3-D Turbulent Near-Wake Structure of a Rectangular Cylinder
in Channel Flow
M. Senda(1) , K. Inaoka, N. Shigemoto and T. Okuno
Department of Mechanical Engineering, Doshisha University, Kyotanabe, Kyoto 610-0321
(1)
E-Mail: msenda@mail.doshisha.ac.jp
Abstract
Turbulent channel flow past a rectangular cylinder has been studied experimentally with a laser Doppler
velocimetry by obtaining the phase-averaged velocity statistics in order to investigate the 3-D near-wake
structure of the shedding vortices from the cylinder. The width-to-height ratio of the cross section of the
cylinder is 0.5 and 1.0. The blockage ratio is 20% and the Reynolds number based on the channel height
and the mean velocity at the inlet of the test section is 15000. The experimental result is illustrated in Fig.
A-1 as a schematic structural view of the shedding vortices. The figure shows the vortex pair shed
successively from the lower and upper sides of the cylinder. In this flow configuration, the wall-ward
flow towards the lower channel wall (〈vp〉< 0) is induced in the region between the shedding vortices.
Associated with the positive and negative vortices, the periodic streamwise velocity〈up〉takes a large
value in the regions swerved from the vortex center, while the turbulent normal stresses〈ut2〉and〈vt2〉
are large in the core regions of the shedding vortices. Although the shedding vortex structure at the
mid-span region of the channel is indeed two-dimensional, the vortices at the spanwise positions remote
from the mid-span are caused to be out of phase, accompanied by an appearance of the spanwise fluid
motion. Near the vortex center, the periodic spanwise velocity〈wp〉is negative and the turbulent normal
stress〈wt2〉has large values.
Figure A-1. Schematic structural view of the shedding vortices.
〈up〉〈vp〉〈wp〉: phase-averaged periodic (coherent) velocities
〈ut2〉
〈vt2〉
〈wt2〉: phase-averaged turbulent (incoherent) normal stresses
1
1. Introduction
Turbulent flows past a rectangular cylinder have been studied extensively by many investigators in fluid
and thermal engineering (e.g. Okajima, 1982; Durao et al., 1988; Taniguchi et al., 1988; Yao et al., 1994;
Lyn et al., 1995;). One of the main features of this flow configuration is the periodic vortex shedding.
Periodic force loading due to the pressure variation on the cylinder caused by the vortex shedding in
uniform flow has been a research subject in fluid engineering. On the other hand, periodic vortices shed
from the cylinder in channel flow and their affect on the channel wall have attracted much attention in
thermal engineering, especially from the view point of heat transfer enhancement.
We have investigated the thermal and flow characteristics in a turbulent channel flow obstructed by a
rectangular cylinder. It has been found from the unsteady heat transfer measurement with a thin-film heat
flux sensor that the wall heat flux fluctuates periodically in phase with the shedding vortices from the
cylinder, and that the vortices induce the wall-ward flow from the core region of the channel which result
in the heat transfer enhancement at the channel wall (Nakagawa et al., 1999a). In order to clarify the
structure of the shedding vortices related closely to the heat transfer enhancement, the two-dimensional
unsteady near-wake structure of the rectangular cylinder has been studied experimentally using a laser
Doppler velocimetry (LDV) by obtaining the phase-averaged velocity statistics as well as the
time-averaged ones (Nakagawa et al., 1999b).
Present paper is an extended study of the near wake structure of the rectangular cylinder related to our
previous work and is aimed at the three-dimensional near-wake structure, since it is well known that
above a critical Reynolds number, vortex shedding from nominally two-dimensional bluff bodies reveals
the three-dimensional behavior (Williamson, 1996).
2. Experimental Apparatus and Procedure
Figure 1 shows the schematic diagram of the test section. Experiments were carried out in a closed water
channel, in which the rectangular cylinder was placed symmetrically. The channel had a 50mm×350mm
cross-section. The height of the cylinder, h, was 10mm (the blockage ratio was fixed as 20% and the
aspect ratio of the cylinder was 35) and its width, b, was changed so that the width-to-height ratios of the
cylinder were b/h=0.5 and b/h=1.0. The origin of the coordinate system is located in the center of the inlet
of the test section, and x* stands for the streamwise distance from the rear surface of the cylinder. The
Reynolds number based on the channel height and the mean velocity at the inlet of the test section was
Re≡UmH/ν＝15000 and the resulting Strouhal numbers, St=fh/Um*, were 0.15 (b/h=0.5) and 0.13
(b/h=1.0) respectively, where Um* is the time-averaged velocity at the cross section where the cylinder
was installed.
Fig. 1.
Schematic of the test section.
2
The basic experimental procedure has been described in detail in our related paper (Nakagawa et al.,
1999b) and will only be briefly outlined. A two-color laser Doppler velocimetry (LDV) was used for the
measurements of the streamwise and normal velocities (U and V) and also of the streamwise and
spanwise velocities (U and W) to study the 3-D turbulent near-wake structure in the region of 1≦x*/h≦9,
-2.4≦y/h≦0 and 0≦z/h≦15, although the measurement region of the velocity component normal to the
wall was restricted to｜y/h｜≦1.8 due to the constraint on optical access near the wall. Because of the
large-scale organized fluid motion around a bluff body, time-varying velocity in the near wake includes a
periodic component and the velocity signal is analyzed in terms of the triple decomposition (Hussain and
Reynolds, 1970), where the instantaneous streamwise velocity is decomposed as U(t)=U+up+ut , in which
up is the periodic or coherent component at constant phase and ut corresponds to the turbulent or
incoherent contribution. The phase-averaged velocity statistics, which are denoted below by a symbol〈 〉,
over any extensive region of the near wake were obtained based on a reference signal. Since the reference
phase becomes less relevant due to phase jitter at distances far from the location of the reference signal,
the wavelet analysis (Farge, 1992; Li, 1998) was applied in the present study to the velocity signal of
LDV at each measurement point and the phase determined from the wavelet coefficients was used as the
reference phase.
The wavelet analysis is to decompose arbitrary signals into localized contributions that can be labeled
by a parameter of scale. The continuous wavelet transform of a function f(t) relative to an analyzing
wavelet ψ(t) can be defined as
∞
Wf (a, b) =
∫
∧
f (t )ψ a ,b (t )dt ,
−∞
where
∧
∧
−1
t −b
ψ a ,b (t ) is the complex conjugate of ψ a ,b (t ) defined as ψ a ,b (t ) = a 2ψ 
.
 a 
a and b are the scale dilation parameter and the translation parameter respectively. The wavelet coefficient
Wf(a,b) can be interpreted as the relative contribution of scale a to the signal at position b. From the
resulting complex-valued wavelet coefficients, we can obtain the modulus WfM, which gives the energy
density, and the phase WfP, which measures the instantaneous frequency.
Wf M =
(Wf R )2 + (Wf I )2
 Wf
Wf P = tan −1  I
 Wf R

 ,

where WfR and WfI are the real and imaginary part of Wf(a,b). In this paper, the Gavor wavelet function
was chosen as an analyzing wavelet ψ(t) and a 3rd-order spline interpolation was adopted for a signal
from LDV to make a function f(t) continuous.
Figure 2 illustrates the result of wavelet analysis of LDV signal f(t). The ordinate of the modulus WfM
and the phase WfP stands for the scale (frequency) and the intensity of the modulus WfM is expressed in
shade-coded. It is seen that the phase change with time corresponds well to the periodic velocity
fluctuation at a scale value of 9.2 where the modulus of wavelet coefficients is high.
The origin of the phase at each measurement point was calibrated with the reference phase determined
by the hot film probe at a fixed position in the flow. A total number of 20 phase bins was used for a
vortex shedding cycle and the number of samples used in the ensemble average was about 3000 per phase
bin.
3
WfP
10.5
9.5
8.5
10.5
WfM
9.5
f(t)
8.5
0
1000
0
2000
1000
t (ms)
(a) b/h=0.5
t (ms)
2000
(b) b/h=1.0
Fig.2. Wavelet transform of LDV signal.
3. Results and Discussion
Figure 3 shows the contours of phase-averaged spanwise vorticity, ＜ωz＞h/Um, for the cylinder of
b/h=0.5 on the mid-span (z/H=0.0), where ω z = ∂ v p ∂x − ∂ u p ∂y . In this figure, phase(0,10) means
that the lower half of the flow region is at phase 0 and the upper half is at phase 10. A total number of 20
phase bins was used for one cycle period and it was assumed that the flow in the upper half could be
obtained by reflecting the measured lower half about the centerline (Lyn et al., 1995). The abscissa of the
Fig. 3. Contours of coherent spanwise vorticity for b/h=0.5.
(a) phase (0,10)
(b) phase (4,14)
4
(c) phase (8,18)
figure is the non-dimensional distance x*/h from the rear surface of the cylinder. Vortices with opposite
signs shed alternately from the upper and lower sides of the rectangular cylinder and their downstream
movement with the passage of the phase are clearly observed in the figure.
It is well known that above a critical Reynolds number, vortex shedding from the nominally
two-dimensional bluff bodies reveals three-dimensional behavior (Williamson, 1996, Saha et al., 2003).
Hydrogen bubble visualizations of the flow both in x-y and x-z planes are shown in Fig. 4 at three
successive different times. A generating bubble wire was strung taut in the spanwise direction at y/h=-0.5
to observe the vortices shed from the lower side of the cylinder. The flow characteristic of x-y plane was
photographed using two-color slit-ray projection at z/H=0.0 (blue) and z/H=2.0 (red). As seen from the
visualization in x-y plane, two vortices with different color do not overlap each other as time passed, and
the spanwise vortex is seen in x-z plane to slant toward the upstream direction near the side wall of the
channel. The vortices at different spanwise positions lead to be out of phase indicating that the near-wake
of the rectangular cylinder has a three dimensional structure.
Figure 5 shows the contour plots of the phase-averaged coherent velocities 〈up〉,〈vp〉 and〈wp〉
at four spanwise positions of z/H=0.0, 1.0, 2.0 and 3.0 both for b/h=0.5 and b/h=1.0. The velocities are
normalized with the mean velocity Um and the abscissa of the figure is the phase. The ordinate of the
figure is the non-dimensional distance y/h and each contour map shows the lower half of the channel (y/h
≦0.0). The solid and broken line stand for the positive and negative value of the coherent velocity
respectively. In the case of b/h=0.5, the shedding vortices retain their two-dimensional structure in the
region of ｜z/H｜≦2.0 at the streamwise position of x*/h=1.5 as seen from the contours of the
streamwise and normal phase-averaged velocities. However, the distribution of the streamwise and
normal velocities becomes to be out of phase at z/H=3.0, which accompanied by an appearance of the
coherent spanwise fluid motion.
x*/h
x*/h
x*/h
Fig. 4. Hydrogen bubble visualization for b/h=0.5.
5
(A)
(b)
(a)
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0
0.4
0
-0.4
0
0
-0.4
0
0
0.4
0
0 0.4
-0.4
-0.4
0
0
5
0
0
10
0.4
15
phase
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
20
0
0
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0.8
0.4
0
-0.8
-0.4
-0.8
-0.4
0
-0.4
0
0.8
0.4
0
5
-0.4
0
1.2
0.8
0.4
0
-0.8
-0.4
0
0
0.8
0.4
0
10
phase
0
0
-0.4
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
15
20
0
0
0
0
0
0
0
0
0
0
0
0
0
5
0
0
10
phase
-0.4
0
0
0
-0.4
-0.4
0.4
0
5
10
15
phase
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0
0
0.4
0.8
0.4
0
0
5
15
20
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0
0.4
-0.8
0
0
-0.4
0
0
-0.4
10
15
phase
5
10
phase
-0.8
-0.4
0
0
0
0
0
0
x*/h=1.5
10
20
0
0
0
0.8
0.4
0.8
0.4
phase
0.8
0.4
0
-0.8
-0.4
0
0
-0.8
-0.4
0
5
0
10
phase
15
0
15
0.4
0.4
0
0
0
0
20
5
0
10
phase
5
0
10
b/h=0.5
15
phase
0
0.4
0
0
-0.4
0
0
-0.4
0
0.4
0
0 -0.4
0.4
5
-0.4
10
15
phase
20
15
20
0
0
0
0
0
0
0
0
0
0
0
0
0
5
10
phase
15
x*/h=4.0
b/h=1.0
Fig. 5. Contours of coherent (periodic) components of velocities at four
spanwise positions of z/H=0.0, 1.0, 2.0 and 3.0 for b/h=0.5 and b/h=1.0.
(A)
<up>/Um, contour interval 0.1
(C)
<wp>/Um, contour interval 0.1
6
(B)
z/H=0.0
20
-0.4
0.4
-0.4
x*/h=1.5
x*/h=4.0
z/H=1.0
0
0
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
0
0
0
0
20
0
0
z/H=2.0
(b)
0
0
0
0
-0.4
0
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0.8
0.4
-0.8
-0.4
z/H=3.0
0
(a)
0
5
15
0
0
(b)
0
0.4 0
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
0
-0.4
0
0
0
20
0
0
0
0
0.4
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0
0
-0.4
0
0.4
0
0.4
-0.4
0.4
0
0.4
-0.4
0
0.4
20
-0.8
0.8
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
(a)
0
-0.4
(b)
0
(b)
-0.4
-0.4
0
0
0.4
(a)
(C)
-0.4
0
0
(b)
-0.8
-0.4
0
0
0
(a)
(B)
(a)
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
<vp>/Um, contour interval 0.1
20
(A)
(a)
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
(b)
0.1
0.1
0.1
0.1
0.2
0.1
0.1
0.1
0
0.2
0.1
5
(B)
10
15
phase
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
20
0.1
0.1
0.1
0.2
0.1
0.1
0
5
10
phase
(a)
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0.2
0.20.4
0.2
0.4
0.2
5
15
20
10
15
phase
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
20
0.1
0.1
0.1
0.1
0.1
0.1
0
5
10
phase
0.2
0.2
0.2
5
10
phase
15
20
z/H=3.0
0.1
0
5
15
0.4
0.4
0.2
0.2
0.4
0.2
0.2
0
20
5
0.1
z/H=1.0
0.1
z/H=0.0
15
20
(b)
0.2
0.2
10
phase
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0.2
0.2
z/H=2.0
0.1
(a)
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0.2
0
(b)
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0.1
0.1
(b)
0.2
0
(a)
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
10
phase
15
20
0.2
0.2
0.2
0
5
10
phase
15
20
(C)
(b)
(a)
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
-2.5
0.2
0.1
0.2
0.1
0.3
0.2
0.1
0.1
0.1
0.1
0
5
10
phase
15
20
(a)
0.1
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0.1
0.1
0.1
0.1
0
x*/h=1.5
5
10
phase
15
0.1
0.1
0
20
(b)
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
0
-0.5
-1
-1.5
-2
5
10
phase
15
20
0.1
0
x*/h=1.5
x*/h=4.0
b/h=0.5
5
10
phase
15
x*/h=4.0
b/h=1.0
Fig. 6. Contours of incoherent (turbulent) components of Reynolds normal stresses
at four spanwise positions of z/H=0.0, 1.0, 2.0 and 3.0 for b/h=0.5 and b/h=1.0.
(A)
<ut2>/Um2, contour interval 0.02
(C)
<wt2>/Um2, contour interval 0.02
(B) <vt2>/Um2, contour interval 0.04
7
20
At a further downstream of x*/h=4.0 for b/h=0.5, the region where the shedding vortices retain their
two-dimensionality becomes narrow in the spanwise direction and the out of phase movement of the
primary vortex appears distinctly at z/H=2.0. Also, the spatial evolution of the spanwise velocity
component can be clearly observed to be widespread both in y and z directions. It is important to note
here that the positive 〈vp〉appears in the phase range between those in which the negative and positive
〈up〉appeare and vice versa. This is a consequence of the fluid motion which is induced toward the
channel wall by the vortex shed from the lower side of the cylinder and the successive one from the upper
side of the cylinder. In the case of b/h=1.0, the three-dimensional fluid motion may appear at a further
upstream position. The contours of the phase-averaged streamwise and normal velocities indicate that the
spanwise region in which the shedding vortices retain their two-dimensional structure is ｜z/H｜≦1.0 at
x*/h=1.5. Since the vorticity in the case of the aspect ratio of b/h=1.0 is weaker than that of the case of
b/h=0.5, the spanwise periodic fluid motion〈wp〉could not be captured clearly.
z/H=3.0
z/H=2.0
z/H=1.0
z/H=0.0
Fig. 7.
Contours of coherent and incoherent components of Reynolds normal
stresses for b/h=0.5 at z/H=0.0 , 1.0, 2.0 and 3.0 .
(a) <up2>/Um2, contour interval 0.04
(c)
<vp2>/Um2, contour interval 0.08
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(b) <ut2>/Um2, contour interval 0.02
(d) <vt2>/Um2, contour interval 0.04
The out of phase structure of the shedding vortices in the spanwise direction is also confirmed in Fig.6
which shows the contours of the phase-averaged turbulent components of Reynolds normal stresses ＜ut2
＞, ＜vt2＞and＜wt2＞normalized with the mean velocity. From the figure, the spanwise position in
which the out of phase movement of the primary vortex appears is seen to agree with the results in Fig.5.
However, it is noteworthy that the extent of the phase where the turbulent normal stress is large deviates
from that of the phase-averaged velocity by about five phase bins (a quarter of shedding vortex period).
Furthermore, in contrast to the contours of the coherent velocity component, the phase ranges, in which
＜ut2＞, ＜vt2＞and＜wt2＞ are high, agree with one another.
Contours of coherent (periodic) and incoherent (turbulent) components of Reynolds normal stresses are
shown in Fig. 7 at phase(0, 10) for the case of b/h=0.5. The figure indicates the contours in the reference
frame moving with the vortices and the reference frame velocity of 0.95Um is determined based on the
movement of vorticity peaks shown in Fig.2. In the figure, the Reynolds normal stresses are normalized
with the mean velocity and the location of the streamline center is denoted by a symbol(×). It is seen that
the streamwise coherent components ＜up2＞ have their peaks at the location swerved from the vortex
center in the direction normal to the flow, whereas the normal coherent components ＜vp2＞ have large
values in the region between the successive vortices shed alternately from the upper and lower sides of
the rectangular cylinder. Both coherent components become weak rapidly as the spanwise position
exceeds z/H=2.0. On the other hand, the incoherent Reynolds normal stresses in the streamwise and
normal directions, ＜ut2＞ and ＜vt2＞, have their peaks near the vortex center and remain large even in
the vortices led to be out of phase at z/H≧2.0.
4. Summary
The unsteady turbulent channel flow past a rectangular cylinder has been studied experimentally with a
laser Doppler velocimetry by obtaining the phase-averaged velocity statistics. The present work is aimed
at the three-dimensional near-wake structure of the shedding vortices from the cylinder. The experimental
results obtained are summarized in Fig. 8 as a schematic structural view of the shedding vortices. The
figure shows the vortex pair shed successively from the lower and upper sides of the cylinder. In this
configuration, the wall-ward flow toward the lower channel wall (＜vp＞＜0) is induced by the vortex
pair in the region between the shedding vortices. Associated with the positive and negative vortices, the
Fig. 8.
Schematic structural view of the shedding vortices.
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streamwise coherent velocity component ＜up＞ takes a large value in the regions swerved from the
vortex center as shown in the figure. In these regions, the coherent (periodic) components of Reynolds
normal stresses have their peak values, while the distribution of their incoherent (turbulent) components
overlaps with the contour of the spanwise vorticity. Although the shedding vortex structure at the
mid-span region is indeed two-dimensional, the vortices at the spanwise positions remote from the
mid-plane of the channel are caused to be out of phase, accompanied by an appearance of the coherent
spanwise fluid motion. Near the vortex center, the phase-averaged spanwise velocity ＜wp＞ is negative
and the turbulent component of the Reynolds nomal stress ＜wt2＞ has large values.
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