Conditional Source-term Estimation Research Articles

An a priori analysis on principal component analysis based conditional source-term estimation model for Sandia jet flames

Data from all spatial locations and two turbulent flames in the Sandia/TUD database are used to explore the feasibility of adopting principal components (PC) as conditional variables in the conditional source-term estimation (CSE) model. Principal component analysis (PCA) is applied to both Flame C and F to generate the new set of controlling variables, PC-scores. Two PCA scaling methods have been adopted, namely Pareto and Auto-scaling (AS). Regardless of the scaling option selected and the flame investigated, it was found that a single principal component score (PC1-score) correlated with temperature accounts for the largest amount of variance. As such, the conditional space fluctuations and normalized RMS of both flames’ reactive scalars around PC1-Pareto and PC1-AS are examined and compared against the ones obtained with the mixture fraction, Z. The results indicate that both PC1-scores are not able to accurately quantify the thermo-chemical state-space of Flame C compared to mixture fraction, in particular for the fuel and all the intermediate species. Interestingly, using a single principal component score for Flame F significantly improved the conditional fluctuations, suggesting that a single PC-score well correlated with temperature can more effectively reduce the spatial gradient and Reynolds numbers effects than mixture fraction. The results are further depicted by comparing the trends obtained for Flame F with two-condition conditional averages around Z and four different progress variable definitions, c. While doubly conditioning enabled to detach the conditional averages of all the scalars from the physical domain, the results obtained with PC1-Pareto and PC1-AS were found to not deviate by much (excluding the mass fractions of CH4). This leads to believe that a conditional moment closure-based model such as CSE, coupled with PCA, can perhaps recover with satisfactory levels of approximation the thermo-chemical state-space of Flame F and separate the conditional manifold from the real domain.

Large eddy simulations of a buoyant turbulent line flame using conditional source-term estimation (CSE)

The objective of this paper is to assess the capabilities of the Conditional Source-term Estimation (CSE) approach applied to a laboratory scaled turbulent buoyant flame without extinction. CSE is coupled with the Large Eddy Simulation (LES) solver, FireFOAM. Tabulated detailed chemistry is included. Radiation is treated using the optically thin model using four different implementations and the effect of subgrid scale (sgs) Turbulence-Radiation Interactions (TRI) is considered. Predictions of time-averaged temperatures and corresponding root mean square (rms) are compared with the experimental measurements at several locations. Further, flame shape, unconditional and conditional species mass fractions, axial velocity and mixture fraction are also examined for qualitative analysis. The predicted temperatures are in good agreement with the experimental data, except very close to the fuel inlet. The predicted temperatures increase more slowly compared to the experimental data in the first 0.09 m above the burner. The temperature rms is in reasonable agreement with experimental data with the peak being overpredicted by approximately 17%. The predicted flame height closely matches the experimental value. The temperature predictions are consistent with previously published results. Best predictions are obtained when the effect of sgs TRI is included. For this flame, the optically thin assumption is found to be valid, in agreement with previous investigations. The calculated radiative fraction is found to be close to the experimental value, but the postprocessed heat release rate is larger than the experimental finding. This first LES-CSE study provides a good foundation for further developments, for example more detailed radiation models, for more complex fire cases.

Numerical simulation of turbulent premixed flames with the conditional source-term estimation model using Bernstein polynomial expansion

Conditional Source-term Estimation (CSE) is a turbulence-chemistry interaction model similar to CMC, except that the conditional scalars are calculated from unconditional ones using an integral equation. This problem is inherently ill-posed and should be regularised. Recently, an efficient regularisation approach based on Bernstein polynomial expansion was proposed by Mahdipour and Salehi (Combust. Flame, 2022) in an a priori analysis using DNS data. This work implements this approach in a reacting flow solver, and two laboratory-scale turbulent premixed flames are simulated in the Reynolds-Averaged Navier-Stokes (RANS) context. The turbulent intensity in the first flame is low, and the results show that, unlike the conventional CSE approach, the new approach can accurately predict the flamelet conditional averages. Furthermore, the predicted averaged velocity field and major and minor species mass fractions compare favourably with the experimental measurements. The turbulent intensity in the second flame is relatively higher, and the predicted conditional averages should deviate from an unstrained laminar flame solution. The new approach can correctly predict this trend as well as the flame height in this flame. The computational cost of the new CSE approach is also substantially reduced compared to the regular CSE approach.

LES study of turbulent ethanol spray flames using CSE coupled with non-adiabatic chemistry tables

Conditional Source-term Estimation (CSE) is applied to three turbulent ethanol spray flames (EtF3, EtF6, and EtF8) in Large Eddy Simulation (LES). The objectives of this paper are to include the heat losses due to spray evaporation and gas radiation in the chemistry tabulation, assess the impact of these changes on the temperature and droplet statistics, and evaluate the performance of LES-CSE for the selected flames. The profiles of gas temperature, spray velocity, velocity root mean square (rms) and droplet size distribution are well reproduced in the simulations compared to available experimental data. Temperature underpredictions near the centreline are observed, in particular, at locations closer to the jet exit for flames with lower jet velocity. A wider flame is predicted in EtF8 compared to the experiment and regions of local extinction are visible. The use of non-adiabatic chemistry library results in a noticeable improvement in the temperature predictions near the peak locations, especially for flames with higher velocity and closer to the jet exit. The heat losses due to evaporation are larger than those from radiation, confirming the importance of including the evaporation effects in the chemistry tables. The droplet velocity is well predicted, except for EtF8 where an underprediction is observed far downstream. The velocity rms is slightly underpredicted at some locations, probably due to the simple stochastic model used. Overall, LES-CSE with non-adiabatic chemistry tables successfully captures the gas-spray quantities in the selected flames.

Investigation of conditional source-term estimation coupled with a semi-empirical model for soot predictions in two turbulent flames

The modelling of soot formation is investigated for two turbulent flames, at atmospheric and 3 atm pressure conditions. For the first time, a semi-empirical soot formulation that accounts for soot inception, coagulation, surface growth, and oxidation processes is coupled with the turbulent combustion model, Conditional Source-term Estimation (CSE) using Reynolds-Averaged Navier–Stokes equations. Detailed chemistry is included and an optically thin radiation model is considered. Non-adiabatic chemistry tabulations are created. Good agreement with the experiments is found for turbulent mixing and temperature fields in both flames, with some discrepancies believed to be due to the turbulence modelling approach. At 1 atm, the soot volume fractions are in reasonable agreement with the experiments, but typically smaller than the measurements with the centerline peak locating closer to the fuel exit. At 3 atm, good agreement between the numerical predictions and experimental data is achieved for the soot volume fraction within the experimental error. The centerline peak location is observed slightly farther downstream. Possible sources of discrepancies are examined and comparison with previously published numerical results is also undertaken. Differential diffusion and modified soot chemistry constants may bring further improvement. Without any particular tuning of soot chemistry, soot modelling within CSE is shown to be a promising approach.

Localized conditional source-term estimation model for turbulent combustion

A novel functional form for approximating the conditional scalars in turbulent reacting flows is introduced based on the Bernstein polynomial. Multi-scalar measurement data of turbulent premixed and non-premixed flames are used to demonstrate that the new functional form provides an excellent reduced-order model for the conditional scalars. This model order reduction technique can be used to improve the accuracy, reduce the computational cost and enhance the spatial localization of the Conditional Source-term Estimation (CSE) model. CSE is a turbulence-chemistry interaction model similar to the Conditional Moment Closure (CMC) model, except that the conditional scalars are estimated from the filtered field in an ensemble of LES cells using an integral equation. An a priori analysis using the DNS data of a series of statistically planar turbulent premixed flames shows that using Bernstein polynomials as the presumed functional form for the conditional scalars provides better regularization than the conventional CSE approach. Furthermore, the ensemble size – that was previously kept on the order of thousands of LES cells in the singly-conditioned CSE – can be reduced to as low as 16 LES cells. This enhanced localization approach reduces the modelling error and the computational cost compared to the conventional CSE approach for tabulated and reduced chemistry models.

Simulations of partially premixed turbulent ethanol spray flames using doubly conditional source term estimation (DCSE)

Singly Conditional Source-term Estimation (CSE) is extended to Doubly Conditional Source-term Estimation (DCSE) to simulate three turbulent ethanol spray flames (EtF1, EtF3 and EtF4) for the first time. Reynolds-Averaged Navier Stokes equations are solved with a two-way coupling Eulerian-Lagrangian method to account for the gas-spray interactions. The objective of this work is to compare the performance of CSE and DCSE in simulating flames with varying levels of pre-evaporation leading to different levels of premixing and combustion modes. Tabulated detailed chemistry is included. The gas phase temperature and spray velocity predictions are compared with experimental data and previously published studies. Both CSE and DCSE can capture the temperature trends adequately. However, DCSE clearly improves the mean temperature predictions over CSE and the predicted temperatures are in close agreement with the experimental data. Close to nozzle exit and centreline, CSE and DCSE systematically predict lower temperatures compared to the experiments. Also noticed in other published studies, this may be due to larger experimental errors at this location, combined with modelling assumptions. Further analysis is conducted by examining the mean heat release rate and mean evaporation rate reproduced in CSE and DCSE. DCSE is shown to better capture some features of the complex structure of these flames. The progress variable predictions are consistent with the temperature predictions. The predicted mean droplets velocity is in good agreement with the experimental data at z/D=10 and z/D=20. However, farther downstream, the mean spray velocity was underpredicted for CSE and DCSE. The liquid volume flux is underpredicted near the centreline indicating higher evaporation rates in the simulations compared to the experiments. The predicted Sauter mean diameter also agrees well with the experiments with some discrepancies for the large droplets. Further investigation is needed to improve the currently available evaporation models.

Simulations of turbulent acetone spray flames using the conditional source term estimation (CSE) approach

Turbulent dilute acetone spray flames are simulated in an Eulerian–Lagrangian framework using Conditional Source-term Estimation (CSE) for closure of the mean reaction rate. The objective of this work is to assess the performance of the coupled CSE-spray approach, investigated for the first time for the selected configuration. One non-reacting case and four non-premixed flames, AcF1–AcF3 and AcF5 are chosen. Detailed tabulated chemistry is included. Predictions of mean axial velocity, gas temperature and spray statistics are compared with available experimental data. Good agreement between the predicted gas temperatures and experimental values is found. The peak temperature tends to be overpredicted and shifted radially outwards in the simulations, in particular farther downstream. Same trends are observed in all flames, but the predictions deteriorate with flame AcF5. Possible sources of discrepancy are inaccurate turbulent mixing field, increased level of premixing for flame AcF5, the neglect of spray effect in the chemistry tabulation and a larger experimental uncertainty in the temperatures in the high-temperature regions. The mean droplet velocity profiles are in good agreement with the experiments for the four flames. However, the velocity was underpredicted, in particular near the centreline, for flames AcF1 and AcF2.

A priori evaluation of the Double-conditioned Conditional Source-term Estimation model for high-pressure heptane turbulent combustion using DNS data obtained with one-step chemistry

An evaluation of the submodels constituting the Double-Conditioned Source-term Estimation (DCSE) method for utilization in Large Eddy Simulation (LES) is presented which makes use of a single realization of a turbulent high-pressure reactive-flow database obtained from Direct Numerical Simulation (DNS). A filtered and coarsened DNS (FCDNS) field is first created to mimic a real LES field, and the FCDNS values for thermodynamic variables (temperature, density and species mass fractions) are used as inputs to the DCSE model to predict the FCDNS reaction rates. It is found that there are significant errors in the predictions of the filtered reaction rate compared to the template. These errors are attributed to three important aspects of the model. First, the conditional-filtered values of the thermodynamic variables are found by an inversion of an integral equation, but this inversion requires the definition of spatial ensembles in which it must be performed; these ensembles are a subset of the entire spatial domain containing the LES volumes. A drastic reduction in the number of ensembles from the number of LES volumes (i.e. the largest possible number of such ensembles) deteriorates the model, whereas a further and smaller such reduction has a small effect. Second, the procedure for the integral inversion introduces problems related to the method selected to regularize the inversion. Third, by far the largest source of error was found to be due to the modelling of the joint probability density function of the conditioning variables: assuming a β-PDF for the marginal PDF of the reaction progress variable is shown to be a particularly poor choice, as is the assumption that the conditioning variables are statistically independent.

Simulation of ECN diesel spray A using conditional source-term estimation

A novel combustion modelling approach is proposed here to study the transient effects of diesel spray. Conditional Source-term Estimation (CSE) is a combustion model which invokes the Conditional Moment Closure (CMC) hypothesis to provide an approximation of the mean chemical source term in an averaged transport equation. Unlike CMC, where transport equations are solved for conditional moments, CSE recovers these conditional moments through the solution of an inverse problem. Integral equations are inverted for the conditional moments, by assuming spatial homogeneity in the conditional averages where Tikhonov regularisation is applied. Previous CSE studies have shown that the model is able to predict the flame characteristics successfully for both premixed and non-premixed combustion modes. However, most of these investigations were based on methane flames. This study will be the first successful application of CSE to a complex hydrocarbon fuel, n-dodecane, under the Engine Combustion Network's (ECN) ‘Spray A’ conditions. Detailed chemistry is included in tabulated form using the Flamelet Generated Manifold (FGM) methodology. The predictions of this study include both the Favre averaged conditional mass fraction of reactive species and temperature. The results are compared with available experimental data and previous numerical results. Both RANS and LES simulations are performed under the same condition. The objectives of the paper are (i) assessment of the application of CSE on igniting diesel spray (ii) comparison of the CSE numerical results with available experimental results and previous numerical simulations. Overall, the combination of a chemical mechanism that has been tuned to predict ‘Spray A’ conditions with the CSE-FGM model is able to successfully predict autoignition delay time and lift-off length of n-dodecane spray within the scatter of the experimental data. CSE-FGM offers a feasible tool for detailed combustion analysis of diesel spray flames. Both RANS and LES can give reasonably good global predictions of the flame.

Large Eddy Simulation of lifted turbulent flame in cold air using Doubly Conditional Source-term Estimation

The present study is focused on Large Eddy Simulation with Doubly Conditional Source-term Estimation (DCSE) applied to a series of lifted turbulent flames in cold air without autoignition. DCSE is formulated for partially-premixed turbulent combustion. Predictions for time-averaged methane concentration and its root mean square (rms), Probability Density Function (PDF), flame width, lift-off height and some qualitative observations are presented and compared with experimental data. The predicted methane concentrations are in good agreement with the experiments. Discrepancies for the rms are observed and several sources of discrepancies are identified. PDF of methane concentration and flammability region are also in good agreement. The lift-off heights are predicted very well. Some edge flame characteristics are visible in the simulated flames. The progress variable scalar dissipation rate model does not have a significant impact on methane concentration predictions, while it has a larger effect on predicted lift-off height and temperature. Inclusion of the non-premixed mode in the closure of the progress variable transport equation has a small effect on the current lift-off height predictions, but has a larger impact on the temperature profiles.

The modeling of the turbulent reaction rate under high-pressure conditions: A priori evaluation of the Conditional Source-term Estimation concept

With the goal of obtaining an accurate model of the turbulent reaction rate for Large Eddy Simulations (LESs), the modeling of turbulence-chemistry-thermodynamic interaction is addressed through an a priori study of a Direct Numerical Simulation (DNS) database representing high-pressure turbulent combustion. The DNS database consists of simulations of a temporal mixing layer in which a single-step chemical reaction occurs. The potential of the single-conditioned Conditional Source-term Estimation (CSE) approach to model the filtered turbulent reaction rate needed for conducting LES is examined. Evaluations conducted with the mixture fraction as a conditioning variable at two filter widths and with the probability density function (PDF) extracted from the DNS database that represents the mixture fraction, show that the deviation between the model and template is large and substantially increases with filter width. To address this deviation, the Double-conditioned CSE (DCSE) approach is explored with two different second conditioning variables, the first conditioning variable still being the mixture fraction; four filter widths are considered. The first choice of the second conditioning variable is a normalized progress variable based on the CO2 mass fraction and the second choice of the second conditioning variable is a normalized temperature. With each second conditional variable, the DCSE results represent a substantial improvement over CSE, by as much as an order of magnitude when measured by the relative error from the DNS-extracted filtered reaction rate. A quantitative test based on a root mean square identifies the reason for the DCSE success compared to CSE: unlike in CSE, the DCSE is able to substantially reduce the fluctuations of the modeled reaction rate from the filtered reaction rate over the entire range of the filtered reaction rate values, and particularly at the larger reaction rate values for which the DCSE model has higher fidelity. The results are shown to be equally favorable for the two conditioning variables. Consideration of DNS realizations at higher free-stream pressure or larger Reynolds number shows that the results are essentially equally successful at other pressure or Reynolds number values.

Spatial gradients of conditional averages in turbulent flames

Data from the two different methane/air jet experimental databases is examined to understand the effects of spatial variations in the conditional averages of reactive scalars in the context of the Conditional Source-term Estimation (CSE) model, with particular emphasis on the double-conditional version, DCSE. Data from all spatial locations and from the four turbulent flames in the Sandia/TUD database are averaged together to generate a global, two-condition (mixture fraction and a species-based reaction progress variable) conditional average; it is found that the conditional fluctuations around this new, two-condition conditional average do not exhibit any significant dependence on either spatial location or even on Reynolds number. Furthermore, the normalized RMS of the conditional fluctuations are found to be dramatically lower than those seen using only one condition. These results indicate that the two-condition conditional averages in the Sandia/TUD piloted methane/air flame series do not vary in space, nor vary with Reynolds number. In the Sydney Swirl burner data, the two-condition conditional averages exhibit modest gradients in space which are attributed to heat transfer with the bluff body in the flow; when a third condition is added – based on the total enthalpy – the spatial gradients in the conditional average of temperature a dramatically reduced. In both of the cases tested, the normalized RMS of the conditional fluctuations around both the two- and three-condition conditional averages are relatively small. These results imply that, in a simulation of these flames using DCSE, one would be better off ignoring spatial variations in the conditional averages and instead collecting all of the spatial data together for the integral equation inversion using either two or three conditioning variables in a single, global ensemble species involved in relatively slow chemistry, such as NO. The results also imply that the underlying assumptions in DCSE – that the conditional variances and the spatial gradients in the conditional averages are both negligible – are valid when sufficient conditioning variables are used, at least for the flames considered.

Comparative study between Conditional Moment Closure (CMC) and Conditional Source-term Estimation (CSE) applied to piloted jet flames

Conditional Moment Closure (CMC) and Conditional Source-term Estimation (CSE) are investigated in a direct side-by-side comparison of the Sandia flames. The objective of the present study is to determine the strengths and weaknesses of each modelling approach given the same initial conditions and computational set-up. The predictions from both CMC and CSE are compared to detailed experimental data. In case of Sandia flame D, the CMC and CSE turbulent flow and mixing fields are similar near the nozzle exit, in agreement with the experimental data, with some differences farther downstream. The CMC and CSE conditional mass fractions for the major species are in good agreement with the experimental data downstream of the nozzle for lean mixtures. For fuel rich mixtures, an underprediction of the conditional mass fractions of methane and corresponding overestimation of the conditional mass fraction of water are observed at some axial locations. The CMC and CSE conditional mass fractions of the minor species and conditional temperature reproduce the general characteristics of the experimental profiles. However, major differences are observed for Sandia flame E. The discrepancies are attributed to the use of RANS and the boundary conditions set in CMC and some approximations made in the chemistry tables in CSE. Favre-averaged profiles are comparable in CMC and CSE. The computational time for each model is investigated and CSE appears to be faster. Further discussion on some of the strengths and weaknesses of each combustion model is also presented. Overall, both CMC and CSE are shown to produce results of comparable quality given the same numerical schemes, mesh, and boundary conditions.

Evaluation of mixture-fraction-based turbulent-reaction-rate model assumptions for high-pressure reactive flows

Several assumptions of atmospheric-pressure (atmospheric-p) single-phase turbulent reaction rate models are examined for high-p reactive flows having turbulent characteristics. The study uses a Direct Numerical Simulation (DNS) database described elsewhere (Bellan, 2017). This database was obtained with a model combining multi-species mixing under high- p conditions, a real-gas equation of state (EOS) and a single-step chemical reaction. The database, created in the configuration of a temporal mixing layer, probes the effect of the initial Reynolds number, Re0, of the initial pressure, p0, and of the initial composition of the two mixing-layer streams. The reaction is initiated in a turbulent flow and in each simulation the computations are pursued past a time when a maximum average-volumetric p is attained, tpp*. The examination of the vorticity and enstrophy equations at a time before reaction initiation and also at tpp* highlights the pivotal role of the high density-gradient magnitude regions (Bellan, 2017) in producing turbulence and the crucial role at tpp* of the baroclinic effect representing the misalignments of gradients of thermodynamic variables. Compared to the classical mixture fraction equation, the results show that the mixture fraction obeys an equation in which there is an additional diffusion term having a larger r.m.s. magnitude than that of the mixture-fraction typical diffusion term. An assessment of the Conditional Source-term Estimation (CSE) model assumptions found that, using an accurate mixture fraction probability density function (PDF), one can obtain a very good representation of the turbulent reaction rate in the most intense reaction regions, however the quality of the predictions of CSE deteriorated significantly when a common model for the PDF, the β -PDF, was used despite the β-PDF being constructed with the accurate moments extracted from the DNS. In the regions of minimal reaction, the CSE model using the exact PDF extracted from the DNS does not provide an accurate representation of the turbulent reaction rate, a fact which is attributed to the combined effect of lack of correlation between fluctuations of the reaction rate and of the mixture fraction, and to the strong correlation of the thermodynamic variables through the real-gas EOS in regions of colder and denser fluid.

Species and temperature predictions in a semi-industrial MILD furnace using a non-adiabatic conditional source-term estimation formulation

A Reynolds-Averaged Navier–Stokes (RANS) simulation of the semi-industrial International Flame Research Foundation (IFRF) furnace is performed using a non-adiabatic Conditional Source-term Estimation (CSE) formulation. This represents the first time that a CSE formulation, which accounts for the effect of radiation on the conditional reaction rates, has been applied to a large scale semi-industrial furnace. The objective of the current study is to assess the capabilities of CSE to accurately reproduce the velocity field, temperature, species concentration and nitrogen oxides (NOx) emission for the IFRF furnace. The flow field is solved using the standard k–ε turbulence model and detailed chemistry is included. NOx emissions are calculated using two different methods. Predicted velocity profiles are in good agreement with the experimental data. The predicted peak temperature occurs closer to the centreline, as compared to the experimental observations, suggesting that the mixing between the fuel jet and vitiated air jet may be overestimated. Good agreement between the species concentrations, including NOx, and the experimental data is observed near the burner exit. Farther downstream, the centreline oxygen concentration is found to be underpredicted. Predicted NOx concentrations are in good agreement with experimental data when calculated using the method of Peters and Weber. The current study indicates that RANS-CSE can accurately predict the main characteristics seen in a semi-industrial IFRF furnace.

Doubly Conditional Source-term Estimation (DCSE) for the modelling of turbulent stratified V-shaped flame

A partially premixed formulation of Conditional Source-term Estimation including doubly conditioned averages, namely DCSE, is applied to several turbulent stratified V-shaped flames using a Reynolds Averaged Navier Stokes (RANS) approach. Three different stratification levels are considered with different turbulence conditions. In contrast to the authors’ initial work (Dovizio et al., 2016), a more elaborate closure of the scalar dissipation rate is implemented in the progress variable variance equation, where a dependence on the local value of the equivalence ratio is considered. Detailed chemistry is included. Good predictions are obtained for the mean axial and transverse velocity components, and progress variable compared to experimental data for the stoichiometric cases, in particular for the lean and rich cases, where the predictions are significantly improved compared to previous numerical studies. The flame thickness is found to be reduced when the stratification increases, consistent with experimental findings. Future work will examine the closure of cross-dissipation terms in the transport equations and the DCSE implementation in LES.

Conditional Source-term Estimation using dynamic ensemble selection and parallel iterative solution

A modified version of the Least-Square QR-factorisation (LSQR) algorithm has been implemented in conjunction with Conditional Source-term Estimation (CSE) for lean, turbulent premixed methane–air combustion via Large Eddy Simulation (LES). The iterative solver can reduce computational times by an order of magnitude during the inversion phase of CSE in comparison with the conventional LU-decomposition method. The advantages of iterative and parallel iterative solvers become more prominent as the size of the system increases. The ensemble selection procedure for computing averages within localised regions of the simulation domain has also been updated to a dynamic routine. This allows for more flexible and efficient allocation of computational resources along with reduced input from the user, especially for complex geometries. Preliminary LES calculations have shown that the implementation of an iterative solver and a dynamic ensemble selection algorithm will reduce computational times significantly with negligible error contribution for one-condition CSE, which is applicable to purely premixed or non-premixed turbulent combustion problems. In addition, these algorithms provide the foundation for exceptional computational cost savings for the inversion in two-condition CSE, or Doubly Conditional Source-term Estimation (DCSE), which has shown promise for predicting partially-premixed combustion. Parallel computation of the inverse solution is particularly beneficial to DCSE as the computational cost of the inversion process is considerably larger than in one-condition CSE.

Application of Conditional Source-term Estimation to two turbulent non-premixed methanol flames

Conditional Source-term Estimation (CSE) is a turbulent combustion model that uses conditional averages to close the chemical source term. Previous CSE studies have shown that the model is able to predict the flame characteristics successfully; however, these studies have only focused on simple hydrocarbon fuels mostly composed of methane. The objective of the present paper is to evaluate the capabilities of CSE applied to turbulent non-premixed methanol flames, which has never been done previously. The current study investigates two different types of methanol flames: piloted and bluff-body flames. For the piloted flame, the standard k–ε model is used for turbulence modelling, while the Shear Stress Transport (SST) k–ω model is applied to the bluff-body case. Different values of empirical constants within the turbulence models were tested, and it was found that Cε1 = 1.7 for the piloted flame and γ2 = 0.66 for the bluff-body flame provided the best agreement with experimental measurements for the mixing field. Detailed chemistry is included in tabulated form using the Trajectory Generated Low-Dimensional Manifold (TGLDM) method. The predictions including both the Favre-averaged and conditional mass fraction of reactive species and temperature are compared with available experimental data and previous numerical results. Overall, the CSE predictions of conditional and unconditional quantities are in good agreement with the experimental data except for hydrogen. Sources of discrepancies are identified such as the chemical kinetics and neglect of differential diffusion. Large eddy simulations may also help to improve the velocity and mixing field predictions.

RANS Simulations of a Series of Turbulent V-Shaped Flames using Conditional Source-term Estimation

In the present study, Reynolds Averaged Navier Stokes (RANS) simulations are applied to a series of turbulent V-shaped flames. Two formulations of Conditional Source-term Estimation (CSE) are developed using singly and doubly conditioned averages for turbulent premixed and partially premixed flames, respectively. Detailed chemistry is included. Conditionally averaged chemical source terms are closed by conditional averaged scalars which are obtained by inverting an integral equation. The objectives are to study a turbulent premixed V-shaped flame using the premixed CSE approach and apply the Doubly Conditional CSE (DCSE) combustion model to a case of stratified combustion. The partially premixed implementation involves double conditioning on two variables, mixture fraction and progress variable. The present study represents the first application of DCSE for a series of turbulent stratified flames. First, CSE is analysed for fully premixed conditions. A sensitivity analysis on the number of CSE ensembles and different scalar dissipation model closures is performed. Good results are obtained in terms of velocity and progress variable profiles. Finally, the partially premixed formulation is applied to the stratified case at three different conditions, corresponding to two different turbulence grids and three different profiles of the equivalence ratio, providing promising results.