Modelling of turbulent lifted jet flames using flamelets: a priori assessment and a posteriori validation

This study focuses on the modelling of turbulent lifted jet flames using flamelets and a presumed Probability Density Function (PDF) approach with interest in both flame lift-off height and flame brush structure. First, flamelet models used to capture contributions from premixed and non-premixed modes of the partially premixed combustion in the lifted jet flame are assessed using a Direct Numerical Simulation (DNS) data for a turbulent lifted hydrogen jet flame. The joint PDFs of mixture fraction Z and progress variable c, including their statistical correlation, are obtained using a copula method, which is also validated using the DNS data. The statistically independent PDFs are found to be generally inadequate to represent the joint PDFs from the DNS data. The effects of Z–c correlation and the contribution from the non-premixed combustion mode on the flame lift-off height are studied systematically by including one effect at a time in the simulations used for a posteriori validation. A simple model including the effects of chemical kinetics and scalar dissipation rate is suggested and used for non-premixed combustion contributions. The results clearly show that both Z–c correlation and non-premixed combustion effects are required in the premixed flamelets approach to get good agreement with the measured flame lift-off heights as a function of jet velocity. The flame brush structure reported in earlier experimental studies is also captured reasonably well for various axial positions. It seems that flame stabilisation is influenced by both premixed and non-premixed combustion modes, and their mutual influences.     

Problems of predicting turbulent burning rates

Different approaches to the modelling of turbulent combustion first are reviewed briefly. A unified, stretched flamelet approach then is presented. With Reynolds stress modelling and a generalized probability density function (PDF) of strain rate, it enables a source term, in the form of a probability of burning function, P_b, to be expressed as a function of Markstein numbers and the Karlovitz stretch factor. When P_b is combined with some turbulent flame fractal considerations, an expression is obtained for the turbulent burning velocity. When it is combined with the profile of the unstretched laminar flame volumetric heat release rate plotted against the reaction progress variable and the PDF of the latter, an expression is obtained for the mean volumetric turbulent heat release rate. Through these relationships experimental values of turbulent burning velocity might be used to evaluate P_b and hence the CFD source term, the mean volumetric heat release rate.

Different theoretical expressions for the turbulent burning velocity, including the present one, are compared with experimental measurements. The differences between these are discussed and this is followed by a review of CFD applications of these flamelet concepts to premixed and non-premixed combustion. The various assumptions made in the course of the analyses are scrutinized in the light of recent direct numerical simulations of turbulent flames and the applications to the flames of laser diagnostics. Remaining problem areas include a sufficiently general combination of strain rate and flame curvature PDFs to give a single PDF of flame stretch rate, the nature of flame quenching under positive and negative stretch rates, flame responses to changing stretch rates and the effects of flame instabilities.     

Flame front analysis of high-pressure turbulent lean premixed methane–air flames

An experimental study on lean turbulent premixed methane–air flames at high pressure is conducted by using a turbulent Bunsen flame configuration. A single equivalence ratio flame at Φ = 0.6 is explored for pressures ranging from atmospheric pressure to 0.9 MPa. LDA measurements of the cold flow indicate that turbulence intensities and the integral length scale are not sensitive to pressure. Due to the decreased kinematic viscosity with increasing pressure, the turbulent Reynolds numbers increase, and isotropic turbulence scaling relations indicate a large decrease of the smallest turbulence scales. Available experimental results and PREMIX code computations indicate a decrease in laminar flame propagation velocities with increasing pressure, essentially between the atmospheric pressure and 0.5 MPa. The u′/S_L ratio increases therefore accordingly. Instantaneous flame images are obtained by Mie scattering tomography. The images and their analysis show that pressure increase generates small scale flame structures. In an attempt to generalize these results, the variance of the flamelet curvatures, the standard deviation of the flamelet orientation angle, and the flamelet crossing lengths have been plotted against which is proportional to the ratio between the integral and Taylor length scales, and which increases with pressure. These three parameters vary linearly with the ratio between large and small turbulence scales and clearly indicate the strong effect of this parameter on premixed turbulent flame dynamics and structure. An obvious consequence is the increase in flame surface density and hence burning rate with pressure, as confirmed by its direct determination from 2D tomographic images.     

Effects of Lewis number,density ratio and gravity on burning velocity and conditional statistics in stagnating turbulent premixed flames

DNS is performed to analyse the effects of Lewis number (Le), density ratio and gravity in stagnating turbulent premixed flames. The results show good agreement with those of Lee and Huh (Combustion and Flame, Vol. 159, 2012, pp. 1576–1591) with respect to the turbulent burning velocity, S_T, in terms of turbulent diffusivity, flamelet thickness, mean curvature and displacement speed at the leading edge. In all four stagnating flames studied, a mean tangential strain rate resulting in a mean flamelet thickness smaller than the unstretched laminar flame thickness leads to an increase in S_T. A flame cusp of positive curvature involves a superadiabatic burned gas temperature due to diffusive–thermal instability for an Le less than unity. Wrinkling tends to be suppressed at a larger density ratio, not enhanced by hydrodynamic instability, in the stagnating flow configuration. Turbulence is produced, resulting in highly anisotropic turbulence with heavier unburned gas accelerating through a flame brush by Rayleigh–Taylor instability. Results are also provided on brush thickness, flame surface density and conditional velocities in burned and unburned gas and on flame surfaces to represent the internal brush structures for all four test flames.     

LES/PDF for premixed combustion in the DNS limit

We investigated the behaviour of the composition probability density function (PDF) model equations used in a large-eddy simulation (LES) of turbulent combustion in the direct numerical simulation (DNS) limit; that is, in the limit of the LES resolution length scale Δ (and the numerical mesh spacing h) being small compared to the smallest flow length scale, so that the resolution is sufficient to perform a DNS. The correct behaviour of a PDF model in the DNS limit is that the resolved composition fields satisfy the DNS equations, and there are no residual fluctuations (i.e. the PDF is everywhere a delta function). In the DNS limit, the treatment of molecular diffusion in the PDF equations is crucial, and both the ‘random-walk’ and ‘mean-drift’ models for molecular diffusion are investigated. Two test cases are considered, both of premixed laminar flames (of thickness δ_L). We examine the solutions of the model PDF equations for these test cases as functions of Δ/δ_L and h/δ_L. Each of the two PDF models has advantages and disadvantages. The mean-drift model behaves correctly in the DNS limit, but it is more difficult to implement and computationally more expensive. The random-walk model does not have the correct behaviour in the DNS limit in that it produces non-zero residual fluctuations. However, if the specified mixing rate Ω normalised by the reaction timescale τ_c is sufficiently large (Ωτ_c ? 1), then the residual fluctuations are less than 10% and the observed flame speed and thickness are close to their laminar values. Away from the DNS limit (i.e. h/δ_L ? 1), the observed flame thickness scales with the mesh spacing h, and the flame speed scales with Ωh. For this case it is possible to construct a non-general specification of the mixing rate Ω such that the flame speed matches the laminar flame speed.     

Conditional moment closure and transient flamelet modelling for detailed structure and NO x formation characteristics of turbulent nonpremixed jet and recirculating flames

This study has been mainly motivated to assess computationally and theoretically the conditional moment closure (CMC) model and the transient flamelet model for the simulation of turbulent nonpremixed flames. These two turbulent combustion models are implemented into the unstructured grid finite volume method that efficiently handles physically and geometrically complex turbulent reacting flows. Moreover, the parallel algorithm has been implemented to improve computational efficiency as well as to reduce the memory load of the CMC procedure. Example cases include two turbulent CO/H₂/N₂ jet flames having different flow timescales and the turbulent nonpremixed H₂/CO flame stabilized on an axisymmetric bluff-body burner. The Lagrangian flamelet model and the simplified CMC formulation are applied to the strongly parabolic jet flame calculation. On the other hand, the Eulerian particle flamelet model and full conservative CMC formulation are employed for the bluff-body flame with flow recirculation. Based on the numerical results, a detailed discussion is given for the comparative performances of the two combustion models in terms of the flame structure and NO _x formation characteristics.     

Anisotropic Reynolds stress tensor representation in shear flows using DNS and experimental data

Tensorial decompositions and projections are used to study the performance of algebraic non-linear models and predict the anisotropy of the Reynolds stresses. Direct numerical simulation (DNS) data for plane channel flows at friction Reynolds number (Re_τ = 180, 395, 590, 1000), and for the boundary layer using both DNS (Re_τ = 359, 830, 1271) and experimental data (Re_τ = 2680, 3891, 4941, 7164) are used to build and evaluate the models. These data are projected into tensorial basis formed from the symmetric part of mean velocity gradient and non-persistence-of-straining tensor. Six models are proposed and their performances are investigated. The scalar coefficients for these six different levels of approximations of the Reynolds stress tensor are derived, and made dimensionless using the classical turbulent scales, the kinetic turbulent energy (κ) and its dissipation rate (ε). The dimensionless coefficients are then coupled with classical wall functions. One model is selected by comparing the predicted Reynolds stress components with experimental and DNS data, presenting a good prediction for the shear and normal Reynolds stresses.     

Fractal modelling of turbulent combustion

In a previous paper we proposed a new model for turbulent flows, called the fractal model (FM), which is applicable both to RANS and LES formulations. Here, the model is extended to the reactive case with the goal of simulating turbulent flames, both premixed and non-premixed.

FM is a subgrid model that describes the physics of the small scales of turbulence building on the phenomenological concept of vortex cascade and on fractal theory. The physics of the small scales is summarized by a turbulent ‘viscosity’ μ_t, to be added to the molecular one. μ_t is zero where the flow is laminar and, in particular, goes to zero at solid walls.

The fundamental assumption in treating combustion in this work is that chemical reactions take place only at the dissipative scales of turbulence, i.e. near the so-called ‘fine structures’ (the eddy dissipation concept). FM predicts the growth of dissipative scales due to heat release; therefore, it enables a local DNS in the hot regions of the flow where the dissipative scale may grow up to the cell dimension. FM can also estimate the volume fraction γ* occupied by the ‘fine structures’; this quantity is critical for modelling the reaction rate, and therefore the source terms in the energy and species equations. FM can also estimate the local surface of the reactive ‘fine structures’, that is, the local flame front area. It also takes into account, although in approximate manner, the formation of islands of unburnt mixture. In this paper, the model (in the isotropic formulation (IFM)) is used in conjunction with a time-dependent LES (but with the limitations of an isotropic model) approach and is validated through a three-dimensional axisymmetric diffusion flame studied experimentally by Correa and Gulati and numerically by many researchers. The time-dependent results obtained are in good agreement with the experiments. Moreover, the IFM solution offers a possible explanation for the stabilization process of this flame, which undergoes local stretching of the order of 46 000 s^?1.     

Prediction of NO in turbulent diffusion flames using Eulerian particle flamelet model

The combustion characteristics for the turbulent diffusion flames using the unsteady flamelet concept have been numerically investigated. The Favre-averaged Navier–Stokes equations are solved by a finite volume method of SIMPLE type that incorporates the laminar flamelet concept with a modified k ? ε turbulence model. The NO formation is estimated by solving the Eulerian particle transport equations in a postprocessing mode. Two test problems are considered: CH₄/H₂/N₂ jet flame and CH₄/H₂ stabilised bluff body flame. The temperature and species profiles are well captured by the flamelet model. Two different chemical mechanisms (GRI 2.11 and 3.0) give nearly identical results for temperature and species except NO. The GRI 3.0 gives significantly higher NO levels compared to the GRI 2.11. This is mainly attributed to the difference in NO formation by the prompt mechanism. The NO formation is sensitive to the number of flamelet particles. The NO levels for two test flames do not change when the flamelet particle number exceeds six.     

Investigation of lengthscales, scalar dissipation, and flame orientation in a piloted diffusion flame by LES

This work investigates the structure of a diffusion flame in terms of lengthscales, scalar dissipation, and flame orientation by using large eddy simulation. This has been performed for a turbulent, non-premixed, piloted methane/air jet flame (Flame D) at a Reynolds-number of 22,400. A steady flamelet model, which was represented by artificial neural networks, yields species mass fractions, density, and viscosity as a function of the mixture fraction. This will be shown to suffice to simulate such flames. To allow to examine scalar dissipation, a grid of 1.97 × 10⁶ nodes was applied that resolves more than 75% of the turbulent kinetic energy. The accuracy of the results is assessed by varying the grid-resolution and by comparison to experimental data by Barlow, Frank, Karpetis, Schneider (Sandia, Darmstadt), and others. The numerical procedure solves the filtered, incompressible transport equations for mass, momentum, and mixture fraction. For subgrid closure, an eddy viscosity/diffusivity approach is applied, relying on the dynamic Germano model. Artificial turbulent inflow velocities were generated to feature proper one- and two-point statistics. The results obtained for both the one- and two-point statistics were found in good agreement to the experimental data. The PDF of the flame orientation shows the tilting of the flame fronts towards the centerline. Finally, the steady flamelet approach was found to be sufficient for this type of flame unless slowly reacting species are of interest.     

Vorticity,backscatter and counter-gradient transport predictions using two-level simulation of turbulent flows

The two-level simulation (TLS) method evolves both the large-and the small-scale fields in a two-scale approach and has shown good predictive capabilities in both isotropic and wall-bounded high Reynolds number (Re) turbulent flows in the past. Sensitivity and ability of this modelling approach to predict fundamental features (such as backscatter, counter-gradient turbulent transport, small-scale vorticity, etc.) seen in high Re turbulent flows is assessed here by using two direct numerical simulation (DNS) datasets corresponding to a forced isotropic turbulence at Taylor’s microscale-based Reynolds number Re_λ ≈ 433 and a fully developed turbulent flow in a periodic channel at friction Reynolds number Re_τ ≈ 1000. It is shown that TLS captures the dynamics of local co-/counter-gradient transport and backscatter at the requisite scales of interest. These observations are further confirmed through a posteriori investigation of the flow in a periodic channel at Re_τ = 2000. The results reveal that the TLS method can capture both the large- and the small-scale flow physics in a consistent manner, and at a reduced overall cost when compared to the estimated DNS or wall-resolved LES cost.     

Modelling compression ignition engines by incorporation of the flamelet generated manifolds combustion closure

Tabulated chemistry models allow to include detailed chemistry effects at low cost in numerical simulations of reactive flows. Characteristics of the reactive fluid flows are described by a reduced set of parameters that are representative of the flame structure at small scales so-called flamelets. For a specific turbulent combustion configuration, flamelet combustion closure, with proper formulation of the flame structure can be applied. In this study, flamelet generated manifolds (FGM) combustion closure with progress variable approach were incorporated with OpenFOAM® source code to model combustion within compression ignition engines. For IC engine applications, multi-dimensional flamelet look-up tables for counter flow diffusive flame configuration were generated. Source terms of non-premixed combustion configuration in flamelet domain were tabulated based on pressure, temperature of unburned mixture, mixture fraction, and progress variable. A new frozen flamelet method was introduced to link one dimensional reaction diffusion space to multi-dimensional Computational Fluid Dynamics (CFD) physical space to fulfill correct modelling of thermal state of the engine at expansion stroke when charge composition was changed after combustion and reaction rates were subsided. Predictability of the developed numerical framework were evaluated for Sandia Spray A (constant volume vessel), Spray B (light duty optical Diesel engine), and a heavy duty Diesel engine experiments under Reynolds averaged Navier Stokes turbulence formulation. Results showed that application of multi-dimensional FGM combustion closure can comprehensively predict key parameters such as: ignition delay, in-cylinder pressure, apparent heat release rate, flame lift-off , and flame structure in Diesel engines.     

Investigating asymptotic suction boundary layers using a one-dimensional stochastic turbulence model

The turbulent asymptotic suction boundary layer is studied using a one-dimensional turbulence (ODT) model. ODT is a fully resolved, unsteady stochastic simulation technique. While flow properties reside on a one-dimensional domain, turbulent advection is represented using mapping events whose occurrences are governed by a random process. Due to its reduced spatial dimensionality, ODT achieves major cost reductions compared to three-dimensional (3D) simulations. A comparison to recent direct numerical simulation (DNS) data at moderate Reynolds number (Re = u_∞ / v₀ = 333, where u_∞ and v₀ are the free stream and suction velocity, respectively) suggests that the ODT model is capable of reproducing several velocity statistics, i.e. mean velocity and turbulent kinetic energy budgets, while peak turbulent stresses are under-estimated by ODT. Variation of the Reynolds number in the range Re ∈ [333,400,500,1000] shows that ODT can reproduce various trends observed as a result of increased suction in turbulent asymptotic suction boundary layers, i.e. the reduction of Reynolds stresses and enhanced skin friction. While up to Re = 500 our results can be directly compared to recent LES data, the simulation at Re = 1000 is currently not feasible through full 3D simulations, hence ODT may assist the design of future DNS or LES simulations at larger Reynolds numbers.     

Investigation of conditional source-term estimation applied to a non-premixed turbulent flame

A turbulent combustion model, Conditional Source-term Estimation (CSE) is applied to a non-premixed turbulent jet methane flame. The conditional chemical source terms are determined on the basis of first order closure and the conditional averaged species concentrations are obtained by inverting an integral equation. The Tikhonov method is implemented for regularisation. Detailed chemistry is tabulated using the trajectory generated low-dimensional manifold method. Radiation due to the gaseous species is included. Reynolds Averaged Navier–Stokes calculations are performed using two different turbulence models. The objectives of the paper are (i) assessment of the impact of the main numerical parameters in CSE and (ii) comparison of the CSE numerical predictions with available experimental data and results from previous simulations for the selected flame. The number of CSE domains and the number of points in each CSE domain are shown to have a significant impact on the results if not selected appropriately. The present CSE calculations always converge to unique and stable predictions. The corrected k–ε model yields mixture fraction profiles in good agreement with the experimental data values for axial locations in the first half of the flame. Farther downstream, the RNG k–ε model performs better. Overall, the current predictions for the mixture fraction are in good agreement with the experimental data. The predicted temperatures using CSE and the k–ε turbulence model with a modified value of C_ε1 = 1.47 are found to be in very good agreement with the experimental data. Further, the current CSE results are of comparable quality with previous simulations using the flamelet model and conditional moment closure. Future work may include further investigation on optimal determination of the regularisation parameter and alternative regularisation techniques, soot modelling within the CSE formulation, and improved formulation of radiation.     

Inner–outer interactions in rough-wall turbulence

Here we revisit the inner–outer interaction model (IOIM) of Marusic et al. (Science, vol. 329, 2010, pp. 193–196) that enables the prediction of statistics of the fluctuating streamwise velocity in the inner region of wall-bounded turbulent flows from a large-scale velocity signature measured in the outer region of the flow. The model is characterised by two empirically observed inner–outer interactions: superposition of energy from outer region large-scale motions; and amplitude modulation by these large-scale motions of a small-scale ‘universal’ signal (u^*), which in smooth-wall flows is Reynolds number invariant. In the present study, the inner–outer interactions in rough-wall turbulent boundary layers are examined within the framework of the IOIM. Simultaneous two-point hot-wire anemometry measurements enable quantification, via the model parameters, of the strengths of superposition and amplitude modulation effects in a rough-wall flow, and these are compared to a smooth-wall flow. It is shown that the present rough-wall significantly reduces the effects of superposition, while increasing the amplitude modulation effect. The former is true even in flows that exhibit outer region similarity. Using the model parameters obtained from the two-point measurements, predictions of inner region streamwise velocity statistics and spectra are compared to measurements over a range of friction and roughness Reynolds numbers. These results indicate that the u^* signal does depend on roughness Reynolds number (k⁺_s), but is robust to changes in friction Reynolds number (δ⁺). Additionally, the superposition strength is shown to be relatively independent of both roughness and friction Reynolds number. The implications of the present results on the suitability of the IOIM as a predictive tool in rough-wall turbulence are discussed.     

A priori investigation of subgrid correlation of mixture fraction and progress variable in partially premixed flames

Subgrid correlation of mixture fraction, Z, and progress variable, c, is investigated using direct numerical dimulation (DNS) data of a hydrogen lifted jet flame. Joint subgrid behaviour of these two scalars are obtained using a Gaussian-type filter for a broad range of filter sizes. A joint probability density function (JPDF) constructed using single-snapshot DNS data is compared qualitatively with that computed using two independent β-PDFs and a copula method. Strong negative correlation observed at different streamwise locations in the flame is captured well by the copula method. The subgrid contribution to the Z–c correlation becomes important if the filter is of the size of the laminar flame thickness or larger. A priori assessment for the filtered reaction rate using the flamelet approach with independent β-PDFs and correlated JPDF is then performed. Comparison with the DNS data shows that both models provide reasonably good results for a range of filter sizes. However, the reaction rate computed using copula JPDF is found to have a better agreement with the DNS data for large filter sizes because the subgrid Z–c correlation effect is included.     

Constrained large-eddy simulation of supersonic turbulent boundary layer over a compression ramp

A supersonic turbulent boundary layer over a compression ramp is numerically investigated using the constrained large-eddy simulation (CLES) method. The compression corner is characterised by a deflection angle of 24°. The free-stream Mach number is Ma_∞ = 2.9, and the Reynolds number based on the momentum thickness of inlet boundary layer is Re_θ = 2300. The mean and statistical quantities, such as mean velocity, wall pressure and Reynolds stresses, are thoroughly analysed and compared with those from traditional large-eddy simulation (LES), experimental measurement and direct numerical simulation (DNS). It turns out that CLES can predict the friction coefficient, wall-pressure distribution, size of separation bubble, Reynolds stresses, etc. more accurately than traditional LES, and the results are in reasonable agreement with the experimental and/or DNS data. Also discussed are the effects of specific parameterisations of the Reynolds constraint and interfacial positions separating the constrained and unconstrained regions on the performance of the CLES method.     

Assessment of dynamic closure for premixed combustion large eddy simulation

Turbulent piloted Bunsen flames of stoichiometric methane–air mixtures are computed using the large eddy simulation (LES) paradigm involving an algebraic closure for the filtered reaction rate. This closure involves the filtered scalar dissipation rate of a reaction progress variable. The model for this dissipation rate involves a parameter β_c representing the flame front curvature effects induced by turbulence, chemical reactions, molecular dissipation, and their interactions at the sub-grid level, suggesting that this parameter may vary with filter width or be a scale-dependent. Thus, it would be ideal to evaluate this parameter dynamically by LES. A procedure for this evaluation is discussed and assessed using direct numerical simulation (DNS) data and LES calculations. The probability density functions of β_c obtained from the DNS and LES calculations are very similar when the turbulent Reynolds number is sufficiently large and when the filter width normalised by the laminar flame thermal thickness is larger than unity. Results obtained using a constant (static) value for this parameter are also used for comparative evaluation. Detailed discussion presented in this paper suggests that the dynamic procedure works well and physical insights and reasonings are provided to explain the observed behaviour.     

DNS assessment of relation between mean reaction and scalar dissipation rates in the flamelet regime of premixed turbulent combustion

The linear relation between the mean rate of product creation and the mean scalar dissipation rate, derived in the seminal paper by K.N.C. Bray [‘The interaction between turbulence and combustion’, Proceedings of the Combustion Institute, Vol. 17 (1979), pp. 223–233], is the cornerstone for models of premixed turbulent combustion that deal with the dissipation rate in order to close the reaction rate. In the present work, this linear relation is straightforwardly validated by analysing data computed earlier in the 3D Direct Numerical Simulation (DNS) of three statistically stationary, 1D, planar turbulent flames associated with the flamelet regime of premixed combustion. Although the linear relation does not hold at the leading and trailing edges of the mean flame brush, such a result is expected within the framework of Bray's theory. However, the present DNS yields substantially larger (smaller) values of an input parameter c_m (or K₂ = 1/(2c_m ? 1)), involved by the studied linear relation, when compared to the commonly used value of c_m = 0.7 (or K₂ = 2.5). To gain further insight into the issue and into the eventual dependence of c_m on mixture composition, the DNS data are combined with the results of numerical simulations of stationary, 1D, planar laminar methane–air flames with complex chemistry, with the results being reported in terms of differently defined combustion progress variables c, i.e. the normalised temperature, density, or mole fraction of CH₄, O₂, CO₂ or H₂O. Such a study indicates the dependence of c_m both on the definition of c and on the equivalence ratio. Nevertheless, K₂ and c_m can be estimated by processing the results of simulations of counterpart laminar premixed flames. Similar conclusions were also drawn by skipping the DNS data, but invoking a presumed beta probability density function in order to evaluate c_m for the differently defined c's and various equivalence ratios.     

Prediction of local extinction and re-ignition effects in non-premixed turbulent combustion using a flamelet/progress variable approach

The flamelet/progress variable approach (FPVA) has been proposed by Pierce and Moin as a model for turbulent non-premixed combustion in large-eddy simulation. The filtered chemical source term in this model appears in unclosed form, and is modeled by a presumed probability density function (PDF) for the joint PDF of the mixture fraction Z and a flamelet parameter λ. While the marginal PDF of Z can be reasonably approximated by a beta distribution, a model for the conditional PDF of the flamelet parameter needs to be developed. Further, the ability of FPVA to predict extinction and re-ignition has also not been assessed. In this paper, we address these aspects of the model using the DNS database of Sripakagorn et al. It is first shown that the steady flamelet assumption in the context of FPVA leads to good predictions even for high levels of local extinction. Three different models for the conditional PDF of the flamelet parameter are tested in an a priori sense. Results obtained using a delta function to model the conditional PDF of λ lead to an overprediction of the mean temperature, even with only moderate extinction levels. It is shown that if the conditional PDF of λ is modeled by a beta distribution conditioned on Z, then FPVA can predict extinction and re-ignition effects, and good agreement between the model and DNS data for the mean temperature is observed.