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Investigation of premixed turbulent combustion in a semi-confined explosion chamber
Experimental Thermal and Fluid Science 27 (2003) 355–361 www.elsevier.com/locate/etfs
Investigation of premixed turbulent combustion in a semi-confined explosion chamber S. Patel a, S.S. Ibrahim a
a,*
, M.A. Yehia
b,*,1
, G.K. Hargrave
a
Department of Aeronautical and Automotive Engineering, Faculty of Engineering, Loughborough University, Loughborough LE11 3TU, UK b Department of Mechanical Power, Faculty of Engineering, Cairo University, 12613 Cairo, Egypt Received 6 January 2002; received in revised form 17 June 2002; accepted 9 July 2002
Abstract A modified flame surface density (FSD) combustion model has been developed and applied to simulate the deflagration of a highly turbulent premixed flame inside a semi-confined explosion chamber. The chamber was of 500-mm length and 150 mm 150 mm cross-section. A stoichiometric methane–air mixture was ignited to initiate the explosion. High turbulence levels were generated through interaction of the propagating flame with three consecutive solid obstacles of rectangular configuration. High speed laser sheet flow visualisation techniques were used to obtain experimental data. Turbulent flow field has been calculated using a compressible eddy viscosity model. The model formulation and results obtained are presented and discussed in terms of model performance at all stages of flame propagation from ignition until venting. The transient developments of flamelet mean, turbulence, and curvature stretch were formulated and implemented in the model. The model predictions for flame shape, speed and pressure history has been compared with the highly resolved experimental data. Experimental and numerical results of spatio-temporal dynamics are found to be in good agreement. Moreover, the predicted flame stretch is presented and discussed. Ó 2003 Elsevier Science Inc. All rights reserved. Keywords: Premixed flame; Turbulent combustion; Flame propagation
1. Introduction Premixed turbulent combustion is of great technological importance particularly for spark-ignition, explosion risk assessment and gas turbine engines. Optical diagnostic measurement techniques have advanced recently to enable a better understanding of the development details of a propagating turbulent premixed flame. The present experimental work provides data in smallscale chamber aimed at investigating the interaction between the flame and multiple obstacles in semi-confined geometry. A total direct numerical simulation (DNS), in which the complete spectrum of scales characterizing the hydrocarbon combustion in a high Reynolds number turbulent flow, is still not close to being possible [1]. A topic *
Corresponding authors. Tel.: +44-1509-223240/7253; fax: +441509-223946/7275. E-mail addresses: [email protected] (S.S. Ibrahim), m.a. [email protected] (M.A. Yehia). 1 Tel.: +44-1509-227247; fax: +44-1509-227575.
of current interest is the incorporation of chemical reaction effects into large Eddy simulations (LES) in which large-scale time-dependent eddies are resolved, while smaller sub-grid scale effects are modeled, [2,3]. However, uncertainties remain on the appropriate mathematical description of a turbulent flame interacting with sub-grid scales, and a critical parameter requiring closure is the mean rate of reaction. Both DNS and LES are useful tools but for engineering problems Reynolds averaged numerical methods have shown to be feasible. Classical, Eddy breakup models (EBU) and their extensions given by [4,5] provide a crude first approach to turbulent premixed combustion simulations when the primary interest is in worst-case estimates. Extensions of the mixing controlled EBU model [4] to include characteristic time scales of chemistry lack a systematic derivation from first principles and should be considered as ad hoc ‘‘fixes’’. With suitable implementation of more realistic chemical time scale [6] one might obtain reasonable agreement with experimental data, however, the predictive capabilities will be limited within the range of simple flow conditions.
0894-1777/03/$ - see front matter Ó 2003 Elsevier Science Inc. All rights reserved. doi:10.1016/S0894-1777(02)00238-8
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S. Patel et al. / Experimental Thermal and Fluid Science 27 (2003) 355–361
Nomenclature A Aij c~ C0 Cs k K L Ms Ps R RL Ts t uL u0 Ui
flamelet surface area, m2 weighting parameter for the mean flame stretch, dimensionless Favre-averaged reaction progress variable constant, dimensionless flamelet curvature stretch, m1 s1 turbulent kinetic energy, m2 s2 Karlovitz number, dimensionless integral length scale, m mean flamelet stretch, m1 s1 flamelet propagation stretch, m1 s1 rate of reaction per unit area, kg m2 s1 turbulence Reynolds number, dimensionless flamelet turbulence stretch, m1 s1 time, s laminar burning velocity, m s1 Reynolds fluctuating velocity, m s1 axial flow velocity, m s1
Flamelet model techniques that are adopted in the formulation of the present modeling work provide means to introduce chemical and turbulence time scales by considering a thin laminar flame in a turbulent flow field. The key goals behind flamelet modeling are to incorporate the effects of fast but finite reaction rates, the inherent quasi-laminar flame dynamics and the intimate coupling between chemical reactions and molecular transport that arises when rapid chemistry enforces very thin flame structures. The alternative modeling strategy has been pursued for the flamelet regime since the first introduction of the Bray–Moss–Libby model (BML). This approach is based on evaluating the flame surface area to volume ratio (flame surface density), which can be computed via an algebraic [7] or through a transport equation [8]. The BML method depends upon a precise model for the flame integral wrinkling length scale, Ly . This is not yet available, and only empirical forms are used [9] thus BML will not be considered here. The transported flame surface density model (FSD) [10] provides more realistic representation of the effects of different stretch due to convection, flame propagation, diffusion, production and destruction on the estimation of flame area where production and destruction terms are treated differently [10]. In this paper, a modified FSD model was used and comparisons were made with highly resolved experimental data for flame front images. 2. Experimental rig The experimental rig (Fig. 1) consists of a simple square-section explosion chamber manufactured from
_ w xj
local mean rate of chemical kg m3 s1 distance in jth direction, m
reaction,
Greek symbols a parameter in the Ts term, dimensionless am constant in the a equation, dimensionless asi constant in the Cs equation, dimensionless asu constant in the Cs equation, dimensionless dL laminar flame thickness, m rate of dissipation of k, m2 s3 R mean flamelet surface density, m1 qu density of unburned mixture, kg m3 CK intermittent net flame stretch function, dimensionless s thermal expansion, dimensionless m kinematic viscosity, m2 s1
polycarbonate to facilitate the application of optical diagnostics. The chamber was 150 mm 150 mm crosssection and 500 mm long. It was closed at one end and a thin plastic diaphragm was used to seal the open end in order to contain the flammable mixture. During the flame propagation, the diaphragm was ruptured at low pressure to reduce excessive over-pressure generation. The flammable mixture used was stoichiometric methane–air. Three obstacles of rectangular cross-section 75 mm 10 mm, providing a blockage ratio of 50% were mounted within the explosion unit centered and separated by 100 mm from the ignition end and from each other. The premixed methane–air gas was purged through the chamber for several minutes to establish a homogeneous mixture. The mixture was then allowed to settle in the chamber before being ignited by a spark source located in the closed end of the chamber. A high-speed, laser-sheet flow visualization system, incorporating a copper-vapor laser and Kodak 4540 high-speed digital video camera, was used to record the progress of the flame. The premixed methane–air entering the explosion chamber was seeded with micronsized droplets of olive oil to act as scattering centres for the laser light. The laser was formed into a laser sheet 150 mm high by 1 mm thick and used to illuminate the region of interest within the explosion chamber. Laser light was scattered by the particles and recorded using the high-speed camera at 9000 frames/s. As the flame front propagates down the combustion chamber the oil droplets are consumed, differentiating the burned/unburned mixture. Image analysis provided data for flame front location at different times after igni-
S. Patel et al. / Experimental Thermal and Fluid Science 27 (2003) 355–361
357
Fig. 1. Schematic diagram of the experimental rig.
tion and the volume of trapped mixture behind the obstacles. 3. The model Transient calculations were carried out using a twodimensional finite difference code (TRF2D) which solves the Favre averaged conservation equations for momentum, turbulence and energy. The Eddy viscosity model due to [11] modified to include compressibility effects was used to close the Reynolds stress and turbulent diffusion terms. The equations are linearized using finite volume techniques, in which Euler time differencing and hybrid spatial differencing are used. The PISO solver due to [12] was used to solve the set of the coupled non-linear equations. The code is second order accurate both in time and space. _ , was calculated The local mean rate of reaction, w using: _ ¼ RR w ð1Þ where R is the rate of reaction per unit area computed from R ¼ qu uL
ð2Þ
and R is the flamelet surface density. Here qu is the density of the unburned mixture. The flamelet surface density, R, is calculated using a transport equation of the form; DR ¼ Ms þ Ts þ Ps þ Cs ð3Þ Dt The term in the left represents the rate of change of the flamelet surface density, R, due to accumulation, convection, and diffusion, and the terms on the right denote
mean, turbulence, propagation, and curavture flamelet stretch terms representing the source/sink term for the R. Previous work compared between different choices in the estimation of R which have a significant effect on the final results of the global model. Such comparative work includes [8,9,13]. For the present work, it was found that the flamelet propagation stretch term, Ps , was negligible compared with mean, turbulence and curvature stretch terms. This conforms with previous findings for other work (see for example [9]). The mean flamelet stretch term, Ms , is closed as [14,15]: oUi Ms ¼ Aij R ð4Þ oxj where Aij is a weighting parameter for the mean flame stretch, which has been set to a value of 0.2 based on typical results observed by [16]. The turbulent flamelet stretch term, Ts , is closed as [17]: 0 u L e R ð5Þ Ts ¼ aCK ; uL dL k where CK is the intermittent net flame stretch function which is a function accounting for the relevant scales of turbulence interacting with the propagating flame obtained through fitting experimental findings. The parameter a controls the weighting for the turbulent stretch. A previous one-dimensional analysis of flame propagation in homogeneous turbulence [18] suggested that for accurate calculation of the flame burning velocity, a should take the following formula; 0 1 u a ¼ am RL0:4 ð6Þ uL
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S. Patel et al. / Experimental Thermal and Fluid Science 27 (2003) 355–361
where RL is the turbulence Reynolds number, RL ¼
u0 L m
ð7Þ
Here RL is the turbulent Reynolds number, u0 is the rms turbulent velocity, L is the integral length scale and m is the kinematic viscosity and am is a constant which is set to a value of 12. One-dimensional flame is an idealization which is difficult (and some believe even impossible) to achieve experimentally [19]. Thus the extension of this postulate and its application to the present two-dimensional work and thus compared to experimental data forms a significant stage of its development. The flamelet curvature stretch term, Cs , is closed [16] as: Cs ¼
ðasu C 0 asi Þ ð1 þ s~ cÞR2 c~ð1 c~Þ
ð8Þ
where C 0 ¼ 0:5. In light of recent experimental evidence [20] on the possibility of three different modes for the curvature stretch term (all positive, positive–negative, all negative) asu and asi have been introduced here to allow for these possible modes. In the present work asu and asi have been set to values of 0.02 and 0.8 respectively. These values were obtained in a sensitivity study realizing their effect on the peak pressures predicted. The flamelet stretch rate is traditionally represented through the dimensionless Karlovitz number, K, which is defined as dL 1 dA K¼ ð9Þ uL A dt where dL is the laminar flame thickness, uL the unstrained laminar burning velocity and A is the flamelet surface area. [21] computed the Karlovitz number as 0 2 u K ¼ 0:145 RL0:5 ð10Þ uL The calculations employed non-uniform finite volume rectangular grid, with a 2 mm resolution in all directions. No slip boundary conditions were imposed along the chamber walls and the internal solid obstacles. The computational mesh was extended by 25% outside the combustion chamber for the release of the charge into the ambient atmosphere. This is done to ensure proper implementation of the transmissive outlet boundary conditions.
4. Results and analysis Experimental and predicted images of flame front propagation are shown in Fig. 2. The experimental
representative sequences of images extracted from the high-speed laser sheet videos are compared to the mathematical model results represented by the progress variable calculations. As shown, flame propagation features as suggested by both the experimental images and predictions are similar; that is the flame flattening as it approaches the first obstacle, jetting past the obstacles, turning behind the obstacles, and the flame reconnection. In particular the two flat flames approaching each other behind the obstacles is well predicted. However, the predictions show that flame curling behind the obstacle is not well reproduced. A recent work by [22] demonstrated that the use of a non-linear eddy viscosity model such as that of [23] would have improved the flame structure. A comparison between the predicted and experimental results on flame speed as a function of distance from the ignition end is shown in Fig. 3. The experimental observation of a drop in flame acceleration between the obstacles and increased acceleration as the flame propagates past the obstacles is well reproduced by the predictions. However, it can be seen that the drop in flame acceleration between the obstacles is more pronounced in the experimental data compared to the predictions, which can be attributed to the ratio of the rate of flame propagation in the lateral to the traverse directions between the obstacles. As shown in Fig. 4, the trend in the variation and time of occurrence of the pressure peaks are in good agreement with the experimental observations. The pressure predictions clearly show a decay in the rate of increase in pressure between times ranging from 20 to 26 ms which can be attributed to the flame jetting. The trough and peak observed in the experimental pressure history between 25 and 34 ms can be attributed to the effects of the blockage imposed by the obstacles, the flame jetting, and the disintegration of the disposable sealing membrane. In Fig. 5 the variation of the predicted flamelet surface density, R, at different times from ignition can be seen. The values of R range from 0 to 150 m1 and agree well with the values observed in the experiments by [9,20,24]. The contribution of the mean (Ms ), turbulence (Ts ), and curvature (Cs ) flamelet stretching mechanisms at different times after ignition are also shown in Fig. 5. It can be seen that the mean flamelet stretch dominates in the flow jetting regions which can be explained by noting the existence of high velocity gradients. Furthermore, for the present simulations it can be seen that the mean flamelet stretch peaks in the jetting region corresponding to the third obstacle from the ignition end with values around 55,000 m1 s1 . Higher values of turbulence flamelet stretch can be seen to dominate closer to the centreline of the combustion chamber compared to the mean flamelet stretch term. However, the maximum value of the Ts term can be seen to occur behind the first obstacle. This is in line with the work of
S. Patel et al. / Experimental Thermal and Fluid Science 27 (2003) 355–361
359
Fig. 2. Comparison between high speed images (top) and predicted (bottom) flame front structure at different times (ms) after ignition.
Fig. 3. Comparison between predicted and measured flame front speed values at different locations from the ignition end. Obstacle locations are shown on the horizontal axis.
[25] where it can be gleaned that rigorous kinematic interaction between the flamelets and turbulence dominates in the corrugated flamelets as opposed to a more highly turbulent premixed flames such as in the thin reaction zones regime. The variation of the flamelet curvature stretch is also shown in Fig. 5. It can be seen that the levels of flamelet curvature stretch imposed on the flamelet do not change significantly at the various
Fig. 4. Comparison between predicted and measured overpressure values at different times after ignition.
stages of flame propagation. However, it can be seen that the zone of Cs , influence increases as the flame propagates past the three obstacles, which can be due to the increasing turbulent flame brush thickness as a result of averaging over the significant variations of instantaneous flamelet wrinkles. Fig. 6 shows the variation of the calculated Karlovitz stretch factor values along the flame front at different times after ignition as based on Eq. (10). Values simulated shows that the regimes of combustion are all
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S. Patel et al. / Experimental Thermal and Fluid Science 27 (2003) 355–361
Fig. 5. Variation of flamelet surface density (R), and mean (Ms ), turbulence (Ts ), and curvature (Cs ) flamelet stretching mechanisms at different times (ms) of flame propagation, after ignition.
in good agreement with experimental data as reported by [26]. It can be seen that the flame stretch at the semi-confinement between the first two obstacles is of lower values suggesting that it lies in the continuous laminar flame combustion regime. As the flame propagates past the second obstacle and before venting starts, the stretch increases fast to reach a combustion regime where the flame starts to break-up due to high flame stretch.
5. Conclusions A study was performed to investigate transient premixed combustion in an explosion chamber. The propagation of the flame front was mapped through the use of high speed laser sheet flow visualisation. A modified FSD model has been developed and implemented in a compressible turbulent reacting flow code and applied to predict the flame propagation past multiple obstacles
S. Patel et al. / Experimental Thermal and Fluid Science 27 (2003) 355–361
361
Fig. 6. Variation of the calculated Karlovitz stretch factor values along the flame front at different times (ms) after ignition.
mounted inside a semi-confined explosion chamber. Results from the various stretching mechanisms at different stages of flame propagation were presented. These show that the turbulence stretch dominates in regions behind the obstacles due to the presence of highly recirculating zones while the mean flamelet stretch is dominant in the flow jetting regions around the obstacles. Numerical results have shown that the values of the flamelet stretch imposed on the flame front ahead of each obstacle is found to increase as the flame propagates from the ignition end towards the venting end. Comparison between numerical and experimental results show that the model yields plausible results for flame structure, pressure time history and flame speed. At present, work is underway to investigate the validity of the model under different flow and mixture conditions. Moreover, work is underway to improve the predictions of the turbulent field through the use of LES models. References [1] K.N.C. Bray, in: Twenty-Sixth Symp. (Int.) Combust., The Combustion Institute, Pittsburgh, 1996, pp. 1–26. [2] S. Menon, W. Jou, Combust. Sci. Technol. 75 (1991) 53–68. [3] M.P. Kirkpatrick, S.S. Ibrahim, S.W. Armfield, A.R. Masri, Third Asia-Pacific Conference on Combustion, Korea, 24–27 June 2001, pp. 1–4. [4] D.B. Spalding, in: Thirteenth Symp. (Int.) Combust., The Combustion Institute, Pittsburgh, 1971, pp. 649–657. [5] B.F. Magnussen, B.H. Hjertager, in: Sixteenth Symp. (Int.) Combust., The Combustion Institute, Pittsburgh, 1976, pp. 719–729. [6] C.A. Catlin, M. Fairweather, S.S. Ibrahim, Combust. Flame 102 (1995) 115–128. [7] K.N.C. Bray, Proc. R. Soc. Lond., Ser. A 431 (1990) 315–335.
[8] J.M. Duclos, D. Veynante, T. Poinsot, Combust. Flame 95 (1993) 101–109. [9] D. Veynante, J.M. Duclos, J. Piana, Experimental and theoretical analysis of flame surface density modelling for premixed turbulent combustion, Direct numerical simulation for turbulent reacting flows, in: T. Baritaud, T. Poinsot, M. Baum (Eds.), Centre De Recherche sur la combustion turbulente, 1996. [10] R.O. Prasad, R.N. Paul, Y.R. Sivathanu, J.P. Gore, Combust. Flame 117 (1999) 514. [11] B.E. Launder, D.B. Spalding, The numerical computation of turbulent flow, Comp. Meth. Appl. Mech. Eng. 3 (1974) 269–281. [12] R.I. Issa, J. Comp. Phys. 62 (1986) 40–65. [13] R.O.S. Prasad, J.P. Gore, Combust. Flame 116 (1999) 1–14. [14] P. Boudier, S. Henriot, T. Poinsot, T. Baritaud, in: TwentyFourth Symp. (Int.) Combust., The Combustion Institute, Pittsburgh, 1992, pp. 503–510. [15] R.S. Cant, S.B. Pope, K.N.C. Bray, in: Twenty Third Symp. (Int.) Combust., The Combustion Institute, Pittsburgh, 1990, pp. 809– 815. [16] D.A. Veynante, A. TrouveÕ, K.N.C. Bray, T. Mantel, J. Fluid Mech. (1997) 332–358. [17] C. Meneveau, T. Poinsot, Combust. Flame 86 (1991) 311–318. [18] S.N.D.H. Patel, S.S. Ibrahim, JSME Int. J. Ser. B Fluids Thermal Eng. 45 (3) (2002) 725–735. [19] F. Fichot, F. Lacas, D. Veynante, S. Candel, Combust. Sci. Tech. 90 (1993) 35–60. [20] S.S. Shy, I.E. Lee, T.S. Yang, Combust. Flame 118 (1999) 606– 618. [21] C.R. Choi, K.Y. Huh, Combust. Flame 114 (1998) 336–348. [22] N.S. Kuo, Y.C. Lin, C.A. Lin, Third Int. Symp. Turbulence, Heat Mass Trans., 2000, pp. 397–404. [23] T.J. Craft, B.E. Launder, K. Suga, Fifth Int. Symp. Refined Flow Modell. Turbulence Measure., 1993, pp. 219–227. [24] B.M. Deschamps, G.J. Smallwood, J. Prieur, D.R. Snelling, O.L. Gulder, in: Twenty-Six Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh, 1996, pp. 427–435. [25] N. Peters, Turbulent combustion, Cambridge Monographs on Mechanics, Cambridge University Press, 2000. [26] R.G. Abdel-Gayed, D. Bradley, F.K.K. Lung, Combust. Flame 76 (1989) 213–218.
Investigation of premixed turbulent combustion in a semi-confined explosion chamber S. Patel a, S.S. Ibrahim a
a,*
, M.A. Yehia
b,*,1
, G.K. Hargrave
a
Department of Aeronautical and Automotive Engineering, Faculty of Engineering, Loughborough University, Loughborough LE11 3TU, UK b Department of Mechanical Power, Faculty of Engineering, Cairo University, 12613 Cairo, Egypt Received 6 January 2002; received in revised form 17 June 2002; accepted 9 July 2002
Abstract A modified flame surface density (FSD) combustion model has been developed and applied to simulate the deflagration of a highly turbulent premixed flame inside a semi-confined explosion chamber. The chamber was of 500-mm length and 150 mm 150 mm cross-section. A stoichiometric methane–air mixture was ignited to initiate the explosion. High turbulence levels were generated through interaction of the propagating flame with three consecutive solid obstacles of rectangular configuration. High speed laser sheet flow visualisation techniques were used to obtain experimental data. Turbulent flow field has been calculated using a compressible eddy viscosity model. The model formulation and results obtained are presented and discussed in terms of model performance at all stages of flame propagation from ignition until venting. The transient developments of flamelet mean, turbulence, and curvature stretch were formulated and implemented in the model. The model predictions for flame shape, speed and pressure history has been compared with the highly resolved experimental data. Experimental and numerical results of spatio-temporal dynamics are found to be in good agreement. Moreover, the predicted flame stretch is presented and discussed. Ó 2003 Elsevier Science Inc. All rights reserved. Keywords: Premixed flame; Turbulent combustion; Flame propagation
1. Introduction Premixed turbulent combustion is of great technological importance particularly for spark-ignition, explosion risk assessment and gas turbine engines. Optical diagnostic measurement techniques have advanced recently to enable a better understanding of the development details of a propagating turbulent premixed flame. The present experimental work provides data in smallscale chamber aimed at investigating the interaction between the flame and multiple obstacles in semi-confined geometry. A total direct numerical simulation (DNS), in which the complete spectrum of scales characterizing the hydrocarbon combustion in a high Reynolds number turbulent flow, is still not close to being possible [1]. A topic *
Corresponding authors. Tel.: +44-1509-223240/7253; fax: +441509-223946/7275. E-mail addresses: [email protected] (S.S. Ibrahim), m.a. [email protected] (M.A. Yehia). 1 Tel.: +44-1509-227247; fax: +44-1509-227575.
of current interest is the incorporation of chemical reaction effects into large Eddy simulations (LES) in which large-scale time-dependent eddies are resolved, while smaller sub-grid scale effects are modeled, [2,3]. However, uncertainties remain on the appropriate mathematical description of a turbulent flame interacting with sub-grid scales, and a critical parameter requiring closure is the mean rate of reaction. Both DNS and LES are useful tools but for engineering problems Reynolds averaged numerical methods have shown to be feasible. Classical, Eddy breakup models (EBU) and their extensions given by [4,5] provide a crude first approach to turbulent premixed combustion simulations when the primary interest is in worst-case estimates. Extensions of the mixing controlled EBU model [4] to include characteristic time scales of chemistry lack a systematic derivation from first principles and should be considered as ad hoc ‘‘fixes’’. With suitable implementation of more realistic chemical time scale [6] one might obtain reasonable agreement with experimental data, however, the predictive capabilities will be limited within the range of simple flow conditions.
0894-1777/03/$ - see front matter Ó 2003 Elsevier Science Inc. All rights reserved. doi:10.1016/S0894-1777(02)00238-8
356
S. Patel et al. / Experimental Thermal and Fluid Science 27 (2003) 355–361
Nomenclature A Aij c~ C0 Cs k K L Ms Ps R RL Ts t uL u0 Ui
flamelet surface area, m2 weighting parameter for the mean flame stretch, dimensionless Favre-averaged reaction progress variable constant, dimensionless flamelet curvature stretch, m1 s1 turbulent kinetic energy, m2 s2 Karlovitz number, dimensionless integral length scale, m mean flamelet stretch, m1 s1 flamelet propagation stretch, m1 s1 rate of reaction per unit area, kg m2 s1 turbulence Reynolds number, dimensionless flamelet turbulence stretch, m1 s1 time, s laminar burning velocity, m s1 Reynolds fluctuating velocity, m s1 axial flow velocity, m s1
Flamelet model techniques that are adopted in the formulation of the present modeling work provide means to introduce chemical and turbulence time scales by considering a thin laminar flame in a turbulent flow field. The key goals behind flamelet modeling are to incorporate the effects of fast but finite reaction rates, the inherent quasi-laminar flame dynamics and the intimate coupling between chemical reactions and molecular transport that arises when rapid chemistry enforces very thin flame structures. The alternative modeling strategy has been pursued for the flamelet regime since the first introduction of the Bray–Moss–Libby model (BML). This approach is based on evaluating the flame surface area to volume ratio (flame surface density), which can be computed via an algebraic [7] or through a transport equation [8]. The BML method depends upon a precise model for the flame integral wrinkling length scale, Ly . This is not yet available, and only empirical forms are used [9] thus BML will not be considered here. The transported flame surface density model (FSD) [10] provides more realistic representation of the effects of different stretch due to convection, flame propagation, diffusion, production and destruction on the estimation of flame area where production and destruction terms are treated differently [10]. In this paper, a modified FSD model was used and comparisons were made with highly resolved experimental data for flame front images. 2. Experimental rig The experimental rig (Fig. 1) consists of a simple square-section explosion chamber manufactured from
_ w xj
local mean rate of chemical kg m3 s1 distance in jth direction, m
reaction,
Greek symbols a parameter in the Ts term, dimensionless am constant in the a equation, dimensionless asi constant in the Cs equation, dimensionless asu constant in the Cs equation, dimensionless dL laminar flame thickness, m rate of dissipation of k, m2 s3 R mean flamelet surface density, m1 qu density of unburned mixture, kg m3 CK intermittent net flame stretch function, dimensionless s thermal expansion, dimensionless m kinematic viscosity, m2 s1
polycarbonate to facilitate the application of optical diagnostics. The chamber was 150 mm 150 mm crosssection and 500 mm long. It was closed at one end and a thin plastic diaphragm was used to seal the open end in order to contain the flammable mixture. During the flame propagation, the diaphragm was ruptured at low pressure to reduce excessive over-pressure generation. The flammable mixture used was stoichiometric methane–air. Three obstacles of rectangular cross-section 75 mm 10 mm, providing a blockage ratio of 50% were mounted within the explosion unit centered and separated by 100 mm from the ignition end and from each other. The premixed methane–air gas was purged through the chamber for several minutes to establish a homogeneous mixture. The mixture was then allowed to settle in the chamber before being ignited by a spark source located in the closed end of the chamber. A high-speed, laser-sheet flow visualization system, incorporating a copper-vapor laser and Kodak 4540 high-speed digital video camera, was used to record the progress of the flame. The premixed methane–air entering the explosion chamber was seeded with micronsized droplets of olive oil to act as scattering centres for the laser light. The laser was formed into a laser sheet 150 mm high by 1 mm thick and used to illuminate the region of interest within the explosion chamber. Laser light was scattered by the particles and recorded using the high-speed camera at 9000 frames/s. As the flame front propagates down the combustion chamber the oil droplets are consumed, differentiating the burned/unburned mixture. Image analysis provided data for flame front location at different times after igni-
S. Patel et al. / Experimental Thermal and Fluid Science 27 (2003) 355–361
357
Fig. 1. Schematic diagram of the experimental rig.
tion and the volume of trapped mixture behind the obstacles. 3. The model Transient calculations were carried out using a twodimensional finite difference code (TRF2D) which solves the Favre averaged conservation equations for momentum, turbulence and energy. The Eddy viscosity model due to [11] modified to include compressibility effects was used to close the Reynolds stress and turbulent diffusion terms. The equations are linearized using finite volume techniques, in which Euler time differencing and hybrid spatial differencing are used. The PISO solver due to [12] was used to solve the set of the coupled non-linear equations. The code is second order accurate both in time and space. _ , was calculated The local mean rate of reaction, w using: _ ¼ RR w ð1Þ where R is the rate of reaction per unit area computed from R ¼ qu uL
ð2Þ
and R is the flamelet surface density. Here qu is the density of the unburned mixture. The flamelet surface density, R, is calculated using a transport equation of the form; DR ¼ Ms þ Ts þ Ps þ Cs ð3Þ Dt The term in the left represents the rate of change of the flamelet surface density, R, due to accumulation, convection, and diffusion, and the terms on the right denote
mean, turbulence, propagation, and curavture flamelet stretch terms representing the source/sink term for the R. Previous work compared between different choices in the estimation of R which have a significant effect on the final results of the global model. Such comparative work includes [8,9,13]. For the present work, it was found that the flamelet propagation stretch term, Ps , was negligible compared with mean, turbulence and curvature stretch terms. This conforms with previous findings for other work (see for example [9]). The mean flamelet stretch term, Ms , is closed as [14,15]: oUi Ms ¼ Aij R ð4Þ oxj where Aij is a weighting parameter for the mean flame stretch, which has been set to a value of 0.2 based on typical results observed by [16]. The turbulent flamelet stretch term, Ts , is closed as [17]: 0 u L e R ð5Þ Ts ¼ aCK ; uL dL k where CK is the intermittent net flame stretch function which is a function accounting for the relevant scales of turbulence interacting with the propagating flame obtained through fitting experimental findings. The parameter a controls the weighting for the turbulent stretch. A previous one-dimensional analysis of flame propagation in homogeneous turbulence [18] suggested that for accurate calculation of the flame burning velocity, a should take the following formula; 0 1 u a ¼ am RL0:4 ð6Þ uL
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S. Patel et al. / Experimental Thermal and Fluid Science 27 (2003) 355–361
where RL is the turbulence Reynolds number, RL ¼
u0 L m
ð7Þ
Here RL is the turbulent Reynolds number, u0 is the rms turbulent velocity, L is the integral length scale and m is the kinematic viscosity and am is a constant which is set to a value of 12. One-dimensional flame is an idealization which is difficult (and some believe even impossible) to achieve experimentally [19]. Thus the extension of this postulate and its application to the present two-dimensional work and thus compared to experimental data forms a significant stage of its development. The flamelet curvature stretch term, Cs , is closed [16] as: Cs ¼
ðasu C 0 asi Þ ð1 þ s~ cÞR2 c~ð1 c~Þ
ð8Þ
where C 0 ¼ 0:5. In light of recent experimental evidence [20] on the possibility of three different modes for the curvature stretch term (all positive, positive–negative, all negative) asu and asi have been introduced here to allow for these possible modes. In the present work asu and asi have been set to values of 0.02 and 0.8 respectively. These values were obtained in a sensitivity study realizing their effect on the peak pressures predicted. The flamelet stretch rate is traditionally represented through the dimensionless Karlovitz number, K, which is defined as dL 1 dA K¼ ð9Þ uL A dt where dL is the laminar flame thickness, uL the unstrained laminar burning velocity and A is the flamelet surface area. [21] computed the Karlovitz number as 0 2 u K ¼ 0:145 RL0:5 ð10Þ uL The calculations employed non-uniform finite volume rectangular grid, with a 2 mm resolution in all directions. No slip boundary conditions were imposed along the chamber walls and the internal solid obstacles. The computational mesh was extended by 25% outside the combustion chamber for the release of the charge into the ambient atmosphere. This is done to ensure proper implementation of the transmissive outlet boundary conditions.
4. Results and analysis Experimental and predicted images of flame front propagation are shown in Fig. 2. The experimental
representative sequences of images extracted from the high-speed laser sheet videos are compared to the mathematical model results represented by the progress variable calculations. As shown, flame propagation features as suggested by both the experimental images and predictions are similar; that is the flame flattening as it approaches the first obstacle, jetting past the obstacles, turning behind the obstacles, and the flame reconnection. In particular the two flat flames approaching each other behind the obstacles is well predicted. However, the predictions show that flame curling behind the obstacle is not well reproduced. A recent work by [22] demonstrated that the use of a non-linear eddy viscosity model such as that of [23] would have improved the flame structure. A comparison between the predicted and experimental results on flame speed as a function of distance from the ignition end is shown in Fig. 3. The experimental observation of a drop in flame acceleration between the obstacles and increased acceleration as the flame propagates past the obstacles is well reproduced by the predictions. However, it can be seen that the drop in flame acceleration between the obstacles is more pronounced in the experimental data compared to the predictions, which can be attributed to the ratio of the rate of flame propagation in the lateral to the traverse directions between the obstacles. As shown in Fig. 4, the trend in the variation and time of occurrence of the pressure peaks are in good agreement with the experimental observations. The pressure predictions clearly show a decay in the rate of increase in pressure between times ranging from 20 to 26 ms which can be attributed to the flame jetting. The trough and peak observed in the experimental pressure history between 25 and 34 ms can be attributed to the effects of the blockage imposed by the obstacles, the flame jetting, and the disintegration of the disposable sealing membrane. In Fig. 5 the variation of the predicted flamelet surface density, R, at different times from ignition can be seen. The values of R range from 0 to 150 m1 and agree well with the values observed in the experiments by [9,20,24]. The contribution of the mean (Ms ), turbulence (Ts ), and curvature (Cs ) flamelet stretching mechanisms at different times after ignition are also shown in Fig. 5. It can be seen that the mean flamelet stretch dominates in the flow jetting regions which can be explained by noting the existence of high velocity gradients. Furthermore, for the present simulations it can be seen that the mean flamelet stretch peaks in the jetting region corresponding to the third obstacle from the ignition end with values around 55,000 m1 s1 . Higher values of turbulence flamelet stretch can be seen to dominate closer to the centreline of the combustion chamber compared to the mean flamelet stretch term. However, the maximum value of the Ts term can be seen to occur behind the first obstacle. This is in line with the work of
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Fig. 2. Comparison between high speed images (top) and predicted (bottom) flame front structure at different times (ms) after ignition.
Fig. 3. Comparison between predicted and measured flame front speed values at different locations from the ignition end. Obstacle locations are shown on the horizontal axis.
[25] where it can be gleaned that rigorous kinematic interaction between the flamelets and turbulence dominates in the corrugated flamelets as opposed to a more highly turbulent premixed flames such as in the thin reaction zones regime. The variation of the flamelet curvature stretch is also shown in Fig. 5. It can be seen that the levels of flamelet curvature stretch imposed on the flamelet do not change significantly at the various
Fig. 4. Comparison between predicted and measured overpressure values at different times after ignition.
stages of flame propagation. However, it can be seen that the zone of Cs , influence increases as the flame propagates past the three obstacles, which can be due to the increasing turbulent flame brush thickness as a result of averaging over the significant variations of instantaneous flamelet wrinkles. Fig. 6 shows the variation of the calculated Karlovitz stretch factor values along the flame front at different times after ignition as based on Eq. (10). Values simulated shows that the regimes of combustion are all
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Fig. 5. Variation of flamelet surface density (R), and mean (Ms ), turbulence (Ts ), and curvature (Cs ) flamelet stretching mechanisms at different times (ms) of flame propagation, after ignition.
in good agreement with experimental data as reported by [26]. It can be seen that the flame stretch at the semi-confinement between the first two obstacles is of lower values suggesting that it lies in the continuous laminar flame combustion regime. As the flame propagates past the second obstacle and before venting starts, the stretch increases fast to reach a combustion regime where the flame starts to break-up due to high flame stretch.
5. Conclusions A study was performed to investigate transient premixed combustion in an explosion chamber. The propagation of the flame front was mapped through the use of high speed laser sheet flow visualisation. A modified FSD model has been developed and implemented in a compressible turbulent reacting flow code and applied to predict the flame propagation past multiple obstacles
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Fig. 6. Variation of the calculated Karlovitz stretch factor values along the flame front at different times (ms) after ignition.
mounted inside a semi-confined explosion chamber. Results from the various stretching mechanisms at different stages of flame propagation were presented. These show that the turbulence stretch dominates in regions behind the obstacles due to the presence of highly recirculating zones while the mean flamelet stretch is dominant in the flow jetting regions around the obstacles. Numerical results have shown that the values of the flamelet stretch imposed on the flame front ahead of each obstacle is found to increase as the flame propagates from the ignition end towards the venting end. Comparison between numerical and experimental results show that the model yields plausible results for flame structure, pressure time history and flame speed. At present, work is underway to investigate the validity of the model under different flow and mixture conditions. Moreover, work is underway to improve the predictions of the turbulent field through the use of LES models. References [1] K.N.C. Bray, in: Twenty-Sixth Symp. (Int.) Combust., The Combustion Institute, Pittsburgh, 1996, pp. 1–26. [2] S. Menon, W. Jou, Combust. Sci. Technol. 75 (1991) 53–68. [3] M.P. Kirkpatrick, S.S. Ibrahim, S.W. Armfield, A.R. Masri, Third Asia-Pacific Conference on Combustion, Korea, 24–27 June 2001, pp. 1–4. [4] D.B. Spalding, in: Thirteenth Symp. (Int.) Combust., The Combustion Institute, Pittsburgh, 1971, pp. 649–657. [5] B.F. Magnussen, B.H. Hjertager, in: Sixteenth Symp. (Int.) Combust., The Combustion Institute, Pittsburgh, 1976, pp. 719–729. [6] C.A. Catlin, M. Fairweather, S.S. Ibrahim, Combust. Flame 102 (1995) 115–128. [7] K.N.C. Bray, Proc. R. Soc. Lond., Ser. A 431 (1990) 315–335.
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