RESEARCH ARTICLE

Flow temporal reconstruction from non time-resolved datapart II: practical implementation, methodology validation,and applications

Mathieu Legrand • Jose Nogueira •

Shigeru Tachibana • Antonio Lecuona •

Sara Nauri

Received: 4 May 2010 / Revised: 12 April 2011 / Accepted: 26 April 2011 / Published online: 7 May 2011

� Springer-Verlag 2011

Abstract This paper proposes a method to sort experi-

mental snapshots of a periodic flow using information from

the first three POD coefficients. Even in presence of tur-

bulence, phase-average flow fields are reconstructed with

this novel technique. The main objective is to identify and

track traveling coherent structures in these pseudo periodic

flows. This provides a tool for shedding light on flow

dynamics and allows for dynamical contents comparison,

instead of using mean statistics or traditional point-based

correlation techniques. To evaluate the performance of the

technique, apart from a laminar test on the relative strength

of the POD modes, four additional tests have been per-

formed. In the first of these tests, time-resolved PIV mea-

surements of a turbulent flow with an externally forced

main frequency allows to compare real phase-locked

average data with reconstructed phase obtained using the

technique proposed in the paper. The reconstruction tech-

nique is then applied to a set of non-forced, non time-

resolved Stereo PIV measurements in an atmospheric

burner, under combustion conditions. Besides checking

that the reconstruction on different planes matches, there is

no indication of the magnitude of the error for the proposed

technique. In order to obtain some data regarding this

aspect, two additional tests are performed on simulated

non-externally forced laminar flows with the addition

of a digital filter resembling turbulence (Klein et al. in

J Comput Phys 186:652–665, 2003). With this information,

the limitation of the technique applicability to periodic

flows including turbulence or secondary frequency features

is further discussed on the basis of the relative strength of

the Proper Orthogonal Decomposition (POD) modes. The

discussion offered indicates coherence between the recon-

structed results and those obtained in the simulations. In

addition, it allows defining a threshold parameter that

indicates when the proposed technique is suitable or not.

For those researchers interested on the background and

possible generalizations of the technique, part I of this

work (Legrand et al. in Exp Fluid (submitted in 2010)

2011) offers the mathematic fundamentals of the general

space–time reconstruction technique using POD coeffi-

cients. Noteworthy, the involved computational time is

relatively small: all the reconstructions have been per-

formed in the order of minutes.

1 Introduction

The most common source of light for PIV analysis of flow

fields are pulsed lasers with repetition rates in the order of

10 Hz. Due to the these low repetition rates, it is usual to

have series of non time-resolved PIV experimental snap-

shots of a periodical flow at random phase instants. This

paper proposes a method to sort such statistically inde-

pendent snapshots using information from the first three

POD coefficients. The technique also provides information

about the phase-instant corresponding to each snapshot. In

relation to computing time, the method is remarkably

efficient.

M. Legrand (&) � J. Nogueira � A. Lecuona

Department of Thermal and Fluids Engineering,

Universidad Carlos III, Madrid, Spain

e-mail: mlegrand@ing.uc3m.es

S. Tachibana

Aerospace Research and Development,

Japan Aerospace Exploration Agency, Tokyo, Japan

S. Nauri

Design Systems & Services, QinetiQ, Farnborough, UK

123

Exp Fluids (2011) 51:861–870

DOI 10.1007/s00348-011-1113-3

Once the snapshots are sorted it is possible to recon-

struct phase-averaged flow fields that reduce the impact of

turbulence and secondary frequencies on the main flow

structures. This way, information on traveling coherent

structures is unveiled in addition to the classical average

and variance statistics of the flow.

This sorting method that uses the first three POD modes

coefficients has been developed for scenarios were the

following hypotheses can be assumed:

(i) Taylor hypothesis: i.e., convective and weak dissi-

pative structures are present in the flow field.

(ii) Integration domain X (region of interest) is much

larger than structures sizes.

(iii) The flow is periodic (or pseudo periodic) and is

dominated by its fundamental frequency f.

Additionally, the following hypothesis further simplifies

the calculations when applicable:

(iv) �Uðr~Þ does not vary much along the convection

direction, for distances in the order of the structures

sizes.

Under hypotheses (i), (ii), and (iii), the diagonalization

of the cross-correlation matrix Cij = 1N

RRXuiðr~Þujðr~ÞdX

(where uiðr~Þ are the snapshot vector fields and N the

number of realizations) results in eigen-values k(k) and

eigen-vectors vðkÞ�i of particular properties indicated in

expression (1) when sorted by its relative magnitude.

Incidentally, in POD analysis vðkÞ�i represents the contri-

bution of each snapshot i to the POD mode k.

kð0Þ[ kð1Þ[ kð2Þ[ kðkÞ for any k [ 2

vð0Þ�i ¼ Qþ þ A cos 2p i� /0

� �

vð1Þ�i ¼ A cos 2p i� /0

� �þ Q�

vð2Þ�i ¼ A sin 2p i� /0

� �

ð1Þ

In this expression, i is the non-dimensional phase-

instant that would allow the sorting of the snapshots if

obtained. It is defined as i ¼ fti, where ti is the phase-time

at which the snapshot has been captured. /0, Q?, A, and

Q- are scalar constants depending on the flow under study.

If the number of snapshots N is large enough to ensure

the measurements are randomly distributed along a period,

Q? and Q- correspond to the average of the N values

obtained for vð0Þ�i and vð1Þ�i , respectively. The peak-to-peak

amplitude A in Eq. 1 may be calculated asffiffiffi2p

times the root

mean square of each function vðkÞ�i . In case the reader is

interested in the mathematical fundamentals that establishes

these relations, part I of this work (Legrand et al. 2011) give

the detailed reasoning. Note also that Q? [ A [ Q-

(demonstrated in the Appendix of Legrand et al. 2011).

Hypothesis (iv) leads to the simplified expression in

Eq. 2, as in Legrand et al. (2010), where the lack of asterisk

in the eigen-vectors notation indicate that they have been

normalized using the value of A.

vð0Þi �ffiffiffiffi1

N

r

vð1Þi � �ffiffiffiffi2

N

r

cos 2p i� p4

� �

vð2Þi � �ffiffiffiffi2

N

r

sin 2p i� p4

� �

ð2Þ

For higher POD modes, i.e., n [ 2, the eigen-functions are

doublets of the form:

vðnÞi ¼ffiffiffiffi2

N

r

cos 2p iþ /n

� �

vðnþ1Þi ¼

ffiffiffiffi2

N

r

sin 2p iþ /n

� �

8>>><

>>>:

ð3Þ

Experimental eigen-vectors ~vðkÞs are identical to the

theoretical onesvðkÞi , but in an a priori different, random

order (s = i). Taking advantage of this, unsorted PIV

snapshots can be sorted along a period by fitting the

experimental ~vðkÞs to the vðkÞi functions given in Eqs. 1 and

2. Each reconstructed phase hs is obtained by minimizing

over s the following expression:

hs ¼2pi

Nif ns ¼

P2k¼0 xðkÞ ~vðkÞs � vðkÞi

� �2

P2k¼0 xðkÞ

¼ minimum

ð4Þ

Here, x(j) are the inverse of the peak-to-peak amplitude

of the normalized vðkÞi i:e:xðkÞ ¼R 1

0vðkÞ�i

� �2

di

�

A

�

.

Then, bin-to-bin phase averaging can be easily performed

from the reconstructed time series. This will be the object

of next sections.

The main purpose here is not to analyze a specific flow

field nor to conduct direct comparison with CFD data, but

to show the viability and reliability of the reconstruction

technique. In order to achieve this, in Sect. 2, the procedure

is applied to CFD time-resolved data (laminar unsteady

Navier–Stokes equations). In Sect. 3, the technique is

applied to a set of time-resolved PIV data of a low-swirl

excited flame (Tachibana et al. 2009). Phase-locked aver-

age data from the experiment allows for a comparison

between real flow and reconstructed phases. Good agree-

ment reveals the technique suitability in planar PIV mea-

surements, even under the additional complication of

combustion. Section 4 presents the same kind of results for

non time-resolved Stereo PIV (S-PIV) in a low-swirl nat-

urally developing premixed flame (Legrand 2008 and

862 Exp Fluids (2011) 51:861–870

123

Legrand et al. 2010). The agreement between the recon-

structed time evolutions at different measurement planes is

a further sign of coherence and verisimilitude of the results.

Finally, the technique applicability is discussed on the basis

of strength of the POD modes.

2 Application to 2-D CFD data

In this section, the methodology presented in the Sect. 1 is

applied to an incompressible laminar Von Karman 2-D

vortex street in the wake of a confined square cylinder of

side h. The present simulation does not pretend to be

highly accurate. Instead, the purpose here is generating a

suitable time-resolved set of 2-D data in order to compare

realistic simulated time information with the reconstructed

one.

The mesh near the rod is square structured, and

unstructured where pressure gradients are not significant,

as shown in Fig. 1. The Fluent� CFD computer code

solved the laminar unsteady Navier–Stokes equations. The

Reynolds number, based on h and far field velocity U?,

was Reh & 630. The solution has been marched in time

until it reached periodic conditions for the whole flow field.

Instantaneous vorticity fields are depicted in Fig. 1b for

four phase-angles u, showing vortex shedding in the wake

of the cylinder at a frequency f & 14.7 Hz. The corre-

sponding Strouhal number is St ¼ hfU1¼ 0:147.

The procedure described is Sect. 1 has been applied to

100 velocity vector fields, taken from the simulation at

every 10 time steps. The total elapsed time corresponds to

about 7 flow periods. POD analysis allows calculating the

actual ~vðkÞi coefficients from the unsorted snapshots. In

Fig. 2a, these ~vðkÞi are displayed for the first 3 modes, while

Fig. 2b shows its adjustment to the theoretical vðkÞi fitting

functions. In this case, functions in Eq. 2 are preferred to

the ones offered in Eq. 1, since the mean flow does not

vary much along the convective direction and ~vð0Þi ¼ const:

Fitting results, shown in Fig. 2b, exhibit a very good

agreement and hence the reconstructed phase-angle hmatches almost perfectly the real phase-angle u. Standard

deviation rh ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

s¼1hs�usð Þ2

N

qhas been estimated as small

as 1 deg. In addition to that the following 2 modes (i.e. for

k = {3, 4}) are also depicted in Fig. 2b to show the con-

tributions of higher order modes and also the good agree-

ment with its theoretical expression in Eq. 3.

Finally, Fig. 3 shows the differences between the CFD

instantaneous vorticity fields and the reconstructed ones for

4 phase-angles. Small discrepancies in the near rod region

have to be mainly inferred from some blurring effect due to

the phase averaging process.

This is a simple test case where the second and third

POD modes are responsible for 96% of the fluctuating

kinetic energy. However, it is useful as comparison with

illustrate the applicability of the method in other examples

like depicted in the following section.

3 Application to time-resolved PIV data

After the promising results obtained in previous section

using ‘‘clean,’’ laminar CFD data, in this section, the

application of the reconstruction technique to time-

resolved experimental PIV data is studied. Again, the

purpose here is not the study of the flow field itself but

rather testing the capability of the procedure described in

Sect. 1 of this paper. The flow under study is the one

reported in Tachibana et al. 2009, whose experimental

setup is briefly described in Fig. 4. Main flow is the mix-

ture of air and methane while secondary air is injected

Fig. 1 a Mesh around the bluff body; b instantaneous vorticity fields at 4 instants. From left–right, top–bottom: u = {0�, 90�, 180�, 270�}

Exp Fluids (2011) 51:861–870 863

123

through four tangential orifices, which confer angular

momentum. The resulting diverging outflow allows for

anchoring a lifted premixed flame above the nozzle (Cheng

1995, Cheng et al. 2000), so that a good optical access to

the flow is possible. A high speed direct drive valve (DDV)

produces the swirl flow oscillations at a fixed frequency of

50 Hz, which results in the flow stretch rate oscillations

after the nozzle exit. Corresponding to the flow oscilla-

tions, the flame front position and the flame brush thickness

vary at the same frequency. Real phase-angle u can be

extracted from the valve controller signal. The PIV mea-

surements are conducted at 5 kHz during 1 s approxi-

mately. Phase averaging is then performed for 32 phases

(each 11.25 deg.) from N = 5,099 PIV snapshots. Thus,

the number of snapshots per phase is about Np = N/

32 & 160, providing statistical convergence for average

calculation (c. f. paragraph 3.iv of part I).

In this particular case, as the flow periodicity follows on

tangential active forcing, there is not an a priori evidence

that the flow is not varying much along convective direc-

tion. This is the reason why the more general expressions

of Eq. 1 are used to perform the fitting for POD modes

contributions, reported in Fig. 5a. Actually, the first mode

coefficient exhibits the fluctuating behavior reported in

Eq. 1 and not found in Eq. 2. Although the fitting in

Fig. 5a is not as clean as the one of Fig. 2b, a good general

agreement can be observed. Thanks to the time-resolved

acquisition, the real phase u can be obtained. This permits

to construct the plot of h as a function of u, as shown in

Fig. 5b. Reconstructed phase standard deviation around the

real phase rh has been calculated from this plot, yielding to

a reasonably small value: rh = 22.4 deg.

Regarding the reconstructed flow fields, the upper row

of Fig. 6 shows the phase averaging for 4 equally separated

Fig. 2 a ~vðkÞi contributions of the ui unsorted snapshots to the first 3 POD modes; b ~vðkÞi coefficients sorted along a period as a function of the

reconstructed phase-angle h

Fig. 3 Top CFD vorticity fields [1/s] for 4 phase-angles corresponding to a–d: u = {0�, 90�, 180�, 270�}. Bottom Difference from reconstructed

vorticity for h = u (zoom in 9 2), same color scale

864 Exp Fluids (2011) 51:861–870

123

angles u. Similarly, the lower row represents the same

result for the reconstruction procedure, with the corre-

sponding h angles. A very good overall conformity can be

appreciated, as the global flow pattern is respected for any

phase-angle. Nevertheless, some discrepancies persist due

to phase dispersion.

For a detailed comparison between real flow and the

reconstructed one, a reference point was chosen (marked

by a black cross in Fig. 6). It is located 10 mm above the

burner exit along the center line. Figure 7 shows both the

real phase-locked average and the reconstructed average at

this point for the axial velocity. In addition, the same graph

Fig. 4 a Configuration of the

jet-type Low-Swirl Burner.

b Experimental setup.

Reproduced from Tachibana

et al. 2009

Fig. 5 a Phase fitting using

Eq. 1. b Reconstructed phase has a function of real phase u

Fig. 6 Axial velocity map and pseudo-streamlines. Top row real phase-locked average flow. Lower row reconstructed flow. (Black cross markthe location of the reference point in Fig. 7)

Exp Fluids (2011) 51:861–870 865

123

shows in-phase rms values. Again, the procedure performs

a good reconstruction, as flow average does not exhibit

discrepancies higher than 10% for any phase-angle. Fur-

thermore, rms trend is well conserved and differences are

relatively small (less than 20% of the real in-phase rms

values for the greatest difference).

Figure 7 also shows the reconstruction using only the first

3 POD modes and their corresponding theoretical coeffi-

cients, vðkÞi (as in Eq. 1). It clearly tends to underestimate

axial velocity, and it shows the disadvantage of reproducing

locally a pure cosine function. These discrepancies illustrate

the loss of POD spatial contents already discussed in para-

graph 3.iii of part I (Legrand et al. 2011). Nevertheless, for

that case, the general tendency is conserved.

4 Application to real 3-D flow S-PIV data

and applicability of the procedure

In the previous sections, the procedure for a space–time

reconstruction of a periodic flow has been described and

applied to different flows whose time evolution was known

(i.e., CFD data in Sect. 2 and time-resolved PIV results in

Sect. 3). That allowed for comparisons between real flow

evolution and its reconstruction in order to show the

capability/reliability of the technique. However, the real

interest of the technique is, on the contrary, to reconstruct

the unknown flow temporal evolution from statistically

independent planar measurements. At this aim, the fol-

lowing subsections present an example of application to a

set of non time-resolved stereo PIV measurements.

It is worth noticing that the flow fields of Sects. 2 and 3

present some characteristics that are not common in flows

of industrial interest. In particular, CFD data are free of

experimental noise and the laminar computation used in

Sect. 2 does not model any turbulence. On the other hand,

the oscillating low-swirl flame is strongly excited tangen-

tially, so that the flow exhibits a clear frequency peak that

widely dominates the flow spectrum. These issues do not

fully support the applicability/suitability of the technique to

real flows, where frequency peaks are generally not so

pronounced and where stochastic turbulence and experi-

mental noise are important. These points are further dis-

cussed in a second subsection.

4.1 Application to real 3-D flow: non time-resolved

S-PIV data

The flow under study corresponds to the one issuing from

a low-swirl burner (LSB) distinct from the one of Sect. 3.

Here, the main body of the burner consists in a cylindrical

plenum. It is fed with a propane/air premixed mixture at

ambient temperature, issuing through two symmetric

opposite pipes with tangential outlets that provide angular

momentum. The swirling flow exits the plenum through

the outer coaxial annulus passage of the central straight

pipes. This passage is connected to the plenum at its

lower end by three symmetrical slot-like ports at 120�azimuthal position, which leaves approximately 90% open

area. The inner axial pipe passes through the back-plate

and does not interact with the plenum neither carry any

swirl. Both flows merge inside the nozzle, of diameter

D = 26 mm, before reaching the exit plane, and come out

to the open atmosphere. Bulk velocity Ub at the exit plane

is about 7.5 m/s. The flame is anchored in a diverging

flow, ensuring its stability above the nozzle exit plane.

Figure 8 briefly describes the experimental setup and the

flame topology obtained (for a Reynolds number

ReD = UbD/v & 14,000, and swirl number S & 0.51, as

defined in Legrand et al. 2010). Detailed information on

the facility and experimental S-PIV setup is available in

Legrand et al. 2010.

Whereas the flow was tangentially excited in Sect. 3;

here, the LSB flame is free of any artificial forcing. Besides

that the flow naturally tends to present a pseudo periodical

behavior that allows for applying the procedure for tem-

poral reconstruction. Fundamental frequency is found to be

around 500 Hz, while S-PIV image pairs are acquired

every 1.5 s approx., ensuring non time-resolved conditions

for these measurements. For the vertical S-PIV measure-

ment plane, Fig. 9a shows the contribution of each of the

1,000 snapshots to the first 3 POD modes, while Fig. 9b

shows the fitting used (Eqs. 1 and 4) in order to sort the

PIV snapshots along a time-phase. Here, the fitting is not as

clean as in Fig. 2b because of the effect of a strong tur-

bulence (turbulent kinetic energy is about 15 m2/s2 in the

shear layers). For the horizontal measurement planes,

Fig. 7 Axial velocity at 10 mm downstream the burner exit in the

center line. Filled squares phase-locked statistics (average and rms);

Open squares reconstructed statistics (phase average and phase rms)

866 Exp Fluids (2011) 51:861–870

123

snapshots contributions and fitting are very similar and for

this reason, they are omitted.

To offer a better view of the mean flow field topology, the

left-hand side of Fig. 10 shows a 3-D rendering of 3 mea-

surement planes: r2/D = 0, z/D = 1.0, and z/D = 3.0. The

null axial velocity contour is depicted with dashed white

lines to outline the recirculation zone boundary. The phase

averaging method for the time history reconstruction has

been used to analyze this LSB flame dynamics. Now, the

correlation matrix Ci,j has been computed from three velocity

components (3C) fields, as stereoscopic PIV measurements

offer this possibility. Results are presented in the right part of

Fig. 10, which depicts four different h time-phases. The axial

velocity is depicted in the top row of images for a vertical

measurement plane and in the bottom row for the horizontal

plane at z/D = 1. Flame front location, depicted as a con-

tinuous white line, has been estimated from seeding density

spatial gradient as in Legrand et al. 2010.

Results in the top row are very consistent with the ones

in the bottom row, even if the fitting has been performed

from 2D-3C velocity data from two different planes

(r2/D = 0, z/D = 1.0). The agreement between the recon-

structed time evolutions in different measurement planes is

a further sign of coherence and verisimilitude of the results

in real flow conditions. Central jet bulk describes a circular

path around the burner axis, close to the nozzle exit. This

perturbs the recirculation zone and the low velocity region

comes down toward the nozzle lip (left-hand side of

Fig. 10b). This low velocity region is rotating, as can be

observed in the horizontal planes, shown in the bottom row

of this figure. The asymmetry exhibited by this motion

confirms the one observable in the left side of Fig. 10,

showing stronger recirculation on the down left position

than on the upper right one for this figure.

4.2 Validation of the methodology

As it can be seen in Fig. 9b, the strength of turbulence

seems to play an important role in the dispersion of the

phase fitting; and thus in the efficiency and accuracy of the

Fig. 8 Stereo PIV setup. a Sketch of the burner and typical flame shape in LSB configuration, dimensions are in mm; b S-PIV setup for vertical

measurement plane; c For horizontal measurement planes. Coordinate system is also indicated

Fig. 9 Snapshots’ contributions

to POD modes (vertical planemeasurement). a Unsorted;

b Sorted along phase-angle h by

fitting as in Eq. 4

Exp Fluids (2011) 51:861–870 867

123

reconstruction. In order to study this effect and subse-

quently to determine a range of applicability of the meth-

odology, a set of different flows has been analyzed varying

their turbulence intensity.

First, the laminar CFD test case of Sect. 3 for a Karman

vortex street behind a square rod has been modified,

overlaying an artificial 3-D stochastic turbulence. To

achieve that, the digital filter proposed by Klein et al. 2003

has been used. The method was originally developed to

generate a suitable time-resolved turbulence to feed the

unsteady boundary conditions of DNS simulations. It has

the advantage of generating velocity fluctuations that pre-

serve the prescribed turbulent kinetic energy spectrum.

Moreover, the instantaneous distribution of fluctuating

velocity is not purely random, but reproduces well the

spatial correlation of turbulence. This way, the generated

turbulence is spatially coherent and the typical structure

size was chosen to be less than 5% of the spatial domain

considered (X). In this case, the authors choose not to use

temporal coherence for the turbulence in order to reproduce

the conditions of non time-resolved measurements (so that

the assumptions made for C3, C4, and C5 terms in Eq. 4 in

Legrand et al. 2011 are fulfilled). Then, different turbu-

lence intensities were used (12 values ranging from 0 to

150% of U?), and they were overlaid on to the N = 100

flow fields of the CFD simulation, in order to test the

efficiency of the time reconstruction technique under more

realistic conditions.

In addition to this test flow field, and to complete the

study, an artificial periodic flow field has been generated. It

corresponds to the straight convection of a vortex in a

rectangular box. When the vortex reaches the end of the

box, it appears again at the beginning, ensuring periodical

conditions for this artificial flow. Tangential velocity dis-

tribution around the vortex core position is as follows:

Vh ¼ 2Vmax r=R0ð Þ1þ r=R0ð Þ2 , where Vmax is the maximum tangential

velocity, located at the vortex radius r = R0. The same

procedure for adding turbulence was then used, as for the

Karman vortex street (12 turbulence intensities from 0 to

1.5 Vmax).

Finally, the time-resolved data from the forced LSB

flame of Sect. 3 were also used, for seven distinct forcing

cases (Tachibana et al. 2009). For high forcing cases, tur-

bulence intensity is relatively low when compared to the

energy contribution of the large-scale oscillations of the

flow. On the contrary, for low forcing cases, turbulence

dominates the flow, while large-scale fluctuations are

almost absent. Those differences are used in order to study

the effect of the turbulence intensity on the accuracy of the

time reconstruction procedure.

In addition, in-phase rms appears to converge for more

than 3,000 snapshots (less than 10% relative error for the

rms estimation rmsN in each phase). Both turbulent fluc-

tuations and random measurement errors contribute to

rms and both converge like 1 ffiffiffiffi

Np

. So using N = 1,000

rms1;000 / 1=ffiffiffiffiffiffiffiffiffiffiffiffi1; 000p� �

and N = 5,099 snapshots

(rms5;099 � rms1;000

ffiffiffi5p

) provides twice more sets of dif-

ferent ‘‘artificial’’ turbulence intensities (14 in total; using

two distinct N and seven different forcing cases). The

effect of adding turbulence is illustrated in Fig. 11a, which

depicts the normalized frequency spectrum. Frequency has

been normalized with the fundamental frequency f, and the

amplitude, with the maximum amplitude, reached for

f. This figure shows how the relative amplitude of the peak

frequency is smaller when turbulence intensity is increased

(either decreasing N or adding artificial turbulence).

For each of the above described test cases (38 in total),

the time reconstruction procedure has been applied. The

reconstruction computational time with a contemporary

Fig. 10 LSB reacting flow. Left: 3-D representation of average axial

velocity. Right: Reconstructed phase average axial velocity from

1,000 PIV snapshots. (top) r2/D = 0; (bottom); z/D = 1. Dashed lines

indicate correspondence for the two planes. a–e h = 0�, b–f h = 90�,

c–g h = 180�, d–h h = 270�. Operating conditions: /0 = 1.5,

S *0.51, ReD *14,000

868 Exp Fluids (2011) 51:861–870

123

standard personal computer ranges from several minutes

for the smaller sets of data, up to 5 h for N = 5,099. This

relatively short computational time, even for large sets of

acquisition series, makes the technique quite attractive in

terms of machine time consumption.

Figure 11b shows the normalized spectrum of the POD

eigen-values k(k), for the same cases than Fig. 11a. The

case of Sect. 4.1 is also represented for comparison. For a

given flow field, it has been observed that the strength of

the second POD mode k(1) relative to the rest of the POD

modes kð1Þ.PN

k¼1 kðkÞ clearly decreases as turbulence

intensity increases (note thatPN

k¼1 kðkÞ is roughly the area

beneath the curves in Fig. 11b). This reveals that higher

POD modes somehow describe small-scale dissipative and

temporally non-correlated structures, characteristic of sto-

chastic turbulence; whereas lower modes (k(1) and k(2) in

particular) represents larger scale, space–time coherent

weakly dissipative structures. As their contribution to the

flow energy decreases, the frequency peak in energy

spectrum gets weaker, so that the temporal reconstruction

may be compromised, because of periodical structures

being embedded in strong uncorrelated turbulence noise.

In order to study this effect on the reliability of the

reconstruction, the standard deviation rh (as defined in

Sect. 2) of the reconstructed phase-angle h around the real

phase u has been calculated since the real time evolution is

known for each test case. rh has been normalized by the

standard deviation of the data if it were completely random

rmax ¼ p=ffiffiffi3p

. Figure 12 depicts rh/rmax versus the relative

strength of the second POD mode kð1Þ.PN

k¼1 kðkÞ.

As it is noticeable in Fig. 12, every point seems to fit the

same exponential tendency, independently of the kind of

periodic flow considered. As turbulence intensity increases,

the relative strength of the second POD mode decreases

and rh/rmax gets larger, reaching almost unity when intense

turbulence dominates the flow energy. This graph allows

for establishing a limit of validity of the application of the

time reconstruction procedure for any periodic flow. When

the parameter K ¼ kð1Þ.PN

k¼1 kðkÞ is less than 5% (i.e.,

when mode 1 and 2 cumulate contribution to fluctuating

flow is less than 10%), the phase standard deviation is too

large (rh/rmax [ 0.7) to guarantee a reliable reconstruction.

Without any a priori knowledge on the flow behavior, this

criterion allows to determine if the reconstruction will be

accurate or not. Noteworthy, for the low-swirl flame of

Sect. 4.1, K * 8%, suggesting the reconstruction is reliable

in that case.

This validation indicates that even when the first POD

modes have a limited contribution to the global instanta-

neous flow field, their eigen-values are still valid for sorting

the real snapshots in case of periodic flow.

The use of the real snapshots instead of the first POD

modes to generate phase-averaged data may result in

better results, as the contents of other appropriate POD

Fig. 11 Influence of turbulence

intensity on the frequency

spectrum (a) and on the eigen-

values spectrum (b); for

different flow fields

Increasing turbulence intensity

Section 2

High forcing, N=5,099 (Fig. 11)Section 3

High forcing, N=1,000 (Fig. 11)

Low forcing, N=5,099 (Fig. 11)

Karman street, high turbulence (Fig. 11)

Fig. 12 Evolution of the normalized phase standard deviation versus

the relative strength of the second POD mode. The particular cases

presented in Sects. 2 and 3 of this paper are remarked in the graph

Exp Fluids (2011) 51:861–870 869

123

modes is automatically included in the average as Fig. 7

indicated.

5 Conclusions

On the basis of the model proposed in Part I of this work,

an algorithm has been developed to identify and track

traveling coherent structures in quasi periodic flows. Phase-

average flow fields are reconstructed with a correlation-

based technique, which involves some similarity with

POD. It uses POD eigen-vectors, but not POD modes.

The mathematical model, aimed at discriminating

space–time coherent structures from proper stochastic tur-

bulence, provides a tool for shedding light on flow

dynamics. Additionally, this stands as an interesting tool

for further experiment/simulation comparisons, allowing

for dynamical contents comparison, instead of using mean

statistics or traditional point-based techniques.

The temporal reconstruction technique has been tested

on ‘‘clean’’ CFD data with excellent results. Its application

on time-resolved PIV measurements from a low-swirl

burner excited tangentially also shows potential. Moreover,

the knowledge of the flow temporal evolution allows for

comparing the real flow with the reconstructed one, leading

to discrepancies in the axial velocity less than 10% of the

real phase-average.

The application to statistically independent measure-

ments on a low-swirl flame and for different measurements

planes shows a consistent reconstruction, although in that

case a comparison with the real temporal evolution is not

available. However, the reconstructed phase-average

coincides for the two perpendicular measurements planes

and this for any of the considered phase-angles. In that

case, the time reconstruction allows for a better under-

standing of the flow dynamics, and this under difficult

experimental conditions (turbulent combustion).

In addition, the relatively short computational time,

even for large sets of acquisition series, makes the recon-

struction technique attractive to recover temporal infor-

mation when the latter is not available from measurements.

Finally, some considerations about the applicability of

the reconstruction technique have been also taken into

account, especially the influence on the reliability of the

reconstruction of a prescribed non time-correlated turbu-

lence intensity. For this study, in addition to the time-

resolved PIV measurements, simulated laminar flows with

the addition of a digital filter resembling turbulence have

been used. Coherence is observed for both kinds of data.

Through the study of the reconstructed phase dispersion,

the accuracy of the procedure has shown to be sensitive to

the relative strength of the POD modes 1 and 2. This is

useful for establishing a limit of validity of the time

reconstruction technique for pseudo periodical flows when

its temporal evolution is unknown. Further checks of the

methodology and evaluation of the reconstruction error by

researchers with other time-resolved PIV measurements or

better numerical simulations is of interest. Based on the

available data, the relative strength of the second POD

mode, K ¼ kð1Þ.PN

k¼1 kðkÞ, is proposed as criterion to

evaluate the application validity with a threshold value in

the order of 7 %.

Acknowledgments This work has been partially funded by the

CoJeN European project, Specific Targeted RESEARCH Project EU

Contract No. AST3-CT-2003-502790; the Spanish Research Agency

grant DPI2002-02453 ‘‘Tecnicas avanzadas de Velocimetrıa por

Imagen de Partıculas (PIV) Aplicadas a Flujos de Interes Industrial’’

and the Spanish Research Agency grant ENE2006-13617

‘‘TERMOPIV: Combustion y transferencia de calor analizadas con

PIV avanzado.’’ We are also grateful to the Japan Aerospace and

Exploration Agency (JAXA) for its collaboration, its time-resolved

PIV data from their swirl-stabilized burner, and for kindly receiving

UC3 M personal during two months. We would like also to express a

special acknowledgment to the laboratory technicians Manuel Santos

and Carlos Cobos, for their help in the burner design and fabrication.

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