Correction: Sub-grid Scale Modeling of Turbulent Spray Flame using Regularized Deconvolution Method

Transcription

1 .c1 AIAA SciTech Forum 8 12 January 218, Kissimmee, Florida 218 AIAA Aerospace Sciences Meeting / c1 Correction: Sub-grid Scale Modeling of Turbulent Spray Flame using Regularized Deconvolution Method Author(s) Name: Qing Wang(1); Xinyu Zhao(2); Matthias Ihme(1) Author(s) Affiliations: 1. Stanford University, Stanford, CA, United States. 2. University of Connecticut, Storrs, Storrs, CT, United States. Correction DOI: / c1 Correction Notice The x and y labels in Fig. 14 should be swapped. The x label should be Da and y label should be Da v. Copyright 218 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

2 AIAA SciTech Forum 8 12 January 218, Kissimmee, Florida 218 AIAA Aerospace Sciences Meeting / Sub-grid Scale Modeling of Turbulent Spray Flame using Regularized Deconvolution Method Qing Wang and Matthias Ihme Stanford University, Stanford, CA, 9435, United States Xinyu Zhao University of Connecticut, Storrs, CT, 6269, United States A new model based on regularized deconvolution methods (RDM) is developed for the closure of sub-grid scale (SGS) terms in turbulent spray combustion. Previous a priori numerical studies of these flames have identified sensitivities to the turbulence-droplet and turbulenceflame interactions. The objective of this work is to examine e ects of the SGS closures though large-eddy simulations (LES) of an acetone spray flame. A series of experiments of turbulent spray flames were conducted at the University of Sydney to establish a benchmark database to support model developments, including the measurements of the target acetone spray flame. Three LES cases using di erent modeling combinations for the SGS turbulence-droplet and turbulence-flame interactions are conducted to assess the performance of the new model. Systematic model evaluations show that the consideration of SGS turbulence-droplet interactions using RDM improves the prediction of gas-phase combustion downstream where the mesh is coarse. Further improvement in temperature prediction is observed in the case where RDM is also used as the closure for turbulence-flame interaction. I. Introduction spray combustion, as seen in internal and aeronautical combustion engines, is critical for power generation Tin the aerospace industry. To improve the e ciency of such engines, numerical studies have been conducted extensively using large-eddy simulation (LES) for these configurations. Due to their multiphase and multiphysics nature, LES simulations of spray combustion are sensitive to the discrete-phase models, the gas-phase models and the two-phase interaction models. In the present study, the sub-grid turbulence-droplets and turbulence-flame interactions are investigated. For this, a new sub-grid model based on regularized deconvolution method (RDM) is developed for the closure of these sub-grid scale (SGS) e ects. For the discrete phase, the RDM based turbulence-droplet interactions model accounts for the fluctuation of both the gas-phase velocity and temperature fields, which provides corrections for the droplet momentum and energy equations. In particular, the fluctuation of temperature can significantly change the distribution of the sub-grid temperature under certain conditions [1, 2]. However, the reconstruction of the gas-phase temperature has rarely been considered in literature [3]. The velocity and temperature fields that are reconstructed using independent Wiener processes are often decorrelated. Therefore, the primary objective of the present study is to provide a comprehensive and consistent SGS closure model for both e ects. For the gas-phase, the RDM based turbulence-flame interactions model is proposed to reconstruct the SGS turbulent flame structure. Sub-grid fluctuations of the reaction source term are represented explicitly on a high-resolution grid in this approach. It is therefore of interest to assess the capability of RDM as a turbulence-flame interaction model for gas-phase combustion, as a secondary objective. To achieve these goals, the turbulent spray flame experiments conducted by Gounder et al. [4] are employed as test cases. The well-quantified boundary conditions in these experiments make them desirable for the development of modeling capabilities to predict turbulent mixing, reaction chemistry, and multiphase transport. This series of flames Graduate student, Department of Mechanical Engineering, Stanford University, 488 Escondido Mall, Stanford, CA 9435, United States, AIAA student member. Associate Professor, Department of Mechanical Engineering, Stanford University, 488 Escondido Mall, Stanford, CA 9435, United States, AIAA member. Assistant Professor, Department of Mechanical Engineering, University of Connecticut, 191 Auditorium Road Unit 3139, Storrs, CT 6269, United States, AIAA member Copyright 218 by authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

3 have been studied by various groups, considering di erent modeling approaches for droplet transport and gas-phase combustion [5 8]. While the majority of the existing research focuses on the e ects of di erent turbulent combustion models, the evaporation models [9], the influence of inlet velocity profiles [1], and the e ects of sub-grid fluctuation on the droplet velocity are also studied [11]. The ethanol-based experiments are more investigated than the acetone-based experiments due to the increasing interest in bio-fuel. However, the acetone/air flame is shown to behave closer to a premixed flame than the ethanol/air flame due to faster evaporation [6]. From a model assessment point of view, the acetone/air flame is more challenging to models, hence is chosen as the test flame in the present study. The rest of the paper first introduces the modeling approaches, highlighting a new RDM based turbulence-droplet and turbulence-flame interactions model. The experimental and numerical setup is introduced next, with quantified boundary and initial conditions. Finally, results obtained from di erent models are compared, and the reasons that lead to the observed di erences are discussed. II. Governing Equations and Modeling In the present study, we analyzed the performance of LES in simulating a spray jet flame. Di erent SGS closure models are considered, and their performance is evaluated. Detailed formulations for both carrier phase and dispersed phase are described in the following subsections. Configurations on the SGS modeling approaches are discussed. A. Governing Equations and Physical Models In the present work, flamelet progress variable (FPV) is used to model the chemical reactions in LES. The governing equations describing conservation of mass, momentum, and scalars in the gas-phase take the ( eu i i =, (1a) eu eu i eu j = ij du d,i dm d + uj (m d + u d,i j dt dt (1b) @Z 1 dm d eu i ez = D Z + i V dt, (1c) @C eu i ec = D C i C +! C, (1d) where Z is the mixture fraction and C = Y CO2 + Y CO + Y H2 O + Y H2 is the progress variable [12]. The sub-grid stress term is described by eu i e)/@x u i )/@x i for = u i, Z, C. The dispersed phase is modeled by a Lagrangian approach where the following equations are solved with the LES equations [13]: dm d dt du d dt dt d dt = Sh gm d 3Sc g d log(1 + B m ), (2) = f 1 d u s, (3) = f 2 Nu g C p,m T s + L v dm d d 3Pr g C l m d C l dt. (4) Here d = l D 2 /(18µ g ) is the droplet relaxation time constant, C p,m = (1 Y v )C p,g + Y v C p,v is the heat capacity of the gas mixture, B m = (Y s Y v )/(1 Y s ) is the Spalding number, and f 1 and f 2 are the correction factors [13]. u s and T s are the unclosed slip velocity and temperature di erence, respectively. In Eqs. 2 to 4, subscript g, d and v denote the gas-phase, dispersed phase and vapor phase respectively. The FPV chemistry table is constructed from one-dimensional counterflow di usion flame simulations. In this FPV-model, liquid fuel is assumed to be pre-vaporized. In this approach, the temperature of the fuel stream is reduced by an amount that corresponds to the latent heat of evaporation in the one-dimensional flamelet computations. This approach is valid for flows with small Stokes numbers [14, 15], which is defined as the ratio of the relaxation time scale d = d dp/(18µ) 2 to the convective time scale. Note that the Stokes number for the configuration considered in this study is St = 1, showing that the pre-vaporization assumption is valid for the current configuration. 2

4 B. Modeling approaches in LES The sub-grid turbulence-droplet and turbulence-chemistry interactions are identified to be two major SGS e ects in multiphase reacting LES. In literature, the sub-grid turbulence-droplet interactions are often neglected in LES [11]. Filtered velocities and temperature from the carrier phase equations are interpolated at the droplet locations and are used to compute the unclosed terms in Eqs. 3 and 4, as u s = eu(x d ) u d, (5) T s = et(x d ) T d. (6) The e ects of sub-grid fluctuation in velocity and temperature can be significant in multiphase reacting flow simulations, especially for sprays with relatively small Stokes numbers [2]. Ignoring the e ect of sub-grid fluctuations can overestimate the slip velocity and temperature di erences. The incorrect characterization of interactions between the dispersed and carrier phase can lead to mis-predictions in flame topology hence heat release. The impact of sub-grid fluctuations on velocity (i.e., sub-grid dispersion) has been investigated by a few studies. However, the sub-grid scalar dispersion is rarely studied or modeled in the literature [3]. To consider both e ects, a regularized deconvolution method (RDM) [16] is proposed in this study. This method reconstructs the sub-grid fluctuations of the flow field, hence can capture the interactions between droplets and flow structures that are smaller than the mesh grid. In practice, for a filtered variable, e, RDM is formulated as:? = arg min e = =P G? k 2 + k = D? k 2, (7) k C? = e, = apple? apple + where? is the deconvolved solution, P is the interpolation matrix that projects the LES solution from the computational grid to a refined grid, G is the filter kernel which is assumed to take a top-hat profile, D is a finite di erence second order derivative matrix, C is the mean preserving constraint matrix, and is the regularization factor which is estimated as = tr( eg T eg)/tr(i) [17]. The solution of Eq. (7) takes the following form: " # " # 1 " #? = =G T G + = DT D = = CT = =G T P =? =C I e, (8) = = where? is the Lagrange multiplier that is discarded in the model formulation. The inequality constraints are enforced by rescaling the deconvolved solution obtained from Eq. (8). The scaling factors are computed as: S + = min S = min + e i, max? j j i e 1 Æ, (9) Æ e i, max e 1 Æ. (1)? i j j Æ By rescaling the deconvolved solution around the LES solution using the minimum scaling factor obtained from Eqs. (9) and (1), the deconvolved solution is updated as:? j,new = min(s +?, S j + e. (11) )( e) With this, the turbulence-droplet interaction terms are modeled as: u s = u? (x d ) u d, (12) T s = T? (x d ) T d. (13) Additionally, for reacting flow LES, a turbulence closure for chemical reaction term is required. The classical approach to this for FPV is to apply the presumed PDF model [18]. In this model, the filtered chemical source term for C is computed as: π π g! C =! C (Z, C)P(C Z) ep(z)dzdc. (14) 3

5 Table 1 Modeling terms for LES cases considered in the present study. SGS modeling term Turbulence-droplet interaction None RDM RDM Turbulence-flame interaction -PDF -PDF RDM In Eq. 14, P(C Z) is the probability density function (PDF) of progress variable conditioned on the mixture fraction, and is modeled as a -PDF. ep(z) is the PDF of the mixture fraction, which is modeling using a -PDF. An alternative approach for the closure of turbulence-flame interaction is to apply RDM. In this approach, the reaction source term is first computed using the reconstructed scalar field and then filtered explicitly. Applying this technique, the sub-grid structures are represented explicitly; therefore the interactions between the flame and the sub-grid eddies are considered per se. The source term is then filtered explicitly using a top-hat filter and projected back onto the LES grid, which is: g! C =! ü C (Z?, C? ) (15) Based on di erent modeling approaches, three sets of LES are conducted, as shown in Table 1. III. Experimental and Computational Setup Numerical simulations are performed for the spray flame that was experimentally investigated at the University of Sydney [4]. In this configuration, fuel is provided by a central turbulent jet. Liquid fuel is atomized by a nebulizer located 215 mm upstream of the exit plane and injected through the central jet. The diameter of the jet is 1.5 mm and surrounded by a 25-mm-diameter premixed stoichiometric pilot that is generated from a mixture of C 2 H 2, H 2, and air. The burner is accommodated in a wind tunnel of diameter 14 mm with ambient air flow at 4.5 m/s. The temperature was measured using thermal couples, and larger uncertainty is expected with the temperature measurement near the centerline [5]. Fig. 1 Experimental configuration for the spray jet flame [4]. In the present study, LES calculations of the operating condition AcF3 are performed, corresponding to a carrier gas velocity of u j = 24 m/s and exit equivalence ratio of = 1.8. The Reynolds number for this flow is 2, 7. The carrier mass flow rate is 15 g/min, and the liquid fuel injection rate is 45 g/min. The LES calculations are conducted in a cylindrical coordinate system on a structured mesh with N x N r N = cells in a domain of size 4d j 15d j 2. The spray jet inflow is generated from a separate simulation 4

6 of the injection tube, and inflow profiles are collected at the location where the gas-phase fuel mass flux and equivalence ratio match the inlet conditions reported in the experiment. The droplet distribution at the inlet is prescribed from a Rosin-Rammler distribution with the scale factor a = 1 and the shape factor b = 1.8, as shown in Fig. 2. Parameters of the PDF are fitted from the droplet diameter distribution obtained from the experiment. Reacting flow LES with modeling specifications listed in Sec. II are conducted and compared against experiments. Results of the simulations are shown in the subsequent sessions..4.3 Rosin-Rammler PDF.2 Fig d[µm] Droplet diameter distribution at the inlet. IV. Results and Discussions A. Instantaneous flow-field results Qualitative comparisons of the LES results with experimental flow field results for OH and CH 3 COCH 3 are provided in Fig. 3. The size of each window in Fig. 3 is 1d j 1d j. Each window is positioned such that one of its vertical edges coincides with the centerline and the x-location of its center is given by the specified coordinate. The left column of each sub-figure shows the PLIF results from the experiment, and the right column shows the instantaneous FRC solution. The top row is OH and the bottom row is CH 3 COCH 3. Qualitative agreement in the flame location, reaction zone thickness, and flame-turbulence coupling between LES and experiments is observed at all locations. The flow topology is well captured by the LES. Figures 4 to 6 show instantaneous iso-contours of the gas-phase flow field of the axial velocity, temperature and reaction source term for the three modeling approaches. The location of the flame front in the center of the jet is di erent for the three cases, which are at x/d j = 2, x/d j = 25, and x/d j = 28 for,, and, respectively. Consequently, the temperature in the jet core is higher for compared to and. B. Analysis of dispersed-phase results Figures 7 to 1 show the mean and root mean square (RMS) velocity of droplets in the axial and radial directions respectively. Good agreement in RMS velocities is observed for all cases with experiment, showing that the sub-grid dispersion model is insensitive in representing these quantities. The mean radial velocity from all LES cases matches with experimental results at x/d j = 2 and 3, but shows and underprediction at x/d j = 1. This is due to the misrepresentation of droplet velocity distribution at the inlet. As the droplets move downstream, the e ect of boundary condition diminishes. At x/d j = 1 and 2, the mean axial velocity of the droplets in all LES cases show good agreement with the experiment. At x/d j = 3 where the grid is relatively coarse compared to upstream, the mean axial velocity is over-predicted in. The overprediction of mean velocity is compensated in and by the application of sub-grid dispersion model, where good agreements with the experiment are observed. As has been shown in [2], the slip velocity of droplets is over-predicted without considering the e ect of sub-grid turbulence-droplet interactions in the a posteriori simulations. As shown in Figs. 4 to 6, the velocity of the carrier phase decays at x/d j 3. Droplets with large inertia supersede the mean flow of the carrier phase and experience a drag force. This causes dissipation of the kinetic energy of the droplets, which leads to decay of droplet velocity. In where only large-scale flow structures are considered when computing the slip velocity, the energy dissipation for droplet moment is underestimated. This leads to an over-prediction of the axial droplet velocity. By considering the sub-grid fluctuation of the carrier flow 5

7 x/d = 5 x/d = 1 x/d = 15 x/d = 2 x/d = 25 x/d = 3 OH CH3 COCH3 OH CH3 COCH3 Experiment Fig. 3 FRC. LES Qualitative comparison of OH and CH3 COCH3 between experiment PLIF results and LES results with u T C Fig. 4 Temperature, streamwise velocity field, and reaction source term for from x/d j = to 4. with RDM, as in case and, the prediction of dispersed phase energy dissipation is improved. Hence the axial component of the mean velocity reduces and agrees with the experiment for and. Figure 11 shows the radial profile of mean droplet Sauter mean diameter (D32 ) at x/d j = 1, 2, and 3. In all three cases, D32 is under-estimated across the shear layer between the jet and pilot, especially for flow at x/d j = 1. Based on results shown in Fig. 8, an underestimation of droplets radial velocity is observed. This suggests that only small droplets can penetrate through the shear layer between the jet and the pilot stream in the simulations. The misrepresentation of inlet droplet velocity distribution directly influences the spreading of the liquid jet at upstream locations. C. Analysis of gas-phase results The net influence of the sub-grid turbulence-droplet and turbulence-flame interactions models to the heat release of the spray flame is assessed in Fig. 12, where the temperature profiles of the carrier phase from the simulations 6

8 u T u [m/s] C Fig. 5 Temperature, streamwise velocity field, and reaction source term for from x/d j = to 4. u T C Fig. 6 Temperature, streamwise velocity field, and reaction source term for from x/d j = to Fig. 7 Mean axial velocity profiles from x/d j = 5 to 3. are compared with those obtained from the experiment. At x/d j = 1 and 2, all cases show good agreement with experimental results. This suggests that the gas-phase flow field is insensitive to the SGS closures in these regions. On the one hand, as we show in Section B, sub-grid dispersion is insignificant in these regions for the dispersed phase. 7

9 v [m/s] Fig. 8 Mean radial velocity profiles at x/d j = 1, 2, and 3. vrms [m/s] D32 [µm] urms [m/s] Fig. 9 RMS axial velocity profiles at x/d j = 1, 2, and 3. Fig. 1 RMS radial velocity profiles at x/d j = 1, 2, and 3. Fig. 11 Mean Sauter mean diameter profiles at x/d j = 1, 2, and 3. Feedbacks from the dispersed-phase to the gas-phase is comparable in all cases. On the other hand, it is observed that the peak of temperature is correlated with the hot reaction product in the pilot stream. The increment of temperature in the centerline of the jet is a result of heat penetration from the pilot stream to the jet. The influence of reaction heat release is secondary in these regions, therefore the di erence in turbulence closure of chemical source term does not a ect the results. At x/d j = 3, we see a clear distinction in temperature profiles for the three cases, especially near the centerline of the jet. In, the centerline temperature is significantly overpredicted. The prediction improves in, after the consideration of droplet dynamics with the sub-grid closures. Further improvement is observed in, where 8

10 RDM is used as the turbulence closure for the chemical reaction term. In this region, the increment in temperature near the centerline of the jet is dominated by heat release from combustion. The presumed-pdf model underestimates the turbulent fluctuation of the scalar field, which leads to over-estimated chemical reaction terms. The RDM model enhances the SGS turbulent fluctuation, which leads to a reduction in reaction source term and improved agreement with the experiment. T T [K] Fig. 12 Mean temperature, progress variable and mixture fraction profiles at x/d j = 1, 2, and 3. To further understand the reasons leading to the observed di erence at x/d j = 3, the mean profiles of the progress variable and the mixture fraction are examined, as shown in Fig. 13. The scatter plot of temperature in the mixture fraction space is provided to understand the e ects of the model on the distribution of the temperature. With increasing complexity of models, a non-monotonic behavior is observed in the mean temperature, mixture fraction, and progress variable profiles, indicating coupled evaporation-turbulence-reaction e ects. In, the production of mixture fraction due to evaporation is enhanced slightly near the centerline; this can be a combined e ect of enhanced evaporation by using RDM and the over-estimation of temperature by using the -PDF assumptions. Interestingly, shows closer mixture fraction profiles to, which again can be a combinational e ect of the reduced temperature due to turbulence-flame interactions and suppressed evaporation due to reduced temperature. The scatter plot obtained from shows no data points after Z =.2 and more scattered data points between.15 and.2. recovers the non-premixed flame feature on the lean side of the flame (i.e., corresponding to the coflow side in the physical space; st =.9), and it displays more premixed flame features on the rich side of the flame, as evident by the scatter of the temperature data. The scatter plots provided by are more consistent with previous studies where similar flames are simulated [19]. Z C T Z Fig. 13 Mean mixture fraction and progress variable profiles, and temperature scatter conditioned on mixture fraction at x/d j = 3. The competition between the evaporation, turbulence, and chemical reactions determines the dynamics of the observed phenomena in Fig. 13, and such competition can be quantified through the time scale analysis. By taking the ratio of turbulence and reaction time scales, a Damköhler number (Da) can be defined. Da is used as an indicator of the extent to which turbulence-flame interactions are expected to be important. Similarly, the ratio (Da v ) of a turbulence time scale to an evaporation time scale can serve as an indicator of the degree of turbulence-evaporation interactions [2, 21]. Da v is defined as Da v = turb / evap, where ed = (D 2 L) 1/3 /u is the turbulence timescale, evap = m /v r is the evaporation timescale, m = d d /(Sh g 2) is the di usion film thickness, and v r = m d /4 r 2 d is the equivalence velocity of evaporation. Below the critical value of unity, the turbulence energy is able to increase the mass transfer. For a higher value of Da v, an inverse behavior is even observed [21]. The Damköhler number Da v directly analyzes the e ect of the evaporation source terms m d, hence provides a direct measure of the e ects of the SGS models. Because the same definition is used for the turbulence timescale in Da and Da v, the ratio of the two Damköhler 9

11 numbers provides a measure of the relative rates of evaporation and chemical reactions. The relations between the three timescales are plotted in Fig. 14. The values of Da v and Da are plotted for each grid points at x/d = 3. Da v obtained from and in the slow chemistry limit (close to Da = and inert flows) extend to a larger range than the ones obtained from, indicating enhanced evaporation rates by using and. However, Da v from all three models are larger than unity, suggesting that evaporation is faster than the turbulence eddy turnover time in the target flame. This is the major reason why minor di erences are observed in mean profiles, even though the SGS models directly lead to di erent distributions of the timescales. Moreover, more points from and fall into the high Da and high Da v zones near the diagonal, showing that the reaction time scales and the evaporation timescales are reduced, compared to those produced by. In particular, the accumulation of the data points obtained from near Da further demonstrates the di erence created by the use of a more realistic turbulence-flame interaction model. 2 Da Da v Fig. 14 Comparison of turbulence, reaction, and evaporation times scales for droplets at x/d j = 3. The discrepancy between and the experiment is approximately 3% near the centerline. Based on the time scale analysis, combustion models that consider the finite-rate chemical reactions should be considered to improve the predictability of the simulation [22, 23]. It should also be pointed out that experimental uncertainty is another contributing factor, especially near the centerline when thermal couples are used. V. Conclusion In the present study, a new model based on regularized deconvolution method (RDM) for the closure of sub-grid turbulence-droplet and turbulence-chemistry interactions in a Eulerian-Lagrangian framework is proposed. To study the e ect of these modeling approaches, three LES cases with di erent model combinations are conducted for a turbulent spray jet flame that is studied experimentally at the University of Sydney. Based on our results, we conclude the following: Sub-grid dispersion can be important in both droplet dynamics and reaction heat release, and RDM model is e cient in representing this e ect For FPV, RDM improves the sub-grid scale turbulence modeling which leads to better agreement to the experiment compared to the presumed-pdf model Modeling of droplet dispersion by the mean flow in the radial direction needs to be improved to capture the dynamics of the dispersed phase correctly Inflow droplet velocity distribution is important in representing the liquid jet spreading upstream Additionally, it is observed that evaporation is faster than the turbulence eddy turnover time in the target flame; therefore the reconstruction of the SGS fluctuation might not a ect the evaporation rates significantly. However, the distributions of the temperature, as well as the timescales, can be significantly impacted, and such changes can potentially be important for the prediction of pollutants emission. 1

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