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NASA NTRS1990

A simple analytic model for the evolution of captured galactic disks

The general analytic solution developed by Steinman-Cameron and Durisen and published in 1988 for the evolution of dissipative nonplanar disks is applied to captured galactic disks in model galaxies with nonspherical, scale-free, logarithmic gravitational potentials. Such potentials produce flat rotation curves, similar to those seen in real galaxies. In this case, the analytic solution yields a self-similar structure for the warps and twists that develop while the disks is settling. Being scale-free in a simple, analytic form, this solution is completely defined by only a few dimensionless fitting parameters. As a result, it can be utilized as a mathematical tool to fit settling disks in real galaxies. The minimum time it takes for a disk to settle into a steady state orientation is also a scale-free quantity when expressed in units of the precession period or the orbit period. For realistic parameters, settling times are on the order of one-half to two periods. The use of the time-dependent structure of settling disks as a probe of the three-dimensional mass distribution of the host galaxies, including dark halos, is discussed.

Steiman-Cameron, Thomas Y.↗

NASA NTRS1978

Unsteady flow in a supersonic cascade with subsonic leading-edge locus

Linearized theory is used to predict the unsteady flow in a supersonic cascade with subsonic axial flow velocity. A closed-form analytical solution is obtained by using a double application of the Wiener-Hopf technique. Although numerical and semianalytical solutions of this problem have already appeared in the literature, this paper contains the first completely analytical solution. It has been stated in the literature that the blade source should vanish at the infinite duct resonance condition. The present analysis shows that this does not occur. This apparent discrepancy is explained in the paper.

Adamczyk, J. J.↗

NASA NTRS1990

A two-step iterative method for evolving nonlinear acoustic systems to a steady-state

A new approach for evolving two-dimensional nonlinear acoustic systems with flow to a steady state is presented. The approach is a two-step iterative method which is tested on a benchmark acoustic problem for which an exact analytical solution is available. Results are also calculated for a nonlinear acoustic problem for which an exact analytical solution is not known. Results indicate that the two-step method represents a powerful, efficient, and stable method for evolving two-dimensional acoustic systems to a steady state, and that the method is applicable to any number of spatial dimensions and to other hyperbolic systems. It is noted that for the benchmark problem only a single iteration on the method is required when the transient and steady-state field are of the same order of magnitude; however, four iterations are required when the steady-state field is several orders of magnitude smaller than the transient field. This method requires six iterations before achieving a steady state for the nonlinear test problem.

Watson, Willie R.↗

NASA NTRS1992

Uniform GTD solution for the diffraction by metallic tapes on panelled compact-range reflectors

Metallic tape is commonly used to cover the interpanel gaps which occur in paneled compact-range reflectors. It is therefore of interest to study the effect of the scattering by the tape on the field in the target zone of the range. An analytical solution is presented for the target zone fields scattered by 2D metallic tapes. It is formulated by the generalized scattering matrix technique in conjunction with the Wiener-Hopf procedure. An extension to treat 3D tapes can be accomplished using the 2D solution via the equivalent current concept. The analytical solution is compared with a reference moment method solution to confirm the accuracy of the former.

Somers, G. A.↗

NASA NTRS1992

Physics of heat pipe rewetting

Although several studies have been made to determine the rewetting characteristics of liquid films on heated rods, tubes, and flat plates, no solutions are yet available to describe the rewetting process of a hot plate subjected to a uniform heating. A model is presented to analyze the rewetting process of such plates with and without grooves. Approximate analytical solutions are presented for the prediction of the rewetting velocity and the transient temperature profiles of the plates. It is shown that the present rewetting velocity solution reduces correctly to the existing solution for the rewetting of an initially hot isothermal plate without heating from beneath the plate. Numerical solutions have also been obtained to validate the analytical solutions.

Chan, S. H.↗

NASA NTRS1996

Element Library for Three-Dimensional Stress Analysis by the Integrated Force Method

The Integrated Force Method, a recently developed method for analyzing structures, is extended in this paper to three-dimensional structural analysis. First, a general formulation is developed to generate the stress interpolation matrix in terms of complete polynomials of the required order. The formulation is based on definitions of the stress tensor components in term of stress functions. The stress functions are written as complete polynomials and substituted into expressions for stress components. Then elimination of the dependent coefficients leaves the stress components expressed as complete polynomials whose coefficients are defined as generalized independent forces. Such derived components of the stress tensor identically satisfy homogenous Navier equations of equilibrium. The resulting element matrices are invariant with respect to coordinate transformation and are free of spurious zero-energy modes. The formulation provides a rational way to calculate the exact number of independent forces necessary to arrive at an approximation of the required order for complete polynomials. The influence of reducing the number of independent forces on the accuracy of the response is also analyzed. The stress fields derived are used to develop a comprehensive finite element library for three-dimensional structural analysis by the Integrated Force Method. Both tetrahedral- and hexahedral-shaped elements capable of modeling arbitrary geometric configurations are developed. A number of examples with known analytical solutions are solved by using the developments presented herein. The results are in good agreement with the analytical solutions. The responses obtained with the Integrated Force Method are also compared with those generated by the standard displacement method. In most cases, the performance of the Integrated Force Method is better overall.

Kaljevic, Igor↗

NASA NTRS2008

Transient Effects in Planar Solidification of Dilute Binary Alloys

The initial transient during planar solidification of dilute binary alloys is studied in the framework of the boundary integral method that leads to the non-linear Volterra integral governing equation. An analytical solution of this equation is obtained for the case of a constant growth rate which constitutes the well-known Tiller's formula for the solute transient. The more physically relevant, constant ramping down temperature case has been studied both numerically and analytically. In particular, an asymptotic analytical solution is obtained for the initial transient behavior. A numerical technique to solve the non-linear Volterra equation is developed and the solution is obtained for a family of the governing parameters. For the rapid solidification condition, growth rate spikes have been observed even for the infinite kinetics model. When recirculating fluid flow is included into the analysis, the spike feature is dramatically diminished. Finally, we have investigated planar solidification with a fluctuating temperature field as a possible mechanism for frequently observed solute trapping bands.

Mazuruk, K.↗

NASA NTRS2008

Transient Effects in Planar Solidification of Dilute Binary Alloys

Mazuruk, Konstantin↗

NASA NTRS2004

Computational Simulation of Acoustic Modes in Rocket Combustors

A combination of computational fluid dynamic analysis and analytical solutions is being used to characterize the dominant modes in liquid rocket engines in conjunction with laboratory experiments. The analytical solutions are based on simplified geometries and flow conditions and are used for careful validation of the numerical formulation. The validated computational model is then extended to realistic geometries and flow conditions to test the effects of various parameters on chamber modes, to guide and interpret companion laboratory experiments in simplified combustors, and to scale the measurements to engine operating conditions. In turn, the experiments are used to validate and improve the model. The present paper gives an overview of the numerical and analytical techniques along with comparisons illustrating the accuracy of the computations as a function of grid resolution. A representative parametric study of the effect of combustor mean flow Mach number and combustor aspect ratio on the chamber modes is then presented for both transverse and longitudinal modes. The results show that higher mean flow Mach numbers drive the modes to lower frequencies. Estimates of transverse wave mechanics in a high aspect ratio combustor are then contrasted with longitudinal modes in a long and narrow combustor to provide understanding of potential experimental simulations.

Harper, Brent↗

NASA NTRSDate not supplied

End-to-End Performance Optimization of a Crewed Lunar Landing Mission Staged from a Near Rectilinear Halo Orbit

A crewed lunar landing mission staged from a Near Rectilinear Halo Orbit (NRHO) presents a unique set of challenges not considered before for the Apollo-type direct missions. The NRHO does not have a fixed orbital plane and its period also changes with time, due to which the mission performance is very sensitive to the timing and location of the NRHO departure and arrival burns for a near-polar surface landing site. For such type of mission staged from the NRHO, an end-to-end optimization approach for the entire mission, from the NRHO departure to surface landing and from surface lift-off to the NRHO insertion, is well-suited for extracting the maximum performance out of this design. In this paper, an integrated approach for performing end-to-end optimization of a NRHO-staged crewed lunar landing mission is proposed that optimizes the in-space and powered flight segments simultaneously using Copernicus trajectory optimization tool. Copernicus is augmented with analytical approximations for the in-space, powered descent, and powered ascent segments in this work to improve the convergence of its multiple-shooting based direct optimization method. New analytical solutions are derived for the different phases of the powered flight segments by solving associated guidance problems. These analytical solutions act as initial guesses for starting the numerical optimization of the integrated mission in Copernicus. An example lunar landing mission is simulated to validate the proposed end-to-end optimization approach and the results are compared with the conventional patched trajectory approach.

Bharat Mahajan↗

NASA NTRS1989

Two-dimensional numerical simulation of a Stirling engine heat exchanger

The first phase of an effort to develop multidimensional models of Stirling engine components is described; the ultimate goal is to model an entire engine working space. More specifically, parallel plate and tubular heat exchanger models with emphasis on the central part of the channel (i.e., ignoring hydrodynamic and thermal end effects) are described. The model assumes: laminar, incompressible flow with constant thermophysical properties. In addition, a constant axial temperature gradient is imposed. The governing equations, describing the model, were solved using Crank-Nicloson finite-difference scheme. Model predictions were compared with analytical solutions for oscillating/reversing flow and heat transfer in order to check numerical accuracy. Excellent agreement was obtained for the model predictions with analytical solutions available for both flow in circular tubes and between parallel plates. Also the heat transfer computational results are in good agreement with the heat transfer analytical results for parallel plates.

Ibrahim, Mounir B.↗

NASA NTRS1994

The oscillations of superfluid neutron stars

The effects of superfluid hydrodynamics on the oscillations of neutron stars are investigated. The equations describing the small-amplitude pulsations of a neutron star's neutron-proton superfluid mixture are reduced to a system of three second-order equations for three scalar potentials. A variational principle is developed from which the frequencies of the modes of these superfluid oscillations may be estimated. These pulsations are studied by finding analytical solutions to the equations for simple uniform models of nonrotating neutron stars and numerical solutions for realistic models (including rotation). These solutions reveal that the lowest frequency modes are almost indistinguishable from those based on ordinary-fluid hydrodynamics. The analytical solutions also reveal the existence of a new set of modes having no ordinary-fluid counterpart.

Lindblom, Lee↗

NASA NTRS1983

Thermal Analysis of Antenna Structures. Part 2: Panel Temperature Distribution

This article is the second in a series that analyzes the temperature distribution in microwave antennas. An analytical solution in a series form is obtained for the temperature distribution in a flat plate analogous to an antenna surface panel under arbitrary temperature and boundary conditions. The solution includes the effects of radiation and air convection from the plate. Good agreement is obtained between the numerical and analytical solutions.

Schonfeld, D.↗

NASA NTRS1975

Approximate solutions of radiative transfer in dusty nebulae. 2: Hydrogen and helium

The ionization structure of hydrogen and helium in dusty nebulae is discussed. The general equations for pure hydrogen nebulae are modified to include helium. An approximate analytic solution is presented for Stroemgren radii of hydrogen and helium in the absence of dust. It is shown that these results, with simple modifications, are also applicable to dusty nebulae where the effective absorption cross section of dust grains varies slowly with frequency in the 1000 to 200A range. No analytic solutions are possible if this cross section varies rapidly with frequency. In this case, however, simple coupled differential equations are derived. Approximate analytic expressions are presented for evaluation of the variation of the fraction of ionizing radiation absorbed by dust and the ratio of the volume emission measures of He II to H II regions with the spectrum of the ionizing source, helium abundance, and absorption properties of dust. The effects of dust on the He III zone are discussed. Results are restricted to spherically symmetric nebulae, but nonuniform gas and dust distributions and clumpiness can be taken into account.

Dana, R. A.↗

NASA NTRS1971

Further studies on the dimensionless parameters associated with the 'in vivo' transport of heat within biological tissue.

Dimensionless parameters appearing in the analysis of transport of heat within living biological tissues are presented and discussed. Dimensional analysis as well as some analytical solutions have been employed to obtain the physically and physiologically most significant dimensionless parameters associated with these types of problems. Some analytical solutions are presented for both steady state and transient cases to illustrate the significance of the derived parameters.-

Shitzer, A.↗

NASA NTRS1991

The free compressible viscous vortex

The present study investigates the effects of compressibility on free (unsteady) viscous heat-conducting vortices. Analytical solutions are found in the limit of large but finite Reynolds number and small but finite Mach number. It is shown that the spreading of the vortex causes a radial flow. This flow is given by the solution of an ordinary differential equation, which gives the dependence of the radial velocity on the tangential velocity, density, and temperature profiles of the vortex. Estimates of the radial velocity found by solving this equation are found to be in good agreement with numerical solutions of the full equations. The equations for the viscous evolution are expanded in powers of Mach number to obtain detailed analytical solutions. It is shown that swirling axisymmetric compressible flows generate negative radial velocities far from the vortex core owing to viscous effects, regardless of the initial distributions of vorticity, density, and entropy.

Colonius, Tim↗

NASA NTRS1976

Approximate solutions of radiative transfer in dusty nebulae. II - Hydrogen and helium

The ionization structure of hydrogen and helium in dusty nebulae is discussed. General equations for pure hydrogen nebulae have been modified to include helium. An approximate analytical solution is presented for Stroemgren radii of hydrogen and helium in the absence of dust. It is shown that these results, with simple modifications, are also applicable to dusty nebulae where the effective absorption cross section of dust grains varies slowly with frequency in the range from 1000 to 200 A. No analytical solutions are possible if this cross section varies rapidly with frequency. In this case, however, simple coupled differential equations are derived which can easily be solved numerically. Approximate analytical expressions are given for evaluating the variation of the fraction of ionizing radiation absorbed by dust and the ratio of the volume emission measures of He II to H II regions with the spectrum of the ionizing source, helium abundance, and absorption properties of dust. The effects of dust on the He III zone are discussed. The present results are restricted to spherically symmetric nebulae, but nonuniform gas and dust distributions and clumpiness can be taken into account by the general results.

Dana, R. A.↗

NASA NTRS2003

Finite Element Modeling of a Cylindrical Contact Using Hertzian Assumptions

The turbine blades in the high-pressure fuel turbopump/alternate turbopump (HPFTP/AT) are subjected to hot gases rapidly flowing around them. This flow excites vibrations in the blades. Naturally, one has to worry about resonance, so a damping device was added to dissipate some energy from the system. The foundation is now laid for a very complex problem. The damper is in contact with the blade, so now there are contact stresses (both normal and tangential) to contend with. Since these stresses can be very high, it is not all that difficult to yield the material. Friction is another non-linearity and the blade is made out of a Nickel-based single-crystal superalloy that is orthotropic. A few approaches exist to solve such a problem and computer models, using contact elements, have been built with friction, plasticity, etc. These models are quite cumbersome and require many hours to solve just one load case and material orientation. A simpler approach is required. Ideally, the model should be simplified so the analysis can be conducted faster. When working with contact problems determining the contact patch and the stresses in the material are the main concerns. Closed-form solutions for non-conforming bodies, developed by Hertz, made out of isotropic materials are readily available. More involved solutions for 3-D cases using different materials are also available. The question is this: can Hertzian1 solutions be applied, or superimposed, to more complicated problems-like those involving anisotropic materials? That is the point of the investigation here. If these results agree with the more complicated computer models, then the analytical solutions can be used in lieu of the numerical solutions that take a very long time to process. As time goes on, the analytical solution will eventually have to include things like friction and plasticity. The models in this report use no contact elements and are essentially an applied load problem using Hertzian assumptions to determine the contact patch dimensions.

Knudsen, Erik↗