Scalar Dissipation Rate Transport Research Articles

A Comparison between Statistical Behaviours of Scalar Dissipation Rate between Homogeneous MILD Combustion and Premixed Turbulent Flames

Three-dimensional Direct Numerical Simulations (DNS) data has been utilised to analyse statistical behaviours of the scalar dissipation rate (SDR) and its transport for homogeneous methane-air mixture turbulent Moderate or Intense Low oxygen Dilution (MILD) combustion for different O2 dilution levels and turbulence intensities for different reaction progress variable definitions. Additional DNS has been conducted for turbulent premixed flames and passive scalar mixing for the purpose of comparison with the SDR statistics of the homogeneous mixture MILD combustion with that in conventional premixed combustion and passive scalar mixing. It has been found that the peak mean value of the scalar dissipation rate decreases with decreasing O2 concentration for MILD combustion cases. Moreover, SDR magnitudes increase with increasing turbulence intensity for both MILD and conventional premixed combustion cases. The profiles and mean values of the scalar dissipation rate conditioned upon the reaction progress variable are found to be sensitive to the choice of the reaction progress variable definition. This behaviour arises due to the differences in the distributions of the species mass fractions within the flame. The strain rate contribution and the molecular dissipation term are found to be the leading order contributors in the scalar dissipation rate transport for MILD combustion; whereas, in conventional premixed flames, the terms rising from density variation and reaction rate gradient also play leading roles in addition to the strain rate and molecular dissipation contributions. By contrast, the terms due to density gradient and reaction rate gradient remain negligible in comparison to the leading order contributors in MILD combustion cases due to small density variation because of moderate temperature rise and small reaction rate gradient magnitudes. Furthermore, the qualitative behaviour of the strain rate contribution to the SDR transport in premixed flames is significantly different to that in the case of MILD combustion and passive scalar mixing. The findings of the current analysis indicate that the scalar dissipation rate statistics in MILD combustion show several qualitative similarities to the passive scalar mixing despite major differences with the SDR transport in conventional turbulent premixed flames. This further suggests that the scalar dissipation rate models, which were originally proposed in the context of passive scalar mixing, have the potential to be applicable for MILD combustion but the models for the premixed turbulent combustion may not be applicable for MILD combustion of homogeneous mixtures.

Spatial characteristics and modelling of mixture fraction variance and scalar dissipation rate in steady turbulent round jets

The spatial characteristics of scalar dissipation rate (SDR) are investigated in a steady-state turbulent round jet using Direct Numerical Simulation (DNS). Radial distribution of mixture fraction is found to agree well with the experimental data. Radial distributions of the turbulent diffusivities and the turbulent Schmidt numbers show different behaviours, based on how these quantities are defined. Point-wise statistics confirm that the log-normal distribution fits the probability distribution functions (PDFs) of the SDR well, except for a small positive skewness attributed to flow anisotropy. Joint PDFs of mixture fraction and SDR are symmetric with respect to mixture fraction. Self-similarity of the mean SDR, as well as its axial, radial and azimuthal components appears after a certain downstream distance. In the self-similar region, centreline SDR scales with downstream distance, x−4. Normalised SDR radial profiles show an increase near the centreline, followed by a steep decrease, after an off-centreline peak. Magnitudes of the axial, radial and azimuthal components of the mean SDR are approximately the same in the self-similar region. The budget analysis of the mean SDR transport equation reveals that the production term by scalar field stretch and the destruction term due to local curvature effects are the two dominant terms. A modification of an existing model for the self-similar mean SDR radial profiles is proposed. This is found to improve the prediction of mean SDR distributions.

Impact of scalar mixing uncertainty on the predictions of reactor-based closures: Application to a lifted methane/air jet flame

This work is devoted to quantify the predictive uncertainty in RANS simulation of a non-premixed lifted flame due to uncertainty in the model parameters of the scalar dissipation rate transport equation. The uncertainty propagation and the global sensitivity analysis of the effect of such parameters on the quantities of interest (QoIs) is performed employing Polynomial Chaos Expansions as surrogate models of the uncertain response. This approach is applied on a lifted methane-air jet flame in vitiated coflow, already experimentally investigated by Cabra et al [1]. The results show the effectiveness of the approach to provide predictions with estimates of uncertainty. It is shown that the the uncertainty in the mixture fraction and temperature is negligible as long as only pure mixing happens, then it becomes significant in the regions where ignition begins, starting from z/D=30. Of the four parameters considered, i.e., CD1, CD2, CP1 and CP2, main and total effect sensitivity indices show that the largest contribution to the uncertainty in the flame temperature is given by the two dissipation parameters CD1 and CD2, while the production parameter CP2 has almost negligible impact on the predictions. Lastly, the surrogate models are used to determine an optimum set of parameters that minimizes the distance with the experimental measures, leading to improved predictions of the QoIs.

Alignment statistics of pressure Hessian with strain rate tensor and reactive scalar gradient in turbulent premixed flames

The relative alignment of the eigenvectors of pressure Hessian with reactive scalar gradient and strain rate eigenvectors in turbulent premixed flames have been analyzed for Karlovitz number values ranging from 0.75 to 126 using a detailed chemistry three-dimensional direct numerical simulations database of H2–air premixed flames. The reactive scalar gradient preferentially aligns with the most extensive strain rate eigendirection for large Damköhler number and small Karlovitz number values, whereas a preferential collinear alignment between the reactive scalar gradient with the most compressive strain rate eigendirection is observed in flames with small Damköhler number and large Karlovitz number. By contrast, the eigenvectors of pressure Hessian do not perfectly align with the reactive scalar gradient, and the net effect of the pressure Hessian on the evolution of the normal strain rate contribution to the scalar dissipation rate transport acts to reduce the scalar gradient in the zone of high dilatation rate. The eigenvectors of pressure Hessian and the strain rate are aligned in such a manner that the contribution of pressure Hessian to the evolution of principal strain rates tends to augment the most extensive principal strain rate for small and moderate values of Karlovitz numbers, whereas this contribution plays an important role for the evolution of the intermediate principal strain rate for large values of Karlovitz number. As the reactive scalar gradient does not align with the intermediate strain rate eigenvector, the influence of pressure Hessian contributions to the scalar–turbulence interaction remains weak for large values of Karlovitz number.

Principal strain rate evolution within turbulent premixed flames for different combustion regimes

The statistical behaviors of the principal strain rates and its evolution in turbulent premixed flames have been analyzed using a three-dimensional direct numerical simulations dataset of statistically planar turbulent premixed flames with different turbulence intensities spanning from the corrugated flamelet regime to the thin reaction zone regime. It has been found that the scalar gradient predominantly aligns collinearly with the most extensive principal strain rate eigendirection within the flame for large values of Damköhler numbers and small values of turbulence intensities and Karlovitz numbers. However, this tendency weakens with the increasing turbulence intensity, which, for a given integral length scale, amounts to a decrease (an increase) in the Damköhler (Karlovitz) number. Moreover, it has been observed that the terms due to molecular diffusion, pressure Hessian, and the correlation between pressure and density gradients play key roles in the evolution of principal strain rates for flames with large values of Damköhler numbers and small values of Karlovitz numbers. However, the relative importance of the terms arising from the correlation between pressure and density gradients and the pressure Hessian relative to the strain rate and vorticity contributions of the principal strain rate transport diminishes with the increasing Karlovitz number and decreasing Damköhler number. The statistical behaviors of the mean values of the terms of the transport equation of the principal strain rate have been explained based on the relative alignments of principal strain rate eigenvectors with vorticity, pressure gradient, and the eigenvectors of the pressure Hessian tensor. The findings of the current analysis suggest that the pressure gradient and pressure Hessian tensor play key roles in the evolution of principal strain rates within premixed turbulent flames, and their influence needs to be accounted for high fidelity modeling of the tangential strain rate and scalar–turbulence interaction terms of the flame surface density and scalar dissipation rate transport equations, respectively. This provides possible explanations for the modification in the alignment of the reactive scalar gradient with local principal strain rates in premixed flames in comparison to that in non-reacting turbulent flows.

Scalar dissipation rate transport conditional on flow topologies in different regimes of premixed turbulent combustion

The Favre-averaged scalar dissipation rate transport conditional on local flow topologies in premixed turbulent flames has been analysed 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 premixed turbulent combustion. The local flow topologies have been categorised by the values of the three invariants of the velocity gradient tensor and the statistical behaviour of the Favre-averaged scalar dissipation rate conditional on these flow topologies has been analysed in detail for different choices of reaction progress variable. The qualitative behaviour of the scalar-turbulence interaction term in the Favre-averaged scalar dissipation rate transport equation has been found to be affected by the regime of combustion, whereas the chemical reaction rate gradient contribution to the scalar dissipation rate transport has been found to be affected by the choice of the reaction progress variable. The topologies, which exist for all values of dilatation rate, contribute significantly to the Favre-averaged scalar dissipation rate transport in premixed turbulent flames for all regimes of combustion. In addition, the flow topologies, which are obtained only for positive values of dilatation rate, contribute significantly to the Favre-averaged scalar dissipation rate transport in the case representing the corrugated flamelets regime combustion. An unstable nodal flow topology, which is representative of a counter-flow configuration, has been found to be a dominant contributor to the Favre-averaged scalar dissipation rate transport for all regimes of combustion irrespective of the choice of reaction progress variable.

Flow topology distribution in head-on quenching of turbulent premixed flame: A Direct Numerical Simulation analysis

The distribution of flow topologies within the flame, and their evolution with flame quenching have been analysed using a Direct Numerical Simulation (DNS) database of head-on quenching of statistically planar turbulent premixed flames by isothermal inert walls for different values of turbulence intensity and global Lewis number. It has been found that dilatation rate plays a key role in determining the flow topology distribution within the flame and this dilatation rate field is significantly affected by the flame quenching in the vicinity of the wall. The influence of the wall on the dilatation rate field in turn affects the statistical behaviour of all three invariants of the velocity gradient tensor and the distribution of flow topologies. The effects of heat release and thermal expansion strengthen with decreasing Lewis number which give rise to an increase in the probability of obtaining topologies which are specific to high positive values of dilatation rate. As the magnitude of positive dilatation rate and the likelihood of obtaining it decrease with flame quenching, the probability of finding the topologies, which are obtained only for positive values of dilatation rate, decreases close to the wall. The interrelation between the flow and flame topologies has been analysed in terms of Gaussian flame curvature and mean of principal flame curvatures. The contributions of individual flow topologies on the mean behaviour of wall heat flux magnitude, and the scalar-turbulence interaction and vortex-stretching terms in the scalar dissipation rate and enstrophy transport equations, respectively have been analysed in detail and dominant flow topologies which dictate the mean behaviours of these quantities have been identified. Detailed physical explanations have been provided for the observed flow topology distribution and its contribution to the scalar-turbulence and vortex-stretching terms. The nodal flow topologies have been found to be the significant contributors to the wall heat flux magnitude during head-on quenching of turbulent premixed flames irrespective of the value of global Lewis number.

A Priori Direct Numerical Simulation Modeling of Scalar Dissipation Rate Transport in Head-On Quenching of Turbulent Premixed Flames

ABSTRACTThe statistical behavior and modeling of scalar dissipation rate (SDR) transport for head-on quenching of turbulent premixed flames by an inert isothermal wall have been analyzed in the context of Reynolds averaged Navier–Stokes simulations based on three-dimensional simple chemistry direct numerical simulation (DNS) data. It has been found that the density variation, scalar-turbulence interaction, reaction rate gradient, molecular diffusivity gradient, and molecular dissipation terms, i.e., , and , respectively, act as leading order contributors to the SDR transport away from the wall and the turbulent transport and molecular diffusion terms remain negligible in comparison to the other terms. The leading order contributors to the SDR transport have been found to be in a rough equilibrium away from the wall before the quenching is initiated but this equilibrium is not maintained during flame quenching. The predictions of the existing models for the unclosed terms of the SDR transport equation have been assessed with respect to the corresponding quantities extracted from DNS data. No existing models have been found to predict the near-wall behavior of the unclosed terms of the SDR transport equation. The models, which exhibit the most satisfactory performance away from the wall, have been modified to account for near-wall behavior in such a manner that the modified models asymptotically approach the existing model expressions away from the wall.

A Priori Assessment of Scalar Dissipation Rate Closure for Large Eddy Simulations of Turbulent Premixed Combustion Using a Detailed Chemistry Direct Numerical Simulation Database

ABSTRACTAlgebraic and transport equation-based closures of Favre-filtered scalar dissipation rate (SDR) of the reaction progress variable, in the context of large eddy simulations, have been assessed using detailed chemistry direct numerical simulation (DNS) data of a stoichiometric H2-air turbulent V flame. The Favre-filtered SDR and the unclosed terms of its transport equation have been extracted by explicitly filtering the DNS data for different choices of the reaction progress variable. An algebraic closure of SDR, which was proposed previously using simple chemistry DNS data, has been found to predict the Favre-filtered SDR satisfactorily for detailed chemistry DNS data for different choices of the reaction progress variable. Similarly, the models of the unclosed sub-grid convection, density variation, scalar-turbulence interaction, reaction rate gradient, molecular dissipation, and diffusivity gradient terms of the Favre-filtered SDR transport equation, which have previously been proposed based on simple chemistry DNS data, have been found to satisfactorily predict both the qualitative and quantitative behaviors of these unclosed terms for a range of filter widths , for two different choices of reaction progress variable in the case of the detailed chemistry DNS dataset considered in this analysis.

Modeling of Lewis number dependence of scalar dissipation rate transport for Large Eddy Simulations of turbulent premixed combustion

ABSTRACTThe influences of differential diffusion of heat and mass on the Favre-filtered scalar dissipation rate (SDR) transport have been analyzed and modeled using a priori analysis of Direct Numerical Simulations (DNS) data of freely propagating statistically planar turbulent premixed flames with different values of global Lewis number, Le. The DNS data has been explicitly filtered using a Gaussian filter to obtain the unclosed terms of the Favre-filtered SDR transport equation, arising from turbulent transport (T1), density variation due to heat release (T2), strain rate contribution due to the alignment of scalar and velocity gradients (T3), correlation between the gradients of reaction rate and reaction progress variable (T4), molecular dissipation of SDR (−D2), and diffusivity gradients f(D). The statistical behaviors of these terms and their scaling estimates reported in a recent analysis have been utilized here to propose models for these unclosed terms in the context of Large Eddy Simulations (LES) and the performances of these models have been assessed using the values obtained from explicitly filtered DNS data. These newly proposed models are found to satisfactorily predict both the qualitative and quantitative behaviors of these unclosed terms for a range of filter widths Δ for all Le cases considered here.

Statistical Behavior of Scalar Dissipation Rate in Head-On Quenching of Turbulent Premixed Flames: A Direct Numerical Simulation Analysis

ABSTRACTThe statistical behavior of the scalar dissipation rate (SDR) and its transport in the context of Reynolds averaged Navier Stokes (RANS) simulations has been analyzed for head-on quenching of turbulent premixed flames using a three-dimensional simplified chemistry based direct numerical simulations (DNS) database. The flame quenching statistics have been analyzed in terms of wall Peclet number (i.e., non-dimensional distance of the flame in the wall normal direction) and non-dimensional wall heat flux magnitude . It has been found that flame wrinkling induces fluctuations of wall heat flux in turbulent cases and the magnitude of the maximum wall heat flux increases with increasing root-mean-square turbulent velocity .The closure of mean reaction rate using the SDR of reaction progress variable in the near wall region has been assessed based on a priori analysis of DNS data. It has been demonstrated that the existing SDR-based reaction rate closure does not work satisfactorily near the wall. A modification to this existing closure has been proposed, which is found to satisfactorily predict the mean reaction rate of reaction progress variable in the near wall region and approaches the existing closure away from the wall. The wall effects on the unclosed terms of the SDR transport equation have been analyzed and the order of magnitude estimates of the leading order contributors to the SDR transport have been utilized to modify an existing algebraic SDR closure to account for the near wall effects. A priori DNS analysis suggests that the proposed modification to the aforementioned SDR closure provides satisfactory prediction both away from and near to the wall.

Analysis of small-scale scalar mixing processes in highly under-expanded jets

In the present study, we numerically investigate turbulent scalar mixing taking place downstream of highly under-expanded jets. The focus is placed on two inter-related issues: (i) the closure of the mean scalar dissipation rate (SDR) and (ii) the turbulence–scalar interaction (TSI) term. It is indeed commonly admitted that the former, i.e., the SDR, which is defined as the product of the scalar diffusivity with the squared scalar gradient, provides a good measure of the mixing efficiency. In turbulent flows, the mean (turbulent) SDR requires a specific closure to be settled. It is generally obtained within the approximation of a linear relaxation of scalar fluctuations or linear relaxation model. We will first evaluate herein the performance of this widely used closure. The analysis is further developed by means of the consideration of the mean SDR transport equation which shows that, in gaseous conditions, the SDR is mainly driven by two terms: (i) a dissipation contribution and (ii) the third-order correlation between the velocity gradient tensor and small-scale scalar anisotropy tensor. The scalar mixing efficiency thus appears to be controlled by the latter quantity, which is often denoted as the TSI term. It can be shown that only the symmetric part (rate of strain) of the velocity gradient tensor contributes to this term; the anti-symmetric part modifies, indeed, the orientation of the scalar gradient but not its magnitude. The classical approach is to analyze this contribution in the eigenframe of the rate of strain tensor. Such analyses show that, in homogeneous isotropic turbulence, the scalar gradient tends to align with the most compressive direction, thus leading to SDR production. However, the present conditions, which are far from homogeneity and involve strong density variations, may modify this classical picture. The present study analyzes this possible influence.

Assessment of Reynolds Averaged Navier–Stokes Modeling of Scalar Dissipation Rate Transport in Turbulent Oblique Premixed Flames

The statistical behavior of scalar dissipation rate (SDR) transport of reaction progress variable and its modeling in the context of Reynolds averaged Navier–Stokes (RANS) simulations has been analyzed based on a-priori analysis of direct numerical simulation (DNS) data for a turbulent premixed V-flame configuration. It has been found that the terms in the scalar dissipation rate transport equation originating from density variation due to heat release, chemical reaction rate gradient, molecular dissipation, and diffusivity gradient are the major contributors to the scalar dissipation rate transport for different downstream axial locations from the flame holder. By contrast, the contribution of turbulent transport of scalar dissipation rate assumes negligible values in comparison to the unclosed terms due to density variation, chemical reaction rate gradient, molecular dissipation, and diffusivity gradient. The statistical behaviors of the unclosed terms arising from density variation, chemical reaction rate gradient, and molecular dissipation for V-flame have been found to be qualitatively consistent with previous results for statistically planar flames. The models for the unclosed turbulent transport, density variation, chemical reaction rate gradient, and molecular dissipation terms of the scalar dissipation rate transport equation, which were proposed based on previous a-priori analysis of statistically planar flames, have been found to predict the corresponding terms satisfactorily at different downstream axial locations in the V-flame configuration considered here. The present analysis shows that the unclosed term in the scalar dissipation rate transport equation originating from diffusivity gradient can assume magnitudes comparable to the density variation contribution to the SDR transport equation, and a model has been proposed for the combined contribution of the chemical reaction rate gradient, molecular dissipation, and diffusivity gradient terms, which has been found to satisfactorily predict the corresponding quantity extracted from DNS data at different axial locations.

Scalar Dissipation Rate Transport and Its Modeling for Large Eddy Simulations of Turbulent Premixed Combustion

Statistical behavior of Favre-filtered scalar dissipation rate (SDR) transport has been analyzed using direct numerical simulations (DNS) data of freely propagating statistically planar turbulent premixed flames. The DNS data has been explicitly filtered using a Gaussian filter to obtain the unclosed terms of the Favre-filtered SDR transport equation, arising from turbulent transport (), density variation due to heat release (), strain rate contribution due to the alignment of scalar and velocity gradients (), correlation between the gradients of reaction rate and reaction progress variable (), molecular dissipation of SDR (), and diffusivity gradients . It has been found that , and are the leading order contributors to the SDR transport and the magnitude of remains smaller than the magnitudes of , and irrespective of the filter width. Models are proposed for these terms in the context of large eddy simulations (LES) and their performances have been assessed with respect to their corresponding values obtained from explicitly filtered DNS data. These newly proposed models are found to satisfactorily predict both the qualitative and quantitative behaviors of these unclosed terms for a range of different values of filter widths , heat release parameter , and turbulent Reynolds number .

Scalar Dissipation Rate Transport in the Context of Large Eddy Simulations for Turbulent Premixed Flames with Non-Unity Lewis Number

The effects of global Lewis number Le on the statistical behaviour of the unclosed terms in the transport equation of the Favre-filtered scalar dissipation rate (SDR) N c have been analysed using a Direct Numerical Simulation (DNS) database of freely propagating statistically planer turbulent premixed flames with Le ranging from 0.34 to 1.2. The DNS data has been explicitly filtered to analyse the statistical behaviour of the unclosed terms in the SDR transport equation arising from turbulent transport T 1, density variation due to heat release T 2, scalar-turbulence interaction T 3, reaction rate gradient T 4, molecular dissipation (−D 2) and diffusivity gradients f(D) in the context of Large Eddy Simulations (LES). It Le has significant effects on the magnitudes of T 1, T 2, T 3, T 4, (−D 2) and f(D). Moreover, both qualitative and quantitative behaviours of the unclosed terms T 1, T 2, T 3, T 4, (−D 2) and f(D) are found to be significantly affected by the LES filter width Δ, which have been explained based on a detailed scaling analysis. Both scaling analysis and DNS data suggest that T 2, T 3, T 4, (−D 2) and f(D) remain leading order contributors to the SDR $\tilde {{N}}_{c} $ transport for LES. The scaling estimates of leading order contributors to the SDR $\tilde {{N}}_{c} $ transport has been utilised to discuss the possibility of extending an existing SDR model for Reynolds Averaged Navier Stokes (RANS) simulation for SDR $\tilde {{N}}_{c} $ closure in the context of LES of turbulent premixed combustion.

Reynolds Number Effects on Scalar Dissipation Rate Transport and Its Modeling in Turbulent Premixed Combustion

The effects of turbulent Reynolds number, Re t , on the transport of scalar dissipation rate of reaction progress variable in the context of Reynolds averaged Navier–Stokes simulations have been analyzed using three-dimensional simplified chemistry-based direct numerical simulation (DNS) data of freely propagating turbulent premixed flames with different values of Re t . Scaling arguments have been used to explain the effects of Re t on the turbulent transport, scalar–turbulence interaction, and the combined reaction and molecular dissipation terms. Suitable modifications to the models for these terms have been proposed to account for Re t effects, and the model parameters include explicit Re t dependence. These expressions approach expected asymptotic limits for large values of Re t . However, turbulent Reynolds number Re t does not seem to have any major effects on the modeling of the term arising from density variation.

Influence of Residence and Scalar Mixing Time Scales in Non-Premixed Combustion in Supersonic Turbulent Flows

In high Mach number turbulent reactive flows, spontaneous ignition appears as a key ingredient for the stabilization of combustion. In our previous analyses devoted to such conditions, chemical kinetics as well as associated finite rate chemistry effects have already received considerable attention. However, the representation of flow time scales, such as residence and mixing time scales, still requires further work. The present modeling study is devoted to this peculiar point. Hence, a transport equation for the quantity of residence time is considered to evaluate the mean residence time associated with both oxidizer and fuel injection streams, while a modeled transport equation for the mean scalar dissipation rate (SDR) is considered to estimate the scalar mixing time scale. This allows us to improve the description of turbulent mixing, including the large-scale engulfment processes through the consideration of the residence time scale, as well as the small-scale molecular mixing processes, the intensity of which is set by the integral scalar (mixing) time scale. Some insights are gained from the analysis of the evolution of the different production terms that appear in the mean SDR transport equation. The model capabilities are evaluated through a comparison between numerical results and the data obtained from experimental studies devoted to supersonic coflowing jets of hydrogen and vitiated air. The first simulated test case corresponds to a detailed experimental database that includes Raman scaterring and laser-induced pre-dissociative fluorescence measurements. Albeit less documented, the second test case is retained to confirm the relevance of the proposed closure. Finally, the comparisons performed with the two distinct sets of experimental data establish that the main physical processes are well-described by the proposed approach.

Determination of three-dimensional quantities related to scalar dissipation rate and its transport from two-dimensional measurements: Direct Numerical Simulation based validation

Three-dimensional compressible Direct Numerical Simulation (DNS) data of freely propagating statistically planar and statistically stationary slot-jet turbulent premixed flames has been used to assess the accuracy of the isotropy-derived correction factors, which relate the two-dimensional projections of the different terms of the Favre-averaged scalar dissipation rate transport equation with their corresponding actual three-dimensional counterparts. The accuracy of these correction factors is assessed using both simplified and detailed chemistry-based DNS data, for a range of values of Karlovitz number Ka, heat release parameter τ, and turbulent Reynolds number Ret. It is shown that the isotropic distribution of the probability density function (pdf) of the angle, ϕ, between the normal vectors of the measurement plane and of the flame surface provides a simple algebraic relation between the scalar dissipation rates evaluated in two and three dimensions (i.e. N∼c and N∼c2D), independent of the considered values of Ka, τ and Ret. The isotropic relations between two-dimensional and three-dimensional counterparts of the curvature and propagation terms in the transport equation of N∼c are also found to work well for all the values of Ka, τ and Ret considered here. However, the relation between the value obtained from two-dimensional projection and the true three-dimensional value for the strain rate term in the N∼c transport equation works well only for large values of Ret and the reasons for this behaviour are explained in detail. It is found that the threshold value of Ret above which the assumption of isotropy yields an accurate relation between two-dimensional projection and three-dimensional values for the strain rate term of the dissipation rate transport equation depends on the regime of the prevailing combustion process.

Effects of Lewis Number on Scalar Dissipation Transport and Its Modeling in Turbulent Premixed Combustion

The effects of Lewis number (Le) on scalar dissipation rate transport of the reaction progress variable have been studied using 3-dimensional DNS data of freely propagating turbulent premixed flames with global Lewis number ranging from 0.34 to 1.2. It has been found that the effects of dilatation rate become increasingly strong with decreasing Le, and is particularly strong for the Le ≪ 1 flames because of greater rate of heat release. Moreover, increasingly strong flame normal acceleration in flames with decreasing Le gives rise to countergradient transport for the turbulent flux of the scalar dissipation rate, whereas this flux exhibits gradient transport for flames with Le ≥ 1. This turbulent flux is observed to correlate with the behavior of the turbulent scalar flux and this behavior is used to propose an algebraic closure for the turbulent flux of the scalar dissipation rate. The modeling of nonunity Le effects on the unclosed terms arising from the density change across the flame front, the turbulence–scalar interaction, the correlation between scalar gradient and the gradient of reaction rate and the molecular dissipation have been addressed. The extent of reaction progress variable gradient alignment with the most extensive (compressive) principal strain rate is found to increase (decrease) with decreasing Le. This alignment behavior gives rise to a sink contribution arising from the turbulence–scalar interaction in the Le ≪ 1 flames. By contrast, the turbulence–scalar interaction acts predominantly as a source term, although this contribution becomes a sink toward the burned gas side for the Le ≥ 1 flames. It has been demonstrated that existing models for the unclosed terms originating from the density change across the flame front and the turbulence–scalar interaction cannot account for strengthening (weakening) of heat release effects when Le < 1 (Le > 1). Suitable modifications to these models have been proposed to account for nonunity Lewis number effects, and validated using direct numerical simulation data. A simple algebraic expression is suggested that successfully relates the scalar dissipation rate of normalized temperature to the scalar dissipation rate of the reaction progress variable in nonunity Lewis number turbulent flames.

The Scalar Gradient Alignment Statistics of Flame Kernels and its Modelling Implications for Turbulent Premixed Combustion

The effects of mean flame curvature on reaction progress variable gradient, \(\nabla c\), alignment with local turbulent strain rate are studied based on three-dimensional Direct Numerical Simulation (DNS) data of turbulent premixed flame kernels with different initial radii under decaying turbulence. A statistically planar flame is also considered in order to compare the results obtained from the kernels with a flame of zero mean curvature. It is found that the dilatation rate effects diminish with decreasing kernel radius due to defocusing of heat in the positively curved regions. This gives rise to a decrease in the extent of reaction progress variable gradient alignment with most extensive principal strain rate with decreasing kernel radius. The modelling implications of the statistics of the alignment of \(\nabla c\) with local strain rate have been studied in terms of scalar dissipation rate transport. A new modelling methodology for the contribution of the scalar-turbulence interaction term in the transport equation for the mean scalar dissipation is suggested addressing the reduced effects of dilatation rate for flame kernels and the diminished value of turbulent straining at the small length scales at which turbulence interacts with small flame kernels. The performance of the new models is found to be satisfactory while comparing to DNS results. The existing models for the dilatation contribution and the combined chemical reaction and molecular dissipation contributions to the transport of mean scalar dissipation, which were originally proposed for statistically planar flames, are found to satisfactorily predict the corresponding quantities for turbulent flame kernels.