Composition Probability Density Function Research Articles

A numerical study of auto-ignition in turbulent lifted flames issuing into a vitiated co-flow

This paper presents a numerical study of auto-ignition in simple jets of a hydrogen–nitrogen mixture issuing into a vitiated co-flowing stream. The stabilization region of these flames is complex and, depending on the flow conditions, may undergo a transition from auto-ignition to premixed flame propagation. The objective of this paper is to develop numerical indicators for identifying such behavior, first in well-known simple test cases and then in the lifted turbulent flames. The calculations employ a composition probability density function (PDF) approach coupled to the commercial CFD code, FLUENT. The in-situ-adaptive tabulation (ISAT) method is used to implement detailed chemical kinetics. A simple k–ϵ turbulence model is used for turbulence along with a low Reynolds number model close to the solid walls of the fuel pipe. The first indicator is based on an analysis of the species transport with respect to the budget of convection, diffusion and chemical reaction terms. This is a powerful tool for investigating aspects of turbulent combustion that would otherwise be prohibitive or impossible to examine experimentally. Reaction balanced by convection with minimal axial diffusion is taken as an indicator of auto-ignition while a diffusive–reactive balance, preceded by a convective–diffusive balanced pre-heat zone, is representative of a premixed flame. The second indicator is the relative location of the onset of creation of certain radical species such as HO2 ahead of the flame zone. The buildup of HO2 prior to the creation of H, O and OH is taken as another indicator of autoignition. The paper first confirms the relevance of these indicators with respect to two simple test cases representing clear auto-ignition and premixed flame propagation. Three turbulent lifted flames are then investigated and the presence of auto-ignition is identified. These numerical tools are essential in providing valuable insights into the stabilization behaviour of these flames, and the demarcation between processes of auto-ignition and premixed flame propagation.

Reconstruction of composite events in neutrino telescopes

Neutrino telescopes detect the emission of Cherenkov light resulting from the tracks and showers of charged and neutral current neutrino interactions. These tracks or showers are reconstructed using a corresponding probability density function (PDF) which depends on measured time and location of the detected photons. We call a composite event the mixed detector response due to the juxtaposition of more than one Cherenkov light source (track or shower). This paper presents the construction of a generic PDF corresponding to a composite hypothesis. This composite PDF is therefore useful to reconstruct an arbitrary event topology and to favor or discard a given event topology hypothesis.

Composite Phase PDF in Gamma Shadowed Nakagami Fading Channel

This paper presents a new Fourier series form probability density function (PDF) for the phase of the received signal over gamma shadowed Nakagami-m fading channel with additive Gaussian noise. This PDF is further used to evaluate the average symbol-error probability of M-ary CPSK. An alternative new closed-form expression for bit error-rate of binary CPSK is also derived and numerically compared with the corresponding result based on the Fourier series form phase PDF. We show that the two expressions are indeed numerically equivalent.

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.

Probability density function modelling of concentration in and above a canopy layer

Non-linear chemical reactions and other important processes depend on average concentration as well as on the fluctuations of concentration of chemical species. Probability density function (PDF) modelling is a powerful tool for evaluating in an exact manner chemical reaction depending on higher order concentration moments. However, only few studies are devoted to investigate applications of PDF modelling to pollutant dispersion and reactions in atmospheric turbulence. In real cases, very often the emissions/reactions of pollutants take place in and above a canopy layer (urban canopy, vegetation canopy); therefore, the vertical turbulence inhomogeneity plays a relevant role. This inhomogeneity can be efficiently taken into account by PDF approach. We implement a simplified one-dimensional model for the velocity composition PDF by coupling a Monte Carlo simulation and a micro-mixing model (IECM). Under a general assumption of horizontally homogeneous extensive plane source the proposed model is able to fairly reproduce all the concentration moments measured in a laboratory experiment on scalar dispersion within a model plant canopy.

PDF modelling of turbulent non-premixed combustion with detailed chemistry

Although the significant advantage for the probability density function (PDF) methods of the exact treatment of chemical reactions in turbulent combustion problems, a detailed chemistry mechanism (e.g., the GRI mechanism) has not been implemented in the practical calculations by now due to the prohibitive computation of PDF methods. In this work, a detailed mechanism (GRI-Mech 3.0, consisting of 53 species and 325 elemental reactions) is firstly incorporated into the PDF calculation of a turbulent non-premixed jet flame (Sandia Flame D). The flow is formulated in the boundary layer form. The joint composition PDF closure level is applied and a multiple-time-scale (MTS) k– ε turbulence model is combined for the closure of turbulent transport terms. The molecular mixing process is modelled by the Euclidean minimum spanning tree (EMST) mixing model. The solutions are obtained by using the space marching algorithm for turbulence equations and node-based Monte Carlo particle method for PDF evolution equation. The chemical reaction source terms are integrated directly. Extensive comparisons between the predictions and the measurements are made, which involve radial profiles of mean and rms (root mean square), conditional mean, scatter plots of scalars and conditional PDF distribution etc. The flame structures are well represented by the present calculation, including intermediate species (e.g. CO and H 2) mass fractions, pollutant NO emission and local extinction.

Pdf calculations of turbulent lifted flames of H2/N2 fuel issuing into a vitiated co-flow

This paper presents detailed calculations of the flow, mixing and composition fields of a simple jet of hydrogen–nitrogen mixture issuing into a vitiated co-flowing stream. The co-flow contains oxygen as well as combustion products and is sufficiently hot to provide an ignition source for a flame that stabilizes at some ten diameters downstream of the jet exit plane. This configuration forms a good model problem for studying lifted flames as well as issues of auto-ignition. The calculations employ a composition probability density function (PDF) approach coupled to the commercial CFD package FLUENT. The in situ adaptive tabulation method is adopted to account for detailed chemical kinetics. A simple k–ε model is used for turbulence along with a low Reynolds number model for the walls. Calculations are optimized to obtain a numerically accurate solution and are repeated for two different H2 mechanisms, each consisting of ten species. The flame is found to be largely controlled by chemical rather than mixing processes. The mechanisms used yield different lift-off heights and compositions that straddle the data. Ignition delays are found to be extremely sensitive to the chemical kinetic rates of some reactions in the mechanisms.

Partially Stirred Reactor Model: Analytical Solutions and Numerical Convergence Study of a PDF/Monte Carlo Method

We investigate the partially stirred reactor (PaSR), which is based on a simplified joint composition probability density function (PDF) transport equation. Analytical solutions for the first four moments of the mass density function (MDF) obtained from the PaSR model are presented. The Monte Carlo particle method with first order time splitting algorithm is implemented to obtain the first four moments of the MDF numerically. The dynamics of the stochastic particle system is determined by inflow-outflow, chemical reaction, and mixing events. Three different inflow-outflow algorithms are investigated: an algorithm based on the inflow-outflow event modeled as a Poisson process, an inflow-outflow algorithm mentioned in the literature, and a novel algorithm derived on the basis of analytical solutions. It is demonstrated that the inflow-outflow algorithm used in the literature can be explained by considering a deterministic waiting time parameter of a corresponding stochastic process, and also forms a specific case of the new algorithm. The number of particles in the ensemble, N, the nondimensional time step, $\Delta t^{*}$ (ratio of the global time step to the characteristic time of an event), and the number of independent simulation trials, L, are the three sources of the numerical error. The split analytical solutions and the numerical experiments suggest that the systematic error converges as $\Delta t^{*}$ and N-1 . The statistical error scales as L-1/2 and N-1/2 . The significance of the numerical parameters and the inflow-outflow algorithms is also studied by applying the PaSR model to a practical case of premixed kerosene and air combustion.

A general mass consistency algorithm for hybrid particle/finite-volume PDF methods

An algorithm is devised to maintain a correct spatial distribution of computational particles in hybrid particle/finite-volume (FV) probability density function (PDF) methods for chemically reacting turbulent flows. The approach is, by design, compatible with existing FV computational fluid dynamics (CFD) codes that are used to model practical engineering flows in complex geometric configurations. The algorithm is suitable for general three-dimensional incompressible or compressible, steady or time-dependent flows using structured or unstructured, stationary or deforming computational meshes. It is compatible with a variety of element shapes commonly used in research and engineering CFD codes including hexahedra, prisms and tetrahedra. Robustness, accuracy and efficiency of the approach are demonstrated via computations for several two- and three-dimensional steady and unsteady flow configurations using computational meshes that vary in element type and in mesh quality. Both composition PDF and velocity PDF methods are employed. This work broadens the accessibility of PDF methods for practical turbulent combustion systems.

A Lagrangian study of scalar diffusion in isotropic turbulence with chemical reaction

Direct numerical simulations are performed of a single-step, nonpremixed, Arrhenius-type reaction developing in isotropic, incompressible, decaying turbulence, for conditions where flame extinction and re-ignition occur. The Lagrangian characteristics of scalar diffusion, information necessary for modeling approaches such as some implementations of probability density function (PDF) methods, are investigated by tracking fluid particles. Focusing on the mixture fraction and temperature as the scalar variables of interest, fluid particles are characterized as continuously burning or noncontinuously burning based upon their recent time history, and noncontinuously burning particles are further characterized based upon their initial regions relative to the flame zone. The behavior of the mixture fraction and temperature fields is contrasted for the different types of particles characterized. Significant differences among these characterized particles are found, for example, in the unclosed conditional expectations of scalar diffusion appearing in the composition PDF equations.

On the behavior of a coupled transient flamelet/pdf transport equation model for nonpremixed turbulent flames

This work introduces a new methodology for the modeling of turbulent flames, based on the combination of two frequently used models: the Eulerian composition probability density function (PDF) transport equation model and the transient laminar flamelet model (TLFM). The current new coupled TLFM/PDF model aims to combine the advantages of both models while avoiding their disadvantages. The flame chosen in this work for the validation was experimentally studied by Barlow and Frank (Effects of Turbulence on Species Mass Fractions in Methane/Air Flames, Proc. Combust. Inst ., 27 , 1087-1095, 1998) and shows visible local extinction at around an axial distance of 30 jet radii and reignition farther downstream. Therefore, this configuration is a good test case to study the capability of numerical models for flames with local extinction and reignition regions. The Eulerian composition PDF transport equation model with a four-step reduced chemistry and the coupled TLFM/PDF model are compared and validated against experiments.

A morphologically optimal strategy for classifier combinaton: multiple expert fusion as a tomographic process

We specify an analogy in which the various classifier combination methodologies are interpreted as the implicit reconstruction, by tomographic means, of the composite probability density function spanning the entirety of the pattern space, the process of feature selection in this scenario amounting to an extremely bandwidth-limited Radon transformation of the training data. This metaphor, once elaborated, immediately suggests techniques for improving the process, ultimately defining, in reconstructive terms, an optimal performance criterion for such combinatorial approaches.

Detailed modeling of soot formation in a partially stirred plug flow reactor

The purpose of this work is to propose a detailed model for the formation of soot in turbulent reacting flow and to use this model to study a carbon black furnace. The model is based on a combination of a detailed reaction mechanism to calculate the gas phase chemistry, a detailed kinetic soot model based on the method of moments, and the joint composition probability density function (PDF) of these scalar quantities.Two problems, which arise when modeling the formation of soot in turbulent flows using a PDF approach, are studied. A consistency study of the combined scalar-soot moment approach reveals that the molecular diffusion term in the PDF-equation can be closed by the IEM and Curl-type mixing models. An investigation of different kernels for the collision frequency of soot particles shows that the influence of turbulence on particle coagulation is negligible for typical flame conditions and the particle size range considered.The model is used as a simple tool to simulate a furnace black process, which is the most important industrial process for the production of carbon blacks. Despite the simplifications in the modeling of the turbulent flow reasonable agreement between the calculated soot yield and data measured in an industrial furnace black reactor is achieved although no adjustments were made to the kinetic parameters of the soot model. The effect of the mixing intensity on soot yield and different soot formation rates is investigated. In addition the influence of different operating conditions such as temperature and equivalence ratio in the primary zone of the reactor is studied.

Application of composition PDF methods in the investigation of turbulence–radiation interactions

The composition probability density function (PDF) method is used to study radiating reactive flows. The method is able to treat turbulence–radiation interactions (TRI) in a rigorous way: many unclosed terms due to TRI in the traditional Reynolds-averaging process can be calculated exactly and all others can be accurately modeled by using the optically thin eddy approximation. The application of the method is demonstrated by considering a simplified methane/air diffusion flame, which shows enhancement of the radiative fluxes as a result of TRI. The importance of considering different TRI terms is investigated, indicating that the absorption coefficient–Plank function correlation is the most important.

CFD Analysis of Premixed Methane Chlorination Reactors with Detailed Chemistry

With the implementation of efficient algorithms for the accurate calculation of reaction source terms, computational fluid dynamics (CFD) is now a powerful tool for the simulation and design of chemical reactors with complex kinetic schemes. The example studied in this work is the methane chlorination reaction for which the detailed chemistry scheme has 152 reactions and 38 species. The adiabatic, jet-stirred chlorination reactor used for the CFD simulations is an insulated right cylinder with a coaxial premixed feed stream at one end. In order for this reactor to remain lit, recirculation of hot products is crucial, and hence, reactor stability is sensitive to both macroscale and microscale mixing. By neglecting density variations, a Lagrangian composition probability density function (PDF) code with a novel chemistry tabulation algorithm (in-situ adaptive tabulation or ISAT) for handling complex reactions is used to simulate the species concentrations and temperature field inside of the reactor. In addition, a reduced mechanism with 21 reactions and 15 species is tested for accuracy against the detailed chemistry scheme, a simplified CSTR model is used to illustrate the shortcomings of zero-dimensional models, and a pair-wise mixing stirred reactor (PMSR) model is used to show the stabilizing effect of micromixing on reactor stability. The CFD simulations are generally in good agreement with results from pilot-scale reactors for the outlet temperature and major species.

Relationship between beach profiles and waves at Duck, North Carolina, determined by canonical correlation analysis

Abstract An 11-year long measurement time series of waves and profiles from Duck, North Carolina, was analyzed using canonical correlation analysis (CCA) in order to determine the covariability between waves and profile response. The main objectives of this analysis were (1) to detect simultaneously occurring patterns in the wave and profile data and, if possible, give such patterns a physical interpretation to increase the insight into the governing processes, and (2) to investigate the possibility to use CCA in a predictive mode with the aim to calculate the profile response from the waves. The profile shape itself and the change between consecutively surveyed profiles were correlated with both the offshore (deep-water) and nearshore wave conditions. In the offshore, the waves were described by composite probability density functions (pdf) derived based on the measured wave conditions prior to a profile survey. Nearshore wave conditions were calculated using a random breaker decay model and averaged local wave properties were used as input to the CCA. The profile response displayed significantly higher correlation with the nearshore wave properties as compared to the offshore waves, and the highest correlation was found between the profile shape and the mean ratio of breaking waves for the time period preceding the profile survey. The CCA using nearshore wave properties indicates a potential for predicting the profile response with an acceptable degree of accuracy once a regression matrix relating the profiles to the waves has been established that represents the typical variability at the site.

PDF modeling of lean premixed combustion using in situ tabulated chemistry

The velocity–composition probability density function (pdf) model coupled with a k-ϵ-based mean flow computational fluid dynamics (CFD) model was used to describe the turbulent fluid flow, heat transfer, chemistry, and their interactions in a bluff-body, lean, premixed, methane–air combustor. Measured data [1, 2] including velocity, temperature, and chemical species concentrations were used to evaluate the model. The chemistry calculations were performed with an in situ look-up tabulation method [3]. A reduced, 5-step chemical mechanism [4] for describing fuel oxidation, CO, and NO chemistry was used in the model. NO formation from thermal, N 2O-intermediate, and prompt pathways was included in the 5-step mechanism. An axisymmetric, unstructured grid was used for solving the Eulerian, mean flow equations and the vertices were used to store mean statistics for solving the Lagrangian, fluid particle equations. Predicted velocity and composition mean statistics were compared to measurements in the bluff-body combustor for a lean equivalence ratio of 0.59. The predictions of major species matched measured and calculated equilibrium values in the recirculation zone. Comparisons of mean CO throughout the combustor were always within an order of magnitude and showed marked improvements over past predictions. Maximum discrepancies between measured and predicted NO concentrations were between 5 and 7 ppm (∼50%). The accessed composition space in this turbulent combustion simulation represented the values of species mole fraction and enthalpy for each fluid particle at each time step and was found to lie in a relatively small, uniquely shaped region that was dictated by the mixing, reaction, and heat transfer in the combustor. This accessed composition region was obtained in situ and required about 35 megabytes of storage once a steady state was reached. This memory requirement was more than three orders of magnitude less than would be needed in a standard, a priori table. The in situ tabulation approach allowed for technically correct and efficient chemical kinetic calculations using the 5-step mechanism in this pdf-method-based, multidimensional combustor model.

A composite PDF of scalar concentration in turbulent flows

Experimental plume data and physical arguments are used to make the case for a composite probability density function to describe scalar concentration at large distances from the source. High concentration tails of the PDF are shown to be well described by a generalized Pareto distribution.

The BMC/GIEM Model for Micromixing in Non-Premixed Turbulent Reacting Flows

A novel beta mapping closure (BMC)/generalized IEM (GIEM) model is proposed to close the micromixing term in the joint composition probability density function (PDF) transport equation for nonpremixed, multiscalar turbulent reacting flows. Because the beta PDF gives an excellent description of binary inert scalar mixing, it is used to approximate the mixture fraction PDF. The proposed model thus effectively extends presumed PDF methods to handle finite-rate chemistry without introducing ad hoc assumptions about the form of the joint composition PDF. The BMC/GIEM model can be solved efficiently using a particle-based Monte-Carlo code. The model is first compared with the amplitude mapping closure (AMC)/GIEM model and homogeneous reaction-diffusion data for scalar mixing with a two-step reaction. Its application to inhomogeneous flows is demonstrated by simulating a reacting scalar mixing layer with a one-step reaction using a Lagrangian composition PDF (LCPDF) code. The linear-mean-square-estimation (LMSE) model is also evaluated for comparison. The results from the BMC/GIEM model are in excellent agreement with experimental data, whereas the LMSE model gives errors in higher-order statistics.

PDF simulation of turbulent non-premixed CH 4/H 2-air flames using automatically reduced chemical kinetics

In this paper, we present numerical simulations of a turbulent non-premixed CH4/H2-air flame based on the solution of the transport equation for the joint velocity and composition probability density function (PDF). A Monte Carlo particle-mesh approach is used to solve the modeled PDF transport equation, in which the mean pressure gradient and turbulent frequency are supplied by a finite-volume mean flow program. The finite-rate chemical kinetics of the CH4/H2-air system is simplified by the method of intrinsic low-dimensional manifolds (ILDM). Simple look-up tables guarantee an easy implementation into the turbulent PDF code. The hybrid solution algorithm is applied to calculate a recently published bluff-body stabilized turbulent non-premixed flame of CH4/H2 by Masri et al. [1]. Two different characteristic flames are presented, where one is close to chemical equilibrium and, for the other, the effects of finite-rate chemistry become significant. The mean temperature distribution and scatter plots for temperature and mass fractions of CO2 and H2O are compared with experimental data. The numerical calculations reproduce the experimentally observed phenomena, such as local extinction, quite well.