Eulerian transported probability density function sub-filter model for large-eddy simulations of turbulent combustion

Reactive flow simulations using large-eddy simulations (LES) require modelling of sub-filter fluctuations. Although conserved scalars like mixture fraction can be represented using a beta-function, the reactive scalar probability density function (PDF) does not follow an universal shape. A one-point one-time joint composition PDF transport equation can be used to describe the evolution of the scalar PDF. The high-dimensional nature of this PDF transport equation requires the use of a statistical ensemble of notional particles and is directly coupled to the LES flow solver. However, the large grid sizes used in LES simulations will make such Lagrangian simulations computationally intractable. Here we propose the use of a Eulerian version of the transported-PDF scheme for simulating turbulent reactive flows. The direct quadrature method of moments (DQMOM) uses scalar-type equations with appropriate source terms to evolve the sub-filter PDF in terms of a finite number of delta-functions. Each delta-peak is characterized by a location and weight that are obtained from individual transport equations. To illustrate the feasibility of the scheme, we compare the model against a particle-based Lagrangian scheme and a presumed PDF model for the evolution of the mixture fraction PDF. All these models are applied to an experimental bluff-body flame and the simulated scalar and flow fields are compared with experimental data. The DQMOM model results show good agreement with the experimental data as well as the other sub-filter models used.     

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.     

Analysis of distances between inclusions in finite binary stochastic materials

A generalized probability density function (PDF) describing the distribution of inter-inclusion distances in finite, isotropic, binary stochastic materials with fixed diameter inclusions has been developed and tested. The new probability density function explicitly accounts for edge effects present in finite two- and three-dimensional stochastic materials. The generalized PDF is shown to include factors that are dependent on both the geometry of the material region as well as the statistical properties of the material. A discussion of the properties and application of this newly developed PDF is provided along with supporting numerical results for case studies in one- and two-dimensions. These numerical results demonstrate the ability of the newly derived PDF to correctly account for edge effects in finite stochastic materials, while still reproducing the expected distribution within the bulk material region.     

Photon time-interval statistics applied to the analysis of laser heterodyne signal with photon counter

In this paper, we report a mathematical derivation of probability density function (PDF) of time-interval between two successive photoelectrons of the laser heterodyne signal, and give a confirmation of the theoretical result by both numerical simulation and an experiment. The PDF curve of the beat signal displays a series of fluctuations, the period and amplitude of which are respectively determined by the beat frequency and the mixing efficiency. The beat frequency is derived from the frequency of fluctuations accordingly when the PDF curve is measured. This frequency measurement method still works while the traditional Fast Fourier Transform (FFT) algorithm hardly derives the correct peak value of the beat frequency in the condition that we detect 80 MHz beat signal with 8 Mcps (counts per-second) photons count rate, and this indicates an advantage of the PDF method.     

Probabilistic solution of nonlinear oscillators excited by combined Gaussian and Poisson white noises

The stationary probability density function (PDF) solution of the stochastic response of nonlinear oscillators is investigated in this paper. The external excitation is assumed to be a combination of Gaussian and Poisson white noises. The PDF solution is governed by the generalized Kolmogorov equation which is solved by the exponential-polynomial closure (EPC) method. In order to evaluate the effectiveness of the EPC method, different nonlinear oscillators are considered in numerical analysis. Nonlinearity exists either in displacement or in velocity for these nonlinear oscillators. The impulse arrival rate, mono-modal PDF and bi-modal PDF are also considered in this study. Compared to the PDF given by Monte Carlo simulation, the EPC method presents good agreement with the simulated result, which can also be observed in the tail region of the PDF solution.     

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 positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations

The Vlasov–Poisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that self-interacts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh. In particular, we develop in this work a method that discretizes the 1 + 1 Vlasov–Poisson system via a high-order discontinuous Galerkin (DG) method in phase space, and an operator split, semi-Lagrangian method in time. Second-order accuracy in time is relatively easy to achieve via Strang operator splitting. With additional work, using higher-order splitting and a higher-order method of characteristics, we also demonstrate how to push this scheme to fourth-order accuracy in time. We show how to resolve all of the Lagrangian dynamics in such a way that mass is exactly conserved, positivity is maintained, and high-order accuracy is achieved. The Poisson equation is solved to high-order via the smallest stencil local discontinuous Galerkin (LDG) approach. We test the proposed scheme on several standard test cases.     

Delayed auger recombination in silicon measured by time-resolved X-ray scattering

We report a new method of measuring the non-radiative recombination rate in bulk Silicon. Synchrotron time-resolved x-ray scattering (TRXS) combines femtometer spatial sensitivity with nanosecond time resolution to record the temporal evolution of a crystal lattice following intense ultrafast laser excitation. Modeling this data requires an Auger recombination time that is considerably slower than previous measurements, which were made at lower laser intensities while probing only a relatively shallow surface depth. We attribute this difference to an enhanced Coulomb interaction that has been predicted to occur in bulk materials with high densities of photoexcited charge carriers.     

Pair distribution function analysis:Fundamentals and application to battery materials

Battery materials are of vital importance in powering a elean and sustainable society.Improving their performance relies on a clear and fundamental understanding of their properties,in particular,structural properties.Pair distribution function(PDF) analysis,which takes into account both Bragg scattering and diffuse scattering,can probe structures of both crystalline and amorphous phases in battery materials.This review first introduces the principle of PDF,followed by its application in battery materials.It shows that PDF is an effective tool in studying a series of key scientific topics in battery materials.They range from local ordering,nano-phase quantification,anion redox reaction,to lithium storage mechanism,and so on.     

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。     

A PDF projection method: A pressure algorithm for stand-alone transported PDFs

In this paper, a new formulation of the projection approach is introduced for stand-alone probability density function (PDF) methods. The method is suitable for applications in low-Mach number transient turbulent reacting flows. The method is based on a fractional step method in which first the advection–diffusion–reaction equations are modelled and solved within a particle-based PDF method to predict an intermediate velocity field. Then the mean velocity field is projected onto a space where the continuity for the mean velocity is satisfied. In this approach, a Poisson equation is solved on the Eulerian grid to obtain the mean pressure field. Then the mean pressure is interpolated at the location of each stochastic Lagrangian particle. The formulation of the Poisson equation avoids the time derivatives of the density (due to convection) as well as second-order spatial derivatives. This in turn eliminates the major sources of instability in the presence of stochastic noise that are inherent in particle-based PDF methods. The convergence of the algorithm (in the non-turbulent case) is investigated first by the method of manufactured solutions. Then the algorithm is applied to a one-dimensional turbulent premixed flame in order to assess the accuracy and convergence of the method in the case of turbulent combustion. As a part of this work, we also apply the algorithm to a more realistic flow, namely a transient turbulent reacting jet, in order to assess the performance of the method.     

An Explicit Family of Probability Measures for Passive Scalar Diffusion in a Random Flow

We explore the evolution of the probability density function (PDF) for an initially deterministic passive scalar diffusing in the presence of a uni-directional, white-noise Gaussian velocity field. For a spatially Gaussian initial profile we derive an exact spatio-temporal PDF for the scalar field renormalized by its spatial maximum. We use this problem as a test-bed for validating a numerical reconstruction procedure for the PDF via an inverse Laplace transform and orthogonal polynomial expansion. With the full PDF for a single Gaussian initial profile available, the orthogonal polynomial reconstruction procedure is carefully benchmarked, with special attentions to the singularities and the convergence criteria developed from the asymptotic study of the expansion coefficients, to motivate the use of different expansion schemes. Lastly, Monte-Carlo simulations stringently tested by the exact formulas for PDF’s and moments offer complete pictures of the spatio-temporal evolution of the scalar PDF’s for different initial data. Through these analyses, we identify how the random advection smooths the scalar PDF from an initial Dirac mass, to a measure with algebraic singularities at the extrema. Furthermore, the Péclet number is shown to be decisive in establishing the transition in the singularity structure of the PDF, from only one algebraic singularity at unit scalar values (small Péclet), to two algebraic singularities at both unit and zero scalar values (large Péclet).     

LES-based Eulerian PDF approach for the simulation of scramjet combustors

A probability density function (PDF) approach to account for turbulence–chemistry interaction in the context of large eddy simulation (LES) based simulation of scramjets is developed. To solve the high-dimensional joint-composition PDF transport equation robustly, the semi-discrete quadrature method of moments (SeQMOM) is employed. The SeQMOM approach addresses key numerical issues in LES related to the inaccuracies in computing filter-scale gradients, enabling an efficient and numerically consistent solution of the PDF transport equation. The computational tool is used to simulate a cavity-stabilized Mach 2.2 supersonic combustor. The LES–SeQMOM approach captures the pressure profiles qualitatively, in spite of the large uncertainties in the boundary conditions. Further, this flow configuration is found to exhibit low-frequency dynamics that lead to oscillatory motions of the shock-system inside the combustor. Detailed analyses of the LES data are used to describe the physical mechanism leading to these oscillations.     

QCD analysis of CMS W+ charm measurements at LHC with s~(1/2)=7 TeV and implications for strange PDF

We calculate cross-sections and cross-section ratios of a charm quark production in association with a W gauge boson at next-to-leading order QCD using MadGraph and CTIONNLO,CT14 NNLO,and MSTW2008 NNLO PDFs.We compare the results with measurements from the CMS detector at the LHC at a center-of-mass energy of 7 TeV.Moreover,we calculate absolute and normalized differential cross-sections as well as differential cross-section ratios as a function of the lepton pseudorapidity from the W boson decay.The correlation between the CT14 NNLO PDFs and Predictions for W+charm data are studied as well.Furthermore,by employing the error PDF updating method proposed by the CTEQ-TEA group,we update CT14 NNLO PDFs,and analyze the impact of CMS 7 TeV W+charm production data to the original CT14 NNLO PDFs.By comparison of the g(x,Q),s(x,Q),u(x,Q),d(x,Q),u(x,Q),and d(x,Q) PDFs at Q=1.3 GeV and Q=100 GeV for the CT14 NNLO and CT14 NNLO+Wc,we see that the error band of the s(x,Q) PDF is reduced in the region x 0.4,and the error band of g(x,Q) PDF is also slightly reduced at region 0.01 x 0.1.     

A LES/PDF simulator on block-structured meshes

A block-structured mesh large-eddy simulation (LES)/probability density function (PDF) simulator is developed within the OpenFOAM framework for computational modelling of complex turbulent reacting flows. The LES/PDF solver is a hybrid solution methodology consisting of (i) a finite-volume (FV) method for solving the filtered mass and momentum equations (LES solver), and (ii) a Lagrangian particle-based Monte Carlo algorithm (PDF solver) for solving the modelled transport equation of the filtered joint PDF of compositions. Both the LES and the PDF methods are developed and combined to form a hybrid LES/PDF simulator entirely within the OpenFOAM framework. The in situ adaptive tabulation method [S.B. Pope, Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation, Combust. Theory Model. 1 (1997), pp. 41–63; L. Lu, S.R. Lantz, Z. Ren, and B.S. Pope, Computationally efficient implementation of combustion chemistry in parallel PDF calculations, J. Comput. Phys. 228 (2009), pp. 5490–5525] is incorporated into the new LES/PDF solver for efficient computations of combustion chemistry with detailed reaction kinetics. The method is designed to utilise a block-structured mesh and can readily be extended to unstructured grids. The three-stage velocity interpolation method of Zhang and Haworth [A general mass consistency algorithm for hybrid particle/finite-volume PDF methods, J. Comput. Phys. 194 (2004), pp. 156–193] is adapted to interpolate the LES velocity field onto particle locations accurately and to enforce the consistency between LES and PDF fields at the numerical solution level. The hybrid algorithm is fully parallelised using the conventional domain decomposition approach. A detailed examination of the effects of each stage and the overall performance of the velocity interpolation algorithm is performed. Accurate coupling of the LES and PDF solvers is demonstrated using the one-way coupling methodology. Then the fully two-way coupled LES/PDF solver is successfully applied to simulate the Sandia Flame-D, and a turbulent non-swirling premixed flame and a turbulent swirling stratified flame from the Cambridge turbulent stratified flame series [M.S. Sweeney, S. Hochgreb, M.J. Dunn, and R.S. Barlow, The structure of turbulent stratified and premixed methane/air flames I: Non-swirling flows, Combust. Flame 159 (2012), pp. 2896–2911; M.S. Sweeney, S. Hochgreb, M.J. Dunn, and R.S. Barlow, The structure of turbulent stratified and premixed methane/air flames II: Swirling flows, Combust. Flame 159 (2012), pp. 2912–2929]. It is found that the LES/PDF method is very robust and the results are in good agreement with the experimental data for both flames.     

Prediction of local extinction and re-ignition effects in non-premixed turbulent combustion using a flamelet/progress variable approach

The flamelet/progress variable approach (FPVA) has been proposed by Pierce and Moin as a model for turbulent non-premixed combustion in large-eddy simulation. The filtered chemical source term in this model appears in unclosed form, and is modeled by a presumed probability density function (PDF) for the joint PDF of the mixture fraction Z and a flamelet parameter λ. While the marginal PDF of Z can be reasonably approximated by a beta distribution, a model for the conditional PDF of the flamelet parameter needs to be developed. Further, the ability of FPVA to predict extinction and re-ignition has also not been assessed. In this paper, we address these aspects of the model using the DNS database of Sripakagorn et al. It is first shown that the steady flamelet assumption in the context of FPVA leads to good predictions even for high levels of local extinction. Three different models for the conditional PDF of the flamelet parameter are tested in an a priori sense. Results obtained using a delta function to model the conditional PDF of λ lead to an overprediction of the mean temperature, even with only moderate extinction levels. It is shown that if the conditional PDF of λ is modeled by a beta distribution conditioned on Z, then FPVA can predict extinction and re-ignition effects, and good agreement between the model and DNS data for the mean temperature is observed.     

辐射温度的数据处理方法

辐射温度是间接驱动惯性约束聚变实验中可以定量测量的重要物理量。引进了一种通过直接计算辐射通量来得到辐射温度的方法,该方法的主要优点是计算简单,避免了某些时刻不能计算辐射温度的情况。由该方法得到的辐射温度与传统方法计算结果符合较好,不确定度均为7%。通过与冲击波测量温度峰值的结果对比确认了数据处理方法的正确性。在此基础上建立了国内某大型激光装置上的新型15道软X光能谱仪的数据处理系统。     

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.     

Efficient implementation of x-ray ghost imaging based on a modified compressive sensing algorithm

Towards efficient implementation of x-ray ghost imaging (XGI), efficient data acquisition and fast image reconstruction together with high image quality are preferred. In view of radiation dose resulted from the incident x-rays, fewer measurements with sufficient signal-to-noise ratio (SNR) are always anticipated. Available methods based on linear and compressive sensing algorithms cannot meet all the requirements simultaneously. In this paper, a method based on a modified compressive sensing algorithm with conjugate gradient descent method (CGDGI) is developed to solve the problems encountered in available XGI methods. Simulation and experiments demonstrate the practicability of CGDGI-based method for the efficient implementation of XGI. The image reconstruction time of sub-second implicates that the proposed method has the potential for real-time XGI.     

X-ray reflectivity data analysis using Bayesian inference: The study of induced Pt magnetization in Pt/Co/Pt

The induced Pt magnetization in a Pt/Co/Pt thin film structure is studied. The normally nonmagnetic Pt acquires a magnetic moment due to the magnetic proximity effect at the Co–Pt interfaces. Element specific Pt structural and magnetic properties are characterized by synchrotron-based resonant x-ray reflectivity and x-ray resonant magnetic reflectivity measurements. An advanced analysis method based on Bayesian inference is used for model fitting of the x-ray data. Using this method, we retrieve the best fit values of material parameters (e.g., thickness, interfacial roughness) from the data. Analysis of x-ray reflectivity data of this specific system shows that the Pt magnetization and Co–Pt interfacial roughness is significantly different between the top and bottom Pt layers, with both values being larger in the top Pt. The successful application of this Bayesian method to study the magnetic and structural properties of a thin film system demonstrates its effectiveness for x-ray reflectivity data analysis.