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Supercritical flows past a square cylinder with rounded corners

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Large-eddy simulations were used to investigate the supercritical aerodynamics of a square cylinder with rounded corners in comparison with those in the subcritical regime. First, the numerical methods, especially the dynamic mixed model, were validated on the basis of their prediction of supercritical flows past a circular cylinder. Then, the supercritical flows past a rounded-corner square cylinder were simulated and systematically clarified. Strong Reynolds number (Re) effects existed in the forces and local pressures as Re increased from o(10⁴) to o(10⁶). Changeover of flow patterns occurred as Re increased. At the supercritical Re, the free stream overall flowed along the cross sections of the cylinder, separated from the leeward corners and generated Karman vortices behind the cylinder. This pattern resulted in a much smaller recirculation region behind the cylinder compared with the subcritical flow. At the micro level, the flow experienced laminar separation and flow reattachment near the frontal corners, followed by the spatial development of turbulent boundary layers (TBLs) on the side faces and turbulent separation near the leeward corners. The feedback by large-scale primary vortex shedding and the small-scale turbulent motions in the high-frequency region with a slope of −5/3 were detected in the TBL. Their interaction affected the spanwise correlations of wall pressure fluctuations. The TBL on the side face differed from the zero-pressure-gradient flat-plate one; it was subjected to pressure gradients varying in space and time.
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Supercritical flows past a square cylinder with rounded corners
Yong Cao, and Tetsuro Tamura
Citation: Physics of Fluids 29, 085110 (2017); doi: 10.1063/1.4998739
View online: http://dx.doi.org/10.1063/1.4998739
View Table of Contents: http://aip.scitation.org/toc/phf/29/8
Published by the American Institute of Physics
PHYSICS OF FLUIDS 29, 085110 (2017)
Supercritical flows past a square cylinder with rounded corners
Yong Caoa) and Tetsuro Tamura
Department of Architecture and Building Engineering, Tokyo Institute of Technology, 4259 Nagatsuta-cho,
Midori-ku, Yokohama, Kanagawa 226-8502, Japan
(Received 24 February 2017; accepted 2 August 2017; published online 23 August 2017)
Large-eddy simulations were used to investigate the supercritical aerodynamics of a square cylinder
with rounded corners in comparison with those in the subcritical regime. First, the numerical methods,
especially the dynamic mixed model, were validated on the basis of their prediction of supercritical
flows past a circular cylinder. Then, the supercritical flows past a rounded-corner square cylinder were
simulated and systematically clarified. Strong Reynolds number (Re) effects existed in the forces and
local pressures as Re increased from o(104) to o(106). Changeover of flow patterns occurred as Re
increased. At the supercritical Re, the free stream overall flowed along the cross sections of the
cylinder, separated from the leeward corners and generated Karman vortices behind the cylinder.
This pattern resulted in a much smaller recirculation region behind the cylinder compared with the
subcritical flow. At the micro level, the flow experienced laminar separation and flow reattachment
near the frontal corners, followed by the spatial development of turbulent boundary layers (TBLs)
on the side faces and turbulent separation near the leeward corners. The feedback by large-scale
primary vortex shedding and the small-scale turbulent motions in the high-frequency region with a
slope of 5/3 were detected in the TBL. Their interaction affected the spanwise correlations of wall
pressure fluctuations. The TBL on the side face differed from the zero-pressure-gradient flat-plate
one; it was subjected to pressure gradients varying in space and time. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4998739]
I. INTRODUCTION
Flow separations and vortex streets play a dominant role
in determining the aerodynamic characteristics of flows past a
two-dimensional bluff body. On the other hand, the cross sec-
tion of a given body significantly influences the time-averaged
flow topology and the time-dependent behavior of the vortex-
shedding process. The aerodynamics of circular and square
cylinders, two typical shapes, has been attracting a great deal
of research interest because of its immense scientific and prac-
tical significance. The flow around a circular cylinder strongly
depends on the Reynolds number (Re = U0D/ν, where U0is the
free-stream velocity, Dis the diameter or width of the cylinder,
and νmeans the kinematic viscosity) because the flow separa-
tion point varies due to the continuous and rounded shape. In
contrast, the aerodynamic characteristics of a smooth square
prism with sharp corners are hardly dependent on Re [at least
Re = U0D/ν=o(104)–o(106)] when the free-stream flow is
uniform because the flow fully separates only at the frontal
sharp corners. Consequently, the primary vortex (or the Kar-
man vortex) behind a square cylinder behaves more vigorously
and oscillates to a greater degree than that behind a circular
cylinder. In the light of this, corner modification has widely
been introduced in engineering applications as a passive flow
control method to reduce the drag force and the fluctuating lift
force due to the vortex shedding in the wake, including but not
limited to rounded corners, chamfered corners, and recessed
corners (Kwok et al.,1988;Kawai,1998; and Tse et al.,2009).
a)Author to whom correspondence should be addressed: cao.y.aa@
m.titech.ac.jp
However, there are many fewer studies on the aerodynamics
of corner-modified square cylinders than on circular or square
cylinders, and hence, a lack of knowledge about them. An
unknown fundamental issue is the Reynolds number depen-
dence for such cross sections based on the fact that most of Re
values investigated so far are not close to those encountered by
actual structures in reality (e.g., if the width of a tall building
is 50 m and the air velocity is 5 m/s, then Re 107).
Previous studies on the aerodynamics of two-dimensional
rounded-corner square cylinders focused on subcritical or
lower Reynolds numbers, mostly due to the limitations of
the experimental facilities or computational power. Most of
them concentrated on the variations of aerodynamic features
brought about by the different radii of the rounded corners.
Park and Yang (2016) recently clarified the effect of rounding
the sharp edges of a square cylinder on the flow instabilities
(respectively, the Hopf bifurcation and the onset of three-
dimensional wakes) by using an immersed boundary method.
It was found that rounding the corners of a square cylinder
tends to stabilize the flow, and the maximum critical Re occurs
when r/D= 0.25 (where ris the projected length of the rounded
corners onto the frontal cylinder wall). Hu et al. (2006) experi-
mentally studied the effect of varying the rounded corner radius
on the near wake of a square prism by using particle imag-
ing velocimetry (PIV), laser Doppler anemometry (LDA), and
hotwire measurements at Re = 2600 and 6000. They found
that as r/Dincreases from 0 (square cylinder with sharp cor-
ners) to 0.5 (circular cylinder), the strength of the shed vortices
attenuates, the Strouhal number (St = fvsD/U0, where fvs is
the frequency of primary vortex shedding) increases by about
60%, and the vortex formation length almost doubles in size.
1070-6631/2017/29(8)/085110/17/$30.00 29, 085110-1 Published by AIP Publishing.
085110-2 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
They suggested that the leading edge corner radius plays a
more important role than the trailing one in influencing the
near wake structure at these Re values. Furthermore, Zhang
and Samtaney (2016) performed a direct numerical simulation
(DNS) at Re = 1000; they emphasized the development of the
separated and transitional flow around the cylinder influenced
by various rounded-corner radii. Their results showed that the
time-averaged wake recirculation length does not monotoni-
cally vary with the corner radius, the maximum value being
at r/D= 0.125. Rounded corners also postponed the transi-
tion onset downstream, occurring at the latest at r/D= 0.125.
Tamura et al. (1998) performed numerical simulations of the
unsteady flow at around Re = o(104) past a square cylinder
with rounded and chamfered corners with r/D= 0.167. The
results showed that the rounded corners reduced mean drag
by about 35% and fluctuating lift by about 30%, together with
an increase in St. Following that, Tamura and Miyagi (1999)
experimentally studied the effects of inflow turbulence at
Re = 3 ×104and found that turbulent inflow contributes to the
flow reattachment to the side faces of the r/D= 0.167 cylinder.
Nevertheless, there is little experimental evidence con-
cerning the Reynolds number effects beyond the subcritical
regime on the rounded-corner square cylinder. An experiment
performed by Delany and Sorensen (1953) about sixty years
ago gave the first insights into Reynolds number effects, where
a rapid jump in the mean drag force was observed at Re 7
×105for r/D= 0.167. Recently, Carassale et al. (2014) experi-
mentally studied the aerodynamic behavior of rounded-corner
square cylinders with r/D= 0.067 and 0.13 for Re values rang-
ing between 1.7 ×104and 2.3 ×105. Interestingly, strong Re
effects were observed for the cylinder with r/D= 0.13 under
a free-stream turbulence intensity of 5%. This result could be
interpreted as indicating that free-stream turbulence has a simi-
lar effect as increasing Re. Nevertheless, one should notice that
the Reynolds numbers tested by Carassale et al. (2014) are rel-
atively low. Considering that the free-stream turbulence cannot
completely replace the effect of increasing Re, a question still
remains about the aerodynamics of such bluff bodies at very-
high Re regardless of the imposition of free-stream turbulence.
Moreover, it should be noted that in contrast with the measure-
ments of Delany and Sorensen (1953) showing a drag crisis as
Re increases, the time-averaged drag force gradually falls from
the high value at Re >5×104. In the previous experiments,
the time-averaged drag decreased within a small Re region,
e.g., from 5 ×104to 1 ×105(Carassale et al.,2014). Like the
classification of Re regimes in flows past a circular cylinder,
we call the Re regions before and after the significant reduc-
tion in mean drag the subcritical and supercritical regimes,
respectively. Despite the significant drag decrease observed
by Delany and Sorensen (1953) and Carassale et al. (2014),
the limited information available is not sufficient to system-
atically understand very-high-Re flow past a rounded-corner
square cylinder. Especially at supercritical Re, information is
lacking on the flow and its interaction with the cylinder.
The increasingly popular and reliable tool, large-eddy
simulation (LES), can provide deep insights into supercritical
flows around a rounded-corner square cylinder. Theoretically
speaking, there is no limitation on Re when the numerical grids
are fine enough. Moreover,as it is a numerical method, LES can
simultaneously provide detailed flow and pressure fields that
would greatly improve the understanding of the aerodynamics
of this bluff body. To the best of our knowledge, no numerical
simulation, especially with a high-accuracy LES technique,
has been conducted with the purpose of predicting supercriti-
cal flows around a rounded-corner square cylinder. However,
accurate simulation of critical or supercritical flows is not a
trivial task, as has been clearly shown in previous numer-
ical tests on flows past a circular cylinder. This is because
the flow experiences the laminar separation, transition of the
separated boundary layer to turbulence, flow reattachment on
the wall, and secondary separation of the turbulent bound-
ary layer. As for supercritical flows past a circular cylinder,
Catalano et al. (2003) used LES with wall modeling to pass
the need for resolving the flow near the wall. The estimated
global aerodynamic characteristics were generally promising
even when using about 2.3 ×106grid points. But obviously,
the accuracy of the predicted complex flow near the cylin-
der was not satisfactory. Since then, computational resources
have become much more powerful, and this means that more
accurate predictions can be made. Consequently, using the
wall-resolving LES method to simulate this typical flow has
attracted much attention [see the studies of Ono and Tamura
(2008), Lehmkuhl et al. (2014), Rodr´
ıguez et al. (2015), Lloyd
and James (2016), and Yeon et al. (2016)]. Generally satisfac-
tory predictions can now be made, including of the flow process
near the wall. More importantly, more experimental data are
available on very-high-Re flows past circular cylinders than on
past rounded-corner square cylinders. In light of this situation,
a circular cylinder would be preferable as a simulation bench-
mark for evaluating the numerical performance of predicting
supercritical flows past a bluff body.
To summarize, the previous studies do not provide ade-
quate knowledge on supercritical flows around a rounded-
corner square cylinder. Moreover, a question remains about
the flows affected purely by Re instead of together with free-
stream turbulence. The present numerical study therefore aims
at elucidating the aerodynamic characteristics of supercritical
flows past a square cylinder with rounded corners under uni-
form inflow and the effect of Re by making comparisons with
the case of the subcritical flow. Specifically, the main goals
of this study include the following: (1) identifying whether
increasing Re causes a variation in the aerodynamic forces
and local pressures for square-section cylinders and describing
these characteristics at different Re levels; (2) describing and
understanding the effect of varying Re on the changeover of the
mean flow pattern and turbulent statistics around the cylinder,
in particular recognizing whether the turbulent transition and
flow reattachment occur at supercritical Re even for a rounded-
corner square cylinder; (3) explaining how the flow patterns
at different Re determine the characteristics of aerodynamic
forces and local pressures; (4) describing the dynamics of the
wake, separated shear layer, and boundary layer and vortical
structures of different scales; (5) understanding the mecha-
nism of interactions between turbulent structures in the reat-
tached boundary layer and primary vortex shedding and how
these interactions affect the spanwise correlations of fluctuat-
ing pressures; and (6) clarifying the features of the reattached
boundary layer at the supercritical Re for this cylinder shape.
085110-3 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
The paper is organized as follows. Section II is a descrip-
tion of the numerical methods. Section III validates the meth-
ods and examines the grid resolution on the basis of simulations
of supercritical flows past a circular cylinder (Re = 6 ×105).
Section IV describes the simulations of flows past a rounded-
corner square cylinder at Re = 2.2 ×104and 1 ×106and
discusses their results, focusing on the aerodynamic features at
supercritical Re compared with those in the subcritical region.
II. NUMERICAL METHODS
The governing equations are solved in the generalized
curvilinear coordinate system. The filtered Navier-Stokes and
continuity equations for incompressible flows are written in
the dimensionless form as
(J¯ui)
t+
ξmJ U m¯ui=
ξm J ξ m
xi
¯p!
+
ξm 1
Re Jξm
xj
ξn
xj
¯ui
ξn+σm
i!,
(1)
(JUm)
ξm
=0, (2)
where Um(=∂ξ m
xiui) is the contravariant velocity perpendicu-
lar to the faces of the grid cells, and J(=
xi
∂ξ m
) is the Jacobian
of the transformation between the physical space and compu-
tational space. The subgrid-scale (SGS) stress is expressed as
σm
i=JUm¯uiJ Umui.
A. SGS modeling
In the dynamic Smagorinsky model (denoted by DSM)
proposed by Germano et al. (1991) and Lilly (1992), the SGS
stress σm
ihas to be modeled using the linear eddy-viscosity
model to relate the residual stress to the filtered rate of strain,
σm
i1
3σk
kδij =Jξm
xjνSGS ¯
Sij
=Jξm
xj 2C¯
2q2¯
Sij ¯
Sij ¯
Sij !, (3)
where νSGS means the eddy viscosity and ¯
is the size of the
grid filter. ¯
Sij =1
2¯ui
∂ξ m
∂ξ m
xj+¯uj
∂ξ n
∂ξ n
xiis the rate-of-strain
tensor. The unknown Smagorinsky coefficient Cis determined
explicitly as
C=DLk
iMk
iE
2¯
2DMk
iMk
iE, (4)
where Lk
i=g
JUk˜
¯uiJ
JUk¯ui,Mk
i=4
H
S
H
¯
Sk
iI
¯
S
¯
Sk
i, and ¯
Sk
i
=J∂ξ k
xj
¯
Sij.
This study takes an advanced approach to SGS modeling,
i.e., the dynamic mixed model (denoted by DMM) proposed by
Zang et al. (1993). The DMM can be regarded as a combination
of the dynamic Smagorinsky model and the scale-similarity
model (Bardina et al.,1983). The three components of SGS
stress are treated differently: the modified Leonard term (Lm
ij )
is calculated explicitly, while the modified cross term and the
modified SGS Reynolds stress are modeled with the linear
eddy-viscosity approximation,
σm
i1
3σk
kδij =Lm
ij +Jξm
xjνSGS ¯
Sij
=Lm
ij +Jξm
xj 2C¯
2q2¯
Sij ¯
Sij ¯
Sij !, (5)
where Lm
ij =J∂ξ m
xj¯ujuiJ∂ξ m
xj¯uj¯uiis the modified Leonard
term.
The unknown coefficient Cis determined by Eq. (6),
which differs from the dynamic Smagorinsky model by adding
the term Hm
iin the numerator,
C=Lm
iHm
iMm
i
2¯
2Mm
iMm
i
, (6)
where Lm
i=
I
JUm˜
¯uiJ
JUm¯ui,Hm
i=
K
J∂ξ m
xj¯uj˜
¯
¯ui
K
J∂ξ m
xj¯uj¯
¯
ui, and
Mm
i=4
˜
¯
S
˜
¯
Sm
iI
¯
S
¯
Sm
i.
As analyzed by Zang et al. (1993), the DMM has several
advantages over the DSM: less modeling requirements because
of the explicit computation of the modified Leonard term; the
potential of energy backscatter to the resolved scales as pro-
vided by the modified Leonard term; no need for aligning the
principal axis of the SGS stress tensor and the resolved strain
rate tensor. One could suppose that the DMM provides more
accurate predictions of supercritical flows; this conjecture will
be examined in Sec. III.
B. Discretization in time and space
The code is based on the finite difference method that has
been successfully applied to flows past bluff bodies at subcrit-
ical Re (Ono and Tamura,2002;Tamura and Ono,2003; and
Cao and Tamura,2016). A non-staggered grid is used herein.
Specifically, the velocity and pressure are stored at the cell cen-
ters, while the contravariant velocity is stored on cell faces. A
fractional step method, originally proposed by Kim and Moin
(1985), is used to advance the velocity and pressure solutions
in time. The time marching of the momentum equation is a
hybrid one; the Crank-Nicolson scheme is applied to the vis-
cous term and an explicit third-order Runge-Kutta method is
used for the convective term. The temporal treatment of the
convective term is to allow a relatively larger time step (Le
and Moin,1991). The detailed procedure is in the work of Cao
and Tamura (2016).
Spatial discretization is generally treated as a second-
order central difference. However, the convective term is
approximated using a fourth-order central difference scheme.
To avoid numerical instability, numerical dissipation is added
through the convective term; the amount is controlled by a
parameter αND. In this study, αND = 0.2 was used, which
means that the numerical dissipation is very small in com-
parison with αND = 3 of the third-order upwind scheme pro-
posed by Kawamura and Kuwahara (1984) and αND = 1 of
the UTOPIA (uniformly third-order polynomial interpolation
algorithm) scheme.
085110-4 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
III. SUPERCRITICAL FLOWS PAST
A CIRCULAR CYLINDER
The numerical experiments examined predictions of
supercritical flows past a bluff body. The flow past a circu-
lar cylinder at Re = 6 ×105was simulated, and the results
were compared with experimental data.
A. Numerical conditions
Four cases, namely, case 1, case 2, case 3, and case 4,
were computed in order to examine the effects of different
grid resolutions and SGS models. The computational domain
was 30 ×24 ×1. Note that all lengths in this study, unless
otherwise stated, have been non-dimensionalized by the cylin-
der diameter or width D. The upstream length between the
inlet and the cylinder center was 10, while the downstream
length from the cylinder center to the outlet was 20. The total
height between the upper and lower boundaries was 24, result-
ing in a blockage ratio (BR) of 4.2%. BR was defined as the
ratio of projected area of the cylinder upon the cross-sectional
area of the computational domain or test section; specifically
for two-dimensional circular or square cylinder, it was the
ratio of the cylinder width or diameter to the height of the
computational domain or test section. The spanwise length was
Lz= 1 for the simulations of supercritical flows around a circu-
lar cylinder, following the studies of Ono and Tamura (2008)
(Lz= 1), Lehmkuhl et al. (2014) (Lz= 0.5π), and Rodr´
ıguez
et al. (2015) (Lz= 0.5πand 1). Lzwas uniformly discretized
into 160 cells in all four cases, resulting in a spanwise resolu-
tion of z= 0.006 25, which can be deemed sufficient on the
basis of LES tests performed by Ono and Tamura (2008) with
z= 0.0067 and Rodr´
ıguez et al. (2015) with z= 0.0078. In
terms of the grid dimensions, cases 1 and 2 used the same grid
system, i.e., 601 ×420 ×161 in (ξ,η,ζ) coordinates with a
total of 40.6 ×106cells. By comparison, cases 3 and 4 had
a grid resolution of 801 ×500 ×161, i.e., 64.5 ×106cells
in total. The grid system of cases 3 and 4 on the (ξ,η) plane
is shown in Fig. 1. The height of the first cell nearest to the
cylinder wall was y= 5.3 ×10 5<0.05/Re1/2 = 6.5 ×10 5.
The maximum non-dimensional wall distance y+was less than
2, and the cylinder-averaged value was around 1. The details
are summarized in Table I. The stretching ratio of cell heights
near the wall was 1.024 for cases 1 and 2 and 1.021 for cases
3 and 4.
The inlet boundary condition was a uniform steady inflow
without any velocity fluctuations, i.e., a constant velocity
FIG. 1. Grid system on the (ξ,η) plane for case 3 and case 4: (a) whole grid; [(b) and (c)] close-ups.
TABLE I. Summary of global quantities including a comparison with previous studies.
Case Re Grid (Lz)y+max y+avg Ncell SGS ¯
CDC0
D¯
CLC0
LSt Lf
Present Case 1 6 ×105601 ×420 ×161 1.91 1.00 40.6M DSM 0.22 0.010 0.0053 0.062 0.441 1.18
Case 2 6 ×105601 ×420 ×161 1.92 1.01 40.6M DMM 0.20 0.010 0.0055 0.062 0.471 1.08
Case 3 6 ×105801 ×500 ×161 1.67 0.88 64.5M DSM 0.24 0.013 0.0077 0.071 0.416 1.16
Case 4 6 ×105801 ×500 ×161 1.68 0.89 64.5M DMM 0.22 0.011 0.0032 0.069 0.444 1.11
Expt. Bearman (1969) 4 ×105... ... ... ... ... 0.23 ... ... ... 0.46 ...
Schewe (1983) 6 ×105. . . . . . . . . . . . . . . 0.22 . . . . . . 0.018 0.47 . . .
LES Ono and Tamura (2008) 6 ×105. . . . . . . . . 13.4M DMM 0.21 . . . . . . 0.08 0.48 . . .
Lehmkuhl et al. (2014) 6.5 ×105... <2 . . . 83.2M WALE 0.232 0.008 0.027 0.076 0.44 1.10
Rodr´
ıguez et al. (2015) 7.2 ×105... <2 . . . 89.4M WALE 0.213 0.009 0.016 0.075 0.45 1.07
Rodr´
ıguez et al. (2015) 8.5 ×105... <2 . . . 105.1M WALE 0.218 0.009 0.007 0.070 0.45 1.10
085110-5 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
FIG. 2. Comparison of present time-
averaged drag and fluctuating lift coef-
ficients with published experiments.
profile u= (U0, 0, 0) was imposed; the cylinder was a no-slip
wall without any wall functions; the spanwise end boundary
conditions were periodic; the upper and lower boundaries were
free-slip. The outlet was set as a convective condition for all
velocity components and pressure. For the other boundaries, a
zero-gradient condition was used for the pressure. With respect
to the effects of the SGS models, cases 1 and 3 used the DSM,
while cases 2 and 4 used the DMM. The dimensionless time
step t*(=tU0/D) was 0.0001 or 0.000 05. The time intervals
for the averaging operation were about 27 shedding cycles.
B. Results and discussion
Table Ishows the global quantities together with those
of the previous experiments and LES studies conducted
at similar Re, including the time-averaged and root mean
square (rms) aerodynamic forces based on the whole spanwise
length ( ¯
CD,C0
D,¯
CL, and C0
L), St, and the time-averaged
formation lengths of the Karman vortex behind the cylin-
der (Lf). Moreover, the mean drags and fluctuating
lifts in Fig. 2are compared with those from the previous
experiments (Wieselsberger,1922;Bursnall and Loftin,1951;
Delany and Sorensen,1953;Achenbach,1968;Jones et al.,
1969;So and Savkar,1981;Cheung,1983;Schewe,1983;
and Blackburn and Melbourne,1996). First, the time-averaged
drags in the experiments are obviously dispersed to a large
degree [Fig. 2(a)]. As pointed out by Zdravkovich (1997),
the influencing parameters (e.g., ratio of the spanwise length
to cylinder diameter Lz/D, BR, and free-stream turbulence
intensity Ti) strongly affect the aerodynamic characteristics.
However, it seems that the ¯
CDvalues from the experiments
were mostly between 0.1 and 0.3. The exception, 0.6, was
measured by Achenbach (1968) with unfavorable influencing
parameters, i.e., Lz/D= 3.3 (too small to avoid the interfer-
ence effect caused by the junction of the cylinder and the
wind-tunnel walls), BR = 16% (much larger than the accept-
able value 5% and tended to augment the blockage effect), and
Ti = 0.7% (relatively high free-stream turbulence intensity).
Although their combined effects are complex, the unfavorable
Lz/Dand BR tend to decrease the base pressures and increase
the mean drag forces according to West and Apelt (1982). The
present LES results are around 0.22, reasonable values for most
of the experiments. The table also indicates that the present ¯
CD
is consistent with the previous well-validated LES predictions
of supercritical flows. Regarding the rms values of the lift,
the present results are in a reasonable region that accounts for
the scatter in the experiments [see Fig. 2(b)]. Moreover, the
present rms lifts (0.07) agree with the previous LES results.
In addition, the drag shows a small degree of fluctuation in time
with C0
D0.01, which is slightly larger than the LES predic-
tions of Lehmkuhl et al. (2014) and Rodr´
ıguez et al. (2015).
The very small time-averaged lifts ¯
CLindicate a symmetric
time-averaged flow topology on both sides of the cylinder that
characterizes the supercritical regime of flow past a circular
cylinder. The time-series of the lifts have well-defined peri-
odic frequencies, matching the findings at supercritical Re
by Bearman (1969), which indicates organized vortex shed-
ding in the wake. Lfis generally consistent with the recircula-
tion lengths obtained by Lehmkuhl et al. (2014) and Rodr´
ıguez
et al. (2015). To summarize, it seems that all four cases tested
give satisfactory prediction accuracy in terms of the global
quantities in comparison with the previous experimental and
numerical studies.
Figure 3(a) shows the time-averaged pressure distribu-
tions along the circumferential direction of the cylinder, where
θ= 0 and 180 correspond to the frontal and base stagna-
tion points, respectively. Overall, the time-averaged pressures
agree well with the experimental values. Note that Flachs-
bart’s experimental data (1929) are taken from the work of
Roshko (1961). The kink in the numerical pressure distribu-
tions is correct. In particular for case 4, there is a near pressure
plateau beginning at θls = 100 and a sharp increase between
θTr = 105 and θr= 110. These points correspond to the sepa-
ration of the laminar boundary layer, transition to turbulence
beyond laminar separation, and reattachment of the turbulent
FIG. 3. Distributions around the cylin-
der: (a) time-averaged pressure; (b)
time-averaged wall shear stress. θ= 0
corresponds to the frontal stagnation
point. θls,θr, and θts are the angu-
lar positions of laminar separation, flow
reattachment, and turbulent boundary
layer separation for case 4.
085110-6 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
boundary layer. They are represented more clearly by the
distribution of the time-averaged wall shear stresses (after non-
dimensionalization) in Fig. 3(b), where τ > 0 in the clockwise
direction. For case 4, the wall shear stress experiences sign
changes at θls = 100, θr= 110, and θts = 114 that correspond
to laminar boundary layer separation, flow reattachment, and
turbulent boundary layer separation, repectively. The angu-
lar positions of laminar separation, turbulence transition, flow
reattachment, and secondary turbulent separation were quanti-
tatively measured by Tani (1964), Achenbach (1968), and Pfeil
and Orth (1990). But the skin friction probe was a miniature
“fence” protruding above the surface in the work of Achenbach
(1968); hence, the separation bubble marking the supercritical
regime was not successfully detected. Tani (1964) used dye
visualization in water and found angles of θls = 102, θTr = 110,
and θr= 117, while Pfeil and Orth (1990) measured the pres-
sure distribution within the boundary layer, finding θls = 100,
θTr = 110, and θr= 114. By comparison, the present results
(say, case 4) reproduced the positions of laminar separation,
in spite of the slight under-prediction of the transition and
reattachment positions of the boundary layer. However, regard-
ing the SGS models, only the DMM provided the obvious
reattachment after the transition to turbulence of the bound-
ary layer; this is based on the fact that the wall shear stress
changes sign from negative to positive at θr= 110 for cases
2 and 4 in Fig. 3(b). Similar conclusions about SGS model-
ing were obtained by Ono and Tamura (2008). In addition, for
the high grid resolutions tested presently, it seems that SGS
models play a more important role than grid resolution does
because the flow reattachment was still not reproduced even
though the grid resolution increased in going from case 1 to
case 3.
Another critical parameter is the separation point of the
turbulent boundary layer (θts). We obtained θts = 114 on the
basis of the sign change of the wall shear stress. Achenbach
(1968) suggested that the turbulent separation of the boundary
layer is at θts = 147 on the basis of skin friction measurements.
This value is widely used to validate numerical results
(Lehmkuhl et al.,2014;Rodr´
ıguez et al.,2015; and Yeon et al.,
2016). However, as mentioned before, the accuracy of the
measurement by Achenbach (1968) is worth further discussion
because of the possible disturbance from the fence probe. Two
factors, at least, are notable in Fig. 5 in the work of Achenbach
(1968). One is the failure of a separation bubble to form, from
the distribution of skin friction at the supercritical Re. The
other is the delay of the starting point of the near plateau of the
base pressure compared with other experimental and numer-
ical studies; that is, the plateau of the base pressure started
at θ150, while most of the other experiments obtained θ
= 125–140 [e.g., Flachsbart’s measurements, as cited in
Roshko (1961), Bursnall and Loftin (1951), Tani (1964), and
Shih et al. (1993) among others]. Thus, it is possible that
Achenbach (1968) overpredicted the separation location of the
turbulent boundary layer. More experimental measurements
are needed to make a numerical validation.
The time-averaged velocity streamlines of case 4 in
Fig. 4(a) shows that the width of the recirculation region
behind the cylinder becomes small because of the delay of
the flow separation compared with the well-known subcritical
flow. Figure 4(b) depicts the vector field of the time-averaged
velocity near the separation point. We can clearly see the pro-
cess of laminar separation, flow reattachment, and turbulent
separation. The closed flow region between separation and
reattachment is called the separation bubble. It is worth noting
that at this Re, two separation bubbles formed symmetrically
on both sides of the cylinder. Beyond the separation bubbles,
reverse flow occurs very near the cylinder wall.
On the whole, the supercritical flow simulations past a
circular cylinder validated well the present numerical schemes
and time-marching algorithm. Grid convergence was obtained
when using the same SGS model because the grid refinement
did not significantly change the global quantities and time-
averaged flow topology. The DMM is recommended over the
DSM because of its accurate prediction of flow reattachment
on the cylinder wall.
IV. SUPERCRITICAL FLOWS PAST A
ROUNDED-CORNER SQUARE CYLINDER
A. Numerical model and grid system
The numerical methods validated in Sec. III were used
for predicting supercritical flows past a rounded-corner square
FIG. 4. Time-averaged velocity field of case 4: (a)
streamlines around the cylinder; (b) the velocity vector
near the separation bubble.
085110-7 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
FIG. 5. Grid system on the (ξ,η) plane
for case E6L: (a) whole grid; [(b) and
(c)] close-ups.
cylinder. Note that only the more advanced DMM was used in
this section. The radii of the rounded corners were r= 0.167
scaled by the cylinder width, which is similar to the val-
ues in the experiments of Delany and Sorensen (1953) and
Carassale et al. (2014). The supercritical regime was repre-
sented by Re = 1.0 ×106, namely, case E6L, which is higher
than Re that causes a dramatic decrease in drag [see the work
of Delany and Sorensen (1953)]. Their computational domain
and grid system on the (ξ,η) plane are shown in Fig. 5. Case
E6L had a spanwise length of Lz= 4 that has been demon-
strated to be sufficient for the objectives of this study (see
the Appendix). The typical subcritical Re = 2.2 ×104was also
taken into consideration, namely, case E4. In similar grid
topologies, case E4 differed from the supercritical cases only
in terms of the grid parameters in Table II. The total cell num-
bers were 21.8 and 384.9 ×106for case E4 and case E6L,
respectively. The spanwise resolutions were z= 0.006 25 for
case E6L (the same as in the simulations of supercritical
flows past a circular cylinder) and z= 0.05 for case E4.
The height of the first cell (y) nearest to the no-slip cylin-
der wall was less than 0.1/Re0.5 for the subcritical case, while
it was less than 0.05/Re0.5 for the supercritical case. The
maximum y+was less than 3.0, and the space-averaged y+
was around 1.0. t*was 0.0005 for case E4 and 0.0001 for
case E6L. The statistics of the aerodynamic characteristics
were computed for a time span of over 20 vortex-shedding
cycles.
B. Aerodynamic force and pressure
This simulation explored how the aerodynamic forces and
local pressure vary with increasing Re from o(104) to o(106).
The statistics of the global quantities are summarized in
Table III, including the time-averaged and rms forces and St.
Overall, the effect of Re is strong on the aerodynamic forces, as
is apparent from the variations in the quantities as Re increases
from 2.2 ×104to 1.0 ×106. The mean drag coefficient ¯
CD
experiences two distinct quantitative levels in the subcriti-
cal and supercritical regimes, i.e., it decreases substantially
from 1.39 to 0.55 as Re increases. These results match the
experimental measurements on similar cylinders conducted
by Tamura et al. (1998), Delany and Sorensen (1953), and
Carassale et al. (2014). The rms values of the forces also
decrease when Re goes beyond the subcritical regime. For
example, the rms lift coefficient C0
Lfalls dramatically by about
60% as Re increases from 2.2 ×104to 1.0 ×106. Moreover,
St nearly doubles over the course of the rise in Re.
Figures 6(a) and 6(b) show the distributions of the time-
averaged and fluctuating pressure coefficients at different Re.
Like the total forces, one can see strong Re effects in the local
pressures not only in the distribution shapes but also in the
quantitative values. hpihas two valleys near the frontal and
leeward corners at Re = o(106). The first one is located at
θmin1 = 50.9 at Re = 1.0 ×106, which moves downstream
as Re increases. The second valley is at θmin2 = 126 in the
upstream part of leeward rounded corners. Between them, hpi
has a plateau on the side wall. In comparison with the obvious
extreme negative values, there is no big change in hpiwith θ
after the frontal corner at the subcritical Re. In particular, the
base pressure of case E6L is much larger than that of case E4.
For this cylinder, the decrease in ¯
CDwith increasing Re to the
supercritical regime is primarily due to the horizontal com-
ponent decomposed from the negative pressure on the frontal
rounded corners and the smaller suction pressure on the rear
face. In Fig. 6(b), the supercritical case E6L shows two sharp
peaks near the frontal and leeward corners, denoted by θmax1
and θmax2. The exact locations of θmax1 and θmax2 are shown in
TABLE II. Summary of grid information for flows past a rounded-corner square cylinder. Lzand zare spanwise
lengths and span resolutions, respectively; yis the height of the first cell nearest to the cylinder wall; y+max and
y+avg are the maximum and span-averaged y+;t*is the non-dimensional time step.
Case Re Grid Ncell LzSGS zy y+max y+avg t*
Case E4 2.2 ×104301 ×300 ×241 21.8 ×10612 DMM 0.05 5.59 ×10 41.13 0.33 0.0005
Case E6L 1.0 ×1061201 ×500 ×641 384.9 ×1064 DMM 0.006 25 4.08 ×10 52.71 1.06 0.0001
085110-8 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
TABLE III. Summary of global quantities for the flows past a rounded-corner square cylinder at subcritical and
supercritical Re. Previous experimental data are also included.
Case Re ¯
CDC0
D¯
CLC0
LSt
LES Case E4 2.2 ×1041.39 0.06 0.0053 0.73 0.144
Case E6L 1.0 ×1060.55 0.04 0.0100 0.30 0.280
Expt. Tamura et al. (1998) 6.0 ×1041.11 . . . . . . 0.39 0.140
Carassale et al. (2014) 3.7 ×1041.45 . . . 0.05 0.52 0.138
Delany and Sorensen (1953) Subcritical 1.24 . . . .. . . . . . . .
Delany and Sorensen (1953) Supercritical 0 .56 . . . . . . . . . . . .
FIG. 6. Distributions of time-averaged
and fluctuating pressures around the
cylinder.
TABLE IV. Main angular positions characterizing the distributions of time-
averaged and fluctuating pressures and wall shear stress. θfand θlare the
ranges of angular positions of the frontal and leeward rounded corners,
respectively.
Case θfθlθmin1 θmin2 θmax1 θmax2 θs1 θrθs2
Case E4 33.7–56.3 123.7–146.3 46 . . . 47 .. . 50 . . . . . .
Case E6L 33.7–56.3 123.7–146.3 51 126 56 130 54 58 127.6
Table IV. They are slightly more downstream than θmin1 and
θmin2. Nevertheless, the above characteristic angular positions
are still within the ranges of the rounded corners. Furthermore,
the magnitude of the σppeak at the leeward corner is slightly
larger than that at the frontal corner. Apart from these peaks,
σpon the side faces of case E6L is much lower than that in
case E4 on average, whereas the σpvalues on the rear faces
are comparable.
C. Statistical flow field
The significant effect of Re on aerodynamic forces and
pressures points to a changeover of flow patterns due to the
increase in Re from subcritical to supercritical levels. This
section will attempt to describe and understand the mean flow
and turbulent statistics at supercritical Re. Meanwhile, the
statistical flow field will be used to explain the characteristics
of the integral forces and local pressures acting on the
cylinder.
The overall flows at the subcritical and supercritical Re
are shown in Fig. 7by using the streamlines of span- and
time-averaged velocity with the stream function ψranging
between 0.2 and 0.2. The flow topology at the supercritical Re
differs substantially from that at the subcritical Re. The most
notable point is that the fluid separates completely from the
frontal corner at subcritical Re, whereas the free stream flows
along the side face and finally separates from the leeward cor-
ner at supercritical Re. Such a flow induces a large curvature
in the streamlines near the rounded corners and a mild cur-
vature near the leeward corners. As a result, the streamlines
at supercritical Re get closer to each other around the frontal
and leeward corners, which leads to valleys of mean pressure
around the corners [see Fig. 6(a)]. Behind the cylinder, the
width of the recirculation region in the vertical direction (dw)
is much smaller than that at subcritical Re. In particular, dw>D
at subcritical Re, but dw<Dat supercritical Re. Meanwhile,
the streamwise length of the recirculation region (denoted by
Lf) at supercritical Re is much smaller as compared with case
E4; specifically, Lffalls from 1.49 at subcritical Re to 1.07
at supercritical Re. The reduction in the size of the recircu-
lation regions from the subcritical to supercritical regime is
FIG. 7. Streamlines of span- and time-
averaged velocity: (a) case E4 and (b)
case E6L.
085110-9 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
FIG. 8. Close-ups of streamlines of span- and time-
averaged velocity: (a) case E4, Re = 2.2 ×104and (b)
case E6L, Re = 1.0 ×106. Note that ψmeans the stream
function.
responsible for the growth in the time-averaged base pressure
and St.
In order to describe the flow features near the wall, Fig. 8
shows close-ups of the time-averaged velocity in the vicinity
of the side face. Moreover, the quantitative distributions of
time-averaged wall shear stress are plotted in Fig. 9. At sub-
critical Re, a large-scale recirculation region (also called the
secondary vortex) exists immediately behind the frontal corner
and beneath the separated shear layer. However, its velocity is
actually very low, as evidenced by the small positive wall shear
stress when θ[60, 90]. At supercritical Re, the free stream
separates from the downstream part of the frontal rounded cor-
ner and then reattaches to the side face. The separation bubble
is formed between the first separation point θs1 and the reat-
tachment point θr. But it is so small-scale that it is invisible on
the scale of the cylinder width. The reattached flow remains
throughout almost the whole side face, before separating again
from the leeward corner at θs2. The exact angular positions of
flow separation θs1, reattachment θr, and secondary separation
θs2 for case E4 and case E6L are listed in Table IV. Taking the
typical angular positions in the mean and rms pressure dis-
tributions into consideration, the following relations can be
easily found: θmin1 < θs1 < θ max1 < θrand θmin2 < θs2 < θmax2 .
One can interpret the first relation simply as that the separation
bubble is exactly in the region with a strongly adverse pressure
FIG. 9. Distributions of time-averaged wall shear stress around the cylinder;
the clockwise direction is positive.
gradient, and σpreaches a maximum in the middle of the
separation bubble, implying oscillation of the separation
bubble in time.
The turbulent statistics of the wake, shear layer, and
attached boundary layer are explored for both subcritical and
supercritical Re in Fig. 10. Figure 10(a) shows the field of
normal Reynolds stress. At subcritical Re, the velocity fluc-
tuation next to the side walls is even larger than that in the
near-wake region; this results from the large flapping degree
of the separated shear layers. In contrast, at supercritical Re,
the fluctuation in velocity is eliminated by the attached bound-
ary layers, and the peaks are located in the near-wake because
of the periodic alternation of the low-speed recirculation flow
and high-speed potential flow entrained by the Karman vor-
tex. Regarding the wake, Fig. 10(b) displays the quantitative
profiles of the rms velocity σuand σvalong the wake center-
line, where the peak value of case E6L is significantly reduced
compared with case E4 and its location is moved upstream.
It indicates that the reduction in the agitation intensity of the
flow structures in the wake is possibly offset by their shorter
distance to the rear face in case E6L. Consequently, the overall
effects on the rear faces are similar in case E4 and case E6L, and
this leads to a comparable σpon the rear faces [see Fig. 6(b)].
Regarding the shear layer or boundary layer at subcritical and
supercritical Re, Fig. 10(c) shows the profiles of σualong
the vertical direction at the stations of x/D=±0.33, ±0.30,
±0.25, ±0.20, ±0.15, ±0.10, ±0.05, and 0, where the vertical
axis is a logarithmic scale in order to display detailed infor-
mation adjacent to the wall. Note that x/D= 0.33 is located
on the flat part of the side face with θ= 56.6 but very close to
the ends of the rounded corners. As expected, the subcritical
case E4 has a larger σuand the heights of the peaks gradually
increase in the xdirection along the curvature of the separated
shear layer. But for case E6L, the peaks of σuare generally
very weak and their heights are very low. This indicates that
the boundary layer is very thin. Surprisingly, σu0.15 at
x= 0.33 is substantial, which possibly indicates oscillation
of the separation bubble near the frontal rounded corner. The
relatively high σuis consistent with the high σpnear the frontal
corners [see Fig. 6(b)].
085110-10 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
FIG. 10. (a) Contours of span-averaged fluctuating streamwise velocity for case E4 and case E6L; (b) profiles of σuand σvalong the wake centerline;
(c) profiles of σuversus the vertical direction in the shear-layer region, where the rule for measuring the magnitude of σuis shown at the top-right
corner.
D. Instantaneous flow field
In this section, the instantaneous flow field is visualized in
order to describe the dynamic behavior of the wake, detached
shear layer, and reattached boundary layer at subcritical and
supercritical Re.
Figure 11 shows the instantaneous flow field for case E4
and case E6L within the same region of x= [ 1, 10] and
y= [ 2, 2], which is represented by the streamwise veloc-
ity (U), magnitude of velocity (magU), and spanwise vorticity
component (Vort Z). They are chosen at the instants having
local maximum values of total lift. It is apparent that periodic
FIG. 11. Instantaneous flow field represented by the streamwise velocity, magnitude of velocity, and spanwise vorticity (from top to bottom) for case E4 (left)
and case E6L (right) at the instants with extreme values of total lift.
085110-11 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
FIG. 12. Close-ups of instantaneous field around the cylinder at the same instants as the counterparts in Fig. 11: (a) case E4 and (b) case E6L; (c) distribution
of instantaneous streamwise velocity component Uon the third mesh layer nearest to the cylinder wall for case E6L.
vortex streets form in the wake at subcritical and supercritical
Re. At this instant, the first primary vortex forms on the upper
side of the wake recirculation region. However, the distance
between any two Karman vortices decreases in going from
case E4 to case E6L, which implies that the vortex shedding
frequency increases in going from case E4 to case E6L. The
width of the vortex streets is reduced by the increase in Re not
only in the near wake but also in the far wake. By comparison,
the Karman vortex at supercritical Re is composed of eddies
on smaller scales than at subcritical Re.
Figure 12 shows close-ups at the same instants shown in
Fig. 11 to get further insight into the dynamics of the shear
layer and boundary layer. Case E4 at Re = o(104) displays
a typical subcritical flow; that is, the flow completely sep-
arates from the frontal rounded corners, and the free shear
layers flap to a large degree on the two sides of the cylin-
der and connect to the primary vortex behind the cylinder.
Regarding the separated shear layers, one can observe the
evolution of the Kelvin-Helmholtz (K-H) instability and tran-
sition to turbulence. Two shear layers flap out of phase with
each other on both sides of the cylinder. In contrast, a distinct
flow image appears at Re = o(106). Overall, the free stream
seems to flow along the shape of cross sections of rounded-
corner square cylinders and finally separates from the leeward
corners. Here, the laminar free stream seems to transit to turbu-
lence near the frontal corner. After that, turbulence boundary
layers (TBLs) with very thin thickness gradually develop on
the side faces. The TBLs then start to separate from the lee-
ward corners. This means that the separated shear layer at
supercritical Re is turbulent, composed of various small-scale
eddies. The small-scale eddies accumulate to form the Karman
vortex behind the cylinder. Furthermore, the detached shear
layers at supercritical Re oscillate at the same frequency as
the vortex shedding. In addition, Fig. 12(c) shows the instanta-
neous existence of a separation bubble near the frontal corners,
indicated by the sign change of Ualong the circumferen-
tial direction on the third mesh layer nearest to the cylinder
wall. It also oscillates periodically with the Karman vortex
shedding.
E. Vortical structures of different scales
and their effects
This section qualitatively and quantitatively examines
the scales of the vortical structures around the cylinder. It
elucidates the interactions between the small-scale turbulent
structures in the attached boundary layer and the large-scale
vortex shedding as well as sheds light on the effects on
spanwise correlations of fluctuating velocity and pressure.
Figure 13 displays the one-dimensional energy spectra
of the fluctuating streamwise velocity for case E4 and case
E6L, where a line with a slope of 5/3 is added for com-
parison. It should be noted that both probe points are at x
= 0, while the vertical positions yin the figures are near
the time-averaged shear layers for case E4 and the attached
boundary layer for case E6L. At Re = o(104), the Strouhal
frequency ( fvs ) caused by the Karman vortex shedding in the
wake and its harmonics are characterized by sharp peaks. The
free shear layers springing from the bluff bodies at such a
high Re relate to the K-H instability. The transition eddies
with a frequency significantly higher than fvs can be measured.
FIG. 13. One-dimensional energy spectra of the streamwise velocity compo-
nent measured above the upper cylinder walls for case E4 and case E6L, where
a line with a slope of 5/3 is added for comparison. The subcritical probe point
is located near the separated shear layer at x= 0, while the supercritical one
corresponds to three vertical positions y= 0.5005, 0.501, and 0.51 at x= 0.
085110-12 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
FIG. 14. Spanwise correlations of fluctuating stream-wise velocity at Re
= 1.0 ×106. Note that the first point to the second one is separated by 10z.
For the rounded-corner square cylinders, the broadband peak
corresponding to the shear-layer instability also occurs at fsl.
Specifically, the frequency ratio of the shear-layer frequency to
the primary vortex is fsl/fvs 20. It is in good agreement with
the generalized power law approximation proposed by Prasad
and Williamson (1997) for circular cylinders ( fsl/fvs = 0.0235
Re0.67 19). In contrast, for supercritical Re, three positions in
the TBL (i.e., y= 0.5005, 0.501, and 0.510) are chosen in order
to plot the energy spectra; they correspond to y+= 14.7, 29.4,
and 294. There is a prominent peak corresponding to primary
vortex shedding. That is, the boundary layer is unceasingly
subjected to unsteadily harmonic oscillation fed back from the
vortex shedding behind the cylinder. Comparing the spectra at
y= 0.5005, 0.5010, and 0.5100 reveals that the energy of the
peak frequency relative to those of the surrounding frequencies
is larger when the probe point is farther from the wall. How-
ever, the frequency of the shear-layer instability cannot be seen
any more. Instead, the slope of 5/3 is observed neatly in the
high-frequency range, which means turbulent motions on a
wide range of small scales in the attached turbulent bound-
ary flow on the side faces. Thus, the velocity quantities in
this TBL could be decomposed into U=hUi+˜
U+u0,
where the angle bracket indicates the time-averaged value,
the tilde indicates the periodic component of vortex shedding,
and the prime indicates the high-frequency turbulent compo-
nents. The high-frequency turbulent components can be seen
in the visualization of the fluctuating stream-wise velocity in
Fig. 19.
Now let us investigate the effects of the different-scale
components on the TBL on the spanwise correlations at dif-
ferent spanwise distances (see Fig. 14). Indeed, the spanwise
distance is scaled by the cylinder size D. The small-scale struc-
tures shown in Fig. 19 result in rapid decay of the spanwise
correlations of fluctuating stream-wise velocity Ru(s) at the
small spanwise distance s. It is worth noting that the first and
second points are separated by 10z= 0.0625 because of the
FIG. 16. Spanwise correlations Rp(s) of fluctuating pressure probed in the
middle of upper and lower cylinder walls, where only half of the spanwise
lengths are shown, considering the symmetry of Rp(s) around the middle
of the spanwise lengths. Note that the first and second points are separated
by 10z.
sampling spatial resolution of time-dependent data, which is
larger than the TBL thickness (δ0.02–0.03). After the rapid
decay, Ru(s) remains non-zero at large salthough the values
are very small. The non-zero Ru(s) is determined by the influ-
ence of the Karman vortex shedding covering the whole span
[see the wake along the span in Fig. 15(a)]. Furthermore, at
large s,Ru(s) tends to be smaller at the points closer to the
wall. This phenomenon is consistent with the lower energy of
the primary vortex shedding at the points closer to the wall.
This is as expected because the impact of the vortex shedding
becomes smaller for the points closer to the wall due to the
increased wall shear effects.
Figure 16 shows the spanwise correlation coefficients
Rp(s) of fluctuating pressure coefficients in the middle of the
upper and lower cylinder walls. Note that only half of the span-
wise lengths are shown, considering the symmetry of Rp(s)
around the middle of the spanwise lengths under the peri-
odic conditions on both spanwise boundaries. There are few
differences between the spanwise correlations based on the
upper and lower walls. First, at subcritical Re, Rp(s) gradually
decreases from unity with s. The time series of the pressure
coefficients versus dimensionless time t*(=tU0/D) shown in
Fig. 17(a) was measured in the middle of the side faces with
a spanwise spacing Ms= 3. Clearly, the variation in pressure
amplitude and the phase lag between the pressures gauged
separating Ms= 3 contribute to the decrease in Rp(s).
Of more interest is the Rp(s) at supercritical Re. It drops
rapidly from unity (note that the probe points have an interval
larger than δ), followed by a smooth reduction tendency with
rising s. Two time series of pressure coefficients with a span-
wise distance of 1 are displayed in Fig. 17(b). Like the velocity
series on the TBL, the total series of pressure is composed of
a time-averaged constant, sine waves, and higher-frequency
components. The pressure field in the wake and on the cylinder
FIG. 15. Instantaneous distributions on
the three surfaces, i.e., the cylinder sur-
face, the mid-span plane, and the mid-
height plane: (a) the primary vortex
shedding covering the whole span; (b)
the relation between pressure field in the
wake and on the cylinder.
085110-13 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
FIG. 17. Time series of pressure coef-
ficients measured in the middle of side
walls for case E4 (left) and case E6L
(right).
is visualized in Fig. 15(b) for the low-frequency sine wave. The
low-frequency pressure oscillation is over the cylinder span,
together with the vortex shedding in the wake. Examination of
the power spectra shows that its frequency is consistent with
the vortex shedding frequency. Generally speaking, the wall
pressure fluctuations in the large scales can originate from the
potential field and the outer layers of the TBL (Choi and Moin,
1990 and Farabee and Casarella,1991). Thus, the reason that
the spanwise correlations of fluctuating pressure did not die out
even at large distances is the feedback from the nearly parallel
Karman vortex shedding, which spans the whole domain. But
at small distances scomparable to the TBL thickness (<<D), a
rapid decrease in spanwise correlations away from unity occurs
at both Rp(s) and Ru(s); it is produced by the superimposed
high-frequency vortical structures in the TBL.
Let us briefly summarize the interaction between the
Karman vortices and the TBLs on the side faces. The flows
attached to the side faces are influenced by the Karman vor-
tices and random turbulent components simultaneously. In
other words, the attached flows are the TBLs under the pres-
sure gradients that vary periodically with the primary vortex
shedding behind the cylinder. However, the quasi-coherent
structures inherent in the TBLs are so small-scale compared
with the Karman vortices that we could regard the TBLs as
two-dimensional thin flow layers. On the scale of the cylin-
der width, the thin flow layer (as a whole) oscillates in time
at the same pace as the alternate vortex shedding, together
with the separation bubbles near the frontal corners. This is
the feedback from the Karman vortex shedding into the TBLs
on the side faces. Conversely, small-scale turbulent structures
developed in the TBLs separate, accumulate, and comprise the
large-scale Karman vortices.
F. Discussion on TBL on side faces
The TBL attached on the side faces might be different
from the standard fully developed TBLs on a flat plate. The
initial profile in the front of the side face exhibits a convexity
unlike that of a standard laminar or turbulent boundary layer
on a flat plate because of the high acceleration around the
rounded corners. Following that, the boundary layer is sequen-
tially influenced by several unsteady external effects including
an adverse pressure gradient (APG), a zero pressure gradient
(ZPG), and a favorable pressure gradient (FPG) in the stream-
wise direction. There is also an APG of a certain degree along
the vertical direction y.
Now let us focus on the effect of the mean pressure gra-
dients, most notably, in the streamwise direction. Figure 18(a)
shows the distributions of time-averaged pressure coefficients
and their gradients ( dhpi
dx ) along x. The region from upstream to
downstream is divided into APG, ZPG, and FPG. The nearly
ZPG regime is bordered by x= 0.1 and 0.2. Figure 18(b)
shows the successive variation of hUiprofiles using outer law
variables. First, let us discuss the profiles in the APG region.
The velocity profile at x= 0.33 immediately downstream
from the corner has a flow reversal in the vicinity of the wall
(small separation bubble) and an obvious convexity when the
distance to the wall is about 0.005 due to the greater upstream
acceleration in this local vertical region [Fig. 12(b)]. But the
convexity progressively becomes smoother as xincreases,
which is a characteristic of APG. In the APG region, the thick-
ness of the boundary layer increases quickly downstream (see
also the flow field). Second, the profiles in the ZPG region
are almost unchanged, which means the TBL becomes fully
developed there and independent of x. However, when mea-
sured using the inner law variables y+and u+, the profiles in the
ZPG region appear laminar-like, i.e., higher than the standard
log-law profile. One of the reasons is possibly the accelera-
tion phase of the superimposed harmonic oscillation cycles,
as observed in the studies by Spalart and Baldwin (1987)
and Akhavan et al. (1991). Moreover, the vertical APG with
increasing ytends to stabilize the turbulent boundary layer
in a similar manner to stable thermal stratification. The flow
FIG. 18. (a) Distributions of time-
averaged pressure coefficients and their
gradients along x, divided by the regions
of APG, ZPG, and FPG. (b) Profiles of
the time-averaged streamwise velocity
using outer law variables. Red solid line:
x= 0.1; blue solid line: x= 0.2.
085110-14 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
FIG. 19. (a) Instantaneous stream-wise velocity field,
where the locations of the velocity profiles are marked.
(b) Distribution of U-hUiat y= 0.501, where the mean
velocity hUiis selected on the basis of the x= 0.05 profile.
acceleration in the FPG region helps the profiles to steepen
and gradually rebuild the convexity when the distance to the
wall is less than 0.005.
In the flow field of instantaneous spanwise vorticity
[Fig. 19(a)], FPG seems to affect the vortical structures in the
TBL. Especially near the leeward corner, there are fewer and
weaker vorticity structures in the outer region. Figure 19(b)
shows the fluctuating component of U-hUion the y= 0.501
plane, where hUiis selected on the basis of the x= 0.05 pro-
file. The turbulent structures in this FPG region are more
ordered, elongated, and aligned in the streamwise direction
upstream of the leeward corners. Such features in acceler-
ating boundary layers were systematically investigated by
Piomelli et al. (2000).
V. CONCLUSIONS
LES was used to investigate the aerodynamics of a square
cylinder with rounded corners at supercritical Re compared
with those at subcritical Re. First, the numerical methods,
especially SGS models, were validated as to their prediction
accuracy of supercritical flows past a circular cylinder. The
results show that the dynamic mixed model can accurately
predict the process of laminar separation, transition to turbu-
lence, flow reattachment, and turbulent separation. After that,
the flow past a rounded-corner square cylinder was simulated
and systematically clarified. The major findings are as follows:
(1) Significant variations of aerodynamic forces and pres-
sure were found as Re increased from o(104) to o(106).
Specifically, the time-averaged drag and St undergo
two distinct quantitative levels in the subcritical and
supercritical regimes. Local pressures change not only
in the shapes of their distribution but also in their
quantitative levels. In particular, extreme maximum
or minimum pressures appear at the frontal and rear
corners.
(2) The changeover of the mean flow patterns brought about
by the increase from the subcritical to supercritical
Re was clarified. Unlike the complete separation from
the frontal corners at subcritical Re, the free stream
overall flows along the cross sections of cylinder, sepa-
rates from the leeward corners, and generates the twin
vortices behind the cylinder at supercritical Re. At a
microlevel, the flow experiences laminar separation and
flow reattachment near the frontal corner, resulting in
a small separation bubble. This type of flow has a
much smaller recirculation region behind the cylinder
and a low turbulent kinetic energy. The flow patterns
explained the characteristics of the forces and local
pressures.
(3) The dynamics of the wake and separated shear layer
and boundary layer were elucidated. Even at super-
critical Re, periodic vortex shedding occurs but with
a shorter distance between neighboring vortices. The
laminar flow starts to transit to turbulence near the
frontal corner, followed by the spatial development of
small-scale eddies in the TBL on the side face and tur-
bulent separation near the leeward corner. The detached
TBL oscillates at the same frequency as the vortex
shedding, and the small eddies accumulate to form the
primary vortex behind the cylinder.
(4) Different-scale vortical structures were qualitatively
and quantitatively examined. At subcritical Re, typi-
cal frequencies were measured in the shear layer: the
primary vortex shedding frequency and its harmonics
and the K-H instability frequency. At supercritical Re,
besides the turbulent motions in the high-frequency
region with 5/3 slope, there is feedback of the pri-
mary vortex shedding covering the whole span in the
TBL. Moreover, there is a large-scale gap between the
turbulent structures in the TBL and the Karman vortex.
The small-scale structures in the TBL are responsible
085110-15 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
TABLE V. Comparison of global quantities at different spanwise lengths of the computational domain at Re = 1.0
×106.
Case Grid LzNcell z y+max y+avg ¯
CDC0
D¯
CLC0
LSt Lf
Case E6 1201 ×500 ×161 1 96.7 ×1060.006 25 2.71 1.07 0.58 0.11 0.0155 0.41 0.253 1.06
Case E6L 1201 ×500 ×641 4 384.9 ×1060.006 25 2.71 1.06 0.55 0.04 0.0100 0.30 0.280 1.07
FIG. 20. Distributions of time-averaged and fluctuating pressures when the spanwise length of the computational domain is Lz= 1 and 4 at Re = 1.0 ×106.
for the rapid drop in the spanwise correlations of the
fluctuating velocity and wall pressure from unity at
very small spanwise distances relative to the cylinder
size. The large-scale vortex shedding leads to span-
wise correlations being non-zero at large distances.
Moreover, the effect of vortex shedding on the TBL
changes depending on the normal distance from the
wall.
(5) The TBL on the side face was found to be different
from the standard zero-pressure-gradient flat-plate
TBL, which is sequentially influenced by several
unsteady external effects, including APG, ZPG, and
FPG in the streamwise direction. The convexity in the
initial velocity profile is progressively smoothened in
the APG region and as the boundary layer thickness
increases. The profiles remain unchanged and appear
laminar-like in the ZPG region. The FPG immediately
upstream the leeward corner rebuilds the convexity of
the profiles, and it elongates and aligns the TBL vortical
structures in the streamwise direction.
This paper reported for the first time a big difference
between the subcritical and supercritical flows around the same
geometry, i.e., a square cylinder with rounded corners. Some
issues remain to be studied in the future: the variation from sub-
critical to supercritical flows around a rounded-corner square
cylinder; the similarities and differences in the transition from
the subcritical to supercritical regime between the case of the
rounded-corner square cylinder and the case of the canonical
circular cylinder.
ACKNOWLEDGMENTS
The computations were conducted on the Earth Simulator
at the Japan Agency for Marine-Earth Science and Technol-
ogy (JAMSTEC). Authors are also grateful to the support of
oversea doctoral study from the China Scholarship Council
(CSC).
APPENDIX: EXAMINATION OF THE SPANWISE
LENGTH OF THE COMPUTATIONAL DOMAIN
AT SUPERCRITICAL RE
Here, let us examine the effects of the spanwise length
of the computational domain for a supercritical-Re flow past a
rounded-corner square cylinder. On the basis of the simulation
results of supercritical flows past a circular cylinder, this study
tests two commonly accepted spanwise lengths, i.e., Lz= 1 and
4. The mesh information is listed in Table V, together with typi-
cal physical quantities. The extension of the spanwise length Lz
from 1 to 4 (i.e., from case E6 to case E6L) slightly influences
the time-averaged drag and lift coefficients and the formation
length within a deviation of about 5%. However, it somewhat
affects the fluctuation-related quantities, i.e., the rms forces
and St. In Fig. 20, the hpidistributions are very similar for the
two spanwise lengths Lz= 1 and 4; this behavior is similar to
that of the mean drag. Even though the shorter Lztends to over-
estimate the fluctuating quantities on the side and rear faces, the
shapes of the rms pressure distributions at supercritical Re are
FIG. 21. One-dimensional energy spectra of the stream-wise velocity fluc-
tuation in the wake (e.g., x= 1.5, y= 0), near the separated shear layer
(e.g., x= 0.71, y= 0.42), and the attached turbulent boundary layer
(e.g., x= 0, y= 0.501). The spectra are shifted along the Euu axis for
clarity.
085110-16 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017)
FIG. 22. Two-point correlation of the spanwise velocity fluctuation in the wake (a), the recirculation region behind the cylinder (b), and the attached turbulent
boundary layer (c). Note that the first and second points are separated by 10z.
generally similar when the spanwise lengths Lzequal 1 and
4. Thus, a small quantitative difference in the rms pressure
is expected when extending the spanwise length. However,
such a small difference should not influence the flow physics.
Moreover, the spanwise resolution zis carefully treated to
be much smaller than those adopted widely for subcritical-
Re flows. The value of zlearns the experiences from the
well-validated LESs of supercritical flows around a circu-
lar cylinder (Ono and Tamura,2008;Lehmkuhl et al.,2014;
and Rodr´
ıguez et al.,2015) and was selected to be smaller
than the above references while the present study follows the
numerical methods of Ono and Tamura (2008) and also has a
negligible amount of numerical dissipation. It was also shown
to be sufficient in Sec. III. Thus, it is reasonable from the
computational point of view to choose Lz= 4 and maintain
the high spanwise resolution to capture the small-scale flow
structures.
On the basis of the supercritical flows around a rounded-
corner square cylinder, the sufficiency of the spanwise res-
olution is further examined. Two ways are used to evaluate
the grid resolution for LES. First, the one-dimensional energy
spectra are computed and plotted in Fig. 21. Three typical
positions are selected: first one is located in the wake (beyond
the mean recirculation zone behind the cylinder); second one
is near the separated shear layer; and the third one is in the
attached turbulent boundary layer. The peaks in Fig. 21 are
reasonably associated with the frequency of the Karman vor-
tex shedding ( fvs) and its harmonic. More notably, just before
the dissipation ranges, the large-extent inertial sub-ranges fea-
tured by a 5/3 slope are predicted accurately for all three
positions. That is, the small-scale turbulent motions are suffi-
ciently reproduced in different flow regions of interest around
the cylinder. Second, the spanwise correlations of fluctuat-
ing velocity provide another measure of spanwise resolutions
as shown in Fig. 22. When the correlations reach zero, one
can judge how many cells cover the largest scale turbulent
motion. As recommended by Davidson (2009), at least 8 cells
should be sufficient for LES. In Fig. 22, the largest scales are
on average covered by about 8, 70, and 100 cells in the attached
boundary layer, near the separated shear layer, and in the wake,
respectively. To summarize, the present spanwise resolution is
believed to be sufficient after the evaluation by the two com-
mon approaches based both on energy spectra and two-point
correlations.
As shown in Fig. 16, the spanwise correlations of the wall
pressure fluctuation are not weak even at the large spanwise
distances of supercritical Re. The reason may be the lack of
a large phase variation in vortex shedding along the span
(e.g., oblique shedding), as shown in the previous studies
(Szepessy and Bearman,1992;Szepessy,1994; and Cao
and Tamura,2015). Nevertheless, out-of-phase vortex shed-
ding occurs intermittently and infrequently compared with
parallel shedding. Thus, there should be no change in the
main flow physics if the domain length is further elongated
from Lz= 4.
This study focused on the three-dimensional vortical
structures in the flow when parallel shedding dominates [see
Fig. 15(a)] (e.g., the various-scale eddies comprising the Kar-
man vortex in the wake, the longitudinal vortices connecting
the Karman spanwise vortices, and the small-scale turbulent
structures in the boundary layer). The spanwise correlations of
the spanwise velocity components are plotted in Fig. 22, which
were measured in the wake, the recirculation region behind the
cylinder, and the attached turbulent boundary layer. They fall
approximately to zero for the distances shorter than half the
total spanwise length. It illustrates that the development of
these three-dimensional vortical structures and their interac-
tion are independent of the spanwise length. To summarize,
case E6L with Lz= 4 is believed to be reasonably sufficient
to discuss the supercritical flow around the rounded-corner
square cylinders.
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... As shown in the previous numerical simulation performed by Tamura et al. (1998), a rounded corner applied to a square cylinder affected the wake separation and reduced the drag and lift coefficients. Subsequently, that the rounded corner of square cylinders can suppress the wake separation and influence the vortex shedding frequency has been repeatedly observed in experiments and simulations (e.g., Hu et al., 2006;Park and Yang, 2016;Zhang and Samtaney, 2016;Cao and Tamura, 2017). Moreover, an application rounding the roof's trailing edge to suppress the wake separation was employed for a hatchback Ahmed body by Thacker et al. (2012). ...
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The wake bi-stability behind notchback Ahmed bodies is investigated by performing wind tunnel experiments and large eddy simulations (LESs). The focus of this study is on the suppression of bi-stable wakes achieved by rounding the roof's trailing edge of the body. The suppression effect is found to depend on the Reynolds number (Re). The wake behind a sharp edge remains bi-stable for all tested Re. However, for a rounded edge with small radius, wake bi-stability at Re=0.5×10^5 and wake symmetrization with 0.75×105≤Re≤1.5×10^5 are observed. Increasing Re with Re≥1.75×10^5, the wake returns to the bi-stable state. Particularly, with Re≥2×10^5, a stable asymmetric wake state with no switches is observed for long periods. Performing LES confirms the expected wake asymmetry at Re=0.5×10^5 and symmetry at Re=1×10^5 for the case of rounded edge with a small radius. Besides, another wake symmetry is observed for the rounded edge with a large radius at Re=0.5×10^5. For the two wake symmetries shown in the LES results, the symmetrization is attributed to wake suppression in the notchback region, forcing the flow separation from the rear roof to attach to the slant on both sides of the body.
... For rounded corners, despite the number of studies assessing the large-Reynolds number case (see for e.g. Lamballais, Silvestrini & Laizet 2008Cao & Tamura 2017), few works have considered the dependence of the onset of the instabilities on the corner curvature. Park & Yang (2016) determined, via linear stability analysis, how the primary two-dimensional and the three-dimensional instabilities are affected by rounding the four corners of a rectangular cylinder with AR = 1, exploring the shapes ranging between the square cylinder with sharp edges and the circular cylinder. ...
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This paper aims to provide a comprehensive review of the literature and to impart a perspective on the current status and future directions for fundamentals of flow around and heat transfer from a square cylinder with modified corners. The focus is on how rounded, chamfered, recessed, and cut corners of a square cylinder impact the wake structure, flow instability, fluid forces, vortex dynamics, wake width, vortex formation length, boundary layer thickness, displacement thickness, transition to turbulence, drag crisis, and heat transfer. A circular cylinder undergoes drag crisis flow while a square cylinder does not. Sufficiently rounded corners can rejuvenate the drag crisis flow. The corner modifications encompassing heat transfer performance from low to high can be sorted out as sharp, cut, rounded, and chamfered, given the same scale of modifications. Based on the review and analysis of data in the literature, the optimal corner modifications are suggested for reduced forces and enhanced heat transfer. The research gap in the literature is identified and put forward for future investigations.
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The flow past a square cylinder at Re = 2.2 × 104 is analyzed by large-eddy simulation (LES) using the fine grids in order to represent details of near-cylinder flows. The accuracy of LES on structured and unstructured grids is assessed from the engineering viewpoint, compared with previous studies. The finite differencing method code with 4th order central scheme for the convective term is used for structured LES, while the open-source finite volume method code (OpenFOAM 2.3.0) with 1st–2nd order schemes is applied for unstructured LES. Typical schemes in OpenFOAM are tested, i.e., “LUST” (blending of linear and linear upwind schemes), “limitedLinear” (TVD) and “linearUpwind” (linear upwind). In this study, three effects are emphasized: numerical schemes for convective terms, meshing strategies, and spanwise resolution and length. On the whole, OpenFOAM can obtain fairly accurate prediction of the time-averaged and r.m.s quantities no matter which numerical scheme is used. “LUST” and “linearUpwind” are suggested. Meshing refinement in wake can be a solution to improve the far-wake velocity distribution and to overcome the earlier energy decay of turbulent motions in the inertial subrange caused by artificial dissipation. Different degrees of instantaneous flow reattachment near the trailing edges are found in the results obtained by OpenFOAM. Flow reattachment is closely associated with the roll-up of shear layer, and the flow topology between the shear layer and the side wall featured by the frontal and leeward secondary vortex. In this regard, the refinement of hexahedra cells near cylinder gives a best solution among all present cases tested compared with the previous DNS result. Generally the Kelvin–Helmholtz instability can be accurately predicted by OpenFOAM under the present numerical conditions. In addition, the spanwise resolution seems no big effect in predicted results when less than 0.05D. By contrast, the increase in spanwise length to 14D from 4D plays an important role in obtaining reasonable spanwise correlation of pressure and consequently the overall fluctuating lifts.
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Direct numerical simulation is performed for flow past an isolated cylinder at . The corners of the cylinder are rounded at different radii, with the non-dimensional radius of curvature varying from (square cylinder with sharp corners) to 0.500 (circular cylinder), in which R is the corner radius and D is the cylinder diameter. Our objective is to investigate the effect of the rounded corners on the development of the separated and transitional flow past the cylinder in terms of time-averaged statistics, time-dependent behavior, turbulent statistics and three-dimensional flow patterns. Numerical results reveal that the rounding of the corners significantly reduces the time-averaged drag and the force fluctuations. The wake flow downstream of the square cylinder recovers the slowest and has the largest wake width. However, the statistical quantities do not monotonically vary with the corner radius, but exhibit drastic variations between the cases of square cylinder and partially rounded cylinders, and between the latter and the circular cylinder. The free shear layer separated from the cylinder is the most stable in which the first roll up of the wake vortex occurs furthest from the cylinder and results in the largest recirculation bubble, whose size reduces as further increases. The coherent and incoherent Reynolds stresses are most pronounced in the near-wake close to the reattachment point, while also being noticable in the shear layer for the square and cylinders. The wake vortices translate in the streamwise direction with a convection velocity that is almost constant at approximately 80% of the incoming flow velocity. These vortices exhibit nearly the same trajectory for the rounded cylinders and are furthest away from the wake centerline for the square one. The flow past the square cylinder is strongly three-dimensional as indicated by the significant primary and secondary enstrophy, while it is dominated by the primary enstrophy ( ) for the rounded cylinders.
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Instabilities in the flow past a rounded square cylinder have been numerically studied in order to clarify the effects of rounding the sharp edges of a square cylinder on the primary and secondary instabilities associated with the flow. Rounding the edges was done by inscribing a quarter circle of radius $r$ in each edge of a square cylinder of height $d$ . Nine cases of rounding were considered, ranging from a square cylinder ( $r/d=0$ ) to a circular cylinder ( $r/d=0.5$ ) with an increment of 0.0625. Each cross-section was numerically implemented in a Cartesian grid system by using an immersed boundary method. The key parameters are the Reynolds number ( Re ) and the edge-radius ratio ( $r/d$ ). For low Re , the flow is steady and symmetric with respect to the centreline. Over the first critical Reynolds number ( $Re_{c1}$ ), the flow undergoes a Hopf bifurcation to a time-periodic flow, termed the primary instability. As Re further increases, the onset of the secondary instability of three-dimensional (3D) nature is detected beyond the second critical Reynolds number ( $Re_{c2}$ ). Rounding the sharp edges of a square cylinder significantly affects the flow topology, leading to noticeable changes in both instabilities. By employing the Stuart–Landau equation, we investigated the criticality of the primary instability depending upon $r/d$ . The onset of the 3D secondary instability was detected by using Floquet stability analysis. The temporal and spatial characteristics of the dominant modes (A, B, QP) were described. The neutral stability curves of each mode were computed depending upon $r/d$ .
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Large-eddy simulation of turbulent flow past a circular cylinder at sub- to super-critical Reynolds numbers is performed using a high-fidelity orthogonal curvilinear grid solver. Verification studies investigate the effects of grid resolution, aspect ratio and convection scheme. Monotonic convergence is achieved in grid convergence studies. Validation studies use all available experimental benchmark data. Although the grids are relatively large and fine enough for sufficiently resolved turbulence near the cylinder, the grid uncertainties are large indicating the need for even finer grids. Large aspect ratio is required for sub-critical Reynolds number cases, whereas small aspect ratio is sufficient for critical and super-critical Reynolds number cases. All the experimental trends were predicted with reasonable accuracy, in consideration the large facility bias, age of most of the data, and differences between experimental and computational setup in particular free stream turbulence and roughness. The largest errors were for under prediction of turbulence separation.
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Large eddy simulations of the flow around a circular cylinder at high Reynolds numbers are reported. Five Reynolds numbers were chosen, such that the drag crisis was captured. A total of 18 cases were computed to investigate the effect of gridding strategy, domain width, turbulence modelling and numerical schemes on the results. It was found that unstructured grids provide better resolution of key flow features, when a ‘reasonable’ grid size is to be maintained. When using coarse grids for large eddy simuation, the effect of the turbulence models and numerical schemes becomes more pronounced. The dynamic mixed Smagorinsky model was found to be superior to the Smagorinsky model, since the model coefficient is allowed to dynamically adjust based on the local flow and grid size. A blended upwind-central convection scheme was also found to provide the best accuracy, since a fully central scheme exhibits artificial wiggles which pollute the entire solution. Mean drag, fluctuating lift and Strouhal number are compared to experiments and empirical estimates for Reynolds numbers ranging from 6.31 × 104 − 5.06 × 105. In terms of the drag coefficient, the drag crisis is well captured by the present simulations, although the other integral quantities (rms lift and Strouhal number) less so. For the lowest Reynolds number, the drag is seen to be most sensitive to the domain width, while at the higher Reynolds numbers the grid resolution plays a more important role.
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In contrast with a wide range of applications concerning flows around a circular cylinder at upper subcritical Reynolds numbers (Re), there is no systematic understanding about the fundamentals of so-called random flow patterns, and their effects on intermittent modulations in the time history of pressure or force, and the decrease in their spanwise correlations. This paper employed the large-eddy simulation (LES) technique to predict flows past a circular cylinder at Re=1.3×105 and to provide images based on flow visualization that can clarify the physical mechanism responsible for these outcomes. A reasonably sufficient spanwise length was adopted for the numerical model by taking into consideration the effect of aspect ratios (the spanwise length to the diameter). We found that even at such high Res, a three-dimensional pattern of vortical field is present in the wake resulting in total force modulation and weak spanwise correlation, e.g., obvious oblique shedding. The whole development process of the three-dimensional wake is exhibited as a universal. The results revealed that local phase variations in primary vortex shedding are the starting points of three-dimensional wake patterns, which are induced by the "irregular" streamwise vortex. The three-dimensional near wake following local phase variations is associated with a successive evolution composed of certain stages in order. Quantitative analyses based on the time series of sectional lift coefficients show that intermittent increase in primary shedding periods and sectional lift streak divisions are closely related to local phase variations and vortex division in the development process of the three-dimensional pattern. In addition to the phase difference along the span, the three-dimensional pattern also weakens vortex shedding in cross sections perpendicular to the axis of the cylinder, resulting in modulation of the sectional lift coefficient.
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Large-eddy simulations (LES) of the flow past a circular cylinder are used to investigate the flow topology and the vortex shedding process at Reynolds numbers $Re=2.5 \times10^5- 8.5\times10^5$. This range encompasses both the critical and super-critical regimes. As the flow enters the critical regime, major changes occur which affect the flow configuration. Asymmetries in the flow are found in the critical regime, whereas the wake recovers its symmetry and stabilizes in the super-critical regime. Wake characteristic lengths are measured and compared between the different Reynolds numbers. It is shown that the super-critical regime is characterised by a plateau in the drag coefficient at about C_D=0.22, and a quasi-stable wake which has a non-dimensional width of d_w/D=0.4. The periodic nature of the flow is analysed by means of measurements of the unsteady drag and lift coefficients. Power spectra of the lift fluctuations are computed. Wake vortex shedding is found to occur for both regimes investigated, although a jump in frequencies is observed when the flow enters the super-critical regime. In this regime, non-dimensional vortex-shedding frequency is almost constant and equal to St=f_{vs} D/U_{ref}=0.44. The analysis also shows a steep decrease in the fluctuating lift when entering the super-critical regime. The combined analysis of both wake topology and vortex shedding complements the physical picture of a stable and highly coherent flow in the super-critical regime.