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

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

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

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

Multi-environment probability density function method for modelling turbulent combustion using realistic chemical kinetics

A computational fluid dynamics (CFD) tool for performing turbulent combustion simulations that require finite-rate chemistry is developed and tested by modelling a series of bluff-body stabilized flames that exhibit different levels of finite-rate chemistry effects ranging from near equilibrium to near global extinction. The new modelling tool is based on the multi-environment probability density function (MEPDF) methodology and combines the following: the direct quadrature method of moments (DQMOM); the interaction-by-exchange-with-the-mean (IEM) mixing model; and realistic combustion chemistry. Using DQMOM, the MEPDF model can be derived from the transport PDF equation by depicting the joint composition PDF as a weighted summation of a finite number of multi-dimensional Dirac delta functions in the composition space. The MEPDF method with multiple reactive scalars retains the unique property of the joint PDF method of treating chemical reactions exactly. However, unlike the joint PDF methods that typically must resort to particle-based Monte-Carlo solution schemes, the MEPDF equations (i.e. the transport equations of the weighted delta-peaks) can be solved by traditional Eulerian grid-based techniques. In the current study, a pseudo time-splitting scheme is adopted to solve the MEPDF equations; the reaction source terms are computed with a highly efficient and accurate in-situ adaptive tabulation (ISAT) algorithm. A 19-species reduced mechanism based on quasi-steady state assumptions is used in the simulations of the bluff-body flames. The modelling results are compared with the experimental data, including mixing, temperature, major species and important minor species such as CO and NO. Compared with simulations using a Monte-Carlo joint PDF method, the new approach shows comparable accuracy.     

A Eulerian PDF scheme for LES of nonpremixed turbulent combustion with second-order accurate mixture fraction

Monte Carlo simulations of joint probability density function (PDF) approaches have been developed in the past largely with Reynolds averaged Navier Stokes (RANS) applications. Current interests are in the extension of PDF approaches to large eddy simulation (LES). As LES resolves accurately the large scales of turbulence in time, the Monte Carlo simulation and the flow field need to be tightly coupled. A tight coupling can be achieved if the consistency between the scalar field solution obtained via finite-volume (FV) methods and that from the stochastic solution of the PDF is ensured. For nonpremixed turbulent flames with two distinct streams, the local reactive mixture is described by the mixture fraction. A Eulerian Monte Carlo method is developed to achieve a second-order accuracy in the instantaneous filtered mixture fraction that is consistent with the corresponding FV. The performances of the proposed scheme are extensively evaluated using a one-dimensional model. Then, the scheme is applied to two cases with LES. The first one is a non-reacting mixing flow of two different fluids. The second case is the Sandia piloted turbulent flame D with a steady state flamelet model. Both results confirm the consistency of the proposed method to the level of filtered mixture fraction.     

湍流扩散燃烧的数值研究-PDF方法和火焰面模型的性能比较

采用标量概率密度函数(PDF)方法、稳态和非稳态火焰面模型三种方法对一个值班湍流CH_4/O_2/N_2射流扩散火焰(Sandia Flame D)进行数值计算,以比较不同燃烧模型的性能。PDF方法通过计算反应标量的PDF输运方程来得到标量分布,而火焰面模型只求解单标量混合物分数的PDF方程,组分和温度分布通过火焰面方程的求解或者火焰面数据库的插值得到。计算结果和实验数据对比表明PDF方法计算结果最好但计算量相当大,稳态火焰面模型则反之。综合而言,非稳态火焰面模型的预测结果相对稳态模型有了非常大的改进,而计算量仍然容易接受,非常适合工程应用。     

Large eddy simulation of a turbulent nonpremixed piloted methane jet flame (Sandia Flame D)

Large eddy simulation (LES) is conducted of the Sandia Flame D [Proc. Combust. Inst. 27 (1998) 1087, Sandia National Laboratories (2004)], which is a turbulent piloted nonpremixed methane jet flame. The subgrid scale (SGS) closure is based on the scalar filtered mass density function (SFMDF) methodology [J. Fluid Mech. 401 (1999) 85]. The SFMDF is basically the mass weighted probability density function (PDF) of the SGS scalar quantities [Turbulent Flows (2000)]. For this flame (which exhibits little local extinction), a simple flamelet model is used to relate the instantaneous composition to the mixture fraction. The modelled SFMDF transport equation is solved by a hybrid finite-difference/Monte Carlo scheme. This is the first LES of a realistic turbulent flame using the transported PDF method as the SGS closure. The results via this method capture important features of the flame as observed experimentally.     

Unstrained and strained flamelets for LES of premixed combustion

The unstrained and strained flamelet closures for filtered reaction rate in large eddy simulation (LES) of premixed flames are studied. The required sub-grid scale (SGS) PDF in these closures is presumed using the Beta function. The relative performances of these closures are assessed by comparing numerical results from large eddy simulations of piloted Bunsen flames of stoichiometric methane–air mixture with experimental measurements. The strained flamelets closure is observed to underestimate the burn rate and thus the reactive scalars mass fractions are under-predicted with an over-prediction of fuel mass fraction compared with the unstrained flamelet closure. The physical reasons for this relative behaviour are discussed. The results of unstrained flamelet closure compare well with experimental data. The SGS variance of the progress variable required for the presumed PDF is obtained by solving its transport equation. An order of magnitude analysis of this equation suggests that the commonly used algebraic model obtained by balancing source and sink in this transport equation does not hold. This algebraic model is shown to underestimate the SGS variance substantially and the implications of this variance model for the filtered reaction rate closures are highlighted.     

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

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

Transported PDF simulation of auto-ignition of a turbulent methane jet in a hot,vitiated coflow

In the current work, the auto-ignition of a turbulent round methane jet is studied numerically by means of a transported probability density function (PDF) method. The methane jet is issued into a hot, vitiated coflow, where it ignites to form a steady lifted flame. For this flame, experimental data of hydroxyl, temperature and mixture fraction are provided in the area where the fuel auto-ignites. To model this experiment, the transport equation for the thermochemical PDF is solved using a hybrid finite volume / Lagrangian Monte-Carlo method. Turbulence is modelled using the k-? turbulence model including a jet-correction. Computational results are compared to experimental data in terms of mean quantities, variances and lift-off height. Moreover, the structure of the one-point, one-time marginal PDF of temperature is analysed and compared to experimental data which are provided in this work. It is found that the transported PDF method in conjunction with the k-? model is capable of reproducing these statistical data very well. In particular the effect of ignition on the marginal PDF of temperature can be well reproduced with this approach. To further analyse the relevant processes in the evolution of the temperature PDF, a statistically homogeneous system is studied both numerically and analytically.     

Large eddy simulation of soot evolution in turbulent nonpremixed bluff body flames

Large Eddy Simulation (LES) is utilized to investigate soot evolution in a series of turbulent nonpremixed bluff body flames featuring different bluff body diameters. The modeling framework relies on recent development in the soot subfilter Probability Density Function (PDF) model that can correctly account for the distribution of soot with respect to mixture fraction, correcting errors in previous soot subfilter PDF models that significantly overpredict soot oxidation. With the previous soot subfilter PDF model, no soot was observed outside of the recirculation zone in past studies on similar bluff body flames. Results obtained with the current LES modeling approach compare favorably with the experimental measurements of the flow field and the soot volume fraction. Notably, the current LES modeling approach correctly predicts large soot volume fractions in the recirculation zone, a decrease in the soot volume fraction through the high-strain neck region, and then an increase again in the downstream jet-like region. Consistent with the experimental measurements, the larger bluff body diameter, with its larger recirculation zone with longer residence times, generates more soot in the recirculation zone and also more soot in the high-strain neck region. Analysis of the soot volume fraction source terms lead to mechanistic understanding of soot evolution in the entirety of the bluff body flames. Most of the soot generated in the recirculation zone is oxidized but some escapes unoxidized and is passively transported through the neck region. Virtually no new soot forms in the downstream jet-like region, and the increase in the soot volume fraction in the jet-like region is due to acetylene-based surface growth of the soot transported through the neck region. This mechanism could not be predicted with the previous soot subfilter PDF model, with the recent soot subfilter PDF model being critical in the understanding of this basic mechanism.     

湍流扩散火焰的非稳态火焰面模拟

采用稳态的和非稳态的火焰面模型同时对一个湍流甲烷射流扩散火焰进行了数值模拟,比较了两者对湍流平均火焰结构、活性自由基和污染物(氮氧化物)排放的模拟效果。速度场采用κ-ε模型计算,守恒标量混合物分数的分布通过其概率密度函数(PDF)输运方程的求解得到。稳态的火焰面结构由查询火焰面数据库得到,而非稳态的火焰面结构由火焰面方程和流场方程耦合求解来计算。采用详细的GRI—Mech 3.0机理描述甲烷的氧化和氮氧化物的形成。数值模拟结果和实验数据作了广泛的对比,验证了火焰面模型对湍流扩散燃烧的定量模拟能力。     

A Fokker–Planck approach to a moment closure for mixing in variable-density turbulence

ABSTRACT

We develop a theory for the cascade mixing terms in a moment closure approach to binary active scalar mixing in variable-density turbulence. To address the variable-density complications we apply, as a principle and constraint, the conservation of the probability density function (PDF) through a Fokker–Planck equation with bounded sample space whose attractor is the beta PDF with skewness. Mixing is related to a single-point PDF as a realisability principle to provide mathematically rigorous expressions for the small scale statistics in terms of largescale moments. The problem of the unknown small-scale mixing is replaced with the determination of the drift and diffusion terms of a Fokker–Planck equation in a beta-PDF-convergent stochastic process. We find that realisability of a beta-convergent process requires the mixing time-scale ratio, taken as a constant in passive scalar mixing, to be a function of the mean mass fraction, mean fluid density, the Atwood number, the density-volume correlation and moments of the density field. We develop and compare the new model with direct numerical simulations data of non-stationary homogeneous variable-density turbulence.     

Large Eddy Simulation of a turbulent lifted flame using multi-modal manifold-based models: Feasibility and interpretability

A general model for multi-modal turbulent combustion is achievable with two-dimensional manifold equations that use the mixture fraction and a generalized progress variable as coordinates. Information about the underlying mode of combustion is encoded in three scalar dissipation rates that appear as parameters in the two-dimensional equations. In this work, Large Eddy Simulation (LES) of a multi-modal turbulent lifted hydrogen jet flame in a vitiated coflow is performed using this new turbulent combustion model, leveraging both convolution-on-the-fly and In-Situ Adaptive Tabulation for computational tractability. The simulation predicts a lifted flame consistent with observations from past experiments. The feasibility of such a model implemented in LES is examined, and the cost per timestep is found to be comparable to conventional one-dimensional manifold-based models describing one asymptotic mode of combustion. Additionally, the model provides clear interpretability, allowing for combustion mode analysis to be performed with ease by evaluating the scalar dissipation rates and generalized progress variable source term. This analysis is used to show that the flame is stabilized by autoignition and has a trailing nonpremixed flame. Furthermore, transport of progress variable from the most reactive mixture fraction towards richer mixtures at the centerline is found to be important.     

A priori analysis of sub-grid variance of a reactive scalar using DNS data of high Ka flames

Direct numerical simulations (DNS) of low and high Karlovitz number (Ka) flames are analysed to investigate the behaviour of the reactive scalar sub-grid scale (SGS) variance in premixed combustion under a wide range of combustion conditions (regimes). An order of magnitude analysis is performed to assess the importance of various terms in the variance evolution equation and the analysis is validated using the DNS results. This analysis sheds light on the relative behaviour among turbulent transport and production, scalar dissipation and chemical processes involved in the evolution of the SGS variance at different Ka. The common expectation is that the variance equation shifts from a reaction-dissipation balance at low Ka to a production–dissipation balance at high Ka with diminishing reaction contribution. However, in large eddy simulation (LES), a high Ka alone does not make the reaction term negligible, as the relative importance of the reaction term has a concurrent increase with filter size. The filter size can be relatively large compared with the Kolmogorov length scale in practical LES of high Ka flames, and as a consequence a reaction–production–dissipation balance may prevail in the variance equation even in a high Ka configuration, and this possibility is quantified using the DNS analysis in this work. This has implications from modelling perspectives, and therefore two commonly used closures in LES for the SGS scalar dissipation rate are investigated a priori to estimate the importance of the above balance in LES modelling. The results are explained to highlight the interplay among turbulence, chemistry and dissipation processes as a function of Ka.     

Generalised scale-by-scale energy-budget equations and large-eddy simulations of anisotropic scalar turbulence at various Schmidt numbers

A necessary condition for the accurate prediction of turbulent flows using large-eddy simulation (LES) is the correct representation of energy transfer between the different scales of turbulence in the LES. For scalar turbulence, transfer of energy between turbulent length scales is described by a transport equation for the second moment of the scalar increment. For homogeneous isotropic turbulence, the underlying equation is the well-known Yaglom equation. In the present work, we study the turbulent mixing of a passive scalar with an imposed mean gradient by homogeneous isotropic turbulence. Both direct numerical simulations (DNS) and LES are performed for this configuration at various Schmidt numbers, ranging from 0.11 to 5.56. As the assumptions made in the derivation of the Yaglom equation are violated for the case considered here, a generalised Yaglom equation accounting for anisotropic effects, induced by the mean gradient, is derived in this work. This equation can be interpreted as a scale-by-scale energy-budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. The equation is evaluated for the conducted DNS, followed by a discussion of physical effects present at different scales for various Schmidt numbers. For an analysis of the energy transfer in LES, a generalised Yaglom equation for the second moment of the filtered scalar increment is derived. In this equation, new terms appear due to the interaction between resolved and unresolved scales. In an a-priori test, this filtered energy-budget equation is evaluated by means of explicitly filtered DNS data. In addition, LES calculations of the same configuration are performed, and the energy budget as well as the different terms are thereby analysed in an a-posteriori test. It is shown that LES using an eddy viscosity model is able to fulfil the generalised filtered Yaglom equation for the present configuration. Further, the dependence of the terms appearing in the filtered energy-budget equation on varying Schmidt numbers is discussed.     

Large Eddy Simulation composition equations for single-phase and two-phase fully multicomponent flows

The Large Eddy Simulation (LES) equations for multicomponent (MC) fuel single-phase (SP) flow and two-phase (TP) flow with phase change are derived from the Direct Numerical Simulation (DNS) equations by filtering the DNS equations using a top-hat filter. Additional to the equations solved for single-component (SC) fuels, composition equations enter the formulation. The species composition is represented through a Probability Distribution Function (PDF), and DNS equations for the PDF moments are solved to find the composition. The TP filtered equations contain three categories of subgrid-scale (SGS) terms: (1) SGS–flux terms, (2) filtered source terms (FSTs) and (3) terms representing the ‘LES assumptions’. For SP flows no FSTs exist. The SGS terms in the LES equations must be either shown negligible or modeled. It is shown that for the composition equations, two equivalent forms of the DNS equations lead to two non-equivalent forms of the LES equations. Criteria are proposed to select the form best suited for LES. These criteria are used in conjunction with evaluations based on a DNS database portraying mixing and phase change, and lead to choosing one of the LES forms which satisfies all criteria. It is shown that the LES assumptions lead to additional SGS terms which require modeling. Further considerations are made for reactive flows.     

钝体后湍流预混燃烧的PDF模拟

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

Modeling of multicomponent-fuel drop-laden mixing layers having a multitude of species

A formulation representing multicomponent-fuel (MC-fuel) composition as a probability distribution function (PDF) depending on the molar mass is used to construct a model of a large number of MC-fuel drops evaporating in a gas flow, so as to assess the extent of fuel specificity on the vapor composition. The PDF is a combination of two Gamma PDFs, which was previously shown to duplicate the behavior of a fuel composed of many species during single drop evaporation. The conservation equations are Eulerian for the flow and Lagrangian for the physical drops, all of which are individually followed. The gas conservation equations for mass, momentum, species, and energy are complemented by differential conservation equations for the first four moments of the gas-composition PDF; all coupled to the perfect gas equation of state. Source terms in all conservation equations couple the gas phase to the drops. The drop conservation equations for mass, position, momentum, and energy are complemented by differential equations for four moments of the liquid-composition PDF. The simulations are performed for a three-dimensional mixing layer whose lower stream is initially laden with drops. Initial perturbations excite the layer to promote the double pairing of its four initial spanwise vortices to an ultimate vortex. The drop temperature is initially lower than that of the surrounding gas, initiating drop heating and evaporation. The results focus on both evolution and the state of the drops and gas when layers reach a momentum-thickness maximum past the double vortex pairing; particular emphasis is on the gas composition. Comparisons between simulations with n-decane, diesel, and three kerosenes show that at same initial Reynolds number and Stokes number distribution, a single-component fuel cannot represent MC fuels. Substantial differences among the MC-fuel vapor composition indicate that fuel specificity must be captured for the prediction of combustion.     

Direct numerical simulations and analysis of three-dimensional n-heptane spray flames in a model swirl combustor

Three-dimensional n-heptane spray flames in a swirl combustor are investigated by means of direct numerical simulation (DNS) to provide insight into realistic spray evaporation and combustion as well as relevant modeling issues. The variable-density, low-Mach number Navier–Stokes equations are solved using a fully conservative and kinetic energy conserving finite difference scheme in cylindrical coordinates. Dispersed droplets are tracked in a Lagrangian framework. Droplet evaporation is described by an equilibrium model. Gas combustion is represented using an adaptive one-step irreversible reaction. Two different cases are studied: a lean case that resembles a lean direct injection combustion, and a rich case that represents the primary combustion region of a rich-burn/quick-quench/lean-burn combustor. The results suggest that premixed combustion contribute more than 70% to the total heat release rate, although diffusion flame have volumetrically a higher contribution. The conditional mean scalar dissipation rate is shown to be strongly influenced, especially in the rich case. The conditional mean evaporation rate increases almost linearly with mixture fraction in the lean case, but shows a more complex behavior in the rich case. The probability density functions (PDF) of mixture fraction in spray combustion are shown to be quite complex. To model this behavior, the formulation of the PDF in a transformed mixture fraction space is proposed and demonstrated to predict the DNS data reasonably well.