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NASA NTRS1981

The helium abundance of Jupiter from Voyager

The helium abundance in the Jovian atmosphere is derived from Voyager 1 data by two methods. The first method uses only infrared spectra from selected locations on the planet while the second method uses a thermal profile independently derived from radio occultation measurements and infrared spectra recorded near the occultation point. A hydrogen mole fraction of 0.897 plus or minus 0.030 is obtained from the first method, while the second method gives 0.880 plus or minus 0.036, corresponding to helium mass fractions of 0.19 plus or minus 0.05 and 0.21 plus or minus 0.06, respectively. The estimated errors for the first method are primarily due to systematic uncertainties in the H2 and He absorption coefficients, while those for the second method result mainly from errors in the radio occultation profile and are less well known. Random errors in the measured infrared spectra are found to be negligible in both cases. The results are consistent with a uniform mix of hydrogen and helium within Jupiter's interior, but a modest amount of helium depletion (Delta Y equal to or less than 0.05) cannot be excluded.

Gautier, D.↗

NASA NTRS1984

Solar Seismology from Space. a Conference at Snowmass, Colorado

The quality of the ground based observing environment suffers from several degrading factors: diurnal interruptions and thermal variations, atmospheric seeing and transparency fluctuations and adverse weather interruptions are among the chief difficulties. The limited fraction of the solar surface observable from only one vantage point is also a potential limitation to the quality of the data available without going to space. Primary conference goals were to discuss in depth the scientific return from current observations and analyses of solar oscillations, to discuss the instrumental and site requirements for realizing the full potential of the seismic analysis method, and to help bring new workers into the field by collecting and summarizing the key background theory. At the conclusion of the conference there was a clear consensus that ground based observation would not be able to provide data of the quality required to permit a substantial analysis of the solar convection zone dynamics or to permit a full deduction of the solar interior structure.

Ulrich, R. K.↗

NASA NTRS1973

Matrix operator theory of radiative transfer. I - Rayleigh scattering.

An entirely rigorous method for the solution of the equations for radiative transfer based on the matrix operator theory is reviewed. The advantages of the present method are: (1) all orders of the reflection and transmission matrices are calculated at once; (2) layers of any thickness may be combined, so that a realistic model of the atmosphere can be developed from any arbitrary number of layers, each with different properties and thicknesses; (3) calculations can readily be made for large optical depths and with highly anisotropic phase functions; (4) results are obtained for any desired value of the surface albedo including the value unity and for a large number of polar and azimuthal angles; (5) all fundamental equations can be interpreted immediately in terms of the physical interactions appropriate to the problem; and (6) both upward and downward radiance can be calculated at interior points from relatively simple expressions.

Plass, G. N.↗

NASA NTRS1975

Analysis of liquid-metal-jet impingement cooling in a corner region and for a row of jets

A conformal mapping method was used to analyze liquid-metal-jet impingement heat transfer. The jet flow region and energy equation are transformed to correspond to uniform flow in a parallel plate channel with nonuniform heat addition along a portion of one wall. The exact solution for the wall-temperature distribution was obtained in the transformed channel, and the results are mapped back into the physical plane. Two geometries are analyzed. One is for a single slot jet directed either into an interior corner formed by two flat plates, or over the external sides of the corner; the flat plates are uniformly heated, and the corner can have various included angles. The heat-transfer coefficient at the stagnation point at the apex of the plates is obtained as a function of the corner angle, and temperature distributions are calculated along the heated walls. The second geometry is an infinite row of uniformly spaced parallel slot jets impinging normally against a uniformly heated plate. The heat-transfer behavior is obtained as a function of the spacing between the jets. Results are given for several jet Peclet numbers from 5 to 50.

Siegel, R.↗

NASA NTRS1984

Transformation of two and three-dimensional regions by elliptic systems

The research during this period continued to expand the class of numerical algorithms that can be accurately and efficiently implemented on overlapping grids. Whereas previous calculations have been used to solve elliptic equations and to find the steady-state solution of parabolic equations, the present work is aimed towards developing time-accurate solution techniques for parabolic and hyperbolic equations. The primary difficulty here is in the correct treatment of the interior boundary nodes that must be updated at each iteration. The implementation of explicit methods is straightforward. However, the common practice of lagging these values when using an implicit methods leads to inconsistencies in the difference equation. One way to avoid this problem is to alternately calculate with an implicit and an explicit method on each subgrid. With this procedure, the explicit method generates boundary values at the next time level which are then used by the implicit step. It can be shown that when a backward implicit method is combined with a forward explicit method, the composite method is second order accurate and unconditionally stable for linear problems. A second area in which progress can be reported is in the distribution of grid points on curves and surfaces.

Mastin, E. W.↗

NASA NTRS2014

Optical Polarization of Light from a Sorghum Canopy Measured Under Both a Clear and an Overcast Sky

Introduction: We tested the hypothesis that the optical polarization of the light reflected by a sorghum canopy is due to a Fresnel-type redirection, by sorghum leaf surfaces, of light from an unpolarized light source, the sun or overcast sky, toward the measuring sensor. If it can be shown that the source of the polarization of the light scattered by the sorghum canopy is a first surface, Fresnel-type reflection, then removing this surface reflected light from measurements of canopy reflectance presumably would allow better insight into the biochemical processes such as photosynthesis and metabolism that occur in the interiors of sorghum canopy leaves. Methods: We constructed a tower 5.9m tall in the center of a homogenous sorghum field. We equipped two Barnes MMR radiometers with polarization analyzers on the number 1, 3 and 7 Landsat TM wavelength bands. Positioning the radiometers atop the tower, we collected radiance data in 44 view directions on two days, one day with an overcast sky and the other, clear and sunlit. From the radiance data we calculated the linear polarization of the reflected light for each radiometer wavelength channel and view direction. Results and Discussion: Our experimental results support our hypothesis, showing that the amplitude of the linearly polarized portion of the light reflected by the sorghum canopy varied dramatically with view azimuth direction under a point source, the sun, but the amplitude varied little with view azimuth direction under the hemispherical source, the overcast sky. Under the clear sky, the angle of polarization depended upon the angle of incidence of the sunlight on the leaf, while under the overcast sky the angle of polarization depended upon the zenith view angle. These results support a polarized radiation transport model of the canopy that is based upon a first surface, Fresnel reflection from leaves in the sorghum canopy.

cloud cover↗

NASA NTRS1992

Diffraction by a symmetric material junction simulated with generalized sheet transistion conditions

When diffraction by a material discontinuity in a thick dielectric/ferrite slab is treated for the dual integral equation approach (a variation of the Weiner-Hopf method), the resulting solution displays unknown constants, characteristic of the incompleteness associated with the generalized transition conditions employed in modeling a thick slab. The constants can be shown to depend on the geometry and properties of the discontinuity. Their explicit determination by enforcing field continuity across the slab junction is treated. The field internal to the slab is determined by first expressing the field as an expansion of discrete and continuous eigenmodes valid in the exterior and interior slab region. The expansion or mode coefficients are then identified by recasting the Weiner-Hopf solution in a suitable form, enabling the solution constants to be numerically computed by matching the field expansions to the left and right of the slab discontinuity at a discrete set of points.

Ricoy, Mark A.↗

NASA NTRS2014

Applying Dynamical Systems Theory to Optimize Libration Point Orbit Stationkeeping Maneuvers for WIND

NASA's WIND mission has been operating in a large amplitude Lissajous orbit in the vicinity of the interior libration point of the Sun-Earth/Moon system since 2004. Regular stationkeeping maneuvers are required to maintain the orbit due to the instability around the collinear libration points. Historically these stationkeeping maneuvers have been performed by applying an incremental change in velocity, or (delta)v along the spacecraft-Sun vector as projected into the ecliptic plane. Previous studies have shown that the magnitude of libration point stationkeeping maneuvers can be minimized by applying the (delta)v in the direction of the local stable manifold found using dynamical systems theory. This paper presents the analysis of this new maneuver strategy which shows that the magnitude of stationkeeping maneuvers can be decreased by 5 to 25 percent, depending on the location in the orbit where the maneuver is performed. The implementation of the optimized maneuver method into operations is discussed and results are presented for the first two optimized stationkeeping maneuvers executed by WIND.

WIND↗

NASA NTRS2007

Reentry Thermal Analysis of a Generic Crew Exploration Vehicle Structure

Comparative studies were performed on the heat-shielding characteristics of honeycomb-core sandwich panels fabricated with different materials for possible use as wall panels for the proposed crew exploration vehicle. Graphite/epoxy sandwich panel was found to outperform aluminum sandwich panel under the same geometry due to superior heat-shielding qualities and lower material density. Also, representative reentry heat-transfer analysis was performed on the windward wall structures of a generic crew exploration vehicle. The Apollo low Earth orbit reentry trajectory was used to calculate the reentry heating rates. The generic crew exploration vehicle has a graphite/epoxy composite honeycomb sandwich exterior wall and an aluminum honeycomb sandwich interior wall, and is protected with the Apollo thermal protection system ablative material. In the thermal analysis computer program used, the TPS ablation effect was not yet included; however, the results from the nonablation heat-transfer analyses were used to develop a "virtual ablation" method to estimate the ablation heat loads and the thermal protection system recession thicknesses. Depending on the severity of the heating-rate time history, the virtual ablation period was found to last for 87 to 107 seconds and the ablation heat load was estimated to be in the range of 86 to 88 percent of the total heat load for the ablation time period. The thermal protection system recession thickness was estimated to be in the range of 0.08 to 0.11 inches. For the crew exploration vehicle zero-tilt and 18-degree-tilt stagnation points, thermal protection system thicknesses of h = {0.717, 0.733} inches were found to be adequate to keep the substructural composite sandwich temperature below the limit of 300 F.

Ko, William L.↗

NASA NTRS1980

Aircraft noise control in the 1980's

It is pointed out that a need exists for the orderly development of technology and engineering methods for noise control of all types of aircraft. The nature and scope of aircraft noise problems are reviewed, and a description is provided of noise control progress made to date. The most serious aircraft noise problems confronting communities in the past two decades have been associated with the subsonic air carrier jet transport. Operational trends related to traffic growth and operational constraints are examined, and noise level trends are considered, taking into account engine cycle developments, the role of Federal noise certification, future noise exposures, and multiple noise sources. Advanced source noise reduction technology developments are discussed along with a noise impact assessment, and advanced operating procedures. Attention is also given to engine noise and sonic boom exposures in connection with supersonic air carrier aircraft, and exterior and interior noise control related to propeller/rotor aircraft.

Hubbard, H. H.↗

NASA NTRS1985

Algebraic grid generation using tensor product B-splines

Finite difference methods are more successful if the accompanying grid has lines which are smooth and nearly orthogonal. The development of an algorithm which produces such a grid when given the boundary description. Topological considerations in structuring the grid generation mapping are discussed. The concept of the degree of a mapping and how it can be used to determine what requirements are necessary if a mapping is to produce a suitable grid is examined. The grid generation algorithm uses a mapping composed of bicubic B-splines. Boundary coefficients are chosen so that the splines produce Schoenberg's variation diminishing spline approximation to the boundary. Interior coefficients are initially chosen to give a variation diminishing approximation to the transfinite bilinear interpolant of the function mapping the boundary of the unit square onto the boundary grid. The practicality of optimizing the grid by minimizing a functional involving the Jacobian of the grid generation mapping at each interior grid point and the dot product of vectors tangent to the grid lines is investigated. Grids generated by using the algorithm are presented.

Saunders, B. V.↗

NASA NTRS2005

Multilayer Composite Pressure Vessels

A method has been devised to enable the fabrication of lightweight pressure vessels from multilayer composite materials. This method is related to, but not the same as, the method described in gMaking a Metal- Lined Composite-Overwrapped Pressure Vessel h (MFS-31814), NASA Tech Briefs, Vol. 29, No. 3 (March 2005), page 59. The method is flexible in that it poses no major impediment to changes in tank design and is applicable to a wide range of tank sizes. The figure depicts a finished tank fabricated by this method, showing layers added at various stages of the fabrication process. In the first step of the process, a mandrel that defines the size and shape of the interior of the tank is machined from a polyurethane foam or other suitable lightweight tooling material. The mandrel is outfitted with metallic end fittings on a shaft. Each end fitting includes an outer flange that has a small step to accommodate a thin layer of graphite/epoxy or other suitable composite material. The outer surface of the mandrel (but not the fittings) is covered with a suitable release material. The composite material is filament- wound so as to cover the entire surface of the mandrel from the step on one end fitting to the step on the other end fitting. The composite material is then cured in place. The entire workpiece is cut in half in a plane perpendicular to the axis of symmetry at its mid-length point, yielding two composite-material half shells, each containing half of the foam mandrel. The halves of the mandrel are removed from within the composite shells, then the shells are reassembled and bonded together with a belly band of cured composite material. The resulting composite shell becomes a mandrel for the subsequent steps of the fabrication process and remains inside the final tank. The outer surface of the composite shell is covered with a layer of material designed to be impermeable by the pressurized fluid to be contained in the tank. A second step on the outer flange of each end fitting accommodates this layer. Depending on the application, this layer could be, for example, a layer of rubber, a polymer film, or an electrodeposited layer of metal. If the fluid to be contained in the tank is a gas, then the best permeation barrier is electrodeposited metal (typically copper or nickel), which can be effective at a thickness of as little as 0.005 in (.0.13 mm). The electrodeposited metal becomes molecularly bonded to the second step on each metallic end fitting. The permeation-barrier layer is covered with many layers of filament-wound composite material, which could be the same as, or different from, the composite material of the inner shell. Finally, the filament-wound composite material is cured in an ov

DeLay, Tom↗

NASA NTRS2001

Light Isotopes and Trace Organics Analysis of Mars Samples with Mass Spectrometry

Precision measurement of light isotopes in Mars surface minerals and comparison of this isotopic composition with atmospheric gas and other, well-mixed reservoirs such as surface dust are necessary to understand the history of atmospheric evolution from a possibly warmer and wetter Martian surface to the present state. Atmospheric sources and sinks that set these ratios are volcanism, solar wind sputtering, photochemical processes, and weathering. Measurement of a range of trace organic species with a particular focus on species such as amino acids that are the building blocks of terrestrial life are likewise important to address the questions of prebiotic and present or past biological activity on Mars. The workshop topics "isotopic mineralogy" and "biology and pre-biotic chemistry" will be addressed from the point of view of the capabilities and limitations of insitu mass spectrometry (MS) techniques such as thermally evolved gas analysis (TEGA) and gas chromatography (GC) surface experiments using MS, in both cases, as a final chemical and isotopic composition detector. Insitu experiments using straightforward adaptations of existing space proven hardware can provide a substantial improvement in the precision and accuracy of our present knowledge of isotopic composition both in molecular and atomic species in the atmosphere and those chemically bound in rocks and soils. Likewise, detection of trace organic species with greatly improved sensitivity from the Viking GCMS experiment is possible using gas enrichment techniques. The limits to precision and accuracy of presently feasible insitu techniques compared to laboratory analysis of returned samples will be explored. The insitu techniques are sufficiently powerful that they can provide a high fidelity method of screening samples obtained from a diverse set of surface locations such as the subsurface or the interior of rocks for selection of those that are the most interesting for return to Earth.

Mahaffy, P.↗

NASA NTRSDate not supplied

Volumetric Assessment of UPRITE Exercises From Marker-Based Motion Capture

BACKGROUND Lack of volumetric data on full-body movement of exercises presents a challenge to ensuring the fit of crew member’s full range of motion on the International Space Station (ISS). The Upright Proprioception Retention via In-flight Training and Evaluation (UPRITE) is a sensorimotor countermeasure device designed for maintaining crew members’ proprioception in a microgravity environment. A footplate—attached to a static base—rotates in two degrees of freedom (pitch and roll) up to a 20 deg angle. An initial volumetric assessment assuming an upright standing posture produced a cone-like shape with a narrow bottom and wide top. Such general volumetric assessments risk creating an overly conservative volume estimate, taking up more space than is necessary on the already limited interior space of the ISS, and neglecting necessary volume due to oversimplifying assumptions. Rather, higher-fidelity volumetric assessments offer more comprehensive insights in an environment where every area counts. The main objective of this work is to provide the spatial parameters of exercises on the UPRITE such that it is placed on the ISS according to its volumetric demands or that usage is adjusted to fit the available space. METHODS In 2023, a data collection was performed originally to inform loads and dynamics of system use and was recently leveraged for volumetric assessment. Three human subjects representing different body types (~63-76 inches in stature) performed a variety of board manipulations using UPRITE with body weight offload. The test collected the 3D positional data of a modified full-body Plug-in Gait marker set [1] via a 16-camera OptiTrack MoCap system. After processing – filling marker gaps and trimming data – in OptiTrack Motive, the recorded marker location data, which included device markers, was exported to a readable trajectory file. To accurately represent the full volume defining landmarks, additional markers were digitally added to an unscaled Modified Full Body Model [2]. The model was then scaled according to its subject parameters upon which an inverse kinematics analysis was performed. A custom plugin yielded model marker location data files. Volumetric analyses were performed on the recorded trajectory and model trajectory files using a custom Python-built tool that extracted the marker location data and plotted it in a 3D space. Concerned with only the maximum volume of the motion, a 3D convex hull analysis was applied to the plot, extracting the vertices or external points of the eventual 3D CAD output, dubbed aptly as a “volume shell”. This overall approach was based on guidance in a NASA-STD-3001 Technical Brief [3]. RESULTS AND DISCUSSION Batch volumetric assessment on the exercises for each subject was performed, producing high-fidelity volume shells in minimal time. Preliminary results highlighted the value in higher-fidelity volumes based on collected data when possible. For example, revolving a single posture in the cone assessment would not have sufficiently captured a single leg stance; rather, it would need to involve swinging the leg both forward and back. Additional observations and the maximal dimensions of the volumes, including those based on scaled data for ISS anthropometric requirements, will be presented at the Human Research Program Investigator’s Workshop. CONCLUSIONS While this work’s primary objective was for the UPRITE-to-ISS integration, the tool built to conduct this analysis has wide applications for future exercise systems as an informational tool for optimal device placement. The tool and its findings also have implications for exercise device design and spacecraft interior considerations on Gateway, the Lunar Pressurized Rover, and beyond. REFERENCES [1] Bell, C. A., et al. (2023) Recent Improvements and Verification of a Full Body Model in OpenSim. NASA Human Research Program Investigator’s Workshop. https://ntrs.nasa.gov/citations/20230001080 [2] Lostroscio, K., et al (2023) The Digital Astronaut Simulation. AHFE International Conference on Human Factors in Design, Engineering, and Computing for All. [3] Exercise Overview. (2023) NASA-STD-3001 Technical Brief. https://www.nasa.gov/wp-content/uploads/2023/12/ochmo-tb-031-exercise-overview.pdf?emrc=9d454c?emrc=9d454c

L D Quinto↗

NASA NTRS2015

A Fully Nonmetallic Gas Turbine Engine Enabled by Additive Manufacturing of Ceramic Composites: Additive Manufacturing and Characterization of Ceramic Composites - Part III

This publication is the third part of a three part report of the project entitled "A Fully Nonmetallic Gas Turbine Engine Enabled by Additive Manufacturing" funded by NASA Aeronautics Research Institute (NARI). The objective of this project was to conduct additive manufacturing to produce ceramic matrix composite materials and aircraft engine components by the binder jet process. Different SiC powders with median sizes ranging from 9.3 to 53.0 microns were investigated solely and in powder blends in order to maximize powder packing. Various infiltration approaches were investigated to include polycarbosilane (SMP-10), phenolic, and liquid silicon. Single infiltrations of SMP-10 and phenolic only slightly filled in the interior. When the SMP-10 was loaded with sub-micron sized SiC powders, the infiltrant gave a much better result of filling in the interior. Silicon carbide fibers were added to the powder bed to make ceramic matrix composite materials. Microscopy showed that the fibers were well distributed with no preferred orientation on the horizontal plane and fibers in the vertical plane were at angles as much as 45deg. Secondary infiltration steps were necessary to further densify the material. Two to three extra infiltration steps of SMP-10 increased the density by 0.20 to 0.55 g/cc. However, the highest densities achieved were 2.10 to 2.15 g/cc. Mechanical tests consisting of 4 point bend tests were conducted. Samples from the two CMC panels had higher strengths and strains to failure than the samples from the two nonfiber reinforced panels. The highest strengths were from Set N with 65 vol% fiber loading which had an average strength of 66 MPa. Analysis of the fracture surfaces did not reveal pullout of the reinforcing fibers. Blunt fiber failure suggested that there was not composite behavior. The binder jet additive manufacturing method was used to also demonstrate the fabrication of turbine engine vane components of two different designs and sizes. The binder jet method has benefits over the conventional manufacturing of CMCs in that prototype and production parts can be fabricated quickly and economically with no tooling and extensive hand layup.

Additive Manufacturing↗

NASA NTRS1980

Numerical generation of two-dimensional grids by the use of Poisson equations with grid control at boundaries

A method for generating boundary-fitted, curvilinear, two dimensional grids by the use of the Poisson equations is presented. Grids of C-type and O-type were made about airfoils and other shapes, with circular, rectangular, cascade-type, and other outer boundary shapes. Both viscous and inviscid spacings were used. In all cases, two important types of grid control can be exercised at both inner and outer boundaries. First is arbitrary control of the distances between the boundaries and the adjacent lines of the same coordinate family, i.e., stand-off distances. Second is arbitrary control of the angles with which lines of the opposite coordinate family intersect the boundaries. Thus, both grid cell size (or aspect ratio) and grid cell skewness are controlled at boundaries. Reasonable cell size and shape are ensured even in cases wherein extreme boundary shapes would tend to cause skewness or poorly controlled grid spacing. An inherent feature of the Poisson equations is that lines in the interior of the grid smoothly connect the boundary points (the grid mapping functions are second order differentiable).

Sorenson, R. L.↗

NASA NTRS1995

Uncertainties for two-dimensional models of solar rotation from helioseismic eigenfrequency splitting

Observed solar p-mode frequency splittings can be used to estimate angular velocity as a function of position in the solar interior. Formal uncertainties of such estimates depend on the method of estimation (e.g., least-squares), the distribution of errors in the observations, and the parameterization imposed on the angular velocity. We obtain lower bounds on the uncertainties that do not depend on the method of estimation; the bounds depend on an assumed parameterization, but the fact that they are lower bounds for the 'true' uncertainty does not. Ninety-five percent confidence intervals for estimates of the angular velocity from 1986 Big Bear Solar Observatory (BBSO) data, based on a 3659 element tensor-product cubic-spline parameterization, are everywhere wider than 120 nHz, and exceed 60,000 nHz near the core. When compared with estimates of the solar rotation, these bounds reveal that useful inferences based on pointwise estimates of the angular velocity using 1986 BBSO splitting data are not feasible over most of the Sun's volume. The discouraging size of the uncertainties is due principally to the fact that helioseismic measurements are insensitive to changes in the angular velocity at individual points, so estimates of point values based on splittings are extremely uncertain. Functionals that measure distributed 'smooth' properties are, in general, better constrained than estimates of the rotation at a point. For example, the uncertainties in estimated differences of average rotation between adjacent blocks of about 0.001 solar volumes across the base of the convective zone are much smaller, and one of several estimated differences we compute appears significant at the 95% level.

Genovese, Christopher R.↗

NASA NTRS1982

Frequency and Time Domain Modeling of Acoustic Liner Boundary Conditions

As part of a research program directed at the acoustics of advanced subsonic propulsion systems undertaken at NASA Langley, Duke University was funded to develop a boundary condition model for bulk-reacting nacelle liners. The overall objective of the Langley program was to understand and predict noise from advanced subsonic transport engines and to develop related noise control technology. The overall technical areas included: fan and propeller source noise, acoustics of ducts and duct liners, interior noise, subjective acoustics, and systems noise prediction. The Duke effort was directed toward duct liner acoustics for the development of analytical methods to characterize liner behavior in both frequency domain and time domain. A review of duct acoustics and liner technology can be found in Reference [1]. At that time, NASA Langley was investigating the propulsion concept of an advanced ducted fan, with a large diameter housed inside a relatively short duct. Fan diameters in excess of ten feet were proposed. The lengths of both the inlet and exhaust portions of the duct were to be short, probably less than half the fan diameter. The nacelle itself would be relatively thin-walled for reasons of aerodynamic efficiency. The blade-passage frequency was expected to be less than I kHz, and very likely in the 200 to 300 Hz range. Because of the design constraints of a short duct, a thin nacelle, and long acoustic wavelengths, the application of effective liner technology would be especially challenging. One of the needs of the NASA Langley program was the capability to accurately and efficiently predict the behavior of the acoustic liner. The traditional point impedance method was not an adequate model for proposed liner designs. The method was too restrictive to represent bulk reacting liners and to allow for the characterization of many possible innovative liner concepts. In the research effort at Duke, an alternative method, initially developed to handle bulk reacting layers as described in Reference [2], was extended to apply to a broad range of liner types. This method included the effect of local gradients along the liner surface, and was particularly appropriate for situations with flow over the liner and grazing incidence acoustic fields. In order to utilize time domain computational methods to solve for the propfan acoustic field, corresponding liner boundary conditions were developed for time domain solutions rather than frequency domain solutions.

Bliss, Donald B.↗