12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

TIME-RESOLVED PIV INVESTIGATION OF GAP RATIO EFFECTS ON
TURBULENT FLOW PAST A SQUARE CYLINDER WITH SMOOTH AND
ROUGH UPSTREAM WALLSHeath Chalmers
Department of Mechanical Engineering
University of Manitoba
Winnipeg, MB, R3T 5V6, Canada
chalmerh@myumanitoba.ca

Xingjun Fang
Department of Mechanical Engineering
University of Manitoba
Winnipeg, MB, R3T 5V6, Canada
fangx@myumanitoba.ca

Mark F. Tachie
Department of Mechanical Engineering
University of Manitoba
Winnipeg, MB, R3T 5V6, Canada
mark.tachie@umanitoba.ca

ABSTRACT

shedding structures in the wake region are suppressed for turbulence intensities larger than 20%. An experimental study on
near-wall square cylinder by Martinuzzi et al. (2003) showedthat the vortex shedding frequency in the wake region is independent of gap ratio, which is at variance with large eddy
simulation performed by Samani & Bergstrom (2015).
The flow around a square cylinder becomes more complex
when a finite gap is introduced beneath the cylinder due to theinteraction between the separated shear layers from either sideof the cylinder and the shear layer emanating from the wall.
In general, vortex shedding is suppressed when the gap ratio
is decreased below a threshold value, nonetheless, the critical
gap ratio is not universal but ranges from G/h = 0.3 (Martinuzzi et al., 2003) to G/h = 0.5 as summarized by Samani &Bergstrom (2015). Samani & Bergstrom (2015) and Shi et al.
(2010) showed that the energy content of the low order proper
orthogonal decomposition (POD) modes is a function of gap
ratio, and the energy of the first two POD modes is reduced
when the cylinder is closer to the wall. Shi et al. (2010) also
reported relatively poor coherence of the trajectory of the first
two POD mode coefficients when the square cylinder is placednear the wall compared to larger gap height cases.
In spite of numerous prior studies, the present understanding of gap ratio effects on the spatio-temporal characteristics of vortical structures around square cylinders is deficient.
Furthermore, past studies were performed over smooth walls.
Since upstream wall roughness significantly alters the turbulent and unsteady characteristics in forward- backward-facing
step flows (Fang & Tachie, 2018), the present study investigates the combined effects of wall roughness and gap ratio on
the turbulence statistics and spatio-temporal characteristics ofcoherent structures around a near-wall square cylinder with upstream wall roughness.

Time-resolved particle image velocimetry is used to investigate turbulent flow separation around a square cylinder
placed at various gap heights away from a smooth and roughupstream wall. The gap heights range from 0.0 to 2.0 times thecylinder height, while the Reynolds number based on the freestream velocity and cylinder height is 12 750. The combined
effects of gap ratio and wall roughness on the turbulence statistics and coherent structures are analysed. In particular, proper
orthogonal decomposition in conjunction with phase averaging are employed to investigate the spatio-temporal characteristics of the vortex shedding motions. The results indicate that
the Reynolds stresses are disproportionately high in the upper
wake region for smaller gap ratios, but are similar in magnitude to those in the lower wake region at high gap ratios. The
vortex shedding motion behind the cylinder is disrupted forgap ratio equal to 0.3 times the step height, particularly whenwall roughness is present. The dominant structure behind thesurface-mounted cylinder reflects its interaction with coherent
structures of the incoming turbulent boundary layer whereas
those behind the offset cases are dictated by the vortex shedding motion. Phase averaging shows that wall roughness reduces streamline curvature and increases the Reynolds stressdecay rate.

Introduction
Turbulent flow past a square cylinder in the vicinity of arough wall exhibits distinct and complex flow characteristics
such as flow separation, acute streamline curvature, and vortex
shedding. Many experimental studies have investigated the effects of turbulence intensity on pressure coefficient and vortex
shedding of a square cylinder in uniform flow. The investigation by Saathoff & Melbourne (1999) showed that increasing
the turbulence intensity from 5% to 12% decreases the meanpressure coefficient near the leading edge of the cylinder, whilea further increase in the turbulence intensity from 12% to 24%
increases the mean pressure coefficient. Meanwhile, the vortex

Experimental Procedure
The experiments were conducted in an open recirculatingwater channel. To investigate the effects of wall roughness on

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
reported by Shi et al. (2010). However, the present values ofmean recirculation length are shorter than those reported previously due to the thicker oncoming turbulent boundary layer
along with higher turbulent intensity induced by the upstream
rough wall in the present study. Table 2 shows the recirculation
length for all test cases investigated. Following the observationfrom Figure 1, the recirculation length decreases both as thegap size increases, and as a result of upstream wall roughness.

the flow physics surrounding a square cylinder, two different
roughness conditions were employed. For the smooth case, theflow was tripped at the channel inlet using a 100 mm long stripof 16-grit sandpaper. This was followed by a smooth acrylic
plate leading up to the square cylinder 3100 mm downstreamof the channel inlet. For the rough case, the flow trip consistedof a 100 mm long toothed barrier placed at the channel inlet
followed by sandpaper roughness leading up to the cylinder
at 4500 mm downstream of the channel inlet. The boundary
layer, δ , and momentum, θ , thicknesses, along with the shapefactor, δ /θ are shown in Table 1 for the smooth and roughincoming boundary layers. The Reynolds numbers based on
the step height and momentum thickness respectively are also
shown. The square cylinder had a body height h, of 25 mm andthe gap height, G, was varied so that the examined gap ratios
G/h were 0.0, 0.3, 0.5, and 2.0.

Table 2.

Lr /hTable 1. Parameters characterizing the state of the incoming
TBLs for different wall conditions.

Smooth TBL

Rough TBL

U∞

0.51

0.51

δ /h

3.6

7.2

θ /h

0.46

1.05

H

1.34

1.56

Reh

12 80012 800Reθ

5 800

13 400Recirculation length, Lr /h, for various test cases.

G/hSmooth Wall

Rough Wall

0.0

10.4

7.8

0.3

2.6

2.1

0.5

1.3

1.7

2.0

0.6

0.7

The contour levels of the u0 u0 show strong turbulence motion in the separated shear layer emanating from the leadingedge of the cylinder as well as from trailing edge of the cylinder at both the upper and lower surfaces. The magnitude of
the Reynolds stress is larger in the shear layer emanating fromthe upper edge of the cylinder when the cylinder is placed near
the wall. The presence of the wall also introduces a region ofstrong velocity fluctuations near the wall immediately downstream of the cylinder trailing edge, as shown in Figure 1a.
Comparison with the smooth wall data (not shown) reveals
that for the G/h = 0.0 and 0.3 cases, the Reynolds stresses arelarger over the rough wall but also decay faster in the downstream direction.

A time-resolved particle image velocimetry (TR-PIV)
system was used to measure the velocity field at the mid- spanof the test section. 10µm silver coated hollow glass spheres
were used as seeding particles. The seeding particles were illuminated using a dual-cavity dual-head neodymium yttrium
lithium fluoride (Nd:YLF) laser emitting green light with amaximum energy of 30 MJ/pulse. Two high-speed complementary metal-oxide semiconductor (CMOS) cameras wereused to capture particle images at a frequency of 807 Hz upstream of, over, and downstream of the square cylinder. 48
000 images were taken for the G/h = 2.0 case while 60 000
images were collected for the remaining test cases. DaVis version 10 supplied by LaVision Inc. was used to calculate thevelocity vectors and an in-house MATLAB script was developed to process the resultant data. The vector calculation ofeach test case was processed using initial and final interrogation areas of 128 pixel × 128 pixel with 50% overlap and 24
pixel × 24 pixel with 75% overlap, respectively.

Results & Discussion
Turbulence Statistics
Figure 1 shows the streamwise Reynolds normal stresses,
u0 u0 , for the G/h = 0.3 and 2.0 cases, with both the mean
streamlines and zero-velocity isopleths superimposed. Fromthe mean streamlines, three distinct recirculation regions arepresent for each test case, one above the cylinder and one emanating from each of the upper and lower edges of the cylinder
in the wake region. The size of the recirculation region is determined using the zero-velocity contour and shortens as the gap
ratio increases from G/h = 0.3 to 2.0. This variation is also

Figure 1. Contours of the u0 u0 Reynolds stresses for therough wall G/h = 0.3 (a) and G/h = 2.0 (b) test cases, respectively. The mean streamlines (solid black) and zero-velocity
contour (dashed red) are superimposed.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Proper Orthogonal Decomposition
In order to further investigate the spatial characteristics
of coherent structures in the cylinder wake, a POD algorithm
was implemented according to the snapshot method proposedby Sirovich (1987) wherein a singular value decomposition is
performed on the matrix ψ = V ΛM T , where the matrices V ,Λ, and M represent the POD modes, energy, and coefficients
respectively. To show a typical pattern associated with vortex
shedding in the wake of a cylinder, Figure 2 displays the first
two POD modes for the G/h = 2.0 case with a rough upstream
wall. The first POD mode shows alternating positive and negative vortices emanating from the upper and lower surfacesof the square cylinder. The second POD mode shows similar
structures as the first POD mode, but with a downstream shiftof approximately 1.5 step heights. Furthermore, the first two
POD modes of this test case contribute 56.2% of the total POD
energy. These observation are in accordance with a POD modepair as defined by Riches et al. (2018) for vortex shedding in
the wake of a cylinder.

Figure 3. Contours of the first two POD modes for thesmooth wall G/h = 0.3 test case.

Figure 2. Contours of the first two POD modes for the roughwall G/h = 2.0 test case.
Figures 3 and 4 show the POD mode contours for
the G/h = 0.3 case with smooth and rough upstream wall
respectively. For the smooth case, alternating positive andnegative vortices are observed in both POD modes, however asignificant incline of vortices away from the wall is noticeable.
Also, the downstream shift of coherent structures from thefirst mode to the second mode is less than one step height.
The rough case exhibits markedly different behaviour. Within
the first POD mode, multiple vortices are clustered near thebottom wall, whereas the second POD mode exhibits twolarge vortices centred at the body centerline downstream of
the cylinder. These observations deviate from the traditional
understanding that von Karman vortices present themselvesin the first two POD modes (Samani Bergstrom, 2015; Shi etal., 2010). These results suggest that upstream wall roughness
significantly modifies the vortical structures downstreamof the gap exit near the wall, and therefore, disturbs theoccurrence of organized vortex shedding motion.

Figure 4. Contours of the first two POD modes for the roughwall G/h = 0.3 test case.

Several researchers have used phase portraits of the first
two POD mode to visualize the coherence of the vortex shedding motions in the wake of an offset cylinder (Shi et al. 2010;
Riches et al. 2018). In this same vain, Figure 5 compares the
temporal trajectory of the first two POD mode coefficients forthe rough wall G/h = 0.3 and the smooth wall G/h = 0.5 cases.
The orbital trajectory is discernible to varying degrees for all
test cases indicating the occurrence of vortex shedding motions in each case. Evidently, the orbital is more organized forthe larger gap ratio and smooth wall cases. This is indicativeof weaker perturbation of incoming turbulence on the vortex
shedding motion when the cylinder is farther away from thewall especially for the upstream smooth wall case. In particular, for the rough wall G/h = 0.3 case, the trajectory of the first3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
two POD mode coefficients is the most random compared withother cases. This randomness is attributed to the intermittencyof vortex shedding motion imposed by the significant incoming turbulence through the gap. Bai & Alam (2018) and Bailey
et al. (2002) observed that for a square cylinder, the regularvortex shedding motion is interrupted as the shear layer intermittently reattaches on the side surfaces. The intermittency ofvortex shedding observed in Figure 5 is likely due to a strong
tendency of the mean flow to reattach on the undersurface ofthe cylinder due to the high incoming turbulence intensity.

Figure 6. Frequency spectra of the first two POD mode coefficients and the velocity fluctuations at the step height inthe corresponding turbulent boundary layer for the G/h = 0.0(a,b), G/h = 0.3 (c,d), and G/h = 0.5 (e,f) test cases.

Figure 5. Phase portraits of the first two POD mode coefficients for the G/h = 0.3 (a,b) and G/h = 0.5 (c,d) test cases.

Figure 5 to perform phase averaging of the flow field. Figures
7 and 8 show the phase-averaged streamwise Reynolds normal
stress, hu0 u0 i, at three different phase ranges for the smoothand rough G/h = 0.3 cases respectively. The phase averaged
streamlines provide insight into the unsteady oscillations of
the turbulent recirculation region and is reminiscent of theunsteady recirculation described by Sherry et al. (2010) over
a forward-facing step using phase averaging. In accordancewith the results from Table 2, the streamwise extent of therecirculation bubble is larger for smooth wall case in Figure 7than in the rough wall case shown in Figure 8. Furthermore,
the height of the recirculation bubble above the cylinder ishigher in the smooth wall case than the rough wall case. This
indicates that wall roughness decreases the vertical amplitudeat which the wake oscillates behind a square cylinder, andreduces the intensity of streamline curvature in the wake.
These observations are in accordance with the findings of
Shi et al. (2010). The same modulation of the wake is notobserved below the square cylinder due to wall confinement
effects. Figure 7 further reveals that at specific phase angles,
a third recirculation bubble exists downstream of the cylinder
near x/h = 4.0. This additional recirculation bubble issuppressed when wall roughness is introduced upstream of
the cylinder.

In order to investigate the effects causing the overall
dampening of vortex shedding at smaller gap ratios, thefrequency spectra of the first two POD mode coefficients arepresented in Figure 6 along with the spectra of the velocity
fluctuations at the cylinder centerline in the upstream turbulent
boundary layer. For the surface-mounted cylinders, the first
two POD mode oscillate at low frequencies between St = 0.02
and St = 0.03 which is similar to that of the incoming flow
conditions. Once the cylinder is offset from the wall, thetwo mode coefficients possess identical dominant frequencies
for each test case, that is distinct from the boundary layer
frequency. In general, the dominant frequency of vortex
shedding motions behind a square cylinder are reported tobe between St = 0.129 and St = 0.140 (Durao et al. 1991;
Bosch et al. 1996; Kumeran & Vengadesan 2007). This range
is consistent with the dominant frequency for the smoothwall G/h = 0.5 case, however the other test cases possess
lower dominant frequencies. Based on the results shownin Figure 6, the dominant frequency becomes smaller withboth smaller gap ratios and increased turbulence intensity
as a result of wall roughness. Overall, it is concluded thatthe dominant structure for the wall-mounted cylinder case isdictated by the coherent structures embedded in the incoming
flow condition. However, the vortex shedding motion behind
an offset cylinder overwhelms the perturbation of incoming
turbulence.

The progression of the phase-averaged Reynolds stresses
through different phase bins depicts the downstream convection of recirculation bubbles. Recirculation regions originating over the step are convected forwards and dissipate in thedownstream direction. Shi et al. (2010) also showed that as
the recirculation bubbles are convected downstream, regions of
low-momentum fluid near the gap exit are carried downstream
as well. During this process, patches of elevated Reynolds

Phase Averaging

 (2) √ 
A phase angle, φ = tan−1 aa(1) √λλ1 , is defined using2
the trajectory map of the first two POD mode coefficients in4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022stresses that originate in the upper and lower shear layers arealso convected downstream. Compared with the smooth caseshown in Figure 7, these patches of hu0 u0 i in the rough wallcase generally decay faster and possess stronger magnitudes in
the upper shear layer. This is why the elevated time averaged
Reynolds normal stress extends further in the downstream direction, and has a smaller magnitude in the upper shear layer
with an upstream smooth wall. Furthermore, the peak hu0 u0 i isfound along the separating streamlines which is in god agreement with both Sherry et al. (2010) and Shi et al. (2010).

Figure 8. Phase-averaged streamwise Reynolds normal
stress, hu0 u0 i, for the rough wall G/h = 0.3 case at three different phase bins.
A proper orthogonal decomposition (POD) was performed to investigate the coherent structures in the wakeregion. Notably, the contribution of the first two POD modes
to the total energy decreases significantly at smaller gap ratios.
This suggests that the coherence of vortical structures in thewake region decreases at smaller gap ratios and with wallroughness. This observation is also noted in the phase portraitsof the first two POD modes, showing that proximity to thewall and wall roughness both act to disrupt vortex shedding
motions in the wake. Additionally, analysis of the frequency
spectra shows that for the wall-mounted case, the fluctuationof the first POD mode coefficient reflects the interaction of
the cylinder with the incoming flow structures. For the offset
cylinders, on the other hand, the first two mode coefficients
possess identical dominant frequencies that are significantly
higher than the dominant frequency of the turbulent boundary
layer, and represent the vortex shedding motions in the wakeregion.

Figure 7. Phase-averaged streamwise Reynolds normal
stress, hu0 u0 i, for the smooth wall G/h = 0.3 case at three different phase bins.

Conclusions
Turbulent flows past square cylinders offset from awall at different distances were studied using time-resolved
particle image velocimetry. The examined ratios of the gap
height underneath the cylinder to cylinder height were 0.0,
0.3, 0.5, and 2.0. Two upstream roughness conditions were
employed to investigate wall roughness effects on the mean
flow, Reynolds stresses, and vortex shedding phenomenon in
the cylinder wake.

Phase averaging reveals that both mean recirculation
bubbles and patches of elevated Reynolds stresses are convected downstream through time. Furthermore, the temporalvariation of the streamlines shows that wall roughness acts
to diminish the intensity of streamline curvature and also
increases the streamwise decay rate of Reynolds stresses in
the wake.

There exists two counter-rotating recirculation bubbles
in the wake of offset cylinders, while only one reverse flow
region exists behind a wall-mounted cylinder. The streamwise
extent of the recirculation bubbles decreases when there
is wall roughness present and at smaller gap ratios. The
streamwise decay rate of Reynolds stresses increases with gap
ratio. Furthermore, the magnitudes of Reynolds stresses arelarger in the upper recirculation region when the gap ratio is
equal to 0.3 times the step height, but is similar to that in thelower recirculation region when the gap size is increased to
2.0 times the step height.

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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