Proceedings of GT2006 ASME Turbo Expo 2006: Power for Land, Sea and Air May 8-11, 2006, Barcelona, Spain GT2006-90764 NUMERICAL SIMULATIONS OF ISOTHERMAL FLOW IN A SWIRL BURNER Manuel Garc´ ıa-Villalba Institute for Hydromechanics University of Karlsruhe 76128 Karlsruhe, Germany Email: [email protected]Jochen Fr ¨ ohlich Institute for Technical Chemistry and Polymer Chemistry University of Karlsruhe 76128 Karlsruhe, Germany Email: [email protected]Wolfgang Rodi Institute for Hydromechanics University of Karlsruhe 76128 Karlsruhe, Germany Email: [email protected]ABSTRACT In this paper, the non-reacting flow in a swirl burner is stu- died using Large Eddy Simulation. The configuration consists of two unconfined co-annular jets at a Reynolds number of 81500. The flow is characterized by a Swirl number of 0.93. Two cases are studied in the paper differing with respect to the axial loca- tion of the inner pilot jet. It was observed in a companion ex- periment (Bender and B¨ uchner, 2005) that when the inner jet is retracted the flow oscillations are considerably amplified. This is also found in the present simulations. Large-scale coherent structures rotating at a constant rate are observed when the inner jet is retracted. The rotation of the structures leads to vigorous oscillations in the velocity and pressure time signals recorded at selected points in the flow. In addition, the mean velocities, the turbulent fluctuations and the frequency of the oscillations are in good agreement with the experiments. A conditional averaging procedure is used to perform a detailed analysis of the physics leading to the low-frequency oscillations. NOMENCLATURE f peak Fundamental frequency of oscillation k Fluctuating kinetic energy p Modified pressure R Outer radius of the main jet Re Reynolds number RMS Root mean square S Swirl number t b Characteristic time based on R and U b U b Bulk velocity of the main jet u x Axial velocity component u r Radial velocity component u θ Tangential velocity component x, r , θ Cylindrical coordinates (axial, radial, tangential) x, y , z Cartesian coordinates x pilot Axial location of the pilot jet exit α Angle used in the conditional-average technique hφi Mean value of the quantity φ φ c Conditional average of the quantity φ φ 00 Instantaneous fluctuation of the quantity φ, φ 00 = φ -hφi ρ Density INTRODUCTION In recent years, there has been increased demand for gas turbines that operate in a lean premixed mode of combustion in an effort to meet stringent emission goals. Highly turbulent swirl-stabilized flames are often used in this context. However, swirling flows are prone to flow instabilities which can trigger combustion oscillations and cause damage to the device. Lean premixed burners in modern gas turbines often make use of a richer pilot flame which is typically introduced near the symme- try axis. In order to prevent the appearance of undesired flow insta- bilities, it is necessary to understand the underlying physical phe- nomena. Several mechanisms have been identified in the lit- erature as potential triggers of combustion instabilities. There is, however, no consensus about the real importance of each of 1 Copyright c 2006 by ASME
Transcript
Proceedings of GT2006ASME Turbo Expo 2006: Power for Land, Sea and Air
May 8-11, 2006, Barcelona, Spain
GT2006-90764
NUMERICAL SIMULATIONS OF ISOTHERMAL FLOW IN A SWIRL BURNER
Manuel Garcıa-VillalbaInstitute for Hydromechanics
University of Karlsruhe76128 Karlsruhe, GermanyEmail: [email protected]
ABSTRACTIn this paper, the non-reacting flow in a swirl burner is stu-
died using Large Eddy Simulation. The configuration consists oftwo unconfined co-annular jets at a Reynolds number of 81500.The flow is characterized by a Swirl number of 0.93. Two casesare studied in the paper differing with respect to the axial loca-tion of the inner pilot jet. It was observed in a companion ex-periment (Bender and Buchner, 2005) that when the inner jet isretracted the flow oscillations are considerably amplified. Thisis also found in the present simulations. Large-scale coherentstructures rotating at a constant rate are observed when the innerjet is retracted. The rotation of the structures leads to vigorousoscillations in the velocity and pressure time signals recorded atselected points in the flow. In addition, the mean velocities, theturbulent fluctuations and the frequency of the oscillations are ingood agreement with the experiments. A conditional averagingprocedure is used to perform a detailed analysis of the physicsleading to the low-frequency oscillations.
NOMENCLATUREfpeak Fundamental frequency of oscillationk Fluctuating kinetic energyp Modified pressureR Outer radius of the main jetRe Reynolds numberRMS Root mean squareS Swirl numbertb Characteristic time based on R and Ub
Ub Bulk velocity of the main jetux Axial velocity componentur Radial velocity componentuθ Tangential velocity componentx,r,θ Cylindrical coordinates (axial, radial, tangential)x,y,z Cartesian coordinatesxpilot Axial location of the pilot jet exitα Angle used in the conditional-average technique〈φ〉 Mean value of the quantity φφc Conditional average of the quantity φφ′′ Instantaneous fluctuation of the quantity φ, φ′′ = φ−〈φ〉ρ Density
INTRODUCTIONIn recent years, there has been increased demand for gas
turbines that operate in a lean premixed mode of combustionin an effort to meet stringent emission goals. Highly turbulentswirl-stabilized flames are often used in this context. However,swirling flows are prone to flow instabilities which can triggercombustion oscillations and cause damage to the device. Leanpremixed burners in modern gas turbines often make use of aricher pilot flame which is typically introduced near the symme-try axis.
In order to prevent the appearance of undesired flow insta-bilities, it is necessary to understand the underlying physical phe-nomena. Several mechanisms have been identified in the lit-erature as potential triggers of combustion instabilities. Thereis, however, no consensus about the real importance of each of
them. Lieuwen et al. [1] suggested that heat-release oscillationsexcited by fluctuations in the composition of the reactive mix-ture entering the combustion zone are the dominant mechanismresponsible for the instabilities observed in the combustor. Otherauthors [2, 3] favour the in-phase formation and combustion oflarge-scale coherent vortical structures. In premixed combustors,these large-scale structures play an important role in combustionand heat-release processes by controlling the mixing between thefresh mixture and hot combustion products [4].
The formation of large-scale coherent structures is a funda-mental fluid-dynamical problem, which must be understood alsoin the absence of combustion. Large Eddy Simulation (LES) is aparticularly suitable approach for studying this problem. It al-lows the treatment of high-Reynolds-number flows and at thesame time the explicit computation of these structures. If prop-erly conducted, LES should have only limited sensitivity to mod-elling assumptions. In the context of swirling flows, LES wasfirst applied by Pierce and Moin [5]. Wang et al. [6] performedLES of swirling flow in a dump combustor and studied the in-fluence of the level of swirl on the mean flow and on turbulentfluctuations. LES has also been used in combination with othertechniques like acoustic analysis; for example, Roux et al. [7]studied the interaction between coherent structures and acousticmodes and found important differences between iso-thermal andreactive cases.
In the present paper LES is used to study the iso-thermalflow in a swirl burner at two different configurations. The ana-lysis of the results focuses on the strength and sensitivity of flowinstabilities generating large-scale coherent structures. In [8, 9]the present authors performed LES of an unconfined annularswirling jet. Instabilities leading to large-scale coherent struc-tures were detected and identified to be responsible for the oscil-lations observed in the corresponding experiment.
Preliminary results for the configurations considered belowwere reported in [10]. These simulations were repeated there-after with an improved inlet condition. Furthermore, the analysispresented below goes into substantially more detail, e.g. by de-termination of conditional averages.
PHYSICAL AND NUMERICAL MODELLINGExperimental configuration
In [11] a co-annular swirl burner was developed which al-lows the change of geometrical features over a wide range. Theburner, depicted in Fig. 1, is composed of two co-annular jets,a central pilot jet and a concentrically aligned main jet, whoseswirl can be adjusted individually. In the experimental condi-tions considered here, a radial swirler was used for generatingthe swirl in the main jet. For the central pilot jet, an axial swirlerwas used to generate a co-rotating flow.
A large number of experiments were performed with thisburner in several configurations including isothermal and reac-
Figure 1. Sketch of the burner (taken from [11]).
tive cases [11, 12]. For the isothermal flow without external for-cing, it was observed that axial retraction of the central jet intothe duct leads to an increased amplitude of flow oscillations re-flected by audible noise. In order to investigate this phenomenonby means of LES, two cases were selected. In the first case theinner jet is not retracted, i.e. both jets exit at the same position.In the second case, the pilot jet is retracted by 40 mm. This re-traction of the pilot jet generates a double expansion for the mainjet (see Fig. 2 below).
In both cases the co-annular jets issue into an ambient of thesame fluid which is at rest in the experiment. The outer radiusof the main jet, R = 55 mm, is used as the reference length. Thereference velocity is the bulk velocity of the main jet Ub = 22.1m/s and the reference time is tb = R/Ub. The inner radius of themain jet is 0.63R. For the pilot jet the inner radius is 0.27R andthe outer radius is 0.51R. The mass flux of the pilot jet is 10% of the total mass flow. The Reynolds number based on thebulk velocity of the main jet Ub and R is Re = 81000. The swirlnumber is defined as
S =
R R0 ρ〈ux〉〈uθ〉r2 dr
RR R
0 ρ〈ux〉2r dr, (1)
where 〈ux〉 and 〈uθ〉 are the mean axial and azimuthal velocitiesrespectively. Its value at the burner exit is S = 0.93.
As mentioned above, two cases have been considered, onewithout retraction of the pilot jet, i.e. xpilot = 0. In the second onethe retraction is xpilot = −0.73R. LDA measurements are avail-able for both cases [11]. In particular, radial profiles of mean andRMS velocities are available at four axial stations in the near fieldof the burner. Only for the case with retraction, power spectra ofvelocity fluctuations were also measured.
Numerical setupThe simulations were performed with the in-house code
LESOCC2 [13], which is a successor of the code LESOCC [14].It solves the incompressible Navier-Stokes equations on curvi-linear block-structured grids. A collocated finite-volume dis-cretization with second-order central schemes for convectionand diffusion terms is employed. Temporal discretization isperformed with a three-stage Runge-Kutta scheme solving thepressure-correction equation in the last stage only. The Rhie andChow momentum interpolation [15] is applied to avoid pressure-velocity decoupling. The dynamic Smagorinsky sub-grid scalemodel [16] was used in the simulations. The model parame-ter was determined using an explicit three-dimensional box filterof width equal to twice the mesh size. The eddy viscosity wasclipped to avoid negative values and was smoothed by temporalrelaxation [17].
The computational domain extends 32R downstream of theburner exit located at x/R = 0 and 12R in radial direction. It alsocovers part of the inlet ducts, Fig. 2. The block-structured meshconsists of about 8.5 million cells with 160 cells in azimuthaldirection. The grid is stretched in both the axial and radial direc-tion to allow for concentration of points close to the nozzle andthe inlet duct walls. The stretching factor is everywhere less than5 %.
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Figure 2. Numerical setup and boundary conditions. Color representsmean axial velocity.
The specification of the inflow conditions for both jets re-quires a strong idealization. For the main jet, the way the swirlis introduced is not so critical because the swirler is located up-stream, far away from the region of interest. Therefore, the flowis prescribed at the circumferential inflow boundary located at thebeginning of the inlet duct (see Fig. 2). At this position steadytop-hat profiles for the radial and azimuthal velocity componentsare imposed. This procedure was validated in [9]. The swirler ofthe pilot jet, on the other hand, is located directly at the jet out-let, Fig. 1. A numerical representation of this swirler would bevery demanding because of the large number of blades and was
therefore not considered in the present investigation. Instead, theinflow conditions for the pilot jet were obtained by performing si-multaneously a separate, streamwise periodic LES of developedswirling flow in an annular pipe (see Fig. 2) using body forcesto generate co-rotating swirl with Spilot = 2 as described in [18],where Spilot is the swirl number of the pilot jet only. Recall thatSpilot has little impact on the swirl number of the entire flow dueto the small mass flux of this stream. No-slip conditions wereapplied at solid walls. The fluid entrained by the jet is fed in bya mild co-flow of 5 % of Ub. By using different values of the co-flow velocity it was shown in [19] that the flow development isnot sensitive to this conditions. Free-slip conditions were appliedat the open lateral boundary. A convective outflow condition wasused at the exit boundary.
In both cases simulated the same boundary conditions wereemployed. Fig. 2 displays a zoom of the inflow region forxpilot = −0.73R. In the case xpilot = 0, not shown here, the in-flow region differs because the wall separating main and pilotjet and the cylindrical centre body reach until x = 0, with theinflow plane for the pilot jet still located at the same positionx/R = −0.73. This is illustrated in Fig. 3 below.
AVERAGE FLOWAfter discarding initial transients, statistics were collected
for 100tb, which is long enough to obtain converged values in thenear field of the burner. The averaging was performed in time andalso along the azimuthal direction. Only resolved fluctuationsare accounted for. It was checked, however, that the modelledsubgrid-scale contributions are negligible [19].
StreamlinesFig. 3 shows the two-dimensional streamlines of the average
flow in an axial plane for both cases. It is well known that at thishigh level of swirl a recirculation zone forms in the central regionof the flow [20]. This phenomenon is related to the presenceof a low pressure region on the symmetry axis. The influenceof the retraction of the inner jet is remarkable. For xpilot = 0the recirculation forms immediately behind the cylindrical centrebody and the length of the recirculation zone is about 9R. Inthe case xpilot = −0.73R the length of the recirculation zone isonly about 5R. The two streams mix before the final expansionand the recirculation is detached from the burner. The maximumwidth of the recirculation bubble is about 0.8R in both cases andit is attained at x/R = 1.2 for xpilot = 0 and at x/R = 1.5 forxpilot = −0.73R. Far downstream of the jet exit, for x/R ≥ 6, themean flow is not fully converged in the vicinity of the symmetryaxis, as indicated by the wavy streamlines. The reason is thatat this position the motions are slower and substantially longeraveraging times would be necessary to obtain a fully-convergedmean flow.
Figure 3. Streamlines of the average flow in an axial plane a) xpilot = 0b) xpilot = −0.73R.
Mean and RMS velocity profilesA comparison of simulations with experiments is reported in
Figs. 4-7, showing radial profiles of mean velocity and turbulentfluctuations at several axial stations for both cases.
The agreement with the experimental data is in general goodfor the mean flow. The case xpilot = 0 is well reproduced in thesimulation, Fig. 4, which is noteworthy in spite of the strongidealization in setting up the inflow conditions for the pilot jet.The limited strength of the pilot jet can be appreciated by themean axial and tangential velocity at x/R = 0.1. In the casexpilot = −0.73R, Fig. 5, a discrepancy is evident at x/R = 0.1;the backflow is overpredicted in the simulation. This implies thatthe recirculation zone in Fig. 3b does not correspond exactly tothe experimental one, which was measured to be slightly furtherdownstream. Nevertheless, other characteristics are very wellpredicted so that this simulation is still close to the experiment.For example, the spreading of the jet is in good agreement withthe experiment and so are the turbulent fluctuations of axial andtangential velocity, Fig. 7. The agreement is also good for theturbulent fluctuations in the case xpilot = 0, Fig. 6.
Some features are common in both cases. Apart from thepresence of a recirculation zone, two complex shear layers sub-ject to curvature effects are present in the flow. The inner shearlayer is formed between the main jet and the recirculation zone.The outer shear layer is formed between the main jet and the sur-rounding co-flow. In the case xpilot = 0, the turbulent fluctuationsgenerated in these layers are clearly visible up to x/R = 1 in theprofiles of RMS fluctuations by corresponding peaks (Fig. 6). Inthe case xpilot = −0.73R this feature is only observed in the pro-file of the axial fluctuations very close to the jet exit 7a. Notealso that the level of fluctuations at x/R = 0.1 is much higher
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Figure 4. Radial profiles of mean velocity xpilot = 0 a) Axial velocity b)Tangential velocity. Symbols, experiments [11]. Lines, LES.
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Figure 5. Radial profiles of mean velocity xpilot = −0.73R a) Axialvelocity b) Tangential velocity. Symbols, experiments [11]. Lines, LES.
for xpilot = −0.73R. In that case the maximum RMS is about0.5Ub while in the case xpilot = 0 it does not reach 0.3Ub. Fur-ther downstream at x/R = 3 this difference has vanished and inboth cases the maximum RMS fluctuation is close to 0.3Ub, al-though the radial spreading of the profiles is larger in the casexpilot = −0.73R.
Fluctuating kinetic energyTo conclude the description of the average flow, Fig. 8 dis-
plays contours of the fluctuating kinetic energy, using the samescale for both cases. It is obvious that the retraction of the pi-lot jet leads to a large increase in the level of the fluctuatingenergy. In the case xpilot = 0 the fluctuating kinetic energy isconcentrated in the two shear layers mentioned above and themaximum level is kmax/U2
b ∼ 0.14. In the case xpilot = −0.73Rthe kinetic energy is concentrated in three regions, just behind
Figure 6. Radial profiles of RMS velocity xpilot = 0 a) Axial velocity b)Tangential velocity. Symbols, experiments [11]. Lines, LES.
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Figure 7. Radial profiles of RMS velocity xpilot = −0.73R a) Axial ve-locity b) Tangential velocity. Symbols, experiments [11]. Lines, LES.
the inner part of the burner, at the beginning of the recirculationbubble (compare Fig. 8b and Fig. 3b) and in the region of theinner shear layer. As evidenced by the RMS profile of tangen-tial velocity fluctuations, Fig 7 at x/R = 0.1 (and radial fluctua-tions, not shown here), showing a pronounced local maximum atthe symmetry axis, these two components (radial and tangential)contribute mainly to the concentration of kinetic energy at thebeginning of the recirculation zone. The features observed herewill be discussed below in connection with the vortical structurespresent in the respective flows.
INSTANTANEOUS FLOW AND SPECTRACoherent structures
For a swirling annular jet, large scale coherent structureswere identified and their evolution and interaction described in[8, 9]. It was shown that two families of structures appear, an in-
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Figure 8. Fluctuating kinetic energy a) xpilot = 0 b) xpilot =−0.73R.
ner one oriented quasi-streamwise and located in the inner shearlayer formed by the jet on its boundary with the recirculationzone (the so-called precessing vortex cores [20]), and an outerone oriented at a larger angle to the axis and situated in theouter shear layer formed on the boundary with the surround-ing co-flow. Fig. 9 shows iso-surfaces of pressure fluctuationsp′′ for both cases visualizing the coherent structures of the flow.Pressure fluctuations are more suitable for the visualization ofcoherent structures than the commonly used instantaneous pres-sure [21] because iso-surfaces of the latter are influenced by thespatially-variable average pressure field which is unrelated to in-stantaneous structures. Figs. 9a,c display two different levels ofp′′ for the case xpilot = 0, namely p′′ = −0.3 and p′′ = −0.15,respectively. Figs. 9b,d show the level p′′ = −0.3 at two dif-ferent instants in time for the case xpilot = −0.73. The color ofthe structures is given by the radial gradient of mean axial veloc-ity. In the inner shear layer ∂〈ux〉/∂r > 0 and the iso-surface iscolored in yellow. In the outer shear layer ∂〈ux〉/∂r < 0 and theiso-surface is colored in red.
Pronounced large-scale coherent structures are observed inthe case of the retracted pilot jet, Fig. 9b,d. As in the case with-out inner jet [9], two structures can be observed in these pictures.Animations have shown that the rotation of the inner structurearound the symmetry axis is very regular. At some instances,however, the inner structure branches leading to two arms asshown in Fig. 9d. The leading one, in the direction of the ro-tation, is faster than the second one and takes over in terms ofstrength. The one behind disappears at the exit in less than halfa rotation period and in the downstream field during another halfperiod. In the case without retraction, xpilot = 0, the structures aresubstantially smaller and more irregular. In fact, if one comparesthe same level of pressure fluctuations p′′ = −0.3, hardly any
Figure 9. Coherent structures visualized using an iso-surface of pres-sure fluctuations. Left, xpilot = 0. Right, xpilot = −0.73R. a,b,d)p−〈p〉 = −0.3. c) p−〈p〉 = −0.15. Color as explained in the text.
structure is visible in the flow, Fig. 9a. Increasing the pressurelevel to p′′ = −0.15, small structures are visible, which exhibitsmall coherence. At this point it should be recalled that the sameflow with just the pilot jet blocked shows substantial coherentstructures, similar to the ones for xpilot = −0.73R but somewhatweaker and less organized [8]. In the case xpilot = 0, hence, thepilot jet destroys the large-scale structures. When the pilot jet isretracted to xpilot = −0.73R this is not observed. The cylindri-cal tube enclosing the main jet prevents the recirculation bubblefrom moving upstream to the central bluff body containing theexit of the pilot jet. This holds for the mean flow (Fig. 3b) as wellas for the conditionally averaged flow discussed below (Fig. 16).The pilot jet therefore only ”hits” the upstream front of the re-circulation bubble but cannot penetrate into the inner shear layerwhere it would be able to impact on the coherent structures. Thedifferent coherent structures observed in both cases explain thedifferent levels of fluctuating kinetic energy encountered close tothe burner exit in Fig. 8. In a theoretical study [22], Juniper andCandel performed a stability analysis for the case of co-annularjets without swirl. They showed how stability is reduced if theinner stream mixes with the outer one before the exit plane of theouter tube, the same trend as reported here. It would be interest-ing to perform a similar stability analysis for cases with swirl.
SpectraIn the experiment [11], time signals of velocity have been
recorded at several radial positions close to the burner exit atx/R = 0.1 for the case xpilot = −0.73R. The case xpilot = 0was not measured because in preliminary tests no instabilitywas observed. During the simulation velocity and pressure sig-
nals were recorded at the same positions for a duration of 80tb.Furthermore, signals were recorded for each of these x− andr−positions at 12 different angular locations over which addi-tional averaging was performed. On the symmetry axis no an-gular averaging is possible and only one signal was recorded,and at r/R = 0.1 and r/R = 0.18 only four angular signals wererecorded with the particular grid used. The spectra were obtainedsplitting each signal in three overlapping segments of length 40tbmultiplying it by a Hanning window and averaging over the seg-ments.
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t/tbFigure 10. Time signal of axial velocity at x/R = 0.1, r/R = 0.73recorded during the simulations. a) xpilot = 0. b) xpilot = −0.73R.
The difference between the time signals of both cases is ev-ident form Fig. 10. In case xpilot = 0, Fig. 10a, the signal ex-hibits the typical irregularity of a turbulent signal. Fig. 10b, onthe other hand shows that in case xpilot = −0.73R a flow instabi-lity has developed wich causes a regular oscillation of the signalwith large amplitude. The low frequency oscillations of this sig-nal produce a pronounced peak in the power spectrum of the axialvelocity fluctuations, Fig 11. The frequency of the principal peakis fpeak = 0.25Ub/R, which in dimensional units corresponds to avalue of fpeak = 102Hz. The amplitude of the peak is very large,covering almost two decades in logarithmic scale. The total fluc-tuating energy is substantially larger than for case xpilot = 0, re-flected by the larger integral under this curve. This is in line withthe fluctuating kinetic energy contours of Fig. 8 and the RMSvalues of Figs. 6 and 7. In the case xpilot = 0, no pronouncedpeak is observed which confirms the preliminary experimentaltests in which no flow instability was detected. The smaller peakwhich appears in case xpilot = 0 at a frequency 0.16Ub/R cannotbe related to the small coherent structures observed in Fig. 9c
because these structures have a shorter time scale which wouldcorrespond to higher frequencies. This finding deserves furtherinvestigation.
A comparison of the spectrum from the LES for xpilot =−0.73R and the corresponding experimental spectrum in Fig 11serves to further validate the simulations. The agreement for bothfrequency and amplitude of the dominant peak is remarkable.Also the second harmonic is well predicted in the simulations.
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Figure 11. Power spectrum of axial velocity fluctuations at x/R = 0.1,r/R = 0.73. Black line, experiment [11] xpilot = −0.73R. Red line,simulation xpilot = −0.73R. Blue line, simulation xpilot = 0.
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Figure 12. Amplitude of the power spectrum at the fundamental fre-quency fpeak at x/R = 0.1 as a function of r/R for case xpilot =−0.73R a) experiment [11]. b) simulation.
The amplitude of the power spectrum at the peak frequencyis now considered. It is quite a sensitive quantity, much morethan the peak frequency itself. Fig. 12 shows this amplitude atthe fundamental frequency fpeak as a function of the radial posi-tion. The shape of the curves is different for the three velocity
components. The simulation reproduces quite well the trends ofthe experiment. Only for r/R < 0.2 the simulation overpredictsthe amplitudes. In that region the impact of azimuthal averagingis small (even non-existent at the axis) so that this could have aneffect. In [9] similar plots were reported for an annular jet. It wasshown there that an important issue is the location of the mini-mum for the amplitude of tangential velocity fluctuations. Thisminimum indicates the mean radial location of the centre of theinner structure. Therefore, it is noteworthy that the minimum iswell reproduced in the simulation.
CONDITIONALLY–AVERAGED FLOWIt has been shown in the previous sections that large-scale
structures rotating around the symmetry axis are present in theflow in the case xpilot = −0.73R. Due to the high level of turbu-lence, the vortical structures are highly irregular as evidenced bythe difference between Figs. 9b and 9d. The pronounced peakin the power spectrum, Fig. 11, indicates that the rotation of thestructure is very regular and allows the calculation of conditionalaverages. The purpose of the latter is to remove the irregularityinduced by the turbulent motions. The method, which is des-cribed in the following section, consists basically in defining acoordinate system y− z with origin at the symmetry axis whichrotates with the structure and perform the averaging procedurein this rotating coordinate system. Note that it is not possible toperform this kind of analysis for the case xpilot = 0 due to a lackof a regular frequency of oscillation, Fig 11 .
ProcedureIn order to investigate the main characteristics of the cohe-
rent structures, 140 instantaneous three-dimensional fields havebeen recorded. They are separated in time by 0.8tb, so thatin each period of rotation of the structure 5 fields have beenrecorded. The time span covered by the fields is 112tb. If theoscillations are truly periodic, the definition of the axes whichrotate with the structure is straightforward, with a fixed angle ofrotation in a fixed time. In the present case, however, the motionof the structure is only quasi-periodic and therefore the methodhas to be more elaborate. The centre of the structure has to bedetermined for each instantaneous field and a subsequent rota-tion of the field is performed, such that the centre of the vortex isalways on the y−axis. This is equivalent to defining a coordinatesystem y− z which rotates with the structure.
The method is illustrated in Fig. 13 and proceeds as follows.The radial location of the dominant inner structures in the trans-verse plane x/R = 0.1 is known from Fig. 12 (minimum of uθ).In the present case r/R = 0.35 is used. The centre of the vortexis identified as the local minimum of the pressure fluctuations atthat position. In the present case the coherent structure is veryregular and the detection is simple. In other cases [9], up to three
Figure 13. Illustration of the conditional average procedure. Color ac-cording to p−〈p〉. The black circle indicates the points where the mini-mum of the pressure is looked for. The red circle indicates the outer radiusof the burner.
of these coherent structures can co-exist at certain instants andin that case the dominant one is selected. Then, the full three-dimensional field is rotated by an angle α, Fig. 13, so that eachfield has the structure at the same location and standard averagingis performed. Midgley et al. [23] used a similar method to ana-lyze two-dimensional data from experiments on fuel injectors.Here, however, three-dimensional fields are available. Note thatthis procedure is fundamentally different from phase-averaging.There is no external trigger or internal frequency which wouldsuggest to divide the rotation period into several phases reducingtherefore the amount of samples in each phase. Instead, all an-gles are statistically equivalent due to the cylindrical symmetryof the problem.
a) b)
Figure 14. Coherent structures obtained using the conditional averageflow field. a) Iso-surface pc − 〈p〉 = −0.1. Color as in Fig. 9. b)Iso-surfaces uc
x −〈ux〉 = 0.3 (green) and ucx −〈ux〉 = 0.3 (blue).
ResultsIn the following, conditionally averaged quantities are in-
dicated with an upper index c. Fig. 14a shows an iso-surfaceof pressure fluctuations, pc −〈p〉 = −0.1 visualizing the cohe-rent structures of the flow using the same style as Figs. 9b,d. Itis clear that using this procedure the large-scale coherent struc-tures have been substantially smoothed. The criterion which hasbeen used to obtain the conditional-averages involves only theinner structure. Nevertheless, the outer structure does not dis-appear with the conditional-averaging but appears at the sameangular position demonstrating its link to the inner structure. Infact, the outer structure is triggered by the inner one as demons-trated by different studies of similar flows [8, 9]. In [9] this is-sue was investigated for a pure annular jet with weaker coherentstructures. Determining the correlation between u′′x and p′′ it wasfound for that case that strong positive u′′x values correlate withnegative p′′, i.e. the inner vortex structure, and in fact advancethese in the sense of rotation. The same behaviour is obviousin the present case from the conditional average. The green sur-face of positive uc
x−fluctuations in Fig. 14b advances the neg-ative pc−fluctuations of the inner structure in Fig. 14a (yellowsurface). When plotting all three iso-surfaces in a single graphit is even more visible that the downstream end of the positiveuc
x−fluctuation surface winds along the connection line betweenthe inner (yellow) and the outer (red) vortex structure of Fig. 14a.This feeds the outer (red) structure turning in clockwise sensearound its own axis in the view of Fig. 14a. Negative fluctua-tions of uc
x, represented by the blue surface in Fig. 14b, occur atthe opposite side of the jet and seem not related to a continuouslypresent rotating large-scale coherent structure.
Figs. 15 and 16 show two-dimensional cuts of the condi-tionally averaged flow. Pseudo-streamlines of this flow field pro-jected onto two planes are displayed. In Fig. 15 the pseudo-streamlines are based on (uc
r ,ucθ) and in Fig. 16 on (uc
x,ucr). The
color represents pc and the thick line indicates the contour lineuc
x = 0. The latter shows that the recirculation region is dis-placed off the symmetry axis in Fig. 15. The pressure mini-mum generated by the inner vortex structure is well visible inthe y− z−plane together with the vortex motion surrounding it.The pressure minimum is off the axis at r/R = 0.35 (a posteriorijustifying the choice of this radius for the conditioning) and bydefinition at the y−axis. The pseudo-streamlines spiral arounda different point closer to the axis. The view in Fig. 15 is indownstream direction and rotation of the flow and the structuresin counter-clockwise direction. The recirculation region hencelags behind the inner structure by about 130◦.
The inner and outer structure of Fig. 14 are also visible inFig. 16. The inner structure shows up through the pressure mini-mum around x/R = 0 and z > 0. The outer structure is reflectedby the recirculation regions and the spiralling or bending stream-lines at the top and the bottom of the figure. From the pseudo-streamlines for x ∼ 0 in this figure it is also clear that the inner
Figure 15. Two-dimensional cut through the plane x/R = 0.1 of theconditional-averaged flow. Color is given by pc. Pseudo-streamlines cal-culated using uc
Figure 16. Two-dimensional cut through the plane y/R = 0 of theconditional-averaged flow. Color is given by pc. Pseudo-streamlines cal-culated using uc
x and ucr . Thick black line uc
x = 0.
structure is correlated with high forward axial velocity, for z > 0,while the low axial velocity is located in the opposite side forz < 0. This is indicated also by the asymmetry of the recircula-tion region (see also Fig. 17 below) .
The previous information is contained in a more quantita-tive way in Fig. 17, which shows mean and conditional-averagedprofiles of pressure and velocity. In Fig. 17a, the strength of thepressure minimum related to the centre of the structure is visibleby comparison to the mean pressure. Note also in Fig. 17b that
the conditionally averaged axial velocity ucx at y = 0, hence in
the inner structure, is higher than the mean axial velocity. Therecirculation zone, as expected from the two-dimensional plots,is displaced towards the opposite side. In Fig. 17c, the radialposition at which the conditionally averaged tangential velocityuc
θ at y = 0 equals the mean velocity, roughly corresponds to theminimum of the pressure. This is also expected because at thecentre of the structure the fluctuations of the tangential velocitycomponent have to vanish, as discussed in [9].
r
p
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
y=0z=0mean
r
ux
-1.5 -1 -0.5 0 0.5 1 1.5
0
0.5
1
y=0z=0mean
r
ut
-1.5 -1 -0.5 0 0.5 1 1.5
-1
-0.5
0
0.5
1y=0z=0mean
a)
b) c)
Figure 17. Conditional-averaged velocity profiles at x/R = 0.1. a)Pressure. b) Axial velocity. c) Tangential velocity.
Finally, Fig. 18 shows the same plane as Fig. 15 butthe color and streamlines are given by the equivalent Reynolds-decomposed quantities, i.e. color by pc − 〈p〉 and streamlinesby (uc
r −〈ur〉,ucθ −〈uθ〉). The thick line again represents uc
x = 0.In this figure the region of low pressure fluctuations correspondsto the inner structure of Fig. 14a. Note that it forms outside theboundary of the recirculation zone.
Yazdabadi et al. [24] performed phase-averaged measure-ments in a cyclone dust separator and obtained similar plots asFig. 15. Their conclusion was that the reverse flow zone dis-places the central vortex core to create the precessing vortex core.The reverse flow zone would then provide feedback for the pre-cessing vortex core, and precess around the central axis behindthe precessing vortex core. In the present case, Figs. 14a and18 suggest an alternative explanation, although perhaps compa-tible with the previous one: The inner structure (precessing vor-
tex core) is formed as an instability of the shear layer (Kelvin-Helmholtz instability). It is therefore formed on the boundaryof the recirculation zone, Fig. 18, and advected by the meanflow. This structure then constrains the motion of the recircula-tion zone which is displaced off the symmetry axis and precessesbehind the structure.
z
y
-1 0 1
-1
-0.5
0
0.5
1 pfluct
0.480.380.280.180.08
-0.02-0.12-0.22-0.32-0.42-0.52-0.62-0.72-0.82
Figure 18. Two-dimensional cut through the plane x/R = 0.1 of theconditional-averaged flow. Color is given by pc − 〈p〉. Pseudo-streamlines calculated using uc
r −〈ur〉 and ucθ −〈uθ〉. Thick black line
ucx = 0.
CONCLUSIONSLarge-eddy simulations of incompressible flow in a swirl
burner have been reported. The influence of geometrical fea-tures of the burner have been investigated by comparison of twocases with a different exit position of the inner jet. The simula-tions have been validated by comparison with corresponding ex-periments and very good agreement was obtained for mean flow,turbulent fluctuations and the frequency of oscillation. The simu-lations have confirmed that the retraction of the pilot jet leads tothe generation of enhanced flow instabilities. The related large-scale coherent structures have been identified and analysed byusing instantaneous plots, spectra and conditional averages. Thelatter provides a precise picture of the characteristics of the large-scale coherent structures by removing the irregularity associatedwith the turbulent motions. These structures are relevant to themixing of heat and species in the near field of the burner, and thetechnique can presumably be applied to the reactive case as well.
ACKNOWLEDGMENTThis work was funded by the German Research Foundation
(DFG) through project A6 in the Collaborative Research CentreSFB-606 at the University of Karlsruhe. The authors are grate-ful to Dr. H. Buchner and Mr. O. Petsch for providing the ex-perimental data. The calculations were carried out on the HPXC6000 Cluster of the University of Karlsruhe Computer Cen-tre.
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