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NASA NTRSDate not supplied

Is LaTeX Use Correlated with the Number of Equations in a Manuscript?

Astronomers often use LaTeX to compose and submit paper manuscripts (1), but its adoption is not universal. Some journals prefer Word submissions, and large organizations with standardized software loads may also pressure their scientists to use Word. Furthermore, text and table transcription has been shown to be more efficient with Word than with LaTeX (2). LaTeX is most favored by fields that are mathematically intensive (1). While Knauff and Nejasmic (2) found Word to be similarly efficient in transcribing equations, users reported greater enjoyment of LaTeX. We suspect that Word may therefore influence its users to include fewer equations, but this underlying hypothesis is not easily tested. We test the hypothesis that manuscripts submitted to the arXiV in LaTeX format have more equations on average than those submitted in PDF format only.

Althea V Moorhead↗

DOE OSTI2025

Voltage Probability Density Function Shaping Control Strategy Considering Grid Operational Uncertainties

It is well-known that power systems operation always affected by various uncertainties which make the bus voltage a random process that can be characterized by its probability density function (PDF) at any time instant. In this context, this paper presents a novel PDF-based voltage control framework for power systems. By modeling voltage as a stochastic process, we formulate a stochastic differential equationthat captures grid uncertainties. The associated Fokker-Planck-Kolmogorov equation is derived to describe the evolution of the voltage PDF, which enables the formulation of a PDF-shaping control strategy. To simplify the PDF control formulation, a B-spline neural network is introduced for real-time estimation and regulation of the voltage distribution. The proposed PDF control law updates voltage references for energy storage systems and synchronous generators using real-time PDF measurements and feedback signals. The proposed method is validated on a modified Kundur’s two-area system. Simulation results demonstrate that the controller can significantly improve the voltage stability under stochastic conditions, highlighting its effectiveness in modern inverter-rich grids.

Gui, Yonghao [ORNL] (ORCID:0000000250435534)↗

NASA NTRS1996

Measurements and Modeling of Soot Formation and Radiation in Microgravity Jet Diffusion Flames

This is a computational and experimental study for soot formation and radiative heat transfer in jet diffusion flames under normal gravity (1-g) and microgravity (0-g) conditions. Instantaneous soot volume fraction maps are measured using a full-field imaging absorption technique developed by the authors. A compact, self-contained drop rig is used for microgravity experiments in the 2.2-second drop tower facility at NASA Lewis Research Center. On modeling, we have coupled flame structure and soot formation models with detailed radiation transfer calculations. Favre-averaged boundary layer equations with a k-e-g turbulence model are used to predict the flow field, and a conserved scalar approach with an assumed Beta-pdf are used to predict gaseous species mole fraction. Scalar transport equations are used to describe soot volume fraction and number density distributions, with formation and oxidation terms modeled by one-step rate equations and thermophoretic effects included. An energy equation is included to couple flame structure and radiation analyses through iterations, neglecting turbulence-radiation interactions. The YIX solution for a finite cylindrical enclosure is used for radiative heat transfer calculations. The spectral absorption coefficient for soot aggregates is calculated from the Rayleigh solution using complex refractive index data from a Drude- Lorentz model. The exponential-wide-band model is used to calculate the spectral absorption coefficient for H20 and C02. It is shown that when compared to results from true spectral integration, the Rosseland mean absorption coefficient can provide reasonably accurate predictions for the type of flames studied. The soot formation model proposed by Moss, Syed, and Stewart seems to produce better fits to experimental data and more physically sound than the simpler model by Khan et al. Predicted soot volume fraction and temperature results agree well with published data for a normal gravity co-flow laminar flames and turbulent jet flames. Predicted soot volume fraction results also agree with our data for 1-g and 0-g laminar jet names as well as 1-g turbulent jet flames.

Ku, Jerry C.↗

NASA NTRS1983

Modeling of Fluctuating Mass Flux in Variable Density Flows

The approach solves for both Reynolds and Favre averaged quantities and calculates the scalar pdf. Turbulent models used to close the governing equations are formulated to account for complex mixing and variable density effects. In addition, turbulent mass diffusivities are not assumed to be in constant proportion to turbulent momentum diffusivities. The governing equations are solved by a combination of finite-difference technique and Monte-Carlo simulation. Some preliminary results on simple variable density shear flows are presented. The differences between these results and those obtained using conventional models are discussed.

So, R. M. C.↗

NASA NTRS2000

NCC: A Physics-Based Design and Analysis Tool for Combustion Systems

The National Combustion Code (NCC) is an integrated system of computer codes for physics-based design and analysis of combustion systems. It uses unstructured meshes and runs on parallel computing platforms. The NCC is composed of a set of distinct yet closely related modules. They are: (1) a gaseous flow module solving 3-D Navier-Stokes equations; (2) a turbulence module containing the non-linear k-epsilon models; (3) a chemistry module using either the conventional reduced kinetics approach of solving species equations or the Intrinsic Low Dimensional Manifold (ILDM) kinetics approach of table looking up in conjunction with solving the equations of the progressive variables; (4) a turbulence-chemistry interaction module including the option of solving the joint probability density function (PDF) for species and enthalpy; and (5) a spray module for solving the liquid phase equations. In early 1995, an industry-government team was formed to develop the NCC. In July 1998, the baseline beta version was completed and presented in two NCC sessions at the 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 1998. An overview of this baseline beta version was presented at the NASA HPCCP/CAS Workshop 98, August 1998. Since then, the effort has been focused on the streamlining, validation, and enhancement of the th baseline beta version. The progress is presented in two NCC sessions at the AIAA 38 Aerospace Sciences Meeting & Exhibit, January 2000. At this NASA HPCCP/CAS Workshop 2000, an overview of the NCC papers presented at the AIAA 38 th Aerospace Sciences Meeting & Exhibit is presented, with emphasis on the reduction of analysis time of simulating the (gaseous) reacting flows in full combustors. In addition, results of NCC simulation of a modern turbofan combustor will also be reported.

Liu, Nan-Suey↗

NASA NTRS1992

Towards understanding turbulent scalar mixing

In an effort towards understanding turbulent scalar mixing, we study the effect of molecular mixing, first in isolation and then by accounting for the effects of the velocity field. The chief motivation for this approach stems from the strong resemblance of the scalar probability density function (PDF) obtained from the scalar field evolving from the heat conduction equation that arises in a turbulent velocity field. However, the evolution of the scalar dissipation is different for the two cases. We attempt to account for these differences, which are due to the velocity field, using a Lagrangian frame analysis. After establishing the usefulness of this approach, we use the heat-conduction simulations (HCS), in lieu of the more expensive direct numerical simulations (DNS), to study many of the less understood aspects of turbulent mixing. Comparison between the HCS data and available models are made whenever possible. It is established that the beta PDF characterizes the evolution of the scalar PDF during mixing from all types of non-premixed initial conditions.

Girimaji, Sharath S.↗

NASA NTRS1988

Analysis of homogeneous turbulent reacting flows

Full turbulence simulations at low Reynolds numbers were made for the single-step, irreversible, bimolecular reaction between non-premixed reactants in isochoric, decaying homogeneous turbulence. Various initial conditions for the scalar field were used in the simulations to control the initial scalar dissipation length scale, and simulations were also made for temperature-dependent reaction rates and for non-stoichiometric and unequal diffusivity conditions. Joint probability density functions (pdf's), conditional pdf's, and various statistical quantities appearing in the moment equations were computed. Preliminary analysis of the results indicates that compressive strain-rate correlates better than other dynamical quantities with local reaction rate, and the locations of peak reaction rates seem to be insensitive to the scalar field initial conditions.

Leonard, A. D.↗

NASA NTRS2003

Large Eddy Simulation Of Gravitational Effects In Transitional And Turbulent Gas-Jet Diffusion Flames

The influence of gravity on the spatial and the compositional structures of transitional and turbulent hydrocarbon diffusion flames are studies via large eddy simulation (LES) and direct numerical simulation (DNS) of round and planar jets. The subgrid-scale (SGS) closures in LES are based on the filtered mass density function (FMDF) methodology. The FMDF represents the joint probability density function (PDF) of the SGS scalars, and is obtained by solving its transport equation. The fundamental advantage of LES/FMDF is that it accounts for the effects of chemical reaction and buoyancy exactly. The methodology is employed for capturing some of the fundamental influences of gravity in equilibrium flames via realistic chemical kinetic schemes. Some preliminary investigation of the gravity effects in non-equilibrium flames is also conducted, but with idealized chemical kinetics models.

Jaberi, Farhad A.↗

NASA NTRS1998

Improved Modeling of Finite-Rate Turbulent Combustion Processes in Research Combustors

The objective of this thesis is to further develop and test a stochastic model of turbulent combustion in recirculating flows. There is a requirement to increase the accuracy of multi-dimensional combustion predictions. As turbulence affects reaction rates, this interaction must be more accurately evaluated. In this work a more physically correct way of handling the interaction of turbulence on combustion is further developed and tested. As turbulence involves randomness, stochastic modeling is used. Averaged values such as temperature and species concentration are found by integrating the probability density function (pdf) over the range of the scalar. The model in this work does not assume the pdf type, but solves for the evolution of the pdf using the Monte Carlo solution technique. The model is further developed by including a more robust reaction solver, by using accurate thermodynamics and by more accurate transport elements. The stochastic method is used with Semi-Implicit Method for Pressure-Linked Equations. The SIMPLE method is used to solve for velocity, pressure, turbulent kinetic energy and dissipation. The pdf solver solves for temperature and species concentration. Thus, the method is partially familiar to combustor engineers. The method is compared to benchmark experimental data and baseline calculations. The baseline method was tested on isothermal flows, evaporating sprays and combusting sprays. Pdf and baseline predictions were performed for three diffusion flames and one premixed flame. The pdf method predicted lower combustion rates than the baseline method in agreement with the data, except for the premixed flame. The baseline and stochastic predictions bounded the experimental data for the premixed flame. The use of a continuous mixing model or relax to mean mixing model had little effect on the prediction of average temperature. Two grids were used in a hydrogen diffusion flame simulation. Grid density did not effect the predictions except for peak temperature and tangential velocity. The hybrid pdf method did take longer and required more memory, but has a theoretical basis to extend to many reaction steps which cannot be said of current turbulent combustion models.

VanOverbeke, Thomas J.↗

NASA NTRS1990

A hybrid Reynolds averaged/PDF closure model for supersonic turbulent combustion

A hybrid Reynolds averaged/assumed pdf approach has been developed and applied to the study of turbulent combustion in a supersonic mixing layer. This approach is used to address the 'laminar-like' treatment of the thermochemical terms that appear in the conservation equations. Calculations were carried out for two experiments involving H2-air supersonic turbulent mixing. Two different forms of the pdf were implemented. In general, the results show modest improvement from previous calculations. Moreover, the results appear to be somewhat independent of the form of the assumed pdf.

Frankel, Steven H.↗

NASA NTRS1994

A 'skewed' lognormal approximation to the probablility distribution function of the large-scale density field

I propose a method to fit the probability distribution function (PDF) of the large-scale density field rho, motivated by a Lagrangian version of the continuity equation. It consists in applying the Edgeworth expansion to the quantity Phi identical with log rho - mean value of log rho. The method is tested on the matter particle distribution in two cold dark matter N-body simulations of different physical sizes to cover a large dynamic range. It is seen to be very efficient, even in the nonlinear regime, and may thus be used as an analytical tool to study the effect on the PDF of the transition between the weakly nonlinear regime and the highly nonlinear regime.

Colombi, S.↗

NASA NTRS2001

Filtered Mass Density Function for Design Simulation of High Speed Airbreathing Propulsion Systems

The objective of this research is to develop and implement new methodology for large eddy simulation of (LES) of high-speed reacting turbulent flows. We have just completed two (2) years of Phase I of this research. This annual report provides a brief and up-to-date summary of our activities during the period: September 1, 2000 through August 31, 2001. In the work within the past year, a methodology termed "velocity-scalar filtered density function" (VSFDF) is developed and implemented for large eddy simulation (LES) of turbulent flows. In this methodology the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function (PDF) of all of the components of the velocity and scalar vectors. An exact transport equation is derived for the VSFDF in which the effects of the unresolved SGS convection, SGS velocity-scalar source, and SGS scalar-scalar source terms appear in closed form. The remaining unclosed terms in this equation are modeled. A system of stochastic differential equations (SDEs) which yields statistically equivalent results to the modeled VSFDF transport equation is constructed. These SDEs are solved numerically by a Lagrangian Monte Carlo procedure. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed by comparison with results obtained by an Eulerian LES procedure in which the corresponding transport equations for the first two SGS moments are solved. The unclosed SGS convection, SGS velocity-scalar source, and SGS scalar-scalar source in the Eulerian LES are replaced by corresponding terms from VSFDF equation. The consistency of the results is then analyzed for a case of two dimensional mixing layer.

Drozda, T. G.↗

DOE OSTI2025

Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker–Planck Equations

The Fokker-Planck (FP) equation is a foundational partial differential equation (PDE) in stochastic processes involving Brownian motions. However, the curse of dimensionality (CoD) poses a formidable challenge when dealing with high-dimensional FP equations. Although Monte Carlo simulation and (vanilla) Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit significant numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The point-wise PDF values tend to decrease exponentially as dimensionality increases, surpassing the precision of numerical simulations and resulting in substantial errors. In addition, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a notoriously difficult Hamilton-Jacobi-Bellman (HJB) equation, which is impractical for PINN learning, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in stochastic differential equations (SDEs). The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling, offering an effective means to overcome the CoD. Three fitting methods, Score Matching (SM), Sliced Score Matching (SSM), and Score-PINN, are introduced, each contributing unique advantages in computational complexity, accuracy, and generality. The proposed score-based SDE solver operates in two stages: first, employing score matching or Score-PINN to acquire the score function; and second, solving the LL via an ordinary differential equation (ODE) using the obtained score function. Comparative evaluations across these methods showcase varying trade-offs. The proposed methodology is evaluated across diverse SDEs, including anisotropic Ornstein-Uhlenbeck processes, geometric Brownian motion, and Brownian motion with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy distributions. The numerical results demonstrate the score-based SDE solver’s stability, speed, and performance across different experimental settings, solidifying its potential as a solution to CoD for high-dimensional FP equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Closure models for the feedback of energetic particles on plasma turbulence

Energetic particles interact with the plasma surrounding them, resonating with certain types of plasma waves to stabilize them while destabilizing others, and changing the character of the background turbulence in ways that have not been fully quantified or understood. Interaction with the turbulent background plasma is key to the acceleration of many types of energetic particles including high-energy cosmic rays, solar energetic particles, and pick-up ions. The acceleration of particles is a process that would ideally be described by a kinetic model, a type of model that follows a probability distribution function (PDF) for all particles in 7-dimensional (x, y, z, v x , v y , v z , t) space. Because of the high dimensionality of a kinetic model, simulations that solve kinetic equations use the largest computational resources currently available, and are yet unable to simulate a realistic number of particles, reach the large scales necessary for astrophysical problems, and use high-precision numerical methods. Two available alternatives to kinetic plasma models have been explored for this problem, with limited success. One is a multi-fluid model produced by a cumulant discarding closure, which evolves coupled equations for the velocity, magnetic field, and internal energy for both the background plasma and the fluid of energetic particles. However, simulations that solve multi-fluid magnetohydrodynamic (MHD) equations are able to include the interaction with energetic particles only in crude ways, typically as an add-on pressure term. The second alternative is to use a hybrid method to couple a fluid description of the background plasma to a kinetic model or a Fokker–Planck model for the energetic particles. These methods are hampered by the physical modeling of the coupling. In this work, we develop a new model, which follows the PDF for all particles; this can be viewed as a step toward physical realism above a multi-fluid MHD model, while also being more computationally efficient than a kinetic model. The equations we develop model both the background plasma and the energetic particles self-consistently. Over the last decade, similar PDF methods have been developed to a high level of sophistication to model reactive flows and turbulent combustion for engineering applications. For treatment of the feedback of the energetic particles on a background plasma, a PDF closure approach should evaluate the mean characteristics, including the density, with better statistical quality than will particle-sampling procedures.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1993

A k-omega multivariate beta PDF for supersonic turbulent combustion

In a recent attempt by the authors at predicting measurements in coaxial supersonic turbulent reacting mixing layers involving H2 and air, a number of discrepancies involving the concentrations and their variances were noted. The turbulence model employed was a one-equation model based on the turbulent kinetic energy. This required the specification of a length scale. In an attempt at detecting the cause of the discrepancy, a coupled k-omega joint probability density function (PDF) is employed in conjunction with a Navier-Stokes solver. The results show that improvements resulting from a k-omega model are quite modest.

Alexopoulos, G. A.↗

NASA NTRS1992

Center for Modeling of Turbulence and Transition (CMOTT): Research Briefs, 1992

The progress is reported of the Center for Modeling of Turbulence and Transition (CMOTT). The main objective of the CMOTT is to develop, validate and implement the turbulence and transition models for practical engineering flows. The flows of interest are three-dimensional, incompressible and compressible flows with chemical reaction. The research covers two-equation (e.g., k-e) and algebraic Reynolds-stress models, second moment closure models, probability density function (pdf) models, Renormalization Group Theory (RNG), Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS).

Liou, William W.↗

DOE OSTI2024

RAIS Preliminary Remediation Goals for Chemicals

Chemical preliminary remediation goals (PRGs) (https://rais.ornl.gov/cgibin/prg/PRG_search?select=chem) are calculated by selecting the applicable chemical(s) together with the applicable media, land use, and exposure route combination. If “site-specific” is selected as the PRG type, the following page will show the equations and exposure parameters used for deriving the PRGs, and some of the parameter values may be changed as necessary. If “default” is selected as the PRG type, the RAIS will proceed directly to the results page. Multiple chemicals can be selected. Results can be downloaded in .xlsx or .pdf formats. Additionally, the session inputs for the PRG calculator can be saved for future use and recalled by the PRG calculator. The derivation of the selected PRG and the applicable equations and exposure parameters can be reviewed in more detail using the RAIS Chemical PRG Calculator User Guide (https://rais.ornl.gov/tools/rais_chemical_prg_guide.html).

Manning, Karessa [Oak Ridge National Laboratory (O↗

DOE OSTI2024

RAIS Preliminary Remediation Goals for Radionuclides

Radionuclide PRGs (https://rais.ornl.gov/cgi-bin/prg/PRG_search?select=rad) are calculated by selecting the applicable radionuclide(s) together with the applicable media, land use, and exposure route combination. If “site-specific” is selected as the PRG type, the following page will show the equations and exposure parameters used for deriving the PRGs, and some of the parameter values may be changed as necessary. If “default” is selected as the PRG type, the RAIS will proceed directly to the results page. Multiple radionuclides can be selected with this tool. Results can be downloaded in .xlsx or .pdf formats. Additionally, the session inputs for the PRG calculator can be saved for future use and recalled by the PRG calculator. The derivation of the selected PRG(s) and the applicable equations and exposure parameters can be reviewed in more detail using the RAIS Radionuclide PRG Calculator User Guide available here: https://rais.ornl.gov/tools/rais_rad_prg_guide.html. html.

Manning, Karessa [Oak Ridge National Laboratory (O↗