Search NASA⌕ Search

SEARCH · Search NASA

Results for “Number theory”

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 199 records · Page 11

Allowables for Structural Composites

In order to obtain the most benefit from building a structure with composite laminates, the strength of the laminate must be known. Based on the "weakest link" theory, the lower strength numbers obtained from testing are the ones to be used for design and analysis. The strength value to be used is determined by a statistical analysis of the test data, and is known as an allowable. MIL-HDBK- 17 outlines procedures to follow for determining these allowables. There are two types of statistically determined allowables, A- Basis and B-Basis. A-Basis is defined as a strength value at which only 1 in 100 specimens will fail with a 95% confidence level. B-Basis is a strength value at which only 10 in 100 specimens will fail with a 95% confidence level. As more specimens are tested a higher value of strength can be used as a valid allowable. Composites are highly process dependent and show much strength variation with environment, so it is critical to test materials and environments that are representative of hardware. Either using data obtained from a previous test series, or extrapolation to a certain temperature is highly discouraged.

Nettles, Alan T.↗

Upset Recovery Human Factors

Accident and incident analyses as well as industry group concerns and recommendations have justified taking a second look at proficiency standards related to upset recovery training and performance. Quite a number of factors and theories have been suggested-- leading the NASA Aviation Safety Program to reconsider manual handling skills in highly automated aircraft particularly in conditions that can potentially lead to Loss of Control events. Our team of Subject Matter Experts (SMEs) first identified 76 Basic Recovery Skills that were important for effective crew response under five different anomaly conditions. In addition to manual handling skills, the skill set included knowledge and cognitive skills, as well as decision making and management skills. Advanced Recovery Skills were identified by combining skills, integrating with crew resource management skills, and developing heuristics for decision making.Using the Advanced Recovery Skill set, the SMEs then developed a generic process flow starting from the problem discovery phase (e.g., identifying an anomaly) through the decision making and management phase (e.g., assessing response options), through the recovery phase (e.g., controlling the aircraft). The generic process flow was refined by testing it against six additional scenarios. The next part of the project was to develop an approach for assessing and revising a generic training curriculum (we used an operators Advanced Qualification Program (AQP) as a framework). Although many of the Basic and Advanced Recovery Skills could be found in the Job Task Listing, they were not always structured or combined in the most effective way. Recommendations were developed for assessing relevant aspects of the Job Task Listing and Continuing Qualification curriculum so that the more comprehensive set of Upset Recovery skillsincluding Human Factors--could be trained and assessed in the most appropriate and effective context. The existing AQP methodology provides a natural way to insert targeted Upset Recovery skills into its system of proficiency objectives, training devices, training activities, and ultimately, into the event sets of a simulator training scenario.

AQP training↗

Simulation of Nonlinear Instabilities in an Attachment-Line Boundary Layer

The linear and the nonlinear stability of disturbances that propagate along the attachment line of a three-dimensional boundary layer is considered. The spatially evolving disturbances in the boundary layer are computed by direct numerical simulation (DNS) of the unsteady, incompressible Navier-Stokes equations. Disturbances are introduced either by forcing at the in ow or by applying suction and blowing at the wall. Quasi-parallel linear stability theory and a nonparallel theory yield notably different stability characteristics for disturbances near the critical Reynolds number; the DNS results con rm the latter theory. Previously, a weakly nonlinear theory and computations revealed a high wave-number region of subcritical disturbance growth. More recent computations have failed to achieve this subcritical growth. The present computational results indicate the presence of subcritically growing disturbances; the results support the weakly nonlinear theory. Furthermore, an explanation is provided for the previous theoretical and computational discrepancy. In addition, the present results demonstrate that steady suction can be used to stabilize disturbances that otherwise grow subcritically along the attachment line.

Joslin, Ronald D.↗

The Goertler instability on an airfoil

Goertler vortices arise in boundary layers along concave surfaces due to centrifugal effects. This paper presents some results of an experiment conducted to study the development of these vortices on an airfoil with a pressure gradient in the concave region where an attached laminar boundary layer was insured with suction through a perforated panel. The sublimating chemical technique was used to visualize Goertler vortices and the velocity field was measured by laser velocimetry. The vortex wavelength clearly varied with Goertler number as predicted by linear theory. Both flow visualization and velocity measurements indicated vortex damping in the convex zone. Secondary instability was observed at the higher Goertler numbers.

Mangalam, S. M.↗

Design optimization for cost and quality: The robust design approach

Designing reliable, low cost, and operable space systems has become the key to future space operations. Designing high quality space systems at low cost is an economic and technological challenge to the designer. A systematic and efficient way to meet this challenge is a new method of design optimization for performance, quality, and cost, called Robust Design. Robust Design is an approach for design optimization. It consists of: making system performance insensitive to material and subsystem variation, thus allowing the use of less costly materials and components; making designs less sensitive to the variations in the operating environment, thus improving reliability and reducing operating costs; and using a new structured development process so that engineering time is used most productively. The objective in Robust Design is to select the best combination of controllable design parameters so that the system is most robust to uncontrollable noise factors. The robust design methodology uses a mathematical tool called an orthogonal array, from design of experiments theory, to study a large number of decision variables with a significantly small number of experiments. Robust design also uses a statistical measure of performance, called a signal-to-noise ratio, from electrical control theory, to evaluate the level of performance and the effect of noise factors. The purpose is to investigate the Robust Design methodology for improving quality and cost, demonstrate its application by the use of an example, and suggest its use as an integral part of space system design process.

Unal, Resit↗

The Goertler Instability on an Airfoil

Goertler vortices arise in boundary layers along concave surfaces due to centrifugal effects. This paper presents some results of an experiment conducted to study the development of these vortices on an airfoil with a pressure gradient in the concave region where an attached laminar boundary layer was insured with suction through a perforated panel. The sublimating chemical technique was used to visualize Goertler vortices and the velocity field was measured by laser velocimetry. The vortex wavelength clearly varied with Goertler number as predicted by linear theory. Both flow visualization and velocity measurements indicated vortex damping in the convex zone. Secondary instability was observed at the higher Goertler numbers.

Mangalam, S. M.↗

Theory of hydromagnetic turbulence

The present state of MHD turbulence theory as a possible solar wind research tool is surveyed. The theory is statistical, and does not make statements about individual events. The ensembles considered typically have individual realizations which differ qualitatively, unlike equilibrium statistical mechanics. Most of the theory deals with highly symmetric situations; most of these symmetries have yet to be tested in the solar wind. The applicability of MHD itself to solar wind parameters is highly questionable; yet it has no competitors, as a potentially comprehensive dynamical description. The purpose of solar wind research require sharper articulation. If they are to understand radial turbulent plasma flows from spheres, laboratory experiments and numerical solution of equations of motion may be cheap alternative to spacecraft. If "real life" information is demanded, multiple spacecraft with variable separation may be necessary to go further. The principal emphasis in the theory so far has been on spectral behavior for spatial covariances in wave number space. There is no respectable theory of these for highly anisotropic situations. A rather slow development of theory acts as a brake on justifiable measurement, at this point.

Montgomery, D.↗

Analytic and Computational Perspectives of Multi-Scale Theory for Homogeneous, Laminated Composite, and Sandwich Beams and Plates

This paper reviews the theoretical foundation and computational mechanics aspects of the recently developed shear-deformation theory, called the Refined Zigzag Theory (RZT). The theory is based on a multi-scale formalism in which an equivalent single-layer plate theory is refined with a robust set of zigzag local layer displacements that are free of the usual deficiencies found in common plate theories with zigzag kinematics. In the RZT, first-order shear-deformation plate theory is used as the equivalent single-layer plate theory, which represents the overall response characteristics. Local piecewise-linear zigzag displacements are used to provide corrections to these overall response characteristics that are associated with the plate heterogeneity and the relative stiffnesses of the layers. The theory does not rely on shear correction factors and is equally accurate for homogeneous, laminated composite, and sandwich beams and plates. Regardless of the number of material layers, the theory maintains only seven kinematic unknowns that describe the membrane, bending, and transverse shear plate-deformation modes. Derived from the virtual work principle, RZT is well-suited for developing computationally efficient, C0-continuous finite elements; formulations of several RZT-based elements are highlighted. The theory and its finite elements provide a unified and reliable computational platform for the analysis and design of high-performance load-bearing aerospace structures.

Tessler, Alexander↗

Analytic and Computational Perspectives of Multi-Scale Theory for Homogeneous, Laminated Composite, and Sandwich Beams and Plates

This paper reviews the theoretical foundation and computational mechanics aspects of the recently developed shear-deformation theory, called the Refined Zigzag Theory (RZT). The theory is based on a multi-scale formalism in which an equivalent single-layer plate theory is refined with a robust set of zigzag local layer displacements that are free of the usual deficiencies found in common plate theories with zigzag kinematics. In the RZT, first-order shear-deformation plate theory is used as the equivalent single-layer plate theory, which represents the overall response characteristics. Local piecewise-linear zigzag displacements are used to provide corrections to these overall response characteristics that are associated with the plate heterogeneity and the relative stiffnesses of the layers. The theory does not rely on shear correction factors and is equally accurate for homogeneous, laminated composite, and sandwich beams and plates. Regardless of the number of material layers, the theory maintains only seven kinematic unknowns that describe the membrane, bending, and transverse shear plate-deformation modes. Derived from the virtual work principle, RZT is well-suited for developing computationally efficient, C(sup 0)-continuous finite elements; formulations of several RZT-based elements are highlighted. The theory and its finite element approximations thus provide a unified and reliable computational platform for the analysis and design of high-performance load-bearing aerospace structures.

Tessler, Alexander↗

Preliminary Tests in the Supersonic Sphere

This report presents preliminary data obtained in the Langley supersonic sphere. The supersonic sphere is essentially a whirling mechanism enclosed in a steel shell which can be filled with either air or Freon gas. The test models for two-dimensional study are of propeller form having the same plan form and diameter but varying only in the airfoil shape and thickness ratio. Torque coefficients for the 16-006, 65-110, and the 15 percent thick ellipse models are presented, as well as pressure distributions on a circular-arc supersonic airfoil section having a maximum thickness of 10 percent chord at the 1/3-chord position. Torque coefficients were measured in both Freon and air on the 15 percent thick ellipse, and the data obtained in air and Freon are found to be in close agreement. The torque coefficients for the three previously mentioned models showed large differences in magnitude at tip Mach numbers above 1, the model with the thickest airfoil section having the largest torque coefficient. Pressure distribution on the previously mentioned circular-arc airfoil section are presented at Mach numbers of 0.69, 1.26, and 1.42. At Mach numbers of 1.26 and 1.42 the test section is in the mixed flow region where both subsonic and supersonic speeds occur on the airfoil. No adequate theory has been developed for this condition of mixed flow, but the experimental data have been compared with values of pressure based on Ackeret's theory. The experimental data obtained at a Mach number of 1.26 on the rear portion of the airfoil section agree fairly well with the values calculated by Ackeret's theory. At a Mach number of 1.42 a larger percentage of the airfoil is in supersonic flow, and the experimental data for the entire airfoil agree fairly well with the values obtained using Ackeret's theory.

Baker, John E.↗

On the theory of H2 rotational excitation

A number of mechanisms for exciting higher-J states in the ground vibrational level are considered. Attention is given to collisions with H atoms and electrons. Other mechanisms involve a cascading down from upper vibrational levels, following absorption and reemission of photons in the Lyman and Werner bands, or a cascading down from upper vibrational levels, following formation of hydrogen molecules in excited vibrational and rotational levels. Theoretical population densities of rotational levels are shown in a graph.

Spitzer, L., Jr.↗

On the Lagrangian description of unsteady boundary layer separation. Part 1: General theory

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

Vandommelen, Leon L.↗

On the Lagrangian description of unsteady boundary-layer separation. I - General theory

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

Van Dommelen, Leon L.↗

Comparison of Several Methods of Predicting the Pressure Loss at Altitude Across a Baffled Aircraft-Engine Cylinder

Several methods of predicting the compressible-flow pressure loss across a baffled aircraft-engine cylinder were analytically related and were experimentally investigated on a typical air-cooled aircraft-engine cylinder. Tests with and without heat transfer covered a wide range of cooling-air flows and simulated altitudes from sea level to 40,000 feet. Both the analysis and the test results showed that the method based on the density determined by the static pressure and the stagnation temperature at the baffle exit gave results comparable with those obtained from methods derived by one-dimensional-flow theory. The method based on a characteristic Mach number, although related analytically to one-dimensional-flow theory, was found impractical in the present tests because of the difficulty encountered in defining the proper characteristic state of the cooling air. Accurate predictions of altitude pressure loss can apparently be made by these methods, provided that they are based on the results of sea-level tests with heat transfer.

Neustein, Joseph↗

Large numbers hypothesis. II - Electromagnetic radiation

This paper develops the theory of electromagnetic radiation in the units covariant formalism incorporating Dirac's large numbers hypothesis (LNH). A direct field-to-particle technique is used to obtain the photon propagation equation which explicitly involves the photon replication rate. This replication rate is fixed uniquely by requiring that the form of a free-photon distribution function be preserved, as required by the 2.7 K cosmic radiation. One finds that with this particular photon replication rate the units covariant formalism developed in Paper I actually predicts that the ratio of photon number to proton number in the universe varies as t to the 1/4, precisely in accord with LNH. The cosmological red-shift law is also derived and it is shown to differ considerably from the standard form of (nu)(R) - const.

Adams, P. J.↗

Theoretical and experimental study of twisted and cambered delta wings designed for a Mach number of 3.5

Data are provided for the evaluation of the aerodynamic performance of a series of twisted and cambered delta wings designed for a Mach number of 3.5. Systematic force and pressure data are also presented for comparison with theory. Force tests were made at Mach numbers of 2.3, 3.0, 3.5, 4.0, and 4.6. Design lift coefficients of 0.0 and 0.1 were employed on the 55 deg and 68 deg sweep wings, and design lift coefficients of 0.0, 0.05, and 0.1 were employed on the 76 deg sweep wings. Pressure tests were conducted on the 55 deg and 76 deg sweep flat wings and on the 0.1 design lift coefficient 76 deg sweep wing. The results indicate that for the sweep angles tested, an increase in the zero-lift pitching-moment coefficient is the primary benefit of twist and camber at a Mach number of 3.5. Comparison of the experimental results with results obtained from several lift theories indicates that the Carlson-Middleton linear theory method gave the best overall agreement. The pressure data indicate, however, that there is a cancellation of error at high angle of attack where the lower surface pressures are significantly underpredicted over the inboard region of the wing and where the upper and lower surface pressures are overpredicted over the outboard region of the wing.

Sorrells, R. B., III↗

Magnetohydrodynamic and gasdynamic theories for planetary bow waves

The observed properties of bow waves and the associated plasma flows are outlined, along with those features identified that can be described by a continuum magnetohydrodynamic flow theory as opposed to a more detailed multicomponent particle and field plasma theory. The primary objectives are to provide an account of the fundamental concepts and current status of the magnetohydrodynamic and gas dynamic theories for solar wind flow past planetary bodies. This includes a critical examination of: (1) the fundamental assumptions of the theories; (2) the various simplifying approximations introduced to obtain tractable mathematical problems; (3) the limitations they impose on the results; and (4) the relationship between the results of the simpler gas dynamic-frozen field theory and the more accurate but less completely worked out magnetohydrodynamic theory. Representative results of the various theories are presented and compared. A number of deficiencies, ambiguities, and suggestions for improvements are discussed, and several significant extensions of the theory required to provide comparable results for all planets, their satellites, and comets are noted.

Spreiter, J. R.↗