Search NASA⌕ Search

SEARCH · Search NASA

Results for “PDF equation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1997

Verification and Improvement of Flamelet Approach for Non-Premixed Flames

Studies in the mathematical modeling of the high-speed turbulent combustion has received renewal attention in the recent years. The review of fundamentals, approaches and extensive bibliography was presented by Bray, Libbi and Williams. In order to obtain accurate predictions for turbulent combustible flows, the effects of turbulent fluctuations on the chemical source terms should be taken into account. The averaging of chemical source terms requires to utilize probability density function (PDF) model. There are two main approaches which are dominant in high-speed combustion modeling now. In the first approach, PDF form is assumed based on intuitia of modelliers (see, for example, Spiegler et.al.; Girimaji; Baurle et.al.). The second way is much more elaborate and it is based on the solution of evolution equation for PDF. This approach was proposed by S.Pope for incompressible flames. Recently, it was modified for modeling of compressible flames in studies of Farschi; Hsu; Hsu, Raji, Norris; Eifer, Kollman. But its realization in CFD is extremely expensive in computations due to large multidimensionality of PDF evolution equation (Baurle, Hsu, Hassan).

Zaitsev, S.↗
NASA NTRS2004

EUPDF-II: An Eulerian Joint Scalar Monte Carlo PDF Module : User's Manual

EUPDF-II provides the solution for the species and temperature fields based on an evolution equation for PDF (Probability Density Function) and it is developed mainly for application with sprays, combustion, parallel computing, and unstructured grids. It is designed to be massively parallel and could easily be coupled with any existing gas-phase CFD and spray solvers. The solver accommodates the use of an unstructured mesh with mixed elements of either triangular, quadrilateral, and/or tetrahedral type. The manual provides the user with an understanding of the various models involved in the PDF formulation, its code structure and solution algorithm, and various other issues related to parallelization and its coupling with other solvers. The source code of EUPDF-II will be available with National Combustion Code (NCC) as a complete package.

Raju, M. S.↗
NASA NTRS1985

Effect of reactive intermediates on the propagation of turbulent premixed flames

The most simple version of the Bray and Libby model of turbulent flame propagation has been extended to include a two-step reaction mechanism in which reactants decompose into reactive intermediates which then form products. A transport equation for the joint pdf of intermediate and product concentrations was derived, the pdf parameterized, and transport equations for the parameters obtained. The one-dimensional concentration equations were then solved for the parameters of the pdf and the turbulent kinetic energy. Flame speed and angle are recovered as eigenvalues of the problem. The results show the essential features observed in experiments. The product pdf is composed of peaks at zero and unity product concentration, with a continuous structure representing partially reacted gas. The intermediates pdf has a peak at zero concentration and a broad bell-shaped distribution at nonzero concentration, representing partially reacted gas. The results also show the effect of partitioning of energy release on the flame thickness and speed.

Azzazy, M.↗
NASA NTRS1987

Numerical simulation of a plane turbulent mixing layer, with applications to isothermal, rapid reactions

A hybrid method has been developed for the numerical prediction of turbulent mixing in a spatially-developing, free shear layer. Most significantly, the computation incorporates the effects of large-scale structures, Schmidt number and Reynolds number on mixing, which have been overlooked in the past. In flow field prediction, large-eddy simulation was conducted by a modified 2-D vortex method with subgrid-scale modeling. The predicted mean velocities, shear layer growth rates, Reynolds stresses, and the RMS of longitudinal velocity fluctuations were found to be in good agreement with experiments, although the lateral velocity fluctuations were overpredicted. In scalar transport, the Monte Carlo method was extended to the simulation of the time-dependent pdf transport equation. For the first time, the mixing frequency in Curl's coalescence/dispersion model was estimated by using Broadwell and Breidenthal's theory of micromixing, which involves Schmidt number, Reynolds number and the local vorticity. Numerical tests were performed for a gaseous case and an aqueous case. Evidence that pure freestream fluids are entrained into the layer by large-scale motions was found in the predicted pdf. Mean concentration profiles were found to be insensitive to Schmidt number, while the unmixedness was higher for higher Schmidt number. Applications were made to mixing layers with isothermal, fast reactions. The predicted difference in product thickness of the two cases was in reasonable quantitative agreement with experimental measurements.

Lin, P.↗
NASA NTRS1990

The pdf approach to turbulent flow

This paper provides a detailed discussion of the theory and application of probability density function (pdf) methods, which provide a complete statistical description of turbulent flow fields at a single point or a finite number of points. The basic laws governing the flow of Newtonian fluids are set up in the Eulerian and the Lagrangian frame, and the exact and linear equations for the characteristic functionals in those frames are discussed. Pdf equations in both frames are derived as Fourier transforms of the equations of the characteristic functions. Possible formulations for the nonclosed terms in the pdf equation are discussed, their properties are assessed, and closure modes for the molecular-transport and the fluctuating pressure-gradient terms are reviewed. The application of pdf methods to turbulent combustion flows, supersonic flows, and the interaction of turbulence with shock waves is discussed.

Kollmann, W.↗
NASA NTRS1991

A study of hydrogen diffusion flames using PDF turbulence model

The application of probability density function (pdf) turbulence models is addressed in this work. For the purpose of accurate prediction of turbulent combustion, an algorithm that combines a conventional CFD flow solver with the Monte Carlo simulation of the pdf evolution equation has been developed. The algorithm has been validated using experimental data for a heated turbulent plane jet. The study of H2-F2 diffusion flames has been carried out using this algorithm. Numerical results compared favorably with experimental data. The computuations show that the flame center shifts as the equivalence ratio changes, and that for the same equivalence ratio, similarity solutions for flames exist.

Hsu, Andrew T.↗
NASA NTRS1991

A study of hydrogen diffusion flames using PDF turbulence model

The application of probability density function (pdf) turbulence models is addressed. For the purpose of accurate prediction of turbulent combustion, an algorithm that combines a conventional computational fluid dynamic (CFD) flow solver with the Monte Carlo simulation of the pdf evolution equation was developed. The algorithm was validated using experimental data for a heated turbulent plane jet. The study of H2-F2 diffusion flames was carried out using this algorithm. Numerical results compared favorably with experimental data. The computations show that the flame center shifts as the equivalence ratio changes, and that for the same equivalence ratio, similarity solutions for flames exist.

Hsu, Andrew T.↗
NASA NTRS1993

Research activities at the Center for Modeling of Turbulence and Transition

The main research activities at the Center for Modeling of Turbulence and Transition (CMOTT) are described. The research objective of CMOTT is to improve and/or develop turbulence and transition models for propulsion systems. The flows of interest in propulsion systems can be both compressible and incompressible, three dimensional, bounded by complex wall geometries, chemically reacting, and involve 'bypass' transition. The most relevant turbulence and transition models for the above flows are one- and two-equation eddy viscosity models, Reynolds stress algebraic- and transport-equation models, pdf models, and multiple-scale models. All these models are classified as one-point closure schemes since only one-point (in time and space) turbulent correlations, such as second moments (Reynolds stresses and turbulent heat fluxes) and third moments, are involved. In computational fluid dynamics, all turbulent quantities are one-point correlations. Therefore, the study of one-point turbulent closure schemes is the focus of our turbulence research. However, other research, such as the renormalization group theory, the direct interaction approximation method, and numerical simulations are also pursued to support the development of turbulence modeling.

Shih, Tsan-Hsing↗
NASA NTRS1998

Current Status on the use of Parallel Computing in Turbulent Reacting Flow Computations Involving Sprays, Monte Carlo PDF and Unstructured Grids

The state of the art in multidimensional combustor modeling as evidenced by the level of sophistication employed in terms of modeling and numerical accuracy considerations, is also dictated by the available computer memory and turnaround times afforded by present-day computers. With the aim of advancing the current multi-dimensional computational tools used in the design of advanced technology combustors, a solution procedure is developed that combines the novelty of the coupled CFD/spray/scalar Monte Carlo PDF (Probability Density Function) computations on unstructured grids with the ability to run on parallel architectures. In this approach, the mean gas-phase velocity and turbulence fields are determined from a standard turbulence model, the joint composition of species and enthalpy from the solution of a modeled PDF transport equation, and a Lagrangian-based dilute spray model is used for the liquid-phase representation. The gas-turbine combustor flows are often characterized by a complex interaction between various physical processes associated with the interaction between the liquid and gas phases, droplet vaporization, turbulent mixing, heat release associated with chemical kinetics, radiative heat transfer associated with highly absorbing and radiating species, among others. The rate controlling processes often interact with each other at various disparate time 1 and length scales. In particular, turbulence plays an important role in determining the rates of mass and heat transfer, chemical reactions, and liquid phase evaporation in many practical combustion devices.

Raju, M. S.↗
NASA NTRS2003

Presumed PDF Modeling of Early Flame Propagation in Moderate to Intense Turbulence Environments

The present paper describes the results obtained from a one-dimensional time dependent numerical technique that simulates early flame propagation in a moderate to intense turbulent environment. Attention is focused on the development of a spark-ignited, premixed, lean methane/air mixture with the unsteady spherical flame propagating in homogeneous and isotropic turbulence. A Monte-Carlo particle tracking method, based upon the method of fractional steps, is utilized to simulate the phenomena represented by a probability density function (PDF) transport equation. Gaussian distributions of fluctuating velocity and fuel concentration are prescribed. Attention is focused on three primary parameters that influence the initial flame kernel growth: the detailed ignition system characteristics, the mixture composition, and the nature of the flow field. The computational results of moderate and intense isotropic turbulence suggests that flames within the distributed reaction zone are not as vulnerable, as traditionally believed, to the adverse effects of increased turbulence intensity. It is also shown that the magnitude of the flame front thickness significantly impacts the turbulent consumption flame speed. Flame conditions studied have fuel equivalence ratio s in the range phi = 0.6 to 0.9 at standard temperature and pressure.

Carmen, Christina↗
NASA NTRS1995

PDF methods for turbulent reactive flows

Viewgraphs are presented on computation of turbulent combustion, governing equations, closure problem, PDF modeling of turbulent reactive flows, validation cases, current projects, and collaboration with industry and technology transfer.

Hsu, Andrew T.↗
DOE OSTI2025

Data-Driven Closures and Assimilation for Stiff Multiscale Random Dynamics

Here, we introduce a data-driven and physics-informed framework for propagating uncertainty in stiff, multiscale random ordinary differential equations (RODEs) driven by correlated (colored) noise. Unlike systems subjected to Gaussian white noise, a deterministic equation for the joint probability density function (PDF) of RODE state variables does not exist in closed form. Moreover, such an equation would require as many phase-space variables as there are states in the RODE system. To alleviate this curse of dimensionality, we instead derive exact, albeit unclosed, reduced-order PDF (RoPDF) equations for low-dimensional observables/quantities of interest. The unclosed terms take the form of state-dependent conditional expectations, which are directly estimated from data at sparse observation times. However, for systems exhibiting stiff, multiscale dynamics, data sparsity introduces regression discrepancies that compound during RoPDF evolution. This is overcome by introducing a kinetic-like defect term to the RoPDF equation, which is learned by assimilating in sparse, low-fidelity RoPDF estimates. Two assimilation methods are considered, namely nudging and deep neural networks, which are successfully tested against Monte Carlo simulations.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1992

An assumed pdf approach for the calculation of supersonic mixing layers

In an effort to predict the effect that turbulent mixing has on the extent of combustion, a one-equation turbulence model is added to an existing Navier-Stokes solver with finite-rate chemistry. To average the chemical-source terms appearing in the species-continuity equations, an assumed pdf approach is also used. This code was used to analyze the mixing and combustion caused by the mixing layer formed by supersonic coaxial H2-air streams. The chemistry model employed allows for the formation of H2O2 and HO2. Comparisons are made with recent measurements using laser Raman diagnostics. Comparisons include temperature and its rms, and concentrations of H2, O2, N2, H2O, and OH. In general, good agreement with experiment was noted.

Baurle, R. A.↗
DOE OSTI2019

A Fokker–Planck approach to a moment closure for mixing in variable-density turbulence

We develop a theory for the cascade mixing terms in a moment closure approach to binary active scalar mixing in variable-density turbulence. To address the variable-density complications we apply, as a principle and constraint, the conservation of the probability density function (PDF) through a Fokker–Planck equation with bounded sample space whose attractor is the beta PDF with skewness. Mixing is related to a single-point PDF as a realisability principle to provide mathematically rigorous expressions for the small scale statistics in terms of largescale moments. The problem of the unknown small-scale mixing is replaced with the determination of the drift and diffusion terms of a Fokker–Planck equation in a beta-PDF-convergent stochastic process. We find that realisability of a beta-convergent process requires the mixing time-scale ratio, taken as a constant in passive scalar mixing, to be a function of the mean mass fraction, mean fluid density, the Atwood number, the density-volume correlation and moments of the density field. We develop and compare the new model with direct numerical simulations data of non-stationary homogeneous variable-density turbulence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

A Novel In Situ Flamelet Tabulation Methodology for the Representative Interactive Flamelet Model

A priori flamelet tabulation methods have been used extensively to model premixed and non-premixed combustion in Reynolds Averaged Navier-Stokes (RANS) and Large Eddy Simulations (LES). Pre-tabulation of flamelet libraries have led to significant cost reduction of CFD simulations. However, these tabulation methods rely on certain simplifying assumptions. The dimensionality of the flamelet manifolds and size of flamelet libraries are also constrained due to memory and computational limitations. This limits the applicability and predictive capability of flamelet models. The traditional representative interactive flamelet (RIF) approach with multiple flamelets on the other hand involves online solution of flamelet equations and presumed PDF integration at each computational cell making it attractive but computationally expensive and not suitable for LES with large cell counts. The current work discusses the development of a novel methodology that combines the advantages of an online flamelet solver along with the computational efficacy of tabulated models. The novel in situ tabulation approach consists of creating a flamelet table at each computational time step using the unsteady flamelet equations. The flamelets are then integrated in real time based on their respective scalar dissipation rate histories. This approach is capable of including history effects and unsteady chemical kinetic effects as it involves the online solution of unsteady flamelets. Implementation and comprehensive validation of the new framework is carried out against gas jet flames as well as spray flames at high pressures. The approach is first validated for a partially premixed methane gas-jet flame in a RANS framework. The temperature and species mass fractions are compared against experimental results at different locations of a lifted flame. The approach is able to capture the transient flame characteristics. The model is then validated against an n -dodecane spray flame at high pressures in a LES framework with a 103 species n -dodecane chemistry mechanism. The model is able to capture ignition delays and flame liftoff of the n -dodecane spray over varying ambient temperature conditions. Due to the offsetting of presumed PDF integration cost, the proposed framework is shown to be faster than the traditional RIF approach and the speedup over RIF increases with an increase in the cell count. Furthermore, this framework enables use of the RIF model for large cell count LES simulations.

42 ENGINEERING↗
NASA NTRS2012

Ensemble Averaged Probability Density Function (APDF) for Compressible Turbulent Reacting Flows

In this paper, we present a concept of the averaged probability density function (APDF) for studying compressible turbulent reacting flows. The APDF is defined as an ensemble average of the fine grained probability density function (FG-PDF) with a mass density weighting. It can be used to exactly deduce the mass density weighted, ensemble averaged turbulent mean variables. The transport equation for APDF can be derived in two ways. One is the traditional way that starts from the transport equation of FG-PDF, in which the compressible Navier- Stokes equations are embedded. The resulting transport equation of APDF is then in a traditional form that contains conditional means of all terms from the right hand side of the Navier-Stokes equations except for the chemical reaction term. These conditional means are new unknown quantities that need to be modeled. Another way of deriving the transport equation of APDF is to start directly from the ensemble averaged Navier-Stokes equations. The resulting transport equation of APDF derived from this approach appears in a closed form without any need for additional modeling. The methodology of ensemble averaging presented in this paper can be extended to other averaging procedures: for example, the Reynolds time averaging for statistically steady flow and the Reynolds spatial averaging for statistically homogeneous flow. It can also be extended to a time or spatial filtering procedure to construct the filtered density function (FDF) for the large eddy simulation (LES) of compressible turbulent reacting flows.

Shih, Tsan-Hsing↗
DOE OSTI2025

Kaon gluon parton distribution and momentum fraction from 2+1+1 lattice QCD with high statistics

We present a high-statistics lattice-QCD determination of the kaon gluon parton distribution function and gluon momentum fraction. We use clover valence fermion action to take 1,296,640 kaon-correlator measurements on a highly improved staggered quark ensemble with 𝑎 ≈ 0.12 fm and 310-MeV pion mass generated by the MILC Collaboration. A detailed investigation into the impact of gauge-link smearing on the gluonic matrix elements indicates that five steps of hypercubic smearing offer an effective balance between signal quality and preservation of long-distance physics. We report a nonperturbatively renormalized kaon gluon momentum fraction of Math output error at 𝜇 = 2 GeV in the Math output error scheme. Using reduced pseudo-Ioffe-time-distribution matrix elements and pseudo-parton-distribution-function (PDF) matching, we extract the kaon gluon PDF and compare with the prediction from the Dyson-Schwinger equation and with the pion PDF obtained from the same ensemble.

First-principles calculations↗
DOE OSTI2022

Stochastic Modeling in a Multimaterial Continuum Mixture Shock Physics Code

Stochastic modelling approaches are presented to capture random effects at multiple time and length scales. Random processes that occur at the microscale produce nondeterministic effects at the macroscale. Here we present three stochastic modeling approaches that describe random processes at microscopic length scales and map these processes to the macroscopic length scale. The first stochastic modeling approach is based upon a particle based numerical technique to solve a Stochastic Differential Equation (SDE) using an arbitrary diffusion process to capture random processes at the microstructural level. The second approach prescribes a Probability Density Function (PDF) for the drift and diffusion of the random variable derived using the forward and backward Kolmogorov equations. This method requires mean and drift evolution PDF transport equations. The third approach is the coupling of multiple random variables which are dependent on each other. The relationship of the PDFs and a coupling function, known as a copula, produces a Joint Probability Density Function (JPDF). These stochastic modeling approaches are implemented into a Multiple Component (MC) shock physics computational code and used to model statistical fracture and reactive flow applications.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗