Eulerian transported probability density function sub-filter model for large-eddy simulations of turbulent combustion

Reactive flow simulations using large-eddy simulations (LES) require modelling of sub-filter fluctuations. Although conserved scalars like mixture fraction can be represented using a beta-function, the reactive scalar probability density function (PDF) does not follow an universal shape. A one-point one-time joint composition PDF transport equation can be used to describe the evolution of the scalar PDF. The high-dimensional nature of this PDF transport equation requires the use of a statistical ensemble of notional particles and is directly coupled to the LES flow solver. However, the large grid sizes used in LES simulations will make such Lagrangian simulations computationally intractable. Here we propose the use of a Eulerian version of the transported-PDF scheme for simulating turbulent reactive flows. The direct quadrature method of moments (DQMOM) uses scalar-type equations with appropriate source terms to evolve the sub-filter PDF in terms of a finite number of delta-functions. Each delta-peak is characterized by a location and weight that are obtained from individual transport equations. To illustrate the feasibility of the scheme, we compare the model against a particle-based Lagrangian scheme and a presumed PDF model for the evolution of the mixture fraction PDF. All these models are applied to an experimental bluff-body flame and the simulated scalar and flow fields are compared with experimental data. The DQMOM model results show good agreement with the experimental data as well as the other sub-filter models used.     

Comparisons of Lagrangian and Eulerian PDF methods in simulations of non-premixed turbulent jet flames with moderate-to-strong turbulence-chemistry interactions

Transported probability density function (PDF) methods have been applied widely and effectively for modelling turbulent reacting flows. In most applications of PDF methods to date, Lagrangian particle Monte Carlo algorithms have been used to solve a modelled PDF transport equation. However, Lagrangian particle PDF methods are computationally intensive and are not readily integrated into conventional Eulerian computational fluid dynamics (CFD) codes. Eulerian field PDF methods have been proposed as an alternative. Here a systematic comparison is performed among three methods for solving the same underlying modelled composition PDF transport equation: a consistent hybrid Lagrangian particle/Eulerian mesh (LPEM) method, a stochastic Eulerian field (SEF) method and a deterministic Eulerian field method with a direct-quadrature-method-of-moments closure (a multi-environment PDF-MEPDF method). The comparisons have been made in simulations of a series of three non-premixed, piloted methane–air turbulent jet flames that exhibit progressively increasing levels of local extinction and turbulence-chemistry interactions: Sandia/TUD flames D, E and F. The three PDF methods have been implemented using the same underlying CFD solver, and results obtained using the three methods have been compared using (to the extent possible) equivalent physical models and numerical parameters. Reasonably converged mean and rms scalar profiles are obtained using 40 particles per cell for the LPEM method or 40 Eulerian fields for the SEF method. Results from these stochastic methods are compared with results obtained using two- and three-environment MEPDF methods. The relative advantages and disadvantages of each method in terms of accuracy and computational requirements are explored and identified. In general, the results obtained from the two stochastic methods (LPEM and SEF) are very similar, and are in closer agreement with experimental measurements than those obtained using the MEPDF method, while MEPDF is the most computationally efficient of the three methods. These and other findings are discussed in detail.     

Simulations of local extinction phenomena in bluff-body stabilized diffusion flames with a Lagrangian reactedness model

Two-dimensional large-eddy simulations of bluff-body stabilized flames of methane and propane, exhibiting significant finite-rate chemistry effects, are presented. A partial equilibrium/two-scalar exponential probability density function (PDF) combustion submodel is applied at the subgrid level. Subgrid scale motions are modelled with a first-order closure employing an anisotropic subgrid eddy-viscosity and two equations for the subgrid turbulent kinetic and scalar energies. Statistical independence of the joint PDF scalars is avoided and the necessary moments are obtained from an extended scale-similarity assumption. Extinction is accounted for by comparing the local turbulent Damköhler number against a ‘critical’ local limit related to the Gibson scalar scale and the reaction zone thickness in mixture fraction space. The post-extinction regime is modelled via a Lagrangian transport equation for a reactedness progress variable which follows a linear deterministic relaxation to its mean value (interaction by exchange with the mean model; IEM).Comparisons between simulations and measurements suggested the ability of the adopted methodology to represent the experimental variations in the momentum and scalar fields at conditions close to the lean or the rich blow-out limit. Favourable agreement was achieved in the calculation of the recirculation lengths and the peak temperature and turbulence levels in the near-wake region. Significant experimental trends, such as the suppression of the large-scale organized motions in the developing wake at low and medium fuel injection rates, and the re-emergence of the quasi-periodic shedding activity close to the lean limit, were also reproduced. Quantitative discrepancies increased in the prediction of major species, but the measured trends due to the effects of partial extinction were adequately recovered.     

A multivariate quadrature based moment method for LES based modeling of supersonic combustion

The transported probability density function (PDF) approach is a powerful technique for large eddy simulation (LES) based modeling of scramjet combustors. In this approach, a high-dimensional transport equation for the joint composition-enthalpy PDF needs to be solved. Quadrature based approaches provide deterministic Eulerian methods for solving the joint-PDF transport equation. In this work, it is first demonstrated that the numerical errors associated with LES require special care in the development of PDF solution algorithms. The direct quadrature method of moments (DQMOM) is one quadrature-based approach developed for supersonic combustion modeling. This approach is shown to generate inconsistent evolution of the scalar moments. Further, gradient-based source terms that appear in the DQMOM transport equations are severely underpredicted in LES leading to artificial mixing of fuel and oxidizer. To overcome these numerical issues, a semi-discrete quadrature method of moments (SeQMOM) is formulated. The performance of the new technique is compared with the DQMOM approach in canonical flow configurations as well as a three-dimensional supersonic cavity stabilized flame configuration. The SeQMOM approach is shown to predict subfilter statistics accurately compared to the DQMOM approach.     

A quadrature-based LES/transported probability density function approach for modeling supersonic combustion

The joint-scalar probability density function (PDF) approach provides a comprehensive framework for large eddy simulation (LES) based combustion modeling. However, currently available stochastic approaches for solving the high-dimensional PDF transport equation can be error prone and numerically unstable in highly compressible shock-containing flows. In this work, a novel Eulerian approach called the direct quadrature method of moments (DQMOM) is developed for evolving the PDF-based supersonic combustion model. The DQMOM technique uses a set of scalar transport equations with specific source terms to recover the PDF. The new technique is coupled to a compressible LES solver through the energy equation. The DQMOM approach is then used to simulate two practical flow configurations: a supersonic reacting jet and a cavity-stabilized supersonic combustor. Comparisons with experimental data demonstrate the predictive accuracy of the method.     

Axis switching in turbulent buoyant diffusion flames

Dynamics of buoyant diffusion flames from rectangular, square, and round fuel sources were investigated using direct numerical simulation (DNS). Fully three-dimensional simulations were performed employing high-order numerical methods and boundary conditions to solve governing equations for variable-density flow and finite-rate Arrhenius chemistry. Significant differences among the different cases were revealed in the vortex dynamics, entrainment rate, small-scale mixing, and consequently flame structures. Mixing and entrainment enhancement in non-circular flames in comparison with circular ones was explained using the Biot–Savart instability theory, which relates vortex dynamics to the local azimuthal curvature. An extension of the theory elucidated why rectangular flames entrain more efficiently and spread wider than square ones, although both configurations have corners. It also provided an explanation for the aspect ratio effects in the near field. In the far field, nonlinear effects were dominant and the general transport equations for vorticity were analyzed in detail. The corner effects and aspect ratio effects were shown to be augmented by the intricate interactions among vortex dynamics, combustion, and buoyancy through the various terms in the equations. The presence of corners in non-circular flames led to concentrated regions of fine-scale mixing and intense reactions centered around the corners. Moreover, the rectangular flames exhibited a different dynamic behavior from even the square one, by creating discrepancies in entrainment, mixing, and combustion between the minor and major axis directions. Increasing the aspect ratio exacerbated such directional discrepancies, and ultimately led to axis switching. It was the first time that axis switching was observed by DNS in a rectangular flame of aspect ratio 3, which raised further questions in combustion prediction and control. Finally, a unified explanation for corner and aspect ratio effects was given on the basis of the Biot–Savart instability theory and the vorticity transport equations.     

On the role of mixing models in the simulation of MILD combustion using finite-rate chemistry combustion models

The present work shows an in-depth analysis about the role of mixing models on the simulation of MILD combustion using a finite-rate combustion model, the Partially Stirred Reactor approach (PaSR). Different approaches of increasing complexity are compared: a simple model based on a fraction of the integral time scale, a fractal-based mixing model and a dynamic mixing model based on the resolution of transport equations for scalar variance and dissipation rate. The approach is validated using detailed experimental data from flames stabilized on the Adelaide Jet-in-Hot Co-flow (JHC) burner at different fuel-jet Reynolds numbers (5k, 10k and 20k) and different co-flow oxygen dilution levels (3%, 6% and 9%). The results indicate the major role of mixing models to correctly handle turbulence/chemistry interactions and clearly indicate the superior performances of the dynamic mixing model over the other tested approaches.     

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.     

PDF方法模拟钝体驻定的湍流扩散火焰

采用标量联合的概率密度函数方法,对钝体驻定的湍流射流扩散Sydney火焰HM1进行数值模拟,结合当地自适应建表方法加速化学反应计算,用修正的LRR-IP雷诺应力模型求解速度场.首次对3种不同规模的甲烷化学反应动力学机理进行研究,并与实验数据进行比较,结果表明,模型和反应机理很好地预测了速度场和标量场的变化及局部熄火现象,而考虑反应机理中的C2化学对火焰HM1的影响不大.     

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.     

Transported PDF simulation of turbulent CH4/H2 flames under MILD conditions with particle-level sensitivity analysis

Transported probability density function (TPDF) simulation with sensitivity analysis has been conducted for turbulent non-premixed CH₄/H₂ flames of the jet-into-hot-coflow (JHC) burner, which is a typical model to emulate moderate or intense low oxygen dilution combustion (MILD). Specifically, two cases with different levels of oxygen in the coflow stream, namely HM1 and HM3, are simulated to reveal the differences between MILD and hot-temperature combustion. The TPDF simulation well predicts the temperature and species distributions including those of OH, CO and NO for both cases with a 25-species mechanism. The reduced reaction activity in HM1 as reflected in the peak OH concentration is well correlated to the reduced oxygen in the coflow stream. The particle-level local sensitivities with respect to mixing and chemical reaction further show dramatic differences in the flame characteristics. HM1 is less sensitive to mixing and reaction parameters than HM3 due to the suppressed combustion process. Specifically, for HM1 the sensitivities to mixing and chemical reactions have comparable magnitude, indicating that the combustion progress is controlled by both mixing and reaction in MILD combustion. For HM3, there is however a change in the combustion mode: during the flame initialization, the combustion progress is more sensitive to chemical reactions, indicating that finite-rate chemistry is the controlling process during the autoignition process for flame stabilization; at further downstream where the flame has established, the combustion progress is controlled by mixing, which is characteristic of nonpremixed flames. An examination of the particles with the largest sensitivities reveals the difference in the controlling mixtures for flame stabilization, namely, the stoichiometric mixtures are important for HM1, whereas, fuel-lean mixtures are controlling for HM3. The study demonstrates the potential of TPDF simulations with sensitivity analysis to investigate the effects of finite-rate chemistry on the flame characteristics and emissions, and reveal the controlling physio-chemical processes in MILD combustion.     

Large eddy simulations of turbulent flames using the filtered density function model

Large eddy simulations (LES) of the Sandia/Sydney swirl burners (SM1 and SMA1) and the Sandia/Darmstadt piloted jet diffusion flame (Flame D) are performed. These flames are part of the database of turbulent reacting flows widely considered as benchmark test cases for validating turbulent-combustion models. In the simulations presented in this paper, the subgrid scale (SGS) closure model adopted for turbulence-chemistry interactions is based on the transport filtered density function (FDF) model. In the FDF model, the transport equation for the joint probability density function (PDF) of scalars is solved. The main advantage of this model is that the filtered reaction rates can be exactly computed. However, the density field, computed directly from the FDF solver and needed in the hydrodynamic equations, is noisy and causes numerical instability. Two numerical approaches that yield a smooth density field are examined. The two methods are based on transport equations for specific sensible enthalpy (h_s) and RT, where R is the gas constant and T is the temperature. Consistency of the two methods is assessed in a bluff-body configuration using Reynolds averaged Navier-Stokes (RANS) methodology in conjunction with the transported PDF method. It is observed that the h_s method is superior to the RT method. Both methods are used in LES of the SM1 burner. In the near-field region, the h_s method produces better predictions of temperature. However, in the far-field region, both methods show deviation from data. Simulations of the SMA1 burner and Flame D are also presented using the h_s method. Some deficiencies are seen in the predictions of the SMA1 burner that may be related to the simple chemical kinetics model and mixing model used in the simulations. Simulations of Flame D show good agreement with data. These results indicate that, while further improvements to the methodology are needed, the LES/FDF method has the potential to accurately predict complex turbulent flames.     

Flow and mixing fields of turbulent bluff-body jets and flames

The mean structure of turbulent bluff-body jets and flames is presented. Measurements of the flow and mixing fields are compared with predictions made using standard turbulence models. It is found that two vortices exist in the recirculation zone; an outer vortex close to the air coflow and an inner vortex between the outer vortex and the jet. The inner vortex is found to shift downstream with increasing jet momentum flux relative to the coflow momentum flux and gradually loses its circulation pattern. The momentum flux ratio of the jet to the coflow in isothermal flows is found to be the only scaling parameter for the flow field structure. Three mixing layers are identified in the recirculation zone. Numerical simulations using the standard k-? and Reynolds stress turbulence models underpredict the length of the recirculation zone. A simple modification to the C₁ constant in the dissipation transport equation fixes this deficiency and gives better predictions of the flow and mixing fields. The mixed-is-burnt combustion model is found to be adequate for simulating the temperature and mixing field in the recirculation zone of the bluff-body flames.     

钝体后湍流预混燃烧的PDF模拟

本文采用PDF方法对矩形燃烧室内钝体后的湍流预混火焰进行了数值模拟。脉动速度-频率-标量联合的PDF输运方程用Monte Carlo方法求解,质量、动量和能量的平均值由基于无结构网格的有限体积法求解,压力通过状态方程获得。PDF方程中所需的平均密度、平均速度和压力由有限体积法提供,并将用Monte Carlo方法求出的雷诺应力、化学反应源项和比热比传递给有限体积法。本文对丙烷和空气燃烧的不同简化化学反应机理进行了研究,并与实验结果进行比较,获得满意的结果。     

PDF calculations of piloted premixed jet flames

A series of piloted premixed jet flames with strong finite-rate chemistry effects is studied using the joint velocity-turbulence frequency-composition PDF method. The numerical accuracy of the calculations is demonstrated, and the calculations are compared to experimental data. It is found that all calculations show good agreement with the measurements of mean and rms mixture fraction fields, while the reaction progress is overpredicted to varying degrees depending on the jet velocity. In the calculations of the flame with the lowest jet velocity, the species and temperature show reasonable agreement with the measurements, with the exception of a small region near the centerline where products and temperature are overpredicted and fuel and oxidizer are underpredicted. In the calculations of the flame with the highest jet velocity, however, the overprediction of products and temperature and underprediction of fuel and oxidizer is far more severe. An extensive set of sensitivity studies on inlet boundary conditions, turbulence model constants, mixing models and constants, radiation treatment, and chemical mechanisms is conducted to show that any parameter variation offers little improvement from the base case. To shed light on these discrepancies, diagnostic calculations are performed in which the chemical reactions are artificially slowed. These diagnostic calculations serve to validate the experimental data and to quantify the amount by which the base case calculations overpredict reaction progress. Improved calculations of this flame are achieved only through artificially slowing down the chemical reaction by a factor of about 10. The mixing model behavior in this combustion regime is identified as a likely cause for the observed discrepancy in reaction progress.     

Large Eddy Simulations of a piloted lean premix jet flame using finite-rate chemistry

A Large Eddy Simulation (LES) model capable of accurately representing finite-rate chemistry effects in turbulent premixed combustion is presented. The LES computations use finite-rate chemistry and implicit LES combustion modelling to simulate an experimentally well-documented lean-premixed jet flame stabilized by a stoichiometric pilot. The validity of the implicit LES assumption is discussed and criteria are expressed in terms of subgrid scale Damköhler and Karlovitz numbers. Simulation results are compared to experimental data for velocity, temperature and species mass fractions of CH₄, CO and OH. The simulation results highlight the validity and capability of the present approach for the flame and in general the combustion regime examined. A sensitivity analysis to the choice of the finite-rate chemistry mechanism is reported, this analysis indicates that the one and two-step global reaction mechanisms evaluated fail to capture the reaction layer with sufficient accuracy, while a 20-species skeletal mechanism reproduces the experimental observations accurately including the key finite-rate chemistry indicators CO and OH. The LES results are shown to be grid insensitive and that the grid resolution within the bounds examined is far less important compared to the sensitivity of the finite-rate chemistry representation. The results are analyzed in terms of the flame dynamics and it is shown that intense small scale mixing (high Karlovitz number) between the pilot and the jet is an important mechanism for the stabilization of the flame.     

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.     

Modeling and simulation of titanium dioxide nanoparticle synthesis with finite-rate sintering in planar jets

Numerical simulations of titanium dioxide nanoparticle synthesis in planar, non-premixed diffusion flames are performed. Titania is produced by the oxidation of titanium tetrachloride using a methane–air flame. The flow field is obtained using the two-dimensional Navier–Stokes equations. The methane–air flame and oxidation of titanium tetrachloride are modeled via one-step reactions. Evolution of the particle field is obtained via a nodal method which accounts for nucleation, condensation, coagulation, and coalescence with finite-rate sintering. The modeling of finite-rate sintering is accomplished via the use of uniform primary-particle size distribution. Simulations are performed at two different jet-to-co-flow velocity ratios as well as with finite-rate and instantaneous sintering models. In doing so we elucidate the effect of fluid mixing and finite-rate sintering on the particle field. Results show that highly agglomerated particles are found on the periphery of the eddies, where the collisions leading to nanoparticle coagulation occur faster than nanoparticle coalescence.     

Improved MMC-LES to compute the structure of a mixed-mode turbulent flame series

An improved version of the Lagrangian multiple mapping conditioning model coupled with large eddy simulation (MMC-LES) is applied to compute the structure of a range of turbulent mixed-mode flames stabilised on the Sydney piloted burner with compositionally inhomogeneous inlet. The calculations are compared with detailed measurements. While premixed MMC-LES models are yet to be developed, the most commonly deployed version and code for MMC-LES uses a mixture-fraction-based approach and a sparse-Lagrangian notional particle method that is most suitable for non-premixed combustion. The improvements reported here are due to (i) a more rigorous density coupling approach, (ii) a more refined mixing time scale, and (iii) selective particle intensification to improve localness in physical space as well as mixture fraction space. This intensification is selectively applied in upstream regions where mixed-mode combustion dominates and is relaxed further downstream as the jet flames transition to a non-premixed flame structure. The computed results show good agreement with experimental data hence confirming the feasibility of the improved MMC-LES approach in being able to account for mixed-mode combustion as well as for finite-rate chemistry effects. These promising results warrant future extension of the selective particle intensification method to become automatically adaptive.