A STEADY PSEUDO-COMPRESSIBILITY APPROACH BASED ON UNSTRUCTURED HYBRID FINITE VOLUME TECHNIQUES APPLIED TO TURBULENT PREMIXED FLAME PROPAGATION

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1 A STEADY PSEUDO-COMPRESSIBILITY APPROACH BASED ON UNSTRUCTURED HYBRID FINITE VOLUME TECHNIQUES APPLIED TO TURBULENT PREMIXED FLAME PROPAGATION W. M. C. Dourado, A-P IAE CTA São José dos Campos SP Brazil P. Bruel, CNRS LCD - Site du CEAT 43 Route de l Aérodrome 8636 Poitiers Cedex Frane Pasal.Bruel@ld.ensma.fr J. L. F. Azevedo, ASE-N IAE CTA São José dos Campos SP Brazil Azevedo@iae.ta.br ABSTRACT A pseudo-ompressibility method for zero Mah number turbulent reative flows with heat release is ombined with an unstrutured finite volume hybrid grid sheme. The spatial disretization is based on an overlapped ell vertex approah. An infinite freely planar flame propagating into a turbulent medium of premixed reatants is onsidered as a test ase. The reourse to a flamelet ombustion modeling for whih the reation rate is quenhed in a ontinuous way ensures the uniqueness of the turbulent flame propagation veloity. To integrate the final form of disretized governing equations, a three-stage hybrid time-stepping sheme is used and artifiial dissipation terms are added to stabilize the onvergene path towards the final steady solution. The results obtained with suh a numerial proedure prove to be in good agreement with those reported in the literature on the very same flow geometry. Indeed, the flame struture as well as its propagation veloity are aurately predited thus onfirming the validity of the approah followed and demonstrating that suh a numerial proedure will be a valuable tool to deal with omplex reative flow geometries. INTRODUCTION Steady and unsteady zero Mah number reative flows in the sense of Majda and Sethian (1985), are often enountered in many systems of pratial interest suh as furnaes or jet engines. In suh ases, the pressure an be onsidered as being thermodynamially onstant e.g. the density variations are uniquely related to the temperature hanges due to the heat released by the ombustion proess. When ompressible flow solvers are employed in suh situations, their onvergene rate is beoming so slow that speifi proedures suh as preonditioning tehniques have to be employed to improve them, but in the limit Ma O even the reourse to suh tehniques is not suffiient to ensure a satisfatory onvergene rate (Dourado and Azevedo, 1996 and 1999; Dourado et al., 2). Aordingly, speifi numerial strategies have to be employed suh as the Simple, the PISO or the pseudoompressibility tehnique (Chorin, 1967). The latter approah presents the immense advantage over the two others that its introdution in a ompressible flow solver is easy and leaves the struture of the omputer program largely unmodified while allowing the alulations of both inert and reating flows (Corvelle, 1998; Corvelle et al.; 1999). On the other hand, during the last two deades, the finite volume unstrutured grid tehnique has shown its ability to ope with a wide range of omplex ompressible flow geometries (Jameson et al., 1986, Barth and Jespersen, 1989) inluding the use of mesh refinement tehniques (Batina, 1989, Trépanier et al., 1991, Nomura and Hughes, 1992) that are appealing for properly resolving the high spatial gradients that are potentially present in the flows under investigation. Thus, the temptation is strong to ombine the advantages of both the pseudo-ompressibility method and the finite volume unstrutured grid tehnique to develop a omputer ode able to deal with low Mah number inert or reating flows. Along these lines, the present work is devoted to the development and the test of suh a ombination that results in the development of a 2-D omputer ode that will be tested on the geometry of an 1-D freely propagating turbulent premixed flame whose veloity is known, thanks to the formulation retained for the ombustion modeling. Engenharia Térmia, nº 4, 23 p

2 PHYSICAL MODELING AND NUMERICAL STRATEGY Combustion Model Sine the present study is mainly onerned with the numerial method of solution of the governing equations rather than the modeling of the ombustion proess itself, only a brief desription of the modeling is given here. The formulation retained is similar to that of Corvelle et al. (1999). This hoie is motivated by the fat that suh a modeling yields a turbulent premixed flame that propagates at a veloity that an be diretly derived from the so-alled KPP analysis (Kolmogorov et al., 1937, Corvelle et al., 1999). Consequently, suh a test ase permits an unambiguous evaluation of the apability of the ode to predit the orret turbulent flame speed whih is intrinsi to the set of equations onsidered. So, the reourse to the omparison with experimental data is not required at this stage as long as it is not the physial modeling itself whih is to be tested. A simplified Bray-Moss-Libby like formulation (Bray and Moss, 1977, Bray et al., 1984, Bray et al., 1988), thereafter referred to as BML is used with a quenhed mean reation rate. Aordingly, the isenthalpi ombustion proess takes plae in infinitely thin-laminar like reative interfaes (flamelets) whose thikness is less than the Kolmogorov sale and whose laminar flame veloity is negligible ompared to the veloity flutuations assoiated with turbulene. An observer reording the temperature signal at a given point within the flow would obtain basially two values of the temperature signal e.g. T T b in the fully burnt produts and T Tr in the unburned mixture. The ombustion proess an then be represented through the evolution of a single bi-valued progress variable ( in the unburned mixture and to 1 in the fully burnt produts) whih plays the role of a redued temperature via the following relation: T Tr (1) T Tb The zero Mah number hypothesis leads to the following equation of state: T rtr btb (2) or alternatively : r (3) 1 where the heat release parameter is defined by: Tb Tr (4) Tr With use of the lassial Favre average (Favre, 1965), the governing equation for the mean progress variable ~ an be written as: L( ~ ) w (5) where the operator L regroups the onvetive and diffusive operators. To ensure the existene of a unique propagation veloity for the turbulent flame brush, a ontinuous quenhed form for the mean reation rate w is hosen (Corvelle, 1998; Corvelle et al., 1999), namely: w C w otherwise Pseudo-Compressibility Tehnique The pseudo or artifiial ompressibility tehnique was proposed initially by Chorin (1967) to deal with inert inompressible flows. It has been extended to steady reative flows by Bruel et al. (1996), and to unsteady reative flows by Corvelle et al. (1999). The basi idea behind this tehnique onsists in modifying the ontinuity equation to introdue a finite sound speed during the ourse of the onvergene proess toward the solution. Thus, the ontinuity equation permits de fato the alulation of the stati pressure and reads as: 1 p u i x i where is the pseudo-ompressibility fator expressed in m 2 / s 2 and is the pseudo-time. This equation is physially meaningful when onvergene is reahed in p e.g. when vanishes. The pseudo-sound speed is given by: a u 2 (8) * and the loal orresponding pseudo-mah number Ma * u defined by Ma is thus always less than one as soon as a is larger than zero. For a 1-D turbulent flame and using an impliit strutured finite differenes approah, Bruel et al. (1996) have shown that i) the value of the pseudoompressibility fator influenes strongly the onvergene rate of the alulations and ii) a value 2 2 5u ref to 1u ref where u ref is the largest onvetive veloity expeted in the flow is found to be a good ompromise in terms of onvergene rate. Mean Governing Equations In spite of the fat that the test ase onsidered here is one-dimensional, the governing equations are ast into a two-dimensional form sine an unstrutured approah for dealing with suh equations annot be formulated otherwise. For the simplified ombustion model retained here, the main mean variables that desribe the turbulent reative flow are the density, the Cartesian veloity omponents u ~ i, the stati pressure p and the progress reation variable ~. The steady governing equations that inlude the pseudo-ompressibility approah are given by: q E e i x i E v i x i S (6) (7) (9) 42 Engenharia Térmia, nº 4, 23 p

3 with x1 x and x1 y, the various vetors an be expressed as: k r 3 2 u r 2 r C.75 k r 32 l i (14) q E v x S p u v E e x xx u u xy u v D x u w u u 2 p uv u E v y E e y v v v 2 p v xy u v yy v v D y u (1) where w is the mean reation rate given by Eq. (6) and ij is the laminar stress tensor expressed for a Newtonian fluid by: ij l u i x j u j x i 2 3 u k x k ij (11) l In Eq. (1), D is the laminar diffusion oeffiient, Sl l the kinemati laminar visosity and S l the laminar Shmidt number. The Reynolds stresses, present in Eq. (1) are losed with a Boussinesq-like expression, namely (Frish, 1995): u i u j t u i x j ~ " " j uj x i 2 3 k t u k x k (12) 1 Where u i k u is the turbulene kineti energy. The 2 turbulent visosity oeffiient an be expressed as a funtion of k ~ and of its dissipation rate ~ by (Launder and Spalding, 1974): t C k 2 (13) with C being equal to.9. In the present work, the turbulene field is supposed to be frozen at the level that prevails in the fresh reatants so that k ~ ~ k and ~ r ~ with: r ' '2 where ur u r and l i, to be presribed, stand for the rms veloity flutuations and the turbulene integral length that prevail in the fresh reatants, respetively. The turbulent mass flux present in Eq. (1) is also expressed through a gradient approximation, namely (Bray et al., 1984): u i D t x i (15) t where Dt is the turbulent diffusion oeffiient, St t t the turbulent kinemati visosity and the turbulent Shmidt number. The losed form of Eqs. (9) is now written in an integral form using the Green's theorem: V V qdv S E e nds S E v nds V SdV (16) where represents the outgoing normal unitary vetor to surfae S of volume V and and $ are the onvetive and diffusion flux vetors whose omponents are given in Eq. (1). In two dimensional Cartesian notation, the produt is equal to y x where and are the basis vetors. Spatial Disretization As the ontrol volume an be onstruted a priori with different type of elements (triangles and/or quadrangles) on hybrid grid topologies, the elementary ell of the grid onsidered here is simply alled an element. Thus, the ontrol volume V is defined by the external boundary surfae of all elements sharing a ommon node, as skethed in Fig. (1). The spatial disretization adopted here is based on a ell enter finite volume method adapted for ell vertex hybrid elements (Jameson et al., 1986). This sheme proved to be effiient for dealing with steady and unsteady inert flows as reported by Mavriplis et al. (1989), Mavriplis and Venkatakrishnan (1995), Dourado and Azevedo (1996), Mavriplis (1998), Dourado and Azevedo (1999) and Dourado et al. (2). The disretized form of Eq. (16) is written as: V q i nf f1 where q 1 V V qdv e x fy e y fx nf f1 is the average main variables vetor, and v x fy v y fx VS (17) (18) Engenharia Térmia, nº 4, 23 p

4 S i=1 i=1 2 i=1 1 θ k Control Volume for i=1 i=1 5 f = 4 4 f = 5 f = 3 6 Control Volume for 3 Control Volume for i=2 i=2 f = 6 7 f = 7 1 f = 1 f = 2 2 S i=2 Figure 1 - Domain of influene of nodes 1, 2 and 3 and its loal strething for hybrid unstrutured grid. S 1 V V SdV (19) alulations presented in this work and the oeffiient A is adjusted aording to the grid strething by: is the average soure term vetor. E is the average flux vetor at fae f alulated as the arithmeti average of the two fluxes loated on nodes whih delimit this fae. Further details on triangular and hybrid elements in unstrutured grids are presented in the studies that have been referred to above. Artifiial Dissipation For high Reynolds number flows, it is frequently required to stabilize the alulations by adding expliitly some artifiial dissipation to damp out numerial osillations. These additional dissipative terms must be kept at a level that does not endanger the auray of the numerial solution. Aordingly, and as suggested by Dourado et al. (2), the dissipation terms introdued in the systems of equations are written as: D a q i nn k1 k i where A k A i 2 (2) q k 4 2 q (21) and nn is the number of nodes loated on the boundary of the ontrol volume i, as it is skethed in Fig. (1). (4) The onstant k whih defines the level of the fourth order dissipation term is taken equal to 1 / 256 for all the A k 1k os 2 k 2k sin 2 k (22) Subsripts i and k refer to the variable evaluation at nodes i and k, respetively. At eah mesh point, the oeffiients 1 and 2 of Eq. (22) are alulated with the maximum eigenvalue assoiated with the onvetive terms by: 1 s 1 s1 2 s 1 s s1 (23) where s represents the magnitude of the strething vetor S 2 / 3 (see Fig. (1)) and ( r) 1 r For an unstrutured mesh, the maximum eigenvalue at eah mesh point is evaluated by the following disrete approximation: nf f1 f y f v f x f f x 2 y 2 (24) where u, v and a are the average values of the veloity omponents and of the sound speed alulated from the two nodes whih delimit the fae f and umulated over all the boundary faes of the ontrol volume onsidered. Finally, sine the ode is edge based database organized, the artifiial dissipation term formulation was enhaned in order to be usable in the ontext of hybrid grids where triangular and quadrilateral elements oexist. One should observe that, although quadrilateral meshes were used in the present simulations, the ode treats all grids in a fully 44 Engenharia Térmia, nº 4, 23 p

5 unstrutured way. The reader should be aware that the definition of an unstrutured grid is not neessary related to the shape of the ontrol volumes but to the form in whih these ontrol volumes are addressed in the ode. Pseudo-time marhing As exposed before, when one has reourse to the pseudo-ompressibility method to solve the flow governing equations, the solution is physially meaningful only when a steady state in pseudo-time is reahed, so, even to seek steady state solutions, a time-stepping proedure has to be used. Thus, the disretized equations system (17) is first written as: V i dq i d Cq i Dvq i D a q i Sq i i 1n (25) where the residuals of the onvetive, diffusive and stabilizing terms are denoted by C (q) D v (q) and D a, respetively, and the soure term is represented by S (q). Then, a Runge-Kutta expliit three-stage hybrid timestepping sheme is adopted due to its satisfatory performane, simpliity and low omputational ost. The sheme proposed by Manzari et al. (1998), to advane the solution in pseudo-time is implemented here under the following form: n q i q i q i V i 1 C q i D v 1 q i 1 S q i D a 1 q i q n1 3 i q i, 1, 2,3 (26) The operators C, D and S are expressed by using the q i v ( 1) vetor properties q i from stage 1, while the added artifiial dissipation term is alulated only in the first stage and held onstant throughout the next stages. The oeffiients appearing in Eq. (26) are given the values 1.6, 2. 6 and X Figure 2 - Computational grid mesh 1 with 355 nodes and 354 quadrilateral elements. RESULTS The 1-D turbulent premixed flame referened by Corvelle (1998) as Case I is onsidered here to test the present numerial approah. The parameters defining suh a slow propagating flame are given in Table (1). * St u r u ' r l i St Cw r (m/s) (m/s) (m/s) (m) m 3 /s kg/ s kg/m 3 / Table 1 Parameters defining the 1-D turbulent premixed flame onsidered as test ase. The boundary ondition imposed at the reatants side e.g. ur St and ~ has been hosen so that the turbulent flame brush will be steady in the omputational domain whih onsists of a strip of.5 m long and.1m in height with only one division in the latter diretion. An example of grid, alled mesh 1 with 355 nodes in the longitudinal diretion and 354 quadrilateral elements is presented in Fig. (2). In order to properly resolve the strong gradients that are to be expeted in the turbulent brush near the quenhing point, the mesh is highly refined in the region loated between.17m x. 33m with a minimum mesh spaing equal to m. xmin Due to the high Reynolds number flow hypothesis, the laminar diffusive transports will be negleted in front of their turbulent ounterparts. To ensure 1-D like onditions on suh a 2-D mesh, a symmetry ondition is imposed at the lower and upper boundary lines parallel to the flow and are applied to the orresponding nodes and faes. Suh a ondition orresponds to the imposition of zero transverse veloity and gradients. At the burned produts side, the pressure is imposed and all other variables are extrapolated. The initial ondition for the progress variable ~ is given by a ramp form funtion that goes from ~ at x. 27m to ~ 1 at x. 32m. The Courant-Frederih-Lewis (CFL) number used in these alulations is equal to.94 and is kept onstant throughout the omputational domain. Aordingly, the pseudo time step is variable in the omputational domain with a minimum value used in the region of greatest refinement. Suh a CFL number is given by: CFL b 4 x min (27) Engenharia Térmia, nº 4, 23 p

6 LRHS, LRHS p, L, LRHS ρ LRHS LRHS p L LRHS ρ Iter Figure 3 - Convergene history of residue using mesh with 75m / s. where b ub ab is the largest onvetive eigenvaluebased on the pseudo-sound speed given by Eq. (8). As a onsequene, the minimum pseudo-time step for 5 mesh 1 is equal to 2. 1 s. The onvergene riterion set 4 to 1 was met after 275 iterations through a onvergene history presented in Fig. (3). The orresponding profiles of pressure gradient, mean progress variable and mass flux obtained on mesh 1 are presented in Fig. (4). The urvature at the foot of the flame brush is extremely strong and this illustrates why suh a highly strethed mesh has to be used to apture properly the progress variable and pressure gradients in that zone. The profile of pressure gradient plotted against the progress reation variable ~ is displayed in Fig. (5). ρu, ρu X Figure 5 - Profiles of mean pressure gradient versus the 2 mean reation progress variable (mesh 1, 75m 2 / s ). Consequently, the riterion onvergene hosen here appears to be orretly adapted to get aurate onverged results. To investigate the sensitivity of our results to the mesh refinement, three other meshes have been onsidered that differ from mesh 1 by their refinement: i) mesh 2 possesses 5 % more divisions in the x -diretion in the 4 interval.17 m x. 33 m with xmin m ; ii) mesh 3 has 1 % more divisions in the x -diretion on the 4 same interval, whih gives xmin m and iii) mesh 4 has the same number of nodes as mesh 3 but the interval of refinement is redued to.22 m x. 3 m, whih gives a minimum mesh disretization of 4 x 1.33 m. min Mesh-1 Corvelle 25 Mesh 1 Mesh 2 Mesh 3 Mesh 4 Corvelle Figure 4 - Profiles of mean reation progress variable, mean mass flux and pressure gradient versus x o-ordinate. A good agreement is obtained between the present results and those of Corvelle (1998) with a differene in the maximum value of the pressure gradient of less than 6 % Figure 6 - Profiles of mean pressure gradient versus the mean reation progress variable for meshes 1, 2, 3 and 4. It should be remembered than in her study, Corvelle (1998) was using an auto-adaptive mesh refinement with a 46 Engenharia Térmia, nº 4, 23 p

7 minimum spaing equal to m. The results xmin obtained with the different grids are presented in Fig. (6) where the profiles of the pressure gradient are plotted versus the mean reation progress variable β-hat = 6 - β-hat = 7 - β-hat = 8 - β-hat = 9 - β-hat = 1 - β-hat = 12 - β-hat = 16 - β-hat = 32 - β-hat = Iter Figure 7 Residue history for the x-diretion momentum equation for several values of ˆ. It an be learly seen that as soon as the mesh is suffiiently refined, no substantial differene an be observed with the results by Corvelle (1998), obtained, it should be stressed with a ompletely different numerial formulation. A short analysis of the influene of the hoie of the pseudoompressibility fator ˆ on the onvergene rate was 2 u b arried out here and Fig. (7) presents the onvergene history of the residue of the momentum equation in the x - diretion obtained for different values of ˆ. 1 The dependene on the value of ˆ of the number of iterations required to meet the onvergene riterion LRHS 1 4 is presented in Fig. (8). The optimum u value of ˆ, around 1, obtained here onfirmed the results reported by Bruel et al. (1996), and shows that suh an optimum is poorly dependent on the numerial approah retained and thus an be onsidered as speifi of the pseudo-ompressibility approah itself. CONCLUDING REMARKS The proposed ombination of the pseudoompressibility method with an unstrutured formulation to solve the governing equations of a turbulent premixed flame proves to work properly and to produe results that ompare well with those obtained in the literature on the very same flow geometry. Due to the entered type of spatial disretization used here, a fourth-order artifiial numerial dissipation has been added to stabilize the alulations but the results show that the auray of the final solution has been preserved. The use of an expliit three-stage hybrid time stepping sheme shows that it is powerful enough to alulate the steady solution in the pseudo-time of the system of disretized governing equations. Nevertheless, the use of suh an expliit time stepping sheme has the drawbak of limiting the maximum possible CFL-number value resulting in an inreased total number of pseudo-time steps required to ahieve onvergene. If needed, an impliit formulation ould be onsidered to speed-up the onvergene proess. Finally, an auto-adaptive Lagrangian-Eulerian approah on unstrutured grid (Trépanier et al., 1991; Nomura and Hughes, 1992) ould greatly help in reduing the total number of grid elements required to resolve properly the high gradients present in the flow while preserving the overall auray of the alulations. Future work will fous on these diretions. Aknowledgments Number of time step (X 1) The authors would like to aknowledge the finanial support from CNPq, and CAPES. A large part of this work has been arried out during the stay of the first author at the Laboratoire de Combustion et de Détonique (Poitiers, Frane) that belongs to the Centre National de la Reherhe Sientifique (CNRS) and in the framework of a ''o-tutelle'' thesis prepared under the o-supervision of the University of Poitiers and ITA. The authors further aknowledge the partial support of CNPq under the Integrated Projet Researh Grant N /96-. REFERENCES β-hat Figure 8 - Dependene on ˆ of the number of time steps required to ahieve onvergene. Barth, T. J., and Jespersen, D. C, 1989, "The Design an Appliation of Upwind Shemes on Unstrutured Meshes", 27 th AIAA Aerospae Sienes Meeting, AIAA Paper Batina, J. T., 1989, "Vortex-Dominated Conial- Flow Computations using Unstrutured Adaptively Refined Engenharia Térmia, nº 4, 23 p

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