towards the prediction of combustion noise in a …...towards the prediction of combustion noise in...

Towards the Prediction of Combustion Noise in a GasTurbine Combustor: Isothermal Flow Case

F. Flemming, C. Olbricht, B. Wegner, A. Sadiki and J. JanickaInstitute for Energy and Powerplant Technology,Petersenstr. 30, D-64287 Darmstadt,Darmstadt University of Technology([email protected])

F. Bake, U. Michel, I. Rohle and B. LehmannDLR – Institute of Propulsion Technology,Muller-Breslau-Str. 8, D-10623 BerlinGerman Aerospace Center

Abstract. In order to evaluate the direct and indirect contributions to the total combustionnoise emission, a combustion chamber consisting of a swirl burner and an exit nozzle of Laval-shape, representing a gas turbine combustor, is investigated by means of experiments and largeeddy simulation. Focused on the isothermal flow case first and encouraged by a good overallagreement between the LES and the experimental data for the flow field, the characterisation ofthe flow with respect to noise sources is performed. To analyse acoustic properties of the flow,time- and lengthscales are evaluated inside the combustor. Furthermore, the evidence for theexistence of a precessing vortex core, typical for configurations with swirl, is revealed. Finally,the effect of the precessing vortex core on the flow inside the Laval-nozzle is discussed.

Keywords: gas turbine combustor, large eddy simulation, combustion noise, lengthscales,precessing vortex core

1. Introduction

In the research area of aero-engine noise the part of the noise gener-ated by combustion has become increasingly important, especially duringlanding approach of modern aircraft. This observation is mainly based onthe achievements in decreasing jet mixing noise and fan noise of modernaero-engines.

The total noise radiated by a combustion chamber system consists of directand indirect combustion noise like shown in a generalised acoustic energyequation by Dowling [6]. The direct noise sources are related to the unsteadycombustion process itself, e.g. to unsteady heat release. The indirect combus-tion noise is generated when fluid with a nonuniform entropy distribution isaccelerated in, or convected through, the nozzle located at the downstreamend of the combustion chamber. The accelerated or decelerated hot spot ra-diates sound due to a fluctuating mass flux. In gas turbines, the inlet guidevanes for the first turbine stage serve as nozzle for the combustion chamber.The flow in this nozzle is choked in aero-engines in practically all relevant

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2004 Kluwer Academic Publishers. Printed in the Netherlands.

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2 Flemming, Olbricht, Wegner, Sadiki, Janicka, Bake, Michel, Rohle, Lehmann

operating conditions. The underlying theory of these acoustic mechanisms isfully described by Marble & Candel [22]. The separation of hydrodynamic,entropy and combustion noise was investigated earlier via cross spectral anal-ysis [23]. However, the contribution to the total noise emission of aero-enginecombustors is not known.

In order to evaluate the respective contributions of both, direct and indi-rect combustion noise, to the total noise emission a model combustor flowis investigated. For this purpose the enclosed flow field, dominated by thestabilising swirl, is analysed by means of simulations and experiments. Thenoise generation inside the combustor is dominated by unsteadiness, as acous-tic sources usually arise from unsteady flow phenomena. Hence a simulationtechnique that takes instationary effects into account is desirable. Further-more, the noise emissions of turbulent flames usually consist of relativelyhigh frequencies.

Due to the unsteady nature of the flow phenomena inside the combustorand the high frequency content of the noise, stationary statistical turbulencemodels based on Reynolds- or Favre-averaged balance equations (RANS)are overstrained by the requirements. The application of unsteady statisti-cal methods (URANS) to such configurations is not conclusively clarifiedyet [7], but might turn out to be a sufficient approach [33]. The large eddysimulation (LES) on the other hand is well suited for the investigation ofthe flow field within the combustion chamber. It captures the instationarymotion of the large scales, while modelling the small structures, which are tosome extent of universal character. Today there is a wide spread agreementon the high potential of the LES method to predict highly instationary andcomplex flows with and without chemical reactions [14], such as investigatedin the presented case. Hence the LES is the desirable tool for the simulationof acoustic sources. The experiments on the other hand are needed to validatesuch results. Here, the flow field, the mixing and the chemistry, as well as theacoustic properties should be considered.

The capability of combustion LES for non-premixed cases in particularhas been clearly demonstrated by rather simple benchmark configurations[2, 9, 16, 24]. But also in combustor-like configurations as well as for realcombustors – isothermal as well as reactive – LES was already successfullyapplied and the mixing as well as combustion phenomena have been inves-tigated [1, 14, 17, 33, 34]. In many of the applications of LES in complexgeometries mentioned above a comparison to experimental results is missingdue to the complexity of the apparatus or the experimental inaccessibility ofthe combustion zone. In the present work, the experiment is optically accessi-ble and therefore a validation of the simulation can be provided. Although thelong term objective of the work aims at predicting the combustion noise, as afirst step this shall be done for the isothermal case, since here the experimentcan provide a well founded basis of results.

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Towards the Prediction of Combustion Noise in a Gas Turbine Combustor 3

2. Experimental Setup

2.1. COMBUSTOR TEST RIG

The setup for the experimental investigation is carefully chosen to replicatecombustion flow characteristics of full scale gas turbines while still permittinganalysis by experimental means. In order to simplify the numerical approachthe system is designed axis-symmetrical.

Figure 1. Isometric view and photo of the experimental facility.

The test rig, shown in figure 1 consists of three sections: combustionchamber, outlet nozzle and exhaust duct. It is driven by a swirled dual air-flow nozzle in order to stabilise the combustion zone. A sketch of the burnerin figure 2 illustrates the inner and the outer co-rotating air-flow which areboth fed from a common plenum chamber. The inner air nozzle has a diameterof 15 mm and the outer annular air nozzle has 17 and 25 mm inner and outerdiameter, respectively. For thermal investigations methane gas can be injectedas fuel through an annular slot between the air streams. In the experiments de-scribed here the air flow field at room temperature discarding the fuel supply

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is analysed. Based on the inner nozzle diameter and the bulk velocity at thenozzle exit, �� m � s, the Reynolds-number is Re �� .

Figure 2. Layout of the dual swirl nozzle.

The combustion chamber itself is made of a fused quartz glass cylinderwith 100 mm inner diameter and 3 mm wall thickness. It has a length of113 mm and is terminated by a convergent–divergent nozzle structure shownin figure 3. The outlet nozzle shape is exchangeable, whereby the throat diam-eters and therewith the outlet Mach numbers are variable. This design is basedon the purpose of investigating the contribution of entropy noise sources tothe total combustion noise emission at different outlet Mach numbers. In thecurrent case an outlet diameter of 17 mm is chosen which results in a Machnumber of �� in the thermal operating point. In the isothermal modethe Mach number is, of course, with �� explicitly lower.

The outlet nozzle is attached to an exhaust duct with the same diameteras the glass combustion chamber. In order to reduce the impedance jumpat the exhaust outlet, an end diffuser is installed. In addition, the diffuseris perforated with holes of � mm diameter with increasing perforation den-sity towards the exit. This exhaust duct termination shall reduce acousticreflections in order to cut down dominant influence of these reflections onthe combustion process. It is designed in consideration of the results byShenoda [29]. For acoustic studies of combustion noise emission the exhaustduct is equipped with heat-proof probe microphones at three axial and fourcircumferential positions, respectively.

The underlying coordinate system of this flow configuration is plotted infigure 3 where the origin is fixed in the outlet plane of the swirl burner nozzleon the centre axis of the combustion chamber.

At the operating point of the subsequent investigations the mass flow rateof air amounts to 13 Nm � /h which corresponds in the thermal case including

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Figure 3. Sketch of the combustion chamber setup used for the model experiments.

1 Nm � /h of methane gas to a thermal capacity of 10 kW with an equivalenceratio of 0.726 and a nominal swirl number of � �!�"�� .2.2. DIAGNOSTIC TECHNIQUES

The velocity measurements were conducted using a commercial LDV system(DANTEC) and an Ar-Ion laser. The optical configuration consists of a two-component setup ( #%$'&)(+*-,.�/�0�� and ��1�� nm). As seeding particles ZrO 2with a particle size up to 10 3 m was used.

Performing LDV measurements in objects with curved surfaces demandscertain attention to refraction effects of the laser beams. Due to the cylindricalshape of the combustion chamber the effects are different for the two com-ponents of the LDV. Necessary corrections are described in detail by Fritz[10]. However, for the measurements along the optical axis of the LDV setuponly the axial velocity component was acquired. Two-component measure-ments were performed at fixed radial positions along axial lines. In order toanalyse the photomultiplier signals two DANTEC burst spectrum analysers(BSA) were used. In the case of two-component measurements coincidentsignals were evaluated which enables the extraction of cross-correlations( 4"�658795;: ).

In the thermal case also the temperature and the acoustic sound field,besides the flow field, are of particular interest. For the measurement of tem-perature fluctuations a laser based method is currently under developmentfor its application in reactive flows. The acoustic field is recorded using

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the above-mentioned installation of heat-proof probe microphones and anadequate data acquisition and analysing system.

3. Modelling and Numerical Setup

3.1. GOVERNING EQUATIONS

Since the flow velocities in most combustion applications are relatively low,the density can be assumed to be independent of the pressure. This yields anincompressible formulation and the low-Mach number approximation of thegoverning equations can be utilised. Still, the density varies strongly due tochemical reactions. The following two equations represent the Favre-filtered(density-weighted filtering: <>=? � < ? ) equations for mass and momentum,subjected to density variations.@ <@ ACB @@ D�E F <G=H EJI �"� (1)@@ A F < =H%K I B @@ D ELF < =H%K =H EMI � (2)@@ D E N < =O N @ =H K@ D E B @ =H�E@ DPKJQSR �� < =O

@ =HUT@ DUT6V K E B <XW (-Y)(K E QZR @ [@ D%KIn the case of non-premixed combustion, usually a fast chemistry approach

is sufficient. Furthermore, the heat loss due to radiation can be neglected inmost technical applications, especially in gas turbine combustors. Thereforethe complete thermo-chemical state of the fluid is governed by the mixture offuel and oxidiser, which is described by means of the mixture fraction \ . Thededuction of this approach is known as the “Shvab-Zeldovich” formalism. Forpure fuel, \ takes the value of \]�^ and for pure oxidiser the value of \_�� , respectively. The mixture fraction can be interpreted as a dimensionlessfuel concentration or enthalpy. The filtered transport equation for the mixturefraction reads as follows.@@ A F < =\ I B @@ D%K F < =\ =HPK I � @@ DPK N < =` @ =\@ DPK B <Ga (-Y)(K Q (3)

By filtering the transport equations for momentum and mixture fraction,two unresolved terms arise in the resulting equations (2) and (3) due tothe non-linearities of the convective terms. These are namely W (-Y)(K E , the sub-grid scale stresses, and a (-Y)(K , the sub-grid scale scalar flux, respectively. Thesub-grid scale stresses are closed by the standard Smagorinsky model [30].

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Hereby, the anisotropic part of the sub-grid scale stress tensor is modelledwith an eddy-viscosity assumption, while the isotropic part is included intothe pressure term.W (-Y)(K E R � W (bY)(TcT V K E �� OJd =� K E with OJd �fecgih j�k1lnmmm =� K E mmm (4)=� K E � � N

@ =H K@ D E B @ =H�E@ D%K Q (5)

The model coefficient g h in equation (4) is obtained by the dynamic pro-cedure – originally proposed by Germano [12] – using the least-squaresapproach following Lilly [20].

The filtering operation is performed implicitly by means of the finite vol-ume discretisation. Therefore, the filter width j coincides with the size ofthe grid cells. Furthermore, no special wall treatment is included into the sub-grid scale model. The approach relies on the dynamic procedure to reproducethe correct asymptotic behaviour of the turbulent flow near the wall, see e.g.Lesieur and Metais [19].

The second unclosed term, namely the sub-grid scale scalar flux a (-Y)(K ,is modelled by an eddy-diffusivity approach, similarly to the sub-grid scalestresses. Here, a constant turbulent Schmidt-number o d relates the turbulentdiffusion coefficient

` d to the turbulent viscosity O d .a (-Y)(K R ` d @ =\@ D%K with` d � O do d and o d "��p� R ��'q (6)

While this approach is the simplest approach available, it has been found togive accurate agreement in the past [2, 9, 16, 24].

In the case of non-premixed combustion, either equilibrium chemistry orsteady flamelets have been successfully employed. In both cases, the tem-perature, species mass fractions, as well as the density and the viscosityare a function of mixture fraction only. The filtered thermo-chemical vari-ables are obtained by an integration over the density weighted sub-grid scaleprobability density function (Pdf) of the mixture fraction =r F \ I .=? F =\s�Lt\ 5 5 l I ��u_vw ? F \ I =r F \ I d \ (7)

The shape of the Pdf in equation (7) is presumed to be a x -Pdf, parameterisedby the filtered mixture fraction =\ and it’s variance t\ 5 5 l as proposed by Cookand Riley [3]. =r F \ I =x F \sy =\ � t\ 5 5 l I (8)

For this well-known approach, the sub-grid scale variance of the mixturefraction has to be described as well. Instead of solving an additional transport

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equation for the variance [4], an algebraic model relying on the equilibriumassumption is used [2].t\ 5 5 l �g * j l N @ =\@ DPK @ =\@ DPK Q with g * ��z� (9)

The long term objective of the numerical part of the present investigationaims at performing incompressible reactive computations to evaluate the tur-bulent density fluctuations and use these as a source to predict the acousticcharacteristics of the configuration. Although such an approach needs a con-sideration of combustion, we restrict ourselves to an isothermal case fromwhich a first insight into the acoustic properties of the flow has to be gained.

3.2. NUMERICAL PROCEDURE

The previously presented governing equations are integrated by the threedimensional finite-volume CFD code FASTEST-3D. This code uses geometry-flexible, block-structured, boundary fitted grids and allows therefore torepresent complex geometries as presented in this contribution. A collocatedand cell-centred arrangement of the variables is used on the grid.

The spatial discretisations are performed by second-order central schemes,especially designed for complex grids by Lehnhauser and Schafer [18], ex-cept for the convective transport in the scalar equation. Here, a flux limiterscheme with total variation diminishing (TVD) properties has been used toensure bounded and non-oscillatory transport as solutions for the mixturefraction [32].

The pressure-velocity coupling is achieved by a SIMPLE similar proce-dure. A semi-implicit Crank-Nicolson method of second order accuracy isemployed for the time integration of the transport equations. Finally, the re-sulting set of linear equations is solved iteratively utilising Stones stronglyimplicit procedure (SIP).

The code is parallelised by domain decomposition using the MPI messagepassing library. For further information on the details of the employed methodrefer to Durst and Schafer [8].

3.3. NUMERICAL CONFIGURATION AND BOUNDARY CONDITIONS

For the numerical investigation two LES on different grids are performed. Inthe first simulation a total number of ��{��}|0~�� grid points is used. Based onthis coarse grid the second calculation is realised on a systematically refinedgrid which doubles the grid points in each logical direction so that a totalnumber of ��|�~�� arises.

To capture the expected hydrodynamic instability and the recirculationzone the computational domain includes the region of the swirler nozzle

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without the inner and outer circumferential channels (see figure 2). Instead,corresponding inlet velocity profiles were prescribed on the lateral swirlersurfaces. The third inlet region consists of the annular slot. In contrast tothe experimental investigation this inlet was fed with a dummy mass flow of;� � �� with no swirl component in order to analyse the mixing behaviour.Therefore the properties of air at ambient temperature and pressure are used.

The whole computational domain, shown in figure 4 consists of the dual-swirler nozzle, the combustion chamber and the Laval-shaped outlet nozzle.The radial direction in the exit plane of the swirler nozzle is discretised withqJ� ( �� ) cells and the circumferential direction with {�� ( z��{ ) cells, respec-tively. Altogether the numerical domain of the swirler nozzle consists ofapproximately q��{�{X|~~� 2 ( {��z�X|~~�� ) grid points. The outlet nozzle is mappedwith roughly ��|0~� 2 ( ��|�~� � ) cells.

A no slip boundary condition is applied for all solid walls in the domain.At the exit plane of the outlet nozzle a Neumann condition (zero gradient) isused.

In both simulations ��p� s of physical real time is discretised with ~��( �� ) time steps, which corresponds to approximately 40 flow throughtimes, based on the bulk velocity at the nozzle exit, �� m � s. Forthe evaluation of the statistics ~�� samples of the whole flow field arerecorded. First and second moments, as well as time- and lengthscales havebeen computed from these results.

4. Results and Discussion

In the following section we shall firstly address the statistical evaluation ofthe flow inside the combustor in order to validate the large eddy simulation.Therefore we present profiles of first and second moments of the main flowproperties, namely the velocity and the mixture fraction, obtained from ex-periments and the simulation. Secondly we will discuss temporal, as well asspatial autocorrelations to extract time- and lengthscales, important for theacoustic analysis of flow inside the configuration. Finally, the evidence forthe existence of a so called precessing vortex core (PVC) and its influenceon the flow through the Laval-nozzle is provided. Throughout the followingsection the outer diameter of the outer air nozzle is used for normalisationpurposes (

` �� mm).

4.1. STATISTICS OF THE FLOW

As the main comparison to experimental data, the mean axial velocity � isconsidered in figure 5. While the first position D �� mm is very close to thenozzle exit, the last profile at D ��~�� mm is situated close to the exit of the

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Figure 4. Two views of the numerical mesh for the coarse LES.

combustion chamber and therefore close to the inlet of the Laval-nozzle (seefigure 11). The overall agreement of the coarse LES with the measurements isvery reasonable, for the mean as well as the fluctuations of the axial velocity.One can clearly recognise the recirculation zone with a negative mean flow atthe axis ( �0� ` �"� I of the combustor. Also noticeable is, that the coarse LESpredicts the length of the recirculation zone too large. The velocity measure-ments at the position D �/�� mm exhibit an already positive mean flow atthe axis. First promising results of the fine LES exhibit differences and willimprove, when more samples for the statistical evaluation are available (sofar only two flow through times are considered).

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-4 0 4 812

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x = 30 mm

0246

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exp.

Figure 5. Profiles of the first and second moment of the axial velocity at different positionsalong the axis of the combustor for two different LES resolutions as well as experiments.

At the near nozzle position ( D �� mm), one can see the influence ofthe nozzle geometry on the flow field. The two swirling flows coming fromthe nozzle result in a double peak structure at this point. The fuel film fromthe airblast-nozzle is injected right into this shear layer. This ensures goodmixing between fuel and oxidiser. The fluctuations of the axial velocity alignwell with the experimental results. Not only the shape of the profiles, but alsotheir magnitude agrees well.

As expected, the flow accelerates towards the Laval-nozzle at the exit ofthe combustor. A mean Mach number of Ma "�� is reached numerically inthe throat of the nozzle. This compares well with the measured Mach-numberof Ma �� . While the mean flow accelerates, the velocity fluctuationsremain rather small, compared to the initial magnitude close to the burnerexit. One can therefore divide the combustor into three regions. The lowerregion, where the flow field is dominated by the swirl burner, resulting in arecirculation zone and the upper region, where the exit nozzle, acceleratingthe fluid, is the major influence factor. A very sensitive position is the interme-diate area between these two sub domains, forming the third region inside thecombustor. Here the mean velocities are rather small and evenly distributed

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along the radius, while the fluctuations are of larger magnitude as the mean,resulting in a nearly homogeneous and highly turbulent flow field.

Due to the experimental setup of the combustor wall via a glass cylinder,the measurement of other velocity components than the axial one is veryrestricted. One has to validate the simulation by existing measured profiles.Based on the good overall agreement obtained, LES predictions for otherquantities are provided nevertheless. Additionally, the LES solution consistsof cartesian velocity components. Hence the ‘radial’ and ‘tangential’ velocitychange their sign across the axis of the axis-symmetric configuration, but arestill named accordingly for clarity.

The following radial profiles presented in figure 6(a) provide a furtherinsight into the flow field of this configuration. The radial velocity � profilesexhibit, that the fluctuations are almost always larger than the mean value.Therefore, no direct conclusion about the instantaneous flow field can bedrawn from the mean profiles.

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a. Radial velocity b. Tangential velocity

Figure 6. Profiles of the ‘radial’ and ‘tangential’ velocities and their fluctuations at differentaxial positions.

The tangential velocity component 7 in figure 6(b) exhibits the expectedbehaviour. The two streams ejecting from the nozzle into the combustorare clearly visible. Furthermore, the swirling character of the flow extendsthroughout the complete inner combustor domain. An interesting feature ofthe mean flow is the almost linear range of the tangential velocity at the axis,indicating a fixed body rotation to some extent. On the other hand, the con-stant value of 4�7 :� �� m � s for the range � �0� ` �% ��~�~�c�� seemsto contradict this observation. Obviously, towards the wall, the tangentialcomponent has to decrease to zero, as does any other velocity component.

A comparison of the axial distribution of the velocity is presented infigure 7 for different radial positions ( ��y R ��y R z� mm). The positionshave been chosen to align with the axis, inside the inner shear layer and at

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the edge of the nozzle. Measurements of the axial and tangential velocitycomponents were possible in the axial direction at these radial positions. Theradial velocity component could not be captured by the experimental setup.In the figure, the axial dimension of the combustion chamber is indicated bytwo vertical dotted lines.

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a. Axial velocity b. Tangential velocity

Figure 7. Profiles of the mean axial and tangential velocity at different radial positions.

Considering the profiles at �� R z� mm, the agreement between theexperimental and numerical data is evident. The flow field is mostly domi-nated by the swirl of the outer slot in the nozzle. The two profiles towardsthe axis show a disagreement. Still, the phenomenological characteristics ofthe flow are reproduced by the LES. The quantitative difference betweenthe measurements and the simulation are probably due to multiple factors.The swirl number of the simulation deviates from the experimental value byapproximately 16%, as will be discussed later. Furthermore, the mass flowrate for the two swirling streams of the experimental setup is given only in aglobal manner – the exact distribution between the two streams is not known– and therefore the LES might have some deviating inflow conditions. Finally,the fuel jet injected into the shear layer between the two swirling air streamswas not present in the experiments. The flow inside the Laval-nozzle on theother hand is behaving as expected. It accelerates until the minimal diameteris reached and decelerates afterwards.

Since for the two-component measurements coincident signals were eval-uated, a comparison of the cross-correlation between the axial and tangentialvelocity component is presented in figure 8. Considering the large scatterin the experimental data, the good agreement between the LES and the ex-periments is evident. This underlines the predictive capabilities of the LESmethod for complex turbulent flows. As a final comment, the flow field inside

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the combustor is reproduced reasonably well by the LES, so further analysisof the simulation can be performed.

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<U’W

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2 /s2 ]

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Figure 8. Profiles of the velocity cross-correlation ��s�� at two different radialpositions.

A key feature in order to simulate non-premixed combustion is the mix-ture of the fuel jet with the swirling air streams. This mixture is describedvia the mixture fraction approach, as described in the previous section. Theradial profiles of the mixture fraction are given in figure 9. One can see thegood mixing properties of a swirl burner, as used in this configuration, sincealready at D ��q mm, the mixture is almost homogeneous. The fuel jet mixesimmediately in the shear layer between the two streams.

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Figure 9. Profiles of the mean mixture fraction and its fluctuations at two different axialpositions close to the burner exit.

For the sake of completeness, the resolved turbulent kinetic energy ¡¢�vl tH 5K H 5K is computed and presented in figure 10. The transition from the regionclose to the nozzle exit, with two distinct peaks of high energy in the shearlayer between the two swirling air streams on each side of the axis towardsthe Laval-nozzle with the highest energy at the centreline is visible.

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00.20.40.60.8

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-3 [m

2 /s2 ]

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coarse LES

Figure 10. Profiles of the resolved turbulent kinetic energy £ at different axial positions alongthe combustion chamber.

4.2. TIME- AND LENGTHSCALES

For the characterisation of the turbulent flow inside the combustor, time-and lengthscales have been evaluated at various positions in the flow field,as indicated in figure 11. This information is important in terms of theacoustic analysis of the configuration. The scales determine the possiblesize and frequency of noise sources inside the combustor. While the exper-imental measurements allow only for temporal analysis of the data, sinceone-point LDV measurements were performed, the LES delivers also spatialinformation.

To compute a timescale from the data, one has to integrate the temporalautocorrelation function. This function is defined for some quantity ? as givenin equation (10).

¤�¥�¥�¦ d F D%K �§j A I � 4 ? 5 F D%K � A I ? 5 F DPK � A B j A I :4 ? 5 l F DPK � A I : (10)

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x [m]

r[m

]

-0.05 0 0.05 0.1 0.15 0.2-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-2 0 2 4 6 8 10 12 14 16 18

<U> [m/s]

T1 T2 T3 T4

<U> [m/s]

Figure 11. Positions of the different autocorrelations. The colouring corresponds to the meanaxial velocity �¨�©� .

For the lengthscale, the spatial autocorrelation is required, which is definedvery similarly as shown in equation (11).¤�¥�¥�¦ ª F D%K �§j D%K I � 4 ? 5 F DPK � A I ? 5 F DPK B j DPK � A I :4 ? 5 l F D%K � A I : v+« l 4 ? 5 l F D%K B j DPK � A I : v+« l (11)

The time- and lengthscale is formally defined as the integral over the com-plete range of the time lag or the spatial lag, respectively [25]. This is shownonly for the spatial autocorrelation, but is fully equivalent for the temporaldata. ¬ ¥�¥ F DPK I � � u ¯®° ® ¤�¥�¥�¦ ª F DPK �§j D%K I d j DPK (12)

For practical purposes and further insight, the integration is split up intothe positive and negative part. By this procedure, one gets more informationon the structure of the turbulent flow into each direction, starting from thefixed position.¬ ¥�¥ � � e ¬ ¥0¥ B ¬ °¥�¥ k ¬ ¥�¥ �9u ¯®w ¤�¥�¥�¦ ª F DPK �§j D%K I d j DPK (13)

As a first result, the temporal evaluation of the simulation at the positionT1 (see figure 11) is presented in figure 12. In all spectra, a “ R �� ” line fromthe theory of homogeneous turbulence [25] has been included, indicating,that the present configuration is only to some extent complying with theory.Still, the cutoff frequency of the implicit LES filter lies well within the inertialsubrange.

The resolved energy spectra of the radial and tangential velocity compo-nent ±³²�² & ±³´U´ , extracted from the time trace at this position, exhibit a sharppeak at a frequency of \µ ��J�� Hz (highlighted by a vertical line in figure 12).

ftc.2004.paper.tex; 26/08/2004; 12:11; p.16


1

0

-1

20100-10-20

∆t [ms]

Ruu

10 0

10-2

10-4

10-6

10 310 210 1

f [Hz]

Euu

[m2 /s

2 ]

f-5/3pos. T1 1

0

-1

20100-10-20

∆t [ms]

Ruu

10 0

10-2

10-4

10-6

10 310 210 1

f [Hz]

Euu

[m2 /s

2 ]

f-5/3pos. T1

a. Axial velocity component

1

0

-1

20100-10-20

∆t [ms]

Rvv

10 0

10-2

10-4

10-6

10 310 210 1

f [Hz]

Evv

[m2 /s

2 ]

340

Hz

f-5/3pos. T1

1

0

-1

20100-10-20

∆t [ms]

Rw

w

10 0

10-2

10-4

10-6

10 310 210 1

f [Hz]

Ew

w [m

2 /s2 ]

340

Hz

f-5/3pos. T1

b. Radial component c. Tangential component

Figure 12. Resolved energy spectrum and temporal autocorrelation functions at the positionT1.

Preliminary evaluations of the finely resolved LES indicate a frequency of\µ "�� Hz in the spectra at the same position. This peak corresponds to therevolution frequency of a coherent structure, namely a precessing vortex core(PVC) and will be discussed in detail in the next section. The axial componenton the other hand does not show this feature. One can conclude, that thestructure is only revolving around the centreline, but not pulsating into thecombustor. The frequency of this coherent structure is also evident from theautocorrelation functions presented in the same figure. While the shape ofthe autocorrelation of the axial velocity is of a somewhat classical turbulentshape, the other two components are typical for swirling flows, exhibiting astrong coherent structure.

The temporal power spectrum of the axial velocity component in figure 13,acquired by experiments at a position D �9{ mm and �¶� R z� mm, exhibitsa frequency of \. �� Hz. This is close to the frequency stemming from thesimulation. The absolute deviation is approximately z��· , which lies wellwithin the range of possible prediction. The position is chosen such, thataverage position of the outer shear layer is captured. Therefore also the axialvelocity component exhibits the frequency of the precessing vortex core.

ftc.2004.paper.tex; 26/08/2004; 12:11; p.17


10-2

10-3

10-4

10 310 210 1

f [Hz]

Euu

[m2 /s

2 ] f-5/3

x = 6 mm, r = -12 mm 398

Hz

Figure 13. Temporal power spectrum of an experimental time trace of the axial velocitycomponent at the position ¸º¹9» mm and ¼½¹�¾À¿�Á mm, exhibiting a distinct frequencyof approximately ÂcÃ)Ä Hz.

Interestingly, the coherent structure can not be seen as clearly at the posi-tions T2 and T3 – the first one is shown in figure 14(a). Since the positionsare situated at the axis and the revolving structure is of a helical type, as canbe seen later, the frequencies are not as dominant at these positions. Still asmall peak is visible at the marked frequency of �J�� Hz.

1

0

-1

20100-10-20

∆t [ms]

Ruu

10 0

10-2

10-4

10-6

10 310 210 1

f [Hz]

Euu

[m2 /s

2 ]

f-5/3pos. T2

1

0

-1

1050-5-10

∆t [ms]

Ruu

10 0

10-2

10-4

10-6

10 310 210 1

f [Hz]

Euu

[m2 /s

2 ]

f-5/3pos. T4

1

0

-1

20100-10-20

∆t [ms]

Rvv

10 0

10-2

10-4

10-6

10 310 210 1

f [Hz]

Evv

[m2 /s

2 ]

f-5/3pos. T2

1

0

-1

1050-5-10

∆t [ms]

Rvv

10 0

10-2

10-4

10-6

10 310 210 1

f [Hz]

Evv

[m2 /s

2 ]

f-5/3pos. T4

1

0

-1

20100-10-20

∆t [ms]

Rw

w

10 0

10-2

10-4

10-6

10 310 210 1

f [Hz]

Ew

w [m

2 /s2 ]

f-5/3pos. T2

1

0

-1

1050-5-10

∆t [ms]

Rw

w

10 0

10-2

10-4

10-6

10 310 210 1

f [Hz]

Ew

w [m

2 /s2 ]

f-5/3pos. T4

a. Position T2 b. Position T4 – Throat Laval-nozzle

Figure 14. Resolved energy spectra functions at the positions T2 and T4.

Further downstream in the Laval-nozzle, the spectra change drastically,as shown in figure 14(b) for the position T4 located in the throat of theLaval-nozzle. Now a frequency band around approximately \� Åz��{�� Hzand its harmonics appear for all velocity components. This is four timesthe frequency of the PVC. The width of this peak in the spectra is large incomparison to one at the position T1.

ftc.2004.paper.tex; 26/08/2004; 12:11; p.18


As a result of the temporal evaluation at the different positions, a crudeestimate of the integral timescale can be provided. The timescale dependson the quantity that is evaluated. For the velocities, the timescale variesfrom �� to �� ms. These numerically obtained results agree well withexperimental findings. At the positions T2 and T3, the timescale of the ax-ial velocity is determined from the experiment to be z�� ms and z�� ms,respectively. Generally, the axial velocity component has a continuously in-creasing timescale as one moves downstream along the centreline. The radialand tangential component on the other hand have the largest timescales inthe intermediate region (see above), where the flow field is relatively homo-geneous. Finally, the timescale of the mixture fraction ranges from �� to�� ms. This increase occurs in the first region, close to the nozzle. Fromposition T3 onwards, the timescale is almost constant at the upper limit.Considering the areas of small timescales, the near nozzle area is the maincontributor to the noise sources. The throat of the Laval-nozzle is the secondarea, where noise is generated.

Further investigations of the spatial autocorrelation function, as defined inequation (11) yield an estimate for the integral lengthscale in the combustor.The autocorrelations along the radius have been integrated in the positive andnegative radial direction at the positions D �"��§�� & ��{ mm – correspondingto positions T1, T2 & T3. This results in a radial distribution of the length-scales as presented in figure 15. If one considers the inner domain of thecombustor ( R ÇÆÈ�0� ` Æ� ) – since here the wall has no large impact on theresults – three major conclusions can be drawn.

First, the profiles are symmetric with respect to the axis. This is necessarilyrequired, since the configuration is axis-symmetric. Small deviations arisefrom a lack of sampling points extracted from the LES. Second, the length-scale of the in-line velocity component

¬GÉ²�² (in-line with the radial direction)is approximately twice as large as the lengthscale of the two componentsperpendicular to the radial direction of evaluation

¬ ÉÊzÊ and

¬ É´U´ . This agreeswell with earlier observations in swirling flows [11, 27, 28]. Third, the length-scales grow with the axial distance from the nozzle. This can be expected,since the flow settles down when moving from the first region towards thesecond region, mentioned before.

Finally, the axial distribution of integral lengthscales inside the combustoris presented in figure 16. Again, the increase in lengthscales with axial dis-tance from the nozzle can be observed as before. Also the in-line component(this time

¬�ÉÊMÊ ) is approximately twice the perpendicular components. Inter-estingly, the positive part of the lengthscales (

¬ ¥�¥ ) increases much faster thanthe negative part. This might be connected to the length of the recirculationzone, which ends around D � ` Ë . Inside this zone, the velocities have asmaller spatial correlation length than downstream of the recirculation zone.Hence, the lengthscale in the positive direction includes these narrow corre-

ftc.2004.paper.tex; 26/08/2004; 12:11; p.19


0 2 4 6 8 L+

ϕϕL-

ϕϕ

Luu x = 0 mm

0 4 8

1216

L [m

m]

Lvv

0 4 8

1216

-2 -1 0 1 2r/D [-]

Lww

0 5

101520 L+

ϕϕL-

ϕϕ

Luu x = 20 mm

0 5

101520

L [m

m]

Lvv

0 5

101520

-2 -1 0 1 2r/D [-]

Lww

0 4 8

1216 L+

ϕϕL-

ϕϕ

Luu x = 56 mm

0 8

162432

L [m

m]

Lvv

0 4 8

1216

-2 -1 0 1 2r/D [-]

Lww

Figure 15. Radial distribution of integral lengthscales in the positive ÌUÍÎ~Î and negative ÌÐÏÎ~Îdirection at the positions ¸³¹µÑMÒbÁ)Ñ & Ó)» mm, corresponding to the positions T1, T2 & T3.

lations only as long as the fixed point lies inside the recirculation zone. Thelengthscales in negative direction on the other hand always include the recir-culation zone and therefore increase only slowly (and almost linearly) withaxial distance. Towards the Laval-nozzle the lengthscales decrease again,which aligns with a positive velocity gradient since the flow accelerates again.Here, turbulence gets enhanced and thus the correlation length decreasesagain.

020406080

100 L+ϕϕ

L-ϕϕ

Luu

01020304050

L [m

m]

Lvv

01020304050

0 1 2 3 4x/D [-]

Lww

Figure 16. Axial distribution of radial lengthscales inside the combustor.

In terms of the noise production of this configuration, there are two mainareas of sources arising from turbulent structures of small scales. These areasare related to high vorticity and therefore strong streamline curvature [26].The first area is close to the nozzle around D � ` �� and extending towardsD � ` , namely the recirculation zone (see figure 7). The second oneis located towards/inside the Laval-nozzle. This might be obvious, since at

ftc.2004.paper.tex; 26/08/2004; 12:11; p.20


these two positions, the cross-sectional area of the combustor varies strongly,yielding large velocity variations. But also the time- and lengthscales reflectthis observation, given that small time- and lengthscales result in strong noisesources.

4.3. PRECESSING VORTEX CORE

As shown in the previous section, power spectra computed from temporalautocorrelations from the LES (figures 12 & 14) exhibit dominant frequencieswith distinct peaks. These frequencies indicate the presence of a so calledprecessing vortex core (PVC), a hydrodynamic instability which is found inmany practical swirl flows. This kind of coherent structure develops for awide range of swirl and Reynolds numbers and is characterised by a preces-sion motion of the centre of swirl about the symmetry axis of the geometry.Depending on the geometry and flow conditions various regimes and shapesof PVC are known to exist. A recent review on this topic has been givenby Lucca-Negro & O’Doherty [21]. Since the portion of the total kineticenergy contained in the motion of the PVC is usually significant (this canbe seen directly in the temporal autocorrelations as well as in the powerspectra), the corresponding velocity amplitudes are non-negligible and thePVC can strongly influence mixing and in turn also combustion. Thereforethe PVC phenomenon is suspected to be at least one factor in the generationof combustion noise as well as combustion instability [31] and deserves someattention.

As recent publications show [5, 33], the method of large eddy simulation iswell suited to predict and to analyse the precessing vortex core phenomenon.Using instantaneous LES information, the PVC can be visualised by meansof some criterion for detecting coherent structures such as the # l -criterion[15]. Figure 17 shows isosurfaces of # l which is defined as the second largesteigenvalue of the following tensor.=� K T =� T E B =Ô K T =Ô T E (14)

Here, =� K E and =Ô K E are the symmetric and antisymmetric parts of the instanta-neous resolved velocity gradient tensor.

It can be seen that in the lower part of the airblast nozzle the vortex core isstraight and in a steady state. After the nozzle contraction, instability sets onand the vortex core is not aligned straight with the symmetry axis anymore,taking on a helical shape. This indicates that we are dealing with a spiral typevortex breakdown. The direction of the resulting precession is the directionof the swirl itself and the precession frequency, as obtained from the LESpower spectra, is \¢ ��J�� Hz. While for the acoustics the absolute value isof direct interest, dimensionless frequencies in terms of the Strouhal numberare relevant from a fluid mechanical point of view. The Strouhal number is

ftc.2004.paper.tex; 26/08/2004; 12:11; p.21


Figure 17. Visualisation of the precessing vortex core (PVC) by means of the Õ0Ö -criterion. Ahelical structure can be seen precessing around the centerline of the configuration. The timelag between the single snapshots is approximately ¿Ø×§Ù of the revolution period of the PVC.

defined as � A � \Ú| ¬� � (15)

where

¬and � are a characteristic length and velocity, respectively. Using

the same values as for the Reynolds number (inner nozzle diameter,

¬ �z� mm, and the bulk velocity at the nozzle exit, �^�� m � s), a value of� A �Û��{� is computed, which is well within the range expected for swirlflow instabilities.

The experiment yielded a somewhat higher frequency of �� Hz, implyinga misprediction of the frequency by the LES of about z��· . From the litera-ture it is known, that the frequency is a (in wide ranges linear) function ofthe swirl number � , where swirl number now means the actual one, not thegeometrical one. This implies that a slight misprediction of the tangential andaxial velocity profiles at the swirler exit can lead to a wrong swirl number andin turn to a different frequency. In the current case, the swirl number from theLES was computed as �]�"��p�0{ using the definition according to Gupta [13]– in this case,

¤corresponds to the outer nozzle radius of z�� mm.

�]��ÜÝw 4È<Ú:Ç4"��:}4Þ7 :È� l d �¤ | ÜÝ w 4ß<Ú:}4"�à: l � d � (16)

This is about z{�· less than the nominal value of �L�á� �� , which isalmost the same relative deviation as between the measured and simulatedfrequencies. Unfortunately, due to limited optical access no measurementsare available close to the swirler exit and hence it could not be checkedwether wrong profiles are responsible for the deviation. But if so, thereare three possible reasons for getting such profiles. Firstly, a non-sufficient

ftc.2004.paper.tex; 26/08/2004; 12:11; p.22


modelling/resolution of the swirler nozzle could lead to inaccurate results.Secondly, as mentioned before, the mass flow rates of the two swirlingstreams are not exactly known. Finally, there might be a non-negligible in-fluence of the fuel gas film on the swirling air flow. While the fuel injectionwas switched off for the isothermal measurements, it was included in thesimulation. Unfortunately, a clear judgement in this question cannot be madedue to the lack of LDV data near to the nozzle.

Inside the throat of the Laval-nozzle, a second dominant frequency arises,as was shown in figure 14(b). The peak lies around \� âz��{�� Hz, which isexactly four times the frequency of the PVC discovered in the simulation.Therefore, the structure in the nozzle is most likely triggered by the PVC.Both, the PVC and the coherent structure in the nozzle have the same di-rection of precession, another hint, that some connection between the twophenomena must exist. The exact phenomenon and mechanisms yieldingsuch a coupling are still not uncovered, but will be a major topic of futureinvestigations.

5. Conclusions

In the present work, a model combustor of a gas turbine has been investigatedexperimentally, as well as numerically by means of large eddy simulation.Extensive comparisons between the experiment and the simulation have beenperformed. So far we have restricted ourselves to the isothermal case first. Thecharacterisation of the flow field with respect to noise was the main target ofthis work. Therefore time- and lengthscales have been evaluated in order todetermine areas of high turbulence, equivalent to small structures.

The overall agreement between the LES and the experimental data isacceptable. Therefore, further investigations of this configuration are rep-resenting the physics inside the combustor and the exit nozzle connectedto the combustion chamber. The major noise source areas are close to theairblast nozzle, inside the recirculation zone, and inside the converging partof the Laval-nozzle at the exit. Here the smallest time- and lengthscales wereobserved.

A precessing vortex core has been found by means of temporal powerspectra and the # l -criterion. The agreement between the experimentally es-timated frequency of the coherent structure and the results of the simulationlies within the same range as the deviation of the swirl numbers. Therefore itcan be concluded, that the simulation captures the instationary and complexflow field well. Interestingly, the PVC triggers a coherent structure inside theLaval-nozzle, that has a precession frequency of almost exact four times theone of the PVC. This phenomenon has to be investigated further in the future.

ftc.2004.paper.tex; 26/08/2004; 12:11; p.23


The results of the isothermal case considered in this work are the first stepto the reactive case simulation. Only then combustion noise can be evaluated.It is necessary though, to validate the isothermal case first, otherwise a sepa-ration of noise arising from combustion and noise arising from the turbulentflow field is not possible. Same holds for the respective contributions of thedirect and indirect noise to the total noise emission in the reactive case. Thiswill obviously be the main task for the future research.

Acknowledgements

The authors gratefully acknowledge the financial support by the GermanResearch Council (DFG) through the Research Unit FOR 486 “CombustionNoise”.

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