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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 613 records · Page 34
All sourcesNASA NTRSDOE OSTI
NASA NTRS2003

Time-Dependent Material Data Essential for the Durability Analysis of Composite Flywheels Provided by Compressive Experiments

Successful spaceflight operations require onboard power management systems that reliably achieve mission objectives for a minimal launch weight. Because of their high specific energies and potential for reduced maintenance and logistics, composite flywheels are an attractive alternative to electrochemical batteries. The Rotor Durability Team, which comprises members from the Ohio Aerospace Institute (OAI) and the NASA Glenn Research Center, completed a program of elevated temperature testing at Glenn' s Life Prediction Branch's Fatigue Laboratory. The experiments provided unique design data essential to the safety and durability of flywheel energy storage systems for the International Space Station and other manned spaceflight applications. Analysis of the experimental data (ref. 1) demonstrated that the compressive stress relaxation of composite flywheel rotor material is significantly greater than the commonly available tensile stress relaxation data. Durability analysis of compression preloaded flywheel rotors is required for accurate safe-life predictions for use in the International Space Station.

Thesken, John C.↗
DOE OSTI2022

Mesoscale simulations of pressure-shear loading of granular tungsten carbide

Numerical simulations of pressure-shear loading of a granular material are performed using the shock physics code CTH. A simple mesoscale model for the granular material is used that consists of a randomly packed arrangement of solid circular or spherical grains of uniform size separated by vacuum. The grain material is described by a simple shock equation of state, elastic perfectly plastic strength model, and fracture model with baseline parameters for WC taken from previous mesoscale modeling work. Simulations using the baseline material parameters are performed at the same initial conditions of pressure-shear experiments on dry WC powders. Except for some localized flow regions appearing in simulations with an approximate treatment of sliding interfaces among grains, the samples respond elastically during shear, which is in contrast to experimental observations. By extending the simulations to higher shear wave amplitudes, macroscopic shear failure of the simulated samples is observed with the shear strength increasing with increasing stress confinement. The shear strength is also found to be strongly dependent on the grain interface treatment and on the fracture stress of the grains, though the variation in shear strength due to fracture stress decreases with increasing stress confinement. At partial compactions, the transverse velocity histories show strain-hardening behavior followed by formation of a shear interface that extends through the transverse dimensions of the sample. Near full compaction, no strain hardening is observed and, instead, the sample transitions sharply from an elastic response to formation of an internal shear interface. Agreement with experiment is shown to worsen with increasing confinement stress with simulations overpredicting the shear strengths measured in experiment. The source of the disagreement can be ultimately attributed to the Eulerian nature of the simulations, which do not treat contact and fracture realistically.

36 MATERIALS SCIENCE↗
NASA NTRS1991

Effect of interlaminar normal stresses on the uniaxial zero-to-tension fatigue behavior of graphite/epoxy tubes

Fatigue tests conducted with and without internal pressure have been found to possess approximately the same fatigue life as (+/-45)s graphite/epoxy tubes for zero-to-tension axial load-controlled conditions on an axial torsion servohydraulic apparatus. These tests therefore cannot be considered as confirmations of the suspected detrimental effect of interlaminar tensile stresses on the fatigue performance of thin-walled tubes. The addition of 90-deg plies on both the inside and the outside is found to significantly improve the tubes' static and fatigue strengths.

Krempl, Erhard↗
DOE OSTI2024

Why Perovskite Thermal Stress is Unaffected by Thin Contact Layers

Metal halide perovskite photovoltaics have emerged as a high efficiency, low-cost alternative that can potentially rival or enhance conventional silicon technology. Despite exceptional initial power conversion efficiencies, achieving compliance with international standards and widespread adoption requires further enhancements to their operational stability. Notably, addressing mechanical strain and stress in brittle perovskites has emerged as a pivotal approach to mitigate chemical degradation and improve reliability during thermal cycling. Here, in this study, a popularized strain engineering strategy is investigated in which a high coefficient of thermal expansion (CTE) hole transport layer (i.e., PDCBT) is cast onto inorganic perovskite (CsPbI 2 Br) at 100 °C. Contrary to previously published results, the X-ray diffraction (XRD):Sin 2 ψ and substrate curvature measurement techniques show that the hole transport layer has no discernible impact on perovskite strain. The accuracy of the XRD:Sin 2 ψ method for measuring strain is highlighted in contrast to an analysis based on shifts of single XRD peaks which can be influenced by multiple artifacts. The findings in this study are in accordance with mechanics theory: thin layers are unable to induce significant strain changes in perovskite thin films as the force they apply is negligible compared to that applied by a thick and stiff substrate.

14 SOLAR ENERGY↗
DOE OSTI2020

Technical Performance of Refractory Liners for Molten Chloride Salt Thermal Energy Storage Systems

A chloride-based molten-salt system that uses a ternary blend of MgCl2/KCl/NaCl is investigated to provide higher temperature thermal energy storage capability. Despite higher thermal stability, molten chlorides present several unique challenges, including the design of internal refractory-ceramic liners to prevent the corrosion and thermal stress of alloy tank shells. This work discusses issues and potential solutions related to containment of molten chloride salt, specifically the optimization of the refractory material at the molten salt interface (hot face). The down-selected hot face candidate limits permeation of salt through the material and forms a highly stable secondary surface phase in equilibrium with the molten salt. A mortar is created using the corrosion resistant hot face brick. Brick and mortar composite are subjected to mechanical stress/strain analysis, in order to calculate composite material properties and better inform thermomechanical models. The U.S. Department of Energy Generation 3 (DOE Gen3) program seeks to develop higher efficiency CSP plants that can provide cost-competitive, flexible power in the U.S. electric grid. The proposed Gen3 Liquid Pathway CSP plant closely resembles the configuration of current nitrate salt power towers with two-tank storage (Gen2). The differences between Gen2 and Gen3 include the types of compatible materials used in salt storage tank construction. Stainless steel loses strength at Gen3 temperatures, and although nickel superalloys would be capable of withstanding sustained high temperatures, these materials are prohibitively expensive at scale. Uninsulated tank shells also pose a significant risk as common steels are highly susceptible to chemical attack from molten chloride salt. To address these concerns, refractory-ceramic based containment materials are proposed to line the inside of the hot and cold storage tanks. In doing so, stainless or carbon steel shells may be used in construction depending on the level of insulation provided. The composition of the internal liner requires careful consideration to maximize the efficacy of multiple parameters including corrosion resistance, strength at operating temperature, durability, and cost. This is particularly true for the material at the interface with the salt, known as the "hot face", which is responsible for protecting the insulating layers between the tank shell and the hot face brick layer. The molten salt in this system is superheated over 300 °C above its freeze temperature. Therefore, unlike other industrial processes which use refractory-lined vessels, it is not expected that a freeze plane will develop in the hot face. Therefore, the hot face must be designed to withstand chemical corrosion and inhibit permeation of molten salt into the insulating layers. A down selection was performed to identify a hot face candidate best equipped to maintain thermal, mechanical, and chemical integrity when exposed to molten salt over extended periods of time. Long-duration chemical capability experiments were conducted with the down selected hot face refractory fully immersed in molten chloride salt for up to 3000 hours. The average salt penetration does not exceed 100 microns. When extrapolated to 20 and 30 years of continuous exposure, the expected salt penetration depth is approximately 2.0 mm and 2.9 mm, respectively. A magnesium-rich secondary phases develops at the salt/refractory interface. X-ray diffraction identifies the material as forsterite (Mg2SiO4), which is reported to form synthetically in molten chloride salt solutions. These results suggest the selected hot face will adequately inhibit salt permeation. While there is optimism that the hot face brick will inhibit salt permeation, mortar joints are typically the weakest point of a refractory brick liner. From a thermochemical perspective, differing thermal expansion coefficients may result in the mortar and brick to grow independent of each other, creating gaps through which molten salt can penetrate. To address this issue, NREL has developed an in-house mortar composed of the down selected hot face brick that has been shown to be compatible with the salt. Compressive stress/strain analyses have been performed on the brick/ mortar composites to generate stress/strain curves. Modulus of elasticity and Poisson's ratio of the composite may be calculated from the stress/strain curves, in order to provide more representative data to finite element mechanical models for accurate approximation of stress on the tank shell and the amount of thermal expansion expected within the tank liner.

41 EE - Solar Energy Technologies Office (EE-4S)↗
NASA NTRS2016

Metabolomic and Genomic Markers of Atherosclerosis as Related to Oxidative Stress, Inflammation, and Vascular Function in Twin Astronauts

Background: Future human space travel will consist primarily of long-duration missions onboard the International Space Station (ISS) or exploration-class missions to Mars, its moons, or nearby asteroids. Astronauts participating in long-duration missions may be at an increased risk of oxidative stress and inflammatory damage due to radiation, psychological stress, altered physical activity, nutritional insufficiency, and hyperoxia during extravehicular activity. By studying one identical twin during his 1-year ISS mission and one ground-based twin, this work extends a current NASA-funded investigation to determine whether these spaceflight factors contribute to an accelerated progression of atherosclerosis. This study of twins affords a unique opportunity to examine the spaceflight-related atherosclerosis risk independent of the confounding factors associated with different genotypes. Purpose: The purpose of this investigation is to determine whether biomarkers of oxidative and inflammatory stress are elevated during and after long-duration spaceflight and determine if a relation exists between levels of these biomarkers and structural and functional indices of atherosclerotic risk measured in the carotid and brachial arteries. These physiological and biochemical data will be extended by using an exploratory approach to investigate the relationship between intermediate phenotypes and risk factors for atherosclerosis and the metabolomic signature from plasma and urine samples. Since metabolites are often the indirect products of gene expression, we will simultaneously assess gene expression and DNA methylation in leukocytes. Hypothesis: We predict that the space-flown twin will experience elevated biomarkers of oxidative stress and inflammatory damage, altered arterial structure and function, accelerated telomere shortening, dysregulation of genes associated with oxidative stress and inflammation, and a metabolic profile shift that is associated with elevated atherosclerosis risk factors. Conversely, these will not be observed in the ground-based twin. Methods: We will measure blood and urine biomarkers of oxidative stress and inflammation as well as arterial structure and function (carotid intima-medial thickness and brachial artery flow-mediated dilation) in one twin astronaut before, during, and after long-duration spaceflight and in his twin serving as a ground-based control. Furthermore, we will measure metabolomics (targeted and untargeted approaches) and genomic markers (DNA methylation, mRNA gene expression, telomere length) to elucidate the molecular mechanisms involved. A panel of biomarkers of oxidative and inflammatory stress will be measured in venous blood samples and 24-hour (in-flight) and 48-hour (pre- and post-flight) urine pools twice before flight, early (flight days 15 and 60) and late (2 weeks before landing) during the mission, and early in the post-flight recovery phase (approximately 3-5 days after landing). Arterial structure, assessed from measures of intima-media thickness, will be measured at the same times. Arterial function will be assessed using brachial flow-mediated dilation, a well-validated measure used to assess endothelium-dependent vasodilation and a sensitive predictor of atherosclerotic risk, only before and after spaceflight. Discussion: Pre- and in-flight data collection is in progress for the space-flown twin, and similar data have been obtained from the ground-based twin. Blood and urine samples will be batch processed when received from ISS after the conclusion of the 1-year mission. Results from these individual subjects will be compared to the larger complement of subjects participating in the companion study currently ongoing in ISS astronauts.

Lee, Stuart M. C.↗
NASA NTRS2017

Metabolomic and Genomic Markers of Atherosclerosis as Related to Oxidative Stress, Inflammation, and Vascular Function in Twin Astronauts

BACKGROUND: Future human space travel will consist primarily of long-duration missions onboard the International Space Station (ISS) or exploration-class missions to Mars, its moons, or nearby asteroids. Astronauts participating in long-duration missions may be at an increased risk of oxidative stress and inflammatory damage due to radiation, psychological stress, altered physical activity, nutritional insufficiency, and hyperoxia during extravehicular activity. By studying one identical twin during his 1-year ISS mission and his ground-based twin, this work extends a current NASA-funded investigation to determine whether these spaceflight factors contribute to an accelerated progression of atherosclerosis. This study of twins affords a unique opportunity to examine spaceflight-related atherosclerosis risk that is independent of the confounding factors associated with different genotypes. PURPOSE: The purpose of this investigation was to determine whether biomarkers of oxidative and inflammatory stress are elevated during and after long-duration spaceflight and determine if a relation exists between levels of these biomarkers and structural and functional indices of atherosclerotic risk measured in the carotid and brachial arteries. These physiological and biochemical data will be extended by using an exploratory approach to investigate the relationship between intermediate phenotypes and risk factors for atherosclerosis and the metabolomic signature from plasma and urine samples. Since metabolites are often the indirect products of gene expression, we simultaneously assessed gene expression and DNA methylation in leukocytes. HYPOTHESIS: We predict that, compared to the ground-based twin, the space-flown twin will experience elevated biomarkers of oxidative stress and inflammatory damage, altered arterial structure and function, accelerated telomere shortening, dysregulation of genes associated with oxidative stress and inflammation, and a metabolic profile shift that is associated with elevated atherosclerosis risk factors. METHODS: In the space-flown twin, a panel of biomarkers of oxidative and inflammatory stress were measured in venous blood samples and in 24-h (in-flight) and 48-h (pre- and post-flight) urine pools collected twice before flight, six times during the mission (~FD15, 75, 180, 240, 300, 335), and early in the post-flight recovery phase (3-5 days after landing). We also measured metabolomic (targeted and untargeted approaches) and genomic markers (DNA methylation, mRNA gene expression, telomere length) in these samples. Arterial structure, assessed from measures of intima-media thickness, also were measured using standard clinical ultrasound at the same time points. Arterial function was assessed using brachial flow-mediated dilation, a well-validated measure used to assess endothelium-dependent vasodilation and a sensitive predictor of atherosclerotic risk, only before and after spaceflight. All of the same measures were obtained in the ground-based twin, but less frequently. DISCUSSION: All data collection has been completed for both the space-flown twin and the ground-based twin. Vascular structure and function measures have been analyzed, blood and urine samples have been batch-processed. Results from these individuals will be compared to each other, to data from other Twin Study investigations, and to the larger complement of subjects participating in the companion study currently ongoing in ISS astronauts.

Lee, Stuart M. C.↗
NASA NTRS2004

A Novel Inter Core-Cladding Lithium Niobate Thin Film Coated Fiber Modulator/Sensor

A fiber modulator/sensor has been fabricated by depositing a lithium niobate sol-gel thin film between the core and cladding of a fiber preform. The preform is then drawn into 125 micron fiber. The proposed design of lithium niobate cylinder fibers can enhance the existing methodology for detecting sound waves under water utilizing the acoustooptic properties of lithium niobate. Upon application of a stress or strain, light propagating inside the core, according to the principle of total internal reflection, escapes, into the cladding because of the photoelastic boundary layer of lithium niobate. Test results of the lithium niobate fiber reveal a reduction in the 1550 nm, 4mW source with applied tension. The source power from an ordinary quartz fiber under the same stress condition remained invariant to applied tension.

Jamison, Tracee L.↗
NASA NTRS1992

Structural Tailoring of Advanced Turboprops (STAT). Theoretical manual

This manual describes the theories in the Structural Tailoring of Advanced Turboprops (STAT) computer program, which was developed to perform numerical optimizations on highly swept propfan blades. The optimization procedure seeks to minimize an objective function, defined as either direct operating cost or aeroelastic differences between a blade and its scaled model, by tuning internal and external geometry variables that must satisfy realistic blade design constraints. The STAT analyses include an aerodynamic efficiency evaluation, a finite element stress and vibration analysis, an acoustic analysis, a flutter analysis, and a once-per-revolution (1-p) forced response life prediction capability. The STAT constraints include blade stresses, blade resonances, flutter, tip displacements, and a 1-P forced response life fraction. The STAT variables include all blade internal and external geometry parameters needed to define a composite material blade. The STAT objective function is dependent upon a blade baseline definition which the user supplies to describe a current blade design for cost optimization or for the tailoring of an aeroelastic scale model.

Brown, K. W.↗
NASA NTRS1994

Thermocryogenic buckling and stress analyses of a partially filled cryogenic tank subjected to cylindrical strip heating

Thermocryogenic buckling and stress analyses were conducted on a horizontally oriented cryogenic tank using the finite element method. The tank is a finite-length circular cylindrical shell with its two ends capped with hemispherical shells. The tank is subjected to cylindrical strip heating in the region above the liquid-cryogen fill level and to cryogenic cooling below the fill level (i.e., under thermocryogenic loading). The effects of cryogen fill level on the buckling temperature and thermocryogenic stress field were investigated in detail. Both the buckling temperature and stress magnitudes were relatively insensitive to the cryogen fill level. The buckling temperature, however, was quite sensitive to the radius-to-thickness ratio. A mechanical stress analysis of the tank also was conducted when the tank was under: (1) cryogen liquid pressure loading; (2) internal pressure loading; and (3) tank-wall inertia loading. Deformed shapes of the cryogenic tanks under different loading conditions were shown, and high-stress domains were mapped on the tank wall for the strain-gage installations. The accuracies of solutions from different finite element models were compared.

Ko, William L.↗
DOE OSTI2025

An Internal Digital Image Correlation Technique for High-Strain Rate Dynamic Experiments

Full-field, quantitative visualization techniques, such as digital image correlation (DIC), have unlocked vast opportunities for experimental mechanics. However, DIC has traditionally been a surface measurement technique, and has not been extended to perform measurements on the interior of specimens for dynamic, full-scale laboratory experiments. This limitation restricts the scope of physics which can be investigated through DIC measurements, especially in the context of heterogeneous materials. The focus of this study is to develop a method for performing internal DIC measurements in dynamic experiments. The aim is to demonstrate its feasibility and accuracy across a range of stresses (up to 650 MPa), strain rates (10 3 - 10 6 s -1 ), and high-strain rate loading conditions (e.g., ramped and shock wave loading). Internal DIC is developed based on the concept of applying a speckle pattern at an inner-plane of a transparent specimen. The high-speed imaging configuration is coupled to the traditional dynamic experimental setups, and is focused on the internal speckle pattern. During the experiment, while the sample deforms dynamically, in-plane, two-dimensional deformations are measured via correlation of the internal speckle pattern. In this study, the viability and accuracy of the internal DIC technique is demonstrated for split-Hopkinson (Kolsky) pressure bar (SHPB) and plate impact experiments. The internal DIC experimental technique is successfully demonstrated in both the SHPB and plate impact experiments. In the SHPB setting, the accuracy of the technique is excellent throughout the deformation regime, with measurement noise of approximately 0.2% strain. In the case of plate impact experiments, the technique performs well, with error and measurement noise of 1% strain. The internal DIC technique has been developed and demonstrated to work well for full-scale dynamic high-strain rate and shock laboratory experiments, and the accuracy is quantified. Here, the technique can aid in investigating the physics and mechanics of the dynamic behavior of materials, including local deformation fields around dynamically loaded material heterogeneities.

36 MATERIALS SCIENCE↗
NASA NTRS1994

Thermally induced damage in composite space structure: Predictive methodology and experimental correlation

A general analysis method is presented to predict matrix cracks in all plies of a composite laminate, and resulting degraded laminate properties, as functions of temperature or thermal cycles. A shear lag solution of the stresses in the vicinity of cracks and a fracture mechanics crack formation criteria are used to predict cracks. Damage is modeled incrementally, which allows the inclusion of the effects of temperature dependent material properties and softening of the laminate due to previous cracking. The analysis is incorporated into an easy-to-use computer program. The analysis is correlated with experimentally measured crack densities in a variety of laminates exposed to monotonically decreasing temperatures. Crack densities are measured at the edges of specimens by microscopic inspection, and throughout the specimen volumes by x-ray and sanding down of the edges. Correlation between the analytical results and the crack densities in the interiors of the specimens was quite good. Crack densities measured at specimen edges did not agree with internal crack densities (or analyses) in some cases. A free-edge stress analysis clarified the reasons for these discrepancies.

Park, Cecelia↗
NASA NTRS1984

Evolution of assumed stress hybrid finite element

Early versions of the assumed stress hybrid finite elements were based on the a priori satisifaction of stress equilibrium conditions. In the new version such conditions are relaxed but are introduced through additional internal displacement functions as Lagrange multipliers. A rational procedure is to choose the displacement terms such that the resulting strains are now of complete polynomials up to the same degree as that of the assumed stresses. Several example problems indicate that optimal element properties are resulted by this method.

Pian, T. H. H.↗
DOE OSTI2020

Transformation-induced fracture toughening in CuAlBe shape memory alloys: A phase-field study

Stress-induced transformation ahead of a crack tip can relax the stress concentration and produce fracture toughness in shape memory alloys (SMAs). In this manuscript, we utilize a nonisothermal phase-field model (PFM) to study the martensitic transformation induced crack tip toughening in CuAlBe SMA. The force-displacement curve, transformation zone and high stress zone in single crystalline samples show high dependency on the grain orientation with respect to the crack alignment. Comparison between isothermal and non-isothermal simulations reveals that the transformation-induced self-heating decreases the toughening capability by increasing the critical transformation stress. Investigating the high stress zones in CuAlBe and the nontransforming CuAl alloy shows that the toughening is obtained by redistributing the stress concentration far from the crack tip, as the high stress zone follows the tip of the transformation zone. Increasing the acuity of the crack tip is found to generate more symmetric martensite wings on both side of the crack axis and a more localized high stress zone. The polycrystal specimen displays higher toughening due to the internal constraints related to the presence of various grains with difference orientations. Depending on the orientation of the grain inclosing the crack tip, the toughening can be lower or higher. Finally, coincidence of the crack tip with a triple junction is found to improve the toughening behavior.

42 ENGINEERING↗
DOE OSTI2026

3D probabilistic fracture mechanics / computational fluid dynamics simulation of a reactor pressure vessel under transient conditions

Reactor pressure vessels (RPVs) are safety-critical light-water-reactor components that, under irradiation, experience long-term material degradation in the form of embrittlement. This can increase their susceptibility to fracture under thermal-shock conditions, which could occur during off-normal transients such as loss-of-coolant accidents (LOCAs). During a LOCA, the most severe conditions for the RPV occur when emergency core cooling water is injected through the cold legs into the water-and-steam-filled RPV. The rapid cooling of the downcomer and internal RPV surface causes decreased temperature and elevated thermally driven tensile stresses in the RPV wall. This, combined with long-term material embrittlement, may cause fracture initiation at pre-existing flaws, challenging the integrity of the RPV. Assessing RPV integrity during transients with a large spatial variation in the coolant temperature requires a modeling approach that considers the effects of spatially varying coolant temperature on the fracture probability of a population of flaws distributed throughout the RPV, accounting for spatially varying embrittlement. Here, the present study addresses this need by demonstrating first-of-its-kind coupling of a high-fidelity 3D computational fluid dynamics code with 3D probabilistic fracture mechanics This was accomplished using representative models of a pressurized-water reactor subjected to small- and medium-break LOCA conditions, both of which can result in large spatial temperature variations. While the observed impact of accounting for 3D effects was minimal under the small-break LOCA this study indicates a significant increase in the probability of fracture initiation under the medium-break LOCA when 3D effects are considered, relative to a spatially uniform cooling scenario.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1971

Moonquakes.

Although the average rate of seismic energy release within the moon appears to be far below that of the earth, over 100 events believed to be moonquakes have been recorded by the two seismic stations installed on the lunar surface during Apollo missions 12 and 14. With few exceptions, the moonquakes occur at monthly intervals near times of perigee and apogee and show correlations with the longer-term (7-month) lunar gravity variations. The repeating moonquakes are believed to occur at not less than 10 different locations. However, a single focal zone accounts for 80 per cent of the total seismic energy detected. This active zone appears to be 600 kilometers south-southwest of the Apollo 12 and 14 sites and deep within the moon. Each focal zone must be small (less than 10 kilometers in linear dimension) and fixed in location over a 14-month period. Cumulative strain at each location is inferred. Thus, the moonquakes appear to be releasing internal strain of unknown origin, the release being triggered by tidal stresses.

Latham, G.↗
NASA NTRS1981

On the formation of protostellar disks

An analysis is presented of the hydrodynamic aspects of the growth of protostellar disks from the accretion of a rotating gas cloud. Streamline equations for the flow onto the disk are derived, including the special case where the disk is very thin. For the radial structure of the disk, detailed equations of conservation of mass, momentum, angular momentum, and energy are given. Exact solutions are found for the case of no internal viscosity. For the more general case where there is viscous stress, hypotheses are formulated with the objective of discovering the main qualitative characteristics of a turbulent system and to estimate the magnitude of the turbulent stress. Solutions are also found for boundary conditions, and the large and small viscosity limits. The relation of energy balance to disk thermal structure is detailed. Finally, trends in the collapse of the dust cloud are discussed in the light of the solutions given.

Cassen, P.↗