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Direct comparison of PDF and scalar dissipation rates between LEM simulations and experiments for turbulent,… Tsui, H. P.; Kamal, M. M.; Hochgreb, S. (Simone); Bushe, W. K. Mar 31, 2016

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June 1, 2015Direct comparison of PDF and scalar dissipation rates between LEMsimulations and experiments for turbulent, premixed methane air flamesH. P. Tsui∗,1, M. M. Kamal2, S. Hochgreb2, and W. K. Bushe11 *Department of Mechanical Engineering, University of British ColumbiaVancouver, BC, V6T 1Z4, Canada2 Department of Engineering, University of CambridgeCambridge, CB2 1PZ, United KingdomAbstractWe present a direct comparison between the predicted and measured probability density functions (PDF)of the reaction progress variable and conditioned values of the scalar dissipation rates (SDR) in premixedturbulent flames. The predictions are based on simulations of premixed flames using the linear-eddymodel (LEM), parameterised by a wide range of integral length scales and turbulent Reynolds numbers.The experimental results are highly spatially resolved temperature and species data from the Cambridge-Sandia swirl burner. The LEM simulations display remarkable accuracy in capturing the features ob-served experimentally. Further, the results reveal that the LEM calculated PDF and SDR for premixedflames remain relatively steady under a variety of turbulent conditions, including variations in the inte-gral length and turbulent Reynolds number. In general, it appears to be practical to use representativepseudo-turbulent PDF and SDR models for a range of turbulence intensities and length scales.1 IntroductionSeveral practical models for turbulent premixed combustion rely on an accurate representation of theprobability density function (PDF) of a reaction progress variable, which is often parameterised by themean and variance of that progress variable [1–6]. These presumed PDF approaches are often imple-mented in conjunction with tabulated chemical variables to achieve detailed chemistry calculations inturbulent combustion simulations. Such models have been developed for both the Reynolds-averagedNavier-Stokes (RANS) and large eddy simulation (LES) paradigms. Previous work has shown that theaccuracy of these methods depends to a considerable extent on the accuracy of the function presumed forthe PDF of the progress variable [1].A number of different presumed PDF models have been previously investigated for premixed com-bustion. The often used β -PDF does recover the extreme properties expected of the true PDF, such as δfunctions at the zero and unity extremes of reaction progress for maximal variance, and single δ functionsat the mean for zero variance. However, it fails to reproduce the shape of the true PDF in more generalcases [1]. The issue is related to the fact that the shape of the true PDF appears to be a function of howthe chemical reaction rates vary as a function of the progress variable; hence, different chemical kineticslead to different shapes of the PDF of progress variable. The form of the β -PDF – which is of courseentirely independent of the chemical kinetics – can lead to significant inaccuracies. Most critically, therecan be a biasing error, as the discrepancies tend to occur at the same values of the progress variable forany particular flame.∗corresponding author, hongtsui@alumni.ubc.caOne of the primary concerns in the design of modern engines is the reduction of harmful pollutants;specifically, the current generation of numerical models must be able to predict both the thermodynamicproperties of the reacting mixture and the formation of these minor species with sufficient accuracy toresolve the parts per million produced. A prominent example is the prediction of prompt flame NOx viathe Fenimore pathway [7]. The predicted NOx values from this mechanism are strongly affected by thechoice of the PDF model, as this pathway is sensitive to the predicted temperatures and flame profile inthe reactive regions.In an attempt to account for the effects of chemistry on the shape of the PDF, Bray et al. [8] proposedusing a premixed laminar flame to model the functional dependence of the PDF on progress variable.The proposed probability dependence of the flame existing at any given state is inversely proportional tothe magnitude of the gradient of the temperature. Their original formulation only provided coverage forflames with very high variance. Jin et al. [1] then proposed a modification to the Bray PDF that extendsthe original formulation to cover all possible mean and variance combinations. This is accomplished bytruncating the PDF shape function as needed to match the mean and variance parameters. It was foundto significantly improve the fit of the PDF to that extracted from Direct Numerical Simulation (DNS)results. A shortcoming of this method is that, at the point of truncation, the model PDF has a sharp dropto zero, whilst the true PDF tends to be more rounded.To address this issue, a one-dimensional turbulent method was proposed to take the place of thetypical laminar flame calculation to tabulate pseudo-turbulent PDF models for RANS and LES closures.The Linear-Eddy Model, an inexpensive one dimensional stochastic mixing model, has demonstratedthe ability to capture important effects from the interaction between chemistry and turbulence on thePDF distributions sufficiently well [9–11]. This model provides us with a mechanism to investigate thegeneral flame characteristics at very high turbulence intensities, much beyond the capability of currentDNS strategies. In turn, it permits us to analyze the behavior of the PDF constructed at these highlyturbulent states.While the LEM has been implemented to investigate the shape of the PDF distributions [9, 10], suchsimulations were performed to analyze the PDF for specific flames. The current study is primarily inter-ested in the tabulation of a PDF lookup table useful for subsequent RANS and LES flame computations(a pre-processing operation not unlike pre-calculating the β -function and storing that in a lookup table).At first, there was a suggestion that one ought to try to match the turbulence statistics in the LEM cal-culations to those that one expects to find in the later turbulent flame calculation. Indeed, a possibility isthat one might need to add a dimension to the lookup table (something like the local turbulent Reynoldsnumber) to account for variations in local turbulence properties in the turbulent flame calculation andtheir effect on the shape of the PDF. This is a large part of the motivation for the LEM work presentedhere: do the turbulence properties affect the shape of the PDF? If so, how? How important is it to matchthe LEM turbulence properties used in generating the PDF lookup table to those that will be found in theturbulent flow to be calculated later?A related question can be asked about the local gradient of progress variable in a premixed turbulentflame which is closely related to the scalar dissipation rate (SDR). For premixed combustion, the SDR,χc, of a temperature-based reaction progress variable is,χc(T,φ) = αc(T,φ)∇c ·∇c, (1)where αc(T,φ) and ∇c are the thermal diffusivity at the local temperature and equivalence ratio and thegradient of the progress variable, respectively.The SDR is an important quantity to both non-premixed and premixed combustion modeling [12, 13].It is an unclosed term that appears in the transport equation for the variance of progress variable, whereχc directly measures the decay rate of fluctuations through turbulent micromixing [13]. Since the burnrate of many combustion processes depends on the contact area and local gradient between the reactants,it is reasonable for most combustion models to assume that the mean burning rate of the flames eitherexplicitly or implicitly depends on the scalar dissipation rate. For example, Conditional Moment Closure(CMC) uses the scalar dissipation rate conditioned by the progress variable to calculate micromixing [13];not surprisingly, modeling the conditional scalar dissipation term χc|c∗ conditioned on the local andglobal progress of reaction emerges as one of the main difficulties in applying CMC to turbulent premixedflames [14].The same LEM method can be used to generate a model for the conditional scalar dissipation, whereone simply conditionally averages the scalar dissipation in the LEM temperature profiles that are used toconstruct the PDF model. The unconditional mean SDR can be obtained by convolving the conditionalSDR with the model PDF. This allows for the construction of pseudo-turbulent PDF and SDR modelsthat are perfectly consistent with one another. While it is possible to obtain PDF and SDR distributionsfor premixed combustion from DNS, the associated cost is generally prohibitive for flows with relevantturbulent conditions [1]. More importantly, studies typically tend to focus on the analysis of the PDFs andSDRs at specific points within the domain. This leads to the problem that although DNS and experimentscan provide valuable insight to the behavior of the PDF and SDR [15–17], they cannot provide usableinput models for subsequent RANS and LES combustion calculations, which is the primary motivationbehind the current work. The incorporation of turbulence characteristics in the PDF distributions couldprovide a solution to the deficiencies seen in a number of current PDF models, such as the ad hoc β -pdf orthe laminar approaches. Having more accurate PDF and SDR models would be beneficial to a number ofRANS and LES strategies as closures for turbulent premixed combustion typically rely on some variantof the PDF or SDR model for the reaction progress variable as the input [12, 13].Recent detailed measurements of species and temperature have been made by the Cambridge-Sandiaswirl burner [16, 18–20]. This swirl burner was designed specifically to explore the influence of stratifica-tion on the flame. However, the very detailed nature of the scalar and velocity measurements have madethe data set attractive as a target for premixed flame model validation as well [21, 22]. In particular, thecomprehensive database allows conditioning on a number of different variables, including equivalenceratio (for the stratified flames), temperature (or progress of reaction) or any other suitable scalar.In this paper we use the experimental dataset to obtain detailed PDFs of the progress of reaction. Thetemperature is used to characterise the extent of reaction, for direct comparison with the PDFs generatedfrom LEM simulations for three different swirl (and turbulence) levels. We consider the measured andcomputed variances, as well as the detailed shape of the PDFs in the comparison. In addition, we arealso able to directly evaluate the unconditional mean SDR values obtained from experiment and LEMsimulations. In the following sections we discuss the numerical approach, followed by a summary of theexperiments and the data treatment used.2 Numerical Conditions: Premixed Combustion2.1 Linear-Eddy ModelThe Linear-Eddy Model has been demonstrated to replicate the flow statistics for simple turbulent con-ditions with acceptable accuracy [23–27]. Given the one-dimensional nature of the model, the compu-tational costs remain relatively low for most practical cases. Here, the LEM is used in a pre-processingmanner for the tabulation of discrete PDF and SDR models, which can be implemented in subsequentRANS and LES applications.The LEM can be divided into two modules. The deterministic component consists of the usual one-dimensional gas dynamics evolutionary equations, whereas the stochastic component consists of randomeddy events. The turbulence concept of LEM postulates a random process that rearranges fluid elementsalong a line in order simulate the chaotic vortices that appear in turbulent fields. These one-dimensionalvortices, known as triplet maps, generate discontinuous fluid motions, which lead to a random walk offluid elements. The eddy event frequency per unit length of the domain is governed by [26],λLEM =545νRetCλ l30(l0/lk)5/3−11− (lk/l0)4/3, (2)where ν , Ret , l0 and lk are the kinematic viscosity, turbulent Reynolds number, the integral length scaleand the Kolmogorov scale, respectively. The empirical parameter Cλ should typically be tuned to theflame in question for LEM studies [10]; however, we are primarily interested in investigating the changesin the PDF models associated with variations in the turbulent fluctuations and integral length scales. Assuch, Cλ is held constant for consistency between cases. The value of 15.0 for Cλ is adopted from [10].Similarly, the parameter used to scale the Kolmogorov length, Nη , which typically requires tuning to casespecific conditions, is also held at a constant value of 1.0 for all of the LEM cases. This parameter isapplied to the inertial scaling law in the following form,lk = Nη l0Re−3/4t . (3)The deterministic and stochastic modules are implemented simultaneously during the simulation toachieve a pseudo-turbulent effect; this coupling between the stochastic advective process and determin-istic evolutionary equations in a one dimensional computational domain permits the LEM to simulateturbulent flows with higher Reynolds numbers than multi-dimensional models. The current LEM variantfollows the formulation of [28] to accommodate premixed turbulent reacting flows.To a large extent, the applicability of LEM depends on the homogeneity of the turbulence and the de-gree of which the problem can be characterised in one dimension within the simulation domain. This sug-gests that the method may fail where there is significant coupling between turbulence non-homogeneityat the small scales and the flow field. For many problems in premixed combustion, the spatial scaleswhere reactions take place tend to be of the order of the laminar flame thickness, and in many cases,experiments show that at such small scales of turbulence, its characteristics have decayed to conditionsthat are reasonably isotropic and uniform.2.2 LEM Simulation MethodsA global, six-step mechanism by Chang et al. [29] designed to simulate premixed methane-air combus-tion is implemented to generate the flame profiles. The six global reaction rates have been reduced byapproximately 8% to achieve the correct unstrained laminar flame propagation speed observed in the ref-erence solution from Cantera’s GRI MECH 3.0 [30] calculation. This extra calibration procedure ensuresincreased accuracy for our value of equivalence ratio of 0.731.All of the species properties are calculated using CHEMKIN-II [31], including specific heats, diffu-sion coefficients, thermal conductivities and enthalpies. The thermodynamic coefficients are based onthe CHEMKIN Thermodynamic Data Base [32]. Figure 1 illustrates the flame solutions from Canteraand the Chang mechanism. Mixture-averaged transport is adapted to reduce computational time. Theinflow mixture was set to atmospheric pressure and 294 K at an equivalence ratio of 0.73. The calculatedlaminar flame speed, SL, under these conditions is 0.214 m/s, and the calculated laminar flame thickness,δ f , is 588 µm. The overall flame thickness, propagation speed, equilibrium temperature and mass frac-tions of major and minor species are sufficiently well matched between the reduced Chang and full GRImechanisms.The LEM simulations are conducted with a mixed first order upwind and second order centeredspatial scheme. Explicit time steps are taken in order to cope with the stochastic nature of the model. Theinstantaneous temperature profiles are recorded at regular time intervals on the order of 1% of a large eddyturnover time for the freely propagating flames. Some 3,500 to 16,000 temperature profiles are stored forpost-simulation construction of the LEM PDF models. The exact number of profiles required to create astatistically converged PDF model is dependent primarily on the turbulence intensity of the flame. Tenprototype turbulent premixed flames with various integral length scales and turbulent Reynolds numbersare tested. The flow parameters are varied from case to case such that the prototype flames adhere topredetermined locations on the Borghi diagram, as illustrated in Figure 2. The LEM parameters used tosimulate the freely propagating flames at the prescribed turbulent conditions are summarised in Table 1.1Reduced mechanisms are typically designed to operate over a range of thermodynamic conditions. However, due to amore restricted degree of freedom from the reduced number of steps, the level of accuracy may not be identical over the rangeof operations. For this numerical study, we are primarily interested at premixed flames with φ = 0.73; thus, we optimize thereduced mechanism for this particular equivalence ratio.·10−3(Minor)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1020406080100120140Velocity(cm/s)x (cm)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120040060080010001200140016001800Temperature(K)x (cm)·10−3(g/cm3)CH4H2OCO2COH2HOHODensity TemperatureFigure 1. Premixed laminar flame solution at an equivalence ratio of 0.73 for the 6-step global mechanism(line) and Cantera [33] (symbols).10−1 100 101 102 10310−1100101102103u′/sLl0/δflaminarflamesbrokenreactionzones4b3a 3b 3c2a 2b 2c1a 1cRe= 1lk = 0.01δflk = δf1bFigure 2. Borghi diagram showing locations of the ten prototype LEM flames (1a to 4b). The LEMtest cases are represented by ’◦’. The experimental flames are represented by triangles: SwB1 (’⊳’),SwB2 (’▽’), and SwB3 (’⊲’). It has been found that increasing the integral length beyond an order ofmagnitude above the laminar flame thickness while holding the turbulent fluctuations constant does notsignificantly alter the PDF profiles for the LEM simulations.Throughout the paper, we use a temperature-based reaction progress variable:c =T −T0Te(φ)−T0, (4)where Te(φ) is the equilibrium temperature at the equivalence ratio φ and T0 is the initial temperature.Case Ret l0 (cm) lk (cm) u′/SL Domain (cm) Min. cells/cm Cells used1a 5 0.02992 8.95·10−3 9.826 1.0 670 1,0001b 50 0.2992 1.59·10−2 9.826 1.0 337 1,0001c 500 2.992 2.83·10−2 9.826 4.0 318 1,2722a 50 0.02992 1.59·10−3 98.26 1.0 3,771 3,7722b 500 0.2992 2.83·10−3 98.26 1.0 2,121 2,1222c 5000 2.992 5.03·10−3 98.26 4.0 1,193 4,7723a 500 0.02992 2.83·10−4 982.6 1.0 21,204 21,2043b 5000 0.2992 5.03·10−4 982.6 1.0 11,924 11,9243c 50000 2.992 8.95·10−4 982.6 4.0 6,706 26,8244b 50000 0.2992 8.95·10−5 9826 1.0 67,053 67,054Table 1. Relevant LEM simulation parameters: l0 and lk are the integral and Kolmogorov scales. Otherconstant parameters are invariant between the cases, including δ f , SL,Cλ and Nη , which are respectively,588 µm, 0.214 cm/s, 15.0 and 1.0. A minimum of 1,000 computational cells are used for each simulation.From the simulations or experiments, it is possible to construct a PDF with the following form,P(c∗;x, t)≈ P(c∗; c¯, c¯′2), (5)where c¯ and c¯′2are the mean and variance of the progress variable, and c∗ is the discretized variable rep-resenting the continuous c space. In another words, the PDF at any space and time within the simulationdomain can be approximated by the mean and variance of the distribution. Such a formulation is com-patible with several current models for the turbulence and chemistry interactions for turbulent premixedcombustion [1–4]. A detailed description of the LEM PDF formulation can be found in [11], while abrief overview of the construction of the LEM PDF is provided in Section 4. Furthermore, we define anormalised variance, c¯′2n, determined by the limiting variance of a perfectly segregated mixture of hot andcold gases as:c¯′2n =c¯′2c¯(1− c¯). (6)This formulation is more suitable for the discrete tabulation of the PDF table, as the boundary valuesconveniently vary between zero and unity.3 Experimental Conditions: Stratified Swirl BurnerThe flames and experimental techniques have been described in previous papers [16, 19, 34], so themethods are summarised briefly here. A turbulent flame is stabilised on a bluff body, with reactants fedthrough two concentric streams at specified inner and outer equivalence ratios (φ ). In the present paper,only premixed cases are considered. Swirl can be added to the flame by splitting the outer stream througha tangential inlet.The global equivalence ratio for the flame is nominally 0.75 (measured via species is 0.73 +/- 0.018),and the range of local equivalence ratios spans 0.375-1.125. Scalar data obtained fromRayleigh/Raman/CO-LIF line measurements at 103 µm resolution allows the behaviour of key combustion species – CH4, CO2,CO, H2, H2O and O2 – to be probed within the instantaneous flame front. Simultaneous cross-planar OH-PLIF is used to determine the orientation of the instantaneous flame normal in the scalar measurementwindow, allowing real gradients of the temperature and hence the progress variable to be obtained.The operating conditions considered in the present analysis are listed in Table 2. Measurements weretaken at six heights along the centerline of the flame, by collecting data in 6 mm linear segments. Thenotation SwBNz is used to denote cases at different locations, where N is the case number and z is theaxial coordinate. We are considering three premixed flames under different swirl conditions.Case SFR(1) SN(2) u′/S(3)LSwB1 0 0 10.2SwB2 0.25 0.26 12.2SwB3 0.33 0.45 14.3Table 2. Operating Conditions: (1) SFR = ratio of split flow to swirlers to total flow, (2) SN = measuredswirl number, ratio of tangential to axial momentum, (3) Maximum total u′/SL at the midpoint of theflame brush at z = 30 mm [20, 35].The bulk velocity in the outer annulus,Uo = 18.7 m/s, is set at more than twice the value of the velocityin the inner annulus, Ui = 8.3 m/s, so as to generate substantial levels of shear, and thus turbulence,between the two flows. Co-flow air was supplied around the outer annulus with a bulk velocity Uco =0.4 m/s to provide well-characterised boundary conditions. The Reynolds numbers derived from the bulkvelocities at the exit geometry are Rei = 5,960 for the inner flow and Reo = 11,500 for the outer flow.Multi-scalar laser diagnostics were applied at the Turbulent Combustion Laboratory in Sandia Na-tional Laboratories, and extensively described in [16, 19, 34]. The diagnostics setup allows for the linemeasurement of temperature (Rayleigh scattering) and major species (Raman scattering and CO-LIF) at103 µm projected pixel resolution with simultaneous cross planar OH-PLIF at 48 µm projected pixelresolution. Signal to noise ratios are of the order of 150 for temperatures, and about 60 for equivalenceratio, with estimated accuracies of 2% and 5%, respectively. As the optical resolution of the Raman-Rayleigh-LIF measurements is smaller than the spatial sampling rate, the resolution of the temperatureand major species measurements is limited by the sampling resolution (103 µm) and the laser beamdiameter (0.22 mm, 1/e2). The optical resolution (1/e2) of the OH-PLIF measurement is between 98and 144 µm and therefore the spatial resolution of the OH-PLIF measurements is limited by the opticalresolution rather than the sampling resolution. The OH-PLIF cross planar technique allows the flamenormal in 3D to be assessed relatively to the measurement line, thus allowing the real gradients to beobtained by projection. Radial profiles were obtained by moving the burner horizontally in 4 mm steps,producing overlapping steps in the relative position of the 6 mm wide measurement window, with 300laser shots taken at each step. Radial profiles were taken at axial increments of 10 mm above the burnerexit to capture changes in flame structure with axial distance. Substantially larger datasets (5,000-30,000shots) were taken at the intersection of the mean flame brush and the mixing layer for stratified flamesonly. Here we confine ourselves to the premixed cases only, within 2.5% of the nominal value of 0.75.4 Construction of PDF Models4.1 LEM Flame ProfilesFigure 3 illustrates a few selected LEM temperature profiles for two prototype flames of varying turbu-lence intensities. It can been seen that the influence of the triplet maps become more pronounced andmodify a larger portion of the flame as the turbulent Reynolds number increases. Consequently, the turbu-lent temperature profiles become more distinguishable from the laminar counterpart, with larger effectiveflame thicknesses. These perturbations propagate through the flame and reduce in strength until theyeither blend into the flame profile or get transported out of the domain.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1200400600800100012001400160018002000Temperature(K)x (cm)(a) Case 2b (Ret = 500).0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1200400600800100012001400160018002000Temperature(K)x (cm)(b) Case 4b (Ret = 50,000).Figure 3. Characteristic LEM temperature profiles of two prototype flames at different Ret . The individ-ual profiles on each graph are separated by at least one large eddy turnover time.At first inspection, it appears that the average temperature gradients should increase with the numberof triplet maps implemented per unit time; however, the diffusion mechanism from the evolutionaryequations quickly diminishes the sharp discontinuities introduced in the slope of the temperature field.As a consequence, the flame begins to broaden as it recovers from this perturbed state. The net effect is todecrease the overall conditional averages of the gradients when considered across the entire width of theflame. The temperature profiles depicted in Figure 3 can then be transformed into the progress variablespace via Equation 4.4.2 LEM PDF ConstructionThe method used to construct the LEM PDF models is first discussed in the context of the modifiedlaminar flamelet PDF, which is then extended to the LEM formulation. The relevant parameters governingthe behavior of the distributions are the mean (c¯) and variance (c¯′2), as previously mentioned by Eq. 5.Truncations of the profiles in the spatial domain are applied to the one-dimensional flame profiles, leadingto changes in both c¯ and c¯′2. The mean of the distribution increases for every point removed from theunburnt mixture (c = 0) boundary, whist the opposite is true for points removed from the burnt mixture(c = 1) boundary. Moreover, the variance decreases for every point removed from either boundary.Figure 4(a) demonstrates an example of these truncations. The symbols correspond to the truncationlimits, (x1,c1) and (x2,c2), where only the cells within the interval are retained. Arbitrary values of meanand variance are set to c¯ = 0.50 and c¯′2n = 0.39 for illustrative purposes. The modified laminar flameletPDFs with the prescribed means and variances can then be constructed with the truncated flame profilesaccording to methods described by previous work [1]. An example of the modifield laminar flameletPDF of c¯ = 0.50 and c¯′2n = 0.39 is shown in Figure 4(b). Crucially, it can be deduced that there exists oneunique modified laminar flamelet PDF for every mean and variance combination.0 0.2 0.4 0.6 0.8 (cm)(x1, c1)(x2, c2)(a) Truncated laminar flame profile0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9∗)c∗(b) Modified laminar flamelet PDFFigure 4. (a) (x1,c1) and (x2,c2) mark the truncation limits; only the cells within the interval are retained.(b) The modified laminar flamelet PDF with c¯ = 0.50 and c¯′2n = 0.39 constructed using the truncated flameprofile is shown.The LEM PDFs are constructed in a similar manner compared to the modified laminar flamelet PDFs.Instead of one unique, steady, laminar temperature profile, the LEM generates transient temperatureprofiles; however, each LEM temperature profile is equally valid and must be equally weighted in anensemble average to arrive at the final PDF model. Consequently, the truncation strategy prescribed forthe modified laminar flamelet profiles to obtain the desired mean and variance is applied to each LEMtemperature profile. Here, when a (x1, x2) pair is fixed in real space, averaging together the many LEMrealizations leads to a particular mean and variance of progress variable. In principle then, one couldvary the positions of x1 and x2 for each temperature profile until the desired mean and variance for thePDF is achieved. In practice, all possible discrete (x1, x2) combinations are used to populate our PDFlookup table, which is stored as a function of the mean and variance. During subsequent RANS and LEScalculations, this table is called and interpolated as needed to obtain the PDF for any required mean andvariance combination.Figure 5(a) illustrates two truncated LEM profiles that correspond to PDFs with c¯ = 0.5 and c¯′2n =0.33, similar to the laminar flame shown in Figure 4(a). The truncation positions are shifted for individualtemperature profiles because the flame changes with time. The final PDF model is acquired by averaginga number of PDFs built from such truncated LEM flames. Figure 5(b) illustrates an example of theconverged solution constructed by averaging 10,000 LEM PDFs at a relatively low turbulent Reynoldsnumber. It is apparent the LEM PDF displays peaks of lower magnitude at each boundary than themodified laminar flamelet PDF model. This smoothing effect is a direct consequence of the applicationof triplet maps to the temperature profiles. The mapping introduces variations in the temperature gradientsat a given temperature (Figure 3(b)); as a result, the probability of the flame existing at any given statevaries with different temperature profiles. When averaged, the PDF decreases to zero at each boundarymore gradually [11].0 0.2 0.4 0.6 0.8 (cm)(a) Truncated instantaneous LEM flame profiles0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9∗)c∗(b) Modified laminar flamelet and LEM PDFsFigure 5. (a) The portions to be retained are within the intervals delimited by the circles. The fourtruncation positions are selected such that the resultant PDF would have c¯ = 0.5 and c¯′2n = 0.39. Thetruncation boundaries are different for each temperature profile because of the transient effects. (b) Themodified laminar flamelet (dash) and LEM (solid) PDFs of similar mean and variance are shown.5 Results5.1 Probability Density FunctionThe LEM PDF models spanning a range of distribution means and variances are illustrated by Figure 6.This study reveals that the characteristics of the PDF models for all ten prototype flames display remark-able similarities across the scope of test scenarios despite a three-order change of magnitude in turbulentfluctuation intensity and length scale on the Borghi diagram. The PDFs tend to become slightly lessbivariate as the turbulence intensities increase. This effect is particularly noticeable for cases with meansand normalised variances having values close to 0.5.The PDF distributions captured at various axial positions for the three experimental flames (SwB1-3)operating at different swirling conditions are illustrated by Figures 7-9. The LEM PDFs tabulated as afunction of c∗ from one particular prototype flame has been superimposed for direct comparison betweenthe model and the swirl burner data2. In general, a high degree of similarity can be observed betweenthe experimental and LEM PDFs in overall features, magnitudes and peak positions. The 3 × 5 panelscorrespond to different distances from the stabilisation point of the flame and radial positions across theflame brush, which are characterised by various combinations of c¯ and c¯′2n.2 LEM prototype flame 1b is selected for this demonstration. However, it is evident from Figure 6 that the location on theBorghi diagram does not significantly influence the resulting LEM PDF for a given variance and mean combination.∗)|c¯=0.11c¯′2n= 0.25 c¯′2n= 0.49 c¯′2n= 0.7500.10.20.3P(c∗)|c¯=0.3100.10.20.3P(c∗)|c¯=0.5100.10.20.3P(c∗)|c¯=0.710 0.2 0.4 0.6 0.8∗)|c¯=0.910.2 0.4 0.6 0.8 0c∗0.2 0.4 0.6 0.8 1Figure 6. PDF models at various c¯ and c¯′2n are shown; each row represents one value of c¯ and threevalues of c¯′2n (ranging from left to right: 0.25, 0.49 and 0.75). Vertical and horizontal axis on each graphrepresent the probability and the progress variable, respectively. Solid: case 4b, dash: case 3b, dash dot:case 2b, dot: case 1b (notation is in accordance with Figure 2).Starting with SwB1 (Figure 7), the model PDFs are well matched for all normalized variance valuesbelow a distribution mean of approximately 0.7. At low c¯ (left columns), we are at the reactant edge of theflame brush, and the PDF peaks at c∗ = 0. This agrees with the calculated LEM PDFs, but the measuredPDFs are somewhat wider than the calculations suggest. At the intermediate values of c¯, we find twolow peaks, one at zero and one around 0.8-0.9, which are well captured by the LEM model, even at lowvariance levels. Closer to the product end of the flame brush (right columns), the PDF peak appears notat unity, but at slightly lower values, between 0.8 and 0.9, much like the experimental values. The peakat lower values than full reaction progress is attributed to the limited time for completion of reaction dueto turbulent mixing of cold and hot gases. However, heat transfer to the base can also contribute to thelowering of the extent of reaction.Proceeding to the first swirl flame (Figure 8), it can be seen that the LEM and experimental PDFsshare as many similarities as with the no swirl case. The model and experiment agree fairly well for mostvalues of variances at a distribution mean of less than approximately 0.7. The experimental PDFs tendto be slightly less bivariate in comparison to the no swirl case, reflected by the decreasing distributionvariance. An interesting difference can be observed for the results having c¯ ≈ 0.7 and c¯′2n < 0.4. Theprobability of c∗ continues to be non-zero towards the 0 boundary for the experimental PDFs whereasthe LEM PDFs tend to go to zero. This could suggest that the swirling flow may prolong the time duringwhich the flame spends in the preheat layer, effectively increasing the probability of finding the flame ata low c∗ state and decreasing the gradient of the progress variable around this region of the flame.For the SwB3 flame (Figure 9), the LEM and experimental PDFs are typically well matched fordistributions with 0.3 < c¯ < 0.7. For PDFs with low values of c¯ (c¯ ≈ 0.11), it can be seen that theexperimental PDF peaks have lower magnitudes and in some cases, are shifted towards a non-zero c∗value (c∗ = 0.05). More so, the distributions appear to be slightly wider than the LEM counterparts. Thissuggests that increasing the swirl number of the flame may have greater impact on the colder regions ofthe flame, where the conditional mean is low – such a behavior is not apparent in the SwB1 and SwB2cases. Overall, the PDFs for all cases, with and without swirl, are still captured for the correspondingcombination of means and variances. However, the agreement is generally better for the intermediatevalues of c¯ than for the extremes.∗)c′2n = 0.618c ≈ 0.11c′2n = 0.664c ≈ 0.31c′2n = 0.652c ≈ 0.51c′2n = 0.609c ≈ 0.71z=60mmc′2n = 0.479c ≈ 0.9100.10.20.3P(c∗)c′2n = 0.560 c′2n = 0.586 c′2n = 0.562 c′2n = 0.509z=30mmc′2n = 0.2680 0.2 0.4 0.6 0.8∗)c′2n = 0.2700.2 0.4 0.6 0.8 0c′2n = 0.3450.2 0.4 0.6 0.8 0c∗c′2n = 0.3460.2 0.4 0.6 0.8 0c′2n = 0.2870.2 0.4 0.6 0.8 1z=10mmc′2n = 0.024Figure 7. PDFs measured from the SwB1 flame at various axial locations conditioned by differentdistribution means, c¯ (solid). The corresponding LEM PDFs of similar c¯ and c¯′2n are superimposed (dash).∗)c′2n = 0.489c ≈ 0.11c′2n = 0.623c ≈ 0.31c′2n = 0.613c ≈ 0.51c′2n = 0.562c ≈ 0.71z=60mmc′2n = 0.449c ≈ 0.9100.10.20.3P(c∗)c′2n = 0.573 c′2n = 0.593 c′2n = 0.563 c′2n = 0.509z=30mmc′2n = 0.2730 0.2 0.4 0.6 0.8∗)c′2n = 0.3900.2 0.4 0.6 0.8 0c′2n = 0.4350.2 0.4 0.6 0.8 0c∗c′2n = 0.4120.2 0.4 0.6 0.8 0c′2n = 0.3410.2 0.4 0.6 0.8 1z=10mmc′2n = 0.112Figure 8. PDFs measured from the SwB2 flame at various axial locations conditioned by differentdistribution means, c¯ (solid). The corresponding LEM PDFs of similar c¯ and c¯′2n are superimposed (dash).∗)c′2n = 0.394c ≈ 0.11c′2n = 0.500c ≈ 0.31c′2n = 0.578c ≈ 0.51c′2n = 0.581c ≈ 0.71z=60mmc′2n = 0.367c ≈ 0.9100.10.20.3P(c∗)c′2n = 0.496 c′2n = 0.567 c′2n = 0.585 c′2n = 0.556z=30mmc′2n = 0.3540 0.2 0.4 0.6 0.8∗)c′2n = 0.4500.2 0.4 0.6 0.8 0c′2n = 0.4980.2 0.4 0.6 0.8 0c∗c′2n = 0.4890.2 0.4 0.6 0.8 0c′2n = 0.4430.2 0.4 0.6 0.8 1z=10mmc′2n = 0.317Figure 9. PDFs measured from the SwB3 flame at various axial locations, conditioned by differentdistribution means, c¯ (solid). The corresponding LEM PDFs of similar c¯ and c¯′2n are superimposed (dash).5.2 Scalar Dissipation RateA relationship can be established between the conditional scalar dissipation rate (χc|c∗) and the PDF ofthe reaction progress variable (P(c∗)) [14],χc =∫ 10(χc|c∗)P(c∗) dc∗, (7)where χc is the unconditional mean scalar dissipation rate, or simply the mean scalar dissipation rate.Equation 7 is particularly interesting to our analysis because the LEM temperature profiles can be used toconstruct both χc|c∗ and P(c∗). A detailed description of the methodology used in constructing the PDFand SDR models can be found in [11].There are two observable behaviors from the conditionally averaged SDR models from the LEM.First, as the characteristic flame properties are shifted upwards in the Borghi diagram (increasing u′/SL),the peak magnitudes of the dissipation rates tend to decrease. Second, if the characteristic flame propertiesare shifted rightward in the Borghi diagram (increasing l0/δ f ), the peak magnitudes of the SDRs tend toincrease towards the limit of a laminar flame. It is difficult to determine from the current simulation resultswhether the flame broadening (increasing u′/SL) or the integral scale increment (increasing l0/δ f ) effectis dominant within a specific combustion regime. It is clear, however, that the processes will competewith one another in general, leading to relatively unchanged conditionally averaged SDR distributionsfor the special case of isotropic, homogeneous flames.For a given chemical mechanism and set of LEM parameters, there should be one pseudo-invariantconditional scalar dissipation rate (cSDR) averaged from the temperature profile datasets. This cSDRcan then be convolved with the companion PDF to arrive at the unconditional mean SDR via Equation 7.To reduce simulation time during practical implementation for complex combustion models, the valuesof the unconditional mean SDR can be pre-tabulated at desired combinations of means and variances ofthe reaction progress variable for efficient retrievals. The same LEM datasets from the PDF study can beused to generate one conditionally averaged SDR model for each of the ten prototype flames. The resultsare presented in Figure 10.0123χc|c∗×δf/SL  1a (5)2a (50)3a (500)  1a (5)1b (50)1c (500)0123χc|c∗×δf/SL  1b (50)2b (500)3b (5000)4b (50000)  2a (50)2b (500)2c (5000)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10123c∗χc|c∗×δf/SL  1c (500)2c (5000)3c (50000)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1c∗  3a (500)3b (5000)3c (50000)Figure 10. Non-dimensionalized conditional average of the SDR models from the ten prototype flames.The left and right columns illustrate changes in the turbulent fluctuations and integral length scales,respectively. The turbulent Reynolds number of each case is recorded in the parentheses. The laminarcase is displayed in light gray as a reference.Figure 11 illustrates the pseudo-invariant conditional SDR distributions from the experiments. Theconditional SDR typically decreases with increasing swirl and appears to be most similar to the laminarlimit at z = 30 mm. This axial location roughly coincides with the intersection of the mixing and shearlayers, promoting the development of a highly turbulent and dissipative region, after which the SDRquickly diminish. The comparison with the LEM model is striking: in the model, there is little sensitivityto the local turbulence characteristics, whereas the experiments indicate that there are significant changesin the SDR values; the magnitude tends to be more sensitive to the effects of swirl. Nevertheless, theshape of the distributions are well captured, with the maximum in the vicinity of c∗ = 0.65 to 0.75. Themaximum generally moves rightward, towards c∗ = 0.75, as the SDR magnitude decreases.00.511.522.53z = 10 mmSwB1z = 20 mm z = 30 mm z = 40 mm z = 50 mm z = 60 mmχc|c∗×δf/SL00.511.522.53SwB2χc|c∗×δf/SL0 0.2 0.4 0.6 0.8 000.511.522.53SwB30.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0c∗0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1χc|c∗×δf/SLFigure 11. Non-dimensionalized conditional average of the SDR from the three experimental flames.Each column represents a unique axial position from the flame stabilisation point while each row rep-resents one swirling condition. The error bars indicate +/- one standard deviation from the mean. Thelaminar case is displayed in light gray as a reference.As mentioned in the Introduction, several models for turbulent combustion relate the local rate of mi-cromixing to the conditional scalar dissipation. A common approach to modelling scalar dissipation [36–38] is to extract the functional dependence on the conditioning variable and model the conditional scalardissipation as,χ|c∗= χ0× f (c∗), (8)where χ0 is then calculated by some other means and is not a function of the conditioning variable. Asusual, the integration of f (c∗) over the progress variable space results in the value of unity. Here, weperform the same decomposition to our results to see if the underlying functional dependence of scalardissipation on the conditioning variable changes – that is, to test whether or not such a decompositionmight make sense.Figure 12 depicts the functional dependence of scalar dissipation ( f (c∗)) at three axial positions fromthe flame stabilisation point. This figure confirms the observation that the cSDRs tend to shift rightwardwith increasing swirl intensity, particularly at the beginning and end of the flame brush, correspondingto the axial positions of 10 mm and 60 mm. The reason for such changes in the shape of the normalisedcSDR is not immediately clear from the current results; rather than speculate on the possible causes, weleave this question open for future work. In general, the LEM results tend to remain close to that of thelaminar flame as the model is unable to emulate the effects of swirl. Furthermore, Figure 13 illustratesthe values of the normalization constant, χ0, in relation to the axial position for the experimental flames.It is interesting that the maximum values of χ0 for all of the experimental flames occur at the intersectionof the mixing and shear layers near z = 30 mm.0 0.2 0.4 0.6 0.8 000.511.522.53c∗χ|c∗/χ010 mm0.2 0.4 0.6 0.8 0c∗30 mm0.2 0.4 0.6 0.8 1c∗60 mmFigure 12. Functional dependence of the scalar dissipation ( f (c∗)) at three axial positions downstreamfrom the flame stabilisation point (10, 30 and 60 mm) with the LEM results superimposed. The experi-mental flames are represented by symbols: SwB1 (’+’), SwB2 (’’), and SwB3 (’◦’). The LEM resultis represented by the solid line. The laminar case is displayed in light gray as a reference.10 20 30 40 50 6000. (mm)χ0=∫1 0χc|c∗dc∗×δ f/SLFigure 13. Values of χ0 in relation to the distance downstream of the flame stabilisation point. Theexperimental flames are represented by symbols: SwB1 (’+’), SwB2 (’’), and SwB3 (’◦’). The LEM(black line) and laminar flame (gray line) results are not dependent on the axial position.The convolution of the pseudo-invariant conditional SDR with the appropriate PDF distribution viaEquation 7 provides the solution for the unconditional mean SDR, χc, or simply the mean SDR. Themodelling of this term remains as one of the final challenges in applying some combustion models topremixed flame calculations [13]. Figure 14(a) illustrates the behavior of the mean SDR as a functionof the mean and variance of the progress variable as calcuated by LEM. The mean SDR peaks around c¯= 0.5 and c¯′2n ≈ 0 as the PDF and cSDR distributions have coinciding maxima for these values of c¯ andc¯′2n. The experimental values can be seen in Figure 14(b), where the magnitudes of the mean SDR areevidently lower than the predicted values. The LEM predictions are best when the swirl intensity of theburner is at its lowest. This suggests that swirling flows induce a decrease in the magnitude of the SDR,both conditional and unconditional, which cannot be replicated by a one dimensional turbulence model.Moreover, recalling the PDF results from the previous section, one can conclude that swirl plays a moreprominent role in the determination of the SDR than the PDF distributions.¯′2nc¯χ¯c(a) LEM00.250.50.751c¯′2 n=0.300.250.50.75c¯′2 n=0.4χ¯c00.250.50.75c¯′2 n=0.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9¯c¯′2 n=0.6(b) Experiments and LEMFigure 14. Non-dimensionalized unconditional mean scalar dissipation rates (χ¯c × δ f /SL) as predictedby (a) LEM and (b) experiments with LEM solutions superimposed at various combinations of c¯ and c¯′2n.Solid: LEM; dash: SwB1; dash-dot: SwB2; dot: SwB3.6 Discussion6.1 Probability Density FunctionThe LEM formulated PDF models display minimal variations in distribution shape and magnitude withchanging turbulent fluctuations and integral lengths. Typically, the PDF distributions become slightly lessbivariate with increasing turbulent fluctuations. Towards the reactant boundary (c∗ = 0), the turbulentfluctuations appear to increase the diffusive mixing in the preheat layer, leading to slightly faster flamedevelopment in this region. In turn, this causes a decrease in the duration of the flame residing at lowvalues of c∗. Towards the products boundary (c∗ = 1), the model suggests that the eddy interactionsimpede the flame from fully oxidizing towards the end of the reaction. This does not imply a reductionin the flame speed, but rather a barrier to reaching full chemical equilibrium. This effect can be observedin Figure 6, where the PDF models constructed from flames with higher turbulent fluctuations have therightward boundaries shifted slightly towards the left. The combination of these phenomena causes theLEM PDF to deviate from the bivariate distribution seen in laminar PDF models as turbulence intensityincreases, in agreement with the experimental results. The model of course neglects any potential heattransfer issues, which are present in the experiments, and may influence the approach to equilibrium atthe base of the flame.The swirl burner results (Figures 7-9) reveal that the shape of the PDFs is dependent on the distancefrom the base of the flame and secondarily on the amount of swirl in the flame. The dependence ondistance from the base arises from the different turbulence characteristics across the flame brush as theflow evolves downstream. The variance increases with distance from the base, and with swirl, and thisdirectly affects the corresponding PDFs. The experimental PDFs are perhaps surprisingly well capturedby the two input parameters of c¯ and c¯′2n. Nevertheless, once the variance and mean are determined, thereare only so many sensible ways for the PDF to be arranged within the LEM context, so perhaps thisjustifies the good agreement.For future implementations of the LEM PDF, where a pre-processing operation is used to populate aPDF lookup table, the data suggests that variations in the turbulent Reynolds number and integral lengthsinduce a rather modest effect on the shape of the PDF. It appears that it may not be necessary to addan additional dimension related to the turbulence intensity to the lookup table. Indeed, while it wouldbe ideal to construct a table using representative conditions, using a model generated from a flame withlower turbulence intensities (which is the less computationally intensive option) may suffice. The numberof temperature profiles required to construct converged LEM PDF models can be markedly reduced whilethe additional error contribution remains small. This is especially true for premixed flames with turbulentReynolds numbers and integral lengths within an order of magnitude of the targeted flame on the Borghidiagram. Further discretion may be required when applying different PDF models for flames exposedto more intense swirling conditions. The experimental results reveal that the overall flame gradients andhence the flame surface density function reduce in magnitude under swirl. However, such changes maynot greatly alter the shape of the PDF of the progress variable because of the normalization procedurerequired in the construction of the LEM PDF models.6.2 Scalar Dissipation RateAs previously mentioned, there are two observable behaviors from the conditionally averaged SDRscomputed by the LEM. First, as the intensity of turbulence increases, the peak magnitudes of the meandissipation rates tend to decrease. Such an effect has been previously observed in DNS studies and isattributed to flame broadening caused by the decrease in peak temperature gradients [39]. This effect isemulated by the LEM via the interactions between the turbulent eddies and the scalar fields. These smalleddies penetrate into the reaction zone, increasing the local gradients of the scalar fields. This is followedby the dissipation of the sharp discontinuities introduced in the slope of the scalar fields due to the coupleddiffusion mechanism. Effectively, the flame is broadened, increasing its thickness and decreasing thegradients. Second, as the turbulent length scale increases at constant turbulent fluctuation intensity, thepeak magnitudes of the SDRs tend in increase towards the limit of a laminar flame3. Such an effectcould be connected to the increasing Kolmogorov length associated with the integral scale increment. Anincrease in the Kolmogorov length means the smallest eddies will become too large to effectively changethe local structure of the flame. In another words, the eddies have more difficulty penetrating the preheatlayer and the reaction zone, thus reducing the interaction between the vortices and the flame. With asmaller number of interactions, the pseudo-turbulent SDR distributions will become more similar to thelaminar SDR distribution.It is important to emphasise that the current numerical results do not include the effect of strain.Additional modelling parameters would need to be introduced to the transport equations for such calcu-lations. Previous studies have shown that regions under highly positive strain are correlated with regionsof increased scalar dissipation, though this is not an exclusive association [40, 41]. For turbulent flamesexposed to high strain rates, neglecting this effect may lead to non-negligible changes to both the PDFand SDR distributions.The pseudo-invariant conditional SDR distributions and mean SDR values from the experimentalflames show a more pronouced decrease in magnitude with increasing swirl effects. It appears that theintersection of the mixing and shear layers near z = 30 mm leads to quickly diminishing gradients of cfor positions downstream of this location. This suggests that additional parameters which quantify thedissipative effects of mixing and swirling may be necessary in order to accurately model the conditionalSDR distributions. Moreover, the entrainment of cold air into the premixed flame caused by the increasein fluid motion may induce changes in the temperature based reaction progress variable that the LEMcannot currently simulate. It is, however, curious that the PDF distributions from both the LEM and ex-periments reveal far less susceptibility to changes when exposed to physical effects, such as entrainmentof air and swirl, in comparison to the conditional SDR distributions.Nevertheless, the quantity that requires closure in combustion models is the unconditional mean SDR,defined by Equation 7. In this regard, it appears that the LEM can reproduce values of χc well within50% of the experimental equivalent, but the accuracy depends primarily on the swirl number of theflow. For most points, the LEM over-estimates the value of χc, which is understandable considering the3 For unstrained flows, the maximum conditionally averaged SDR distribution corresponds to the laminar flame with thesame chemistry; any turbulence interactions decrease the magnitude of the SDR distribution with respect to the laminar limit.conditional SDR distributions from the model indicate higher magnitudes compared to the experiments.Further research could explore the possibility of including additional modelling parameters into the LEMformulated conditional SDR model according to physical effects governing the flow field. In this case,perhaps a post-processing correction to account for the entrainment of air induced by the swirl. This couldamount to implementing a simple correction factor to the magnitude of the cSDR, which is calculated asa function of the local anisotropy.7 ConclusionThis study investigates the effects of variation in swirl and turbulence intensity on the probability dis-tribution of the reaction progress variable for a series of globally lean, turbulent premixed flames, bothexperimentally and numerically.The PDF models constructed from the LEM simulations indicate that increasing turbulence intensityhas a rather modest impact on the distributions. The minor changes are only observed at medium values(approximately 0.5) of means and normalized variances. The PDFs from the experimental flames showa bivariate distribution for all cases, with peaks in the unburned and burned regions. The thermal flamethickness is generally larger than the unstrained (and strained) laminar flame thickness. The data thereforesuggests that the thermal gradient is smeared by turbulent diffusion at scales on the order of the flamethickness. The LEM PDF model demonstrates good agreement with the experimental results in termsof nominal changes with turbulence. Particularly, it is able to capture the overall shape and the effect ofsmoothing towards the left and right boundaries of the PDF distributions.The LEM results suggest that the conditional SDR would decrease towards the upper regions ofthe Borghi diagram and would increase in peak magnitudes towards the unstrained laminar limit withincreasing integral scales. The experimental conditional SDR for all axial locations of the flame brushshows a greater decrease in magnitude than predicted by the numerical model, though this could bepartially caused by an interaction between shearing and mixing layers within a specific region of the swirlflames, leading to greater dissipation downstream of this location. The values for the unconditional meanSDR predicted by LEM are typically within 50% of the experimental values. Moreover, the accuracy ofthe numerical model seems to be dependent on the swirl number of the flow.Overall, the results point towards the idea that the PDF for premixed flames remains relatively steadyunder a variety of turbulent conditions, including changes in the integral length, velocity fluctuation andswirl number. As a consequence, it appears to be practical to use a representative pseudo-turbulent PDFmodel for a range of turbulent conditions. Future studies could focus on understanding the physical inter-pretation behind the relative invariance in the PDF model with variations in both the turbulent fluctuationsand integral length scales. Another possible direction is to determine if this invariance also extends toother fuels in the premixed combustion regime. Swirl appears to have greater influence on the local flamegradients; this, in turn, leads to more substantial changes in the SDR, which the model presented herecannot capture in its current form. Perhaps a practical solution in the near future would be to implementa correction factor to the LEM formulated SDR models based on the local anisotropy of the flow field.8 AcknowledgementsThe authors wish to express their gratitude to the Natural Science and Engineering Research Council ofCanada for partial funding of the numerical work described herein. The measurements at Sandia Na-tional Labs were sponsored by the United States Department of Energy, Office of Basic Energy Sciences,Division of Chemical Sciences, Geosciences and Biosciences. We thank Dr. Robert Barlow of San-dia National Laboratories for allowing the use of the database acquired at his laboratory, and for usefulcomments on the initial draft manuscript. M. 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