The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
Experimental investigation on the aerodynamic behavior of
square cylinders with rounded corners
Luigi Carassale a, Andrea Freda a, Michela Marrè Brunenghi a,
Giuseppe Piccardo a, Giovanni Solari a
a
Dept. of Civil Environmental and Geotechnical Engineering, University of Genova, Italy
ABSTRACT: The influence of corner shaping on the aerodynamic behavior of square cylinders
is studied through wind tunnel tests. Beside the sharp-edge corner condition considered as a
benchmark, two different rounded-corner radii (r/b = 1/15 and 2/15) are studied. Global forces
and surface pressure are simultaneously measured in the Reynolds number range between 2.5˜104
and 1.8˜105. The measurements are extended to angles of incidence between 0 and 45°, but the
analysis and the discussion presented herein is focused on the low angle of incidence range. It is
found that the critical angle of incidence where the flow reattaches on the lateral face exposed to
the flow decreases as r/b increases and that an intermittent flow condition exists. In the case of
turbulent incoming flow, two different aerodynamic regimes governed by the Reynolds number
have been recognized.
KEYWORDS: Square cylinder, rounded corners, intermittence, Reynolds number effect
1 INTRODUCTION
A careful modeling of the corners shape of building and structural elements exposed to the wind
has become a major objective for a wind-response-oriented optimal design. The introduction of
rounded or chamfered corners has often the positive effect of reducing the drag force and the
fluctuation of the transversal force due to vortex shedding [e.g. 1, 2], but can produce a relatively
complicated aerodynamic behavior whose physical or numerical modeling may be challenging.
In contrast to sharp-edge bodies, the absence of fixed separation points can introduce significant
dependencies on the Reynolds number and on the characteristics of the incoming flow that must
be taken into account during the design stage [e.g. 2 ± 4].
The simplest and probably the most commonly investigate sharp-edge body is the square cylinder, thus it is the natural candidate to investigate the effect of corner shaping on the aerodynamic behavior of bluff bodies. With the twofold purpose of investigating the basic behavior of
rounded corners and providing technical information useful for wind engineers, a series of windtunnel tests on square cylinders with rounded corners has been carried out. Rigid models of cylinders with two corner radii (r/b = 1/15 and 2/15), beside the sharp-corner case, have been realized and tested measuring the global forces and the pressure field along a cross-section. The
considered Reynolds number range is between 2.5˜104 and 1.8˜105. Two levels of turbulence intensity (0.2% and 6%) in the incoming flow have been considered.
Section 2 provides a brief review of the current knowledge on the aerodynamic behavior of
sharp-edge square cylinders, with particular reference to the qualitative description of the flow
field for different angles of incidence. These concepts are then used as a guide for the interpretation of the experimental results presented in Section 3 and discussed in Section 4. Due to the
space limitation, the discussion is focused on two specific phenomena that have been observed:
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1) the intermittence of the vortex-shedding regime at the critical angle of incidence in smooth
flow; 2) the inversion of the lift coefficient slope at zero angle of incidence in turbulent flow.
2 AERODYNAMIC BEHAVIOUR OF SHARP-CORNER SQUARE CYLINDERS
Sharp-edge square cylinders have been studied by several researchers who focused their attention
on numerous aerodynamic parameters including pressure distribution, drag and lift forces, vortex
shedding properties, as well as near wake velocity field [e.g. 5 ± 11]. The flow pattern around a
square cylinder is strongly dependent on the angle of incidence, D, and at least two characteristic
flow regimes are clearly identified and separated by a critical angle of incidence, Dcr located
about 12-15° [8]. For D < Dcr the boundary layer on the lateral faces is completely separated,
while for D > Dcr the flow reattaches on the lateral face exposed to the wind forming a separation
bubble [e.g. 10, 11]. A further sub-classification of the two mentioned regimes is possible [8],
but is not relevant for the purpose of the present study. The subcritical regime (D < Dcr) is characterized by negative slope of the lift coefficient, which changes sharply to positive as D becomes
grater than Dcr. Besides, the transition from the subcritical to the supercritical regime produces a
rapid increment of the Strouhal number that corresponds to the reduction of the wake width due
to the flow reattachment [6]. In the critical regime both the drag coefficient and the fluctuating
lift coefficient have a minimum value. This scenario is not strongly modified by the Reynolds
number and the characteristics of the incoming flow, even if some significant influence does exist. The thickening of the shear layers due to a small-scale free-stream turbulence promotes the
formation of the separation bubble which tends to appear for smaller angle of incidence and to
shrink towards the leading edge [6]. This mechanism is probably influenced by the Reynolds
number as suggested in [4] for a the case of a 2:1 rectangular cylinder, however no effect has
been revealed on square cylinders [9].
3 EXPERIMENTAL RESULTS
The experimental tests are carried out in the closed-circuit wind tunnel at the University of Genova with cross section 1700 u 1350 mm. The models are realized through the assemblage of
aluminum plates, have a span length l = 500 mm and are mounted in cross-flow configuration on
a force balance realized by six resistive load cells. End plates are installed at the extremities of
the models to maintain a two-dimensional flow condition and separate the model from the boundary layer on the wind-tunnel walls. The mid-span cross section of the models is instrumented by
a ring of N pressure taps (N ranges from 20 to 44 for the different tested models) connected
through short tubes to 32-channel PSI pressure scanners mounted inside the model. Figure 1
shows the cross section of the tested cylinders and the reference system used for the presentation
of the results. Beside the sharp-edge square cross section used as a benchmark test, two roundedcorner configurations with r/b = 1/15 and 2/15 (r and b being, respectively, the corner radius and
the square size, Fig. 1) have been considered.
The force balance measurements are analyzed calculating the steady aerodynamic drag and lift
coefficients (CD and CL) the Strouhal number (St) and the fluctuating lift coefficient (CLrms) defined as:
CD
E > D@
0.5UblU 2
CL
E > L@
0.5UblU 2
St
ns b
U
940
CLrms
2
E ª L E > L @ º
¬
¼
0.5UblU 2
0.5
(1)
The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
where D and L are, respectively, the measured drag and lift forces (Fig. 1), U the air density, U
the undisturbed mean wind velocity; E[x] is the statistic average operator that is implemented as
a time average adopting the hypothesis of ergodic behavior; ns is the vortex shedding peak frequency estimated by fitting the Power Spectral Density (PSD) function of L through a Gaussian
function in the neighborhood of its peak. Analogously, steady and fluctuating pressure coefficients are defined as
Cp
E > p p0 @
0.5UU
2
C prms
2
E ª p E > p @ º
¬
¼
2
0.5UU
0.5
(2)
where p is the pressure measured on the body surface and p0 is the static wind-tunnel pressure.
The influence of the Reynolds number (Re) is explored both varying the wind-tunnel velocity
U in the range 5 ± 25 m/s, as well as the body size b in the range 60 ± 150 mm. No blockage correction has been adopted since the focus of the paper is the qualitative description of the behavior, which is not affected by slight modifications of the estimated numerical values.
Two conditions for the incoming flow have been considered: 1) smooth flow condition characterized by a turbulence intensity about 0.2%; 2) turbulent flow condition, produced through a
grid realized by square bars, characterized by the intensity of the longitudinal turbulence Iu about
5% and integral length scale Lu about 20 mm. These parameters are substantially stable through
the whole considered wind velocity range (Fig. 2).
b
r/b=0
b
r/b=1/15
b
r/b=2/15
b
Figure 1. Experimental setup
Figure 3 shows the variation of CD (a), CL (b), St (c) and CLrms (d) with respect to the angle of
incidence for the three considered models in smooth flow. No significant influence of the Re has
been observed in the range between 2.69˜104 and 1.65˜105. It can be observed that the three models have a similar qualitative behavior characterized by the inversion of the lift slope for a critical
angle of incidence, Dcr, that decreases as r/b increases. For D = Dcr the steady drag and the fluctuating lift coefficients have a minimum value, while the Strouhal number increases sharply. For
the sharp-corner model Dcr was found about 12° and is in substantial accord to previous experimentations (see [11] for a review of previous results). In the case of rounded-corner models it
was found Dcr = 7° and 5° for r/b = 1/15 and 2/15, respectively. Tamura & Miyagi [2], working
in similar flow conditions, found Dcr = 4° for r/b = 2.5/15.
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6
Iu (%) , Lu (cm)
5
4
3
2
Iu
1
Lu
0
0
5
10
15
U (m / s)
20
25
Figure 2. Intensity and integral length scale of the longitudinal turbulence in turbulent flow condition.
2.4
0.4
(a)
(b)
0.2
2.2
0
2
CL
CD
-0.2
1.8
-0.4
1.6
-0.6
r/b = 0
r/b = 1/15
1.4
0
10
20
D (°)
30
40
0.18
r/b = 0
0.17
-1
50
r/b = 2/15
0
10
20
D (°)
30
40
1.2
(c)
r/b = 0
1
r/b = 1/15
50
(d)
r/b = 1/15
r/b = 2/15
r/b = 2/15
0.16
0.8
CL
St
rms
0.15
0.14
0.6
0.4
0.13
0.2
0.12
0.11
r/b = 1/15
-0.8
r/b = 2/15
1.2
r/b = 0
0
10
20
D (°)
30
40
50
0
0
10
20
D (°)
30
40
50
Figure 3. Steady drag (a) and lift (b) coefficients, Strouhal number (c) and fluctuating lift coefficient (d) in smooth
flow condition.
Figure 4 shows the variation of CD (a), CL (b), St (c) and CLrms (d) with respect to the angle of
incidence, for r/b = 2/15, in turbulent flow for Re between 2.49˜104 and 1.81˜105. For the tests
corresponding to Re = 2.49˜104, 5.14˜104, 7.81˜104, a model with b = 75 mm was used, while a
larger model with b = 150 mm was adopted for the other tests. A very strong dependency on Re
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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
is observed for small angle of incidence. In particular two distinct behaviors can be observed,
which are separated by a critical Reynolds number Recr 5˜104. These two regimes are referred
to as subcritical (Re < Recr) and supercritical (Re > Recr).
2.5
0.8
(a)
(b)
0.6
0.4
2
CL
CD
0.2
1.5
0
-0.2
1
-0.4
-0.6
0.5
-10
0
10
D (°)
20
0.3
-0.8
-10
30
0
10
D (°)
20
30
0.9
(c)
(d)
0.8
0.25
0.7
2.49 10 4
5.10 10 4
0.6
CL
St
rms
0.2
0.15
5.14 10 4
0.5
7.81 10 4
0.4
1.15 10 5
1.01 10 5
1.28 10 5
1.42 10 5
0.3
0.1
1.54 10 5
1.68 10 5
0.2
1.81 10 5
0.05
-10
0
10
D (°)
20
30
0.1
-10
0
10
D (°)
20
30
Figure 4. Steady drag (a) and lift (b) coefficients, Strouhal number (c) and fluctuating lift coefficient (d) in turbulent
flow condition for r/b = 2/15 and different Reynolds numbers.
The results obtained for Re = 2.49˜104 and 5.10˜104 are practically coincident even if are obtained using two different models (b = 75 mm and 150 mm, respectively) and are substantially
similar to the results observed in smooth-flow condition. The steady drag coefficient for D = 0 is
about 1.3, (slightly lower than in smooth flow). The minimum value of the steady lift coefficient
is about -0.7 and is obtained for D = 3 - 4° (slightly before than Dcr in smooth flow); the Strouhal
number is about 0.13 for D = 0 and rapidly increases up to 0.18 for D = 4° (again not far from the
values observed in smooth flow).
As Re increases, the steady lift slope for D = 0 switches from negative to positive; further increments of Re have the result of extending the range of D for which a positive slope is observed.
This trend seems quite regular with respect to Re, even if some small discrepancies between the
behavior of the two models (different b) have been observed and may be related to the effect of
the turbulence scale parameter Lu/b. The maximum value of the steady lift coefficient increases
as Re increases and is obtained for larger and larger angles of incidence; for Re = 1.82˜105 the
maximum CL is about 0.45 and appears for D = 4°. In the region where the steady lift slope is
positive the Strouhal number is about 0.27 (approximately twice the value obtained in smooth
flow) and is practically insensitive to Re. As the slope of CL becomes negative, St drops to a very
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low value about 0.10 and then recovers the trend observed in smooth flow for D > 10°. The minimum value of the steady drag coefficient is reached for the angle of incidence corresponding to
the maximum vales for CL and St.
The supercritical regime has been observed only for r/b = 2/15, while for r/b = 1/15 only the
subcritical regime has been observed for Re up to 2.32˜105.
4 DISCUSSION
During the experimentation described in Section 3 two main issues clearly emerged and require a
discussion. The former regards the sharp discontinuity of the aerodynamic behavior observed for
D = Dcr in smooth-flow condition. The latter issue is related to the two distinct flow behaviors
observed for different Re in turbulent flow for r/b = 2/15.
4.1 Discontinuity of the aerodynamic behavior at the critical angle of incidence in smooth flow
For the case of sharp-corner square cylinders, the existence of two flow regimes separated by the
critical angle of incidence has been clearly documented through accurate flow visualization techniques [8, 10]. The qualitative similarity of the behavior observed for the three tested models
suggests that a modification of the corner geometry (within the considered limits) does not produce qualitative variations of the flow pattern, but rather modifies the limit of existence of the
flow regimes known for the sharp corner case. In particular the increment of r/b produces a reduction of Dcr.
The transition between these two regimes is due to the reattachment of the mean flow on the
lateral face exposed to the wind, generating a separation bubble; however tRWKHDXWKRUV¶NQRwledge the unsteady nature of this transition has never been carefully investigated. Figure 5 shows
a time-frequency analysis of the lift force measured in smooth flow on the sharp-edge model (a)
and rounded-corner models with r/b = 1/15 (b) and 2/15 (c) for D = Dcr. The colormaps represent
the amplitude of the wavelet transform of L, plotted with respect to the time t and the frequency n
non-dimensionalized through b and U; the mother wavelet used for the analysis is an analytic
Morlet type; the lift coefficient is reported below the maps for reference. The length of the considered time windows is 800 non-dimensional time units, which roughly correspond to 120 cycles
of vortex shedding. From the wavelet maps it can be observed that the vortex-shedding peak frequency changes with time fluctuating between the values reported in Figure 3 for angles just before and after Dcr. This result suggests that the wavelet maps can be employed to study, from a
qualitative point of view the stability of the two concurrent flow regimes (subcritical and supercritical) that appear in the neighborhood of Dcr, as well as the transition from one regime to the
other. The comparison of the three wavelet maps reveals that the flow around the sharp-corner
cylinder tends to have smooth and frequent transitions between the two flow regimes giving rise
to a behavior that may be identified as irregular vortex shedding (Fig. 5a). On the contrary, in the
case of rounded-corner cylinder with r/b = 2/15, a proper intermittent behavior has been found as
documented in Figure 5c, where the transition between supercritical to subcritical, as well as
from subcritical to supercritical regime is clearly visible. The transitions are very sharp and the
time in which the system remains stable in a regime is relatively long (in Figure 5c the length of
the subcritical phase corresponds to about 50 cycles of vortex shedding).
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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
(a)
(b)
(c)
Figure 5. Wavelet map and time history of the lift coefficient for sharp corners (a), r/b = 1/15, r/b = 2/15 (c).
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4.2 Inversion of the lift slope at small angle of incidence in turbulent flow
The analysis of the results obtained in turbulent flow for the rounded cylinder with r/b = 2/15
suggests the existence of two flow regimes for small angle of incidence, whose transition is governed by the Reynolds number.
On the basis of aerodynamic coefficients and Strouhal number, it may be deduced that in the
subcritical Re regime the aerodynamic behavior is similar to the one observed in smooth-flow
condition, even if the sharp transition with intermittence for D = Dcr described in Section 4.1 does
not take place. This analogy (which implies the conclusion that the free-stream turbulence has a
limited effect in the subcritical Re regime) can be verified by comparing the mean and fluctuating
pressure coefficients obtained in smooth and turbulent flow for D = 0 and different Re (Fig. 6). In
particular it can be observed that Cp obtained in turbulent flow and subcritical regime (Fig. 6c, Re
= 2.49˜104 and 5.10˜104) is very similar to Cp obtained in smooth flow (Fig. 6a), which is substantially independent of Re; the same analogy also applies to the fluctuating pressure coefficients (Fig. 6b and d).
For small angles of incidence the passage to the supercritical Re regime deeply changes the
aerodynamic behavior: the steady drag coefficient drops to values about 0.7 (Fig. 4a), the steady
lift slope switches to positive (Fig. 4b) and the Strouhal number doubles (Fig. 4c). The steady
pressure coefficient (Fig. 6c) is strongly dependent on Re, whose increment has two major effects: 1) the suction near the leading edges increases; 2) the point on the lateral faces where the
base pressure is recovered moves upstream. As far as the fluctuating pressure coefficient is concerned (Fig. 6d) an increment of Re produces: 1) a reduction of the pressure fluctuation by the
midpoint of the lateral faces; 2) an increment of the pressure fluctuation near the trailing edges;
3) an increment of the pressure fluctuation on the leeward face. The above observations suggest
that in the supercritical Re regime the flow reattaches on the lateral faces even for D = 0 forming
two the separation bubbles that shrink towards the leading edges as Re increases.
Figure 7 shows the steady pressure coefficient for smooth flow (Fig. 7a) and turbulent flow
(Fig. 7c), as well as the respective fluctuating coefficients (Fig. 7b and d) for D = 4°. It can be observed that while in smooth flow condition the pressure field is only slightly modified by passing
from D = 0 to D = 4°, in case of turbulent flow the modifications are relevant and can be summarizes as: 1) near the lower (in Fig. 7) leading edge the pressure field is independent of Re and the
separation bubble tends to shirk as D increases likewise it happens for the sharp-corner cylinder
for D > Dcr; 2) near the upper leading edge the steady pressure field is strongly dependent on Re
and the point where the base pressure is recovered shifts downstream. A further increment of D
prevents the flow reattachment on the leeward lateral face and produces a flow pattern similar to
the one observed in smooth flow for D > Dcr. The angle of incidence for which this transition
arises corresponds to the maximum of the steady lift coefficient and is strongly dependent on Re.
946
The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
(a)
(b)
2.49 10 4
5.14 10 4
1.01 10 5
1.15 10 5
1.28 10 5
1.42 10 5
1.54 10 5
1.68 10 5
(c)
(d)
1.81 10 5
Figure 6. Pressure coefficients for D = 0. Smooth flow Cp (a) and Cprms (b); turbulent flow Cp (c) and Cprms (d).
(a)
(b)
2.49 10 4
5.14 10 4
1.01 10 5
1.15 10 5
1.28 10 5
1.42 10 5
1.54 10 5
1.68 10 5
(c)
(d)
1.81 10 5
Figure 7. Pressure coefficients for D = 4°. Smooth flow Cp (a) and Cprms (b); turbulent flow Cp (c) and Cprms (d).
947
5 CONCLUSIONS
The effects of corner shaping on the aerodynamic behavior of square cylinders have been investigated through the analysis and discussion of wind-tunnel experiments. It has been found that
rounded corners promote the reattachment of the flow on the lateral faces producing 1) a reduction of the critical angle of incidence Dcr, 2) an intermittent behavior for D = Dcr, 3) the inversion
of the lift slope for small angles of incidence. This latter effect has been observed only for r/b =
2/15, however it cannot be excluded that it may appear also for smaller r/b ratios at sufficiently
high Reynolds numbers.
The technical implications of the mentioned effects are important due to the large variation of
the aerodynamic coefficients, as well as in relation to galloping instability. From the results obtained in smooth flow it can be concluded that the classical galloping model (based upon a quasisteady assumption) cannot be adopted due to the observed intermittent behavior. On the other
hand, the results obtained in turbulent flow reveal that the lift coefficient or rounded corner cylinders can be negative or positive for different Re and that, in supercritical Re regime, the necessary condition for galloping (negative lift slope) does not appear at D = 0, but in a range between
5° and 10°.
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