Probability Density Function Model Research Articles

Analytical and Numerical Solution of the Equation for the Probability Density Function of the Particle Velocity in a Turbulent Flow

A study has been made of the random motion of inertial particles in a homogeneous isotropic turbulent gas flow. Fluctuations of the gas velocity along the particle path were modeled by the Gaussian random process with a finite time of degeneracy of the autocorrelation function. A closed equation has been obtained for the probability density function of the particle velocity, for which two methods of numerical solution have been proposed: using the finite-difference scheme and using one based on direct numerical modeling of an empirical probability density function. The empirical probability density function was obtained as a result of the averaging of random particle paths, which are a solution of a system of ordinary stochastic differential equations. The results of numerical calculation have been compared with the analytical solution describing the dynamics of the probability density function of the particle-velocity distribution.

A priori analysis of a power-law mixing model for transported PDF model based on high Karlovitz turbulent premixed DNS flames

Accurate modeling of mixing in large-eddy simulation (LES)/transported probability density function (PDF) modeling of turbulent combustion remains an outstanding issue. The issue is particularly salient in turbulent premixed combustion under extreme conditions such as high-Karlovitz number Ka. The present study addresses this issue by conducting an a priori analysis of a power-law scaling based mixing timescale model for the transported PDF model. A recently produced DNS dataset of a high-Ka turbulent jet flame is used for the analysis. A power-law scaling is observed for a scaling factor used to model the sub-filter scale mixing timescale in this high-Ka turbulent premixed DNS flame when the LES filter size is much greater than the characteristic thermal thickness of a laminar premixed flame. The sensitivity of the observed power-law scaling to the different viewpoints (local or global) and to the different scalars for the data analysis is examined and the dependence of the model parameters on the dimensionless numbers Ka and Re (the Reynolds number) is investigated. Different model formulations for the mixing timescale are then constructed and assessed in the DNS flame. The proposed model is found to be able to reproduce the mixing timescale informed by the high-Ka DNS flame significantly better than a previous model.

Coupling of mixing models with manifold based simplified chemistry in PDF modeling of turbulent reacting flows

For general reacting flows the numerical simulation faces two main challenges. One is the high dimensionality and stiffness of the governing conservation equations due to detailed chemistry, which can be solved by using simplified chemical kinetics. The other one is the difficulty of modeling the coupling of turbulence with thermo-chemical source term. The probability density function (PDF) method allows to calculate turbulent reacting flows by solving the thermal-chemical source term in closed form. Usually, the PDF method for turbulent processes such as mixing processes and the reduction method for chemical kinetics are developed separately. However, coupling of both processes plays an important role for the numerical accuracy. To investigate the importance of coupling between turbulence and simplified chemistry, two different coupling strategies for mixing and reduced chemistry are discussed and tested for the well-known Sandia Flames E and F, in which there is a strong interaction between turbulence and chemical kinetics. The EMST mixing model is chosen for turbulent mixing, while the Reaction-Diffusion Manifolds (REDIMs) is used as simplified chemistry. However, the proposed strategies are also valid for other mixing models and manifold based simplified chemistry.

Probability density function (PDF) models for particle transport in porous media

Mathematical models based on probability density functions (PDF) have been extensively used in hydrology and subsurface flow problems, to describe the uncertainty in porous media properties (e.g., permeability modelled as random field). Recently, closer to the spirit of PDF models for turbulent flows, some approaches have used this statistical viewpoint also in pore-scale transport processes (fully resolved porous media models). When a concentration field is transported, by advection and diffusion, in a heterogeneous medium, in fact, spatial PDFs can be defined to characterise local fluctuations and improve or better understand the closures performed by classical upscaling methods. In the study of hydrodynamical dispersion, for example, PDE-based PDF approach can replace expensive and noisy Lagrangian simulations (e.g., trajectories of drift-diffusion stochastic processes). In this work we derive a joint position-velocity Fokker–Planck equation to model the motion of particles undergoing advection and diffusion in in deterministic or stochastic heterogeneous velocity fields. After appropriate closure assumptions, this description can help deriving rigorously stochastic models for the statistics of Lagrangian velocities. This is very important to be able to characterise the dispersion properties and can, for example, inform velocity evolution processes in continuous time random walk dispersion models. The closure problem that arises when averaging the Fokker–Planck equation shows also interesting similarities with the mixing problem and can be used to propose alternative closures for anomalous dispersion.

Impact of probability density function model choice in intensity-modulation direct-detection free-space optical communication analyses

Performance metrics such as mean-bit-error rate and probability of fade for free-space optical communication (FSOC) applications using intensity-modulation direct detection are theoretically calculated based on probability density functions (PDFs) describing irradiance fluctuations. Theoretical calculations using common PDF models can result in significant errors in prediction of performance metrics. In particular, these models do not consider the change in skewness of the distribution as the aperture size increases, and often positively skewed distributions (right tailed) are used to model scenarios where the true statistics are negatively skewed (left tailed). We evaluate the magnitude of errors in the prediction of bit-error rate and probability of fade based on simulation data for a collimated Gaussian beam in a realistic FSOC scenario for two strengths of turbulence and varying aperture sizes. The PDF models considered are lognormal, gamma–gamma, inverse-Gaussian-gamma, fractional exponential, exponentiated-Weibull, 3-parameter-Weibull, and normal distributions.

Comparison of probability density functions for aperture-averaged irradiance fluctuations of a Gaussian beam with beam wander.

Over the years, there has been much interest in the use of optical wavelengths for communication because of the potential for high data rates. However, the performance of these systems can become significantly degraded due to turbulence-induced signal fluctuations. These fluctuations can be minimized by enlarging the receiving aperture, thereby averaging the fluctuations. There is extensive interest in developing probability density functions (PDFs) describing these intensity fluctuations so as to accurately predict system performance. This work examines several PDF models that have been suggested to represent fluctuations by comparing them to simulations of realistic propagation scenarios of a collimated Gaussian beam with centroid wander. Unlike with an infinite plane or spherical wave, the empirical PDF shape in these simulations changed significantly with increased aperture, going from a positively skewed to a negatively skewed distribution; therefore, the PDF model that describes it also must change. This change in skew can have serious consequences on the metrics of an optical communication system, e.g., number of fades and bit-error rate. In this work, we examine the evolution of the empirical PDF with aperture size and the fit of potential PDF models under various strengths of turbulence. We show that the change in skew with increasing aperture size occurred for all strengths of turbulence and that the currently used PDF models do not adequately characterize this for realistically sized apertures where beam wander is present.

Optimal Power Flow Incorporating FACTS Devices and Stochastic Wind Power Generation Using Krill Herd Algorithm

This paper deals with investigating the Optimal Power Flow (OPF) solution of power systems considering Flexible AC Transmission Systems (FACTS) devices and wind power generation under uncertainty. The Krill Herd Algorithm (KHA), as a new meta-heuristic approach, is employed to cope with the OPF problem of power systems, incorporating FACTS devices and stochastic wind power generation. The wind power uncertainty is included in the optimization problem using Weibull probability density function modeling to determine the optimal values of decision variables. Various objective functions, including minimization of fuel cost, active power losses across transmission lines, emission, and Combined Economic and Environmental Costs (CEEC), are separately formulated to solve the OPF considering FACTS devices and stochastic wind power generation. The effectiveness of the KHA approach is investigated on modified IEEE-30 bus and IEEE-57 bus test systems and compared with other conventional methods available in the literature.

LES/PDF modeling of swirl-stabilized non-premixed methane/air flames with local extinction and re-ignition

Turbulent non-premixed flames with local extinction and re-ignition exhibit multiple combustion modes including ignition waves, diffusion flames, partially premixed flames, and ignition-assisted partially premixed flames. The mechanisms of local extinction and re-ignition are not well understood and numerical modeling of multi-mode combustion is a challenging task. In this work, a specially designed swirl-burner was used to study local extinction and re-ignition of non-premixed turbulent methane/air flames. High speed Particle Image Velocimetry (PIV) and laser induced fluorescence of OH radicals (OH-PLIF) measurements along with Large Eddy Simulation (LES) were carried out to investigate the mechanisms of extinction and re-ignition processes in the burner. LES is based on a transported probability density function model within the framework of Eulerian Stochastic Fields (PDF-ESF). It is found that local extinction occurs when the scalar dissipation rate around the stoichiometric mixture fraction is high. The characteristic time scale for local extinction and re-ignition in the present flames is an order of magnitude longer than the characteristic time scale of diffusion/extinction of laminar flamelets. There are two mechanisms for flame hole re-ignition in the present flames. First, under low degree of local extinction conditions (i.e., for small flame holes surrounded by flames) the flame hole re-ignition is due to the mechanism of turbulent flame folding. Second, under high degree of extinction conditions (i.e., with large regions of extinction and lifted flames), re-ignition of the locally extinguished flame is due to the mechanism of ignition assisted partially premixed flame propagation. The results show that the PDF-ESF model is capable of simulating the quenching and re-ignition process found in the experiments.

Design of single and dual-mode companding scalar quantizers based on piecewise linear approximation of the Gaussian PDF

This paper proposes a novel companding quantizer design for Gaussian source by using piecewise linear approximation of the probability density function. As the optimal companding technique represents very effective tool for quantizing signals with non-linear distribution, such as speech, the lack of optimal closed-form solution for Gaussian source represents an issue that motivated us to provide an approximation model, supported with closed-form designing expressions, suitable for applications requiring low processing delays. We provide the derivation of closed-form expressions for approximate compressor and inverse compressor functions, which are used to define representation levels and boundary segments, as the optimal compandor design requires integral equations solving. Also, we analyze how the probability density function within a dual-mode framework behaves and we propose a novel model of probability density function. System performances are analyzed for various values of configuration parameters and estimated theoretical performances are compared with appropriate simulations. Moreover, the results are compared with performance of the optimal compandor to justify the usefulness of dual-mode technique and it is demonstrated that the model we propose outperforms it, making the model competitive for speech applications where the system complexity is significant or critical aspect alongside achieved performances.

Bimodal distribution of wind pressure on windward facades of high-rise buildings induced by interference effects

Understanding the characteristics of wind pressures on structural façades is of great importance for wind-resistant design of claddings and glass curtains of buildings. This article presents some interesting findings about the behavior of wind pressure on the façades of a high-rise building influenced by an interference building via wind tunnel tests. The bimodal probability distribution (BPD) of wind pressure acting on the windward side of the principal building is highlighted, which has been rarely reported and investigated in previous studies. This BPD featured wind pressure consists of both positive and negative components that alternate along time in a consistent way to the vortex shedding downwind of the interference building. By changing the relative position between the principal building and interference building, it is observed that the bimodal distribution can be transformed to a unimodal form, mostly with a long tailed on the left-hand-side of the probability-density-function (PDF). Based on the above findings, this study proposes a mixture PDF model which is actually a combination of the Gaussian model and the generalized-extreme-value (GEV) model. While the Gaussian PDF accounts for the component dominated by the background approaching flows, the GEV one is associated with the aerodynamic interference effects. Parametric analysis is then conducted for the BPD feature based on the proposed mixture model, and detailed process for the transformation of different PDF patterns (from varied bimodal patterns to both Gaussian and tailed unimodal statuses) is reproduced. The findings can provide useful insights for the wind-resistant design of claddings and curtain walls of high-rise buildings, and for further analyzing non-Gaussian characteristics of wind pressure, such as for estimating extreme pressure values.

Statistical analysis of the accuracy of water content-based electrical properties tomography.

Water content-based electrical properties tomography (wEPT) can retrieve electrical properties (EPs) from water content maps, thereby eliminating the need for B1 field measurement in the traditional magnetic resonance electrical properties tomography method. The wEPT is performed by conventional MR scanning, such as T1 -weighted spin-echo imaging, and thus can be directly applied to clinical settings. However, the random noise propagation involved in wEPT causes inaccuracy in EP mapping. To guarantee the EP estimates desired for clinical practice, this study statically investigates the noise-specific uncertainty of wEPT through probability density function models. We calculated the probability distribution of EP maps with different noise levels and examined the effects of scan parameters on reconstruction accuracy with various flip angles (FAs) and repetition time (TR) settings. The theoretical derivation was validated by Monte Carlo simulations and human imaging experiment at 3 T. Results showed that a serious deviation could occur in tissues with large conductivity value at a low signal-to-noise ratio and quantitatively demonstrate that such deviation could be mitigated by increased FAs or TRs. This study provided useful information for the setup of scan parameters, evaluation of accuracy of the wEPT under specific SNR levels, and promote its clinical applications.

GMM-based Automatic Defect Recognition Algorithm for Pressure Vessels Defect Detection through ECPT

In order to realize the automatic identification of pressure vessel defects, an improved adaptive defect recognition feature extraction algorithm through ECPT (Eddy current pulsed thermography) is proposed. The proposed feature extraction algorithm consists of five elements: thermal image data segmentation, variable interval search, probability density function modeling, data classification, and reconstructed image acquisition. The combination of data block selection and variable interval search can reduce the double counting. And the KG-EM (Kmeans-GMM-EM) algorithm is proposed to obtain the Gaussian mixture model corresponding to the classification, and thus the corresponding probability is obtained to classify the TTRs (Transient Thermal Response). The reconstructed thermal image is obtained by the classified TTRs. This method can extract the main information of the image accurately and efficiently. Experimental results are provided to demonstrate their effectiveness.

Multi-Objective Optimal Power Flow With Integration of Renewable Energy Sources Using Fuzzy Membership Function

In recent past, to meet the growing energy demand of electricity, integration of renewable energy resources (RESs) in an electrical network is a center of attention. Furthermore, optimal integration of these RESs make this task more challenging because of their intermittent nature. Therefore, in the present study power flow problem is treated as a multi-constraint, multi-objective optimal power flow (MOOPF) problem along with optimal integration of RESs. Whereas, the objectives of MOOPF are threefold: overall generation cost, real power loss of system and carbon emission reduction of thermal sources. In this work, a computationally efficient technique is presented to find the most feasible values of different control variables of the power system having distributed RESs. Whereas, the constraint satisfaction is achieved by using penalty function approach (PFA) and to further develop true Pareto front (PF), Pareto dominance method is used to categorize Pareto dominate solution. Moreover, to deal with intermittent nature of RES, probability density function (PDF) and stochastic power models of RES are used to calculate available power from RESs. Since, objectives of the MOOPF problem are conflicting in nature, after having the set of non-dominating solutions fuzzy membership function (FMF) approach has been used to extract the best compromise solution (BCS). To test the validity of developed technique, the IEEE-30 bus system has been modified with integration of RESs and final optimization problem is solved by using particle swarm optimization (PSO) algorithm. Simulation results show the achievement of proposed technique managing fuel cost value long with the optimal values of other objectives.

Comparison of Combustion Models for Lifted Hydrogen Flames within RANS Framework

Within the framework of a Reynolds averaged numerical simulation (RANS) methodology for modeling turbulence, a comparative numerical study of turbulent lifted H2/N2 flames is presented. Three different turbulent combustion models, namely, the eddy dissipation model (EDM), the eddy dissipation concept (EDC), and the composition probability density function (PDF) transport model, are considered in the analysis. A wide range of global and detailed combustion reaction mechanisms are investigated. As turbulence model, the Standard k-ε model is used, which delivered a comparatively good accuracy within an initial validation study, performed for a non-reacting H2/N2 jet. The predictions for the lifted H2/N2 flame are compared with the published measurements of other authors, and the relative performance of the turbulent combustion models and combustion reaction mechanisms are assessed. The flame lift-off height is taken as the measure of prediction quality. The results show that the latter depends remarkably on the reaction mechanism and the turbulent combustion model applied. It is observed that a substantially better prediction quality for the whole range of experimentally observed lift-off heights is provided by the PDF model, when applied in combination with a detailed reaction mechanism dedicated for hydrogen combustion.

A two-scale Langevin PDF model for Richtmyer–Meshkov turbulence with a small Atwood number

In this article, we derive a Langevin probability density function (PDF) model in order to predict turbulent mixing zone evolutions when generated by the Richtmyer–Meshkov instability. The aim of the model is to account for the permanence of large eddies which is observed in these flows when the density contrast, as measured by the Atwood number, is small. To this end, a two-scale decomposition of the velocity field is proposed and used to adapt existing Langevin models. In addition, the role played by pressure fluctuations on the transport of kinetic energy is also discussed. A closure for this turbulent process is added to the two-scale Langevin model. Finally, large-eddy simulations of Richtmyer–Meshkov turbulent flows are performed and used to validate the different closures proposed in this work.

Eulerian conditional statistics of turbulent flow in a macroscale multi-inlet vortex chemical reactor

The conditional velocity time averages (⟨Ui|ξ⟩) and conditional mixture fraction time averages (⟨Φ|ωi⟩) were computed based on the Eulerian approach from the experimental data measured in a macroscale multi-inlet vortex chemical reactor. The conditioning events were determined by equally sized intervals of the sample space variable for the mixture fraction (ξ) and the velocity vector (ωi). The experimental data, which consisted of instantaneous velocities and concentration fields for two Reynolds numbers (Re = 3250 and 8125), were acquired using the simultaneous stereoscopic particle image velocimetry (stereo-PIV) and planar laser induced fluorescence techniques. Two mathematical models, the linear approximation and probability density function (PDF) gradient diffusion, were validated by experimental results. The results of the velocity conditioned on the mixture fraction demonstrated that the linear model works well in a low turbulence region away from the reactor center. Near the reactor center, high velocity gradients coupled with low concentration gradients reduce the accuracy of the linear model predictions. Nevertheless, an excellent agreement was found for the conditional events within ±2Φrms (mixture fraction root mean square). Due to lower concentration gradient in the tangential direction, the linear model better predicted the tangential velocity component for all locations investigated. The PDF model with an isotropic turbulent diffusivity performed inadequately for the tangential and axial velocity components. A modified version of the PDF model that considers the three components of the turbulent diffusivity produced a better agreement with the experimental data especially in the spiral arms regions of significant concentration gradients. Furthermore, the mixture fraction conditioned on the velocity vector components showed a more linear behavior near the reactor center, where the PDF of the mixture fraction is a Gaussian distribution. As the concentration gradients became prominent away from the reactor, ⟨Φ|ωi⟩ also deviated from the linear pattern. This was especially remarkable for the mixture fraction conditioned on the tangential velocity. The overall prediction of ⟨Φ|ωi⟩ improves at higher Reynolds number as the fluid mixing is enhanced.

Probability density function based modeling of spatial feature variation in capsule endoscopy data for automatic bleeding detection

Wireless capsule endoscopy (WCE) is a video technology to inspect abnormalities, like bleeding in the gastrointestinal tract. In order to avoid a complex and long duration manual review process, automatic bleeding detection schemes are developed that mainly utilize features extracted from WCE images. In feature-based bleeding detection schemes, either global features are used which produce averaged characteristics ignoring the effect of smaller bleeding regions or local features are utilized that cause large feature dimension. In this paper, pixels of interest (POI) in a given WCE image are determined using a linear separation scheme, local spatial features are then extracted from the POI and finally, a suitable characteristic probability density function (PDF) is fitted over the resulting feature space. The proposed PDF model fitting based approach not only reduces the computational complexity but also offers more consistent representation of a class. Details analysis are carried out to find the best suitable PDF and it is found that fitting of Rayleigh PDF model to the local spatial features is best suited for bleeding detection. For the purpose of classification, the fitted PDF parameters are used as features in the supervised support vector machine classifier. Pixels residing in the close vicinity of the POI are further classified with the help of an unsupervised clustering-based scheme to extract more precise bleeding regions. A large number of WCE images obtained from 30 publicly available WCE videos are used for performance evaluation of the proposed scheme and the effects on classification performance due to the changes in PDF models, block statistics, color spaces, and classifiers are experimentally analyzed. The proposed scheme shows satisfactory performance in terms of sensitivity (97.55%), specificity (96.59%) and accuracy (96.77%) and the results obtained by the proposed method outperforms the results reported for some state-of-the-art methods.

Quantitative evaluation of liver diseases using multi-PDF models

We have proposed the triple Rayleigh probability density function (TR-PDF) model as the in vivo QUS method of liver fibrosis. Additionally, the double Nakagami probability density function (DN-PDF) model was proposed to evaluate the amount and distribution condition of fat droplets in the fatty-liver livers. The DN-PDF permitted to detect early and moderate fatty livers in previous ex vivo studies with over 15 MHz ultrasound. In our recent studies, we are examining clinical data to determine whether two multi-PDF models can evaluate each disease even at low frequencies. In this talk, the results of in vivo studies for liver fibrosis and fatty liver will be introduced.We have proposed the triple Rayleigh probability density function (TR-PDF) model as the in vivo QUS method of liver fibrosis. Additionally, the double Nakagami probability density function (DN-PDF) model was proposed to evaluate the amount and distribution condition of fat droplets in the fatty-liver livers. The DN-PDF permitted to detect early and moderate fatty livers in previous ex vivo studies with over 15 MHz ultrasound. In our recent studies, we are examining clinical data to determine whether two multi-PDF models can evaluate each disease even at low frequencies. In this talk, the results of in vivo studies for liver fibrosis and fatty liver will be introduced.

Deep learning for presumed probability density function models

In this work, we use machine learning (ML) techniques to develop presumed probability density function (PDF) models for large eddy simulations (LES) of reacting flows. The joint sub-filter PDF of mixture fraction and progress variable is modeled using various ML algorithms and commonly used analytical models. The ML algorithms evaluated in the work are representative of three major classes of ML techniques: traditional ensemble methods (random forests), deep learning (deep neural network (DNN)s), and generative learning (conditional variational autoencoder (CVAE)). The first two algorithms are supervised learning algorithms, and the third is an unsupervised learning algorithm. Data from direct numerical simulation (DNS) of the low-swirl burner[1] are used to develop training data for sub-filter PDF models. Models are evaluated on predictions of the sub-filter PDF as well as predictions of the filtered reaction rate of the progress variable, computed through an integral of the product of the sub-filter PDF and the conditional means of the reaction rate. This a-priori modeling study demonstrates that deep learning models for presumed PDF modeling are three times more accurate than analytical β–β PDF models and linear regression models. These models are as accurate as random forest models while using five times fewer trainable parameters and being 25 times faster for inference. In this work, conditional unsupervised learning did not present additional advantages beyond supervised learning with a feed-forward neural network. We illustrate how models generalize to other regions of the flow and develop criteria based on the Jensen–Shannon divergence to quantify the performance of a model on new data.

Enhanced Cross Entropy Method for Composite Power System Reliability Evaluation

Cross entropy method (CEM) can effectively accelerate the reliability evaluation of power system. It is mainly utilized to conduct importance sampling (IS) on discrete component state variables and independent or fully correlated continuous variables. To carry out IS on continuous variables with complicated correlation, this paper proposes an enhanced CEM (ECEM) that addresses the following two problems: the first one is the probabilistic modeling of the joint probability density function (PDF) for the correlated continuous variables, and the second one is the IS on the modeled joint PDF. For the first one, a Gaussian mixture model (GMM) is utilized to estimate the original joint PDF of correlated continuous variables. For the second one, the parameters of the GMM based joint PDF are optimized directly by a direct-updating-rule based method (DURM) to obtain the optimal IS-PDF. Furthermore, as an alternative to DURM, a Nataf transformation based method (NTM) is also presented to distort the parameters of the GMM-based joint PDF indirectly. Compared with NTM, which is suitable for normal copula correlation, DURM has a much wider application scope without any restriction on the correlation type. Finally, an improved unbiased estimator for the reliability index is presented to further accelerate ECEM. Test studies on RTS79 and MRTS with wind farm verify the effectiveness of ECEM.