Computations of turbulent lean premixed combustion using conditional moment closure

Conditional Moment Closure (CMC) is a suitable method for predicting scalars such as carbon monoxide with slow chemical time scales in turbulent combustion. Although this method has been successfully applied to non-premixed combustion, its application to lean premixed combustion is rare. In this study the CMC method is used to compute piloted lean premixed combustion in a distributed combustion regime. The conditional scalar dissipation rate of the conditioning scalar, the progress variable, is closed using an algebraic model and turbulence is modelled using the standard k–? model. The conditional mean reaction rate is closed using a first order CMC closure with the GRI-3.0 chemical mechanism to represent the chemical kinetics of methane oxidation. The PDF of the progress variable is obtained using a presumed shape with the Beta function. The computed results are compared with the experimental measurements and earlier computations using the transported PDF approach. The results show reasonable agreement with the experimental measurements and are consistent with the transported PDF computations. When the compounded effects of shear-turbulence and flame are strong, second order closures may be required for the CMC.     

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.     

A LES-CMC formulation for premixed flames including differential diffusion

A finite volume large eddy simulation–conditional moment closure (LES-CMC) numerical framework for premixed combustion developed in a previous studyhas been extended to account for differential diffusion. The non-unity Lewis number CMC transport equation has an additional convective term in sample space proportional to the conditional diffusion of the progress variable, that in turn accounts for diffusion normal to the flame front and curvature-induced effects. Planar laminar simulations are first performed using a spatially homogeneous non-unity Lewis number CMC formulation and validated against physical-space fully resolved reference solutions. The same CMC formulation is subsequently used to numerically investigate the effects of curvature for laminar flames having different effective Lewis numbers: a lean methane–air flame with Le_eff = 0.99 and a lean hydrogen–air flame with Le_eff = 0.33. Results suggest that curvature does not affect the conditional heat release if the effective Lewis number tends to unity, so that curvature-induced transport may be neglected. Finally, the effect of turbulence on the flame structure is qualitatively analysed using LES-CMC simulations with and without differential diffusion for a turbulent premixed bluff body methane–air flame exhibiting local extinction behaviour. Overall, both the unity and the non-unity computations predict the characteristic M-shaped flame observed experimentally, although some minor differences are identified. The findings suggest that for the high Karlovitz number (from 1 to 10) flame considered, turbulent mixing within the flame weakens the differential transport contribution by reducing the conditional scalar dissipation rate and accordingly the conditional diffusion of the progress variable.     

Modelling of turbulent scalar flux in turbulent premixed flames based on DNS databases

A transport equation for scalar flux in turbulent premixed flames was modelled on the basis of DNS databases. Fully developed turbulent premixed flames were obtained for three different density ratios of flames with a single-step irreversible reaction, while the turbulent intensity was comparable to the laminar burning velocity. These DNS databases showed that the countergradient diffusion was dominant in the flame region. Analyses of the Favre-averaged transport equation for turbulent scalar flux proved that the pressure related terms and the velocity–reaction rate correlation term played important roles on the countergradient diffusion, while the mean velocity gradient term, the mean progress variable gradient term and dissipation terms suppressed it. Based on these analyses, modelling of the combustion-related terms was discussed. The mean pressure gradient term and the fluctuating pressure term were modelled by scaling, and these models were in good agreement with DNS databases. The dissipation terms and the velocity–reaction rate correlation term were also modelled, and these models mimicked DNS well.     

Large Eddy Simulation of a turbulent lifted flame using multi-modal manifold-based models: Feasibility and interpretability

A general model for multi-modal turbulent combustion is achievable with two-dimensional manifold equations that use the mixture fraction and a generalized progress variable as coordinates. Information about the underlying mode of combustion is encoded in three scalar dissipation rates that appear as parameters in the two-dimensional equations. In this work, Large Eddy Simulation (LES) of a multi-modal turbulent lifted hydrogen jet flame in a vitiated coflow is performed using this new turbulent combustion model, leveraging both convolution-on-the-fly and In-Situ Adaptive Tabulation for computational tractability. The simulation predicts a lifted flame consistent with observations from past experiments. The feasibility of such a model implemented in LES is examined, and the cost per timestep is found to be comparable to conventional one-dimensional manifold-based models describing one asymptotic mode of combustion. Additionally, the model provides clear interpretability, allowing for combustion mode analysis to be performed with ease by evaluating the scalar dissipation rates and generalized progress variable source term. This analysis is used to show that the flame is stabilized by autoignition and has a trailing nonpremixed flame. Furthermore, transport of progress variable from the most reactive mixture fraction towards richer mixtures at the centerline is found to be important.     

A priori analysis of sub-grid variance of a reactive scalar using DNS data of high Ka flames

Direct numerical simulations (DNS) of low and high Karlovitz number (Ka) flames are analysed to investigate the behaviour of the reactive scalar sub-grid scale (SGS) variance in premixed combustion under a wide range of combustion conditions (regimes). An order of magnitude analysis is performed to assess the importance of various terms in the variance evolution equation and the analysis is validated using the DNS results. This analysis sheds light on the relative behaviour among turbulent transport and production, scalar dissipation and chemical processes involved in the evolution of the SGS variance at different Ka. The common expectation is that the variance equation shifts from a reaction-dissipation balance at low Ka to a production–dissipation balance at high Ka with diminishing reaction contribution. However, in large eddy simulation (LES), a high Ka alone does not make the reaction term negligible, as the relative importance of the reaction term has a concurrent increase with filter size. The filter size can be relatively large compared with the Kolmogorov length scale in practical LES of high Ka flames, and as a consequence a reaction–production–dissipation balance may prevail in the variance equation even in a high Ka configuration, and this possibility is quantified using the DNS analysis in this work. This has implications from modelling perspectives, and therefore two commonly used closures in LES for the SGS scalar dissipation rate are investigated a priori to estimate the importance of the above balance in LES modelling. The results are explained to highlight the interplay among turbulence, chemistry and dissipation processes as a function of Ka.     

Simulation of flame lift-off on a diesel jet using a generalized flame surface density modeling approach

A generalized flame surface density modelling approach is presented to simulate the transient ignition and flame stabilization of a diesel jet flame, for which experimental data are available. The approach consists of four submodels: a mixing model, a generalized flame surface density model, a generalized progress variable model, and a chemistry model. A database containing the laminar model reaction rates per unit generalized flame surface density is generated by solving the unsteady flamelet equations. The RANS-CFD code solves for the mean flame surface density and mean progress variable. The coupling of the models is done via the progress variable and the scalar dissipation rate. The proposed approach is found to be adapted to simulate such a lifted flame and yields good trend agreement for ignition delay and flame lift-off vs. liquid penetration. These first promising results are encouraging to further explore and to apply this method to a more industrial configuration such as a diesel engine.     

Detailed measurements of local scalar front structures in stagnation-type turbulent premixed flames

Local scalar front structures of OH mole fraction, reaction progress variable, and its three-dimensional gradient have been measured in stagnation-type turbulent premixed flames. The reaction progress variable front is observed to change with increasing turbulence from parallel iso-scalar contours but reduced progress variable gradients, called the lamella-like front, to disrupted non-parallel iso-contours that deviate substantially from those of wrinkled laminar flamelets, called the non-flamelet front. This transition is attributed to the different scales of interaction between the flame internal structure and a spectrum of turbulence extending from the integral scale to the Kolmogorov scale. The lamella-like front pattern occurs when the length scales of interaction are smaller than the laminar flame thickness but the time scales are greater than the flame residence time. The non-flamelet front pattern occurs when the length scales of interaction are greater than the laminar flame thickness but the time scales are smaller than the flame residence time. This difference corresponds to the change of combustion regime from complex-strain flame front to turbulent flame front on a revised regime diagram. A correlation is also proposed for the turbulent flame brush thickness as a function of turbulent Reynolds number and heat release parameter. The heat release parameter is considered to arise from the non-passive effects of flame-surface wrinkling.     

Conserved scalar mixing in a confined-opposed-jets flow

The focus of this paper is on the mixing of a conserved passive scalar for Sc = 1 (Sc is the Schmidt number) in axisymmetric turbulence for which the initial injections of turbulent kinetic energy and scalar variance are similar. Two confined-opposed-jets (COJ) are experimentally studied through simultaneous PIV (particle image velocimetry) and PLIF (planar laser induced fluorescence) measurements, for different flow regimes. One-point transport equation for the scalar variance is assessed through experimental data, along the common axis of the two opposed jets, and different physical phenomena are revealed (production, diffusion, dissipation). The production of scalar variance is equilibrated by the diffusion term (～75%) and the mean dissipation of the scalar variance (～25%). To further assess the scalar behaviour at each scale in this anisotropic, but axisymmetric, flow, a scale-by-scale scalar variance budget equation is derived for axisymmetric turbulence. This equation reduces to Yaglom's 4/3 law, under additional restrictions. The equation is assessed through experimental data, in the impingement region between the two COJ. In particular, the anisotropic energy transfer along different directions is quantified. It is shown that for scales smaller than the size of the central region, Δ, the cascade of the scalar variance is completely inhibited, independently of the particular direction. For scales larger than Δ, the apparent aspect of the energy transfer is that of an inverse cascade, with positive values of the scalar variance transfer. Nonetheless, inhomogeneity of the flow and mixing at those scales is directly responsible for these positive values.     

Scalar dissipation rates in isothermal and reactive turbulent opposed-jets: 1-D-Raman/Rayleigh experiments supported by LES

Isothermal and reactive turbulent opposed flows are presented, which are appropriate to test the applicability and performance of models for turbulence, mixing, chemical reaction, and turbulence-chemistry interaction. Transient flow and scalar fields are measured using laser Doppler velocimetry and one-dimensionally resolved Raman/Rayleigh spectroscopy. Aside of statistical moments of temperature, mean species, and velocity components, scalar dissipation rate across the mixing and reaction layer is determined on a single-shot base. Using large eddy simulation in connection with a steady flamelet model, it is shown how numerical data can serve to estimate the influence of experimental noise upon a measured quantity, such as scalar dissipation. As a key result, it is shown that an increase in scalar rate of dissipation by chemical reactions is caused by a significant increase in the mixture fraction diffusivity, which outweighs the decrease in mixture fraction gradients. In mixture fraction space, local maxima of scalar dissipation rate are found on the rich side, which cannot be correctly reproduced by the steady flamelet model assuming equal species diffusivity. Furthermore, the impact of experimental noise on conditional probability density functions of scalar dissipation rate is shown (exemplary) to lead to erroneous conclusions from experimental data.     

Second-order conditional moment closure modeling of a turbulent CH4/H2/N2 jet diffusion flame

The second-order CMC model for a detailed chemical mechanism is used to model a turbulent CH₄/H₂/N₂ jet diffusion flame. Second-order corrections are made to the three rate limiting steps of methane–air combustion, while first-order closure is employed for all the other steps. Elementary reaction steps have a wide range of timescales with only a few of them slow enough to interact with turbulent mixing. Those steps with relatively large timescales require higher-order correction to represent the effect of fluctuating scalar dissipation rates. Results show improved prediction of conditional mean temperature and mass fractions of OH and NO. Major species are not much influenced by second-order corrections except near the nozzle exit. A parametric study is performed to evaluate the effects of the variance parameter in log-normal scalar dissipation PDF and the constants for the dissipation term in conditional variance and covariance equations.     

A self-contained progress variable space solution method for thermochemical variables and flame speed in freely-propagating premixed flamelets

Flamelet models for premixed combustion, which are based on equations formulated and solved in progress variable space, have been proposed in the past, but have not been adopted for chemistry reduction methods. This is due to one limitation of these models: they need a closure for both the magnitude and the shape of the gradient (or scalar dissipation rate) of the progress variable, which is essential for an accurate prediction of the flame displacement speed. So far, solution methods for the aforementioned models require gradient information as an input, which is either modelled and non-generic, or extracted from a previous physical space flame solution for the analogous problem. The objective of this work is to provide a self-contained solution method for freely-propagating premixed flamelets in progress variable space, by solving an additional flamelet equation for the gradient of the progress variable. With this, the novel method provides both magnitude and shape of the gradient. Studying hydrogen-air and methane-air configurations, it is demonstrated that an accurate prediction of the laminar flame speed without the necessity for further input parameters can be obtained.     

Turbulent scalar flux transport in head-on quenching of turbulent premixed flames: a direct numerical simulations approach to assess models for Reynolds averaged Navier Stokes simulations

The statistical behaviour and the modelling of turbulent scalar flux transport have been analysed using a direct numerical simulation (DNS) database of head-on quenching of statistically planar turbulent premixed flames by an isothermal wall. A range of different values of Damköhler, Karlovitz numbers and Lewis numbers has been considered for this analysis. The magnitudes of the turbulent transport and mean velocity gradient terms in the turbulent scalar flux transport equation remain small in comparison to the pressure gradient, molecular dissipation and reaction-velocity fluctuation correlation terms in the turbulent scalar flux transport equation when the flame is away from the wall but the magnitudes of all these terms diminish and assume comparable values during flame quenching before vanishing altogether. It has been found that the existing models for the turbulent transport, pressure gradient, molecular dissipation and reaction-velocity fluctuation correlation terms in the turbulent scalar flux transport equation do not adequately address the respective behaviours extracted from DNS data in the near-wall region during flame quenching. Existing models for transport equation-based closures of turbulent scalar flux have been modified in such a manner that these models provide satisfactory prediction both near to and away from the wall.     

Zone conditional analysis of a freely propagating one-dimensional turbulent premixed flame

The zone conditional conservation equations are derived and validated against the DNS data of a freely propagating one-dimensional turbulent premixed flame. Conditional flow velocities are calculated by the conditional continuity and momentum equations, and a modeled transport equation for the Reynolds average reaction progress variable. An asymptotic formula for turbulent burning velocity is obtained with the effects of a finite Damköhler number accounted for as an additional factor. It is shown that flame generated turbulence is primarily due to correlations between fluctuating gas velocities and fluctuating unit normal vector on a flame surface. More investigation is required to validate general predictive capability of the derived conditional conservation equations and the relationships modeled for closure.     

Unstrained and strained flamelets for LES of premixed combustion

The unstrained and strained flamelet closures for filtered reaction rate in large eddy simulation (LES) of premixed flames are studied. The required sub-grid scale (SGS) PDF in these closures is presumed using the Beta function. The relative performances of these closures are assessed by comparing numerical results from large eddy simulations of piloted Bunsen flames of stoichiometric methane–air mixture with experimental measurements. The strained flamelets closure is observed to underestimate the burn rate and thus the reactive scalars mass fractions are under-predicted with an over-prediction of fuel mass fraction compared with the unstrained flamelet closure. The physical reasons for this relative behaviour are discussed. The results of unstrained flamelet closure compare well with experimental data. The SGS variance of the progress variable required for the presumed PDF is obtained by solving its transport equation. An order of magnitude analysis of this equation suggests that the commonly used algebraic model obtained by balancing source and sink in this transport equation does not hold. This algebraic model is shown to underestimate the SGS variance substantially and the implications of this variance model for the filtered reaction rate closures are highlighted.     

Experimental study of velocity-scalar filtered joint density function for LES of turbulent combustion

The velocity-scalar filtered joint density function (FJDF) used in large eddy simulation (LES) of turbulent combustion is experimentally studied. Measurements are made in the fully developed region of an axisymmetric turbulent jet using an array consisting of three X-wires and resistance-wire temperature sensors. Filtering in the cross-stream and streamwise directions is realized by using the array and by invoking Taylor’s hypothesis, respectively. The means of the FJDF conditional on the subgrid-scale (SGS) turbulent kinetic energy and the SGS scalar variance at a given location range from close to joint normal to bimodal with the peaks separated in both velocity and scalar spaces, which correspond to qualitatively different mixing regimes. For close to joint normal FJDFs, the SGS fields are well mixed. For bimodal FJDFs, the conditionally filtered scalar diffusion and dissipation strongly depend on the SGS velocity and scalar, consistent with a combination of diffusion layers and plane strain in the SGS fields, which is similar to the counter-flow model for laminar flamelets. The results suggest that in LES, both mixing regimes could potentially be modeled accurately. The velocity field affects the SGS variance and the filtered scalar dissipation rate primarily by changing the degree of nonequilibrium of the SGS scalar and the SGS time scale, respectively. This study further demonstrates the importance of including velocity in mixing models.     

A DNS assessment of linear relations between filtered reaction rate,flame surface density,and scalar dissipation rate in a weakly turbulent premixed flame

Linear relations between (i) filtered reaction rate and filtered flame surface density (FSD) and (ii) filtered reaction rate and filtered scalar dissipation rate (SDR), which are widely used in Large Eddy Simulation (LES) research into premixed turbulent combustion, are examined by processing DNS data obtained from a statistically 1D planar flame under weakly turbulent conditions that are most favourable for the two approaches (flamelet combustion regime, single-step chemistry, equidiffusive mixture, adiabatic burner, and low Mach number). The analysis well supports the former approach provided that the filtered reaction rate is combined with filtered molecular transport term. In such a case, both the RANS and LES FSD approaches are based on local relations valid within weakly perturbed flamelets. Accordingly, simply recasting RANS expressions to a filtered form works well. On the contrary, while the FSD and SDR approaches appear to be basically similar at first glance, the analysis does not support the latter one, but shows that a ratio of the filtered reaction rate to the filtered SDR is strongly scattered within the studied flame brush, with its conditionally mean value varying significantly with Favre-filtered combustion progress variable. As argued in the paper, these limitations of the LES SDR approach stem from the fact that it is based on a relation valid after integration over weakly perturbed flamelets, but this relation does not hold locally within such flamelets. Consequently, when a sufficiently small filter is applied to instantaneous fields, the filter may contain only a part of the local flamelet, whereas the linear relation holds solely for the entire flamelet and may not hold within the filtered flamelet volume. Thus, the present study implies that straightforwardly recasting well-established RANS equations to a filtered form is a flawed approach if the equations are based on integral features of local burning.     

Conditional statistics of nonreacting and reacting sprays in turbulent flows by direct numerical simulation

Conditional statistics concerning evaporation and combustion of a spray are investigated in homogeneous, isotropic, and decaying two-dimensional (2D) turbulence. Randomly distributed, polydisperse droplets of n-heptane go through single-step combustion chemistry. Attention is focused on parametric effects of initial Sauter mean radius (SMR), turbulence level and droplet velocity in both reacting and nonreacting cases. A simple linear model for the conditional evaporation rate is proposed and validated against DNS data. A conventional β-probability density function (pdf) is shown to be valid with no peak occurring on the fuel side. The amplitude mapping closure (AMC) model works well for the conditional scalar dissipation rate with evaporating and reacting sprays. Parametric study shows that initial SMR and droplet velocity are major factors affecting conditional flame structures, whereas the effect of reaction is not significant except during autoignition.     

Influence of advective heat flux on extinction scalar dissipation rate and velocity in negative edge flames

A counterflow flame geometry, which has previously been experimentally shown to produce stable negative edge flames, was studied using numerical simulations. In this geometry, the flame edge is formed off the counterflow centreline owing to a local increase in scalar dissipation rate. Hot products from the stable nonpremixed flame on the centreline flow through the edge at velocities of ～ 1–5 m/s. The size of the counterflow burner and the gas flowrates are varied in the simulations to alter the flame strength and velocity at the flame edge. The advection of products through the edge is shown to extend the flame extinction to higher scalar dissipation rates than required for centreline extinction. For high velocities, the scalar dissipation rate required for flame extinction can be related to the centreline extinction value by considering only the effect of energy addition to the flame edge via advection. However, for lower edge flame velocities, the effects of increased thermal and species diffusion through the edge must also be included. Since the advection at the edge is a product of both the local velocity and temperature gradient, a single correlation between the scalar dissipation rate and the negative edge flame velocity does not exist.