Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Boundary value problem”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 919 records · Page 51
All sourcesNASA NTRSDOE OSTI
NASA NTRS1948

The Problem of Torsion in Prismatic Members of Circular Segmental Cross Section

The problem is solved by approximation, by setting up a function complying with the differential equation of the stress function, and determining the coefficients appearing in it in such a way that the boundary condition is fulfilled as nearly as possible. For the semicircle, for which the solution is known, the method yields very accurate values; the approximated stress distribution is in good agreement with the accurately computed distribution. Stress and strain measurements indicate that the approximate solution is in sufficiently exact agreement with reality for segmental cross sections.

Weigand, A.↗
NASA NTRS1947

Some Recent Contributions to the Study of Transition and Turbulent Boundary Layers

The first part of this paper reviews the present state of the problem of the instability of laminar boundary layers which has formed an important part of the general lectures by von Karman at the first and fourth Congresses and by Taylor at the fifth Congress. This problem may now be considered as essentially solved as the result of work completed since 1938. When the velocity fluctuations of the free-stream flow are less than 0.1 percent of the mean speed, instability occurs as described by the well-known Tollmien-Schlichting theory. The Tollmien-Schlichting waves were first observed experimentally by Schubauer and Skramstad in 1940. They devised methods of introducing controlled small disturbances and obtained measured values of frequency, damping, and wave length at various Reynolds numbers which agreed well with the theoretical results. Their experimental results were confirmed by Liepmann. Much theoretical work was done in Germany in extending the Tol1mien-Schlichting theory to other boundary conditions, in particular to flow along a porous wall to which suction is applied for removing part of the boundary layer. The second part of this paper summarizes the present state of knowledge of the mechanics of turbulent boundary layers, and of the methods now being used for fundamental studies of the turbulent fluctuations in turbulent boundary layers. A brief review is given of the semi-empirical method of approach as developed by Buri, Gruschwitz, Fediaevsky, and Kalikhman. In recent years the National Advisory.Commsittee for Aeronautics has sponsored a detailed study at the National Bureau of Standards of the turbulent fluctuations in a turbulent boundary layer under adverse pressure gradient sufficient to produce separation. The aims of this investigation and its present status are described.

Dryden, Hugh L↗
NASA NTRS1992

Computational aspects of helicopter trim analysis and damping levels from Floquet theory

Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated.

Gaonkar, Gopal H.↗
NASA NTRS1993

Helicopter trim analysis by shooting and finite element methods with optimally damped Newton iterations

Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used, and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated.

Achar, N. S.↗
NASA NTRS1994

Transfer function modeling of damping mechanisms in distributed parameter models

This work formulates a method for the modeling of material damping characteristics in distributed parameter models which may be easily applied to models such as rod, plate, and beam equations. The general linear boundary value vibration equation is modified to incorporate hysteresis effects represented by complex stiffness using the transfer function approach proposed by Golla and Hughes. The governing characteristic equations are decoupled through separation of variables yielding solutions similar to those of undamped classical theory, allowing solution of the steady state as well as transient response. Example problems and solutions are provided demonstrating the similarity of the solutions to those of the classical theories and transient responses of nonviscous systems.

Slater, J. C.↗
NASA NTRS1991

On the solution of a lubrication problem with particulate solids

The lubrication characteristic of a fluid with solid particles is studied using the continuum theory of mixtures. The governing equations are formulated and appropriate boundary conditions are introduced for an arbitrary-shaped lubricant film thickness. As a special case, closed-form analytical perturbation solutions for pressure and shear stress are obtained for a mixture of a conventional oil and solid particles with small values of solid-volume fraction sheared in the clearance space of an infinitely long slider bearing. It is found that when compared with a pure fluid, the mixture of the fluid and solid generates a higher pressure and therefore a higher load-carrying capacity with the added advantage of a reduction in the coefficient of friction.

Dai, F.↗
NASA NTRS1986

Sensitivity of an atmospheric photochemistry model to chlorine perturbations including consideration of uncertainty propagation

Models of stratospheric photochemistry are generally tested by comparing their predictions for the composition of the present atmosphere with measurements of species concentrations. These models are then used to make predictions of the atmospheric sensitivity to perturbations. Here the problem of the sensitivity of such a model to chlorine perturbations ranging from the present influx of chlorine-containing compounds to several times that influx is addressed. The effects of uncertainties in input parameters, including reaction rate coefficients, cross sections, solar fluxes, and boundary conditions, are evaluated using a Monte Carlo method in which the values of the input parameters are randomly selected. The results are probability distributions for present atmosheric concentrations and for calculated perturbations due to chlorine from fluorocarbons. For more than 300 Monte Carlo runs the calculated ozone perturbation for continued emission of fluorocarbons at today's rates had a mean value of -6.2 percent, with a 1-sigma width of 5.5 percent. Using the same runs but only allowing the cases in which the calculated present atmosphere values of NO, NO2, and ClO at 25 km altitude fell within the range of measurements yielded a mean ozone depletion of -3 percent, with a 1-sigma deviation of 2.2 percent. The model showed a nonlinear behavior as a function of added fluorocarbons. The mean of the Monte Carlo runs was less nonlinear than the model run using mean value of the input parameters.

Stolarski, R. S.↗
NASA NTRS1995

The initial-value problem for viscous channel flows

Plane viscous channel flows are perturbed and the ensuing initial-value problems are investigated in detail. Unlike traditional methods where traveling wave normal modes are assumed for solution, this works offers a means whereby completely arbitrary initial input can be specified without having to resort to eigenfunction expansions. The full temporal behavior, including both early time transients and the long time asymptotics, can be determined for any initial disturbance. Effects of three-dimensionality can be assessed. The bases for the analysis are: (a) linearization of the governing equations; (b) Fourier decomposition in the spanwise and streamwise directions of the flow; and (c) direct numerical integration of the resulting partial differential equations. All of the stability data that are known for such flows can be reproduced. Also, the optimal initial condition can be determined in a straight forward manner and such optimal conditions clearly reflect transient growth data that is easily determined by a rational choice of a basis for the initial conditions. Although there can be significant transient growth for subcritical values of the Reynolds number using this approach it does not appear possible that arbitrary initial conditions will lead to the exceptionally large transient amplitudes that have been determined by optimization of normal modes. The approach is general and can be applied to other classes of problems where only a finite discrete spectrum exists, such as the boundary layer for example.

Criminale, W. O.↗
NASA NTRS1988

The causes for geographical variations in OS187/OS186 at the Cretaceous-Tertiary boundary

Researchers at Yale has approached the problem of the osmium isotopic composition of marine deposits formed in contact with both oxidized and reduced bottom waters. The measured (187) Os/(186) Os ratios of modern bulk sediment can be explained using mixing equations involving continental detrital, volcaniclastic, cosmogenic and hydrogeneous components. These studies show that sediments deposited under reducing marine conditions contain a hydrogenous component which is enriched in Re and has a radiogenic (187) Os/(186) Os ratio. The presence of such a hydrogenous component in the marine fish clay at Stevns Klint can account for the elevation of its (187) Os/(186) Os ration above the expected meteoritic value. Mass balance considerations require the Re/Os ratio of the phase precipitated from the terminal Cretaceous sea at Stevns Klint to have been about one tenth the value observed in contemporary deposits in the Black Sea, assuming Re has not been lost (or Os gained) subsequent to precipitation. In continental sections, the elevation of the (187) Os/(186) Os ratio in boundary layers may be due to precipitation from continental waters of crustally-derived radiogenic osmium either contemporaneous with the meteoritic (or mantle) osmium deposition or later during diagenesis.

Turekian, K. K.↗
NASA NTRS1988

Bayesian inference in geomagnetism

The inverse problem in empirical geomagnetic modeling is investigated, with critical examination of recently published studies. Particular attention is given to the use of Bayesian inference (BI) to select the damping parameter lambda in the uniqueness portion of the inverse problem. The mathematical bases of BI and stochastic inversion are explored, with consideration of bound-softening problems and resolution in linear Gaussian BI. The problem of estimating the radial magnetic field B(r) at the earth core-mantle boundary from surface and satellite measurements is then analyzed in detail, with specific attention to the selection of lambda in the studies of Gubbins (1983) and Gubbins and Bloxham (1985). It is argued that the selection method is inappropriate and leads to lambda values much larger than those that would result if a reasonable bound on the heat flow at the CMB were assumed.

Backus, George E.↗
NASA NTRS1992

N-point correlation functions in the CfA and SSRS redshift distribution of galaxies

Using counts in cells, we estimate the volume-average N-point galaxy correlation functions for N = 2, 3, and 4, in redshift samples of the CfA and SSRS catalogs. Volume-limited samples of different sizes are used to study the uncertainties at different scales, the shot noise, and the problem with the boundaries. The hierarchical constants S3 and S4 agree well in all samples in CfA and SSRS, with average S3 = 194 +/- 0.07 and S4 = 4.56 +/- 0.53. We compare these results with estimates obtained from angular catalogs and recent analysis over IRAS samples. The amplitudes SJ seem larger in real space than in redshift space, although the values from the angular analysis correspond to smaller scales, where we might expect larger nonperturbative effects. It is also found that S3 and S4 are smaller for IRAS than for optical galaxies. This, together with the fact that IRAS galaxies have smaller amplitude for the above correlation functions, indicates that the density fluctuations of IRAS galaxies cannot be simply proportional to the density fluctuations of optical galaxies, i.e., biasing has to be nonlinear between them.

Gaztanaga, Enrique↗
NASA NTRS1933

An Airfoil Spanning an Open Jet

Proceeding from the fundamental problem on the mutual relation of a wing and free boundaries the distribution of the circulation is determined for an airfoil spanning an open jet of rectangular section at different aspect ratios, and then for an open jet of circular section. The solution is obtained by means of a Fourier series and computations have been performed for different values of the variables. The second part describes the experiments performed for the purpose of proving the theory. The results confirm the theory. In conclusion it defines the induced drag of a wing extending across an open jet and compares it with the drag of a monoplane having a span equal to the jet width at equal total lift.

Stuper, J↗
NASA NTRS1994

LSENS: A General Chemical Kinetics and Sensitivity Analysis Code for homogeneous gas-phase reactions. Part 1: Theory and numerical solution procedures

LSENS, the Lewis General Chemical Kinetics and Sensitivity Analysis Code, has been developed for solving complex, homogeneous, gas-phase chemical kinetics problems and contains sensitivity analysis for a variety of problems, including nonisothermal situations. This report is part 1 of a series of three reference publications that describe LENS, provide a detailed guide to its usage, and present many example problems. Part 1 derives the governing equations and describes the numerical solution procedures for the types of problems that can be solved. The accuracy and efficiency of LSENS are examined by means of various test problems, and comparisons with other methods and codes are presented. LSENS is a flexible, convenient, accurate, and efficient solver for chemical reaction problems such as static system; steady, one-dimensional, inviscid flow; reaction behind incident shock wave, including boundary layer correction; and perfectly stirred (highly backmixed) reactor. In addition, the chemical equilibrium state can be computed for the following assigned states: temperature and pressure, enthalpy and pressure, temperature and volume, and internal energy and volume. For static problems the code computes the sensitivity coefficients of the dependent variables and their temporal derivatives with respect to the initial values of the dependent variables and/or the three rate coefficient parameters of the chemical reactions.

Radhakrishnan, Krishnan↗
NASA NTRS1994

LSENS: A General Chemical Kinetics and Sensitivity Analysis Code for homogeneous gas-phase reactions. Part 3: Illustrative test problems

LSENS, the Lewis General Chemical Kinetics and Sensitivity Analysis Code, has been developed for solving complex, homogeneous, gas-phase chemical kinetics problems and contains sensitivity analysis for a variety of problems, including nonisothermal situations. This report is part 3 of a series of three reference publications that describe LSENS, provide a detailed guide to its usage, and present many example problems. Part 3 explains the kinetics and kinetics-plus-sensitivity analysis problems supplied with LSENS and presents sample results. These problems illustrate the various capabilities of, and reaction models that can be solved by, the code and may provide a convenient starting point for the user to construct the problem data file required to execute LSENS. LSENS is a flexible, convenient, accurate, and efficient solver for chemical reaction problems such as static system; steady, one-dimensional, inviscid flow; reaction behind incident shock wave, including boundary layer correction; and perfectly stirred (highly backmixed) reactor. In addition, the chemical equilibrium state can be computed for the following assigned states: temperature and pressure, enthalpy and pressure, temperature and volume, and internal energy and volume. For static problems the code computes the sensitivity coefficients of the dependent variables and their temporal derivatives with respect to the initial values of the dependent variables and/or the three rate coefficient parameters of the chemical reactions.

Bittker, David A.↗
NASA NTRS1994

LSENS, a general chemical kinetics and sensitivity analysis code for homogeneous gas-phase reactions. 2: Code description and usage

LSENS, the Lewis General Chemical Kinetics Analysis Code, has been developed for solving complex, homogeneous, gas-phase chemical kinetics problems and contains sensitivity analysis for a variety of problems, including nonisothermal situations. This report is part 2 of a series of three reference publications that describe LSENS, provide a detailed guide to its usage, and present many example problems. Part 2 describes the code, how to modify it, and its usage, including preparation of the problem data file required to execute LSENS. Code usage is illustrated by several example problems, which further explain preparation of the problem data file and show how to obtain desired accuracy in the computed results. LSENS is a flexible, convenient, accurate, and efficient solver for chemical reaction problems such as static system; steady, one-dimensional, inviscid flow; reaction behind incident shock wave, including boundary layer correction; and perfectly stirred (highly backmixed) reactor. In addition, the chemical equilibrium state can be computed for the following assigned states: temperature and pressure, enthalpy and pressure, temperature and volume, and internal energy and volume. For static problems the code computes the sensitivity coefficients of the dependent variables and their temporal derivatives with respect to the initial values of the dependent variables and/or the three rate coefficient parameters of the chemical reactions. Part 1 (NASA RP-1328) derives the governing equations describes the numerical solution procedures for the types of problems that can be solved by lSENS. Part 3 (NASA RP-1330) explains the kinetics and kinetics-plus-sensitivity-analysis problems supplied with LSENS and presents sample results.

Radhakrishnan, Krishnan↗
NASA NTRS1994

LSENS, A General Chemical Kinetics and Sensitivity Analysis Code for Homogeneous Gas-Phase Reactions: Code Description and Usage - Part 2

LSENS, the Lewis General Chemical Kinetics and Sensitivity Analysis Code, has been developed for solving complex, homogeneous, gas-phase chemical kinetics problems and contains sensitivity analysis for a variety of problems, including nonisothermal situations. This report is part II of a series of three reference publications that describe LSENS, provide a detailed guide to its usage, and present many example problems. Part II describes the code, how to modify it, and its usage, including preparation of the problem data file required to execute LSENS. Code usage is illustrated by several example problems, which further explain preparation of the problem data file and show how to obtain desired accuracy in the computed results. LSENS is a flexible, convenient, accurate, and efficient solver for chemical reaction problems such as static system; steady, one-dimensional, inviscid flow; reaction behind incident shock wave, including boundary layer correction; and perfectly stirred (highly backmixed) reactor. In addition, the chemical equilibrium state can be computed for the following assigned states: temperature and pressure, enthalpy and pressure, temperature and volume, and internal energy and volume. For static problems the code computes the sensitivity coefficients of the dependent variables and their temporal derivatives with respect to the initial values of the dependent variables and/or the three rate coefficient parameters of the chemical reactions. Part I (NASA RP-1328) derives the governing equations and describes the numerical solution procedures for the types of problems that can be solved by LSENS. Part III (NASA RP-1330) explains the kinetics and kinetics-plus-sensitivity-analysis problems supplied with LSENS and presents sample results.

Radhakrishnan, Krishnan↗
NASA NTRS1996

Mixing and Transition Control Studied

Considerable progress in understanding nonlinear phenomena in both unbounded and wallbounded shear flow transition has been made through the use of a combination of high- Reynolds-number asymptotic and numerical methods. The objective of this continuing work is to fully understand the nonlinear dynamics so that ultimately (1) an effective means of mixing and transition control can be developed and (2) the source terms in the aeroacoustic noise problem can be modeled more accurately. Two important aspects of the work are that (1) the disturbances evolve from strictly linear instability waves on weakly nonparallel mean flows so that the proper upstream conditions are applied in the nonlinear or wave-interaction streamwise region and (2) the asymptotic formulations lead to parabolic problems so that the question of proper out-flow boundary conditions--still a research issue for direct numerical simulations of convectively unstable shear flows--does not arise. Composite expansion techniques are used to obtain solutions that account for both mean-flow-evolution and nonlinear effects. A previously derived theory for the amplitude evolution of a two-dimensional instability wave in an incompressible mixing layer (which is in quantitative agreement with available experimental data for the first nonlinear saturation stage for a plane-jet shear layer, a circular-jet shear layer, and a mixing layer behind a splitter plate) have been extended to include a wave-interaction stage with a three-dimensional subharmonic. The ultimate wave interaction effects can either give rise to explosive growth or an equilibrium solution, both of which are intimately associated with the nonlinear self-interaction of the three dimensional component. The extended theory is being evaluated numerically. In contrast to the mixing-layer situation, earlier comparisons of theoretical predictions based on asymptotic methods and experiments in wall-bounded shear-flow transition have been somewhat lacking in one aspect or another. The current work strongly suggests that the main weakness is the underlying asymptotic representation of the linear "part" of the problem and not the explicit modeling of the nonlinear/wave-interaction effects. Consequently, the long-wave-length/high-Reynolds-number asymptotic limit for the Blasius boundary-layer stability problem was reexamined, and a new dispersion relationship for the instability waves that is uniformly valid for both the upper- and lower branch regions to the required order of approximation was obtained. A comparison with numerical results, obtained by solving the Orr-Sommerfeld stability problem, shows that the asymptotic formula provides surprisingly good results, even for values of the frequency parameter usually encountered in experimental investigations. This is particularly evident in the dynamically important upper-branch region, where much of the nonlinear interactions in transition experiments are believed to take place. The result is important in that it can be used to greatly improve the accuracy of weakly nonlinear critical-layer-based theories, and a consistent nonlinear theory is currently under evaluation.

Source record↗