Application of the conditional source-term estimation model for turbulence–chemistry interactions in a premixed flame

Conditional Source-term Estimation (CSE) is a closure model for turbulence–chemistry interactions. This model uses the first-order CMC hypothesis to close the chemical reaction source terms. The conditional scalar field is estimated by solving an integral equation using inverse methods. It was originally developed and has been used extensively in non-premixed combustion. This work is the first application of this combustion model for a premixed flame. CSE is coupled with a Trajectory Generated Low-Dimensional Manifold (TGLDM) model for chemistry. The CSE-TGLDM combustion model is used in a RANS code to simulate a turbulent premixed Bunsen burner. Along with this combustion model, a similar model which relies on the flamelet assumption is also used for comparison. The results of these two approaches in the prediction of the velocity field, temperature and species mass fractions are compared together. Although the flamelet model is less computationally expensive, the CSE combustion model is more general and does not have the limiting assumption underlying the flamelet model.     

Simulation of transient turbulent methane jet ignition and combustion under engine-relevant conditions using conditional source-term estimation with detailed chemistry

The ignition and combustion processes of transient turbulent methane jets under high-pressure and moderate temperature conditions were simulated using a computationally efficient combustion model. Closure for the mean chemical source-terms was obtained with Conditional Source-term Estimation (CSE) using first conditional moment closure in conjunction with a detailed chemical kinetic mechanism, which was reduced to a Trajectory-Generated Low-Dimensional Manifold (TGLDM). The accuracy of the manifold was first validated against the direct integral method by comparing the predicted reactive scalar profiles in three methane–air reaction systems: a laminar premixed flame, a laminar flamelet and a perfectly stirred reactor. Detailed CFD simulations incorporating the CSE-TGLDM model were able to provide reasonably good predictions of the experimental ignition delay and initial ignition kernel locations of the methane jets reported in the literature with relatively low computational cost. Nitrogen oxides formed in the methane jet flame were found to be underpredicted by the model by as much as a factor of 2. The discrepancy may be attributable to the inability of the simulation to account for the effects of the rarefaction wave in the shock-tube experiments.     

A LES/PDF simulator on block-structured meshes

A block-structured mesh large-eddy simulation (LES)/probability density function (PDF) simulator is developed within the OpenFOAM framework for computational modelling of complex turbulent reacting flows. The LES/PDF solver is a hybrid solution methodology consisting of (i) a finite-volume (FV) method for solving the filtered mass and momentum equations (LES solver), and (ii) a Lagrangian particle-based Monte Carlo algorithm (PDF solver) for solving the modelled transport equation of the filtered joint PDF of compositions. Both the LES and the PDF methods are developed and combined to form a hybrid LES/PDF simulator entirely within the OpenFOAM framework. The in situ adaptive tabulation method [S.B. Pope, Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation, Combust. Theory Model. 1 (1997), pp. 41–63; L. Lu, S.R. Lantz, Z. Ren, and B.S. Pope, Computationally efficient implementation of combustion chemistry in parallel PDF calculations, J. Comput. Phys. 228 (2009), pp. 5490–5525] is incorporated into the new LES/PDF solver for efficient computations of combustion chemistry with detailed reaction kinetics. The method is designed to utilise a block-structured mesh and can readily be extended to unstructured grids. The three-stage velocity interpolation method of Zhang and Haworth [A general mass consistency algorithm for hybrid particle/finite-volume PDF methods, J. Comput. Phys. 194 (2004), pp. 156–193] is adapted to interpolate the LES velocity field onto particle locations accurately and to enforce the consistency between LES and PDF fields at the numerical solution level. The hybrid algorithm is fully parallelised using the conventional domain decomposition approach. A detailed examination of the effects of each stage and the overall performance of the velocity interpolation algorithm is performed. Accurate coupling of the LES and PDF solvers is demonstrated using the one-way coupling methodology. Then the fully two-way coupled LES/PDF solver is successfully applied to simulate the Sandia Flame-D, and a turbulent non-swirling premixed flame and a turbulent swirling stratified flame from the Cambridge turbulent stratified flame series [M.S. Sweeney, S. Hochgreb, M.J. Dunn, and R.S. Barlow, The structure of turbulent stratified and premixed methane/air flames I: Non-swirling flows, Combust. Flame 159 (2012), pp. 2896–2911; M.S. Sweeney, S. Hochgreb, M.J. Dunn, and R.S. Barlow, The structure of turbulent stratified and premixed methane/air flames II: Swirling flows, Combust. Flame 159 (2012), pp. 2912–2929]. It is found that the LES/PDF method is very robust and the results are in good agreement with the experimental data for both flames.     

Predicting the ignition delay of turbulent methane jets using Conditional Source-term Estimation

A predictive simulation of the autoignition process of non-premixed methane in a turbulent jet configuration was performed. Closure for the chemical source-term was obtained using Conditional Source-term Estimation with Laminar Flamelet Decomposition (CSE-LFD). The ambient oxidizer conditions – the high pressure and moderate temperatures characteristic of compression ignition engines – were chosen with the intent to validate the combustion model used under engine-relevant conditions. Validation was obtained by comparison of the predicted ignition delay to experimental results obtained from a shock-tube facility at several initial temperatures. Overall, the combination of full chemistry that has been carefully tuned to predict autoignition of premixed methane–air mixtures under similar temperature/pressure conditions with the CSE-LFD model is able to successfully predict the autoignition delay time of methane–air jets well within the scatter in the experimental data.     

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

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

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 C_ε1 = 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.     

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.     

A direct approach to generalised multiple mapping conditioning for selected turbulent diffusion flame cases

This work presents a direct and transparent interpretation of two concepts for modelling turbulent combustion: generalised Multiple Mapping Conditioning (MMC) and sparse-Lagrangian Large Eddy Simulation (LES). The MMC approach is presented as a hybrid between the Probability Density Function (PDF) method and approaches based on conditioning (e.g. Conditional Moment Closure, flamelet, etc.). The sparse-Lagrangian approach, which allows for a dramatic reduction of computational cost, is viewed as an alternative interpretation of the Filtered Density Function (FDF) methods. This work presents simulations of several turbulent diffusion flame cases and discusses the universality of the localness parameter between these cases and the universality of sparse-Lagrangian FDF methods with MMC.     

Filtered density function simulator on unstructured meshes

A new computational filtered density function (FDF) methodology is developed for large eddy simulation (LES) of turbulent reacting flows. This methodology is based on a Lagrangian Monte Carlo (MC) FDF solver constructed on a domain portrayed by an unstructured mesh. The base filtered transport equations on this mesh are solved by a finite-volume (FV) method. The consistency of the hybrid FV-MC solver and the realizability of the simulated results are demonstrated via LES of a temporally developing mixing layer. The overall performance of the model is appraised by comparison with direct numerical simulation (DNS) data. The algorithmic implementation in the commercial software ANSYS-FLUENT facilitates future FDF-LES of turbulent combustion in complex configurations.     

Effects of combined dimension reduction and tabulation on the simulations of a turbulent premixed flame using a large-eddy simulation/probability density function method

A turbulent lean-premixed propane–air flame stabilised by a triangular cylinder as a flame-holder is simulated to assess the accuracy and computational efficiency of combined dimension reduction and tabulation of chemistry. The computational condition matches the Volvo rig experiments. For the reactive simulation, the Lagrangian Large-Eddy Simulation/Probability Density Function (LES/PDF) formulation is used. A novel two-way coupling approach between LES and PDF is applied to obtain resolved density to reduce its statistical fluctuations. Composition mixing is evaluated by the modified Interaction-by-Exchange with the Mean (IEM) model. A baseline case uses In Situ Adaptive Tabulation (ISAT) to calculate chemical reactions efficiently. Its results demonstrate good agreement with the experimental measurements in turbulence statistics, temperature, and minor species mass fractions. For dimension reduction, 11 and 16 represented species are chosen and a variant of Rate Controlled Constrained Equilibrium (RCCE) is applied in conjunction with ISAT to each case. All the quantities in the comparison are indistinguishable from the baseline results using ISAT only. The combined use of RCCE/ISAT reduces the computational time for chemical reaction by more than 50%. However, for the current turbulent premixed flame, chemical reaction takes only a minor portion of the overall computational cost, in contrast to non-premixed flame simulations using LES/PDF, presumably due to the restricted manifold of purely premixed flame in the composition space. Instead, composition mixing is the major contributor to cost reduction since the mean-drift term, which is computationally expensive, is computed for the reduced representation. Overall, a reduction of more than 15% in the computational cost is obtained.     

Visualisation of results from LES of combustion

We examine the role of visualisation in the context of LES simulations of premixed turbulent combustion. The physical processes involved in premixed turbulent combustion are extremely complex, and the modelling of both the turbulence (via LES) and the combustion (via flame-wrinkling models) is difficult. Appropriate visualisation is required to understand the behaviour of the models, and ultimately to understand better the flow processes which are important in many industrial applications. We examine visualisations of two specific cases; simple flame kernel growth in a box of turbulence, and combustion behind a backward-facing step. A number of visualisation techniques are used to produce results that are similar to experimentally determined Schlieren and Mie photography for the flame kernel. In addition, isosurfaces of the reaction regress variable coloured by the laminar flame speed and sub-grid wrinkling are also plotted in an attempt to gain deeper insight into the physics of turbulent combustion in the context of these particular cases. Finally we discuss the role of the WWW in the continuing development of scientific visualisation techniques.     

Paradigms in turbulent combustion research

The development of the basic conceptual viewpoints, or paradigms, for turbulent combustion in gases over the last 50 years is reviewed. Significant progress has been made. Recent successes in the prediction of pollutant species and extinction/re-ignition phenomena in non-premixed flames are seen as the result of close interaction between experimentalists, theoreticians, and modellers. Premixed turbulent flames seem to be dependent on a much wider range of factors, and predictive capabilities are not so advanced. Implications for large eddy simulation (LES) and partially premixed combustion are outlined.     

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.     

Inverse analysis and regularisation in conditional source-term estimation modelling

Conditional Source-term Estimation (CSE) obtains the conditional species mass fractions by inverting a Fredholm integral equation of the first kind. In the present work, a Bayesian framework is used to compare two different regularisation methods: zeroth-order temporal Tikhonov regulatisation and first-order spatial Tikhonov regularisation. The objectives of the current study are: (i) to elucidate the ill-posedness of the inverse problem; (ii) to understand the origin of the perturbations in the data and quantify their magnitude; (iii) to quantify the uncertainty in the solution using different priors; and (iv) to determine the regularisation method best suited to this problem. A singular value decomposition shows that the current inverse problem is ill-posed. Perturbations to the data may be caused by the use of a discrete mixture fraction grid for calculating the mixture fraction PDF. The magnitude of the perturbations is estimated using a box filter and the uncertainty in the solution is determined based on the width of the credible intervals. The width of the credible intervals is significantly reduced with the inclusion of a smoothing prior and the recovered solution is in better agreement with the exact solution. The credible intervals for temporal and spatial smoothing are shown to be similar. Credible intervals for temporal smoothing depend on the solution from the previous time step and a smooth solution is not guaranteed. For spatial smoothing, the credible intervals are not dependent upon a previous solution and better predict characteristics for higher mixture fraction values. These characteristics make spatial smoothing a promising alternative method for recovering a solution from the CSE inversion process.     

Combustion modeling using Principal Component Analysis: A posteriori validation on Sandia flames D,E and F

The present work shows the first application of the PC-transport approach in the context of Large Eddy Simulation (LES) of turbulent combustion. Detailed kinetic mechanisms, together with advanced computational tools, are needed to advance our knowledge of turbulent reacting systems. However, the cost related to high-fidelity simulations of turbulent reacting flows is still prohibitive for realistic configurations. Therefore, there is a need to reduce the complexity of the problem by identifying low-dimensional manifolds. To this end, the potential offered by Principal Component Analysis (PCA) in parameterizing the thermo-chemical state-space is very appealing. The present paper extends the PC-transport framework to three-dimensional Large Eddy Simulation (LES), coupling PCA with Gaussian Process Regression (GPR). To demonstrate the potential of the method, LES simulations of Sandia flames D, E and F are shown. Results show the great potential of the PC-GPR model, as indicated by the accuracy of the simulation results when compared with experimental data, using only 2 principal components. The sensitivity to the kinetic mechanism and subgrid closure model is also investigated.     

Three-dimensional electromagnetic nonlinear inversion in layered media by a hybrid diagonal tensor approximation: Stabilized biconjugate gradient fast Fourier transform method

This paper presents an efficient three-dimensional nonlinear electromagnetic inversion method in a multilayered medium for radar applications where the object size is comparable to the wavelength. In the first step of this two-step inversion algorithm, the diagonal tensor approximation is used in the Born iterative method. The solution of this approximate inversion is used as an initial guess for the second step in which further inversion is carried out using a distorted Born iterative method. Since the aim of the second step is to improve the accuracy of the inversion, a full-wave solver, the stabilized biconjugate-gradient fast Fourier transform algorithm, is used for forward modelling. The conjugate-gradient method is applied at each inversion iteration to minimize the functional cost. The usage of an iterative solver based on the FFT algorithm and the developed recursive matrix method combined with an interpolation technique to evaluate the layered medium Green's functions rapidly, makes this method highly efficient. An inversion problem with 32 768 complex unknowns can be solved with 1% relative error by using a simple personal computer. Several numerical experiments for arbitrarily located source and receiver arrays are presented to show the high efficiency and accuracy of the proposed method.     

Three-dimensional electromagnetic nonlinear inversion in layered media by a hybrid diagonal tensor approximation: Stabilized biconjugate gradient fast Fourier transform method

This paper presents an efficient three-dimensional nonlinear electromagnetic inversion method in a multilayered medium for radar applications where the object size is comparable to the wavelength. In the first step of this two-step inversion algorithm, the diagonal tensor approximation is used in the Born iterative method. The solution of this approximate inversion is used as an initial guess for the second step in which further inversion is carried out using a distorted Born iterative method. Since the aim of the second step is to improve the accuracy of the inversion, a full-wave solver, the stabilized biconjugate-gradient fast Fourier transform algorithm, is used for forward modelling. The conjugate-gradient method is applied at each inversion iteration to minimize the functional cost. The usage of an iterative solver based on the FFT algorithm and the developed recursive matrix method combined with an interpolation technique to evaluate the layered medium Green's functions rapidly, makes this method highly efficient. An inversion problem with 32 768 complex unknowns can be solved with 1% relative error by using a simple personal computer. Several numerical experiments for arbitrarily located source and receiver arrays are presented to show the high efficiency and accuracy of the proposed method.     

LES modelling of turbulent non-premixed jet flames with correlated dynamic adaptive chemistry

Large eddy simulations (LES) for turbulent flames with detailed kinetic mechanisms have received growing interest. However, a direct implementation of detailed kinetic mechanisms in LES modelling of turbulent combustion remains a challenge due to the requirement of huge computational resources. An on-the-fly mechanism reduction method named correlated dynamic adaptive chemistry (CoDAC) is proposed to overcome this issue. A LES was conducted for Sandia Flame-D, with the reaction mechanism of GRI-Mech 3.0 consisting of 53 species and 325 reactions. The reduction threshold used in LES was obtained a priori by using auto-ignition model and partially stirred reactor (PaSR) with pairwise mixing model. LES results with CoDAC are in good agreement with experimental data and those without reduction. The conditional mean of the number of selected species indicates that a large size of locally reduced mechanism is required in the reaction zone where CH₄ is destructed. A computational time analysis shows that the PaSR model predicts better than the auto-ignition model on the wall time reduction with CoDAC in LES.     

Numerical simulation of turbulent combustion: Scientific challenges

Predictive simulation of engine combustion is key to understanding the underlying complicated physicochemical processes, improving engine performance, and reducing pollutant emissions. Critical issues as turbulence modeling, turbulence-chemistry interaction, and accommodation of detailed chemical kinetics in complex flows remain challenging and essential for high-fidelity combustion simulation. This paper reviews the current status of the state-of-the-art large eddy simulation (LES)/prob-ability density function (PDF)/detailed chemistry approach that can address the three challenging modelling issues. PDF as a subgrid model for LES is formulated and the hybrid mesh-particle method for LES/PDF simulations is described. Then the development need in micro-mixing models for the PDF simulations of turbulent premixed combustion is identified. Finally the different acceleration methods for detailed chemistry are reviewed and a combined strategy is proposed for further development.