An age extended progress variable for conditioning reaction rates

Combustion instability due to thermo-acoustic interactions in high-speed propulsion devices such as gas turbines and rocket engines result from pressure waves with very large amplitudes propagating back and forth in the combustion chamber. Exposure to the pressure fluctuations over a long period of time can lead to a cataclysmic failure of engines. The underlying physics governing the generation of the thermo-acoustic instability is a complex interaction among heat release, turbulence, and acoustic waves. Currently, it is very difficult to accurately predict the expected level of oscillations in a combustor. Hence development of strategies and engineering solutions to mitigate thermo-acoustic instability is an active area of research in both academia and industry. In this work, we carry out numerical modeling of thermo-acoustic instability in a self-excited, laboratory scale, model rocket combustor developed at Purdue University. Two different turbulent combustion models to account for turbulence-chemistry interactions are considered in this study, the flamelet model and the transported probability density function (PDF) method. <br>In the flamelet modeling approach, detailed chemical kinetics can be easily incorporated at a relatively low cost in comparison to other turbulent combustion models and it also accounts for turbulence-chemistry interactions. The flamelet model study is divided into two parts. In first part, we examine the effect of different numerical approaches for implementing the flamelet model. In advanced modeling and simulations of turbulent combustion, the accuracy of model predictions is affected by physical model errors as well as errors that arise from the numerical implementation of models in simulation codes. Here we are mainly concerned with the effect of numerical implementation on model predictions of turbulent combustion. Particularly, we employ the flamelet/progress variable (FPV) model and examine the effect of various numerical approaches for the flamelet table integration, with presumed shapes of PDF, on the FPV modeling results. Three different presumed-PDF table integration approaches are examined in detail by employing different numerical integration strategies. The effect of the different presumed-PDF table integration approaches is examined on predictions of two real flames, a laboratory-scale turbulent free jet flame, Sandia Flame D and the self-excited resonance model rocket combustor. Significant difference is observed in the predictions both of the flames. The results in this study further support the claims made in previous studies that it is imperative to preserve the laminar flamelet structure during integration while using the flamelet model to achieve better predictions in simulations. In the second part of the flamelet modeling study, computational investigations of the coupling between the transient flame dynamics such as the ignition delay and local extinction and the thermo-acoustic instability developed in a self-excited resonance combustor to gain deep insights into the mechanisms of thermo-acoustic instability. A modeling framework that employs different flamelet models (the steady flamelet model and the flamelet/progress variable approach) is developed to enable the examination of the effect of the transient flame dynamics caused by the strong coupling of the turbulent mixing and finite-rate chemical kinetics on the occurrence of thermo-acoustic instability. The models are validated by using the available experimental data for the pressure signal. Parametric studies are performed to examine the effect of the occurrence of the transient flame dynamics, the effect of artificial amplification of the Damkohler number, and the effect of neglecting mixture fraction fluctuations on the predictions of the thermo-acoustic instability. The parametric studies reveal that the occurrence of transient flame dynamics has a strong influence on the onset of the thermo-acoustic instability. Further analysis is then conducted to localize the effect of a particular flame dynamic event, the ignition delay, on the thermo-acoustic instability. The reverse effect of the occurrence of the thermo-acoustic instability on the transient flame dynamics in the combustor is also investigated by examining the temporal evolution of the local flame events in conjunction with the pressure wave propagation. The above observed two-way coupling between the transient flame dynamics (the ignition delay) and the thermo-acoustic instability provides a plausible mechanism of the self-excited and sustained thermo-acoustic instability observed in the combustor.<br>The second turbulent combustion model considered in this study is the transported PDF method. The transported PDF method is one of the most attractive models because it treats the highly-nonlinear chemical reaction source term without a closure requirement and it is a generalized model for a wide range of turbulent combustion problems.Traditionally, the transported PDF method has been used to model low-Mach number, incompressible flows where the pressure is assumed to be thermodynamically constant. Since there is significant pressure fluctuations in the model rocket combustor, the flow is highly compressible and it is necessary to account for this compressibility in the transported PDF method. In the past there has been very little work to model compressible reactive flows using the transported PDF and no effort has been made to model thermo-acoustic instability using the transported PDF method. There is a pressing need to further examine and develop the transported PDF method for compressible reactive flows to broaden our understanding of physical phenomenon like thermo-acoustic instability, interaction between combustion and strong shock and expansion waves, coupling between acoustic and heat release which are observed in high-speed turbulent combustion problems. To address this, a modeling framework for compressible turbulent reactive flows by the using the transported PDF method is developed. This framework is validated in a series of test cases ranging from pure mixing to a supersonic turbulent jet flame. The framework is then used to study the thermo-acoustic interactions in the self-excited model rocket combustor.

A new perspective on modelling passive scalar conditional mixing statistics in turbulent spray flames

The mapping closure of Chen et al. [Phys. Rev. Lett., 63, 1989] is a transported probability density function (PDF) method that has proven very efficient for modelling of turbulent mixing in homogeneous turbulence. By utilizing a Gaussian reference field, the solution to the mapping function (in homogeneous turbulence) can be found analytically for a range of initial conditions common for turbulent combustion applications, e.g. for binary or trinary mixing. The purpose of this paper is to investigate the possibility of making this solution a presumed mapping function (PMF) for inhomogeneous flows. The PMF in turn will imply a presumed mixture fraction PDF that can be used for a wide range of models in turbulent combustion, e.g. flamelet models, the conditional moment closure (CMC) or large eddy simulations. The true novelty of the paper, though, is in the derivation of highly efficient, closed algebraic expressions for several existing models of conditional statistics, e.g. for the conditional scalar dissipation/diffusion rate or the conditional mean velocity. The closed form expressions nearly eliminates the overhead computational cost that usually is associated with nonlinear models for conditional statistics. In this respect it is argued that the PMF is particularly well suited for CMC that relies heavily on manipulations of the PDF for consistency. The accuracy of the PMF approach is shown with comparison to DNS of a single scalar mixing layer to be better than for the β-PDF. Not only in the shape of the PDF itself, but also for all conditional statistics models computed from the PDF.

Three-dimensional compressible simplified chemistry-based direct numerical simulations (DNS) of statistically planar turbulent stratified flames at global equivalence ratios ⟨φ⟩ = 0.7 and ⟨φ⟩ = 1.0 have been carried out to analyze the statistical behavior of the transport of the variance of the Favre fuel mass fraction fluctuations (where , and are Reynolds average, Favre mean, and Favre fluctuation of a general quantity q where ρ is the gas density) in the context of Reynolds-averaged Navier-Stokes (RANS) simulations. It has been found that algebraic expressions are inadequate for predicting in low Damköhler number combustion, and a transport equation for may need to be solved. The statistical behaviors of and the unclosed terms of the transport equation (i.e., the terms originating from turbulent transport T 1 , reaction rate T 3 , and molecular dissipation D 2) have been analyzed in detail. It has been found, that T 3 and D 2 remain leading order contributors to the transport in all cases, whereas, the contribution of T 1 remains small in ⟨φ⟩ = 1.0 cases, but plays a more important role in ⟨φ⟩ = 0.7 cases. Through an a-priori DNS analysis, the modeling of T 1, T 3, and D 2 has been addressed in detail. A model has been identified for T 1, which provides satisfactory agreement with the DNS data. The models for T 3 and D 2, which were originally proposed for high Damköhler number flames, have been modified for low Damköhler combustion in the context of RANS simulations, and the predictions of the modified models are found to be in good agreement with the corresponding quantities extracted from the DNS data.

In this paper, a new "presumed" Probability Density Function (PDF) approach coupled with a Lagrangian tracking method is proposed for turbulent combustion modeling. The test and the investigation of the model are conducted by comparing the model results with DNS data for a premixed flame subjected in a decaying turbulent field. The newly constructed PDF, which incorporates the instantaneous chemical reaction term, demonstrates consistent improvement over conventional assumed PDF models. It has been found that the time evolution of the mean scalar, the variance and the mean reaction rate are strongly influenced by a parameter deduced by a Lagrangian equation which takes into account explicitly the local reaction rate. Tests have been performed for a moderate Damk?hler number, and it is expected the model may cover a broader range of Damk?hler number. The comparison with the DNS data demonstrates that the proposed model may be promising and affordable for implementation in a moment-equation solver.

Hybrid multiple mapping conditioning modeling of local extinction

A simplified CMC approach based on tabulated reaction rates applied to a lifted methane–air jet flame

Conditional moment closure for turbulent combustion

Strained flamelets for turbulent premixed flames, I: Formulation and planar flame results

This work presents an analysis of the external structure of diffusion flames established with liquid fuel vaporisation (ethanol) in a counterflow configuration. The model assumes that part of the fuel is in the liquid phase (droplets) dispersed in the gas phase, which presents an initial fuel mass fraction. The spray–flamelet model is distinguished from the flamelet model by the presence of droplets in the reaction zone. There is no relative velocity between the droplets and the gas phase and the droplets vaporise completely before the stagnation plane. The mathematical description of the dilute spray–flamelet is made by the conservation equations with source terms corresponding to the fuel droplets. In the fuel species conservation equation, the source term has a strength proportional to the droplets vaporisation rate. In the energy conservation equation, the source term (heat sink) has a strength proportional to the product of the droplet vaporisation rate and the latent heat. The model exhibits a parameter defined as the spray combustion parameter M that combines the chemical reaction, flow field and spray properties involved in the problem. An analysis is performed to determine how the dilute spray–flamelet is affected by the liquid fuel injection position, the spray combustion parameter M and the temperature and fuel mass fraction Φ in the incoming fuel stream. A uniform distribution is considered to describe the droplets. The results confirm the expected influence of the spray combustion parameter M and the ambient temperature on the spray–flamelet. Effects on the temperature profile can be observed on the spray–flamelet by varying the liquid fuel injection position and the fuel mass fraction Φ in the incoming fuel stream. The results also indicate that, depending on the value of M, there is a value of Φ above which a burning regime is imposed that is almost totally controlled by the incoming fuel stream conditions.

The application of machine learning algorithms to model subgrid-scale filtered density functions (FDFs), required to estimate filtered reaction rates for Large Eddy Simulation (LES) of chemically reacting flows, is discussed in this chapter. Three test cases, i.e., a low-swirl premixed methane-air flame, a MILD combustion of methane-air mixtures, and a kerosene spray turbulent flame, are presented. The scalar statistics in these test cases may not be easily represented using the commonly used presumed shapes for modeling FDFs of mixture fraction and progress variable. Hence, the use of ML methods is explored. Particularly, deep neural network (DNN) to infer joint FDFs of mixture fraction and progress variable is reviewed here. The Direct Numerical Simulation (DNS) datasets employed to train the DNNs in each test case are described. The DNN performances are shown and compared to typical presumed probability density function (PDF) models. Finally, this chapter examines the advantages and caveats of the DNN-based approach.

Conditional statistics of a laboratory-scale lifted turbulent H2/N2 flame using direct numerical simulation

Direct numerical simulations (DNS) of statistically planar flames at moderate and high Karlovitz number (Ka) have been used to perform an a priori evaluation of a presumed-PDF model approach for filtered reaction rate in the framework of large eddy simulation (LES) for different LES filter sizes. The model is statistical and uses a presumed shape, based here on a beta-distribution, for the sub-grid probability density function (PDF) of a reaction progress variable. Flamelet tabulation is used for the unfiltered reaction rate. It is known that presumed PDF with flamelet tabulation may lead to over-prediction of the modelled reaction rate. This is assessed in a methodical way using DNS of varying complexity, including single-step chemistry and complex methane/air chemistry at equivalence ratio 0.6. It is shown that the error is strongly related to the filter size. A correction function is proposed in this work which can reduce the error on the reaction rate modelling at low turbulence intensities by up to 50%, and which is obtained by imposing that the consumption speed based on the modelled reaction rate matches the exact one in the flamelet limit. A second analysis is also conducted to assess the accuracy of the flamelet assumption itself. This analysis is conducted for a wide range of Ka, from 6 to 4100. It is found that at high Ka this assumption is weaker as expected, however results improve with larger filter sizes due to the reduction of the scatter produced by the fluctuations of the exact reaction rate.

Combustion in a CI engine is initiated by self-ignition of fuel–air mixture caused by high pressure and high temperature; a process known as autoignition. Autoignition is a challenging problem to simulate as the temperature increases from initial temperature to the adiabatic flame temperature in a very short duration. Numerical study of a turbulent flow using RANS/LES encounters a closure problem. Accuracy of closure models can be improved through experimental results, theoretical reasoning and direct numerical simulation (DNS) data. A review of DNS of autoignition in a turbulent non-premixed medium is presented in this chapter. As observed from DNS study, autoignition sites in a turbulent non-premixed medium are not randomly distributed but follow a pattern in the mixture fraction-scalar dissipation rate space. Turbulent flow is always three-dimensional in nature. 2D DNS of autoignition shows that ignition delay time increases with increase in initial turbulence intensity, which contradicts with the experimental observation. 3D DNS of autoignition resolves this conflict. The conflict is mainly due to the absence of vortex-stretching phenomenon in 2D DNS. Homogeneous charge compression ignition (HCCI) is being considered as one of the strategies toward improving performance of conventional CI engines. However, HCCI engines suffer from drawbacks like lack of control over combustion and limited operating regime. One of the modifications suggested in the HCCI technology to overcome these drawbacks is the use of stratification in the fuel–air mixture. Therefore, a few DNS studies on autoignition in the stratified medium have been discussed here. Further, discussion on the conditional moment closure (CMC) model and its validation using DNS data has been presented. Ignition delay time predicted by CMC was found to be in good agreement with DNS predictions.

A new method for the approximation of multivariate scalar probability density functions (PDFs) in turbulent reacting flow by means of a joint presumed discrete distribution (jPDD) is presented. The jPDDs can be generated with specified mean values and variances as well as covariances. Correlations between variables – e.g. fluctuating mixture fractions and/or reaction progress – can thereby be taken into account. In this way the new approach overcomes an important limitation of ordinary presumed PDF methods, where statistical independence between the variables is often assumed. Different methods are presented to generate discrete distributions, based either on biased random number generators or on mixing models familiar from PDF transport models. The new approach is extensively validated on a turbulent flow configuration with simultaneous mixing and reaction. Large eddy simulation data as well as results from a transported PDF model are used for the validation of the jPDD approach. The comparison shows that in particular distributions generated with mixing models are able to predict mean reaction rates accurately. For the configuration considered, the neglect of correlations results in significant underestimation of reaction rates. Moreover it is found that higher statistical moments (e.g. the skewness) can influence reaction rates. The consequences for the generation of jPDDs are discussed. In summary, the new jPDD model has the potential to be significantly more accurate than established presumed PDF methods, because correlations between fluctuating variables can be taken into account. At the same time, the new approach is nearly as efficient as standard presumed PDF formulations, since mean rates are computed in a pre-processing step and stored in look-up tables as a function of the first and second moments of the relevant variables.