Flame Surface Density Transport Research Articles

Modelling of the Flame Surface Density Transport During Flame-Wall Interaction of Premixed Flames within Turbulent Boundary Layers

ABSTRACT A priori Direct Numerical Simulation (DNS) analysis of the modelling of the generalised flame surface density (FSD) transport equation in the framework of Reynolds Averaged Navier-Stokes (RANS) simulations has been conducted with a focus on flame-wall interaction (FWI) within turbulent boundary layers. The analysis was performed using two different scenarios: one involving the unsteady head-on interaction of a statistically planar premixed flame as it propagates through a turbulent boundary layer, and the other addressing statistically stationary oblique-wall interaction of a V-shaped premixed flame within a turbulent channel flow. Flame-wall interaction has been observed to exert a significant influence on the statistical behavior of the terms in the FSD transport equation. Throughout all phases of flame-wall interaction in both configurations, the primary source and sink terms in the FSD transport equation are consistently associated with the tangential strain rate and curvature terms. The relative importance of the contributions of the propagation and turbulent transport terms in the FSD transport equation have been found to be influenced by the flame-wall interaction configuration. Existing models for the tangential strain rate term in the FSD transport equation have been identified as inadequate based on a priori DNS assessment. Hence, adjustments to these models have been proposed to address the impact of flame orientation and near-wall effects specifically in the context of flame-wall interaction within turbulent boundary layers. The alignment between the flame normal vector and the wall normal vector has been identified as a crucial factor in modelling the contributions of unresolved dilatation rate and unresolved normal strain rate to the FSD transport equation. New models for these terms have been demonstrated to exhibit satisfactory agreement with the corresponding terms extracted from DNS data.

Turbulence Effects on the Statistical Behaviour and Modelling of Flame Surface Density and the Terms of Its Transport Equation in Turbulent Premixed Flames

The influence of the ratio of integral length scale to flame thickness on the statistical behaviours of flame surface density (FSD) and its transport has been analysed using a Direct Numerical Simulation database of three-dimensional statistically planar turbulent premixed flames for different turbulence intensities. It has been found that turbulent burning velocity based on volume-integration of reaction rate and flame surface area increase but the peak magnitudes of the FSD and the terms of the FSD transport term decrease with an increase in length scale ratio for a given turbulence intensity. The flame brush thickness and flame wrinkling increase with an increase in length scale ratio for all turbulence intensities. However, the qualitative behaviours of the unclosed terms in the FSD transport equation remain unaltered by the length scale ratio and in all cases the tangential strain rate term and the curvature term act as leading order source and sink, respectively. A decrease in length scale ratio for a given turbulence intensity leads to a decrease in Damköhler number and an increase in Karlovitz number. This has an implication on the alignment of reactive scalar gradient with local strain rate eigenvectors, which in turn increases positive contribution of the tangential strain rate term with a decrease in length scale ratio. Moreover, an increase in Karlovitz number increases the likelihood of negative contribution of the curvature term. Thus, the magnitude of the negative contribution of the FSD curvature term increases with a decrease in length scale ratio for a given turbulence intensity. The model for the tangential strain rate term, which explicitly considers the scalar gradient alignment with local principal strain rate eigenvectors, has been shown to be more successful than the models that do not account for the scalar gradient alignment characteristics. Moreover, the existing model for the curvature and propagation term needed modification to account for greater likelihood of negative values for higher Karlovitz number. However, the models for the unclosed flux of FSD and the mean reaction rate closure are not significantly affected by the length scale ratio.

Displacement speed, flame surface density and burning rate in highly turbulent premixed flames characterized by low Lewis numbers

Direct numerical simulation data obtained from four pairs of turbulent, lean hydrogen–air, complex-chemistry flames are analysed to explore the influence of molecular diffusion on flame surface density, displacement speed $S_d$ and the flame surface density transport equation terms. Each pair involves (i) a flame where mixture-averaged molecular diffusivities are adopted and Lewis number $Le$ is significantly less than unity and (ii) an equidiffusive flame where all molecular diffusivities are set equal to molecular heat diffusivity of the mixture and $Le=1$ , with other things being equal. Reported results show that significantly higher turbulent burning rates simulated in the former flames result mainly from an increase in the local fuel consumption rate, whereas an increase in flame surface area plays a secondary role, especially in more intense turbulence. The rate increase stems from (i) an increase in the peak local fuel consumption rate and (ii) an increase in a width of a zone where the rate is significant. The latter phenomenon is of more importance in richer flames and both phenomena are most pronounced in the vicinity of the flame leading edges, thus indicating a crucial role played by the leading edge of a premixed turbulent flame in its propagation. Moreover, mean displacement speed differs significantly from the laminar flame speed even in the equidiffusive flames, varies substantially across flame brush and may be negative at the leading edges of highly turbulent flames.

Effects of Body Forces on the Statistics of Flame Surface Density and Its Evolution in Statistically Planar Turbulent Premixed Flames

Body forces such as buoyancy and externally imposed pressure gradients are expected to have a strong influence on turbulent premixed combustion due to the considerable changes in density between the unburned and fully burned gases. The present work utilises Direct Numerical Simulation data of three-dimensional statistically planar turbulent premixed flames to study the influence of body forces on the statistical behaviour of the flame surface density (FSD) and its evolution within the flame brush. The analysis has been carried out for different turbulence intensities and normalised body force values (i.e., Froude numbers). A positive value of the body force signifies an unstable density stratification (i.e., body force is directed from the heavier unburned gas to the lighter burned gas) and vice versa. It is found that for a given set of turbulence parameters, flame wrinkling increases with an increase in body force magnitude in the unstable configuration. Furthermore, higher magnitudes of body force in the unstable density stratification configuration promote a gradient type transport of turbulent scalar and FSD fluxes, and this tendency weakens in the stable density stratification configuration where a counter-gradient type transport is promoted. The statistical behaviours of the different terms in the FSD transport equation and their closures in the context of Reynolds Averaged Navier–Stokes simulations have been analysed in detail. It has been demonstrated that the effects of body force on the FSD and the terms of its transport equation weakens with increasing turbulence intensity as a result of the diminishing relative strength of body force in comparison to the inertial force. The predictions of the existing models have been assessed with respect to the corresponding terms extracted from the explicitly averaged DNS data, and based on this evaluation, suitable modifications have been made to the existing models to incorporate the effects of body force (or Froude number).

On the validity of Damköhler's second hypothesis in statistically planar turbulent premixed flames in the thin reaction zones regime

The validity of Damköhler's second hypothesis, which was originally proposed for premixed combustion for conditions where the large scale turbulent timescale remains smaller than the chemical timescale and the integral length scale remains smaller than the flame thickness, has been assessed for a range of Damköhler and Karlovitz numbers (i.e. 0.2≤Da≤3.0 and 0.58≤Ka≤33.34) using a three-dimensional DNS database of statistically planar turbulent premixed flames subjected to forced unburned gas turbulence. It has been found that Damköhler's first hypothesis remains valid for all cases considered in this work irrespective of the combustion regime. By contrast, Damköhler's second hypothesis does not hold in the strict sense for the thin reaction zones regime flames with small values of Damköhler number. However, the ratio of the turbulent flame speed normalised by laminar burning velocity and the square root of the ratio of turbulent diffusivity to mass diffusivity converges to a value of the order of unity for the flames with high turbulence intensity and low Damköhler number in the thin reaction zones regime. Under this condition, Damköhler's second hypothesis, in an order of magnitude sense, can be considered to be valid alongside Damköhler's first hypothesis. A scaling analysis has been carried out using the equilibrium of the tangential strain rate and curvature terms in the Flame Surface Density transport equation to demonstrate that Damköhler's second hypothesis can be expected to be valid only in an order of magnitude sense for the thin reaction zones regime flames with small values of Damköhler number.

Flame Surface Density Transport Statistics for High Pressure Turbulent Premixed Bunsen Flames in the Context of Large Eddy Simulation

ABSTRACT The implications of elevated pressure on the statistical behavior of the flame surface density (FSD) transport statistics together with the behavior of selected established sub-models of the unclosed terms of the FSD transport equation have been analyzed in the context of large eddy simulation. For this purpose, five turbulent premixed Bunsen flames have been considered from an existing database. Four of the Bunsen flames are characterized by different pressure levels and are located on the boundary of the wrinkled and the corrugated flamelet (CF) regimes to allow for the possibility and clear identification of combustion instabilities that are often observed at high pressures. The fifth flame is in the thin reaction zones regime and serves as a reference case for the purpose of comparison. For a given filter width to flame thickness ratio, the terms of the FSD transport equation and their models behave in a qualitatively similar manner for different pressure levels. However, as the flame thickness decreases with increasing pressure, it is unlikely that a high pressure flame will be simulated for the same spatial resolution (i.e. LES filter width) normalized by flame thickness as that of an atmospheric flame. It is more likely that the spatial resolution remains constant and in this case, the modeling becomes much more challenging for higher pressures: the magnitudes of the sub-grid contributions increase for larger filter width to flame thickness ratio and thus the accuracy of sub-grid modeling is expected to play a more important role in determining the fidelity of the simulations. The flames considered in this work feature a relatively low ratio of turbulent velocity fluctuations to laminar flame speed, and under these conditions, positive values of the sub-grid curvature term and negative values of the strain rate term are observed toward the leading edge of the flame brush. This behavior cannot be captured by the well-established existing models for the sub-grid curvature term as these model expressions yield deterministically negative values. Similarly, existing models for the tangential strain term yield deterministically positive values. Detailed explanations have been provided for the observed behaviors of the unclosed terms of the FSD transport equations and their respective model predictions.

Insights into the Bending Effect in Premixed Turbulent Combustion Using the Flame Surface Density Transport

ABSTRACTThe bending effect of turbulent flame speed variation (i.e., the deviation from the linear increase of flame speed with increasing root-mean-square turbulent velocity fluctuation) has been investigated based on a Direct Numerical Simulation database of statistically planar turbulent premixed flames propagating into forced unburned gas turbulence. The validity of Damköhler’s first hypothesis has been utilized to analyze the bending effect in terms of generalized Flame Surface Density (FSD) evolution. The volume-integrated value of the tangential strain rate term of the FSD transport equation remains positive, whereas the volume-integrated value of the curvature term assumes negative values. Under statistically stationary state, the positive value of the volume-integrated tangential strain rate term remains in equilibrium with the negative value of the volume-integrated curvature term. It has been found that the contribution of the normal strain rate to the flame surface area remains negative for small turbulence intensities, which eventually become positive for large turbulence intensities. This is a consequence of the change of collinear alignment of the reaction progress variable gradient from the most extensive principal strain rate direction to the direction of the eigenvector associated with the most compressive principal strain rate with increasing turbulence intensity. An increase in turbulence intensity increases the width of the probability density functions of flame curvature, and thereby increases the surface-averaged curvature squared values. This eventually makes the FSD curvature term due to the tangential diffusion component of displacement speed as the major contributor to the negative contribution of the volume-integrated curvature term in the FSD transport equation for large turbulence intensities. However, the negative contribution of the volume-integrated FSD curvature term does not increase indefinitely with increasing turbulence intensity and the inner cut-off scale, which also limits the maximum possible value of the volume-integrated FSD strain rate term under statistically stationary state, governs the maximum possible destruction of flame surface area. It has been argued that the upper limits of the flame surface area generation and destruction are responsible for the bending effects in the variations of turbulent flame speed and flame surface area.

Effects of Reaction Progress Variable Definition on the Flame Surface Density Transport Statistics and Closure for Different Combustion Regimes

ABSTRACTThe implications of the choice of reaction progress variable on the performances of the flame surface density (FSD) based mean reaction rate closure and the well-established sub-models of the FSD transport have been analysed in context of Reynolds Averaged Navier Stokes simulations. For this purpose, a detailed chemistry direct numerical simulation (DNS) database of freely-propagating statistically planar air flames (with an equivalence ratio of 0.7) spanning the corrugated flamelets (CF), thin reaction zones (TRZ) and broken reaction zones (BRZ) regimes of premixed turbulent combustion has been considered. The FSD and the unclosed terms of its transport equation have been analysed for reaction progress variables defined based on normalised and mass fractions and temperature. The performances of the closures for turbulent flux of FSD, and tangential strain rate term have been found to be mostly unaffected by the choice of reaction progress variable. However, the well-established existing models for the unresolved tangential strain rate term have been found not to perform well for the cases representing the CF and TRZ regimes of premixed combustion. The performance of a well-established existing model for the combined propagation and curvature terms has been found to be significantly dependent on the choice of reaction progress variable. Furthermore, the surface-averaged value of the density-weighted displacement speed cannot be approximated by the corresponding unstretched laminar flame value especially for the flames in the BRZ regime. Detailed explanations have been provided for the observed behaviours of the FSD based reaction rate closure and sub-models for the unclosed terms of the FSD transport equation in different combustion regimes for different choices of reaction progress variable.

Generalized flame surface density transport conditional on flow topologies for turbulent H2-air premixed flames in different regimes of combustion

The generalized flame surface density (FSD) transport conditional on local flow topologies in premixed turbulent flames has been analyzed based on a detailed chemistry direct numerical simulation database of statistically planar turbulent hydrogen-air premixed flames with an equivalence ratio of 0.7 representing the corrugated flamelets, thin reaction zones and broken reaction zones regimes of combustion. The local flow topologies have been categorized by the values of the three invariants of the velocity gradient tensor and the statistical behaviors of the generalized FSD and different terms of its transport equation conditional on these flow topologies have been analyzed in detail for different choices of the reaction progress variable. The qualitative behavior of the different terms of the generalized FSD transport equation has been found to be similar for different choices of reaction progress variable but the statistical behaviors of the tangential strain rate term and its components have been found to be affected by the regime of combustion. The topologies, which exist for all values of dilatation rate, contribute significantly to the generalized FSD transport in premixed turbulent flames for all regimes of combustion. An unstable nodal flow topology, which is representative of a counter-flow configuration, has been found to be a dominant contributor to the FSD transport for all regimes of combustion irrespective of the choice of reaction progress variable. Moreover, a focal topology which is obtained only for positive values of dilatation rate, has been found to contribute significantly, especially to the curvature and propagation terms of the FSD transport equation for all regimes of combustion including the broken reaction zones regime. However, the contributions of the flow topologies to the turbulent transport and tangential strain rate term, which are obtained only for positive dilatation rates, have been found to weaken from the corrugated flamelets to the broken reaction zones regime.

Flame surface density based modelling of head-on quenching of turbulent premixed flames

The near-wall behaviour of the generalised flame surface density (FSD) transport in the context of Reynolds Averaged Navier–Stokes (RANS) simulations has been analysed for different values of global Lewis number using three-dimensional Direct Numerical Simulation (DNS) data of head-on quenching of statistically planar turbulent premixed flames by an isothermal inert wall. It has been found that the statistical behaviour of the FSD based reaction rate closure and the terms of the FSD transport equation are significantly affected by the presence of the wall and by the global Lewis number. The near-wall predictions of the standard FSD based mean reaction rate closure and existing sub-models for the unclosed terms of the FSD transport equation have been found to be inadequate based on a-priori DNS assessment, and modifications to these models have been suggested so that the predictions of modified models for reaction rate closure and FSD transport remain satisfactory, both close to the wall and away from it over a wide range of global Lewis number.

Flamelet perturbations and flame surface density transport in weakly turbulent premixed combustion

DNS data obtained under conditions of weak turbulence that are well associated with the flamelet combustion regime are analysed in order (i) to assess the widely-accepted linear relation between the mean mass rate of product creation and the mean Flame Surface Density (FSD) and (ii) to investigate transport of the FSD and the role played by local flamelet perturbations in the FSD transport. While, in line with common expectations, a ratio of is found to be close to the unperturbed laminar flame speed S0L within the largest part of the mean flame brush, this ratio is significantly smaller (larger) than S0L at the leading (trailing) edge of the flame brush. Nevertheless, under the conditions of the present study, this difference in and can be disregarded when computing burning velocity by integrating over the flame brush, provided that is extracted from the DNS data. Even in the case of weak turbulence addressed here, the FSD transport is substantially affected by the difference between local density-weighted displacement speed ρSd/ρu and S0L. This difference is associated with local perturbations of flamelet structure by turbulent eddies, with the local flamelet curvature (strain rate) playing a significantly more (less) important role in the FSD transport under the conditions of the present study. While the difference between ρSd/ρu and S0L in the FSD transport equation can be approximated with a linear function of the local flamelet curvature by processing the DNS data, Markstein lengths associated with such an approximation (i) are scattered, (ii) vary within the mean flame brush, and (iii) differ significantly from the counterpart laminar Markstein length.

Modeling of the Strain Rate Contribution to the Flame Surface Density Transport for Non-Unity Lewis Number Flames in Large Eddy Simulations

The strain rate contribution in the generalized flame surface density (FSD) transport equation remains a leading order unclosed source term, which plays a pivotal role in the modeling of transport for all filter widths in the context of large eddy simulations (LES). To date, most FSD-based closures have been proposed for flames without differential diffusion effects of heat and mass, characterized by a global Lewis number equal to unity (i.e., ). The effects of differential diffusion arising due to non-unity Lewis number on the FSD transport have rarely been analyzed in existing literature. In the present analysis, the statistical behaviors of the strain rate term of the FSD transport equation have been analyzed using a DNS database of freely propagating statistically planar turbulent premixed flames with a global Lewis number ranging from 0.34 to 1.2 (i.e., = 0.34–1.2). The FSD strain rate term has been split into components originating from the gradients of Favre-filtered velocity components (i.e., ), strain rate contribution due to chemical heat release (i.e., ) and sub-grid processes (i.e., ). The contributions of and assume positive values throughout the flame brush for all values of . The performances of the existing models for and have been assessed for flames with different values of global Lewis number . The contribution of remains positive throughout the flame brush for the = 0.6, 0.8, 1.0, and 1.2 flames but the variation of towards the burned gas side of the flame brush for the flame remains qualitatively different in comparison to the other cases considered here. The effects of on the local alignment of reaction progress variable gradient with principal strain rates are responsible for the observed influences of Lewis number on the sub-grid strain rate term . The existing models have been found to be inadequate for the purpose of capturing the qualitative behaviors of the sub-grid strain rate term for flames. Here a new model has been proposed based on a-priori analysis of explicitly filtered DNS data, which has been demonstrated to capture both the qualitative and quantitative behaviors of for all values of for flames with ranging from 0.34 to 1.2.

Modelling of the Curvature Term in the Flame Surface Density Transport Equation: A Direct Numerical Simulations Based Analysis

A simple chemistry based three-dimensional Direct Numerical Simulations (DNS) database of freely propagating statistically planar turbulent premixed flames with a range of different values of Karlovitz number Ka, turbulent Reynolds number Re t, heat release parameter τ and global Lewis number Le has been used for the modelling of the curvature term of the generalised Flame Surface Density (FSD) transport equation in the context of Reynolds Averaged Navier Stokes (RANS) simulations. The curvature term has been split into the contributions arising due to the reaction and normal diffusion components of displacement speed (i.e. T1) and the term arising due to the tangential diffusion component of displacement speed (i.e. T2). Subsequently, the sub-terms (i.e. T1 and T2) of the curvature contribution to the FSD transport have been split into the closed (i.e. T1 r and T2 r) and unclosed (i.e. T1 ur and T2 ur) components. It has been found that T2 remains deterministically negative throughout the flame brush. However, the qualitative behaviour of T1 changes significantly depending upon the values of Ka, Re t and Le. Detailed physical explanations have been provided for the observed behaviours of the components of the curvature term. Moreover, it has been observed that the closed contributions of T1 and T2 (i.e. T1 r and T2 r) remains negligible in comparison to the unclosed contributions (i.e. T1 ur and T2 ur). Suitable model expressions have been identified for T1 ur and T2 ur in the context of RANS simulations, which are shown to perform satisfactorily in all cases considered in the current analysis, accounting for the variations in Ka, Re t, τ and Le.

INVESTIGATION OF FLAME STRETCH IN TURBULENT LIFTED JET FLAME

DNS data of a laboratory-scale turbulent lifted hydrogen jet flame has been analyzed to show that this flame has mixed mode combustion not only at the flame base but also in downstream locations. The mixed mode combustion is observed in instantaneous structures as in earlier studies and in averaged structure, in which the predominant mode is found to be premixed combustion with varying equivalence ratio. The non-premixed combustion in the averaged structure is observed only in a narrow region at the edge of the jet shear layer. The analyzes of flame stretch show large probability for negative flame stretch leading to negative surface averaged flame stretch. The displacement speed-curvature correlation is observed to be negative contributing to the negative flame stretch and partial premixing resulting from jet entrainment acts to reduce the negative correlation. The contribution of turbulent straining to the flame stretch is observed to be negative when the scalar gradient aligns with the most extensive principal strain rate. The physics behind the negative flame stretch resulting from turbulent straining is discussed and elucidated through a simple analysis of the flame surface density transport equation.

Modelling of the Tangential Strain Rate Term in the Flame Surface Density Transport Equation in the Context of Reynolds Averaged Navier Stokes Simulations: A Direct Numerical Simulation Analysis

A direct numerical simulation (DNS) database of freely propagating statistically planar turbulent premixed flames with a range of different values of Karlovitz number Ka, turbulent Reynolds numberRet, heat release parameterτ, and global Lewis number Le has been used to assess the models of the tangential strain rate term in the generalised flame surface density (FSD) transport equation in the context of Reynolds averaged Navier Stokes (RANS) simulations. The tangential strain rate term has been split into contributions arising due to dilatation rateTDand flame normal strain rate (-TN). Subsequently,TDand (-TN) were split into their resolved (i.e.,TD1and (-TN1)) and unresolved (TD2and (-TN2)) components. Detailed physical explanations have been provided for the observed behaviours of the components of the tangential strain rate term. This analysis gave way to the modelling of the unresolved dilatation rate and flame normal strain rate contributions. Models have been identified forTD2and (-TN2) for RANS simulations, which are shown to perform satisfactorily in all cases considered, accounting for the variations in Ka,Ret,τand Le. The performance of the newly proposed models for the FSD strain rate term have been found to be either comparable to or better than the existing models.

A posteriori testing of the flame surface density transport equation for LES

Flame Surface Density (FSD) models for Large Eddy Simulation (LES) are implemented and tested for a canonical configuration and a practical bluff body stabilised burner, comparing common algebraic closures with a transport equation closure in the context of turbulent premixed combustion. The transported method is expected to yield advantages over algebraic closures, as the equilibrium of subgrid production and destruction of FSD is no longer enforced and resolved processes of strain, propagation and curvature are explicitly accounted for. These advantages might have the potential to improve the ability to capture large-scale unsteady flame propagation in situations with combustion instabilities or situations where the flame encounters progressive wrinkling with time. The initial study of a propagating turbulent flame in wind-tunnel turbulence shows that the Algebraic Flame Surface Density (FSDA) method can predict an excessively wrinkled flame under fine grid conditions, potentially increasing the consumption rate of reactants to artificially higher levels. In contrast, the Flame Surface Density Transport (FSDT) closure predicts a smooth flame front and avoids the formation of artificial flame cusps when the grid is refined. Five FSDA models and the FSDT approach are then applied to the LES of the Volvo Rig. The predicted mean velocities are found to be relatively insensitive to the use of the FSDT and FSDA approaches, whereas temperature predictions exhibit appreciable differences for different formulations. The FSDT approach yields very similar temperature predictions to two of the tested FSDA models, quantitatively capturing the mean temperature. Grid refinement is found to improve the FSDT predictions of the mean flame spread. Overall, the paper demonstrates that the apparently complicated FSD transport equation approach can be implemented and applied to realistic, strongly wrinkled flames with good success, and opens up the field for further work to improve the models and the overall FSDT approach.

Two-dimensional direct numerical simulation evaluation of the flame-surface density model for flames developing from an ignition kernel in lean methane/air mixtures under engine conditions

Flame surface density (FSD) models are often used in premixed turbulent combustion modeling to provide closure in large eddy simulations (LES) and Reynolds-averaged simulations. In the present study, data obtained from direct numerical simulation, with reduced chemistry, of flames developing from ignition kernels in lean methane-air mixtures is used to study the accuracy of the FSD modeling terms for LES applications. This study is conducted at elevated temperature and pressure conditions relevant to lean-burn natural gas engines. The closure models for four terms in the FSD transport equation are studied. It is observed that the propagation term as resolved by the LES grid adequately models the actual propagation term without additional sub-grid scale modeling. On the other hand, the closure of the curvature term requires a sub-grid scale model. Two sub-grid scale curvature models are studied and it is seen that there are differences between the actual and the modeled curvature terms during the early evolution of the ignition kernel. The agreement improves as the developing flame approaches a statistically steady-state. The modeling of the sub-grid convection term is analyzed and differences between the actual and the modeled terms are observed to grow during the transient evolution of the ignition kernel. A sub-grid scale model for the closure of the strain rate term shows disagreement at both early and late stages of kernel growth. In general, the sensitivity of the strain rate and curvature models to the filter width is found to be greater than for convection and propagation.

Turbulent Reynolds number dependence of flame surface density transport in the context of Reynolds averaged Navier–Stokes simulations

The influence of turbulent Reynolds number Ret on the statistical behaviour and modelling of flame surface density (FSD) transport has been analysed using a direct numerical simulation (DNS) database of freely propagating turbulent premixed flames with values of Ret ranging from 20 to 100. The variation of Ret is brought about by modifying the Damköhler and Karlovitz numbers independently of each other. The findings indicate that the qualitative behaviour of the various terms of the FSD transport equation remains unaffected by the changes in turbulent Reynolds number, but their relative contributions to the transport of FSD are affected to some extent. The effects of turbulent Reynolds number on the modelling of the turbulent transport term, the tangential strain rate term and the combined propagation and curvature terms are addressed in detail. It is demonstrated that model parameters for the turbulent transport and tangential strain rate term exhibit turbulent Reynolds number dependence for small values of Ret and assume asymptotic values for Ret⩾50. By contrast, the model parameters for the combined propagation and curvature term are found to be insensitive to the variation of turbulent Reynolds number. Existing models for the turbulent transport term and the tangential strain rate term are modified to account for the observed turbulent Reynolds number dependence.

A PDF combustion model for turbulent premixed flames

In this paper we propose a novel model for turbulent premixed flames in the corrugated flamelet regime. The model combines the premixed flamelet approach with a BML ansatz in a PDF framework. For this purpose we introduce a progress variable, which is zero for unburnt and equal to one for reacting and burnt state. The probability that the progress variable switches from zero to one depends on the flame surface density (FSD). The FSD itself is modeled based on a modified flame residence time starting when the progress variable switches to one. To account for flame curvature, collapse and cusp formation, a stretch factor consistent with existing FSD transport equation source terms is employed to modify the flame residence time. To evolve the sensible enthalpy, conditional mixing is applied, which prevents mixing across the flame front. If the progress variable is one, a premixed flamelet is employed. With this approach, some model difficulties related to solving transport equations for the mean FSD and progress variable are overcome. For example, counter-gradient diffusion and FSD transport due to flame propagation appear in closed form. The combination of the BML approach with a flamelet model leads to more accurate temperature predictions, which is demonstrated for the Aachen flames F1–F3.

Modelling of the Curvature Term of the Flame Surface Density Transport Equation for Large Eddy Simulations

A simplified chemistry based three-dimensional Direct Numerical Simulation (DNS) database of freely propagating statistically planar turbulent premixed flames with a range of different values of turbulent Reynolds number has been used for the a priori modelling of the curvature term of the generalised Flame Surface Density (FSD) transport equation in the context of Large Eddy Simulation (LES). The curvature term has been split into the contributions arising due to the reaction and normal diffusion components of displacement speed and the term originating from the tangential diffusion component of displacement speed. Subsequently, these contributions of the curvature term have been split into the resolved and subgrid contributions. New models have been proposed for the subgrid curvature terms arising from the combined reaction and normal diffusion components and the tangential diffusion component of displacement speed. The performances of the new model and the existing models for the subgrid curvature term have been compared with the corresponding quantity extracted from the explicitly filtered DNS data. The new model for the subgrid curvature term is shown to perform satisfactorily in all cases considered in the current study, accounting for wide variations in LES filter size.