Experimental study of highly turbulent isothermal opposed-jet flowsGianfilippo Coppolaa� and Alessandro Gomezb�

Department of Mechanical Engineering, Yale Center for Combustion Studies,Yale University, New Haven, Connecticut 06520-8286, USA

�Received 27 January 2010; accepted 29 July 2010; published online 14 October 2010�

Opposed-jet flows have been shown to provide a valuable means to study a variety of combustionproblems, but have been limited to either laminar or modestly turbulent conditions. With theultimate goal of developing a burner for laboratory flames reaching turbulence regimes of relevanceto practical systems, we characterized highly turbulent, strained, isothermal, opposed-jet flows usingparticle image velocimetry �PIV�. The bulk strain rate was kept at 1250 s−1 and specially designedand properly positioned turbulence generation plates in the incoming streams boosted the turbulenceintensity to well above 20%, under conditions that are amenable to flame stabilization. The datawere analyzed with proper orthogonal decomposition �POD� and a novel statistical analysisconditioned to the instantaneous position of the stagnation surface. Both POD and the conditionalanalysis were found to be valuable tools allowing for the separation of the truly turbulentfluctuations from potential artifacts introduced by relatively low-frequency, large-scale instabilitiesthat would otherwise partly mask the turbulence. These instabilities cause the stagnation surface towobble with both an axial oscillation and a precession motion about the system axis of symmetry.Once these artifacts are removed, the longitudinal integral length scales are found to decrease as oneapproaches the stagnation line, as a consequence of the strained flow field, with the correspondingouter scale turbulent Reynolds number following a similar trend. The Taylor scale Reynolds numberis found to be roughly constant throughout the flow field at about 200, with a value virtuallyindependent of the data analysis technique. The novel conditional statistics allowed for theidentification of highly convoluted stagnation lines and, in some cases, of strong three-dimensionaleffects, that can be screened, as they typically yield more than one stagnation line in the flow field.The ability to lock on the instantaneous stagnation line, at the intersection of the stagnation surfacewith the PIV measurement plane, is particularly useful in the combustion context, since the flame isaerodynamically stabilized in the vicinity of the stagnation surface. Estimates of the ratio of themean residence time �inverse strain rate� to the vortex turnover time yield values greater than unity.The conditional mean velocity gradient suggests that, in contrast to the existing literature, thehighest gradients are around the system centerline, which would result in a higher probability offlame extinction in that region under chemically reacting conditions. The compactness of the domainand the short mean residence time render the system well suited to direct numerical simulation,more so than conventional jet flames. © 2010 American Institute of Physics.�doi:10.1063/1.3484253�

I. INTRODUCTION

The stagnation flow produced by two opposed coaxialjets under laminar �steady and unsteady� and, more recently,turbulent conditions has been extensively studied in theliterature.1–10 The interest was initially motivated by the factthat it provides flame anchoring in a wall-free environment,which simplifies the analysis of fundamental combustion re-search. It is also relevant to modeling turbulent flames inter-preted as an ensemble of laminar flamelets with a distribu-tion of scalar dissipation rates and stoichiometric mixturefractions.11 In recent work, we demonstrated the generationof intense turbulence in jet flows with specially designed and

properly positioned plates.12 We leapfrogged to demonstrat-ing this system in counterflow flames.13 We also proposedthe turbulent counterflow flame as a benchmark for testingand validating computational models in regimes of practicalcombustion systems. After legitimizing the effort from acombustion viewpoint, we now step back and examine sys-tematically the isothermal stagnation flow field under intenseturbulence.

Understanding the isothermal flow field is a prerequisiteto detailed studies of the reactive flow counterpart. It isalso the limit field of a flame close to extinction. All thestudies on this configuration, either numerical or experimen-tal, have been carried out at low or modest Reynoldsnumbers.1,2,4,14–20 The reason for this is that flames extin-guish when the characteristic convective-diffusive time is ofthe same order as a characteristic chemical time, that is,when the Damkholer number is of unity order.21 Theconvective-diffusive time is the inverse of the scalar dissipa-

a�Present address: Department of Bioengineering, Imperial College, London,UK.

b�Author to whom correspondence should be addressed. Electronic mail:alessandro.gomez@yale.edu. Telephone: �203� 432-4384. Fax: �203� 432-7654.

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tion rate and is related to the strain rate, while the chemicaltime is affected by the fuel-oxidizer composition and is re-lated to a simplified one-step chemistry that captures theflame heat release. Therefore, for a given fuel-oxidizer com-position, there will be a strain rate which will bring the flameto extinction. The extinction strain rate limits the achievableReynolds numbers, by setting an upper limit to flow rates fora given geometry. This limit and the Reynolds number canbe increased by burning in oxygen-rich environment. Evenso, boosting of the turbulent Reynolds number by an effec-tive turbulence generation scheme remains inevitable.

The most common approach to turbulence generation isto use a perforated plate downstream of a contracting nozzle.This approach allowed Kostiuk et al.2 to achieve a �12%turbulence intensity, an integral scale L�2.25 mm, and aturbulent Reynolds number based on the integral scale,ReL=165, as measured close to the jet exit. This work re-mains the most comprehensive description of isothermalflows of this nature, albeit at the reported modest Reynoldsnumbers. These authors also were the first to report a signifi-cant increase in the rms velocity as the flow stagnates.Mastorakas et al.22 discussed low-frequency flame motion ina combustion study in a similar configuration, which resultedin higher rms temperature values, as revealed by high-passfiltering of the data. The presence of large-scale intermittenceof the mixing layer in the axial direction was also recognizedby Geyer et al.23 without commenting on the origin of thisinstability. The study of Lindstedt et al.4 on opposed isother-mal flows using both particle image velocimetry �PIV� dataand modeling makes no comments on this effect.

Of relevance to the present discussion and not widelyrecognized is the fact that the opposed-jet configuration is ahydrodynamic unstable configuration even at high Re andstrain rates, as it is characterized by a pitchfork bifurcationwith two stable solutions with the stagnation line positionedclose to either nozzles and one unstable solution, with thestagnation line half way between the nozzles.5,7,24,25 A prob-able cause of the instability, similarly to sudden-expansionflows,26–29 is the presence of strong recirculation bubbles andtheir dynamic coupling either in the vicinity of the turbu-lence generation plates within the nozzle or at the exit of thejets. One may question whether this instability is distinctfrom true turbulence, in which case it would artificiallymasks the turbulent features of these flows, as detected, forexample, by rms measurements. As a result, a robust statis-tical analysis of the data is necessary to assess this point.

The goal of the present contribution is twofold. First, wedevelop an appropriate data analysis to unveil turbulencecharacteristics in the vicinity of the stagnation plane, usingproper orthogonal decomposition �POD� and some novelconditional statistics. The reason for focusing on the fluiddynamics in the vicinity of this plane is that it plays a criticalrole to the flame anchoring, which inevitably occurs within avery short distance �less than 1 mm� from it. Therefore,tracking the motion and the topology of this plane will alsobe useful in subsequent combustion studies. Second, we ana-lyze key features of isothermal flows in this configurationunder conditions of intense turbulence �O�1000� Ret�, unlikeall previous studies in the literature to date.

II. EXPERIMENTAL SETUP

The experimental apparatus �Fig. 1� is composed of twoidentical nozzles, aligned along their axis of symmetry andfacing each other at a distance of 1.5Dout. Each nozzle issurrounded by a larger one providing a coflowing stream toshield the main flow from the surrounding environment. N2

�O2� enters the bottom �top� nozzle and then flow throughtwo honeycombs �not shown�, positioned 38 mm apart. Eachhoneycomb is 19 mm thick with a 0.8 mm cell size. Theuniform flow leaving this flow-conditioning section, with thelast honeycomb positioned �38 mm upstream of the turbu-lence generating plate, is then forced through this plate12 andeventually exits through a 12.7 mm diameter �Dout� openingat the nozzle exit section. The selection and position of theplate ��42 mm from the nozzle exit section� is aimed atmaximizing the turbulent intensity, with typical values ofu� /U at 0.22 �as measured by hot-wire anemomentry� andthe uniformity of the flow at the nozzle exit. Different gasesare used as the system, along with the gas supply, is designedfor fundamental research in turbulent combustion. The flowrate is controlled by digital flow controllers �Teledyne-Hastings 300D series� providing a �0.5% readout accuracyand a +0.2% full scale one. The PIV system is operated inthe cross-correlation mode using a multipass cross-correlation algorithm with interrogation window shifting anddeformation to maximize resolution and minimize vectordrop out. The system optical magnification is about 1:1,which results in a field of view of about 15 mm�15 mm.The final window size is 32 by 32 pixels, equivalent to aphysical size of 0.24�0.24 mm which, with 50% overlap,produced a vector every �0.12 mm. Further details of thePIV setup are in Ref. 13.

The turbulence generator plate, when present, is placedat its optimal distance from the nozzle exit12 in each nozzle.

FIG. 1. Schematic of the opposed-jet assembly.

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Several experiments are conducted: a flow baseline case, inthe absence of plates in either stream �NN case�, the top-bottom �TB� case with identical plates in both nozzles, andthe TN and NB cases with only one plate present in the topnozzle and bottom one, respectively. The last three cases areof all of relevance to combustion applications. Unless explic-itly stated otherwise, most of the presented data pertain to theTB case. The bulk strain rate, SR=Ub / �H /2� is set to�1250 s−1, equivalent to a bulk flow velocity on the oxygenside Ub�11.9 m /s and to an engineering Reynolds number,Re�10 000. The turbulent integral and Taylor lengthscales, L and �, are measured 0.5 mm above the bottomnozzle exit section by hot-wire anemometry and are found tobe, respectively, �4.5 and �0.6 mm, resulting in ReL

�960 and Re��190, on the basis of the integral and Taylorscale, respectively. The corresponding values extracted fromthe PIV data will be discussed in Sec. III.

All quantities of interest are derived by analyzing largesets of double exposure images �785 for each of the NB, TNand TB cases; 471 for the NN case�. The derivatives wereestimated using Richardson’s finite difference scheme with-out smoothing the gradients of the measured field.30

III. RESULTS AND DISCUSSION

Figure 2 shows the phase plot of the instantaneous ve-locity components, u and v in three cases: NN with no tur-bulence generation plates, TN with one plate in the topburner, and TB with the plates on both sides. The figureshows all the �u,v� states accessed by the system at a fewpoints along the centerline, measured from the midpoint be-tween the two burners: z= �3 mm and z=0 mm. In the NNcase �Fig. 2�a��, as expected, all the points fall close to eachother in a very narrow domain, indicating that at each loca-tions, there are only small velocity fluctuations. Still, thestagnation plane, z=0 mm, at the centerline is accessingboth positive and negative values, implying that thez=0 mm point finds itself on either side of the stagnationplane. Case TN �Fig. 2�b�� shows a cloud of data point witha “centroid” shifted toward the side with the plate, where thelargest spread is observed. Focusing on the u spread, weobserve that now even the point at z= �3 mm is accessingboth positive and negative values, with a larger spread on theside of the turbulence generator, indicating a possibly largeshift of the stagnation plane. In the TB case �Fig. 2�c��, thissituation is repeated, but now the spread is symmetric. Theseresults are in qualitative agreement with previousexperiments,2,3 but here the oscillations are significantlymore pronounced. The very small velocity fluctuations in theNN configuration suggest that the stagnation plane fluctua-tions may also be a consequence of small fluctuations in thegas supply. In the other two cases, on the other hand, stag-nation plane oscillations are certainly caused by the large-scale structures generated by the plate.

A concern arises that these oscillation may be associatedwith a stagnation flow intrinsic instability.7,8,24 The “bounc-ing around” of the stagnation plane due to these instabilitiesmay affect the statistics and specifically the velocity fluctua-tions even though it may not be strictly classified as part of

the turbulent motion. Indeed, visual observation confirmedthe presence of low-frequency �O�100 Hz� or less� bouncingup and down and tilting of the stagnation plane. This phe-nomenon could help explain, at least in part, the disagree-ment between Reynolds stress models and experiments re-ported in Refs. 4 and 19. In fact, in the classicalnonconditional statistics, time averaging at fixed spatial lo-cations in a laboratory frame of reference implies averaging“different” flows because of the different positions of theinstantaneous stagnation plane. Reynolds-averaged Navier–

FIG. 2. Phase plot of velocity components at different axial locations alongthe centerline. Triangles: z=0 mm; circles: z=3 mm; squares: z=−3 mm.The reference location at z=0 mm corresponds to the mean stagnationplane and the negative sign indicates locations between the mean stagnationplane and the bottom nozzle. Case NN �a�; case TN �b�; case TB �c�.

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Stokes �RANS� models are not capable by design to dealwith such situations and alternative approaches have to bepursued for numerical simulation and/or experimental dataanalysis, possibly by extending the simulation domain to theturbulence generation plates.

A statistical data analysis that takes into account the sys-tem large-scale/low-frequency dynamics and separates itfrom the turbulence fluctuations is needed to provide insightinto the systems physics and suggest appropriate modelingapproaches. In the ensuing discussion, we present three dif-ferent tools to analyze the flow field: �a� conventional ornonconditional statistics, to compare the data with those inliterature, by taking a fixed coordinate system anchored atthe midpoint between the two burners, that is relevant to thesteady mean flow field features; �b� POD that may help filterthe large-scale instabilities unrelated to turbulence; and �c� amethod to condition the statistics to the instantaneous stag-nation plane, or, more properly, line in two-dimensionalmeasurements.

A. Nonconditional statistics

A cylindrical coordinate system �r, z� is used with originat the intersection between the mean stagnation plane and thesystem geometric centerline. Positive z indicates locationsbetween the stagnation plane and the top nozzle �Fig. 1�. ThePIV snapshots were time-averaged to extract mean and rmsflow variables. The available number of instantaneous snap-shots, for each case, produced well converged mean and ac-ceptable fluctuating components. The statistical uncertaintywas estimated assuming a general distribution for thesamples31 and 95% confidence interval. Error propagationrules were applied to estimate the uncertainties in the derivedquantities and found the maximum error in the region ofinterest �near centerline� at �0.35 m /s and �0.21 m /s forthe mean axial and radial velocity components, �0.3 m /sand �0.19 m /s for the axial and radial rms velocity, and�570 s−1 and �430 s−1 for the mean gradients dU/dz anddV/dr.

Figure 3�a� shows the profiles of the mean axial andradial velocity components along the radial coordinate atthree different axial locations with respect to the mean stag-nation plane. Measurements were performed with andwithout the turbulence generation plate, but no significantdifferences were observed in the mean measurements. Con-sequently only one set of measurements is presented for thetop burner in the TB case, since the bottom burner producesvirtually identical data. Trends are conforming to expecta-tions. Further from the stagnation plane, near the burner exit,the axial velocity profiles exhibit the characteristic M-shape1

�z=−7.0 mm�, with values close to the nozzle wall over-shooting the rather flat profile in the vicinity of the axis ofsymmetry. When velocity measurements are performed on asingle nozzle in a free jet configuration, no overshoot isdetected.12 The finite separation between the burners is re-sponsible for this change in the inlet velocity profile. Thisfinding can be explained qualitatively from Bernoulli’sprinciple.32 On the axis of symmetry at the mean stagnationsurface, the mean axial velocity is zero and the mean static

pressure attains its maximum. As a consequence, the stream-lines in the proximity of the axis are subject to a larger ad-verse mean pressure gradient as compared to those near thenozzle walls. To make quantitative comparisons betweencomputational and experimental results, it is clearly indis-pensable to have measurements of the nozzle velocityprofiles.

The degree of uniformity increases as the stagnationplane is approached and the overshoot disappears for theaxial component. The radial component is linear with theradial coordinate through most of the domain, until mixingwith shroud gas becomes significant. The slope also in-creases as the stagnation plane is approached, indicating anincrease in the strain rate, which is consistent with the be-havior of opposed-jet flows with finite separation betweenthe two streams. Figure 3�b� shows the axial profiles alongthe centerline of the rms values of both velocity components.The profiles are in qualitative agreement with previousmeasurements,2,4 with a strong peak at the stagnation planefor u� and a less pronounced one for v�. This peak will haveto be probed further by other data analysis techniques toensure that it is indeed a feature of the turbulence, as op-posed to some instability artifact.

FIG. 3. �a� Radial profiles of mean axial and radial velocity components�TB case� at different distances from the mean stagnation plane assumed asz=0. Circles: z=−0.5 mm; squares: z=−3.0 mm; triangles: z=−7.0 mm.�b� rms values along the burner centerline.

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B. POD analysis

We now use the POD �Ref. 33� of individual PIV veloc-ity field realizations to decompose the flow field in a numberof modes, such that the total amount of kinetic energy in asnapshot associated with a specific mode is proportional tothe corresponding eigenvalue. Individual modes can be inter-preted as flow fields statistically relevant, although not nec-essarily existing �modes have physical meaning only coupledwith the respective coefficients�, and their effect on the flowcan be selectively analyzed. It should be kept in mind, how-ever, that rare but dynamically important events are not theideal target for POD, since it is based on time-averaged cor-relations. In line with standard practice, the fluctuating flowfield for case TB was decomposed by this technique andrevealed a well defined flow structure. Because of memorylimitations, a slightly smaller domain along x and y wasprocessed. POD was also used as an inhomogeneous filter,34

which is particularly well suited to the present flow. Thestatistical uncertainty estimated as for the nonconditional sta-tistics is virtually the same as in that case.

Figure 4 shows the cumulative percentage of the totalkinetic energy of the flow versus the number of modes in asemilog graph to evidence the role of the first few modes. Itis clear that modes 1 and 2 are responsible for approximately25% and 15% of the total kinetic energy of the system, re-spectively, and are significantly stronger than the othermodes. Mode 1 �Fig. 5�a�� has the structure of a symmetricaxial flow, while mode 2 �Fig. 5�b�� has the structure of avortex of size comparable with the nozzle diameter. Recon-structing the flow field accounting only for mode 1 added tothe time-averaged field, that is, mode 0, shows that this mode

is responsible for large-scale oscillations of the stagnationline as a whole. Figures 5�c� and 5�d� show a sequence of theoscillation of the stagnation plane captured by reconstructingthe flow field with only mode 0 and mode 1. The stagnationline fluctuates from a mean position at approximatelyz=2 mm �c� to z=−2 mm �d�. The reconstruction of theflow field with only mode 2 added to mode 0 shows that thismode is responsible for the stagnation line tilting about itscenter �Figs. 5�e� and 5�f��. These snapshots were chosenamong the most extreme cases to underline the effect of themode structure. By considering the system axial symmetry, aprecession motion about the mean flow centerline seems tobe a plausible description of this type of instability superim-posed on the mean turbulent flow.35 As a result, the stagna-tion line motion can be tentatively decomposed in an axialoscillation, a precession motion about the mean centerlineand the turbulent contribution.

A phase plot �Fig. 6�, in which the axes represent thePOD coefficient relative to modes 1 and 2, can be used todescribe the relationship between these two modes.36 If thetwo modes of oscillation �axial and processional� were inphase, the scatter plot would present a circular shape, while adistortion to circular orbits of different radii36 would indicatea phase difference. The plot in Fig. 6 indicates that the twomodes of oscillation are not correlated, therefore suggestingthat they may be representative of different flow dynamics.As an additional bonus from the POD treatment, we cananalyze the large-scale oscillation frequency separating theaxial oscillations �reconstructing the flow with only mode 0and mode 1� from the tilting motion �reconstructing the flowwith only mode 0 and mode 2�. Visual observations suggestthat the oscillations are a low-frequency phenomenon, withcharacteristic frequency on the order of 100 Hz, as revealedby high-speed movies. The power spectrum as measured byhot-wire, 0.5 mm above the bottom nozzle exit section wasreported not to show peculiar peaks at any frequency.13 Thismay suggest that the oscillations are chaotic in nature, inagreement with previous findings,8 and it seems plausible toassume that they may be triggered by the dynamics of therecirculation zones in the vicinity of the turbulence genera-tion plate also in view of the results in Fig. 2.

C. Stagnation line conditional statistics „SLCS…

We now describe a third and novel method to analyzethe database on conditional statistics with respect to the po-sition of the instantaneous stagnation surface. Such a surfaceplays a key role in the flow field especially from a flamestabilization perspective, since under burning conditions andat large strain rates, the flame will inevitably anchor in itsimmediate vicinity ��1 mm�. An analysis of criticalpoints37–39 was used to identify the instantaneous stagnationline, that is, the intersect of the stagnation surface with thetwo-dimensional PIV plane, relevant critical points �saddle,vortices, etc.� and three-dimensional effects. Instantaneousand conditional mean strain rates at the instantaneous stag-nation line were also calculated since they are of direct rel-evance to flame extinction. The estimated statistical uncer-tainties are at �0.23 and �0.24 m /s for the mean axial and

FIG. 4. Percentage of the total turbulent kinetic energy vs POD mode �TBcase�.

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radial velocities, �0.2 and �0.27 m /s for the axial and ra-dial rms velocities, and �550 and �430 s−1 for the condi-tional mean gradients dU/dz and dVs /ds. After introducingthe method and showing samples of instantaneous flow field,we present averages and rms results from this perspectiveand conclude with a comparison of the relative merits of thistechnique with respect to POD.

1. Stagnation line tracking and critical point analysis

The SLCS analysis is based on the identification andtracking of the instantaneous stagnation line in each PIVsnapshot. The instantaneous stagnation line is, as its time-averaged counterpart, a flow separatrix, i.e., a line that di-

vides the flow in two regions, the top nozzle flow region, andthe bottom nozzle one. The approach implemented in thiswork exploits the concepts of flow topology and criticalpoints analysis. A critical point is defined as a point in theflow where the velocity is zero in an appropriate frame ofreference. The slope of the streamline, at the point, isindeterminate38 and the critical point can be classified by thenature of the eigenvalues of the local velocity gradienttensor.38,39 Critical points are classified in nodes, foci,saddles and bifurcation lines or asymptotic trajectories,38

which, if closed, are called limits cycles. While the full 3Dinformation is needed to describe the flow topology,38 PIVmeasurements only provide information in a 2D plane, which

FIG. 5. Vector representations of modes 1 �a� and 2 �b� for case TB. Snapshots of the vector field reconstructed with only modes 0 and 1 �c and d�. Snapshotsof the vector field reconstructed with only modes 0 and 2 �e and f�. TB case. Vector fields are undersampled for readability.

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could lead to misinterpretation.38,39 Nevertheless, as demon-strated by many authors,37–42 sectional streamlines can pro-vide useful insight into the flow structure. Details of thetracking algorithm to identify the stagnation line are in theAppendix.

As an example, we present an instantaneous snapshot inFig. 7. The stagnation line is represented by the thick grayline. The saddle points are represented by the black circlesand the nodes/foci �vortices� by the white circles. If one isable to find all the saddle points belonging to the stagnationline, then by deploying the associated sectional streamlinesin the stretching direction, it is possible to identify the stag-nation line. In reality, the stagnation line is also defined byother critical points such as nodes and/or foci �white circlesin Fig. 7�, representing sectional streamlines end points inthis specific circumstance �e.g., sinks� and their effect is toconvolve the stagnation line. These critical points can onlybe sinks �points where sectional streamlines end� since, bydefinition, if a source �point where sectional streamlines de-part from� were present, it would necessarily be surrounded�in our case� by two stagnation lines, as verified in what

follows. The algorithm was tested for a random sample ofsnapshots of different complexities by computing a largenumber of sectional streamlines with origin at the top and/orbottom edges of the PIV domain and verifying that the iden-tified streamline was indeed a flow separatrix. The algorithmperformed very well, producing the required stagnation linesin 783 out of 785 snapshots for the TB case, demonstratingits robustness. We also notice that the saddle points can beclassified in primary and secondary points: a primary point isdefined as a point whose sectional streamlines, deployedalong the eigendirection associated with the positive eigen-values, belong to the stagnation line and/or whose sectionalstreamlines, integrated backward in time along the directionassociated to the negative eigenvalue, end on opposite sides�top and bottom� of the flow field. A secondary point is anypoint not satisfying this criterion. Our algorithm shows thatthe stagnation lines, defined as the shortest path between twoextreme points, are also the lines joining the primary stagna-tion points.

The algorithm produced multiple paths �stagnation lines�in some instances, as shown in Fig. 8, which, at first sight,may seem unphysical. We verified all these “unusual” casesand found that they correspond typically to conditions withstrong three-dimensional effects, implying a flow conditionin which the flow is at large angles with respect to the PIVplane, and resulting in a number of nodes/foci, which dra-matically alter the flow in limited regions of the domain, asshown in Fig. 8. The two-dimensional cut through the flowdomain, resulting from the 2D PIV technique, prevents thissituation to be properly described. This type of situation wasverified in fewer than 15% of the 785 available snapshots forcase TB, that were excluded from the statistics.

The difficulty in accounting for these cases lies in ourlack of understanding of the impact of such flow conditionon the flow and, for burning situations, flame dynamics.Nevertheless, to verify that the discarded snapshots do notaffect the statistics, the average value �between the two in-tersection points at the two stagnation lines� of the quantity

FIG. 6. Phase plot of the POD coefficients, a1�t� and a2�t�, relative to mode1 �abscissa� and mode 2 �ordinate�, respectively. TB case.

FIG. 7. �Color online� Snapshot of the velocity field, represented by thevelocity vectors. Superimposed are the flow sectional streamlines originat-ing from the top nozzle and from the bottom nozzle �thin lines�. The thickgray line is stagnation line, the white circles are nodes/foci, and the blackcircles are stagnation points.

FIG. 8. �Color online� Instantaneous sectional streamlines superimposed tovector field in the presence of strong three-dimensional effects. Superim-posed are the flow sectional streamlines originating from the top nozzle andfrom the bottom nozzle �thin lines�. The thick gray line is stagnation line,the white circles are nodes/foci, and the black circles are stagnation points.

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of interest was also considered, and no noticeable changewas noticed in results. It should be pointed out that, in manyinstances, the fluid element “entering” the PIV domain mayleave from only one side, generating a more local stagnationline surrounding the region of flow injection and ejection.Our algorithm is capable of capturing all these situations byexploiting graph theory. In fact, it suffices to find the shortestpath using the starting and end points from the same side ora standard algorithm to find cycles. All these cases have notbeen removed from the statistics, since they produce a singleinstantaneous stagnation line and the presence of a 3D regionin the flow would not produce results different from whatcould be achieved with a standard technique, such as POD ornonconditional statistics.

In some respect, the capability of clearly identifying 3Deffects may be an important strength of the algorithm, as wenow have a simple way to characterize them and study theirimpact on the flame physics, without the need to resort to themuch more involved 3D-PIV. Of relevance are also caseswith vortices embedded in the stagnation line, resulting in aroll up of the stagnation line itself �Fig. 8�, which, along withthe three-dimensional effects, raises the general issue of howto define a highly convoluted stagnation line as flow separa-trix or flow limiting line.

Once the stagnation line is identified, it is possible totrack it, analyze its motion, and perform statistics relative tothe stagnation line. Figure 9 shows the histograms of theintersection of the instantaneous stagnation line and themean centerline for the same three cases: NN, TN, and TB.The width of the histogram grows progressively in the threecases, with a �0.5 mm excursion in the NN case, growingto about �5.0 and �6.5 mm in the TN and TB cases, re-spectively. Case TN exhibits also some skewness with re-spect to the symmetric TB case as a result of the asymmetryof the two burners and the penetration depth of the turbu-lence fluctuations in the opposite half of the flow field. Onemay draw the conclusion that the motion is triggered by aninstantaneous momentum imbalance, likely caused by the in-teraction of random large-scale eddies with the stagnationsurface. In reality, since the integral scale is on the order of a

few millimeters �see below�, while the radial extension of theplane is on the order of the nozzle diameter, i.e., �12.0 mm,an instantaneous momentum imbalance would just deformthe plane, possibly creating v-shaped stagnation lines.

One has to take into account the fact that, as discussedearlier, the opposed-jet configuration is a hydrodynamic un-stable configuration5,7,24,25 even at such high Reynolds num-bers and strain rates. Our experiments without coflow showthat under the same experimental conditions, but without tur-bulence generators, the stagnation surface gets steadily at-tached to either nozzle outlet, which happens also underlaminar conditions.5,7,24,25 Introducing the coflow brings thestagnation surface back at about the mid-distance betweenthe nozzles, and small oscillations can still be observed �Fig.9�. When one or both turbulence generators are in place, theamplitude of the oscillations increases dramatically �Fig. 9,see also Figs. 2�b� and 2�c��. This finding suggests that thecoflow brings some stability to the system by pushing radi-ally outward the recirculation bubbles, and thus “weakening”the cause of the instability. The main flow and coflow masssources are decoupled, but the radial jet generated by theopposed-jets impingement acts as a coupling element in thesystem. One can argue that the instantaneous perturbationscaused by the incoming eddies alter the direction of the ra-dial jet, which is temporarily shifted toward one or the othernozzle, forcing the coflow impingement surface away fromits neutral position. This effect is transient, producing a tem-porary oscillation of the stagnation surface away from the“stable” position, soon overcome by the stabilizing effect ofthe main flow and the coflow.

2. Conditional statistics coordinate system

We can now define a curvilinear coordinate system salong the instantaneous stagnation line �Fig. 10�a��, with ori-gin at the left of the domain. In the new coordinate system,the velocity at the stagnation surface has only a componentalong s, Vs, corresponding to the local tangent to the stream-line and zero component along the local normal. All the dataare then presented in terms of the components �u,v� of Vs asa function of the radial component of s, r, to facilitate com-parison with the nonconditional counterpart. In line with theconventional nonconditional approach, we measured thequantities of interest along the mean stagnation streamline�Fig. 10�a��, but placing the origin of the axial coordinate atthe intersection of such a streamline with the instantaneousstagnation line �Fig. 10�a��. The analysis is again “translated”in terms of axial coordinate z. With this choice of origin, theaxial distance z can no longer be seen as coincident with thegeometric midpoint between the nozzles. In fact, as the ori-gin moves with the instantaneous stagnation line �Fig.10�b��, it moves closer to either nozzle, and it is easy to seethat under the extreme case of the instantaneous stagnationline shifted very close to, say, the top nozzle, one can extractdata samples only from the opposite side of the line, goingtoward the bottom nozzle, for a distance equal to the nozzleseparation. Under this conditions, the newly defined axialcoordinate z can extend, in principle, from z=−19 mm toz=19 mm, producing an effective nozzle separation of

FIG. 9. Histograms of the intercept of the stagnation line with the centerlinefor the three cases: NN, no plate �plain line�, TN, plate in top burner �dashedline�, and TB plate on both sides �dotted line�.

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38 mm. The amplitude of the stagnation line motion dependson the presence of the turbulence generators �Figs. 2 and 9�,increasing from NN to TN to TB, implying that the effectivenozzle separation would be increasing accordingly. Themean stagnation streamline is almost parallel to the geomet-ric centerline, therefore, the projection of the velocity alongthis line is a close approximation to the axial velocity, and inthe ensuing discussion, we will make no distinction betweenthe two.

D. Comparison among the three data processing

We now make a direct comparison among three sets ofrms data: the data from the nonconditional statistics, thosefiltered using the POD technique by removing modes 0, 1,and 2, as per discussion of Fig. 5, and the data from theconditional statistics with respect to the stagnation line. Wewill start by revisiting the rms profiles along the mean stag-nation streamline that had been shown only for the noncon-ditional statistics in Fig. 3�b�. Mean stagnation streamline

and stagnation line should not be confused: the first on av-erage is roughly coincident with the burner axis and the sec-ond is the intersection of the stagnation plane with the lasersheet used in PIV.

Figure 11 shows the comparison of the rms axial com-ponent along the mean stagnation streamline �burner axis�.The removal of mode 1 and, to a much lesser extent mode 2,induces a dramatic drop in u�. The v� profile, on the otherhand, is not greatly affected, but shows a pronounced peakaround z�−0.9. It exhibits a short region of decay, beforeincreasing again at z� �5. This profile, up to the point ofmaximum before the dip, is in good agreement with theliterature.2 The conditional statistics offers a more extremepicture, to some extent, than the one showed by the POD,with a more pronounced dip in u� and peak in v�. The ve-locity rms along the mean stagnation line conforms to thepicture of stagnation flow against a plate.43–45 In fact, theaxial fluctuations u� tend to decrease approaching the z=0point �representative of the stagnation point�, while the radialfluctuations v� tend peak at the same point. Unlike the stag-nation point flow onto a flat surface, here the u� values donot fall to zero because the instantaneous stagnation line ishighly convoluted and intersects the mean stagnation stream-line at angles other than 90°, introducing nonzero u and ventries in the statistics. This finding may help explaining thedifference between POD-filtered data and conditional statis-tics. With the present choice of coordinate system for theconditional statistics, the flow can probably be described bya stagnation point flow over a continuously wobbling sur-face. This is further supported by the v� fluctuating compo-nent, which is in fairly good agreement with the noncondi-tional one, away from z=0. The large difference in the levels

FIG. 10. �Color online� Coordinate system for conditional statistics showing�a� the mean stagnation streamline and its intersection with the instanta-neous stagnation line �z=0� and �b� three instantaneous realizations of thestagnation lines, some velocity vectors Vs tangent to the stagnation line andtheir components along x and z, u and v, respectively.

FIG. 11. Comparison of the three data analyses. Axial �a� and radial �b�velocity fluctuations along the centerline �burner axis�: nonconditional sta-tistics �solid black�, POD �dotted gray�, and SLCS �dashed black�.

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of u� between conditional and POD data with respect to non-conditional data underline the large contribution of the large-scale/low-frequency oscillation of the stagnation line to thestatistics. The stagnation line instability is therefore respon-sible for masking the truly turbulent effects by artificiallyincreasing the turbulent fluctuations. The turbulence-inducedstagnation line convolution is responsible for the differencesbetween conditional and POD profiles. Also, moving awayfrom the center, z=0, one can see that the differences in u�levels tend to decrease, suggesting that the oscillation doesnot affect the entire domain.

Figure 12�a� shows the rms axial and radial velocity pro-files evaluated by the nonconditional statistics at a few axiallocations as a function of the radial coordinate. The presenceof the turbulence generator produces the desired effect ofboosting dramatically the turbulence levels as compared tovalues on the order of 0.2 m/s in the absence of the plates.The profiles show good radial homogeneity near the center-line. The radial uniformity for u� tends to deteriorate as thestagnation plane is approached, with the growth rate of theaxial fluctuations decreasing at the centerline as compared tolarger radial positions, probably as a consequence of a mix-ing layer instability. Whereas the v� are reasonably uniformregardless of the distance from the mean stagnation plane, u�increases as the stagnation plane is approached regardless ofradial location. Turning to the POD-filtered data �Fig. 12�b��,we note that subtracting mode 1 and mode 2 has a modesteffect on v� and fairly dramatic one on u�, that now becomesmuch more homogeneous throughout the flow field. Gener-ally, in view of the modest difference between the filtereddata of u� and v�, the level of isotropy of the field is alsogood. This is yet further proof that the POD filtering ap-proach manages to remove the artifacts of the opposed-jetinstability that artificially boost the velocity fluctuations be-yond those caused by turbulence. To avoid excessive clutter-ing of the plot, we show the comparison between noncondi-tional data and the SLCS ones in a separate graph �Fig.12�c��. The comparison is performed just along the stagna-tion line. When compared to the nonconditional counterpart,we see definitely lower absolute values of u�, n underliningthe artifact of the large-scale fluctuations and turbulent jetsinteraction �stagnation line convolution� on the velocity rmsestimates, and the effectiveness of also the SLCS approach toseparate them. Also in this case, the large-scale fluctuationshave a much more pronounced effect on u� rather than v�,suggesting that the fluctuations result mostly in the “bounc-ing” up and down of the stagnation line. This observationsuggests that the RANS models are unable to predict thecorrect fluctuations profiles in the opposed-jet configurationbecause of their intrinsic inability to model an unsteady field.Generally, the conditional statistics data are in reasonableagreement with the POD-filtered ones �Fig. 12�b�� with somediscrepancy in u� around r=0. This is because the z=0 originfor the conditional statistics is on the instantaneous stagna-tion line, therefore following it in its motion, while the z=0 origin in the POD statistics is fixed with respect to thelaboratory frame of reference and at each instant can be oneither side of the instantaneous stagnation line.

Turning our attention to the turbulent length scales, we

now follow the evolution of length scales in the strainingfield and relate them to the measured gradients. Integralscales can be estimated by integration of the two-point one-time correlation coefficients, Ri,j, by full-field approaches,such as 2D spectral analysis or 2D autocorrelationestimate,46 or by spatial filtering techniques,34 possiblycoupled to vortex detection schemes to estimate the vortexsize. The analysis of the two-point one-time correlation co-efficients is widely accepted and allows for comparison withthe literature. Therefore, it will be used for the three dataanalysis methodologies. The 2D spectral analysis or autocor-

a)

b)

c)

FIG. 12. Comparison of the three data analyses for the radial profiles ofaxial and radial velocity fluctuations. ��a� and �b�� Radial profiles at differentdistances from the mean stagnation line assumed as z=0. Circles: z=0.5 mm; squares: z=3.0 mm; triangles: z=7.0 mm. �a� Nonconditionalstatistics. �b� POD filtered statistics. �c� Conditional statistics �gray� vs non-conditional statistics �black� along the stagnation line. Circles: axial velocityfluctuation; triangles: radial velocity fluctuations.

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relation requires a region where the length scales involvedare statistically homogeneous, and the present flow field isnot the ideal target, in view of the axial inhomogeneity. Thetwo-point correlation coefficients were estimated along theaxial distance, starting from the edge of the PIV domaincloser to the nozzle exit section �z=−8� to the other side. Atevery point, integration was taken from the point of interestto the point where the coefficient decreases to a value of 0.1,toward the stagnation line. Although unusual, this value waschosen in place of the first zero crossing, to account for thelack of zero crossing near the stagnation line, possibly as aconsequence of the larger measurement error, and at thesame time to have an estimate comparable to the previousmeasurements.13

The longitudinal integral length scale and transversal in-ner length scale are identified respectively as Lu and �v

�Figs. 13�a� and 13�b��. Near the nozzle exit section, z�−8,the nonconditional velocity longitudinal integral length scale,Lu�5.0 mm, while the equivalent length scales estimatedby the POD filtered velocity and by the SLCS data analysisare both �3.0 mm. This suggests that the nonconditionallength scale is overestimated, since the two-point correlationis affected by the stagnation line large-scale/low-frequencyinstabilities, notwithstanding the fact that there is an inherentunderestimation in the calculation since the correlation coef-ficient does not drop to zero because of the limited domainsize.47 Furthermore, evidence in literature47 suggests that thePIV domain size to integral length scale ratio is an indicatorof length scale accuracy and a ratio in the range of 2–8 issuggested for accurate estimates, a criterion that is met in thepresent work, more so for the conditional and POD filteredestimates. Notice that the conditional integral scales do notchange significantly if estimated from the original or POD-filtered fields �not shown�, confirming the consistency of theconditional statistics. Plotting the integral scale along theburner centerline �Fig. 13�a�� through the mean stagnationpoint shows that the nonconditional Lu first increases to amaximum at z� �5 mm and then decreases approachingthe mean stagnation line at z=0. The initial increase may be

an artifact of the large-scale oscillations, as it is not presentin either the POD filtered or the conditional estimates. Thedecrease is in line with the idea that the large-scale structuresare deformed by the straining field, as they are convected bythe flow, in fair agreement with previous results.2 In reality adifferent picture appears from the analysis of the conditionalor POD filtered data, where there is a clear decay of thelongitudinal integral length scale approaching the stagnationline at z=0, and the decay is more pronounced in the case ofthe conditional length scale, reaching a value of �0.5 mm,than for the POD-filtered length scale, reaching a value of�1.5 mm. The transversal integral scale �not shown� de-creases from about 1.6 mm at z�−8 to about 1.1 mm atz�0, to a good approximation, independently of the datafiltering and conditioning and can be considered approxi-mately constant for practical purpose. The conditional orPOD filtered results suggest that structures elongated in theaxial direction downstream the nozzle exit section deforminto quasicircular structures at the stagnation line, as a con-sequence of the straining field, in agreement with the ob-served near stagnation line vortices �Figs. 7 and 8�. The dif-ferences between conditional and POD-filtered longitudinallength scale profiles are a consequence of the stagnation lineconvolution, unaccounted for in the POD-filtered statistics.

The inner or Taylor scale �Fig. 13�b�� is estimated fromthe velocity rms and the velocity gradients rms.48 The spatialderivatives are computed using the second order accurateRichardson scheme.30 The transverse Taylor scales are rela-tively constant and about 0.7–0.8 mm in all the cases.

The nonconditional estimates again show some differ-ences that can be associated with the large-scale fluctuationsaffecting the estimate and increasing considerably toward thestagnation line. The integral length scale behavior results in alarge overestimate of the associated turbulent Reynolds num-ber �Fig. 13�c�� for the nonconditional data near the nozzlemouths, at about 1100, versus a consistent estimate of about600 in the other cases. Furthermore, in agreement with lit-erature, the nonconditional statistics shows that the Re�Lu�increases toward the mean stagnation line, while both POD

FIG. 13. Comparison of the three data analyses for theaxial profiles for the nonconditional �plain line�, POD�dotted line�, and conditional �dashed line� estimates ofthe longitudinal integral scale �a�, transversal Taylorscale �b�, integral scale based Reynolds number �c�, andTaylor scale based Reynolds number �d�.

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filtered and conditional estimates show a more or less pro-nounced decrease, implying that the actual turbulent fieldexperienced by a flame near the stagnation line is not neces-sarily well described by estimates at the nozzle exit section.The Taylor scale Reynolds number �Fig. 13�d�� is insteadquite unaffected and on the order of 200 regardless of thedata analysis. As a result, it is a more robust indicator oflevel of turbulence as compared to the outer scale Reynoldsnumber.

Next, we consider the strain rate that, as discussed ear-lier, plays a critical role in combustion applications with re-spect to flame extinction. The axial velocity gradient in thez-direction and the radial velocity gradient in the r-direction,�u /�z and �v /�r, are plotted as a function of the radial co-ordinate and the axial one in Figs. 14 and 15, respectively. InFig. 14 the measurements are performed along the stagnationline, whereas in Fig. 15 they are performed along the stag-nation streamline. In both figures, the plots on the top rowresult from the nonconditional statistics, those on the centerrow from the POD-filtered approach and those at the bottomfrom the SCSL approach. Consistently with mass conserva-tion, the gradient of the axial velocity component, that iscoincident with the strain rate, is approximately twice theradial component gradient near the mean stagnation line. The

nonconditional mean gradient profiles �Fig. 14�a�� are rela-tively flat, but with a considerable spread that is probablyreflective of the large measurement uncertainty. Mode 1 andmode 2 contribute modestly to the straining field since thedata �Fig. 14�b�� are similar to those for the nonconditionalstatistics �Fig. 14�a��. We notice here that in these cases themeasurement uncertainty is of the same order as the varia-tions. If we now turn to the conditional statistics, using thecurvilinear coordinate system as previously defined, we canestimate the instantaneous velocity gradient experienced by afluid element at the stagnation line as �vs /�s, where vs is theinstantaneous velocity along s. Figure 14�c� shows the meanand rms values of such velocity gradient, providing a differ-ent picture from the conventional or POD statistics. Themean velocity gradient is higher at the center of the domainthan at its periphery, and the velocity gradient standard de-viation is again constant, suggesting that higher instanta-neous velocity gradients will be experienced closer to thecenterline, consistently with the fact that the extinction holesare found mostly in the center of the domain under burningconditions.13 The velocity gradient rms absolute values arealso much lower than the nonconditional ones �not shown�,reflecting the effect of turbulence structures interacting at orwith the stagnation line, purged of the large-scale fluctua-

FIG. 14. Profiles of axial and radial velocity gradients along the stagnationline. �a� Nonconditional statistics. �b� POD filtered statistics. �c� Conditionalmean and rms velocity gradient.

FIG. 15. Profiles of axial and radial velocity gradients along the stagnationstreamline. �a� Nonconditional statistics. �b� POD filtered statistics. �c� Con-ditional mean and rms velocity gradient.

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tions. In addition, the lower data scattering is probably in-dicative of the robustness of the data analysis.

Turning to the profiles of the same variables as a func-tion of the axial coordinate �Fig. 15�a��, we notice that thereis little difference between the nonconditional and POD-filtered profiles, and the bell shaped profiles along z peakaround the mean stagnation point, suggesting that large strainrates are more frequent when the stagnation line is aroundthat point. The conditional statistics allow us to estimate theinstantaneous velocity gradient along the stagnation stream-line, �u /�z, which is expected to be the same as the noncon-ditional one. In reality, the statistics are substantially differ-ent, since with the present choice of origin, the statisticalquantities are calculated with respect to the instantaneousstagnation line. The conditional mean and rms �Fig. 15�c��present a sharp peak at z=0, with values for the mean reach-ing almost 5000 s−1, quite different from the previous re-sults. The sharp peak and the very high value imply that thesampling is being performed in a region �around the stagna-tion line� characterized by higher velocity gradients than therest of the field. Overall, the strain/stretch rates peaks are atthe stagnation line �Fig. 14� and are centered on the meanstagnation streamline �Fig. 15�, suggesting that in this regionthere is a higher probability of extinction.

We can now estimate the ratio, �r /�e, between the meanresidence time �r and the eddy turnover time �e. The meanresidence time, �r�1 /Sb�1.0 ms, where Sb�dV /dr�1000 s−1 �at r ,z=0�, as we have seen, independently ofthe filtering procedure. The eddy turnover time, �e�L /k0.5

may depend on the filtering procedure, since the integrallength scale L and the turbulent kinetic energy k are greatlyaffected by the removal of the large-scale oscillations, asseen previously. Using the POD filtered data, we estimatedk0.5�3.4 m /s and L�2.8 mm, therefore �e�0.8 ms, andthe ratio �r /�e�1.2. We conclude that the present configura-tion has, to some extent, sidestepped the limitation of youngturbulence and, at the same time, that rapid distortion theoryarguments do not apply to scales on the order of the integrallength scale because the requirement �r /�e�1 is not satis-fied. Furthermore, in view of the millisecond residence timeand the compactness of the spatial domain, with the flowfield resulting in a disk with vertical dimensions of approxi-mately 1.9 cm and transverse dimensions of less than 10 cm,the system should be better-suited for fully resolved directnumerical simulation studies as compared to turbulent jets atcomparable turbulence level.

High-speed movies of particle-seeded flows revealedthat the large-scale low-frequency instabilities with the ensu-ing oscillation and precession of the gas stagnation surface�or line� occur with characteristic times on the order of tensof milliseconds or larger, that is, much longer than the eddyturnover time, regardless of whether such a time is computedfrom the filtered or unfiltered data. As a result, these insta-bilities are unquestionably distinct from turbulence and theirmasking effects need to be screened out to assess the turbu-lence features of these flows, by using the procedures out-lined in this article in the preceding sections.

It is interesting, at this point, to consider the effect of theturbulent flow field on the structure of the local instantaneous

stagnation points defining the instantaneous stagnation line.The statistics are now conditional to the local instantaneousstagnation point �Fig. 16�a��, i.e., every such point is local-ized in each PIV snapshot, and the velocity is measuredalong its characteristic directions, up to some distance �Fig.16�b��. All the collected velocity measurements are then sta-tistically analyzed. The stagnation points are of interest be-cause they are closely related to turbulence pair separationand scalar mixing,49 both relevant to combustion and, spe-cifically, to extinction/reignition problems. To this end, wedefine a local curvilinear coordinate system �Fig. 16�b�� withorigin on the instantaneous stagnation point and axes �s, n�,tangent, respectively, to the sectional streamlines related tothe eigendirection of the specific stagnation point. The veloc-ity along s and the velocity along n will be described interms of their Cartesian components u and v, while the ve-locity gradients are, respectively, dVs /ds and dVn /dn.

The relevant quantities are then time-averaged at fixedcurvilinear distance from the origin. This coordinate has no

FIG. 16. �Color online� Coordinate system for local conditional statistics.�a� One instantaneous realization of the stagnation line, two stagnationpoints, the stagnation point local sectional streamlines, and vortex. �b� Localstagnation line, local stagnation streamline, the stagnation point local sec-tional streamlines �thin lines�, stagnation point �black circle�, local coordi-nate system �s,n�, and respective direction vectors �black�.

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simple or intuitive relationship with the standard coordinatesystem �r, z�, although, statistically speaking, the s coordi-nate will roughly align with r and the n coordinate with z.The velocity rms levels reflect the presence of the turbulencegenerators �Fig. 17�. The zero value of the fluctuations isconsistent with the definition of critical points as points ofzero velocity. We also notice that under these conditions, thestagnation points present isotropic conditions around the ori-gin, up to a few millimeters, along s. The velocity gradientsalong s and n �Fig. 18� show sharp peaks in the mean andsharp dips in the rms at z=0. The mean velocity gradients�Fig. 18� are on the order of �5500 and �10 000 s−1, re-spectively, for dVs /ds and dVn /dn. Such high mean valuessuggest that the stagnation points are points where extinctionis likely to happen.

E. Relative merits of POD versus SLCS

The newly developed algorithm has some advantages ascompared to POD, as a very natural and intuitive approachbeing based on critical point analysis. In contrast to POD,which requires a number of fields to guarantee convergedstatistics, our approach works on instantaneous velocity fieldand is able to capture instantaneous events. An interesting,and possibly important, feature is the capability to identifyand screen 3D effects. The main drawback with respect toPOD is the higher computational cost. Furthermore, PODhas a wider range of applicability.

The ability to lock on the instantaneous stagnation sur-

face represents a highly valuable tool for studying turbulentcounterflow flames, as the stagnation surface is an approxi-mate indicator of the flame position in diffusion and partiallypremixed flames. Instantaneous strain rates, conditioned onthe position of the stagnation surface, give access to data,which could otherwise be obtained only by more costly anddifficult techniques, as by combining planar laser inducedfluorescence and PIV measurements. The new algorithm canalso be regarded as complementary to POD analysis, since itcan be applied to a POD-filtered field.

IV. CONCLUSIONS

A systematic experimental study was conducted onhighly strained, isothermal, opposed-jet flows under intenseturbulence using PIV. The bulk strain rate was kept at1250 s−1 and specially designed and properly positioned tur-bulence generation plates in the incoming streams boostedthe turbulence intensity to 22%, under conditions that aresuitable for flame stabilization. The turbulent flow field wasanalyzed by three different data reduction techniques: non-conditional, POD, and stagnation line conditional statistics�SLCS� analyses. The conditional analysis is the result of anovel approach relying on the identification and tracking ofthe instantaneous stagnation line �the two-dimensional coun-terpart of a stagnation surface in three dimensions�, with re-spect to which the statistical analysis is conditioned. It islabeled SLCS. The ability to lock on the instantaneous stag-nation line, at the intersection of the stagnation surface withthe PIV measurement plane, is particularly useful in the com-

FIG. 17. Axial and radial velocity rms profiles along the local instantaneousstagnation line �a� and local instantaneous stagnation streamline �b�.

FIG. 18. Velocity gradients along the local instantaneous stagnation line �a�and local instantaneous stagnation streamline �b�.

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bustion context, since the flame is aerodynamically stabilizedin the vicinity of the stagnation surface. Principal conclu-sions follow.

• The POD analysis shows the existence of two modesof oscillation, an axial oscillation, and a precessionmotion about the system axis of symmetry. High-speedmovies confirmed that the existence of large-scalelow-frequency instabilities are chaotic in nature. How-ever, they are distinct from turbulence, as revealed bya comparison of their characteristic time with the inte-gral turbulent timescale, the first being at least oneorder of magnitude larger than the latter. The ensuingmotion of the instantaneous stagnation line may maskthe truly turbulent fluctuations and needs to be prop-erly filtered.

• The SLCS algorithm identifies highly convoluted stag-nation lines and even roll up in the presence of vorti-ces. It can also identify and screen three-dimensionaleffects that may lead to more than one stagnation linein a two-dimensional velocity field.

• The Taylor scale Reynolds number remains relativelyconstant from nozzle to nozzle at a value of approxi-mately 200 and shows also good consistency amongthe different data reduction techniques, suggesting thatit is a more reliable measure of turbulence than theReynolds number based on the integral scale.

• The use of POD and SLCS revealed that the POD-filtered integral length scale decreases as one ap-proaches the stagnation line, as a consequence of thestraining field, with the corresponding turbulent Rey-nolds number following a similar trend.

• The SLCS suggests that the highest mean velocity gra-dients are around the system centerline, which resultsin a higher probability of flame extinction in that re-gion under chemically reacting conditions.

• Estimates of the mean residence time compared to thevortex turnover time yielded values greater than unitysuggest that the turbulence is reasonably well devel-oped in the present flows.

ACKNOWLEDGMENTS

We are indebted to Dr. Kailasnath Purushothaman fortechnical discussions, to Mr. Bruno Coriton for technical dis-cussions and for providing high-speed movies of the flow forour examination, and to Nick Bernardo for technical assis-tance in the construction of the hardware. The support ofDARPA under Grant No. DAAD19-01-1-0664 �Dr. RichardJ. Paur, Contract Monitor� is gratefully acknowledged.

APPENDIX: SLCS ALGORITHM

The SLCS algorithm to identify the instantaneous stag-nation line follows the following main steps.

• Finding critical points. The time-averaged flow field issubtracted from the instantaneous ones before identi-fying the zero velocity points. In our case, because ofthe nature of the flow, we could localize the critical

points by looking directly for the zero velocity condi-tion in each snapshot.

• Identifying type of critical points. At each of these

points we estimate the velocity gradient tensor �v� .The velocity tensor invariants �p, q, ��, as defined byZhou et al.,39 define the critical point topology: saddle�also stagnation� point for q�0; node for �0 andq0; focus for ��0 and p�0; center for p=0 andq0. Also, a node/focus is attracting if p�0 and re-pelling if p0.39

• Obtaining sectional streamlines end points. Criticalpoint eigenvalues and eigenvectors are estimated toidentify the saddle points’ principal directions. Wethen calculate the characteristic sectional streamlinesfor each stagnation point and store the correspondingend points.

• Building adjacency matrix. The stagnation line can beseen as the line starting at one edge of the PIV domain�e.g., left� and reaching the opposite edge �e.g., right�and connecting one or more stagnation points �andnodes/foci/limit cycles�. If one now computes the endpoints of the sectional streamlines associated with thepositive eigenvalues of each stagnation point and iden-tifies them by integer numbers, the system can be de-scribed as a graph,50 where the end points are thenodes and the sectional streamlines are the edges. Thegraph is represented by the adjacency matrix. At thispoint, the problem becomes that of finding the pathfrom two points on opposite ends of the PIV domain.To this end, we used the end point identifiers to buildthe adjacency matrix50 of the graph.

• Applying Dijkstra algorithm. To find the shortest pathbetween any two nodes, we used a Matlab routine51

based on the Dijkstra algorithm.50 All the sectionalstreamlines end points �associated with the stagnationpoints� belonging to the left and right edge of the PIVdomain were taken as starting and end points of thepaths, respectively.

• Identifying instantaneous stagnation line. The previ-ous step can return multiple paths between any twopoints �left and right�. We assumed the shortest path tobe representative of a possible stagnation line. The as-sumption is confirmed by the results. A stagnation flowby definition is characterized by at least one saddle/stagnation point. In a turbulent counterflow configura-tion, the stagnation line can be quite convoluted andmore than one saddle point can be defining it.

• Identifying three-dimensional flow regions. The previ-ous step can still return multiple stagnation lines. Thissituation requires a flow source within the PIV plane,which can be caused by strong three-dimensional ef-fects. To automatically deal with such situations, weexploited the fact that stagnation lines cannot intersecteach other, therefore, of all the stagnation line endpoints, we consider only the two points with the lowestand highest values of the axial coordinate. We then

105101-15 Experimental study of isothermal opposed-jet flows Phys. Fluids 22, 105101 �2010�

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impose that the starting point with high axial coordi-nate can only connect to the end point with the equiva-lent high coordinate, to avoid “crossing.”

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105101-16 G. Coppola and A. Gomez Phys. Fluids 22, 105101 �2010�

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