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Vorticity characterization in a turbulent boundary layer using PIV and POD analysis

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4th International Symposium on Particle Image VelocimetryG¨ottingen, Germany, September 17-19, 2001 PIV’01 Paper 1184
Vorticity characterization in a turbulent boundary layerusing PIV and POD analysis
A. Liberzon, R. Gurka, G. Hetsroni
Abstract
We investigate coherent structures in turbulent boundary layer ﬂow using Particle ImageVelocimetry(PIV) technique. Flow coherent patterns have been identiﬁed and characterizedusing Proper Orthogonal Decomposition technique, applied to the instantaneous vorticityﬁelds. Spatial characteristics, previously reported in the literature, such as non-dimensionallength scale (
x
+
≈
1000), and the average expansion angle (
∼
9
◦
), have been found fromlinear combination of three ﬁrst POD modes.
1 Introduction
Wall bounded turbulent ﬂows have been extensively discussed in the literature regarding thecoherent structures in the near-wall region (see for example Panton, 1997). The investigation of the mechanisms that produce and control this phenomenon in a turbulent boundary layer requiresthe identiﬁcation of the coherent patterns.Vorticity is one of the main parameters that characterize coherent structures and it plays adominant role in the dynamics of turbulence in general and in the energy cascade in particular(Tsinober, 2000). A high spatial resolution demand for the vorticity calculation has been achievedusing Particle Image Velocimetry (PIV) measurement technique (Raﬀel et al., 1998).It had been recognized for a long time, that raw data does not help to characterize the coherentstructures. Without subjective conditional sampling techniques, usually applied to measurementdata, neither vorticity nor another ﬂow characteristic parameter, would provide a mechanism todistinguish between the coherent and non-coherent parts of the ﬂow.In the present work, the unbiased statistical analysis method, Proper Orthogonal Decompo-sition (POD) (Berkooz et al., 1993) has been applied to the PIV data measured in a ﬂume. Inaddition to the common POD analysis of the velocity maps, we propose to analyze the modesof the instantaneous vorticity ﬁeld in order to extract the most energetic parts of the vorticityﬂuctuations.
2 Experimental analysis
The investigated ﬂow ﬁeld is a turbulent boundary layer in a ﬂume. The ﬂume is made fromglass and has dimensions of 4
.
9
×
0
.
32
×
0
.
1
m
. A schematic diagram of the conﬁguration is shownin ﬁgure 1. The Reynolds number was 20000 based on the water height. The velocity ﬁeld wasmeasured 2.5 m from the entrance and the ﬁeld of view was 0
.
1
×
0
.
1
m
at the wall-normal (X-Y)plane. A PIV system was utilized for this work. Hollow glass sphere particles were seeded with anaverage diameter of 10
µm
. A choice of 32
×
32 pixels square interrogation areas, a 100
µm/pixel
A. Liberzon, R. Gurka, G. Hetsroni, Mechanical Engineering Department, Technion,Haifa, IsraelCorrespondence to:Roi Gurka, Mechanical Engineering Department, Technion, Haifa 32000, Israel.E-mail: gurka@tx.technion.ac.il Tel: +972-48-29-38611
PIV’01 Paper 1184
Figure 1: Schematic drawing of the experimental setupratio, and repeating the PIV analysis every 8 pixels resulted in 11500 vectors in a given ﬁeld of view. The calculations were based on 150 image pairs resulting in 150 vector maps.
2.1 Vorticity ﬁeld calculation
The calculation of a vorticity ﬁeld based on PIV velocity measurements requires the applica-tion of numerical diﬀerentiation scheme. In this work, ﬁve numerical algorithms of the vorticitycalculation (see e.g. Raﬀel et al., 1998) were tested using Monte-Carlo numerical simulations basedon two-dimensional Taylor-Green vortex, shown in ﬁgure 2. The instantaneous ﬂow ﬁeld of theTaylor-Green vortex ﬂow is a good choice to evaluate the vorticity and derivatives calculationalgorithms because it includes strong rotation. The simulation results show that the least squaremethod is the most robust and accurate for PIV data. Moreover, this second order diﬀerencescheme provides the smallest error in instantaneous vorticity ﬁelds. Figure 3 presents the rel-ative error propagation versus additive noise levels (i.e., 2%, 5%, and 7.5% from r.m.s. of theTaylor-Green velocity magnitude) for three number of simulation runs.In order to investigate coherent structures in shear layers one probably needs the vorticityﬂuctuations that are calculated by removing the average vorticity ﬁeld. The average vorticity isrelated as a background masking noise and has to be removed from each instantaneous vorticitymap. Therefore, the accuracy of the averaging is critical and it was tested by comparison betweenthe averaged vorticity based on the averaged velocity and by averaging the instantaneous vorticityﬁelds. Figure 5 demonstrates that there is no diﬀerence between these two average vorticity ﬁelds.
3 Identiﬁcation analysis
3.1 Proper Orthogonal Decomposition
Proper orthogonal decomposition (POD), has been proposed by Lumley (1970) as a methodthat could provide an unbiased identiﬁcation of coherent structures in turbulent ﬂow. This tech-nique is based on second-order statistical properties which result in a set of optimal eigenfunctions.2
PIV’01 Paper 1184
Figure 2: Taylor-Green vortex ﬂow ﬁeldFigure 3: Relative error of the vorticity value versus diﬀerent levels of the additive noise for
N
=a) 100, b) 500, and c) 1000 numerical runs.3
PIV’01 Paper 1184
Specially, the method makes use of eigenvalues and eigenvectors of the covariance matrix, indicat-ing the dispersion of the variable distribution,The covariance matrix of random variable
X
is approximated by:
C
= 1
M
M
i
=1
ˆ
X
i
ˆ
X
T i
(1)where ˆ
X
are deﬁned as the ﬂuctuation vectors around the average, ¯
X
.Due to computational problems, the covariance matrix has been estimated using method of snapshots, as described by Sirovich (1987a). Using this method, the covariance matrix becomes:
C
ij
=
ˆ
X
i
,
ˆ
X
j
, i,j
= 1
,...,M
(2)where
·
,
·
denotes the usual Euclidean inner product. The matrix is now
M
×
M
instead of
N
×
N
and, assuming
M < N
, we can compute its eigenvalues and eigenvectors more easily. Sincethe covariance matrix is symmetric, we know that its eigenvalues,
λ
i
, are nonnegative and itseigenvectors,
φ
i
,
i
= 1
,...,M
, form a complete orthogonal set. The orthogonal eigenfunctions of the data are deﬁned as:Ψ
[
k
]
=
M
i
=1
φ
[
k
]
i
ˆ
X
i
, k
= 1
,...,M
(3)where
φ
[
k
]
i
is the
i
-th component of the
k
-th eigenvector. Lumley refers to these eigenfunctions ascoherent structures of the data. Whether or not they appear as spatial structures in a laboratoryexperiment is questionable Sirovich (1987b). Nevertheless, there is a common agreement that theywill be present at least indirectly. One of the possibilities is that a coherent pattern will consistof a linear combination of eigenfunctions.The “energy” of the data is deﬁned as being the sum of the eigenvalues of the covariancematrix:
E
=
M
i
=1
λ
i
(4)To each eigenfunction we assign an energy percentage based on the eigenfunction’s associatedeigenvalue:
E
k
=
λ
k
E
(5)Using only the ﬁrst
K
(
K < M
) most energetic eigenfunctions, we can construct an approximationto the data:
X
≈
¯
X
+
K
i
=1
a
i
Ψ
[
i
]
(6)
3.2 POD and vorticity
Usually, POD analysis is applied to velocity ﬁelds under the assumption that the decomposedvelocity eigenfunctions could be referred as coherent structures of the data. This straightforwardPOD analysis of the PIV data in the present work has not provided additional information aboutthe coherent part of the ﬂow in the ﬂume. Nevertheless, the ﬁrst modes accurately recover theinstantaneous velocity ﬁelds (see ﬁgure 4. Assuming that the vorticity in turbulent ﬂows is more4
PIV’01 Paper 1184
Figure 4: a) Original and b) reconstructed velocity magnitude ﬁelds of an instantaneous map.meaningful property in characterization of coherent structures then the velocity (e.g., Hunt andVassilicos, 2000) we proceeded to the investigation of the vorticity ﬂuctuations.The combination of POD and vorticity is not straightforward and requires clariﬁcation. Similarto the interpretation of the POD modes as the energetic part of the ﬂow for velocity ﬁelds, onecan relate the ﬁrst decomposed vorticity modes as the energetic part of the instantaneous vorticityﬁeld. In the present work we subtracted the average part from each instantaneous vorticity map,following by consequential POD analysis. Both reconstructed vorticity map and linear combinationof the ﬁrst three modes are shown in the following section.
4 Results
The averaged vorticity is shown in ﬁgure 5. Beside of the clear evidence of three regions of theﬂow (outer, entrainment and near wall layers, from top to bottom), it is not possible to extract anyadditional information regarding the coherent part of the ﬂow. Figure 6 presents one instantaneousvorticity ﬁeld together with a reconstructed vorticity ﬁeld using ten POD modes. Although, fewsmall features (probably included in higher modes) do not appear in the reconstructed ﬁeld, theoverall map could be successfully reconstructed by only several ﬁrst modes. The linear combinationof three ﬁrst POD modes of the vorticity ﬂuctuations is shown in ﬁgure 7. Two large-scale coherentpatterns with positive and negative vorticity values are located side-by-side in the near-wall regionof the ﬂow. Repeated measurements and analysis have shown the consistent spatial characteristicsof those features: the streamwise non-dimensional length (
x
+
1000) and the average expansionangle (8
.
5
0
).5

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