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1 University of Huddersfied Repository Hardaupas, Y., Souopouos, N., Stetsyuk, V. and Tayor, A. M. K. P. Experimenta Assessment of subgrid scae Probabiity Density Function Modes for Large Eddy Simuation Origina Citation Hardaupas, Y., Souopouos, N., Stetsyuk, V. and Tayor, A. M. K. P. (2012) Experimenta Assessment of subgrid scae Probabiity Density Function Modes for Large Eddy Simuation. In: 16th Internationa Symposium on Appications of Laser Techniques to Fuid Mechanics, 9 12 Juy, 2012, Lisbon, Portuga. This version is avaiabe at The University Repository is a digita coection of the research output of the University, avaiabe on Open Access. Copyright and Mora Rights for the items on this site are retained by the individua author and/or other copyright owners. Users may access fu items free of charge; copies of fu text items generay can be reproduced, dispayed or performed and given to third parties in any format or medium for persona research or study, educationa or not for profit purposes without prior permission or charge, provided: The authors, tite and fu bibiographic detais is credited in any copy; A hyperink and/or URL is incuded for the origina metadata page; and The content is not changed in any way. For more information, incuding our poicy and submission procedure, pease contact the Repository Team at: E.maibox@hud.ac.uk.

2 Experimenta Assessment of subgrid scae Probabiity Density Function Modes for Large Eddy Simuation Yannis Hardaupas 1,*, Nikoaos Souopouos 1, Viachesav Stetsyuk 1 and Aex M. K. P. Tayor 1 1: Imperia Coege London, Department of Mechanica Engineering, South Kensington Campus, London, UK * corresponding author: y.hardaupas@imperia.ac.uk Abstract Fitered density functions (FDFs) of mixture fraction are quantified by anaying experimenta data obtained from two-dimensiona panar aser-induced fuorescence scaar measurements in the isotherma swiring fow of a combustor operating at a Reynods number of 28,662 for three different swir numbers (0.3, 0.58 and 1.07). Two-dimensiona fitering using a box fiter was performed on the measured scaar to obtain the fitered variabes used for presumed β -FDF and top-hat FDF for Large Eddy Simuations (LES). A dependant variabe from the measured scaar, which was a pre-computed temperature, was integrated over the experimentay obtained FDF as we as over the presumed beta or top-hat FDFs and a reative error in temperature prediction was cacuated. The experimentay measured FDFs depended on swir numbers and axia and radia positions in the fow. The FDFs were unimoda in the regions of ow variance and bimoda in the regions of high variance. The infuence of the fiter spatia dimension on the measured FDF was evauated and consequences for Subgrid modeing for LES discussed. 1. Introduction In Large Eddy Simuations (LES) of turbuent reacting fows, spatiay-averaged versions of the equations describing motion and chemica reaction are usuay soved on a grid, which is coarse reative to the smaest scaes of the fuid motion. The effects of the smaest scaes are modeed in order to cose the LES equations. Modes of the LES of combusting fows rey on the probabiity density function (PDF) of the sub-grid scae (SGS) species mass fractions, where sub-grid refers to motions that are smaer than the computationa grid sie. The fitered quantity f (veocity, mixture fraction etc.) in physica space is simpy defined as a convoution of non-fitered fied with a function or convoution kerne as. ' ' ' (1) f() x = f( x) G( x x) dx In a non-premixed fame, fue and oxidier diffuse into the reaction one where they undergo chemica reaction and are converted to combustion products. In fames that are near equiibrium the reaction rate is much faster than the diffusion rate, which is governed by the mixture fraction gradients. In turbuent reacting fows, the fame is stretched and wrinked due to turbuent motions so that the temperature, mixture fraction and species gradients increase. These gradients increase the diffusion out of the reaction one and the oca non-equiibrium increases. If the assumption of oca equiibrium is vaid then, e.g. a aminar fameet mode can be used to mode such regime. The fameet approach can be seen as a way of decouping chemistry from the CFD sover. In adiabatic systems fame temperature, species mass fractions etc. are parameters of the mixture fraction and the scaar dissipation rate that can be pre-computed. Therefore, it is possibe to obtain so caed ook up tabes with vaues of temperature, species mass fractions etc. as a function of scaar dissipation rate, the mixture fraction or both. Since in LES ony fitered vaues of the fow fied are known, it is impossibe to integrate the ook-up tabes in order to obtain the - 1 -

3 temperature and the species mass fractions because no subgrid distribution is known. A genera approach to integrate the ook-up fameet tabes over the subgrid scaes in computationa modeing is based on the soution of a transport equation for the FDF, which is simiar to the transport equation for the Reynods Averaged Navier Stokes (RANS) PDF approach. However, the most simpified and commony used approach is based on presumed fitered density functions, which are usuay approximated to be β or top-hat functions and parameteried by the first two moments of the mixture fraction, namey the fitered mixture fraction and its variance. The success of transported PDF methods (Pope, 1985) in the RANS context promoted the deveopment of the transported FDF concept in the LES context. Transported FDF methods have been deveoped for instance by Gao & O'Brien (1993), Coucci et a. (1998), Sheikhi et a. (2003), Raman et a. (2006). The vaidity and appicabiity of presumed FDF methods were aso investigated by severa authors using DNS data of nonpremixed reacting fows (Cook & Riey 1994; Jimene et a. 2000; Foyd et a. 2009). The resuts confirmed that the presumed FDF methods, based on beta-function, provided a good estimate for the FDF of the mixture fraction. However, Tong (2001) and Tong et a. (2005) showed that the rea FDF can substantiay deviate from the beta-function approximation. If an assumption of inearity within the LES ces is made, the FDF can aso be approximated by so caed top-hat function (Borghi and Moreau, 1977). The top-hat function, spanning from minimum mixture fraction vaue a to maximum vaue b, and the beta function are defined as: B ''2 (, ) = (, ) Π = a 1 ( 1 ) ( 1 ) ''2 1 b 1 1 a 1 b 1 d 0 b ; a ; a = ; 2 ( 1 ) a= 1 ''2 b ''2 = + ; = 12 (2) 2 ; b ( ) ( 1 ) = 1 1 ''2 where the parameters of the above equations are defined in Section 6 of the paper. The fitered mixture fraction is determined by the soution of the transport equations and its variance is usuay modeed by different agebraic modes. If the FDF is known then a dependant variabes can be computed. For instance, if chemistry is assumed to be infinitey fast then oca equiibrium is reached and mass fractions, a reaction rate or a temperature are functions of mixture fraction ony. In this case a dependant variabes (e.g. a temperature) can be computed. The concept of the FDF can aso be extended to finite-rate chemistry. In this case, the FDF is a joint function of mixture fraction and its scaar dissipation rate, but this is beyond the scope of the present work. Generay, the FDF can be constructed from any sub-fiter scaar variabe that is obtained from experiments as foows. At a given spatia ocation, a box fiter of known sie is appied and the scaar vaues, e.g. of mixture fraction, are extracted and binned into bins, which then can easiy be converted into the FDF by counting frequencies and dividing them by a reative bin width (bin width times the cardinaity of a set). Mathematicay the fitered density function of a scaar f is given according to the foowing formua (Tong, 2001): ( ) ˆ ( ' ) (3) f ( ˆ φ φ; x, t) = δ φ x, t φ G x x dx (4) The shape of the fitered density function may change the resoved dependant variabes such as temperature, species mass fraction, density etc. In this work the resoved dependant variabe was chosen to be the resoved temperature due to the interest of this work to modes for turbuent reacting fows. The beta and top-hat functions were parameteried by the first two moments of - 2 -

4 mixture fraction, i.e. resoved mean and its variance. The resoved dependant mean Temperature can be computed then according to the foowing formuas: ( ) T f = T ( ) f x, t d (5) 0 ''2 ( ) T β = T( ) B, d (6) 1 0 ''2 ( ) T Π = T( ) Π, d (7) The subgrid scae variance, used in the presumed FDF modeing approach, was computed according to the foowing formua: ''2 { } 2 ' = ( x) ( x) G( x x) dx (8) In order to compute the resoved mean temperature from the measured mixture fraction, the reationship between temperature and mixture fraction must be known. This assumption is vaid in the case of fast chemistry and is used in this work to convert the measured scaar concentration into temperature. In the case of finite chemistry a simiar approach to compute the resoved mean temperature can be adopted even though it is not considered here. The reationship between fame temperature and mixture fraction was computed by using the commercia software CHEMKIN-Pro. CHEMKIN-Pro is made of different FORTRAN subroutines that aow modeing of different reacting systems under different operating conditions. The software uses detaied chemica mechanism to compute chemicay reacting fows, which is by defaut the GRI-3.0. However, in this work the San Diego mechanism, thermodynamic database and the transport database were chosen. This data were pubished by University of Caifornia at San Diego (UCSD). In the San Diego mechanism, the number of species and reaction is kept to the appropriate minimum needed to accuratey describe the modeed system, thus minimiing the uncertainties from the chemica reaction rate parameters. This approach significanty differs from other data sets, for exampe GRI-3.0. The other data sets attempt to incude a potentiay reevant eementary steps that can ead to the uncertainties, especiay when arge number of steps is used. A subroutine (OPPDIF) was used to compute the fame temperature of methane/air diffusion fame at atmospheric pressure and temperature. OPPDIF simuates the fame structure for axisymmetric or panar diffusion fames between two opposed fow noes. This procedure is equivaent to generating aminar-fameet ibraries to be used with CFD codes. The OPPDIF subroutine is based on a mode that was originay proposed by Kee et a. (1988), where the detaied derivation of the governing equations that are used in OPPDIF can aso be found and are not presented here. The remaining paper is structured as foows. Section 2 describes the swiring air fow burner and the optica instrumentation used to measure the scaar, which is then converted to the temperature according to the expanation provided above. Section 3 describes the resuts and discusses the findings in terms of the differences between the measurements and the resuts from the presumed FDF approaches used in LES. The paper ends with a summary of the main concusions. 2. Experimenta setup The burner used in this study is shown in Figure 1 and was based on the design of Dixon et a. (1983) and repicated in the work of Miosavjevic (1993). The burner comprised two concentric pipes with the annuus suppying air and the centra pipe deivering air seeded with acetone vapor, which is the measured scaar quantity. The centra pipe had an inner diameter D f of 15mm - 3 -

5 and an outer diameter of 18 mm, was 0.75m ong and was ocated concentricay in the outer pipe of inner diameter, D, 50.8 mm and ength downstream of the swir chamber of 264mm. The annuar air stream was divided into two separatey metered streams. One (`tangentia air') passed through a static Swirer containing six tangentia sots to impart anguar momentum. The static Swirer was ocated in a penum chamber in which the Swiring air was combined with the second stream, which deivered 'axia air'. Meta pates were instaed in the axia and tangentia air sections of the penum chamber to ensure that the axia and tangentia air streams were distributed uniformy upstream of the inets into the annuar air suppy stream of the burner, where they were combined to contro the strength of Swir at the burner exit. The tangentia and axia air fowrates were metered by fowmeters after correction to atmospheric pressure and temperature. The axia direction, parae to the fow propagation, was denoted as y and the radia direction, perpendicuar to the main direction of the fow, as x. The swir number is defined as foows: S = 2G Θ (9) G Z D where S is the Swir number, G Θ is the axia fux of anguar momentum and G is the axia fux of axia momentum and are determined beow as: G Θ = 2πρ G Z = 2πρ R r =r i R r =r i WrUrdr UUrdr where U and W are the axia and tangentia veocity components and r i and R the radius of the inner and outer pipe respectivey. The swir number of the secondary air was cacuated in the work of Miosavjevic (1993) using measured axia and tangentia veocity profies with a aser Dopper anemometer at 0.04D downstream of the exit pane of the burner and its vaue is presented in Tabe 1 aong with operating conditions used during experiments. Tabe 1: Operating conditions S V axia, V swir, V fue, U fue, U air, /min /min /min m/s m/s Re, air Re, fue Note: U denotes veocity, S swir number and V- voumetric fow rate in iters per minute A cass-iv high power Q-switched Nd:YAG frequency-quadruped 266nm aser from Continuum Inc., USA was used to excite the acetone vapor fuorescence. The aser beam was produced at the fundamenta waveength of 1064nm (1050mJ) and passed through frequency douber to produce 532nm (green ight, 455mJ) then passed through a fourth harmonic generator in order to produce 266nm ( : 120mJ) at the exit port of the aser. The aser puse width was in the range of 5: 7 ns at Fu Width at Haf Maximum (FWHM). A singe diachronic mirror was used to decrease the amount of 532nm ight reaching the burner and to steer the aser beam from the aser exit port towards the experimenta setup and sheet forming optics. The sheet forming optics consisted of a singe cyindrica positive ens with foca ength of 300mm. The ens was used to focus the aser beam into a thin waist, with foca point ocated at the axis of symmetry of the fow. The average aser energy at the test section was 70mJ/puse. Actua aser energy fuctuated during experiments from puse to puse and was equa to mj/puse

6 Instantaneous corrections of the sheet profie intensity variations were not performed. The mean aser sheet profie was acquired and was used in the image correction procedure. Figure 1: Atmospheric burner used for scaar measurements under isotherma conditions, aser sheet ocations for optica measurements and optica instrumentation arrangement. A doube frame Charge Coupe Device (CCD) (Imager Intense) camera from LaVision Inc. was used in this work. This camera was equipped with a Nikkor ens with foca ength f = 50mm and f#1.4 (where f# is the f-number). A 13mm extension ring was aso used after the camera ens for tighter focusing. A BG3 bandpass Schott fiter was used in front of camera ens in order to bock any remaining 532nm ight that might interfere with the acquired signa. Any scattered 266nm UV ight that might come from the test voume was bocked by the gass materia of the camera ens. The array sie of the CCD camera was 1376 x 1040 pixes and the pixe sie was 6.45 µ m. The optica magnification was determined by using a caibration target pate and was found to be mm/pixe. The aser sheet thickness was determined to be 0.15mm. Both the aser and the CCD camera were eectronicay controed from software package DaVis 7.2, provided by LaVision Inc. USA, instaed on a computer running Windows XP. The computer, which ran the LaVision software package (version 7.2) was used for a aspects of the experiment, incuding tuning the system, caibration and acquiring the experimenta data. Data communication ines from the aser and camera were input to the computer. Post-processing was performed by using commercia software package MATLAB R2009b in order to quantify the acetone vapor concentration. This was converted to Temperature using the approach described in Section 1. The FDFs were cacuated from the instantaneous imaging for two box fiter sies, namey 0.3 and 1.0mm. The ratios of the fiter sie to turbuent dissipative scaes are summaried in Tabe 2. Tong et a. (2007) pointed out that fiter sies shoud be significanty arger compared to the dissipative scaes to ensure that the resuts are reevant to LES. However, it is not yet cear what the optima ratio of typica LES fiter sie to dissipative scaes is. As noted by Meyers et a. (2008), a good LES is amost DNS. In genera, the LES resuts shoud be in better agreement with DNS if the LES fiter sie Δ is chosen to be as cose as possibe to Komogorov scae and when Δ< η the turbuence is fuy resoved. Larger fiter sie, i.e. arger than 1.0mm in the current fow, was not used because the FDF was found to be bimoda even at this fiter sie. S η, m Tabe 2: Fiter to dissipative scae ratio. Two box fiters were used, i.e. 0.3 and 1.0mm. λ, m / / / Δ / µ B µ Δ η Δ= 0.3 Δ λb Δ= 0.3 Δ η Δ= 1.0 λb Δ=

7 The process of cacuation of the FDF from two-dimensiona data invoved binning the scaar data in equay spaced bins. The discrete vaues of scaar data that fe in each bin were counted so that frequency of occurrence was cacuated. The vaue of FDF was then obtained by normaiing estimated frequencies by the corresponding bin width and the tota number of sampes in a bins. The tota number of sampes depends on the box fiter sies. For instance, if the box fiter is 1mm, which corresponds to 38 pixes on the images, then the tota number of sampes wi be 38x38. The fitered mixture fraction fied in physica space was simpy obtained as a convoution of non-fitered fied with a box fiter according to Eqn. (1). The subgrid scae variance, which was used in the presumed FDFs, was computed according to Eqn. (8) as foows. A box fiter was appied, e.g. 1.0mm, which occupied an area of 38 pixes by 38 pixes on the instantaneous images of scaar concentration. Then, by pacing this spatia fiter at one ocation of an instantaneous image the average vaues were computed as the sum of a sampes within the fiter box divided by a number of sampes in the bin. This mean vaue, which is defined as resoved mean, was assigned at the center of the box fiter and then the fiter was transated by 1 pixe increment steps aong the instantaneous image, thus performing the fitering procedure. The subgrid scae variance vas computed by subtracting the mean (computed from the box fiter) of each instantaneous mixture fraction sampe that was within the fiter area. The resuting vaues were squared, averaged and assigned to the centre of the fiter box. Nine different spatia ocations at each aser sheet axia position were studied. The specific positions are shown in Figure 2 where the effects of swir are to be examined as a function of degree of mixing. The six spatia ocations were chosen to be within the shear ayer, where highy segregated fow within the fiter is observed, and three at the burner centerine, where neary homogeneous fow fied is expected. In fact, windows 1, 4, 7 and 3, 6, 9 demonstrate simiar trends and were chosen to decrease possibe random errors. Figure 2: Time-averaged mixture fraction distribution (computed from 1500 reaiations) for axia distance from the burner exit of 1D f and S=0.3 and seected spatia ocations used to compute the FDFs. The spatia ocation used to compute the FDF was chosen within the shear ayer and the burner centerine, which is denoted as vertica dashed ine. The figure is presented on non-dimensiona coordinates, namey dimensioness centerine (axia) positions and dimensioness radia positions. The radia distance from the centreine was normaised by the burner radius R and the axia distance from the burner exit was normaised by the centra pipe diameter D f. 3. Resuts and discussion Exampes of measured instantaneous (before any spatia fitering) mixture fraction spatia distributions at different axia distances from the burner exit, i.e. 7D f, 5D f, 3D f, 1Df (from top to bottom respectivey) and S=0.58 are shown in Figure 3. The diameter of the centra pipe is Df=15mm and the sma short vertica ines at ±0.3 x/r denote its position. The vertica dashed ine denotes the burner centreine. As expected the mixture fraction is maximum cose to the centreine and reduces with radia distance, since the scaar concentration is ower at the edge of the centra jet. The mixing process, which is enhanced by the presence of swir, ensures that the scaar is we mixed away from the burner exit. Therefore, the spatia variation of the mixture fraction is sma at 7D f, which resuts in sma subgrid scae variance after fitering

8 Figure 3: Exampes of measured instantaneous mixture fraction fieds at different axia distances from the burner exit, i.e. 7D f, 5D f, 3D f, 1D f (from top to bottom) and S=0.58. At 7D f the variation of the mixture fraction is sma, which resuts in sma subgrid scae variance. This in fact wi indicate that at this ocation the FDF is bimoda. Vertica dashed ine denotes the burner centerine. D f =15mm, R = 25.4mm. Sma short vertica ines at ±0.3 x/r denote the position of the centra pipe

9 Figure 4 shows an exampe of two measured FDFs of mixture fraction estimated from the presented singe reaiation (instantaneous scaar fied) at window 1 of Figure 2 for two different spatia fiter sies of 1.0mm and 0.3mm (shown as two back boxes on the figure) at 1D f and S=0.3. This figure demonstrates the quaitative change of the shape of the FDF as a function of box fiter sie. It is cear that for arge sie fiter of 1.0mm, the FDF is bimoda and it becomes unimoda when the sie of the spatia fiter is reduced to 0.3mm. In addition, if the box fiter overaps with an area that demonstrates steep gradient of mixture fraction, the FDF can be bimoda. In contrast, if the box fiter is chosen to be sma enough (i.e. cose to Komogorov or Batcheor scaes) then it resoves homogeneous mixture fraction regions, in which the SGS variance is sma. Due to space imits, the findings for other window positions are not supported by graphica representations. However, at window 2 and for S=0, the FDF is approaching a deta function for the both box fiter sies, i.e. 0.3 and 1.0mm, due to the fact that the mixture fraction is aways one at that region. Simiar resuts can be observed at window 2 (as we as at windows 5 and 8) for S=0.3, because mixing is good and the fied is neary homogeneous. It shoud be noted that windows 1, 4 and 7 mirror the behaviour of the FDF in windows 3, 6 and 9. In fact, a nine spatia ocations in the fow were considered to ensure that the FDF behaviour is not affected by the data processing method or random errors. Figure 4: Exampe of two measured FDFs estimated from the presented singe reaiation (instantaneous scaar fied) at window 1 for two different spatia fiter sies of 1.0mm and 0.3mm (shown as two back boxes on the figure) at 1D f and S=0.3. Vertica scae is not provided on the instantaneous scaar image because the figure aims to iustrate the quaitative change of the shape of the FDF as a function of box fiter sie. Severa measured FDFs for different arge and sma SGS variances of mixture fraction are presented in Figures 5 and 6 as a function of axia position in the fow and fow swir number. It shoud be noted that the FDF computations are presented here for one ocation ony (at window 1 of Figure 2) due to imitations on the ength of the paper. The conserved scaar FDFs exhibit two ceary visibe regimes, which are denoted as homogeneous (Figure 6) and nonhomogeneous (Figure 5). The term homogeneous is reated to the unimoda distribution of the FDF, whie the term non-homogeneous is reated to bimoda distribution of the FDF. The two different shapes of FDF distributions were observed in fow regions where high and ow mixture fraction segregations were present (see Figures 5 and 6). Bimoda distributions - 8 -

10 occur for both fue-rich and fue-ean vaues of mixture fraction, when SGS variance is arge. Unimoda distributions are found when SGS variance is sma. When SGS variance decreases, the corresponding beta-fdf and top-hat functions are cose to each other and in a imiting case become deta-functions. Figures 5 and 6 aso present the corresponding shapes of presumed β -FDF and top-hat FDF, constructed from the resoved mean and the measured SGS variance (according to Eqn. 1 and 8). Figure 5 presents the FDF measurements from window 1 at 5D f, 3D f, 1D f for two fow regimes with S=0 (axia jet without cofow of air) and S=0.3 and for box fiter sie of 1.0mm. Figure 6 presents measurements at the same axia positions for S=0.58 and S=1.07. Figure 5 iustrates that when fow is segregated within the fiter voume the FDF becomes bimoda, which is due to non-mixed regime. On the other hand in Figure 6 unimoda distributions are ceary observed. The effect of swir number can be seen in Figure 5 (for S=0 and S=0.3) and Figure 6 (for S=0.58 and S=1.07) as an increase in the eve of mixing, which increases homogeneity of the fow fied. That is, at the same spatia ocation in the fow, but for different swir numbers the FDF can be either bimoda or unimoda. The resuts confirm that higher swir numbers increase unimodaity, especiay at arge axia distances, i.e. 5Df and 7D f. A unimoda distribution was observed at a nine spatia ocations for 1500 instantaneous reaiations at 7D f and swir number of 1.07 due to sma subgrid scae variance. This is aso consistent with Tong et a. (2007), who measured the conditiona fitered mass density functions (FMDF) in reacting fows. They pointed out that the conditiona FMDF was found to be unimoda, and not far from Gaussian, when the subgrid scae variance was sma. The SGS variance represents how far mixture fraction vaues can deviate from the mean and hence the SGS variance can characterie the rate and the eve of mixing within one LES ce. For arger variance, the FMDF was found to be bimoda. Larger subgrid scae variance indicates that fue and air are essentiay segregated within fiter voume. The SGS variance depends on the LES fiter width and unmixedness and therefore is directy reated to the oca scaar mixing. It shoud be noted that the actua spatia resoution of the mixture fraction and the actua box fiter sie are arger than the nomina vaues due to optica system distortions and imperfections. Extra pixes shoud be added to each spatia fiter to compensate for imperfections of the optica system. It is generay assumed that the difference between nomina and actua fiter sie wi make itte discrepancy in the FDF. The integrated resoved mean temperature obtained from CHEMKIN soution is summaried in Tabe 3. The tabe describes severa temperature integrations as the dependant variabe for severa SGS variances. The temperature integration was performed for severa SGS variances, so that arge and sma variances were considered. For arge variance and hence high bimodaity, the difference between the temperature computed with actua measured FDF and the beta-fdf was ranging from 5% to 0.2%. In most cases, the difference in the resoved temperature cacuation computed by different presumed FDFs is not significant (ess or equa to 1%). This impies than the actua source of error in LES computations may be due to subgrid variance modes and not due to presumed FDF approaches. This is aso consistent with the concusions from the LES cacuations of Foyd et a. (2009). However, in some cases the difference can be as high as 18% for the top-hat FDF. On the other hand, there is no evidence that arge SGS variance can introduce significant errors because there is no direct correation between the measured SGS variance and the computationa error, computed from the dependant variabes. This is in fact appied to beta-fdf, where arge temperature error can be found for smaer SGS variance and in turn smaer errors can be found for arger SGS variance. The top-hat FDF can potentiay introduce significant errors, especiay when arge SGS variance is observed. It shoud be pointed out that the beta-fdf function can provide an adequate approximation of the true FDF of the mixture fraction. Even in case of arge variances, it does not automaticay impy that bimodaity can be the source of error. It aso shoud be pointed out that when SGS variance is sma, the beta-fdf and the reativey simpe top-hat FDF are cose to each other

11 Figure 5: Measured FDFs computed from singe reaiations at window 1 of Figure 2 potted for different arge variances; β -FDF and top-hat FDF are shown as green dashed and bue dashed ines. The bimoda shape indicates that the fue and air are essentiay segregated within the box fiter voume indicating nonmixed regime. The segregation can be due to insufficient mixing or due to the fiter sie, i.e. for the same spatia ocation two different fiter sies produce different FDFs. It shoud be noted that the vaues of the FDF printed aong y-axis are not the same. From top to bottom are shown the FDFs at 1D f, 3D f and 5D f, where on the eft is the fow for S=0 (axia), on the right is the fow for S=0.3 for box fiter sie of 1.0mm

12 Figure 6: Measured FDFs from instantaneous reaiation potted for different sma variances and box fiter of 1.0mm; β - and top-hat FDFs are shown as green dashed and bue dashed ines respectivey. The unimoda shape indicates that the subgrid scae variance is sma, which is due to either we-mixed regime or due to non-mixed regime as, for exampe, in a potentia core of a jet. It shoud be noted that the vaues of the FDF printed aong y-axis are not the same. From top to bottom are shown for 1D f, 3D f and 5D f, where eft coumn corresponds to S=1.07 and eft to S=

13 16th Int Symp on Appications of Laser Techniques to Fuid Mechanics Tabe 3: Resoved mean temperature integrated over the measured and presumed beta and top-hat FDFs ''2 T,K T β,k T,K Δ,% Δ,% f , Note: Computationa error is cacuated according to the foowing formuas Tf T β Tf T Δ β = 100% (beta-fdf); Δ = 100% (top-hat FDF) T T f f β 4. Concusions This paper presents an attempt to anaye the properties of the fitered density function of mixture fraction, measured in an isotherma swiring fow after fitering at two different scaes, and compare it with assumed beta-fdf and top-hat FDF, which are commony used in LES computations. The experiments were carried out on the swiring isotherma air fows seeded with acetone and confirmed the theory that the shape of the FDF was highy dependent on the subgrid scae variance. It was proven that unimodaity or bimodaity can be observed at the same spatia ocation for different spatia fiter sies, different axia ocations and different swir numbers. On the other hand a direct correation between the FDF shape and dependant variabe integration error was not found. This means that even when FDF is highy bimoda, the presumed top-hat or β -FDF integrate the resoved mean temperature correcty. However, in some cases with sma SGS variance, the temperature might not be integrated correcty either by β -FDF or top-hat FDF. Therefore, the main error that may arise in LES simuations is an adequate prediction of the SGS variance. Nevertheess, the concept of the fitered density function appied to LES computations is sti an adequate choice for soving combustion probems given that the SGS variance is modeed accuratey. 5. Acknowedgment This work was partiay supported by Aan Howard schoarship for Energy Futures

14 6. Nomencature Mixture fraction (instantaneous unfitered) ''2 Mean vaue of resoved Mixture fraction T Variance of resoved mixture fraction f T β Resoved mean temperature computed by integration of measured FDF, [K] T Π Resoved mean temperature computed by integration of presumed β -FDF, [K] Δ Resoved mean temperature computed by integration of presumed top-hat FDF, [K] Π Δ β Integration temperature computationa error between β -FDF and the measured FDF, [%] Δ Π Integration temperature comp. error between top-hat FDF and the measured FDF, [%] f Measured FDF B Presumed β -FDF Π Presumed top-hat FDF η Komogorov ength scae, [ µ m ] λ Batcheor ength scae, [ µ m ] B Δ Box fiter width, [mm] S Swir number 7. References 1. J. Foyd, A. M. Kempf, A. Kronenburg and R. H. Ram. A simpe mode for the fitered density function for passive scaar combustion LES. Combustion Theory and Modeing, 13:4, pp , A. Kempf, W. Maaasekera, K. K. J. Ranga-Dinesh and O. Stein. Large eddy simuations of swiring non-premixed fames with fameet modes: A comparison of numerica methods. Fow Turbuence Combust, W. Koman and J. Janicka. The probabiity density function of a passive scaar in turbuent shear fows. Physics of Fuids, 25 (10), pp , J. Meyers et a. (eds). Quaity and Reiabiity of Large-Eddy Simuations, Springer Science+Business Media B.V p V.D. Miosavjevic. Natura gas, kerosene and puveried fue fired swir burners. PhD thesis. University of London, Juy S. B. Pope. Sef-conditioned fieds for arge-eddy simuations of turbuent fows. J. Fuid Mech., Vo. 652, pp , S.B. Pope. Turbuent fows. Cambridge University press, S. B. Pope. Ten questions concerning the arge-eddy simuation of turbuent fows. New Journa of Physics 35, 16 March N. Souopouos. Experimenta investigation of scaar mixing in unsteady turbuent jets. PhD thesis, University of London Apri C. Tong, J. Wyngaard and J. Brasseurs. Experimenta Study of the Subgrid-Scae Stresses in the Atmospheric Surface Layer. J. of Atmospheric sciences, Vo Juy C. Tong. Measurements of conserved scaar fitered density function in a turbuent jet. Physics of Fuids, Vo. 13, No. 10, October C. Tong. Measurements of conserved scaar fitered density function in a turbuent jet. Physics of Fuids, Vo. 13, No. 10, October