An elementary model for the validation of flamelet approximations in non-premixed turbulent combustion

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1 Combust. Theor Modelling 4 () 89. Printed in the UK PII: S ()975- An elementar model for the validation of flamelet approimations in non-premied turbulent combustion A Bourliou and A J Majda Département de Mathématiques et Statistique, Université de Montréal, CP 68 succ. Centre-Ville, Montréal, Québec H3C 3J7, Canada Courant Institute, New York Universit, 5 Mercer Street, New York, NY, USA Received November 999, in final form 5 Ma Abstract. The fundamental soundness of three flamelet models for non-premied turbulent combustion is eamined on the basis of their performance in an idealized model problem that merges ideas from the laminar asmptotic theor for non-premied flames and rigorous homogenization theor for the diffusion of a passive scalar. The overall flame configuration is stabilized b a mean gradient in the passive scalar: large Damköhler number asmptotics results are available for the laminar case to quantif the finite-rate effects that cause the flame to depart from its equilibrium state; the same results can also be used to incorporate higher-order corrections in the approimation of the reactive variables in terms of the passive scalar. The use of such flamelet approimations has been etended well beond the laminar regime as the lie at the core of practical strategies to simulate non-premied flames in the turbulent regime: the flamelet representation avoids the problem of turbulence closure for the reactive variables b replacing it b the presumabl much simpler closure problem for a passive scalar. It is precisel the validit of this substitution outside the laminar regime that is addressed here in the idealized contet of a class of small-scale periodic flows for which etensive rigorous results are available for the passive scalar statistics. Results for this simplified problem are reported here for significant wide ranges of Peclet and Damköhler numbers. Asmptotic convergence is observed in terms of the Damköhler number, with a convergence rate that is found to match the laminar predictions and appears relativel insensitive to the Peclet number. The passive scalar dissipation plas a ke role in achieving higher-order corrections for the finite-rate case: replacing its pointwise value b an averaged value is convenient practicall and can be rigorousl motivated for the class of flows studied here, but while it does achieve an overall improvement over the lower-order equilibrium model, the simplification compromises the higher asmptotic convergence observed with the original finite-rate flamelet model with eact local dissipation. (Some figures in this article are in colour onl in the electronic version; see Introduction In turbulent non-premied flames in the flamelet regime, reaction between the fuel and oidizer occurs on a length scale (the so-called flame thickness) that is tpicall much smaller than one could reasonabl hope or care to resolve in a practical computation concerned with the behaviour of the flow at much larger scales. The unresolved reaction term must therefore be included via a model, as is done for the unresolved turbulent terms. One approach to formulating such a model is to utilize the passive scalar formulation. Let Y be an reactive variable (for eample, the fuel or oidizer mass fraction) whose evolution is governed b an advection diffusion reaction equation for which there is no obvious turbulence closure available because of the reaction term. In turbulence flamelet models, this problem is replaced b that of the turbulence closure for a passive scalar Z (that is, one that obes an advection diffusion equation with no reaction source term, so that closure is in some sense much simpler) //89+$3. IOP Publishing Ltd 89

2 9 A Bourliou and A J Majda The simplest version of such a procedure epresses the reactive scalar Y as a function of the passive scalar Z: Y(,t) = Y flamelet (Z(, t)). Computing Y P, the average value of Y over the domain P, can then be performed if one knows PDF P (Z), the detailed probabilit densit function for the passive scalar Z on that domain: Y P = Y(Z(,t)) P = Y flamelet (Z) PDF P (Z) dz. A practical turbulent flamelet model therefore consists of three parts: (a) an algorithm to epress a reactive scalar Y in terms of a passive scalar Z; (b) a presumed form for the PDF of Z in terms of its average and other moments; (c) a closed-form evolution equation for an statistics of Z needed in the presumed epression of PDF P (Z). Two statements can be made at this point which are directl relevant to this tpe of approach: Statement. The assumption underling (a) in the three-step procedure above is rigorousl satisfied in the asmptotic sense for laminar flow fields. (see [5] and section. below). In the turbulent case, it is a common ad hoc assumption that turbulence will not affect the flame structure if the flame thickness is small compared with the length scale representative of the smallest energ containing turbulent eddies so that the laminar relationship between Y and Z would still hold in some sense to be made more precise later. Statement. There are a large number of theoretical results for the turbulent diffusion of a passive scalar that could be directl eploited to validate practical modelling strategies used in (b) and (c) above (see [7]). Those results correspond to a hierarch of increasingl comple, turbulent-like, multiple-scale velocit fields for which rigorous answers can be provided to questions such as, for eample, the validit of the turbulent diffusion model (including the validit of specific classes of approaches to estimate it in a computation) and the PDF of the passive scalar including a possible departure from Gaussian statistics. The results reported here constitute the first step in a sstematic stud to verif the relevance of those rigorous results in assessing the validit of turbulent flamelet strategies as outlined above. For passive scalar results to indeed be relevant to turbulent non-premied flames, one must first verif whether statement (persistence of the laminar flamelet structure) as assumed in the first step of the flamelet model is indeed valid for the tpe of multiple-scale velocit fields tpicall used in the studies in statement (turbulent diffusion theor). The answer is shown to be es, and precise asmptotic convergence rates will be obtained from the computations with three approimations for the flamelet structure (equilibrium model, finite-rate model with eact pointwise dissipation or with mean dissipation) for a wide range of Peclet and Damköhler numbers and several small-scale turbulent flow geometries. Such a sstematic asmptotic stud for a variet of flows is possible because the model problem which we set up here leads to a two-dimensional stead solution, so that reliable and computationall affordable results are available (partial analtic results for some cases, cheap numerical results otherwise): this is achieved b imposing a mean gradient on the passive scalar (a similar devise has also been proposed independentl in [], although there solutions are stead onl in the mean). This set-up is somewhat different from the usual counterflow configuration for laminar non-premied flamelets for which the precise asmptotic structure (including the effects of temperature dependence) has been described in great detail b Liñan [6], and whose application to the modelling of turbulent flame has been reviewed thoroughl b Peters [].

3 An elementar model for the validation of non-premied flamelets 9 The mean gradient configuration was selected here for computational convenience and for the direct link with turbulent diffusion theor, as described later in this paper. Moreover, despite the simplicit of the model problem introduced here, realistic flame configurations are obtained with results believed to be sufficientl generic that the theor-based approach used in this stud constitutes a novel alternative for the validation of turbulent flamelet models, to complement recent efforts using direct numerical simulation (DNS) (for eample Cook et al [ 3], Jimenez et al [4], Leonard and Hill [5], Mell et al [9], or the review b Vervisch and Poinsot [4]); such DNS efforts necessaril are limited to a narrow range of Peclet and Damköhler numbers.. Set-up for the model problem.. Basic equations and passive scalar formulation This stud focuses on the stead-state solution of the following sstem of advection diffusion reaction equations for the two reactive scalars Y and Y (in non-dimensional form): Y + Pe v Y = Y Da Y Y () t Y + Pe v Y = Y Da Y Y. () t Here, v = v(, ) is taken to be a non-dimensional incompressible stead two-dimensional field of period P = in both dimensions (to be described in more detail in section.3 below). The Peclet number is defined as Pe = Pλ/D where λ is some constant with the dimensions of velocit which measures the magnitude of the velocit field, and where D is the diffusivit of both scalars. The Damköhler number Da is defined as Da = kp /D with k is the reaction source term constant. If Y and Y represent, respectivel, the fuel and oidizer mass fractions, the source term corresponds to a one-step irreversible reaction Y + Y Y p (where Y p is the mass fraction of the product) at the finite reaction rate ω = ky Y. One of the two reactive equations can be replaced b the simpler advection diffusion equation if one introduces the passive scalar Z = (Y Y )/ with the evolution equation Z + Pe v Z = Z. (3) t The sstem of equations (), () or equivalentl (), (3) was considered in earl work of O Brien [] to illustrate basic ideas behind the PDF approach to turbulent closure for reactive flows. In this paper, the sstem is solved for Y i (, ), Z(, ) in a rectangular domain with L G / L G / and P, where L G is taken to be much larger than P (in the present calculations, L G /P = ). Precise boundar conditions for all three scalars are stated net, even though their motivation will become clear onl later. The passive scalar Z(,) is assumed to be the sum of a mean value Z G = /L G, corresponding to a mean gradient in with slope /L G, and of a bi-periodic perturbation Z P of mean zero and period P = : Z(,) = /L G + Z P (, ). (4) This is achieved b imposing the following boundar conditions: Z(, = ) = Z(, = ) for L G / L G / (5) Z( L G /,)= Z(L G /,) for. (6) The reactive scalars Y and Y are also periodic in and satisf the following boundar conditions in at A = L G / and L G /: Y i ( A,)= Y i,eq ( A,) (7)

4 9 A Bourliou and A J Majda where the equilibrium values Y i,eq are given b Y,eq (, ) = Z(,) + Z(,) Y,eq (, ) = Z(,) + Z(,) (8) as discussed in the following section... The reference laminar stead flame The interpretation of the above sstem in terms of a non-premied flame is best eplained b first considering the special case where v =. It is clear then that Z = /L G,Z P = isa trivial stead solution of equation (3), in which case the entire problem reduces to the stead one-dimensional laminar flame set-up introduced in the classical paper b Williams [5] for the asmptotic stud of a non-premied flame which we briefl review net. If one defines the relevant Damköhler group Da G = kl G /D (therefore Da G = Da(L G /P ), with Da the Damköhler number used in the previous section), the asmptotic results can be summarized as follows. In the limit of Da G, Y and Y cannot coeist since the react completel as soon as the become mied through diffusion at the stoichiometric level Z = (located at = ). To the right of this level, there is onl fuel (Y,eq = Z and Y,eq = ), to the left, there is onl oidizer (Y,eq = Z and Y,eq = ) so that one recovers the epressions in equation (8). For large but finite values of Da G, the departure of Y (Z) and Y (Z) from their equilibrium values, denoted dy (Z, Da G ) and dy (Z, Da G ), is small (amplitude Da /3 G ) and confined to a thin region of thickness Da /3 G around the stoichiometric level Z = (in Z-coordinates; in -space, the thickness is Da /3 G L G). This can be seen from the following asmptotic result: pick a reference Damköhler number Da ref which is sufficientl large for the asmptotic model to appl, and compute once for all the solutions of equation () at that particular Da ref number: the solution at an other Damköhler number in the asmptotic regime can be obtained directl from the computed values b eploiting that asmptoticall, the data points (Z norm, dy norm ) defined as ((Da G /Da ref ) /3 Z, (Da G /Da ref ) /3 dy)are independent of Da G and equal to their precomputed values at Da ref..3. Perturbation velocit fields: description and summar of relevant results from turbulent diffusion theor The epression for Y i = Y i,eq +dy i (Z, Da G ) = Y i (Z, Da G ) as a function of Z and Da G as described in the previous section defines the laminar flamelet structure corresponding to a mean gradient in the passive scalar. Rewriting in the laminar case L G = / Z so that Da G = k/d Z, the flamelet model epresses Y i as a function of Z and its dissipation D Z. We are now interested in eamining how useful this remains once we have introduced the non-zero velocit perturbation at the scale P = intermediate between the passive scalar gradient length scale L G and the thin flame thickness L G Da /3 G. The velocit fields v() considered in this stud are special cases of a class of velocit fields that have been used etensivel [7, 8] to analse homogenization theories for the turbulent diffusion of a passive scalar. We will consider two such flow fields, writing v(, ) = (u(, ), v(, )). Case A: simple horizontal shear with constant vertical flow u(, ) = u() = sin(π/p) v(,) = constant = λ where λ is the non-dimensional ratio between the velocit perturbation and the mean flow. In the computations below, λ was taken to be λ = 5/λ (so that dimensionall, the mean flow intensit is constant and equal to λ = 5).

5 An elementar model for the validation of non-premied flamelets 93 Case A : simple shear + mean flow Case B : δ= Case C : δ=.5 Case D : δ= Figure. Streamlines of the velocit fields (four periodic cells shown for each case). Cases B, C, D: Childress Soward flows. Define the stream function F(,), F(,) = K P ( ( ) ( ) ( ) ( )) π π π π sin sin + δ cos cos. π P P P P The velocit field is then computed as u(, ) = F v(,) = F. Three values for δ will be used: case B corresponds to δ =, case C corresponds to δ =.5 and case D corresponds to δ =. For each of these values, the stream function constant K was chosen so that the average kinetic energ over a periodic cell is unit. The stream function corresponding to these four flows are shown in figure. In case A, the mean flow with λ non-zero blocks the streamlines in the -direction associated with the small-scale shear, while case B is a shear flow tilted at 45. Case D is a small-scale turbulent flow consisting of an arra of eddies, while case C represents an intermediate case with both shear and eddies.

6 94 A Bourliou and A J Majda 3 Case A: simple shear + mean flow 6 Case B: δ= Case C: δ=.5 3 Case D: δ= Figure. Passive scalar (with zero-level) for Pe =..4. Turbulent diffusion theor predictions We now epress the passive scalar equation, equation (3), in terms of the stead perturbation Z P, resulting in the equation Pe v Z P Z P = Pe (P /L G )v e. (9) Equations of this tpe pla a central role in rigorous homogenization theor to represent asmptoticall the effects of the small-scale periodic velocit field v on the large-scale passive scalar field in terms of an effective diffusivit matri [7, 8]. A detailed summar of the basic rigorous Renolds averaging theor for these problems as well as the theor for enhanced diffusivit including man eplicit eamples can be found in sections. and. of [7]. In particular, the rigorous effective turbulent diffusivit is + κ T where the enhanced diffusivit κ T can be epressed as κ T = L G Z P Z P P () where g P represents the averaging of the function g over the periodic cell. For the present case, it is rigorousl equivalent to the following epression: κ T = L G uz P P. ()

7 An elementar model for the validation of non-premied flamelets 95 3 High Pe ; High Da 3 High Pe ; Low Da Low Pe ; High Da 3 Low Pe ; Low Da Figure 3. Case C (δ =.5), reaction rate for various Pe and Da. The fact that equation () is equivalent to equation () involves several manipulations using integration b parts; the details of this calculation can be found in equations () and (5) and the subsequent discussion on pp 5 of [7]. In the present case therefore, the mean gradient Z G is itself unaffected at large scale b the presence of the velocit perturbation, but its diffusive flu is enhanced and as a consequence, combustion is enhanced. The enhanced turbulent diffusivit for the four small-scale periodic flows in cases A D ehibits a wide range of different scaling regimes as the Peclet number is varied [7, 8]. In particular, cases B and C ehibit strongl enhanced turbulent diffusion as the Peclet number increases while the enhanced diffusion is ver modest at large Peclet numbers for cases A and D (see table below)..5. Summar for the set-up and non-dimensionalization In summar, there eists a stead solution to the sstem of equations (), () or (), (3) with stead velocit fields and the boundar conditions described above and the calculations in this stud will compute it directl b solving in non-dimensional form the appropriate equations for the passive mean gradient perturbation and for the reactive variables. The computations

8 96 A Bourliou and A J Majda Table. Turbulent diffusivities κ T. Flow tpe Pe = Pe = 5 Pe = Pe = Case A.97E Case B Case C Case D with results reported net correspond to the following non-dimensional numbers: L G /P = Da = 38 [, 8, 4 3, 8 3 ]. When there is no turbulent velocit field, those values correspond to asmptotic laminar flame thicknesses given, respectivel, b P, P/, P/4 and P/8. Pe =, 5,,. λ = 5 (case A onl). 3 Pe= Da Local Dissipation 3 Pe= Da Mean Dissipation.5.5 dy norm norm Z norm 3 Pe= Da Local Dissipation Z norm 3 Pe= Da Mean Dissipation.5.5 dy norm dy norm Z norm Z norm Figure 4. Case C (δ =.5): scatter plots for Pe = and two different Damköhler numbers. Top, large Da (laminar flame thickness of 8 ); bottom, low Da (laminar flame thickness of ). Data on the left from the finite-rate flamelet with eact pointwise dissipation; data on the right from the model with cell-averaged dissipation.

9 An elementar model for the validation of non-premied flamelets 97 3 Pe= Da Local Dissipation 4 3 Pe= Da Mean Dissipation dy norm dy norm Z norm Pe= Da Local Dissipation Z norm 5 Pe= Da Mean Dissipation dy norm.6.4 dy norm Z norm Z norm Figure 5. Case C (δ =.5): same as in figure 4 but with Pe =. The problem therefore contains three relevant length scales: the large length scale L G which is related to the mean gradient in the passive scalar (L G = in the present computations), the cell size for the velocit perturbation with a unit period P = and the small length scale representing the laminar flame thickness. The objective of the stud is to quantif the effect of the model problem velocit field at the intermediate length scale on the asmptotic results obtained when it is equal to zero. 3. Direct numerical simulation results 3.. Passive scalar Cases A and B: eplicit formulae for the perturbation and for the turbulent diffusivit. Case A is a special case of a more general periodic shear for which eplicit formulae for the solution and the enhanced diffusivit are reviewed in [7]. Define Pe, the Peclet number associated with the mean flow, as Pe = Pe λ/λ, the solution can then be written as Z P () = Pe/L G π(4π + Pe ) / sin(π θ) () with the phase angle θ given b cos θ = π/(4π + Pe ) / and sin θ = Pe/(4π + Pe ) /. This shows that the presence of a mean flow (θ ) has two effects: it shifts the phase of Z P b the angle θ and it reduces its amplitude.

10 98 A Bourliou and A J Majda Figure 6. Case C (δ =.5) Pe =, Da large (laminar flame thickness = 8 ): dissipation and error in reactive scalar using the flamelet model with either the eact pointwise dissipation or with the cell-averaged dissipation. The turbulent diffusivit is obtained using the formula in equation (): Pe κ T = (4π + Pe. (3) ) One recovers the result originall documented in [7, 8] that contrasts the tremendous boost in diffusivit that can occur if Pe = with the case where Pe is large and the enhancement is much smaller due to streamline blocking b the mean flow. A ver similar result can be obtained directl for case B (Childress Soward flow with δ = ) b noticing that the solution reduces to Z P (, ) = Z P ( ) which is obtained as for case A b replacing in equation () b, Pe b Pe/ with Pe identicall zero.

11 An elementar model for the validation of non-premied flamelets 99 Figure 7. Case C (δ =.5): asmptotic convergence of reactive scalar as Da for the four values of Pe: circles, comparison between equilibrium model; triangles, flamelet with eact dissipation; stars, flamelet with mean dissipation. Cases C and D: numerical solution. There are no closed-form solutions for those cases but accurate numerical solutions are readil available [7, 8]. The solution here was obtained numericall b integrating equation (3) using centred differences. The resulting sstem was solved using the generalized minimum-residual (GMRES) method with diagonal scaling as implemented in the routine DSDGMR from the public domain librar SLATEC. Again, enhanced diffusivit is computed through the solution Z P (, ) using equation () and the values are reported in table. Colour maps for Z(,) = /L G + Z P (, ) are shown in figure along with the stoichiometric level (shown in black) Z(,) = for the case Pe =. Reactive scalars Once the solution for Z is known (either analticall or numericall as described above), one needs to solve one of the two equations in (), (), and obtain the other scalar b taking into account the fact that Y Y = Z in equations () and (): Pe u Y + Pe v Y = Y Da Y (Y Z) Pe u Y + Pe v Y = Y Da (Z + Y )Y

12 A Bourliou and A J Majda Figure 8. Case C (δ =.5): asmptotic convergence eponent β (summar of figure 7) as a function of Pe for all three models. with periodic boundar conditions in the -direction and Dirichlet boundar conditions in the -direction corresponding to the equilibrium conditions in equation (8). Those stead nonlinear equations are discretized using centred differences and solved with Newton s method. Convergence is ver fast if one uses as an initial guess the equilibrium solution in the entire domain for Y and Y (as in equation (8)); an even more accurate initial guess involves the finite-rate flamelet model to be discussed in section 4.3 below. Which of Y and Y should be solved for b integrating the nonlinear equation is irrelevant from the analtic point of view, but can affect the numerical accurac. Since the solution is a small departure from equilibrium, Y and Y hardl coeist and one is tpicall much smaller than the other. For round-off error control, it is then important to integrate the nonlinear equation for the smallest of the two and obtain the largest from Z rather than the other wa around, where the deficient species would be obtained with ver poor accurac as the ver small difference between two numbers. Results are illustrated for case C onl (hbrid case, δ =.5), with similar results for the other three flow fields. Colour maps for the reaction rate are shown in figure 3 for four combinations of Pe and Da. The high Peclet number is Pe = and the low Peclet number is Pe =. The high Damköhler number corresponds to a laminar flame thickness of 8 and the low Damköhler number corresponds to a laminar flame thickness of (normalized with the periodic cell size). Three periods in are shown, while onl a fraction of the domain is shown in since relaation to equilibrium occurs within the fraction of the domain displaed here. At high Damköhler number (plots on the left), the reaction zone is indeed seen to be concentrated around the stoichiometric level (shown as a thin black line) with roughl a flame thickness comparable to the laminar value of, at least for the low Peclet case. At low Damköhler 8 number, the flame thickness is much larger, again close to the laminar value of for the low Peclet number case, even somewhat larger for the large Peclet case, with large departure from the laminar flamelet structure in that last case.

13 An elementar model for the validation of non-premied flamelets Figure 9. Case A (simple shear with a transverse mean): asmptotic convergence of reactive scalar as Da for the four values of Pe: circles, comparison between equilibrium model; triangles, flamelet with eact dissipation; stars, flamelet with mean dissipation. 4. Description of the turbulent flamelet models As stated in the introduction, it will be assumed in what follows that the detailed solution for the passive scalar Z is available (computed as described in section 3). Although it is customar to reserve the appellation of flamelet to models that incorporate finite-rate effects, it is used loosel here to designate an model that relates the reactive scalars to the passive scalar according to a thin-flame structure. 4.. Equilibrium model The equilibrium model is ver simple: if the Damköhler number is sufficientl large, the reactive scalars can be approimated b their equilibrium values Y i,eq (Z) (see equation (8)) with a small error of order Da /3 G. 4.. Finite-rate flamelet with pointwise dissipation A better approimation at large but finite Damköhler number is obtained b incorporating the net term in the asmptotic epansion for the reactive scalars in term of Da /3 G. This can be done efficientl b reling on the results summarized in section.. Assume that the

14 A Bourliou and A J Majda Figure. Case A: asmptotic convergence eponent β (summar of figure 9) as a function of Pe for all three models. asmptotic function dy norm (Z norm ) has been pre-computed once for all values. Those data can be used to obtain efficientl an approimation for Y i = Y i,eq +dy i using the following algorithm: compute the local dissipation D Z and use it to compute a local Damköhler number Da G (, ); obtain the normalized z-coordinate using Z norm (, ) = (Da G (, )/Da ref ) /3 Z(,); interpolate from the previousl computed values to obtain the corresponding normalized departure from equilibrium dy norm (Z norm ); rescale to obtain the local departure from equilibrium: dy i (, ) = (Da G (, )/Da ref ) /3 dy norm (Z norm ). The resulting approimation should agree asmptoticall with the eact value for the reactive scalar with an error of the order of Da /3 G Finite-rate flamelet with cell-averaged dissipation In practical computations, the detailed values for the passive scalar Z are not eplicitl computed at the small turbulent scales and various approimations to build the PDF of Z are used instead. An investigation of the effect of substituting an approimate PDF of Z to its detailed knowledge (or equivalentl, to its eact PDF) will be reported in another paper. For now, we onl consider the impact of the following approimation: while the passive scalar itself is assumed to be known in detail, we could restrict ourselves to using onl a cell-averaged value for its dissipation, so that in the procedure above for the finite-rate flamelet, one would use an average value Da G P instead of the pointwise value Da G (, ). According to equation (), the cell-averaged dissipation is actuall directl proportional to the total diffusivit + κ T. Indeed, Z = Z G + Z P =/L G + κ T /L G. (4)

15 An elementar model for the validation of non-premied flamelets 3 Figure. Case B (δ = ): asmptotic convergence of reactive scalar as Da for the four values of Pe: circles, comparison between equilibrium model; triangles, flamelet with eact dissipation; stars, flamelet with mean dissipation. The cell-averaged passive scalar dissipation can therefore be epressed eplicitl in terms of the turbulent diffusivit of the passive scalar, which one would need to compute anwa in a practical computation. For the models considered here, this confirms a suggestion to that effect in [3], where it was proposed that turbulence subgrid models for the diffusivit and for the mean dissipation should be related. 5. Validation with scatter plots A qualitative wa to assess the validit of the finite-rate flamelet models, with the eact pointwise dissipation or with the cell-averaged dissipation, is to produce scatter plots in the (Z norm, dy norm ) coordinates. Given both Y and Z from the direct numerical simulation, compute dy = Y Y,eq (Z) = Y Z Z and also a local turbulent Damköhler number Da = Da G (, ) = D( Z ) or its cell-averaged value. If indeed the data belong to a laminar flamelet structure, then the reduced data point ((Da G /Da ref ) /3 Z, (Da G /Da ref ) /3 dy ) should lie on the universal pre-computed graph (Z norm, dy norm ). Results are illustrated for case C, with similar results for the other cases. The scatter plots are reported in figure 4 for Pe = and in figure 5 for Pe =. In those plots, Da corresponds to a large Damköhler number and corresponding flame thickness of 8, while Da corresponds to a small

16 4 A Bourliou and A J Majda Figure. Case B: asmptotic convergence eponent β (summar of figure ) as a function of Pe for all three models. Damköhler number and the corresponding laminar flame thickness of, so that in this latest case, the flamelet approimation is less likel to appl. The reference curve, precomputed once for all, is shown as a thick white line. The conclusions of this qualitative analsis are as follows: at low Peclet number (Pe = ) and large Damköhler number, there is little scatter for either model, which means that the assumption of a laminar flamelet structure is good. As to be epected, the model with the eact pointwise dissipation is somewhat better than that with the cell-averaged dissipation. For the same low Peclet number, at lower Damköhler number, there is more scatter. For this problem, where the laminar flame thickness is of the order of the velocit perturbation periodic cell size, the model with the averaged dissipation performs better. Given the relationship between the passive scalar mean dissipation and its total diffusivit, this amounts to saing that the flame structure resembles more closel that of a laminar flame with diffusivit given b the total diffusivit (molecular and turbulent) than one with the original molecular diffusivit. At large Peclet number (figure 5), there is more scatter because of the larger amplitude in perturbation velocit (the scale of the plot has been adjusted to accommodate the scatter) but the trends from the low Peclet cases are confirmed: more scatter at low Da than at large Da; at large Damköhler number, better performance of the model with the pointwise dissipation, and the reverse at lower Damköhler number. 6. Asmptotic convergence More precise statements regarding the performance of the models can be obtained b computing their pointwise errors. At each point (, ) with passive scalar Z(, ) from the direct simulation, one can compute estimates for Y according to the three models in section 4 and compare them with the eact value Y from the direct simulation. Figure 6 displas colour maps for the errors in Y for the finite-rate flamelet with eact pointwise dissipation (top) and with the cell-averaged dissipation (middle) along with a colour map of the pointwise dissipation itself (bottom) for case C with Pe = and a Damköhler number corresponding

17 An elementar model for the validation of non-premied flamelets 5 Figure 3. Case D (δ = ): asmptotic convergence of reactive scalar as Da for the four values of Pe: circles, comparison between equilibrium model; triangles, flamelet with eact dissipation; stars, flamelet with mean dissipation; diamonds, flamelet with optimal dissipation. to a laminar flame thickness of (the stoichiometric level Z = is also shown as a black 8 line on the dissipation colour map). As epected, pointwise errors are concentrated for both models in the thin reaction zone, but particularl in the areas with strong vortices. From the dissipation map, it is seen that those vortices correspond to areas with little dissipation. Replacing the pointwise dissipation b its cell-averaged value causes the flamelet model with averaged dissipation to sstematicall overestimate the reaction rate. The flamelet model with eact pointwise dissipation will be seen in the net figures to have much smaller errors; in this case, the dominant errors also come from the vortices, but there appears to be some error cancellation on either side of the flame in those areas. Global errors corresponding to the different models are computed as the L norm over the entire domain of the pointwise errors in Y. Results are reported in figure 7 for case C where the errors are displaed as a function of the inverse of the Damköhler number, each plot corresponding to one of the four Peclet numbers used in the stud. As epected, for an given method, the error increases as the Damköhler number decreases and it also increases if the Peclet number increases. It is interesting to compare at an given Peclet number, the respective performance of the three methods. The equilibrium model alwas leads to the largest error for all cases. The flamelet model with the eact pointwise dissipation is usuall the most accurate method, and with an apparent larger rate of convergence. The onl eception is observed for

18 6 A Bourliou and A J Majda Figure 4. Case D: asmptotic convergence eponent β (summar of figure 3) as a function of Pe for all four models. the case with the lowest Damköhler number, where the laminar flame thickness is equal to the velocit perturbation cell size: then the flamelet model with the cell-averaged dissipation actuall outperforms that with the eact pointwise dissipation. This confirms the conclusion from the scatter-plot analsis: if the flame thickness is as wide as the velocit perturbation scale, then the flame appears to be closer to a laminar flame with the total diffusivit as assumed b the mean-dissipation flamelet model, than to the original flame with onl the molecular diffusivit. The overall trend of the mean-dissipation flamelet model appears, however, to follow the rate of the equilibrium model more than the eact dissipation flamelet model. This is confirmed in figure 8 which shows the eponential fit of the error in terms of Da /3 for each Peclet number. The laminar asmptotic theor predicts that the L norm for the equilibrium model should depend quadraticall on Da /3 (the amplitude of the error scales like Da /3, and also the flame thickness, so that the integral of the domain should scale like Da /3 ). Similarl, the flamelet model with pointwise dissipation includes the Da /3 term from the asmptotic epansion in the computation so that one should gain one order of accurac asmptoticall, with a cubic convergence. This is indeed what is approimativel observed as reported in figure 8, where the eponent β is reported (an eponential fit based on the three largest Damköhler numbers, ecluding the laminar flame thickness of ). The flamelet model with eact dissipation leads to a convergence eponent of.8, and the equilibrium model, of. There is not much influence of the Peclet number on the eponent (even though the overall magnitude of the errors does depend ver much on the Peclet number). It is also seen that the flamelet model with mean dissipation converges quadraticall, so that it resembles more the convergence of an equilibrium model than a complete flamelet model. Those trends are confirmed for the other flow field cases. Figures 9 and for the simple shear with transverse mean also displa the cross-over between the mean-dissipation and the eact dissipation models at low Da. Note that for this case, errors are ver small (overall turbulent diffusivities are small). Great care was needed with the numerical processing of the data to avoid spurious errors not related to the asmptotic convergence, but it does seem to affect the eact-dissipation model for the low Peclet, large Damköhler case, which eplains the questionable convergence

19 An elementar model for the validation of non-premied flamelets 7 Figure 5. Case C (δ =.5): profiles of the errors in Y m, the cell-averaged Y with the three models, compared with Y,eact, the eact cell-averaged value for Y from the direct simulation. Pe = (low Pe), (high Pe) and laminar flame thickness of (low Da) and 8 (high Da). eponent of for Pe =, while it is seen to be (at least) 3 for the other values of Peclet numbers, as epected from the theor. Figures and correspond to case B (δ = ) with much larger turbulent diffusivities. For this case, the eact dissipation model is alwas the best, even at low Damköhler numbers. Again, the equilibrium model converges quadraticall, regardless of the Peclet number; the flamelet model with eact pointwise dissipation converges (almost) cubicall, while the same model with the cell-averaged dissipation converges just slightl better than quadraticall. Figure 3 and figure 4 correspond to case D (δ = ) where the flow field induces patterns of vortices which present a special challenge to flamelet models. This is particularl obvious for the flamelet model with cell-averaged dissipation which at the lowest Damköhler number can lead to errors larger than the equilibrium model. This poor convergence is confirmed in the convergence eponent. For this case, a best-case scenario was also studied b defining an optimal dissipation obtained using the following procedure which identifies the constant value for the cell dissipation (to substitute for the cell-averaged dissipation) that leads to the smallest error. This is done numericall b eperimenting with the range of pointwise dissipations actuall observed in the domain. For each case, the optimal mean dissipation was selected that corresponded to the smallest global error for the flamelet model with constant dissipation. The convergence properties with this optimized model are somewhat better than

20 8 A Bourliou and A J Majda Figure 6. Case C (δ =.5): asmptotic convergence for the cell-averaged reactive scalar as Da for the four values of Pe: circles, comparison between equilibrium model; triangles, flamelet with eact dissipation; stars, flamelet with mean dissipation. the original trend, but still remain at best comparable to the equilibrium model at sufficientl large Damköhler number. Finall, the asmptotic convergence analsis is applied to the cell-averaged value of Y. Cell-averaging the data is performed at each point b replacing its value b its mean value over a two-dimensional cell centred around the point, as commonl done in a priori validation of large-edd simulations via the filtering process. Cell-averaging is therefore seen to reduce the original two-dimensional field to a one-dimensional profile in. Profiles for the (signed) errors in the cell-averaged values for Y are shown for case C in figure 5. The equilibrium model is seen to sstematicall underestimate the fuel concentration (it assumes instantaneous burning at the stoichiometric level). Again, the flamelet model with eact dissipation leads to the smallest discrepancies, ecept at low Peclet and low Damköhler number where the model with cell-averaged dissipation provides the least inaccurate values. L norms for the errors in the cell-averaged values and their eponential fit are computed as was done above for the pointwise errors. The are shown in figures 6 and 7 where the trends are somewhat similar to the corresponding plots for the errors in the unaveraged Y (figures 7 and 8). For instance, there is a cross-over of the two flamelet models for low Damköhler numbers and a higher asmptotic convergence rate is observed for the eact-dissipation flamelet model compared with the other two models. A first notable difference, however, is that the flamelet model with cell-averaged dissipation performs here with a much better error constant than the equilibrium model: this is because the equilibrium model sstematicall underpredicts Y (which eplains

21 An elementar model for the validation of non-premied flamelets 9 Figure 7. Case C: asmptotic convergence eponent β for the cell-averaged reactive scalar (summar of figure 6) as a function of Pe for all three models. wh the magnitude of the error has hardl changed between the averaged and unaveraged case from figure 7), while the cell-averaging of Y results in some error cancellation for the flamelet model with the cell-averaged dissipation. While for those two models, the convergence eponent remains, a second noticeable difference due to cell-averaging the data is that the convergence eponent for the flamelet model with eact dissipation has now improved from 3 to 4; this is likel to also be due to error cancellation during the cell-averaging procedure on Y as a result of the smmetr in the problem. 7. Conclusions A model problem with a rigorous analtical basis has been introduced here to stud the effect of idealized small-scale turbulence on the performance of flamelet models for non-premied combustion. Flamelet models with pointwise and averaged dissipation as well as an equilibrium model have been studied for a variet of small-scale flow geometries and a wide range of Peclet and Damköhler numbers. The computations performed in this stud indicate that the equilibrium model and the flamelet model with the eact pointwise dissipation converge asmptoticall with a rate, respectivel, quadratic and cubic in the L norm, as predicted b the laminar theor. Convergence rates appear to be actuall rather insensitive to the intensit of the perturbation velocit at the intermediate scale, as measured b the flow Peclet number, although the overall magnitude of the errors does depend on the Peclet number. Resorting to the cell-averaged dissipation is a ver attractive option for the flows studied here because the cell-averaged dissipation is simpl proportional to the turbulent diffusivit and could therefore be obtained at no additional cost in a simulation. The finite-rate flamelet model that uses this cell-averaged dissipation actuall outperforms that with the eact pointwise dissipation when the Damköhler number is ver low but at larger Damköhler numbers, it converges onl marginall better than the equilibrium model in the sense that the error constant is usuall slightl smaller, but

22 A Bourliou and A J Majda the asmptotic convergence rate is quadratic, the same as that of the equilibrium model. In all cases, however, all three flamelet models do converge and the soundness of the flamelet strateg of epressing the reactive scalars in terms of the passive scalar as in a laminar flame is confirmed, even for finite (large) Damköhler numbers and significant turbulent miing. The main challenge for the success of their practical implementation is therefore to be able to obtain a sufficientl accurate description of the passive scalar via its PDF. There is ongoing work b the authors to rigorousl characterize the passive scalar PDF and its dissipation in the presence of a mean gradient for the class of flows studied here as well as generalizations to suitable time periodic and random flows. The merit of the practical computational strategies to fit the scalar PDF and its dissipation will then be assessed in terms of their contribution to the overall flamelet model errors for test cases similar to those presented here, for which one can afford to obtain results for a wide range of Peclet and Damköhler numbers and for which a sstematic validation is possible b comparison with rigorous theoretical results and well resolved direct simulations. These results will be presented elsewhere in the near future. Acknowledgment This research is based upon work supported in part b the US Arm Research Office under grant nos DAAG and DAAG References [] Cook A W and Rile J J 998 Subgrid-scale modeling for turbulent reacting flows Combust. Flame [] Cook A W and Rile J J 994 A subgrid model for equilibrium chemistr in turbulent flows Phs. Fluids [3] Cook A W, Rile J J and Kosal G 997 A laminar flamelet approach to subgrid-scale chemistr in turbulent flows Combust. Flame 9 33 [4] Jimenez J, Liñan A, Rogers M M and Higuera F J 998 A priori testing of subgrid models for chemicall reacting non-premied turbulent shear flows J. Fluid Mech [5] Leonard A D and Hill J C 997 Scalar dissipation and miing in turbulent reacting flows Phs. Fluids A [6] Liñan A 974 The asmptotic structure of counterflow diffusion flames for large activation energies Acta Astronaut [7] Majda A J and Kramer P R 999 Simplified models for turbulent diffusion: theor, numerical modelling and phsical phenomena Phs. Rep. 34(4 5) [8] Majda A J and McLaughlin R 993 The effect of mean flows on enhanced diffusivit in transport b incompressible periodic velocit fields Stud. Appl. Math [9] Mell W E, Nilsen V, Kosal G and Rile J J 994 Investigation of closure models for non-premied turbulent reacting flows Phs. Fluids [] O Brien E E 98 The probabilit densit function (pdf) approach to reacting turbulent flows Turbulent Reacting Flows ed P A Libb and FAWilliams (Berlin: Springer) pp 85 8 [] Overholt M R and Pope S B 999 Direct numerical simulation of a statisticall stationar, turbulent reacting flow Combust. Theor Modelling [] Peters N 984 Laminar diffusion flamelet models in non-premied models in turbulent combustion Prog. Energ Combust. Sci. 39 [3] Pierce C D and P Moin 998 A dnamic model for subgrid-scale variance and dissipation rate of a conserved scalar Phs. Fluids 34 4 [4] Vervisch L and Poinsot T 998 Direct numerical simulation of non-premied turbulent flames Ann. Rev. Fluid Mech [5] Williams F 97 Theor of combustion in laminar flows Ann. Rev. Fluid Mech