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About 2,224 papers for “joint probability density function”

The Joint Scalar Probability Density Function Method

W.P. Jones · 2002

The purpose of this chapter is to review the joint probability density function, (PDF), evolution equation approach to calculating the properties of turbulent flames. In this approach the closure of the micromixing term represents a central difficulty and for this, various proposals including the linear mean square estimation closure, coalescence-dispersion models and Langevin, binomial, mapping and Euclidean minimum spanning tree closures are appraised. Results obtained with the PDF equation using a modified coalescence-dispersion model and a simple global reaction mechanism demonstrate that the method is capable of reproducing measured mixture fraction, fuel and CO2 profiles in jet diffusion flames and counterflow premixed and partially premixed turbulent flames. The levels of carbon monoxide are somewhat overpredicted in all the flames considered but it is argued that this is primarily a consequence of limitations in the simplified chemical reaction mechanism used. The need for improved models to represent micromixing is also suggested.

Cited by 8

Generalized Joint Probability Density Function Formulation inTurbulent Combustion using DeepONet

Rishikesh Ranade, Kevin M. Gitushi, Tarek Echekki · arXiv (Cornell University) · 2021

Joint probability density function (PDF)-based models in turbulent combustion provide direct closure for turbulence-chemistry interactions. The joint PDFs capture the turbulent flame dynamics at different spatial locations and hence it is crucial to represent them accurately. The jointPDFs are parameterized on the unconditional means of thermo-chemical state variables, which can be high dimensional. Thus, accurate construction of joint PDFs at various spatial locations may require an exorbitant amount of data. In a previous work, we introduced a framework that alleviated data requirements by constructing joint PDFs in a lower dimensional space using principal component analysis (PCA) in conjunction with Kernel Density Estimation (KDE). However, constructing the principal component (PC) joint PDFs is still computationally expensive as they are required to be calculated at each spatial location in the turbulent flame. In this work, we propose the concept of a generalized joint PDF model using the Deep Operator Network (DeepONet). The DeepONet is a machine learning model that is parameterized on the unconditional means of PCs at a given spatial location and discrete PC coordinates and predicts the joint probability density value for the corresponding PC coordinate. We demonstrate the accuracy and generalizability of the DeepONet on the Sandia flames, D, E and F. The DeepONet is trained based on the PC joint PDFs observed inflame E and yields excellent predictions of joint PDFs shapes at different spatial locations of flamesD and F, which are not seen during training

Cited by 7Open access

Exact Joint Density-Current Probability Function for the Asymmetric Exclusion Process

Martin Depken, R B Stinchcombe · Physical Review Letters · 2004

We study the asymmetric simple exclusion process with open boundaries and derive the exact form of the joint probability function for the occupation number and the current through the system. We further consider the thermodynamic limit, showing that the resulting distribution is non-Gaussian and that the density fluctuations have a discontinuity at the continuous phase transition, while the current fluctuations are continuous. The derivations are performed by using the standard operator algebraic approach and by the introduction of new operators satisfying a modified version of the original algebra.

Cited by 22Open access

Nonlinear characterization of aeroelastic systems through the Joint Probability Density Function

Antônio Ernando Carlos Ferreira Júnior · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · 2021

A indústria aeronáutica realiza investimentos significativos na manutenção de aeronaves, devido a diversos fenômenos aeroelásticos que podem ocorrer ao longo das suas vidas úteis, incluindo danos à integridade estrutural. A mantenabilidade é um campo de operação responsável por analisar tal integridade por meio de estudos do ciclo de um componente, desenvolvendo processos ótimos de manutenção de uma estrutura, a fim de evitar possíveis acidentes. Para garantir a confiabilidade de um sistema, a previsibilidade de eventuais problemas ao longo de sua operação torna-se fundamental na pretensão de disponibilidade e desempenho de uma aeronave, a partir do monitoramento de sinais obtidos por sensores distribuídos pela estrutura. Uma alternativa de atingir tal requisito é a análise dos sinais por meio da Função Densidade de Probabilidade Conjunta (J-PDF), um método estatístico que realiza mensurações do comportamento de variáveis conjuntas ao longo de um intervalo de tempo. Logo, a probabilidade da variável em determinado estado pode indicar a possibilidade de futuras anormalidades no sistema. A presente pesquisa propôs a análise da J-PDF e a avaliação da sua eficiência na área da Aeroelasticidade por meio de sinais não lineares oriundos de estruturas aeronáuticas, evidenciando a potencialidade da J-PDF para a previsão de anormalidades em uma aeronave e apresentando as vantagens que tal metodologia estatística possui, não apenas nos processos de mantenabilidade, mas também em diversas áreas de pesquisa. Os resultados demonstram a eficiência da J-PDF quando aplicada em sistemas aeroelásticos não lineares, em séries temporais numéricas e experimentais, de forma que o espaço de fase gerado demonstra a probabilidade do comportamento das variáveis ao longo do tempo. Com isso, é possível detectar futuras anormalidades e demais efeitos quando o sistema é submetido a condições específicas.

Cited by 0Open access

JPResUnet: A Joint Probability Density Function Translation Model in Partially Premixed Flames

Hanying Yang, James C. Massey, N. Swaminathan · ArXiv.org · 2025

Machine learning (ML) models are often constrained by their limitations in extrapolation, which restricts their applicability in engineering contexts. Conversely, while exhibiting broad generality, many established scientific models seem to lack the necessary accuracy. This study addresses these challenges by introducing JPResUnet (Joint PDF Residual U-net), a novel model that integrates the strengths of both ML and traditional scientific approaches to predict sub-grid joint probability density functions (PDFs) in partially premixed flames. JPResUnet employs a residual U-Net architecture to translate classic $β$-PDFs to sub-grid PDFs. The model is trained using direct numerical simulation (DNS) data from methane-air moderate or intense low-oxygen dilution (MILD) combustion and is initially tested through a priori assessments on out-of-sample data. Comparative analyses against an artificial neural network (ANN) and the $β$-PDF approach demonstrate that JPResUnet consistently outperforms these methods in capturing complex sub-grid features with greater accuracy and robustness for both box and Gaussian kernels of varying widths, and for more extrapolated cases. Subsequent a posteriori assessment involves two versions of JPResUnet with different output PDF resolutions, which are deployed for large eddy simulation (LES) of a multi-regime burner through the look-up table (LUT) approach. The higher resolution model yields improvements in temperature estimates compared to the conventional LUT method. This highlights the potential of the JPResUnet model for robust and accurate LES of reacting flows with ML.

Cited by 0Open access

Calculations of Swirl Combustors Using Joint Velocity-Scalar Probability Density Function Method

M. S. Anand, Andrew Hsu, Stephen B. Pope · AIAA Journal · 1997

Calculations are reported for recirculating swirling reacting flows using a joint velocity-scalar probability density function (PDF) method. The PDF method offers signiflcant advantages over conventional finite volume, Reynolds-average-based methods, especially for the computation of turbulent reacting flows. The PDF calculations reported here are based on a newly developed solution algorithm for elliptic flows, and on newly developed models for turbulent frequency and velocity that are simpler than those used in previously reported PDF calculations. Calculations are performed for two different gas-turbine-like swirl combustor flows for which detailed measurements are available. The computed results are in good agreement with experimental data.

Cited by 26

Adaptive Nonparametric Kernel Density Estimation Approach for Joint Probability Density Function Modeling of Multiple Wind Farms

Nan Yang, Yu Huang, Dengxu Hou et al. · Energies · 2019

The uncertainty of wind power brings many challenges to the operation and control of power systems, especially for the joint operation of multiple wind farms. Therefore, the study of the joint probability density function (JPDF) of multiple wind farms plays a significant role in the operation and control of power systems with multiple wind farms. This research was innovative in two ways. One, an adaptive bandwidth improvement strategy was proposed. It replaced the traditional fixed bandwidth of multivariate nonparametric kernel density estimation (MNKDE) with an adaptive bandwidth. Two, based on the above strategy, an adaptive multi-variable non-parametric kernel density estimation (AMNKDE) approach was proposed and applied to the JPDF modeling for multiple wind farms. The specific steps of AMNKDE were as follows: First, the model of AMNKDE was constructed using the optimal bandwidth. Second, an optimal model of bandwidth based on Euclidean distance and maximum distance was constructed, and the comprehensive minimum of these distances was used as a measure of optimal bandwidth. Finally, the ordinal optimization (OO) algorithm was used to solve this model. The scenario results indicated that the overall fitness error of the AMNKDE method was 8.81% and 11.6% lower than that of the traditional MNKDE method and the Copula-based parameter estimation method, respectively. After replacing the modeling object the overall fitness error of the comprehensive Copula method increased by as much as 1.94 times that of AMNKDE. In summary, the proposed approach not only possesses higher accuracy and better applicability but also solved the local adaptability problem of the traditional MNKDE.

Cited by 44Open access

Joint probability density function of a scalar and its gradient in isotropic turbulence

Feng Gao, Edward E. O’Brien · Physics of Fluids A Fluid Dynamics · 1991

The joint probability density function (pdf) of a scalar and its gradient are investigated. Under isotropic conditions, a relation is derived between this pdf and the joint pdf of a scalar and its x derivative. Physical properties of the scalar derivative are analyzed quantitatively to show that its pdf conditional on a given scalar value is close to a Gaussian distribution. The conditional pdf of the scalar derivative is then expanded according to the Gram–Charlier expansion. The coefficients of this expansion are evaluated using direct numerical simulation (DNS) of stationary turbulent scalar fields. Results obtained from a number of simulations support the assumption of a near-Gaussian distribution for the scalar derivative. Finally, the intermittency of the scalar gradient is discussed.

Cited by 23

Joint-velocity-scalar probability density function method for modelling high-speed turbulent combustion

Tin-Hang Un · Spiral (Imperial College London) · 2025

This thesis presents a joint-velocity-scalar probability density function (VSPDF) model aimed at improving dual-mode ramjet engine simulations. This model stands out as a unified closure for sub-grid turbulence interactions in large eddy simulations (LES), requiring minimal coefficients and making no assumptions about specific flame structure or combustion regime. The VSPDF is transported using a conservative and thermodynamically consistent stochastic fields (SF) formulation, ensuring stability in shocked turbulent flows. The Eulerian nature of SF allows direct implementation of a high-order numerical scheme and adaptive mesh refinement (AMR). Various cases showcase the model’s unique capabilities, such as the ability to distinguish between laminar shear and turbulence, and to capture extreme events such as autoignition. Two improvements to the model are proposed. The first uses a tensor-based neural network to replace existing sub-grid mixing model. By respecting physical symmetries and dimensional consistency, this approach outperforms both the original simplified Langevin model and a conventional neural network. The second improvement targets the computational cost for resolving boundary layers by integrating an ODE-based wall model with the SF, producing accurate wall fluxes with 10 to 50 times coarser grids. A coupling procedure is proposed to maintain stability and ensure correct sub-grid fluctuation profiles. Simulations of a supersonic channel agree well with direct numerical simulations, showing significant improvement over LES without a wall model. Finally, the wall-modelled SF is applied to simulate a dual-mode ramjet engine. The method shows good agreement with experimental results in terms of shock train length, pressure profile, and flame position, in two fundamentally different conditions. This demonstrates the model's wide applicability, allowing for confident use in the design and optimisation of ramjet and scramjet engines. This study represents the first application of VSPDF in simulating ramjet engines, as well as the first solver to integrate SF, AMR, and wall modelling.

Cited by 0Open access