Analysis on Flow Around a Rectangular Cube by Means of LDA, PIV
and the Numerical Simulation by PCC Method
- In the Case of Inclined Cube Tomokazu Nomura*1, Yasushi Takahashi*2, Tsuneaki Ishima*1 and Tomio Obokata*1
*1 Dept. of Mechanical System Engineering, Gunma University
1-5-1, Tenjin, Kiryu, 376-8515 JAPAN
*2 Honda R&D Co., Ltd. Asaka R&D Center, CIS
ABSTRACT
Analysis on flow around a rectangular cube set on the floor in a rectangular channel is described. The study is carried
out using two-dimensional laser Doppler anemometry (2D-LDA) with fiber optics, Particle Image Velocimetry (PIV)
and numerical simulation by PCC (Partial Cells in Cartesian coordinate) method. In this study, the rectangular cube
(65mm x 58mm x 31mm) is set with 10 degrees of offset angle to the flow direction. The mean flow velocity and the
relative turbulence intensity are 4.0m/s and 1.5%, respectively in both two experiments and calculation. Under this
condition, the flow Reynolds number based on the rectangular cube height and the mean velocity is about 8.8 x 103.
Firstly, comparisons between LDA and PIV results with numerical results are made for the mean velocity U, the
turbulent intensity u’ and the Reynolds stress u’w’. Figure 1 shows distributions of U and u’ at the back of the
rectangular cube. It shows a good agreement between both LDA and PIV measurement results. The results by the
numerical simulation show little differences from the experimental results. Secondly, from the instantaneous velocity
fields obtained by PIV measurements, the visualizations of vortices distributions by the Reynolds decomposition and
the Galilean decomposition methods are performed. Figure 2 shows the Reynolds decomposed fluctuations at the side
tip of rectangular cube. Many vortices are visualized along the separation line. The results analyzed from another
techniques are described in this paper. Finally, the integral length scale measurements are performed by both LDA and
PIV. For the LDA measurements, length scale is estimated from the assumption of Taylor’s hypothesis. Figure 3 shows
the comparison of length scale distributions between LDA and PIV measurements at the back of rectangular cube. It is
pointed out that both results show the big differences in the higher turbulence regions of u’/U > 20%.
60
80
z mm
60
40
y mm
LDA
PIV
CFD
z
20
50
40
x
4.0 m/s
0
2 4
U m/s
6 0
0.5
1
1.5
30
u’
Fig. 1 Vertical distributions of mean velocity U and
turbulence intensity u’ at x = 334mm, y = 0mm.
240
250
270
Fig. 2 Reynolds decomposed fluctuations.
16
10000
PIV
LDA
u'/U
12
Lx mm
260
x mm
1000
8
100
4
10
0
0
20
40
60
z mm
80
u'/U %
-2 0
1
100
Fig. 3 Comparison of length scale between LDA and PIV results at x = 334mm, y = 0mm.
1. INTORODUCTION
The flow around an obstacle is one of the fundamental studies in the fluid dynamics. In that flow, the flow
field is complex because there are highly turbulent recirculation zones, shear layer and near wall flows. In the industrial
area, the flow structures with obstacle are frequently observed, e.g. wake flows behind a car or a motorcycle, flows
around buildings or bridges. Since this kind of flow has enough difficulty, in other words this kind of flow is simple but
consists of recirculation flow, re-attachment, shear layer, vortex shedding and boundary layer, it is interesting to apply
the newer measuring techniques and newer numerical methods to analyze the flows. In the present study, the flow
around a rectangular cube with off-set angle has been studied, because the obstacle has a certain angle to the flow in the
many actual situations. The measuring techniques of a laser-Doppler anemometer (LDA) and a particle image
velocimetry (PIV) are applied for estimating the flow field.
The PIV is useful to visualize the flow fields and to evaluate the velocity and the turbulence characteristics
such as turbulent intensity, Reynolds stress and length scale. In order to visualize the vortices and to analyze the spatial
vortex distribution, decompositions of instantaneous fluctuation of the velocity and vorticity are applied well for the
velocity fields. For estimation of the instantaneous fluctuation of the velocity, several techniques such as Galilean
decomposition and LES decomposition are used (Adrian et al., 1998, 2000). Moreover, PIV is effective to estimate the
integral length scale, since PIV can measure the two-dimensional instantaneous velocity filed. On the other hand, when
using point measurement such as LDA, determination of the length scale has complicated. One of the possibilities of
measuring the length scale with LDA is a two point measurements (Obokata et al., 1989), but the optics of the
two-points measurement LDA is also complicated. Then, the results from autocorrelation of turbulent fluctuation and
using Taylor's hypothesis, the length scale can be estimated (Elavarasan et al., 1996; Valentino et al., 1996). However,
under the flow with highly turbulence, the Taylor’s hypothesis gives less accuracy (Elavarasan et al., 1996). Evaluation
of the accuracy of the Taylor’s hypothesis in the wake region is one of the importances of the fluid engineering. In this
study, both of the measuring methods of PIV and LDA are applied to obtain the length scale in the recirculation zone.
In recent years, large size numerical simulations (large number of mesh points, three dimensional calculation
and so on) become possible by the remarkable developments in personal computer. However, in the large size
simulations need some kinds of the turbulence models, such as k-ε model. The simple flow is useful for the verification
of the numerical simulation. The LDA and the PIV are provided with enough high spatial resolution and accuracy for
the verification of the numerical code.
The purpose of the study is to know the turbulence characteristics in the flow using both of the measuring
techniques of PIV and LDA. The flow around the rectangular cube with 10 degree of the offset angle has been studied.
A numerical simulation with PCC (Partial Cells in Cartesian coordinate) method is also applied to obtain the velocity
field. The experimental and simulation results are compared with each other. In the comparison, mean velocity and
turbulence intensity are used for evaluation of the accuracy of the numerical code. By using the advantage of the PIV,
several analyzing methods of Reynolds and Galilean decompositions are applied (Adrian et al., 1998, 2000). These
methods can evaluate the spatial positions of the vortices. Comparison of the length scales obtained by PIV and LDA
has been also carried out. To discuss the difference between the results in the length scales is the other topic of the
study.
2. EXPERIMENTAL APPRATUS AND NUMERICAL SIMULATION
2.1 Laser Doppler Anemometry
Figure 4 shows the experimental setup for LDA measurements. In this study, the test section was a
rectangular duct whose length = 1000mm, width = 400mm and height = 200mm, and the rectangular cube (length =
65mm, width = 58mm, and height = 31mm) was set with 10 degrees of offset angle to the flow direction. The origin of
coordinate was set at the center of the nozzle exit and x-, y-, and z-axes were set to the flow direction, transverse
direction, and vertical direction, respectively. The mean flow velocity and turbulent intensity were set to 4.0m/s and
1.5%, respectively and the flow Reynolds number based on the rectangular cube height and the mean flow velocity was
about 8.8 x 103.
The measuring positions of LDA are shown in Figure 5. At all points, the measurements were made from z =
0 to 100mm. In order to measure the velocities of x-direction and z-direction at the same point simultaneously, the study
was carried out using the two-dimensional LDA with fiber optics. The sending optics of the two-dimensional LDA were
consisted of Ar+ gas laser (λ=488nm) which had double Bragg cell of 80 and 85 MHz and He-Ne gas laser (λ=632.8nm)
which had double Bragg cell of 80 and 83 MHz with a transmitter lens of focal length of 300 mm and full beam angle
of 3.10 degree and 5.82 degree respectively. The intersection volumes of incident beams were 122µm x 4.51mm and
198µm x 3.91mm. However, the actual intersection volumes were 122µm x 0.4mm and 198µm x 0.35mm because of
the forward scattering system with off set angle of 30 degrees. The water particles, the mean diameter of about 6 µm,
generated by humidifier were used as seeding particle. The scattered light from the seeding particles was detected by
means of two Photomultipliers. Two BSAs (Burst Spectrum Analyzer) were used for processing the Doppler signals.
220 258. 295.5 334 374
434
200 240 277 314 354 394
474 mm
Frequency Shifters
Argon gas laser
Computer
He-Ne gas laser
Beam splitter
Optical Fiber
y
x-z
table
Probe
Nozzle
55
35
y 17.5
0
-17.5
-35
-55
c
d
e
x
Motor
0
a b
Fig.5 Measuring points for LDA.
x
Nd:YAG laser
Rectangular Cube
Rectangular Cube
Laser sheet
z
Test section
x
Computer
CCD Camera
Burst Spectrum
Analyzer
Oscilloscope
Fig.4 Experimental setup for two dimensional laser
Doppler anemometry measurements.
Computer
PIV Processor
Fig.6 Experimental setup for particle image
velocimetry measurements.
Statistic values of mean velocity, turbulence intensity and Reynolds stress were calculated from the acquired time series
data by personal computer.
2.2 Particle Image Velocimetry (PIV)
The Particle Image Velocimetry was used for measure the instantaneous velocity fields in this study. Figure 6
illustrates the experimental setup for PIV measurements. The PIV system consists of an Nd:YAG laser (New Wave
Research: Model SOLO III-15), a high-resolution CCD camera (KODAK MEGAPLUS ES-1.0), a PIV processor
(DANTEC Flow Map 2000) and a personal computer. The light pulse pairs were generated by Nd:YAG laser, whose
power was 50 mJ and wavelength of 532 nm. The laser beams were transformed into light sheets using a cylindrical
lens. The high-resolution CCD camera with 1008 x 1018 pixels positioned in a direction normal to the laser sheet. The
CCD camera was synchronized with laser pulses and save two images corresponding to the laser pulses. The image was
divided by the interrogation areas of 32 x 32 pixels and was processed 50% x 50% overlapping. From each pair of
images, two-dimensional flow field around the rectangular cube was obtained using a cross correlation method. The
error vectors were removed from the vector maps by post processing and statistic values of mean velocity, turbulence
intensity and Reynolds stress were calculated from these vector maps by the personal computer.
2.3 Numerical simulation by PCC method
The PCC (Partial Cells in Cartesian coordinate) method CFD code (Takahashi et al., 1994) was used for
calculating the flow field around a rectangular cube. The PCC method was developed to reduce the input data
preparations by using Cartesian coordinate and partial cells for any geometry. Figure 7 shows examples of partial cells.
A partial cell is a fluid cell with missed portions by walls from a hexahedron, so many different shapes are existed.
Because of existing the partial cells, a combination of a low Reynolds number k-ε model and log law is applied for the
boundary conditions of the turbulence model. The SIMPLE algorithm and hybrid scheme are used (Patankar, 1980).
The mass conservation equation is expressed as:
∂ ρ ∂ (ρ u ) ∂ ( ρ v ) ∂ (ρ w)
+
+
+
=0
∂t
∂x
∂y
∂z
(1)
The momentum equation for x direction is expressed as:
∂ ( ρ u ) ∂ ( ρ u u ) ∂ ( ρ v u ) ∂ ( ρ wu ) ∂ p
+
+
+
+
∂t
∂x
∂y
∂z
∂x
=
where,
∂u  ∂ 
∂u  ∂ 
∂ u  1 ∂   ∂u ∂v ∂w  2 ∂ (ρ k )
∂ 
µ
+
µ
+
µ
+
 −
+
+
µ 
∂ x  e ∂ x  ∂ y  e ∂ y  ∂ z  e ∂ z  3 ∂ x  e  ∂x ∂y ∂z  3 ∂ x
k2
(2)
µ e = µ + µ t = µ + Cµ ρ f µ
ε
(3)

+ 
 y + 


 1 − exp − y 
f µ = 1 − exp −


 87 
 50.5 





(4)
1
y+ =
1
ρ y C µ4 k 2
(5)
µ
The momentum equations in y and z direction are expressed similarly to the above equation.
The turbulent kinetic energy is expressed as:
∂(ρ k ) ∂(ρ u k ) ∂(ρν k ) ∂(ρwk)
+
+
+
∂t
∂x
∂y
∂z
=
where,
∂  µ t ∂ u  ∂  µ t ∂ u  ∂  µ t ∂ u 
+
+
+ Pk − ρε
∂ x  σ k ∂ x  ∂ y  σ k ∂ y  ∂ z  σ k ∂ z 
(6)
 ∂ u  2  ∂ v  2  ∂ w  2 
 
 +
P k = 2 µ t   + 
 ∂ x   ∂ y   ∂ z  
1

2
2
2 
2
2
2  2












∂u ∂ w
∂ v ∂ u  ∂ w ∂ v
∂u ∂ w
∂ v ∂ u 
 ∂ w ∂v 
 + 
 
 + 
 
+ µ t 
+
+ 
+
+
−  + 
−
−
 ∂ y ∂z 
x
y
∂
∂
z
x
y
z
z
x
x ∂ y  
∂
∂
∂
∂
∂
∂
∂














−
2  ∂u ∂v ∂ w 

+
µ  +
3 t  ∂ x ∂ y ∂ z 
1
in the partial cell,
Pk =τ w
2
(7)
1
C µ4 k 2
(8)
κ y
y
x
0
Partial cell
Fig.7 Examples of partial cells.
z
0
x
Fig.8 Computational mesh around a rectangular cube.
1
where, 10 < y + < 300 ,
or
y < 10 ,
where,
ρ
u = u 2 + v 2 + w2
+
ρ
u
τw=µ
(
)
k u
1
τ w = ρ C µ4 k 2
(
ln E y +
(9)
)
(10)
y
1
2
(11)
The eddy dissipation rate equation is expressed as:
∂ (ρ ε ) ∂ (ρ u ε ) ∂ (ρν ε ) ∂ (ρ wε )
+
+
+
∂t
∂x
∂y
∂z
∂  µ t ∂ ε  ∂  µ t ∂ ε  ∂  µ t ∂ ε  ε
+
+
+ (C1 P k − C 2 f 2 ρ ε )
∂x σ ∂x  ∂ y σ ∂ y  ∂z σ ∂z  k
 ε

 ε

 ε

f = 1 − exp(−0.2 y + )
=
where,
(12)
(13)
2
3
if y + >
or y + <
1
κ
1
κ
,
ε=
,
ε=
3
C µ4 ρ k 2
(14)
κy
Cµ ρk2
(15)
µ
Model constants are shown in Table 1. All model constants in PCC methods were the same as those of normal
k-ε model. In this study, the flow field was calculated using non-uniform computational mesh, which is 175 x 135 x 90.
Figure 8 shows the computational mesh around the rectangular cube. The inlet velocity and the relative turbulent
intensity were set to 4.0 m/s and 1.5%, respectively, such as experiments.
Table 1 Model coefficients in PCC method.
Cµ
C1
C2
σk
σε
E
κ
0.09
1.44
1.92
1.0
1.3
8.91
0.42
3. RESULTS AND DISCUSSIONS
3.1 Mean velocity distribution
Figures 9 (a)-(e) show the comparisons of vertical distributions of the mean velocity U along the centerline
among LDA, PIV measurements and the numerical simulation results by PCC method. The measuring points (a) to (e)
are illustrated in Fig.5. LDA and PIV results are shown by black and white symbols respectively and solid lines show
the numerical simulation results. For PIV results, since there are too many measuring points, only the halves of these
points are plotted. In front of the rectangular cube at x = 220mm, good agreements are obtained from these three results,
as shown in Figure 9 (a). In Figure 9 (b), here is the region that the flow collides with the rectangular cube and small
recirculation zone is observed near the floor. Three results are almost same distributions at this position. Figure 9 (c)
(a) x = 220 mm
(b) x = 240 mm
(c) x = 277 mm
(e) x = 354 mm
(d) x = 334 mm
80
LDA
PIV
CFD
z mm
60
40
20
0
-2
0
2
4
6
-2
0
2
4
6
-2
0
2
4
6
-2
0
2
4
6
-2
0
2
4
6
U m/s
U m/s
U m/s
U m/s
U m/s
Fig. 9 Comparisons of mean velocity U distributions on the centerline among LDA, PIV measurements
and numerical simulation.
80
(a) x = 220 mm
(b) x = 240 mm
(c) x = 277 mm
(d) x = 334 mm
(e) x = 354 mm
LDA
PIV
CFD
z mm
60
40
20
0
-2
0
2
-2
0
-2
2
0
2
-2
0
2
-2
0
2
W m/s
W m/s
W m/s
W m/s
W m/s
Fig. 10 Comparisons of mean velocity W distributions on the centerline among LDA, PIV measurements
and numerical simulation.
(a) x = 220 mm
(b) x = 240 mm
(c) x = 277 mm
(d) x = 334 mm
(e) x = 354 mm
80
LDA
PIV
CFD
z mm
60
40
20
0
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
u’ m/s
u’ m/s
u’ m/s
u’ m/s
u’ m/s
Fig. 11 Comparisons of turbulence intensity u’ distributions on the centerline LDA, PIV measurements
and numerical simulation.
shows the results measured upon the rectangular cube at x = 277mm, both two measurements and numerical simulation
results are almost same profiles. Small recirculation zone due to the separation from the rectangular tip is observed by
LDA and PIV measurements, however numerical results differ slightly from the measurements. Figures 9 (d) and (e)
show the results in the recirculation zone behind the rectangular cube and show the good agreement between both LDA
and PIV results. However numerical simulation results show smaller velocity than experimental results in the
recirculation zone, and it means that the calculated recirculation zone is larger than that of experiments. It is well known
phenomenon that smaller recirculation zone was calculated than experimental results at the flow over the back facing
step (Kobayashi, 1995). The opposite tendency was observed here and more improvements are requested in the
turbulent model for calculation.
Figures 10 (a)-(e) illustrate the comparisons of vertical distributions of the mean velocity W along the
centerline. The distributions of the mean velocity W show the good agreements between LDA and PIV measurements at
all positions. Behind the rectangular cube with x = 334 and 354 mm, numerical results differ slightly from
measurements, such as U distributions.
3.2 Turbulence intensity and Reynolds stress distributions
Figures 11 (a)-(e) show comparisons of vertical distributions of turbulence intensity u’ along the centerline
among LDA, PIV measurements and numerical simulation. At the numerical simulation, u' was calculated from the
assumption of isotropic turbulence as follow;
u '=
2
k
3
(16)
where, k is kinetic turbulence energy.
In front of the rectangular cube at x = 220mm, PIV measurements and numerical simulation results are
smaller than LDA measurements as shown in Fig 11(a). At x = 240mm, the values of u’ are smaller than that values at x
= 220mm since the flow behavior was restricted by the rectangular cube. LDA and PIV results are almost same on the
(a) x = 220 mm
(b) x = 240 mm
(c) x = 277 mm
(d) x = 334 mm
(e) x = 354 mm
80
LDA
PIV
CFD
z mm
60
40
20
0
-0.6 -0.4 -0.2 0
2
u’w’ m /s
0.2 -0.6 -0.4 -0.2 0
2
2
u’w’ m /s
0.2 -0.6 -0.4 -0.2 0
2
2
u’w’ m /s
0.2 -0.6 -0.4 -0.2 0
2
2
u’w’ m /s
0.2 -0.6 -0.4 -0.2 0
2
2
u’w’ m /s
0.2
2
Fig. 12 Comparisons of Reynolds stress u’w’ distributions on the centerline among LDA, PIV
measurements and numerical simulation.
rectangular cube and numerical results are about the half values of the measurements. Behind a rectangular cube, the all
profiles about the turbulence intensity show the good agreement among LDA, PIV and numerical results as shown in
Figs 11(d) and (e). However, the PIV results are little smaller than the LDA results.
Figures 12 (a)-(e) show the comparisons of vertical distributions of Reynolds stress along the centerline
among the LDA, PIV measurements and the numerical simulation. In general, the profiles and values about Reynolds
stress are almost same between both measurements and numerical simulations except x = 240 mm. At only x = 240 mm,
numerical results indicated the opposed sign form PIV measurements near the floor. In the numerical simulation, the
velocity gradient in x-direction seems to have large error because that the line is next to partial cells.
3.3 Spatial structure of vortices
The Particle Image Velocimetry can measure the instantaneous 2-dimensional velocity field. Then, it is useful
to understand the spatial vortex structure. In the present study, the PIV was applied for visualization of the instantaneous
velocity field and observation of the spatial distribution of the vortices. In order to visualize the vortices, the vorticity
distribution method, Reynolds decomposition, Galilean decomposition and LES decomposition are used widely. In this
study, the vorticity method, the Reynolds decomposition and the Galilean decomposition are applied for the analyses on
the flow around the rectangular cube.
Measuring area
Rectangular cube
x
31
60
50
50
mm
60
y
y mm
5
flow direction
40
30
4.0 m/s
240
250
260
x mm
Fig. 13 Instantaneous velocity fields.
270
40
30
4.0 m/s
240
250
260
x mm
Fig. 14 Mean velocity field.
270
Figures 13-17 illustrate the results from PIV measurements. These figures are obtained at the same measuring
area. In this case, total measuring area was 31 x 31 mm, and the actual interrogation area size was 0.97 x 0.97mm in
order to obtain the enough spatial resolution. The vectors were displayed at the intervals of 0.48mm because of applying
50% x 50% overlaps. In this case, the pulse interval of two laser shots was set to 30µs. Figure 13 shows an
instantaneous velocity field passing the side of the rectangular cube at z = 5 mm from the floor. The separation line is
formed from the corner of the rectangular cube. Then, the shear layer and recirculation flow are observed. The figure
60
60
50
50
y mm
y mm
ω 1/s
40
30
40
4.0 m/s
240
250
260
x mm
-1.20 -0.93 -0.67 -0.40 -0.13 0.13 0.40 0.67 0.93 1.20
30
270
240
50
50
y mm
60
40
260
x mm
4.0 m/s
40
4.0 m/s
(a)
30
240
250
260
x mm
270
(b)
30
240
250
260
x mm
60
50
40
4.0 m/s
(c)
30
240
270
Fig. 16 Vorticity distribution.
60
y mm
y mm
Fig. 15 Reynolds decomposed fluctuations.
250
250
260
x mm
270
Fig. 17 Fluctuation velocity fields around a rectangular cube by Galilean decomposition.
(a) Uc=2.0 m/s, Vc=1.6 m/s; (b) Uc=2.0 m/s, Vc=3.0 m/s; (c) Uc=4.0 m/s, Vc=1.6 m/s.
270
shows there are many small scale vortices around the rectangular cube (e.g. x = 267mm and y = 33mm and around x =
266mm and y = 42mm). The boundary, called as high shear line, meanders because of the effects of small scale vortices.
Figure 14 shows the mean velocity field obtained by averaging over about 100 instantaneous vector maps
from the PIV measurements. The separated flow area can be seen as small vectors near the rectangular cube, and only
one recirculating flow is existing in the area.
For evaluating the some small scale vortices in the turbulent flow, decomposition methods are applied widely.
The decomposition means separating and classification of vortices in the instantaneous velocity field from the total
vector map. One of the famous decomposition is Reynolds decomposition. It is defined as
u = U + u'
(17)
where, u is an instantaneous velocity, U is mean velocity and u’ is Reynolds decomposed velocity fluctuation (Tennekes
et al., 1972). According to the paper (Adrian et. al., 1998, 2000), there are some possibilities to obtain the mean value of
U. In this study, the U was set to the mean velocity at each point. The u’ which obtained by the above equation is simply
called Reynolds decomposed velocity fluctuation. The Reynolds decomposed fluctuation velocity at each point was
obtained by subtracting mean velocity, shown in Figure 14, from the instantaneous velocity at each point. Figure 15
shows the results of the Reynolds decomposed velocity fluctuation. The result indicates that some small vortices are
visualized in the main flow over the separation line. These small vortices are not recognized in the instantaneous
velocity fields. Also inside the separated flow region, the vortices are more clearly visualized than that of the
instantaneous velocity field. The vortex street is standing along the shear layer, and the size of the vortex becomes
larger in the downstream.
Figure 16 illustrates the vorticity contours. The vorticity was obtained by
ω=
∂v ∂u
−
∂x ∂y
(18)
These values are calculated from the same instantaneous velocity field as shown in Figure 13. The positions with large
absolute values are corresponded to the positions where the vortices are observed by the Reynolds decomposition
velocity fluctuation field. The large vorticities exist along the inner edge of the shear layer. There are also small
vorticities in the mainstream. Note that no vorticity can be observed inside the shear layer in this case.
Figure 17 shows the Galilean decomposition, which subtracted constant convection velocity from the
instantaneous velocity fields. This is expressed as
u = U c + uc
(19)
where, u is an instantaneous velocity, Uc is constant convection velocity, and uc is Galilean decomposed velocity
fluctuation. In Figure 17(a), convection velocities are Uc = 2.0m/s and Vc = 1.6m/s, which are the mean velocities by
averaging a measuring region. In this case, two vortices are observed at about x = 252mm, y = 42mm and x = 267mm, y
= 48mm. Figure 17(b) shows the case that convection velocities are Uc = 2.0m/s and Vc =3.0m/s. Some small vortices
are visualized. Finally, Figure 17(c) shows the case that convection velocities are Uc = 4.0m/s and Vc = 1.6m/s. Some
vortices are exposed at the region with 255mm < x < 265mm and 48mm < y < 55mm in this case. Using the Galilean
decomposed method, it is possible to provide the several scale of the vortices which have the different conventional
velocities.
3.4 Length scale measurements
The PIV is useful to evaluate the integral length scale because it has the two-dimensional and instantaneous
velocity information. Figure 18 shows the spatial correlation curve calculated from PIV results. The spatial correlation
f(x) was calculated as
f ( x) =
u (ξ )u (ξ + x)
u ' (ξ )u ' (ξ + x)
(20)
where, x is the distance between two measurement points, u is fluctuation velocity and u’ is turbulent intensity. The
transversal correlation has the negative values in the larger separation points although the longitudinal correlation has
only positive values in the figure. The results show that the longitudinal correlation value is higher than that of
transversal one. This feature is commonly observed, however the report (Josefsson et al., 2001) showed opposed
tendency.
Figure 19 shows the comparison of integral length scale distributions between the LDA and the PIV
measurements in the flow field behind the rectangular cube on the centerline of x = 334 mm. In the figure, relative
turbulence intensities measured by the LDA are also indicated. For the LDA measurements, length scale is estimated
from Taylor’s hypothesis as,
Lx = Lt × U
(21)
where, Lx is integral length scale, Lt is integral time scale and U is mean velocity.
The integral time scale at the LDA measurements is obtained by
1.2
longitudinal
correlation
transversal
correlation
1.0
Correlation
0.8
0.6
0.4
0.2
0.0
-0.2
-10
0
10
20
30
40
Separation mm
Fig.18 The longitudinal and transversal spatial correlation curves measured at
x = 320mm and z = 5mm.
10000
PIV
LDA
u'/U
Lx mm
12
1000
8
100
4
10
0
0
20
40
60
80
u'/U %
16
1
100
z mm
Fig. 19 Comparison of length scale between LDA and PIV measurements at the centerline of a
rectangular cube and at x = 334 mm.
u (ξ )u (ξ + t )
u ' (ξ )u ' (ξ + t )
f (t ) =
(22)
where f(t) is the auto-correlation function obtained from time series of LDA data.
For the PIV measurements, length scale is calculated by the next equation.
∞
Lx = ∫ f ( x)dx
0
(23)
The order of both values is the same. This means the spatial correlation functions which estimated by the PIV
and LDA may be suitable each other. The PIV has a possibility to measure both of the spatial correlation and length
scale. The positions of the local peak in the length scale have small difference between the PIV and the LDA results, but
the positions are not so far in the figure. The peak values are also close to each other. By the detail comparison between
the results from LDA and PIV, the both results are different in the region of z = 0 to 25mm where the mean velocity has
the negative value. In the region where the velocity is negative, the relative turbulence intensity is large. Taylor’s
hypothesis can be used in the region where the relative turbulence intensity is small enough. This is the reason why the
results between the PIV and the LDA are different in the region of z < 25 mm. In the region, the LDA is not able to
estimate the length scale correctly. In the region over z = 30mm, where the relative turbulence intensity is lower than
20%, both distribution is similar and it is considered that Taylor’s hypothesis is applicable within the limitation.
4. SUMMARY
A flow field around a rectangular cube was analyzed by LDA, PIV and the numerical simulation by PCC method. These
three results were compared in the mean velocity, turbulence intensity and Reynolds stress. The visualizations of
vortices distributions by Reynolds and Galilean decompositions were applied to the instantaneous velocity fields
obtained by PIV measurements. Moreover, the integral length scale estimations were carried out by LDA and PIV and
the comparison of integral length scale distributions between both results was made. The concluding remarks are
followings;
1.
LDA and PIV results about the mean velocities, the turbulence intensity and Reynolds stress are in good
agreement in the whole experimental region.
2.
The numerical simulation results and the experimental results about the mean velocity distributions are similar.
However the numerical simulation results estimated the larger recirculation zone than that of measurements.
The numerical simulation results about the turbulence intensity and Reynolds stress show the almost same
tendency as the experimental results.
3.
Applications of the Reynolds and the Galilean decomposition methods to the PIV results are effective to
analyze the spatial structure of vortices along the shear layer.
4.
Integral length scales estimated by LDA and PIV are similar with excepting the recirculation zone where the
relative turbulence intensity is over 20 %. In the area with high relative turbulence intensity, Taylor’s
hypothesis cannot estimate the integral length scale correctly.
Acknowledgement
The authors want to give special thanks to Mr. Nakamura and Dr. Araki for their help and advice in this study.
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