Experiments in Fluids 33 (2002) 854–862
DOI 10.1007/s00348-002-0497-5
Turbulent flow around a surface-mounted obstacle
using 2D-3C DPIV
J.M.M. Sousa
854
Abstract The mean turbulent flow structure around a
cube mounted on the surface of an open-surface water
channel was studied using a two-dimensional implementation of digital particle image velocimetry (DPIV). The
out-of-plane velocity component was obtained by the use
of the concept of continuity applied to two-dimensional
velocity fields recorded in parallel planes. Various methods were used for the identification and localization of
large-scale vortical structures in the three-dimensional
flow around the surface-mounted obstacle. The results
show the feasibility of its application to three-dimensional
PIV data and the superior performance of recent
identification techniques (namely swirling strength and
normalized angular momentum), over the classical
vorticity-based criterion.
1
Introduction
Most experimental studies of the flow over sharp-edged
obstacles mounted in channels focus on two-dimensional
(2D) geometries. These are easier to handle, which explains
the large number of investigations on two-dimensional
surface-mounted obstacles reported in the literature. By
contrast, quantitative data for turbulent flows over threedimensional (3D) obstacles are still scant. Castro and
Robins (1977) and Hunt et al. (1978) have studied the
structure and topology of the flow around cuboids for a
boundary-layer-type of approach flow. On the other hand,
Martinuzzi and Tropea (1993) paid special attention to the
turbulence characteristics of the flow around prismatic
obstacles mounted in a plane channel. Other aspects such
as the recovery of the turbulent boundary layer downstream of the obstacle and the effect of flow angle-of-attack
have also been studied respectively by Schofield and Logan
(1990) and Natarajan and Chyu (1994).
Despite their geometrical simplicity, these flows exhibit
remarkable topological complexity, making them very attractive for the use of particle image velocimetry (PIV).
However, in order to capture all the relevant flow structures
Accepted: 25 April 2002
Published online: 17 July 2002
Springer-Verlag 2002
J.M.M. Sousa
Instituto Superior Técnico,
Mechanical Engineering Department,
Av. Rovisco Pais, 1049–001 Lisboa, Portugal
E-mail: msousa@alfa.ist.utl.pt
in three-dimensional problems, the three velocity components (3C) should be made available. As reviewed by
Hinsch (1995) and Royer and Stanislas (1996), an increasing variety of approaches may be employed to achieve
this objective. Possible options are stereoscopic recording
using two cameras or a single camera, holographic recording and dual-plane PIV, to name a few of today’s most
widely used methods. The concept of continuity may also
be used to recover the third velocity component by integration of the continuity equation over a set of 2D velocity
fields recorded in parallel planes. Robinson and Rockwell
(1992) made a critical assessment of this procedure using
synthetic flow fields. Brücker (1995) used the method with
success in the characterization of starting flow around a
short cylinder, using a scanning light sheet. However, the
technique presents a few limitations, namely the requisites
of flow incompressibility and a priori knowledge of the outof-plane velocity component at the starting locations of
integration. Furthermore, the region of interest in the flow
must be scrutinized with many closely spaced cut planes,
and in order to minimize the errors in this approach, a finer
discretization must be used in areas where the out-of-plane
velocity attains larger magnitudes. Nevertheless, this solution may provide reliable results for incompressible flows
containing symmetry planes and/or well-specified boundary conditions. In addition, it presents the paramount advantage of demanding only a simple 2D PIV arrangement
to retrieve the three-dimensional structure of a flow.
Once the three velocity components have been obtained,
the identification and localization of large-scale vortical
structures is of capital importance to accomplish a proper
analysis of the flow around a surface-mounted obstacle.
Again, various methodologies have been proposed to meet
this purpose. Most of these methods were first designed for
application to direct and large-eddy simulation data, but
the same concepts may, in principle, be applied to PIV data
as well. Alternatives to the classical use of simple vorticitybased criteria are, for example, the computation of the
normalized angular momentum, presented by Michard
et al. (1997), and the determination of the swirling strength,
as suggested by Adrian et al. (2000). Nevertheless, it should
be noted that the application of any of the above-mentioned methods to PIV data aimed at the retrieval of 3D
vortical structures is rarely reported in the literature.
In the present work, the mean turbulent flow structure
around a cube mounted on the surface of an open-surface
water channel was investigated using a 2D implementation
of digital particle image velocimetry (DPIV). Threedimensional flow maps were derived by the use of the
concept of continuity applied to 2D velocity fields recorded in parallel planes. The large-scale vortical structures in the flow were identified and localized by the
application of various methods. A critical assessment of
the performance of these methods was conducted as well.
Reliable turbulent characteristics were obtained by collecting a statistically significant number of measurements
at each spatial location.
2
Experimental arrangement and data processing
In order to study the flow around a three-dimensional
surface-mounted obstacle, a cube was placed in the bottom
of an open-surface water channel. The mass flow rate recirculating inside the channel produced a surface velocity
U0 = 0.07 m/s. The unperturbed flow upstream of the
obstacle was a turbulent boundary layer characterized by a
Reynolds number based on the surface velocity and momentum thickness Reh=770. A detailed description of the
water channel facility employed in these experiments was
given by Freek et al. (1997). The surface-mounted obstacle
was a cube made of transparent Perspex with size
L=40 mm. As a result, the presently investigated obstacle
flow was characterized by a Reynolds number ReL=3210.
Additional details about the test section can be found in
Sousa et al. (2000).
The digital 2D PIV system employed to map the flow
field is based on CCD camera recording and Ar-ion laser
illumination. MJPEG compression of the PIV images was
also used to make it easier to handle the large quantities of
data. A thorough description of the present system as well
as a comprehensive discussion on the advantages and
drawbacks of using MJPEG compression was presented by
Freek et al. (1999).
A total of 28 sequences containing 2000 PIV images was
collected for this study. The data sets were formed by 23
vertically cut planes, spanning 3.5 times the size of the
cube (14/4 L), as shown in Fig. 1. The spacing between the
cut planes was 5 mm, except for the six planes located at
the spanwise borders, for which a spacing of 10 mm was
used. In addition, data sets formed by five horizontal cut
Fig. 1. Flow configuration and measurement locations
planes at selected locations were also collected during the
measurement campaign.
Digital image compression allowed on-line recording of
the data at a sampling frequency of 25 Hz (only 80 s to
acquire each data set), keeping storage needs low. Flow
maps were obtained by off-line processing of PIV images
interrogated in sub-images of size IA=32·32 pixels. The
spacing between interrogation windows was prescribed as
16 pixels (i.e. 50% IA overlap), producing 47·35 vectors
per vector field. An overview of the relevant experimental
parameters is shown in Table 1.
The mean (U,V) flow and other turbulence statistics
were calculated for each data set by using single-point
ensemble averaging. According to Raffel et al. (1998), the
total measurement error in displacement PIV vectors can
be written as the sum of a bias error and a measurement
uncertainty (random error). Unbiased values of the correlation function were obtained by dividing out the corresponding weighting factors prior to the estimation of the
fractional displacement, as suggested by Westerweel
(1993). A Gaussian estimator of fractional displacement
Table 1. Summary of relevant experimental parameters
Channel:
Type
Length (m)
Width (m)
Open surface
2.5
0.20
Flow:
Fluid
Temperature (C)
Depth (m)
Surface velocity U0 (m/s)
Obstacle height L (mm)
Reynolds number Reh
Reynolds number ReL
Water
28
0.11
0.07
40
770
3210
Seeding:
Type
Nominal diameter (lm)
Polyamid
60
Illumination:
Type
Source
Maximum power (W)
Thickness (mm)
Pulse separation (ms)
Number of exposures
Light sheet
Ar+ laser (cw)+Bragg cell
5
3
10
2
Recording:
Type
Resolution (pixels)
Lens focal length (mm)
Numerical aperture
Image magnification
Image compression
Electronic (CCD)
576·768
26
2.0
0.21
5.5
Interrogation:
Resolution IA (pixels)
Spacing (pixels)
Multigrid levels
32·32
16
2
Data sets:
Number
Size/data set (Mb)
Images/data set
Vectors/image (no mask)
23+5
160
2000
1645
855
856
was used to achieve sub-pixel accuracy, leading to
negligible values of rms tracking error associated with the
estimator. For the recorded particle image sizes (typically
3 pixels), the foregoing error is expected to be smaller than
0.02 pixels and, in fact, evidence of peak locking was not
found in histograms of displacement data obtained at selected locations. Increased values of dynamic spatial range
and data yield were obtained by the implementation of
window offset coupled with a multigrid/pass procedure
(see, e.g. Raffel et al. 1998), which also allowed further
reductions in the measurement noise, in conformity with
the findings of Westerweel et al. (1995). Employing synthetic images and Monte Carlo simulations, Freek et al.
(1998) showed that, in such idealized conditions, the
present PIV evaluation algorithm may reduce the error in
mean displacement to values below 0.01 pixels when
choosing IA=32·32 pixels. In addition, high seeding densities and a thicker-than-usual light sheet (see Table 1)
were used in the present experiments in order to minimize
the errors from the presence of displacement gradients in
the images and from the effect of out-of-plane motion,
respectively. Finally, spurious measurements were removed from all vector fields by employing a median-based
procedure similar to that indicated by Westerweel (1994).
As a consequence of the above assertions, the dominant
source of error in the present results is believed to arise
from the statistical evaluation of the turbulent flow
quantities. Hence, based on the analysis proposed by
Yanta and Smith (1973), the statistical uncertainties in the
mean and variance values for a 95% confidence level are
estimated to lie below 1% and 3%, respectively.
The out-of-plane component W of the mean flow was
computed from the two in-plane components by using the
continuity equation for incompressible flow applied to the
time-averaged velocity field
@U @V @W
þ
þ
¼ 0:
@x @y
@z
ð1Þ
Integration of Eq. (1) in the spanwise z-direction yields the
expression to retrieve the W-component
Wðxi ; yj ; zk Þ ¼Wðxi ; yj ; zk1 Þ
Zzk @V @U xi ; yj ; z þ
xi ; yj ; z dz;
@x
@y
zk1
ð2Þ
discretized in the nodes i, j, k of a three-dimensional grid.
The derivatives in the integral of Eq. (2) were computed in
the x–y-plane, employing (second-order) central differences where possible. In the vicinity of solid and open
boundaries either forward or backward (first-order) differences were used. A simple four-neighbor smoothing
kernel was applied to the planar data before the derivations, in the same manner as described by Robinson and
Rockwell (1993). However, in contrast to the work of the
preceding authors, the numerical integrations were carried
out here via direct computation of the integral in Eq. (2)
by the mid-point rule. The integrations started at k=2, and
W=0 was assumed for k=1 (symmetry plane) and for k=kw
(solid wall). The final outcome of this procedure was the
three-dimensional (U, V, W) mean flow inside the volume
defined by the 23 cut planes.
3
Results and discussion
3.1
Reconstruction of the three-dimensional velocity field
As described in Sect. 2, the three components of the mean
flow velocity were obtained using the concept of continuity
to derive the out-of-plane velocity field. The procedure
brings uncertainties to the values of W, which are mainly
related to the propagation of uncertainties in the base
values of (U, V) as well as with the presence of discontinuities in the domain of integration of Eq. (2). In addition,
it should be noted that these two sources of uncertainty are
often correlated, further increasing the error magnitude in
these areas. As explained before, efforts were made to
minimize the former source of uncertainty during the
measurement and processing stages. On the other hand,
Robinson and Rockwell (1993) carried out a systematic
investigation of the latter type of uncertainties, comparing
their results with exact solutions. In the present flow
problem an exact solution is obviously not available.
Therefore a few selected horizontal cut planes were used as
reference for comparison with the reconstructed data obtained for the same y-locations.
Figure 2 displays velocity vector fields in the x–z-cut
planes for directly measured 2D data (left) and reconstructed 3D data (right). The represented cut planes cover
the flow area from the vicinity of the bottom wall up to a
region where the presence of the obstacle causes only a
mild deviation of the streamlines, in a range of the
y-coordinate between 3 and 50 mm. It can be seen that
both the velocity magnitude and the flow pattern in the
x–z-planes are reproduced fairly well by the reconstruction procedure. Discrepancies between the two sets seem
to be found mainly near the obstacle borders, where surface discontinuities appear. A more quantitative perception of the errors resulting from the reconstruction
procedure can be obtained from Fig. 3. This figure depicts
error fields (absolute value) obtained by subtracting the
measured W-velocity fields from the corresponding ones
derived from the concept of continuity. All values were
previously normalized by the surface velocity, and linear
interpolation has been used to collocate all the data in the
grid used for the direct measurements. The error maps
confirm that larger error values are ordinarily obtained
near the obstacle corners. In addition, it can be seen that
the areas surrounding the front corners are particularly
critical as a consequence of the sharp velocity gradients.
Furthermore, the large amount of outflow through lateral
boundaries occurring in the cut plane neighboring the wall
is correlated with large errors as well. As expected, a
correlation is also found between larger (absolute) errors
and higher magnitudes of the W-velocity, which generally
decreases with rising values of the y-coordinate (see also
Fig. 2). The asymmetry in the maps illustrated in Fig. 3
857
Fig. 2. Directly measured (left)
and reconstructed (right) velocity vector fields in x–z-planes
may originate from a small misalignment of the flow,
which could not be reproduced in the reconstructed data
due to the assumption of symmetry with respect to the line
z=0. Finally, it should be mentioned that a reduction in the
errors described above could be achieved by the use of a
larger spatial resolution in the PIV measurements at the
expense of the size of the investigated flow domain.
A full view of the reconstructed three-dimensional
mean flow field around the surface-mounted obstacle is
given in Fig. 4. Significant curling of the streaklines may
be observed in various areas of the flow, indicating the
presence of several large-scale vortical structures (the
grayscale code in the volume ribbons represents the
magnitude of the vorticity vector). The identification and
localization of these areas in the constructed 3D PIV data
will be the next step in this study.
In this study, the large-scale (three-dimensional) vortices were identified and localized in the reconstructed
mean velocity field by the use of distinct methodologies.
The most usual procedure to reach this goal consists of the
computation of the vorticity field. However, in 3D flow the
vorticity vector also has three components as follows:
@W @V ~
@U @W ~
@V @U ~
~
~
~
XrV¼
iþ
jþ
k:
@y @z
@z @x
@x @y
ð3Þ
Hence the modulus of the vorticity vector must be calculated in this case. Again, the first derivatives of the velocity
field in Eq. (3) were computed from central differences
where possible. As the planar fields had been previously
filtered by a smoothing kernel for the calculation of the outof-plane velocity component, the above procedure was
preferred to the use of Stokes’ theorem to evaluate vorticity.
3.2
In order to visualize the vortices, surfaces of constant
Identification of three-dimensional vortical structures
vorticity magnitude were constructed, as shown in Fig. 6,
The main features of the flow around surface-mounted
obstacles characterized by a small aspect ratio were ana- by the use of off-the-shelf graphical visualization software.
lyzed in detail using smoke and oil-film visualization by The vector field and the vorticity contours at the centerplane and in the vicinity of the bottom wall are also porHunt et al. (1978) and Martinuzzi and Tropea (1993).
Based on the foregoing investigations and on the present trayed in the figure. It can be concluded from the results
results, a model was constructed to illustrate the topology that the areas where the large-scale vortices should appear
have been identified. Nevertheless, the localization of the
of the most significant vortical structures in the flow.
Figure 5 shows a schematic form of this model, identifying vortices is difficult because the structures were not individualized by this procedure. A reduction in the level of
four principal structures: A, B, C and D.
858
Fig. 3. Error fields for the outof-plane velocity component in
the reconstructed data; contour
spacing: 0.05U0
Fig. 4. Perspective view of the
flow (streaklines) around the
surface-mounted obstacle
vorticity characterizing the surface had the consequence of
missing part of the structures. It can be seen that the shear
layers developing from the frontal sharp edges of the obstacle and the boundary layer are regions of high vorticity,
which are not correlated with the presence of vortices. Such
high values of vorticity tend to hide the presence of the
large-scale vortical motions embedded in these regions.
Better results were expected to be obtained by the use of
a methodology based on the analysis of the (mean) local
velocity gradient tensor and its corresponding eigenvalues.
Once more central differences were used where possible to
compute velocity gradients from the previously filtered
data. Generally, in 3D flow the local velocity gradient tensor
has one real eigenvalue and a pair of complex conjugate
eigenvalues, as follows from the characteristic equation:
h
i
!
r V k ¼ 0 , k1 ¼ kR 2 R; k2;3 ¼ kcr kci 2 C:
ð4Þ
By computing the value of kci2 obtained from Eq. (4) for
each location in the flow, the strength of the local swirling
motion (denoted by swirling strength) can thus be quantified.
a volume W surrounding a general point P in the flow as
follows:
Z
1
ð~
x ~
xP Þ ~
Vð~
xÞ
~
d~
f ð~
xP Þ ¼
x:
ð5Þ
~
X ~x2X j~
x Þ
x ~
xP j Vð~
This function is called the normalized angular momentum (NAM), and its modulus varies in the range (0,1). In
order to apply the above definition to the PIV field,
Eq. (5) must be discretized for a general point Pi and its
neighbors Mn in the surrounding volume, thus yielding
Fig. 5. Schematic representation of the principal vortical structures
around a surface-mounted obstacle
The new results show an improvement in the identification of the large-scale vortices (Fig. 7). The method
seems to separate the regions characterized by the presence of local swirling motion from those exhibiting only
high shear, thereby facilitating the localization of the
vortices in the flow. However, the legs of the horseshoe
vortex (A) encircling the obstacle could not be clearly visualized. An explanation for this may be found by noting
that the strength of horseshoe vortex decreased considerably as it was convected downstream. The turbulence
generated by the presence of the obstacle certainly played
an important role in this process, augmenting the action of
dissipation mechanisms.
A third method for the identification of the three-dimensional vortical structures in the flow around the
surface-mounted obstacle was implemented in the course
of this work. In contrast with the previous methodologies, which both make use of quantities related to velocity gradients, this method is based on purely
geometrical considerations regarding the flow structure.
It demands the computation of a vector function ~
f ð~
xP Þ on
X ½~
1
xðPi Þ ~
VðMn Þ
xðMn Þ ~
~
;
f ðPi Þ ¼
3
~
xðPi Þj VðMn Þ
xðMn Þ ~
ð2N 1Þ n j~
ð6Þ
where N is the number of layers defining the abovementioned volume. As a result, the value of N is an
important parameter that controls the efficiency of this
technique. In the present case, it was found that N=3 was
the best choice, though there were not significant differences for values in the range (2,4). It follows from
Eq. (6) that this method has the important advantage of
requiring numerical integrations instead of numerical
differentiations. However, the corresponding computational effort is larger in this case as well.
The outcome of this procedure may be observed in
Fig. 8, which shows surfaces of constant modulus of NAM.
All the large-scale vortical structures can be identified in
the figure as when visualizing swirling strength. However,
the surfaces are smoother than before, which may facilitate
the localization of the vortices. The horseshoe vortex (A)
now seems to extend beyond the obstacle, although it is
partially hidden by the wake vortex (D). Moreover, the
roof vortex (B) appears much smaller than before, which
represents a more realistic view. Nevertheless, a clear
Fig. 6. Surfaces of constant value of vorticity magnitude and cut planes
859
860
Fig. 7. Surfaces of constant value of swirling
strength and cut planes
superiority of this method over the latter should not be
claimed as a general conclusion.
Finally, surfaces of constant turbulent kinetic energy
around the surface-mounted obstacle are presented in
Fig. 9. It must be noted that the spanwise component of
the normal stresses was estimated from the planar com-
ponents u02 and v02 by computing their average at each
point in the grid, resulting in the following formula for
turbulent kinetic energy q:
q¼
3 02
u þ v02 :
4
ð7Þ
Fig. 8. Surfaces of constant value of normalized
angular momentum modulus and cut planes
861
Fig. 9. Surfaces of constant value of turbulent kinetic energy and cut planes
The above approximation is expected to provide realistic
results in the absence of direct measurements of the
spanwise component of the normal stresses. However, the
data gathered by Martinuzzi and Tropea (1993) for surface-mounted cubical obstacles in channel flow indicate
that significant deviations from Eq. (7) may occur, especially inside recirculations.
The large-scale unsteadiness of structure (A) upstream
from the obstacle is known to contribute strongly to the
turbulent kinetic energy q. Sousa et al. (2000) found that
though for the current Reynolds number the horseshoe
vortex (A) does not constitute the dominant source of
unsteadiness in the flow, this region is far from quiescence.
As a consequence, the vortex can easily be identified in the
figure. However, the remaining principal vortical structures are immersed in shear layers, thus hindering their
identification. It can also be seen that the legs of the
horseshoe vortex cross a region on the sides of the obstacle
displaying high values of q, which seems to support the
above assertion that turbulence dissipation was an explanation for the fading of the legs.
4
Conclusions
Measurements of the turbulent flow around a surfacemounted cube were performed using a two-dimensional
implementation of digital PIV. The concept of continuity
applied to two-dimensional velocity fields recorded in
parallel planes was used to derive the out-of-plane velocity
component. As a result, three-dimensional PIV maps of
the mean flow containing more than 30,000 vectors were
obtained. Selected transverse planes extracted from the
constructed flow volume were compared with their counterparts obtained by direct measurement. In the present
case, it was observed that reliable results could be
obtained, despite the presence of geometrical discontinuities in the flow. These are clearly the main sources of
uncertainty in the reconstruction of the flow field, as efforts were made to minimize other effects.
Various methodologies were used for the identification
of the principal large-scale vortical structures in the threedimensional flow around the surface-mounted obstacle:
the classical vorticity-based criterion, the determination of
the swirling strength and the calculation of the normalized
angular momentum. The results show the feasibility of its
application to three-dimensional PIV data and the superior performance of the latter two methods over the first.
The availability of higher-order turbulence statistics from
the PIV measurements also allowed observation of the
correlation between regions characterized by large values
of turbulent kinetic energy and those occupied by the
large-scale vortices in the flow.
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