A hybrid finite volume – transported joint probability density function (FV/JPDF) method is used to model piloted flames with inhomogeneous inlets. The flames were experimentally investigated using a retractable central tube within the main burner to control the degree of mixing at the exit. A five-gas (C2H2, H2, CO2, N2, air) co–flow pilot located outside the burner was used to match the composition and adiabatic temperature of a stoichiometric methane/air flame. The applied hybrid method features a flow field calculation using a time-dependent finite-volume based method closed at the second-moment level with the scalar field obtained at the joint-scalar (JPDF) level. The current methodology is applicable to both premixed combustion and diffusion-dominated regions without assumption regarding the inclusion of the chemistry. Results show that the current method can accurately capture the stratified premixed flame mode near the burner exit as well as the diffusion-dominated flame far downstream. The transition between the combustion modes occurs around ten tube diameters downstream of the burner exit and it is observed that the flame structure is very sensitive to the prediction of the flow field in this region

Lindstedt, RP

Tian, L

English

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Transported PDF Modelling and Analysis of Partially Premixed FlamesL. Tian, R. P. Lindstedt∗Department of Mechanical Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK.AbstractA hybrid finite volume – transported joint probability density function (FV/JPDF) method is used to model piloted flames withinhomogeneous inlets. The flames were experimentally investigated using a retractable central tube within the main burner tocontrol the degree of mixing at the exit. A five-gas (C2H2, H2, CO2, N2, air) co–flow pilot located outside the burner was usedto match the composition and adiabatic temperature of a stoichiometric methane/air flame. The applied hybrid method featuresa flow field calculation using a time-dependent finite-volume based method closed at the second-moment level with the scalarfield obtained at the joint-scalar (JPDF) level. The current methodology is applicable to both premixed combustion and diffusion-dominated regions without assumption regarding the inclusion of the chemistry. Results show that the current method can accuratelycapture the stratified premixed flame mode near the burner exit as well as the diffusion-dominated flame far downstream. Thetransition between the combustion modes occurs around ten tube diameters downstream of the burner exit and it is observed thatthe flame structure is very sensitive to the prediction of the flow field in this region.1. IntroductionPartially premixed flames caused by compositional inhomo-geneities are common in practical combustion devices. Exam-ples include flames in gas turbines, gas direct injection enginesand industrial burners [1–3]. The intrinsic nature of multi-modecombustion presents difficulties for the application of regimedependent approaches such as the laminar flamelet concept [4]and conditional moment closures [5]. The transported PDFmethod has the potential to resolve such problems as the chem-ical reaction source term appears in closed form [6].An excellent experimental database of partially premixedCH4-air flames has been made available through the Sydneypiloted burner with inhomogeneous inlets [7, 8]. The ge-ometry features a concentric retractable inner tube that con-trols the degree of mixing at the burner exit plane. Subse-quent measurements of temperature and species concentrationshave been made at Sandia [8] and can be used to analysethe physics behind the transition from stratified-premixed todiffusion-dominated combustion. Attempts to develop rathersimple combustion models have been made [9, 10] with signif-icant deficiencies for highly inhomogeneous cases.The current study considers a strongly inhomogeneousSydney–Sandia case with a complex transition between com-bustion modes. The inner tube with pure methane was recessedinto the outer air-supply tube by 75 mm (10 × the main tubediameter Dj). A five-gas (C2H2, H2, CO2, N2, air) pilot waslocated outside the central burner assembly with the composi-tion and adiabatic temperature matching that of a stoichiomet-ric methane/air flame. A joint-scalar transported PDF methodwith the velocity field closed at the second moment level [11] iscombined with a systematically reduced H/C/N/O mechanismfeaturing 300 reactions, 20 solved and 28 steady-state speciesfor the thermochemistry and the modified Curl’s model [12].Computational results are compared with measurements alongthe radial direction at different axial positions.2. Computational ModelThe methodology used in the present study is basedon a hybrid finite volume/joint transported PDF (FV/JPDF)method [11] with an extensively validated reduced H/C/N/O∗Corresponding author: p.lindstedt@imperial.ac.ukProceedings of the European Combustion Meeting 2017chemical mechanism [11, 13]. The flow field is solved using anelliptic time-dependent, two-dimensional axi-symmetric com-pressible flow solver closed at the second-moment level. Thegeneralised Langevin model of Haworth and Pope [14] is usedfor the pressure redistribution terms and combined with a stan-dard closure for the turbulent kinetic energy dissipation rate(ε̃) [15] with Cε2 = 1.80. The JPDF method provides the scalarfield and the density for the finite-volume part. The joint-scalarPDF is given below [6]:∂ρ̄P̃ (Ψ)∂t+∂ρ̄ũlP̃ (Ψ)∂xl+N∑α=1∂∂Ψα(ρ̄Sα(Ψ)P̃ (Ψ))=− ∂∂xl(ρ̄〈u′′l |φ = Ψ〉P̃ (Ψ))+N∑α=1∂∂Ψα(〈1ρ∂Jl,α∂xl∣∣∣∣φ = Ψ〉) (1)The above composition (and enthalpy) PDF equation is solvedusing a Monte–Carlo method featuring Lagrangian particles.The turbulent transport of the joint PDF is closed through a gra-dient diffusion approximation with the turbulent Prandtl numberset to 0.7 and the mixing term is modelled using the modifiedCurl’s model [12] using the standard scalar time-scale [11, 16]ratio defined in Eq. (2) with Cφ = 2.3 [11, 13].τ−1φ =ε̃φφ̃′′2 =Cφ2ε̃k̃(2)3. Case Configuration3.1 Experimental setupThe case (FJ200-5GP-Lr75-80) discussed below was inves-tigated experimentally by Barlow et al. [8] using the configura-tion shown in Fig. 1. For the current study, the four concentricstreams are, starting from the centreline, pure methane, air, pi-lot and co-flow air. Following the same order, the diametersof three tubes are 4, 7.5 and 18 mm, where Dj = 7.5 mm isdefined as the main tube/burner diameter. The volumetric flowrate of the methane to air jets (VF /VA) is 1:2 with the fuel jetis recessed by Lr = 75 mm. Hence, the composition distri-bution varies along the burner exit. The average velocity at theburner exit was 80 m/s. The pilot has an unburnt bulk velocityFigure 1: Schematic of the burner with dimensions in mm [8].of 3.72 m/s (burnt velocity ∼ 24.14 m/s) and matches the C/Hratio and adiabatic temperature of a stoichiometric methane/airflame. Co-flow air is provided at a velocity of 15 m/s.3.2 Computational setupThe present study is focussed on the turbulence-chemistryinteractions above the burner exit. Hence, a 300 mm (axial) ×113 mm radial computational domain was used starting fromthe burner exit plane. The domains was discretised using 283(axial) × 144 (radial) control volumes with a near burner reso-lution of 0.15 mm (radial) and 0.5 mm (axial).The boundary conditions for the velocity field at the burnerexit (current inflow) were obtained from the Large Eddy Sim-ulation (LES) database shared by Mueller et al. [17]. Meanand fluctuating velocities of the co-flow follow the measure-ments [7] using the the same setup although with slightly dif-ferent burner exit velocities. The mean velocity of the pilotwas set to 24.1 m/s with a boundary layer of 1 mm thicknessincluded for the shear layers. The fluctuating velocity was esti-mated using a linear interpolation between the main tube jet andthe co-flow. The integral length scale was prescribed as 1/8thof the main tube diameter, 0.07 of the hydraulic diameter in thepilot and from L = κy in the co-flow (κ = 0.41 is the vonKarman constant and y is the perpendicular distance from thepilot surface). The dissipation rate was approximated by,ε̃ =√32(k̃2/3L) (3)A transmissive boundary condition was for the perpendicu-lar velocity at the outlet of the domain (x = 300 mm) with theremaining variables assigned using a von Neumann condition.A symmetric boundary was used at r = 0 mm. The profile ofmixture fraction along the burner exit was also obtained fromthe LES database [17]. Approximately 120 stochastic particleswere used each cell in the transported PDF method [11, 13].The PDF calculation was initiated from a flamelet solu-tion where the initial transients had vanished. It is noted thatpremixed unreacted mixtures is present and the use of a con-ventional β − PDF based diffusion flame approach tends tooverestimate the temperature in the burner region. As shownin Fig. 2a, measurements [8] also confirm that the fuel jet isnot ignited. It was also observed in the present study that anerroneous estimation of the burner temperature obtained usingflamelet data cannot be washed away by the transported PDF(a)0 0.1 0.2 0.3ξ [−]03006009001200150018002100T[K](b)Figure 2: Instantaneous scatter data for temperature againstmixture fraction at x/Dj = 1 (a) [8] and a schematic of“cut-off” applied to the flamelet simulation (b). The nominalflammability limits of 0.035 ≤ ξ ≥ 0.095 are indicated via thered dashed lines.57 67 77 870500100015002000t [ms]T̃[K]x = 13 mmx = 30 mmx = 65 mmx = 120 mmFigure 3: History of temperature at different axial locations, atradial position r = 4 mm. Lines as legends.method. Hence, as illustrated in Fig. 2b, a gradual “cut-off”criterion between the stoichiometric value and the rich flamma-bility limit was used to generate the flamelet look-up table.4. ResultsThe case studied (FJ200-5GP-Lr75-80) was extensively dis-cussed at the 13th TNF workshop [18]. It is noted in the pro-ceedings [18] that most of simulations fail to capture the mix-ing between fuel and air, leading to large discrepancies down-stream the burner exit. Hence, radial profiles of mixture fractionand species are presented at four different axial locations down-stream of the burner exit. As illustrated in Fig. 3, the currentcomputations suggest that the flame burns stably and time av-erages of statistics between 58 ms and 85 ms are presented inFigs. 4 and 5.4.1 Computed and measured mean scalar dataFigure 4a shows a comparison of the current predictionsand measurement of the mean mixture fraction. It can be ob-served that the transported PDF method provides good agree-ment along the radial direction at different axial positions exceptfor a slight discrepancy close to the centreline at x/Dj = 15.However, small discrepancies can have a significant impact onthe flame structure. As shown in Fig. 4b, the predicted temper-ature agrees well with the measurement with the exception ofx/Dj = 15. The over-estimation of mixture fraction close to2the centreline at x/Dj = 15 contributes to the slight under-prediction of temperature at the same location, whereas, theslight over-estimation of mixture fraction on the lean side leadsto a substantial over-prediction of temperature. It is noted thatthe mixture fraction on the lean side is close to the lean flamma-bility limit and hence a slight flow field discrepancy induces amuch more significant difference in temperature due to ignition.It is also important to note that the flame undergoes a transitionof combustion modes around x/Dj = 15. Overall, the pre-mixed and diffusion flame modes are well captured with the dy-namics of the transition between the two modes exceptionallysensitive to minor discrepancies in the flow field.The production rate of CO is also very sensitive to the com-bustion regime. A very large scatter in results can be observedfor CO among different groups/methods presented at the TNFworkshop [18]. Radial profiles of CO obtained in the currentstudy are shown in Fig. 4c. It can be observed the predictionsagree well with experimental data at all axial locations althoughdifferences in the flow field leads to a modest over-predictionof CO at x/Dj = 15. Overall, Fig. 4c suggests that the cur-rent closure of the thermochemistry adequately represents thespecies concentrations for the different combustion modes.4.2 Computed and measured scalar variancesRadial profiles of the mixture fraction variance are shownin Fig. 5a. The values are under-estimated at the burner exit butaccurately picks up the location of two peaks. The agreementclose to the centreline recovers further downstream. Hence, theunder-estimation of fluctuating mixture fraction are likely in-fluenced by the boundary conditions, especially the prescribedlength scale. The predicted temperature variance is compared tothe experiment in Fig. 5b. The transported PDF result capturesthe two-peak characteristic of temperature variance close to theburner exit and also the “jet-like” distribution at x/Dj = 30.Similarly to the profiles of mean temperature, the discrepancyat x/Dj = 15 is attributed to the reproduction of the flow fieldduring the combustion mode transition (e.g. early ignition). Ra-dial profiles of CO variance are shown in Fig. 5c. The predic-tions agrees well with the experiment at all axial locations.5. ConclusionsThe present study has explored the behaviour of Syd-ney/Sandia piloted flames with inhomogeneous jets using ahybrid FV/JPDF method. The PDF method was initialisedby flamelet results based on a gradual “cut-off” criterionwith boundary conditions at the burner exit obtained from aLES database. Results suggest that the applied thermochem-istry combined with the transported PDF method can cap-ture features of both stratified-premixed combustion close tothe burner exit and diffusion-dominated combustion far down-stream. However, it is also noted that the dynamics of the tran-sition between the different combustion modes is exceptionallysensitive to flow field predictions and, potentially, to boundaryconditions. It is hence suggested that further experimental andcomputational studies would be fully justified for this very in-teresting case.AcknowledgementsAuthors gratefully acknowledge the sharing of the LESdatabase by Bruce Perry and Michael Mueller from PrincetonUniversity. Lu Tian would like to acknowledge the financialsupport of Imperial College PhD Scholarship.References[1] G. Staffelbach, L. Gicquel, G. Boudier, T. Poinsot, Pro-ceedings of the Combustion Institute 32 (2009) 2909–2916.[2] M.C. Drake, D.C. Haworth, Proceedings of the Combus-tion Institute 31 (2007) 99–124.[3] N. Syred, J.M. Ber, Combustion and Flame 23 (1974)143–201.[4] N. Peters, Progress in Energy and Combustion Science 10(1984) 319–339.[5] A.Y. Klimenko, R.W. Bilger, Progress in energy and com-bustion science 25 (1999) 595–687.[6] S.B. Pope, Physics of Fluids 24 (1981) 588–596.[7] S. Meares, V.N. Prasad, G. Magnotti, R.S. Barlow, A.R.Masri, Proceedings of the Combustion Institute 35 (2015)1477–1484.[8] R.S. Barlow, S. Meares, G. Magnotti, H. Cutcher, A.R.Masri, Combustion and Flame 162 (2015) 3516–3540.[9] S. Galindo, F. Salehi, M. Cleary, A. Masri, Proceedings ofthe Combustion Institute 36 (2017) 1759–1766.[10] H. Wu, M. Ihme, Fuel 186 (2016) 853–863.[11] T.S. Kuan, R.P. Lindstedt, Proceedings of the CombustionInstitute 30 (2005) 767–774.[12] J. Janicka, W. Kolbe, W. Kollmann, Journal of Non-Equilibrium Thermodynamics 4 (1979) 47–66.[13] K. Gkagkas, R.P. Lindstedt, T.S. Kuan, Flow, Turbulenceand Combustion 82 (2009) 493–509.[14] D.C. Haworth, S.B. Pope, Physics of Fluids 29 (1986)387–405.[15] W.P. Jones, B.E. Launder, International Journal of Heatand Mass Transfer 15 (1972) 301–314.[16] R.P. Lindstedt, E.M. Váos, Combustion and Flame 145(2006) 495–511.[17] B.A. Perry, M.E. Mueller, A.R. Masri, Proceedings of theCombustion Institute 36 (2017) 1767–1775.[18] Proceedings of the TNF13 Workshop, Thirteenth Inter-national Workshop on Measurement and Computationof Turbulent Flames, Seoul, Korea, 7.28-7.30, 2016.http://www.sandia.gov/TNF/13thWorkshop/TNF13.html.300.10.20.30.40.50.60.70.80.91x/D = 100.10.20.30.40.50.60.70.80.9 x/D = 5f̃[−]r [mm]00.10.20.30.40.50.60.70.80.9 x/D = 150 0.5 1 1.5 2 2.5 3 3.5r/D00.10.20.30.40.50.60.70.80.9 x/D = 30(a)05001000150020002500x/D = 10500100015002000x/D = 5T̃[K]0500100015002000x/D = 150 0.5 1 1.5 2 2.5 3 3.5r/D0500100015002000x/D = 30(b)00.020.040.060.08x/D = 100.020.040.06x/D = 5ỸCO[−]00.020.040.06x/D = 150 0.5 1 1.5 2 2.5 3 3.5r/D00.020.040.06x/D = 30(c)Figure 4: Radial profiles of mean mixture fraction (a), temperature (b) and CO mass fraction (c). Symbols and lines: (◦) Experiment;(—–) Transported PDF method.400.050.10.150.2x/D = 1r/D00.050.10.15x/D = 5f′′[−]r/D00.050.10.15x/D = 150 0.5 1 1.5 2 2.5 3 3.5r/D00.050.10.15x/D = 30(a)0200400600800x/D = 1r/D0200400600x/D = 5T′′[K]r/D0200400600x/D = 150 0.5 1 1.5 2 2.5 3 3.5r/D0200400600x/D = 30(b)00.010.020.03x/D = 1r/D00.010.02x/D = 5Y′′ CO[−]r/D00.010.02x/D = 150 0.5 1 1.5 2 2.5 3 3.5r/D00.010.02x/D = 30(c)Figure 5: Radial profiles of fluctuating mixture fraction (a), temperature (b) and CO mass fraction (c). Symbols and lines: (◦)Experiment; (—–) Transported PDF method.5

Transported PDF modelling and analysis of partially premixed flames

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Transported PDF modelling and analysis of partially premixed flames

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