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Dutta2008.pdf
Experimental Investigation of Flow Past a Square Cylinder
at an Angle of Incidence
Sushanta Dutta1; P. K. Panigrahi2; and K. Muralidhar3
Abstract: Flow past a square cylinder placed at an angle to the incoming flow is experimentally investigated using particle image
velocimetry, hot wire anemometry, and flow visualization. The Reynolds number based on cylinder size and the average incoming velocity
is set equal to 410. Data for four cylinder orientations 共␪ = 0, 22.5, 30, and 45°兲 and two aspect ratios 关AR= 16 and 28兴 are reported.
Results are presented in terms of drag coefficient, Strouhal number, time averaged velocity, stream traces, turbulence intensity, power
spectra, and vorticity field. In addition, flow visualization images in the near wake of the cylinder are discussed. The shape and size of the
recirculation bubble downstream of the cylinder are strong functions of orientation. A minimum in drag coefficient and maximum in
Strouhal number is observed at a cylinder orientation of 22.5°. The v-velocity profile and time-average stream traces show that the wake
and the separation process are asymmetric at orientations of 22.5 and 30°. The corresponding power spectra show additional peaks related
to secondary vortical structures that arise from nonlinear interaction between the Karman vortices. The flow visualization images show the
streamwise separation distance between the alternating vortices to be a function of cylinder orientation. Further, the flow approaches three
dimensionality early, i.e., closer to the cylinder surface for the 22.5° orientation. The drag coefficient decreases with an increase in aspect
ratio, while the Strouhal number is seen to increase with aspect ratio. The turbulence intensity is higher for the large aspect ratio cylinder
and the maximum turbulence intensity appears at an earlier streamwise location. The overall dependence of drag coefficient and Strouhal
number on orientation is preserved for the two aspect ratios studied.
DOI: 10.1061/共ASCE兲0733-9399共2008兲134:9共788兲
CE Database subject headings: Cylinders; Drag; Coefficients; Experimentation.
Introduction
The study of bluff body wakes is important for applications in
aerodynamics, wind engineering, and electronics cooling. Bluff
body cross sections that are often employed are circular and rectangular 共especially, square兲. The flow details behind these geometries depend on Reynolds number, blockage ratio, and free
stream turbulence. For square/rectangular cross section geometries, the orientation with respect to the mean flow is another
important parameter. At low Reynolds number, aspect ratio and
end conditions play a significant role in determining the flow
properties. Flow past a square cylinder resembles flow past a
circular cylinder as far as instabilities are concerned. But the
separation mechanism and the consequent dependence of lift,
drag, and Strouhal number on the Reynolds number are significantly different. The separation points are fixed for the square
cylinder either at the leading edge or the trailing edge, depending
on the Reynolds number. The vortex formation region is signifi1
Lecturer, Dept. of Mechanical Engineering, National Institute of
Technology Silchar, Silchar 788010 India. E-mail: [email protected]
2
Professor, Dept. of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016 India 共corresponding author兲. E-mail:
[email protected]
3
Professor, Dept. of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016 India. E-mail: [email protected]
Note. Associate Editor: Brett F. Sanders. Discussion open until
February 1, 2009. Separate discussions must be submitted for individual
papers. The manuscript for this paper was submitted for review and possible publication on May 22, 2007; approved on February 1, 2008. This
paper is part of the Journal of Engineering Mechanics, Vol. 134, No. 9,
September 1, 2008. ©ASCE, ISSN 0733-9399/2008/9-788–803/$25.00.
cantly broader and longer for a square cylinder compared to the
circular. The study on the effect of aspect ratio in the literature
has been limited to a circular cylinder. The present study reports
on experimental measurements of flow patterns in the wake of
square cylinder cross section. Its sensitivity to aspect ratio and
orientation with respect to the mean flow 共namely, the incidence
angle兲 are examined. Relevant studies from the literature are reviewed below.
Obasaju 共1983兲 used hotwire anemometry to study the effect
of cylinder orientation 共0, 10, 13.5, 20, and 45°兲 at high Reynolds
number. A reduction in drag coefficient and a sharp rise in Strouhal number was seen at an angle close to 13.5°. This effect was
attributed to the shear layer reattachment over one of the edges of
the cylinder. Knisely 共1990兲 experimentally investigated the
variation of Strouhal number of rectangular cylinders with side
ratios in the range of 0.04– 1 and angles of incidence from 0 to
90°. A sharp rise in Strouhal number for a small angle of incidence was reported.
Lee and Budwig 共1991兲 studied the effect of aspect ratio for a
circular cylinder at low Reynolds number using flow visualization
and hotwire anemometry. For an aspect ratio greater than 60, a
discontinuity in the Strouhal number value in the Reynolds number range 共64⬍ R ⬍ 130兲 was reported. There is a stabilization
effect on the wake for a small aspect ratio cylinder. The wake
width increases with a reduction in aspect ratio. Stäger and Eckelmann 共1991兲 studied the effect of end plates on the shedding
frequency of circular cylinders in the intermediate range of Reynolds numbers 共300–5,000兲. Near the end plate, the shedding frequency is lower than that at midspan. The end effect faded away
with the increase in Reynolds number. König et al. 共1993兲 studied
the Strouhal number–Reynolds number relationship at various as-
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pect ratios and end conditions. The discontinuity in the relationship was attributed to the oblique shedding angle and transition in
discrete shedding modes.
Szepessy and Bearman 共1992兲 studied the effect of aspect ratio
共0.25– 12兲 and end plates for flow past a circular cylinder at high
Reynolds number 共8 ⫻ 103 ⬍ R ⬍ 1.4⫻ 105兲. With appropriate end
plates they showed that wake flow is two dimensional. Norberg
共1993兲 reported the Strouhal numbers for rectangular cylinders of
various side ratios 共1–5兲 and incidence angles 共0 – 90° 兲 for Reynolds numbers in the range 400– 3 ⫻ 104 from hotwire anemometry measurements. For intermediate angles, Strouhal number and
drag coefficient were found to be nearly constant when based on
the projected dimension of the cylinder. Norberg 共1994兲 studied
the effect of aspect ratio for flow past a circular cylinder over a
wide range of Reynolds numbers starting from laminar to transition and ultimately turbulent flow. The critical Reynolds number
for the onset of vortex shedding was found to be constant for
aspect ratios 共ARs兲 larger than 40 and was delayed for lower
aspect ratios. At an aspect ratio larger than 100 the Strouhal number was found to be independent of aspect ratio.
Mansy et al. 共1994兲 reported laser Doppler velocimeter 共LDV兲
data based on an investigation of flow past a circular cylinder
over a range of Reynolds numbers. The writers observed the
three-dimensional structures to be strongest close to the cylinder
during formation of primary vortices. The larger scales are amplified and the smaller scales attenuated in the downstream direction. Brede et al. 共1996兲 did a particle image velocimetry 共PIV兲
study for flow past a circular cylinder at Reynolds numbers between 160 and 500. Two modes of secondary vortices 共A and B兲
with different wavelengths were observed. They showed the effect of aspect ratio on the spanwise wavelength and circulation of
streamwise eddies. Williamson 共1997兲 reviewed the nature of
flow past a circular cylinder at low Reynolds numbers. The flow
three dimensionality in nominally two dimensional geometry at
low Reynolds numbers was attributed to the vortex dislocation.
One other phenomenon responsible for three dimensionality was
shown to be the oblique shedding.
Dutta et al. 共2003兲 reported on the sensitivity of the wake
behind a square cylinder to the angle of incidence 共␪ = 0 – 60° 兲
and high Reynolds numbers 共1,340, 4,990, and 9,980兲 based on
hot-wire anemometry measurements and smoke visualization.
They observed reduced drag coefficient and higher Strouhal number with an increase in the orientation angle. Oudheusden et al.
共2005兲 studied the vortex shedding characteristics in the near
wake of a square cylinder at incidence to the mean flow for Reynolds numbers of 4,000, 10,000, and 20,000 using PIV. The writers used proper orthogonal decomposition 共POD兲 to reconstruct
the phase-averaged flow field from time uncorrelated data. The
pattern of individual POD modes was found to be a function of
the incidence angle. For a cylinder at incidence a striking difference in vortex formation from the upper and lower side of the
body was evident from various phase angles.
The literature review indicates the importance of cylinder orientation for bluff bodies at low Reynolds numbers. There is an
additional influence of aspect ratio that contributes to three dimensionality, but has been experimentally investigated only for
circular cylinders. For a square cylinder, a joint study of cylinder
orientation and aspect ratio is not available. The present work
reports measurements in the wake of a square cylinder at an intermediate Reynolds number of 410. Four cylinder orientations
共0 – 45° 兲 and two aspect ratios, namely 16 and 28, are considered.
The blockage ratio was less than 7% and is not an influential
parameter in the study. The velocity field is mapped using PIV
and hotwire anemometry 共HWA兲. Flow visualization images compare the flow structures that appear in the wake. The sensitivity of
the flow properties to cylinder orientation and aspect ratio are
reported.
Apparatus and Instrumentation
Experiments have been carried out in a vertical test cell made of
Plexiglas with air as the working fluid 共Fig. 1兲. The test cell has
two optical windows, one for the passage of the laser sheet and
the other for recording by the camera. The cross section of the test
cell is 9.5⫻ 4.8 cm2 and the overall length is 2 m. The active
length of the test cell where measurements have been carried out
is 0.3 m. A contraction ratio of 10:1 ahead of the test section has
been used. Cylinders of square cross section 共3 and 3.4 mm edge兲
used for experiments are made of Plexiglas and machined for
sharp edges. Therefore, the aspect ratio in our experiment is equal
to 16 共48/3兲 and 28 共95/3.4兲. Limited data at an aspect ratio of 60
have been obtained in a separate wind tunnel 共cross section 300
⫻ 160 mm兲 using a square cylinder of 5 mm. The cylinder is
mounted horizontally with its axis perpendicular to the flow direction. It is supported along the two side walls. With reference to
Fig. 1, the x axis is vertical and aligned with the mean flow
direction. The z axis coincides with the cylinder axis and the y
axis is perpendicular to x and z. The distance of the cylinder axis
from the outlet of the contraction is at least ten times the cylinder
edge to ensure adequate decay of free stream disturbances 共Sohankar et al. 1998兲.
End plates have been used in the past experiments to minimize
the wall boundary layer effect, suppress three dimensionality, and
for parallel vortex shedding. Measurements were carried out with
and without end plates for the square cylinder over a Reynolds
number range of 200–600. The endplate parameters were based
on that of Norberg 共1994兲 and Stansby 共1974兲. For configurations
with and without end plates, the difference in the velocity profile
at different spanwise z locations was found to be minimal. Hence,
the effect of endplates on the velocity profile was insignificant in
both spanwise and streamwise direction. The Strouhal number
was also measured under these conditions. The difference in
Strouhal number with and without endplates was less than ⫾2%.
The low turbulence level in the test section of the present study
and the proximity of the cylinder to the inlet of the test section are
responsible for the thin boundary layer and hence a small endwall effect. The data reported in the present study do not utilize
end plates.
The flow in the test section is set up by a suction pump driven
by a single phase motor at the outlet. The power supply to the
pump was stabilized to provide a practically constant input voltage to the motor. The free stream turbulence level in the approach
flow was quite small; it was found to be less than the background
noise of the anemometer 共⬍0.05% 兲. Flow parallelism in the approach flow was better than 98% over 95% of the width of the
test cell. Stable velocities in the range of 0.5– 3 m / s were realized
in the test cell to cover the Reynolds number range of 100–800.
Measurements of the velocity field over selected planes were
carried out using a PIV system. It comprises a double pulsed
Nd:YAG laser 共new wave lasers, ␭ = 532 nm兲, 15 mJ/ pulse, a
Peltier-cooled 12 bit charge-coupled device 共CCD兲 camera 共PCO
Sensicam兲 with a frame speed of 8 Hz, a synchronizer, frame
grabber, and a dual processor PC. The CCD consists of an array
of 1,280⫻ 1,024 pixels. A Nikon 50 mm manual lens with
f # = 1.4 was attached to the CCD camera for covering the field of
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Fig. 1. 共a兲 Schematic drawing of experimental setup; 共b兲 schematic drawing of prismatic cylinder and associated vortices
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Table 1. Comparison of Drag Coefficient with Literature for Flow Past Square Cylinder
Writers
Davis et al. 共1984兲
Sohankar et al. 共1999兲
Saha et al. 共2003兲
Li and Humphrey 共1995兲
Present
Nature of
study
Aspect ratio
Blockage
Numerical 共2D兲/experimental
Numerical 共3D兲
Numerical 共3D兲
Numerical 共2D兲
Experimental
1
0.167
6
0.055
6–10
0.100
6.44
0.192
16
0.030
28
0.070
Note: Values within parenthesis include the contribution of velocity fluctuations on the time averaged drag.
interest. The field of view for PIV measurements was 40 mm by
35 mm. Velocity vectors were calculated from particle traces by
the adaptive cross-correlation method 共Panigrahi et al. 2005兲. The
final interrogation size was 16⫻ 16 pixels from an initial starting
value of 64⫻ 64 and 5,561 velocity vectors were obtained with a
spatial resolution of 0.5 mm. Inconsistent velocity vectors were
eliminated by local median filtering. The laser pulse width was
20 ␮s and the time delay between two successive pulses was
varied from 40 to 200 ␮s depending on the fluid velocity 共Keane
and Adrian 1990兲. The time-averaged velocity field was obtained
by averaging a sequence of 200 velocity vector images, corresponding to a total time duration of 50 s. Proper seeding is essential for good PIV images, particularly when recirculation regions
are to be imaged. In the present work, seed particles were added
to the main flow by a number of copper tubes upstream of the
honeycomb section. Multiple holes were drilled in these copper
tubes to make the seeding uniform over the entire test cross section. Laskin nozzles were used to produce seeding particles from
corn oil. The mean diameter of the oil particles was estimated to
be 2 ␮m. The instantaneous flow visualization images were recorded using the PIV system itself with a reduced particle density
with a laser pulse width of 25 ␮s and an exposure time of 4 ms.
Local time-averaged velocity and velocity fluctuations were
measured using a hotwire anemometer 共DANTEC兲. An X-wire
probe was used for measuring two components of velocity. With
the square cylinder placed horizontally, the X configuration was
formed in the vertical plane. The two wires of the probe were
calibrated against a pitot-static tube connected to a digital manometer 共FURNESS CONTROLS, 19.99 mm H2O兲. The anemometer output voltage was collected in a PC through a data
acquisition card 共National Instruments兲 with LabVIEW software.
The measurement procedure adopted in the present work is similar to that presented by the writers elsewhere 共Dutta et al. 2003,
2007兲. In the low velocity regime, measurements with the pitotstatic tube as well as the hotwire anemometer are prone to errors.
These can arise from higher order physical phenomena including
free convection and probe interference effects. The errors can be
controlled by using a pitot-static tube of small diameter 共3 mm in
the present study兲; in addition the hotwire probe in the present
work was operated at a lower temperature 共of around 150° C兲
leading to minimization of free convection and radiation errors,
without loss of sensitivity. The power spectra of the velocity
fluctuations were determined using the fast Fourier transform
共FFT兲 algorithm. The sampling frequency used was 1,000 Hz, the
signal length being 20 s. A band pass filter in the range of
0.1 Hz– 1 kHz was additionally used.
Flow visualization was carried out in the test cell using a light
sheet of a pulsed Nd:YAG laser. The images were recorded with a
CCD camera synchronized with the firing of the laser. The light
sheet and the camera were perpendicular to each other. The par-
R
C̄D
470
400
400
500
410
420
1.95
1.67
2.10
1.98
2.29 共2.60兲
2.03 共2.57兲
ticle density was adjusted by lowering the Laskin nozzle pressure
to obtain a higher sensitivity of the images of the flow structure.
Uncertainty and Repeatability
The seeding of flow with oil particles, calibration, laser light reflection, background illumination, image digitization, cross correlation calculation, velocity gradients, and out-of-plane particle
motion affect the accuracy of PIV measurements. Tracer particles
need to follow the main air flow without any lag. For the particle
size utilized and the range of frequencies in the wake, an expected
slip velocity error of 0.3– 0.5% relative to the instantaneous local
velocity is expected. A second source of error in velocity measurement is due to the weight of the particle. In the present experiments, the weight effect on the seed particles was examined
by conducting experiments at a fixed Reynolds number by varying the size of the cylinder. The streamline plot and the dimensionless size of the recirculation region were found to be identical
in each case, and independent of the fluid velocity. The noise due
to background light was minimized by using a band-pass filter 共at
the wavelength of the laser兲 before the camera sensor. The x and
y component velocity profiles from PIV measurements compared
very well with those from the hotwire in the far field region,
confirming the proper implementation of both the techniques and
the measurement procedure. From repeated measurements at the
zero degree cylinder orientation 共with Reynolds number kept constant to within ⫾1%兲, the uncertainty in drag coefficient was
determined to be within ⫾5%. The uncertainty in Strouhal number was ⫾2%.
Validation of Experimental Data
Measurements have been compared against published results in
terms of drag coefficient, Strouhal number, and vorticity patterns.
Table 1 presents a comparison of time-averaged drag coefficient
of a square cylinder for zero angle of incidence. For threedimensional numerical simulation, the drag coefficient is a value
averaged over the entire span of the cylinder. The experimental
value is that of the midplane, obtained by a wake survey method.
The drag coefficient from all studies are in the range of 1.95–2.6.
Fig. 2 compares the Strouhal number from the present work with
that of other investigations at similar Reynolds number and a zero
degree orientation. There is a satisfactory match between our data
共AR= 28兲 and that of Okajima 共1982兲 at lower Reynolds number
共R ⬍ 300兲. At higher Reynolds number, there is a reasonable
match between the present data and that of Norberg 共1994兲. The
comparison of Strouhal number data with other investigations
共Davis et al. 1984; Sohankar et al. 1999; Sohankar et al. 2003兲
show intrinsic uncertainty in measurements at intermediate Rey-
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Fig. 2. Strouhal number versus Reynolds number value comparison
with literature
nolds numbers. Against this background, the match shown in Fig.
2 can be taken to be satisfactory.
Fig. 3 shows the instantaneous streamwise vorticity contours
共␻x兲 above a circular cylinder obtained in the present work using
PIV. The axis of the vorticity component is along the flow direction. The Reynolds number of the present work is 330. At this
Reynolds number, secondary 共streamwise兲 vortices are generated
along with the spanwise 共Karman兲 vortices. The secondary vortices are arranged along a line parallel to the cylinder axis. The
sense of rotation changes along the spanwise direction. The spanwise wavelength of the secondary vortices is around one cylinder
diameter, the spacing between the vorticity peaks being, in general, a constant. This compares well with the PIV results of Brede
et al. 共1996兲. The results of Brede et al. 共1996兲 are at Reynolds
number 290. Overall, the favorable comparison of vorticity patterns, Strouhal number, and drag coefficient value with the literature confirms the correct implementation of the experimental
procedure in the present study.
Results and Discussion
Results have been presented for a square cylinder oriented at
various angles to the incoming flow. The ranges of parameters
considered in the study are as follows: incidence angle 0 – 45°;
Reynolds number 410; and aspect ratio 16 and 28. Mirror images
of the 0 – 45° flow fields are produced for cylinder angles between
45 and 90°. The drag coefficient and Strouhal number are based
on B, the cylinder dimension irrespective of the cylinder angle. It
is expected that an increase in the projected dimension due to the
cylinder orientation will increase the drag coefficient and reduce
the Strouhal number. Experiments show that drag coefficient decreases, attains a minimum, and then increases with an increase in
the orientation angle. The Strouhal number shows an opposite
trend. The reasons for this behavior have been explored in the
study.
The change in the angle of the cylinder affects the wake primarily due to the following two factors: 共1兲 change in the projected dimension normal to the flow; and 共2兲 movement of the
point of separation and hence the position of the dividing streamline. The dividing streamlines are symmetric for 0 and 45°, but
evolve unsymmetrically at all other angles. The loss of symmetry
is felt in the time-averaged velocity distribution. The projected
dimension affects the minimum streamwise velocity in the near
wake. The lowest u velocity is to be expected for an angle of 45°,
in comparison to all other angles. For a square cylinder, the points
of separation are fixed at the upstream corners. There is a possibility of flow detaching from the upstream corner, closing in on
the cylinder, and separating once again from the rear corners.
When the square cylinder is inclined to the mean flow, only one
pair of corners contribute to flow separation. Change in aspect
ratio influences the degree of three dimensionality of the flow
field. A vigorous third component of velocity along the cylinder
axis is indicative of a weakened wake in the main flow direction.
The thrust of the present experimental study is to examine the
sensitivity of the wake of the cylinder with respect to orientation
and aspect ratio. Accordingly, the results of the present study have
been discussed in the following sequence: 共1兲 drag coefficient and
Strouhal number; 共2兲 flow visualization; 共3兲 time-averaged stream
traces; 共4兲 time-averaged velocity profiles; 共5兲 recovery of centerline velocity; 共6兲 time-averaged velocity fluctuation; 共7兲 timeaveraged vorticity; and 共8兲 power spectra.
Strouhal Number and Drag Coefficient
The Strouhal number has been calculated from the location of the
spectral peak of the velocity trace as recorded by the hotwire
anemometer. The time-averaged drag coefficient has been calculated using the wake survey method based on applying a momentum balance approach over a control volume around a cylinder.
Drag coefficient reported here arises from the combined effect of
momentum deficit and time-averaged turbulent stresses at the outflow plane of the wake. It has been determined as a time-averaged
quantity from a PIV data set of 200 images. The drag coefficient
has been determined from the profiles of velocity and velocity
fluctuations across the entire test cell at a streamwise location of
x = 10. Farther downstream 共x = 15兲, the time-averaged velocity
field was significantly distorted by the presence of the wall and
was not preferred. Since the plane x = 10 is not sufficiently far
away from the cylinder, the correction arising from turbulent
stresses is expected to be significant. The drag coefficient has
been calculated from the extended formula
冕 冉 冊 冕冉
+⬁
CD = 2
−⬁
Fig. 3. Instantaneous vorticity contours on y-z plane above circular
cylinder
u
u
1−
dy + 2
U
U
+⬁
−⬁
冊
v ⬘2 − u ⬘2
dy
U2
共1兲
The first term is the momentum deficit of the time-averaged flow
field and the second term is the contribution of the turbulent fluc-
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Fig. 4. Variation of Strouhal number with cylinder orientations for
three aspect ratios at R = 410
tuations. In all experiments, the second term was found to be
10–15% of the total drag.
Fig. 4 shows the Strouhal number variation with cylinder orientation at R = 410. The trend with respect to the incidence angle
is similar for all aspect ratios with a maximum Strouhal number at
the 22.5° orientation. The Strouhal number increases from the 0°
incidence angle to 22.5°, with a subsequent drop for an increase
in the incidence angle. From experiments, Chen and Liu 共1999兲
observed an increase in the Strouhal number with respect to the
incidence angle for a square cylinder until 17° 共at which S was
0.187兲 followed by a marginal drop to an asymptotic value of
about 0.175. The Reynolds number range considered was 2,000–
21,000. The incidence angle that produces a maximum in Strouhal number was attributed to the onset of flow reattachment to the
side face of the cylinder. The overall trend in the Strouhal number
variation with respect to the orientation is quite similar for the
two studies. The difference in the magnitudes of the Strouhal
number between the present study and those of Chen and Liu
共1999兲 can be attributed to the difference in Reynolds number.
The Strouhal number variation with incidence angle is related
to an increase in the projected dimension of the cylinder with
respect to the incoming flow. The vortex shedding frequency is
influenced by the width between two free shear layers and the free
stream velocity. The increase in the incidence angle leads to an
increase in the distance between the two free shear layers. Therefore, an increase in the incidence angle results in a reduced interaction between the two shear layers and a drop in the Strouhal
number. This explanation is applicable for angles beyond 22.5°. It
is also possible that the separating shear layer on one side draws
the neighboring vortex in the opposite shear layer towards it,
leading to an increase in the Strouhal number. Thus, a maximum
seen at 22.5° incidence angle is due to competing effects of an
increased projected dimension and a shorter vortex roll-up
distance. These aspects are examined further in the following
sections.
The Strouhal number increases with aspect ratio for all angles
共Fig. 4兲. The Strouhal number increases from 0.124 to 0.145 with
an increase in aspect ratio from 16 to 28 at 0° orientation. There
is, however, only a small difference in the Strouhal number between AR= 28 and 60. Norberg 共1994兲 observed a jump in the
Strouhal number from about 0.14 to 0.16 for a circular cylinder at
an aspect ratio equal to 30. Lee and Budwig 共1991兲 observed a
Fig. 5. Variation of time-averaged drag coefficient with cylinder
orientation for two aspect ratios at R = 410
similar behavior of the Strouhal number with respect to aspect
ratio for a circular cylinder with a maximum in the Strouhal number attained at about AR= 35. For a subsequent increase in aspect
ratio, no change in the Strouhal number value was observed. The
difference in magnitudes of Strouhal number and the critical aspect ratio among these studies can be attributed to the differences
in the Reynolds number and the geometry of the bluff body. Overall, it can be concluded that the effect of aspect ratio on the
Strouhal number is similar for circular and square cylinders.
The total drag coefficient can be obtained from the complete
velocity profile measurements in all spanwise planes due to the
three dimensionality of the flow field. The full-cylinder drag coefficient will be lower than that of the midplane. From the velocity measurements in the spanwise direction, it was observed that
the drag coefficient diminishes uniformly towards the side walls.
Thus the trends seen in the midplane carry over to the entire
cylinder.
Fig. 5 shows the effect of cylinder orientation on the timeaveraged drag coefficient for two aspect ratios. A minimum in
drag coefficient is observed at 22.5° for both aspect ratios. The
magnitude of drag coefficient is higher at the lower aspect ratio
共AR= 16兲 compared to the higher 共AR= 28兲. This trend is opposite to that of the Strouhal number 共Fig. 4兲, where the Strouhal
number at the higher aspect ratio is also higher. A higher Strouhal
number is an indication of the positive interaction between the
opposed shear layers that feed the fluid to the growing vortex and
cause early shedding. It also leads to weakening of the wake by
increased mixing and therefore a reduction in the drag coefficient.
The influence of aspect ratio on the Strouhal number and drag
coefficient can be explained in terms of the strength of flow along
the length of the cylinder. The secondary flow can be seen as a
mechanism by which the overall pressure difference across the
cylinder 共and hence CD兲 is reduced. Fig. 5 shows that a minimum
in drag coefficient occurs at 22.5°; the corresponding strength of
the secondary flow would be a maximum. The particle traces
support this expectation. Since interaction of shear layers is predominantly a two-dimensional phenomenon, secondary flow does
not alter Strouhal number and an asymptotic limit is reached for
increasing aspect ratios.
The trends seen in Figs. 4 and 5 are explained in terms of the
detailed flow fields in the following sections.
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Fig. 6. Near wake instantaneous particle traces behind square cylinder for different orientations at R = 410: 共a兲 AR= 16; 共b兲 AR= 28;
10 mm scale is included with images
Flow Visualization
Particle traces for different cylinder orientations 共0, 22.5, 30, and
45°兲 are shown in Fig. 6. The interest here is in examining the
possibility of shear layer reattachment over the cylinder surface
and the distance over which the shear layer rolls up. To examine
these effects, the camera was held at a small inclination 共⯝3 ° 兲
with respect to the cylinder axis. The images shown are instantaneous. For angles of 0 and 45°, flow separation is from the upstream corners of the cylinder. The separated shear layers diverge
in the streamwise direction and the possibility of shear layer reattachment is absent for both aspect ratios. At other angles 共22.5
and 30°兲, flow separation is asymmetric with the shear layer from
one side diverging away from the cylinder midplane. At the other
side, the shear layer leaving a corner remains close to the cylinder
surface that acts as an after-body. There are two asymmetric recirculation zones behind the two downstream edges of the cylinder. The shear layers do not allow an abrupt change in the slope of
the streak lines. It is hence to be concluded that experiments do
not show reattachment followed by a second separation. The distance over which the shear layers roll up is a measure of the time
taken for instability to set in, and hence inversely to Strouhal
number. For 0 and 45°, the shear layers on each side of the cylinder roll up over similar distances, and a well-defined Strouhal
number is obtained. At other orientations, the shear layers on each
side are markedly different, and two vortex shedding frequencies
are realized. It is also possible that the vortex shedding frequency
corresponding to the vortex that rolls up over a shorter distance
drives the unsteadiness in the wake. In this context, a single
Strouhal number is once again realized. At other angles, the
Strouhal number is determined by the vortex that rolls up over a
shorter distance 共though modulated by the second longer shear
layer兲. This point has been clarified from the power spectra discussed in the “Power Spectra” section.
The flow visualization images of Fig. 6 also explain the reduction in drag coefficient at certain angles. The drag coefficient is
expected to increase continuously with angle due to increased
blockage of flow. Simultaneously, the relatively longer shear layer
formed at angles other than 0 and 45° results in asymmetry of the
wake, larger transverse velocities, better mixing, and hence a
higher base pressure. The distance between the vortices in the
opposite shear layer in the flow visualization images 共Fig. 6兲 confirm this expectation. Thus, at an intermediate angle 共22.5° in the
present study兲, a minimum in drag coefficient is realized. The
effect of aspect ratio on the Strouhal number and drag coefficient
共Figs. 4 and 5兲 is also explained from the flow visualization pictures of Fig. 6. The separation between the vortices of opposite
shear layers is lower for AR= 28 compared to AR= 16 共Fig. 6兲.
Thus the interaction between alternating vortices is lower for the
small aspect ratio cylinder leading to reduction in Strouhal number and increase in drag coefficient.
Fig. 7 shows the flow visualization pictures in the x-z plane at
different cylinder orientations for aspect ratios of 16 and 28. The
objective is to show the spanwise variation and hence the three
dimensionality of flow behind the square cylinder. Mushroom
type vortical pair structures are seen in the visualization images.
These structures are similar to the Mode B type seen behind a
cylinder wake 共Williamson 1996兲. However, these structures are
comparatively more irregular in the spanwise plane for the
present study possibly due to higher Reynolds number and vortex
dislocation. The three dimensionality appears early for the low
aspect ratio cylinder compared to that of the large aspect ratio.
For AR= 28, the three-dimensional vortex structures appear at
streamwise x locations that depend on the cylinder orientation.
The three dimensionality shows up very close to the cylinder for
an orientation equal to 22.5° 共AR= 28兲. The early appearance of
three dimensionality 共3D兲 may also be correlated to the minimum
drag at this orientation. The x location at which three dimensionality appears is also related to the evolution of streamwise turbu-
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Time-Averaged Stream Traces
Fig. 8 compares the time-averaged stream traces in the wake of
the cylinder at Reynolds numbers of 410 for two aspect ratios 共16
and 28兲. The stream traces are plotted at the midplane 共z = 0兲 of
the cylinder. Two bubbles with opposed direction of circulation
form behind the cylinder. At all cylinder orientations, the size of
the recirculation bubble in the streamwise as well as the transverse direction is greater for the low aspect ratio cylinder compared to the high. The larger recirculation bubble for low aspect
ratio indicates a greater formation length of the vortex. The increase in formation length at the lower aspect ratio can be related
to the early appearance of spanwise variation, namely three dimensionality of the flow field. It correlates with the higher drag
coefficient shown in Fig. 5. Similarly, the wake width is comparatively smaller for the high aspect ratio cylinder. The smaller recirculation bubble size and wake width for the high aspect ratio
cylinder relates to the corresponding lower drag coefficient of
Fig. 5.
The recirculation bubbles of Fig. 8 also demonstrate that flow
is asymmetric when the orientation of the cylinder is 22.5 or 30°.
The separating streamlines on each side of the recirculation
bubble are oriented at an angle with respect to the mean flow,
indicating greater interaction between the two. The core location
and transverse extent of the recirculation bubbles are different for
these cylinder orientations. The effect is clearly evident for the
higher aspect ratio.
Of the two aspect ratios tested, the size of the recirculation
region and the drag coefficient were higher for the lower aspect
ratio. Thus, the effect of lowering aspect ratio is to stabilize the
shear layers, enlarge the wake, and lead to higher drag on the
cylinder. The stabilization effect in-turn, lowers the Strouhal number. The intensifying of the flow in the third dimension at the
lower aspect ratio has only a secondary influence on the average
flow properties.
Time-Averaged Velocity Profiles
Fig. 7. Spanwise near-field particle traces in x − z plane at y = 0.5
behind square cylinder at different cylinder orientations at R = 410:
共a兲 AR= 16; 共b兲 AR= 28; 10 mm scale is included with images
lent intensity. As discussed in a later section, turbulence intensity
was observed to peak earlier at a cylinder angle of 22.5°. There is
no spanwise variation in the onset of 3D flow structures for any of
the cylinder orientations. This indicates parallel vortex shedding
behind the square cylinder. At low aspect ratio 共AR= 16兲, the
three dimensionality appears closer to the cylinder; in addition
this location is insensitive to the cylinder orientation.
Profiles of two components of the time-averaged velocity are
compared in Fig. 9 for the four incidence angles 共0, 22.5, 30, and
45°兲. The comparison is presented for four downstream locations
共x = 2, 4, 6, and 8兲. These streamwise locations have been selected
to include important regions of interest, namely the prerecirculation bubble 共x = 2兲, the core recirculation bubble 共x = 4 , 6兲, and
postrecirculation region 共x = 8兲. Velocity has been nondimensionalized with that of the incoming stream and x-y coordinates
are nondimensionalized with the cylinder edge. With an increase
in the downstream distance, the x component of the centerline
velocity recovers towards the free-stream value, while the
y-component velocity approaches zero. The wake becomes
broader due to entrainment of the fluid into the wake. In near
wake, the wake size increases rapidly because of displacement by
the two oppositely oriented eddies. In the far downstream, the
wake size reaches a limiting value since the eddies are weakened
by viscous dissipation and diffusion.
Fig. 9 shows that the u- and v-velocity profiles have a strong
dependence on aspect ratio. For the streamwise velocity, recovery
is much faster for the higher aspect ratio cylinder when compared
to the lower. When the aspect ratio is high, the initial recovery of
streamwise velocity is faster when compared to the low aspect
ratio. This is related to the size of the recirculation bubble in the
streamwise direction and its location with respect to the x-z midplane. The u-velocity profile at 22.5° orientation shows minimal
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Fig. 8. Time-averaged stream traces in wake of square cylinder at R = 410; influence of cylinder orientation; first row, AR= 16; second row
AR= 28
velocity deficit compared to other cylinder angles for both aspect
ratios. This trend is in conformity with the lowest drag coefficient
seen at this orientation 共Fig. 5兲. The streamwise velocity profile
for the low aspect ratio cylinder continues to develop until the
location of x = 8. In contrast, the streamwise velocity profile for
high aspect ratio at 22.5 and 30° orientations attain a self-similar
profile at x = 8.
Both positive and negative transverse velocities are seen above
and below the midplane of the cylinder. Thus, fluid particles are
entrained from each side into the wake. Compared to streamwise
velocity, the transverse v velocity shows greater variation with
respect to the incidence angle and aspect ratio. At x = 2, the
v-velocity profile shows similar variation for all incidence angles
at AR= 16. This is because x = 2 is a location upstream of the
recirculation bubble for all cylinder orientations at this aspect
ratio. In contrast, the v-velocity profile shows significant variation
as a function of incidence angle for the high aspect cylinder
共AR= 28兲. This is because x = 2 is a location upstream of the recirculation bubble at 0° and downstream of the recirculation
bubble at 22.5° 共Fig. 8兲. Therefore, on an average, the direction of
v velocity at 22.5° is opposite that at 0°. Similarly, the v velocity
at x = 6 shows an opposite trend when compared to x = 2 for AR
= 16. Immediately upstream of the core of the recirculation
bubble, the v-velocity profile shows dual peaks at the x = 4 location for all incidence angles and AR= 16. Similar dual peaks are
observed at x = 2 and ␪ = 45° for AR= 28. This is due to the rapid
change streamwise velocity near the core location. The v-velocity
profile is asymmetric for 22.5 and 30° orientations. Therefore, the
local instability modes and their amplification rate are altered for
these orientations of the square cylinder leading to the appearance
of additional harmonics in the flow fluctuations. These are shown
in power spectra section in the form of power spectra. These
modes are not as clearly evident at AR= 16 due to an early influence of three dimensionality 共Fig. 7兲. The asymmetric separation
processes for 22.5 and 30° are clear from the v-velocity profile
when compared to the u velocity. At the end of the recirculation
bubble, the transverse velocity magnitudes reduce, leading to a
slow but definite recovery of the streamwise velocity.
Recovery of Centerline Velocity
Centerline recovery of the streamwise velocity and decay of the
transverse velocity for various angles are compared in Fig. 10.
The transverse velocity has been plotted along the x axis at a
particular offset location from the cylinder centerline 共y = 1兲. The
u velocity is zero on the cylinder surface and is negative 共in the
time-averaged sense兲 up to a certain downstream distance. Later,
it increases with the x coordinate and reaches an asymptotic
value.
Two factors play a major role in determining centerline recovery of velocity. One is the wake size and the second is base
pressure on the rear surface of the cylinder. With downstream
distance, the wake becomes broader due to entrainment of the
fluid and the average pressure difference between the wake and
the outer flow diminishes. Hence, the pressure difference between
the core of the wake and the external flow determines the initial
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Fig. 9. Time-averaged u and v velocity profiles at four downstream locations 共x = 2, 4, 6, and 8兲 for two aspect ratios 共16 and 28兲 and four
cylinder orientations 共0, 22.5, 30, and 45°兲 at R = 410
recovery. Downstream recovery in velocity depends on flow readjustment in the form of the fluid drawn into the wake from the
external flow. The first factor is a strong function of cylinder
orientation. Hence the base region of the cylinder carries these
characteristics; at longer distances downstream, these factors are
less significant.
Fig. 10 shows that centerline recovery is faster for the high
aspect ratio cylinder. For both aspect ratios, the centerline velocity reaches an asymptotic value in the range of 0.6–0.65. The
decay of transverse velocity is also faster for the high aspect ratio
cylinder. The asymptotic limit of u velocity is reached at around
x = 15 and 7, respectively, for the low and high aspect ratios. This
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Fig. 10. Centerline recovery of streamwise and transverse velocity component for four cylinder orientations 共0, 22.5, 30, and 45°兲 at two aspect
ratios 共16 and 28兲 at R = 410. Third column shows wake size for AR= 16 and 28.
is in accordance with the larger recirculation bubble for the lower
aspect ratio experiment. The greatest negative value attained
by the streamwise velocity is higher for AR= 16 compared to
AR= 28. Similarly, the magnitude of the largest transverse velocity is higher for the low aspect ratio cylinder. The average centerline velocity is lower inside the recirculation zone, where the v
velocity is high. The minimum in centerline velocity occurs
around the core of the recirculation bubble. The streamwise location where v velocity attains a minimum takes place at the end of
the recirculation bubble.
Fig. 10 also shows the growth rate of the wake as a function of
incidence angle at two different aspect ratios. The wake width 共w兲
is calculated as the cross stream 共y兲 separation of y location corresponding to 50% of velocity deficit, i.e., difference between
freestream and minimum velocity. Fig. 10 shows the dependence
of wake width on both aspect ratio and cylinder orientation angle.
The average wake width is higher at low aspect ratio 共AR= 16兲
compared to that at higher aspect ratio 共AR= 28兲. Similarly, the
wake width is minimal at 22.5° orientation angle in the downstream region. It may be noted that this angle corresponds to
minimum drag. The wake reaches an asymptotic state after an
initial increase and decrease in its growth rate. The overall wake
growth rate can be correlated to the stream trace results of Fig. 8.
The wake width increases until the core of the recirculation
bubble from downstream of the cylinder and subsequently re-
duces until the end of the recirculation zone. This is followed by
asymptotic increase in wake growth. The smaller recirculation
zone size for the AR= 28 case in Fig. 8 corresponds to early
saturation in growth rate 共see Fig. 10兲.
Time-Averaged Velocity Fluctuation
Fig. 11 compares the total turbulence intensity fields at different
cylinder orientations. The streamwise growth of the shear layer
shows up in the figure with broadening of the high turbulent
zones in the transverse direction. The turbulence intensity is
higher for the large aspect ratio cylinder when compared to the
low. A greater wall effect along with early onset of three dimensionality for the low aspect ratio cylinder is responsible for dampening the turbulent fluctuations. In addition, the turbulence
production is higher for the large aspect ratio cylinder due to the
smaller recirculation bubble that leads to a higher velocity gradient in the shear layer. The turbulence intensity increases both in
the streamwise and transverse direction from the edge of the cylinder and midplane of the cylinder axis, respectively, with a subsequent drop after reaching a maximum value. The maximum
turbulence intensity zone is located at a farther streamwise location for the low aspect ratio when compared to the high. This is in
accordance with the larger size of the recirculation bubble for the
low aspect ratio cylinder 共Fig. 8兲.
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Fig. 12. Turbulence intensity development along centerline for
aspect ratios of 16 and 28 at different orientation angle of cylinder
2
Fig. 11. Contour plot of percentage turbulence intensity 共urms
2 0.5
+ vrms兲 / U兲 in wake of square cylinder at Reynolds number of 410
for two aspect ratios 关16 共a兲; 28 共b兲兴 and four cylinder orientations.
Cylinder center is at x = 0.
Fig. 12 shows the streamwise variation of the resultant velocity fluctuations at the central midplane 共y = 0兲 for two aspect ratios
共AR= 16 and 28兲 and different cylinder orientations. The objective of this plot is to demonstrate differences in the formation
length of the vortices. For both aspect ratios, the turbulence intensity increases in the streamwise direction, reaching a maximum value followed by slow decay. The maximum turbulence
intensity appears at a later streamwise location for the lower aspect ratio cylinder. The turbulence intensity for the higher aspect
ratio cylinder with orientations of 0 and 45° peaks at a different
streamwise location when compared to 22.5 and 30°. This is because at 22.5 and 30°, the flow field is asymmetric when compared to the 0 and 45° angles. For the asymmetric flow field, the
turbulence intensity increases faster when compared to the sym-
metric wake. No significant difference in the turbulence intensity
between different incidence angles is observed in the near-field
region 共x ⬍ 2兲 of the low aspect ratio cylinder. This is possibly
due to the early appearance of three dimensionality 共Fig. 7兲. However, the maximum turbulent intensity continues to be observed
slightly earlier for the asymmetric cases of 22.5 and 30° when
compared to the symmetric 共0 and 45°兲.
Time-Averaged Vorticity
Spanwise vorticity 共namely the vorticity component whose axis is
parallel to the cylinder axis兲 is discussed in the present section.
Vorticity values are determined over a grid in terms of circulation
per unit area as well as by direct differentiation of the velocity
field. The contours plotted on the basis of these two definitions
are practically identical. The scale for nondimensionalizing vorticity is U / B.
Fig. 13 shows the contours of time averaged spanwise vorticity 共␻z兲 at R = 410. Contours corresponding to the maximum and
minimum values are also shown. The influence of orientation on
vorticity can be understood in light of the discussion in the sections “Flow Visualization” and “Time-Averaged Stream Traces.”
Fig. 13 shows opposed vorticity of equal strength at both the
cylinder corners. The maximum nondimensional value of spanwise vorticity is observed at the cylinder corner, where the shear
layer is initiated. In the immediate vicinity of the cylinder, the
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Fig. 13. Time-averaged spanwise vorticity 共␻z兲 contours for four cylinder orientations and two aspect ratios 共16, first row and 28, second row兲
at R = 410. Vorticity profiles are plotted at x = 2 and 4. Dashed lines show negative vorticity while solid lines represent positive vorticity;
⌬␻z = 0.5.
vorticity is primarily due to shear in the velocity as the vortices
are formed and shed after a certain formation length. The transverse spread of the vortices generated at the corners of the cylinder determines the wake size and is controlled by two opposing
factors. First is the lower average pressure within the wake. This
is balanced by momentum transport normal to the main flow direction. The latter comprises viscous diffusion along with trans-
port by the time-dependent transverse velocity. These factors lead
to a smaller recirculation bubble for the higher aspect ratio cylinder 共Fig. 8兲. Consequently, for a high aspect ratio, the spreading
of vortices in the streamwise and transverse directions is smaller
when compared to the low aspect ratio cylinder. The peak in the
spanwise vorticity component correlates with the minimum in
pressure in the near wake, and hence the drag coefficient. Based
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Fig. 14. Power spectra of x component of velocity at R = 410 for four different cylinder orientations: x = 5, 15; y = 1.2; 共left兲 AR= 16; 共right兲
AR= 28. Inset shows time trace of the transverse component of velocity.
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on Fig. 5, higher spanwise vorticity is expected for the lower
aspect ratio cylinder. Fig. 13, however, shows that vorticity magnitudes are quite similar for the two aspect ratios. An explanation
is the stronger three dimensionality of the flow field for the
smaller aspect ratio, that in turns equalizes pressure in the spanwise direction and lowers peak spanwise vorticity values.
3.
Power Spectra
Fig. 14 shows the power spectra of u velocity in the near field
共x = 5兲 and the far field 共x = 15兲. The velocity time trace has been
inserted in each spectrum plot. The velocity trace confirms the
periodic nature of the flow. Spectra were determined by a twowire hot-wire probe placed at an offset location of y = 1.2. A clear
spectral peak is observed for both low and high aspect ratio cylinders and the peak locations correspond to the vortex shedding
frequency. The frequency is higher for AR= 28 when compared to
AR= 16. It confirms the Strouhal number trends reported in Fig.
2. The peak at the vortex shedding frequency is sharper for the
high aspect ratio cylinder. The diminished strength at the low
aspect ratio is due to a greater quasi-periodicity and jitter in the
shedding pattern.
In the near-field region, a second peak is seen for AR= 28 at
incidence angles equal to 22.5 and 30°. The appearance of these
harmonics can be attributed to the nonlinear interaction between
the Karman vortices due to flow asymmetry at angles of 22.5 and
30°. The structures of vortices, seen in the flow visualization pictures of Fig. 6, are different for low and high aspect ratio cylinders and support the shapes of the power spectra. The velocity
time trace shown as an inset in Fig. 14 also confirms the frequency doubling of the vortex structures. The separation distances
between two consecutive shed vortices are different for the high
aspect ratio 共AR= 28兲 when compared to the low 共AR= 16兲. This
translates into differing frequency contents in the high aspect ratio
experiment.
In the far-field region 共x = 15兲, the power spectra for AR= 28
contain only the fundamental contrary to the presence of an additional harmonics in the near field. The power spectra for higher
aspect ratio is less noisy compared to that of the lower aspect
ratio in both near and far field regions. The spectra in the inertial
subrange have a higher negative slope, closer to −5 / 3, for the low
aspect ratio. It confirms an early approach to three dimensionality
共Fig. 7兲.
Conclusions
The effect of orientation of a square cylinder and the role of
aspect ratio in determining the wake properties have been experimentally investigated. Particle image velocimetry and hot-wire
anemometry have been used for flow measurement. Four cylinder
orientations 共␪ = 0, 22.5, 30, and 45°兲 and two aspect ratios 共AR
= 16 and 28兲 are studied. The Reynolds number, based on the
cylinder size and average upstream velocity, is set equal to 410.
Drag coefficient, Strouhal number, centerline recovery, total velocity fluctuations, velocity spectra, stream traces, and vorticity
contours are reported. The following conclusions have been arrived at in the present work:
1. A minimum in the time-averaged drag coefficient is seen at
22.5°. At this angle, the Strouhal number is a maximum;
2. The main reason for a minimum in drag coefficient at 22.5°
is wake asymmetry originating from shear layers of unequal
lengths on each side of the cylinder. The v-velocity profile
4.
5.
6.
also bring out the extent of asymmetry in the flow field. The
loss of symmetry of the wake increases the transverse velocity, increases the base pressure, and lowers drag. This factor
is counterbalanced by an increase in the projected area, but
the minimum in the drag coefficient at an orientation of 22.5°
reveals that the former has an overall stronger influence at
small cylinder angles;
Stronger three dimensionality of the flow field at 22.5° additionally contributes to higher base pressure and lower drag;
The shorter shear layer on one side of the cylinder governs
wake unsteadiness and hence Strouhal number;
The visualization images show that the separation distance
between the alternating Karman vortices is a function of the
cylinder orientation; and
Aspect ratio: the size of the recirculation bubble is smaller at
the higher aspect ratio. Hence the drag coefficient is correspondingly smaller. At the lower aspect ratio, the shear layers
are stabilized, resulting in higher drag and lower Strouhal
number. The stronger three dimensionality of flow at the
lower aspect ratio has only a marginal overall effect. The
turbulence intensity is higher for the larger aspect ratio cylinder since the size of the recirculation size is reduced. Correspondingly, the maximum turbulence intensity appears at
an earlier streamwise location. However, the overall dependence of the wake properties on cylinder orientation is seen
at both aspect ratios.
Notation
The following symbols are used in this paper:
B ⫽ edge of square cylinder 共m兲;
CD ⫽ drag coefficient based on average upstream velocity
and B, ⫽drag per unit length/ 1 / 2␳U2B;
f ⫽ dimensionless frequency variable, frequency 共Hz兲
⫻ B / U;
L ⫽ length of square cylinder 共m兲;
R ⫽ Reynolds number, ␳UB / ␮;
S ⫽ Strouhal number based on v signal, f vB / U;
U ⫽ upstream velocity 共m/s兲;
u ⫽ x-component velocity 共m/s兲;
v ⫽ y-component velocity 共m/s兲;
w ⫽ wake width 共m兲;
X , Y ⫽ dimensionless coordinates from cylinder center 共m兲;
x , y ⫽ dimensionless coordinates from cylinder center
scaled by B. x coordinates is along flow direction;
z ⫽ dimensionless coordinates parallel to cylinder axis;
␪ ⫽ orientation of cylinder with respect to incoming
flow;
␮ ⫽ dynamic viscosity 共Pa s兲;
␳ ⫽ fluid density 共kg/ m3兲; and
␻z ⫽ spanwise component of velocity scaled by U / B.
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