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Results for “LAGRANGIAN FUNCTION”

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DOE OSTI2020

Fifth-degree elastic energy for predictive continuum stress-strain relations and elastic instabilities under large strain and complex loading in silicon

Materials under complex loading develop large strains and often phase transformation via an elastic instability, as observed in both simple and complex systems. Here, we represent a material (exemplified for Si I) under large Lagrangian strains within a continuum description by a 5th-order elastic energy found by minimizing error relative to density functional theory (DFT) results. The Cauchy stress—Lagrangian strain curves for arbitrary complex loadings are in excellent correspondence with DFT results, including the elastic instability driving the Si I → II phase transformation (PT) and the shear instabilities. PT conditions for Si I → II under action of cubic axial stresses are linear in Cauchy stresses in agreement with DFT predictions. Such continuum elastic energy permits study of elastic instabilities and orientational dependence leading to different PTs, slip, twinning, or fracture, providing a fundamental basis for continuum physics simulations of crystal behavior under extreme loading.

Levitas, Valery↗

DOE OSTI2021

Numerical Investigation of Enhanced Dehumidification Processes By Using Dielectrophoresis Principles in Moist Airflows

Dispersed particle-laden flows are encountered in many building and industrial applications, such as flow in a fluidized bed, hydrocarbon transportation in pipelines, and the fouling of air-cooled heat exchangers (Kuruneru et al., 2016; Ray et al., 2019; Wang et al., 2019). Computational fluid dynamic (CFD) models have been developed in recent years to depict particle-fluid and particle-particle interactions in laminar or turbulent flows with increasing accuracy and stability. One particular particle-laden system of interest for moisture control is electrically-enhanced condensation in air and water droplet flows. Electrically-enhanced condensation consists of the use of highly charged water droplets injected in the moist air. The droplets become electric seeds that attract polar water vapor molecules to their surfaces and promote condensation. The nucleation and growth of the charged droplets deplete the vapor phase near a droplet, which is compensated for by the dielectrophoresis flow and diffusion. Dielectrophoresis flow involves surrounding vapor at a distance of about 10 to 100 nm for droplets charged by an electrospray compared to ~2 nm for a single electron charge in a droplet. As the vapor molecules collapse on the surface of the droplets, their initial electrical charge decreases with time due to the neutralization of the ions. While the physics of this phenomena is well known, engineering models for predicting the condensation rates are not available. This work computationally investigates dehumidification of moist airflow in a converging rectangular duct. The objective is to develop an engineering model that predicts water vapor condensation by employing dielectrophoresis principles. We construct a Computational Fluid Dynamics (CFD) model of the duct with electrically-enhanced condensation. The model is implemented in the open-source software OpenFOAM. We utilize the Multi-Phase Particle-In-Cell (MP-PIC) method coupled with a Population Balance Equation (PBE) approach to simulate the particle-laden system. This methodology is an Eulerian-Lagrangian approach used to simulate the droplets' behavior in the humid air. The MP-PIC approach (Andrews and O'Rourke, 1996) mitigates the computational cost by parceling several fundamental particles with similar properties (such as types, sizes, and temperature) into one computational particle. Thus, the billions of particles can be substituted by millions of computational particles without significant loss of information. The PBE was considered with the Lagrangian frame to combine the particle distribution function used in MP-PIC (Kim et al., 2020). This approach preserves mass and energy conservation between the phases in the Eulerian and Lagrangian structures. The PBE in this procedure was directly linked to the discrete parcels, making the simulation of the particle distribution computationally efficient and robust. The MP-PIC-PBE approach used in the present work was applied to the dehumidification of air. Water droplets were injected in the air stream and forced to grow according to experimentally derived correlation. The experiments were conducted on a converging duct with the same geometry and boundary conditions used to build the CFD model. This approach enabled us to approximate the effect of dielectrophoresis phenomena on the droplet and air interface. This presentation will discuss the details of the new CFD model built for the duct, the implementation of the model in OpenFOAM CFD programming language, and the experimental validation of the newly developed model. The results revealed a moderate yet measurable increase in droplet diameter due to water vapor condensation at the vapor-liquid interface of the electrically charged droplets' surface. The seed water droplet particles grew in size by capturing the water vapor in the surrounding air. The OpenFOAM model predicted reductions of humidity in the air from 5 to 10 percent.

Yel Mahi, Maliha↗

NASA NTRS1975

Turbulence in a gaseous hydrogen-liquid oxygen rocket combustion chamber

The intensity of turbulence and the Lagrangian correlation coefficient for a LOX-GH2 rocket combustion chamber was determined from experimental measurements of tracer gas diffusion. A combination of Taylor's turbulent diffusion theory and a numerical method for solving the conservation equations of fluid mechanics was used to calculate these quantities. Taylor's theory was extended to consider the inhomogeneity of the turbulence field in the axial direction of the combustion chamber, and an exponential function was used to represent the Lagrangian correlation coefficient. The results indicate that the value of the intensity of turbulence reaches a maximum of 14% at a location about 7" downstream from the injector. The Lagrangian correlation coefficient associated with this value is given by the above exponential expression where alpha = 10,000/sec.

Lebas, J.↗

NASA NTRS1974

Experimental determination of turbulence in a GH2-GOX rocket combustion chamber

The intensity of turbulence and the Lagrangian correlation coefficient for a gaseous rocket combustion chamber have been determined from the experimental measurements of the tracer gas diffusion. A combination of Taylor's turbulent diffusion theory and Spalding's numerical method for solving the conservation equations of fluid mechanics was used to calculate these quantities. Taylor's theory was extended to consider the inhomogeneity of the turbulence field in the axial direction of the combustion chamber. An exponential function was used to represent the Lagrangian correlation coefficient. The results indicate that the maximum value of the intensity of turbulence is about 15% and the Lagrangian correlation coefficient drops to about 0.12 in one inch of the chamber length.

Tou, P.↗

NASA NTRS1978

Measurement of the turbulence in a gaseous rocket combustion chamber

The intensity of turbulence and the Lagrangian correlation coefficient in a gaseous hydrogen (GH2)-gaseous oxygen (GOX) rocket combustion chamber are studied. A helium tracer gas was injected at an upstream point on the combustion chamber centerline. Samples collected downstream were analyzed for the spreading of the tracer gas concentration. Using Taylor's turbulent diffusion theory and Gosman's numerical method, turbulence parameters were found. In order to evaluate turbulence field inhomogeneity in the axial direction, Taylor's theory was extended. An exponential function was used to approximate the Lagrangian correlation coefficient. It was found that turbulence intensity varies from 15% near the injector to 4% at the nozzle entrance, and that the Lagrangian coefficient rapidly decreases when the dispersion time difference increases.

Tou, P. P.↗

NASA NTRS1978

Stochastic inverse problem in the radiation of noise

The reported investigation is concerned with a stochastic inverse radiation problem in a uniform medium. The problem is illustrated with the aid of a simple model consisting of an array of point sources. The entropy functional is chosen to be the structural functional in determining the source distribution. A general theory for the stochastic inverse problem is introduced. It is shown that the general procedure yields the methods of the Lagrangian multiplier, when the structural and residual functionals are specialized. Tihonov's regularization and a method related to generalized or pseudoinverses are also obtained. Examples considered for purposes of illustration are related to a continuous source with the least noise intensity, a continuous source with a potential, and an axisymmetric line source.

Chow, P. L.↗

DOE OSTI2026

Global symmetry and integral constraint on superconformal lines in four dimensions

We study properties of point-like impurities preserving flavor symmetry and supersymmetry in four-dimensional 𝒩 = 2 field theories. At large distances, such impurities are described by half-BPS superconformal line defects. By working in the AdS 2 × S 2 conformal frame, we develop a novel and simpler way of deriving the superconformal Ward identities relating the various two-point functions of flavor current multiplet operators in the presence of the defect. We use these relations to simplify a certain integrated two-point function of flavor current multiplet operators that, in Lagrangian theories, can be computed using supersymmetric localization. The simplification gives an integral constraint on the two-point function of the flavor current multiplet superconformal primary with trivial integration measure in the AdS 2 × S 2 conformal frame. We provide several consistency checks on our Ward identities.

extended supersymmetry↗

NASA NTRS1998

Effects of Helicity on Lagrangian and Eulerian Time Correlations in Turbulence

Taylor series expansions of turbulent time correlation functions are applied to show that helicity influences Eulerian time correlations more strongly than Lagrangian time correlations: to second order in time, the helicity effect on Lagrangian time correlations vanishes, but the helicity effect on Eulerian time correlations is nonzero. Fourier analysis shows that the helicity effect on Eulerian time correlations is confined to the largest inertial range scales. Some implications for sound radiation by swirling flows are discussed.

Rubinstein, Robert↗

DOE OSTI2023

Lagrangian large eddy simulations via physics-informed machine learning

High-Reynolds number homogeneous isotropic turbulence (HIT) is fully described within the Navier–Stokes (NS) equations, which are notoriously difficult to solve numerically. Engineers, interested primarily in describing turbulence at a reduced range of resolved scales, have designed heuristics, known as large eddy simulation (LES). LES is described in terms of the temporally evolving Eulerian velocity field defined over a spatial grid with the mean-spacing correspondent to the resolved scale. This classic Eulerian LES depends on assumptions about effects of subgrid scales on the resolved scales. Here, we take an alternative approach and design LES heuristics stated in terms of Lagrangian particles moving with the flow. Our Lagrangian LES, thus L-LES, is described by equations generalizing the weakly compressible smoothed particle hydrodynamics formulation with extended parametric and functional freedom, which is then resolved via Machine Learning training on Lagrangian data from direct numerical simulations of the NS equations. The L-LES model includes physics-informed parameterization and functional form, by combining physics-based parameters and physics-inspired Neural Networks to describe the evolution of turbulence within the resolved range of scales. The subgrid-scale contributions are modeled separately with physical constraints to account for the effects from unresolved scales. We build the resulting model under the differentiable programming framework to facilitate efficient training. We experiment with loss functions of different types, including physics-informed ones accounting for statistics of Lagrangian particles. We show that our L-LES model is capable of reproducing Eulerian and unique Lagrangian turbulence structures and statistics over a range of turbulent Mach numbers.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2023

Code for the manuscript "Lagrangian Large Eddy Simulations vis Physics-informed Machine Learning"

We would like to disclose a code based on Python/Pytorch, in which we implement the data-driven physics-informed machine learning of Lagrangian large eddy simulation that is described in the manuscript "Lagrangian Large Eddy Simulations via Physics-Informed Machine Learning ". High-Reynolds number Homogeneous Isotropic Turbulence is fully described within the Navier-Stokes (NS) equations, which are notoriously difficult to solve numerically. Engineers, interested primarily in describing turbulence at a reduced range of resolved scales, have designed heuristics, known as Large Eddy Simulation (LES). LES is described in terms of the temporally evolving Eulerian velocity field defined over a spatial grid with the mean-spacing correspondent to the resolved scale. This classic Eulerian LES depends on assumptions about the effects of sub-grid scales on the resolved scales. Here, we take an alternative approach and design novel LES heuristics stated in terms of Lagrangian particles moving with the flow. Our \emph{Lagrangian LES}, thus L-LES, is described by equations generalizing the weakly compressible Smoothed Particle Hydrodynamics formulation with extended parametric and functional freedom, which is then resolved via Machine Learning training on Lagrangian data from Direct Numerical Simulations of the NS equations. The L-LES model includes physics-informed parameterization and functional form, by combining physics-based parameters and physics-inspired Neural Networks to describe the evolution of turbulence within the resolved range of scales. The sub-grid scale contributions are modeled separately with physical constraints to account for the effects from un-resolved scales. We build the resulting model under the Differentiable Programming framework to facilitate efficient training. We experiment with loss functions of different types, including physics-informed ones accounting for statistics of Lagrangian particles. We show that our Lagrangian LES model is capable of reproducing Eulerian and unique Lagrangian turbulence structures and statistics over a range of turbulent Mach numbers.

Tian, Yifeng↗

NASA NTRS2014

A Regional CO2 Observing System Simulation Experiment for the ASCENDS Satellite Mission

Top-down estimates of the spatiotemporal variations in emissions and uptake of CO2 will benefit from the increasing measurement density brought by recent and future additions to the suite of in situ and remote CO2 measurement platforms. In particular, the planned NASA Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS) satellite mission will provide greater coverage in cloudy regions, at high latitudes, and at night than passive satellite systems, as well as high precision and accuracy. In a novel approach to quantifying the ability of satellite column measurements to constrain CO2 fluxes, we use a portable library of footprints (surface influence functions) generated by the WRF-STILT Lagrangian transport model in a regional Bayesian synthesis inversion. The regional Lagrangian framework is well suited to make use of ASCENDS observations to constrain fluxes at high resolution, in this case at 1 degree latitude x 1 degree longitude and weekly for North America. We consider random measurement errors only, modeled as a function of mission and instrument design specifications along with realistic atmospheric and surface conditions. We find that the ASCENDS observations could potentially reduce flux uncertainties substantially at biome and finer scales. At the 1 degree x 1 degree, weekly scale, the largest uncertainty reductions, on the order of 50 percent, occur where and when there is good coverage by observations with low measurement errors and the a priori uncertainties are large. Uncertainty reductions are smaller for a 1.57 micron candidate wavelength than for a 2.05 micron wavelength, and are smaller for the higher of the two measurement error levels that we consider (1.0 ppm vs. 0.5 ppm clear-sky error at Railroad Valley, Nevada). Uncertainty reductions at the annual, biome scale range from 40 percent to 75 percent across our four instrument design cases, and from 65 percent to 85 percent for the continent as a whole. Our uncertainty reductions at various scales are substantially smaller than those from a global ASCENDS inversion on a coarser grid, demonstrating how quantitative results can depend on inversion methodology. The a posteriori flux uncertainties we obtain, ranging from 0.01 to 0.06 Pg C yr-1 across the biomes, would meet requirements for improved understanding of long-term carbon sinks suggested by a previous study.

remote sensing↗

NASA NTRS1972

Stability of orthotropic stiffened composite plates.

It is pointed out that the buckling problem of a plate with restrictions resulting from boundary conditions is suited for an application of Lagrangian multipliers. An approach used by Budiansky and Hu (1946) is extended to the study of orthotropic stiffened plates. The variations of the total energy with respect to the three deformation components are considered. The equilibrium equations and the boundary conditions are obtained. It is shown that the deflection functions must satisfy the equilibrium equations. The Lagrangian multipliers are then used to solve stability problems on the basis of the total energy involved, the equilibrium equations, and the boundary conditions. The theoretical analysis is supplemented by an experimental investigation.

Chiu, K. D.↗

NASA NTRS1995

Modified Interior Distance Functions (Theory and Methods)

In this paper we introduced and developed the theory of Modified Interior Distance Functions (MIDF's). The MIDF is a Classical Lagrangian (CL) for a constrained optimization problem which is equivalent to the initial one and can be obtained from the latter by monotone transformation both the objective function and constraints. In contrast to the Interior Distance Functions (IDF's), which played a fundamental role in Interior Point Methods (IPM's), the MIDF's are defined on an extended feasible set and along with center, have two extra tools, which control the computational process: the barrier parameter and the vector of Lagrange multipliers. The extra tools allow to attach to the MEDF's very important properties of Augmented Lagrangeans. One can consider the MIDFs as Interior Augmented Lagrangeans. It makes MIDF's similar in spirit to Modified Barrier Functions (MBF's), although there is a fundamental difference between them both in theory and methods. Based on MIDF's theory, Modified Center Methods (MCM's) have been developed and analyzed. The MCM's find an unconstrained minimizer in primal space and update the Lagrange multipliers, while both the center and the barrier parameter can be fixed or updated at each step. The MCM's convergence was investigated, and their rate of convergence was estimated. The extension of the feasible set and the special role of the Lagrange multipliers allow to develop MCM's, which produce, in case of nondegenerate constrained optimization, a primal and dual sequences that converge to the primal-dual solutions with linear rate, even when both the center and the barrier parameter are fixed. Moreover, every Lagrange multipliers update shrinks the distance to the primal dual solution by a factor 0 less than gamma less than 1 which can be made as small as one wants by choosing a fixed interior point as a 'center' and a fixed but large enough barrier parameter. The numericai realization of MCM leads to the Newton MCM (NMCM). The approximation for the primal minimizer one finds by Newton Method followed by the Lagrange multipliers update. Due to the MCM convergence, when both the center and the barrier parameter are fixed, the condition of the MDF Hessism and the neighborhood of the primal ninimizer where Newton method is 'well' defined remains stable. It contributes to both the complexity and the numerical stability of the NMCM.

Polyak, Roman A.↗

NASA NTRSDate not supplied

Modeling of Cavity Residence Time via Modeled Lagrangian Particle Tracking

One of the critical components in a scramjet is the cavity flame holder. In evaluating the sizing of this component, the residence time of the fuel in the cavity is a common parameter used. However, to obtain the residence time via CFD is an expensive undertaking. One method is to perform an LES simulation of the cavity and use Lagrangian particle tracking to obtain both the mean and probability density function of the time that the particles remain in the cavity. While this method affords a significant amount of information, it is very costly in CPU run time. A simpler method involves performing a time accurate RANS simulation of the cavity, and when converged, overlay a traceable scalar in the cavity region, and then record the decay of this scalar’s flux at a downstream location. The time scale of the decay rate can then be estimated and a residence time inferred. This method is less computationally expensive than the LES approach, but still requires a time-accurate solution while only providing a mean cavity residence time evaluation. In this paper a method of determining the mean cavity residence time as well as the Probability Density Function (PDF) of the residence time is proposed and demonstrated. This method is based on Lagrangian particle tracking modeled by the Generalized Langevin Equation and is implemented as a post-processing step for a steady RANS solution. Computational cost is on the order of minutes and the results are in agreement with values obtained by the LES and scalar decay methods.

Langevin↗

DOE OSTI2025

Tropical amplitudes for colored Lagrangians

Recently a new formulation for scattering amplitudes in Tr(Φ 3 ) theory has been given based on simple combinatorial ideas in the space of kinematic data. This allows all-loop integrated amplitudes to be expressed as “curve integrals” defined using tropical building blocks — the “headlight functions”. This paper shows how the formulation extends to the amplitudes of more general Lagrangians. We will present a number of different ways of introducing tropical “numerator functions” that allow us to describe general Lagrangian interactions. The simplest family of these “tropical numerators” computes the amplitudes of interesting Lagrangians with infinitely many interactions. We also describe methods for tropically formulating the amplitudes for general Lagrangians. One uses a variant of “Wick contraction” to glue together numerator factors for general interaction vertices. Another uses a natural characterization of polygons on surfaces to give a novel combinatorial description of all possible diagrams associated with arbitrary valence interactions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Electric and magnetic black holes in a new nonlinear electrodynamics model

Highlights: • We introduce a new NED model which is comparable with the BI model in the weak-field limit. • We find an electric black hole solution in the context of Einstein’s gravity minimally coupled with the new NED model. • We study the thermal stability of the electric black hole solution. • Modified Smarr’s formula consistent with the first law of black hole thermodynamics is obtained. A new nonlinear electrodynamics (NED) model is introduced in the form of a nonpolynomial Lagrangian which admits static spherical and also plane wave solutions in a flat space. Upon coupling with gravity the electric field is finite and comparable with the Born–Infeld counterpart. The electric and magnetic black hole solutions in the Einstein’s gravity coupled with this NED model are presented. The solutions give both asymptotically and in the weak field limit Reissner–Nordström (RN) black hole and unlike the other known models our electric solution is expressed in terms of elementary functions in a closed form. We study the first law and derive the modified Smarr’s formula for the electric type extension of our model. Considerable rich structure, especially thermodynamic ones, ranging from first to the second order phase transitions are added to the RN black hole of linear electrodynamics with this NED model. Having the exact solution for the metric function at our disposal we investigate the stability of the electric black hole from both the thermodynamical and causal points of view.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Gauge invariance of radiative jet functions in the position-space formulation of SCET

In subleading powers of soft-collinear effective theory (SCET), the Lagrangian contains couplings between soft quarks and hard-collinear quarks. Matrix elements of the hard-collinear parts of these couplings are radiative jet functions. In the position-space formulation of SCET, the Lagrangians are constructed from operators that appear to be gauge invariant. Nevertheless, we find violations of gauge invariance arise in the hard-collinear sector because gauge transformations can shift the momentum of a hard-collinear quark field from the hard-collinear sector to the soft sector, where the hard-collinear fields, by definition, have no support. The violations of gauge invariance are manifested in perturbation theory in the hard-collinear sector through the absence of certain Feynman diagrams that would be present in full QCD. A consequence of the absence of these diagrams is that the radiative jet functions that follow directly from the position-space Lagrangians are not gauge invariant, and we demonstrate this through explicit calculations in lower-order perturbation theory. We obtain gauge-invariant Lagrangians by adding to existing position-space Lagrangians terms that are proportional to the soft-quark equation of motion. These gauge-invariant Lagrangians are valid for nonzero, as well as zero, quark masses. We also remark briefly on the gauge invariance of certain Lagrangians that have been constructed in the label-momentum formulation of SCET. Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1991

Compressible flows with periodic vortical disturbances around lifting airfoils

A numerical method is developed for solving periodic, three-dimensional, vortical flows around lifting airfoils in subsonic flow. The first-order method that is presented fully accounts for the distortion effects of the nonuniform mean flow on the convected upstream vortical disturbances. The unsteady velocity is split into a vortical component which is a known function of the upstream flow conditions and the Lagrangian coordinates of the mean flow, and an irrotational field whose potential satisfies a nonconstant-coefficient, inhomogeneous, convective wave equation. Using an elliptic coordinate transformation, the unsteady boundary value problem is solved in the frequency domain on grids which are determined as a function of the Mach number and reduced frequency. The numerical scheme is validated through extensive comparisons with known solutions to unsteady vortical flow problems. In general, it is seen that the agreement between the numerical and analytical results is very good for reduced frequencies ranging from 0 to 4, and for Mach numbers ranging from .1 to .8. Numerical results are also presented for a wide variety of flow configurations for the purpose of determining the effects of airfoil thickness, angle of attack, camber, and Mach number on the unsteady lift and moment of airfoils subjected to periodic vortical gusts. It is seen that each of these parameters can have a significant effect on the unsteady airfoil response to the incident disturbances, and that the effect depends strongly upon the reduced frequency and the dimensionality of the gust. For a one-dimensional (transverse) or two-dimensional (transverse and longitudinal) gust, the results indicate that airfoil thickness increases the unsteady lift and moment at the low reduced frequencies but decreases it at the high reduced frequencies. The results show that an increase in airfoil Mach number leads to a significant increase in the unsteady lift and moment for the low reduced frequencies, but a significant decrease for the high reduced frequencies.

Scott, James R.↗