Low-dimensional manifolds in direct numerical simulations of premixed turbulent flames

Transcription

1 Proceedings of the Combustion Institute 3 (27) Proceedings of the Combustion Institute Low-dimensional manifolds in direct numerical simulations of premixed turbulent flames J.A. van Oijen *, R.J.M. Bastiaans, L.P.H. de Goey Mechanical Engineering, Technische Universiteit Eindhoven, P.O. Box 53, 56 MB Eindhoven, The Netherlands Abstract Direct numerical simulation (DNS) is a very powerful tool to evaluate the validity of new models and theories for turbulent combustion, but the application of detailed chemistry is limited. In this paper, a dimension-reduction technique called the flamelet-generated manifold (FGM) method is considered. In this method a manifold is created by solving a set of one-dimensional flamelet equations. The use of low-dimensional FGM s in DNS of premixed turbulent flames in the thin reaction zones regime is investigated. A three-dimensional (3D) DNS is performed of a spherically expanding, premixed, turbulent, methane air flame. D and 2D FGM s are created and used in simulations of flamelets which are subjected to stretch and curvature effects derived from the 3D DNS results. The results are compared with results from flamelet simulations with detailed chemistry. The results show that deviations from the D FGM due to stretch and curvature effects are significant, but they appear to be embedded in a 2D manifold. This 2D manifold corresponds well with 2D FGM s that are created in different ways, but it shows large differences with a 2D manifold based on chemical kinetics alone. This indicates that an attracting low-dimensional manifold exists which is not solely determined by chemical kinetics. As a consequence, the results of the flamelet simulations using 2D FGM s are more accurate than when a D FGM is applied: the mean error in the burning velocity is almost an order of magnitude smaller. Ó 26 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Manifolds; Flamelets; Turbulent premixed flames; Direct numerical simulation. Introduction * Corresponding author. Fax: address: j.a.v.oijen@tue.nl (J.A. van Oijen). Direct numerical simulation (DNS) is a very powerful tool to test the validity of new models and theories for premixed turbulent combustion (see e.g., [ 3]). The disadvantage of using threedimensional (3D) DNS is that the application of detailed chemical kinetic mechanisms is still limited. It is well recognized that methodologies are needed, which radically decrease the computational burden imposed by the use of detailed chemical kinetics. Very successful are dimension-reduction techniques, which are reviewed in [4]. The most commonly used dimension-reduction methods are based on the quasi steady-state assumption (QSSA). In this methodology, the n chemical species are divided in n s slow species (or controlling variables) and n f = n n s fast species. For each fast species a QSSA is invoked and the differential equation describing its evolution is replaced by an algebraic one. In the n-dimensional species or composition space, these n f algebraic equations define an n s -dimensional manifold. Since all compositions in a reactive system are assumed to lie /$ - see front matter Ó 26 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:.6/j.proci

2 378 J.A. van Oijen et al. / Proceedings of the Combustion Institute 3 (27) on this manifold, the system can be described by n s controlling variables and a dimension reduction is achieved. A well-known method is the intrinsic low-dimensional manifold (ILDM) method [5], which uses an eigenvalue analysis of the jacobian of the chemical reactive system to identify fast and slow species. Recently, manifold methods have been introduced that are not only based on reaction kinetics. These methods, which are called the flamelet-generated manifold (FGM) method [6] and flame prolongation of ILDM (FPI) [7], combine the ideas of the flamelet and manifold approach. In these methods, a manifold is formed by the compositions found in a D premixed flame. This has the advantage that the main effects of convective and diffusive transport are taken into account in the manifold as well. FGM with only a single controlling variable (apart from additional controlling variables to take variations in enthalpy and stoichiometry into account) has proven to be successful for partially premixed laminar flames [8 ]. In[], FGM has been applied in DNS of turbulent flames in the corrugated flamelet regime [2]. In this paper, we study the use of FGM in DNS of premixed turbulent flames in the thin reaction zones regime. In such highly turbulent flames, deviations from D flamelet behavior due to flame stretch and curvature are important and a D manifold may not suffice. Extension of FGM to more dimensions is therefore investigated and the most important questions are: does a low-dimensional manifold exist for these premixed turbulent flames? And if so, can it be represented by a low-dimensional FGM? To answer these questions, we first introduce a flamelet description for premixed flames, which is the basis for the FGM method. In Section 3, it is explained how multi-dimensional FGM s are created. The flamelet description is also used to investigate flame stretch and curvature effects in 3D DNS of spherically expanding turbulent methane air flames. A similar analysis has been carried out in [], but then for turbulent flames in the corrugated flamelet regime. The coupling of flame stretch and curvature with non-unity Lewis number transport processes is an important issue in premixed turbulent flames (see e.g., [3]). In previous publications [8,9] we have shown that the combined effect of flame stretch and preferential diffusion can be modeled with FGM. In this paper, however, a unity Lewis number approach is adopted to simplify the analysis. Furthermore, it is expected that preferential diffusion effects are negligible in the stoichiometric methane air flames studied here, because the Lewis number of methane is close to unity. This has been demonstrated recently by de Swart et al. [4]. The 3D DNS is performed by using FGM, because the use of detailed kinetics is currently too costly from a computational point of view. As a result, a direct comparison between FGM and detailed chemistry in DNS is not possible in this paper. Nevertheless, to compare the FGM method with detailed kinetics, D flamelets subjected to stretch and curvature fields derived from the 3D DNS results are simulated with detailed kinetics. The compositions in these detailed flamelet simulations are analyzed in composition space and they are compared with 2D ILDM and FGM s. Furthermore, D and 2D FGM s are used to compute the same flamelets and results for the mass burning rate are compared with results using detailed chemistry. 2. Flamelet description The flamelet description for premixed flames introduced by De Goey and Ten Thije Boonkkamp [5] has been derived in a systematic way by decomposing the system of combustion equations in three parts: () a flow and mixing part without chemistry, (2) a kinematic equation for the flame motion, including internal flame dynamics and (3) a flamelet part describing the inner structure and propagation speed of the flame structure. The flame, including internal structure, is described in terms of iso-surfaces of a progress variable Y, which can be any linear combination of species mass fractions. The motion of each iso-surface of Y is described by the kinematic equation [6] oy ot þ~v f ry ¼ oy ot þ~v ry s LjrYj ¼; ðþ where ~v f ¼ ~v þ s L ~n is the local velocity of a flame surface being the sum of the fluid velocity ~v and the local burning velocity s L ~n and where the unit normal vector ~n on the flame surface, directed towards the unburnt mixture, can be written as ~n ¼ ry=jryj. Note that Eq. () is introduced for each iso-surface in the complete flame region where < Y <. As a result ~v f and s L are field variables, which vary throughout the flame region. In some other publications (e.g., [2]) s L is referred to as displacement speed. The stretch field K inside the flame zone is defined as the relative rate of change of the mass M in a small part of the flame, enclosed by a small, flame volume V, moving with velocity ~v f : K ¼ M dm dt with Z M ¼ V ðtþ q dv : ð2þ From this definition of K and the kinematic Eq. () the following set of flamelet equations for the conservation equations of mass and Y

3 J.A. van Oijen et al. / Proceedings of the Combustion Institute 3 (27) can be derived rigorously in a orthogonal flameadapted coordinate system [5]: o ðrmþ ¼ rqk; os ð3þ o oy rmy rqd Y ¼ r _x Y rqky; ð4þ os os where s is the arc length through a flamelet, locally perpendicular to the iso-planes of Y and r the surface area through which the transport processes in the flame-adapted system take place. Note that the local curvature j of the flame surfaces is related to the surface area r by j ¼r~n ¼ or r os : ð5þ In Eq. (3) the mass burning rate m = qs L is introduced with q the density of the gas mixture. D i and _x i are the diffusion coefficient and chemical source term of species i, respectively. The diffusive fluxes are assumed to be proportional to the gradient of the species mass fraction. The equations for the other species mass fractions Y i have a similar form as Eq. (4) o r os rmy i rqd i oy i os _x i ¼ qky i þ Q i ; but with an extra term Q i defined as ð6þ Q i ¼ dy i ds þrðqd ir k Y i Þ; ð7þ with $ i the gradient operator in direction parallel to the flame surface only. The first term on the right-hand side accounts for unsteady effects in the flame-adapted reference frame: o ¼ os o þ ~v ot f r. The second term accounts for diffusive transport processes due to the fact that iso-planes of different species are not parallel. These extra terms are small and neglected inside the laminar flamelet regimes [5]. The set of flamelet Eqs. (3), (4) and (6) together with a similar equation for the enthalpy, describes the internal flame structure, dynamics and the eigenvalue for the mass burning rate m(s) for a flamelet with a particular Q i (s), stretch field K(s), and curvature field j(s). 3. Flamelet-generated manifolds The flamelet equations derived in the previous section are used to generate FGM s. If we assume that all perturbations from local D flat flame behavior can be neglected, then we can use K =, j = (or equally r = ), and Q i =. The set of equations then describes an unstretched, D, flat flamelet: the most elementary premixed flame structure. Its solution Y i (s) can be considered as a D manifold in composition space. This D FGM can be parameterized by a single controlling variable Y.In[6] it is shown that the D FGM corresponds to the D ILDM at high temperatures, where chemistry is dominant. At moderate temperatures, molecular transport is important as well and the FGM is closer to compositions found in a premixed flame. In turbulent flames large deviations from the D manifold may arise due to flame stretch and curvature effects. To take this into account the dimension of the manifold can be increased. A larger degree of freedom makes a better description of the composition possible and therefore it increases the accuracy of the model. A multidimensional FGM is generated by computing a series of flamelets, which form together a multidimensional surface in composition space. There are two ways to compute the different flamelets. First, the boundary condition at the unburnt side (s fi ) can be changed. This method has been introduced by Pope and Maas [7] to generate multi-dimensional manifolds and it was used in [6] to create a 2D FGM. In this paper, we use a 2D FGM generated by converting a part of the initial CH 4 and O 2 into H 2 and CO, while keeping the element mass fractions and the enthalpy of the mixture constant. Second, a multi-dimensional FGM can be generated by including terms in the flamelet equations, which have been neglected in the D FGM approach. In this way more physics is taken into account, which results in a more accurate description of the composition. Effects of flame stretch, for instance, can be included by modeling it with a constant stretch rate K = const in the flamelet equations. By applying a range of stretch rates a 2D manifold is created. Differently, flame curvature can be taken into account by assuming constant curvature j = const. In this paper, these two obvious choices are considered, but many others could be thought of as well. In Section 6, these two 2D FGM s and the one created by changing the inlet condition will be compared. It s worth mentioning, that all three 2D manifolds in this paper are parameterized in the same way, i.e., by two controlling variables Y and Y 2, and not by using the stretch rate or curvature. 4. Direct numerical simulation with FGM Freely expanding flames are modeled in a turbulent flow field by using DNS. The fully compressible combustion equations are solved in a cubic 3D domain with a length of 5. mm and 27 grid points uniformly distributed in each direction. All boundaries of the domain are modeled as outlet boundaries to prevent pressure build-up in the domain. The initial field is

4 38 J.A. van Oijen et al. / Proceedings of the Combustion Institute 3 (27) obtained from detailed simulations of laminar spherically expanding flames with the D flame code CHEMD [8]. A stoichiometric methane air mixture is considered at atmospheric pressure with an unburnt temperature T u = 3 K. The initial flame is a sphere with a radius of approximately mm at 698 K and its center located at the center of the computational domain. This initial field is superimposed onto a turbulent flow field, which is modeled to be homogeneous and isotropic. This yields T /d L = with d L =.5 mm the laminar flame thickness based on the maximum gradient of the temperature. The turbulent velocity scale u =s L ¼ 4: with s L the burning velocity of a flat stretchless flame with respect to the unburnt mixture. The turbulent Reynolds number is Re ¼ u T =s L d L ¼ 4:. In this case, combustion takes place in the thin reaction zones regime. The DNS is performed using a D FGM based on the GRI reaction mechanism 3. [9]. The use of detailed kinetics is currently not feasible due to the extremely high computational demands. Applying a multi-dimensional FGM is unnecessary, because a direct comparison with detailed chemistry is not possible anyway. The mass fraction of carbon dioxide, which is monotonously increasing, is used as single controlling variable Y ¼ Y CO2 =Y burnt CO 2. Since pressure, enthalpy and element mass fractions are constant in these flames, they are not needed as additional controlling variables. When a D FGM is applied, transport equations do not have to be solved for all species mass fractions. Instead a differential equation is solved for the controlling variable only. Since the reaction layer for this, slowly changing variable is thicker than for radicals such as CH, a relatively coarse grid is sufficient to resolve the structure of the flame. A more detailed description and a validation of the numerical method are given in [2]. A cross section through the center of the flame at t = T /u is shown in Fig.. After this time no significant changes in the statistics are found and effects of the artificial initial condition have disappeared. Velocity vectors and isotherms corresponding to the position of the unburnt side, the inner layer and the burnt side of the flame (T = 35, 698 and 849 K) are displayed. The inner layer is defined as the position where the chemical source term of the progress variable _x Y reaches its maximum value along s. The temperature at the burnt side corresponds to the position where _x Y has decreased to % of its maximum value. The distance between the isotherms T = 698 and 849 K gives a good indication of the thickness of the reaction layer. The preheat zone of the flame is the region between the isotherms T = 35 and 698 K. It can be seen that the average radius of the flame ball has hardly changed during one eddy turn-over time Fig.. Cross sections through the center of the flame at tu / T =. The solid lines are isotherms corresponding to the unburnt side, the inner layer and the burnt side of the flame (T = 35, 698 and 849 K). The vectors represent the gas velocity (u, v). Six flamelets (thick gray lines) are projected on the plane z =. The spatial coordinates are given in mm. However, the local flame front is clearly distorted by the turbulent flow. Turbulent eddies have distorted the preheat zone of the flame, but the reaction layer remains intact. This behavior is expected in the thin reaction zones regime. 5. Flamelet analysis To perform the flamelet analysis, D flame paths~xðsþ are reconstructed from the DNS results, by integrating d~x ¼ ~ndn in the direction normal to the iso-surfaces of the progress variable. A number of these flame paths (or flamelets) is shown in Fig.. The flamelets are projected on the plane z = through the center of the flame and cross the plane of projection at the position of their inner layer. Since the flame paths are curved, the other parts of the flamelets do not lie in this plane and therefore they do not seem to reach the unburnt boundary. Once the flame paths are constructed, the profiles of the relevant variables along these paths can be computed. In Fig. 2 the profiles of the temperature T, the dimensionless flame stretch rate Kd L =s L and curvature jd L are shown for the different flamelets displayed in Fig.. The arc-length s is scaled with the flame thickness d L. The profiles of T in the preheat zone are changed significantly by the flow, but they are almost undisturbed near the inner layer s = s il. The gradient of T at the inner layer is nearly constant. The dimensionless stretch rates are O(), locally reaching values up to.

5 J.A. van Oijen et al. / Proceedings of the Combustion Institute 3 (27) Fig. 2. Profiles of temperature T, the dimensionless flame stretch rate Kd L =s L and curvature jd L along the flame paths displayed in Fig Fig. 3. Scatter plot of the scaled mass burning rate m il =m il at the inner layer computed with D FGM (yaxis) and detailed chemistry (x-axis). makes the system steady and D. Extinction and re-ignition phenomena are therefore not included in these flamelet simulations. The set of equations is solved similar to a flat, stretchless, D flame, but now with prescribed K and r field. The simulations are carried out using a D FGM and detailed chemistry, which makes a direct comparison at turbulent conditions possible without performing a DNS with detailed kinetics. The mass burning rate m il at the inner layer of O( 3 ) flamelets is determined for both models and the results are compared in Fig. 3. The results from the simulations with FGM agree quite well with those computed using detailed chemistry. The difference is larger at small values of m il, because these flamelets experience the highest stretch rates. The average deviation is 5%. This means that the stretch rates are comparable to the laminar flame time scale and thus not negligible. The same holds for curvature, because the dimensionless curvature jd L of the flame surface is also O(). It is expected that such high stretch and curvature rates cause significant deviations in composition space from the D manifold approach. This cannot be checked directly because the DNS is carried out with a D FGM, which does not allow such deviations. Therefore, the influence of flame stretch and curvature on the kinetics is investigated by computing flamelets with detailed chemistry that are subjected to the stretch and curvature effects occurring in the turbulent flame. In these simulations the flamelet Eqs. (3), (4) and (6) are solved for the stretch and surface fields (K(s) and r(s)) taken from the DNS data as described above. The term Q i in (6) is neglected, which 6. Manifold analysis The accuracy of the FGM model can be improved by increasing the dimension of the manifold. In order to assess whether this approach can be successful, the dimension of the accessed space in composition space is investigated for the flamelet simulations with detailed chemistry as presented in the previous section. As a first step we constructed scatter plots of species mass fractions versus Y ¼ Y CO2 =Y burnt CO 2 and compare the result to the D FGM. The top row of Fig. 4 shows that the species mass fractions Y i (i =H 2, O and CO) remain clustered around the D FGM, but that deviations of % occur. The deviations DY i ¼ Y i Y D i of the species mass fractions from the D FGM are not random, but appear to be strongly coupled. This can be seen in the second and third row of Fig. 4, in which scatter plots of DY i versus DY 2 are shown conditioned at

6 382 J.A. van Oijen et al. / Proceedings of the Combustion Institute 3 (27) x x x Fig. 4. Top row: scatter plots of species mass fraction Y i versus first controlling variable Y. The solid line is the D FGM. The dashed line denotes the scaled chemical source term of Y. Second and third row: scatter plots of variations in the species mass fraction DY i versus variations in the second controlling variable Y 2 at Y ¼ :6 and.8, respectively. The lines are the 2D ILDM (solid) and the 2D FGM created by including stretch (dashed). Bottom row: mean error of different manifold approaches as function of Y. Dash dotted, D FGM; solid, 2D ILDM; dashed, 2D FGM (stretch); solid gray, 2D FGM (curvature); dashed gray, 2D FGM (initial condition); left column, H 2 ; middle column, O; right column, CO.

7 J.A. van Oijen et al. / Proceedings of the Combustion Institute 3 (27) Y ¼ :6 and.8, respectively. Y ¼ :6 corresponds to the point at which _x Y is maximum (see top-left graph of Fig. 4), while Y ¼ :8 is much closer to chemical equilibrium. The mass fraction of OH is used as second controlling variable Y 2 ¼ Y OH and the deviations are scaled with the maximum value of the variable in the D FGM. In these scatter plots, cross sections of 2D manifolds are shown as well. The 2D ILDM shown in these plots is in fact the 2D plane spanned by the right eigenvectors of the Jacobian matrix J ij ¼ o _x i =oy j corresponding to the two slowest chemical processes [5]. At Y ¼ :8 the scatter corresponds well with the 2D ILDM, but at Y ¼ :6 relatively large deviations occur. The 2D FGM created by applying a constant stretch, agrees well with the scatter from the detailed chemistry simulations at both values of Y. The other 2D FGM s are not shown because they almost coincide with the 2D FGM created by stretch and the plots then become cluttered. A quantitative analysis is performed by computing the mean deviation between the scatter and the different manifolds. These results are plotted as function of Y in the bottom row of Fig. 4. The 2D ILDM improves the D manifold results only a little at high values of Y. The 2D FGM s created by applying stretch and curvature are almost the same and result in a significant reduction of the error in the main part of the domain, where chemical source terms are important. At Y ¼ :27 a small peak arises, because Y 2 can not parameterize the 2D manifold there: DY 2 DY i. Small variations in Y 2 then result in large changes in Y i. Another choice of controlling variable(s) might solve this problem. The 2D FGM created by changing the inlet condition, gives comparable results, although somewhat poorer than the other 2D FGM s. Near the unburnt mixture, this method results in a large error for H 2 and CO, again because Y 2 fails to parameterize the 2D manifold there. Note that this 2D FGM has been created by adding H 2 and CO to the inlet composition, while Y OH remained zero. Scatter plots for other species show similar behavior than those shown in Fig. 4. This a priori analysis demonstrates that the accuracy of FGM can be increased significantly by increasing the dimension of the manifold to two. Three or more dimensions are not investigated here because the error for a 2D FGM is already in the same order of numerical errors due to interpolation. In order to investigate whether the use of a 2D manifold actually improves the accuracy of flame simulations, a posteriori analysis is performed. To that end the flamelets introduced before are computed with the 2D FGM generated by including stretch. In this case, transport equations similar to Eq. (4) are solved for both controlling variables. The results are shown in Fig. 5 as a scatter plot of m il computed with the 2D FGM versus m il computed with detailed chemistry. The mean deviation is now %, while it was 5% for a D FGM. Similar results are achieved when using the other 2D FGM s. 7. Conclusions Fig. 5. Scatter plot of the scaled mass burning rate m il =m il at the inner layer computed with 2D FGM (yaxis) and detailed chemistry (x-axis). The results in this paper show that deviations from ideal D flamelet behavior due to stretch and curvature effects in premixed turbulent flames can be significant in the thin reaction zones regime. These deviations are not random, but appear to be embedded in a 2D manifold for the case studied here. This is in favor of using low-dimensional manifold methods like ILDM and FGM. However, the deviations are not imbedded in the 2D ILDM for a large range of Y. On the other hand, all three 2D FGM s are able to capture the composition variations quite accurately. It is interesting to note that the way the 2D FGM is created, has only little influence on the final accuracy. The manifolds created by including stretch or curvature are nearly the same: the difference between these manifolds is smaller than the difference with results from detailed simulations. This indicates that for the flames studied here, an attracting low-dimensional manifold exists, which is not governed by chemical kinetics only. The DNS is carried out with a D FGM. As a result, the stretch and curvature fields obtained from the DNS results are not exactly the same as they would appear in a DNS with detailed chemistry. However, they will also be not much different, because the error introduced by FGM is approximately 5%. Furthermore, the coupling with the flow field is via the density, which is hardly influenced by changes in the species composition and therefore well described by the D FGM. Another consequence of using FGM in

8 384 J.A. van Oijen et al. / Proceedings of the Combustion Institute 3 (27) the DNS is that extinction and re-ignition phenomena are not included in the current analysis. Although they are not likely to occur at the conditions studied in this paper, they will be relevant at higher turbulence levels. To model these phenomena with FGM, a manifold with more than two dimensions might be needed. A DNS using detailed chemistry can clarify these aspects. Acknowledgment The financial support of the Dutch technology foundation STW is gratefully acknowledged. References [] J.B. Bell, M.S. Day, J.F. Grcar, Proc. Combust. Inst. 29 (22) [2] E.R. Hawkes, J.H. Chen, Proc. Combust. Inst. 3 (25) [3] D. Thévenin, Proc. Combust. Inst. 3 (25) [4] J.F. Griffiths, Prog. Energy Combust. Sci. 2 (995) [5] U. Maas, S.B. Pope, Combust. Flame 88 (992) [6] J.A. van Oijen, L.P.H. de Goey, Combust. Sci. Technol. 6 (2) [7] O. Gicquel, N. Darabiha, D. Thévenin, Proc. Combust. Inst. 28 (2) [8] J.A. van Oijen, F.A. Lammers, L.P.H. de Goey, Combust. Flame 27 (2) [9] J.A. van Oijen, L.P.H. de Goey, Combust. Theory Modelling 6 (22) [] J.A. van Oijen, L.P.H. de Goey, Combust. Theory Modelling 8 (24) [] J.A. van Oijen, G.R.A. Groot, R.J.M. Bastiaans, L.P.H. de Goey, Proc. Combust. Inst. 3 (25) [2] N. Peters, Turbulent Combustion, Cambridge University Press, 2. [3] A.N. Lipatnikov, J. Chomiak, Prog. Energy Combust. Sci. 3 (25) 73. [4] J.A.M. de Swart, G.R.A. Groot, J.A. van Oijen, J.H.M. ten Thije Boonkkamp, L.P.H. de Goey, Combust. Flame 45 (26) [5] L.P.H. de Goey, J.H.M. ten Thije Boonkkamp, Combust. Flame 9 (999) [6] M. Matalon, Combust. Sci. Technol. 3 (983) [7] S.B. Pope, U. Maas, Simplifying chemical kinetics: Trajectory-generated low-dimensional manifolds, Tech. Rep. FDA 93-, Cornell University (993). [8] CHEMD, A one-dimensional laminar flame code, Eindhoven University of Technology. URL < [9] G.P. Smith, D.M. Golden, M. Frenklach, N.W. Moriarty, B. Eiteneer, M. Goldenberg, C.T. Bowman, R.K. Hanson, S. Song, W.C. Gardiner Jr., V.V. Lissianski, Z. Qin. URL < [2] R.J.M. Bastiaans, J.A. van Oijen, S.M. Martin, L.P.H. de Goey, H. Pitsch, CTR Annual Research Briefs (24)