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

SPATIO-TEMPORAL CHARACTERISTICS OF TURBULENT FLOW SEPARATIONSAROUND RECTANGULAR AND TRAPEZOIDAL PRISMS IN UNIFORM FLOW
Sedem Kumahor
Xingjun Fang
Department of Mechanical Engineering
Department of Mechanical Engineering
University of Manitoba
University of Manitoba
Winnipeg, Manitoba, R3T 5V6, Canada
Winnipeg, Manitoba, R3T 5V6, Canada
kumahors@myumanitoba.ca
fangx@myumanitoba.ca
Mark F. Tachie
Department of Mechanical Engineering
University of Manitoba
Winnipeg, Manitoba, R3T 5V6, Canada
mark.tachie@umanitoba.ca
ABSTRACT
Turbulent flow separations over rectangular prisms of
streamwise aspect ratios of 1 (AR1) and 2 (AR2), and a
trapezoidal prism were experimentally studied using timeresolved particle image velocimetry at a Reynolds number(based on free-stream velocity and prism height) of 16200.
The trapezoidal prism has a 45° inclined trailing face and its
upper and lower streamwise lengths are, respectively,
identical to those of the AR1 and AR2 prisms. The mean flow,
vortex shedding motions and spatial characteristics of thesymmetric and asymmetric wakes produced by the rectangularand trapezoidal prisms, respectively, are investigated in termsof turbulent statistics, frequency spectrum and two-point
autocorrelation. The results indicate that the mean
recirculation bubble and magnitudes of the Reynolds stresses
in the wake region of the trapezoidal prism are in close
agreement with AR2, indicating the dominance of the shear
layer from the longer side of the trapezoidal prism. The
fundamental shedding frequency for the trapezoidal prism andits harmonic are intermediate to those observed for therectangular prisms. Two-point autocorrelation analyses reveal
a higher structural coherency between the wake vortex andsmall-scale vortices formed at the leading edge of AR2 prism
compared to AR1 and the trapezoidal prism. Spectral proper
orthogonal decomposition analyses also reveal the coherent
structures that occur at the dominant and sub-dominant
shedding frequencies.

(a)

(b)

(c)

Figure 1: Cross-sectional view of the AR1 prism (a),trapezoidal prism (b) and AR2 prism.
attributed to the intermittent reattachment of the shear layer onto the body surface (Okajima, 1982; Nakagawa et al. 1999).
Consequently, there is significant narrowing and widening of
the wake vortices shed from the trailing edge of the AR2
prism. Recent experimental investigations by Mohebi et al.
(2017) for AR = 1.92 also showed a significantly larger wakerecirculation region compared to the square cylinder (AR1).
They argued that the trailing edge flow dynamics plays animportant role in the vortex formation and shedding process
around rectangular prisms.
In this study, the sharp trailing edge of the AR2 prism isreplaced by a 45° inclined face, creating a trapezoidal prismas shown in Figure 1(b). This modification results in a prismwith upper and lower streamwise lengths corresponding to 1
and 2 times the prism heights, respectively, and an
introduction of geometrically-imposed asymmetry on the flow
dynamics. The objective of the present study is to investigate
the spatio-temporal features of symmetric wake flows
produced over rectangular prisms of aspect ratios of 1 (Figure1(a)) and 2 (Figure 1(c)), and an asymmetric wake flow overa trapezoidal prism (Figure 1(b)) using time-resolved particle
image velocimetry (TR-PIV). The results are analysed interms of the mean velocity, Reynolds stresses, frequency

INTRODUCTION
Transport phenomena and vortex shedding motions inturbulent flows over symmetric bluff bodies such asrectangular prisms in uniform flows have been criticallyexamined over the past decades. The turbulent statistics, dragcharacteristic and vortex shedding motions strongly depend on
the streamwise aspect ratio (AR = 𝐿 / ℎ , where 𝐿 and ℎrepresent the streamwise length and vertical height of theprism, respectively). The fundamental vortex shedding
frequency (St = 𝑓ℎ/𝑈𝑒 , where 𝑓 is the dominant frequency
and 𝑈𝑒 is the free-stream velocity), for example, decreases
from 0.13 to 0.08 as the aspect ratio increases from 1 to 2
(Okajima, 1982, Knisely, 1990; Norberg, 1993). Meanwhile,
the peak Reynolds normal stresses in the near wake region ofa square prism are an order of magnitude higher than thecorresponding values for a rectangular prism with an aspectratio of 2 (Nakagawa et al. 1999).
The differences in the vortex shedding dynamics andmean flow topology for AR2 compared to AR1 have been1

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

spectra and two-point correlation. The large-scale coherent
flow structures are also extracted and analysed using spectral
proper orthogonal decomposition (SPOD)(a)

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 6.0 m, 0.6 m and 0.45 m, respectively. Two rectangularprisms and a trapezoidal prism with vertical height, ℎ =0.03 m and spanwise width, 𝐵 = 0.58 m were investigated. As
shown in Figure 1, the streamwise lengths of the rectangularprisms are 0.03 m (AR1) and 0.06 m (AR2) while thestreamwise lengths of the upper and lower sides of the
trapezoidal prism are 0.03 and 0.06 m, respectively. The water
depth was kept constant at 0.43 m and the prisms werepositioned such that the prism mid-height was 0.215 m above
the bottom wall. The measured boundary layer thickness was
0.065 m and hence the prism was exposed to uniform
oncoming flow. The free-stream velocity was set to 𝑈𝑒 =0.540 m/s and the Reynolds number was 16200.
Time-resolved particle image velocimetry (TR-PIV) wasused to perform velocity measurements at the mid-span of thechannel. The water was seeded with 10 μm silver coatedhollow glass spheres of specific gravity, 1.4. A diode pumped
dual-cavity high-speed Neodymium-doped yttrium lithium
fluoride (Nd:YLF) laser with maximum pulse energy of 30
mJ/pulse for each cavity was used to illuminate the seeding
particles. Two high speed 12-bit complementary metal oxide
semiconductor cameras of resolution, 2560 pixel × 1600 pixel
positioned side-by-side were used to image the flow in two
fields of view (FOV) simultaneously. The dimensions of the
FOVs over the rectangular prisms 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. For the trapezoidal prism, the FOVs over theprism and in the wake region were, respectively, 2.7ℎ × 1.7ℎ
and 7.3ℎ × 4.6ℎ, resulting in digital image resolutions of 31.8
pixel/mm and 11.7 pixel/mm, respectively. These spatial
resolutions were chosen such that the flow interactions
between the leading and trailing edges are resolved in the first
FOV and the large wake recirculation region are resolved in
the second FOV.
The sampling frequency was set to 800 Hz and 96000
images were captured for each test case. Data acquisition andimage post-processing were performed using commercial
software (DaVis version 10.0.5) supplied by LaVision Inc.
Velocity vectors were calculated using a GPU-accelerated
multi-pass cross-correlation algorithm with a single initialpass of 128 pixel × 128 pixel interrogation area (IA) with 50%
overlap followed by four final passes of 24 pixel × 24 pixel IAwith 75% overlap.

(b)

(c)

Figure 2: Streamwise mean velocity (𝑈) superimposed withrepresentative mean streamlines and the isopleth of 𝑈 =0. Symbol + marks the peak with the value written above.
Locations A and B marked using × represent the locationsof peak ̅̅̅̅̅
𝑢′𝑢′ and ̅̅̅̅̅
𝑣′𝑣′, respectively.
reveal distinct mean recirculation bubbles over the rectangularprisms, but not over the trapezoidal prism. The streamwise
extent of mean flow reversal within the wake region (LR), as
indicated by the isopleth of 𝑈 = 0, is significantly shorter inthe AR1 case (LR/ℎ = 1.6) compared to AR2 and trapezoidal
prims (LR/ℎ ~ 4.2). The value of LR/ℎ = 1.6 for AR1 is in good
agreement with that reported by Trias et al. (2015). Near theupper trailing edges, the maximum magnitudes of the mean
backflow were 𝑈/𝑈𝑒 = 0.08, 0.29 and 0.25 for the AR1, AR2
and trapezoidal prisms, respectively. This backflow toward
the leading edge of the prisms creates an adverse pressure
gradient leading to the formation of a small secondary
recirculation bubble near the leading edge of the prisms (Trias
et al., 2015; Kumahor and Tachie, 2022). The well-resolved
secondary recirculation bubbles close to the leading edge ofthe prisms are indicative of the fine spatial resolution adopted
in this study.
̅̅̅̅̅) aroundThe contours of the Reynolds shear stress (𝑢′𝑣′the prisms are compared in Figures 3. A distinct region of
positively valued ̅̅̅̅̅
𝑢′𝑣′ is evident close to the leading edge
(𝑥/ℎ < 0.5), regardless of prism geometry. This is induced by
misalignment of the Cartesian coordinate with local mean
streamlines (Fang and Tachie, 2019). There exist dual local
peaks of ̅̅̅̅̅
𝑢′𝑣′ within the wake region. It is noted that themagnitudes of peak Reynolds shear stress and Reynolds
normal stresses (not shown) in the AR1 case are consistently
larger than the other two prisms. The stronger Reynolds
stresses in the AR1 case promote entrainment of free-stream
flow into the wake flows and thus reduce the size of the mean
recirculation bubble as shown in Figure 2(a). Figures 2 and 3
clearly show that the mean flow topology including therecirculation length as well as the magnitude of Reynolds

RESULTS AND DISCUSSION
Contour plots of the streamwise mean velocity
superimposed with the mean streamlines are shown in Figure
2 to visualize the mean flow over the prisms. The approachflow accelerates past the prisms with local maximum velocity
at a location approximately above the highest point of themean separating streamline. The value of maximum velocity
over the trapezoidal prism is lower compared to therectangular prisms due to the topological differences in themean flow over the prisms. Specifically, the mean streamlines
212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022

(a)

(a)

(b)

(b)

(c)

(c)
̅̅̅̅̅) superimposed with
Figure 3: Reynolds shear stress (𝑢′𝑣′
the mean separating streamline. Symbol + marks the peakwith the value written beside it.
stresses and their streamwise locations for the trapezoidal
prism are in closer agreement with AR2 than the AR1 prism.
These similarities between the trapezoidal and AR2 prisms
suggest the dominance of the shear layer from the longer sideof the trapezoidal prism on the mean flow and turbulent
statistics in the near wake region.
Figure 4 presents the frequency spectra of the streamwise
and vertical fluctuating velocities at the peak locations of ̅̅̅̅̅
𝑢′𝑢′̅̅̅̅̅ (point B). All spectra exhibit the -5/3 slope(point A) and 𝑣′𝑣′
over a wide range of frequencies, reflecting a well-resolved
inertial subrange at high Reynolds numbers. At the peaklocation of ̅̅̅̅̅
𝑢′𝑢′ , the fundamental von-Kármán (VK) vortex
shedding frequencies for the AR1, AR2 and trapezoidal
prisms are 𝑆𝑡𝑉𝐾 = 0.138, 0.084 and 0.104, respectively. The
dominance of these frequencies is also evident in 𝜙𝑣𝑣 at the̅̅̅̅̅ . The dominant frequencies for the
peak location of 𝑣′𝑣′
rectangular prisms agree well with fundamental shedding
frequencies reported for AR1 and AR2 in previous
investigations (Okajima, 1982; Knisely, 1990; Norberg,1993). Meanwhile, the 𝑆𝑡𝑉𝐾 = 0.104 for the trapezoidal prismis midway between the corresponding values for the tworectangular prisms. The spectra also revealed peaks at the 2 nd
and 3rd harmonics, although the peaks at the harmonics are
more prominent for AR1 compared to the other prisms. For
the AR2 case, a peak value at St = 0.142 (which is lower thanthe 2nd harmonic) is also observed which is consistent with thefindings by Knisely (1990) and Norberg (1993). The existenceof these two frequencies for the AR2 case have beenpreviously attributed to the intermittent reattachment of theseparated shear layer onto the prism surface. Consequently,
there is a dual vortex shedding process evidenced in thesignificantly larger wake recirculation region observed in
Figure 2.
Figure 5 examines the spatial structures using two-point
autocorrelations of the streamwise fluctuating velocity. Here,Figure 4: Frequency spectra of the streamwise fluctuating
velocity (𝜙𝑢𝑢 ) and vertical fluctuating velocity (𝜙𝑣𝑣 ) at thelocations of peak ̅̅̅̅̅
𝑢′𝑢′ (A) and peak ̅̅̅̅̅
𝑣′𝑣′ (B) as marked in
Figure 2 for the AR1 (a), trapezoidal (b) and AR2 (c)prisms.
the reference point is chosen to be 0.1ℎ above the trailing edge
and the mean separating streamline are superimposed to
examine the feedback influence of the large-scale wake vortex
on the small-scale vortices formed near the leading edge of theprisms. Downstream of the reference point, the isopleths ofcorrelation coefficient (𝑅𝑢𝑢 ) extend towards the symmetry
plane but decorrelate more rapidly in the wake of AR2 prism.
This suggests that the vortices passing the reference point
tends to maintain their structural integrity so as to interlink the
dynamics of unsteadiness near the separating point and wakeregion in the AR1 and trapezoidal cases. In the AR2 case, on
the other hand, the flow structure experiences a longer
developing distance over the bluff body and thus dominates
over the influence of the structure in the wake region.
Upstream of the reference point, the correlation coefficient
decays more slowly over the AR2 prism compared to AR1 andtrapezoidal prisms. Additionally, the levels of correlations
near the separating streamline over the AR1 and AR2 prisms
are significantly higher compared to the trapezoidal prism. In
the case of the trapezoidal prism, for instance, thedecorrelation point along the mean separating streamline(which is defined as the intersection of the mean streamline
and the isopleth of 𝑅𝑢𝑢 = 0) is only 0.5ℎ (i.e., 𝑥/ℎ = 0.5) from3

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

(a)

(b)

(b)

(c)

(c)
Figure 5: Two-point autocorrelation of the streamwise
fluctuating velocity (𝑅𝑢𝑢 ) for the AR1 (a), trapezoidal (b)and AR2 (c) prisms. with the reference points 0.1ℎ above
the trailing edge. Blue continuous line represents the mean
separating streamline and the contours levels shown are in
0.1 increments.
the self-correlation point compared to 0.95ℎ (i.e., 𝑥/ℎ = 0.05)
for the AR1. For the AR2 prism, correlation coefficient as highas 𝑅𝑢𝑢 = 0.4 are evident in the vicinity of the leading edge.
These observations suggest that the interaction between wakevortex and vortices near the separating point is the weakest in
the trapezoidal case. In fact, the correlations in both AR1 andAR2 cases are still significant upstream of the separating
point, especially for the former one.
Spectral proper orthogonal decomposition (SPOD),
proposed by Sieber et al. (2016) was performed for the flow
around the tested prisms. This method, which is an extensionof the snapshot POD method (Sirovich, 1987) decomposes thephenomena occurring at multiple frequencies and energies
using a spectral filter. This enables a clear separation of
spatially and temporally coherent structures into a series ofspectral modes that represent the flow physics. Detailed
formulation and application of the technique to various TRPIV datasets are presented in Sieber et al. (2016).
The premultiplied frequency spectra of the first two spatial
mode coefficients are presented in Figure 6. Consistent withthe findings by Sieber et al. (2016) for periodic vortex
shedding motions, the dominant frequencies in the spectra of
the first SPOD mode correspond to the fundamental vonKármán (VK) vortex shedding frequencies presented in Figure
4 for the rectangular prisms. Comparing the spectral energy
content for the first and second modes, all the spectral peaks
for the first mode are more than tenfold of the second mode.
This suggests that the major flow coherency is presented in thefirst mode and hence attention is focused solely on the firstmode hereinafter. For AR1, the second and third harmonicfrequencies of the fundamental vortex shedding frequency(𝑆𝑡𝑉𝐾 = 0.138) are evident as sub-dominant peaks in the

Figure 6: Premultiplied frequency spectra of the first two
spatial POD mode coefficients (𝜙𝑎(𝑖) ) for the AR1 (a),trapezoidal (b) and AR2 (c) prisms. The high frequency
variations are shown clearly using figure inserts at the top
right corner.
spectra (see Figure 6(a)). For AR2, two distinct shedding
frequencies are observed, with dominance at 𝑆𝑡𝑉𝐾 = 0.084 andsub-dominance at 𝑆𝑡 = 0.142. The existence of the dual vortex
shedding frequencies corroborates the findings by previous
investigations (Okajima, 1982; Knisely, 1990; Norberg,1993). For the trapezoidal prism, the dominant frequency (𝑆𝑡= 0.140) is weakly observed in Figure 4(b), meanwhile, thefundamental shedding frequency (𝑆𝑡𝑉𝐾 = 0.084) is the subdominant peak in the spectra presented in Figure 6(b). Thisobservation suggests that the trapezoidal prism also undergoes
a dual vortex shedding process, similar to the AR2 prism. It isinteresting to note that the dominant peak (𝑆𝑡 = 0.140) is
similar to fundamental shedding frequency for AR1 (𝑆𝑡𝑉𝐾 =0.138) and the sub-dominant peak for AR2 (𝑆𝑡 = 0.142).
The SPOD modes at selected phases at the dominant andsub-dominant frequencies are presented in Figure 7. For AR1,
the coherent structure at 𝑆𝑡𝑉𝐾 = 0.138 represents the alternate
vortex shedding process which is symmetric about the prism
centerline (Figure 7(a)). At 2𝑆𝑡𝑉𝐾 , the structures resulting
from vortex break up and an alteration of the vortex shedding
process are observed in Figure 7(b). In this case, the vortex

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: Contours of the first two dominant SPOD modes using the streamwise velocity fluctuations (𝑢’) for AR1 (a,b), trapezoidal(c,d) and AR2 (e,f) prisms. The symbol (+) represents the vortex centers.

cores are off-centered and the wavelength (defined as thestreamwise distance between the vortex cores) isapproximately half that at 𝑆𝑡𝑉𝐾 . For AR2, a large coherent
structure emanating from the leading edge is convected into
the wake region at 𝑆𝑡𝑉𝐾 = 0.084 (Figure 7(e)). Near the
trailing edge, the flow is observed to bump upward and flow
upstream, indicating that the separated shear layer is not
reattached. Thus, the flow structure occurring at 𝑆𝑡𝑉𝐾 = 0.084
is shown to correspond to the instance when the wake width iswide, and the separated shear layer is not reattached. At 𝑆𝑡 =0.142, the spatial modes emanate from the trailing edge andare concentrated within the wake region, suggesting that thevortex shedding is occurring from the trailing edge and theseparated shear layer is reattached onto the surface of theprism. The evidently higher spectral energy content of 𝑆𝑡𝑉𝐾 =0.084 (Figure 6(c)) and the significantly larger wavelength is
responsible for the large recirculation region around the AR2
prism. It is important to note that, the vortex cores for both
frequencies are located along the prism centerline, indicating
the symmetric nature of the dual vortex shedding process forAR2. For the trapezoidal prism, the dominant peak ( 𝑆𝑡 =0.140) reveals coherent structures that are aligned at an angleof 45°, corresponding to the angle of inclination of the trailingface. At 𝑆𝑡 = 0.104, the coherent structures emanate from theleading edge and the vortex cores are apparently aligned at
approximately the prism mid-height. Furthermore, thewavelengths in both Figures 7(c,d) are identical.

2 (AR2), and a trapezoidal prism in uniform flow. The
trapezoidal prism had a 45° inclined trailing face that resulted
in its upper and lower streamwise lengths corresponding tothose of the AR1 and AR2 prisms, respectively. The Reynolds
number based on the free-stream velocity and prism heightwas 16200.
Contours of the streamwise mean velocity show mean
flow separation at the sharp leading edge for all tested prisms,
meanwhile, the value of maximum velocity over the
trapezoidal prism is lower compared to the rectangular prisms.
The mean streamlines also showed that the mean recirculation
bubble over the trapezoidal prism is suppressed due to themodified trailing edge. Two-point correlation analyses showthat the interaction between wake vortex and vortices near theleading edge is weakest over the trapezoidal prism. The
stronger magnitude of Reynolds stresses in the AR1 casepromotes entrainment of free-stream flow into the wake flows
and hence reduces the size of the mean recirculation bubble.
On the other hand, the recirculation length of the mean
recirculation bubble in the wake for AR2 and the trapezoidal
prism are similar.
Dual vortex shedding frequencies are observed over theAR2 and trapezoidal prisms, while a single dominant shedding
frequency is observed for AR1. Spectral proper orthogonal
decomposition shows that the dominant and sub-dominantfrequencies in AR2 correspond to a wide and narrow wake
width, respectively, which are a consequence of theintermittent reattachment of the shear layer onto the prism. For
the trapezoidal prism, the dominant and sub-dominantfrequencies show coherent structures that are aligned at a 45°
angle and in the streamwise direction, respectively. The sizeof the vortices formed at the dominant frequency for AR2 andthe trapezoidal prism are similar, suggesting the dominance of

CONCLUSION
A time-resolved particle image velocimetry technique wasused to investigate the turbulent flow characteristics overrectangular prisms of streamwise aspect ratios of 1 (AR1) and5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
the shear layer from the longer (AR2) side of the trapezoidal
prism on the wake flow dynamics.

ACKNOWLEDGMENT
The funding support from the Natural Sciences andEngineering Research Council of Canada (NSERC), to MFT
is gratefully acknowledged. We are also grateful to Canada
Foundation for Innovation for funding for the experimental
facility.

REFERENCES
Fang, X., and Tachie M. F., 2019a, “Flows over surfacemounted bluff bodies with different spanwise widths
submerged in a deep turbulent boundary layer”, Journal of
Fluid Mechanics, Vol 877, pp. 717-758
Knisely, C. W., 1990, “Strouhal numbers of rectangularcylinders at incidence: a review and new data”, Journal of
Fluids and Structures, Vol. 4, pp. 371-393.
Kumahor, S., and Tachie M. F., 2022, “Turbulent flow
around rectangular cylinders with different streamwise aspectratios”. Journal of Fluids Engineering, Vol. 144, pp. 051304.
Lander, D. C., Moore, D. M., Letchford, C. W., andAmitay, M., 2018, “Scaling of square-prism shear layers”,
Journal of Fluid Mechanics, Vol. 849, pp. 1096-1119.
Mohebi, M., Plessix, P., Martinuzzi, R. J., and Wood, D.
H., 2017, “Effect of thickness-to-chord ratio on the wake of
two-dimensional rectangular cylinders”. Physical Review
Fluids, Vol 2, pp. 064702.
Nakagawa, S., Nitta, K., and Senda, M., 1999, “Anexperimental study on unsteady turbulent near wake of arectangular cylinder in a channel flow”, Experiments in
Fluids, Vol. 27, pp. 284-294.
Norberg, C., 1993, “Flow around rectangular cylinders:
Pressure forces and wake frequencies”, Journal of Wind
Engineering and Industrial Aerodynamics, Vol. 49, pp. 187196.
Okajima, A., 1982, “Strouhal numbers of rectangularcylinders”, Journal of Fluid Mechanics, Vol 123, pp. 379-398.
Sieber, M., Paschereit, C. O., and Oberleithner, K., 2016,“Spectral proper orthogonal decomposition”. Journal of Fluid
Mechanics, Vol. 792, pp. 798-828.
Sirovich, L., 1987, “Turbulence and the dynamics ofcoherent structures. Part I, II and III”. Quarterly of Applied
Mathematics, Vol. 45, pp. 561-571.
Trias, F. X., Gorobets, A., and Olivia, A., 2015,“Turbulent flow around a square cylinder at Reynolds number
22,000: A DNS study”, Computers and Fluids, Vol. 123, pp.
87-98

6