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POD of vorticity fields: A method for spatial characterization of coherent structures

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POD of vorticity ﬁelds: A method for spatial characterizationof coherent structures
Roi Gurka
a
, Alexander Liberzon
b,*
, Gad Hetsroni
c,1
a
Department of Mechanical and Materials Engineering, University of Western Ontario, London, Canada
b
Institute of Environmental Engineering, ETH Zurich, CH-8093 Zurich, Switzerland
c
Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel
Received 4 January 2005; received in revised form 23 December 2005; accepted 10 January 2006Available online 6 March 2006
Abstract
We present a method to identify large scale coherent structures, in turbulent ﬂows, and characterize them. The method is based on thelinear combination of the proper orthogonal decomposition (POD) modes of vorticity. Spanwise vorticity is derived from the two-dimen-sional and two-component velocity ﬁelds measured by means of particle image velocimetry (PIV) in the streamwise–wall normal plane of a fully developed turbulent boundary layer in a ﬂume. The identiﬁcation method makes use of the whole data set simultaneously, throughthe two-point correlation tensor, providing a statistical description of the dominant coherent motions in a turbulent boundary layer. Theidentiﬁed pattern resembles an elongated, quasi-streamwise, vortical structure with streamwise length equal to the water height in theﬂume and inclined upwards in the streamwise–wall normal plane at angle of approximately 8
.
2006 Elsevier Inc. All rights reserved.
PACS:
47.27.Nz; 47.54.+r
Keywords:
Boundary layer; Vorticity; Proper orthogonal decomposition; Coherent structures; Identiﬁcation
1. Introduction
Turbulent boundary layers have been extensively inves-tigated, since this is the place where the important phenom-ena of momentum and heat transfer, energy productionand dissipation occur. One of the most important featuresin a turbulent boundary layer is the existence of coherentpatterns (Kline et al., 1967; Panton, 1997). The role of coherent structures in the aforementioned phenomena,the interaction between the structures, and their relationto the properties of a turbulent ﬂow, remain unclear (e.g.,Tsinober, 2000). There is even no general consensus aboutthe geometry and the spatial characteristics of the coherentstructures. Extensive experimental and numerical eﬀortswere devoted to this problem during the past years (Robin-son, 1991; Panton, 1997) and several models of the coher-ent structures were proposed, such as hairpin vortexpackets (Zhou et al., 1999), horseshoe vortex (Theodorsen,
1952), funnel (Kaftori et al., 1994), and near-wall longitu-
dinal vortices (Schoppa and Hussain, 2000), among others.Despite their diﬀerent names, the spatial characteristics of the structures in all these models exhibit a remarkable sim-ilarity. For example, the individual coherent structures inthe numerical simulations (e.g., Zhou et al., 1999) werefound to grow upwards in the streamwise–wall normalplane at the angle of 8–12
, as it was observed in the exper-iments of Head and Bandyopadhyay (1981) and Kaftoriet al. (1994).A study of coherent structures demands an objective,unbiased, statistical method of identiﬁcation of the coher-
0142-727X/$ - see front matter
2006 Elsevier Inc. All rights reserved.doi:10.1016/j.ijheatﬂuidﬂow.2006.01.001
*
Corresponding author.
E-mail addresses:
rgurka@eng.uwo.ca (R. Gurka), liberzon@ihw.baug.ethz.ch (A. Liberzon), hetsroni@tx.technion.ac.il (G. Hetsroni).
1
Tel.: +972 4 829 2058; fax: +972 48 23 8101.
www.elsevier.com/locate/ijhﬀ
International Journal of Heat and Fluid Flow 27 (2006) 416–423
ent patterns in the multi-dimensional data sets. Numerousidentiﬁcation techniques have been proposed and imple-mented to the results of numerical simulations and experi-ments (see, for example Bonnet et al., 1998). Theidentiﬁcation methods are associated with one of the phys-ically meaningful ﬂow quantities, such as turbulent velocityor velocity derivatives. Vorticity, which plays a dominantrole in the dynamics of turbulent ﬂows (e.g., Klewicki,1997; Tsinober, 2000), and linked directly to the coherentstructures, is also one of the best choices for the identiﬁca-tion (Bonnet et al., 1998; Gunes and Rist, 2004). We pro-posed in Liberzon et al. (2005) to use the linearcombination of the proper orthogonal decomposition(Lumley, 1970) modes of the numerically simulated three-dimensional vorticity ﬁelds in order to identify and charac-terize the coherent structures in a turbulent channel ﬂow.This method is distinct from the previous studies of theproper orthogonal decomposition (POD) (see reviews of Berkooz et al., 1993; Holmes et al., 1996), that have mostlyanalyzed velocity data, and have not used the combinationsof POD modes in order to evaluate the spatial properties of the coherent structures.In the present study we propose to use the same statisti-cal, unbiased characterization method, and in addition, uti-lize the recent developments of the particle imagevelocimetry (PIV) experimental technique (Adrian, 1991;Raﬀel et al., 1998). PIV provides measurements of thetwo-dimensional, two-component velocity ﬁelds with a spa-tial resolution, which is suﬃciently high for the estimationof the out-of-plane component of vorticity. We apply ourmethod to the ensemble of the instantaneous two-dimen-sional scalar ﬁelds of spanwise vorticity, experimentallyobtained in a turbulent ﬂow in a ﬂume. The goal is to iden-tify the most essential features of the turbulent ﬂow whichare associated with high enstrophy, and characterize theirgeometry through the linear combination of the dominantPOD modes.Section 2 describes the ﬂow facility and the PIV measur-ing system, along with the experimental results of the tur-bulent boundary layer. Section 3 is devoted to thepresentation of the identiﬁcation methodology and the dis-cussion of the characterization results. Section 4 comprisesof some general remarks for conclusion.
2. The experiment
2.1. Experimental setup
The experiment was performed in a ﬂume with dimen-sions of 4.9
·
0.3
·
0.1 m, shown in Fig. 1. A detaileddescription of the ﬂume is given in Liberzon et al. (2003)and Gurka et al. (2004), and here it is described onlybrieﬂy. The entrance and the following part of the ﬂume(up to 2.8 m downstream) are made of glass in order tomake ﬂow visualization and PIV measurements possible.All necessary precautions were taken to reproduce thesame experimental conditions as in Hetsroni et al. (1997):(i) the eddies and recirculating currents were damped bymeans of grids in the inlet tank (as presented by dashedlines in Fig. 1), (ii) baﬄes were installed in the pipe portionof the tank, the inlet to the channel was a converging chan-nel in order to have a smooth entrance, and (iii) the pumpwas isolated from the system by means of rubber joints ﬁt-ted to the intake and discharge pipes. The pump was a 0.75HP, 60 RPM centrifugal pump. Flowmeter, with an accu-racy of 0.5% of the measured ﬂow rate, continuouslyrecorded the ﬂow rate. In order to make the measurementarea long enough and avoid the ﬂow depth decrease at theend of the ﬂume, an array of cylinders restricted the ﬂowbefore the outlet. The measurements have been performedwith treated and ﬁltered tap water.The PIV system, shown in Fig. 2, was composed of adouble, pulsed, Nd:YAG laser (170 mJ/pulse, 15 Hz,532 nm), optics for forming light sheet, and the CCD cam-era (8 bit, 1024
·
1024 pixels) with a recording rate of 30frames-per-second. The camera was located 0.05 m fromthe side wall of the ﬂume, normal to the laser light sheetformed in the mid-plane of the ﬂume, measuring stream-wise and wall normal components of the velocity vector.Time separation between the two laser pulses was adjustedto 3 ms according to the free-stream streamwise velocity
U
1
= 0.21 m/s, and 150 successive velocity realizationswere measured for the total time of 10 s. The Reynoldsnumber, based on the ﬂow height
h
= 0.08 m and the kine-matic viscosity
m
= 0.8
·
10
6
m
2
/s, was 21,000. Hollowglass spherical particles with an average diameter of 11
l
m, were used for seeding. The calibration procedure
Outlet tankInlet tankGrid2.1 m2.8 mTest areaPumpRegulatorFlowmeterGlass flume
Fig. 1. Schematic view of the experimental facility.
R. Gurka et al. / Int. J. Heat and Fluid Flow 27 (2006) 416–423
417
and PIV cross-correlation analysis were performed byusing Insight 5.1 software, with 64
·
64 pixels interrogationareas and 50% overlapping. Spatial resolution of the cam-era was 80
l
m per pixel which provided a ﬁeld of view of approximately 80
·
80 mm
2
. The analysis produced about1000 vectors in each realization, ﬁltered by using the stan-dard median and global outlier ﬁlters. During the post-pro-cessing analysis, 5% of the vectors were found to beerroneous. These vectors were removed and the gaps wereﬁlled with linear interpolations of the nearest neighborpoints.
2.2. Experimental results
Wemeasurethetwo-dimensional,two-componentveloc-ity ﬁelds,
~
u
1
~
u
2
in the streamwise–wall normal plane,
x
1
–
x
2
(subscripts 1, 2 and 3 correspond to the streamwise, wallnormal, and spanwise coordinates, respectively). Spanwisevorticity,
~
x
3
is calculated through a numerical diﬀerentia-tion of the velocity ﬁelds. We denote the instantaneousﬁelds with a tilde,
~
, the capital letters refer to the meanquantities, such as average vorticity
X
3
¼
f
x
3
, the small let-ters denote the ﬂuctuating ﬁelds (
x
3
¼
~
x
3
X
3
), and theroot-mean-square values are indicated by the apostrophe,for example
x
0
3
¼
ﬃﬃﬃﬃﬃﬃ
x
2
p
.An example of the ﬂuctuating velocity vector ﬁeld{
u
1
,
u
2
} in the streamwise–wall normal (
x
1
–
x
2
) plane isshown as a vector plot in Fig. 3a. The abscissa is thestreamwise coordinate,
x
1
, and the ordinate is the wall nor-mal normalized coordinate,
x
2
, both normalized by thewater height
h
.Despite the masking eﬀect of the strong mean shearin the turbulent boundary layer on the underlying coher-ent structures, in the instantaneous ﬁelds we regularlyobserve the large patterns of concentrated vorticity, elon-gated in the streamwise direction and inclined upwardsfrom the wall. These patterns represent, to the best of our understanding, the footprints of the large scale coher-ent structures, previously reported in the literature (e.g.,Kaftori et al., 1994; Klewicki, 1997; Bonnet et al., 1998).An example of an instantaneous footprint of a structurecould be seen in the contour plot of instantaneous vorticityﬁeld in Fig. 3b (we emphasize its envelope with a thickline).The statistical properties of the turbulent boundarylayer are given in Fig. 4. These include the proﬁles of ther.m.s. of velocity components,
u
0
,
v
0
, (
u
=
u
1
,
v
=
u
2
) theReynolds stresses
uv
, the mean streamwise velocity
U
(allnormalized to the free-stream velocity,
U
1
), and r.m.s. of spanwise vorticity
x
0
3
, normalized by the friction velocity,
u
*
. The friction velocity
u
*
is estimated according to Kaf-tori et al. (1994). The curves (1–4) (for the mean velocityand stresses) that refer to the results obtained by Klebanoﬀ (1954) and reproduced in Schlichting (1979), are given for
comparison. An additional curve (5) that shows the r.m.s.of spanwise vorticity is taken from Kim et al. (1987). Ourresults are presented by symbols and error bars. The sym-bols and the bars represent the average values and theuncertainty of the data, respectively.
0.25 0.5 0.75 1 1.250.05 m/s 0.050.150.250.35
x
2
/ h x
2
/ h
x
1
/h x
1
/h
0.25 0.5 0.75 1.0 1.250.050.150.250.35
-10 -5 0 5 10 15 20
3
w ~
(a)(b)
Fig. 3. Instantaneous ﬂow ﬁeld
x
1
–
x
2
plane: (a) velocity vector ﬁeld and(b) contours of the spanwise vorticity
~
x
3
. The ﬂow is from left to right.
Laser sheetLaser sheet
x
FlowFlowNd:YAGlaser
2
x
1
x
1
x
3
CCDcameraFront viewTop view
0.3m0.15m0.08m
Fig. 2. Schematic drawing of the experimental setup. The measurementplane is the streamwise–wall normal plane,
x
1
–
x
2
, at the middle plane of the ﬂume.418
R. Gurka et al. / Int. J. Heat and Fluid Flow 27 (2006) 416–423
3. Characterization of vorticity ﬁelds
3.1. Methodology
In order to insure an objective identiﬁcation process, wesuggested in our previous work (Liberzon et al., 2005) toadopt the following guidelines:
•
Data analysis has to be performed without thresholdoperations, and the same ﬁlters must be applied to allthe data.
•
Data has to be statistically signiﬁcant in order to charac-terize the existing structures, over a period of time.
•
Analysis is based on a ﬂow characteristic which stronglyrepresents turbulence.The suggested method is, in some sense, a combinationof the ‘‘characteristic eddy’’ concept of Lumley (1970) withthe reconstruction technique which is similar to one pro-posed by Gordeyev and Thomas (2002). The ‘‘large scalestructure’’ or alternatively the ‘‘characteristic eddy’’ areidentiﬁed through a linear combination of the dominantmodes of the proper orthogonal decomposition (POD) of vorticity:
^
x
i
ð
x
Þ ¼
X
N n
¼
1
a
n
/
ni
ð
x
Þ
i
¼
1
;
2
;
3
;
ð
1
Þ
where
/
ni
ð
x
Þ
is an eigenfunction of order
n
of the one of thecomponents, denoted by subscript
i
.
a
n
denotes the corre-sponding coeﬃcient. An overview of the POD procedureand the way we apply it to the PIV results are given inAppendix A.We infer that the proposed identiﬁcation method, whichis deﬁned in the above expression (Eq. (1)) and which isusing the rigorously proven optimal presentation throughthe POD, is an objective procedure and it satisﬁes theguidelines listed at the beginning of this section. Moreover,we show in the following that in addition to the identiﬁca-tion, it also provides the spatial characterization of thecoherent structures in turbulent ﬂows.
3.2. Characterization and discussion
The analysis, based on the POD, has been implementedin certain types of ﬂows, such as jets, boundary layers,backward facing step ﬂows (e.g., Holmes et al., 1996;Gordeyev and Thomas, 2002). In most of the studies theﬂuctuating velocity ﬁelds were analyzed, assuming thatthe large-scale coherent structures contain the main frac-tion of the turbulent kinetic energy. However, it was notedin experimental studies of Kostas et al. (2001) and Liberzonet al. (2001) and shown in our recent study of the numeri-cally simulated ﬂow (Liberzon et al., 2005), that the vortic-ity ﬁelds are more pertinent for the identiﬁcation of coherent motions. This is mainly due to the fact that vortic-ity is Galilean invariant and, therefore, it is insensitive tothe variations of the streamwise velocity that otherwisecause the so-called ‘‘jitter eﬀect’’ and smear the boundariesof the coherent pattern.We present a direct comparison of the vector PODmodes of the instantaneous velocity,
/
u
, with the scalarmodes of spanwise vorticity,
/
x
3
. In order to comparethe vector ﬁelds with the scalar ﬁelds, we calculate the curlof the velocity POD mode and equate it with the PODmode of spanwise vorticity. In Fig. 5a the ﬁrst velocitymode is shown as a vector plot, and its curl is speciﬁedby the contour lines. There is an evidence of a large scale
Fig. 4. Mean proﬁles of turbulent stresses (i.e.,
u
0
,
v
0
, and
ð
uv
Þ
), and meanstreamwise velocity
U
, all normalized by the free-stream streamwisevelocity,
U
1
, and r.m.s. of the spanwise vorticity
x
0
3
, normalized by thefriction velocity,
u
*
. Symbols and error bars represent the average and thevariance of the data, respectively. For the comparison, the results of Schlichting (1979), (1–4) and of Kim et al. (1987) (5) are shown as curves.
-0.06-0.06 -0.06-0.060.06
0.25 0.5 0.75 1.0 1.250.25 0.5 0.75 1.0 1.250.050.150.250.350.050.150.250.35
x
2
/ h x
2
/ h
x
1
/h
(a)(b)
Fig. 5. (a) The vector plot of the ﬁrst POD mode of the ﬂuctuatingvelocity, and contour plot of the curl of the POD mode. (b) The ﬁrst PODmode of the vorticity component
/
(1)
(
x
3
).
R. Gurka et al. / Int. J. Heat and Fluid Flow 27 (2006) 416–423
419
pattern in Fig. 5a, which is elongated in the streamwisedirection and inclined upwards in the streamwise–wall nor-mal plane. We observe this pattern, more clearly in Fig. 5b,in which the boundaries of the pattern are less smeared out,it is easier identiﬁed in respect to the background vorticityand apparently also less contaminated with the noise. Weinfer that this is due to the weak inﬂuence of the streamwisevelocity variations on vorticity, which is, by deﬁnition, aGalilean invariant.In addition, in Fig. 6 we demonstrate that the 10 domi-nant POD modes of vorticity are suﬃcient for the recon-struction of any instantaneous vorticity ﬁeld with areasonable accuracy. The reconstruction means ‘‘partialreconstruction’’, deﬁned in Appendix A (Eq. (9),
K
= 10).A relative error, based on the mean square diﬀerencebetween the srcinal and the reconstructed ﬁelds, is alsodeﬁned in Appendix A (Eq. (10)) and depicted in Fig. 7,
along with the relative contributions of the single PODmodes of spanwise vorticity. It is clear that the contribu-tion of the ﬁrst modes is much higher than of the following,higher modes, and the relative error is below 8%. We alsoobserve that the convergence of the cumulative contribu-tion of POD modes towards the 100% is relatively slow,as it was expected for the case of the small scale quantity,such as ﬂuctuating vorticity (e.g., Liberzon et al., 2005).The physical signiﬁcance of vorticity in turbulent ﬂowsand the aforementioned technical details such as Galileaninvariance, a sharper image and the small reconstructionerror, are the main arguments for us to utilize vorticityeigenmodes in our identiﬁcation method. Since a singlePOD mode of spanwise vorticity does not reproduce acoherent pattern typically observed in our measurements(e.g., Fig. 3b), we suggest to use the linear combinationof the dominant POD modes in order to identify and char-acterize its spatial structure.In Fig. 8a–c we demonstrate the linear combinations of the 3, 5, and 10 dominant POD modes of spanwise vortic-ity, respectively. For the sake of comparison we show thecombination of all the available 150 modes in Fig. 8d.The ascertained picture in Fig. 8a–d is of the streamwise-elongated pattern of concentrated vorticity (emphasizedby a thick contour line). We also realize that the shape,the streamwise length and the inclination angle of this pat-tern, outlined in Fig. 8a, have been not signiﬁcantly alteredby introducing the higher modes.When the higher modes have been added to the linearcombination, the pattern preserves its overall shape and
0.050.150.250.350.050.150.250.350.25 0.5 0.75 1.0 1.250.25 0.5 0.75 1.0 1.25
x
2
/ h x
2
/ h
x
1
/h x
1
/h
Fig. 6. Spanwise vorticity
e
x
3
ﬁelds, srcinal (top) and reconstructed bymeans of 10 POD modes (bottom).Fig. 7. Relative contribution of the single POD modes (dashed line), itscumulative summation (solid line) and relative error
e
(chain line), versus anumber of POD modes.
-0.06-0.04-0.020
0.050.150.250.350.050.150.250.350.050.150.250.350.050.150.250.350.25 0.5 0.75 1.0 1.250.25 0.5 0.75 1.0 1.250.25 0.5 0.75 1.0 1.250.25 0.5 0.75 1.0 1.25
0.020.040.060.080.1
streamwise length
angle
x
2
/ h x
2
/ h x
2
/ h x
2
/ h
x
1
/h
(d)(c)(b)(a)
Fig. 8. Linear combination of the POD modes of the ﬂuctuating vorticity
x
3
component, (a) 3, (b) 5, (c) 10, and (d) 150 modes, respectively.420
R. Gurka et al. / Int. J. Heat and Fluid Flow 27 (2006) 416–423

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