identification and analysis of coherent structures in the near field of a

PHYSICS OF FLUIDS 18, 055103 �2006�

Identification and analysis of coherent structures in the near fieldof a turbulent unconfined annular swirling jet using large eddy simulation

M. García-Villalba, J. Fröhlich, and W. RodiSFB606, University of Karlsruhe, Kaiserstrasse 12, 76128 Karlsruhe, Germany

�Received 20 July 2005; accepted 22 March 2006; published online 18 May 2006�

Large eddy simulations of incompressible turbulent flow in an unconfined annular swirling jet atReynolds number 81500 are reported, based on the outer radius of the jet. The results are inexcellent agreement with experimental data for mean flow, turbulent statistics, and power spectraldensities of velocity fluctuations. Two dominant families of large-scale coherent structures areidentified in the flow. Both are orthogonal to the mean three-dimensional streamlines, whichsuggests that they are formed as the result of a Kelvin-Helmholtz instability. Instantaneous vortexstructures as well as different types of spectra and two-point correlations are presented to furtherelucidate the properties of the flow. © 2006 American Institute of Physics.�DOI: 10.1063/1.2202648�

I. INTRODUCTION

Swirling flows are widely used in many industrial appli-cations, such as cyclone separators, heat exchangers, or jetpumps. One of the most important industrial applications ofswirling flows is their use in combustion devices. Here, theyserve to stabilize the flame near the burner exit through arecirculation zone generated by the imposed swirl. This zonecan appear far from any boundary which is advantageous asthe walls are then remote from the flame, thus reducing theirheat load �Gupta et al., Ref. 1, Chap. 1�.

Recirculation in a swirling flow generally is the result ofso-called vortex breakdown which occurs when, with in-creasing swirl, the pressure on the axis decreases such thatthe regular spiraling motion becomes unstable and developsa steady or unsteady stagnation point on or near the axis.2

This subject has attracted considerable attention over manyyears. A comprehensive review is given by Lucca-Negro andO’Doherty,3 where different regimes are described and theinfluence of the various parameters is discussed. Numerousexperimental and numerical investigations have been pub-lished on this issue, but mostly for the laminar case.4–7 Re-cently, stability analyses have been conducted for such casesby Gallaire and Chomaz,8 but again for the laminar case.Despite all these efforts no conclusive explanation for theonset of vortex breakdown is currently available.3,8

Combustor flows are characterized by high Reynoldsnumbers and broadband turbulent fluctuations. The extensionof laminar studies of vortex breakdown to turbulent swirlingflows is a delicate issue as illustrated by the “conical” vortexbreakdown observed by Sarpkaya.9 Paschereit et al.10 haveattempted a linear stability analysis in the turbulent casestarting from a given experimentally observed average turbu-lent flow field. Experimental investigations of turbulentswirling flows have been performed most often for pipe flowwith a sudden expansion.11–14 The region of interest, thedump after the expansion, hence is confined and for some ofthe cited configurations the expansion rate is relatively small.
Experiments with turbulent swirling jets, i.e., expansions
1070-6631/2006/18�5�/055103/17/$23.00 18, 05510

Downloaded 07 Jun 2006 to 129.13.222.108. Redistribution subject to

with large or infinite aspect ratio, are more scarce.15 Corre-sponding simulations are more delicate due to the larger do-main and the issue of defining far-field boundary conditions.

Turbulent swirling flows, in particular for high swirlnumbers, typically feature pronounced coherentstructures.16,17 These are influenced by several effects likeinternal velocity gradients in form of shear layers and swirl.The latter enhances the occurrence of azimuthal instabilitieswhile nonswirling jets mostly exhibit axisymmetric coherentstructures.18 The most prominent coherent structures inswirling flows are the “precessing vortex cores” �PVC� ori-ented at a low angle with respect to the axis of rotation�Gupta et al., Ref. 1, p. 191�. For low swirl numbers coherentstructures are relatively weak,19 while with higher swirlnumbers their intensity increases substantially �Gupta et al.,Ref. 1, Chap. 4�. For applications, coherent structures playan important role as they influence mixing of heat and spe-cies to a large extent and hence the entire reaction process inswirl burners.18,20

Large eddy simulation �LES� is a particularly suitableapproach to investigate the generation and evolution of co-herent structures in turbulent swirling flows. It allows treat-ment of high Reynolds number flows and explicit computa-tion of these structures, and has only limited sensitivity tomodeling assumptions. In the present paper, LES is appliedto study unconfined swirling flows generated by an annularjet as described below. The use of LES for swirling flows isrelatively recent, presumably caused by the requirement ofspecifying unsteady turbulent inflow conditions for stream-wise nonperiodic flows. The first LES pertinent to the presentconfiguration was performed by Akselvoll and Moin21 for anonswirling confined coannular jet. Pierce and Moin22 sub-sequently accomplished a corresponding LES with swirl andsimulated the experiments of Roback and Johnson11 andSommerfeld and Qiu.12 Schlüter23 performed LES of com-bustor flows using a compressible formulation in the con-stant density regime and observed PVC without, however, a
corresponding pronounced peak in the computed spectra.
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http://dx.doi.org/10.1063/1.2202648

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055103-2 García-Villalba, Fröhlich, and Rodi Phys. Fluids 18, 055103 �2006�

McIlwain and Pollard24 used LES to compute a turbulent jetwith a low swirl number. The swirl was found to increase thenumber of streamwise braids and therefore it enhanced thebreakdown mechanisms of the vortex rings. Apte et al.25

simulated the confined configuration of Sommerfeld andQiu12 and included the transport of Lagrangian particles. Theresults for the fluid phase are in good agreement with theexperiment but no detailed analysis of coherent structureswas performed. Wegner et al.26,27 have performed LES andunsteady Reynolds averaged Navier-Stokes �URANS� simu-lations of an unconfined swirling flow and obtained spiralingvortex structures. The agreement with the corresponding ex-perimental data for mean flow and fluctuation, however,seems not entirely satisfactory. Lu et al.28 performed LES ofa turbulent round jet issuing into a dump combustor. Vortexstructures were however only little addressed but analyzed interms of their interaction with acoustic modes of the com-bustor. Wang et al.29 investigated the unsteady flow evolutionof a swirl injector. The configuration was very complex in-volving three radial swirlers, with the flow in one of themcounter-rotating with respect to the others. Two swirl num-bers were investigated and it was found that for the higherone the flow structures became much more complicated.

LES for reactive swirling flows was performed by Pierceand Moin.22 Menon and co-workers30,31 performed severalsuch simulations. Sankaran and Menon31 observed a strongunsteady vortex core in the cold flow and quantified the im-pact of combustion by simulating reactive and nonreactiveflow in the same configuration. Huang et al.32 investigatedthe interaction between turbulent flow motions and oscilla-tory combustion of a swirl-stabilized combustor. The flowfield exhibited a very complex structure, including thebubble and spiral modes of vortex breakdown and a PVC.

The present paper is concerned with LES of constant-density flow of a free annular jet at high swirl number andhigh Reynolds number. The first purpose is to demonstrateby comparison to a companion experiment that the simula-tion technique can simulate flows of this kind with high qual-ity. The second and main intention then is to gain under-standing of the complex dynamics of the coherent structuresin the flow studied, to provide quantitative results on statis-tics and to address possible mechanisms of instability.

In Secs. II and III the computational setup is described,together with its relation to the companion experiment. Sec-tion IV presents results for time-averaged quantities and vali-dation of the LES technique by means of comparison withthe measurements. Section V focuses on the analysis of co-herent structures and is supplemented by the spectral analy-ses in Sec. VI, while Sec.VII summarizes and concludes thepaper.

II. EXPERIMENTAL CONFIGURATION

The configuration considered in the present paper wasinvestigated experimentally by Hillemanns.33 New measure-ments for the same configuration were performed recently byBüchner and Petsch.34 Both sets of measurements were ob-tained using laser Doppler anemometry �LDA�. The configu-
ration consists of an annular swirling jet issuing into an am-

bient of the same fluid which is at rest. The swirl is generatedwith a movable block swirl generator,35 in which flow is fedradially into the swirl-generating device and radial and tan-gential vane angles are altered to adjust the desired level ofswirl. The reported measurements are confined to the nearfield of the jet. They include first- and second-order momentsof the velocity at several axial stations downstream of the jetexit. The second set of experiments34 also includes powerspectral densities of velocity fluctuations at some pointsclose to the nozzle. The experiments do not provide anymeasurements of the flow in the region upstream of the jetoutlet.

III. COMPUTATIONAL SETUP

A. Equations, subgrid-scale model, and discretization

The governing equations are the filtered incompressible

Navier-Stokes equations for the resolved velocity ui and theresolved pressure divided by the density, which is denoted p̄.These equations read

�ui

�xi= 0, �1�

�ui

�t+

�ui¯ uj

¯

�xj= −

�p̄

�xi+

��2�Sij��xj

−��ij

�xj, �2�

where � is the molecular viscosity and Sij =12 ��ui /�xj

+�uj /�xi� the filtered strain-rate tensor. The term �ij =uiuj

−ui¯ uj

¯ results from the unresolved subgrid-scale contributionsand needs to be modeled by a subgrid-scale �SGS� model.

The dynamic Smagorinsky subgrid-scale model first pro-posed by Germano et al.,36 with the modification of Lilly,37

is used in the present investigation. It models the anisotropicpart of the SGS term via

�ij −1

3�ij�kk = − 2�tSij , �3�

while the trace �kk is lumped into a modified pressure. Theeddy viscosity is given by

�t = C�2�S̄�, �S̄� = �2Sij¯ Sij

¯ �1/2, �4�

with �= ��x�y�z�1/3. In the dynamic model, the model pa-rameter C is determined using an explicit box filter of widthequal to twice the mesh size. The eddy-viscosity �t is clippedto avoid negative values and smoothed by temporalrelaxation,38

�tn+1 = ��t

* + �1 − ��tn, �5�

with the relaxation factor �=5·10−4 and �t* the value deter-

mined by the original model.In the sequel, overbars denoting the resolved quantities

will be dropped for clarity. For the same reason Cartesianvelocity components are converted to cylindrical ones, usingthe notation ux, ur, and u�, for the axial, radial, and tangentialvelocity component, respectively. The simulations were per-formed with the in-house code LESOCC2. This is a successor
of the code LESOCC �Ref. 38� and is described in its most

055103-3 Identification and analysis of coherent structures Phys. Fluids 18, 055103 �2006�

recent status by Hinterberger.39 The code solves Eqs. �1� and�2� on body-fitted, curvilinear grids using a cell-centered fi-nite volume method with collocated storage for the Cartesianvelocity components. Second-order central differences areemployed for the convection as well as for the diffusiveterms. A fractional step method is used with a Runge-Kuttapredictor and the solution of a pressure-correction equationin the final step as a corrector.40 The Rhie and Chow momen-tum interpolation41 is applied to avoid pressure-velocity de-coupling. The Poisson equation for the pressure increment issolved iteratively by means of the “strongly implicitprocedure.”42 Parallelization is implemented via domain de-composition, and explicit message passing is used with twohalo cells along the interdomain boundaries for intermediatestorage.

B. Computational domain, boundary conditions,and grid

The geometry of the computational domain is shown inFig. 1. It features an annular jet issuing into a large cylindri-cal domain with bulk velocity Ub. The reference length isR=D /2, where D is the outer diameter of the annular jet. Theinner radius is 0.5R. The computational domain also includesa crude representation of the inlet duct upstream of the jetexit. No-slip boundary conditions are applied at the walls. Inpreliminary studies, alternative modeling using wall func-tions was employed in simulations of an annular jet issuinginto a confined domain. The effect of the type of boundaryconditions on the flow was found to be minor. At the exitboundary, a convective outflow condition is used, reading43

�ui

�t+ Uconv

�ui

�n= 0, �6�

with n the outward normal coordinate and Uconv a convectivevelocity, in this case the coflow velocity. Free-slip conditionsare applied at the lateral boundary, which is placed far awayfrom the region of interest at 12R. Note that in the figure thelateral and the downstream boundaries are not positioned toscale. The fluid to be entrained is fed in by a mild coflowingstream of 5% of the bulk velocity in the plane x=0. Thecoflow was not present in the experiment. It is usually intro-

44,45
duced in the simulation of free jets to avoid the forma-

tion of a large, unphysical recirculation zone in the outerregion. In order to address the influence of the coflow, twoextra simulations on a coarser grid have been performed withcoflow velocities of 2% and 10% of Ub. In the near field ofthe jet outlet, no major differences were observed with thesechanges. This approach is further justified to some extent bythe experimental configuration in which the annular jet exitsfrom a pipe and not from an opening in a wall.

The specification of inlet boundary conditions is nottrivial in the present case. It was found in previoussimulations46 that the prescription of inflow conditions rightat the nozzle outlet substantially impacts on the result.Hence, the inlet of the computational domain has to be lo-cated far enough upstream of this point, but no experimentalinformation is available there. On this background the fol-lowing strategy was applied. The flow is prescribed at thecircumferential inflow boundary located at the beginning ofthe inlet duct. The flow comes in radially having an azi-muthal component. At this position steady top-hat profilesfor the radial and azimuthal velocity components are im-posed. This is not an ideal approach because for LES un-steady inflow boundary conditions are usually required. Itwill be demonstrated, however, that this procedure does notaffect the results because turbulence readily develops in theduct upstream of the jet exit. Figure 2 shows the develop-ment of the flow in the inlet duct by means of the instanta-neous axial velocity and the instantaneous turbulent kineticenergy. The inlet swirl was adjusted such that the computedswirl number at the jet outlet matches the experimentalvalue. This procedure, in spite of being a strong idealization,yields the correct statistics and coherent structures in the in-terior of the domain46 without the costly representation of theswirl-generating device.

Various views of the block-structured grid employed areshown in Fig. 3. The grid was generated with the commercialsoftware ICEM-CFD Hexa and comprises about 6 million hexa-hedral cells. It is stretched in both the axial and the radialdirection to allow for concentration of points close to the jetexit and the inlet duct walls. In the azimuthal direction 160grid points are used. The stretching factor is everywhere lessthan 5%. The minimum axial spacing is located at the jet

FIG. 1. Geometry of the computa-tional domain and applied boundaryconditions.

outlet and is �x=0.02R. In the vicinity of the walls, the



minimum radial spacing is �r=0.008R. As shown in Fig.3�d�, the region near the symmetry axis consists of a qua-sisquare mesh, which eliminates the centerline singularitypresent in grids which use cylindrical coordinates.

C. Parameters

The following reference quantities are used throughoutthe paper: the outer radius of the annulus R=50 mm forlengths; the bulk velocity Ub=25.5 m/s for velocities; andtb=R /Ub for times. The Reynolds number of the flow basedon the bulk velocity Ub and the outer radius of the jet isRe=81 500. The swirl number is defined as

FIG. 2. Development of the flow in the inlet duct. �a� Instant

FIG. 3. Computational grid. �a� Symmetry plane, zoom near the jet exit. Eve
with all grid lines shown. �c� Same as �b� but zoomed around the symmetry axis

S =

�0

R

�uxu�r2dr

R�0

R

�ux2rdr

. �7�

In the present calculation its value is S=1.2 at x /R=−2 andS=0.9 at x /R=0. This decay in the swirl number is causedby a decay of the tangential velocity in the inlet duct and asubstantial change in the shape of the profiles of both theaxial and the tangential velocity components. It has beenobserved also by Sankaran and Menon.31 Since S represents

s axial velocity. �b� Instantaneous fluctuating kinetic energy.

urth grid line is shown. �b� Cut in the y−z plane covering the entire domain

aneou

ry fo
. �d� Strong zoom around the axis.


an integrated quantity, two swirling jets of completely differ-ent velocity distributions may have the same swirl number.Although the swirl number hence only provides an incom-plete description of the properties of a swirling jet, it is com-monly used in the literature and therefore also employed inthe present text.

IV. TIME-AVERAGED RESULTS

A. Averaging procedure

The simulation was run for several time units tb to elimi-nate the effects of the initial conditions. After this period,statistical quantities were collected for 150 tb, which is longenough to obtain converged values in the near field of the jetexit. The averaging was performed in time and also along theazimuthal direction. Average quantities are denoted by angu-lar brackets and corresponding fluctuations by a doubleprime. Far downstream of the jet exit, for x /R�6, the sta-tistical quantities are not fully converged in the vicinity ofthe symmetry axis because the motions are slower in thatregion and the impact of azimuthal averaging is low near theaxis. Substantially longer averaging times would be neces-sary to improve in this respect. However, the focus of thispaper is the near field of the jet outlet extending roughly upto x /R=3. In that region the averaging period is sufficient.This is supported not only by the smoothness of the resultspresented below and their good agreement with the experi-mental data, but also by the following reasoning. At x /R=3the characteristic tangential velocity is 0.5Ub and can be

FIG. 4. Average streamlines in the symmetry plane obtained from thesimulation.

FIG. 5. Vector plots in the symmetry plane. �a� Average flow. �b� In


found at r /R=1. Hence, an averaging period of 150 timeunits yields an average over 12 revolutions of the flow, fur-ther supplemented by azimuthal averaging.

B. Regions of the flow

First, a general picture is presented by a brief descriptionof the different regions of the flow.

Figure 4 shows the computed time-averaged two-dimensional streamlines. A long recirculation zone is ob-served in the central region. The length of this zone is about6R, and its maximum diameter 1.4R. Note that as the jet isannular, a recirculation zone is also generated without swirlbecause the cylindrical center body acts as a bluff body.47 Inthis reference it has been shown that in the case without swirlthe recirculation is very short for the present geometry, itslength being roughly R, and the jet recovers very quickly. Along recirculation zone is typical for flows with a high levelof swirl �Gupta et al., Ref. 1, Chap. 4�. This phenomenon isrelated to the presence of a low-pressure region on the sym-metry axis of the flow. Further information on the averageflow is provided by the plot of velocity vectors in Fig. 5�a�.It shows that the jet has an outward radial component due tothe centrifugal forces and widens substantially further down-stream.

The jet produces two shear layers, an inner one on itsborder with the recirculating flow and an outer one on itsborder with the surrounding coflow. This can be seen betterin Fig. 6�a�, which shows the mean axial velocity profile atx /R=0.2. The shear layers underlay substantial curvature ef-fects due to the swirl. It is worth noting that the two shearlayers formed by the jet are spatially very close to each other.The inner one is wider than the outer one and exhibits alarger change in velocity due to the negative velocity in therecirculation zone. This issue is addressed further in Fig. 11below.

In the outer region, the jet entrains fluid from the ambi-ent in a very intermittent manner, which is visible in the plotof instantaneous velocity vectors of Fig. 5�b�. This is a com-plex phenomenon, which is addressed further below. Thevanishing slope of the streamlines remote from the jet atx /R=0 and rR in Fig. 4 is due to the coflow boundarycondition. The velocity at this position, however, is only 5%

stantaneous flow. �Values are interpolated to a Cartesian grid.�



of the jet axial velocity, so that the influence on the region ofinterest is negligible. The low velocity in the outer part caneasily be appreciated in Figs. 5 and 6.

The swirling jet spreads radially outwards much fasterthan in the case of a nonswirling jet.21 In the far field down-stream of the recirculation zone, the fluctuating motions aremuch slower than in the near field. As stated above, thepresent integration time is not long enough to study this re-gion, which deserves further investigation.

C. Profiles of mean velocity and fluctuations:Comparison with experimental data

Figure 6 shows mean velocity profiles at four axial mea-surement stations in the near field of the jet, ranging fromx /R=0.2 to x /R=3. Measurements were performed along a

FIG. 6. Radial profiles of mean velocity components at x /R=0.2, 1.0, 2.0, anopen symbols, experimental data from Ref. 34; closed symbols, experimen�u�� /Ub. Right, radial velocity, �ur� /Ub. To enhance readability the range of

radial line on both sides of the axis. All velocity data are


plotted versus the radial coordinate so that measurementpoints on opposite sides of the symmetry axis appear to-gether. This provides an estimation of the experimental un-certainty. The overall agreement between experiment andsimulation is very good at most stations. Particularly good isthe agreement at x /R=0.2, Figs. 6�a�–6�c�. This suggests thatthe experimental inflow conditions are modeled properly.The available experimental data are confined to the rangebetween x /R=0.2 and x /R=3. Therefore, the length of therecirculation zone cannot be compared, but the mean axialvelocity profiles in Fig. 6 show that until x=3R the shape ofthe recirculation zone is well predicted. The spreading of thejet is also in good agreement with the experimental data. Theradial velocity component was the most difficult componentto measure in the experiment,34 which is reflected by the

�top to bottom as indicated in the right-most column�. Solid line, simulation;ata from Ref. 33. Left, axial velocity, �ux� /Ub. Center, tangential velocity,vertical axis has been adjusted individually.

d 3.0tal dthe

scatter in these data. The agreement close to the jet outlet is



quite good, as seen in Figs. 6�c� and 6�f�, whereas a fairlysignificant difference is observed at x /R=2 and x /R=3, Figs.6�i� and 6�l�. In order to investigate this issue, the continuityequation for the averaged flow

��ux��x

+1

r

��r�ur��r

= 0, �8�

can be used. Defining q1=−r��ux� /�x and q2=��r�ur�� /�r, itreads q1=q2. These terms are estimated from the experimen-tal data. At x=2R, q1 was estimated using central differencesand the experimental profiles of �ux� in Figs. 6�d� and 6�j�.The profile of �ur� in Fig. 6�i� was used to estimate q2. Figure7 shows the profiles of q1 and q2 at x=2R. It is clear thatthese data do not fulfill continuity. In this figure, q1 obtainedfrom the simulation is also shown and the agreement withthe experiment is remarkable, showing that the measurementof the radial velocity component at this location contains asystematic error.

Figure 8 shows the turbulent fluctuations of the threevelocity components �only the resolved part for the LES�.The qualitative agreement between experimental and compu-tational results is good for all three components. As with themean velocities, the agreement is better for the axial andazimuthal intensities than for the radial component. Althoughthe qualitative agreement is good, the calculation tends toslightly underpredict the turbulent intensities. As the eddyviscosity in the present formulation of the dynamic model isclipped to avoid negative values, backscatter is not ac-counted for, and this results in an excess of dissipation, es-pecially in the region of the jet where the fluid is entrained. Itis well known that eddy-viscosity models are deficient insuch a region because the entrainment is a very intermittentphenomenon. It should be stressed, as in the case of the meanvelocities, that the agreement at x=0.2R is very good, Figs.8�a�–8�c�. Hence, in spite of the strong idealization appliedto the inlet geometry, avoiding the swirl-generating device,and the specification of steady laminar inflow conditions atthe entry of the inlet duct, this procedure yields the correctturbulent fluctuations at the inlet into the main domain, i.e.,

FIG. 7. Radial distribution of the terms in the continuity equation for theaveraged flow at x=2R. Solid line q1=−r��ux� /�x estimated from the ex-periment. Dashed line q1 from the simulation. Symbols q2=��r�ur�� /�r es-timated from the experiment.

at the jet exit. The fluctuations induced in the shear layers


can be observed in Figs. 8�a�–8�c�. The inner layer generatessubstantial fluctuations between r /R0.5 and r /R0.9,while in the central region of the domain the level ofturbulence is much smaller. The outer layer is much thinnerthan the inner one, but there is also a peak of turbulent in-tensity associated with it around r /R1.1. To characterizethe amount of subgrid-scale modeling, note that at x=0.2Rthe maximum eddy viscosity is found in the outer shear layerat r /R1.1, and assumes a value of �t /�45. Similar val-ues of this quantity are observed in the whole near field ofthe jet.

V. INSTANTANEOUS FLOW

A. Dominant structures

In this section the instantaneous flow is discussed withspecial emphasis on coherent structures. Figure 5, alreadydiscussed above, shows velocity vectors in the symmetryplane of the time-averaged flow and a snapshot of the instan-taneous flow. The comparison of these two plots gives animpression of the high level of turbulence, especially in thenear field of the jet.

It is well known that coherent structures are associatedwith local minima of the pressure field.48,49 Previousstudies50 have shown that the pressure fluctuation is moresuitable for the visualization of coherent structures than thecommonly used instantaneous pressure. Isosurfaces of thelatter are influenced by the spatially variable average pres-sure field which is unrelated to instantaneous structures. Thisis avoided when �p� is subtracted from the instantaneousvalue. Figure 9�a� shows an isosurface of the instantaneouspressure fluctuation p�= p̄− �p̄�. In order to facilitate the in-terpretation, additional smoothing was applied to p� in post-processing. This was achieved through two consecutive ap-plications of a three-dimensional box filter of size 2�, whichis in fact the test filter used in the dynamic procedure. Figure9�b� displays the smoothed field and shows that the filteringprocedure does not affect the large-scale coherent structures,which validates this procedure for the present case.

Two families of structures are visible in Fig. 9. In theinner region of the jet, elongated helical vortices are ob-served �labeled I� while those in the outer region �labeled O�are also helical but oriented at a substantially larger anglewith respect to the x axis. In this snapshot two structures ofeach type coexist. One of the inner vortices �labeled I1� ismore pronounced than the other �labeled I2�. In animationsof the flow it has been observed that up to three of thesevortices can coexist at certain instants in time �see Fig. 13below�. Most of the time, however, a single vortex is domi-nant as in Fig. 9�b�. Both families of vortices are rapidlydamped further downstream. At x /R2 they are alreadyvery weak. It is worth noting that the inner structures extendconsiderably upstream of the jet outlet into the annular pipe.The prescription of inlet conditions at the position of the jetexit such as fully developed turbulent annular pipe flow istherefore not feasible for the present case. Rather, the inflowconditions need to be specified substantially upstream of the

46
exit.


In order to show the evolution of the vortices in time,Fig. 10 displays the same isosurface as Fig. 9�b� at threedifferent instants. For clarity, views from two differentangles are provided. The structures rotate around the symme-try axis at a constant rate, clockwise when looking upstream.The pictures in the figure cover an interval of roughly half arotation period in time.

The inner and outer vortices in Figs. 9 and 10 are veryclose to each other and it is difficult to discern if they areactually two structures or just one complex vortex. As dis-cussed above, the annular jet creates two shear layers, one onits border with the recirculation zone and the other one on itsborder with the surrounding ambient fluid, and those are bothprone to the Kelvin-Helmholtz instability. Figure 6�a� showsthe mean axial velocity at x /R=0.2. At this axial station, the

FIG. 8. Radial profiles of rms velocity fluctuations at x /R=0.2, 1.0, 2.0, andopen symbols, experimental data from Ref. 34; closed symbols, experimentalu�

rms/Ub. Right, radial component, urrms/Ub.

inner shear layer extends from r /R0.4 to r /R1 and the


outer one from r /R1 to r /R1.2. Both shear layers canbe distinguished by the radial derivative of the mean axialvelocity, i.e., ��ux� /�r. In the inner shear layer this quantity ispositive and in the outer one it is negative. Therefore, theisosurface in Fig. 10 has been colored with the sign of��ux� /�r. The fact that the inner and outer structures are soclose is due to the small separation between the shear layers.To support this interpretation, an extra simulation was per-formed with a swirl number of S=0.85 at x /R=−2, i.e., 30%below the value used for the main simulation discussed here.In the case with lower S, the shear layers are more separatedthan in the original case. This can be seen in Fig. 11, whichshows the mean axial velocity profile at x /R=0.2 for bothsimulations, together with a rough estimation of the widths

top to bottom as indicated in the right-most column�. Solid line, simulation;from Ref. 33. Left, axial component, ux

rms/Ub. Center, tangential component,
3.0 �data
of the shear layers and the radial separation between the


ld co


inner and the outer one for each case. The two kinds ofvortex structures should therefore be further apart. This isindeed observed in Fig. 12, which shows an instantaneoussnapshot of an isosurface of pressure fluctuations for the casewith S=0.85.

B. Precessing vortex cores

In the literature on the subject, the inner structures areknown as precessing vortex cores �PVC�.1,17 The motion ofthe PVC can be decomposed into two components. Theswirling motion of the main flow rotates the vortex corearound the symmetry axis at the same time the vortex spinsaround its own axis. An idealized sketch of this motion isshown in Fig. 13�d�. As the vortex is oriented preferentially

FIG. 9. Isosurface of pressure fluctuations, p− �p�=−0.3. �a� Original fie

FIG. 10. Isosurface of filtered pressure fluctuations, �p− �p��filt=−0.3 atthree instants in time. The color is given by: ��ux� /�r�0, bright;��ux� /�r�0, dark. Swirl is clockwise when looking upstream in axial
direction.

in the streamwise direction for this level of swirl �see Fig.10�d��, its presence generates radial and tangential velocityfluctuations. In addition to this motion a high level of axialvelocity fluctuations is observed in the interior of the PVC.This is visualized in Fig. 13, which shows contour plots ofaxial velocity fluctuations and pressure fluctuations close tothe outlet at x=0.2R. This plane was chosen since plots likethose in Fig. 10 show that the inner structures are strongestin this part of the domain. The PVC can be identified as theregion of low-pressure fluctuations in Fig. 13�b�, which ismarked with an arrow. At the same position, a region ofpositive axial velocity fluctuations can be seen in Fig. 13�a�,while a large region of negative fluctuations is also presentwith a phase difference of approximately 180°. Figure 14shows time signals of pressure and axial velocity close to thejet exit and Fig. 15 their corresponding cross correlation. Inthe signals the minima of the pressure represent the quasip-eriodic passing of the PVC through the point in which thesignal has been recorded. As Fig. 13 already suggests, thereis a strong correlation between the minima of the pressure

mputed in the simulation. �b� Filtered field employed for visualization.

FIG. 11. Mean axial velocity component �ux� /Ub at x /R=0.2. Dashed line,original simulation. Solid line, simulation with a lower level of swirl, S=0.85. The arrows indicate an estimation of the width of the inner and theouter shear layers for both simulations. The patches show the radial separa-tion between the inner and the outer shear layer. Light patch, original simu-
lation. Dark patch, simulation with lower swirl.


FIG. 12. Coherent structures from a simulation with a lower level of swirl, S=0.85 at x /R=−2, visualized by instantaneous pressure fluctuations �un-smoothed�, colored as in Fig. 10. �a� Side view. �b� View from downstream.

FIG. 13. �a�–�c� Contour plots in a transverse plane at x /R=0.2 and the same instant in time. The straight arrows point at the dominant PVC observed at thisinstant. The curved arrows indicate the sense of rotation. �a� Instantaneous axial velocity fluctuations. �b� Instantaneous pressure fluctuations. �c� Instantaneousaxial velocity component. The white line is a pressure fluctuation contour p− �p�=−0.7 extracted from the data shown in �b�. The black line is the boundary
of the recirculation zone ux=0. �d� Idealized sketch of the PVC motion in the transverse plane.
Downloaded 07 Jun 2006 to 129.13.222.108. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp


and the maxima of the axial velocity. Observe the large am-plitude of the velocity signal and the occurrence of negativevalues. This issue will be discussed further below.

The PVC was detected experimentally some 30 yearsago,16 but the origin of these structures it is still not entirelyclear. In a series of experimental studies,51–53 a possiblemechanism for the formation of the PVC in a confined con-figuration was proposed. It is not clear, however, why thePVC is also observed in open configurations, and why it ispossible that several PVCs coexist at the same instant. Other

FIG. 14. Time signal at x /R=0.1, r /R=0.6. �a� Axial velocity. �b� Pressure.

FIG. 15. Temporal cross correlation of axial velocity and pressure fluctua-
tions �ux��t�p��t+�t�� at x /R=0.1, r /R=0.6.

explanations have also been proposed in the literature.Schlüter23 suggests that the vortex could develop as a Taylor-Görtler instability, or as a shear layer instability in the cir-cumferential direction, like in the Kelvin-Helmholtz instabil-ity. It is known that in the Taylor-Görtler instability thevortex axis is parallel to the mean flow,54 while in theKelvin-Helmholtz the vortex axis is perpendicular to themean flow.

Figure 16 shows the inner coherent structures in thepresent LES and selected three-dimensional streamlines ofthe mean flow. The origin of the latter has been positioned inthe inner region of the jet. The streamlines were computedonly downstream. Obviously, the vortex axis is not parallelto the streamlines but orthogonal to them. This fact supportsthe interpretation that the precessing vortex core is generatedby a shear layer instability, i.e., a Kelvin-Helmholtz instabil-ity, of the inner shear layer.

C. Fluctuations of axial velocity component

Phase-averaged measurements in a typical swirl-burnerconfiguration �nonannular� have shown that the distributionof axial velocity was asymmetric.52 The highest forward ve-locities were measured near the outer edge of the swirlingflow, but the forward velocities in the region containing thePVC were up to 50% larger than the velocities in the dia-metrically opposite region. The region of reverse flow waslocated on the opposite side of the core from this region ofhighest forward velocity. Figure 13�c� shows a contour plotof an instantaneous axial velocity field together with linesshowing the position of the PVC and the recirculation zone.This plot confirms the findings of Froud et al.52 However, bycomparison of Figs. 13�a� and 13�c� it is observed that theregion of highest forward velocity does not correspond to theregion of highest forward velocity fluctuations ux�; the high-est values of forward velocity fluctuations in Fig. 13�a� areremote from the outer edge of the annular jet, while thelargest values of ux in Fig. 13�c� are observed very close tothe outer edge �see also Figs. 6�a� and 8�a��. The axial ve-locity fluctuations are very large, with velocity differences ofup to 2.5Ub. A typical time signal is shown in Fig. 14�a�. In

FIG. 16. Coherent structures of the instantaneous flow as in Fig. 10 but thelevel of the isosurface is p− �p�=−0.5. Only the inner region rR is shown.The black lines represent stream ribbons of the averaged flow issued atdifferent positions in the inner region of the jet around r /R0.5.

order to see that the region of highest positive velocity fluc-



tuations is indeed associated with the PVC, Fig. 15 showsthe temporal cross correlation between the axial velocity andthe pressure �ux��t�p��t+�t��. The negative correlation showsthat the low pressure, which is related to the vortex core, iscorrelated with the high forward velocity. The maximum cor-relation is not achieved for �t / tb=0 but for �t / tb=0.16. Thisis due to the fact that the pressure is minimum at the vortexcore while the velocity fluctuation is maximum at the outeredge of the PVC and lags somewhat behind it. The time2�t / tb=0.32 gives an estimation of the time which a PVCtakes to pass through a point at this radial location.

Figure 17 shows spatial two-point autocorrelations of thethree velocity components with respect to the angular sepa-ration �� at x /R=0.1, r /R=0.6. The axial velocity autocor-relation is different with respect to the other two compo-nents. It does not go to zero when the angular separationincreases but saturates at a negative value of −0.2 for ��=120° to 180°. This is explained by the two regions of posi-tive and negative axial fluctuations which were observed inFig. 13�a�. The radial location at which the correlations arecomputed, r /R=0.6, is positioned slightly radially outwardsof the center of the PVC �see also Fig. 24�a� below�. There-fore, as illustrated by the sketch in Fig. 13�d�, u�� is alwayspositive and the tangential velocity autocorrelation does notbecome negative. On the other hand, the radial fluctuationsdecay from positive to zero values at ��=35° and remainnegative for larger angles approaching zero. The minimum ofthe radial velocity autocorrelation at ��=60° provides anestimate of the angular size of the vortex.

D. Outer structures

The outer structures are mentioned less frequently in theliterature than the precessing vortex core. Gupta et al. �Ref.1, p. 192� described them as a large eddy in the radial-axialdirection which is shed in a continuous process behind thepassing PVC. Figure 18 shows the coherent structures andselected three-dimensional streamlines of the mean flowpassing through the outer structures. The origins of thestreamlines are at x /R=0, r /R=0.95, i.e., in the outer regionof the jet, at four different angles. It can be seen, as in the

FIG. 17. Spatial two-point autocorrelation with angular separation �in de-grees� at x /R=0.1, r /R=0.6. R11, axial velocity. R22, radial velocity. R33,tangential velocity.

case of the inner structures, that the structures are orthogonal


to the average streamlines. This explains the creation of theouter structures by a Kelvin-Helmholtz instability of theouter shear layer, as described above for the inner structures.In Fig. 10 and corresponding animations, it is possible to seethat the appearance of the outer structures is indeed “locked”to the presence of the inner ones. A possible explanation isthat the inner structures are associated with high axial veloci-ties and, as they pass, the high-velocity region triggers theformation of the outer structures.

E. Secondary structures

To conclude the analysis of the coherent structures, Fig.19 shows an isosurface of unfiltered pressure fluctuations.The color in the figure just represents the radial distancefrom the symmetry axis and has been included to facilitatethe visibility of the vortices discussed. It can be seen that thelarge-scale coherent structures generate secondary instabili-ties oriented in the streamwise direction and located at theouter boundary of the outer spirals. The smoothed plot inFig. 18 exhibits “blobs” reminiscent of these, accidentallynear the crossings of the average streamlines plotted in thisfigure. These secondary structures resemble the structures in

FIG. 18. Coherent structures of the instantaneous flow as in Fig. 10. Theblack lines represent stream ribbons of the averaged flow issued at x /R=0,r /R=0.95, and four angular positions with an angular distance of ��=90°.

FIG. 19. Isosurface of unfiltered pressure fluctuations p�=−0.3. It has beencolored according to the radial coordinate to highlight secondary structuresin streamwise direction which are located at the outer boundary of the outer
spiraling vortices.


a plane shear layer where counter-rotating vortex pairs, ori-ented preferentially in the streamwise direction and usuallyknown as “braids,” are formed between the spanwise pri-mary structures due to the stretching of the flow. In thepresent simulation, the secondary vortices observed seem toappear at a relatively large spacing, but so far quantitativestatements cannot be made.

VI. ANALYSIS OF SPECTRA

A. Computational procedure and general shape

In the experiment,34 time signals of velocity have beenrecorded close to the jet exit at x /R=0.1, r /R=0,0.6,0.8.During the simulation, velocity and pressure signals wererecorded at the same positions for a duration of 115 tb. Fur-thermore, signals were recorded for each of these positions inx and r at 12 different angular locations over which addi-tional averaging was performed. Figure 14 shows a smallpart of a time signal of the axial velocity and the pressure atx /R=0.1, r /R=0.6. The precessing vortex cores discussed inthe previous section are expected to pass right through thispoint. Indeed, the presence of the structures is indicated bythe large oscillations observed in the time signals of bothaxial velocity and pressure. An analysis of the power spectraldensity is used to obtain the frequencies of rotation. Theanalysis was performed using a windowed Fourier transformwith a Hanning window55 and segments of length 211, i.e.,spanning a length of 32.8 time units. The full signal wasdecomposed into eight such overlapping segments overwhich averaging was performed. These parameters were se-lected so as to obtain the best possible compromise betweensmoothness of the spectra and width of the frequency win-dows covered.

Figure 20 shows the power spectral density of the axialvelocity fluctuations at x /R=0.1, r /R=0.6, i.e., very close tothe jet exit and in the region of the inner shear layer. Itexhibits several ranges. At low frequencies several peaks ap-pear which represent the energetic content of the coherentstructures. This is followed by a region in which a regular

FIG. 20. Power spectral density of axial velocity fluctuations at x /R=0.1,r /R=0.6. The straight line has a slope of −5/3.

decay of slope −5/3 takes place over about one decade in


frequency. Finally, there is a high-frequency region of fasterdecay which is related to the effective filter of the LES. Inthe experiment it was not possible to measure frequencies inthe turbulent inertial subrange. The maximum resolved fre-quency was about fR /Ub=2. In the graphs below, the com-parison with the experiment is therefore performed only inthis range.

B. Dominant frequencies

Figure 21 shows a comparison of power spectral densi-ties of velocity fluctuations between experiment and simula-tion at x /R=0.1, r /R=0.6 �later it will be important that inthe simulation the position actually was r /R=0.605�. Theinner structures are expected to pass through this point, asevidenced by the previous visualizations. Indeed, they gen-erate a pronounced peak and higher harmonics. All powerspectral densities are normalized with the energy of the sig-nal up to the highest frequency available in the experiment.As the energy at higher frequencies is not negligible in thesimulation, its inclusion in the normalization would affectthe comparison with the experiment. The agreement is excel-lent for the axial and radial velocity components in terms ofthe dominant frequency, even with respect to the amplitudeof the corresponding peaks. Concerning the tangential veloc-ity component, the frequency is in good agreement, but theshape of the spectrum is different with respect to the ampli-tude of the peaks. This will be addressed further below. Theprecessing frequency of the vortices corresponds to the firstpeak, and the higher harmonics corresponds to the number ofvortices present in the flow field. In visualizations as dis-played in Figs. 10 and 13, up to three precessing vortex coresof different strengths were observed at certain instants. Mostof the time, however, only one PVC is present and this iswhy the first peak is more pronounced. In the spectrum ofthe radial velocity, Fig. 21�b�, the first two peaks have simi-lar amplitude. This is due to the following effect. For theaxial fluctuations ux�, a dominant structure such as seen inFig. 13�a� yields a single temporal increase of the signalmeasured at a point like Q in Fig. 13�d�. For the radial fluc-tuations ur�, a single passage of a vortex first produces nega-tive and then positive radial fluctuations as sketched in Fig.13�d�; hence, the spectrum has a peak with twice the fre-quency. The simulation predicts very accurately the frequen-cies of the principal peak: fpeak=0.31Ub /R in the LES andfpeak=0.32Ub /R in the experiment. In the experiment, thelatter corresponds to a dimensional value of f =160 Hz.

Figure 22 shows the power spectral density of velocityfluctuations at x /R=0.1 on the symmetry axis. At this posi-tion only two spectra are shown because all directions per-pendicular to the centerline are statistically equivalent. Also,at this point the simulation is in very good agreement withthe experiment although the agreement is not as impressiveas above. This is due to the fact that on the axis azimuthalaveraging is not possible and therefore a longer time signalwould be needed to provide similar averaging quality. Thespectrum of fluctuations orthogonal to the centerline, Figs.21�b� and 21�d�, exhibits a pronounced peak at the same
frequency as the precessing vortex core. In the spectrum of


the axial velocity fluctuations at this position no pronouncedpeak is observed, Figs. 21�a� and 21�c�. At this location, theaxial component is blocked by the wall, suppressing the pe-riodic oscillations.

FIG. 21. Power spectral density of velocity fluctuations at x /R=0.1, r /R=fluctuations. �b� and �e� Radial velocity fluctuations. �c� and �f� Azimuthal v


As mentioned above in relation to Fig. 21, the agreementwith the experimental data is excellent for the axial and ra-dial velocity components and less good for the tangentialvelocity component. To investigate this issue, time signals

a�–�c� Simulation. �d�–�f� Experiment �Ref. 34�. �a� and �d� Axial velocityty fluctuations.

FIG. 22. Power spectral density of ve-locity fluctuations at x /R=0.1, r /R=0. �a�–�b� Simulation. �c�–�d� Ex-periment �Ref. 34�. �a� and �c� Axialvelocity. �b� and �d� Radial velocity.�Note that all directions orthogonal tothe centerline are statistically equiva-lent. Therefore, only two spectra areshown in this figure.�

0.6. �eloci



have been recorded at x /R=0.1 and x /R=0.4, at several ra-dial positions but each time only at one azimuthal locationfor technical reasons. Therefore, no azimuthal averaging ispossible for these time series and the spectrum is lesssmooth. Figure 23 shows the PSD of tangential velocity atx /R=0.1 and r /R=0.594. Its shape is closer to the experi-mental spectrum, Fig. 21�f�, than the spectrum obtained atr /R=0.605, Fig. 21�c�. For this component the spectrum isvery sensitive to the radial location. This issue is discussed inthe following section

C. Relation between the spectra and the coherentstructures

Figure 24 shows the amplitude of the power spectrum atthe fundamental frequency fpeak as a function of the radialposition. This figure provides information on the organizedfluctuations, while in Fig. 8 the organized fluctuations aremixed with the turbulent fluctuations. It is apparent that thepower spectrum amplitude of the tangential velocity fluctua-tions changes rapidly with radial position around r /R=0.6,explaining the sensitivity to the radial location observed inFigs. 21�c� and 23.

FIG. 23. PSD of tangential velocity fluctuations u�� at x /R=0.1, r /R=0.594. Compare to Figs. 21�e� and 21�f�.

FIG. 24. Value of the power spectrum at the fundamental freque


The shape of the curves shown in Fig. 24 is different forthe three velocity components. In order to understand thefigure it is necessary to consider the motion of the PVC inthe transverse plane x /R=0.1. An idealized sketch is shownin Fig. 13�d�. In the ideal case of a vortex core rotating at aregular rate around the symmetry axis, the spectrum of azi-muthal velocity fluctuations u�� at a point like Q in Fig. 13�d�should not contain any peak, because the center of the vortexis passing always through this point. Therefore, the minimumin the curve for u� observed at r /R=0.55 in Fig. 24�a� indi-cates the mean radial location of the PVC center. For r /R0.5, the axial velocity component is blocked by the centerbody wall while the radial and tangential components arenot. This blocking is reduced further downstream due to thelarger distances from the wall. The radial and tangentialcomponents have a similar behavior for r /R0.5, and theyconverge at the centerline as expected due to axisymmetry.There is a decay in the spectrum amplitude of the radialfluctuations for r /R0.6. Due to the regions of positive andnegative axial fluctuations observed in Fig. 13�a�, the ampli-tude of the axial component is significantly higher than theamplitude of the other components for r /R0.5.

Once the mean radial location of the PVC center hasbeen established, it is interesting to note that at x /R=0.1 andr /R=0.6 there is an accumulation of energy in the spectrumof tangential velocity fluctuations at low frequencies, oneorder of magnitude lower than the fundamental, precessingfrequency, which does not appear in the other components.This can be observed both in the experiment and the simu-lation �see Figs. 21�c� and 21�f�� and is related to the positionof the vortex core. In the mean, its radial position at x /R=0.1 is given by the minimum in Fig. 24�a�, but it is notconstrained and its low-frequency oscillation in the radialdirection is most likely responsible for the accumulation ofenergy at low frequencies.

Finally, as the inner structures rotate at a constant rate,an estimation of the precessing velocity of the structures canbe made using the precessing frequency of the vortices,uprec=�precrc=2 rcfpeak, where �prec and uprec are the angularvelocity and precessing velocity of the inner structures, re-spectively, and rc is the radial position of the vortex center,

ncy fpeak as a function of r /R. �a� x /R=0.1; �b� x /R=0.4.



Fig. 13�d�. In Gupta et al. �Ref. 1, p. 198�, for a configurationwhich exhibited similar phenomena as the present one, it wasfound that at the radius of the center of the vortex core theprecessing velocity of the PVC was equal to the measuredmean tangential velocity at that radius. However, the presentresults point in a different direction. In the present simulationat x /R=0.1, uprec=2 rcfpeak1.15Ub. This precessing veloc-ity is higher than the mean tangential velocity at this point,which is �u��=0.65Ub. The structures are hence movingfaster than the mean flow at the center of the shear layer. Anexplanation can be obtained from the mean tangential veloc-ity profile in Fig. 6�b�. In high-velocity regions �at rR� thearea �2 r� is larger than in low-velocity ones �at r0.5R�,and therefore the center of inertia of the velocity distributionthrough the shear layer is displaced towards higher speed.Furthermore, curvature of the shear layer may also play arole. To clarify this issue, stability analyses using the meanflow as base flow may be helpful but they are beyond thescope of the present paper.

VII. CONCLUSIONS

Swirl flows constitute a challenging and practically rel-evant class of flows. In this paper, a large eddy simulation ofan unconfined annular swirling jet has been reported. Theadequacy of the LES method and the boundary conditionsemployed for studying this problem has been demonstratedby various comparisons with a corresponding experiment: Inthe near field of the jet, the agreement with the experimentaldata is excellent for the mean flow and the turbulent fluctua-tions. Also, the low-frequency range of the power spectraldensity of velocity fluctuations has been found to be in goodagreement with the experiments. This suggests that the large-scale organized motions have been well captured in thesimulation, allowing a more detailed physical interpretationof these motions than is possible from the experimental re-sults.

Analyses of the different vortex structures in the nearfield of the jet have been performed by a variety of postpro-cessing techniques. Two- and three-dimensional visualiza-tions, filtering, different types of averaging, spectra, autocor-relation functions, etc. were employed depending on thespecific question to be answered. These data yielded a clearidentification of the structures and important hints to the re-lated mechanisms of instability. The Kelvin-Helmholtz insta-bility could be identified as the major source for the genera-tion of the coherent vortices observed. Furthermore, a clearconnection between the velocity power spectra and the co-herent vortices could be established.

Further investigations are underway concerned withpractically relevant modifications of the flow investigated,such as the introduction of a pilot jet near the axis.47,56

ACKNOWLEDGMENTS

This work was funded by the German Research Founda-tion �DFG� through Project A6 in the Collaborative ResearchCenter SFB 606 at the University of Karlsruhe. The calcula-tions were carried out on the IBM RS/6000 SP-SMP high-
performance computer of the University of Karlsruhe. The

authors are grateful to Dr. H. Büchner and O. Petsch forproviding the experimental data and to Dr. C. Hinterbergerfor his help in generating the grid. The first author would liketo thank Dr. J. Wissink and Dr. J. C. del Álamo for illumi-nating discussions.

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identification and analysis of coherent structures in the near field of a

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