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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
NASA NTRS2004

Meridional Variations of C2H2 and C2H6 in Jupiter's Atmosphere from Cassini CIRS Infrared Spectra

The abundances of hydrocarbons such as acetylene (C2H2) and ethane (C2H6) in Jupiter's atmosphere are important physical quantities, constraining our models of the chemical and dynamical processes. However, our knowledge of these quantities and their vertical and latitudinal variations has remained sparse. The flyby of the Cassini spacecraft with Jupiter at the end of 2000 provided an excellent opportunity to observe the infrared spectrum with the Composite Infrared Spectrometer (CIRS) instrument, mapping the spatial variation of emissions from 10-1400 cm-1. CIRS spectra taken at the highest resolution (0.5 cm-1) in early December 2000 have been analysed to infer atmospheric temperatures in the stratosphere at 0.5-20 mbar via the v4 of CH4, and in the troposphere at 100-400 mbar, via the hydrogen collision-induced continuum absorption at 600-800 cm. Simultaneously, we have searched for meridional abundance variations in C2H2 and C2H6 via the v5 and vg bands respectively. Tropospheric absorption and stratospheric emission are highly anti-correlated at the CIM resolution, introducing a non-uniqueness into the retrievals, which means that vertical gradient and column abundance cannot be simultaneously found without additional constraints. If we assume the profile shapes from photochemical model calculations, we show that the column abundance of C2H2 must decrease sharply towards the poles, while C2H6 is constant or slightly increasing. The relevance of these results to current photochemical and dynamical knowledge of Jupiter's atmosphere is discussed.

Nixon, C. A.↗
NASA NTRS2015

Numerical Computation of a Continuous-thrust State Transition Matrix Incorporating Accurate Hardware and Ephemeris Models

A significant body of work exists showing that providing a nonlinear programming (NLP) solver with expressions for the problem constraint gradient substantially increases the speed of program execution and can also improve the robustness of convergence, especially for local optimizers. Calculation of these derivatives is often accomplished through the computation of spacecraft's state transition matrix (STM). If the two-body gravitational model is employed as is often done in the context of preliminary design, closed form expressions for these derivatives may be provided. If a high fidelity dynamics model, that might include perturbing forces such as the gravitational effect from multiple third bodies and solar radiation pressure is used then these STM's must be computed numerically. We present a method for the power hardward model and a full ephemeris model. An adaptive-step embedded eight order Dormand-Prince numerical integrator is discussed and a method for the computation of the time of flight derivatives in this framework is presented. The use of these numerically calculated derivatieves offer a substantial improvement over finite differencing in the context of a global optimizer. Specifically the inclusion of these STM's into the low thrust missiondesign tool chain in use at NASA Goddard Spaceflight Center allows for an increased preliminary mission design cadence.

Low Thrust↗
NASA NTRSDate not supplied

Q-Law for Rapid Assessment of Low Thrust Cislunar Trajectories Via Automatic Differentiation

Q-Law is a Lyapunov-based control law used to determine optimal controls for a low thrust trajectory. One major issue with its use is the difficult derivatives re-quired for calculating optimal controls at a given time. In this paper, an implementation of Q-Law with automatic differentiation via a Python package called JAX is applied. With automatic differentiation, the difficult derivatives for Q-Law’s optimal controls are calculated with ease, and derivatives of final states with respect to Q-Law’s weights are found enabling gradient-based optimization of Q-Law for the first time. Different search and optimization methods for finding optimal weights are then compared using the LEO to GEO problem, and it was found that gradient-free methods like design of experiments and genetic algorithm produced the best results, but they took the longest time to get a solution, while the gradient-based method found a locally optimal result in a much faster time. Overall, the run time for a single propagation is manageable and well-suited for a mission designer to use as an initial guess generator for trajectory optimization, or for simple orbit transfer analysis.

Nathan Steffen↗
DOE OSTI2015

When the Desert Beetle Met the Carnivorous Plant: A Perfect Match for Droplet Growth and Shedding

Phase change of vapor followed by coalescence and transport on ubiquitous bumped or curved surfaces is of fundamental importance for a wide range of phenomena and applications from water condensation on cold beverage bottles, to fogging on glasses and windshields, self-cleaning by jumping droplets, weathering, self-assembly, desalination, latent heat transfer, etc. Over the past decades, many attempts to understand and control the droplet growth dynamics and shedding of condensates on textured surfaces have focused on finding the role of micro/nanotexture combined with wettability. In particular, inspired by the Namib desert beetle bump structure, studies tested the effect of topography on the preferential condensation. However, like the preferential condensation observed on flat surfaces, hybrid wettability rather than texture plays a major role; the role of bump topography on local preferential condensation has been unexplored and still not clearly understood. In addition, given that not only facilitating the droplet growth but also transporting the condensed droplets toward the desired reservoir is essential to make fresh sites for renucleation and regrowth of the droplets for enhancing condensation efficiency, the current hybrid-wettability- based design is not efficient to transport the condensates due to the high contact angle hysteresis created by highly wettable pinning points. Here we show that beetle-inspired bump topography leads faster localized condensation and transport of water. Employing simple analytic and more complicated numerical calculations, we reveal the detailed role of topography and predict the focused diffusion flux based on the distortion of concentration gradient around convex surface topography. We experimentally demonstrate the systematic understanding on the unseen effect of topographical parameters on faster droplet growth dynamics on various bump geometries. Further rational design of asymmetric topography and synergetic combination with slippery coating simultaneously enable both faster droplet growth and transport for applications including efficient water condensation.

36 MATERIALS SCIENCE↗
DOE OSTI2015

Production of the thermal scattering law and neutron scattering cross sections for ice from first principles - 14503

Water ice can exist in many polymorphic phases as a result of the wide range of geometric ordering possible for the hydrogen bonds between H{sub 2}O molecules. Ice Ih, which can form under normal atmospheric pressure at 273 K, is the most common type. Ab initio density functional theory, incorporating the generalized gradient approximation (GGA) for the exchange-correlation energy, is used to simulate the hexagonal structure of ice Ih and calculate phonon dispersion relations with harmonic lattice dynamics. While the unit cell for ice Ih contains four molecules with particular alignments of the H{sub 2}O molecules possible, the structure has globally disordered proton arrangement. To improve capturing this random distribution, a 12-molecule unit cell is incorporated along with a 2x2x2 supercell. The total phonon density of states is determined by randomly sampling dispersion relations in the first Brillouin zone. The translational, librational, bending, and stretching modes are produced with high-quality resolution and are consistent with the optical peaks and ranges observed in Raman, infrared, and inelastic neutron scattering experiments [1]. The phonon spectrum is used to calculate the S(α,β) thermal scattering law for ENDF [2] publication in the incoherent approximation, which is appropriate due to the relatively small coherent nuclear scattering cross section for H{sub 2}O. These theoretical results are compared to experimental S(α,β) at 268 K [3] and trends are found to be in good agreement. Finally, total scattering cross sections are calculated by integrating over S(α,β) to determine the incoherent inelastic cross sections and then separately adding the absorption and incoherent elastic components. These results are compared to experimental total cross sections for ice Ih at 115 K over a range of incident energies [4] and found to provide excellent agreement. (authors)

42 ENGINEERING↗
DOE OSTI2026

ZEUS: An Efficient GPU Optimization Method Integrating PSO, BFGS, and Automatic Differentiation

We introduce a novel, efficient computational method, ZEUS, for numerical optimization, and provide an open-source implementation. It has four key ingredients: (1) particle swarm optimization (PSO), (2) the use of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, (3) automatic differentiation (AD), and (4) GPUs. Our approach addresses the computational challenges inherent in high-dimensional, non-convex optimization problems. In the first phase of the algorithm, we get a potentially good set of starting points using PSO. Thereafter, we run BFGS independently in parallel from these starting points. BFGS is one of the best-performing algorithms for numerical optimization. However, it requires the gradient of the function being optimized. ZEUS integrates automatic differentiation into BFGS thus avoiding the need for the user to calculate derivatives explicitly. The use of GPUs allows ZEUS to speed up the calculations substantially. We carry out systematic studies to explore the trade-offs between the number of PSO iterations taken, starting points, and BFGS iteration depth. We show that a handful of iterations of PSO can improve global convergence when combined with BFGS. We also present performance studies using common test functions. The source code can be found at https://github.com/fnal-numerics/global-optimizer-gpu.

Soos, Dominik [Old Dominion U.]↗
DOE OSTI2014

A single-gradient junction technique to replace multiple-junction shifts for craniospinal irradiation treatment

Craniospinal irradiation (CSI) requires abutting fields at the cervical spine. Junction shifts are conventionally used to prevent setup error–induced overdosage/underdosage from occurring at the same location. This study compared the dosimetric differences at the cranial-spinal junction between a single-gradient junction technique and conventional multiple-junction shifts and evaluated the effect of setup errors on the dose distributions between both techniques for a treatment course and single fraction. Conventionally, 2 lateral brain fields and a posterior spine field(s) are used for CSI with weekly 1-cm junction shifts. We retrospectively replanned 4 CSI patients using a single-gradient junction between the lateral brain fields and the posterior spine field. The fields were extended to allow a minimum 3-cm field overlap. The dose gradient at the junction was achieved using dose painting and intensity-modulated radiation therapy planning. The effect of positioning setup errors on the dose distributions for both techniques was simulated by applying shifts of ± 3 and 5 mm. The resulting cervical spine doses across the field junction for both techniques were calculated and compared. Dose profiles were obtained for both a single fraction and entire treatment course to include the effects of the conventional weekly junction shifts. Compared with the conventional technique, the gradient-dose technique resulted in higher dose uniformity and conformity to the target volumes, lower organ at risk (OAR) mean and maximum doses, and diminished hot spots from systematic positioning errors over the course of treatment. Single-fraction hot and cold spots were improved for the gradient-dose technique. The single-gradient junction technique provides improved conformity, dose uniformity, diminished hot spots, lower OAR mean and maximum dose, and one plan for the entire treatment course, which reduces the potential human error associated with conventional 4-shifted plans.

61 RADIATION PROTECTION AND DOSIMETRY↗
NASA NTRS1985

Analysis of data from NASA B-57B gust gradient program

Statistical analysis of the turbulence measured in flight 6 of the NASA B-57B over Denver, Colorado, from July 7 to July 23, 1982 included the calculations of average turbulence parameters, integral length scales, probability density functions, single point autocorrelation coefficients, two point autocorrelation coefficients, normalized autospectra, normalized two point autospectra, and two point cross sectra for gust velocities. The single point autocorrelation coefficients were compared with the theoretical model developed by von Karman. Theoretical analyses were developed which address the effects spanwise gust distributions, using two point spatial turbulence correlations.

Frost, W.↗
NASA NTRS1998

Preston Probe Calibrations at High Reynolds Number

The overall goal of the research effort is to study the performance of two Preston probes designed by NASA Langley Research Center across an unprecedented range of Reynolds number (based on friction velocity and probe diameter), and perform an accurate calibration over the same Reynolds number range. Using the Superpipe facility in Princeton, two rounds of experiments were performed. In each round of experiments for each Reynolds number, the pressure gradient, static pressure from the Preston probes and the total pressure from the Preston probes were measured. In the first round, 3 Preston probes having outer diameters of 0.058 inches, 0.083 inches and 0.203 inches were tested over a large range of pipe Reynolds numbers. Two data reduction methods were employed: first, the static pressure measured on the Preston probe was used to calculate P (modified Preston probe configuration), and secondly, the static pressure measured at the reference pressure tap was used to calculate P (un-modified Preston probe configuration). For both methods, the static pressure was adjusted to correspond with the static pressure at the Preston probe tip using the pressure gradient. The measurements for Preston probes with diameters of 0.058 inches, and 0.083 inches respectively were performed in the test pipe before it was polished a second time. Therefore, the measurements at high pipe Reynolds numbers may have been affected by roughness. In the second round of experiments the 0.058 inches and 0.083 inches diameter, un-modified probes were tested after the pipe was polished and prepared to ensure that the surface was smooth. The average velocity was estimated by assuming that the connection between the centerline velocity and the average velocity was known, and by using a Pitot tube to measure the centerline velocity. A preliminary error estimate suggests that it is possible to introduce a 1% to 2% error in estimating the average velocity using this approach. The evidence on the errors attending the second data set is somewhat circumstantial, and the measurements have not been repeated using a better approach, it seems probable that the correlation given applies to un-modified Preston probes over the range 6.4 less than x* less than 11.3.

Smits, Alexander J.↗
DOE OSTI2022

Electronic Structure Calculations of the Fundamental Interactions in Solvent Extraction Desalination

Solvent extraction desalination (SED) is a promising membrane-free desalination technology, but new solvents are still needed that can improve the molecular-level water selectivity, while minimizing solvent contamination in the aqueous phase. Improvements in solvent design for SED require a fundamental understanding of the molecular interactions governing solvation behavior. Herein, quantum chemical calculations are used to delineate different fundamental interactions in water-solvent systems involving 38 different solvents, with a particular focus on the role of the electrostatic potential characteristics. Furthermore, these interactions are rigorously analyzed using the independent gradient model based on Hirshfeld partition (IGMH), Symmetry-Adapted Perturbation Theory (SAPT), as well as the energy decomposition analysis based on force field (EDA-FF) to compare against a standard force field model. The solvation free energies and partition coefficients are also calculated to estimate water extraction performance in a bulk solvent system.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1960

Effects of Sweep Angle on the Boundary-Layer Stability Characteristics of an Untapered Wing at Low Speeds

An investigation was conducted in the Ames 12-Foot Low-Turbulence Pressure Tunnel to determine the effects of sweep on the boundary-layer stability characteristics of an untapered variable-sweep wing having an NACA 64(2)A015 section normal to the leading edge. Pressure distribution and transition were measured on the wing at low speeds at sweep angles of 0, 10, 20, 30, 40, and 50 deg. and at angles of attack from -3 to 3 deg. The investigation also included flow-visualization studies on the surface at sweep angles from 0 to 50 deg. and total pressure surveys in the boundary layer at a sweep angle of 30 deg. for angles of attack from -12 to 0 deg. It was found that sweep caused premature transition on the wing under certain conditions. This effect resulted from the formation of vortices in the boundary layer when a critical combination of sweep angle, pressure gradient, and stream Reynolds number was attained. A useful parameter in indicating the combined effect of these flow variables on vortex formation and on beginning transition is the crossflow Reynolds number. The critical values of crossflow Reynolds number for vortex formation found in this investigation range from about 135 to 190 and are in good agreement with those reported in previous investigations. The values of crossflow Reynolds number for beginning transitions were found to be between 190 and 260. For each condition (i.e., development of vortices and initiation of transition at a given location) the lower values in the specified ranges were obtained with a light coating of flow-visualization material on the surface. A method is presented for the rapid computation of crossflow Reynolds number on any swept surface for which the pressure distribution is known. From calculations based on this method, it was found that the maximum values of crossflow Reynolds number are attained under conditions of a strong pressure gradient and at a sweep angle of about 50 deg. Due to the primary dependence on pressure gradient, effects of sweep in causing premature transition are generally first encountered on the lower surfaces of wings operating at positive angles of attack.

Boltz, Frederick W.↗
NASA NTRS1994

Quasi-horizontal transport and mixing in the Antarctic stratosphre

The quasi-horizontal transport and mixing properties of the Antarctic stratosphere are investigated with a simi-Lagrangian transport model and a 'contour advection' technique for the winter and spring of 1992 using analyzed winds from the United Kingdom Meteorological Office data assimiliation system. Transport calculations show that passive tracers are well mixed inside the polar vortex as well as in the midlatitude 'surf zone.' A the vortex edge, strong radial gradients in the tracer fields are well preserved, and their evolutions follow that of the potential vorticity until some time after the breakdown of the polar vortex. In the middle stratosphere there is little tracer exchange across the vortex edge in August and September. Some vortex air is eroded into the surf zone in filamentary form in October, and very strong exchange of air occurs between high and middle latitudes in November. In the lower stratosphere the vortex is not so isolated from the midlatitudes as in the middle stratosphere, and there is more mass exchange across the vortex edge. Calculations of the lengthening of material contours using the contour advection technique show that in the middle stratosphere, strong stirring (i.e., stretching and folding of material elements) occurs in the inner vortex, while the strongest stirring occurs in the midlatitude surf zone and the weakest occurs at the vortex edge. In the lower strtosphere, strong stirring occurs in the inner vortex. Stirring is moderate at the vortex edge and in the midlatitudes.

Chen, Ping↗
DOE OSTI2017

Models of Warm Jupiter Atmospheres: Observable Signatures of Obliquity

We present three-dimensional atmospheric circulation models of a hypothetical “warm Jupiter” planet, for a range of possible obliquities from 0° to 90°. We model a Jupiter-mass planet on a 10 day orbit around a Sun-like star, since this hypothetical planet sits at the boundary between planets for which we expect that tidal forces should have aligned their rotation axes with their orbital axes (i.e., ones with zero obliquity) and planets whose timescale for tidal alignment is longer than the typical age of an exoplanet system. In line with observational progress, which is pushing atmospheric characterization for planets on longer orbital periods, we calculate the observable signatures of obliquity for a transiting warm Jupiter: in orbital phase curves of thermal emission and in the hemispheric flux gradients that could be measured by eclipse mapping. For both of these predicted measurements, the signal that we would see depends strongly on our viewing geometry relative to the orientation of the planet’s rotation axis, and we thoroughly identify the degeneracies that result. We compare these signals to the predicted sensitivities of current and future instruments and determine that the James Webb Space Telescope should be able to constrain the obliquities of nearby warm Jupiters to be small (if ⩽10{sup ∘}) or to directly measure them if significantly non-zero (⩾30{sup ∘}) using the technique of eclipse mapping. For a bright target and assuming photon-limited precision, this could be done with a single secondary eclipse observation.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2004

On the Measurement of Elemental Abundance Ratios in Inner Galaxy H II Regions

Although abundance gradients in the Milky Way Galaxy certainly exist, details remain uncertain, particularly in the inner Galaxy, where stars and H II regions in the Galactic plane are obscured optically. In this paper we revisit two previously studied, inner Galaxy H II regions: G333.6-0.2 and W43. We observed three new positions in G333.6-0.2 with the Kuiper Airborne Observatory and reobserved the central position with the Infrared Space Observatory's Long Wavelength Spectrometer in far-infrared lines of S++, N++, N+, and O++. We also added the N+ lines at 122 and 205 microns to the suite of lines measured in W43 by Simpson et al.. The measured electron densities range from approx. 40 to over 4000 per cu cm in a single HII region, indicating that abundance analyses must consider density variations, since the critical densities of the observed lines range from 40 to 9000 per cu cm. We propose a method to handle density variations and make new estimates of the S/H and N/H abundance ratios. We find that our sulfur abundance estimates for G333.6-0.2 and W43 agree with the S/H abundance ratios expected for the gradient previously reported by Simpson et al., with the S/H values revised to be smaller owing to changes in collisional excitation cross sections. The estimated N/H, S/H, and N/S ratios are the most reliable because of their small corrections for unseen ionization states (< or approx. 10%). The estimated N/S ratios for the two sources are smaller than what would be calculated from the N/H and S/H ratios in our previous paper. If all low excitation H II regions had similar changes to their N/S ratios as a result of adding measurements of N+ to previous measurements of N++, there would be no or only a very small gradient in N/S. This is interesting because nitrogen is considered to be a secondary element and sulfur is a primary element in galactic chemical evolution calculations. We compute models of the two H II regions to estimate corrections for the other unseen ionization states. We find, with large uncertainties, that oxygen does not, have a high abundance, with the result that the N/O ratio is as high (approx. 0.35) as previously reported. The reasons for the uncertainty in the ionization corrections for oxygen are both the non-uniqueness of the H II region models and the sensitivity of these models to different input atomic data and stellar atmosphere models. We discuss these predictions and conclude that only a few of the latest models adequately reproduce H II region observations, including the well-known, relatively-large observed Ne++/O++ ratios in low- and moderate-excitation H II regions.

Simpson, Janet P.↗
DOE OSTI2014

Rare earth element components in atmospheric particulates in the Bayan Obo mine region

The Bayan Obo mine, located in Inner Mongolia, China, is the largest light rare earth body ever found in the world. The research for rare earth elements (REEs) enrichment in atmospheric particulates caused by mining and ore processing is fairly limited so far. In this paper, atmospheric particulates including total suspended particulate (TSP) matter and particles with an equivalent aerodynamic diameter less than 10 μm (PM{sub 10}) were collected around the Bayan Obo mine region, in August 2012 and March 2013, to analyze the levels and distributions of REEs in particles. The total concentrations of REEs for TSP were 149.8 and 239.6 ng/m{sup 3}, and those for PM{sub 10} were 42.8 and 68.9 ng/m{sup 3}, in August 2012 and March 2013, respectively. Enrichment factor was calculated for all 14 REEs in the TSP and PM{sub 10} and the results indicated that REEs enrichment in atmosphere particulates was caused by anthropogenic sources and influenced by the strong wind in springtime. The spatial distribution of REEs in TSP showed a strong gradient concentration in the prevailing wind direction. REE chondrite normalized patterns of TSP and PM{sub 10} were similar and the normalized curves inclined to the right side, showing the conspicuous fractionation between the light REEs and heavy REE, which supported by the chondrite normalized concentration ratios calculated for selected elements (La{sub N}/Yb{sub N}, La{sub N}/Sm{sub N}, Gd{sub N}/Yb{sub N}). - Highlights: • TSP and PM{sub 10} samples were collected to analyze the levels and distributions of REE. • Enrichment factors indicated that REE enrichment was caused by anthropogenic sources. • The distribution of REEs showed a strong gradient in the prevailing wind direction. • Obvious fractionation between LREEs and HREEs is observed in atmospheric particulates.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2014

Density functional calculations for structural, electronic, and magnetic properties of gadolinium-oxide clusters

Gadolinium-oxide clusters in various sizes and stoichiometries have been systematically studied by employing the density functional theory with the generalized gradient approximation. The clusters in bulk stoichiometry are relatively more stable and their binding energies increase with the increasing size. Stoichiometric (Gd{sub 2}O{sub 3}){sub n} clusters of n = 1–3 prefer cage-like structures, whereas the clusters of n = 4–30 prefer compact structures layered by wedge-like units and exhibit a rough feature toward the bulk-like arrangement with small disorders of atomic positions. The polyhedral-cages analogous to carbon-fullerenes are stable isomers yet not the minimum energy configurations. Their stabilities can be improved by embedding one oxygen atom or a suitable cage to form core-shell configurations. The mostly favored antiferromagnetic couplings between adjacent Gd atoms are nearly degenerated in energy with their ferromagnetic couplings, resulting in super-paramagnetic characters of gadolinium-oxide clusters. The Ruderman-Kittel-Kasuya-Yosida (RKKY)-type mechanism together with the superexchange-type mechanism plays cooperation role for the magnetic interactions in clusters. We present, as a function of n, calculated binding energies, ionization potential, electron affinity, and electronic dipole moment.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1992

Effects of weather on the retrieval of sea ice concentration and ice type from passive microwave data

Effects of wind, water vapor, and cloud liquid water on ice concentration and ice type calculated from passive microwave data are assessed through radiative transfer calculations and observations. These weather effects can cause overestimates in ice concentration and more substantial underestimates in multi-year ice percentage by decreasing polarization and by decreasing the gradient between frequencies. The effect of surface temperature and air temperature on the magnitudes of weather-related errors is small for ice concentration and substantial for multiyear ice percentage. The existing weather filter in the NASA Team Algorithm addresses only weather effects over open ocean; the additional use of local open-ocean tie points and an alternative weather correction for the marginal ice zone can further reduce errors due to weather. Ice concentrations calculated using 37 versus 18 GHz data show little difference in total ice covered area, but greater differences in intermediate concentration classes. Given the magnitude of weather-related errors in ice classification from passive microwave data, corrections for weather effects may be necessary to detect small trends in ice covered area and ice type for climate studies.

Maslanik, J. A.↗
NASA NTRS1977

Interactive calculation procedure for supersonic flows

An interactive procedure was developed for supersonic viscous flows that can be used for either two-dimensional or axisymmetric configurations. The procedure is directed to supersonic internal flows as well as those supersonic external flows that require consideration of mutual interaction between the outer flow and the boundary layer flow. The flow field is divided into two regions: an inner region which is highly viscous and mostly subsonic and an outer region where the flow is supersonic and in which viscous effects are small but not negligible. For the outer region a numerical solution is obtained by applying the method of characteristics to a system of equations which includes viscous and conduction transport terms only normal to the streamlines. The inner region is treated by a system of equations of the boundary layer type that includes higher order effects such as longitudinal and transverse curvature and normal pressure gradients. These equations are coupled and solved simultaneously in the physical coordinates by using an implicit finite difference scheme. This system can also be used to calculate laminar and turbulent boundary layers using a scalar eddy viscosity concept.

Tassa, Y.↗