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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.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Edge ML for CAN bus intrusion detection in AVs

Autonomous Vehicles (AVs) are revolutionizing transportation, but their reliance on interconnected cyber-physical systems exposes them to unprecedented cybersecurity risks. This study addresses the critical challenge of detecting real-time cyber intrusions in self-driving vehicles by leveraging a dataset from the Udacity self-driving car project. We simulate four high-impact attack vectors, Denial of Service (DoS), spoofing, replay, and fuzzy attacks, by injecting noise into spatial features (e.g., bounding box coordinates) to replicate adversarial scenarios. We develop and evaluate two lightweight neural network architectures (NN-1 and NN-2) alongside a logistic regression baseline (LG-1) for intrusion detection. The models achieve exceptional performance, with NN-2 attaining an AUC score of 93.15% and 93.15% accuracy, demonstrating their suitability for edge deployment in AV environments. Through explainable AI techniques, we uncover unique forensic fingerprints of each attack type, such as spatial corruption in fuzzy attacks and temporal anomalies in replay attacks, offering actionable insights for feature engineering and proactive defense. Visual analytics, including confusion matrices, ROC curves, and feature importance plots, validate the models' robustness and interpretability. This research sets a new benchmark for AV cybersecurity, delivering a scalable, field-ready toolkit for Original Equipment Manufacturers (OEMs) and policymakers. By aligning intrusion fingerprints with SAE J3061 automotive security standards, we provide a pathway for integrating machine learning into safety-critical AV systems. Our findings underscore the urgent need for security-by-design AI, ensuring that AVs not only drive autonomously but also defend autonomously. This work bridges the gap between theoretical cybersecurity and life-preserving engineering, offering a leap toward safer, more secure autonomous transportation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Automated Classification of Vehicle Movements at Signalized Intersections Using Vehicle Trajectories

Accurate vehicle movement classification through signalized intersections is of paramount importance to the analysis of intersection performance and the optimization of traffic control strategies. Conventional techniques for tracking vehicle turning movements depend on infrastructure-based strategies like human counts, loop detectors, and video analytics, all of which are costly, prone to errors, and spatially constrained. High-frequency trajectory data can be utilized to determine vehicle movement patterns in a scalable and infrastructure-independent method due to the adoption of connected vehicles (CVs). In recent years, several studies have utilized connected vehicle data to generate performance measures. Most of the trajectory-based performance measures approaches, however, require map matching-i.e., extracting geospatial references from maps to identify the movements that individual vehicles make at a signalized intersection. These approaches are often time-consuming and hinder scalability since geographic features need to be provided for an analysis to be conducted. Map matching methods are prone to errors as different map versions change these geographic features. This research presents a novel automatic classification pipeline that uses CV trajectory data to classify vehicle movements at signalized crossings, specifically pass-through left-turn and right-turn maneuvers. The process starts by filtering trips that cross a spatial bounding box that has been defined at the target intersection. Approach and departure headings for each trajectory crossing the boundary are computed and are clustered together to identify dominant movements. The proposed algorithm is used to classify the movement of vehicles at 10 intersections in the state of California, and the results indicate that the algorithm can classify movements at these intersections with varying traffic volumes and road network configurations, all in a map-less framework with no need for conflation of vehicle trajectories to a digital base map.

24 POWER TRANSMISSION AND DISTRIBUTION↗
NASA NTRS2008

Advanced Unstructured Grid Generation for Complex Aerodynamic Applications

A new approach for distribution of grid points on the surface and in the volume has been developed and implemented in the NASA unstructured grid generation code VGRID. In addition to the point and line sources of prior work, the new approach utilizes surface and volume sources for automatic curvature-based grid sizing and convenient point distribution in the volume. A new exponential growth function produces smoother and more efficient grids and provides superior control over distribution of grid points in the field. All types of sources support anisotropic grid stretching which not only improves the grid economy but also provides more accurate solutions for certain aerodynamic applications. The new approach does not require a three-dimensional background grid as in the previous methods. Instead, it makes use of an efficient bounding-box auxiliary medium for storing grid parameters defined by surface sources. The new approach is less memory-intensive and more efficient computationally. The grids generated with the new method either eliminate the need for adaptive grid refinement for certain class of problems or provide high quality initial grids that would enhance the performance of many adaptation methods.

Pirzadeh, Shahyar Z.↗
NASA NTRS2010

Advanced Unstructured Grid Generation for Complex Aerodynamic Applications

A new approach for distribution of grid points on the surface and in the volume has been developed. In addition to the point and line sources of prior work, the new approach utilizes surface and volume sources for automatic curvature-based grid sizing and convenient point distribution in the volume. A new exponential growth function produces smoother and more efficient grids and provides superior control over distribution of grid points in the field. All types of sources support anisotropic grid stretching which not only improves the grid economy but also provides more accurate solutions for certain aerodynamic applications. The new approach does not require a three-dimensional background grid as in the previous methods. Instead, it makes use of an efficient bounding-box auxiliary medium for storing grid parameters defined by surface sources. The new approach is less memory-intensive and more efficient computationally. The grids generated with the new method either eliminate the need for adaptive grid refinement for certain class of problems or provide high quality initial grids that would enhance the performance of many adaptation methods.

Pirzadeh, Shahyar↗
NASA NTRS2011

Land and Atmosphere Near-Real-Time Capability for Earth Observing System

The past decade has seen a rapid increase in availability and usage of near-real-time data from satellite sensors. The EOSDIS (Earth Observing System Data and Information System) was not originally designed to provide data with sufficiently low latency to satisfy the requirements for near-real-time users. The EOS (Earth Observing System) instruments aboard the Terra, Aqua and Aura satellites make global measurements daily, which are processed into higher-level 'standard' products within 8-40 hours of observation and then made available to users, primarily earth science researchers. However, applications users, operational agencies, and even researchers desire EOS products in near-real-time to support research and applications, including numerical weather and climate prediction and forecasting, monitoring of natural hazards, ecological/invasive species, agriculture, air quality, disaster relief and homeland security. These users often need data much sooner than routine science processing allows, usually within 3 hours, and are willing to trade science product quality for timely access. While Direct Broadcast provides more timely access to data, it does not provide global coverage. In 2002, a joint initiative between NASA (National Aeronautics and Space Administration), NOAA (National Oceanic and Atmospheric Administration), and the DOD (Department of Defense) was undertaken to provide data from EOS instruments in near-real-time. The NRTPE (Near Real Time Processing Effort) provided products within 3 hours of observation on a best-effort basis. As the popularity of these near-real-time products and applications grew, multiple near-real-time systems began to spring up such as the Rapid Response System. In recognizing the dependence of customers on this data and the need for highly reliable and timely data access, NASA's Earth Science Division sponsored the Earth Science Data and Information System Project (ESDIS)-led development of a new near-real-time system called LANCE (Land, Atmosphere Near-Real-Time Capability for EOS) in 2009. LANCE consists of special processing elements, co-located with selected EOSDIS data centers and processing facilities. A primary goal of LANCE is to bring multiple near-real-time systems under one umbrella, offering commonality in data access, quality control, and latency. LANCE now processes and distributes data from the Moderate Resolution Imaging Spectroradiometer (MODIS), Atmospheric Infrared Sounder (AIRS), Advanced Microwave Scanning Radiometer Earth Observing System (AMSR-E), Microwave Limb Sounder (MLS) and Ozone Monitoring Instrument (OMI) instruments within 3 hours of satellite observation. The Rapid Response System and the Fire Information for Resource Management System (FIRMS) capabilities will be incorporated into LANCE in 2011. LANCE maintains a central website to facilitate easy access to data and user services. LANCE products are extensively tested and compared with science products before being made available to users. Each element also plans to implement redundant network, power and server infrastructure to ensure high availability of data and services. Through the user registration system, users are informed of any data outages and when new products or services will be available for access. Building on a significant investment by NASA in developing science algorithms and products, LANCE creates products that have a demonstrated utility for applications requiring near-real-time data. From lower level data products such as calibrated geolocated radiances to higher-level products such as sea ice extent, snow cover, and cloud cover, users have integrated LANCE data into forecast models and decision support systems. The table above shows the current near-real-time product categories by instrument. The ESDIS Project continues to improve the LANCE system and use the experience gained through practice to seek adjustments to improve the quality and performance of the system. For example, anGC-compliant Web Map Service (WMS) will be added shortly that will allow users to download geo-referenced MODIS images for arbitrary bounding boxes. Further, an OGC-compliant Web Coverage Service (WCS) will be added later this year that will expedite user access to arbitrary data subsets or re-formatted products. AIRS images are now served through WMS and available in multiple formats (PNG, GeoTIFF, KMZ). NASA has established a LANCE User Working Group to steer the development of the system and create a forum for sharing ideas and experiences that are expected to further improve the LANCE capabilities. The LANCE system has proved a success by satisfying the growing needs of the applications and operational communities for land and atmosphere data in near-real-time. NASA's Earth Sciences Division was able to leverage existing science research capabilities to provide the near-real-time community with products and imagery that support monitoring of disasters in a timely manner.

Murphy, Kevin J.↗
NASA NTRS2023

Parameterization of Features on Spherical Surfaces

NASA’s Planetary Systems Laboratory (PSL) at the Goddard Spaceflight Center in Greenbelt, Maryland is tasked with the study of solar system objects. This work is frequently based on analysis of imagery from visiting spacecraft, orbiting telescopes, and ground-based sensors. A consequence of this mission is the need to characterize features on the spherical surface of planets and moons, such characterization including the overall area of the feature, its center, its moments of inertia, its perimeter, its compactness, its major axis, its bounding box, its aspect ratio and its North-South orientation. Accurate assessment of these measures from a “flat” two-dimensional image is a challenge, since the scale of distance observed at the center of a spherical object distorts in any direction towards the horizon. PSL has developed a technique to guarantee accurate and automated measurement at any point on the visible surface, by overlaying an imaginary grid of equal-area cells onto the two-dimensional image of the sphere. The technique is named Grid-Oriented Normalization for Analysis of Spherical Areas (GONASA). This paper discusses the construction of the normalization grid and describes specific algorithms for feature parameterization. The algorithms are implemented in an accompanying Excel spreadsheet, both for clarity and ease of adoption, and to emphasize their aptness for automation. Finally, a concrete use case is offered where the GONASA grid and algorithms are used to characterize methane clouds on the Saturnian moon Titan.

Planetary science↗
NASA NTRS2022

A Comparative Evaluation of Snowflake Particle Shape Estimation Techniques Used By the Precipitation Imaging Package (PIP), Multi-Angle Snowflake Camera (MASC), and Two-Dimensional Video Disdrometer (2DVD)

Measurements of snowflake particle shape are important for studying snow microphysics. While a number of instruments exist that are capable of measuring particle shape, this study focuses on the measurement techniques of three digital video disdrometers: the Precipitation Imaging Package (PIP), the Multi-Angle Snowflake Camera (MASC), and the Two-Dimensional Video Disdrometer (2DVD). To gain a better understanding of the relative strengths and weaknesses of these instruments and to provide a foundation upon which comparisons can be made between studies using data from different instruments, we perform a comparative analysis of the shape measurement algorithms employed by each of the three instruments by applying the algorithms to snowflake images captured by PIP during the ICE-POP 2018 field campaign. Our analysis primarily focuses on the measurement of the aspect ratio of either the particle itself, in the case of PIP and MASC, or of the particle bounding box, in the case of PIP and 2DVD. Both PIP and MASC use shape-fitting algorithms to measure aspect ratio. While our analysis of the MASC aspect ratio suggests that the measurements are reliable, our findings indicate that both the ellipse and rectangle aspect ratios produced by PIP underperformed considerably due to the shortcomings of the PIP shape-fitting techniques. We also demonstrate that reliable measurements of aspect ratio can be retrieved from PIP by reprocessing the raw PIP images using either the MASC ellipse-fitting algorithm or a tensor-based ellipse-fitting algorithm. Because of differences in instrument design, 2DVD produces measurements of particle horizontal and vertical extent rather than length and width. Furthermore, the 2DVD measurements of particle horizontal extent can be contaminated by horizontal particle motion. Our findings indicate that, although the correction technique used to remove the horizontal motion contamination performs remarkably well with snowflakes despite being designed for use with raindrops, the 2DVD measurements of particle horizontal extent are less reliable than those measured by PIP.

Charles N Helms↗
NASA NTRS2023

Object Detection and Pose Estimation Public Competition Sample Data

This data was created with open-source tooling for the purpose of running a public competition. This dataset is a sample of what can be generated for the commissioned competition runner to be able to set up their testing infrastructure. It contains images of publicly available spacecraft models in a Blender scene composed of a light source, a background image, and the spacecraft model with bounding box labels.

James Berck↗
DOE OSTI2021

Transparent Object Tracking Benchmark

Visual tracking has achieved considerable progress in recent years. However, current research in the field mainly focuses on tracking of opaque objects, while little attention is paid to transparent object tracking. In this paper, we make the first attempt in exploring this problem by proposing a Transparent Object Tracking Benchmark (TOTB). Specifically, TOTB consists of 225 videos (86K frames) from 15 diverse transparent object categories. Each sequence is manually labeled with axis-aligned bounding boxes. To the best of our knowledge, TOTB is the first benchmark dedicated to transparent object tracking. In order to understand how existing trackers perform and to provide comparison for future research on TOTB, we extensively evaluate 25 state-of-the-art tracking algorithms. The evaluation results exhibit that more efforts are needed to improve transparent object tracking. Besides, we observe some nontrivial findings from the evaluation that are discrepant with some common beliefs in opaque object tracking. For example, we find that deeper features are not always good for improvements. Moreover, to encourage future research, we introduce a novel tracker, named TransATOM, which leverages transparency features for tracking and surpasses all 25 evaluated approaches by a large margin. By releasing TOTB, we expect to facilitate future research and application of transparent object tracking in both the academia and industry.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Neutrino Interaction Identification for the DUNE Trigger

The DUNE will be a long baseline neutrino oscillation experiment using a high purity muon neutrino beam and near detector, both located at the FNAL, and a far detector hosted 1300 km downstream at the SURF. The 10 kt fiducial mass of LAr will allow DUNE to have a rich off-beam neutrino physics programme, including the study of neutrino signals from core collapse supernovae. The SP DUNE FD module will read out ionisation data at a rate of 1.2 TB\textsuperscript{-1} whilst only a total data volume of 30 PB per year can be permanently stored. DUNE will make use of FPGA resources in the front-end of the DAQ as part of the necessitated triggering system. This thesis presents a validation study of the FPGA-based TPG in the front-end DAQ using data collected by the ProtoDUNE experiment hosted at the CERN. The FPGA-based TPG was utilised as the first stage of a baseline SNB trigger whose performance was evaluated using simulated neutrino interactions for a $11.2 m_\odot$ progenitor star. The efficiency of the baseline SNB trigger was determined to have a lower limit of $97.7\substack{+ 0.2\\ -0.3}\%$ for supernovae at a distance of 20kpc, achieving the technical requirements set out for DUNE. To improve the performance of the SNB trigger at greater SNB distances, the use of a bounding box proposal network, YOLOv3, was explored. This was found to improve the efficiency of the SNB trigger to 100\% up to the far side of the Milky Way galaxy and to $92.5\substack{+ 0.5\\ -0.5}\%$ at the Large Magellanic Cloud.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

A Review on the State of the Art of Machine Learning and Satellite Imaging: Detecting Scene Changes in Selected Nuclear Fuel Cycle Datasets

The timely detection of clandestine nuclear facilities is one of the greatest challenges faced by the International Atomic Energy Agency’s (IAEA). Idaho National Laboratory is currently applying machine learning (ML) to existing satellite imagery (SI) datasets to find facilities within the nuclear fuel life cycle, with primary focus placed on identifying critical predecessor (i.e., fuel fabrication and fuel enrichment) and successor (i.e., nuclear power plants) facilities. This could provide a satellite image methodology that the IAEA could leverage to discover clandestine facilities. The work presented in this paper describes the evolution of a workflow developed by this team for object detection related to critical infrastructure by expanding that workflow for the purpose of identifying nuclear fuel cycle components and automating dependency assessments. This will be done using two methods housed within a single pipeline. The first method involves implementing a DenseNet161 convolutional neural network to classify the images and explain the results using Local Interpretable Model-Agnostic Explanations (LIME). The second method implements You Only Look Once version 5 (YoloV5), to detect objects within images, provide a probability for the detection, and provide a bounding box that corresponds to the object of interest. The results of this work are anticipated to provide a clear picture of this portion of the nuclear fuel cycle and perform as a stand-alone tool for image assessment that can be expanded to additional fuel cycle components and implemented in international safeguards and national security domains. This capability addresses the IAEA’s need to detect undeclared nuclear materials and activities within a state while encompassing the entire nuclear fuel cycle.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Interactive Rotated Object Detection for Novel Class Detection in Remotely Sensed Imagery

In this paper we propose IRTR-DETR an Interactive and Real-Time Rotated DEtection TRansformer that extends IRTDETR to predict rotated bounding boxes. IRTR-DETR maintains the Human-In-The-Loop (HIL) workflow of IRTDETR but introduces rotation-aware heads for improved detection of objects with arbitrary orientations. Similarly to IRTDETR IRTR-DETR can be trained with a small labeled sample set in an interactive setting but we show that it can also be pretrained on related but not identical data--such as a building damage dataset--before being applied to tasks like identifying buildings under construction. We demonstrate the efficacy of our approach on the publicly available Tiny-DOTA and xBD dataset as well as two study-cases on proprietary datasets of greenhouses and houses under construction ("waffle homes"). Detecting greenhouses is highly relevant in the context of damage assessment while "waffle homes" aid understanding typical floorplans and building codes in different areas both thereby supporting population modeling emergency response and policy planning. Our method outperforms the state of the art in interactive rotated object detection on the Tiny-DOTA dataset by 5.7 percent and improves upon the non interactive RTDETR by 7.85 to 19.39 percent (depending on the number of provided samples) while maintaining its real-time efficiency.

Burges, Marvin [ORNL] (ORCID:0000000312690769)↗
DOE OSTI2024

Light Detecting and Ranging (LIDAR) for in-situ heliostat optical error assessment

A system and method for optical assessment of a heliostat includes obtaining a point cloud data representing an image of the heliostat; isolating the data; filtering and fitting the filtered heliostat data to a bounding box; translating the heliostat data to a plane to aid in segmentation; segmenting a plurality of facets of the heliostat fitting each of the segmented facets to a respective plane; generating normal vectors characterizing each of the plurality of facets; and calculating a canting angle associated with each respective facet of the plurality of facets. A heliostat with mirrored facets and a scanner are provided. The scanner captures point cloud data representing the heliostat, which is segmented for each facet. Normal vectors characterize the facets and a canting angle is calculated for the respective facet.

Small, Daniel E.↗
DOE OSTI2025

Active Learning Meets Foundation Models: Fast Remote Sensing Data Annotation for Object Detection

Object detection in remote sensing demands extensive, high-quality annotations—a process that is both labor-intensive and time-consuming. In this work, we introduce a real-time active learning and semi-automated labeling framework that leverages foundation models to streamline dataset annotation for object detection in remote sensing imagery. For example, by integrating a Segment Anything Model (SAM), our approach generates mask-based bounding boxes that serve as the basis for dual sampling: (a) uncertainty estimation to pinpoint challenging samples, and (b) diversity assessment to ensure broad data coverage. Furthermore, our Dynamic Box Switching Module (DBS) addresses the well-known cold start problem for object detection models by replacing its suboptimal initial predictions with SAM-derived masks, thereby enhancing early-stage localization accuracy. Extensive evaluations on multiple remote sensing datasets plus a real-world user study, demonstrate that our framework not only reduces annotation effort, but also significantly boosts detection performance compared to traditional active learning sampling methods. The code for training and the user interface will be made available.

Burges, Marvin [ORNL] (ORCID:0000000312690769)↗
DOE OSTI2023

Charged-Particle Bound States in Periodic Boxes

We consider the binding energy of a two-body system with a repulsive Coulomb interaction in a finite periodic volume. We define the finite-volume Coulomb potential as the usual Coulomb potential, except that the distance is defined the shortest separation between the two bodies in the periodic volume. We investigate this problem in one and three-dimensional periodic boxes and derive the asymptotic behavior of the volume dependence for bound states with zero angular momentum in terms of Whittaker functions. We benchmark our results against numerical calculations and show how the method can be used to extract asymptotic normalization coefficients for charged-particle bound states. Furthermore, the results we derive here have immediate applications for calculations of atomic nuclei in finite periodic volumes for the case where the leading finite-volume correction is associated with two charged clusters.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS2019

Bounding Methods for Heterogeneous Lidar-derived Navigational Geofences

Safe Unmanned Aerial Vehicle (UAV) operations near the ground require navigation methods that avoid fixed obstacles such as buildings, power lines and trees. Aerial lidar surveys of ground structures are available with the precision and accuracy to geolocate obstacles, but the high volume of raw survey data can exceed the compute power of onboard processors and the rendering ability of ground-based flight planning maps. Representing ground structures with bounding polyhedra instead of point clouds greatly reduces the data size and can enable effective obstacle avoidance, as long as the bounding geometry envelopes the structures with high spatial fidelity. This report describes in detail four methods to compute bounding geometries of ground obstacles from lidar point clouds. The four methods are: 1) 2.5D Maximum Elevation Box, 2) 2.5D Ground Map Extrusion, 3) 3D Bounding Cylinder, and 4) 3D Bounding Box. The methods are applied to five point cloud datasets from lidar surveys of UAV flight research sites in Georgia and Virginia with an average point spacing that ranges from 0.1m to 0.6m. The methods are assessed using survey areas with geometrically heterogeneous ground structures: buildings, vegetation, power lines, and sub-meter structures such as road signs and guy wires. The 2.5D Maximum Elevation Box method is useful for simple structures. The 2.5D Ground Map Extrusion method efficiently encloses vegetation, but requires handdrawn ground footprints. The 3D Bounding Cylinder method excels at enclosing linear structures such as power lines and fences. The 3D Bounding Box method excels at enclosing planar structures such as buildings. The methods are compared on the basis of data compression and boundary fidelity on selected areas. The 2.5D methods yield the highest data compression but the polyhedra produced by them enclose significant amounts of empty space. Boundary fidelity is superior for the 3D methods, though this fidelity comes at the cost of a roughly thirtyfold lower data compression ratio than the 2.5D Maximum Elevation Box method. A mix of these output geometries is proposed for autonomous UAV navigation with limited on-board computing. Both the accuracy and spatial detail of emerging satellite-based survey technology lower than that of aerial lidar scanning survey technology. Sub-meter structures and thin linear structures are not reliably mapped at present by satellite-based surveys.

Moore, Andrew J.↗
NASA NTRS2003

Time as an Observable in Nonrelativistic Quantum Mechanics

The argument follows from the viewpoint that quantum mechanics is taken not in the usual form involving vectors and linear operators in Hilbert spaces, but as a boundary value problem for a special class of partial differential equations-in the present work, the nonrelativistic Schrodinger equation for motion of a structureless particle in four- dimensional space-time in the presence of a potential energy distribution that can be time-as well as space-dependent. The domain of interest is taken to be one of two semi-infinite boxes, one bounded by two t=constant planes and the other by two t=constant planes. Each gives rise to a characteristic boundary value problem: one in which the initial, input values on one t=constant wall are given, with zero asymptotic wavefunction values in all spatial directions, the output being the values on the second t=constant wall; the second with certain input values given on both z=constant walls, with zero asymptotic values in all directions involving time and the other spatial coordinates, the output being the complementary values on the z=constant walls. The first problem corresponds to ordinary quantum mechanics; the second, to a fully time-dependent version of a problem normally considered only for the steady state (time-independent Schrodinger equation). The second problem is formulated in detail. A conserved indefinite metric is associated with space-like propagation, where the sign of the norm of a unidirectional state corresponds to its spatial direction of travel.

Hahne, G. E.↗
NASA NTRS1993

Transition to turbulence in an elliptic vortex

We study the three dimensional instability and nonlinear growth of the two dimensional flow described by the stream function Psi = (A sin b1 x sin b2 y)/(b1(exp 2) + b2(exp 2)) where b1 = pi/L1, b2 = pi/L2. This is a swirling flow in a box which is bounded by 0 less than x less than L1, 0 less than y less than L2 and is infinite in the z direction. This flow is a solution of the Navier-Stokes equation with A = exp(-v(b1(exp 2) + b2(exp 2))t) which slowly decays. We seek a viscous solution which starts near this one and slips along but does not penetrate the bounding walls. The vorticity of the basic flow is w(sub z) = A sin b1 x sin b2 y which has maximum value A at the center of the box and drops to zero at the boundaries. We can think of the resulting flow as that of a captive vortex.

Lundgren, Thomas S.↗