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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

On the Nature of Navier-stokes Turbulence

Several turbulent and nonturbulent solutions of the Navier-Stokes equations are obtained. The unaveraged equations are used numerically in conjunction with tools and concepts from nonlinear dynamics, including time series, phase portraits, Poincare sections, largest Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a low-Reynolds-number fully developed turbulence are compared. The solutions, separate exponentially with time, having a positive Liapunov exponent. Thus the turbulence is characterized as chaotic. In a search for solutions which contrast with the turbulent ones, the Reynolds number is reduced. Several qualitatively different flows are noted. These are, fully chaotic, complex period, weakly chaotic, simple periodic, and fixed-point. Of these, only the fully chaotic flows are classified as turbulent. Those flows have both a positive Liapunov exponent and Poincare sections without pattern. By contrast, the weakly chaotic flows have some pattern in their Poincare sections. The fixed-point and periodic flows are nonturbulent, since turbulence, is both time-dependent and aperiodic. Turbulent solutions are obtained in which energy cascades from large to small-scale motions. In general, the spectral energy transfer takes place between wavenumber bands that are considerably separated. The special transfer can occur either as a result of nonlinear turbulence self-interaction or by interaction of turbulence with mean gradients. Turbulent systems are compared with those studied in kinetic theory. The two types of systems are fundamentally different (continuous and dissipative as opposed to discrete and conservative), but there are similarities. For instance, both are nonlinear and show sensitive dependence on initial conditions. Also, the turbulent and molecular stress tensors are identical if the macroscopic velocities for the turbulent stress are replaced by molecular velocities.

Deissler, Robert G.↗
DOE OSTI2015

Impulse position control algorithms for nonlinear systems

The article is devoted to the formalization and description of impulse-sliding regime in nonlinear dynamical systems that arise in the application of impulse position controls of a special kind. The concept of trajectory impulse-sliding regime formalized as some limiting network element Euler polygons generated by a discrete approximation of the impulse position control This paper differs from the previously published papers in that it uses a definition of solutions of systems with impulse controls, it based on the closure of the set of smooth solutions in the space of functions of bounded variation. The need for the study of such regimes is the fact that they often arise when parry disturbances acting on technical or economic control system.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Reduced MHD in Astrophysical Applications: Two-dimensional or Three-dimensional?

Originally proposed as an efficient approach to computation of nonlinear dynamics in tokamak fusion research devices, reduced magnetohydrodynamics (RMHD) has subsequently found application in studies of coronal heating, flux tube dynamics, charged particle transport, and, in general, as an approximation to describe plasma turbulence in space physics and astrophysics. Given the diverse set of derivations available in the literature, there has emerged some level of discussion and a lack of consensus regarding the completeness of RMHD as a turbulence model, and its applicability in contexts such as the solar wind. Some of the key issues in this discussion are examined here, emphasizing that RMHD is properly neither 2D nor fully 3D, being rather an incomplete representation that enforces at least one family of extraneous conservation laws.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

Numerical and Experimental Investigation of Turbulent Transport Control via Shaping of Radial Plasma Flow Profiles (Final Technical Report)

The strong coupling between the different physical variables involved in the plasma transport phenomenon and the high complexity of its dynamics call for a model-based, multivariable approach to profile control where those predictive models could be exploited. The overall objective of this project has been to extend the existing body of work by investigating numerically and experimentally active control of unstable fluctuations, including fully developed turbulence and the associated cross-field particle transport, via manipulation of flow profiles in a magnetized laboratory plasma device. Fluctuations and particle transport can be monitored by an array of electrostatic probes, and E x B flow profiles can be controlled via a set of biased concentric ring electrodes that terminate the plasma column. The goals of the proposed research have been threefold: i- to develop a predictive code to simulate plasma transport in the linear HELCAT (HELicon-CAThode) plasma device at the University of New Mexico (UNM), where the experimental component of the proposed research has been carried out; ii- to establish the feasibility of using advanced model-based control algorithms to control cross-field turbulence-driven particle transport through appropriate manipulation of radial plasma flow profiles, iii- to investigate the fundamental nonlinear dynamics of turbulence and transport physics. Lehigh University (LU), including Prof. Eugenio Schuster and one full-time graduate student, has been primarily responsible for control-oriented modeling and model-based control design. Undergraduate students have also participated in this project through the National Science Foundation Research Experience for Undergraduate (REU) program. The main goal of the LU Plasma Control Group has been to study the feasibility of controlling turbulence-driven transport by shaping the radial poloidal flow profile (i.e., by controlling flow shear) via biased concentric ring electrodes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

Stationary to nonstationary transition in crossed-field devices

The previous results based on numerical simulations showed that a cold electron beam injected in a crossed field gap does not reach a time independent stationary state in the space charge limited regime [P. J. Christenson and Y. Y. Lau, Phys. Plasmas 1, 3725 (1994)]. In this work, the effect of finite injection temperature in the transition from stationary to nonstationary states is investigated. A fully kinetic model for the electron flow is derived and used to determine the possible stationary states of the system. It is found that although there is always a stationary solution for any set of parameters, depending on the injection temperature the electron flow becomes very sensitive to fluctuations and the stationary state is never reached. By investigating the nonlinear dynamics of a characteristic electron, a theory based on a single free parameter is constructed to predict when the transition between stationary and nonstationary states occurs. In agreement with the previous numerical results, the theory indicates that for vanishing temperatures the system never reaches the time independent stationary state in the space charge limited regime. Nevertheless, as the injection temperature is raised it is found a broad range of system parameters for which the stationary state is indeed attained. By properly adjusting the free parameter in the theory, one can be able to describe, to a very good accuracy, when the transition occurs.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2013

Electrons trajectories around a bubble regime in intense laser plasma interaction

Some typical electrons trajectories around a bubble regime in intense laser plasma interaction are investigated theoretically. By considering a modification of the fields and ellipsoid bubble shape due to the presence of residual electrons in the bubble regime, we study in detail the electrons nonlinear dynamics with or without laser pulse. To examine the electron dynamical behaviors, a set of typical electrons, which locate initially at the front of the bubble, on the transverse edge and at the bottom of the bubble respectively, are chosen for study. It is found that the range of trapped electrons in the case with laser pulse is a little narrower than that without laser pulse. The partial phase portraits for electrons around the bubble are presented numerically and their characteristic behaviors are discussed theoretically. Implication of our results on the high quality electron beam generation is also discussed briefly.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2015

POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation

This work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order solution is that reduced order Karush–Kuhn–Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based are developed for proper orthogonal decomposition (POD) ROM data assimilation using both Galerkin and Petrov–Galerkin projections. For the first time POD, tensorial POD, and discrete empirical interpolation method (DEIM) are employed to develop reduced data assimilation systems for a geophysical flow model, namely, the two dimensional shallow water equations. Numerical experiments confirm the theoretical framework for Galerkin projection. In the case of Petrov–Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Dynamical systems and nonlinear transient rheology of the far-from-equilibrium Bjorken flow

In relativistic kinetic theory, the one-particle distribution function is approximated by an asymptotic perturbative power series in the Knudsen number which is divergent. For the Bjorken flow, we expand the distribution function in terms of its moments and study their nonlinear evolution equations. The resulting coupled dynamical system can be solved for each moment consistently using a multiparameter transseries which makes the constitutive relations inherit the same structure. A new nonperturbative dynamical renormalization scheme is born out of this formalism that goes beyond the linear response theory.We show that there is a Lyapunov function, also known as dynamical potential, which is, in general, a function of the moments and time satisfying Lyapunov stability conditions along renormalization group flows connected to the asymptotic hydrodynamic fixed point. As a result, the transport coefficients get dynamically renormalized at every order in the time-dependent perturbative expansion by receiving nonperturbative corrections present in the transseries. The connection between the integration constants and the UV data is discussed using the language of dynamical systems. Furthermore, we show that the first dissipative correction in the Knudsen number to the distribution function is not only determined by the known effective shear viscous term but also a new high-energy nonhydrodynamic mode. It is demonstrated that the survival of this new mode is intrinsically related to the nonlinear mode-to-mode coupling with the shear viscous term. Finally, we comment on some possible phenomenological applications of the proposed nonhydrodynamic transport theory.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Non-autoregressive time-series methods for stable parametric reduced-order models

Advection-dominated dynamical systems, characterized by partial differential equations, are found in applications ranging from weather forecasting to engineering design where accuracy and robustness are crucial. There has been significant interest in the use of techniques borrowed from machine learning to reduce the computational expense and/or improve the accuracy of predictions for these systems. These rely on the identification of a basis that reduces the dimensionality of the problem and the subsequent use of time series and sequential learning methods to forecast the evolution of the reduced state. Often, however, machine-learned predictions after reduced-basis projection are plagued by issues of stability stemming from incomplete capture of multiscale processes as well as due to error growth for long forecast durations. To address these issues, we have developed a non-autoregressive time series approach for predicting linear reduced-basis time histories of forward models. In particular, we demonstrate that non-autoregressive counterparts of sequential learning methods such as long short-term memory (LSTM) considerably improve the stability of machine-learned reduced-order models. Further, we evaluate our approach on the inviscid shallow water equations and show that a non-autoregressive variant of the standard LSTM approach that is bidirectional in the principal component directions obtains the best accuracy for recreating the nonlinear dynamics of partial observations. Moreover-and critical for many applications of these surrogates-inference times are reduced by three orders of magnitude using our approach, compared with both the equation-based Galerkin projection method and the standard LSTM approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2012

MAGNETIC FIELD STRUCTURES TRIGGERING SOLAR FLARES AND CORONAL MASS EJECTIONS

Solar flares and coronal mass ejections, the most catastrophic eruptions in our solar system, have been known to affect terrestrial environments and infrastructure. However, because their triggering mechanism is still not sufficiently understood, our capacity to predict the occurrence of solar eruptions and to forecast space weather is substantially hindered. Even though various models have been proposed to determine the onset of solar eruptions, the types of magnetic structures capable of triggering these eruptions are still unclear. In this study, we solved this problem by systematically surveying the nonlinear dynamics caused by a wide variety of magnetic structures in terms of three-dimensional magnetohydrodynamic simulations. As a result, we determined that two different types of small magnetic structures favor the onset of solar eruptions. These structures, which should appear near the magnetic polarity inversion line (PIL), include magnetic fluxes reversed to the potential component or the nonpotential component of major field on the PIL. In addition, we analyzed two large flares, the X-class flare on 2006 December 13 and the M-class flare on 2011 February 13, using imaging data provided by the Hinode satellite, and we demonstrated that they conform to the simulation predictions. These results suggest that forecasting of solar eruptions is possible with sophisticated observation of a solar magnetic field, although the lead time must be limited by the timescale of changes in the small magnetic structures.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

A simple, stable, and accurate linear tetrahedral finite element for transient, nearly, and fully incompressible solid dynamics: A dynamic variational multiscale approach [A simple, stable, and accurate tetrahedral finite element for transient, nearly incompressible, linear and nonlinear elasticity: A dynamic variational multiscale approach]

Here, we propose a new approach for the stabilization of linear tetrahedral finite elements in the case of nearly incompressible transient solid dynamics computations. Our method is based on a mixed formulation, in which the momentum equation is complemented by a rate equation for the evolution of the pressure field, approximated with piece-wise linear, continuous finite element functions. The pressure equation is stabilized to prevent spurious pressure oscillations in computations. Incidentally, it is also shown that many stabilized methods previously developed for the static case do not generalize easily to transient dynamics. Extensive tests in the context of linear and nonlinear elasticity are used to corroborate the claim that the proposed method is robust, stable, and accurate.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2013

On the propagation of hypersonic solitons in a strained paramagnetic crystal

Nonlinear dynamics of a subnanosecond transverse elastic pulse in a low-temperature paramagnetic crystal placed into a magnetic field and statically strained in the same direction is investigated. Paramagnetic impurities implanted into the crystal have an effective spin of 3/2, and the pulse propagates at right angles to the magnetic field. In the general case, the structure of the pulse is such that the approximation of slowly varying envelopes, which is standard for quasi-monochromatic signals, is inapplicable. Under certain conditions, the pulse propagation in the 1D case is described by the Konno-Kameyama-Sanuki integrable wave equation for strain, which is transformed into the Hirota equation for the envelope of the given strain in the quasi-monochromatic limit. The effect of transverse perturbations on extremely short and quasi-monochromatic solitons is studied in detail. The conditions and features of self-focusing and defocusing of acoustic solitons in the form of extremely short pulses and envelope solitons are revealed. The propagation of an extremely short “half-wave” hypersonic pulse in the “acoustic bullet” regime in the medium with a quasiequilibrium population of quantum sublevels of effective spins is predicted.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Fault Detection and Diagnosis In Hall–Héroult Cells Based on Individual Anode Current Measurements Using Dynamic Kernel PCA

Individual anode current signals in aluminum reduction cells provide localized cell conditions in the vicinity of each anode, which contain more information than the conventionally measured cell voltage and line current. One common use of this measurement is to identify process faults that can cause significant changes in the anode current signals. While this method is simple and direct, it ignores the interactions between anode currents and other important process variables. This paper presents an approach that applies multivariate statistical analysis techniques to individual anode currents and other process operating data, for the detection and diagnosis of local process abnormalities in aluminum reduction cells. Specifically, since the Hall–Héroult process is time-varying with its process variables dynamically and nonlinearly correlated, dynamic kernel principal component analysis with moving windows is used. The cell is discretized into a number of subsystems, with each subsystem representing one anode and cell conditions in its vicinity. The fault associated with each subsystem is identified based on multivariate statistical control charts. The results show that the proposed approach is able to not only effectively pinpoint the problematic areas in the cell, but also assess the effect of the fault on different parts of the cell.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Simulation-based characterization of the variability of earthquake risk to buildings in the near-field

Recent advancements in high performance computing platforms and computational workflow for regional-scale simulations are enabling unprecedented modeling of fault-to-structure earthquake processes. Regional simulations resolving ground motions at frequencies relevant to engineered systems are becoming computationally viable and provide a new capability to improve understanding of the geographical distribution and intensity of risk to buildings and critical infrastructure. As computational capabilities advance, it is essential to move beyond illustrative single rupture realizations for scenario earthquake events towards the development of a full suite of rupture realizations that appropriately characterize the range of risk to building systems. The work described in this article investigates the application of a suite of fault rupture realizations with the objective of assessing near-fault, site-specific seismic demand variability for building structures. A representative high-performance regional-scale computational model is utilized to execute ground motion and building response simulations based on 18 kinematic rupture realizations of an M7 strike-slip scenario earthquake. The fault rupture models for the scenario earthquake are created by systematically perturbing the hypocenter location and stochastically generating rupture parameters (slip, rise time, rake angle) to represent a breadth of ground motion intensities resulting from the spatial and temporal variabilities of an earthquake rupture process. The resulting seismic demand variability for three-story (short period) and forty-story (long period) steel moment-resisting frame buildings is characterized in terms of the median and distribution of peak inter-story drift ratio for a range of near-fault sites. The full suite of 18 fault rupture realizations and approximately 280,000 nonlinear dynamic building simulations indicate that the three-story building undergoes higher median seismic demand and significantly greater variability of demand at a given site than the forty-story building, which has important implications for the level of certainty in predicting building performance during an earthquake. The simulations performed provide deeper insight into the relationship between fault rupture parameterization and building response, which is essential information for developing a representative suite of rupture realizations for specific earthquake scenarios.

58 GEOSCIENCES↗
DOE OSTI2024

A large deformation multiphase continuum mechanics model for shock loading of soft porous materials

A large deformation, coupled finite-element (FE) model is developed to simulate the multiphase response of soft porous materials subjected to high strain-rate loading. The approach is based on the theory of porous media (TPM) at large deformations. Simplifications to the one-dimensional regime studied in the numerical simulations follow. An overview of several different time integration schemes is presented for the purpose of solving the nonlinear dynamic coupled balance of momenta (mixture and fluid) and balance of mass of the mixture equations. Numerical examples are presented for (i) verification against closed-form analytical solutions assuming small loads, (ii) demonstrating large deformation effects at high strain-rate, and (iii) showing differences in deformations between a single-phase elastodynamics model with occluded compressible pore fluid and a multiphase poroelastodynamics model at high strain-rate. The multiphase model shows that the relative motion of the pore fluid significantly dampens the deformation response of the solid skeleton as compared to the single-phase model, and makes it possible to extract quantitative values for the stresses of the different constituents, thereby allowing one to form preliminary conclusions about the onset of damage in the solid skeleton. The novelty of the current work is developing a multiphase, large deformation, mixture theory numerical model for high strain-rate loading of soft porous materials. It was discovered that explicit, adaptive time-stepping Runge–Kutta schemes offer high accuracy at relatively low cost when compared to traditional implicit or explicit central difference time-stepping schemes for shock-like loadings. Here, shock viscosity is added to the mixture momentum balance equation to regularize the shock front, and a stabilization term is added to the mixture mass balance equation to stabilize equal order interpolation finite elements for the coupled finite element solution of multiphase materials.

Engineering↗
DOE OSTI2018

An efficient, partitioned ensemble algorithm for simulating ensembles of evolutionary MHD flows at low magnetic Reynolds number

Studying the propagation of uncertainties in a nonlinear dynamical system usually involves generating a set of samples in the stochastic parameter space and then repeated simulations with different sampled parameters. The main difficulty faced in the process is the excessive computational cost. In this paper, we present an efficient, partitioned ensemble algorithm to determine multiple realizations of a reduced Magnetohydrodynamics (MHD) system, which models MHD flows at low magnetic Reynolds number. The algorithm decouples the fully coupled problem into two smaller subphysics problems, which reduces the size of the linear systems that to be solved and allows the use of optimized codes for each subphysics problem. Moreover, the resulting coefficient matrices are the same for all realizations at each time step, which allows faster computation of all realizations and significant savings in computational cost. We prove this algorithm is first order accurate and long time stable under a time step condition. Numerical examples are provided to verify the theoretical results and demonstrate the efficiency of the algorithm.

Jiang, Nan↗
DOE OSTI2018

{ital N}* Structure and Strong QCD

In attempting to match QCD with Nature, it is necessary to confront the many complexities of strong, nonlinear dynamics in relativistic quantum field theory, e.g. the loss of particle number conservation, the frame and scale dependence of the explanations and interpretations of observable processes, and the evolving character of the relevant degrees-of-freedom. The peculiarities of QCD ensure that it is also the only known fundamental theory with the capacity to sustain massless elementary degrees-of-freedom, gluons and quarks; and yet gluons and quarks are predicted to acquire mass dynamically so that the only massless systems in QCD are its composite Nambu-Goldstone bosons. All other everyday bound states possess nuclear-size masses, far in excess of anything that can directly be tied to the Higgs boson. These observations highlight fundamental questions within the Standard Model: what is the source of the mass for the vast bulk of visible matter in the Universe, how is its appearance connected with confinement; how is this mass distributed within hadrons and does the distribution differ from one hadron to another? This contribution sketches insights drawn using modern methods for the continuum bound-state problem in QCD, and how they have been informed by empirical information on the hadron spectrum and nucleon-to-resonance transition form factors.

Roberts, Craig D.↗
DOE OSTI2022

Conservative Numerical Schemes with Optimal Dispersive Wave Relations: Part II. Numerical Evaluations

A new energy and enstrophy conserving scheme (EEC) for the shallow water equations is proposed and evaluated using a suite of test cases over the global spherical or bounded domain. The evaluation is organized around a set of pre-defined properties: accuracy of individual operators, accuracy of the whole scheme, conservation of key quantities, control of the divergence variable, representation of the energy and enstrophy spectra, and simulation of nonlinear dynamics. The results confirm that the scheme is between the first and second order accurate, and conserves the total energy and potential enstrophy up to the time truncation errors. Here, the scheme is capable of producing more physically realistic energy and enstrophy spectra, indicating that it can help prevent the unphysical energy cascade towards the finest resolvable scales. With an optimal representation of the dispersive wave relations, the scheme is able to keep the flow close to being non-divergent, and maintain the geostrophically balanced structures with large-scale geophysical flows over long-term simulations.

54 ENVIRONMENTAL SCIENCES↗