CAN A ONE-DIMENSIONAL STRAINED FLAME MODEL COMBUSTION IN FLAME-VORTEX INTERACTIONS?

Transcription

1 CAN A ONE-DIMENSIONAL STRAINED FLAME MODEL COMBUSTION IN FLAME-VORTEX INTERACTIONS? Youssef Marzouk *, Habib Nam, and Ahmed Ghoniem * ymarz@mit.edu, hnnam@sandia.gov, ghoniem@mit.edu * Massachusetts Institute of Technology, Cambridge, MA 02139, USA Sandia National Laboratories, Livermore, CA 94550, USA Corresponding author: Ahmed Ghoniem Massachusetts Institute of Technology Room Massachusetts Avenue Cambridge, MA USA ghoniem@mit.edu phone: fax: Desired colloquium: Flame Structure and Dynamics, or Turbulent Combustion and Modeling.

2 CAN A ONE-DIMENSIONAL STRAINED FLAME MODEL COMBUSTION IN FLAME-VORTEX INTERACTIONS? Youssef Marzouk *, Habib Nam, and Ahmed Ghoniem * * Massachusetts Institute of Technology, Cambridge, MA 02139, USA Sandia National Laboratories, Livermore, CA 94550, USA Corresponding author, ghoniem@mit.edu ABSTRACT We present a study of the flame-embedding concept, introduced in [1] as an efficient approach for large-eddy simulation of turbulent combustion at high Reynolds and Damköhler numbers. In flame embedding, the combustion zone is modeled as an unsteady flame modified by the local tangential strain rate, with the latter extracted every time step from the flow field simulations. The flame structure is approximated locally as being one-dimensional, evolving in the flow field generated by a stagnation-point flow whose characteristic parameter is the applied flow strain. Two-dimensional flame-vortex interaction simulations are used as the benchmark for this study. The time-dependent strain rate at the flame front is extracted from the two-dimensional simulations and used in the one-dimensional, unsteady strained flame structure calculations. We present results for a matrix of parameters that define the impact of the flow on the flame in a flame-vortex interaction the vortex pair size, separation and circulation and use the instantaneous burning rate to determine the accuracy of the model. We show that the onedimensional unsteady flame calculation can capture the impact of the applied flow strain on the burning rate if the former is taken as the corresponding non-reacting flow strain rate at the flame reaction zone, and not the applied flow strain at the flame leading edge. The distinction is important in cases where the flame front thickness approaches that of the flow field, a situation that is likely to be encountered when small vortices interact with lean flames. Introduction Direct numerical simulation of turbulence has repeatedly shown that a typical turbulent field is composed of a number of vortical structures which take the form of sheets and worms, with length scales and circulations that depend on the flow Reynolds number. Since it is not yet feasible to perform direct numerical simulations of turbulent combustion in three dimensions while incorporating detailed chemistry and multi-species transport, we will limit our concerns here to two-dimensional simulations. In the latter, the vortical structures are essentially cross sections of rectilinear vortices that can be characterized by their size, circulations and separation distance. Two-dimensional simulations of turbulent premixed combustion, performed at high Damköhler numbers, i.e., with relatively weak turbulence whose length scales are larger that characteristic combustion scales such as a typical flame thickness, show that the local combustion zone consists essentially of single or multiple flame fronts convoluted around the existing vortices, with the flame structure determined by several mechanisms including the

3 curvature of the front, the local strain rate, its history, its proximity to other flames or to boundaries, etc. Several high-resolution experimental visualization studies conducted under similar conditions confirm these observations qualitatively. Such simulations are extremely expensive as they pose the challenge of integrating the unsteady reacting Navier-Stokes equations, coupled with multi-species transport, where a large spectrum of time and length scales are encountered, leading to numerical stiffness and the need for complex numerical treatment. Moreover, while it is possible to perform direct simulation for a number of generic, idealized problems within very small domains, it is unlikely that applying the same strategy to more practical engineering flows will become feasible soon, irrespective of how powerful computational hardware becomes. The challenge therefore is to look for approaches that retain, as much as possible, the underlying physics, while taking advantage of the insight gained from experimental study and numerical simulations to simplify the computations. Many physical mechanisms that have been recognized as contributing substantially to the evolution of turbulent combustion flows are inherently unsteady, e.g., ignition and extinction, interaction with vortical structures, impact of strong strains, et cetera. It is therefore necessary that one form of so-called large-eddy simulation be adopted in turbulent combustion calculations. Approaches for incorporating combustion into these simulations are under development. One such approach is what we call flame embedding. In this approach, the combustion zone structure is assumed to be that of a flame subected to a time varying strain, and possibly curvature. The flame internal profiles are assumed to be locally one-dimensional in the direction normal to the flame surface. The flame itself is assumed to be immersed in a stagnation-point flow whose characteristic strain is the applied flow strain. In order to apply such an approach in a large eddy simulation, it is necessary to track a flame surrogate, that is, a surface representing the interface between reactants and products. This surface is then divided into segments, and each segment is considered as an elemental flame whose structure is modeled using the stagnation point flow flame calculations. The similarity between this approach and flamelet approaches lies in the fact that both treat the combustion zone as being a one-dimensional flame. One difference is that in the flame embedding approach, we retain the unsteadiness of the interactions, and hence preserve the impact of the Lagrangian flow history on combustion. Each elemental flame structure calculation is performed separately using the strain rate computed along its traectory. One step towards formulating this model is to assume that the local flame element is essentially flat and strained, and to use the Lagrangian strain history, which we call the applied flow strain, in the multi-dimensional simulations to determine the impact of the flow on the flame. In order to determine whether the local flame structure can indeed to be computed using this elemental flame approximation, it is important to compare results of a multi-dimensional simulation of turbulent combustion to those obtained using a flame strained at the equivalent strain rate, and determine what that equivalent strain rate is. In this paper, the former is locally approximated using flame-vortex interactions. The latter is computed using the approaches described in [2, 3]. In the remaining sections of the paper, the formulation of the elemental flame model, and the numerical treatment of the models are briefly described. Results of the two dimensional simulations are then presented, and results of the elemental flame model are compared with these simulations in terms of the instantaneous burning rate.

4 Model Formulation The stagnation-point flow strained flame model [3-8] is used, while allowing the strain rate parameter a(t) to vary as a function of time. The one-dimensional governing equations are obtained by applying the boundary-layer approximation across the flame, and a solution is considered along the stagnation streamline x = 0. (Note that the flame is parallel to the x-axis). The stagnation-point flow velocity profile u =a(t)x, v =-a(t)y yields the pressure gradient as a function of the imposed strain: p u a = ρuua ρu (1) x a Introducing the notation U u/ u, V ρ v and substituting the pressure gradient expression into the momentum equation in the boundary layer, we obtain the following equations for species, energy, momentum, and mass conservation, respectively: ρ Y V Y k k + + ( y y ρ YkVk) w kwk = 0 (2) ρ T λ V T K K 1 T 1 T ρcpk, YkV k + wh k k = 0 (3) y cp y y cp k y cp k ρ U 1 a ρ ρ µ U Ua V U U a ρu a a y y y a = 0 (4) ρ V + + ( + 1) ρua= 0 (5) y The diffusion velocity is obtained as V = 1 k k km X D X, where the mixture-averaged diffusion k y 1 Yk coefficient is Dkm =. In the continuity equation (5), =1 for the axisymmetric case, K X D k k and =0 for the planar case. Y k is the mass fraction of species k, while W k and ẇ k are the molar weight and molar production rate, respectively. In the remaining equations, c p is the specific heat of the mixture, λ is the thermal conductivity, h k is the specific enthalpy of the k-th species, H k is the corresponding molar enthalpy, ρ u is the density of the reactants mixture, and µ is the dynamic viscosity of the mixture. Note that thermal diffusion velocities are neglected. The low Mach number assumption has been employed, and hence the density is calculated as a function of the temperature, species mass fractions, and thermodynamic pressure via the ideal gas equation of state. Boundary conditions for the species and energy equations are: y = : Yk = Yk,, T = T y =+ : Yk = Yk, +, T = T + (6) The continuity equation requires only one boundary condition; typically, this boundary condition would specify zero velocity at the stagnation point, V(y=0) = 0. Numerical considerations

5 discussed in the next section, however, suggest that we impose a boundary condition at y =, and let the stagnation point definition fix the origin of the y-axis. The momentum equation requires two boundary conditions. At an unburned stream, u=u, or U=1. Setting the spatial gradients in (4) to zero yields an ODE for the boundary condition on the burned stream. This far-field boundary condition places an important requirement on the size of the computational domain; the flame must be far enough from the + and boundaries for spatial gradients in U to vanish. For unsteady strain, we integrate the following ODE for U U( y =+ )at the burned-stream boundary: b U b a ρu a = Ua b Ub a a + ρ a b (7) where ρ b is the density of the burned mixture. To accurately characterize the flame structure and response, detailed transport and chemistry are used to evaluate transport coefficients and reaction terms in the model. Transport properties (µ, λ, D km ) are evaluated using Sandia s TRANSPORT libraries [9]. Chemical source terms and mixture properties are evaluated using CHEMKIN [10]. We use a C 1 kinetic model for methane-air combustion, consisting of 46 reactions among 16 species [11]. Numerical Solution Numerical solution of the governing equations is obtained via a fully-implicit finite difference method. A first-order backward Euler formulation is used, and the full set of governing equations, including the continuity constraint, is solved simultaneously. A first-order upwind discretization is applied to all convective terms, while diffusion terms are approximated using a central, second-order accuracy scheme. An upwind scheme is also used in the continuity equation, taking the positive sign outside V y to suggest a positive upwind velocity so as to add dissipation of the appropriate sign: ρ V + + ρua =0 ρ n+ 1 ρ n n+ 1 n+ 1 V V 1 n+ 1 n+ 1 n ρ U a = 0 y y t y y 1 (8) This representation of V y prevents direct implementation of a stagnation-point boundary condition. Instead, a boundary value on the mass flux V is chosen at y =. The boundary value is arbitrary provided that it is large enough for the flame to stabilize at a lower mass flux, since V decreases in the direction of the stagnation point. If the boundary value is too large, on the other hand, the flame may not fall within the computational domain. The boundary value can thus be set to any reasonable number based on the size of the computational domain and the strain rate. The solution to the problem matches the mass flux profile to the flame location, as reflected in the profiles of T, Y k, ρ and U. In computations with unsteady strain, this boundary condition on V must be updated regularly. The strain rate parameter a can easily vary one or two orders of magnitude in a given

6 computation; such a change in strain, with a fixed boundary value on the mass flux through the flame, causes the flame to migrate rapidly with respect to the grid. As the flame approaches the boundary of the computational domain, successive regridding become necessary. This scenario is cumbersome and computationally taxing. To correct for this, while retaining the boundary condition V( y = ), we implement a proection method to update the mass flux profile in the case of unsteady strain. At the start of time step n+1, an initial guess for V n+1 is obtained by integrating the continuity equation with a n+1, U n, ρ n : n n ρ ρ t V + 1 n + 1, guess n + 1, guess 1 y V y 1 n n n+ 1 + ρ U a = 0 (9) In a single step, this proection updates the boundary value on V at y = and generates a new guess for V n+1. Updating the boundary value on V thus minimizes flame translation for fast convergence. Spatial discretization is performed on a non-uniform adaptive grid, permitting a dynamic clustering of grid points in regions where spatial gradients are high, and hence ensuring adequate resolution through the reaction-diffusion zone over all the integration time. Grid refinements were enforced at each time step to maintain the difference between neighboring values of any scalar and its gradients below a threshold, while enforcing certain uniformity requirements on the grid [2, 3]. For the chemical kinetic model used in these calculations, the minimum grid spacing was approximately 10 µm. The time step for integration is constant, typically chosen in the range of 1 µs. At each time step, discretization reduces the governing PDEs to a set of nonlinear algebraic equations. The nonlinear system is solved using an inexact Newton iteration [12] which converges to the solution of the nonlinear equations but avoids solving the Newton condition accurately far away from a solution, when the linear model of Newton's method may be poor. In our implementation, the inexact Newton method is coupled with a safeguarded backtracking globalization to improve its domain of convergence [13]. If the step s i of the inexact Newton condition does not sufficiently reduce the residual norm, F, the step is reduced by a scalar factor θ, essentially backtracking along the search direction. Backtracking continues until the condition on F is met, for in a sufficiently small neighborhood of the trial solution x i, the linear model must indicate the correct downward path; the Newton equation is consistent. Solution of the linear system at each Newton iteration proceeds via a Krylov subspace method, BiCGSTAB [14]. Because the stiffness of detailed chemistry renders our linear system ill-conditioned, BiCGSTAB must be accelerated with an incomplete LU factorization preconditioner; here, we implement ILUTP, a refinement of ILU preconditioning developed by Saad [15]. ILUTP provides an LU-factorization of the Jacobian matrix F but regulates fill-in of L and U based on a threshold parameter; it also provides for pivoting, in which F is permuted to insure diagonal dominance. The result is a preconditioned system with a smaller and more uniformly distributed eigenvalue spectrum.

7 Results To capture some of the mechanisms by which turbulence impacts combustion, nominally at the limit of high Damköhler number or in the thin flame regime, we used results of flame-vortex interaction simulations. In these simulations a pair of counter-rotating vortices are allowed to move towards the flame surface at their mutually induced velocity. Turbulence-combustion interactions are manifested in the form of curvature and strong strain rates induced by the vortex pair onto the flame. Flame-vortex interactions have been computed for 1the matrix of parameters given in Table 1, using the circulation Γ, the vortex characteristic dimension (here the Gaussian vortex width δ), and the separation between the two vortices CL as parameters that can be varied to control the induced flow field. case Γ [cm 2 /s] δ [cm] CL [cm] I II III IV Table 1: Flow parameters used in four cases of flame-vortex interaction: vortex circulation Γ, Gaussian vortex width δ, vortex-pair separation CL. A detailed description of the formulation and numerical scheme used in the flame-vortex simulations may be found in [16 18]. The reactants mixture is stoichiometric methane-air at 298 K, diluted with 20% by volume N 2, thus by lowering the unstrained flame temperature to 1965 K; the thermodynamic pressure is 1 atm. The same 46-reaction mechanism used in [16] is incorporated here, with identical transport properties. The initial state corresponds to an unstrained flame introduced at a distance where the impact of the vortex pair is negligibly small. For the purpose of the current study, we only show two cases that are representative of the two extremes for the values of the vortex-induced strains. These are cases I and II; Figures 1 and 2 depict these cases in terms of vorticity contours and flame temperature contours. In the first case, the strain rates at the intersection of the flame surface and the centerline are relatively low, O(300 1/s), rising slowly due to the weak distribution of vorticity within each vortex and the relatively large distance between each vortex and the centerline; in the other case, the corresponding strain rates rise more quickly, finishing an order of magnitude higher. For both cases, successive snapshots in Figures 1 and 2 show the early as well as the later stages of the transient flame response to the advancing vortices. The early stage, before the vortex penetrates the plane of the original flat flame, is characterized by relatively low and slowly rising strain rates, while the later stages, following the intrusion of the vortex into the curving flame mushroom, show higher and faster rise in the strain. In these later stages, the flame surface curves around the outer perimeter of the vortices, as expected. Note that due to the symmetry of the problem, we only show one half of the domain. Since some sections of the flame surface experience substantial curvature, which has not yet been included in our elemental flame model, we focus on the relatively flat flame element moving along the symmetry line and evaluate the applied flow strain rate history there. This is

8 the input to the unsteady strained flame model described earlier i.e., we use the applied flow strain as the parameter a(t) used to characterize the stagnation point flow. Selecting where to evaluate the applied flow strain with respect to the flame surface turns out to be a non-trivial task; as suggested by Figures 1 and 2, due to the length scale of these vortices, the strain rate may vary within the flame front, making it difficult to find a representative, single value for the applied strain rate. Moreover, given that the length scales of the vortex, and hence the strain, may be of the same order of magnitude as that of the flame, these cases may not fit into the simple definition of a thin flame regime. The question remains, though, of whether it is still possible to apply the same thin-flame concept but with a different definition for the applied flow strain that reflects the impact of the flow on the combustion zone. As discussed below, our calculations show that this is indeed possible. Since our obective is to compare the results of two-dimensional simulations with those of the one-dimensional model, we must carefully choose where to evaluate the applied flow strain which characterizes the stagnation point flow field in the elemental flame model. If we choose to use the strain rate at the leading edge of the flame in the two-dimensional simulations, e.g., on the reactants side along the contour where methane is at 99% of its free stream concentration, we may be overestimating the strain rate. This is because the vortex-induced non-reacting strain rate falls monotonically as one traverses the flame away from the vortices. (By non-reacting strain rate we mean the value of the strain rate which would have been measured without the presence of the flame, which induces an extra flow field due to volumetric expansion.). Therefore, a scheme must be devised to adust the value of the flow strain rate downward from its value at the leading edge before using it to compute the elemental flame structure. The following scheme was devised for that purpose. One plausible obective is to estimate what the two-dimensional simulation s non-reacting strain rate would have been at the reaction zone (i.e., in the absence of a flame). This is motivated by the fact that, in the 1-D flame, the strain rate parameter a(t) is equal to the strain rate u r r= 0 at y =, i.e., at the cold side of the flame. If we can construct a functional representation for the strain rate variation ahead of the flame leading edge in the two-dimensional simulations, we can extrapolate that non-reacting strain rate into the flame structure to evaluate its value at the reaction zone. One may use a simple polynomial to extrapolate the strain ahead of the flame leading edge in the 2-D simulation, or use the fact that the non-reacting flow field is induced by two Gaussian vortices to construct a functional representation for the dependence of the strain on the location with respect to the distance from the flame. In either approximation, the applied flow strain, used to parameterize the strain rate a(t) in the stagnation point flow, is selected to be the non-reacting strain which would have been induced at the 2-D flame reaction zone. Another approach for evaluating the applied flow strain at the reaction zone is based on extracting the actual strain rate from the two-dimensional simulations at the two-dimensional flame s reaction zone. (By actual strain rate, we simply mean the measured strain rate in the 2- D flow, containing contributions from the both the vortex-induced velocity field and the flameinduced velocity field.) In practice, we compute this strain rate by averaging the actual strain rates at leading and trailing contours of the flame surface: the 99% contour of CH 4 and the 90% contour of CO 2 ; denote the resulting value by ε. This strain rate ε () t is used to back out an applied flow strain a(t) as follows: The strain rate in the one-dimensional elemental flame model

9 is, by definition, u r = a U. This value rises from a(t) on the cold side of the flame to r= 0 a(t)u b (t) at the products side. We can thus approximate the strain rate in the reaction zone of the one-dimensional flame with an analogous arithmetic mean, at () ( 1+ Ub()/ t) 2. Putting the latter value equal to the average strain rate in the two-dimensional flame, ε () t, we can compute a(t) at any timestep. The resulting expression and ODE, derived from equation (7), are as follows: 2ε at ()= (10) 1 + U b where U ρ b b 2 1 ε ρ u 1 ε = 2Ub ε Ub( 1+ Ub) + 2ε + ( 1+ Ub) (11) ρ + ρ ε ρ + ρ ε u b In effect, this scheme matches the strain rate in the reaction zone of the one-dimensional flame with the strain rate in the reaction zone of the two-dimensional flame, integrating the momentum equation across the 1-D flame and computing the correct applied flow strain a(t) to achieve this matching. Results using the three approximations are now presented. Figures 3 and 4 show the integrated heat release rate along the symmetry line for the cases shown in Figures 1 and 2, respectively first, evaluated from the 2-D simulations, and next, using the elemental flame model while the applied flow strain a(t) is taken as (i) the strain rate at the flame leading edge, (ii) the extrapolated non-reacting strain at the 2-D flame reaction zone, or (iii) the strain evaluated from the actual strain rate at the 2-D flame reaction zone. As expected, using the 2-D leading edge strain to parameterize the stagnation point flow leads to the least accurate results, since the variation of the strain rate within the flame structure is substantial and the value at the leading edge is much higher that the value at the reaction zone. The most accurate result turns out to be that derived from the reacting flow strain at the reaction zone, choice (iii). This is also expected since the role of strain is to change the convection-diffusion balance at the reaction zone, i.e. to change the flux at the reaction zone. On the other hand, in a flame embedding calculation, this value is not available a priori and this choice cannot be implemented. The second option, in which an approximation for the strain rate ahead of the flame is constructed and used to obtain the non-reacting value at the reaction zone, shows reasonably good agreement with the 2-D simulation results and demonstrates that one can indeed model this 2-D flame, which does not conform neatly to the thin-flame assumption, using a stagnation point flame. It is worthwhile mentioning here that the predictions using the model are more accurate in the case of a weak, slowly varying strain than in the other case. This is consistent with the weak dependence of the burn rate on the strain at low values of the latter. In this case, the vortex size is also somewhat larger than the flame front thickness, i.e., the thin flame approximation is more applicable, and the vorticity remains relatively further away from the flame front. Meanwhile, in case II, it is observed that by the time vorticity penetrates into the flame structure, a sharp drop in the heat release rate should be expected, heralding the onset of extinction. u b

10 0.0 ms 2.0 ms 4.0 ms 6.0 ms 8.0 ms Figure 1: Premixed flame interaction with a counter-rotating vortex pair, case I. Colored contours indicate temperature, with darker shading corresponding to burned combustion products. Solid/dashed contours delineate levels of positive/negative vorticity. 0.0 ms 1.0 ms 2.0 ms 3.0 ms 4.0 ms Figure 2: Premixed flame interaction with a counter-rotating vortex pair, case II. Contour definitions are analogous to Figure 1.

11 4E E E E Q[erg/cm 2 s] 2E strain rate [1/s] 1.5E E+08 Q, 2-D Q, match mean ε r Q, match mean ε nr 5E+07 Q, match leading edge ε a, match mean ε r a, match mean ε nr a, match leading edge ε time [ms] Figure 3: Integrated heat release rate histories for various definitions of the applied flow strain, case I. Circles denote instantaneous Q for the two-dimensional flame element. Lines correspond to the one-dimensional computation. Left ordinate corresponds to heat release rate; right ordinate shows applied flow strain rate, a(t).

12 3.5E E E Q[erg/cm 2 s] 2E E+08 1E+08 Q, 2-D Q, match mean ε r Q, match mean ε nr Q, match leading edge ε a, match mean ε r a, match mean ε nr a, match leading edge ε strain rate [1/s] 5E time [ms] 0 Figure 4: Integrated heat release rate histories for various definitions of the applied flow strain, case II. Plot definitions are analogous to Figure 3. Conclusions A stagnation point flame was used to model the evolution of premixed combustion in a flame subected to the impact of a pair of advancing vortices. The applied flow strain, which characterizes the stagnation point flow, was extracted from two-dimensional simulations of the flame-vortex interaction using several different criteria. It was shown that reasonable agreement with the heat release rate extracted from two-dimensional simulations could be achieved if the applied flow strain is defined as the non-reacting flow strain at the reaction zone. The latter was computed by extrapolating the strain rate profile upstream of the flame leading edge into the reaction zone. It was also shown that the predictions are more accurate when conditions are such that the flame is thinner with respect to the vortex size and vorticity does not penetrate into the flame structure. One important conclusion for the flame embedding approach is that the flame surrogate should be the flame reaction zone surface, not the flame leading edge surface.

13 References [1] Petrov, C., Numerical Simulation of Reacting Flows with Complex Chemistry Using Flame Embedding, Ph.D. thesis, MIT Dept of Mechanical Engineering, Feb [2] Marzouk, Y.M., Ghoniem, A.F., and Nam, H.N., A Flame Embedding Model for Turbulent Combustion Simulation, AIAA paper , AIAA, Jan AIAA J., sub udice. [3] Marzouk, Y.M., The Effect of Flow and Mixture Inhomogeneity on the Dynamics of Strained Flames, S.M. thesis, MIT Dept of Mechanical Engineering, Aug [4] Marzouk, Y.M., Ghoniem, A.F., and Nam. H.N., Proc. Combust. Inst 28: (2000). [5] Stahl, G. and Warnatz, J., Combust. Flame 85: (1991). [6] Darabiha, N., Combust. Sci. Tech. 86: (1992). [7] Egolfopoulos, F.N., Proc. Combust. Inst. 25: (1994). [8] Petrov, C. and Ghoniem, A.F., Combust. Flame 102: (1995). [9] Kee, R.J., Dixon-Lewis, G., Warnatz, J., Coltrin, M.E., and Miller, J.A., A Fortran Computer Code Package for the Evaluation of Gas-Phase Multicomponent Transport Properties, Sandia Report No. SAND [10] Kee, R.J., Rupley, F., Meeks, E., and Miller, J.A., Chemkin-III: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical and Plasma Kinetics, Sandia Report No. SAND UC-405. [11] Smooke, M.D., Puri, I.K., and Seshadri, K., Proc. Combust. Inst. 21: (1986). [12] Pernice, M. and Walker, H.F., SIAM J. Sci. Comput. 19: (1998). [13] Eisenstat, S.C. and Walker, H.F., SIAM J. Optimization 4: (1994). [14] van der Vorst, H.A., SIAM J. Sci. Stat. Comput. 13: (1992). [15] Saad, Y., Iterative Methods for Sparse Linear Systems, PWS Publishing, Boston, [16] Nam, H.N., Paul, P.H., Mueller, C.J., and Wyckoff, P.S., Combust. Flame 113: (1998). [17] Nam, H.N., Wyckoff, P.S., and Knio, O.M., J. Comput. Phys. 143: (1998). [18] Knio, O.M., Nam, H.N., and Wyckoff, P.S., J. Comput. Phys. 154: , (1999).