Experimentsin Fluids 17 (i994) 307 314 © Springer-Verlag1994
Stochastic estimation and proper orthogonal decomposition: Complementary
techniques for identifying structure
J. P. Bonnet, D. R. Cole, J. Delville, M. N. Glauser, L. S. Ukeiley
3o7
Abstract The Proper Orthogonal Decomposition (POD) as
introduced by Lumley and the Linear Stochastic Estimation
(LSE) as introduced by Adrian are used to identify structure in
the axisymmetric jet shear layer and the 2-D mixing layer. In this
paper we will briefly discuss the application of each method,
then focus on a novel technique which employs the strengths of
each. This complementary technique consists of projecting the
estimated velocity field obtained from application of LSE onto
the POD eigenfunctions to obtain estimated random coefficients. These estimated random coefficients are then used in
conjunction with the POD eigenfunctions to reconstruct the
estimated random velocity field. A qualitative comparison
between the first POD mode representation of the estimated
random velocity field and that obtained utilizing the original
measured field indicates that the two are remarkably similar, in
both flows. In order to quantitatively assess the technique, the
root mean square (RMS) velocities are computed from the
estimated and original velocity fields and comparisons made. In
both flows the RMS velocities captured using the first POD mode
of the estimated field are very close to those obtained from the
first POD mode of the unestimated original field. These results
show that the complementary technique, which combines LSE
and POD, allows one to obtain time dependent information from
the POD while greatly reducing the amount of instantaneous
data required. Hence, it may not be necessary to measure the
instantaneous velocity field at all points in space simultaneously
to obtain the phase of the structures, but only at a few select
spatial positions. Moreover, this type of an approach can
possibly be used to verify or check low dimensional dynamical
systems models for the POD coefficients (for the first POD mode)
which are currently being developed for both of these flows.
1
Introduction
In order to perform the projection required to obtain the time
dependent random coefficients (the building blocks of the
turbulent flow) from the Proper Orthogonal Decomposition
(POD), it is necessary to have knowledge of the flow field at all
points in space simultaneously. From an experimental point of
view this requires that the flow domain be measured simultaneously on a sufficient spatial grid so as to minimize the
effects of spatial aliasing as discussed by Glauser and George
[12]. This is extremely difficult and can require hundreds of hot
wire probes or full field measurement techniques which, as of
yet, do not provide the necessary capability at high Reynolds
number. The spatial two-point velocity correlation tensor on the
other hand, a statistical quantity, can be obtained on a sufficient
spatial grid with as few as two hot wire probes. In this paper
a technique is proposed which uses the spatially resolved
statistical quantity, the two-point correlation tensor, in
conjunction with the instantaneous information at only a few
select points, to obtain estimates of the time dependent random
coefficients. The complementary technique is composed of three
main steps. First, the eigenfunctions and eigenvalues are
obtained from direct application of the POD to the two-point
spectral tensor in both flows (see for example Glauser et al. [13],
[14] and [15] for the jet and Delville et al. [7], [9] and [lo] for
the plane mixing layer). Second, the Linear Stochastic Estimation
(LSE) [1] is applied to the cross-correlation tensor and multipoint estimates of the random vector field are computed as
described by Cole et al. [5].
Third, the estimated velocity field obtained from step two is
projected onto the eigenfunctions obtained from step one to
obtain the estimated random coefficients. The estimated random
coefficients are then used in conjunction with the POD eigenfunctions to reconstruct the random velocity field. In this study,
Received: 27 September~Accepted: 2 May 1994
only time and the strongly inhomogeneous direction are examined for both flows. The authors recognize that coherent strucJ. P. Bonnet, J. Delville
Centre D'Etudes Aerodynamiques et Thermiques, F-86ooo Poitiers, France tures are 3 dimensional in nature and that it can be quite misleading to extract information from 2 dimensional slices. HowD. R. Cole, M. N. Glauser, L. S. Ukeiley
ever, in this paper the thrust is not to extract physics, but to
Department of Mechanical and Aeronautical Engineering,
demonstrate the usefulness of the complementary technique in
Clarkson University, Potsdam, NY 13699, USA
2 flows where the instantaneous velocities are available simulCorrespondence to: M. N. Glauser
taneously at a cross section in the flow on a sufficient spatial grid.
This allows for comparisons to be made between the reconstructed
The authors wish to thank DRET Grant # 9o/171, the French Embassy
measured velocity field obtained from direct application of the POD
Chateaubriand Fellowship (CIES)program for L. Ukeiley,NASA/LewisGSRP
to
that obtained from application of the complementary technique.
for D. Cole, NASA/AmesDryden, NASA/Langleyand NSF/CNRSthrough the
It
should
be emphasised that the complementary technique allows
international travel grants program for funding various portions of this
for unique applications of the POD, in particular, to real 3D
work.
simultaneously at each x2 location. The numerical
instantaneous fidds. In this section, POD and LSE are briefly
reviewed and the complementary technique introduced.
approximations of Eq. 2 and 4 involves replacing the integrals by
an appropriate quadrature rule (in this study a trapezoidal rule)
as detailed by Glauser et al. [13].
1.1
POt) theo~
308
The POD was first proposed by Lumley [18] in 1967, as
a mathematically unbiased technique for extracting structures
from turbulent flows. Lumley proposed that the coherent
structure be that structure which has the largest mean square
projection on the velocity field. This maximization leads to an
integral eigenvalue problem which has as its kernel the
cross-correlation tensor, Rij(~, U, t, t'). This technique has
recently become more popular with the development of more
powerful computers and advanced data acquisition equipment.
For example, Moin and Moser [20] applied the POD to the full
correlation tensor which was generated numerically from
simulations of turbulent channel flow and Ukeiley et al. [23]
applied it to an experimental data base obtained in the complex
flowfield downstream of a lobed mixer. For a recent comprehensive review of the POD see Berkooz et al. [4]. The POD
reduces to the harmonic orthogonal decomposition in time
since both flows are stationary, hence the eigenfunctions used in
time are Fourier Modes. Only the radial direction (r) in the jet
and the cross-stream direction (y) in the mixing layer, which are
both strongly inhomogeneous in their respective flows, are
examined with the POD.
For the following analysis a coordinate system of xl, x 2 and x3
is used. For the jet; xi = z the streamwise direction, Xa= r, the
radial direction and x3 = 0, the azimuthal direction. For the
mixing layer; x~--x, the streamwise direction x2=y, the
cross-stream direction and x3 = z, the spanwise direction. Given
the afore-mentioned conditions, the spectral tensor may be
defined by the following equation,
t
0 0
So(xz, x2,fXl,
X3)=~Rij(Xz ' x2,, r, xl,o x3o ) e-i2'q~ dz,
(1)
,
Rij (x2, x2,, "c, xx,0 x 30 ) = u i (x2, t, x l0, x 30 ) uj (X2,
t + Z, X~, X~). In
the above equation, f denotes frequency, ~ is the separation in
time and x~ and x ° represent the azimuthal and streamwise
locations in the jet, and spanwise and streamwise locations in the
mixing layer where the correlation tensors were measured. In
this formulation, S0 becomes the kernel in the integral
eigenvalue prbblem which is written as:
where
(z)
The $'s and 2(")(f) are the frequency dependent eigenfunctions
and eigenspectra respectively. Note that dx'2 = r' dr' for the jet
and dy' for the mixing layer. The Fourier Transform of the
velocity can be reconstructed in terms of the ~'s as;
hi= (x2,f) = ~ a,,( f)l/I}'O(xz,f),
(3)
n=l
where
an(f) = 5f*i(x2, f ) ~ f~)(x2, f)dx2.
In 1975, Adrian [1] proposed that stochastic estimation could be
applied to unconditional correlation data. This method uses
what Adrian calls a "conditional eddy". This eddy is a candidate
structure used to detect, within certain limits, other structures of
similar type. Stochastic estimation uses the conditional
information specified about the flow at one or more locations in
conjunction with its statistical properties to estimate the
information at the remaining locations.
Adrian [1] studied conditional flow structures in isotropic
turbulence by computing estimates of the velocity u(x', t) given
that the velocity at (x, t) assumes some specified value u(x, t).
He found that this simple flow, when sampled in a statistical
sense, shows the existence of organized structures. He used
a second order stochastic estimation technique, but concluded
that first order (linear) stochastic estimation (LSE) would have
resulted in nearly identical estimates. This indicates that the
second order term contributed little to the overall estimate.
Tung and Adrian [2z] studied the influence of the third and
fourth order terms on the estimate as well as the second order
term. Their results confrmed the insignificance of the higher
order terms on the overall estimate. Moin, Adrian and Kim [19]
applied stochastic estimation, in order to approximate
conditional vector fields, to a numerically simulated channel
flow. One of their most interesting results, but only discussed
briefly, was that they found good agreement between the LSE and
Lumley's characteristic eddy. This was further examined by
Moser [21] but comparisons were difficult because of the
ambiguity in domain selection for the application of the POD
(i.e., different subdomains in the boundary layer). For further
discussion on the Stochastic Estimation theory see Adrian and
Moin [z] and Guezennec [17].
As discussed above, Tung and Adrian [z2] have shown that
linear stochastic estimation produces reasonable qualitative
estimates and little is to be gained by using second order or
higher. Linear stochastic estimation yields an estimate
~i(x') = Ao(x')uj(x)
~s~j(x~, x;,f, x °, x~) ~,~")(xl, f, 4 , x ° )dx;
_ ; ( , ) ( f ) O,(,) (x2,f,x,,x3).
o o
1.2
Stochastic estimation theory
(4)
So(x2, x'2,f, x°,, x~) is obtained from the experimental
measurements in both flows and used in conjunction with
Eq. z to extract the eigenvalues and eigenfunctions. Note: To
compute a~(f) using Eq. 4, ~i(x2,f) must be available
(5)
with &k computed from,
uj(x) Uk(X) aik(X') = Uj(X)Ui(X')
(6)
where uj(x)Uk(X) and uj(x)ui(x') are the Reynolds stress and
two-point correlation tensors respectively.
For the u, v jet and mixing layer data (u = ul and v = Uz), the
matrices that result from the expansion of Eq. 6 for a two probe
estimate are:
First System:
ry
-~r
,
,
Ur~Vr~
v2,
l U~2Ur~ IXr~!Yr~
Lv~2Ur~Vrz!/rt
I
(7)
Second System:
using the original measured instantaneous velocity data as given
by the inverse Fourier transform of Eq. 3. A flow chart which
compares the steps involved in the complementary technique to
those for a direct application is presented in Fig. 1.
I;r U" UFUUrq F4
V2rl
Vr~Zr2
~
U,'zVr t
bi2rz
bT,
~]r21]r~ Vrzbir2 ~r2
I
jlA r ,/
(8)
2
Experiments
L~M U~s/
where rl and r2 refer to reference probes 1 and a respectively,
and p refers to the probe number. It should be noted, that for
these systems of equations, only the two-point space-time
correlation data is utilized. These systems are not a function of
the condition being investigated. The estimated velocity
components for the two probe reference case can then be found
from the expansion of Eq. 5,
~
rl
rl
r2
big __
-- AilpUCr~
+ AlzpVCrl
-[- A r21]pucr2+ AlzpVCr2
(9)
and
"
-. . . .
Y2
Y2
•
Vp -A2ipUCr,
+ A22pVCr,+ A21pUCr~
+ A22pVCT~
(10)
It is in these estimated velocity equations that the condition
selected plays a role (i.e., through u G, u % v G and vG). A
single probe estimate is obtained by merely setting all terms
containing r2 = o. Without much trouble this system can easily
be expanded to include estimates for all of the probes. This
should result in the estimated velocities being exactly the same
as the actual velocities. This property can then be used as
a check. In this paper it is not the intent of the authors to discuss
what number of probes or their respective positions are the most
appropriate to obtain the best estimate of the velocity field.
These issues are discussed in Cole et al. [5] for the jet and in
Delvile et al. [8] for the mixing layer.
1.3
Complementary technique
The complementary technique utilizes the POD eigenfunctions,
as described in the POD Theory section, and the LSE of the
velocity field, as described in the Stochastic Estimation Theory
section, to obtain estimates of the random coefficients from
which the velocity field can be reconstructed. Mathematically the
stochastic estimates of the random coefficients are calculated
from
ann
est( f ) = S~ St(Xz,f ) ~P}")*(X2,f ) d x ,
(11)
where fi~St(xa,f) is either a single or multipoint (in this study,
2 point) linear stochastic estimate of the velocity field (from the
time Fourier Transform of Eq. 9 and 10) and 0~")*(x2,f) is
obtained from the original POD eigenvalue problem. Note the
similarity to Eq. 4, here however the actual velocity field is
replaced by that estimated from the 2 point linear stochastic
estimate. The estimated u or v velocity can then be reproduced
in Fourier space by
^ est (xz, f )
ui
=
anest(f)Oi (~)(x2'f)
The jet shear layer and 2-D mixing layer experiments, which are
used in this study, are flows where the instantaneous velocities
are available simultaneously at a cross section in the flow on
a sufficient spatial grid. This allows for comparisons to be made
between the reconstructed measured velocity field obtained from
direct application of the POD to that obtained from application
of the complementary technique. Each of these experiments
is briefly described below.
2.1
Jet
The experiment conducted to obtain the data in the
axisymmetric jet was first reported by Glauser and George [13]
and Glauser et al. [14]. The jet had an exit diameter (V) of
0.098 m with a centerline exit velocity of 2o m/s. The Reynolds
number based on exit diameter was 11o,ooo with a 0.35% turbulence intensity in the core region. The data was collected by
two rakes each containing 4 "X" wires. The rakes were placed
3 jet diameters downstream. In the original experiment the rakes
were traversed through 25 azimuthal locations, however in this
study only the azimuthal position of 0 = o is examined. The
hot wires were spaced by approximately lo.9 mm, making the
total distance spanned 76.2 mm, which is approximately twice
the vorticity thickness. The sensing wire used were 5 gm in
diameter and had a sensing length ofl.2 ram. In order to achieve
the ensemble averages necessary to calculate the correlation
tensor 3oo blocks of lO24 samples were collected. The data was
low pass filtered at 800 Hz while being sampled at 2000 Hz. The
data acquisition system was based around a 15 bit, 16 channel
A/D converter with simultaneous sample and hold capability.
2.2
Mixing layer
The experiment to obtain data in the subsonic plane mixing
layer was performed at CEAT/LEA in Poitiers, France. The
subsonic turbulent plane mixing layer had a high speed velocity
of 42.8 m/s and a low speed velocity equal to 25.2 m/s. All the
measurements were taken at 6oo mm downstream of the trailing
edge of the splitting plate, where the vorticity thickness was
27.6 mm. A rake of 12 equally spaced "X" wires was utilized
to obtain the data. The probes were placed symmetrically about
the mixing layer axis and the separation between them was
6 mm. The diameter of the wires was 2.5 gm with a sensing
length of 0.5 mm. The data was simultaneously sampled at
lO kHz using constant temperature anemometers built from
a TSI 175o. For further information the reader is referred to
Delville et al. [7].
(12)
n=l
and then inverse transformed to obtain u~St(x 2, t). Comparisons
are then made, for both flows, between the reconstructed
estimated velocity field as described by Eq. lZ and those obtained
3
Results
As was discussed in the introduction, in this study only time and
the strongly inhomogeneous direction are examined in both
309
Complementary Teeh.
Direct Method
Measure Ui(~,,t) at all
positions in space simultaneaously
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M_easuse~J
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Note: requires a minimum of two probes
Note: requires a significant amount
of probes
J
Compute Rij and extract
POD eigenmodes
31o
Extract POD eigenmodes
I
I
Use LSE to obtain estimate of
Ui~,t) at all positions in space
Project Ui(x,t) onto POD eigenfunctions
to compute the random coefficients
I
Project estimated Ui(x,t) onto POD
eigenfunctions to obtain an estimate
of the random coefficients
t Rebuild original field at all space location t
I Rebnild estimated field at all space locations I
I
0 90 ~-~
•
~
. . ~ ~ K ' ~ . . ~ , , . _
_
~
.-'---T'r'C'~
~"-YZ'~'~--~,¢'/I///,,,"If~
0.13
~""-::
-~~
.....
"~'-"~
" ...............
:'~.~-
a. . . . . . . . . . . . .
. . . . * ......
~ ~ , ,
" .......
""'""~
""~-''"~
,~_;"
Fig. 1. Flow chart showing
comparison between complementary
technique and direct application
"-,,w,+~
..........
Fig. 2. Instantaneous velocity vector plots of the jet seen in a frame of reference moving at uc= 12 m/s
flows. More complete analysis of these data bases, including
multi-dimensional analysis, have been reported by Glauser et al.
[15] and Delville et al. [8] for the jet and mixing layer
respectively.
Vector plots are used here to qualitatively examine the flow
fields. Figure 2 presents an original measured velocity field for
the axisymmetric jet shear layer at one azimuthal position seen
in a frame of reference moving at 12 m/s. This record contains
15o time steps, which corresponds to o.073 sec. Figure 6 presents
an original measured velocity field for the plane mixing layer
at one spanwise location seen moving in a reference frame of
33.7 m/s. In this case there are also 15o time steps, but the
corresponding time is only o.o15 sec. Note that, in the times
selected for the respective flows, several large scale motions are
observed.
Vector plots of the contribution from the first POD mode for
the jet shear layer and the plane mixing layer are shown in Figs.
3 and 7 respectively, using the original measured instantaneous
velocity field in each projection. These are obtained by direct
application of the inverse Fourier transform of Eq. 3. For both
flows, the first mode exhibits most of the large scale features
observed in the original velocity fields (Figs. 2 and 6 for the jet
and mixing layer respectively) although they do not have the
same spatial extent in the visualization• Although not presented
here, the spatial information fills in when more modes are
retained as shown by Glauser et al. [15] and Ukeiley et al. [24]
for the jet and Delville et al. [9] for the mixing layer.
Figures 4 and 8 show two-point linear stochastic estimates
of the original measured velocity field for the jet mixing layer
and the plane mixing layer, respectively. In these cases
instantaneous information at two points is utilized as the
condition from which the remaining information is estimated.
The probes which supply the condition are equally spaced on
either side of the centerline for both flows• Compare Figs. 4
and 8 to Figs. 2 and 6, respectively• Note how much of the
structure observed in the visualizations of the original measured
signals is contained in these two-point estimates and, in particular, how the phase information is preserved. The structures
Fig. 3. A ~ POD mode reconstruction of the jet
311
Fig. 4. StochasticEstimated field of the jet using the wires indicated by the arrows as reference
I
T
T
F
~
I
L
I
.
~
.
.
I
.
~
2
~
£
~
2
~
'
~
.
~
.
~
......
;
~
~
~
;
~
2
~
;
~
1
~
T
~
:
~
2
~
Fig. 5. Complementarytechnique: A 1 POD mode reconstruction of the jet using the stochasticallyestimated field
..
,4~
;'1 18
Fig. 6. Instantaneous velocityvector plots of the mixing layer seen in a frame of reference moving at Uc=33m/s
characteristics are slightly underpredicted. As would be
expected, at the conditional probe positions, all of the features
are captured. It is also seen that the estimates using the data
taken in the plane mixing layer produce better defined
structures. It was shown by Cole et al. [5] that a single point
reconstruction for the jet shear layer is inadequate and that the
best estimates occurred when the condition utilized information
from both sides of the jet shear layer. It was found in the
plane mixing layer, however, that when using a probe placed at
the outer region of the shear layer, the large scale motions were
fairly well represented as discussed by Delville et al. [8]. These
differences can be attributed to the difference in integral
length scale between the two flows.
In Figs. 5 and 9 the results from the application of the
complementary technique are presented for the jet mixing layer
and the plane mixing layer, respectively. The estimated fields
(Application of Eqs. n and 12 using the estimated data presented
in Figs. 4 and 8), are projected onto eigenfunctions obtained
from direct application of the POD. Reasonable estimates of the
large scale structure are obtained, but only a small percentage of
the original measured instantaneous data (25% for the iet and
17% for the mixing layer) has been usedK In fact, one sees that
Figs. 5 and 9 compare quite well to Figs. 3 and 7 respectively,
which were computed using the full measured instantaneous
velocity field.
It should be noted that more features are recovered away
from the center of both the jet shear layer and the mixing layer
from the first POD mode reconstruction of the estimated field
than the estimated field contains itself. This can be seen by
comparing Figs. 5 and 9 to Figs. 4 and 8 respectively. It is
apparent that this effect is most dominant for the mixing layer.
Evidently, the first POD eigenfunction, used in the projection,
contains a significant amount of knowledge of the velocity field
(in the averaged mean square sense) and hence provides the
22.'~22'" ' ................ !222! ........... ~2!~....... ! ! ! 2 ; ! 2 ! . : ! : : : 2 ! ~ .............. 5'-2!!:!~'.22................ : ~ ' 2 Y '
Fig. 7. A 1 POD mode reconstruction of the mixing layer
~::'~:
::
: : : : : : : : : : : : : : : : : : : : :
======================================
. ~ _ , : . . ~ .
~
-
. . . ~ 1
.-
.
.,~
"-',l~r
Fig. 8. Stochastic Estimated field of the mixing layer using the wires indicated by the arrows as reference
--
,
. . . . . . . . .
.
....
===========================================================
.........
,
...........
:-:x
.......
~:::::
..........
:::;
......
......................................... -",," .... : ........ ::2..... ' .......... ~..............
;;:to:
............
" ..........
~. . . . . .
Fig. 9. Complementary technique: A 1 POD mode reconstruction of the mixing layer using the stochastically estimated field
additional information. It has been observed by Ukeiley et al.
[24] that more spatial information is filled in when additional
POD modes are included. They also note that the 3 POD mode
representations of the estimated fields do not compare as well to
the 3 POD mode representation of the original measured field
when compared to the 1 POD representations (i.e., the results
differ more as additional POD modes are used). This could be
interpreted as a surprising result, but this is not unexpected
since the sum of all the terms in Eq. n will result in the estimated
field being recovered and not the original measured field.
These results are summed up quantitatively in Figs. lo and 11
for the jet shear layer and plane mixing layer respectively. These
figures present comparisons between the original measured
and estimated streamwise RMS velocities, and 1 POD mode
representations of each of them. What is seen in Fig. lo, for the
jet shear layer, and in Fig. 11, for the plane shear layer is that
the complementary technique captures almost as much of the
RMS streamwise velocity in a single POD mode, as the direct
application of the POD to the original measured velocity. It is
also evident from Fig. al, as was observed in the instantaneous
plots and discussed in the previous paragraph, that the
complementary technique does a significantly better job at
predicting the flow characteristics away from the center of the
plane mixing layer than the LSE alone. Recently, Ewing [11] has
examined the question of aliasing for the complementary
technique. Although not shown, the eigenspectra for the first
mode computed from the estimated coefficients
0.20
L
i
i
I
i
i
J
• Original
- - -~ S t o c h a s t i c Est.
.....
1 POD m o d e
+ - - - - - + Comp. Tech,
.
o.15
I 0.10
l
"=
/
/
,
0.05
'
p//.¢
\
\
~
> ,S/
•
/
13"
0
o.,
I
&
,
'
o's
riD
---..--
Fig. lo. RMS comparisons for the jet
016
01.7
018
0.9
0.100 ~~ Original
/k
] . . . . ~ StochasticE s t . /
I .....
1 POD mode /
~----.Comp.~,,~
0.075
t
l
q
f
\1
ff
0.050
"
\ \\
'L
, ,///
/ 15':
, l/
0.025
0-1.2
--' - -u-,-0.8
f
j
-0[4
0
y/&~
I
I
0.4
0.8
1.2
=
experimentalists to avoid using Taylor's "Frozen Field"
hypothesis in the streamwise direction. At the present time, the
correlation tensor is typically obtained at one downstream
location and Taylor's hypothesis used to infer the streamwise
dependence of the correlation tensor. This procedure has been
implemented mainly to avoid flow blockage affects of the
upstream rakes of hot wires on the downstream measurements.
The new approach involves measuring apriori, the correlation
tensor at several downstream positions independently. The
experiment is then repeated with a minimal amount of
strategically placed probes, all sampled simultaneously, at each
of the streamwise locations where the correlation tensor has
been measured. The LSE can then be implemented using this
instantaneous data, in conjunction with the well resolved
correlation measurements available at each downstream
location, to obtain an estimate of the entire spatial and temporal
velocity field. Finally, an estimate of the streamwise evolution of
the correlation tensor can be computed from the estimated
velocity feld. The streamwise evolution of the POD
eigenfunction can then be extracted, hence avoiding the use of
Taylor's hypothesis.
Fig. 11. RMS comparisons for the mixing layer
References
(2~st(f) =a~St(f)a~ est (f)) are very close to the original
eigenspectra in both flows. This indicates that the effects of
aliasing are minimal for the two applications presented here.
4
Conclusions and future work
The POD and LSE have been combined in a novel fashion
utilizing the global nature of the POD and the local nature of the
LSE. The two-point LSE estimated instantaneous field is
projected onto the eigenfunctions obtained from the direct
application of POD and a 1 POD mode representation computed
for both the jet shear layer and plane mixing layer. These results
are remarkably similar to a 1 POD mode representation of the
original measured instantaneous field. Hence, the complementary technique retains the phase information of the POD
modes, from the instantaneous signal obtained at a "few" select
spatial locations and knowledge of the instantaneous field at
all spatial positions is not necessarily required. RMS velocity
plots, which are used to quantify the effectiveness of the complementary technique, confirm this as well. They show that a
1 POD mode representation of the RMS estimated field is very
close to that obtained from a 1 POD mode representation of the
RMS original measured field.
In this work the complementary technique was used to obtain
estimates of the random coefficients in the strongly inhomogeneous directions for both the jet shear layer and plane mixing
layer. In the future, the technique can be used in a similar
manner as described above to obtain estimates of the random
coefficients in the remaining directions as well. Hence this type
of an approach can possibly be used to verify or check low
dimensional dynamical systems models for the POD coefficients
(for the first POD mode) which have been developed in the
boundary layer by Aubry et al. [3], in the jet by Glauser et al. [16]
and which are currently under development for the plane mixing
layer by the authors. An additional useful application of the LSE
and POD in tandem holds forth the possiblity of allowing
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lnnouncemen
CALL FOR PAPERS
Symposium on Flow Visualization and Image Processingof Multiphase Systems
1995 Fluid Engineering Division Summer Meeting
(ASME/EALA Sixth International Conference on Laser Anemometry)
American Society of Mechanical Engineers Hilton Head, South Carolina U.S.A.
August 13-18, 1995
This symposium is sponsored by the Multiphase Flow Committee of the
ASME Fluids Engineering Division. It will be a part of the ASME/EALASixth
International Conference on Laser Anemometry under the auspicious of the
ASME Fluids Engineering Division summer Meeting ('95 FEDSM).
Flow visualization is an important experimental methodology which has
been instrumental in promoting and establishing modern science and
technology. It is, presently, extensively employed in various scientific and
high-technology fields. With rapid advances in computer and image
processing techniques, visualization science has become a dynamic
multidisciplinary field of learning not only in the past and present but also in
the future. Its applications cover practically all areas in science and
technology. This symposium is intended for visualizing multiphase flows and
obtaining quantitative information through image processing.
Prospective contributors are requested to submit three copies of a 300
word abstract. The abstract should clearly state the method, results and
indicates the name, address, phone number, and fax number of the
corresponding author. Final acceptance of the papers will be based upon the
review of the complete manuscript according to ASME standards. All
accepted papers will be published in a symposium volume that will be
available at the meeting.
DEADLINES
Submission of abstract to Symposium Chair: August 31, 1994
Notification of preliminary acceptance: October 14, 1994
Full-length papers due to Symposium Chair: November 28, 1994
Notification of final acceptance and sent mats: February 1o, 1995
Final typed mats due to Symposium Chair: April lo, 1995
Symposium Organizers:
Wen-Jei Yang, Chair
Dept. of Mech. Eng. & Appl. Mech.
University of Michigan
215o G.G. Brown Bldg.
Ann. Arbor, MI 481o9, U.S.A.
F. Yamamoto
Dept. of Mech. Eng.
Fukui University
Fukui 91o, Japan
F. Mayinger
Institut ffir Thermodynamik A
Technische Universit~it Miinchen
Postfach zoz4zo
D-8o333 Miinchen, Germany
Tel: (313) 764-991o
Fax: (313) 747-317o
Tel: o776-27-8534
Fax: 0776-27-8748
Tel: 49-89-ZLO5-3451
Fax: 49-89-ZlO5-2ooo