Problem adapted reduced models based on Reaction-Diffusion Manifolds (REDIMs)

In this work a novel modification of the REDIM method is presented. The method follows the main concept of decomposition of time scales. It is based on the assumption of existence of invariant slow manifolds in the thermo-chemical composition space (state space) of a reacting flow. A central point of the current modification is its capability to include both transport and thermo-chemical processes and their coupling into the definition of the reduced model. This feature makes the method more problem oriented, and more accurate in predicting the detailed system dynamics. The manifold of the reduced model is approximated by applying the so-called invariance condition together with repeated integrations of the reduced model in an iterative way. The latter is needed to improve the estimate of gradients of the reduced model parameters (coordinates which define the reduced manifold locally). To verify the approach one-dimensional stationary laminar methane/air and syngas/air flames are investigated. In particular, it is shown that the adaptive REDIM method recovers the full stationary system dynamics governed by detailed chemical kinetics and the molecular transport in the case of a one dimensional reduced model and, therefore, includes the so-called flamelet method as a limiting case.     

Issues arising in the construction of QSSA mechanisms: the case of reduced n-heptane/air models for premixed flames

A model reduction methodology, based on the quasi steady-state approximation (QSSA), is employed for the construction of reduced mechanisms in the case of an n-heptane/air premixed flame. Several issues related to the construction of these reduced mechanisms are discussed; such as the influence of the size of the starting skeletal mechanism, the stiffness reduction, and the truncation/simplification of (i) the expressions of the global rates and (ii) the steady-state relations. The starting point for the reduction is two skeletal mechanisms that involve 177/768 and 66/326 species/reactions, respectively [J. Prager, H.N. Najm, M. Valorani, and D.A. Goussis, Skeletal mechanism generation with CSP and validation for premixed n-heptane flames, Proc. Combust. Inst. 32 (2009), pp. 509–517] and which were derived from the detailed mechanism of Curran et al. [H.J. Curran, P. Gaffuri, W.J. Pitz, and C.K. Westbrook, A comprehensive modeling study of iso-octane oxidation, Combust. Flame 129 (2002), pp. 253–280], which involves 561/2538 species/reactions. From these two skeletal mechanisms, a number of reduced mechanisms of various sizes are produced and analysed. The validity of the reduced mechanism with the minimum size is demonstrated by considering its accuracy regarding the mass fractions of major and minor species, the temperature, and the flame speed, over a wide range of equivalence ratios and pressures.     

Comparative analysis of methods for heat losses in turbulent premixed flames using physically-derived reduced-order manifolds

Heat losses have the potential to substantially modify turbulent combustion processes, especially the formation of pollutants such as nitrogen oxides. The chemistry governing these species is strongly temperature sensitive, making heat losses critical for an accurate prediction. To account for the effects of heat loss in large eddy simulation (LES) using a precomputed reduced-order manifold approach, thermochemical states must be precomputed not only for adiabatic conditions but also over a range of reduced enthalpy states. However, there are a number of methods for producing reduced enthalpy states, which invoke different implicit assumptions. In this work, a set of a priori and a posteriori LES studies have been performed for turbulent premixed flames considering heat losses within a precomputed reduced-order manifold approach to determine the sensitivity to the method by which reduced enthalpy states are generated. Two general approaches are explored for generating these reduced enthalpy states and are compared in detail to assess any effects on turbulent flame structure and emissions. In the first approach, the enthalpy is reduced at the boundary of the one-dimensional (1D) premixed flame solution, resulting in a single enthalpy deficit for a single premixed flame solution. In the second approach, a variable heat loss source term is introduced into the 1D flame solutions by mimicking a real heat loss to reduce the post-flame enthalpy. The two approaches are compared in methane–air piloted turbulent premixed planar jet flames with different diluents that maintain a constant adiabatic flame temperature but experience different radiation heat losses. Both a priori and a posteriori results, as well as a chemical pathway analysis, indicate that the manner by which the heat loss is accounted for in the manifold is of secondary importance compared to other model uncertainties such as the chemical mechanism, except in situations where heat loss is unphysically fast compared to the flame time scale. A new theoretical framework to explain this insensitivity is also proposed, and its validity is briefly assessed.     

A comparative study of turbulence models for non-premixed swirl-stabilized flames

ABSTRACT

The accuracy of turbulent swirl-stabilized flame simulation strongly depends on the choice of turbulence model. In this study, four 3D unsteady turbulence closures, including large eddy simulation, scale-adaptive simulation, and two detached eddy simulation variants, along with four RANS models, including RNG k??, SST k?ω, transition SST, and RSM, are examined for moderate- and high-swirl case studies. It is observed that the scale-adaptive simulation provides the most accurate results for almost all variables and both swirl conditions in the reactive flow. Only the 3D unsteady models predict the vortex breakdown bubble and flame attachment state correctly. However, based on our error analysis, the flow and composition fields predicted by the RANS models are in acceptable agreement with the experimental fields, especially the ones of transition SST when higher swirl number cases or minor species concentration are of interest. Moreover, it is concluded that the viscosity ratio criterion is a better measure of the local LES quality than the turbulent kinetic energy ratio, and the accuracy of a hybrid simulation may be much more dependent on the ability of the model to operate close to the RANS mode where the grid resolution is not sufficient for a resolving simulation than the fraction of the resolved kinetic energy. Finally, the propriety of the base (RANS) model of a DES for the application of interest is important, such that DES with realizable k?? outperforms the commonly used DES with SST k?ω model.     

Comparative analysis of symmetries for the models of mechanics of nonuniform fluids

The symmetries for the main models of mechanics of binary fluids are calculated by the methods of the theory of continuous groups. The fundamental system of the differential form of the main conservation laws is characterized by the ten-parametric Galilean group. The Navier-Stokes set of equations possesses an extended set of symmetries with infinite dimensionality. Simplification of the model changes the order of the set of equations of motion, which leads to the impossibility to take into account the complete set of boundary conditions and formation of discontinuities in solutions for reduced models.     

Comparative analysis of approximate and exact models in inflationary cosmology

A method of constructing and analyzing exact solutions for inflationary cosmology models with a self-action scalar by introducing an effective self-action potential is suggested. On the basis of exact solutions for complete and “shortened” equations obtained in the “slow-descent” approximation, their comparative analysis is made. It is shown that the results obtained for approximate models that are conventionally used for comparison with experimental data may differ greatly from those for exact models because of the structural instability of models with inflation. Ul'anovsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 14–20, January, 2000.     

Adjoint-based sensitivity analysis of flames

Simulation of chemically reacting flows using detailed chemistry introduces a large number of chemistry model parameters. While not all significantly affect the target outcomes of a simulation, the parameters that do are not always known a priori. In order to improve simulations for specified target outcomes, termed quantities of interest (QoIs), the sensitivity of these QoIs to the model parameters are needed. However, evaluating the sensitivities is computationally expensive, especially for complex fuels that may involve many parameters. For these simulations, the forward sensitivity method requires the solution of an additional number of governing equations proportional to the number of parameters. Here, an adjoint sensitivity approach is formulated where the computational cost scales as the number of QoIs and not the number of parameters. Specifically, adjoint equations are derived for laminar, incompressible, variable density reacting flow and applied to hydrogen flame simulations. From the solution of the corresponding adjoint equations, sensitivity of the QoIs to chemistry model parameters is calculated. The one-dimensional simulation results show that the adjoint sensitivity results closely match those of forward sensitivity methods, thus providing validation of the adjoint method. The two-dimensional simulation results indicate the most sensitive parameters for two QoIs, flame tip temperature and NOx emission. For these tests, the adjoint method reduces computational expense compared to forward sensitivity methods by a factor proportional to the number of QoIs over the number of parameters, here 2/172. Such savings can be more drastic for cases that involve complex fuels, such as combustion of jet fuel, requiring thousands of chemistry model parameters. Further, this sensitivity information can be used in development of experiments by pointing out which are the critical chemistry model parameters.     

高光谱数据基于流形的半监督特征选择

传统的高光谱数据特征选择方法分为监督和无监督模式,然而在高光谱数据实际处理中,大量无标记和少量有标记数据并存.此外,传统方法忽视了真实数据嵌入在高维空间中的流形结构.本文提出一种基于流形的半监督特征选择算法,定义一个合理的特征评判准则,考虑标记样本的先验信息以及高维数据局部和非局部结构的不变特性.通过对标记数据类间方差和类内方差的极大化和极小化,优化数据的判别结构;同时通过构建局部graph和非局部graph,挖掘高维数据的流形结构.然后,选择一组有效的特征子集,实现高维数据的特征选择.最后,通过对特征选择后的真实高光谱数据进行分类实验,结果显示本文方法可以很好地对高光谱数据实现降维并且保留数据的主要结构.     

A DNS evaluation of mixing and evaporation models for TPDF modelling of nonpremixed spray flames

In the present work, nonpremixed temporally evolving planar spray jet flames are simulated using both direct numerical simulation (DNS) and the composition transported probability density function (TPDF) method. The objective is to assess the performance of various mixing and evaporation source term distribution models which are required to close the PDF transport equation in spray flames. Quantities which would normally be provided to the TPDF solver by spray models and turbulence models are provided from the DNS: the mean flow velocity, turbulent diffusivity, mixing frequency, and cell-mean evaporation source term. Two cases with different Damköhler numbers (Da) are considered. The low Da case (Da-) features extinction followed by reignition while extinction in the high Da case (Da+) is insignificant. The TPDF modelling considers two mixing models: interaction by exchange with the mean (IEM) and Euclidean minimum spanning trees (EMST). Three models for distribution of the evaporation source terms are considered: EQUAL which distributes them in proportion to notional particles’ mass weight, NEW which creates new particles of pure fuel, and SAT which distributes the sources preferentially to notional particles close to saturation. It is found that the IEM model overpredicts the extinction when used with any evaporation model for both Da- and Da+ cases. The EMST model captures well the trend for extinction and reignition for the Da- case when it is coupled with the EQUAL evaporation model, but it overpredicts the extinction when coupled with the NEW or SAT evaporation model. For the Da+ case, all evaporation models reasonably capture the flame dynamics when coupled with EMST. The flame temperature in the mixture fraction space was examined to further assess the model performance. In general the EMST model results in narrow PDFs with little conditional fluctuation, while the IEM model produces bimodal PDFs with burning and partial extinction branches.     

Stable Galerkin reduced order models for linearized compressible flow

The Galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations. The numerical stability of Galerkin models is shown to depend on the choice of inner product for the projection. For the linearized Euler equations, a symmetry transformation leads to a stable formulation for the inner product. Boundary conditions for compressible flow that preserve stability of the reduced order model are constructed. Preservation of stability for the discrete implementation of the Galerkin projection is made possible using a piecewise-smooth finite element basis. Stability of the reduced order model using this approach is demonstrated on several model problems, where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation.     

Reduced-order models for parameter dependent geometries based on shape sensitivity analysis

The proper orthogonal decomposition (POD) is widely used to derive low-dimensional models of large and complex systems. One of the main drawback of this method, however, is that it is based on reference data. When they are obtained for one single set of parameter values, the resulting model can reproduce the reference dynamics very accurately but generally lack of robustness away from the reference state. It is therefore crucial to enlarge the validity range of these models beyond the parameter values for which they were derived. This paper presents two strategies based on shape sensitivity analysis to partially address this limitation of the POD for parameters that define the geometry of the problem at hand (design or shape parameters.) We first detail the methodology to compute both the POD modes and their Lagrangian sensitivities with respect to shape parameters. From them, we derive improved reduced-order bases to approximate a class of solutions over a range of parameter values. Secondly, we demonstrate the efficiency and limitations of these approaches on two typical flow problems: (1) the one-dimensional Burgers’ equation; (2) the two-dimensional flows past a square cylinder over a range of incidence angles.     

Rate-ratio asymptotic analysis of non-premixed methane flames

The asymptotic structure of laminar, non-premixed methane flames is analysed using a reduced four-step chemical-kinetic mechanism. Chemical reactions are presumed to take place in two layers: the inner layer and the oxidation layer. In the inner layer the fuel reacts with radicals and the main compounds formed are the intermediate species CO and H₂. These intermediate species are oxidized in the oxidation layer. The structure of the oxidation layer is described by two second-order differential equations: one for CO and the other for H₂. Two limiting cases are considered. At one limit the global step CO+H₂?CO₂+H₂ is presumed to maintain partial equilibrium everywhere in the oxidation layer except in a thin layer adjacent to the inner layer. At the other limit the steady-state approximation is introduced for H₂ everywhere in the oxidation layer except in a thin layer adjacent to the inner layer. This limit, called ‘slow CO oxidation’, has not been analysed previously. The structure of the inner layer is described by two second-order differential equations: one for the fuel and the other for the H radicals. This is a significant improvement over previous models in which either a steady-state approximation is introduced for the H radicals in the inner layer, or the reaction between the fuel and radicals is presumed to be very fast. The chain-breaking elementary reaction CH₃+H+M→CH₄+M is found to have a significant influence on the structure of the inner layer and on the scalar dissipation rates at extinction. The influence of this reaction was either neglected in previous models or was included as a perturbation to the principal elementary reactions taking place to the leading order in the inner layer. Using the results of the asymptotic analysis the scalar dissipation rates at extinction are calculated at a pressure of 1 bar. They are found to agree well with those calculated numerically using a chemical-kinetic mechanism made up of elementary reactions.     

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For simplifying the calculation the magnetic distribution on tokamak, some two-dimensional analytic models including the effect of the iron core were established, such as the infinite long iron core model and the spool model. The assumptions of these two-dimensional analytic models lead to different results with the actual magnetic field due to the distinctive boundary condition. In order to accurately calculate the three-dimensional magnetic field distribution in the tokamak with iron core, a three-dimensional numerical finite element model was established based on J-TEXT tokamak. In two conditions, where the total toroidal current is nonzero or zero respectively, more comparison were carried out between the derived results of two-dimensional models and the results at different toroidal positions in three-dimensional models. Furthermore, the toroidal asymmetry of the magnetic field distribution of the three-dimensional model of tokamak with iron core was investigated. The results indicate that the three-dimensional construction of iron core causes the toroidal asymmetric poloidal magnetic field and the difference between the two- and three-dimensional models in the condition with total current of nonzero. However, in the condition with total current of zero, the intensity of toroidal asymmetric is reduced and the difference between the two- and three-dimensional models is smaller.     

Combustion and emission modelling of a direct-injection spark-ignition engine by combining flamelet models for premixed and diffusion flames

The combustion and emission production processes of a DISI (direct-injection spark-ignition) engine were modelled by combining flamelet models for premixed and diffusion flames. A new surrogate fuel was proposed to approximate the complicated composition of real gasoline. In contrast to simpler conventional models, the fuel was modelled as a ternary mixture of three hydrocarbons: iso-octane, n-heptane and toluene. Turbulent flame propagation in a partially premixed field was modelled by a premixed flamelet model. The mass fractions of the detailed composition of species in burnt gas were predicted by a diffusion flamelet model. For the pollutant formation modelling, a two-step oxidation of CO and H₂ was used to simulate the secondary diffusion flame. The extended Zeldovich mechanism was used to model NOx formation, while a phenomenological model was used to model soot formation. This model was initially applied to a simple geometry to investigate the fundamentals of the model's behaviour, after which three-dimensional computational fluid dynamic (CFD) simulations were performed in a realistic engine geometry.     

铁芯托卡马克中二维和三维极向磁场计算模型的对比分析

为方便计算托卡马克磁场分布,建立了一些二维解析铁芯模型。由于前提假设的不同而给磁场分布的计算带来了不同的边界条件,因而得到的磁场分布计算结果与实际情况有所偏差。为了获得铁芯托卡马克的极向磁场三维分布,建立了带铁芯的极向磁场线圈三维数值模型,计算铁芯托卡马克的三维磁场,与不同铁芯模型的磁场计算结果进行比较,并且研究铁芯托卡马克的三维磁场的极向分量在环向上的不对称性。     

The analysis of the electret-depolarization based on various models

This paper gives an analysis of the process of depolarization of an electret, based on models which allow for volume charges of both signs which have been injected into the dielectric and held in traps. It is shown that an inversion of the direction of the discharge current can be observed during electret depolarization.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 104–110, June, 1974.     

A method for solving stochastic equations by reduced order models and local approximations

A method is proposed for solving equations with random entries, referred to as stochastic equations (SEs). The method is based on two recent developments. The first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments. The second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector, referred to as stochastic reduced order model (SROM), and uses it to construct a SROM for the solution of this equation.The proposed method is a direct extension of these two methods. It uses SROMs to select expansion points, rather than selecting these points by geometrical considerations, and represents the solution by linear and/or higher order local approximations. The implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic/differential equations. The method is conceptually simple, non-intrusive, efficient relative to classical Monte Carlo simulation, accurate, and guaranteed to converge to the exact solution.     

Uncertainty-based experimental validation of nonlinear reduced order models

This paper presents the first results of a combined experimental–computational investigation focused on the validation of reduced order models of geometrically nonlinear structures in the presence of uncertainty. The validation approach considered here is based on the premise that the model is valid if the experimental results can be considered as random sample responses of the stochastic system of which the reduced order model is the mean. For the situation considered here, the power spectra of the experiments should lie within the 2nd and 98th percentiles of the response (forming the uncertainty band) of the stochastic model. Nominally clamped–clamped beams are considered to demonstrate the entire process. The construction of two mean reduced order models and their stochastic counterparts are first performed. Then, the validation effort is carried out by comparing experimentally obtained power spectra and their corresponding computational uncertainty bands. This process leads, for both reduced order models, to a very good representation of the important upper envelope (98th percentile) of the experimental data but a less good fit of the lower envelope (2nd percentile).