The Seventh Asia-Pacific Conference on
Wind Engineering, November 8-12, 2009,
Taipei, Taiwan
THREE DIMENSIONAL STRUCTURES OF FLOW BEHIND A
SQUARE PRISM
Hiromasa Kawai1, Yasuo Okuda2 and Masamiki Ohashi3
Professor, Disaster Prevention Research Institute, Kyoto University
Gokasyo, Uji, Kyoto 611-0011, Japan, [email protected]
2
Chief Research Engineer, Building Research Institute
Tachihara 1, Tsukuba, Ibaraki, 305-0802, Japan, [email protected]
3Senior Research Engineer, National Institute for Land and Infrastructure Management
Tachihara 1, Tshukuba, Ibaraki, 305-0802, Japan, [email protected]
1
ABSTRACT
Three dimensional structures of steady and unsteady flows behind a square prism with aspect ratio of 2.7 in a
smooth flow, are investigated by PIV technique synchronizing with velocity measurement by a hot wire
anemometer. It was seen that an arch-type vortex is formed behind the prism. A stagnation point is observed in
the wake at a location of 1.75D behind the prism and 0.3H above the floor, where D is width and H is height of
the prism. Therefore flow over the prism does not attach on the floor but runs away some distance above the
floor. When separated shear layers are rolled up to form Karman vortex in a wake, the arch-type vortex is still
keeping its form in the formation region though the vortex line is twisted very much. As the vortex is growing
and moving downstream, the vortex line is stretched in a stream-wise direction near the tip of the vortex which
does not move and locates stationary just behind the prism at any instant.
KEYWORDS: WAKE, PIV, KARMAN VORTEX, 3D PRISM, ARCH-TYPE VORTEX
Introduction
Flow structures behind various buildings have been investigated as CFD and PIV
techniques have been developing and using widely. However, as three dimensional flows
around buildings are very complicated, detail of the flow structure is still not so clear.
Particularly, an interaction between a flow over a top of a building and vortices formed by
separated shear layers from side faces is still uncertain until now.
Three dimensional flow structures behind a square prism in smooth flow are discussed
in the paper based on results of flow measurements by PIV technique synchronized with
velocity measurement. According to the investigation, vortex lines of vortices from the both
sides of the prism connect to that of a vortex formed by the flow over the top of the prism to
organize an arch- type vortex in a wake. The arch-type vortex is clearly seen not only in an
ensemble averaged mean flow but also is still keeping during Karman vortex formation in
unsteady flow. When Karman vortex is glowing and moving in the formation region, the
vortex line of the arch-type vortex is stretched to the stream wise direction near the tip of the
vortex, but the position of the tip does not move and locates just behind the prism at any
instant.
Experimental arrangements
Wind tunnel experiments were carried out at a wind tunnel of Building Research
Institute, Japan. In order to avoid an effect of naturally developing boundary layer on the
tunnel floor, a temporally floor was set at 30cm above the tunnel floor. A square prism of
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
z/H, Height
z/H, Height
0.8
0.8
which section is 50mm
x 50mm and height is
0.7
0.7
135mm was set at a
position of 80cm from
0.6
0.6
the windward edge of
the temporally floor.
0.5
0.5
Figure 1 shows profiles
of mean velocity and
0.4
0.4
turbulence intensity at
the position of the
0.3
0.3
prism. Effect of a
boundary layer deve0.2
0.2
loped on the temporally
floor is observed up to
0.1
0.1
40mm above the floor.
The flow is uniform
0
0
and the turbulence
0
0.05
0.1
0.15
0 0.2 0.4 0.6 0.8 1 1.2
intensity is below 1%
Iz, Turbulence intensity
Ｕ／Ｕ z/H =0.7, Mean velocity
beyond 40mm. Mean
wind velocity is 3m/s
Figure 1 Profiles of mean velocity and turbulence intensity
in the experiment.
Two
dimensional
Hot wire
PIV measurements were
z=65mm
Y
carried out in 17 horizontal
sections of which height is
15mm
Laser sheet
10mm to 170mm every
10mm interval, and 17
D=50mm
X
vertical sections from the
center line of the prism to
±40mm every 5mm interval.
In the measurements, instanttaneous velocity vectors at
63 x 63 points were calculated from two images,
Figure 2 Experimental arrangement
which are taken by a highspeed video camera 0.25msec interval. 508 sets of the images are taken for 30Hz rate in a run.
In order to investigate an unsteady flow pattern during Karman vortex is been forming,
velocity fluctuation was measured at a position of 15mm apart from the windward corner of
the prism by a hot-wire anemometer synchronizing with the PIV measurement as is shown in
Figure 2. Each set of the images is sampled conditionally based on the velocity signal, and
then is ensemble averaged.
Structure of time averaged flow
Figure 3 shows patterns of ensemble averaged mean stream lines for the typical
horizontal sections. A pair of vortices and a divided point of stream lines are observed when
z/H is 0.074~0.96. When z/H is 1.04, effect of the vortices is still observed just behind the
prism, but is not so obvious when z/H is 1.11.
A flow just above the top face of the prism at z/H=1.11 shows very an attractive
pattern of stream lines. Flow coming from upwind is drawn into the face at a center of
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
the face. This means than the z/H=0.07
laser sheet should crosses the
separated shear layer at the
point. The crossing point moves
downward near the side ridges,
so the separated shear layer is
approaching to the face toward
z/H=0.3
the side ridge. On the other
hand, the pattern of the stream
line at z/H=1.04 is very
complicate, which shows the
flow in the separated shear layer.
Many singular points exist in
the pattern.
z/H=0.59
Figure 4 shows patterns
of ensemble averaged mean
stream lines for the typical
vertical sections. A vortex with
a horizontal axis is formed near
the top just behind the prism.
The dividing stream line for the
flow over the top meets the
z/H=0.89
dividing stream line from the
floor at z=0.3H and x=1.75H in
the center section to be a
stagnation point in the wake.
Therefore, the flow over the top
does not attach on the floor. The
dividing stream lines are
z/H=0.96
approaching the rear face of the
prism when y/D is increasing,
which correspond with a cross
line of the dividing stream lines
in the horizontal section shown
in Fig.3 and the vertical
z/H=1.04
sections. When y/D= 0.6, the
vortex near the top still exists,
and the other vortex-like flow
pattern is seen near the bottom.
When y/D is 0.6 and 0.8, flow
patterns in the separated shear
z/H=1.11
layer of the side face are
observed. The patterns of the
stream line are so complicated
that it is not so easy to image
the flow structure. According to
the pattern of the singular lines,
the flow may be consisted by Figure 3 Stream lines of
mean flow in horizontal
three or four cells of the shear
sections.
y/D=0
y/D=0.2
y/H=0.4
y/D=0.6
y/D=0.8
y/D=1.0
Figure 4 Stream lines of
mean flow in vertical
sections.
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
y/D, Across wind direction
z/H, Height
layer or the shear layer
Center of vortices
is waving in the
Divided p oint of stream lines
vertical direction.
1
Figure 5 shows
the vertical profiles of
x/D=11/6
7/12
0.8
1/12
the downstream positions of the vortex
center and the dividing
x/D=1/12
0.6
stream line crossing
point shown in the
horizontal section of
0.4
Fig.3. When z/H is
0.07 to 0.22, x/D of
0.2
the vortex center is
0.75. The vortex is
approaching to the
0
prism when z/H is
0
0.5
1
1.5
2
0.22 to 0.37. When
z/H is 0.37 to 0.55, the
x/D, Distance from the prizm
x/D=7/12
position of the vortex
Figure
5
Stream-wise
position
of
is immovable of which
center of vortex and crossing
x/D is constant of 0.55.
point of dividing stream lines.
When z/H is larger 0.8,
the
vortex
is
1
approaching rapidly to
the
prism.
The
0.8
position
of
the
crossing point of the
0.6
divided stream lines is
0.4
nearly constant of
1.75D when z/H is
z/H=0.96
x/D=11/6
0.2
smaller than 0.5, then
z/H=0.07
is approaching to the
0
prism.
0
0.25
0.5
0.75
1
Stream wise
x/D,
Along
wind
direction
position of the vortex
is plotted against the
Figure 6 Projective position of
lateral position for
center of vortex.
various heights in
Fig.6. The vortex locates inside the projected area of the
prism near the bottom and the top, and it locates outside
around the central height. According to Fig.6, a line of the
vortex center may be spiral.
Figure 7 Velocity vectors in
Fig.7 shows velocity vectors in three typical cross
cross sections perpendicular
sections perpendicular to flow. It can be seen that flow
to flow.
moves upward and diverges outside just behind the prism.
Flow moves downward from the top of the prism when x/D=7/12 where is the center of the
vortex. At the wake stagnation point, velocity of the downward flow becomes very strong.
Fig.8 shows contour maps of longitudinal vorticity in a section at x/D=11/6 perpendicular to
flow and that in stream wise sections at y/D=8/5. Strong vorticity areas with opposite sign are
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
seen in the perpendicular
section outside of the
prism.
The
strong
vorticity
area
is
distributed outside the
wake region of the prism
as is shown in the map of
the stream-wise section.
x/D=11/6
y/D=5/8
Flow structure during
Karman vortex
formation
Across wind direction, y/D
Wind velocity (m/s)
In
order
to
investigate the three Figure 8 Contour maps of longitudinal vorticity in cross section
perpendicular to flow and that in stream-wise cross sections.
dimensional structure of
Dark part shows large vorticity region in clock-wise direction.
unsteady flow from 2D
PIV
measurement,
conditional technique has
been employed in the
analysis. Figure 9 shows
a conceptual diagram of
the conditional sampling
of PIV images. Velocity
3.4
vector
is
sampled
synchronizing with the
3.2
hot wire signal at a
3
particular
phase
of
2.8
Karman vortex shedding
2.6
and
is
ensemble40
45
50
55
60
65
70
75
80
averaged.
Number of frame
Figure 10 which
is shown in a next page, Figure 9 Conceptual diagram of the conditional sampling of
PIV images
shows
ensemble
averaged stream line
0.8 T =0
upper vortex
T =1/5
patterns in a cycle of the formation and
0.6
lower vortex
the shedding of Karman vortex. Left
T =2/5
0.4
figures show those in a horizontal section
T =3/5
at z/H=0.3. Right figures show those in a
0.2
vertical section at the center of the prism
0
of y/D=0. Rolling up of the separated
T =1/5
T =0
shear layer begins near a leeward side
-0.2
corner of the prism, then the vortex is
-0.4
T =5/5
growing and moving downstream to the
T =4/5
centerline, and sheds into the wake. Each
-0.6
T =3/5
picture is taken for 30Hz interval and the
-0.8
mean velocity is 3m/s, so Strouhal
0 0.2 0.4 0.6 0.8 1 1.2 1.4
number is 0.083. As is seen in the stream
line pattern in the vertical cross section,
Along wind direction, x/D
the vortex near the top of the prism exists
at any instant for the Karman vortex Figure 11 Position of Karman vortex during its
formation in a cycle.
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
formation and does not
0/5
T=0/5
0/5
T=0/5
move at all. The stream line
pattern in the vertical cross
section is different little
from one instant to the
other instant. Circles in the
horizontal sections show
1/5
1/5
1/5
position of the center of
1/5
Karman vortex. Figure 11
which is shown in the
previous
page,
shows
projected positions of the
Karman vortex during a
cycle at z/H=0.3. The
2/5
2/5
2/5
2/5
vortex locates outside of
the prism when the
separated
shear
layer
begins to roll up, and
moves very slowly. Then
the vortex moves rapidly to
3/5
3/5
the center line of the wake,
3/5
3/5
and sheds into the wake.
Figure 12 shows
stream line in horizontal
sections at various heights
at
T=0/5
which
corresponds with the top
4/5
4/5
4/5
4/5
figures in Figure 10. The
vortex at the upper side is
formed close to the center
line near the floor, and it is
away from the center line
to the top of the prism. The
vortex at the lower side
5/5
5/5
5/5
5/5
disappears when z/H is
larger than 0.74.
Figure 13 shows the
stream line in vertical
sections at the instant of
T=0/5. Left and right
figures correspond with the Figure 10 Stream lines in horizontal section at z/H=0.3 and
in vertical section at y/D=0 during Karman vortex
upper and the lower sides
formation in a cycle.
of the center line of the
wake in the horizontal section of Figure 12. When the separated shear layer begins to roll up
at the upper side, the vortex with the horizontal axis can be seen near the top of the prism
even at y=0.6D where is outside of the side face of the prism. On the other hand, the vortex
with the horizontal axis is becoming to collapse apart from the center line in the lower side,
then is separated to two vortices at y=0.4D, and disappears at y=0.6D.
Figure 14 shows velocity vectors of three cross sections perpendicular to flow in
Karman vortex formation at T=0/5. When the separated shear layer rolls up to form the vortex
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
Z/H=0.
Y
y/D=0
Y=0m
X
Z/H=0.3
y/D=0.2
Z/H=0.
Z/H=0.5
x/D=2
1
y/D=-0.2
y/D=0.4
y/D=-0.4
y/D=0.6
Y/D=-0.6
y/D=0.7
y/D=-0.7
1/3
Z/H=0.7
Z/H=0.8
Figure 12 Stream line in
horizontal sections at
various heights at T=0/5.
x/D=1/3
Figure 13 Stream line in vertical sections at various transverse
positions at T=0/5. Right figures show the upper side and left
figures show the lower side of Figure 12.
x/D=1
x/D=2
Figure 14 Velocity vectors of cross sections perpendicular to flow in Karman vortex
formation at T=0/5. Positions of sections show in Figure 12.
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
at the left side as is shown in
x/D=2
y/D=0.7
the left figure of Figure 14,
flow moves upward outside of
the shear layer. On the other
hand, flow moves downward
from the left side to form
vortex like flow at a half
height of the right side of the
prism in the section near the
right side vortex as is shown
in the middle figure of Figure
14. When Karman vortex
sheds into wake, flow begins Figure 15 Contour maps of longitudinal vorticity in
perpendicular and stream-wise cross sections. Dark part
to move downward to right
shows a large vorticity region in clock-wise direction.
from the left side near the top
of the prism, and then turns its direction to left as is
shown in the middle figure of Figure 14.
Figure 15 shows contour maps of
longitudinal vorticity in a cross section
perpendicular to flow at x/D=2 and stream-wise
cross sections at y/D=0.7. It can be seen a large
clock-wise vorticity area in the perpendicular
section. The area corresponds with the position
where the down-flow turns from right to left.
Conclusions-three dimensional structure of wake
According to the results shown in the
previous chapters, 3D structures are imaged and
their sketch are drawn in Figure 16 and Figure 17.
An arch- type vortex is formed behind the prism
both in the mean flow and in the unsteady flow
during Karman vortex formation and shedding.
When the Karman vortex is glowing and moving in
the formation region, the vortex line is stretched to
the stream-wise direction near the tip of the archtype vortex, but the position of the tip does not
move and is located just behind the prism at any
instant.
Figure 16 Schematic diagram of
wake of mean flow behind 3D
square prism
References
Hunt J. C. R., C. J. Abell, J. A. Petreka, and H. Woo, Kinematical
Studies of the Flows around Free or Surface-mounted Obstacles;
Applying Topology to Flow Visualization, Journal of Fluid
Mechanics, vol.86, part 1, pp.179-200, 1978
Shinji Ito, Yasuo Okuda, Masamiki Ohashi, Yasuhito Sasaki,
Tetsuo
Matsuyama,
Hitomitsu
Kikitsu:
Synchronous
Measurement of Wind Flow and Wind Pressure around a Cubic
Model -Instantaneous Reattachment on Surface, Proceedings of
11th International Conference on Wind Engineering (ICWE XI),
2005.2
Figure 17 Schematic diagram of
wake of unsteady flow during
Karman vortex formation behind
3D square prism