Numerical simulations of turbulent lifted jet diffusion ...

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Proceedings of the Combustion Institute 000 (2018) 1–8 www.elsevier.com/locate/proci

Numerical simulations of turbulent lifted jet diffusion flames in a vitiated coflow using the stochastic multiple mapping conditioning approach Sanjeev Kumar Ghai a, Santanu De a,∗, Andreas Kronenburg b a Department

of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India für Technische Verbrennung, University of Stuttgart, Stuttgart 70174, Germany

b Institut

Received 30 November 2017; accepted 10 May 2018 Available online xxx

Abstract Lifted turbulent jet diffusion flames of H2 /N2 issued into a hot coflowing stream of combustion products from a lean premixed H2 /air mixture are simulated using the stochastic multiple mapping conditioning (MMC) approach within the Reynolds-averaged Navier–Stokes (RANS) framework. The underlying turbulent flow is modeled using the two-equation k–ɛ model with modified constants. In the MMC approach, large-scale turbulent fluctuations are emulated by introducing suitable reference variables. A single reference variable is used here to describe the evolution of mixture fraction via a mapping function. The modified Curl’s model has been adapted to model the micro-mixing term. A detailed chemical kinetic mechanism involving 9 species and 21 reversible reactions is used for H2 /O2 combustion. The computed results from the present simulations are found to be in excellent agreement with the available experimental data. Numerical results are found to be sensitive to change in the minor dissipation timescale, which certainly controls the conditional fluctuations around the conditional mean. Also, the predicted flame lift-off heights obtained using the minor mixing time constant Cmin = 0.25 are found to be in close proximity with the experimentally observed values for the entire range of coflow temperatures. © 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Multiple mapping conditioning; Turbulent mixing; Vitiated coflow; Lifted flame; Stochastic processes

1. Introduction An accurate description of highly non-linear reaction rates and modeling of turbulence–chemistry interactions plays a pivotal role towards the development of turbulent combustion models. Direct numerical simulation (DNS) of the practical com∗

Corresponding author. E-mail address: [email protected] (S. De).

bustion systems is not feasible till date. Towards engineering applications, Reynolds-averaged Navier Stokes (RANS) and large eddy simulation (LES) offer solutions for a realistic combustion system within available computational resources. In both of these approaches, the reaction rate term appears as an unclosed quantity, which requires modeling. Comprehensive reviews on the turbulent combustion models are available in Ref. [1]. In general, the turbulent combustion models are classified into

https://doi.org/10.1016/j.proci.2018.05.043 1540-7489 © 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Please cite this article as: S.K. Ghai et al., Numerical simulations of turbulent lifted jet diffusion flames in a vitiated coflow using stochastic multiple mapping conditioning approach, Proceedings of the Combustion Institute (2018), https://doi.org/10.1016/j.proci.2018.05.043

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two categories: one is based on utilizing the mixture fraction and/or progress variable, e.g., flamelet [2] and conditional moment closure (CMC) [3], while another is based on the modeling of the joint probability density function (PDF) method [4]. The latter has the advantage over the former as the reaction rate term appears naturally in closed form. However, the mixing term appearing in the joint PDF method requires modeling. Moreover, PDF methods are more general in their formulation and can be applied irrespective of the combustion regime. However, the same mixing model may not remain applicable to all combustion regimes. Conventional PDF methods are computationally more expensive compared to other turbulent combustion models due to requirements of high memory and computational cost. Again for a practical system which involves hundreds of species, the applicability of joint PDF methods tends to be problematic or impractical. Recently, the multiple mapping conditioning (MMC) model [5] has emerged as a new computational tool for modeling of turbulent combustion. It may be seen as a logical combination of conventional PDF methods and the CMC model with the amalgamation of the concepts of mapping closure [6]. The concept of reference space has been utilized in the formulation of MMC. The shape of the PDF in the reference space is either known a priori or may be simulated by some means, such as Markov process. Scalars are usually divided into major and minor groups. The fluctuations of the minor scalars are completely restricted or the minor scalars are allowed to fluctuate jointly or relative to the major scalars depending on the formulation of the MMC model. The deterministic formulation of MMC closely resembles the CMC method, wherein the fluctuations around the conditional means are completely neglected. On the other side, the probabilistic formulation is associated with the stochastic PDF method in which the minor scalars are allowed to fluctuate relative to the major scalars. Although there is a demarcation between the major and minor scalars, a single transport equation ensures the evolution of all scalars in the reference space. The key difference between standard PDF methods and stochastic MMC is the implementation of the turbulent mixing term: localness of mixing in composition space is a desired feature of any mixing model and MMC ensures such localness by conditioning the turbulent mixing on the reference space. Applications of stochastic MMC have been reported recently within both RANS [7–10] and LES [11] framework. Exhaustive numerical studies have been performed on pilot-stabilized Sandia D–F series using RANS based stochastic MMC simulations [7–10]. In this present work, we extend RANS-MMC to simulate the more complex turbulence–chemistry interaction problem of lifted jet diffusion flames in a vitiated coflowing stream of oxidizer for the first time. The objectives are to

assess the performance of the RANS-MMC approach for lifted flames, analyze the effect of the minor mixing time constant on flame stabilization, and capture the lift-off height (LOH) vs. coflow temperature variation. In the subsequent sections, the MMC approach is briefly described (Section 2), followed by the numerical approach (Section 3). Results from the numerical simulations are presented and discussed in Section 4 and the major conclusions are drawn in the final section. 2. Mathematical formulation The complete theory and the basic concepts including the derivations of the transport equations are given in Ref. [5]. In this work, the stochastic version of MMC is implemented using a single reference variable, ξ ∗ , for the emulation of mixture fraction represented through mapping function XI (ξ; x, t). The standard Gaussian distribution with zero mean and unity variance has been used to represent the distribution of the reference space. The Lagrangian notional particles are used to derive the stochastic formulation of MMC. The corresponding characterized systems of equation are given by Klimenko and Pope [5] d x∗ = U (ξ ∗ , x∗ , t )dt,

(1)

d ξk∗ = A0k (ξ ∗ , x∗ , t )d t + bkl (ξ ∗ , x∗ , t )d ωl∗ ,

(2)

d XI∗ = WI∗ + SI∗ d t,

(3)

SI∗ |ξ ∗ = ξ, x∗ = 0.

(4)

In the above set of equations, A0k = Ak + +

2 ∂ Bkl Pξ ∂ Bkl =− + Bkl ξl ˜ ∂ ξ ∂ ξl Pξ l

1 2 ∂ Bkl Pξ ρU ¯ (1 ) + , ρ¯ P˜ξ ∂ ξl

bki bli = 2Bkl

(5)

(6)

Here, the asterisk is used to denote the stochastic quantities. Equation (1) represents transport in the physical space, where the location of the particle is represented by x∗ . Equations (2) and (3) account for transport in the reference space and in the composition space, respectively. In the above equations, A and B represent drift and diffusion coefficients, respectively, while WI∗ is the chemical source term, SI∗ is a mixing operator which is used to dissipate the minor fluctuations, and dωl∗ is the increment in the Wiener process. In this stochastic formulation of MMC [5], Bkl is modeled independent of ξ , (Bkl = Bkl (x, t )) and it can be related to the unconditional scalar dissipation, N˜ Z as given by


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Bkl (x, t )

∂Z ∂ξ

2 = N˜ Z .

(7)

Adopting Corrsin’s suggestions [12], the unconditional scalar dissipation is modeled as the ratio of decay rate of passive scalar variance to the major timescale, τ φ , which is given as 2 Z N˜ Z = 0.5 τφ

(8)

with τφ = τt /Cφ and the timescale τt = k/ε is the turbulent timescale. The ratio of turbulent and major timescales Cφ depends on the type of flow and other geometrical constraints, and in PDF simulations usually taken as a value between 1.5 and 3.0 [13]. In the present case we used Cφ = 3.0. A more detailed discussion on the conditional dissipation rate modeling including the modeling of dissipation timescales can be found in Wandel [14]. The modified Curl’s mixing model [15] is used here to model the mixing operator SI∗ that appears explicitly in Eq. (3). To ensure localness of mixing in the composition space, particle pairs are not selected randomly, but only those particle pairs within a CFD cell are selected which are close in the reference space instead of the physical space. This is achieved by sorting the particles according to their reference values within a CFD cell, and thereafter particles which are the closest neighbors in the reference space are allowed to mix. More details about the mixing model and the particle management can be found in our previous publication [7]. Here we are only providing the gist of the model due to brevity. Due to the conditioning of the mixing operator on the reference space, the mixing operator controls the conditioned fluctuations and can be represented in MMC as SI∗ =

d X ∗,s X p,q − X ∗,s ≈λ dt τmin

(9)

Where ‘s’ represents particle p or q and Xp, q is the weighted mean of particles (p,q), w p+ wq N w p+ wq N λ = 1 − exp − ≈ . (10) W 2 W 2 Here wp, wq are the weights of the particles p and q, respectively, and W = N i=1 wi is the total weight of particles. The number of particles per CFD cell is denoted by N. The minor timescale, τ min , controls the conditional fluctuation of X about the conditional mean. The major dissipation timescale, τ φ , controls the major fluctuations of the mapping function relating to the mixture fraction, Z through Eq. (2) via the unconditional scalar dissipation rate, N˜ Z . The two timescales have the following relation τmin = Cmin .τφ , where Cmin is the minor mixing time constant and literature [16] suggests Cmin = 0.25 as an adequate value for the mixing time constant.

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This value is obtained using DNS study of homogeneous reacting flows. Recently, Varna et al. [10] corroborated that finding against Sandia D–F series results. It is noted that the value for Cmin reported here is also consistent with Cmin = 0.125 obtained from DNS [16]. The factor of 2 is the direct consequence of a different definition of unconditional scalar dissipation but does not reflect a change in the proportionality constant between τ φ and τ min .

3. Numerical approach In the present work, turbulent lifted jet diffusion flames of H2 /N2 (1:3 by volume) issued into a hot coflowing stream of vitiated lean premixed H2 /air combustion products [17] is considered. The central jet has a diameter of 4.57 mm surrounded by a hot coflow of 210 mm diameter. The jet and coflow velocities are 107 m/s and 3.5 m/s, respectively. More details on the experimental setup and inlet conditions are available in Ref. [17]. Dirichlet boundary conditions are implemented for mixture fraction, velocity, temperature and species mass fractions at the inlet, whereas Neumann boundary conditions are specified at the outlet for all flow variables except pressure. The flow field is solved using a RANS code which is fully coupled to the stochastic MMC model [7–9]. Turbulence is modeled by the k − ε turbulence model with the following set of constants: Cμ = 0.09, Cε1 = 1.53 and Cε2 = 1.85. A detailed chemical mechanism with 9 species and 21 reversible reactions is used for H2 /O2 combustion [18]. A cylindrical domain which extends 50D in the axial and 15D in the radial directions is used. The computational domain is discretized into 100 finite volume cells in both directions. The cells in the radial direction are refined near the jet centerline and within the shear layer between the jet and coflow. At steady state, approximately 0.5 million Lagrangian particles are present within the computational domain with an average of 50 particles per cell. During simulations, the maximum and minimum numbers of particles per CFD cell are restricted to 75 and 25, respectively. The grid sensitivity study has been carried out and numerical predictions of flow fields are found to be independent of further refinement of both numbers of grid and Lagrangian particles.

4. Results and discussions In this section, the numerical results obtained from present RANS-MMC simulations are presented and compared with the published experimental data of turbulent lifted H2 /N2 jet diffusion flames [17]. The effects of variation of minor dissipation timescale on unconditional and conditional statistics are investigated. Towards the end of the


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Fig. 1. Radial profiles of mean axial velocity at different axial locations using three different minor mixing time constants. Lines represent RANS-MMC simulation: Cmin = 0.25 (black solid lines), Cmin = 0.35 (blue dashed lines) and Cmin = 0.5 (red dashed-dotted lines), whereas symbols represent experimental data [17,19]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

Fig. 2. Radial profiles of mean mixture fraction at different axial locations with three different minor mixing time constants. The symbols and the lines are the same as in Fig. 1.

section, the effect of coflow temperature on the LOH is also discussed. 4.1. Unconditional statistics Figures 1–3 represent the radial profiles of Favre mean axial velocity, mean and rms of mixture frac-

Fig. 3. Radial profiles of rms mixture fraction at different axial locations. The symbols and the lines are the same as in Fig. 1.

tion at several downstream locations. The results are presented for three different minor dissipation timescales (Cmin = 0.25, 0.35 and 0.50) and are compared with the experimental data. The predicted results show excellent agreement with the available experimental data over the entire range for all the minor timescales. There is no appreciable difference in the predicted mean profiles of velocity and mixture fraction on the choice of minor mixing timescale. However, some differences between the results are apparent from the radial profiles of rms of mixture fraction where the higher values of minor timescale clearly overpredict the variance of mixture fraction at all axial locations. The case with Cmin = 0.25 produces the best possible agreement with the published experimental data at all axial locations. The same value of Cmin has been suggested in literature and recently Varna et al. [10] also corroborated the similar findings for Sandia D–F flame series results. The present configuration of lifted jet flames offer more complex turbulence–chemistry interactions and it is found that same constant Cmin = 0.25 produces the best possible agreement for these flames. The radial profiles of unconditional temperature and its rms are shown in Fig. 4. The effect of variation of minor timescale ratio is clearly evident from the present set of results. Numerical results from the case with Cmin = 0.25 yields an excellent agreement with the available experimental data for both mean and rms of temperature at all downstream locations. The Favre mean temperature remains underpredicted for values other than Cmin = 0.25 at almost all downstream locations. From the results with Cmin = 0.25, it is clearly


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Fig. 4. Radial profiles of mean temperature and its rms at different axial locations with three different minor mixing time constants. Black square symbols represent the experimental data for mean temperature, and white circle symbols corresponds to rms of temperature [17]. The lines are the same as in Fig. 1.

evident that up to x/D = 8 the temperature profiles only show mixing between the jet and the hot coflowing air stream. Up to x/D = 10, Favre mean temperature within the shear layer increases due to molecular and turbulent mixing, however, the temperature remains almost unchanged at the jet centerline. The centerline profiles of Favre mean values of mixture fraction, temperature, OH and H2 O mass fractions are provided in the supplementary section. Around x/D = 10, ignition occurs within the shear layer (1 < r/D< 2) accompanied by a slight rise in temperature in that region. The same trend was reported from experiments [17]. Beyond x/D = 10, the Favre mean temperature begins to increase at the centerline. Finally, the peak in temperature occurs at the centerline around x/D = 26. Figure 5 represents the radial profiles of OH and H2 O mass fractions at different axial locations. Predicted species mass fractions from present RANSMMC simulations agree reasonably well with the experimental data for Cmin = 0.25 with slight overprediction of OH and H2 O mass fractions near the flame base at x/D = 9. This indicates a somewhat early onset of radical formation which is also consistent with the slight overprediction of unconditional temperature at x/D = 10. Numerical results from other minor timescale ratios (Cmin = 0.35 and 0.5) grossly underpredict species mass fractions within the flame region. Following the crite-

Fig. 5. Radial profiles of OH and H2 O mass fractions of at different axial locations. The symbols and the lines are the same as in Fig. 1.

rion suggested in Ref. [17], we determined the LOH as the first axial location where Favre averaged OH mass fraction reaches 600 ppm. For the base case of 1045 K, the LOH is found to be 10.2D, 12D and 15D for Cmin = 0.25, 0.35 and 0.50, respectively, whereas the experimentally observed value of LOH is 10D [17]. The LOH predicted by Cmin = 0.25 is found to be closest to the experimentally observed value and it is within the 10% experimental uncertainty in measurement of OH mass fraction [17]. The close resemblance of unconditional radial profiles of temperature at all the axial locations attributes to the almost exact match of LOH with the experiment. 4.2. Conditional statistics Conditional mean and rms of temperature and OH mass fractions are shown in Figs. 6 and 7 at different axial locations. The conditional profiles also clearly demonstrate the effect of variation in Cmin . Up to x/D = 8, only a linear mixing profile is observed with negligible variance within the preflame region. Beyond this, ignition starts and the flame develops at downstream locations. The flame base stabilizes in the lean mixture, which is clearly


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Fig. 6. Conditional mean (left) and rms (right) profiles of temperature at different axial locations. The symbols and the lines are the same as in Fig. 1.

Fig. 7. Conditional mean (left) and rms (right) profiles of OH mass fraction at different axial locations. The symbols and the lines are the same as in Fig. 1.

evident from the conditional temperature profiles at x/D = 11. At x/D = 11, the peak in T|η occurs around η = 0.25, thereafter the peak gradually shifts towards the stoichiometric mixture fraction of 0.473. The flame under consideration stabilizes at a location where scalar dissipation rate is low and the mixture is lean, which corresponds to the “most reactive mixture fraction” [20]. The similar trend is also observed for YOH |η as shown in Fig. 7. The conditional statistics show an excellent match with the experimental data for Cmin = 0.25. A higher value of Cmin (e.g., 0.35 and 0.5) delays the ignition, and subsequently the flame stabilizes at locations further downstream. This results in severe underprediction of the conditional means (e.g., T|η, YOH |η) at all downstream locations within the reaction zone. Scatter plots of temperature vs. mixture fraction for Tc = 1045 K and Cmin = 0.25 are presented in Fig. 8. Within the preflame region, the scatter plots converge into a single mixing profile. The transition

from mixing to burning profile begins beyond the flame base (around x/D= 10). These scatter plots closely follow the experimentally observed trend. A minor deviation is observed beyond η > 0.7 around x/D= 14. Similar discrepancies are also reported in previous PDF results [21], which may be due to the interaction between mixing and reaction or slow reaction rates prevailing on the fuel-rich side. Uncertainty in measurement on the rich side could be another reason for the mismatch. In the fuel lean region, the reaction rates are somewhat faster due to the high coflow temperature and predictions in this region are in complete agreement with the experiment. At x/D= 11, both mixing and burning modes are observed. As we move downstream, the unreacted fuel–oxidizer parcels on the lean side begin to disappear at a faster rate compared to those available on the rich side. Around x/D= 26, both experiment and RANS-MMC show only burning profile. However, a somewhat narrower profile is observed in simulations compared to the experiment [17].


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Fig. 9. Variation in LOH with coflow temperature. CMC simulations of Patwardhan et al. [20], PDF simulations of Cao et al. [21], experimental measurements of Cabra et al. [17] and Wu et al. [22] are also plotted.

5. Conclusions

Fig. 8. Scatter plots of temperature at three different axial locations against mixture fraction. The left column represents the experiment [17], whereas the right column represents the RANS-MMC simulations with Cmin = 0.25.

4.3. LOH vs. coflow temperature The LOH is found to be highly sensitive to the change in coflow temperature because the ignition process is primarily dependent on it. Variation in LOH for a range of coflow temperatures are presented in Fig. 9 and compared with experiments [17,22] and those reported in previous numerical simulations [20,21]. The experimental data of Cabra et al. [17] has 3% uncertainty of temperature measurement. A small deviation of 10 K in Tc doubles the LOH. This makes it very difficult to capture the LOH accurately. In order to investigate the dependence of LOH on coflow temperature, simulations are performed for a range of coflow temperatures. Present simulations can quantitatively capture the change in LOH over the entire range of coflow temperature which most of the previous numerical models failed to predict [20]. The LOH is found to be highly sensitive to Cmin . With the twofold increase in minor dissipation time, LOH increases by almost a factor of 1.5. This suggests that a change in minor mixing timescale strongly effects the turbulence mixing and hence ignition. For Cmin = 0.25, an almost exact match with the experiment data [17] is obtained for Tc = 1045 K. Further, the predicted LOH closely follow the experimentally observed values at other coflow temperatures [22].

A RANS based stochastic MMC approach has been used to simulate turbulent lifted jet diffusion flames of H2 /N2 mixture issued into a vitiated coflow. The RANS-MMC model could accurately predict the radial profiles of Favre mean velocity, mean and variance of mixture fraction, mean temperature and OH mass fraction at all axial locations. Further, the conditional mean and rms values of the reactive scalars are also found to be in excellent agreement for the base case of 1045 K. A series of numerical simulations is performed using different values of minor mixing time constant Cmin . Conditional and unconditional statistics obtained for the base case with Cmin = 0.25 yield an excellent agreement with the published experimental data. This value of Cmin is exactly the same as reported in the recent literature [10]. Complementing this base case, a series of numerical simulations is performed for different coflow temperatures using Cmin = 0.25. At low coflow temperatures, most of the stateof-the-art combustion models could not capture the LOH vs. coflow temperature trend quantitatively. However, the variation in LOH predicted by present RANS-MMC simulations is found to be in excellent quantitative agreement for the entire range of coflow temperatures investigated here.

Acknowledgement The authors would like to acknowledge the computer center at IITK (www.iitk.ac.in/cc) for providing the computing facility. SD gratefully acknowledges the financial support from the research initiation grant (IITK/ME/2014319) and the Science and Engineering Research Board


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(SERB)-India Early Carrer Research Award (ECR/2016/001996). AK appreciates the support by the Deutsche Forschungsgemeinschaft (DFG grant number KR3684/7-1) with thanks. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10. 1016/j.proci.2018.05.043. References [1] D. Veynante, L. Vervisch, Prog. Energy Combust. 28 (2002) 193–266. [2] N. Peters, Prog. Energy Combust. 10 (1984) 319–339. [3] A.Y. Klimenko, Phys. Fluids 10 (1998) 922–927. [4] S.B. Pope, Prog. Energy Combust. 11 (1985) 119–192. [5] A.Y. Klimenko, S.B. Pope, Phys. Fluids 15 (2003) 1907–1925. [6] S.B. Pope, Theor. Comput. Fluid Dyn. 2 (1991) 255–270. [7] C. Straub, S. De, A. Kronenburg, K. Vogiatzaki, Combust. Theor. Model. 20 (2016) 894–912. [8] K. Vogiatzaki, A. Kronenburg, S. Navarro– Martinez, W.P. Jones, Proc. Combust. Inst. 33 (2011) 1523–1531.

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