see 7b). Generally speaking, a sampling extent of 30 nondimensional time units is found to
be required in order to assume stationarity affected by the impulsive initial condition, nondimensional
of the signal. Since the first ten units are strongly each simulation was conservatively extended to 50
time units.
In what follows, the main statistical
parameters
of the aerodynamic
behaviour
of the deck are
discussed. ‘D-‘L
St cD med
0.00
.211 ‘9.
I
I St,
o
-0.20
0
10
20
30
40
t
50
10
30
20 (b)
(a)
Figure 7. Optimization
of the extent of sampling-time.
4o
At
So
2D Numerical Simulations
.085
809
St - y,=2.0e-3
.256
St - y,=2.2e-4 Figure 8. Effects of grid spacing on Strouhal number
The power spectral lam-2
and lam-3)
number).
densities
The PSD obtained
the higher
in number.
The corresponding
mechanisms
that
numbers
(Stz/Stl
that
that the physical
shown in Figure have
8 versus
to frequencies
This means
flow visualizations
of the lift coefficient
in Figure
been
frequency
Strouhal
numbers above.
quite different
with the experimental
value is indicated by Figure
by means
of error bars.
5 is qualitatively
coefficients
results
in accordance
complete
obtained
between
Even though
in the computational deviation
the description
,102
t-
I
:
:
: :
:: -
:.
are
the mean
the mean values of the
:
2.Oe-3
,.., +.05
,,: ::
1.3e-2
:
:
,076
simulations around
CL
2.2e-4
,_---J~I_~-L1_~__k___. : “ ‘::: :
,(.,
:“’
Strouhal
x 1.67).
with experiments.
CD
2.0e-3
1.3e-2
shedding
of the flow field provided
and refers to a very likely situation,
are not in close agreement
the main
one (St,/%,
9. The standard
(i.e..
with the
from lam-3 are fewer
due to the different
the ratio
from the experimental
in Figure
In effect
of the first frequency
unique
are physically
(models
(i.e., Strouhal
by three main peaks.
multiples remains
However,
solutions
frequency
5. The mean peaks in the PSD obtained
described
z 2) remains
are integer
The mean values of the drag and lift coefficients compared
for the unsteady
the nondimensional
from the model lam-2 is characterized
peaks correspond
superharmonics).
(PSD)
are plotted
+.Ol
:
2.2e-4
1
-3”” .,.
I
:
!i.
:
‘,
-.08
:‘.
_______’
~_____~--___---,_____--------_--_
‘.., ,,..
:
:
.’
-.23
:8%
::
‘.
T
..:.
:“.’ t -3 I
.’
I
.-.
-.29
,023
-.43 0.001
yfl
0.0001
0.01
(a) Figure 9. Effects of grid spacing on aerodynamic
0.001 (b) coefficients.
Yfi
0.0001
810
L. BRUNO AND
S.
KHRIS
The mean value of the drag coefficient increases consistently with the shear-layer separation and the vortex shedding around the deck. In particular, the drag component involved in the flow circulation in the near-wake area is generally predominant in such quasibluff geometries. In order to relate the drag value and the characteristics
of the wake, Figure 10 presents the
effect of grid spacing on the across-wind extent of the wake and on the velocity defect at two locations in the wake (0.005B any recirculation,
and 0.5B from the trailing edge). The coarsest grid does not predict
so that the deck behaviour is close to the one of a fully streamlined
and is characterized are quite similar.
by a very low drag value. The profiles predicted using the other models The model lam-3 further reduces the velocity defect in the neighbourhood
of y/B = 0 and emphasizes the wake extent in the range -0.2 wider separation
< y/B
< -0.1
because of the
bubble at the lower surface of the deck. It may be noted that the results are
in good agreement the &
section
with the computational
value approaches the section-model
prediction of Larsen [18] using DVM. Even though result, it remains somewhat underestimated.
This is
certainly due to the absence of deck equipment in the models. I y,+1.3e_2
=1.3~_2
YB
I
I .._._
:
I ’
’
+0.150
_ -
+0.072 +0.043
I
I
I
I
1
0.9
1
Larsen [ 181 a yw = 1.3e_2 _.._..._.___._.. y,=2.Oe-3 ------yw = 2.2e-4 -
0.0 -0.097 -0.150
0.8
0.9
1
0.0
I
I
0.4
0.8
(b) U&Jo - x/B
(a)
1.2
OS
= 1.005.
0.6
0.7
0.8
(c) I&/U0 - x/B
= 1.5.
Figure 10. Effects of grid spacing on mean z-velocity defect in the wake. Instead,
mesh refinement
considerably
reduces the lift forces acting on the deck.
noted that as the grid spacing becomes smaller, the computed & the measured values, and its standard
deviation increases.
It may be
moves out of the range of
Hence, mesh refinement enables a
qualitative reproduction of the vortex-shedding process but also appears to reduce accuracy in predicting the mean lift force. The same apparent discrepancy may be noted in other 2D simulations
(e.g., [21]; see Table 1). Even though Kuroda assumed the same grid spacing near
the wall, the error regarding the lift coefficient is somewhat different from the present results. This difference points t,o the dependence of numerical damping both on grid spacing and on the convective scheme. To clarify this aspect, the levels of performance of the finest grid are evaluated in the case where second-order upwind and QUICK schemes are employed. The distributions obtained for the mean Cp and for the std (Cp) are compared in Figure 11. The agreement between the lift coefficient measured and the lift coefficient obtained from the “Yw zz 1.3e - 2-2nd upw” model is seen to be merely fortuitous. From a comparison of the unsteady solutions it is found that as the mesh becomes more refined and the order scheme is higher, the diffusion effect is progressively reduced and does not overcome the very small viscous diffusion of the flow. Hence, the extent of the separation bubble at the lower surface is shortened, and the separated layer past the upper leading edge is better predicted. Despite the major gain
in precision that is obtained,
the size of the separation
bubbles remains somewhat larger than
the measured value. Two main limitations remain in 2D laminar simulations. First, it is clear that further refinements of the near-wall mesh or the adoption of higher-discretization schemes do not represent
2D Numerical Simulations lower surface
I
’
+0.6 1
‘n
y+3e-2
upwd
- 2nd
y,=Z.Oe-3 - 2ndupwd
0
0.2
0.6
0.4
8
smuotbflow: tautship model [I 51 hubulent flow: taut strip model [15] turbulent flow: fidi-scale [I 61
I
L
---.--
yw=2.2e-4-2ndupwd ----y,=2.%-4 - quick Kmuda [Zl] - yw=2.0c-4 - 5th upwd - .-.
upper smfacc
1
1
1
’
811
0 . *
-.
0.8
x/B
1
0.2
0
0.4
0.8
0.6
x/B
1
Figure 11. Effects of numerical diffusion on mean Cp and std (Cp) distributions. realistic
solutions.
thickness
In fact,
(see Figure
nificant
improvement.
diffusive
schemes
the stability
On the other
of the aerodynamic that
hand,
In particular,
mechanism
towards
further
higher
near-wall about
damping
Second,
in the spanwise
frequencies
as a result
of vortex
in the laminar
simulations
is retained
cell
any sig-
by the highest
11 “yW = 2.2e - 4-QUICK”
seems to be no longer assured. diffusion
versus
does not bring
of numerical
(see Figure
the well-known
coefficients
refinement
the reduction
is such that wiggles appear
of the computations
effects remains. transfer
the evolution
9) seems to indicate
model),
and
the failure to model 3D direction
stretching
and the energy[7] continue
to be
neglected. 4.2.
Statistical
Approach
The most refined grid selected paper.
in the sequel of the present
The grid spacing
the two-layer obtained
model
by applying
near the wall complies with the requirements the (wall unit y+ = 1). Table 4 summarizes the standard
k - E model
(STD),
for correct application of main integral parameters
the RNG model
and the Reynolds
stress
model (RSM). Both the STD and RNG models fail to predict the unsteadiness of the flow. This error can no longer be chiefly put down to the numerical damping that results from the combined effect of grid size and discretization that
the reasons
formulation
scheme
on account
for the lack of lift fluctuations
of the turbulence
models
of the previous
(std (Cn)
optimization.
= 0) are to be sought
adopted.
Table 4. Statistical approach: integral parameters. Model
CD
CL
std(C~)
St
It follows rather
in the
812
L. BRUNO AND S. KHRIS
Recently, Shimada and Ishihara [3] have proposed an interesting explanation of the severe limitations of these models in correctly simulating the fluctuating pressure field around elongated rectangular quantity
sections.
Franke
and Rodi [31] argue that
4 can be separated
in vortex-shedding
flows an instantaneous
into C/.)(t)= G + 6 + Cj’
(14)
9’ Indicating by @’ the total fluctuation value $‘, its variance can be expressed
(i.e., periodic as
6 plus stochastic
4’) around
the time-mean
a& = CT;+ a& In particular,
the above equation
(15) can be rewritten
(15) for the velocity
component
Vi as follows:
depends
component of the velocity is evaluated by means of the turbulent kinetic energy k to the turbulent viscosity ut. It follows that the value of the variance strictly on the reliability of one of the weakest and most debatable assumptions of the model,
namely,
the isotropic
pressure
is not explicitly
The stochastic and then related
turbulent
viscosity
modelled
itself.
Furthermore,
in any RANS
model,
the stochastic
component
of the
so that
and std (CL )RANS < std (CL). The statistical periodic where
approach
fluctuations the relative
yields
predominate. contribution
good results
this case the very small eddies observed the stochastic
field by the statistical
eddy viscosity
around
an extent
in the case of massively
without
approach.
turbulence
modelling
As a result,
these
the deck and in the wake. This may damp
as to cause the observed
The RSM model furnishes
separated
flow in which
These conditions do not apply to the present application, of the stochastic component becomes noticeable. Moreover, in are probably
models
assumed
yield higher
the fluctuating
in
levels of
motion
to such
lack of unsteadiness.
an unsteady
solution,
even though
one characterized
by an extremely
reduced value of the standard deviation of the lift coefficient and a unique Strouhal number (Figure 12a). This frequency content is due to the single eddy in the wake of the deck (Figure 12b).
CL
I
I
I
I
I
RSM +0.036 +0.026
5
I
I
I
I
I
5.25
5.5
5.75
6
6.25
,,‘;_p-,
,,,’
.____
+a010
t
(a) Figure 12. MM-time
,/ ,_,’
history of lift coefficient (a) and instantaneous
(b) streamlines
(b).
-- _
tcl;3
2D Numerical Simulations the RSM model
As a result, pared
to the two-equation
Leschziner
affords
closure.
[34] for modelling
In fact, the above approach
to a case that
adopted
by Lien and
turbulent
viscosity
term
= & Ic (_!$!?J$).
is once more based upon the concept
and the value of the free constant model
due to the formalism
the turbulent-diffusion D;
diffusion
improvement in the simulation of the flow as com-
scant
This is probably
Crk = 0.82 is obtained
is rather
different
of isotropic
by applying
from the present
the generalized
one, i.e., a plane
it.
gradient-
homogeneous
shear flow. 4.3.
The Large
The adoption substantial
Simulation
involved
albeit
somewhat
in simulating
burdensome,
the turbulent
is necessary
flow around
to overcome
the GBEB
the
deck using the
approach. of computational
Optimization
The effects interval
of two computational
for time advancement
Table 5 summarizes
parameters
the main integral
results
by the mesh
adopted.
required
the particular
than
On the other
hand,
(-40%)
scheme
in what
that change.
The results
in At markedly
convergence
the value At = 1.e - 2 is adopted
vortex
simulated
the number
for residuals
in the subsequent
using
the largest
directly
reduces
(threshold
the
param-
obtained
It follows that
of the smallest
the increase
to reach
namely, terms.
the above-mentioned
differ from one another. time
follows,
for the convective
by varying
parameters
the characteristic
at each step
5.e - 4). For this reason
are investigated obtained
les-1 and les-2 do not significantly
step is still smaller
iterations
parameters
At and the discretization
eters; the bold style highlights the models time
Approach
of the LES approach,
difficulties
statistical 4.3.1.
Eddy
of
fixed at
simulations.
Table 5. Computing conditions and integral parameters. Model
At
Scheme
C,
les- 1 les-2
l.e-3 l.e-2
2nd up 2nd up
0.10 0.10
les-3
1.e -
QUICK
0.10
les-4
1.e - 2
2nd cnt
0.10
In a way similar
2
to the one adopted
!
St
CD
+ 0.067 + 0.066
0.272 - 0.113
+ 0.075
0.202 -- 0.101
+ 0.071
0.124 - 0.164
0.272 - 0.107
0.197 - 0.292
for simulations
without
turbulence
models,
the remaining
part of this subsection is devoted to clarifying the performance of various discretization in conjunction with LES and to selecting the most efficient scheme for the ensuing compared
to the second-order
lift force to be increased
upwind
scheme,
and the mean
the experimental data. This trend is clarified and the distribution of its standard deviation The differences least as regards
between
the QUICK
the mean Cp distribution
the QUICK
computed
scheme enables
the fluctuations
value of the lift force to be brought
by the distribution in Figure 13.
scheme and the second
of the mean pressure central
scheme
of the closer to
coefficient
are less marked,
at the lower surface of the deck. However,
bubble at the upper surface is only detected by the second central remains somewhat underpredicted. Another two characteristics
schemes runs. As
at
the separation
scheme, even though the suction of the simulated flow argue in
favour of the latter convective scheme. The first is the transverse extension of the wake downstream of the deck (see Figure 14a). The profiles obtained from applying the second upwind scheme and the QUICK scheme do
814
L. BRUNO AND S. KHRIS upper surface
lower surface +0.6
i-I
I
I
+l.O
I
smoothflow: tautstrip model[lS] turbulent flow: tautstripmodel [15]
Znd&ind --.-.-’ stick ----2ndcentral -
turbulentflow: full-scale [16]
0 ??
+
+O.S
0.0
u”
-0.5
0
-1.0 0
0.2
0.4
0.6
0.8
La x/B 1
-
0.2
0
Figure 13. Cp and std (Cp) distributions 3
Y/B
0.4
0.6
0.8
X/B
1
on the deck. 0.124
0.164
0.197
St - 2nd cent
2.5
+0.150
1 2 ICE 0.0
2
1.5 1
-0.097 -0.150
OS
0.8
0.9
1
0.8
0.9 uJuo
x/B
1
0
0.202
J
St _ quick
- x&3=1.5
(a)
Figure 14. Mean r-velocity
@I
defect in the wake (a); Strouhal number (b).
not, substantially differ from one another. Instead, the second central scheme emphasizes the contribution of the vortices ~1 to the velocity defect. The frequency content of the lift force predicted by means of the second central scheme (see Table 5 and Figure 14b) is spread over a wider range (0.1 < St 2 0.2) than the one predicted by the other schemes. This result is hardly surprising if the dispersive effect of the odd-order leading term is borne in mind. Furthermore, the main Strouhal number in the second central simulation is the lower one (St = 0.124), which is related to merging between the vortices at the lower surface and at the leeward edges. Instead, the upwind schemes attribute most of the energy to the higher shedding frequency (St x 0.2), which is exclusively due to the vertical structures emerging at the leading edge. All the above-mentioned characteristics are due to the smaller numerical diffusivity related to the second-order central scheme, and thus, to its capability for describing the effect of the small vertical structures at the lower surface of the deck. For these reasons, the second-order central
815
2D Numerical Simulations scheme
seems to be the most adequate
conclusions
reached
by Breuer
[42] concerning
the case of flow past a circular 4.3.2. The
Effects
of the
subgrid
model
namely,
the so-called
application.
the numerical
This result
effects on large-eddy
confirms
the
simulation
in
cylinder.
Smagorinsky developed
constant by Smagorinsky
introduces Consequently,
only one free input parameter: the arbitrariness of the model
as compared to the statistical approach k-E model). However, the Smagorinsky
(five semiempirical constants are constant plays a very important
Smagorinsky
is substantially reduced required in the standard
one for the present
constant
Cs.
role in LES because its value may considerably affect the turbulent viscosity. The value of Cs has been optimized by several authors in the case of 3D simulations generally subgrid
been set at Cs = 0.1. On the other hand, model
Generally
spect to direct 2D simulations tuning
has not been taken
speaking,
simulation
of a subgrid
effects of the vertical
of the subgrid
structures
model
and potentially
in the case of homogeneous
of the free parameter
c-s
into due consideration
the presence
numerical
it is our opinion
generally
spanwise
and has
the optimization
of the
in 2D simulations. makes
enables
a substantial
application
fluctuations.
model could enable observed
that
with
re-
of the LES approach
difference
to
In particular,
reproduction
in 3D configurations
Mean Streamlines
an adequate
of the characteristic [S]. In order
to verify
CL
19
21
23
25
21
I
29 +0.20
Wh, ’
0.050
p%l -0.192
-OS0
0.075
0.100
0.125
Figure
15.
Time histories of CL and mean streamlines.
816
L. BRUNO
the advisability of such a procedure,
AND S. KHRIS
a number of values of the Smagorinsky constant in the
range 0.05 I Cs i: 0.2 were tested for the present case study. Figure 15 compares the time histories of the lift coefficient in the range 19 5 t 5 29 and the mean streamlines obtained by applying a selected number of Cs values.
The value Cs = 0.1
represents a watershed for the quantities examined. As the value of the Smagorinsky constant exceeds 0.1-0.125, (i) the separation bubble at the lower surface is abruptly reduced (0.125 5 Cs < 0.15); (ii) the merging of the vortices wl and 212(see Figure 1) vanishes; (iii) the mean value of the lift coefficient increases; (iv) the oscillations of the the lift coefficient diminish; and (v) the time history of the lift coefficient approaches a perfect sinusoid (Cs = 0.2). Instead, as the value of the Smagorinsky constant becomes smaller than 0.125-0.1, the evolution previously observed is not altogether respected. In particular,
(9
the length of the separation bubble at the lower surface gradually reduces (0.05 < Cs
0.125); (ii) the recirculating zone downstream of the upper windward edge appears; (iii) the mean value of the lift coefficient increases; (iv) the fluctuations of the lift coefficient diminish; and (v) the frequency content of the lift coefficient appears to spread over a larger range in t;he direction of the higher frequencies. To provide a better description of these trends and to clarify the reasons behind them, the step in the variation of the value of Cs is further reduced.
0.10
_________-_____-____---------________-_
0.09
0.081~
I 0,102
actual computation 0 section model [ 141 ~ tautstrip model [15] ------
Ticks
The evolution of the drag and lift
+O.l
I
3 f
T,TT
-0.1
-0.3
:ImTl fji
-0.5 .050
.075
.lOO
,125
.150
(b) CL.
(a) CD.
0 0
0
m:o 0
80
Q
~____________~______-_-_? 0
00
??
0.158
• ___~~~~~~~~__
0
00 _---~_a-a_____---_Q_~-___________-____. 0.109 0
0
.050
.075
.lOO
.125
,150
cs
.200
(c) St Figure 16. Effects of CS on the aerodynamic bers.
+O.Ol
__ ___ _ _ _ _________________________-0.08
coefficients
and on the Strouhal num-
cs
.200
2D Numerical
coefficients The
and of the Strouhal
discontinuities
so revealing
a significant
range increases. the simulated
Strouhal
numbers
positive
At the lowest
delineate range
three
variation
16.
of Cs the
in the global
the values
coefficients:
measured
that
bound
as
of thP
for Cs > 0.140
in wind-tunnel
tests.
17 and 18 makes it possible
aerodynamic
behaviour
to identify
of the deck and to
for the Cs value. can be observed.
the aerodynamic emerging
and the interaction
confirms
in the range 0.075 < Cs < 0.095.
main ranges
due to the vortices
at high frequencies,
values
out and the upper
of the aerodynamic
do not at all match is obtained
of the pressure
lift forces characterize
in Figure
at Cs z 0.145.
The diagram
(0.2 5 Cs 5 0.145) the flow is fully attached
fluctuations
flow is exclusively
is more spread
on the deck surface in Figures
for the aforesaid
In the upper
content
with the diagrams
the best agreement
schematically no significant
parameter.
appear
the best agreement with the experimental measurements. The is obtained by evaluating the reduced frequencies corresponding
the frequency
Consistently
The mean Cp distribution the reasons
to this
are plotted
clearly
peaks in the PSD of the lift force for each simulation.
the value of Cs decreases,
Once again,
constant
of CD and CL evolutions
case sensitivity present numbers
817
versus the Smagorinsky
in the evolutions
aerodynamic parameters evolution of the Strouhal to the main
numbers
Simulations
behaviour
of the deck. The unsteadiness
at the downstream
between
to the wall of the deck, and
Low values of the drag coefficient corners.
eddies is restricted
The vortices
to the upper
and of the
are shed
and lower ones
in the near wake. At values of the Smagorinsky
constant
in the range 0.140 5 Cs < 0.115, the separation
bubble
at the lower surface is predicted even though its length is largely overestimated. For Cs = 0.140 the value of std (Cp) is very low at x/B = 0.8, which indicates that there is not yet any interaction between
the vortices
interaction
w1 and 212. As the value of the Smagorinsky
is set up and increases
progressively;
however,
constant
if we examine
is further
reduced.
the Strouhal
numbers
we find that vortex merging does not yet represent the main mechanism in vortex shedding. The flow at the upper surface still remains attached to the wall. The drops in the pressure distributions,
again
on the upper
surface,
can be explained
by considerations
that
regard
the
upper surface
1
I 1
smooth flow: taut’strip model ‘[15] turbulent flow: taut strip model [15] turbulent flow: full-scale
-1.5 I
0
I 0.2
1
I
0.4
Figure
0.6
0.8
xlB
1
17. Cp and std (Cp) distributions
I
I
0
0.2
on the deck-Q
0.4
2 0.1
0.6
[161
O8
x/B’
818
L. BRUNO AND S. KHRIS lowermface
uppersurface
+O.b 3
i
+0.4 +0.2 -
,
smooth flow: taut stripmodel [151 0 turbulent flow:tautstripmodel[IS]
c,=0.050 -.-.-. CszO.065 ---.-C, = 0.075 C,=O.O85 -----c, =0.095 ----c, =O.loo -
0
0.2
0.4
0.8
0.6
xlB
??
turbulent flow: full-scale [lb]
0
1
Figure 18. Cp and std (Cp) distributions
energy conservation
of the flow. Finally,
Cs = 0.05, the suction measurements. subranges
of shedding
experimental
From mind
an overall
may be identified:
0.05 < Cs < 0.1; the upper analysis
constant.
the expression
Smagorinsky
frequencies
constant
The
of the results
above
of the subgrid-scale increases,
results,
observed
0.8
I
a
5 0.1.
constant
is reduced it matches
past the leading
the first range
from Cs = 0.1 to the experimental edge.
corresponds
Three closely
main to the
range
(St x 0.2) remains quite constant as Cs bound of the third range constantly increases it is possible
to interpret
for Cs > 0.1 are hardly
eddy viscosity
the subgrid
0.6
so much that
the flow is separated
one (0.11 < St < 0.16); the second
varies in the range as Cs decreases. Smagorinsky
surface,
diminishes
0.4
on the deck-Cs
as the Smagorinsky
at the lower surface
On the upper
0.2
+
uses
eddy viscosity
the effects
surprising
(equations
(9) and (10)).
is progressively
of the
if we bear
in
As the
overestimated,
and
its diffusion effect obscures the small-scale eddies at the root of the unsteadiness of the flow. It should be pointed out that the effects of the Smagorinsky constant in this range of values are similar to the effects of the numerical diffusion due to the grid spacing and the discretization schemes investigated above. For Cs < 0.1, the subgrid
eddy
viscosity
is progressively
underestimated.
As a result,
not
only the diffusive effects fail to conceal the small eddies but also higher frequency fluctuations are introduced in the 2D simulation. The spectrum of the 2D computations, which is generally confined within a narrow band [7], is artificially extended over a wider band. This obviously leads to the above-mentioned higher Strouhal number, but also affects the energy-transfer mechanism. In fact, in 3D simulations a number of three-dimensional features of the flow (such as, vortex stretching and longitudinal eddies in the wake) play an important role in the energy cascade, subtracting an important amount of energy from the transverse flow. Of course, these energytransfer mechanisms cannot be simulated exactly in 2D analysis, but their overall effects can be replaced by a reduced value of the Smagorinsky constant. On the other hand, this approach introduces into the flow field a number of small-sized high-frequency disturbances (wavelets) clearly revealed by the distribution of the mean pressure coefficient (see Figure 18, in particular for Cs = 0.05). In this sense, the effects of the smallest value of Cs are similar to those involved in the numerical dispersion of certain discretization schemes.
2D Numerical Simulations
+0.15 0.0 -0.15
Figure 19. Effects of Cs on the mean velocity defect in the wake x JB = 1.5.
The evolution constant
of the mean
(see Figure
maximum
amount
(Cs < 0.1).
of subgrid
in 2D simulations
hand,
a direct
here proposed
4.4.
the above
x/B = 1.5 versus the Smagorinsky explanation.
(Cs > 0.1) nor dispersed
phenomena
The wake reaches is neither
in a wider
have the same effect on the mean between
it follows that for the case study.
diffused band
its
by an
spectrum
velocity
in the
the curves.
a Cs value
in the range
However,
the simplified
0.65 < Cs < 0.75 is more geometry
of the deck does
a final verdict to be issued. On the one hand, the small number of nodes obtained by the barriers has made possible a number of extended and useful parametric studies.
On the other enable
eddy viscosity both
obtained
in the wake at
at Cs = 0.1, i.e., when the flow energy
from the correspondence
From the results not enable neglecting
defect
seem to confirm
extension
In particular,
wake, as emerges suitable
19) would
across-wind
excessive
velocity
Effects
the failure
comparison
to meet the experimental
with the experimental
or a clear singling-out
of Deck
conditions
data,
for the equipment
a complete
of the effects of the barriers
validation
on the physics
does not
of the approach of the flow.
Equipment
In order to overcome
the above-mentioned
difficulties,
in what
follows the deck equipment
is
introduced into the model. The close-up view of the mesh generated around the fully equipped section is shown in Figure 20b. The near-wall cell thickness is set at yW = 3.le - 4B. Modelling of the barriers
involves
an important
increase
in the number
of cells (76564, +68%
as compared
to
the lam-3 grid). To enable an appreciation of the computational resources involved, Figure 20a summarizes the evolution of CPU time versus cell number in LES simulations (HP j6000-1 proc. PA8600 552 MHz, SPECfp2000
l.oetS
(4
l.o&
433). The 2D detailed
model
(11 days for a computer
l.cle+7
(b)
Figure 20. CPU time versus cell number (a); grid system near the detailed deck (b).
run involv-
820
L. BRUNO AND S. KHRIS
turbulentflow: raut stripmodel[IS] turbulent flow: full-scale[16]
-1.0 0
0.2
0.4
0.6
0.8
xlB
1
0
0.2
0.4
0.6
0.8
x/B
I
Figure 21. Effects of CS on Cp and std (Cp) distributions on the detailed deck
ing 5000 time steps) is certainly more time-consuming
than the basic one (six days). It follows
that the advisability of introducing deck equipment must be carefully evaluated against the gain in accuracy. On the other hand, the extension of the same grid system in the spanwise direction (3D simulation, spanwise dimension of the computational increases the required computational computational
domain set at 1 chord length B) markedly
resources (CPU time of 44 years). It is evident that such a
commitment is out of the question in the case of parametric studies or industrial
applications. The above data confirm the importance of developing computational alternative to 3D simulations in these fields.
approaches
First of all, the parametric study on the effect of the Smagorinsky constant in 2D-LES computations is applied to the detailed deck in order to detect with a greater degree of precision the optimal Cs value for the case study. The analysis is restrictedly performed on the most suitable values of Cs previously selected in the basic simulations. Figure 21 shows the distribution of the mean pressure coefficient and the distribution of its standard deviation on the deck. The effects of the Smagorinsky constant previously observed in the basic geometry (see Figures 17 and 18) is generally confirmed. However, the results in some crucial areas of the deck indicate that the value Cs = 0.075 is the most suitable one. In fact, at Cs = 0.065 the length of the recirculating area on the lower surface is underestimated, and instabilities increase behind the barriers at the upper surface. The disagreement between the experimental measurements and the computational results remains at the lower surface in the range 0 < x/B < 0.2. If it is borne in mind that the anisotropy of the turbulence in the forebody region is probably the cause of this disagreement, it is hardly surprising that the Smagorinsky subgrid scale model proves its insufficiency. The models that yielded the most satisfactory results in the previous simulations (namely, the laminar approach, the Reynolds stress model, and the large eddy simulation with Cs = 0.075) are applied in what follows to the detailed deck. Table 6 compares the main integral parameters of the flow resulting from models with and without barriers. The results obtained using the RSM model are of less interest than the validation of the computational approach and do not provide a significant physical insight into the flow. Nevertheless,
821
2D Numerical Simulations Table 6. Integral parameters with and without barriers.
Figure 22. RSM model: lines (b).
they are briefly regarding history
commented
the weakness
upon
of the deck.
These
small-amplitude
only
It follows that
pates the periodic The comparison from barrier equipment
phenomena.
the hypothesis streamlines
by a transient
phase
shedding
the statistical
The stochastic induced
perturbations
(St = 4.68). The
final
of the leeward
side
proves
its capability
for
from the ensemble-averaged
complex
considerably
22a),
in the wake
solution.
downstream approach
by small-scale,
the eddy viscosity
in the near-
(see Figure
shedding
in the stationary
formulated
22 shows the time
and only the high-frequency
remain
due to vortex
Once again,
as the perturbations
field.
to highlight
22b.
vanish,
fluctuations
stream-
12, Figure
view of the instantaneous
is characterized
progressively
= 0.001)
in Figure
periodic
fluctuation-such turbulent
fluctuations
As in Figure
of the lift force are due to vortex
of the flow is exclusively
as shown
detecting
fluctuations
(std (CL)
unsteadiness
solution
(a) and instantaneous
follows in order to confirm approach.
and the close-up
The computational
in which the low-frequency
barriers,
in what
of the statistical
of the lift coefficient
wake region.
time track of lift coefficient
eddies-are
increases
assumed
in the
and progressively
dissi-
fluctuation. of the results
the differences modelling.
obtained
related
from the laminar
to flow modelling,
As may be readily
on the aerodynamic
coefficients
and LES approaches
as well as the common
appreciated
from Table
may be summarized
makes it possible trend
6, the main
that
results
effects of the
as follows:
(a) rise in the drag force; (b) drop in the mean value of the lift force and its fluctuations; (c) wider band
of the lift spectrum
(St).
A previous work [12] has demonstrated that the contribution of small-sized items of equipment to bridge aerodynamics is primarily due to interference effects between the items of equipment and the deck. In what follows, these interference effects (i.e., regarding
the neighbourhood
phenomena
will be further
of the items of equipment)
distinguished
and global
as local
effects.
As for the mean drag coefficient, the increase in its value due to the barriers is approximately the same in both the laminar and LES approaches (ACD x 30%). Such a major increase cannot be related to the direct contribution due to the details. Instead, we shall focus our attention on the contribution of the base region of the deck.
822
L. BRUNO AND S. KHRIS
Y/B . Larsen [I S]
+0.150 +0.072 +0.043 0.0
-0.097 -0.150 Ian-,-b=
-.-.--
lam_det
les-bas ----les-de.t -
-0.300 0.8
0.9
0.0
1
Figure 23. Velocity
Figure wake. width
23 shows the mean velocity
The
details
increase
of the deck.
sense, the presence
of the details
it is to be pointed
approaches
leads to rather
between
drag value,
0.8
0.9
1
UJU, - x/B=l.S
the width
between the overall
the introduction
predictions
lines at different
locations
in the
of the wake at x/B = 1.005 and the
vortices
which closely
increases
out that
different
0.7
defect along two straight
the distance
a higher
0.8
- x/B=1.005
defect in the wake: basic and detailed deck
the ratio
As a result,
the deck experiences Finally,
0.4
UJU,
x/B
of different
matches
sign is augmented,
the experimental
degree of bluffness of the barriers
of the abscissa
data.
and In this
of the section. in the laminar
of the local maximum
and LES
defect along
the straight line at x/B = 1.5. In the laminar simulations the velocity defect due to the barriers is not diffused along the wake, and the local velocity minimum is aligned with the upper surface. Instead,
in LES the modelling
and a less regular the same
cause
of the barriers
involves
a downturn
of the local velocity
minimum
profile of the mean velocity as the reduction
U,. The above phenomenon would appear to have in the lift coefficient and in the lower bound of the Strouhal
range. In order
to provide
the longitudinal
a comprehensive
component
explanation
of the velocity
of the above
results,
U, and of the vorticity
the mean
w along
x/B = 0.328 are shown in Figure 24. The profile of the mean longitudinal upper surface (y/B > 0.039) in Figure 24b confirms’that the velocity defect to the barriers,
which also shed a certain
amount
of vorticity
in their
profiles
the straight
of line
velocity over the in the wake is due
wake (Figure
24~).
The most surprising results concern the lower side area. On account of the blockage effect of the upwind side barriers, the velocity increases over the lower side of the deck as compared to the case of the basic geometry (see Figure 24d). As the velocity U, outside the outer boundary layer at the separation point increases, the instantaneous flux of vorticity also increases according to the relation
where
ub is the corresponding
velocity
from the base to the same
point.
Hence,
the vorticity
shed downstream of the separation bubble is higher and more concentrated (see Figure 24e). Consequently, the vorticity, convected along the lower surface of the deck, moves downstream of the maximum velocity defect in the wake. Furthermore, the increased velocity of the flow involves a deeper suction on the lower surface of the deck (see Figure 25) both in the laminar simulation (0.55 < x/B < 0.65) and in the LES simulation (0.2 < x/B < 0.4). The more concentrated vorticity reduces the length of the separation bubble and enables the LES simulation to make Both phenomena are indirectly a correct prediction of the experimental pressure distribution.
823
2D Numerical Simulations
j :
lesdet les-bas
-----+0.072 +0.039 0.0 -
I
I
I
I
0
0.1
0.2
I
I
I
0.328 0.4
0.5
x/B
(4
H I 1:
,’
__-’
+0.039 0.0 0.5 1 UJU, - xAM.328
-75
6.1 -
(b) 0.0 I ,,I: ‘-,-
-0.097
-0.15
r?%
/’ :
0.0 +7s x/B=O.328
(c) 0.5
1 I
-75
i :
:
1, ,
-0.15
+75
___4-----___c : ---___
-0.097
---- -_ --y_ : -\
j
0.0
,A-
. _* .’
I--::1
UJU, - xAkO.328
o - x/B=O.328
(4
(e)
Figure 24. Effects of Cs on the aerodynamic coefficients and on the Strouhal numbers. involved
by the barriers
the barriers
in the pressure upwind
and contribute
edge,
distribution. reducing
In particular,
the pressure
std (C,) distribution on the upper the local and the global effects. Finally, numbers.
to reducing
also have local effects on the upper
the upwind
fluctuations surface).
the value of the lift coefficient.
surface,
which involve side barrier
predicted
The mean
local maxima
reattaches
in the basic
streamlines
the flow past the
configuration
in Figure
Of course, and minima (see the
26a confirm
both
the barriers also affect the mechanism of the vortex-formation process and the Strouhal Once again it is possible distinguish direct (i.e., local) phenomena and interaction The former effects can be identified in the classical Karman vortex (i.e., global) ph enomena. streets in the wake of the circular barriers. This kind of shedding is probably responsible for the higher frequency content of the lift force (see Figure 26b) in the detailed configuration (St from 0.317 to 0.333). However, somewhat surprisingly the barriers also affect the lower bound of the Strouhal number range, namely, the shedding frequency related to the interaction between the
a24
L. BRUNO AND S. KHRIS
+0.6
I
I
I
1
flow:tautshipmodel
StKdl
0
0.2
0.4
0.6
0.8
xlB
0
1
Figure 25. Cp and std (Cp) distributions
0.2
0.4
0.6
0
[ 151
0.8
x/B
I
1
on the basic and detailed deck
les-has
.105
.141
,186 ,211 ,235
,317
St - h-bps
0.8s
b
0.6
1 bl bu
les-det
B
0.4
0.0 B98.122
.186.211
(a)
vr that
,333
St - ksdct
(b)
Figure 26. Mean streamlines deck.
vortices
,260
(a) and frequency content of CL (b): basic and detailed
are shed from the separation
bubble
and the vortices
vz that
emerge
past the
leeward edge. As a result of the reduction in the length of the bubble b&.. (see Figure 26a), the vortices ~1 translate along a greater length to reach the leeward edge and to merge into the vortices reduced
~2. Thus, the shedding period becomes longer and the associated Strouhal number from 0.105 2 St < 0.141 (without barriers) to 0.098 2 St < 0.122 (with barriers).
4.5. Flow Around
the
Deck
with
is
Angle of Attack
The goal of the simulations presented herein is to provide a further validation of the proposed 2D approach based upon the modified value of the Smagorinsky constant Cs = 0.075. The aerodynamic behaviour of the GBEB deck has been experimentally angles of attack. The mean pressure distribution on the deck measured by Reinhold et al. [14] is available in the literature 125) for an incidence below).
investigated at various in the wind-tunnel tests of 6 degrees (wind from
-
2D Numerical
8 2 I,
Simulations upper surface
0
0.2
0.4
0.6
0.8
0.2
0
x/B 1
0.4
0.6
0.8
I
I
x/E
I
lower surface
I
-I
lower surface
(b)
(4 Figure
27. Cp and std (Cp) distributions
The same setup was assumed the Smagorinsky
subgrid
by Enevoldsen
scale model.
on the basic
and detailed
deck Q = + 6’
et al. [25] in both 2D and 3D LES simulations
In the computational
simulations,
the turbulence
using level of
the incoming flow was set at zero, and the value of the Smagorinsky constant was set at 0.18. Although Thorbek [28] states that the details were modelled in both 2D and 3D simulations. the local effects of the barriers Figure
cannot
be found in the pressure
27a). The 3D simulation
Figure
27b compares
out barriers.
substantially
the results
It follows that
of the details-combined
improves
of the present
an adequate
study
distribution
the results obtained
scheme
constant
and a refined
surface
(see
using the 2D model.
using 2D models
value of the Smagorinsky
with a numerical
at the upper
obtained
with and with-
and careful
grid yielding
modelling
low dissipation-can
overcome the limitations of 2D LES simulation. In particular, the 2D model ensures the same level of accuracy as 3D simulations, while the computational costs are considerably reduced.
5. CONCLUSIONS The aim of the present vertical
structures
The first
around
paper
has been to discuss
a quasi-bluff
part was devoted
bridge
to a critical
This was followed by a closer investigation flow around
the Great
Belt East Bridge
the reliability
review of currently into the various deck.
of 2D numerical
simulation
of
deck. available
studies
mechanisms
The numerical
simulations
on the topic.
involved
in the unsteady
have shed some light
826
L. BRUNO AND S. KHRIS
on the essential Strouhal
The third studies,
physical
numbers
of the capacity of a number
approach,
a reduced speaking,
the small-scale, models assume
4.3.1.
(vortex
cell thickness dependence
dissipates
the periodic
for the convective Moreover, enabling
stretching,
The overall
upon
In particular,
the case study. model
accuracy
fluxes
in detail.
schemes
are required
to
diffusion. do not properly
the relevance
simulate
was confirmed
by the parametric
of the value of the Smagorinsky
vortices)
spanwise
of using low-diffusive
makes it possible
that
are generally
to take into due account Notwithstanding
2D model
to fix an optimal
the effects of physical
observed
is comparable
effort is considerably
reduced.
in
value for
phenomena
in the case of 3D separated with the accuracy
In principle,
the encouraging
of 3D sim-
a postetiori
the method by Diez and Egozcue [43]. Additional properties can be recovered in [44].
the global effects of the barriers
to separated-type
reported
in the subgrid
fluctuations.
of the optimised
the the proposed
discretiza-
analysis constant
Finally, the important role of deck equipment in bridge aerodynamics the increased computational effort, modelling of the barriers is strongly
research
the
In the laminar
fluctuations
The investigation
estimates can be developed exploiting concerning convergence and stability
this approach
of parametric
For each approach,
of the turbulence
the LES 2D model to take into account
while the computational
in extending
mechanisms.
was examined.
diffusive
numerical
models
the multiple
vortex-shedding
by means of a number
and high-order
approach,
the influence
longitudinal
flows with homogeneous ulations,
to simulate
the ensemble-averaged
scale model was studied this constant,
to a clarification,
of the computational
to the large eddy simulation
schemes
Section
formation. to distinct
complex eddies at the root of the vortex-formation process. The RANS equation such eddies into the turbulent stochastic field. It follows that the augmented eddy
markedly
Referring tion
near-wall
of vortex
were related
of some models of parameters
the flow field without
Generally
viscosity
of the process measured
part of the paper was devoted
influence calculate
features
experimentally
results
obtained
2D LES approach cylinder
is emphasized. recommended
Despite in order
on the flow.
in this
to other
and streamlined
error
estimates
study, classes
sections
caution of flow. represents
must The
be exercised application
a further
of
possible
development.
REFERENCES 1. G. Buresti, Vortex-shedding from bluff bodies, In Wind Effects on Buildings and Davemport), Balkema, Rotterdam, (1998).
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2. H. Nakaguchi, K. Hashimoto and S. Muto, An experimental study on aerodynamic cylinders, J. Japan Sot. of Aeronautical and Space Sci. 16 (168), 1-5, (1968). 3. K. Shimada and T. Ishihara Prediction of aeroelastic J. Aerospace Eng. 4 (12), 122-135, (1999). 4. A. Okajima, Flow around a rectangular 17, 1-19, (1983).
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6. J.H. Walter, Discrete vortex methods in bridge aerodynamics and prospects for parallel computing techniques, In Bridge Aerodynamics, (Edited by A. Larsen and S. Esdahl), pp. 301-311, Balkema, Rotterdam, (1998). 7. S. Murakami, W. Rodi, A. Mochida and S. Sakamoto, 2D square cylinder, J. Wind Eng. 55, 79-80, (1993).
Large eddy simulation
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8. T. Tamura, I. Otha and K. Kuwahara, On the reliability of two-dimensional simulation around a cylinder-type structure, J. Wind Eng. Ind. Aerodyn. 35, 275-298, (1990). 9. S. Murakami and A. Mochida, On turbulent vortex-shedding J. Wind Eng. Ind. Aerodyn. 54, 191-211, (1995). 10. B. Bienkiewice, Wind-tunnel J. Wind Eng. Ind. Aerodyn.
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11. R.H. Scanlan, N.P. Jones, P.P. Sarkar and L. Singh, The effect of section model details on aeroelastic parameters, J. Wind Eng. Ind. Aerodyn. 54/55, 45-53, (1995). 12. L. Bruno, S. Khris and M. Marcillat, Numerical simulation of the effect of section details and partial streamlining on the aerodynamics of a long-span bridge deck, Wind and Structures 4, 315-332, (2001).
2D Numerical Simulations 13. A. Larsen, Aerodynamic aspects of the final design of the 1624-m suspension J. Wind Eng. Ind. Aerodyn. 48, 261-285, (1993).
827 bridge across the Great Belt,
14. T.A. Reinhold, M. Brinch and A. Damsgaard, Wind-tunnel tests for the Great Belt Link, In Aerodynamacs of Large Bridges, (Edited by A. Larsen), pp. 255-267, Balkema, Rotterdam, (1992). 15. G.L. Larose, The response of a suspension bridge deck to the turbulent wind: The taut-strip model approach, M. Eng. SC., The University of Western Ontario, Ontario, (1992). 16. J.B. Frandsen, Simultaneous pressures and accelerations measured full-scale on the Great Belt East suspension bridge, J. Wind Eng. Znd. Aerodyn. 89 (l), 95-129, (2001). 17. A. Larsen and J.H. Walter, Aeroelastic analysis of bridge girder sections based on discrete vortex simulations. J. Wind Eng. Ind. Aerodyn. 67/68, 253-265, (1997). 18. A. Larsen and S. Esdahl, Discrete vortex simulations of fluid-structure interaction in bridge engineering, In Computational Methods for Fluid-Strmture Interaction, (Edited by T. Kvamsdal), Tapir Forlag, Trondheim, (1999). 19. I J. Taylor and M. Vezza, Analysis of the wind loading on bridge-deck sections using a discrete vortex method, In Wind Engineering into the 81 st Century, (Edited by A. Larsen et al.), pp. 1345-1352. Balkema. Rotterdam, (1999). 20. J.B. Frandsen, Comparison of numerical predictions and full-scale measurements of vortex-induced oscillations, In Proc. 4 th International ColloquCm on Bluff Body Aerodynamics d Applications, (Edited by H. J Niemann et al.), Ruhr-University of Bochum, Bochum, (2000). 21. S. Kuroda, Numerical simulation of flow around a box girder of a long-span suspension bridge, J. Wand Eng. Ind. Aerodyn. 67/68, 239-252, (1997). 22. R. Panneer Selvam and H.R. Bosch, Finiteelement modelling of flow around bridges, In Wind Engineenng into the 21 st Century, (Edited by A. Larsen et al.), pp. 1321-1327, Balkema, Rotterdam, (1999). 23. R.P. Selvam, Computational procedures in grid-based computational bridge aerodynamics, In Btidge Aerodynamics, (Edited by A. Larsen and S. Esdahl), pp. 327-336, Balkema, Rotterdam, (1998). 24. C.B. Jenssen and T. Kvamsdal, Computational methods for FSI simulations of slender bridges on high performance computers, In Computational Methods for Fluid-Stmcture Interaction, (Edited by T. Kvamsdal), Tapir Forlag, Trondheim, (1999). 25. I. Enevoldsen, C. Pedersen, S.O. Hansen, T.L. Thorbek and T. Kvamsdal, Computational wind simulations for cable-supported bridges, In Wind Engineering into the 21 at Century, (Edited by A. Larsen et al.), pp. 1265-1270, Balkema, Rotterdam, (1999). 26. M. Pecora, L. Lecce, F. Marulo and D.P. Coiro, Aeroelastic behaviour of long-span bridges with “multibox” type deck sections, J. Wind Eng. Ind. Aerodyn. 48, 343-358, (1993). 27. J.B. Frandsen and F.A. McRobie, Computational aeroelastic modelling to guide long-span bridge cross-section design, In Wind Engineering into the 21 st Century, (Edited by A. Larsen et al.), pp. 1277-1284, Balkema. Rotterdam, (1999). 28. T.L. Thorbek, Numerical simulations of the vortex shedding past the Great Belt East Bridge deck, Private communications, (2001). 29. T. Sarpkaya, Computational methods with vortices-The 1988 Freeman scholar lecture, ASME J. Fluids Eng. 111, 5-52, (1989). 30. S Lee, J.S. Lee and J.D. Kim, Prediction of vortex-induced wind loading on long-span bridges. J. Wind Eng. Ind. Aerodyn. 67/68, 267-278, (1997). 31. R. Franke and W. Rodi, Calculation of vortex shedding past a square cylinder with various turbulence models, In Proc. Eighth Symposium on 7krbulent Shear Flows, pp. 20-1-1-20-I-6, Technical University of Mumch, Munich, (1991). 32. B.E. Launder and D.B. Spalding The numerical computation of turbulent flows, Computer Methods zn Applied Mechanics and Engineering 3, 269-289, (1974). 33. V. Yakhot, S.A. Orszag, S. Thangam, T.B. Gatski and G.C. Speziale, Development of turbulence models for shear flows by a double expansion technique, Phys. Fluids A4 (7), 151@1520, (1992). 34. F.S. Lien and M.A. Leschziner, Assessment of turbulent transport models including nonlinear RNG eddy viscosity formulation and second-moment closure, Computers Fluids 23 (8), 983-1004, (1994). 35. M.M. Gibson and B.E. Launder, Ground effects on pressure fluctuations in the atmospheric boundary layer. J. Fluid Mech. 86, 491-511, (1978). 36. J. Smagorinsky, General circulation experiments with the primitive equations. I. The basic experiment. Month. Weather Rev. 91, 99-164, (1963). 37. H.C. Chen and V.C. Patel, Near-wall turbulence models for complex flows including separation, AIAA Journal 26 (6), 641-648, (1988). 38. R I. Issa, Solution of implicitly discretized fluid-flow equations by operator splitting, J. Comput. Phys. 62, 40-65, (1986). 39. R.F. Warming and R.M. Beam, Upwind second-order difference schemes and application in aerodynamic flows, AZAA Journal 9 (14), 1241-1249, (1976). 40. B.P. Leonard, A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comp. Meth. Appl. Mech. Eng. 19, 59-98, (1979). 41. W. Shyy, S. Thakur and J. Wright, Second-order upwind and central difference schemes for recirculating flow computation,
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42. M. Breuer, Numerical and modelling influences on large-eddy simulation for the flow past a circular cylinder, In Proc. 12 th Symposium on Z’kbulent Shear Flows, Grenoble, (1997). 43. P. Diez and J.J. Egozcue, Probabilistic analysis of an a posteriori error estimator for finite elements, Mathematical Models and Methods in Applied Sciences 5 (ll), 841-854, (2001). 44. K. Gerdes, J.M. Melenk, C. Schwab and D. Schotzau, The hp-version of the streamline diffusion finite element method in two space dimensions, Mathematical Models and Methods in Applied Sciences 2 (ll), 301-337, (2001).
PERGAMON
and Computer
Modelling
37 (2003) 795-828 www.elsevier.com/locate/mcm
The Validity of 2D Numerical Simulations of Vertical Structures Around a Bridge Deck L. BRUNO* Department of Structural Engineering Politecnico di Torino Viale Mattioli 39, 10126 Torino, Italy [email protected]
S. KHRIS OptiFlow, Consulting Company in Numerical Fluid Mechanics IMT-TechnopBle de Chateau-Gombert, Cedex 20, 13451 Marseille, France khrisQoptiflow.fr (Received February 2002; accepted April 2002) Abstract-This paper deals with a computational study for evaluating the capability of 2D numerical simulation for predicting the vertical structure around a quasibluff bridge deck. The laminar form, a number of BANS equation models, and the LES approach are evaluated. The study was applied to the deck section of the Great Belt East Bridge. The results are compared with wind-tunnel data and previously conducted computational simulations. Sensitivity of the results with regard to the computational approaches applied for each model is discussed. Finally, the study confirms the importance of safety-barrier modelling in the analysis of bridge aerodynamics. @ 2003 Elsevier Science Ltd. All rights reserved.
Keywords-Computational wind engineering, Bridge aerodynamics, methods, Turbulence modelling.
Vortex shedding, Grid-based
1. INTRODUCTION The study of aerodynamic characteristics plays a prominent role in the design of long-span bridges. In particular, the vortex-induced forces acting on the bridge deck are of great interest from the practical point of view. In fact, this phenomenon occurs at a relatively low range of wind speeds and can markedly affect the durability and serviceability of the structure. Traditionally, these studies are experimentally carried out by means of wind-tunnel tests, which are economically
burdensome and in some cases not realistic, owing to the lack of aerodynamic
similitude of the model.
In order to respect the conditions of aerodynamic similarity without.
using cumbersome models, wind-tunnel tests often employ section models in the preliminary bridge design. In fact, rigidly mounted models provide a good description of the complex flows that develop around bluff-shaped cylinders at high Reynolds numbers. The first author wishes to express his appreciation to the Fluent Italia and the Hewlett Packard Italia Corporations *Author to whom all correspondence should be addressed. 0895-7177/03/$ - see front matter @ 2003 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(03)00087-S
Typeset
by d,@‘IEX
L. BRUNO AND S. KHRIS
796
A number of fundamental studies, critically reviewed by Buresti [l], have investigated the unsteady
flow around
basic
bluff sections,
such as circular,
results obtained from such studies represent a useful of industrial fields, such as bridge aerodynamics. Generally
speaking,
ticity and by massive
flows of this kind are characterized separation
which involves
and in its wake.
important content.
role in the production of aerodynamic Three-dimensional structures ultimately
spite the nominally
Both of these phenomena,
two-dimensional
geometry
or rectangular
ones.
applications
motion
of the vertical
which are strictly
The
in a number
by shear layers presenting
unsteady
the body
square,
basis for further
marked
structures
vor-
around
interconnected,
play an
forces in terms of mean value and frequency appear in the process of wake formation de-
of the body,
as a result
of the vortex
lines being
distorted. The case study of the rectangular section best applies to the shape of long-span bridge decks. In regard
to these
cylindrical
bodies,
it is widely
recognized
that
their
width-depth
B/D
ratio
strongly affects the characteristics of the separated shear layer that is generated at the leading edge [2]. From this point of view, such sections are classified into two categories, namely, the “separated-type”
sections
cross-sections) latter
(compact
[3]. The former
present
more complex
cross-sections)
are generally
phenomena,
and B/D = 6 or the double
mode
and the ‘?eattached-type”
prone to Karman-type
such as discontinuities
in the lift fluctuations
sections
vortex
shedding,
in Strouhal
the
at B/D = 2.8
number
due to unsteady
(elongated whereas
reattachment
of the
separated shear layer on the side surfaces at 2.0 < B/D < 2.8 [4]. Likewise, most of the bridge decks are quite elongated (B/D z 7) so that occurrence of a reattached shear layer calls for a proper evaluation of other characteristics of the flow, such as the extent of the separation the convection of the vortices along the surfaces of the deck, and the interaction between shed from the windward During
the last
plementary efficiency tunnel
and leeward
decade,
computational (accuracy-cost
tests.
ratio)
To achieve
is ,required
to the potential
flow (see, for example,
Reynolds-averaged A number accuracy time,
imposed
mensions, markedly
The latter
Navier-Stokes
(RANS)
boundary
and grid size. inflates
(NSEs).
numerical
memory
of numerical
equation
parameters
in grid-based conditions,
procedures
simplified
introduced decks.
that
a com-
An increased
as against
have been proposed.
methods
(panel involve
may be further
(DNS),
large-eddy
windThese
methods)
applied
numerical
solution
classified
simulation
as regards (LES),
and
models.
have to be taken methods:
extension
On the other
category
simulation
of bridge
to be attractive
[6]), as well as approaches
numerical
simulation
has gradually
analysis
approach
categories:
into direct
of other
of the
into two main
equations
modelling
computers
for this
this goal, a number
classified
turbulence
of powerful
to the aerodynamic
can be broadly
of the Navier-Stokes
edges [5].
the advent approach
bubble, vortices
into consideration
discretization
of the computational
schemes
to increase
in both
domain
space
in two or three
the and di-
hand,
3D analysis and refinement of spatial resolution Murakami [i’] reports that usage and processing time. For instance,
LES computation of vortex-shedding flow past a square section requires only 60 hours for the 2D model but more than 42 days for the 3D model on a convex ~240 machine. As may be readily understood, even when possible the extension of 3D simulations to industrial applications characterized by complex geometries and high Reynolds numbers (as in the case of long-span bridge decks) is likely to prove extremely burdensome and excessively time-consuming as compared to wind-tunnel tests. In order to reduce the computational costs, many authors have examined the applicability of the above-mentioned approaches to the 2D simulations of vortex-shedding flow past a square cylinder. According to the authors of the present paper, three main studies obtained the most representative results. Tamura et al. [8] discussed the reliability of two-dimensional direct numerical simulation for They emphasized the importance of the mix unsteady flows around cylinder-type structures. between grid size and discretization an excessive amount of numerical
schemes diffusion.
of the nonlinear convection term According to the above authors,
in order to avoid two-dimensional
797
2D Numerical Simulations calculations
cannot
in the process
limitations
properly
simulate
of 2D direct
More recently, square cylinder
the three-dimensional
Consequently,
of wake formation. numerical
simulation
50% with respect
with the Smagorinsky
order to compensate
for the limited
LES computations
with the experimental
results,
discrepancies.
out a wide spectrum
square
According
other
mechanism,
out a critical
sections
enables
hand,
and extended
two-dimensional
an incompleteness
present
by 2D computations
review
the spanwise model
number
analysis,
related
this
through-
R.ANS simula-
diffusion
sections.
process
They
by an eddy
region.
On the
in the underesti-
cross sections. further
to such flows is very high
respecting
using
rectangular
was detected
study of unsteady flows around bridge decks introduces both to physical and to computational aspects. Reynolds
cert,ain
it is fundamentally
even in the high Reynolds-number
The related
the
present
mechanism
performed
lift fluctuations
First,
as well as with those
a close correspondence
because
to elongated
mated
in computational
of 2D and 3D
a 3D phenomenon.
of the tests
reattached-type
b?
(Cs = O.lOj in
of the 2D LES computations
this approach
analyses
was increased
The results
using RANS models,
of the ensemble-averaged
for stationary
direction.
which is essentially
and
class of flows.
et al., the energy-transfer
to Murakami be reproduced
appear
flow past a two-dimensional Smagorinsky-type subgrid
for 2D simulations
of the 3D computations the results
ultimately
used in 3D simulations
in the spanwise
that the k - E model, which incorporates
demonstrated viscosity,
whereas
et al. [3] carried
Shimada tions around
generally
with those obtained
range cannot
due to the vortex-stretching
constant
constant
The results
which
be paid to the reliability
for the above-mentioned
diffusion
were compared
from experiments.
significant
must
Murakami et al. [9] predicted the vortex-shedding using large-eddy simulation (LES). The standard
model was used, and the value of the Smagorinsky
obtained
structures
attention
number
involves
important
difficulties
(1.e + 5 < Re < 1.e + 08):
particular
requirements
in spatial
discretization. Second, acterized
the most widely
used deck sections
by intermediate
currently
levels of “bluffness”
adopted
(elongated
sections
and leeward). Consequently, the flow around them is generally bubbles and shedding of vertical structures that are extremely Third,
the actual
finished
and crash barriers.
bridge
According
be disregarded.
Using
itatively
showed
the influence
response
of bridge
deck carries
to various
flow visualization
decks.
of partial
Adopting
several
authors,
for long-span
bridges
and faces inclined
windward
characterized by small separation variable in frequency and size.
items of equipment,
such as side railings
the effects of such section-model
techniques
in wind-tunnel
streamlining
and traffic
the same approach,
are char-
Scanlan
tests, barriers
details
Bienkiewicz
cannot
[lo] qual-
on the vortex-induced
et al. [ll] pointed
out the criti-
cal dependence of bridge-flutter derivatives upon even minor details, such as deck railings They likewise identified a critical component of the experimental approach in terms of accuracy and similitude
requirements
ers has recently the case of steady To provide present
paper.
in the modelling
been confirmed flow around
an example The
Great
of the section
by computational a streamlined
Belt East
The noticeable performed
influence
of barri-
using RANS models
in
deck [12].
of such peculiarities, Bridge
significant benchmark from the standpoint peculiarities mentioned above fully apply
details.
simulations
we shall introduce (GBEB)
the study
deck is here considered
case adopted
in the
as a particularly
of vortex shedding for three main reasons. First, the to this deck. Second, large vortex-induced oscillat,ions
were measured on the main span just after the deck’s completion. Third, the deck has been the subject of various experimental measurements [13] and computational studies, and thus, provides a significant data base to be used for assessing new simulations. Figure 1 is a schematic illustration of the flow pattern around the deck at Re = UoB/u = 1.5e+5 (where B is the deck width) and o = 0” using the computed instantaneous streamlines. The instants tr and tz correspond to a local maximum value and a local minimum value of the lift force, respectively. The three main characteristics may be easily recognized:
L.
798
BRUNO AND S. KHRIS
vg vortex street behind
the barriers
v3 vortex shedding v, advected boundary separation
vortices
v2 lower vortex
at the lower side
layer point
in the deck wake
interaction reattachment
points
vi-2 coalescence Figure
(a) the strong
1.
interaction
vertical
between
vortices
structures
around
deck.
bubble
Independently,
downstream another
of interaction.
of the lower sharp recirculation
The barriers First,
is accelerated. edge and prevent
perturbations
the flow passing
between
The side barriers
reattach
a larger
vi e
v2
corner,
travel
separation
bubble
and in the near
across
region
from forming.
on account
layer downstream
Second,
the vortex
of the close to
corner,
of two main
of the deck and the lateral
shear
from the
the lower surface
vi passes the lower leeward
surface
the separated
and
vl, which develop
in the near-wake
in the flow pattern
the upper
the vortices;
at its lower surface
zone (~2) is present
seem to enhance
between
The vortices
the lower slope. When the advected procession of vortices coalesces with the vortex 212and is shed in the wake. nomena.
of vortices
vi, ~2, and ~4, ~5;
the deck are located
wake. We shall focus on their mechanism separation
v4 e 3
Flow pattern around the deck: instantaneous streamlines.
(b) the differences in size, intensity, and shedding frequency (c) the flow perturbations induced by the safety barriers. The main
between
due to the barriers
it
phe-
barriers
of the upwind street
us behind
the trailing barriers seems to interact with the vortex v4 in the deck wake. Table 1 summarizes the wind-tunnel setup and the conditions of computation of the most relevant studies. The results obtained are expressed in terms of the following integral parameters: the mean values of the drag coefficient CD = D/(1/2 pUsB) and lift coefficient C:L = L/(1/2 pUaB), and the Strouhal number St = fc,D/U,,where D is the deck depth. Early section-model tests were conducted by Reinhold et al. [14] on a number of candidate cross-sectional shapes. The 1:80 modei selected, designated as H9.1 and characterized by a width-depth ratio Bf D x 7.2, was tested using equipment fully described in [14]. Steady-state wind-load coefficients were subsequently measured on the same cross section using a 1:300 tautstrip model 1151. The overall agreement between the results obtained from models of different size indicates that Reynolds-number influence on the global aerodynamic
effects, even though they are present, do not have a marked behaviour of the deck. The most important discrepancy regards the value of the lift coefficient. In the opinion of the authors of the present study, this
79!l
2D Numerical Simulations Table 1. Outline of published studies-zero Author Reinhold
[141
[151
Larose
Frandsen
Model
Method
[161
Larsen [17,18]
angle-of-attack
(cx = 0’).
Geom.
Re
CD
CL
St
1.e + 5
0.08
+ 0.01
0.109 - 0.158
0.10
- 0.08
0.11
-
_
0.08 - 0.15
0.06
+ 0.06
0.100 - 0.168
Exp.
Section
det
Exp.
Taut-strip
det
7.e + 4
Exp.
Full scale
det
1.7e+7
DVM
pot. + w
2Dba.s
1.e + 5
Taylor [191
DVM
pot. + w
2Dbas
1.e + 5
0.05
+ 0.07
0.16 - 0.18
Frandsen [20]
DVM
pot. + w
2Dbss
1.6e + 7
0.08
+ 0.06
0.09
FEM
NSE lam
2Dbas
1.6e + 7
0.06
- 0.09
0.25
FDM
NSE lam
2Dba.s
3.e + 5
0.07
- 0.19
0.101 - 0.168
Selvam [22,23]
FEM
NSE les
2Dba.s
l.e+5
0.06
- 0.34
0.168
Jenssen [24]
FVM
NSE les
3Ddet
4.5e + 4
0.06
+ 0.04
0.16
FEM
NSE les
PDdet
7.e + 4
0.07
+ 0.08
0.17
Kuroda
[21]
Bnevoldsen
difference model.
[25]
may be related
to the unachieved
In fact, as demonstrated
the higher reduced.
flow velocity The various
and the stronger
vortex-shedding
the spread frequency content The full-scale measurements of the flow. The results lower bound
similitude
suction
of the barriers
along the lower surface.
mechanisms
in the 1:300
blockage effect of the railings
highlighted
in Figure
involves
The lift force is thus 1 probably
account
for
of the lift force that was found experimentally by Reinhold et al. [14]. obtained by Frandsen [16] substantially confirm this characteristic
obtained
of the range
aerodynamic
in [12], the most important
by Larose
[15] seem to indicate
has the most important
that the Strouhal
effects on the structural
number
response
at the
of the deck.
Most of the computational simulations assume the same Reynolds number used by Reinhold et al. [14]; exceptions are represented by [26,27]. On the other hand, only Enevoldsen et al. [25,28] state
that
account
they
include
the deck equipment
the inflow turbulent
Three
general
conclusions
may be drawn
the first place, the underestimated the simulations.
in the computational
level experimentally
from an overall
prediction
This error is certainly
model.
No author
takes into
set at It M 7.5%. analysis
of the drag coefficient
due to the absence
of barriers
of the results is a common
obtained.
feature
and to the incoming
In
in all of smooth
flow. Second, most of the simulations fail to highlight the spread frequency content of the lift force, indicating only one Strouhal number. This difficulty mainly concerns the approaches that solve the Navier-Stokes equations using grid-based methods. Finally, almost all the simulations are conducted
in the 2D computational
of the fact that excessively A number
3D simulations
burdensome
domain.
remain
for industrial
of applications
This common
scarcely
applicable
approach
in extended
provides
confirmation
parametric
studies
and
applications.
are based on the so-called
discrete
vortex
method
(DVM),
which is
comprehensively reviewed in [29]. This approach is widely adopted in bridge aerodynamics (see. for example, [6]) for two main reasons. First, the Lagrangian nature of the method markedly reduces
the difficulties
encountered
in grid-based
methods
(mesh
quality,
numerical
diffusion,
modelling of small details of the deck section). Second, the description of the flow field is obtained by an essentially potential-flow method, so that the computational effort is significantly reduced as compared
to the methods
involving
solution
of the Navier-Stokes
equations.
On the other hand, the above advantages are obtained thanks to a fundamental assumption regarding the physical feature of the flow, namely that “for high Reynolds numbers, viscous effects may not be essential in the development of the wake, as the vorticity that is introduced in the wake from the separation
point
remains
confined
to narrow
regions
and its dynamics
is
mainly dominated by convection, with viscous diffusion only playing a secondary role” [11. From the numerical point of view, this assumption requires that the amount of vorticity w shed from the separation point (which must be known a priori) be introduced by means of particles of finite core size in each time step. This approach has been applied to the study case by Larsen
L. BRUNO AND S. KHRIS
800
et al. [17,18], who obtained
an impressive
agreement
between
data, even if compared with more complex approaches. all applications of vortex models to unsteady separated have shown
that
the
circulation
of the vortices
their
results
and the experimental
According to Sarpkaya [29], “practically flows past two-dimensional bluff bodies
should
be reduced
as a function
of time
and
space in ad hoc manner in order to bring the calculated forces, pressure and circulations in close agreement with those measured”. As a result, the evaluation of the performance of this approach and of its suitability of the aforesaid
for application
numerical
the drag coefficient
by addition
“care must
be taken
Other
approach, of a flat plate
when
analysing
effect is largely
authors
have predicted using
of the barriers.
treated these
element
finite volume
as [ .
results
] porosity
the Great
to these
are
authors.
effects are neglected”; the nature
e.g., finite
difference
i.e.,
of the flow
Belt East Bridge by solving
methods,
value of
the side railings
According
in such a way as to change
of discretization
In the framework
the underestimated
Consequently,
as a solid geometry.
the flow around
a number
very problematical.
and Vezza [19] attribute
overestimated
Stokes equations method,
Taylor
to the lack of modelling
introduced the blockage completely.
to other shapes remains
the Navier-
method,
finite
method.
Among the various works that neglect turbulence modelling (laminar tained by Kuroda [21] are in good agreement with the wind-tunnel data.
form), the results obIt seems that the error
in approximating
balances
turbulence between
the convection
damping. numerical
order to justify
diffusion,
More recently,
obtained
paid to boundary-layer
damping studies
and the scheme
that
are reported
adopted
the absence
of
on the relationship
(fifth-order
upwinding)
in
in a 2D simulation.
on the mesh-density modelling.
a numerical
no parametric
et al. [27] applied
Frandsen study
hand,
grid resolution,
the accuracy
case. A parametric
term provides
On the other
the FEM-without-turbulence sensitivity
As a result,
was initiated
shedding-frequency
model
to the study
with particular predictions
attention
were generally
inaccurate even for the most refined grid so that the authors of the paper concluded that “further studies are needed to establish the model requirements”. Subsequently, Frandsen [20] tested the DVM in order to compare other studies. To the knowledge study.
Only
the two different
of the present
authors,
Lee et al. [30] adopted
loading
on a fully bluff bridge
without
law of the wall.
approaches
deck.
RANS
a RANS
with one another
models
approach
have never
for predicting
The renormalization
The quadratic
upwind
group
interpolation
scheme was employed to approximate the convection term. indirectly validated by performing dynamic structural analysis at the bridge any conclusions
midspan
with the full-scale
to be drawn
In the last few years,
measurements.
on the reliability
the LES approach
been
applied
to the case
the vortex-induced
(RNG)
of
k - E model
wind was used
for convective
kinetics
The unsteady and comparing
wind loading was the displacements
This approach
of the statistical
has been applied
and with the results
clearly
(QUICK)
does not permit
approach. to the case study.
Selvam
[23] dis-
cussed the reliability of the 2D simulation, considering the deck section of the Great Belt East Bridge approach span. Both 2D and 3D computational models were used, neglecting the barriers. No parametric studies were performed. The author assumed the value of the Smagorinsky constant without
proposed by Murakami any further investigations.
drag coefficient
with reasonable
[9] for the square cylinder (C, = 0.15 in 2D, C, = 0.10 in 3D), According to Selvam, only 3D models are able to capture the accuracy.
In a subsequent
paper,
the same author
[22] employed
two different levels of grid refinement and two near-wall treatments (i.e., low Reynolds number and law of the wall) in the 2D model of the Great Belt East Bridge main span. The results of the study indicate an impressive sensitivity of the mean flow to the parameters mentioned no (ACD M 50%, ACL M 400%, ASt M 20%). B ecause of the limited number of computations, conclusions were drawn regarding this sensitivit,y. In the case of nonperiodic flows and spread frequency contents of the aerodynamic forces, the integral parameters only provide a partial physical insight into the flow. Figure 2 compares the
2D Numerical
upper surface
I
I
I
,, Ic
X01
Simulations
I
I
<
0.4
0
0.2
..
0.4
0.6
0.7
0.6
0.8
_
. +
turbulent flow: taut ship model [ 1fl turbulent flow: full-scale [16]
x/B 1
Kuroda(fdm) [21] ~ Jenssen (fvm) [24] -----Frandsen (fern) [ 203 - - - - Frandsen (dvm) [To] .-..-. ..... u”
-1.0 -
I
I
1
,
I
I
lower surface Figure
distributions a number
of the mean of simulations.
the computational First,
2. Published
pressure
minor
thus
imperfections
estimated
both
on the deck
on the deck obtained
discrepancies
clearly
0 I x/B 5 0.2 the experimental of the leading
overestimating
may be related
Second,
coefficient
Two main
lower surface just downstream They
Cp distribution
mean
appear
(a = 0’)
in wind-tunnel between
tests
and in
the experimental
and
distributions.
in the range
phenomenon,
studies:
to the high anisotropy of the leading
the lengths is restricted
simulations.
bubbles
suction
simulations
in the forebody
model cannot tests
on the
ignore this are not clear.
region
[31], but
be ruled out a priori.
at the lower and upper
In the wind-tunnel
the range
a moderate
for this discrepancy
of the turbulence
edge of the geometrical
within
reveal
All the computational The reasons
of the separation
in the computational
the lower surface
point.
the pressure.
data
surfaces
are over-
the recirculation
zone at
0.2 < z/B 5 0.35, and recovery
of the pressure
rapidly occurs within the range 0.35 5 x/B 5 0.45. Except for the DVM approach, all the other numerical methods predict a pressure recovery in the range 0.6 5 x/B < 0.7 and underestimate the suction
peak.
It is to be pointed
out that
the phenomena
occurring
in this region
play a
major role in the vortex-formation process, as emerges from Figure 1. Consequently, the lack of accuracy in this region deeply affects the overall reliability of the simulations. The reason for these errors is not clear. However, if it is taken into account that the characteristics of the incoming flow are rather similar, the differences between computations different computational procedures adopted in the studies. The overestimated
length
of the bubble
at the upper
surface
seem to be related
seems to be directly
to the
related
t,o
the interference effects of the windward side barrier. These effects have a minor impact on the dynamics of the flow but markedly affect the mean value of the lift force. Jenssen [24] has modelled such details, obtaining the local reattachment of the shear layer. Nevertheless, the
802
L. BRUNO AND S. KHRIS
constant value of the pressure zone that is not experimentally From the present primarily
address
critical
downstream detected.
of the railings
review of the studies
the question
of validation
suggests
published,
the presence
it follows that
of the codes employed.
fail to provide any closer insight into the various physical thermore, the effects of various computational parameters
of a recirculation
most of the researches
In many
cases, the results
mechanisms involved in the flow. Furon the solutions is not systematically
clarified. The purpose
of the present
to verify the suitability To do this,
particular
parameters
to be taken
LES approach regards
attention
so enabling
not only
effects of the details
The incompressible,
unsteady,
study.
in the model
models.
In particular, but
The suitability
both to complete
the
also as
of each of
model.
Finally,
the experimental
measurements,
and deck.
computational
parameters
model.
of the aerodynamic
setup,
and to point
up the
field.
EQUATIONS
two-dimensional
The nondimensional
suitable
on the basis of the optimized
2. GOVERNING in the present
turbulence
to be made with wind-tunnel
on the characteristics
the most
subgrid
studies
flow past a bluff bridge
the above-mentioned
is discussed
is included
comparisons
of the previous
unsteady
in the various
as regards
of the Smagorinsky
to turbulence
of the section
correct
the incompleteness
for predicting
has been paid to determining
parameter
approaches
the equipment
is to reduce
into consideration
is optimized
the free input
the various
paper
of 2D simulations
Navier-Stokes
instantaneous
equations
continuity
of motion
and momentum
are solved equations
are div u = 0 and dU
dt where
u, p, and t are, respectively,
by the reference The Reynolds
velocity number
lence-modelling closure
Re results
vector,
pressure,
and
turbulence
are applied.
modelling)
In the framework
a second-order
models
are the standard
(STD)
model.
The Reynolds-averaged
closure model [32] and th e renormalization
incompressible
E$+U.!L!&-L 3 axj
axi
approach
According
to the standard by means
nondimensionalized viscosity
k - E model
u.
$+&!p2
equations
(p_+g1+2& P
turbu-
both first-
are expressed
as
C”+ vt) Sij,
3
[32], the turbulent
axj
approach,
are adopted. The first-order closure group (RNG) (331 forms of the Ic - E
Navier-Stokes
of their transportation
LAM and various
of the statistical
where ut is the isotropic eddy viscosity given as vt = pC,k2/E, and SQ is the mean flow strain rate expressed as follows:
rate E are computed
and time
p, and the kinematic
from these variables.
(i.e., without
approaches models
the velocity
+ & Au,
Us, the deck chord B, the air density
Both the laminar-state order
+ u. grad u = -gradp
k is the turbulent
kinetic
energy
kinetic
energy,
k and its dissipation
equations
[(‘+z)&]+Pk-E
and (5)
803
2D Numerical Simulations where Pk is the production
constant
term of the turbulent
are the conventional
kinetic
energy,
and the values
of the empirical
ones [32].
In order to cope with the well-known
excessive
production
of turbulent
kinetic
energy
model in nonequilibrium situations, the k - E form suggested by the renormalization is also applied. This model introduces a further production term into the transport the dissipation
of this
group [33] equation of
rate E in the form c/J13 (1 - rllrlo) E2 lfP$ X-’
Ii-=
(6)
where
and the constants In order
70, ,0, uE, ok, C,,,
to account
better
CEz, C, are given in [33].
for important
Reynolds stresses, diffusive transport
the energy transfer mechanisms-F’ranke
to the calculation
of vortex
Abandoning solving
the isotropic
transport
such equations turbulent
past
a square
transport
and modelling term
of the flow-such
as the anisotropy
and qbtained
the RSM closes the
Reynolds assumptions
interesting RANS
Rij = q.
stresses
are required.
and the pressure-strain
of the
flow and the mean flow, or the the Reynolds stress model (RSM)
cylinder
hypothesis,
for the individual
are unknown,
diffusive
shedding eddy-viscosity
equations
aspects
between the turbulent and Rodi [31] applied
term,
equations
Several
In the present
are respectively,
results. by
terms
in
study,
the
expressed
by
the models proposed by Lien and Leschziner [34] and by Gibson and Launder [35]. Model equations for the large eddy simulation (LES) are expressed in the form
and a?!&
aii,
--
dt+fijFdz
where the overline
indicates
scale model
subgrid-scale
(8)
dXi
3
subgrid
d
the filtered
value of the variable.
[36]: ksGs indicates expressed by
is employed
eddy viscosity
USGS =
The subgrid-scale tations
mixing
length
PLSGS
Ls~s is related
component
Smagorinsky-type of k, and USGS the
2/2SijSij.
19)
to the Smagorinsky
constant
Cs in 2D compu-
by LSGS = min
where
The standard
the subgrid
n = 0.42, A is the cell surface,
(
ted, C,A1/2
and d the distance
>
,
between
(10) the cell and the nearest
wall
boundary. 2.1.
Near- Wall Turbulence Modelling
Near-wall modelling wall-bounded flows. A suitable for application However, Franke et al.
has a significant impact upon the reliability of numerical simulations of number of works that use various turbulence models render these models throughout the boundary layer by means of the wall-function approach. [31] have demonstrated that “the assumptions of a logarithmic velocity
distribution and of local equilibrium of turbulence are violated in separated flows, especially near separation and reattachment regions” [31]. Bearing in mind the purpose of the present study, the low Reynolds number one-equation model (“two-layer model”) has been adopted in the viscosity-affected region (Re, = p&y/v > 200) in
804
L. BRUNO AND S. KHRIS
conjunction with the statistical approach. This approach was successfully applied by Franke and Rodi [31] in the study case of the square cylinder and by Shimada and Ishihara (31 in predicting vortex
shedding
past rectangular
are here calculated
cylinders.
using the turbulent
ut = The length
The turbulent
kinetic
energy
viscosity
rate E
k from the relations k3/2
pc, JiFl,,
scales I, and 1, are expressed
ut and the dissipation
(11)
by the following
relations:
(12) where the values of the constant When strain
the LES model
in the formulas
is applied,
are taken
from the laminar
stress-
relationship ii
-_=%
where u, = (~~/p)‘.~
is the friction
PWY
Computations method.
were carried
In the grid-based
on fundamental
coupling
unstructured
In particular,
out
method
using
PROCEDURES
the FLUENT
simulation
importance. grid type
~5.4 code,
of flow around
In this study,
and structured
the mapped
LJ'
velocity.
3. NUMERICAL
takes
from Chen and Pate1 [37].
the wall shear stress ru is obtained
mesh types
based
complex
the computational by means
simulations
computations are shown
and complies
using turbulent in Figure
with the mesh requirements
models.
The computational
3. The size of the domain
parametric study reported in [12]. The pressure-velocity coupling is achieved with splitting equation,
of operators
whilst
enforcing
the continuity
of the total
of a parametric
and the boundary
consistently
by means of the pressure-implicit approach
study
approach
SIMPLE
fine for
in the
conditions
with the results
to advance
by grid.
in order to achieve
of the two-layer
domain
is adopted
[38], using a predictor-corrector
are generated
of substructuring
is used in the near solid boundary
volume
mesh generation
domains
resolution of the flow structure in this region. The cell thickness y,,, adjacent to the solid wall forms the subject laminar
on the finite
geometries,
of the
algorithm
the momentum
equation.
-.-.-.-.-.-.-.-.-.-.-.~.~.~.~.~.~.-.~.~.~ periodic (LES) I U=l; v=o; I =o I
U=l;
v=o;I =o
periodic (LEb) ,.,.,.,.-.-.-.-.-.-.-.~.~.~.~,~..m.~.~......a 15B Figure 3. Analytical
I 30B
domain and boundary
conditions.
805
2D Numerical Simulations
For time discretization, mensional
the fully implicit
time step in the laminar
At* < 2.e- 2 with respect
vergence criteria. initial conditions
to the expected
derivative
cept for the nonlinear (second-order
terpolation formula
Euler
and statistical frequency
scheme
approach
content
is adopted.
The nondi-
varies in the range 5.e - 4 <
of the phenomenon
and to the con-
The time advancement in LES undergoes optimization during the study. The are set equal to the inlet boundary conditions (impulsively started simulation).
All the spatial schemes
second-order
approach
terms
are discretized
by the second-order
convection
terms.
The convection
terms
central
difference,
second-order
upwind
kinetics-QUICK
[40]) that
for convective
central
are discretized
differencing, according
[39], and quadratic
are generally
expressed
ex-
to threcl
upwind
according
in-
to the
an d d:,f The weights
= -~e.fe
for the discretization
schemes
adopted
Table 2. Coefficients I
As reviewed
of a signal-without
diffusive schemes Notwithstanding assertions
extent
diffusion.
have a diffusion the signal
the relative
of different
physical
problems
and to numerical
effects (markedly
dispersion
effects
they
On the other
hand,
it is not
and according
affecting
stabilizes
appropriate
Scheme
to cause phase
Second-Order
Upwind
QUICK
universal may vary
time-marching
schemes,
structures errors
of convective schemes.
I---Central
to make
performance
flow seems to be significantly
Leading Term
Second-Order
the computation.
[8]), and computational-domain
frequencies). Table 3. Accuracy
terms
of the above-mentioned
since their
the small vertical
(i.e., it appears
(as in content
even leading-order
to different
(see, for example,
terms
alter the frequency
The accuracy
of these schemes
(i.e., 2D or 3D). The physics of the investigated diffusion
IEe
effect, which generally
grid density
I
I KP
[41]), the odd leading-order
in amplitude.
performance
(see [7,8,42]),
ku
effects-i.e.,
is summarized in Table 3. the above general considerations,
models
to numerical shedding
numerical
but reducing
regarding
in the presence turbulent
scheme)
wiggles
have dispersive
schemes. I
I
&Jw
(see, for example,
schemes)
involving
(as in the QUICK removing
I
authors
are given in Table 2.
of convective
Scheme
by various
the case of second-order
Kwfw- Kwwfww
- ~p.fp -
Effect
sensitive
around
between
both
the deck)
the different
806
L. BRUNO AND S. KHRIS
However, even though a number of studies have been devoted to the evaluation of these schemes, in the present paper a parametric approach.
study is performed using both the laminar form and the LES
4. APPLICATION 4.1.
Simulations
Without
Turbulence
AND
RESULTS
Modelling
The simulations assume the experimental conditions adopted by Reinhold et al. [14] during the section-model tests (see Table l), except for incoming turbulence intensity It = 0. The angle of incidence of the flow is zero. 4.4.1.
Effects
of grid spacing
and discretization
schemes
on the flow field
The effects of the near-wall cell thickness yw in 2D simulations without turbulence are discussed in this section. characterized
To do this, three grid systems are taken into account.
by a nondimensional
grid spacing adjacent
modelling These are
to the solid boundary equal to yw =
1.3e - 2, yu, = 2.0e - 3, and yw = 2.2e - 4 (width B = 1) and will be designated in what follows respectively by lam-l, lam-2, and lam-3. The performance of these three systems is first evaluated in conjunction
with a second-order upwind scheme for convection terms. cellnumber
1.3~2
2.Oe-3
2.2e-4
45512
28767 25865 0.01
0.001
Y$
o.ooo1
(aI
@I
Figure 4. Near-wall cell thickness versus total number of cells (a); grid system near the deck (b).
The reduction of the first cell width involves an increased number of control volumes in the neighbourhood of the wall. The outer part of the computational domain retains the same grid. Figure 4a shows the evolution of the total number. of cells of the model versus near-wall cell thickness. The close-up view of the grid system near the deck is also shown in the case of the most refined mesh (lam-3 Figure 4b). In order to establish a relationship
between the grid spacing and the simulated instantaneous
flow pattern, Figure 5 shows the time history of the lift force in the range 25.6 < At < 27.6, where the nondimensional time is expressed as t = TUo/B. A number of instants are selected on the time track, and the instantaneous streamline patterns are plotted for each model. The coarsest mesh provides a steady solution, i.e., without fluctuations of the lift force. The flow remains completely attached to the deck on the side surfaces, as shown by the streamline pattern just below the diagram. The other grid systems enable observation of major fluctuations of the lift force. The time history of the lift coefficient CL related to the medium level of refinement appears very regular, suggesting that just one mechanism plays a dominant role in the vortex-shedding process. The flow-pattern visualizations in the left-hand side column confirm the above hypothesis. The separation bubble at the lower surface is very large, but the level of vorticity in the recirculation region seems too low to involve vortex shedding. Consequently, the vortex does not interact with the vortex developing in the near-wake region. Local maxima in the lift track (t = 26.0
807
2D Numerical Simulations I I I I,
0 I I I r 1 ’ I * 1 r
yw=l .3e-2 y,=2.0e3 ---.--y,12&4 ------
CL
+0.055
-
I m
:
cL lmed
yw = 1.3e - 2
yw = 2.Oe - 3
Yw = 2.2e - 4 0
A
t = 26.0
t = 26.2
t = 26.2
t = 26.4
t = 26.4
t = 26.7
t = 26.7 Figure 5. Time history of the lift coefficient and instantaneous
and t = 26.4)
are regularly
observed
(t = 26.2 and t = 26.7) are related
streamlines.
when this vortex is shed in the wake. Conversely, to the maximum
development
of the vortex
the minima
close to the wall.
The grid spacing lam-3 enables detection of a fundamental feature of the flow, namely the interaction and merging between the vortices travelling along the lower surface and the vortices (t = 25.9) and minimum (t = 26.2) emerging at the downstream edges. The first maximum substantially
match
those simulated
by means
of the grid lam-2.
But subsequently
the lift force
in the lam-2 model performs one period of oscillation between the instants t = 26.2 and t = 26.7: whereas the CL~ just covers a half period (local maximum at t = 26.7). The streamlines clearly show the merging bubble
of the vortices
downstream
at that moment.
surface
and the clockwise
in the upper area of the deck wake. However, the role played shedding process does not seem to be of primary importance.
by these structures
The pressure in Figure
of the windward
The most refined grid also detects
distribution
6. These
results
edge at the upper
on the lower surface confirm
the differences
at the above-mentioned between
the models
the separation circulating
instants lam-2
flow
in the vort,exis presented
and lam-3.
In the
the pressure fluctuations are always separated by the sharp corner at x/B = 0.8 When the finest grid is used, the pressure fluctuations move backward, without interaction. reducing the extent of the separation bubble. Owing to the larger amount of vorticity produced at the separation point, the pressure at x/B = 0.8 does not remain constant, but reaches a deep trough when the vortices at the lower surface pass the edge. It should be noted that as former
model
L. BRUNO AND S. KHRIS
808 1
1
CP
CP
0.5
0.5
0 -0.5 -1
4
-1.5
-1
‘.
-1.5
J
0
0.2
0.6
0.4
0.8
?IB
1
t , 0
0.2
Figure 6. Instantaneous surface.
approximation with sharp
is enlarged,
case of semibluff
distributions
major
characterized
sections,
of the pressure coefficient
the model just
does not involve corners
simulates
problems
by massive
XII3
1
process,
thus
in the simulation
of flow around
and large-sized
conceals
leading
Cp on the lower
one large eddy on the side surface.
separation
a coarse mesh completely
basis of the vortex-formation
0.8
(b) yw = 2.2e - 4.
(a) yU = 2.0e - 3.
the grid spacing
0.6
0.4
vortices. error.
bluff cylinders Instead,
in the
structures
at the
A number
of grid
the small vertical
to a fundamental
This
points is required both in order to ensure a cell size smaller than the smallest vortex length and to prevent an excessive amount of numerical diffusion related to the discretization scheme adopted for the diffusive
terms.
In order to extend
the discussion
treatment
of the results
the model
lam-3.
is called for. Figure
As may be readily
of this sort, as has already very important extent, Figure
to the integral
for the extraction
in time and space, a statistical
7a shows the CD and CL time traces
appreciated,
been pointed
flow parameters the signals
out by Taylor
of meaningful
are completely
[19], the extent
statistical
simulated
random.
of the sampling
parameters.
To optimize
with
In a case window
is
the sampling
the computations run as long as the lower Strouhal number Sti remains constant (see 7b). Generally speaking, a sampling extent of 30 nondimensional time units is found to
be required in order to assume stationarity affected by the impulsive initial condition, nondimensional
of the signal. Since the first ten units are strongly each simulation was conservatively extended to 50
time units.
In what follows, the main statistical
parameters
of the aerodynamic
behaviour
of the deck are
discussed. ‘D-‘L
St cD med
0.00
.211 ‘9.
I
I St,
o
-0.20
0
10
20
30
40
t
50
10
30
20 (b)
(a)
Figure 7. Optimization
of the extent of sampling-time.
4o
At
So
2D Numerical Simulations
.085
809
St - y,=2.0e-3
.256
St - y,=2.2e-4 Figure 8. Effects of grid spacing on Strouhal number
The power spectral lam-2
and lam-3)
number).
densities
The PSD obtained
the higher
in number.
The corresponding
mechanisms
that
numbers
(Stz/Stl
that
that the physical
shown in Figure have
8 versus
to frequencies
This means
flow visualizations
of the lift coefficient
in Figure
been
frequency
Strouhal
numbers above.
quite different
with the experimental
value is indicated by Figure
by means
of error bars.
5 is qualitatively
coefficients
results
in accordance
complete
obtained
between
Even though
in the computational deviation
the description
,102
t-
I
:
:
: :
:: -
:.
are
the mean
the mean values of the
:
2.Oe-3
,.., +.05
,,: ::
1.3e-2
:
:
,076
simulations around
CL
2.2e-4
,_---J~I_~-L1_~__k___. : “ ‘::: :
,(.,
:“’
Strouhal
x 1.67).
with experiments.
CD
2.0e-3
1.3e-2
shedding
of the flow field provided
and refers to a very likely situation,
are not in close agreement
the main
one (St,/%,
9. The standard
(i.e..
with the
from lam-3 are fewer
due to the different
the ratio
from the experimental
in Figure
In effect
of the first frequency
unique
are physically
(models
(i.e., Strouhal
by three main peaks.
multiples remains
However,
solutions
frequency
5. The mean peaks in the PSD obtained
described
z 2) remains
are integer
The mean values of the drag and lift coefficients compared
for the unsteady
the nondimensional
from the model lam-2 is characterized
peaks correspond
superharmonics).
(PSD)
are plotted
+.Ol
:
2.2e-4
1
-3”” .,.
I
:
!i.
:
‘,
-.08
:‘.
_______’
~_____~--___---,_____--------_--_
‘.., ,,..
:
:
.’
-.23
:8%
::
‘.
T
..:.
:“.’ t -3 I
.’
I
.-.
-.29
,023
-.43 0.001
yfl
0.0001
0.01
(a) Figure 9. Effects of grid spacing on aerodynamic
0.001 (b) coefficients.
Yfi
0.0001
810
L. BRUNO AND
S.
KHRIS
The mean value of the drag coefficient increases consistently with the shear-layer separation and the vortex shedding around the deck. In particular, the drag component involved in the flow circulation in the near-wake area is generally predominant in such quasibluff geometries. In order to relate the drag value and the characteristics
of the wake, Figure 10 presents the
effect of grid spacing on the across-wind extent of the wake and on the velocity defect at two locations in the wake (0.005B any recirculation,
and 0.5B from the trailing edge). The coarsest grid does not predict
so that the deck behaviour is close to the one of a fully streamlined
and is characterized are quite similar.
by a very low drag value. The profiles predicted using the other models The model lam-3 further reduces the velocity defect in the neighbourhood
of y/B = 0 and emphasizes the wake extent in the range -0.2 wider separation
< y/B
< -0.1
because of the
bubble at the lower surface of the deck. It may be noted that the results are
in good agreement the &
section
with the computational
value approaches the section-model
prediction of Larsen [18] using DVM. Even though result, it remains somewhat underestimated.
This is
certainly due to the absence of deck equipment in the models. I y,+1.3e_2
=1.3~_2
YB
I
I .._._
:
I ’
’
+0.150
_ -
+0.072 +0.043
I
I
I
I
1
0.9
1
Larsen [ 181 a yw = 1.3e_2 _.._..._.___._.. y,=2.Oe-3 ------yw = 2.2e-4 -
0.0 -0.097 -0.150
0.8
0.9
1
0.0
I
I
0.4
0.8
(b) U&Jo - x/B
(a)
1.2
OS
= 1.005.
0.6
0.7
0.8
(c) I&/U0 - x/B
= 1.5.
Figure 10. Effects of grid spacing on mean z-velocity defect in the wake. Instead,
mesh refinement
considerably
reduces the lift forces acting on the deck.
noted that as the grid spacing becomes smaller, the computed & the measured values, and its standard
deviation increases.
It may be
moves out of the range of
Hence, mesh refinement enables a
qualitative reproduction of the vortex-shedding process but also appears to reduce accuracy in predicting the mean lift force. The same apparent discrepancy may be noted in other 2D simulations
(e.g., [21]; see Table 1). Even though Kuroda assumed the same grid spacing near
the wall, the error regarding the lift coefficient is somewhat different from the present results. This difference points t,o the dependence of numerical damping both on grid spacing and on the convective scheme. To clarify this aspect, the levels of performance of the finest grid are evaluated in the case where second-order upwind and QUICK schemes are employed. The distributions obtained for the mean Cp and for the std (Cp) are compared in Figure 11. The agreement between the lift coefficient measured and the lift coefficient obtained from the “Yw zz 1.3e - 2-2nd upw” model is seen to be merely fortuitous. From a comparison of the unsteady solutions it is found that as the mesh becomes more refined and the order scheme is higher, the diffusion effect is progressively reduced and does not overcome the very small viscous diffusion of the flow. Hence, the extent of the separation bubble at the lower surface is shortened, and the separated layer past the upper leading edge is better predicted. Despite the major gain
in precision that is obtained,
the size of the separation
bubbles remains somewhat larger than
the measured value. Two main limitations remain in 2D laminar simulations. First, it is clear that further refinements of the near-wall mesh or the adoption of higher-discretization schemes do not represent
2D Numerical Simulations lower surface
I
’
+0.6 1
‘n
y+3e-2
upwd
- 2nd
y,=Z.Oe-3 - 2ndupwd
0
0.2
0.6
0.4
8
smuotbflow: tautship model [I 51 hubulent flow: taut strip model [15] turbulent flow: fidi-scale [I 61
I
L
---.--
yw=2.2e-4-2ndupwd ----y,=2.%-4 - quick Kmuda [Zl] - yw=2.0c-4 - 5th upwd - .-.
upper smfacc
1
1
1
’
811
0 . *
-.
0.8
x/B
1
0.2
0
0.4
0.8
0.6
x/B
1
Figure 11. Effects of numerical diffusion on mean Cp and std (Cp) distributions. realistic
solutions.
thickness
In fact,
(see Figure
nificant
improvement.
diffusive
schemes
the stability
On the other
of the aerodynamic that
hand,
In particular,
mechanism
towards
further
higher
near-wall about
damping
Second,
in the spanwise
frequencies
as a result
of vortex
in the laminar
simulations
is retained
cell
any sig-
by the highest
11 “yW = 2.2e - 4-QUICK”
seems to be no longer assured. diffusion
versus
does not bring
of numerical
(see Figure
the well-known
coefficients
refinement
the reduction
is such that wiggles appear
of the computations
effects remains. transfer
the evolution
9) seems to indicate
model),
and
the failure to model 3D direction
stretching
and the energy[7] continue
to be
neglected. 4.2.
Statistical
Approach
The most refined grid selected paper.
in the sequel of the present
The grid spacing
the two-layer obtained
model
by applying
near the wall complies with the requirements the (wall unit y+ = 1). Table 4 summarizes the standard
k - E model
(STD),
for correct application of main integral parameters
the RNG model
and the Reynolds
stress
model (RSM). Both the STD and RNG models fail to predict the unsteadiness of the flow. This error can no longer be chiefly put down to the numerical damping that results from the combined effect of grid size and discretization that
the reasons
formulation
scheme
on account
for the lack of lift fluctuations
of the turbulence
models
of the previous
(std (Cn)
optimization.
= 0) are to be sought
adopted.
Table 4. Statistical approach: integral parameters. Model
CD
CL
std(C~)
St
It follows rather
in the
812
L. BRUNO AND S. KHRIS
Recently, Shimada and Ishihara [3] have proposed an interesting explanation of the severe limitations of these models in correctly simulating the fluctuating pressure field around elongated rectangular quantity
sections.
Franke
and Rodi [31] argue that
4 can be separated
in vortex-shedding
flows an instantaneous
into C/.)(t)= G + 6 + Cj’
(14)
9’ Indicating by @’ the total fluctuation value $‘, its variance can be expressed
(i.e., periodic as
6 plus stochastic
4’) around
the time-mean
a& = CT;+ a& In particular,
the above equation
(15) can be rewritten
(15) for the velocity
component
Vi as follows:
depends
component of the velocity is evaluated by means of the turbulent kinetic energy k to the turbulent viscosity ut. It follows that the value of the variance strictly on the reliability of one of the weakest and most debatable assumptions of the model,
namely,
the isotropic
pressure
is not explicitly
The stochastic and then related
turbulent
viscosity
modelled
itself.
Furthermore,
in any RANS
model,
the stochastic
component
of the
so that
and std (CL )RANS < std (CL). The statistical periodic where
approach
fluctuations the relative
yields
predominate. contribution
good results
this case the very small eddies observed the stochastic
field by the statistical
eddy viscosity
around
an extent
in the case of massively
without
approach.
turbulence
modelling
As a result,
these
the deck and in the wake. This may damp
as to cause the observed
The RSM model furnishes
separated
flow in which
These conditions do not apply to the present application, of the stochastic component becomes noticeable. Moreover, in are probably
models
assumed
yield higher
the fluctuating
in
levels of
motion
to such
lack of unsteadiness.
an unsteady
solution,
even though
one characterized
by an extremely
reduced value of the standard deviation of the lift coefficient and a unique Strouhal number (Figure 12a). This frequency content is due to the single eddy in the wake of the deck (Figure 12b).
CL
I
I
I
I
I
RSM +0.036 +0.026
5
I
I
I
I
I
5.25
5.5
5.75
6
6.25
,,‘;_p-,
,,,’
.____
+a010
t
(a) Figure 12. MM-time
,/ ,_,’
history of lift coefficient (a) and instantaneous
(b) streamlines
(b).
-- _
tcl;3
2D Numerical Simulations the RSM model
As a result, pared
to the two-equation
Leschziner
affords
closure.
[34] for modelling
In fact, the above approach
to a case that
adopted
by Lien and
turbulent
viscosity
term
= & Ic (_!$!?J$).
is once more based upon the concept
and the value of the free constant model
due to the formalism
the turbulent-diffusion D;
diffusion
improvement in the simulation of the flow as com-
scant
This is probably
Crk = 0.82 is obtained
is rather
different
of isotropic
by applying
from the present
the generalized
one, i.e., a plane
it.
gradient-
homogeneous
shear flow. 4.3.
The Large
The adoption substantial
Simulation
involved
albeit
somewhat
in simulating
burdensome,
the turbulent
is necessary
flow around
to overcome
the GBEB
the
deck using the
approach. of computational
Optimization
The effects interval
of two computational
for time advancement
Table 5 summarizes
parameters
the main integral
results
by the mesh
adopted.
required
the particular
than
On the other
hand,
(-40%)
scheme
in what
that change.
The results
in At markedly
convergence
the value At = 1.e - 2 is adopted
vortex
simulated
the number
for residuals
in the subsequent
using
the largest
directly
reduces
(threshold
the
param-
obtained
It follows that
of the smallest
the increase
to reach
namely, terms.
the above-mentioned
differ from one another. time
follows,
for the convective
by varying
parameters
the characteristic
at each step
5.e - 4). For this reason
are investigated obtained
les-1 and les-2 do not significantly
step is still smaller
iterations
parameters
At and the discretization
eters; the bold style highlights the models time
Approach
of the LES approach,
difficulties
statistical 4.3.1.
Eddy
of
fixed at
simulations.
Table 5. Computing conditions and integral parameters. Model
At
Scheme
C,
les- 1 les-2
l.e-3 l.e-2
2nd up 2nd up
0.10 0.10
les-3
1.e -
QUICK
0.10
les-4
1.e - 2
2nd cnt
0.10
In a way similar
2
to the one adopted
!
St
CD
+ 0.067 + 0.066
0.272 - 0.113
+ 0.075
0.202 -- 0.101
+ 0.071
0.124 - 0.164
0.272 - 0.107
0.197 - 0.292
for simulations
without
turbulence
models,
the remaining
part of this subsection is devoted to clarifying the performance of various discretization in conjunction with LES and to selecting the most efficient scheme for the ensuing compared
to the second-order
lift force to be increased
upwind
scheme,
and the mean
the experimental data. This trend is clarified and the distribution of its standard deviation The differences least as regards
between
the QUICK
the mean Cp distribution
the QUICK
computed
scheme enables
the fluctuations
value of the lift force to be brought
by the distribution in Figure 13.
scheme and the second
of the mean pressure central
scheme
of the closer to
coefficient
are less marked,
at the lower surface of the deck. However,
bubble at the upper surface is only detected by the second central remains somewhat underpredicted. Another two characteristics
schemes runs. As
at
the separation
scheme, even though the suction of the simulated flow argue in
favour of the latter convective scheme. The first is the transverse extension of the wake downstream of the deck (see Figure 14a). The profiles obtained from applying the second upwind scheme and the QUICK scheme do
814
L. BRUNO AND S. KHRIS upper surface
lower surface +0.6
i-I
I
I
+l.O
I
smoothflow: tautstrip model[lS] turbulent flow: tautstripmodel [15]
Znd&ind --.-.-’ stick ----2ndcentral -
turbulentflow: full-scale [16]
0 ??
+
+O.S
0.0
u”
-0.5
0
-1.0 0
0.2
0.4
0.6
0.8
La x/B 1
-
0.2
0
Figure 13. Cp and std (Cp) distributions 3
Y/B
0.4
0.6
0.8
X/B
1
on the deck. 0.124
0.164
0.197
St - 2nd cent
2.5
+0.150
1 2 ICE 0.0
2
1.5 1
-0.097 -0.150
OS
0.8
0.9
1
0.8
0.9 uJuo
x/B
1
0
0.202
J
St _ quick
- x&3=1.5
(a)
Figure 14. Mean r-velocity
@I
defect in the wake (a); Strouhal number (b).
not, substantially differ from one another. Instead, the second central scheme emphasizes the contribution of the vortices ~1 to the velocity defect. The frequency content of the lift force predicted by means of the second central scheme (see Table 5 and Figure 14b) is spread over a wider range (0.1 < St 2 0.2) than the one predicted by the other schemes. This result is hardly surprising if the dispersive effect of the odd-order leading term is borne in mind. Furthermore, the main Strouhal number in the second central simulation is the lower one (St = 0.124), which is related to merging between the vortices at the lower surface and at the leeward edges. Instead, the upwind schemes attribute most of the energy to the higher shedding frequency (St x 0.2), which is exclusively due to the vertical structures emerging at the leading edge. All the above-mentioned characteristics are due to the smaller numerical diffusivity related to the second-order central scheme, and thus, to its capability for describing the effect of the small vertical structures at the lower surface of the deck. For these reasons, the second-order central
815
2D Numerical Simulations scheme
seems to be the most adequate
conclusions
reached
by Breuer
[42] concerning
the case of flow past a circular 4.3.2. The
Effects
of the
subgrid
model
namely,
the so-called
application.
the numerical
This result
effects on large-eddy
confirms
the
simulation
in
cylinder.
Smagorinsky developed
constant by Smagorinsky
introduces Consequently,
only one free input parameter: the arbitrariness of the model
as compared to the statistical approach k-E model). However, the Smagorinsky
(five semiempirical constants are constant plays a very important
Smagorinsky
is substantially reduced required in the standard
one for the present
constant
Cs.
role in LES because its value may considerably affect the turbulent viscosity. The value of Cs has been optimized by several authors in the case of 3D simulations generally subgrid
been set at Cs = 0.1. On the other hand, model
Generally
spect to direct 2D simulations tuning
has not been taken
speaking,
simulation
of a subgrid
effects of the vertical
of the subgrid
structures
model
and potentially
in the case of homogeneous
of the free parameter
c-s
into due consideration
the presence
numerical
it is our opinion
generally
spanwise
and has
the optimization
of the
in 2D simulations. makes
enables
a substantial
application
fluctuations.
model could enable observed
that
with
re-
of the LES approach
difference
to
In particular,
reproduction
in 3D configurations
Mean Streamlines
an adequate
of the characteristic [S]. In order
to verify
CL
19
21
23
25
21
I
29 +0.20
Wh, ’
0.050
p%l -0.192
-OS0
0.075
0.100
0.125
Figure
15.
Time histories of CL and mean streamlines.
816
L. BRUNO
the advisability of such a procedure,
AND S. KHRIS
a number of values of the Smagorinsky constant in the
range 0.05 I Cs i: 0.2 were tested for the present case study. Figure 15 compares the time histories of the lift coefficient in the range 19 5 t 5 29 and the mean streamlines obtained by applying a selected number of Cs values.
The value Cs = 0.1
represents a watershed for the quantities examined. As the value of the Smagorinsky constant exceeds 0.1-0.125, (i) the separation bubble at the lower surface is abruptly reduced (0.125 5 Cs < 0.15); (ii) the merging of the vortices wl and 212(see Figure 1) vanishes; (iii) the mean value of the lift coefficient increases; (iv) the oscillations of the the lift coefficient diminish; and (v) the time history of the lift coefficient approaches a perfect sinusoid (Cs = 0.2). Instead, as the value of the Smagorinsky constant becomes smaller than 0.125-0.1, the evolution previously observed is not altogether respected. In particular,
(9
the length of the separation bubble at the lower surface gradually reduces (0.05 < Cs
0.125); (ii) the recirculating zone downstream of the upper windward edge appears; (iii) the mean value of the lift coefficient increases; (iv) the fluctuations of the lift coefficient diminish; and (v) the frequency content of the lift coefficient appears to spread over a larger range in t;he direction of the higher frequencies. To provide a better description of these trends and to clarify the reasons behind them, the step in the variation of the value of Cs is further reduced.
0.10
_________-_____-____---------________-_
0.09
0.081~
I 0,102
actual computation 0 section model [ 141 ~ tautstrip model [15] ------
Ticks
The evolution of the drag and lift
+O.l
I
3 f
T,TT
-0.1
-0.3
:ImTl fji
-0.5 .050
.075
.lOO
,125
.150
(b) CL.
(a) CD.
0 0
0
m:o 0
80
Q
~____________~______-_-_? 0
00
??
0.158
• ___~~~~~~~~__
0
00 _---~_a-a_____---_Q_~-___________-____. 0.109 0
0
.050
.075
.lOO
.125
,150
cs
.200
(c) St Figure 16. Effects of CS on the aerodynamic bers.
+O.Ol
__ ___ _ _ _ _________________________-0.08
coefficients
and on the Strouhal num-
cs
.200
2D Numerical
coefficients The
and of the Strouhal
discontinuities
so revealing
a significant
range increases. the simulated
Strouhal
numbers
positive
At the lowest
delineate range
three
variation
16.
of Cs the
in the global
the values
coefficients:
measured
that
bound
as
of thP
for Cs > 0.140
in wind-tunnel
tests.
17 and 18 makes it possible
aerodynamic
behaviour
to identify
of the deck and to
for the Cs value. can be observed.
the aerodynamic emerging
and the interaction
confirms
in the range 0.075 < Cs < 0.095.
main ranges
due to the vortices
at high frequencies,
values
out and the upper
of the aerodynamic
do not at all match is obtained
of the pressure
lift forces characterize
in Figure
at Cs z 0.145.
The diagram
(0.2 5 Cs 5 0.145) the flow is fully attached
fluctuations
flow is exclusively
is more spread
on the deck surface in Figures
for the aforesaid
In the upper
content
with the diagrams
the best agreement
schematically no significant
parameter.
appear
the best agreement with the experimental measurements. The is obtained by evaluating the reduced frequencies corresponding
the frequency
Consistently
The mean Cp distribution the reasons
to this
are plotted
clearly
peaks in the PSD of the lift force for each simulation.
the value of Cs decreases,
Once again,
constant
of CD and CL evolutions
case sensitivity present numbers
817
versus the Smagorinsky
in the evolutions
aerodynamic parameters evolution of the Strouhal to the main
numbers
Simulations
behaviour
of the deck. The unsteadiness
at the downstream
between
to the wall of the deck, and
Low values of the drag coefficient corners.
eddies is restricted
The vortices
to the upper
and of the
are shed
and lower ones
in the near wake. At values of the Smagorinsky
constant
in the range 0.140 5 Cs < 0.115, the separation
bubble
at the lower surface is predicted even though its length is largely overestimated. For Cs = 0.140 the value of std (Cp) is very low at x/B = 0.8, which indicates that there is not yet any interaction between
the vortices
interaction
w1 and 212. As the value of the Smagorinsky
is set up and increases
progressively;
however,
constant
if we examine
is further
reduced.
the Strouhal
numbers
we find that vortex merging does not yet represent the main mechanism in vortex shedding. The flow at the upper surface still remains attached to the wall. The drops in the pressure distributions,
again
on the upper
surface,
can be explained
by considerations
that
regard
the
upper surface
1
I 1
smooth flow: taut’strip model ‘[15] turbulent flow: taut strip model [15] turbulent flow: full-scale
-1.5 I
0
I 0.2
1
I
0.4
Figure
0.6
0.8
xlB
1
17. Cp and std (Cp) distributions
I
I
0
0.2
on the deck-Q
0.4
2 0.1
0.6
[161
O8
x/B’
818
L. BRUNO AND S. KHRIS lowermface
uppersurface
+O.b 3
i
+0.4 +0.2 -
,
smooth flow: taut stripmodel [151 0 turbulent flow:tautstripmodel[IS]
c,=0.050 -.-.-. CszO.065 ---.-C, = 0.075 C,=O.O85 -----c, =0.095 ----c, =O.loo -
0
0.2
0.4
0.8
0.6
xlB
??
turbulent flow: full-scale [lb]
0
1
Figure 18. Cp and std (Cp) distributions
energy conservation
of the flow. Finally,
Cs = 0.05, the suction measurements. subranges
of shedding
experimental
From mind
an overall
may be identified:
0.05 < Cs < 0.1; the upper analysis
constant.
the expression
Smagorinsky
frequencies
constant
The
of the results
above
of the subgrid-scale increases,
results,
observed
0.8
I
a
5 0.1.
constant
is reduced it matches
past the leading
the first range
from Cs = 0.1 to the experimental edge.
corresponds
Three closely
main to the
range
(St x 0.2) remains quite constant as Cs bound of the third range constantly increases it is possible
to interpret
for Cs > 0.1 are hardly
eddy viscosity
the subgrid
0.6
so much that
the flow is separated
one (0.11 < St < 0.16); the second
varies in the range as Cs decreases. Smagorinsky
surface,
diminishes
0.4
on the deck-Cs
as the Smagorinsky
at the lower surface
On the upper
0.2
+
uses
eddy viscosity
the effects
surprising
(equations
(9) and (10)).
is progressively
of the
if we bear
in
As the
overestimated,
and
its diffusion effect obscures the small-scale eddies at the root of the unsteadiness of the flow. It should be pointed out that the effects of the Smagorinsky constant in this range of values are similar to the effects of the numerical diffusion due to the grid spacing and the discretization schemes investigated above. For Cs < 0.1, the subgrid
eddy
viscosity
is progressively
underestimated.
As a result,
not
only the diffusive effects fail to conceal the small eddies but also higher frequency fluctuations are introduced in the 2D simulation. The spectrum of the 2D computations, which is generally confined within a narrow band [7], is artificially extended over a wider band. This obviously leads to the above-mentioned higher Strouhal number, but also affects the energy-transfer mechanism. In fact, in 3D simulations a number of three-dimensional features of the flow (such as, vortex stretching and longitudinal eddies in the wake) play an important role in the energy cascade, subtracting an important amount of energy from the transverse flow. Of course, these energytransfer mechanisms cannot be simulated exactly in 2D analysis, but their overall effects can be replaced by a reduced value of the Smagorinsky constant. On the other hand, this approach introduces into the flow field a number of small-sized high-frequency disturbances (wavelets) clearly revealed by the distribution of the mean pressure coefficient (see Figure 18, in particular for Cs = 0.05). In this sense, the effects of the smallest value of Cs are similar to those involved in the numerical dispersion of certain discretization schemes.
2D Numerical Simulations
+0.15 0.0 -0.15
Figure 19. Effects of Cs on the mean velocity defect in the wake x JB = 1.5.
The evolution constant
of the mean
(see Figure
maximum
amount
(Cs < 0.1).
of subgrid
in 2D simulations
hand,
a direct
here proposed
4.4.
the above
x/B = 1.5 versus the Smagorinsky explanation.
(Cs > 0.1) nor dispersed
phenomena
The wake reaches is neither
in a wider
have the same effect on the mean between
it follows that for the case study.
diffused band
its
by an
spectrum
velocity
in the
the curves.
a Cs value
in the range
However,
the simplified
0.65 < Cs < 0.75 is more geometry
of the deck does
a final verdict to be issued. On the one hand, the small number of nodes obtained by the barriers has made possible a number of extended and useful parametric studies.
On the other enable
eddy viscosity both
obtained
in the wake at
at Cs = 0.1, i.e., when the flow energy
from the correspondence
From the results not enable neglecting
defect
seem to confirm
extension
In particular,
wake, as emerges suitable
19) would
across-wind
excessive
velocity
Effects
the failure
comparison
to meet the experimental
with the experimental
or a clear singling-out
of Deck
conditions
data,
for the equipment
a complete
of the effects of the barriers
validation
on the physics
does not
of the approach of the flow.
Equipment
In order to overcome
the above-mentioned
difficulties,
in what
follows the deck equipment
is
introduced into the model. The close-up view of the mesh generated around the fully equipped section is shown in Figure 20b. The near-wall cell thickness is set at yW = 3.le - 4B. Modelling of the barriers
involves
an important
increase
in the number
of cells (76564, +68%
as compared
to
the lam-3 grid). To enable an appreciation of the computational resources involved, Figure 20a summarizes the evolution of CPU time versus cell number in LES simulations (HP j6000-1 proc. PA8600 552 MHz, SPECfp2000
l.oetS
(4
l.o&
433). The 2D detailed
model
(11 days for a computer
l.cle+7
(b)
Figure 20. CPU time versus cell number (a); grid system near the detailed deck (b).
run involv-
820
L. BRUNO AND S. KHRIS
turbulentflow: raut stripmodel[IS] turbulent flow: full-scale[16]
-1.0 0
0.2
0.4
0.6
0.8
xlB
1
0
0.2
0.4
0.6
0.8
x/B
I
Figure 21. Effects of CS on Cp and std (Cp) distributions on the detailed deck
ing 5000 time steps) is certainly more time-consuming
than the basic one (six days). It follows
that the advisability of introducing deck equipment must be carefully evaluated against the gain in accuracy. On the other hand, the extension of the same grid system in the spanwise direction (3D simulation, spanwise dimension of the computational increases the required computational computational
domain set at 1 chord length B) markedly
resources (CPU time of 44 years). It is evident that such a
commitment is out of the question in the case of parametric studies or industrial
applications. The above data confirm the importance of developing computational alternative to 3D simulations in these fields.
approaches
First of all, the parametric study on the effect of the Smagorinsky constant in 2D-LES computations is applied to the detailed deck in order to detect with a greater degree of precision the optimal Cs value for the case study. The analysis is restrictedly performed on the most suitable values of Cs previously selected in the basic simulations. Figure 21 shows the distribution of the mean pressure coefficient and the distribution of its standard deviation on the deck. The effects of the Smagorinsky constant previously observed in the basic geometry (see Figures 17 and 18) is generally confirmed. However, the results in some crucial areas of the deck indicate that the value Cs = 0.075 is the most suitable one. In fact, at Cs = 0.065 the length of the recirculating area on the lower surface is underestimated, and instabilities increase behind the barriers at the upper surface. The disagreement between the experimental measurements and the computational results remains at the lower surface in the range 0 < x/B < 0.2. If it is borne in mind that the anisotropy of the turbulence in the forebody region is probably the cause of this disagreement, it is hardly surprising that the Smagorinsky subgrid scale model proves its insufficiency. The models that yielded the most satisfactory results in the previous simulations (namely, the laminar approach, the Reynolds stress model, and the large eddy simulation with Cs = 0.075) are applied in what follows to the detailed deck. Table 6 compares the main integral parameters of the flow resulting from models with and without barriers. The results obtained using the RSM model are of less interest than the validation of the computational approach and do not provide a significant physical insight into the flow. Nevertheless,
821
2D Numerical Simulations Table 6. Integral parameters with and without barriers.
Figure 22. RSM model: lines (b).
they are briefly regarding history
commented
the weakness
upon
of the deck.
These
small-amplitude
only
It follows that
pates the periodic The comparison from barrier equipment
phenomena.
the hypothesis streamlines
by a transient
phase
shedding
the statistical
The stochastic induced
perturbations
(St = 4.68). The
final
of the leeward
side
proves
its capability
for
from the ensemble-averaged
complex
considerably
22a),
in the wake
solution.
downstream approach
by small-scale,
the eddy viscosity
in the near-
(see Figure
shedding
in the stationary
formulated
22 shows the time
and only the high-frequency
remain
due to vortex
Once again,
as the perturbations
field.
to highlight
22b.
vanish,
fluctuations
stream-
12, Figure
view of the instantaneous
is characterized
progressively
= 0.001)
in Figure
periodic
fluctuation-such turbulent
fluctuations
As in Figure
of the lift force are due to vortex
of the flow is exclusively
as shown
detecting
fluctuations
(std (CL)
unsteadiness
solution
(a) and instantaneous
follows in order to confirm approach.
and the close-up
The computational
in which the low-frequency
barriers,
in what
of the statistical
of the lift coefficient
wake region.
time track of lift coefficient
eddies-are
increases
assumed
in the
and progressively
dissi-
fluctuation. of the results
the differences modelling.
obtained
related
from the laminar
to flow modelling,
As may be readily
on the aerodynamic
coefficients
and LES approaches
as well as the common
appreciated
from Table
may be summarized
makes it possible trend
6, the main
that
results
effects of the
as follows:
(a) rise in the drag force; (b) drop in the mean value of the lift force and its fluctuations; (c) wider band
of the lift spectrum
(St).
A previous work [12] has demonstrated that the contribution of small-sized items of equipment to bridge aerodynamics is primarily due to interference effects between the items of equipment and the deck. In what follows, these interference effects (i.e., regarding
the neighbourhood
phenomena
will be further
of the items of equipment)
distinguished
and global
as local
effects.
As for the mean drag coefficient, the increase in its value due to the barriers is approximately the same in both the laminar and LES approaches (ACD x 30%). Such a major increase cannot be related to the direct contribution due to the details. Instead, we shall focus our attention on the contribution of the base region of the deck.
822
L. BRUNO AND S. KHRIS
Y/B . Larsen [I S]
+0.150 +0.072 +0.043 0.0
-0.097 -0.150 Ian-,-b=
-.-.--
lam_det
les-bas ----les-de.t -
-0.300 0.8
0.9
0.0
1
Figure 23. Velocity
Figure wake. width
23 shows the mean velocity
The
details
increase
of the deck.
sense, the presence
of the details
it is to be pointed
approaches
leads to rather
between
drag value,
0.8
0.9
1
UJU, - x/B=l.S
the width
between the overall
the introduction
predictions
lines at different
locations
in the
of the wake at x/B = 1.005 and the
vortices
which closely
increases
out that
different
0.7
defect along two straight
the distance
a higher
0.8
- x/B=1.005
defect in the wake: basic and detailed deck
the ratio
As a result,
the deck experiences Finally,
0.4
UJU,
x/B
of different
matches
sign is augmented,
the experimental
degree of bluffness of the barriers
of the abscissa
data.
and In this
of the section. in the laminar
of the local maximum
and LES
defect along
the straight line at x/B = 1.5. In the laminar simulations the velocity defect due to the barriers is not diffused along the wake, and the local velocity minimum is aligned with the upper surface. Instead,
in LES the modelling
and a less regular the same
cause
of the barriers
involves
a downturn
of the local velocity
minimum
profile of the mean velocity as the reduction
U,. The above phenomenon would appear to have in the lift coefficient and in the lower bound of the Strouhal
range. In order
to provide
the longitudinal
a comprehensive
component
explanation
of the velocity
of the above
results,
U, and of the vorticity
the mean
w along
x/B = 0.328 are shown in Figure 24. The profile of the mean longitudinal upper surface (y/B > 0.039) in Figure 24b confirms’that the velocity defect to the barriers,
which also shed a certain
amount
of vorticity
in their
profiles
the straight
of line
velocity over the in the wake is due
wake (Figure
24~).
The most surprising results concern the lower side area. On account of the blockage effect of the upwind side barriers, the velocity increases over the lower side of the deck as compared to the case of the basic geometry (see Figure 24d). As the velocity U, outside the outer boundary layer at the separation point increases, the instantaneous flux of vorticity also increases according to the relation
where
ub is the corresponding
velocity
from the base to the same
point.
Hence,
the vorticity
shed downstream of the separation bubble is higher and more concentrated (see Figure 24e). Consequently, the vorticity, convected along the lower surface of the deck, moves downstream of the maximum velocity defect in the wake. Furthermore, the increased velocity of the flow involves a deeper suction on the lower surface of the deck (see Figure 25) both in the laminar simulation (0.55 < x/B < 0.65) and in the LES simulation (0.2 < x/B < 0.4). The more concentrated vorticity reduces the length of the separation bubble and enables the LES simulation to make Both phenomena are indirectly a correct prediction of the experimental pressure distribution.
823
2D Numerical Simulations
j :
lesdet les-bas
-----+0.072 +0.039 0.0 -
I
I
I
I
0
0.1
0.2
I
I
I
0.328 0.4
0.5
x/B
(4
H I 1:
,’
__-’
+0.039 0.0 0.5 1 UJU, - xAM.328
-75
6.1 -
(b) 0.0 I ,,I: ‘-,-
-0.097
-0.15
r?%
/’ :
0.0 +7s x/B=O.328
(c) 0.5
1 I
-75
i :
:
1, ,
-0.15
+75
___4-----___c : ---___
-0.097
---- -_ --y_ : -\
j
0.0
,A-
. _* .’
I--::1
UJU, - xAkO.328
o - x/B=O.328
(4
(e)
Figure 24. Effects of Cs on the aerodynamic coefficients and on the Strouhal numbers. involved
by the barriers
the barriers
in the pressure upwind
and contribute
edge,
distribution. reducing
In particular,
the pressure
std (C,) distribution on the upper the local and the global effects. Finally, numbers.
to reducing
also have local effects on the upper
the upwind
fluctuations surface).
the value of the lift coefficient.
surface,
which involve side barrier
predicted
The mean
local maxima
reattaches
in the basic
streamlines
the flow past the
configuration
in Figure
Of course, and minima (see the
26a confirm
both
the barriers also affect the mechanism of the vortex-formation process and the Strouhal Once again it is possible distinguish direct (i.e., local) phenomena and interaction The former effects can be identified in the classical Karman vortex (i.e., global) ph enomena. streets in the wake of the circular barriers. This kind of shedding is probably responsible for the higher frequency content of the lift force (see Figure 26b) in the detailed configuration (St from 0.317 to 0.333). However, somewhat surprisingly the barriers also affect the lower bound of the Strouhal number range, namely, the shedding frequency related to the interaction between the
a24
L. BRUNO AND S. KHRIS
+0.6
I
I
I
1
flow:tautshipmodel
StKdl
0
0.2
0.4
0.6
0.8
xlB
0
1
Figure 25. Cp and std (Cp) distributions
0.2
0.4
0.6
0
[ 151
0.8
x/B
I
1
on the basic and detailed deck
les-has
.105
.141
,186 ,211 ,235
,317
St - h-bps
0.8s
b
0.6
1 bl bu
les-det
B
0.4
0.0 B98.122
.186.211
(a)
vr that
,333
St - ksdct
(b)
Figure 26. Mean streamlines deck.
vortices
,260
(a) and frequency content of CL (b): basic and detailed
are shed from the separation
bubble
and the vortices
vz that
emerge
past the
leeward edge. As a result of the reduction in the length of the bubble b&.. (see Figure 26a), the vortices ~1 translate along a greater length to reach the leeward edge and to merge into the vortices reduced
~2. Thus, the shedding period becomes longer and the associated Strouhal number from 0.105 2 St < 0.141 (without barriers) to 0.098 2 St < 0.122 (with barriers).
4.5. Flow Around
the
Deck
with
is
Angle of Attack
The goal of the simulations presented herein is to provide a further validation of the proposed 2D approach based upon the modified value of the Smagorinsky constant Cs = 0.075. The aerodynamic behaviour of the GBEB deck has been experimentally angles of attack. The mean pressure distribution on the deck measured by Reinhold et al. [14] is available in the literature 125) for an incidence below).
investigated at various in the wind-tunnel tests of 6 degrees (wind from
-
2D Numerical
8 2 I,
Simulations upper surface
0
0.2
0.4
0.6
0.8
0.2
0
x/B 1
0.4
0.6
0.8
I
I
x/E
I
lower surface
I
-I
lower surface
(b)
(4 Figure
27. Cp and std (Cp) distributions
The same setup was assumed the Smagorinsky
subgrid
by Enevoldsen
scale model.
on the basic
and detailed
deck Q = + 6’
et al. [25] in both 2D and 3D LES simulations
In the computational
simulations,
the turbulence
using level of
the incoming flow was set at zero, and the value of the Smagorinsky constant was set at 0.18. Although Thorbek [28] states that the details were modelled in both 2D and 3D simulations. the local effects of the barriers Figure
cannot
be found in the pressure
27a). The 3D simulation
Figure
27b compares
out barriers.
substantially
the results
It follows that
of the details-combined
improves
of the present
an adequate
study
distribution
the results obtained
scheme
constant
and a refined
surface
(see
using the 2D model.
using 2D models
value of the Smagorinsky
with a numerical
at the upper
obtained
with and with-
and careful
grid yielding
modelling
low dissipation-can
overcome the limitations of 2D LES simulation. In particular, the 2D model ensures the same level of accuracy as 3D simulations, while the computational costs are considerably reduced.
5. CONCLUSIONS The aim of the present vertical
structures
The first
around
paper
has been to discuss
a quasi-bluff
part was devoted
bridge
to a critical
This was followed by a closer investigation flow around
the Great
Belt East Bridge
the reliability
review of currently into the various deck.
of 2D numerical
simulation
of
deck. available
studies
mechanisms
The numerical
simulations
on the topic.
involved
in the unsteady
have shed some light
826
L. BRUNO AND S. KHRIS
on the essential Strouhal
The third studies,
physical
numbers
of the capacity of a number
approach,
a reduced speaking,
the small-scale, models assume
4.3.1.
(vortex
cell thickness dependence
dissipates
the periodic
for the convective Moreover, enabling
stretching,
The overall
upon
In particular,
the case study. model
accuracy
fluxes
in detail.
schemes
are required
to
diffusion. do not properly
the relevance
simulate
was confirmed
by the parametric
of the value of the Smagorinsky
vortices)
spanwise
of using low-diffusive
makes it possible
that
are generally
to take into due account Notwithstanding
2D model
to fix an optimal
the effects of physical
observed
is comparable
effort is considerably
reduced.
in
value for
phenomena
in the case of 3D separated with the accuracy
In principle,
the encouraging
of 3D sim-
a postetiori
the method by Diez and Egozcue [43]. Additional properties can be recovered in [44].
the global effects of the barriers
to separated-type
reported
in the subgrid
fluctuations.
of the optimised
the the proposed
discretiza-
analysis constant
Finally, the important role of deck equipment in bridge aerodynamics the increased computational effort, modelling of the barriers is strongly
research
the
In the laminar
fluctuations
The investigation
estimates can be developed exploiting concerning convergence and stability
this approach
of parametric
For each approach,
of the turbulence
the LES 2D model to take into account
while the computational
in extending
mechanisms.
was examined.
diffusive
numerical
models
the multiple
vortex-shedding
by means of a number
and high-order
approach,
the influence
longitudinal
flows with homogeneous ulations,
to simulate
the ensemble-averaged
scale model was studied this constant,
to a clarification,
of the computational
to the large eddy simulation
schemes
Section
formation. to distinct
complex eddies at the root of the vortex-formation process. The RANS equation such eddies into the turbulent stochastic field. It follows that the augmented eddy
markedly
Referring tion
near-wall
of vortex
were related
of some models of parameters
the flow field without
Generally
viscosity
of the process measured
part of the paper was devoted
influence calculate
features
experimentally
results
obtained
2D LES approach cylinder
is emphasized. recommended
Despite in order
on the flow.
in this
to other
and streamlined
error
estimates
study, classes
sections
caution of flow. represents
must The
be exercised application
a further
of
possible
development.
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