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

EFFECTS OF STREAMWISE ASPECT RATIO ON SPATIO-TEMPORALCHARACTERISTICS OF RECTANGULAR CYLINDERS IN UNIFORM FLOWS
Sedem Kumahor
Department of Mechanical Engineering
University of Manitoba
Winnipeg, Manitoba, R3T 5V6, Canada
kumahors@myumanitoba.ca
Mark F. Tachie
Department of Mechanical Engineering
University of Manitoba
Winnipeg, Manitoba, R3T 5V6, Canada
mark.tachie@umanitoba.ca
ABSTRACT
Turbulent flows over rectangular prisms with streamwise
aspect ratios of AR = 1, 2 and 4 were experimentally studied
using time-resolved particle image velocimetry at a Reynolds
number (based on free-stream velocity and prism height) of
16200. These aspect ratios represent prisms for which themean separated shear layers from the leading edges are sheddirectly into the wake region (AR1), reattach over the prism(AR4), and an intermediate case (AR2). The effects of aspectratio on the mean flow are investigated using the streamwisemean velocity and wake recirculation length. The unsteadycharacteristics of the small-scale Kelvin-Helmholtz vortices
and the energetic large-scale von-Kármán vortices are
examined using frequency spectra of the streamwise velocity
fluctuations, reverse flow area and proper orthogonal
decomposition (POD). The ratio between the frequency
associated with the Kelvin-Helmholtz vortices to thefundamental von-Kármán shedding frequency is shown tovary significantly with aspect ratio. The mean flow topology
reveals a massive recirculation bubble in the wake of the AR2
prism, compared to AR1 and AR4. The differences in thevortex formation and shedding process are revealed in the
dynamics of the first POD mode pair. A dual vortex shedding
instability is manifested over the AR2 prism, which is shown
to be intimately connected with the flow dynamics at thetrailing edge. For AR1 and AR4, the first POD mode paircontain similar energy content, representing the large-scale
von-Kármán vortices.

et al., 2015; Lander et al., 2018; Moore et al., 2019), and thefollowing correlation for the dependency of their frequency
ratio on Reynolds number was proposed by Lander et al.
(2018): 𝑓𝐾𝐻 /𝑓𝑉𝐾 = 0.18𝑅𝑒 0.60 . A rectangular prism with an
aspect ratio of 5 was also investigated by Moore et al. (2019)
and their results showed that 𝑓𝐾𝐻 /𝑓𝑉𝐾 is significantly lower
than corresponding values for a square prism at similar
Reynolds numbers. This observation demonstrates that thedegree of interaction between the Kelvin-Helmholtz and vonKármán vortices is dependent on their proximity or aspectratio of the prisms.
The alternate shedding of the von-Kármán vortices resultsin the formation of coherent and chaotic fluctuating motions
that interact non-linearly. Many techniques, including phaseaveraging (Hussain and Reynolds, 1970), linear stochastic
estimations (Adrian, 1977) and proper orthogonal
decomposition (Lumley, 1967) have been developed to extractand investigate the dynamics of the coherent structures. In
recent years, proper orthogonal decomposition (POD) has
gained popularity in extracting coherent structures fromwhole-field velocity data obtained using techniques such as
particle image velocimetry (PIV). POD provides the optimal
approximation of the flow field into a given number of modes.
The first mode contains the highest turbulent kinetic energy
content and in increasing mode number, a decrease in energy
content. For flow over an Ahmed body, Thacker et al. (2013)
showed that the first POD mode is associated with the low
frequency flapping of the recirculation region over the rearslanted region. A similar observation was reported by Fang
and Tachie (2019) and Kumahor et al. (2021) for flows over a
forward-facing step. For flows over circular cylinders in
uniform flow, Perrin et al. (2007) and Riches et al. (2018) haveshown that the first POD mode pair is associated with thefundamental vortex shedding frequency. Similarly, vanOudheusden et al. (2005) and Mohebi et al. (2018)
investigated coherent structures around rectangular prisms in
uniform flow and showed that the first POD mode pair reveals
the dominant flow features of the vortex shedding process.
The above-mentioned studies have shown that the dynamics
of the coherent structures can be described using POD.
However, the interesting variations in the von-Kármán vortex
shedding process around rectangular prisms in uniform flow
over a wide range of streamwise aspect ratios is still not wellunderstood.
Thus, the goal of the present study is to investigate theturbulent statistics as well as the spatial and temporal featuresof flows over a square prism (AR1) and rectangular prisms of

INTRODUCTION
Separated shear layers induced by sharp-edged bluff
bodies in uniform flows have been the subjected of extensive
research over the past decades due to their prevalence in manyindustrial applications. At high Reynolds numbers (Re > 150),
the flow separates at the sharp leading edge and either
reattaches onto the prism if the streamwise aspect ratio (AR =𝐿/ℎ, where 𝐿 is streamwise length and ℎ is the prism height)
is larger than 3.5 or shed directly into the wake region for the
low aspect ratio prisms (AR < 2.0). The separated shear layer
at the leading edge is characterized by small-scale KelvinHelmholtz (KH) vortices with characteristic high frequency
that manifest as broadband peaks. The footprint of theenergetic large-scale von-Kármán (VK) vortices that are shedalternately into the wake region manifests as dominant peaks
at low-frequency. The interaction between the KelvinHelmholtz and von-Kármán vortices for a square prism has
been examined in considerable detail (Brun et al., 2008; Trias112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
streamwise aspect ratio 2 (AR2) and 4 (AR4) using a timeresolved particle image velocimetry. Furthermore, POD
analysis is performed to extract the dynamics of the coherent
structures around the prisms. These aspect ratios are selected
to ensure a direct shedding into the wake region (AR1), a mean
reattachment onto the prism (AR4) and an intermediate case(AR2) where the separated shear layer does not reattach but
exhibits a more complex dynamics compared to the squareprism. In each case, velocity measurements were performed
over the prism and near-wake region simultaneously toevaluate the interaction between the shear layer from theleading edge and the von-Kármán vortices.

(a)

(b)

EXPERIMENTAL SET-UP
The experiments were performed in an open recirculatingwater channel at the University of Manitoba. The streamwise
length, spanwise width and vertical height of the test section
are 6000 mm, 600 mm and 450 mm, respectively. Three
rectangular prisms with fixed vertical height, ℎ = 30 mm andspanwise width, 𝐵 = 580 mm were machined from smoothacrylic plates, with streamwise lengths of 30 mm (AR1), 60
mm (AR2) and 120 mm (AR4). The water depth was kept
constant at 430 mm and the prisms were positioned such thatthe prism mid-height was 215 mm above the bottom wall. The
free-stream velocity was set to 𝑈𝑒 = 0.540 m/s and Reynolds
number based on cylinder height and free-stream velocity was
16200. The turbulence intensity (𝑇𝑢 = √̅̅̅̅̅
𝑢′𝑢′/𝑈𝑒 ) measured
at the channel mid-height in the absence of the cylinder was
1.2%.
Time-resolved particle image velocimetry was used toperform velocity measurements at the mid-span of thechannel. The water was seeded with 10 μm silver coatedhollow glass spheres with density of 1400 kg/m3. A diodepumped dual-cavity high-speed Neodymium-doped yttrium
lithium fluoride (Nd:YLF) laser with maximum pulse energyof 30 mJ/pulse for each cavity was used to illuminate theseeding particles. Two high speed 12-bit complementary
metal oxide semiconductor cameras of resolution, 2560 pixel× 1600 pixel positioned side-by-side were used tosimultaneously image the flow in two fields of view (FOV).
Since the flow is symmetric about the prism centerline, theFOVs were chosen such that the region over the prism and theupper half of the wake region were captured. The dimensionsof the FOVs over the prism and wake region were,respectively, 3.3ℎ × 2.1ℎ and 4.6ℎ × 2.9ℎ, resulting in digital
image resolutions of 25.0 pixel/mm and 18.4 pixel/mm,
respectively. The sampling frequency was set to 800 Hz and
96000 images were captured for each test case. Dataacquisition and image post-processing were performed using
commercial software (DaVis version 10.0.5) supplied by
LaVision Inc. Velocity vectors were calculated using a GPUaccelerated multi-pass cross-correlation algorithm with asingle initial pass of 128 pixel × 128 pixel interrogation area(IA) with 50% overlap followed by four final passes of 24
pixel × 24 pixel IA with 75% overlap. The vector spacing was
0.20 mm (0.007h) and 0.33 mm (0.011h) for FOVs over theprisms and in the wake region, respectively. The resulting
measurement uncertainty within 95% confidence level was
2.55% and 6.88% in the mean velocities and Reynolds
stresses, respectively, at the locations of high local turbulence
levels (Kumahor and Tachie, 2022).

(c)

Figure 1: Mean streamwise velocity (𝑈) for AR1 (a), AR2(b) and AR4 (c), superimposed with the mean streamlines(black continous lines), isopleth of U = 0 (red continuouslines) and symbol + marks the local peak over the cylinder
with the respective magnitude written above. The insert in
the top left corner shows contours of the turbulent kinetic
energy near the leading edge.
RESULTS AND DISCUSSION
Figure 1 shows contours of the mean streamwise velocity(𝑈) superimposed with mean streamlines and the isopleth of
𝑈 = 0. The separated shear layer is directly shed into the wakeregion for AR1 and AR2, while mean reattachment occurs at
𝑥/ℎ = 3.6 over AR4. The recirculation lengths, which were
determined as the streamwise distance between the trailingedge to location of 𝑈 = 0 along the symmetry plane, were 𝑥/ℎ= 0.60, 2.40 and 0.60, respectively, for AR1, AR2 and AR4.
It was observed that instantaneous reattachment occurs overthe cylinders 35%, 20% and 80% of the time for AR1, AR2
and AR4, respectively. Instantaneous flow visualization
reveals that the backflow deflected upward at the trailing edgeof AR2 significantly attenuates the reattachment of theseparated shear layer. Consequently, the shear layer isstreamwise elongated and descends further downstream into
the wake region. Over the prisms, the maximum value of 𝑈
decreases monotonically with an increase in AR. Although notshown, the turbulent kinetic energy also decreases in′ 𝑢′ +
̅̅̅̅̅̅
magnitude as AR increases, with peak values of 0.5(𝑢2′′̅̅̅̅̅̅
𝑣 𝑣 )/ 𝑈𝑒 = 0.504, 0.202 and 0.072, respectively, for AR1,AR2 and AR4.
Figure 2 shows the premultiplied frequency spectra of
streamwise fluctuating velocity at selected locations along theshear layers. Dominant peaks at Strouhal numbers
corresponding to the fundamental von-Kármán shedding
frequency for the three aspect ratios are evident in each plot,reflecting the global nature of the von-Kármán instabilities.
The Strouhal number of 𝑆𝑡𝑉𝐾 = 0.138, 0.084 and 0.134 for2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022(a)AR1, AR2 and AR4, respectively, are in good agreement withprior studies (Okajima, 1982; Mohebi et al., 2017; Lander etal., 2018). Broadband high frequency peaks reminiscent ofsmall-scale Kelvin-Helmholtz instabilities are also observed,
however, these peaks are comparatively lower than the low
frequency peaks and do not emerge until 𝑥/ℎ = 0.2 (for AR1
and AR2) and 𝑥/ℎ = 0.4 (for AR4). The frequencies
associated with the Kelvin-Helmholtz instability for AR1 andAR2 are similar but relatively higher than for AR4. The valuesof 𝑆𝑡𝐾𝐻 /𝑆𝑡𝑉𝐾 = 24, 38 and 20 clearly demonstrate that theinteraction between the small-scale and large-scale energetic
vortices is not a linear function of aspect ratio. The spectral
migration to lower frequencies observed in the spectra are dueto the pairing of smaller vortices to form successively larger
vortices with characteristic lower frequencies.
Contours of premultiplied frequency spectra of thestreamwise fluctuating velocity at 𝑥/ℎ = 0.20 from the leadingedge are shown in Figure 3. In addition to the peaks
corresponding to the fundamental von-Kármán shedding
frequencies and Kelvin-Helmholtz instabilities, distinct peaks
are also observed at 𝑆𝑡 = 2.24 and 7.11, irrespective of aspectratio. However, these higher frequency peaks do not exhibit
the broadband characteristics of the Kelvin-Helmholtz
instability. Previous studies over square cylinders reported
broadband peaks at St = 6 – 8 (Trias et al., 2015; Lander et al.,2018; Moore et al., 2019), and the limited information
available in the literature suggests that the Kelvin-Helmholtz
frequency varies with Reynolds number, turbulence intensityand blockage ratio (Brun et al., 2008).
Proper orthogonal decomposition (POD) is performed to
extract the dominant coherent structures and examine theircontributions to the vortex shedding process. In the current
implementation of POD, the snapshot POD method proposedby Sirovich (1987) and detailed in Fang and Tachie (2019) is
employed. The POD analysis is performed for all cases within
the area 𝑥/ℎ × 𝑦/ℎ ∈ [0.0, 6.3] × [−0.5, 1.5], subtracting thearea determined as the prism. This implementation area waschosen to encompass the regions of elevated Reynolds stresses
over and behind the prisms. As remarked by van Oudheusden
et al. (2005) and Riches et al. (2018), two POD modes are
typically required for a low-order representation of thecoherent structures associated with the von-Kármán vortex
formation and shedding process. The two modes, often
referred to as the mode pair, must possess similar energy
content. A cross plot of their temporal coefficients should
form an ellipse and their spatial modes must be shifted by onequarter wavelength (𝜆/4) of the coherent structures. With thisframework, the POD mode information is examined toelucidate the influence of streamwise aspect ratio on the
dynamics of the coherent structures.
Figure 4 shows the energy content of the first six mostenergetic POD modes. The total energy contribution of thefirst six modes is 75.2%, 68.6% and 63.3%, respectively, for
AR1, AR2 and AR4. Meanwhile, the first mode paircontribute 66.3%, 54.2% and 54.7%, respectively, for AR1,AR2 and AR4. Consistent with the observations by van
Oudheusden et al. (2005), Mohebi et al. (2018) and Riches etal. (2018), the first mode pair possess similar energy content
for AR1 and AR4, indicating the dominance of the periodic
von-Kármán vortices shed around both cylinders. On the other
hand, the first POD mode is 43% higher than the second POD
mode for AR2. The lower energy content in the second POD
mode is attributed to the loss of coherence and the existenceof a dual vortex shedding process, which was previously

(b)

(c)

Figure 2: Premultiplied frequency spectra of thestreamwise fluctuating velocity (Euu) for AR1 (a), AR2 (b)and AR4 (c) at selected streamwise locations along theouter edge of the shear layer.
(a)

(b)

(c)

Figure 3: Contours of the premultiplied frequency spectraof the streamwise fluctuating velocity (Euu) at 𝑥/ℎ = 0.20
AR1 (a), AR2 (b) and AR4 (c). Horizontal dashed lines
represent the vertical location of the maximum turbulent
kinetic energy.
312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
(a)

(a)

(b)
(b)

(c)

(c)

Figure 5: Cross plot of the temporal POD mode
coefficients a(1) and a(2) for AR1 (a), AR2 (b) and AR4 (c).
structures are significantly more organized in AR1 and AR4,compared to AR2. This indicates significant cycle to cycle
variation in the vortex shedding patterns over and behind theAR2 prism. Frequency spectra of the first four mode
coefficients (𝑎(𝑖) , where 𝑖 is the mode number) and radius
calculated from the orbits in Figure 5 are presented in Figure
6. The results show dominant peaks at the fundamental
shedding frequency for 𝑎(1) and 𝑎(2) , irrespective of aspectratio, suggesting that the first two modes are indeed associatedwith the von-Kármán vortices. These two modes also show
subdominant peaks at the second harmonic for AR1 and AR4,and third harmonic in the case of AR4. The spectra of 𝑎(3) and𝑎 (4) reveal peaks at the fundamental shedding frequency andsecond harmonic for AR1, or the second and third harmonics
for AR4. The peaks observed at these harmonics are areflection of the breakup of large-scale vortices associatedvon-Kármán shedding vortices into smaller and smaller
vortices.
The spectra of 𝑎(1) to 𝑎(4) for AR2 on the other hand,reveal a sub-dominant peak at a second frequency
corresponding to 𝑆𝑡 = 0.145. The peaks occurring at 𝑆𝑡 =0.084 and 0.145 is consistent with the dual vortex shedding
frequencies reported by Okajima (1982) and Knisely (1990)
for the AR2 prism. Reminiscing from Figure 2(b), it is evident
that the peak at 𝑆𝑡 = 0.145 is not associated with the separated
shear layer but only manifests within the wake region. Thissuggests that the state of the developing boundary layer near
the trailing edge significantly impacts the dynamics of thevortex shedding process over the rectangular prisms. The
radius calculated from the orbits of the first mode pair peak at

Figure 4: Relative contribution (%) to the TKE by thestreamwise (𝑢’) and vertical (𝑣’) fluctuating components ofthe first six POD modes for AR1 (a), AR2 (b) and AR4 (d).
Figure inserts present the percentage contribution of 𝑣’ toeach mode.
reported in the literature (Okajima, 1982; Knisely, 1990;
Mohebi et al., 2017). As shown in Figure 1(b) for AR2, thewake recirculation region is significantly larger due to the
suppression of flow reattachment by the strong backflow
emanating from the trailing edge. This suppression
mechanism may lead to the formation of dual vortex shedding
patterns over and behind the AR2 prism, which deviates fromthe conventional von-Kármán shedding process. Since thevon-Kármán vortex shedding motions are dominated byvertical velocity fluctuations (Nakagawa et al., 1999;
Kumahor and Tachie, 2022), the relative contributions of 𝑣’ tothe energy content in each mode are presented by the red
shaded regions of the bars and subsequently as inserts in
Figure 4. The vertical fluctuations contribute to more than
65% of the energy content in the first mode pair for AR1 andAR4 but only 50% for AR2.
The cross plots of the temporal POD mode coefficients ofthe first mode pair are shown in Figure 5. The well-organizedelliptical shape for AR1 and AR4 reveal that the coherent

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
both the fundamental shedding frequency and its second
harmonic for AR1 and AR4, meanwhile, no discernible peakis observed at either dual peaks for AR2.
Following Fang and Tachie (2019) and Kumahor et al.
(2021), the temporal variation of the reverse flow area aroundthe prisms are investigated, and the frequency spectra ispresented in Figure 7. The results show that the enlarging andshrinking of the recirculation bubbles formed around theprisms occur at frequencies consistent with that observed in
the first POD mode pair. Therefore, it is corroborated that thefirst POD mode pair is associated with the large-scale vonKármán vortices shed over the rectangular prisms. It is worth
noting that, the average reverse flow areas (𝐴̅/ℎ2 ) are 0.55,
1.53 and 1.13, respectively, for AR1, AR2 and AR4 indicating
the massive size of the recirculation bubble around the AR2
prism (see Figure 1(b)).
Figure 8 shows the spatial contributions of the first POD
mode pair by the vertical velocity fluctuations. Alternating
positive and negative zones, representative of the von-Kármán
shedding motions are evidenced in both modes. The vortex
cores are observed near the symmetry plane within the wakeregion, signifying the symmetrical nature of the large-scale
von-Kármán vortices. The distance between the negative andpositive zo↓nes obtained from the first POD mode, which isrepresentative of the half wavelength (𝜆/2) of the coherent
structures, is typically used to calculate the convective
velocity (Riches et al., 2018). The resulting convective
velocity ( 𝑈𝑐 /𝑈𝑒 ) is 0.83 and 0.48 for AR1 and AR4,
respectively. In the case of AR2, dual convective velocities
𝑈𝑐 /𝑈𝑒 = 0.35 and 0.61 that correspond to St = 0.084 and0.145, respectively, are obtained.

(a)

(b)

(c)

CONCLUSION
The effects of streamwise aspect ratio (AR) on the
turbulent flow around rectangular cylinders in uniform flow
were experimentally investigated using a time-resolved
particle image velocimetry system. The streamwise aspectratios (streamwise length to height ratio) tested were AR =1,
2 and 4, corresponding to cases when the mean separated shear
layers from the leading edges are shed directly into the wakeregion (AR1), reattach over the prism (AR4), and anintermediate case (AR2). The Reynolds number based on freestream velocity and prism height was 16200.
The ratio between the frequency associated with the smallscale Kelvin-Helmholtz vortices occurring near the leadingedge and the energetic large-scale von-Kármán vortices varies
significantly with aspect ratio. This suggests that theinteraction between the small-scale vortices that are formed in
the shear layer at the leading edge and the large-scale vortices
shed into the wake region is strongly dependent on theirproximity. The mean flow features a massive recirculation
region in the wake of the AR2 prism, compared to AR1 and
AR4. Using proper orthogonal decomposition and the reverse
flow area, the results for the AR2 prism show the existence ofa dual vortex shedding process which is intimately tied to theflow dynamics occurring near the trailing edge. The strong
back flow near the trailing edge for AR2 significantly
attenuates the reattachment of the separated shear layer. For
AR1 and AR4, the fundamental vortex shedding frequency isthe dominant peak and the first POD mode pair contain similarenergy percentages. This suggests regular von-Kármán
shedding patterns over the AR1 and AR4 prisms.

Figure 6: Frequency spectra of the first four POD mode
coefficients and radius calculated from a(1) and a(2) for AR1(a), AR2 (b) and AR4 (c).

Figure 7: Frequency spectra of the reverse flow area

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

(b)

(c)

(d)

(e)

(f)

Figure 8: Contours of the first two POD modes using the vertical velocity fluctuations (𝑣’) for AR1 (a,b), AR2 (c,d) and AR4 (e,f). The
symbol (+) represents the vortex centers
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ACKNOWLEDGMENT
The funding support from the Natural Sciences andEngineering Research Council of Canada to MFT is gratefully
acknowledged. We are also grateful to Canada Foundation for
Innovation for funding for the experimental facility.
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