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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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NASA NTRS1984

Variational objective analyses for cyclone studies

The basic analysis equations, i.e., the two horizontal momentum equations, the hydrostatic equation, and the integrated continuity equation were derived for the nonlinear vertical coordinate, nondimensionalized, and expressed in finite differences on a staggered grid. Special care was taken to transform the hydrostatic equation and the pressure gradient terms of the horizontal momentum equations to nearly eliminate truncation error over steeply sloping terrain. This formulation also eliminated explicit reference to orographically induced variations in the thermodynamic variables so that the variational adjustments are on the scale of the meteorological perturbations. The analysis equations were subjected to the Euler-Lagrange operations as expressed for finite differences and an additional set of five partial differential equations was derived, bringing to nine the number of equations in Model I. Higher order terms, terms containing observed quantities, and terms containing none of the variables to be adjusted were grouped into forcing functions and the equations were solved for the zero order terms. Zero order variables were eliminated between these equations and there resulted two diagnostic equations which take the form of general linear second order partial differential equations with nonconstant coefficients.

Achtemeier, G. L.↗
NASA NTRS1982

Description of the 3 MW SWT-3 wind turbine at San Gorgonio Pass, California

The SWT-3 wind turbine, a microprocessor controlled three bladed variable speed upwind machine with a 3MW rating that is presently operational and undergoing system testing, is discussed. The tower, a rigid triangular truss configuration, is rotated about its vertical axis to position the wind turbine into the prevailing wind. The blades rotate at variable speed in order to maintain an optimum 6 to 1 tip speed ratio between cut in and fated wind velocity, thereby maximizing power extraction from the wind. Rotor variable speed is implemented by the use of a hydrostatic transmission consisting of fourteen fixed displacement pumps operating in conjunction with eighteen variable displacement motors. Full blade pitch with on-off hydraulic actuation is used to maintain 3MW of output power.

Rybak, S. C.↗
NASA NTRS1995

Evaluation of monthly rainfall estimates derived from the special sensor microwave/imager (SSM/I) over the tropical Pacific

Estimates of monthly rainfall have been computed over the tropical Pacific using passive microwave satellite observations from the special sensor microwave/imager (SSM/I) for the period from July 1987 through December 1990. These monthly estimates are calibrated using data from a network of Pacific atoll rain gauges in order to account for systematic biases and are then compared with several visible and infrared satellite-based rainfall estimation techniques for the purpose of evaluating the performance of the microwave-based estimates. Although several key differences among the various techniques are observed, the general features of the monthly rainfall time series agree very well. Finally, the significant error sources contributing to uncertainties in the monthly estimates are examined and an estimate of the total error is produced. The sampling error characteristics are investigated using data from two SSM/I sensors and a detailed analysis of the characteristics of the diurnal cycle of rainfall over the oceans and its contribution to sampling errors in the monthly SSM/I estimates is made using geosynchronous satellite data. Based on the analysis of the sampling and other error sources the total error was estimated to be of the order of 30 to 50% of the monthly rainfall for estimates averaged over 2.5 deg x 2.5 deg latitude/longitude boxes, with a contribution due to diurnal variability of the order of 10%.

Berg, Wesley↗
NASA NTRS2002

Interannual Variability of Martian Global Dust Storms: Simulations with a Low-Order Model of the General Circulation

We present simulations of the interannual variability of martian global dust storms (GDSs) with a simplified low-order model (LOM) of the general circulation. The simplified model allows one to conduct computationally fast long-term simulations of the martian climate system. The LOM is constructed by Galerkin projection of a 2D (zonally averaged) general circulation model (GCM) onto a truncated set of basis functions. The resulting LOM consists of 12 coupled nonlinear ordinary differential equations describing atmospheric dynamics and dust transport within the Hadley cell. The forcing of the model is described by simplified physics based on Newtonian cooling and Rayleigh friction. The atmosphere and surface are coupled: atmospheric heating depends on the dustiness of the atmosphere, and the surface dust source depends on the strength of the atmospheric winds. Parameters of the model are tuned to fit the output of the NASA AMES GCM and the fit is generally very good. Interannual variability of GDSs is possible in the IBM, but only when stochastic forcing is added to the model. The stochastic forcing could be provided by transient weather systems or some surface process such as redistribution of the sand particles in storm generating zones on the surface. The results are sensitive to the value of the saltation threshold, which hints at a possible feedback between saltation threshold and dust storm activity. According to this hypothesis, erodable material builds up its a result of a local process, whose effect is to lower the saltation threshold until a GDS occurs. The saltation threshold adjusts its value so that dust storms are barely able to occur.

Pankine, A. A.↗
NASA NTRS2004

Aerodynamic Shape Optimization Based on Free-form Deformation

This paper presents a free-form deformation technique suitable for aerodynamic shape optimization. Because the proposed technique is independent of grid topology, we can treat structured and unstructured computational fluid dynamics grids in the same manner. The proposed technique is an alternative shape parameterization technique to a trivariate volume technique. It retains the flexibility and freedom of trivariate volumes for CFD shape optimization, but it uses a bivariate surface representation. This reduces the number of design variables by an order of magnitude, and it provides much better control for surface shape changes. The proposed technique is simple, compact, and efficient. The analytical sensitivity derivatives are independent of the design variables and are easily computed for use in a gradient-based optimization. The paper includes the complete formulation and aerodynamics shape optimization results.

Samareh, Jamshid A.↗
NASA NTRS2008

Using LDPC Code Constraints to Aid Recovery of Symbol Timing

A method of utilizing information available in the constraints imposed by a low-density parity-check (LDPC) code has been proposed as a means of aiding the recovery of symbol timing in the reception of a binary-phase-shift-keying (BPSK) signal representing such a code in the presence of noise, timing error, and/or Doppler shift between the transmitter and the receiver. This method and the receiver architecture in which it would be implemented belong to a class of timing-recovery methods and corresponding receiver architectures characterized as pilotless in that they do not require transmission and reception of pilot signals. Acquisition and tracking of a signal of the type described above have traditionally been performed upstream of, and independently of, decoding and have typically involved utilization of a phase-locked loop (PLL). However, the LDPC decoding process, which is iterative, provides information that can be fed back to the timing-recovery receiver circuits to improve performance significantly over that attainable in the absence of such feedback. Prior methods of coupling LDPC decoding with timing recovery had focused on the use of output code words produced as the iterations progress. In contrast, in the present method, one exploits the information available from the metrics computed for the constraint nodes of an LDPC code during the decoding process. In addition, the method involves the use of a waveform model that captures, better than do the waveform models of the prior methods, distortions introduced by receiver timing errors and transmitter/ receiver motions. An LDPC code is commonly represented by use of a bipartite graph containing two sets of nodes. In the graph corresponding to an (n,k) code, the n variable nodes correspond to the code word symbols and the n-k constraint nodes represent the constraints that the code places on the variable nodes in order for them to form a valid code word. The decoding procedure involves iterative computation of values associated with these nodes. A constraint node represents a parity-check equation using a set of variable nodes as inputs. A valid decoded code word is obtained if all parity-check equations are satisfied. After each iteration, the metrics associated with each constraint node can be evaluated to determine the status of the associated parity check. Heretofore, normally, these metrics would be utilized only within the LDPC decoding process to assess whether or not variable nodes had converged to a codeword. In the present method, it is recognized that these metrics can be used to determine accuracy of the timing estimates used in acquiring the sampled data that constitute the input to the LDPC decoder. In fact, the number of constraints that are satisfied exhibits a peak near the optimal timing estimate. Coarse timing estimation (or first-stage estimation as described below) is found via a parametric search for this peak. The present method calls for a two-stage receiver architecture illustrated in the figure. The first stage would correct large time delays and frequency offsets; the second stage would track random walks and correct residual time and frequency offsets. In the first stage, constraint-node feedback from the LDPC decoder would be employed in a search algorithm in which the searches would be performed in successively narrower windows to find the correct time delay and/or frequency offset. The second stage would include a conventional first-order PLL with a decision-aided timing-error detector that would utilize, as its decision aid, decoded symbols from the LDPC decoder. The method has been tested by means of computational simulations in cases involving various timing and frequency errors. The results of the simulations ined in the ideal case of perfect timing in the receiver.

Jones, Christopher↗
NASA NTRS1989

A least-squares finite element method for incompressible Navier-Stokes problem

Most finite element schemes for solving the Navier-Stokes equations can be categorized into the Galerkin mixed method and the penalty method. The mixed method leads to a saddle-point problem. In order to guarantee the existence of a solution, the combination of velocity and pressure interpolations requires satisfaction of the Ladyzhenskaya Babuska Brezzi (LBB) consistency condition which precludes the use of equal order interpolations and many seemingly natural pairs of velocity and pressure elements. In a previous paper a least-squares finite element method based on the first order velocity-pressure-vorticity formulation for the Stokes problem was proposed. This method leads to a minimization problem. The choice of combination of elements is thus not subject to the LBB condition. The numerical experiments exhibit the optimal rate of convergence for all variables with equal order interpolations. A theoretical error analysis supports the numerical results. In this paper the least-squares finite element method is extended to solving the incompressible Navier-Stokes problem.

Jiang, Bonan↗
DOE OSTI2021

Modeling household online shopping demand in the U.S.: a machine learning approach and comparative investigation between 2009 and 2017

Despite the rapid growth of online shopping and research interest in the relationship between online and in-store shopping, national-level modeling and investigation of the demand for online shopping with a prediction focus remain limited in the literature. Here, this paper differs from prior work and leverages two recent releases of the U.S. National Household Travel Survey (NHTS) data for 2009 and 2017 to develop machine learning (ML) models, specifically gradient boosting machine (GBM), for predicting household-level online shopping purchases. The NHTS data allow for not only conducting nationwide investigation but also at the level of households, which is more appropriate than at the individual level given the connected consumption and shopping needs of members in a household. We follow a systematic procedure for model development including employing Recursive Feature Elimination algorithm to select input variables (features) in order to reduce the risk of model overfitting and increase model explainability. Among several ML models, GBM is found to yield the best prediction accuracy. Extensive post-modeling investigation is conducted in a comparative manner between 2009 and 2017, including quantifying the importance of each input variable in predicting online shopping demand, and characterizing value-dependent relationships between demand and the input variables. In doing so, two latest advances in machine learning techniques, namely Shapley value-based feature importance and Accumulated Local Effects plots, are adopted to overcome inherent drawbacks of the popular techniques in current ML modeling. The modeling and investigation are performed at the national level, with a number of findings obtained. The models developed and insights gained can be used for online shopping-related freight demand generation and may also be considered for evaluating the potential impact of relevant policies on online shopping demand.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2026

Changing-look Active Galactic Nuclei Powered by Disk Tearing

Changing-look active galactic nuclei (CLAGN) feature order-of-magnitude variability in both the continuum and broad line luminosities on months-to-years-long timescales, and are currently unexplained. Simulations have demonstrated that rotating black holes (BHs) sometimes tear apart tilted accretion disks. These tearing events violently restructure the disk on timescales much shorter than a viscous timescale, hinting at a connection to CLAGN. Here, we show that disk tearing can power changing-look events. We report synthetic observations of an extremely high-resolution, three-dimensional, general-relativistic magnetohydrodynamic simulation of a geometrically thin, tilted accretion disk around a rapidly rotating, 10 8 M ⊙ BH. We perform ray-tracing calculations that follow the thermal disk emission to both a line-of-sight camera and to a distribution of cameras in a prescribed torus-like broad-line region (BLR). The continuum photoionizes the BLR, and we calculate the resulting spectrum. Both the continuum and line luminosities undergo order-of-magnitude swings on months-to-years-long timescales. We find shorter, week-long variability driven by the geometric precession of the inner disk and an intraday quasiperiodic oscillation driven by radial breathing of the inner disk. When the torn disk precesses, it causes asymmetric illumination of the BLR, driving time-evolving red-to-blue asymmetries of the broad emission lines that may be a smoking gun for disk tearing. We also make predictions for future photometric observations from ULTRASAT and Vera Rubin Observatory, given a Shakura–Sunyaev–like effective temperature profile, both of which may play an important role in detecting future changing-look events. Future work should consider nonthermal emission, especially from the time-variable corona.

accretion↗
NASA NTRS2004

A Mechanism For Solar Forcing of Climate: Did the Maunder Minimum Cause the Little Ice Age?

The mechanism we wish to demonstrate exploits chemical, radiative, and dynamical sensitivities in the stratosphere to affect the climate of the troposphere. The sun, while its variability in total radiative output over the course of the solar cycle is on the order of 0.1%, exhibits variability in the UV output on the order of 5%. We expect to show that a substantially decreased solar UV output lessened the heating of the Earth's stratosphere during the Maunder Minimum, through decreased radiative absorption by ozone and oxygen. These changes in stratospheric heating would lead to major changes in the stratospheric zonal wind pattern which would in turn affect the propagation characteristics of planetary-scale waves launched in the winter hemisphere. Until recently, there was no quantitative data to relate the changes in the stratosphere to those at the surface. There is now empirical evidence from the NCEP Reanalysis data that a definitive effect of the solar cycle on climate in the troposphere exists. Our recent work is summarized as follows (see complete list of publications in later part of this report).

Yung, Yuk L.↗
DOE OSTI2024

Robust Combined Heat and Hybrid Power (CHHP) for High Electrical Efficiency Cogeneration

Georgia Tech (Prime Recipient), the University of Texas at El Paso (UTEP) and the National Energy Technology Laboratory (NETL) investigated a hybrid fuel cell/ gas turbine system concept as a combined heat and hybrid power (CHHP) system for both robust and high power-to-process heat ratio cogeneration. The novelty of the proposed system entailed the distinct, elevated electrical efficiencies it maintains while simultaneously supporting a broad span of heating needs (e.g., supply temperatures) demanded across variable heat loads. The scope included: 1) leveraging a pioneering national lab facility configured for dynamic system operability development of hybrid fuel cell/gas turbine cycles; 2) enabling technology development to adjust and modulate the quality and quantity of thermal supply to bottoming heat loads via novel extreme temperature gas bypass valves. Hybrid fuel cell/gas turbine systems have primarily been reduced-to-practice in a constrained (e.g., initial proof-of-concept) manner and have still demonstrated considerable electrical efficiencies. However, these pre-pilot systems have focused upon electrical efficiencies and electrical power generation as the exclusive energy demand. Such hybrid systems had not been extensively researched or developed for flexible and variable operation consisting of both power and heat demands; however, these variable combined power and heat demands are characteristic of many types of manufacturers such as animal/poultry processing, bakeries and milk/flour/pastry manufacturing, textile mills, and electrochemical processing. Commercially, developing the system into a working combined heat and power system benefits these types of manufacturers by allowing them to meet their power and heat demands at a lower cost, higher efficiency, and/or through onsite generation. Therefore, the technical scope of this project was largely to study and facilitate these hybrid systems as combined heat and hybrid power (CHHP) systems that include dynamic operability for variable heat and power loads and/or grid dynamics for various types of manufacturers. Simulation results were used to predict the performance of the CHHP system and conceptually develop it to achieve desired dynamic operability. Experimentally, the primary goal was to design, manufacture, and experiment upon a high-temperature bypass valve. Experimental data included air mass flow rates through the valve orifice when the valve was changed to variable extent between fully closed and fully open. The experimental data was then used to create a semi-empirical computational model of the bypass valve. Concluded simulation goals for the research included developing computational heat exchanger models for the hybrid system inclusive of the bottoming heat exchanger and the recuperative heat exchanger, and then combining the computational recuperator model with the computational valve model. Afterwards, the computational models were then integrated to predict the dynamic operation of hybrid fuel cell/gas turbine cycles throughout a design space and reporting such. The scope stated in the preceding paragraph was packaged into five specific goals: 1) enabling the simulation of dynamic combined heat and power through the creation of computational, modular heat exchanger models; 2) simulation and exploration of the CHHP system’s performance by integrating the heat exchanger models with the national lab’s pre-existing hybrid system (computational) simulation, but without the recuperator bypass valve concept in order to initially determine how the (baseline) system behaves and can be controlled in order to meet variable heat and power demands; 3) development and initial deployment of the high-temperature recuperator bypass valve technology in order to confirm and characterize the approach; 4) usage of the experimental data for the valve to create a semi-empirical computational model for the bypass valve which could then be combined with the heat exchanger computational models; 5) repeat of the second task of simulating and exploring the system’s performance, but this time including the bypass valve to resolve its efficacy. Tasks were successfully completed, and the general notion of flexibly operating, high electrical efficiency CHHP was further corroborated. Supportive details are provided in the report.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
NASA NTRS2016

Isotopic Ordering in Atmospheric O2 as a Tracer of Ozone Photochemistry and the Tropical Atmosphere

The distribution of isotopes within O2 molecules can be rapidly altered when they react with atomic oxygen. This mechanism is globally important: while other contributions to the global budget of O2 impart isotopic signatures, the O(3P) + O2 reaction resets all such signatures in the atmosphere on subdecadal timescales. Consequently, the isotopic distribution within O2 is determined by O3 photochemistry and the circulation patterns that control where that photochemistry occurs. The variability of isotopic ordering in O2 has not been established, however. We present new measurements of 18O18O in air (reported as delta36 values) from the surface to 33 km altitude. They confirm the basic features of the clumped-isotope budget of O2: Stratospheric air has higher delta36 values than tropospheric air (i.e., more 18O18O), reflecting colder temperatures and fast photochemical cycling of O3. Lower delta36 values in the troposphere arise from photochemistry at warmer temperatures balanced by the influx of high-delta36 air from the stratosphere. These observations agree with predictions derived from the GEOS-Chem chemical transport model, which provides additional insight. We find a link between tropical circulation patterns and regions where delta36 values are reset in the troposphere. The dynamics of these regions influences lapse rates, vertical and horizontal patterns of O2 reordering, and thus the isotopic distribution toward which O2 is driven in the troposphere. Temporal variations in delta36 values at the surface should therefore reflect changes in tropospheric temperatures, photochemistry, and circulation. Our results suggest that the tropospheric O3 burden has remained within a +/-10 percent range since 1978.

oxygen↗
NASA NTRSDate not supplied

High Temperature Material Property Data and Challenges to Thermal Process Model Predictions and In-Situ/Ex-Situ Measurements for Metallic Additive Manufacturing

Understanding and predicting performance properties of parts produced by metallic additive manufacturing has improved significantly over the past decade; however, difficult to measure material properties and process outcomes continue to be challenges. The qualification or certification of aerospace parts require extensive measures to quantify variable part properties in order to buy down the risk of component failure. The variability, inherent to the additive manufacturing, process adds unwanted uncertainty in the production of load critical structural components. Process modeling has proven valuable in providing predictions and context for understanding outcomes of the additive manufacturing process; however, these physically informed process models require material properties at temperatures that are difficult to measure and rarely available. Further, calibrating or validating such models is difficult because the process itself is challenging to measure. This talk will explore some of the challenges resulting from difficult to acquire input data by relating thermal process model predictions to in-situ and ex-situ optical microscopy measurements.

Process Model↗
NASA NTRS2007

Comparison of Fixed and Variable Time Step Trajectory Integration Methods for Cislunar Trajectories

Due to the nonlinear nature of the Earth-Moon-Sun three-body problem and non-spherical gravity, CEV cislunar targeting algorithms will require many propagations in their search for a desired trajectory. For on-board targeting especially, the algorithm must have a simple, fast, and accurate propagator to calculate a trajectory with reasonable computation time, and still be robust enough to remain stable in the various flight regimes that the CEV will experience. This paper compares Cowell s method with a fourth-order Runge- Kutta integrator (RK4), Encke s method with a fourth-order Runge-Kutta- Nystr m integrator (RKN4), and a method known as Multi-Conic. Additionally, the study includes the Bond-Gottlieb 14-element method (BG14) and extends the investigation of Encke-Nystrom methods to integrators of higher order and with variable step size.

Weeks, ichael W.↗
DOE OSTI2021

A Canonical Transformation to Eliminate Resonant Perturbations. I

We study dynamical systems that admit action-angle variables at leading order, which are subject to nearly resonant perturbations. If the frequencies characterizing the unperturbed system are not in resonance, the long-term dynamical evolution may be integrated by orbit-averaging over the high-frequency angles, thereby evolving the orbit-averaged effect of the perturbations. It is well known that such integrators may be constructed via a canonical transformation, which eliminates the high-frequency variables from the orbit-averaged quantities. An example of this algorithm in celestial mechanics is the von Zeipel transformation. However, if the perturbations are inside or close to a resonance, i.e., the frequencies of the unperturbed system are commensurate; these canonical transformations are subject to divergences. We introduce a canonical transformation that eliminates the high-frequency phase variables in the Hamiltonian without encountering divergences. This leads to a well-behaved symplectic integrator. We demonstrate the algorithm through two examples: a resonantly perturbed harmonic oscillator and the gravitational three-body problem in mean motion resonance.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Practical gyrokinetics

We report thousands of gyrokinetic papers have been published since the introduction of the gyrokinetic change of variables. The intent here is not to review the field, but rather the goal is to present a tutorial providing insight into why gyrokinetics is an appropriate description of turbulent transport. The focus is on turbulent transport in axisymmetric tokamaks, but many of the ideas and techniques are applicable to stellarators and other magnetic fusion devices with nested surfaces of constant pressure. Besides the origins of gyrokinetics and recent insights, gyrokinetic orderings and gyrokinetic variables are summarized. Then a compact, but careful, derivation of the simplest electrostatic gyrokinetic equation for tokamaks is presented, along with a brief mention of gyrokinetics for stellarators. The advantages of assuming scale separation between the finer spatial scales and faster time variation of the turbulence and the global behaviour and slow temporal evolution of the background are stressed. Moreover, the procedure for removing the adiabatic or Maxwell–Boltzmann response is emphasized. Scale separation allows the near Maxwellian behaviour of the background and the rapid variation of the turbulence to be described by separate, local gyrokinetic equations. The turbulent fluctuations are found by solving the nonlinear gyrokinetic equation for the non-adiabatic portion of the fluctuating distribution function. Also, generalizations of the gyrokinetic equation so that it retains electromagnetic and flow modifications are considered in detail along with some symmetry properties. In addition, ambipolarity, particle transport and heat transport are addressed, along with a discussion of the complications associated with describing momentum transport and profile evolution.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Practical Gyrokinetics

Thousands of gyrokinetic papers have been published since the introduction of the gyrokinetic change of variables. The intent here is not to review the field, but rather the goal is to present a tutorial providing insight into why gyrokinetics is an appropriate description of turbulent transport. The focus is on turbulent transport in axisymmetric tokamaks, but many of the ideas and techniques are applicable to stellarators and other magnetic fusion devices with nested surfaces of constant pressure. Besides the origins of gyrokinetics and recent insights, gyrokinetic orderings and gyrokinetic variables are summarized. Then a compact, but careful, derivation of the simplest electrostatic gyrokinetic equation for tokamaks is presented, along with a brief mention of gyrokinetics for stellarators. The advantages of assuming scale separation between the finer scales and faster time variation of the turbulence and the global behavior and slow temporal evolution of the background are stressed. Moreover, the procedure for removing the adiabatic or Maxwell-Boltzmann response is emphasized. Scale separation allows the near Maxwellian behavior of the background and the rapid variation of the turbulence to be described by separate local gyrokinetic equations. The turbulent fluctuations are found by solving the nonlinear gyrokinetic equation for the non-adiabatic portion of the fluctuating distribution function. Also, generalizations of the gyrokinetic equation so that it retains electromagnetic and flow modifications are considered in detail along with some symmetry properties. In addition, ambipolarity, particle transport, and heat transport are addressed, along with a discussion of the complications associated with describing momentum transport and profile evolution.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1975

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitation and rotation terms in the equations are of first order in the space variables, the pressure gradient terms are of second order, and the turbulent viscosity term is of third order. The presence of a turbulent viscosity insures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the initial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial inflow plays an important role and allows collapse to occur even when the rotation is large. An approximate solution of the governing partial differential equations is also given; the equations are used to study the spacial distributions of the density and velocity.

Deissler, R. G.↗