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

MEAN WAKE OF SURFACE-MOUNTED FINITE-HEIGHT SQUARE PRISMS OF SMALLASPECT RATIO

Barbara L. da Silva
Department of Mechanical Engineering
University of Saskatchewan
57 Campus Drive, Saskatoon, Saskatchewan,
S7N 5A9, Canada
barbara.silva@usask.ca

Dylan G. H. Hahn
Department of Mechanical Engineering
University of Saskatchewan
57 Campus Drive, Saskatoon, Saskatchewan,
S7N 5A9, Canada
dylan.hahn@usask.ca

David Sumner
Department of Mechanical Engineering
University of Saskatchewan
57 Campus Drive, Saskatoon, Saskatchewan,
S7N 5A9, Canada
david.sumner@usask.ca

Donald J. Bergstrom
Department of Mechanical Engineering
University of Saskatchewan
57 Campus Drive, Saskatoon, Saskatchewan,
S7N 5A9, Canada
don.bergstrom@usask.ca

ABSTRACTThe mean wake of surface-mounted finite-height squareprisms, or blocks, with aspect ratios AR = 0.5, 0.7 and 1 wasinvestigated in a low-speed wind tunnel. Velocity
measurements were made using both a seven-hole probe andthe particle image velocimetry technique. The Reynolds
number based on the blocks’ width D was Re = 7.5 × 104,
and two different incoming flow conditions with boundary
layer thicknesses of δ/D = 0.6 and 2.0 were considered. The
mean wake behaviour changed with AR, and important
trends were identified based on critical points in thesymmetry plane. The mean streamwise vorticity distribution
revealed different flow patterns in the wake, depending on
both AR and δ/D. For AR = 1 and both boundary layers, thevorticity distribution is similar to that of prisms with AR > 1.
For AR = 0.7, a transitional behaviour is observed, while for
AR = 0.5 the wake shows an additional pair of streamwise
vorticity regions. This pattern was found to be related to flow
reattachment on the free end of the block, which becomes
more symptomatic for a thicker boundary layer.

The wake of this type of bluff body follows some of thesame trends observed for surface-mounted finite-heightsquare prisms below the critical AR = 3.5 ~ 4.5, such as
having weaker vortex shedding and no base vortices(Unnikrishnan et al., 2017; Heng and Sumner, 2020). Uniquefeatures have, however, been identified in the literature forprisms of AR < 1, here referred to as blocks. Okamoto et al.
(1995) reported a change in the flow behaviour for AR ≥ 0.7
when compared with AR = 0.5, for which flow reattachment
was observed. The nature of the flow structures aroundblocks has not been extensively explored, however, andcontradictions exist in the literature about this type of flow.
For example, periodic vortex shedding for blocks with
AR < 1 was reported by some researchers (Calluaud et al.,2000), while others did not detect significant periodicity(da Silva et al., 2022).
Sumner et al. (2016) and da Silva et al. (2022) reported
the existence of two pairs of streamwise vorticity regions in
the mean wake of a block with AR = 0.5. The outer vorticity
pair, located closer to the ground plane, has the same
vorticity signs as the ones found for taller prisms. The innervorticity pair has opposite signs of rotation compared withthe outer pair, which is associated with the existence of two

INTRODUCTION
Surface-mounted finite-height square prisms arecommonly found in engineering applications, which
motivates the study of the flow around this type of structure.
The flow around these prisms is more complex than the one
around two-dimensional or “infinite” square prisms, due tothe three-dimensionality added by the surface-mounted
finite-height square prism’s free end and junction with theground plane (Fig. 1). The aspect ratio AR of the prism,
defined as the ratio between the prism height H and its width
D, AR = H/D, is an important parameter in the analysis of
the flow. However, most studies have been dedicated toprisms of high aspect ratio AR (e.g., Yauwenas et al., 2019).
Despite their common applications such as in the study offlow around low-rise buildings and roughness elements, theflow around square prisms of very small aspect ratio(AR < 1) has not been as extensively studied.

Figure 1. Schematic of the flow around a surface-mounted
finite-height square prism of small aspect ratio (block).

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

Figure 2. Mean streamwise vorticity distribution and in-plane velocity vectors (from the seven-hole pressure probe measurements) ina y-z plane located at x/D = 3 for surface-mounted blocks with AR = 0.5 (left), 0.7 (middle) and 1 (right). (a) Thin boundary layer
with δ/D = 0.6 and (b) thick boundary layer with δ/D = 2.0. Every second vector in the y and z directions is shown, for clarity.

downwash peaks in the wake. This observation is in contrast
to the single downwash peak found on the mean wakecentreline for taller prisms.
The purpose of this study is to further investigate themean wake of surface-mounted square blocks and thechanges in the mean streamwise vorticity patterns by
experimentally measuring the wakes of blocks with
AR = 0.5, 0.7 and 1. In addition, the effects of two different
boundary layers with thicknesses of δ/D = 0.6 and 2.0 will beassessed, as this parameter is expected to significantly affectthe development of the wake (Heng and Sumner, 2020).

measurements at x/D = 2 were not possible for AR = 0.7 and1 due to reverse flow near the ground plane. Measurementsin each plane were made for y/D = −3 to 3 and z/D = 0.117 to
AR + 1, with intervals of 5 mm (or 0.083D) between the
points. Samples were taken during 10 s at a rate of 1 kHz.
The uncertainty of the seven-hole probe measurements is lessthan 5% for the velocity magnitude and less than 3° for theflow angle.
Since the seven-hole probe is limited to a flow angle
range of ±72.9°, additional velocity field measurements werecarried out with particle image velocimetry (PIV) in the x-z
symmetry plane where reverse flow takes place. PIV
measurements were carried out only for the thin boundary
layer (δ/D = 0.6). The flow was seeded by atomized
propylene glycol droplets with diameters of less than 5 μm.
A 200-mJ/pulse dual Nd:YAG Solo PIV 200XT laser (New
Wave Research) was used to illuminate the droplets in theflow field. Image pairs were obtained with a Powerview Plus
2MP camera (TSI), for fields of view of dimensions 0.91D ×
0.67D, approximately. An ensemble of 2100 image pairs was
sampled at a rate of 15 Hz for each field of view, and theInsight 3G software (TSI) was used for data acquisition andprocessing. The average uncertainty in the velocity
measurements with PIV was evaluated as 4.5%.

EXPERIMENTAL SET-UP
Experiments were performed using the low-speed windtunnel of the University of Saskatchewan, with a set-up
similar to that of Unnikrishnan et al. (2017) and da Silva etal. (2022). Hollow aluminum square blocks with sharp edges
and black smooth surfaces were installed normal to the windtunnel ground plane, with their front face normal to theincoming flow. The blocks had a constant width of
D = 60 mm and aspect ratios of AR = 0.5, 0.7 and 1.
The freestream velocity was U∞ = 21.1 m/s, giving a
Reynolds number of Re = U∞D/ν = 7.5 × 104 (where ν is the
kinematic viscosity of the fluid). Two boundary layers wereconsidered: a “thin” fully developed turbulent boundary layerof thickness δ/D = 0.6, and a “thick” developing turbulent
boundary layer with δ/D = 2, produced by a trip near theleading edge of the ground plane. The corresponding
boundary layer thicknesses relative to the blocks’ height(δ/H) vary from δ/H = 0.6 for AR = 1 and the thin boundary
layer to δ/H = 4.0 for AR = 0.5 and the thick boundary layer.
Velocity measurements outside of the recirculation
region of the wake were performed with a seven-holepressure probe connected to a Scanivalve ZOC-17IP/8Px
pressure scanner. By using a traversing system, the probe
was positioned in y-z planes located at x/D = 2 (for AR = 0.5
only), 3 and 4, where x is in the streamwise direction, y is in
the transverse direction, z is normal to the ground plane, andthe origin of the coordinate system is located at the centre of
the junction of the block with the ground plane. Note that

RESULTS AND DISCUSSION
Results will be presented separately for the seven-holeprobe and particle image velocimetry wake measurements.

Seven-Hole Probe Results
Figure 2 presents the mean streamwise vorticity(ωxD/U∞) in the y-z plane located at x/D = 3 for AR = 0.5,
0.7 and 1, and for δ/D = 0.6 (Fig. 2a) and 2.0 (Fig. 2b). Its
distribution in the wake of the block of AR = 1 (therightmost images in Fig. 2) follows the same pattern
identified for surface-mounted finite-height square prisms
below the critical AR by Unnikrishnan et al. (2017). The
streamwise vorticity concentrations near the ground plane are
either due to the trailing arms of the horseshoe vortex, or due

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 3. Mean streamwise vorticity distribution and in-plane velocity vectors (from the seven-hole pressure probe measurements) ina y-z plane located at x/D = 4 for surface-mounted blocks with AR = 0.5 (left), 0.7 (middle) and 1 (right). (a) Thin boundary layer
with δ/D = 0.6 and (b) thick boundary layer with δ/D = 2.0. Every second vector in the y and z directions is shown, for clarity.
to an entrainment mechanism close to the ground plane (daSilva et al., 2022). These concentrations extend to the upper
part of the wake. There, they are caused by the interaction of
the free shear layers with the downwash, which is thestrongest near the centre of the wake. By increasing theboundary layer thickness, the same pattern is found, but the
downwash and the upper vorticity regions are weakened. In
addition, some induced vorticity near the ground plane andthe centre of the wake is found for AR = 1, with the thinboundary layer only.
The wake of the block with AR = 0.5 (the leftmost
images in Fig. 2) presents the characteristics identified in
Sumner et al. (2016) and da Silva et al. (2022). The outer
vorticity regions near the ground plane, that were found for
AR = 1 and reported in the literature for taller prisms, arestill present. The downwash is, however, weaker in thecentre. It shows instead two peaks (one at each y side of thewake) that are responsible for forming the inner vorticity
regions with opposite signs of rotation, compared with theupper streamwise vorticity that is typical in the wake of
AR = 1 and taller prisms. The same pattern occurs for bothboundary layers and with approximately the same vorticity
intensity, although the thick boundary layer caused the innervorticity regions to be flattened.
The mean streamwise vorticity in the wake of the blockwith AR = 0.7 (the central images in Fig. 2) shows distinct
patterns for the two boundary layer thicknesses. With the thinboundary layer, neither the upper vorticity pair of AR = 1 orinner vorticity pair of AR = 0.5 is found. With the thickboundary layer, a small inner vorticity pair is present. This
aspect ratio represents a transition state between the twomean vorticity patterns, where the upper vorticity has been
weakened but the two downwash peaks are not yet strong
enough to cause the appearance of the inner vorticity regions
for the thin boundary layer case. By increasing the boundary
layer thickness, the transition from the upper vorticity to theinner vorticity pattern is accelerated.
The streamwise evolution of the wake of the blocks canbe analysed through the mean streamwise vorticity contours
shown in Fig. 3 for the y-z plane located at x/D = 4. The
wake has spread wider in the y direction for all cases, inagreement with Sumner et al. (2016). For AR = 1 (therightmost images in Fig. 3), the upper vorticity regions havelower intensity, due to the weakened downwash at x/D = 4.
The inner vorticity regions found for AR = 0.5 (the leftmost
images in Fig. 3) have decreased in size for both boundary
layers, while the inner vorticity pair found for AR = 0.7 andthe thick boundary layer at x/D = 3 (centre of Fig. 2b) has
disappeared at x/D = 4 (centre of Fig. 3b).
The behaviour presented in Fig. 3 indicates that thestrength of the vorticity patterns – the typical upper vorticity
or the inner vorticity pair with opposite senses of rotation – isrelated to the strength of the downwash, which decreases
downstream in the wake. The mean streamwise vorticity
distribution in the y-z plane with x/D = 2 shown in Fig. 4 for
AR = 0.5 further confirms this observation. In this plane, theinner vorticity pair shows a large magnitude, especially in thelower part of the wake where the flow entrainment
mechanism is more pronounced for this AR. The vorticity
intensity is about the same with the thick boundary layer(Fig. 4b), but the size of the inner vorticity pair is reduced

Figure 4. Mean streamwise vorticity distribution and inplane velocity vectors (from the seven-hole pressure probe
measurements) in a y-z plane located at x/D = 2 for thesurface-mounted block with AR = 0.5. (a) Thin boundary
layer with δ/D = 0.6 and (b) thick boundary layer with
δ/D = 2.0. Every second vector in the y and z directions isshown, for clarity.

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

Figure 5. Mean transverse vorticity distribution and in-plane velocity streamlines (from PIV measurements) in the x-z symmetry
plane (y/D = 0) for surface-mounted blocks with (a) AR = 0.5, (b) 0.7 and (c) 1, and a thin boundary layer with δ/D = 0.6. Theorange, purple and green circles indicate the location of the free end vortex, vortex Bt and the approximate wake reattachment point,respectively. The red circle indicates the approximate reattachment point on the free end of the block with AR = 0.5.

stays at approximately z/H = 1, moving slightly closer
toward the ground plane for smaller AR.
The approximate wake reattachment point also
corresponds to the maximum longitudinal length of the wakefor the three aspect ratios. Table 1 and Figure 5 show that
this length increases with AR, in contrast to theapproximately constant maximum longitudinal length
reported by Wang and Zhou (2009) for AR = 1 to 3. Thisbehaviour further suggests that different flow trends takeplace for the range of AR ≤ 1, in comparison with otherprisms below the critical AR.
The location of the free end vortex also changed
significantly with AR. It moved upstream and downward forsmaller AR, being located at x/D < 0 for AR = 0.5. Other
notable flow features for AR = 0.5 are the two high-vorticity
regions in the wake of this block, in Fig. 5a. The upper one isrelated to the shear layer that separated from the leading edgeof the block, which is also observed for AR = 0.7 (Fig. 5b)
and 1 (Fig. 5c). The lower region is found for AR = 0.5 only,
originating due to the flow that has reattached on the block’s
free end subsequently separating from the trailing edge.
The red circle in Fig. 5a indicates the approximate
reattachment point on the free end of the AR = 0.5 block, at
x/D = 0.25. For the AR = 0.7 and 1 blocks, no reattachment
was observed on the free end. This phenomenon is related
not only to the smaller AR, but also to the higher probabilityof flow reattachment on the free end for larger boundary
layer thickness relative to the prism’s height (Castro andRobins, 1977). For the block with AR = 0.5 and the thinboundary layer, the boundary layer thickness relative to theblock height is δ/H = 1.2, meaning that this block is the only
one of the three that is completely immersed in the boundary
layer.
The present results suggest, therefore, that the occurrenceof the inner vorticity pattern in the wake of the block with
AR = 0.5 is connected to the reattachment of the flow overits free end. This observation is supported by the wake of theblock with AR = 0.7 and the thick boundary layer (δ/H = 2.9

Table 1. Main critical points in the x-z symmetry plane.
Critical point
Free end vortex centre
Centre of vortex Bt
Approx. wakereattachment point

AR = 0.5
x/D = -0.11
z/D = 0.59
x/D = 0.82
z/D = 0.40

AR = 0.7
x/D = 0.03
z/D = 0.84
x/D = 0.88
z/D = 0.62

AR = 1x/D = 0.17
z/D = 1.18
x/D = 1.00
z/D = 0.96

x/D = 1.50

x/D = 1.69

x/D = 2.19

when compared with Fig. 4a. This behaviour is in agreementwith the results obtained at x/D = 3 and 4 for AR = 0.5(shown in Fig. 2 and 3).

Particle Image Velocimetry Results
Figure 5 presents the mean transverse vorticity
component (ωyD/U∞) in the x-z symmetry plane of the wakeof the blocks with AR = 0.5 (Fig. 5a), 0.7 (Fig. 5b) and 1(Fig. 5c), for the thin boundary layer. The location of someof the main critical points in the wake are indicated by thecoloured circles: the orange circle indicates the focus point
corresponding to the centre of the free end vortex; the purple
circle indicates the focus point corresponding to the centre ofvortex Bt, following the nomenclature of Sumner et al.
(2017) and Krajnović (2011); and the green circle indicatesthe approximate mean reattachment point of the wake. Notethat the reattachment point is approximate as PIV
measurements were carried out as close as 2 mm (0.03D)from the ground plane, to avoid excessive reflections. The
locations of these points are presented in Table 1.
The wakes of all three blocks show common features towakes of prisms below the critical aspect ratio, such as theabsence of vortex Nw and the predominance of the large
vortex Bt (Sumner et al., 2017). The location of the centre ofvortex Bt moves closer to the block for smaller AR, but it4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
versus δ/H = 0.9 for the thin boundary layer), for which theinner vorticity pattern also occurred, at least at x/D = 3.
However, since this pattern was not observed for AR = 1 andthe thick boundary layer (δ/H = 2.0), it is expected thatreattachment on the free end of the block and thedevelopment of the inner vorticity pattern in the wake is not
only a function of δ/H, but also of AR.

REFERENCES
Calluaud, D., David, L., and Texier, A., 2000, “Study ofthe laminar flow around a square cylinder”, Proceedings of
the Ninth International Symposium on Flow Visualization,
Edinburgh, UK, Aug. 22–25, p. 402.
Castro, I. P., and Robins, A. G., 1977, “The flow arounda surface-mounted cube in uniform and turbulent streams”,
Journal of Fluid Mechanics, Vol. 79, pp. 307-335.
da Silva, B. L., Sumner, D., and Bergstrom, D. J., 2022,“Mean and dynamic aspects of the wakes of a surfacemounted cube and block”, Journal of Fluids Engineering,Vol. 144, p. 011302.
Heng, H., and Sumner, D., 2020, “Wind loading of afinite prism: aspect ratio, incidence and boundary layer
thickness effects”, Wind and Structures, Vol. 31, pp. 255267.
Krajnović, S., 2011, “Flow around a tall finite cylinder
explored by large eddy simulation”, Journal of Fluid
Mechanics, Vol. 676, pp. 294-317.
Okamoto, S., Tsunoda, K., Takagi, T., Okada, E., andKitani, K., 1995, “Near wake behind square cylinder of finite
length on ground plane”, Transactions of the JSME Series B,
Vol. 61, pp. 3105-3113.
Sumner, D., Unnikrishnan, S., Teng, M., Beitel, A., Das,
A., and Fulton, M., 2016, “The mean wake of low-aspectratio surface-mounted finite-height square prisms and theeffects of incidence angle”, Proceedings of the 8thInternational Colloquium on Bluff Body Aerodynamics andApplications, Boston, MA, USA. June 7-11.
Sumner, D., Rostamy, N., Bergstrom, D.J., and Bugg,
J.D., 2017, “Influence of aspect ratio on the mean flow fieldof a surface-mounted finite-height square prism”,
International Journal of Heat and Fluid Flow, Vol. 65, pp. 120.
Unnikrishnan, S., Ogunremi, A., and Sumner, D., 2017,“The effect of incidence angle on the mean wake of surfacemounted finite-height square prisms”, International Journalof Heat and Fluid Flow, Vol. 66, pp. 137-156.
Wang, H.F., and Zhou, Y., 2009, “The finite-length
square cylinder near wake”, Journal of Fluid Mechanics,
Vol. 638, pp. 453-490.
Yauwenas, Y., Porteous, R., Moreau, D. J., and Doolan,
C. J., 2019, “The effect of aspect ratio on the wake structureof finite wall-mounted square cylinders”, Journal of Fluid
Mechanics, Vol. 875, pp. 929-960.

CONCLUSIONS
The mean wake of surface-mounted blocks wasinvestigated for aspect ratios AR = 0.5, 0.7 and 1, Reynolds
number Re = 7.5 × 104 and boundary layer thicknesses
δ/D = 0.6 and 2.0.
Different mean flow patterns were identified in the wakeof the blocks. For the boundary layer thickness with
δ/D = 0.6, a smaller aspect ratio was associated with a shorter
longitudinal length of the wake and a free end vortex centre
located closer to the leading edge of the block. The mean
streamwise vorticity distribution had different patterns thatwere found to depend on both AR and δ/D. The upper vortex
pair, typical of tall surface-mounted finite-height squareprisms, was only found for AR = 1. It was weakened andreplaced by an inner pair of vorticities with opposite sign ofrotation for smaller AR and larger δ/D.
The occurrence of this inner vorticity pair appears to berelated to flow reattachment on the free end of the block.
These phenomena were found to be a function of both ARand δ/D (or δ/H), so further studies will be necessary toinvestigate in detail the relationship of the inner vorticity pairand flow reattachment on the free end with the flow
parameters.

ACKNOWLEDGEMENTS
Financial support from the National Sciences andEngineering Research Council of Canada (Discovery Grants
Program numbers RGPIN 05103-2016 and 2018-03760) andthe University of Saskatchewan's Dean's Scholarshipprogram is gratefully acknowledged. The technical supportof S. Reinink, J. Bugg and Engineering Shops is also
recognized.

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