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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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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2011

Kummel Disease Treatment by Unipedicular Vertebral Augmentation Using Curved Injection Cannula

Purpose: This study was designed to evaluate the efficacy of the blunt-tipped curved injection needle (BCN) AVAflex (Care Fusion) for vertebral augmentation in cases of Kummel's disease. Methods: We performed 25 vertebral augmentation procedures on 25 consecutive patients (11 men/14 women; mean age, 67 years) with Kummel's disease using the blunt-tipped curved injection needle with PMMA cement. We performed all 25 procedures by unipedicular left approach with patients in prone position under local anesthesia and mild sedation. In all cases, an intravertebral cleft was evident on preprocedural imaging. We evaluated pain intensities by Visual Analogic Scale (VAS) before and at first day, 6 months, and 1 year after procedure. Results: In all cases the curved injection cannula permitted the filling of the clefts and surrounding cancellous bone without any complication. A significant reduction of kyphotic deformities of the treated vertebral bodies was evident. A significance decrease in VAS values at 1 year also was evident (mean decrease 7.2). At plain dynamic postprocedural X-rays checks, there was no sign of pathologic intravertebral motion as evidence of optimal stabilization. Conclusions: BCN AVAflex is a safe and effective device for targeted vertebral augmentation in cases of Kummel's disease. Its distinctive characteristic is the curved injection cannula, which enables targeting the cement injection to areas far off the trajectory of the straight access cannula, thus providing excellent cement spread throughout the entire volume of vertebral body.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
NASA NTRS1978

Application of the finite element method to rotary-wing aeroelasticity

Recent research in rotary-wing aeroelasticity has indicated that all fundamental problems in this area are inherently nonlinear. The non-linearities in this problem are due to the inclusion of finite slopes, due to moderate deflections, in the structural, inertia and aerodynamic operators associated with this aeroelastic problem. In this paper the equations of motion, which are both time and space dependent, for the aeroelastic problem are first formulated in P.D.E. form. Next the equations are linearized about a suitable equilibrium position. The spatial dependence in these equations is discretized using a local Galerkin method of weighted residuals resulting in a finite element formulation of the aeroelastic problem. As an illustration the method is applied to the coupled flap-lag problem of a helicopter rotor blade in hover. Comparison of the solutions with previously published solutions establishes the convergence properties of the method. It is concluded that this formulation is a practical tool for solving rotary-wing aeroelastic stability or response problems.

Friedmann, P.↗
NASA NTRS1979

Sunspots and the physics of magnetic flux tubes. I - The general nature of the sunspot. II - Aerodynamic drag

Analysis of the dynamical stability of a large flux tube suggests that the field of a sunspot must divide into many separate tubes within the first 1000 km below the surface. Buoyancy of the Wilson depression at the visible surface and probably also a downdraft beneath the sunspot hold the separate tubes in a loose cluster. Convective generation of Alfven waves, which are emitted preferentially downward, cools the tubes. Aerodynamic drag on a slender flux tube stretched vertically across a convective cell is also studied. Since the drag is approximately proportional to the local kinetic energy density, the density stratification weights the drag in favor of the upper layers. Horizontal motions concentrated in the bottom of the convective cell may reverse this density effect. A downdraft of about two km/sec through the flux tubes beneath the sunspot is hypothesized.

Parker, E. N.↗
DOE OSTI2019

Radial Velocities, Space Motions, and Nearby Young Moving Group Memberships of Eleven Candidate Young Brown Dwarfs

We present new radial velocity (RV) measurements for 11 candidate young very-low-mass stars and brown dwarfs, with spectral types from M7 to L7. Candidate young objects were identified by features indicative of low surface gravity in their optical and/or near-infrared spectra. RV measurements are derived from high-resolution (R =λ/Δλ = 20,000) J-band spectra taken with NIRSPEC at the Keck Observatory. We combine RVs with proper motions and trigonometric distances to calculate three-dimensional space positions and motions and to evaluate membership probabilities for nearby young moving groups (NYMGs). We propose 2MASS J00452143+1634446 (L2β, J = 13.06) as an RV standard given the precision and stability of measurements from three different studies. We test the precision and accuracy of our RV measurements as a function of spectral type of the comparison object, finding that RV results are essentially indistinguishable even with differences of ±5 spectral subtypes. We also investigate the strengths of gravity-sensitive K i lines at 1.24–1.25 μm and evaluate their consistency with other age indicators. We confirm or reconfirm four brown dwarf members of NYMGs—2MASS J00452143+1634446, WISE J00470038+6803543, 2MASS J011747483403258, and 2MASS J193555952846343—and their previous age estimates. We identify one new brown dwarf member of the Carina-Near moving group, 2M2154−10. The remaining objects do not appear to be members of any known NYMGs, despite their spectral signatures of youth. These results add to the growing number of very-low-mass objects exhibiting signatures of youth that lack likely membership in a known NYMG, thereby compounding the mystery regarding local, low-density star formation.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Advancing Understanding of Deep Convective Anvil Clouds

In this project we use ground-based DOE ARM observations, satellite observations, and Weather Research and Forecasting (WRF) model simulations to arrive at a more complete understanding (or, new theory) of how atmospheric heating profiles in thunderstorm cores drives how quickly air rises in the storms. The rate at which air motion increases or decreases with altitude directly relates to how much cloud mass is ejected laterally into the adjacent atmosphere, which then influences the development and maintenance of “stratiform raining cloud anvils”. Our work has led to new equations connecting the heating in these thunderstorms to these stratiform clouds, with the connection modulated by the stability structure of the atmosphere (i.e., if more stable, then a more rapid change in thunderstorm heating with altitude results in more stratiform cloud, independent of whether over land or ocean, or how organized the thunderstorms are). Improving knowledge of stratiform clouds is important, since they contribute to substantial accumulated rainfall (important to agriculture and local hydrology) as well as radiation, given their large areal extents and lifetime. The radiation impacts likely impact simulated cloud feedback in all Earth System models (ESMs).

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2013

Real-Time Tumor Tracking in the Lung Using an Electromagnetic Tracking System

Purpose: To describe the first use of the commercially available Calypso 4D Localization System in the lung. Methods and Materials: Under an institutional review board-approved protocol and an investigational device exemption from the US Food and Drug Administration, the Calypso system was used with nonclinical methods to acquire real-time 4-dimensional lung tumor tracks for 7 lung cancer patients. The aims of the study were to investigate (1) the potential for bronchoscopic implantation; (2) the stability of smooth-surface beacon transponders (transponders) after implantation; and (3) the ability to acquire tracking information within the lung. Electromagnetic tracking was not used for any clinical decision making and could only be performed before any radiation delivery in a research setting. All motion tracks for each patient were reviewed, and values of the average displacement, amplitude of motion, period, and associated correlation to a sinusoidal model (R{sup 2}) were tabulated for all 42 tracks. Results: For all 7 patients at least 1 transponder was successfully implanted. To assist in securing the transponder at the tumor site, it was necessary to implant a secondary fiducial for most transponders owing to the transponder's smooth surface. For 3 patients, insertion into the lung proved difficult, with only 1 transponder remaining fixed during implantation. One patient developed a pneumothorax after implantation of the secondary fiducial. Once implanted, 13 of 14 transponders remained stable within the lung and were successfully tracked with the tracking system. Conclusions: Our initial experience with electromagnetic guidance within the lung demonstrates that transponder implantation and tracking is achievable though not clinically available. This research investigation proved that lung tumor motion exhibits large variations from fraction to fraction within a single patient and that improvements to both transponder and tracking system are still necessary to create a clinical daily-use system to assist with actual lung radiation therapy.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2016

The role of spray-enhanced swirl flow for combustion stabilization in a stratified-charge DISI engine

Implementing spray-guided stratified-charge direct-injection spark-ignited (DISI) engines is inhibited by the occurrence of misfire and partial burns. Engine-performance tests demonstrate that increasing engine speed induces combustion instability, but this deterioration can be prevented by generating swirling flow during the intake stroke. In-cylinder pressure-based heat-release analysis reveals that the appearance of poor-burn cycles is not solely dependent on the variability of early flame-kernel growth. Moreover, cycles can experience burning-rate regression during later combustion stages and may or may not recover before the end of the cycle. Thermodynamic analysis and optical diagnostics are used here to clarify why swirl improves the combustion repeatability from cycle to cycle. The fluid dynamics of swirl/spray interaction was previously demonstrated using high-speed PIV measurements of in-cylinder motored flow. It was found that the sprays of the multi-hole injector redistribute the intake-generated swirl flow momentum, thereby creating a better-centered higher angular-momentum vortex with reduced variability. The engine operation with high swirl was found to have significant improvement in cycle-to-cycle variations of both flow pattern and flow momentum. This paper is an extension of the previous work. Here, PIV measurements and flame imaging are applied to fired operation for studying how the swirl flow affects variability of ignition and subsequent combustion phases. PIV results for fired operation are consistent with the measurements made of motored flow. They demonstrate that the spark-plasma motion is highly correlated with the direction of the gas flow in the vicinity of the spark-plug gap. Without swirl, the plasma is randomly stretched towards either side of the spark plug, causing variability in the ignition of the two spray plumes that are straddling the spark plug. Conversely, swirl flow always convects the spark plasma towards one spray plume, causing a more repeatable ignition. The swirl decreases local RMS velocity, consistent with an observed reduction of early-burn variability. Broadband flame imaging demonstrates that with swirl, the flame consistently propagates in multiple directions to consume fuel–air mixtures within the piston bowl. In contrast, operation without swirl displays higher variability of flame-spread patterns, occasionally causing the appearance of partial-burn cycles.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2022

Hydrogen induced dislocation core reconstruction in bcc tungsten

Dislocation, playing a crucial role in the plastic deformation of metals, can be significantly affected by introducing solute elements. Hydrogen (H) embrittlement is one such example, while the underlying mechanism for H affected dislocation structural stability and mobility remains unclear and the role of H has been controversial. Here, using first-principles calculations, we demonstrate that the effect of H on screw dislocation in bcc tungsten (W) is H concentration-dependent, signified by a H-induced transition of SD core structure. At low concentrations of H segregation, dislocation maintains the intrinsic easy-core structure, and H atoms are attached to the “periphery” of dislocation to enhance dislocation motion. In contrast, at high H concentrations, dislocation transforms into a hard-core, metal hydride-like structure, as H atoms become the “body” of dislocation to significantly reduce the dislocation mobility in W. Further, such local easy-to-hard transition can also be induced by the other solutes, including helium, carbon, nitrogen and oxygen. Our work sheds new light on the H-dislocation interactions in bcc W, having broad implications in the interstitial solute-related phenomena.

36 MATERIALS SCIENCE↗
DOE OSTI2019

Turbulent nonpremixed cool flames: Experimental measurements, Direct Numerical Simulation, and manifold-based combustion modeling

Turbulence, low-temperature chemistry, and their interactions in the form of turbulent cool flames are critical to understanding and improving advanced engines. Design of such engines requires tractable simulations which in turn necessitate turbulent combustion models that can account for cool flames. While manifold-based turbulent combustion models are an attractive option for hot flames, their applicability to cool flames is not yet fully understood. This is partially due to the lack of turbulent cool flame experiments, which has made model validation difficult. This work addresses these points with a combined experimental and computational investigation of turbulent nonpremixed dimethyl ether cool flames. First, a Co-flow Axisymmetric Reactor-Assisted Turbulent (CARAT) burner is developed and tested. Turbulent cool flames are studied using the formaldehyde planar laser-induced fluorescence (PLIF), acetone PLIF, and planar Rayleigh scattering techniques. The acetone PLIF signals are converted into mixture fraction values, and measurements of time-averaged temperature are derived from the Rayleigh scattering signals by taking advantage of the similarities between cool flames and unburned mixtures. Second, a Direct Numerical Simulation (DNS) of the CARAT burner is conducted. The flame is shown to be sensitive to thermal boundary conditions such that the stabilization method is dependent on a 10 K change in inlet temperature. Comparisons of first- and second-order statistics of temperature, mixture fraction, and formaldehyde between the DNS and experiment show good agreement. The validity of manifold-based turbulent combustion models in turbulent cool flames is then explored by first studying the effective Lewis numbers in the flame using a local differential diffusion parameter analysis. Turbulent cool flames are shown to have effective unity Lewis number transport even at a much lower Reynolds number than typically required for hot flames. Good agreement is shown for temperature and formaldehyde between DNS conditional means and one-dimensional nonpremixed flame solutions. Additionally, the motion of the cool flame in mixture fraction space is shown to be well described by one-dimensional nonpremixed flame solutions. These points indicate that the fundamental physics of turbulent cool flames are analogous to turbulent hot flames, implying that the same reduced-order manifold-based modeling approaches utilized for turbulent hot flames can be utilized for turbulent cool flames.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1989

Strain accumulation and surface deformation along the San Andreas, California

The goal of this project remains to be the achievement of a better understanding of the regional and local deformation and crustal straining processes in western North America, particularly the effect of the San Andreas and nearby faults on the spatial and temporal crustal deformation behavior. Construction of theoretical models based on the mechanics of coupled elastic plate/viscoelastic foundation and large scale crack mechanics provide a rational basis for the interpretation of seismic and aseismic anomalies and expedite efforts in forecasting the stability of plate boundary deformations. In the present period, special focus is placed on the 3-D effect of irregular fault locked patches on the ground measured deformation fields. Specifically, use is made of a newly developed 3-D boundary element program to analyze the fault slip and vertical ground motion in the Parkfield area on the San Andreas.

Li, Victor C.↗
DOE OSTI2023

Local structural mechanism for enhanced energy storage properties in heterovalent doped NaNbO 3 ceramics

In recent years, there is a growing interest for new lead-free oxides with reversible antiferroelectric (AFE)-ferroelectric (FE) phase transition for high-power energy-storage applications. NaNbO 3 -based ceramics are particularly attractive due to their easy synthesis and cost-effectiveness. In order to stabilize reversible AFE-FE phase transition, NaNbO 3 is doped with a combination of heterovalent substitutions, although the underlying structural mechanism for the same is poorly understood. Here, we investigated local and average structures of Ca/Zr doped NaNbO 3 using neutron total scattering. Here, we show that Ca/Zr doping increases the average AFE phase (Pbma) fraction, however, the material remains as a composite of both FE (P2 1 ma) and AFE regions. Analysis of local structure suggests that increase in the long-range AFE phase results from more extensive twinning of local FE regions, due to introduced charge disorder. We propose that enhanced energy-storage properties of Ca/Zr-doped NaNbO 3 arises from localized twin boundary motion between the defect-induced pinning centers.

25 ENERGY STORAGE↗
DOE OSTI2021

Observation of Defect-Assisted Magnetic Vortex Core Reversal at Ultralow Critical Velocity

The stability of structures with nontrivial topology is a subject of great interest, both from fundamental and technological perspectives. The topology of a magnetic vortex, for example, results in high stability of the vortex core polarity, which is attractive for applications. This high stability, however, greatly impedes the required deterministic polarity switching, which is typically only achieved with large magnetic fields or strong dynamic driving. Here, we show that the interaction between the vortex core and manufactured nanoscale defects in the magnetic material lowers the required driving strength for core polarity reversal by more than an order of magnitude. We excite vortex dynamics in thin permalloy disks, and map the twodimensional (2D) vortex core trajectory using 3D time-resolved Kerr microscopy. In pristine samples, we observe normal gyrotropic motion of the vortex core. After laser-induced generation of defects, however, we observe repeated vortex core reversal at much-reduced driving strength. Micromagnetic simulations reveal how local reduction of exchange coupling and saturation magnetization can create vortex core reversal sites for deterministic vortex core switching.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS1999

Comparison of AGE and Spectral Methods for the Simulation of Far-Wakes

Turbulent flow simulation methods based on finite differences are attractive for their simplicity, flexibility and efficiency, but not always for accuracy or stability. This report demonstrates that a good compromise is possible with the Advected Grid Explicit (AGE) method. AGE has proven to be both efficient and accurate for simulating turbulent free-shear flows, including planar mixing layers and planar jets. Its efficiency results from its localized fully explicit finite difference formulation (Bisset 1998a,b) that is very straightforward to compute, outweighing the need for a fairly small timestep. Also, most of the successful simulations were slightly under-resolved, and therefore they were, in effect, large-eddy simulations (LES) without a sub-grid-scale (SGS) model, rather than direct numerical simulations (DNS). The principle is that the role of the smallest scales of turbulent motion (when the Reynolds number is not too low) is to dissipate turbulent energy, and therefore they do not have to be simulated when the numerical method is inherently dissipative at its resolution limits. Such simulations are termed 'auto-LES' (LES with automatic SGS modeling) in this report.

Bisset, D. K.↗
DOE OSTI2025

Atomic-Scale Tracking of Topological Defect Motion and Incommensurate Charge Order Melting

Charge order pervades the phase diagrams of quantum materials where it competes with superconducting and magnetic phases, hosts electronic phase transitions and topological defects, and couples to the lattice generating intricate structural distortions. Incommensurate charge order is readily stabilized in manganese oxides, where it is associated with anomalous electronic and magnetic properties, but its nanoscale structural inhomogeneity complicates precise characterization and understanding of its relationship with competing phases. Leveraging atomic-resolution variable-temperature cryogenic scanning transmission electron microscopy, we characterize the thermal evolution of charge order as it transforms from its ground state in a model manganite system. We find that mobile networks of discommensurations and dislocations generate phase inhomogeneity and induce global incommensurability in an otherwise lattice-locked modulation. Driving the order to melt at high temperatures, the discommensuration density grows, and regions of order locally decouple from the lattice periodicity. Published by the American Physical Society 2025

Schnitzer, Noah (ORCID:0000000210421146)↗
NASA NTRSDate not supplied

Extinction of Solid Diffusion Flame in Microgravity: Details of Quenching and Blowoff Processes

Long duration microgravity experiments aboard the International Space Station determine diffusion flame extinction limits of PMMA spheres. Upon ignition from an electrically-heated coil, the 4-cm-diameter samples are exposed to forced flows ranging from 0.2 to 80 cm/s and an oxygen ranging from 13 to 28% in one atmosphere total pressure. Extinction is reached as oxygen concentration gradually decreases by natural depletion. Five extinction tests are presented at different flow velocities and oxygen concentrations. Quenching at low velocity is observed with the flame tip shrinking upstream and blowoff is observed when a hole forms in the flame at the forward flow stagnation point. However, these processes are not quasistatic. The quenching motion involves periodic flame tip pulsating toward downstream and shrinking upstream with a continually decreasing flame size at the end of each cycle. In blowoff, the flame base pulsates between the flame hole and the downstream location, although with far fewer cycles compared to quenching. The pulsations appear to be the result of a premixed flame front spreading into a combustible mixture. Two specific cases are discussed. In the first, at very low flow velocity (~ 0.4 cm/s) and elevated oxygen, self-sustained flame tip cyclic pulsations are observed for a lengthy period (~15 minutes). In the second case with a higher flow velocity (50 cm/s), the diffusion flame is stabilized at the shoulder of the spherical sample after local stagnation point blowoff. With steady decrease in ambient oxygen due to depletion, spinning flamelets are formed. The long-duration microgravity environment makes it possible to observe. these interesting and detailed extinction processes.

Flame quenching↗
DOE OSTI2013

NON-BAROTROPIC LINEAR ROSSBY WAVE INSTABILITY IN THREE-DIMENSIONAL DISKS

Astrophysical disks with localized radial structure, such as protoplanetary disks containing dead zones or gaps due to disk-planet interaction, may be subject to the non-axisymmetric Rossby wave instability (RWI) that leads to vortex formation. The linear instability has recently been demonstrated in three-dimensional (3D) barotropic disks. It is the purpose of this study to generalize the 3D linear problem to include an energy equation, thereby accounting for baroclinity in three dimensions. Linear stability calculations are presented for radially structured, vertically stratified, geometrically thin disks with non-uniform entropy distribution in both directions. Polytropic equilibria are considered but adiabatic perturbations assumed. The unperturbed disk has a localized radial density bump, making it susceptible to the RWI. The linearized fluid equations are solved numerically as a partial differential equation eigenvalue problem. Emphasis on the ease of method implementation is given. It is found that when the polytropic index is fixed and adiabatic index increased, non-uniform entropy has negligible effect on the RWI growth rate, but pressure and density perturbation magnitudes near a pressure enhancement increase away from the midplane. The associated meridional flow is also qualitatively changed from homentropic calculations. Meridional vortical motion is identified in the nonhomentropic linear solution, as well as in a nonlinear global hydrodynamic simulation of the RWI in an initially isothermal disk evolved adiabatically. Numerical results suggest that buoyancy forces play an important role in the internal flow of Rossby vortices.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

Three-dimensional relativistic pair plasma reconnection with radiative feedback in the Crab Nebula

The discovery of rapid synchrotron gamma-ray flares above 100 MeV from the Crab Nebula has attracted new interest in alternative particle acceleration mechanisms in pulsar wind nebulae. Diffuse shock-acceleration fails to explain the flares because particle acceleration and emission occur during a single or even sub-Larmor timescale. In this regime, the synchrotron energy losses induce a drag force on the particle motion that balances the electric acceleration and prevents the emission of synchrotron radiation above 160 MeV. Previous analytical studies and two-dimensional (2D) particle-in-cell (PIC) simulations indicate that relativistic reconnection is a viable mechanism to circumvent the above difficulties. The reconnection electric field localized at X-points linearly accelerates particles with little radiative energy losses. In this paper, we check whether this mechanism survives in three dimension (3D), using a set of large PIC simulations with radiation reaction force and with a guide field. In agreement with earlier works, we find that the relativistic drift kink instability deforms and then disrupts the layer, resulting in significant plasma heating but few non-thermal particles. A moderate guide field stabilizes the layer and enables particle acceleration. We report that 3D magnetic reconnection can accelerate particles above the standard radiation reaction limit, although the effect is less pronounced than in 2D with no guide field. We confirm that the highest-energy particles form compact bunches within magnetic flux ropes, and a beam tightly confined within the reconnection layer, which could result in the observed Crab flares when, by chance, the beam crosses our line of sight.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2008

Magnetic Compensation for Second-Order Doppler Shift in LITS

The uncertainty in the frequency of a linear-ion-trap frequency standard (LITS) can be reduced substantially by use of a very small magnetic inhomogeneity tailored to compensate for the residual second-order Doppler shift. An effect associated with the relativistic time dilatation, one cause of the second-order Doppler shift, is ion motion that is attributable to the trapping radio-frequency (RF)electromagnetic field used to trap ions. The second-order Doppler shift is reduced by using a multi-pole trap; however it is still the largest source of systematic frequency shift in the latest generation of LITSs, which are among the most stable clocks in the world. The present compensation scheme reduces the frequency instability of the affected LITS to about a tenth of its previous value. The basic principles of prior generation LITSs were discussed in several prior NASA Tech Briefs articles. Below are recapitulated only those items of basic information necessary to place the present development in context. A LITS includes a microwave local oscillator, the frequency of which is stabilized by comparison with the frequency of the ground state hyperfine transition of 199Hg+ ions. The comparison involves a combination of optical and microwave excitation and interrogation of the ions in a linear ion trap in the presence of a nominally uniform magnetic field. In the current version of the LITS, there are two connected traps (see figure): (1) a quadrupole trap wherein the optical excitation and measurement take place and (2) a 12-pole trap (denoted the resonance trap), wherein the microwave interrogation takes place. The ions are initially loaded into the quadrupole trap and are thereafter shuttled between the two traps. Shuttling ions into the resonance trap allows sensitive microwave interrogation to take place well away from loading interference. The axial magnetic field for the resonance trap is generated by an electric current in a finely wound wire coil surrounded by magnetic shields. In the quadrupole and 12-pole traps, the potentials are produced by RF voltages applied to even numbers (4 and 12, respectively) of parallel rods equally spaced around a circle. The polarity of the voltage on each rod is opposite that of the voltage on the adjacent rod. As a result, the amplitude of the RF trapping field is zero along the centerline and increases, with radius, to a maximum value near the rods.

Burt, Eric↗