Computational Models for Turbulent Reacting Flow
Abstract
This book presents the current state of the art in computational models for turbulent reacting flows, and analyzes carefully the strengths and weaknesses of the various techniques described. The focus is on formulation of practical models as opposed to numerical issues arising from their solution. A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods are covered in detail. An introduction to the theory of turbulent and turbulent scalar transport is provided for completeness. The book is aimed at chemical, mechanical, and aerospace engineers in academia and industry, as well as developers of computational fluid dynamics codes for reacting flows.
... Additionally, the rate of self-assembly of lipids is not fast compared with the rate of mixing of water and ethanol at the SGS, which is referred to as micromixing [172]. These kind of flows, where the time scales for turbulent mixing and self-assembly are comparable and micromixing effects are important, are typically modeled using a Probability Distribution Function (PDF) approach for turbulent mixing [173,174]. The PDF (f ϕ ) is a function of the lipid concentrations, water and ethanol volume fractions, as well as the space and time coordinates; from which the individual species concentrations and the self-assembly source term may be obtained [173]. ...
... These kind of flows, where the time scales for turbulent mixing and self-assembly are comparable and micromixing effects are important, are typically modeled using a Probability Distribution Function (PDF) approach for turbulent mixing [173,174]. The PDF (f ϕ ) is a function of the lipid concentrations, water and ethanol volume fractions, as well as the space and time coordinates; from which the individual species concentrations and the self-assembly source term may be obtained [173]. Solution methods for the PDF include transported PDF models, where we explicitly solve transport equations for the PDF, for details see [175]. ...
... Another approach is the finite-mode presumed PDF approach, where the PDF is presumed to be a composition of a finite number of Dirac delta functions [173], ...
Nucleic acids such as mRNA have emerged as a promising therapeutic modality with the capability of addressing a wide range of diseases. Lipid nanoparticles (LNPs) as a delivery platform for nucleic acids were used in the COVID-19 vaccines and have received much attention. While modern manufacturing processes which involve rapidly mixing an organic stream containing the lipids with an aqueous stream containing the nucleic acids are conceptually straightforward, detailed understanding of LNP formation and structure is still limited and scale-up can be challenging. Mathematical and computational methods are a promising avenue for deepening scientific understanding of the LNP formation process and facilitating improved process development and control. This article describes strategies for the mechanistic modeling of LNP formation, starting with strategies to estimate and predict important physicochemical properties of the various species such as diffusivities and solubilities. Subsequently, a framework is outlined for constructing mechanistic models of reactor- and particle-scale processes. Insights gained from the various models are mapped back to product quality attributes and process insights. Lastly, the use of the models to guide development of advanced process control and optimization strategies is discussed.
... Macro mixing was governed by the transfer equations and the k-ε turbulence model 26 . The crystal size distribution is numerically modeled using a High-Resolution, Finite-Volume, Semi-discrete Central Schemes and micro mixing with a hypothetical multi-environment PDF model [24][25][26][27] . Details of the model are described in the following sections. ...
... A multi-environment model of micromixing (PDF) taken from 21 was used to model the mixing effects 24 . In this method, each CFD cell is divided into three probability states or environments: (1) solution environment that contains the mixture of methanol and lovastatin, (2) water environment as the anti-solvent, and (3) a combination of the two that represents the mixed state. ...
... It is assumed that the three environments are in thermal equilibrium in each cell 24 . This means that a single energy equation is considered for all the three environments. ...
Continuous crystallization of lovastatin from a lovastatin-methanol solution and water as the anti-solvent in an impinging jet crystallizer is investigated using a computational fluid dynamics model. To capture the important phenomena, the model is coupled with micro-mixing, population balance, and related energy balance equations. It is implemented in OpenFOAM and validated against experimental data, where a fairly good agreement is found. The effects of key process parameters on the crystallization performance are also studied using the validated model. The results show that increasing the inlet jet velocity from 1 to 4 m/s yields a much narrower size distribution and 70% reduction in the mean crystal size. The four-fold increase in the inlet jet velocity also reduces the crystal production rate by one order of magnitude. Also, it is found that increasing the inlet supersaturation ratio from 6.8 to 8.8 nearly doubles the mean crystal size. Moreover, it results in a wider size distribution and a six-fold increase in the crystal production rate. The simulations also confirm that lower solution to anti-solvent mass flow ratios yield a wider size distribution, a larger mean crystal size and a higher crystal production rate. Increasing this ratio from 0.5 to 2 reduces the production rate by two orders of magnitude.
... The complexity of turbulent motions may be represented as an energy spectrum composed of different length and time scales [Kolmogorov, 1941;Pope, 2000], such as the integral, the Taylor λ and the Kolmogorov η scales. When modelling flames, dealing with chemistry requires additional efforts as chemical processes are associated with timescales differing by several orders of magnitudes [Fox, 2003], i.e. from fast reactions of radicals species (∼ 10 −9 s) to the slow formation of nitrogen oxides NOx or soot (∼ 10 −2 s). ...
... Reactor-based combustion modelling approaches have recently drawn interest for their ability to treat finite-rate chemistry at an affordable cost [Fox, 2003;Poinsot and Veynante, 2005;Bilger et al., 2005]. Notably, these models handle individual multi-species diffusion without requiring any Lewis (Le) number relationships. ...
... Where more complex chemistry is implemented, the definition of an indicative chemical reaction rate becomes significantly more complex. Approaches may include the selection of a surrogate single-step reaction combining local concentrations of reactants and products calculated using more complex chemistry, the inverse of the formation rate of one or several pre-selected species which limit the local reaction rate [Chomiak and Karlsson, 1996], or a more sophisticated approach which may be employed to interpret eigenvalues of the chemical Jacobian matrix [Fox, 2003]. Reactor-based models can also benefit from mechanism reduction techniques to speed up the simulation from chemical kinetics reduction. ...
In flames, turbulence can either limit or enhance combustion efficiency by means of strain and mixing.
The interactions between turbulent motions and chemistry are crucial to the behaviour of combustion processes.
In particular, it is essential to correctly capture non-equilibrium phenomena such as localised ignition and extinction to faithfully predict pollutant formation.
Reactor-based combustion models --- such as the Eddy Dissipation Concept (EDC) or Partially Stirred Reactor (PaSR) --- may account for turbulence-chemistry interactions at an affordable computational cost by calculating combustion rates relying upon canonical reactors of small fluid size and timescale.
The models may include multiscale mixing, detailed chemical kinetic schemes and high-fidelity multispecies diffusion treatments.
Although originally derived for conventional, highly turbulent combustion, numerous recent efforts have sought to generalise beyond simple empirical correlations using more sophisticated relationships.
More recent models incorporate the estimation of scales based on local variables such as turbulent Reynolds and Damköhler numbers, phenomenological descriptions of turbulence based on fractal theory or specific events such as extinction.
These modifications significantly broaden the effective range of operating conditions and combustion regimes these models can be applied to, as in the particular case of Moderate or Intense Low-oxygen Dilution (MILD) combustion.
MILD combustion is renown for its ability to deliver appealing features such as abated pollutant emissions, elevated thermal efficiency and fuel flexibility.
This review describes the development and current state-of-the-art in finite-rate, reactor-based combustion approaches.
Recently investigated model improvements and adaptations will be discussed, with specific focus on the MILD combustion regime.
Finally, to bridge the gap between laboratory-scale canonical burners and industrial combustion systems, the current directions and the future outlook for development are discussed.
... The sole purpose of presenting these equations here is to elucidate the various couplings illustrated in Figure 1 in a mathematical sense. In a vectorial conservative form, the conservation equations for a single-phase gaseous system are written as [10][11][12]: ...
... In reacting systems, it is conventional [12] to include the thermochemical energy as part of the enthalpy of individual species (in addition to the sensible enthalpy), such that ...
... The same two TRI terms also appear in the radiative source term for the overall energy equation. This is easily seen by time-averaging Equation (11), followed by the use of Equation (12), to obtain ...
Modeling thermal radiation in combustion environments can be extremely challenging for two main reasons. First, the radiative transfer equation (RTE), which is the cornerstone of modeling radiation in such environments, is a five-dimensional integro-differential equation. Second, the absorption and scattering coefficients of molecular gases and particulates prevalent in combustion environments oscillate strongly with the wavenumber (or wavelength), i.e., the medium is strongly nongray, requiring the solution of the RTE for a large number of wavenumbers. This article reviews the progress that has been made in this area to date with an emphasis on the work performed over the past three decades. Progress in both deterministic and stochastic (Monte Carlo) solutions of the RTE is reviewed, in addition to the review of the treatment of the spectral properties of gases, soot, and fuel droplets that dominate combustion environments, i.e., spectral or nongray models. The application of the various state-of-the-art nongray models and RTE solution methods to flames (particularly turbulent), fires, combustors, and other combustion systems are summarized along with a critical discussion of the pros and cons of the models and methods. Finally, the challenges that remain in modeling thermal radiation in combustion systems are highlighted and future outlooks are shared.
... As a reminder the turbulent time scale is expressed as function of the turbulent kinetic energy (k) and the mean rate of its dissipation = / . The mean kinetic energy (K) and the turbulent kinetic energy (k) describe the dynamics of turbulence [28][29][30] in the combustion chamber. In this study a stochastic approach has been retained to model the LTC combustion mode, hence a zero-dimensional energy cascade model applied for a compression and expansion cycle was used. ...
... • is a model constant fixed equal to 1 in this study [21,30]. • is model constant fixed equal to 0.09 [30]. ...
... • is a model constant fixed equal to 1 in this study [21,30]. • is model constant fixed equal to 0.09 [30]. • L is the representative geometric length scale. ...
Low Temperature Combustion (LTC) is a relevant process for internal combustion engines (ICE). This combustion mode is based on premixed fuel/air and fuel lean in-cylinder mixture allowing reduction in NOx and PM emissions while maintaining higher thermal efficiency. In order to investigate the effect of engine operating conditions on the behavior of LTC mode, including OH radical evolution, optical measurements and numerical simulations were performed on a transparent CR diesel engine. The homogeneity of the engine charge was obtained by using very early injection timings. In this study, varying injection strategies were investigated for different engine speeds. In parallel to the experimentation, simulations of LTC mode for the same experimental operations were carried out. The model used in this study is based on a stochastic reactor model. This model includes a turbulence (k-ε) model based on a zero-dimensional energy cascade to calculate the turbulent time scale during the cycle. On the other hand, due to the stochastic approach and to reduce initial heterogeneities of the mixture, a confidence parameter was introduced in the global model to consider the real variation ranges of engine. This latter was modeled as a function of the Reynolds number allowing to initiate heterogeneities of temperature and of species mass. OH radicals were estimated with high spatial and temporal resolution using chemiluminescence measurements. Simulated in-cylinder pressure and the OH radical rate were compared to the experimental data. A good agreement was observed in terms of in-cylinder pressure trace and ignition delay times, meaning that the confidence coefficient model is accurate to describe the initial heterogeneities of the mixture. The simulated OH rate profile has the same shape as the measured OH trace and the main ignition occurs at the same time. This study corroborates that the OH radical is an appropriate tool to identify combustion stages.
... The described mathematical and numerical models may be used to solve a wide range of flows, such as reactive ones. In fact, that hybrid method has great advantage when applied to reactive flows since it models in a closed way the chemical reaction term of species transport equations, [1][2][3]. To that extent, in this paper, the authors aim to expose the development of a parallel Lagrangian composition FDF code and its merging with the in-house built AMR3D code( [4] for reactive, multispecies flows. ...
... We used a system of stochastic differential equations equivalent to Eq. (3), which gives us all composition statistics. Although we can handle these equations based on Eulerian and Lagrangian frameworks, we chose the latter approach because it is more accurate than the former due to minimum numerical diffusion [2]. The complete mathematical and numerical Eulerian framework used in the present work can be seen elsewhere in [10,11] and [4]. ...
... where the superscript * refers to any particle, is a composition variables vector comprised of the species mass fraction and the temperature, and Ω is the subgrid frequency of mixing. The term that comprises this frequency is the IEM (Interaction by Exchange with the Mean) micromixing model [2]. We numerically treated Eqs. ...
... Scalar dissipation rate (SDR) is a quantity of fundamental importance in the analysis of turbulent reacting flows, as it characterises the rate of micromixing [1] and can be used for the reaction rate closure in turbulent premixed combustion [2]. Therefore, it is worthwhile to consider the evolution process of SDR within the flame for its closure in turbulent reacting flows. ...
... Therefore, it is worthwhile to consider the evolution process of SDR within the flame for its closure in turbulent reacting flows. Although SDR statistics for turbulent premixed [2] and non-premixed [1,3] combustion have been analysed extensively, limited effort has been directed to the analysis of the SDR statistics for Moderate or Intense Low oxygen Dilution (MILD) combustion despite the importance of SDR in its modelling [4][5][6][7][8]. ...
... Moreover, in the absence of heat release, the SDR transport of reactive scalars in MILD combustion may exhibit similarities with the SDR transport of the passive scalar in the case of pure mixing [10]. However, the similarities and differences in the SDR statistics between MILD combustion, conventional premixed turbulent flames and pure scalar mixing are yet to be analysed in the existing literature; however, this information is important for modelling of micromixing in MILD combustion [1][2][3][4][5][6][7][8]. This gap in the literature is addressed in the current study by considering Direct Numerical Simulation (DNS) data of homogeneous methane-air mixture combustion under MILD conditions for different turbulence intensities and oxygen dilution levels to analyse the statistical behaviour of the SDR of reactive scalars and the terms of their transport equations. ...
Three-dimensional Direct Numerical Simulations (DNS) data has been utilised to analyse statistical behaviours of the scalar dissipation rate (SDR) and its transport for homogeneous methane-air mixture turbulent Moderate or Intense Low oxygen Dilution (MILD) combustion for different O2 dilution levels and turbulence intensities for different reaction progress variable definitions. Additional DNS has been conducted for turbulent premixed flames and passive scalar mixing for the purpose of comparison with the SDR statistics of the homogeneous mixture MILD combustion with that in conventional premixed combustion and passive scalar mixing. It has been found that the peak mean value of the scalar dissipation rate decreases with decreasing O2 concentration for MILD combustion cases. Moreover, SDR magnitudes increase with increasing turbulence intensity for both MILD and conventional premixed combustion cases. The profiles and mean values of the scalar dissipation rate conditioned upon the reaction progress variable are found to be sensitive to the choice of the reaction progress variable definition. This behaviour arises due to the differences in the distributions of the species mass fractions within the flame. The strain rate contribution and the molecular dissipation term are found to be the leading order contributors in the scalar dissipation rate transport for MILD combustion; whereas, in conventional premixed flames, the terms rising from density variation and reaction rate gradient also play leading roles in addition to the strain rate and molecular dissipation contributions. By contrast, the terms due to density gradient and reaction rate gradient remain negligible in comparison to the leading order contributors in MILD combustion cases due to small density variation because of moderate temperature rise and small reaction rate gradient magnitudes. Furthermore, the qualitative behaviour of the strain rate contribution to the SDR transport in premixed flames is significantly different to that in the case of MILD combustion and passive scalar mixing. The findings of the current analysis indicate that the scalar dissipation rate statistics in MILD combustion show several qualitative similarities to the passive scalar mixing despite major differences with the SDR transport in conventional turbulent premixed flames. This further suggests that the scalar dissipation rate models, which were originally proposed in the context of passive scalar mixing, have the potential to be applicable for MILD combustion but the models for the premixed turbulent combustion may not be applicable for MILD combustion of homogeneous mixtures.
... 1,2 Therefore, LES is often combined with other numerical methods to simulate the reactive flows. [3][4][5][6] The present study considers LES combined with Lagrangian particle simulation (LPS) in which the reaction source terms appear in a closed form. 7 LPS solves transport equations of reactive scalars with spatially-distributed Lagrangian notional particles. ...
... At each time step, statistics are evaluated from the particle field by taking ensemble averages of variables assigned to the particles. 4,21 The angle bracket ⟨f ⟩ represents the averages of particles obtained as a function of y and time. Other statistics are also evaluated by sampling the particles as functions of y and time. ...
Large eddy simulation (LES) coupled with Lagrangian particle simulation (LPS) is applied to investigate high‐speed turbulent reacting flows. Here, LES solves a velocity field while LPS solves scalar transport equations with notional particles. Although LPS does not require sub‐grid scale models for chemical source terms, molecular diffusion has to be modeled by a so‐called mixing model, for which a mixing volume model (MVM), that is originally proposed for an inert scalar in incompressible flow, is extended to reactive scalars in compressible flows. The extended model is based on a relaxation process toward the average of nearby notional particles and assumes a common mixing timescale for all species. LES/LPS with the MVM is applied to a temporally‐evolving compressible turbulent planar jet with an isothermal reaction and is tested by comparing the results with direct numerical simulation (DNS). The results show that LES/LPS well predicts the statistics of mass fractions. As the jet Mach number increases, the reaction progress delays due to the delayed jet development. This Mach number dependence is also well reproduced in LES/LPS. The mean molecular diffusion term of the product calculated as a function of its mass fraction also agrees well between LES/LPS and DNS. An important parameter for the MVM is the distance among particles, for which the requirement for accurate prediction is presented for the present test case. LES/LPS with the MVM is expected to be a promising method for investigating compressible turbulent reactive flows at a moderate computational cost.
... They employed a projected fields method to solve the joint composition probability density function (PDF) transport equation in conjunction with a method of moments with interpolative closure (MOMIC) for aerosol dynamics. Here, it should be mentioned that this combined method was first suggested by Fox (2003) under the name DQMOM-IEM (the direct quadrature method of moments using the interaction by exchange with the mean micromixing model). ...
... Note that in Eq. (28) the scalar formation/consumption term appears in closed form while the last term on the right-hand side in Eq. (28) represents sub-grid scale mixing and requires modelling. There are several modelling proposals for closing the micromixing term ( (Fox, 2003). Following Jones and Prasad (2010), the linear mean square estimation (LMSE) micromixing model (Dopazo and O'Brien, 1974;Dopazo, 1975Dopazo, , 1979 is adopted in the present study: ...
In the present study, our recently proposed extended population balance equation (PBE) model for aggregation and sintering is incorporated into a Large-Eddy Simulation - Probability Density Function (LES-PDF) modelling framework to investigate synthesis of silica nanoparticles in a turbulent diffusion flame. The Eulerian stochastic field method is employed to solve the LES-PBE-PDF equations, characterising the influence of the unresolved sub-grid scale motions, and accounting for the complex interaction between turbulence, chemistry and particle dynamics. A different sub-grid mixing time scale is introduced to the model to account for the low diffusivity of nanoparticles. The models for gas-phase chemistry and aerosol dynamics are the same as those recently used by the authors to simulate silica synthesis in a laminar flame (Tsagkaridis et al. 2023). Thus, by retaining the same kinetics without any adjustments in parameters, we focus on the modelling issues arising in silica flame synthesis. The LES results are compared with the detailed experimental in-situ SAXS data of of Camenzind et al (2008). Good agreement is found between numerical predictions and experimental data for temperature along the centreline. However, the LES model underestimates the SAXS data for primary particle diameters by a factor of two. Possible reasons for this discrepancy are discussed by leveraging knowledge acquired from the aforementioned laminar-flame simulations.
... The second term on the right-hand side of the equations is the production of scalar flux due to mean gradients, where the constant C g is equal to 2.86 [46,47]. The last term on the right-hand side represents the scalar dissipation rate, and C d here assumes a value of 2.0 [47,48]. ...
... In this work, a β-PDF distribution for the temperature is assumed, as this type of PDF is considered more accurate in the context of combustion [48]. The mean and the variance of the PDF are estimated from CFD, considering the average temperature T and the temperature variance T′′2 described before in equation (5). ...
Predicting the thermal and environmental performances of combustion systems can be difficult and computationally expensive. Chemical Reactor Networks (CRN) represent an appealing solution for performing faster numerical simulations since they can carry out calculations with detailed kinetics in a significantly reduced amount of time. However, the design of such models is challenging, as it usually requires a considerable amount of expertise from the user. In this work, we present a novel, automatic methodology for the design of CRN models, which consists of the post-processing of CFD data via a combination of unsupervised clustering and graph scanning algorithms. The methodology was tested on a semi-industrial furnace which can operate in Moderate or Intense Low oxygen Dilution (MILD) combustion. The furnace operates at a nominal power input of 15 kW and is fed with several CH4-H2 mixtures, with two different air injector diameters. First, RANS simulations of the different cases were performed and validated against experimental data. Subsequently, different CRNs were extracted singularly for each case and simulated using detailed kinetics mechanisms. The different CRNs showed good performances in predicting NO emissions for the entire range of cases, as the results were in reasonable agreement with experimental data. Last, the capability of a single CRN to extrapolate towards different operating conditions was also assessed by using one single CRN for the simulation of different fuel mixtures. The approach showed a good level of generalization since the NO predictions were close to the experimental values.
... The PBE is commonly used to model the particle size distribution. It depends solely on particle motion, coagulation, and breakup [14][15][16], wherein the internal variables are the spatial coordinates and velocities. PBE-based models have successfully modeled flowing soot during combustion processes [17], aerosol sprays [18], and more. ...
... Following the usual procedure, a finite set of raw moments µ represent f per (8) [14]. The specific moments that make up µ depend on the inversion algorithm and are defined in the appendices A ...
The dynamics of cavitation bubbles are important in many flows, but their small sizes and high number densities often preclude direct numerical simulation. We present a computational model that averages their effect on the flow over larger spatiotemporal scales. The model is based on solving a generalized population balance equation (PBE) for nonlinear bubble dynamics and explicitly represents the evolving probability density of bubble radii and radial velocities. Conditional quadrature-based moment methods (QBMMs) are adapted to solve this PBE. A one-way-coupled bubble dynamics problem demonstrates the efficacy of different QBMMs for the evolving bubble statistics. Results show that enforcing hyperbolicity during moment inversion (CHyQMOM) provides comparable model-form accuracy to the traditional conditional method of moments and decreases computational costs by about ten times for a broad range of test cases. The CHyQMOM-based computational model is implemented in MFC, an open-source multi-phase and high-order-accurate flow solver. We assess the effect of the model and its parameters on a two-way coupled bubble screen flow problem.
... Algebraic methods define chemical timescales based on reaction-rate constants, the net-production rate of a species and species mass-fractions [2][3][4]. Eigenvalue-based methods define the chemical timescales based on the Jacobian describing the reacting flow system [5][6][7][8][9][10]. ...
... Algebraic methods such as RPTS and RTS are defined based on species concentrations and the net production rate ( ! ) as discussed in Equations [9][10][11] shown in Section 2. As the system approaches steady state the net production term tends to zero. The species concentrations and net production rates of trace species and radicals which are formed and destroyed quickly during the combustion process can be very small (10 -50 < ! ...
Turbulent reacting flows occur in a variety of engineering applications such as chemical reactors and power generating equipment (gas turbines and internal combustion engines). Turbulent reacting flows are characterized by two main timescales, namely, flow timescales and chemical (or reaction) timescales. Understanding the relative timescales of flow and reaction kinetics plays an important role, not only in the choice of models required for the accurate simulation of these devices but also their design/optimization studies. There are several definitions of chemical timescales, which can largely be classified as algebraic or eigenvalue-based methods. The computational complexity (and hence cost) depends on the method of evaluation of the chemical timescales and size of the chemical reaction mechanism. The computational cost and robustness of the methodology of evaluating the reaction times scales is an important consideration in large-scale multi-dimensional simulations using detailed chemical mechanisms. In this work, we present a computational efficient and robust methodology to evaluate chemical timescales based on the algebraic method. Comparison of this novel methodology with other traditional methods is presented for a range of fuel-air mixtures, pressures and temperatures conditions. Additionally, chemical timescales are also presented for fuel-air mixtures at conditions of relevance to power generating equipment. The proposed method showed the same temporal characteristics as the eigenvalue-based methods with no additional computational cost for all the 1cases studied. The proposed method thus has the potential for use with multidimensional turbulent reacting flow simulations which require the computation of the Damkohler number.
... T s T φ . The aforementioned scalar time scales are generally assumed to exhibit a very weak dependence on the Schmidt number (Fox 2003), which is usually neglected in scalar dispersion models (see Cassiani et al. 2020). ...
We investigate the concentration fluctuations of passive scalar plumes emitted from small, localised (point-like) steady sources in a neutrally stratified turbulent boundary layer over a rough wall. The study utilises high-resolution large-eddy simulations for sources of varying sizes and heights. The numerical results, which show good agreement with wind-tunnel studies, are used to estimate statistical indicators of the concentration field, including spectra and moments up to the fourth order. These allow us to elucidate the mechanisms responsible for the production, transport and dissipation of concentration fluctuations, with a focus on the very near field, where the skewness is found to have negative values – an aspect not previously highlighted. The gamma probability density function is confirmed to be a robust model for the one-point concentration at sufficiently large distances from the source. However, for ground-level releases in a well-defined area around the plume centreline, the Gaussian distribution is found to be a better statistical model. As recently demonstrated by laboratory results, for elevated releases, the peak and shape of the pre-multiplied scalar spectra are confirmed to be independent of the crosswind location for a given downwind distance. Using a stochastic model and theoretical arguments, we demonstrate that this is due to the concentration spectra being directly shaped by the transverse and vertical velocity components governing the meandering of the plume. Finally, we investigate the intermittency factor, i.e. the probability of non-zero concentration, and analyse its variability depending on the thresholds adopted for its definition.
... More recently, some studies have also extended the FSP simulation with the DNS method. 15 However, the broad range of length scales and timescales 16 with violent turbulence make it not feasible to apply DNS to the full process and full-scale simulation in FSP. In contrast, LES can strike a satisfactory balance between computational accuracy and cost, which will be one of the focal points in the present work. ...
A two‐step simulation approach combining large eddy simulation (LES) for turbulent flame and multidimensional/multivariate population balance Monte Carlo (PBMC) for nanoparticle dynamics is proposed to explore the detailed flame fields and particle dynamics of zirconia nanoparticles in a flame spray pyrolysis (FSP) reactor. The turbulent combustion of gas and spray are simulated by the nonlinear LES‐partially stirred reactor model. Thereafter, the differentially weighted PBMC is applied for the first time to describe the spatially resolved formation and growth of nanoparticles using the flame fields derived from LES as an input. An efficient submodel for particle spatial transport in multidimensional grids is adopted and tested to account for the effect of thermophoresis, convective and diffusion of the nanoparticles. This methodology is successfully applied to an FSP case, demonstrating a reliable level of fidelity when compared to experiments and other model. Simulation results are discussed, in particular the flame fields and the size, morphology, as well as polydisperse primary particle and aggregates size distribution.
... These concepts were later extended to large-eddy simulation (LES) for predicting the distribution of chemical species including finite rate chemistry effects (Jimenez et al. 1997;Pierce & Moin 2004;Domingo, Vervisch & Veynante 2008;Lecocq et al. 2011;Nambully et al. 2014;Mesquita, Mastorakos & Zedda 2023). Therefore, accurate estimates of the scalar variance are required at various levels in the simulation of turbulent combustion, but also more generally in reactive flow modelling (Fox 2003). Classic models for the scalar variance in LES can roughly be divided into two distinct categories: transport-equation-based models (Jimenez et al. 2001;Pera et al. 2006;Kolla et al. 2009;Keil, Klein & Chakraborty 2021) and algebraic models (Cook & Riley 1994;Girimaji & Zhou 1995;Pierce & Moin 1998Balarac, Pitsch & Raman 2008). ...
The unresolved scalar variance in large-eddy simulations of turbulent flows is a fundamental physical and modelling parameter. Despite its importance, relatively few algebraic models have been developed for this important variable with the most prominent models to date being the classic scale-similarity and gradient models. In this work a new generalized modelling framework based on reconstruction has been developed, which in contrast to classic modelling approaches allows the construction of base static variance models of arbitrary accuracy. It is demonstrated that higher-order reconstructions naturally lead to base static variance models of increased accuracy, and that the classic scale-similarity and gradient models are subsets of more general and higher-order models. The classic scale-similarity assumption for developing dynamic models is also revisited, and it is demonstrated that this can essentially be reinterpreted as a two-level reconstruction approach. Based on this result, a new general methodology is proposed that allows the construction of dynamic models for any given base static model, and a corresponding general reconstruction operator, algebraic or iterative. Consequently, improved static and dynamic models for the scalar variance are developed. The newly developed models are then thoroughly tested a priori using two high-fidelity direct numerical simulation databases corresponding to two substantially different flame and flow configurations, and are shown to outperform classic algebraic models for the variance.
... q (k) rad = 0 adiabatic system, < 0 existence of radiation in which ⟨ ⃗ ⟩ the mean value of the thermo-kinetic states of N p particles. Note that although the LMSE is simple and gives reasonable prediction for various cases (Cao et al. 2007;Raman and Pitsch 2007), the principle deficiency of LMSE model is the fact that the shape of the scalar PDF remains unchanged due to the absence of mean scalar gradient and thus never relaxes to Gaussian distribution (Celis and Silva 2015;Fox 2003). Therefore, many other advanced mixing models have been developed. ...
The Partially Stirred Reactor (PaSR) model is carried out for the ammonia-air combustion system by means of stochastic modeling, namely by solving the transport equation for the joint Probability Density Function (PDF). The turbulent mixing is accounted for by the Linear Mean-Square Estimation (LMSE) mixing model. Notwithstanding the simplified nature of the PaSR modeling, the transported-PDF method enables capturing the effect of mixing frequency on the combustion system, especially the NOx emission. Since the chemical source term is in a closed form in the transported-PDF method, it allows us to apply different chemical mechanisms to explore, whether the set of elementary reactions that are identified as important for the prediction of NOx in the PaSR model is sensitive to the choice of chemical mechanisms. Furthermore, the effect of the residence time in the PaSR model has also been studied, and compared with those in the Perfectly Stirred Reactor (PSR) model (infinite large mixing frequency). Moreover, since the ammonia under oxygen enrichment shows some similar combustion behaviors in terms of e.g. laminar burning velocity as the ammonia under hydrogen enrichment, how large the difference of thermo-kinetic states (e.g. temperature and NOx emission) predicted by PaSR models and in laminar premixed flame configuration is also investigated. A further discussion focuses on the effect of thermal radiation, where the radiative heat loss roles in the prediction of NOx for the turbulent simulation is examined. By using the optically thin approximation model, it is shown that the thermal radiation exhibits little effect on the considered combustion systems within a typical turbulent time-scale.
... The eddy dissipation concept (EDC) species model, which is widely used for analysis of coflow flameless combustion, has several limitations when used for simulation of the RAI method, owing to the reaction rates being limited by the turbulence dissipation rates, simplified chemical reaction mechanisms, and the inability to accurately capture downstream strain and dilution effects [16]. We attempted to use the equilibrium probability distribution function (PDF) model and composition PDF model [17][18][19][20] (also called the transport PDF model) instead of the commonly used EDC model to predict the species reaction mechanisms in RAI flameless combustion, with the objective of identifying the reasons and mechanisms underlying species reactions by comparing experimental and CFD results [21]. ...
Flameless combustion has advantages such as low pollution and uniform temperature in the combustion chamber, making it an excellent option for heat exchangers. Previous studies have focused solely on the flameless combustion phenomenon, without considering its interaction with the target being heated. In this study, we conducted experimental and computational fluid analyses on a cylindrical reformer for reverse air injection flameless combustion. Typically, small-scale reformers of 10 kW or less are coaxial triple-tube cylindrical reformers. In contrast, multitubular reformers are used for larger-scale applications, since the heat transfer rate in single-burner cylindrical reformers decreases sharply as the scale increases. Flameless combustion, with high heat transfer efficiency, helps overcome the limitation of premixed burner. Compared with conventional premixed burners, flameless burner decreases the combustion gas outlet temperature by 30% at 25 kW while reducing energy consumption by 24% (owing to the high heat transfer rate) for a given cooling fluid outlet temperature. Furthermore, it is shown that introducing a ring at the combustion chamber exit can enhance combustion gas recirculation. The experimental result was confirmed through computational fluid analysis. It is concluded that for reverse air injection flameless combustion, the combustion gas recirculation rate in the combustion chamber is strongly related to the heat transfer.
... The scalar dissipation rate ε ϕ is defined as 26 : ...
A “pre‐dissolving ammonia” intensification method for continuous production of diethyl methylphosphite (DEMP) was proposed based on kinetics study of methyl dichlorophosphine (MDP) substitution and DEMP acidolysis. The method was verified in a continuous synthesis system consisting of a small jet reactor. Compared with the selectivity of ~80% obtained in a stirred flask, the selectivity reached 98% by the intensification method. Eliminating the resistance of gas–liquid mass transfer and enhancement of mixing enabled rapid removal of the produced HCl, thus, the DEMP selectivity was greatly increased. The high DEMP selectivity obtained was insensitive to reaction temperature, reactor size and ammonia to chlorine ratio in the studied range. A computational fluid dynamics (CFD) model was proposed for this liquid turbulent reactive flow. Numerical simulations revealed that HCl could be removed rapidly by the neutralization reaction, which was limited by micromixing efficiency. The CFD simulation further indicates the feasibility of “pre‐dissolving ammonia” for industrial production.
... A nice review on the discussion about various mixing models can be found in Refs. [39,40]. In this work, the EMST mixing model is used. ...
The present work focuses on the five different chemical mechanisms coupled with probability density function (PDF) model to represent the local extinction and re-ignition flame characteristics of the well-known Sandia Flames D–F. These five mechanisms span from the Foundational Fuel Chemistry Model (FFCM) mechanism involving 38 species to the Glarborg mechanism involving 150 species. The coupled computational fluid dynamics (CFD) and transported-PDF method are used for the turbulence modeling, and the reaction–diffusion manifolds (REDIMs) are used as an advanced technique for the simplification of chemical kinetics and to speed up the numerical computation. It is demonstrated that these chemical mechanisms have an ability to represent the degree of local extinction and re-ignition accurately. Furthermore, the sensitivity analysis shows that the degree of local extinction is very sensitive to only several key elementary reactions, and an analysis on the turbulence–chemistry interaction investigates the influence of these elementary reactions.
... Note that in Eq. (29) the scalar formation/consumption term appears in closed form while the last term on the right-hand side in Eq. (29) represents sub-grid scale mixing and requires modelling. There are several modelling proposals for closing the micromixing term (Fox 2003). Following Jones and Prasad (2010), the linear mean square estimation (LMSE) micromixing model (Dopazo and O'Brien 1974;Dopazo 1975Dopazo , 1979) is adopted in the present study: ...
In the present study, a recently proposed extended population balance equation (PBE) model for aggregation and sintering is incorporated into a large eddy simulation-probability density function (LES-PDF) modelling framework to investigate synthesis of silica nanoparticles in a turbulent diffusion flame. The stochastic field method is employed to solve the LES-PBE-PDF equations, characterising the influence of the unresolved sub-grid scale motions and accounting for the interactions between turbulence, chemistry and particle dynamics. The models for gas-phase chemistry and aerosol dynamics are the same as those recently used by the authors to simulate silica synthesis in a laminar flame (Tsagkaridis et al. in Aerosol Sci Technol 57(4):296–317, 2023). Thus, by retaining the same kinetics without any adjustments in parameters, we focus on the modelling issues arising in silica flame synthesis. The LES results are compared with experimental in-situ small-angle X-ray scattering (SAXS) data from the literature. Good agreement is found between numerical predictions and experimental data for temperature. However, the LES model underestimates the SAXS data for the primary particle diameter by a factor of two. Possible reasons for this discrepancy are discussed in view of the previous laminar flame simulations.
... The explicit form of this correlation term is unknown without further modelling, a condition known as the closure problem (see e.g. Fox 2003 , for more details on the closure problem). The two flux components in equation ( 50 ) have independent dynamics, requiring additional equations to describe their evolution. ...
Turbulence in protoplanetary disks, when present, plays a critical role in transporting dust particles embedded in the gaseous disk component. When using a field description of dust dynamics, a diffusion approach is traditionally used to model this turbulent dust transport. However, it has been shown that classical turbulent diffusion models are not fully self-consistent. Several shortcomings exist, including the ambiguous nature of the diffused quantity and the nonconservation of angular momentum. Orbital effects are also neglected without an explicit prescription. In response to these inconsistencies, we present a novel Eulerian turbulent dust transport model for isotropic and homogeneous turbulence on the basis of a mean-field theory. Our model is based on density-weighted averaging applied to the pressureless fluid equations and uses appropriate turbulence closures. Our model yields novel dynamic equations for the turbulent dust mass flux and recovers existing turbulent transport models in special limiting cases, thus providing a more general and self-consistent description of turbulent particle transport. Importantly, our model ensures the conservation of global angular and linear momentum unconditionally and implicitly accounts for the effects of orbital dynamics in protoplanetary disks. Furthermore, our model correctly describes the vertical settling-diffusion equilibrium solutions for both small and large particles. Hence, this work presents a generalized Eulerian turbulent dust transport model, establishing a comprehensive framework for more detailed studies of turbulent dust transport in protoplanetary disks.
... The study employs a mathematical model that couples the Reynolds-Averaged Navier-Stokes equations (RANS) and the κ-ε turbulence model, the effects of micromixing were considered using the probability density function (PDF) model multi-environment proposed by Fox [6]. The present model captures micromixing at the small mesh scale with a population balance equation, which models the evolution of the crystal size distribution. ...
This work investigates how the configuration of the geometric parameters of a radial crystallizer influences the results of the crystallization of lovastatin by antisolvent and using a multi-scale computational fluid dynamics (CFD) model. The OPENFOAM open-source software uses macro and micromixing expressions for flow, and complete energy and population equilibrium equations during nucleation and crystal growth. The model is based on the Reynolds-Averaged-Navier-Stokes (RANS) equation, along with a multi-environment probability density function (PDF) model and the spatially semi-discretized population equilibrium equation, operating a high-resolution finite volume method. The variation crystallizer construction parameters provided another crystallizer design, and analyses demonstrated improved performance and effects on crystal distribution.
... The new method performs verification calculations and compares them with results from an Euler-Lagrange analysis. From the works [17][18][19], it becomes clear that the computational effort for the DQMOM is lower than for the Euler-Lagrange method. Pigou et al. [20] extended QMOM to solve the publishing balance equations. ...
Accurately prediction of dispersion and polydispersity of droplet flow is not a trivial task due to the complex behavior of the droplet size distribution (DSD) and the strong state of the instantaneous velocity of a droplet on the shape and size of the droplet. Describing the distribution of sizes and velocities of droplets initially formed in sprays is an essential piece of information needed in spray modeling, which is used to define the initial state of the spray droplets in the downstream two-phase flow fields’ predictive computations. The predictive model for the droplet size and velocity distributions in sprays is formulated as the droplet’s velocity magnitude has a power–law relationship with the droplet in this study. The present model incorporates the deterministic and stochastic aspects of the spray formation process. The quadrature method of moments (QMOM) is applied to solve numerically the transport equations of the probability density function coupled with conserved source terms incompressible Navier-Stokes equations for the liquid phase. The sub-models are connected by different source terms signifying the liquid-gas interaction. Equations of transport for spray moments are derived from DSD, and closure is attained using a gamma distribution. The integer spray moments concerning the volume are used to construct the continuous distribution of QMOM. In contrast, the velocity moments are used to determine the droplet velocity as a constant function of the droplet diameter. The model is first applied to simulate a diesel spray tip penetration under nonreactive conditions with different droplet velocity profiles to validate the approach with experimental data. An additional case of a liquid nitrogen spray is applied to show the gamma distribution’s ability to describe spray drop size distribution. The model generally predicts reasonable agreement with the experimental for both cases.
... Because of the relative simplicity of implementation and low computational costs, the moment method has become a powerful tool for investigating the SCE [8], such as Gauss Quadrature method of moment [12,13], moment-conservative fixed pivot technique [14], and direct quadrature spanning tree method [15], etc. In the moment method, the kth order moment (M k ) of PSD is defined as ...
The governing equation of self-preserving size distribution can be obtained by similarity transformation from the Smoluchowski coagulation equation. To deal with the variable upper limit of convolution, the one parameter group transformation is introduced to get the invariant form of the governing equation. However, whether the equation has an invariant solution depending on asymptotic property of the coagulation kernel at the lower-end boundary. For homogeneous kernel, it can be discussed with three classes, type I has no similarity solution, type II has similarity solution which should be analyzed, and type III has similarity solution. Under the existence of similarity solution, an improved iDNS algorithm is developed with fourth order Runge–Kutta method to get the invariant solution of Smoluchowski coagulation equation more efficiently. Based on the invariant solution, a new definition of entropy in statistical physics is proposed to distinguish the convergence of Smoluchowski coagulation equation mathematically. The new entropy is a simple function of algebraic mean volume and has a maximum. At long time, the entropy production approaches to zero, which means the convergence of solution for Smoluchowski coagulation equation mathematically, and the Cercignani conjecture is almost true for Smoluchowksi coagulatin equation.
... Refs. [39][40][41][42]). The scalar dissipation rate (SDR) was modeled as N ≡ D ξ ∇ξ ⋅ ∇ξ ε∕khξ 02 i (D ξ is the diffusivity of ξ) by assuming a linear relaxation model [40]. ...
This paper presents a method for evaluating the risk of autoignition for the canonical problem of an enclosed hydrogen jet in cross-flow (JICF) which is highly relevant to the design of mixing ducts. The proposed method is based on the separation of the underlying mixing pattern from the evolution of the chemical reactions, while the effect of mixing is maintained on the latter with the purpose of creating a reliable yet computationally efficient design tool for hydrogen gas turbines. Two variants of the Incompletely Stirred Reactor Network (ISRN) approach are proposed that provide the evolution of pre-ignition radicals and autoignition kernel location, leveraging a non-reacting CFD solution or an analytical mixing pattern. The ISRN governing equations include all the salient features of hydrogen transport and lead to a conservative estimate of autoignition risk. Application to a few model problems with varied operating conditions suggests that radical build-up in the JICF can lead to autoignition in the vicinity of a most reactive mixture fraction, consistent with other laminar or turbulent hydrogen flows. However, radical formation and autoignition kernel location strongly depend on the prediction of the underlying mixing field and the amount of differential diffusion within the JICF, which here primarily favours lower values of the composite mixture fraction and transport of hydrogen and radicals away from the jet trajectory.
... Taylor-scale vortices are in the area where energy is transmitted from large vortices to small-scale vortices [41]. Within this area, the energy spectrum has a power functional of 5=3 of the wave number that is proportional to the inverse of the vortex length or the frequency. ...
... The resulting Peclet number Pe is defined as the ratio of diffusion time scale to the convection time scale. In the case of turbulent flow systems, the molecular diffusion term can be neglected as diffusion occurs much longer than convection, that is, Pe ) 1. 9 For closure of the turbulent dispersion term, the scalar flux is modeled as the following 22 : ...
While computational fluid dynamics (CFD)‐based compartment models have gained popularity as a cost‐effective alternative to full CFD modeling of complex mixing systems, model development involves significant trial‐and‐error efforts. This work presents a generalized zoning framework for the streamlined development of CFD‐based compartment models with detailed characterization of the reduced‐order representation of the flow physics. With a stirred tank as an illustrative case, reduced‐order model for species transport and heat transfer with turbulent flow is derived, followed by introducing the generalized zoning framework to demonstrate how a reduced‐order compartment model can be constructed based on a simplified CFD simulation. A test case of mixing two miscible thermal fluids is used to evaluate the CFD‐based compartment model. The results demonstrate that the proposed zoning framework exhibits an accurate representation of the CFD simulation of hydrodynamics and demonstrates capabilities of capturing species and heat transfer in turbulent flow systems with complex geometric configurations.
... More details on the historical developments of the TFDF methods may be found in Refs. [14,56,65]. The modelled transport equation of the FDF may be written as [54], ...
Numerical simulations of a partially-premixed, turbulent jet diffusion flame stabilised in a hot vitiated co-flow are performed. For auto-igniting flames, an accurate prediction of flame stabilisation, which depends on a delicate balance between turbulent transport and chemical kinetics at the flame base, poses an enormous challenge to conventional turbulent combustion models. Multiple mapping conditioning/large eddy simulation (MMC-LES), a promising tool for modelling turbulence-chemistry interactions, has been successfully applied to simulate a variety of combustion applications involving gaseous, liquid, and solid fuels. MMC-LES is a full probability density function (PDF) method where MMC plays the role of the mixing model, emulating molecular mixing phenomenon. MMC attempts to produce accurate molecular mixing by localising mixing in an independent, composition-like reference space. Due to this enforced localised mixing, a sparse distribution of stochastic Lagrangian particles may be used for the Monte-Carlo simulation of the sub-grid joint-composition PDF equation. In the present study, we employ MMC-LES to investigate the auto-igniting methane/air flames of UC Berkeley. A sparse resolution of 1 particle per 10 Eulerian finite-volume cells is used in this study, which offers much cheaper computing expenses in comparison to the conventional transported PDF approach. A skeletal chemical mechanism, based on GRI 3.0, containing 30 species and 184 reactions, represents the oxidation of methane. The time-scale of molecular mixing is modelled using the recently published dyn-aISO model to assess its performance in an auto-igniting configuration. The conditional and unconditional profiles of compositional scalars are compared with the data from experimental measurements. Moreover, the sensitivity of flame lift-off with different co-flow temperatures is investigated as well. MMC-LES is able to produce reasonably the location of the flame base as well as other flow and combustion characteristics.
... The subgrid variations in the conditioning variables about their filtered values are represented by the filtered density function (FDF). The FDF can generally be obtained by solving its transport equations using various approaches, e.g., Lagrangian particles (Pope 1985), Eulerian stochastic fields (Jones and Kakhi 1998), and multienvironment (Fox 2003). However, these approaches are computationally expensive and thus using a presumed FDF can be chosen (Pitsch 2006;Pope 2013) to save computational costs. ...
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.
... Numerous combustion models are derived using the one-point one-time PDF approach because no further assumption is required to describe the ame statistics. For LES, the pioneer work of Dopazo [74] introduced a modelled equation for the joint composition PDF, which was further elaborated by Pope [234,235] (for a detailed treatment comprehending the RANS context, consult [120,92,237]). The formulation of the P equation was later on derived from Gao [101] and Jaberi [134]. ...
Combustion technologies based on hydrocarbons will still play a relevant role in the long-
term scenario, especially in the transportation sector. Numerical tools for the computation
of turbulent reacting flows are of significant importance in the development phase of such technologies. Nevertheless, simulations of hydrocarbon combustion are not without difficulties. By increasing the dimensions of the chemical reaction differential equations, the stiffness of the chemical system increases, posing a limit to the available computational resources. An additional challenge is given by wall-confined turbulent flows at high Reynolds numbers, requiring wall treatment. Particular care is needed for the near-wall modelling if the prediction of thermal loads at the combustor wall is of interest, as key parameter to quantify thermomechanical limits.
The main contribution of this work is the investigation of suitable combustion models
for LES applications of methane flames. Reduced chemistry mechanisms and tabulated
chemistry databases are chosen to speed-up the computation. The turbulence-chemistry
interaction (TCI) on the small scales is investigated for CH4/air partially premixed flame
configurations. Two transported PDF approaches are implemented to deal with reaction-
diffusion manifolds (REDIM), respectively the Eulerian Stochastic Fields (ESF) and the
Multiple Mapping Conditioning (MMC).
Wall heat flux predictions are investigated on a sub-scale rocket combustion chamber
operated with gaseous CH4/O2. The best trade-off is sought in order to correctly represent the heat transfer at wall, while keeping the computation cheap. Flamelet-based chemistry databases including heat losses are used to model the effects of Flame/Wall Interaction (FWI). Additional validation of the reduced chemistry models is performed on an experimental configuration featuring near-wall reactions of CH4 in a crossflow of hot combustion gases, where an autoignition delay is observed.
This work shows that the implemented models based on chemistry databases significantly reduce the requirements on computational power, providing a satisfactory accuracy in the results. Strong extinction/re-ignition effects can be well represented by the ESF-REDIM model, the table also showing potential for predicting autoignition delays. When using finite rate chemistry, the use of MMC is found to be advantageous compared to ESF, although satisfactory predictions are already obtained by neglecting the TCI model. Flamelet databases including enthalpy losses provide satisfactory wall heat flux predictions for a variation of chemical mechanisms and near-wall treatments, if the flame is not subjected to autoignition phenomena.
... Turbulent Schmidt and Prandtl numbers are set to the constant values. The fast chemistry eddy dissipation concept by Magnussen and Hjertager (1977) is used for the formulations of the turbulence-chemistry interaction with the single-step reaction for the fuel and oxidizer combustion using the lumped species method, Fox (2003), discussed in McGrattan et al. (2013). ...
O período entre 2018 e 2022 mostrou-nos que o problema dos incêndios à escala global não está a diminuir, antes pelo contrário. Parece que as consequências das alterações climáticas já estão a afectar a ocorrência de incêndios florestais em várias partes do Mundo, de uma forma que só esperaríamos que acontecesse vários anos mais tarde. Em muitos países do Sul da Europa, bem como em algumas regiões dos EUA, Canadá e Austrália, onde estamos habituados a enfrentar a presença de incêndios muito grandes e devastadores, continuamos a ter eventos que quebram recordes. Alguns países, como os da Europa Central e do Norte, que não estavam habituados a ter grandes incêndios, experimentaram-nos durante estes anos. Os anos anteriores foram muito exigentes para todo o Mundo, também noutros aspectos que nos afectaram a todos. Referimo-nos às restrições impostas pela pandemia que limitaram as nossas reuniões e viagens, afectando em muitos casos a saúde dos membros da Comunidade Científica Wildfire. Felizmente, conseguimos encontrar novas formas de comunicação, ultrapassar essas limitações e manter-nos em contacto uns com os outros. Durante semanas e meses, para muitos de nós, as reuniões pessoais e o trabalho de grupo foram substituídos por ligações em linha. Apesar da economia de dinheiro e tempo, e da facilidade de reunir uma grande variedade de pessoas que estas reuniões desde que nos apercebêssemos de que não substituem as reuniões presenciais, que trazem consigo outras dimensões inestimáveis, que fazem parte da comunicação pessoal e ajudam a construir uma comunidade científica.
... The task of a combustion model is to provide a description of the unresolved scales based on the information available during a simulation. Combustion models have been often classified into two categories, the flamelet-like 2 [95,114,236,248,249] and PDF-like approaches 3 [98,129,253]. Besides them, we can also mention reactor-based 4 [55,94,182,199] and conditional-moment approaches (CMC) 5 [157]. ...
This review covers the new developments in machine learning (ML) that are impacting the multi-disciplinary area of aerospace engineering, including fundamental fluid dynamics (experimental and numerical), aerodynamics, acoustics, combustion and structural health monitoring. We review the state of the art, gathering the advantages and challenges of ML methods across different aerospace disciplines and provide our view on future opportunities. The basic concepts and the most relevant strategies for ML are presented together with the most relevant applications in aerospace engineering, revealing that ML is improving aircraft performance and that these techniques will have a large impact in the near future.
The standard statistical approach is covered in this chapter, starting with the definition of the coarse-grained C(umulative)d(istribution)f(function) and the derivation of its transport pde using the Heaviside step function and the Dirac pseudo-function. The role of pressure is analyzed on the basis of Poisson’s pde and Green’s functions. Multidimensional transport equations for Cdfs are derived and the individual terms related to the underlying physics using simple flow situations as examples. The popular Pdf equations are established as a derivative of the Cdf equation. Homogeneous turbulence is solved for a simple example.
Transfer of chemicals between phases is an extremely important indicator for metallurgical systems. At the same time, for high-quality modeling, it is necessary to ensure the similarity of the models, that is, a certain criterion of chemical similarity, which will ensure the adequacy of the modeling, becomes relevant. Different criteria can be used to achieve chemical similarity, the choice of which depends on the importance of one or another aspect that affects the actual chemical reaction. The possibility of using the similarity of Gibbs energies as an indicator of the possibility of the reaction at the temperatures of metallurgical processes and the similarity of the initial concentrations was proposed and studied in the work during high-temperature modeling of the oxidation of silicon from hot metal. It was also mandatory to use the modified Froude criterion. Due to the proposed method of similitude modeling, brass with a mass fraction of zinc of 1%, which has a lower melting point than cast iron, was chosen as a model environment, that is, it requires less modeling costs. Comparison of the obtained results with the results of purging in similar conditions of hot metal showed the closeness of the data, which indicates the feasibility of the proposed method.
The present study aims to increase the selectivity of C2H2 in the partial oxidation process of methane, employing design of experiments (DOE) and computational fluid dynamics (CFD). Central composite design is used to design tests, and analysis of variance is performed to evaluate the percentage of contribution of operating factors on system performance. The operating factors considered in the analysis are preheating temperature, O2/CH4 ratio, and inlet velocity. The system responses are selectivity of C2H2 and conversion of CH4. Furthermore, an optimization method using response surface methodology is utilized to determine the optimal values of operating factors that lead to the best system performance. The findings indicate that increasing the preheating temperature and O2/CH4 ratio boosts the selectivity of C2H2 and reduces the methane conversion percentage, while increasing the inlet velocity has the opposite effect. The optimization method indicates that the maximum selectivity of C2H2 is achieved with conversion of CH4 of 95% under optimal conditions, namely preheating temperature of 1151.13 K, inlet velocity of 222.8 m/s, and O2/CH4 ratio of 0.59.
Turbulent mixing is an omnipresent phenomenon that constantly affects our everyday life and plays an important role in a variety of industrial applications. The simulation of turbulent mixing poses great challenges, since the full resolution of all relevant length and time scales is associated with an immense computational effort. This limitation can be overcome by only resolving the large‐scale effects and completely model the sub‐grid scales. The development of an accurate sub‐grid mixing model is therefore a key challenge to capture all interactions in the sub‐grid scales. At this place, the hierarchical parcel‐swapping (HiPS) model formulated by A.R. Kerstein [J. Stat. Phys. 153, 142–161 (2013)] represents a computationally efficient and scale‐resolving turbulent mixing model. HiPS mimics the effects of turbulence on time‐evolving, diffusive scalar fields. In HiPS, the diffusive scalar fields or a state space is interpreted as a binary tree structure, which is an alternative approach compared to the most common mixing models. Every level of the tree represents a specific length and time scale, which is based on turbulence inertial range scaling. The state variables are only located at the base of the tree and are treated as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by turbulent time scales associated with the sub‐trees. The mixing only takes places between adjacent fluid parcels and at rates consistent with the prevailing diffusion time scales. In this work, the HiPS model formulation for the simulation of passive scalar mixing is detailed first. Preliminary results for the mean square displacement, passive scalar probability density function (PDF) and scalar dissipation rate are given and reveal the strengths of the HiPS model considering the reduced order and computational efficiency. These model investigations are an important step of further HiPS advancements. The integrated auxiliary binary tree structure allows HiPS to satisfy a large number of criteria for a good mixing model. From this point of view, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.
To provide computational fluid dynamics (CFD)–grade experimental data for studying stratification, measurements on the High-Resolution Jet (HiRJet) facility at the University of Michigan have been conducted with density differences of 1.5%
and −1.5%
, respectively. Fluid with a density different from the fluid initially present in the HiRJet tank was injected, and the propagation of the time-dependent density stratification was captured on a two-dimensional plane with the aid of the wire-mesh sensor technique for Reynolds numbers near 5000 and Richardson numbers near 0.29. Direct numerical simulations (DNSs) of the two cases have also been conducted to expand the multifidelity database. The novel experimental and DNS data were then used to assess the predictive capabilities of the Standard k−ε
(SKE) model and the Reynolds Stress Transport (RST) model. In particular, the propagation speed and thickness of the stratification fronts were assessed by comparing the CFD results against the experimental and DNS data. It was found that the general trends of the stratified density fronts were well predicted by the CFD simulations; however, slight overprediction of the thickness of the stratification layer was found with the SKE model while the RST model gave a larger overprediction of the mixing. Examination of the turbulent statistics showed that the turbulent viscosity was largely overpredicted by the RST model compared to the SKE model.
Stationary combustion systems are a largely employed technology in a wide variety of industrial applications. To satisfy its energy needs, the industry mostly relies on the combustion of fossil fuels, becoming a significant contributor to the global worldwide carbon dioxide emissions. For this reason, decarbonising highly energy-intensive industrial sectors is of strategic importance for fighting climate change. The introduction of hydrogen as a carbon-free fuel to replace fossil sources appears an attractive solution to reduce the carbon footprint of stationary combustion systems, but many technical challenges need to be addressed and they have been object of extensive research in the last few years. The scope of this chapter is to review the current state of hydrogen combustion in stationary combustion systems, from an experimental and numerical perspective. In particular, the features of hydrogen and hydrogen-enriched fuels are analyzed ranging from laboratory-scale burners, where innovative concepts can be benchmarked, up to quasi-industrial and full-scale industrial applications. The main numerical approaches employed to model hydrogen combustion are also described. To conclude, research trends future directions are identified, with a particular focus on innovative concepts such as machine learning and digital twins, which can represent an exciting opportunity for new approaches in stationary combustion systems design.
To support industrial sectors in facing the challenges from the energy transition, it is essential to develop accurate Computational Fluid Dynamics (CFD) models for combustion applications.
In particular, turbulence-chemistry interactions (TCI) models that can predict pollutant emissions and energy efficiency from simulated systems are sought.
Among many combustion closures, reactor-based models have recently drawn interest for their ability to treat chemistry with detailed descriptions at an affordable cost.
Originally derived from the intermittency theory of highly turbulent reacting flows, such models have evolved to provide a valuable solution to any type of flame, up to non conventional combustion technologies.
In numerical simulations employing the Partially Stirred Reactor (PaSR) approach as combustion closure, chemical processes are assumed to take place in sub-grid flow regions of typical length scale smaller than numerical control volumes.
Each computational cell is then partitioned into two locally uniform regions, namely the inert surroundings, solely driven by turbulent mixing, and the reacting fine structures.
The mean reaction rates, contributing to the transport equations of the chemical species, are estimated as the reaction rates from the fine structures multiplied by the cell reacting fraction, i.e., the volume fraction of the cell occupied by the fine structures.
Characteristic time scales for mixing and chemistry are used to estimate the cell reacting fraction.
Many time scales formulations exist and modelling efforts are to be put in selecting the most suitable candidates.
Among the available computational approaches, Large Eddy Simulation (LES) has recently gained interest for its ability to provide accurate numerical solutions up to industrial scale problems.
Conversely to costly Direct Numerical Simulation (DNS) where all scales are resolved, LES solves only the scales down to the computational grid resolution level, the shorter scales and their interactions being modelled.
Nevertheless, DNS of turbulent combustion can supply key information on turbulence-chemistry interactions occurring at the smallest scales.
\textit{A priori} testing is an example of modelling routes making use of DNS data for the development and validation of LES combustion models.
The present Thesis reports modelling advances of the Partially Stirred Reactor combustion approach by means of \textit{a priori} investigations on DNS data of turbulent reacting flows.
First, a layer decomposition of the Partially Stirred Reactor combustion model was conducted.
Predictions of the chemical source terms and heat release rates from several combustion models have been compared.
Various formulations of the chemical time scale for the PaSR model were considered.
A class of mathematical functions was constructed to provide modelling guidance.
Several potential modelling improvements were identified throughout the discussion and were grouped in three categories, namely parameter selection, cell reacting fraction reformulation and deep model revision.
Parameter selection aims at finding an optimal set of parameters or submodels to improve model accuracy.
Such a study was conducted on the PaSR model in the context of Moderate or Intense Low-oxygen Dilution (MILD) combustion, an appealing non conventional combustion technology in terms of fuel flexibility, pollutant emissions, and thermal efficiency.
An optimal set of sub models was found for the specific modelling of MILD combustion where turbulence-chemistry interactions are naturally strengthened.
Also, the layer decomposition demonstrated that finding an optimal submodel can be a local concept, i.e., depending on the local flow region.
In this context, a data-driven methodology employing supervised clustering algorithms has been proposed for the local estimation of the optimal chemical time scale formulation in the PaSR model.
A binning operation was used to partition the data into clusters of similar thermo-chemical states.
Within each cluster, the best formulation was found by means of distance minimisation.
Coupling the PaSR model with clustered solutions yielded a systematic modelling error cut-off.
The method was found applicable to any type of flames.
Besides parameter selection, the functional form of the cell reacting fraction in the PaSR model has been investigated more carefully.
A methodology was proposed to extract sub-grid quantities from DNS data matching the physical representation of the cell reacting volume fraction.
Each cell from the DNS was supposed extinct or reacting depending on the local intensity of heat release.
Down-sampling on coarser LES grids, the extracted cell reacting fractions were given by the proportion of active DNS cells within the larger LES control volumes.
Questioning the true representation of the modelled cell reacting fractions, the extracted quantities may cope with modelling needs by means of algebraic fraction forms.
This illustrated the complexity of remodelling the cell reacting fraction form by hand.
Within this context, machine learning and sparse-promoting techniques have been used to explore broad libraries of potential functional forms.
Such approaches returned the solution best balancing accuracy and modelling complexity.
An original functional form of the cell reacting fraction was found and provided higher accuracy results with respect to standard approaches.
The results were validated on combustion datasets operating at different regimes.
Lastly, the PaSR combustion model has been revised more in-depth to cope with the fundamental limitation of relying on a unique cell reacting fraction for all chemical species.
In particular, tools from the machine learning community and arguments from the Computational Singular Perturbation theory have been employed to respectively derive two modelling frameworks.
In the first approach, the closure of a progress variable transported equation with a PaSR approach was developed.
Such equation left an unclosed term, namely the fine structures progress variable, that required modelling.
To this purpose, neural networks (NN) have been trained and tested on DNS data of different turbulent flames.
Great generalisation capabilities were obtained while regressing the subgrid scale variable from information at grid level.
This methodology has showed its ability to be a valuable alternative to the classic closures for the source term of a progress variable transported equation.
The second framework consisted in the integration of multiple chemical time scales in a PaSR approach.
Abandoning the concept of the fine structures, a modal decomposition of the Jacobian matrix of the chemical source terms was performed.
Each mode contributing to the final estimation of the mean reaction rates was multiplied by its modal coefficient resembling a cell reacting fraction.
This innovative framework provided promising results on a rather simple test case, requiring further modelling attention.
Turbulent/turbulent interfacial (TTI) layers are investigated with direct numerical simulation of temporally evolving grid turbulence. The present study considers a temporally evolving wake of two parallel-bar grids with different mesh sizes, which generate homogeneous isotropic turbulent regions with large and small turbulent kinetic energies (TKE). A shearless mixing layer of turbulence forms between the large- and small-TKE regions. The TTI layer bounded by the large- or small-TKE region is identified with a passive scalar field, and the flow statistics are evaluated as functions of a position with respect to the TTI layer. Statistics of a velocity gradient tensor suggest that the center and edges of the TTI layer are dominated by vortex sheets and vortex tubes, respectively. Because of the configuration of these vortical structures, the flow toward the TTI layer in the layer-normal direction generates a compressive strain, which is important to sustain the thin layer structure. The mean velocity jump due to the compressive strain is about [Formula: see text] and is observed over a length of about [Formula: see text], where [Formula: see text] and η are the Kolmogorov velocity and length scales, respectively. The thickness of the TTI layer is about [Formula: see text], which hardly depends on time. The TTI layer has a large surface area when it is bounded by the large-TKE region. Consequently, the shearless mixing layer tends to entrain more amount of fluid from the large-TKE region than from the small-TKE region although the entrainment rate per unit surface area normalized by the Kolmogorov velocity is similar for both regions.
Turbulent mixing plays an important role in a variety of applications ranging from astrophysics to combustion and even pollutant dispersion. The Direct Numerical Simulation (DNS) that resolves all scales is not feasible for most engineering applications since the flow has a wide range of length and time scales, which yields extremely high resolution requirements. Large Eddy Simulations (LES) overcome this limitation by modeling the sub‐grid scale effects. In transported Probability Density Function (PDF) methods, the key challenge is to develop an accurate mixing model. At this point, the Hierarchical Parcel‐Swapping (HiPS) model, introduced by A.R. Kerstein [J. Stat. Phys. 153, 142‐161 (2013)], is an attractive candidate. It is characterized by a computationally efficient representation of the effects of turbulence on a time‐evolving structure of diffusive scalar fields. The interpretation of the diffusive scalar fields or a state space as a binary tree structure is an alternative approach compared to the most common mixing models. The characteristic feature of HiPS is that every level of the tree corresponds to a specific length and time scale, which is based on turbulence inertial range scaling. The state variables only reside at the base of the tree and are understood as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by prevailing turbulent time scales associated with the sub‐trees. The mixing of adjacent fluid parcels is done either instantaneously or at rates consistent with the corresponding diffusion time scales. In this work, HiPS is detailed for the simulation of passive scalar mixing first. Preliminary results for the scalar power spectra, mean square displacement and scalar dissipation rate are shown and reveal a reasonable agreement with experimental findings. Furthermore, the integrated binary tree structure allows to satisfy a large number of criteria for a good mixing model. Considering the reduced order and computational efficiency, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.
The quadrature-based method of moments (QMOM) offers a promising class of approximation techniques for reducing kinetic equations to fluid equations that are valid beyond thermodynamic equilibrium. In this work, we study a particular five-moment variant of QMOM known as HyQMOM and establish that this system is moment-invertible over a convex region in solution space. We then develop a high-order discontinuous Galerkin (DG) scheme for solving the resulting fluid system. The scheme is based on a predictor–corrector approach, where the prediction is a localized space-time DG scheme. The nonlinear algebraic system in this prediction is solved using a Picard iteration. The correction is a straightforward explicit update based on the time-integral of the evolution equation, where the space-time prediction replaces all instances of the exact solution. In the absence of limiters, the high-order scheme does not guarantee that solutions remain in the convex set over which HyQMOM is moment-realizable. To overcome this, we introduce novel limiters that rigorously guarantee that the computed solution does not leave the convex set of realizable solutions, thus guaranteeing the hyperbolicity of the system. We develop positivity-preserving limiters in both the prediction and correction steps and an oscillation limiter that damps unphysical oscillations near shocks. We also develop a novel extension of this scheme to include a BGK collision operator; the proposed method is shown to be asymptotic-preserving in the high-collision limit. The HyQMOM and the HyQMOM-BGK solvers are verified on several test cases, demonstrating high-order accuracy on smooth problems and shock-capturing capability on problems with shocks. The asymptotic-preserving property of the HyQMOM-BGK solver is also numerically verified.
The hybrid method solves the modeled transport equation for the joint PDF of velocity, turbulence frequency, and compositions for turbulent reactive flows. A finite-volume (FV) method is used to solve the mean conservation equations for mass, momentum, and energy and the mean equation of state; and a particle method is used to solve the modeled PDF equation. The method is completely consistent at the level of the governing equations solved by the FV and particle algorithms. In this work, the conditions to be fulfilled for full consistency at the numerical solution level are examined and the independent consistency conditions are identified. Then correction algorithms are developed to enforce these independent consistency conditions to achieve full consistency at the numerical solution level. In addition, a new formulation of the energy equation and the equation of state is developed which is both general and simple. The hybrid method is applied to a non-premixed piloted-jet flame. The numerical results show that the correction algorithms are completely successful in achieving consistency. The convergence of the method is demonstrated and, in particular, it is shown that the bias error is dramatically reduced (compared to that in previous PDF calculations). In addition, the results are shown to be in a good agreement with some earlier PDF calculations and also with the available experimental data. Because of the substantially reduced numerical error (for given grid size and number of particles), the present hybrid method represents a significant advance in the computational efficiency of particle/mesh method for the solution of PDF equations.