Multiple Mapping Conditioning Model Research Articles

A consistent MMC-LES approach for turbulent premixed flames

A multiple mapping conditioning (MMC) approach is coupled with large eddy simulations (LES) for the prediction of the flame propagation, the flame structure and finite rate chemistry effects including NO formation in turbulent premixed flames. The mixing term in MMC relies on a reference field that characterizes the chemical composition within the flame and it is typically taken to be the reaction progress variable. In LES, this reference field needs to be artificially thickened to be resolved on the LES grid. A novel MMC mixing time scale model that had been developed with the aid of direct numerical simulation (DNS), has been adapted to account for the thickening of the reference field. Equally, the particle selection mechanism for particle mixing has been modified to ensure localness of mixing in composition space and to preserve the prediction of a realistic (thin) flame thickness by the stochastic particles. A set of freely propagating turbulent premixed flames with varying equivalence ratios is used to test the performance of the MMC model. Results of MMC-LES are compared with reference DNS that include full chemistry. Additional simulations were carried out where MMC is coupled with quasi-DNS where the flow field is fully resolved but the reaction progress variable field is thickened and its source term is closed with a flamelet approach. This allows to isolate the different modelling assumptions invoked in MMC-LES. Deviations from the exact solution can then be associated with the MMC approach and with thickening of the reference field separately and they are small. The MMC-LES predictions of the turbulent premixed flame propagation speed and flame structure are very good and finite rate chemistry effects can be captured well which is demonstrated by good agreement of predicted NO with DNS data.

Coupling the Multiple Mapping Conditioning Mixing Model with Reaction-diffusion Databases in LES of Methane/air Flames

ABSTRACT The Multiple Mapping Conditioning (MMC) model for sparse Lagrangian Large-Eddy Simulations (LES) is coupled with tabulated chemistry based on reaction-diffusion manifolds (REDIM). In addition to assessing the reduction in computational cost compared to direct integration on the fly, a comparison with the previously implemented REDIM solver based on the Eulerian Stochastic Fields (ESF) is performed. The methane/air Sandia flames D and E are selected as a benchmark, which are amenable to a 2D REDIM database having a passive scalar and one progress variable as dimensions. A comparison between direct integration ESF and MMC shows that the latter gives better predictions of the conditional means and fluctuations, offering at the same time a significantly reduced computational cost. When coupled with REDIM, MMC and ESF models provide similar results, indicating the significant sensitivity of the predictions to the thermochemical states which are restricted to the same REDIM manifold. When the states are projected back onto the REDIM surface in the direction where the progress variables remain almost constant, accuracy is lost in the cold fuel-lean region with MMC. Furthermore, the computational performance gain in MMC compared to ESF is reduced when the REDIM tabulation is used, pointing to non-chemistry related computational bottlenecks in the MMC algorithms.

Parametric sensitivities of the generalized binomial Langevin–multiple mapping conditioning model

The binomial Langevin model (BLM) predicts mixture fraction statistics including higher moments excellently, but imposing boundedness for the large scalar spaces typically associated with chemically reacting flows becomes intractable. This central difficulty can be removed by using the mixture fraction as the reference variable in a generalized multiple mapping conditioning (MMC) approach. The resulting probabilistic BLM–MMC formulation has several free parameters that impact the turbulence–chemistry interactions in complex flows: the dissipation timescale ratio, the locality in selecting pairs of particles for mixing, and the fraction of particles mixed per time step. The impact of parametric variations on the behavior of the BLM–MMC model is investigated for a complex flow featuring auto-ignition to determine model sensitivities and identify optimal values. It is shown that only the mixture fraction rms is sensitive to the dissipation timescale ratio with the expected behavior of an increased ratio leading to a reduction in rms. Controlling locality by increasing the maximum possible distance between paired particles in reference space has a similar impact. Increasing the fraction of particles mixed only affects reacting scalars by advancing ignition. The modified Curl's model is used for the mixing process and the specified amount of mixing principally controls the local extinction and reignition behavior. It is further shown that the standard value of the dissipation timescale ratio is satisfactory; the amount of mixing should be half that specified by Curl's model; and the distance between particle pairs in reference space should be proportional to the diffusion length scale.

Modelling of a turbulent premixed flame series using a new MMC-LES model with a shadow position reference variable

A new multiple mapping conditioning model with a shadow position reference variable for localising mixing is implemented in a large eddy simulation solver and validated against three experimental turbulent premixed flames. The shadow position used here has diffusion coefficient which is linked to the turbulent propagation speed. Localisation of mixing produces a flamelet-like inner structure in shadow position space and the structure in physical space is determined by the subgrid dispersion of shadow and physical positions. The lengths of the predicted flames match the trends observed for the experimental flames, and the predicted inner structures are consistent with what is expected for these flames in the thin reaction zones regime. The statistical predictions are in very good quantitative agreement with the experimental data for velocity and major species. Errors for reaction progress variable, turbulent kinetic energy and minor species (carbon monoxide) are generally larger yet mostly within the ranges reported in the literature.

Numerical investigation of flow and scalar fields of piloted, partially-premixed dimethyl ether/air jet flames using stochastic multiple mapping conditioning approach

Turbulent piloted partially-premixed jet flames of dimethyl ether/air have been investigated using the Reynolds-Averaged Navier-Stokes (RANS) based stochastic multiple mapping conditioning (MMC) approach. The MMC model combines the advantages of mapping closure concept, conditional moment closure, and probability density function (PDF) methods. In the present work, the evolution of the mapping function in the reference space is described using a single reference variable based on the mixture fraction. To achieve localness in the composition space, the turbulent mixing is conditioned on the reference space. In addition to localness, MMC preserves the various characteristics of a good mixing model, such as linearity, Gaussianity, independence, and boundedness. A skeletal chemical mechanism for dimethyl ether is used, which consists of 28 species and 24 reversible reactions. Modified Curl's mixing model is adapted to model the micro-mixing term, and standard two-equation k−ε turbulence model is used to model the Reynolds stress term with a modified set of constants. The computed conditional and unconditional statistics demonstrate an excellent agreement with the available experimental data even for flame displaying a large degree of local extinction and re-ignition. For these flames, radical species distribution conditioned on mixture fraction confirms the physical separation between the OH and CH2O species, which was earlier reported using simultaneous laser-induced fluorescence measurements of these radicals. Predicted unconditional scalar dissipation rate for DME-D and DME-F flames reaffirm that near the inlet an overlap between OH and CH2O radicals are observed where the scalar dissipation rate is fairly high. Subsequently, a distinct separation between these radicals becomes evident at downstream locations where the scalar dissipation rate decreases. For the flames investigated here, a strong correlation is noticed between the peak heat release rate and the reaction rate indicator, ROH based on the product of concentrations of OH and CH2O radicals.

A mixture-fraction-based hybrid binomial Langevin-multiple mapping conditioning model

Generalized Multiple Mapping Conditioning (MMC) allows for the use of any physical quantity to represent the required reference variable provided that it delivers the desired behavior. The binomial Langevin model (BLM) has been shown to predict higher statistical moments with good accuracy. However, joint-scalar modeling for many scalars becomes problematic because scalar bounds must be specified as conditional on other scalars to preserve elemental balances. The resulting volumes in state space become exceptionally complex for realistic problem sizes. In the current work, this central difficulty is avoided by using only velocity and mixture fraction statistics from the BLM with the latter used as the MMC reference variable. The principal advantage of this method is that the implementation of the binomial Langevin mixture fraction is relatively straightforward and provides a direct physical link to MMC. The MMC model is closed using an augmented modified Curl’s model where the selection of particle pairs for (turbulent) mixing ensures proximity in reference space and a corresponding closeness in physical space. The method is evaluated for a lifted methane jet flame undergoing auto-ignition in a vitiated coflow. Most of the major features of the flow are well reproduced and found to generally outperform other modeling approaches, including Large Eddy Simulations using simplified treatments of turbulence–chemistry interactions such as unsteady flamelet/progress variable descriptions.

Application of a multiple mapping conditioning mixing model to ECN Spray A

The engine combustion network (ECN) Spray A is modelled using the Reynolds-averaged Navier–Stokes-transported probability density function (RANS-TPDF) approach to validate the application of a new multiple mapping conditioning (MMC) mixing model to multiphase reactive flows. The composition TPDF equations are solved using a Lagrangian stochastic approach and the spray is modelled with a discrete particle approach. The model is first validated under non-reacting conditions (at 900 K) using experimental mixture-fraction data. Reactive simulations are then performed for three different ambient temperatures (800, 900, 1100 K) and oxygen concentrations (13, 15, 21%) at an ambient density of 22.8 kg/m3. The MMC mixing model is compared with the interaction by exchange with the mean (IEM) mixing model. The ignition delay predictions are not sensitive to the mixing model and are predicted well by both the mixing models under all the tested ambient conditions. The IEM model overpredicts the flame lift-off length (FLOL) at high temperature and high oxygen conditions with a mixing constant Cϕ=2. The MMC model with Cϕ=2 and a target correlation coefficient rt=0.935 between the mixture fraction and a reference variable used to condition mixing predicts good FLOL under all the conditions except 800 K. It is demonstrated that the lift-off length is controllable by changing the target correlation coefficient, while Cϕ and therefore the mixing fields are held fixed. In comparison to the MMC model, the IEM model predicts a higher variance of temperature conditioned on mixture fraction near the flame base owing to its lacking the property of localness. The mixing distance between the notional TPDF particles in the composition space is also higher with the IEM model and it is demonstrated that by changing rt, different levels of mixing locality can be achieved.

A multiple mapping conditioning mixing model with a mixture-fraction like reference variable. Part 2: RANS implementation and validation against a turbulent jet flame

In Part I of this paper, a novel multiple mapping conditioning (MMC) mixing model is proposed and verified using ideal flow test cases. In Part II, the MMC mixing model is validated in the context of inhomogeneous turbulent reactive flows using Sandia flame D–F experimental data. The model is implemented alongside a standard k−ϵ CFD solver in the OpenFOAM code called mmcFoam. The MMC model has two main model constants; Cϕ which has a strong effect on the unconditional statistics and rt which controls the localness in mixture fraction space and hence the level of conditional fluctuations. A detailed series of tests explores the sensitivity of the model predictions towards variations in Cϕ and rt with Cϕ=2 and rt=0.935 found to work well in flame F and subsequently in flames D and E as well. With these settings the model accurately predicts the increasing level of localised extinction with increasing jet Reynolds number. Sensitivity to the number of particles per cell and the chemical kinetics scheme is also investigated.

A PDF approach to thin premixed flamelets using multiple mapping conditioning

This work discusses the principles and application of the multiple mapping conditioning (MMC) framework to the simulation of turbulent premixed combustion. While it is clear that, conceptually, MMC allows for the enforcement of the consistency of probability density function models with the flamelet limit, the implementation of this idea is non-trivial and requires substantial analysis. This work focuses on the principles of selecting reference variables suitable for the simulation of general premixed combustion cases. Reference variables based on the level set, shadow positions and progress variable are considered. The concept of second conditioning introduces a hierarchy of MMC models to combine both generality and computational efficiency. The paper presents stochastic simulations to demonstrate these principles.

Multiple mapping conditioning for silica nanoparticle nucleation in turbulent flows

In this paper a study of the sparse-Lagrangian multiple mapping conditioning (MMC) model for the synthesis of silica nanoparticles is presented. This is the first attempt using the model to simulate the formation of kinetically limited solid state species. A simplified differential diffusion model, involving a modified mixing timescale, is used to account for the much lower diffusion of the solid species. The model is validated against DNS of a counterflowing double shear layer of silane and hot combustion products. The reduced reaction mechanism involves silane decomposition and clustering leading to silica. The model is implemented in the context of large eddy simulations (LES). The MMC model allows a sparse distribution of stochastic particles for computing the reactive scalar field, including silica number density, by enforcing the property of localness in mixture fraction space onto the mixing model. Gas phase mixture fraction and fast reactive species are predicted with acceptable accuracy. The prediction of silica number density is qualitatively correct although the magnitude is underpredicted somewhat and the rms is greatly underpredicted. The effects of differential diffusion are clearly visible in the modelling and the simplified model correctly predicts the trend of increasing silica formation when differential diffusion is included.

MMC-LES simulations of turbulent piloted flames with varying levels of inlet inhomogeneity

This paper presents simulations of piloted turbulent flames where compositional inlet conditions vary from homogeneous to inhomogeneous. The combustion model is sparse-Lagrangian multiple mapping conditioning (MMC) coupled with large eddy simulation (LES). This is a first attempt using MMC-LES to model flows where multi-modes of combustion may occur. The studied burner has two concentric tubes surrounded by an annular pilot. The central tube carrying methane can slide within the outer air tube to induce compositional inhomogeneity at the burner exit plane. This leads to enhanced flame stability at some optimal inner fuel tube recess distance. For comparison purposes, model results are presented firstly for a case with compositional homogeneity at the inlet plane and secondly for the inhomogeneous case. The MMC-LES model shows a very good agreement with the experiment for the homogeneous-inlet case where the prevalence of non-premixed combustion is observed. The inhomogeneous-inlet case features a transition from a premixed flame structure close to the burner to a diffusion flame further downstream. It represents an extreme test of the chosen MMC model which incorporates the enforcement of mixing locally in an extended space comprised of the LES reference mixture fraction and physical location. The computed results show a fair agreement with the data close to the nozzle; closely approaching but not quite achieving the premixed structure. The return to a broad, distributed diffusion flame away from the nozzle is, however, captured very well.

A multiple mapping conditioning model for differential diffusion

This work introduces modeling of differential diffusion within the multiple mapping conditioning (MMC) turbulent mixing and combustion framework. The effect of differential diffusion on scalar variance decay is analyzed and, following a number of publications, is found to scale as Re−1/2. The ability to model the differential decay rates is the most important aim of practical differential diffusion models, and here this is achieved in MMC by introducing what is called the side-stepping method. The approach is practical and, as it does not involve an increase in the number of MMC reference variables, economical. In addition we also investigate the modeling of a more refined and difficult to reproduce differential diffusion effect – the loss of correlation between the different scalars. For this we develop an alternative MMC model with two reference variables but which also makes use of the side-stepping method. The new models are successfully validated against DNS results available in literature for homogenous, isotropic two scalar mixing.

Hybrid multiple mapping conditioning modeling of local extinction

Multiple mapping conditioning (MMC) explicitly includes a link between the physical velocity and the conditioning variable. The usual method of obtaining values for the conditional physical velocity is to define the value of the conditioning variable and use a model for the velocity. In contrast to the conditional moment closure (CMC) approach, the conditioning variable is not a real quantity and hence provides greater flexibility in its definition. The hybrid binomial Langevin–MMC model has the advantage of naturally incorporating velocity–scalar interactions through the binomial Langevin model and the joint probability density function (PDF) is here used to define a reference variable for the MMC part of the model. The current approach has the advantage that difficulties encountered with the binomial Langevin model in modeling scalars with non-elementary bounds are removed. The formulation of the closure leads to locality in scalar space and permits the use of simple approaches for transport in the reference space. The evaluation of the technique is here extended to a jet flame with moderately-high levels of extinction (Sandia Flame E). A direct comparison with experimental data and the EMST (Euclidean minimum spanning tree) model for temperature, species concentrations and burning indexes suggests that the approach performs well.

Multiple mapping conditioning of turbulent jet diffusion flames

The multiple mapping conditioning (MMC) approach is applied to two non-piloted CH 4 / H 2 / N 2 turbulent jet diffusion flames with Reynolds numbers of Re = 15,200 and 22,800. The work presented here examines primarily the suitability of MMC to simulate CH 4 / H 2 flames with varying Re numbers. The equations are solved in a prescribed Gaussian reference space with only one stochastic reference variable emulating the fluctuations of mixture fraction. The mixture fraction is considered as the only major species on which the remaining minor species are conditioned. Fluctuations around the conditional means are ignored. It is shown that the statistics of the mapped reference field are an accurate model for the statistics of the physical field for both flames. A transformation of the Gaussian reference space introduced in previous work on MMC is used to express the MMC model in the same form as CMC. The most important advantage of this transformation is that the conditionally averaged scalar dissipation term is in a closed form. The corresponding temperature and reactive species predictions are generally in good agreement with experimental data. The application to real laboratory flames and the assessment of the new conditional scalar dissipation model for the closure of the singly conditioned CMC equation is the major novelty of this paper. The results are therefore primarily examined with respect to changes of the conditionally averaged quantities in mixture fraction space.

A sparse-Lagrangian multiple mapping conditioning model for turbulent diffusion flames

A sparse-Lagrangian multiple mapping conditioning (MMC) model for turbulent diffusion flames is presented and tested against experimental data for a piloted methane/air jet diffusion flame (Sandia Flame D). The model incorporates a large eddy simulation for the flow field and a stochastic multiple mapping conditioning (MMC) approach for the reactive scalars. The stochastic MMC models the filtered density function of the scalar composition field. The numerical implementation involves a sparse-Lagrangian particle scheme in which there are fewer particles than there are LES grid cells. Predictions of similar accuracy to previously published Flame D simulations are achieved using only 35,000 particles (of these only 10,000 are chemically active). Sub-filter conditional dissipation is modelled by interactions between pairs of particles which are closely located in a reference mixture fraction space interpolated from the underlying Eulerian filtered field. A model is developed for the mixing time-scale which is proportional to the distance between mixing particles. It is shown that the time-scale can be adjusted to achieve good predictions for time-averaged mean and fluctuating statistics of passive and reactive scalars.

A Generalised Multiple Mapping Conditioning Approach for Turbulent Combustion

This paper follows the evolution in understanding of the multiple mapping conditioning (MMC) approach for turbulent combustion and reviews different implementations of MMC models. As the MMC name suggests, the original version represents a consistent combination of CMC-type conditional equations (conditional moment closure) and generalised mapping closure. It seems that the strength of the MMC model, and especially that of its stochastic version, lies in a more general (and much more transparent) interpretation. In this new generalised interpretation, we can replace complicated derivations by physical reasoning and the model appears to be a natural extension of modelling approaches developed in recent decades. MMC can be seen as a methodology for enforcing certain known characteristics of turbulence on a conventional mixing model. This is achieved by localising the mixing operation in a reference space. The reference space variables are selected to emulate the properties of a turbulent flow which have a strong effect on reactive quantities. The best and simplest example is an MMC model which has a single reference variable emulating the mixture fraction. In diffusion flames turbulent fluctuations of reacting quantities are strongly correlated with fluctuations of the mixture fraction. By making mixing local in the reference mixture fraction space a CMC-type mixing closure is enforced. In the original interpretation of MMC the reference variables are modelled as Markov processes. Since the reference variables should emulate properties of turbulent flows as realistically as possible the next step, and the basis of generalised MMC, is to remove the Markovian restriction and set reference variables equal to traced Lagrangian quantities within DNS or LES flow fields. Indeed, no Markov value can emulate the mixture fraction better than the mixture fraction itself. (Using a Markov vector process of dimension higher than the number of conditioning variables represents a more economical alternative for producing reference variables in generalised MMC.) The generalised MMC approach effectively incorporates the mixture fraction-based models, the PDF methods and LES/DNS techniques into a single methodology with possibility of blending useful features developed previously for conventional models. The generalised approach to MMC stimulates a more flexible understanding of simulations using sparsely placed Lagrangian particles as tools that may provide accurate joint distributions of reactive scalars at relatively low computational cost. The physical reasoning behind the new interpretation of MMC is supported by example computations for a partially premixed methane/air diffusion flame (Sandia Flame D). The scheme utilises LES for the dynamic field and a sparse-Lagrangian filtered density function method with MMC mixing for the scalar field. Two different particle mixing schemes are tested. Simulations are performed using only 35,000 Lagrangian particles (of these only 10,000 are chemically active) on a single workstation. The relatively low computational cost allows the use of realistic chemical kinetics containing 34 reactive species and 219 reactions.

‘Hybrid’ multiple mapping conditioning on passive and reactive scalars

A two-dimensional, hybridized multiple mapping conditioning (MMC) model is used to model local extinction and reignition phenomena in homogeneous, isotropic decaying turbulence. The equations are solved in a prescribed, jointly Gaussian reference space with stochastic reference variables emulating the fluctuations of the mixture fraction and normalized sensible enthalpy conditioning variables. In “pure” MMC the joint PDF of the conditioning scalars is a solved quantity. Here we use a hybrid method where the time evolution of the marginal PDF for mixture fraction is solved and a presumed β-PDF is used for the conditional distribution of the normalized sensible enthalpy. Model results are compared with DNS in three flame cases with varying levels of local extinction, up to global extinction. Results for principal chemical species are in very good agreement with DNS and those for intermediate species are also satisfactory. The doubly conditioned MMC yields results which are considerably more accurate than those by modeling with conditioning on mixture fraction alone. A transformation of the Gaussian reference space casts the MMC model in the same form as conditional moment closure (CMC). The great advantage is that the MMC model contains the doubly conditioned scalar dissipation terms in closed form and these are generally found to be in good agreement with the DNS data.

Matching conditional moments in PDF modelling of nonpremixed combustion

The rate of generation of fluctuations with respect to the scalar values conditioned on the mixture fraction, which significantly affects turbulent nonpremixed combustion processes, is examined. Simulation of the rate in a major mixing model is investigated and the derived equations can assist in selecting the model parameters so that the level of conditional fluctuations is better reproduced by the models. A more general formulation of the multiple mapping conditioning (MMC) model that distinguishes the reference and conditioning variables is suggested. This formulation can be viewed as a methodology of enforcing certain desired conditional properties onto conventional mixing models. Examples of constructing consistent MMC models with dissipation and velocity conditioning as well as of combining MMC with large eddy simulations (LES) are also provided.

The modeling of turbulent reactive flows based on multiple mapping conditioning

A new modeling approach—multiple mapping conditioning (MMC)—is introduced to treat mixing and reaction in turbulent flows. The model combines the advantages of the probability density function and the conditional moment closure methods and is based on a certain generalization of the mapping closure concept. An equivalent stochastic formulation of the MMC model is given. The validity of the closuring hypothesis of the model is demonstrated by a comparison with direct numerical simulation results for the three-stream mixing problem.