Piv-data validation algorithm based on statistical analysis
for ensemble of turbulent velocity fields
by
O.M. Heinz, B.B. Ilyushin, D.M. Markovich
Institute of Thermophysics
Siberian Branch of Russian Academy of Sciences,
Lavrentyev Ave., 1, Novosibirsk, 630090, Russia
E-Mail: dmark@itp.nsc.ru
ABSTRACT
The results of elaboration, testing and approbation of statistical method for filtration of the ensemble of measured
instant velocity fields in turbulent flows are presented. The method is based on the analysis of velocity fluctuations
histograms in such an ensemble. As a main assumption the hypothesis is used about exponential decay of the tails of
PDF for each spatial point of the test area. The testing of algorithm was performed on the basis of generated artificial
PDFs with imposed random noise. Developed method is applied for filtration of measured ensemble of instant
velocity fields in turbulent impinging jet.
a
b
2000
2000
1500
1500
1000
1000
500
500
20
20
0
0
-1
0
1
-1
0
1
Fig. 1. Histograms for ensemble of radial velocity fluctuations in mixing layer of an axisymmetric jet.
(a) – before application of filtration, (b) – after application of filtration.
1. INTRODUCTION
The application of PIV technique for study the statistical structure of turbulent flows assumes the sequence of
operations has to be fulfilled: images evaluation, several steps of filtration for removal of “outliers” and, as a final
step, the statistical processing of determined ensemble of instant velocity fields. Intermediate procedures for outliers
rejecting (after some chosen approach for correlation peak detection and validation) usually are based on the analysis
of each instant velocity distribution. Most prevalent among them are: the procedure of range validation (cutting off all
the velocity vectors which exceed the defined threshold for physically realizable values in a tested flow) and different
methods of spatial filtration based on the local smoothness and differentiability of spatial distributions. The first class
of filtration methods does not allow removing all outliers without any a priori knowledge about flow structure, second
one can be correctly applied only at very high spatial resolution of used PIV release. The statistical processing of the
ensemble of velocity fields assumes as a first step the calculation of the single point statistical characteristics –
moments of different order, PDFs etc. Even small amount of outliers in considered spatial point (relatively to whole
number of instant spatial realizations) can result the large errors in calculated high order statistical moments. Fig. 1, a
presents the histogram for pulsations of radial velocity component in the center of mixing layer of submerged
axisymmetric jet after processing with the aid of standard range validation procedure. The ensemble of realizations
consists of 20,000 instant velocity fields. It is clearly seen from the Figure that after such filtration the non-physical
tails remain in the ensemble (shown by arrows). In spite of amount of non-rejected outliers is not large (the mean
velocity value is almost non-sensitive to them because of large array of data), calculated high order moments are
essentially higher than true values: dispersion (2nd moment) – in 2.5 times, skewness factor (3rd moment) – in 7 times
and kurtosis (4th moment) – in 18 times. As a result, the spatial distributions of high order moments are qualitative
incorrect (see below Figs 6 - 7). Therefore the main goal of developing methods of data validation for large statistical
ensembles of velocity fields is finding a way to cut correctly the tails of PDF velocity fluctuations for each spatial
point of flow.
2. ALGORITHM OF BASIC PDF CONSTRUCTION
As a basis of statistical filtration method the construction of a basic function (approximation of real PDF) with
exponential decay of the tails is underlain. This procedure assumes taking into account the physical essence of testing
flow and has to reflect its statistical structure. The assumption is being used that for substantially non-Gauss
distributions (large values of skewness factors and kurtosis) the shape of PDF distribution is influenced in a general
degree by vortex formations from the large-wave interval of turbulent spectrum. This assumption was considered as
valid because the turbulent pulsations from the inertial interval obey the Kolmogoroff universal statistical law with
normal distribution of probabilities, and dissipative interval contains only small part of turbulent energy. At first step
of construction of basic PDF the Gauss distribution is considered with known values of first and second statistical
moments (mean value and dispersion). However if the ensemble of realizations is found to be substantially nonGauss, the second step of PDF construction assumes the using of another distribution. In this new distribution the
following properties have to be realized: exponentially decaying tails; accounting the asymmetry of studied ensemble
(which is resulting by vortex structures); analytical form (for effective use during the processing of large arrays of
data).
Such PDF of turbulent fluctuations of the velocity radial or vertical component (for example vertical) is represented
as a superposition of two independent distributions: inertial interval of turbulent spectrum (background turbulence)
Pb u and large-wave region of PDF spectrum Pc v . Here u and v are the vertical velocity fluctuations of
background turbulence and the vertical velocity field of coherent structures, correspondingly (see Ilyushin, 2001):
P (u, v ) Pb (u )Pc (v ) a
£¦ u 2 ²
£
£
¦
¦ (m v )2 ²
¦ a
¦ (m v )2 ²
¦¯°
1
¦
¦
¡ exp ¦
exp ¦¤ 2 ¦
exp ¦
»
¤
¤
»°
2 »
2
¡
¦¥¦ 2Tb ¦
¦
¦
2QTb
2(Tc ) ¦
2(Tc ) ¦
Tc
Tc
¦
¦ ¦
¦ ¥
¥
¼¦
±°
¢¡
¼
¼
background turbulence PDF
downflow
upflow
Large Scale Eddy Formation PDF
(1)
where σb is the dispersion of background turbulence; a+ and a- are the weight coefficients, σc + and σ c - are the
dispersions, m+ and m- are the centers of distributions of upflow and downflow of coherent structures,
correspondingly. Considering the total turbulent velocity fluctuation w as the sum of u and v , one can receive the
total PDF:
P (w ) ¨
£ (m w )2 ¦
²
£ (m w )2 ¦¦
²
¦
¦
a
a
¦
¦
exp ¦
exp
¤
»
¤
»,
2
2
¦
¦
¦¦
¦¼
2QT
2T
2T
¦
¦ 2QT
¦
¥
¥
¼
P (u, v )E(w u v )dudv R
(2)
where σ+2=(σc+)2+σb2 , σ-2=(σc-)2+σb2.
From the conditions
¨ P(w )dw 1,
R
( T w2
12
¨ wP(w )dw 0,
¨ w P(w )dw w
2
R
2
,
R
is the dispersion and Sw w 3
w2
32
¨ w P(w )dw S
3
w
T 3,
(3)
R
is the skewness factor) the connections for a+ , a - , σ+2, σ-2, m+
and m- are the following:
a a 1,
a m a m 0,
a ¢(m )2 T2 ¯± a ¢(m )2 T2 ¯± T 2 ,
(4)
a ¢(m )3 3m T2 ¯± a ¢(m )3 3m T2 ¯± Sw T 3 .
The conditions for σ+2 and σ-2 are found from the assumption that the square of dispersions σ+2 and σ-2 can be equal
to the square of centers of distribution (m+)2 and (m-)2 correspondingly (De Baas et al., 1986):
T2 (m )2 ,
T2 (m )2 .
(5)
This assumption means that flux directed positively (negatively) remains mainly positive (negative) taking into
account a possible scatter. Necessary condition for closure of the equations set (for σb) is found from the wavelet
model (Tennekes and Lamley, 1972). Here the supposition is used that the main part of the energy-containing interval
of the fluctuations spectrum is defined by a single main wavelet with the typical wave number corresponding to the
lg E(k)
kmax
wavelet
Ec
kmax
long-wave
interval
inertial
interval
Eb=1/2σb
2
Fig. 2. Presentation of the long-wave part of the turbulent fluctuations spectrum as the wavelet.
spectrum maximum (see Fig.2). The simplified eddy with the typical wave number kv is considered in a form of the
localized perturbation of energy in the wave number space (wavelet) with the energy E v = E v (kv ) ¸ kv . This
consideration ensured the cascade transfer of the turbulent energy from large-scale eddies to dissipative eddies. In the
present work the coherent structures in the flow are assumed to be the wavelets containing the energy
2 3
23
Ec =a F k max
(a=1.6 ± 0.02 is the Kolmogoroff constant), k max is the maximum of the spectrum of the turbulent
energy (see Fig. 2). The total turbulent energy is equal to
E kmax
d
kmin
i 0
œ Ev (kv ) x Ev (k max )œ ¢¡32 / 3 ¯±°
i
x 2Ec .
(6)
The analogous result can be obtained by integration of inertial interval (without accounting of the deviation from the
Kolmogoroff spectrum in dissipation region) from k max to the infinity and with appendix the half of the wavelet
energy:
d
Ex
¨ aF
k
kmax
or Tb2 E
1
dk Ec 2Ec ,
2
2 / 3 5 / 3
(7)
( (Tb2 Tc2 ) 2 E ). Regarding this result, k max is expressed as k max 2a E
Mmax 2Q kmax . We used the value of the ratio Tb2 w 2
32
F and then
is equal to 1/3 in the mixing layer (Ilyushin, 2001). This
simple assumption allows determine the PDF 's parameters analytically (see below). It is important for elaboration of
a program code for statistical processing of the large databases. So, the necessary condition for the dispersion of
background turbulence σb is:
Tb2 1
w2 .
3
(8)
Equations (4-5), (8), with accounting the condition for the sign of values m+>0 and m-<0 (which correspond to
upflow and downflow) have exclusive solution:
m T S S 2 8 ¯° ;
±
4 ¡¢
(Tc )2 m T S S 2 8 ¯° ;
±
4 ¡¢
2
1
T2 S S 2 8 ¯° - w 2 ;
¡
¢
±
16
3
a (Tc )2 S S2 8
2 S2 8
;
a S S2 8
2
1
T2 S S 2 8 ¯° - w 2 .
¡
¢
±
16
3
2 S2 8
;
(9)
The results of reconstruction of PDF using this algorithm, its testing and application for numerical modeling of
substance transfer in atmospheric boundary layer is described in the paper of Ilyushin, 2001. In the present work we
apply such approach for modeling the substantially non-Gaussian PDFs for measured ensemble of velocity fields.
3. ALGORITHM OF VALIDATION
Validation procedure is being applied to the ensemble of instant realizations GvGN (GuN ; GvN ) : (u1 !uN ; v1 !vN ) of
G
velocity vector components v (u; v ) . In this paper we consider two-dimensional velocity field. For each spatial
point the ensemble GwN of instant velocity values is being analyzed ( w - is the velocity component: u or v ), the
histogram : N is being built for the velocity range <w min ; w max > with the step equal to 2% (see Fig. 3), where % is
Fig. 4. Histograms and base functions calculated at
each step during filtration of test ensemble.
Distributions
a (Cartesian) – a (logarithmic)
and
e (Cartesian) – e (logarithmic)
are
completely
equivalent (Cartesian and logarithmic coordinates for
nonfiltered and finally filtered distributions
correspondingly). (b) – (d) – different steps of
filtration.
(a)
(b)
Fig.5. Statistical moments calculated at different steps I of filtration. (a) – mean value and dispersion, (b) –
skewness factor and kurtosis.
the accuracy of velocity measurements. The statistical moments are calculated: mean value ˜w § , dispersion
T ˜w 2 § , skewness factor S ˜w 3 § T 3 and kurtosis E ˜w 4 § T 4 3 , where
˜w § 1
N
N
œw ,
i 1
i
˜w 2 § 1
N
N
œ w
i 1
2
i
˜w §
,
˜w 3 § 1
N
N
œ w
i 1
3
i
˜w §
,
˜w 4 § 1
N
N
œ w
i 1
4
i
˜w §
.
(10)
On the basis of these parameters the basic function is being determined. At the first step the Gaussian distribution is
considered:
F1(w ) £ w ˜w §
2 ¦
²
¦
1
¦
exp ¦
¤
»
2
¦
¦
2QT
2
T
¦
¦
¥
¼
(11)
for concerned velocity interval <w min ; w max > of experimentally obtained ensemble. After comparison of built
histogram : N with the first-step basic function F1(w ) it is possible to determine the sub-ensembles Yw1 , Yw2 , !, Ywk
for those the number of realizations with the velocity magnitudes from the range <wi %; wi % > i 1!N differ
from value F1(wi ) more than in B times (after set of tests the optimal value of this parameter was found to be
B 1000 ). Such a procedure is being performed for both velocity components. The velocity vector is rejected from
the statistical processing for considered spatial point when each of velocity components belongs to determined subensembles Yw1 , Yw2 , !, Ywk . Such realizations are not taking into account further. For each following step of filtration
the ensemble truncated at the previous step GvGM (GuM ; GvM ) : (u1 !uM ; v1 !vM ) , ( M N k ) is being analyzed
according algorithm described above. In this case the new functions F2 (w ) (formula 2) are used as basic ones
(instead Gauss), with the parameters (9) calculated according (10) for truncated ensemble. This procedure recurs up
to the conditions when the quantity of wrong vectors defined according such criteria will be equal to zero.
4. TESTING OF ALGORITHM
For the testing of proposed filtration algorithm the generated ensembles of realizations were processed with
predetermined distribution P (w ) and imposed noise f (w ) . As a distribution P (w ) the Gram-Charlier series, cut on
the third term, was used:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 6. Calculated distributions of mean values and dispersions in axisymmetric turbulent impinging jet.
(a), (c), (e), (g) – u, v, Tu , Tv for filtered data correspondingly; (b), (d), (f), (h) - u, v, Tu , Tv for nonfiltered data correspondingly (scales coincide with left column) .
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 7. Calculated distributions of mean values and dispersions in axisymmetric turbulent impinging jet. (a), (c),
(e), (g) – Su , Sv , Eu , Ev for filtered data correspondingly; (b), (d), (f), (h) - Su , Sv , Eu , Ev for non-filtered data
correspondingly.
P (w ) £¦ w 2 ²¦ 1
1
1
¯
exp ¤¦ »¦ ¡1 S w 3 3w E w 4 6w 2 3
° .
¦¥¦ 2 ¦¼¦ ¡¢
°±
2Q
3!
4!
(12)
When the parameters E and S are equal to zero, the series (12) represents the Gaussian function. The “noise” f (w )
representing the harmonic addition with amplitude A (changed from 1 % to 5 % relatively to maximum of
distribution) was imposed onto distribution (12). The wavelength β of this harmonic noise was changed in the range
C 0.01 u 0.1
T , where T is the dispersion of P (w ) :
f (w ) A(1 sin(Cw )).
(13)
Testing procedure includes the calculation of mean values, dispersion, skewness factor and kurtosis for initial
ensemble P (w ) - without noise. Then the ensemble is being built for the superposition P (w ) f (w ) . During
filtration all vectors corresponding to sub-ensembles Yw1 , Yw2 , !, Ywm are considered as erroneous and rejected. Filtered
histograms and corresponding statistical moments were compared with the initial values at each step of filtration
cycle. As a rule, it is necessary to use up to 5 - 8 iterations to reach the true values. The comparison of histograms is
presented in Fig. 4. (for this case the following values of parameters have been chosen: S 0.6; E 2.0 , for (12)
and A 0.01 , C 5 for (13)). The logarithmical coordinates are used here for better viewing of small values of
P (w ) .
It is seen that during each step of filtration the noisy additions to PDF are rejected and resulting histogram (Fig.4, e)
has exponential character for tails decay. Figure 5 shows the relative values of statistical moments, calculated for each
step of filtration. It is obvious that at first step the difference between noisy and true values is substantial. During
filtration these values approach to each other asymptotically. Testing was performed for the noise (with different
amplitudes and β ) imposed both on the Gaussian base distribution and on the different types of non-Gaussian base
distributions (12) with various parameters S и E . The results showed that developed algorithm restores statistical
moments with the accuracy up to 10 % (when the noise amplitude does not exceed 5 %).
5. APPLICATION OF VALIDATION ALGORITHM
The results of application of proposed algorithm to PIV – measurements in axisymmetric turbulent impinging jet are
presented in Figs 6 and 7 for ensemble of instant velocity fields consisting of 5000 realizations. For comparison, both
filtered (a) and unfiltered (b) distributions of statistical moments are shown. It is seen that difference between nonfiltered and filtered distributions grows rapidly with the increasing of the order of moment. Thus, mean values are
calculated correctly without any additional filtration. At the same time the calculated distributions of kurtosis are
physically incorrect for unfiltered case and differ from the true (filtered) values by the several orders. Statistical
moments calculated for filtered arrays are more correct and reflect the physical nature of the impinging jet flow. It is
necessary to note that wrong vectors yield the general contribution into the errors of calculated statistical moments in
the flow regions with large velocity gradients (let mixing layer) and in the vicinity of the jet axis where longitudinal
velocity pulsations are very small and PDF of velocity pulsations is very narrow. The complete set of experiments
processed with described filtration algorithm is presented in the author's work (Bilsky et al., 2001).
The filtration algorithm is realized in the form of program code written in the C++ language and is inbuilt into the
general process of the data acquisition, processing and analysis for PIV-measurements. The developed code is
adapted for both two- and tree-dimensional velocity fields.
6. SUMMARY
The statistical method for filtration of the ensemble of measured instant velocity fields in turbulent flows has been
developed and tested. The method is based on the assumption of exponential decay of PDF tails for velocity
fluctuations in turbulent flows. The possibility of substantial non-Gaussian character of PDFs for turbulent
fluctuations has been taking into account. Such approach allows rejecting the large-amplitude erroneous velocity
vectors and correct calculation of the high-order statistical moments. The testing of algorithm was performed on the
basis of generated artificial PDFs with imposed random noise. Developed method was applied for filtration of
measured ensemble of instant velocity fields in turbulent impinging jet. The physically correct spatial distributions of
statistical moments up to forth order have been obtained.
ACKNOWLEDGMENTS
This work was supported by RFBR grant N 01-02-17662.
REFERENCES
De Baas, A.F., van Dop, H., and Nieuwstadt, F.T.M. (1986). "An application of Langevin equation for
inhomogeneous conditions to dispersion in a convective boundary layer", Quart. J. R. Met. Soc., 112, pp.165-180.
Bilsky, A.V., Heinz, O.M., Ilyushin, B.B., Markovich, D.M., Vasechkin V.N. (2001). "Turbulence statistics and
conditional averaging in axisymmetric impinging jet", CD-ROM Proc. of 4th Int. Symposium on Particle Image
Velocimetry, Gottingen, Germany, 17-19 September, 2001, Paper N 1175, 13 pages.
Ilyushin B.B. (2001). "Use of higher moments to construct pdf's in stratified flows", in: "Closure Strategies for
Turbulent and Transitional Flows". Eds. B.E. Launder, N. D. Sandham. Cambridge University Press, pp.683-699.
Tennekes, H., Lumley, J.L. (1972). "A First Course in Turbulence", MIT Press, Cambridge, Massachusetts.