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 145 records · Page 8

Error in Airspeed Measurement Due to Static-Pressure Field Ahead of the Wing Tip of a Swept-Wing Airplane Model at Transonic Speeds

As part of a study of means of airspeed measurement at transonic speeds the use of static orifices located ahead of the wing tip has been investigated for possible application to service or research airspeed installations. The local static pressure and local Mach number have been measured at a distance of 1 tip chord ahead of the wing tip of a model of a swept-wing fighter airplane at true Mach numbers between 0.7 and 1.08 by the NACA wing-flow method. All measurements were made at or near zero lift. The local Mach number was found to be essentially equal to the true Mach numbers less than about 0.90. The local Mach number was found to be about 0.97 at a true Mach number of 0.95, and to be about 1.04 at a true Mach number of 1.08. The local Mach number provided a reasonable sensitive measure of true Mach number except for a restricted region near a true Mach number of 1.0 where the local Mach number did not change appreciably with true Mach number. The linear theory was found to predict qualitatively the effect of the fuselage on the static pressure ahead of the wing time but gave a reasonable prediction of the effect of the wing on the static pressure only at Mach numbers below 0.95.

Danforth, Edward C.B.↗

Free-Flight Aerodynamic-Heating Data to Mach Number 10.4 for a Modified Von Karman Nose Shape

Aerodynamic-heating data have been obtained on a modified fineness-ratio-5.0 Von Karman nose shape at free-stream Mach numbers up to 10.4 with a rocket-propelled model. Transient skin temperatures were measured at one station, 26.6 inches behind the tip of a nose 31.6 inches long. A maximum skin temperature of 1,663 deg R was measured soon after the maximum Mach number was obtained. During the periods for which experimental Stanton numbers were presented, flow parameters just outside the boundary layer at the temperature measuring station varied as follows: the local Mach number varied in the range between 0.8 and 9.0 and the local Reynolds number varied in the range between 0.8 x 10(exp 6) and 35.5 x 10(exp 6). The ratio of skin temperature to local static temperature varied between 1.0 and 3.6. The experimental Stanton numbers agreed well with Van Driest's turbulent theory while the local Reynolds number was high - that is, while the local Reynolds number varied in a range above 6.8 x 10(exp 6). For local Reynolds numbers less than 3.5 x 10(exp 6) the experimental Stanton numbers were of the magnitude predicted by Van Driest's laminar theory. Transition from turbulent to laminar flow at the temperature measuring station, as indicated by the change in the magnitude of the Stanton number, occurred as the local Reynolds number decreased from 6.8 x 10(exp 6) to 3.5 x 10(exp 6) at essentially a constant local Mach number of about 9.0.

Bland, William M., Jr.↗

Acoustic shocks in a variable area duct containing near sonic flows

Acoustic shock waves in a variable area duct which contains near sonic flows are considered. The problem is modeled after an aeroengine inlet. Area variation of a duct and high Mach number mean the flow reduces acoustical energy yielding substantial noise reduction. One possible reason for this is acoustic shock. The use of an explicit accurate numerical method which captures shocks is described. Comparison of the results are made with an existing asymptotic theory for Mach numbers close to unity. When shock occurs reduction of sound pressure levels are shown by example.

Hariharan, S. I.↗

Acoustic shocks in a variable area duct containing near sonic flows

Acoustic shock waves in a variable area duct which contains near sonic flows are considered. The problem is modeled after an aeroengine inlet. Area variation of a duct and high Mach number mean the flow reduces acoustical energy yielding substantial noise reduction. One possible reason for this is acoustic shock. The use of an explicit accurate numerical method which captures shocks is described. Comparison of the results are made with an existing asymptotic theory for Mach numbers close to unity. When shock occurs reduction of sound pressure levels are shown by example.

Hariharan, S. I.↗

The analysis and simulation of compressible turbulence

Compressible turbulent flows at low turbulent Mach numbers are considered. Contrary to the general belief that such flows are almost incompressible, (i.e., the divergence of the velocity field remains small for all times), it is shown that even if the divergence of the initial velocity field is negligibly small, it can grow rapidly on a non-dimensional time scale which is the inverse of the fluctuating Mach number. An asymptotic theory which enables one to obtain a description of the flow in terms of its divergence-free and vorticity-free components has been developed to solve the initial-value problem. As a result, the various types of low Mach number turbulent regimes have been classified with respect to the initial conditions. Formulae are derived that accurately predict the level of compressibility after the initial transients have disappeared. These results are verified by extensive direct numerical simulations of isotropic turbulence.

Erlebacher, Gordon↗

The analysis and simulation of compressible turbulence

Compressible turbulent flows at low turbulent Mach numbers are considered. Contrary to the general belief that such flows are almost incompressible (i.e., the divergence of the velocity field remains small for all times), it is shown that even if the divergence of the initial velocity field is negligibly small, it can grow rapidly on a nondimensional time scale which is the inverse of the fluctuating Mach number. An asymptotic theory which enables one to obtain a description of the flow in terms of its divergence-free and vorticity-free components has been developed to solve the initial-value problem. As a result, the various types of low Mach number turbulent regimes have been classified with respect to the initial conditions. Formulae are derived that accurately predict the level of compressibility after the initial transients have disappeared. These results are verified by extensive direct numerical simulations of isotropic turbulence.

Erlebacher, Gordon↗

Relaxation processes in a low-order three-dimensional magnetohydrodynamics model

The time asymptotic behavior of a Galerkin model of 3D magnetohydrodynamics (MHD) has been interpreted using the selective decay and dynamic alignment relaxation theories. A large number of simulations has been performed that scan a parameter space defined by the rugged ideal invariants, including energy, cross helicity, and magnetic helicity. It is concluded that time asymptotic state can be interpreted as a relaxation to minimum energy. A simple decay model, based on absolute equilibrium theory, is found to predict a mapping of initial onto time asymptotic states, and to accurately describe the long time behavior of the runs when magnetic helicity is present. Attention is also given to two processes, operating on time scales shorter than selective decay and dynamic alignment, in which the ratio of kinetic to magnetic energy relaxes to values 0(1). The faster of the two processes takes states initially dominant in magnetic energy to a state of near-equipartition between kinetic and magnetic energy through power law growth of kinetic energy. The other process takes states initially dominant in kinetic energy to the near-equipartitioned state through exponential growth of magnetic energy.

Stribling, Troy↗

An improved exceedance theory for combined random stresses

An extension is presented of Rice's classic solution for the exceedances of a constant level by a single random process to its counterpart for an n-dimensional vector process. An interaction boundary, analogous to the constant level considered by Rice for the one-dimensional case, is assumed in the form of a hypersurface. The theory for the numbers of boundary exceedances is developed by using a joint statistical approach which fully accounts for all cross-correlation effects. An exact expression is derived for the n-dimensional exceedance density function, which is valid for an arbitrary interaction boundary. For application to biaxial states of combined random stress, the general theory is reduced to the two-dimensional case. An elliptical stress interaction boundary is assumed and the exact expression for the density function is presented. The equations are expressed in a format which facilitates calculating the exceedances by numerically evaluating a line integral. The behavior of the density function for the two-dimensional case is briefly discussed.

Lester, H. C.↗

A Viscoplastic Constitutive Theory for Monolithic Ceramic Materials

With increasing use of ceramic materials in high temperature structural applications such as advanced heat engine components, the need arises to accurately predict thermomechanical behavior. This paper, which is the first of two in a series, will focus on inelastic deformation behavior associated with these service conditions by providing an overview of a viscoplastic constitutive model that accounts for time-dependent hereditary material deformation (e.g., creep, stress relaxation, etc.) in monolithic structural ceramics. Early work in the field of metal plasticity indicated that inelastic deformations are essentially unaffected by hydrostatic stress. This is not the case, however, for ceramic-based material systems, unless the ceramic is fully dense. The theory presented here allows for fully dense material behavior as a limiting case. In addition, ceramic materials exhibit different time-dependent behavior in tension and compression. Thus, inelastic deformation models for ceramics must be constructed in a fashion that admits both sensitivity to hydrostatic stress and differing behavior in tension and compression. A number of constitutive theories for materials that exhibit sensitivity to the hydrostatic component of stress have been proposed that characterize deformation using time-independent classical plasticity as a foundation. However, none of these theories allow different behavior in tension and compression. In addition, these theories are somewhat lacking in that they are unable to capture creep, relaxation, and rate-sensitive phenomena exhibited by ceramic materials at high temperature. When subjected to elevated service temperatures, ceramic materials exhibit complex thermomechanical behavior that is inherently time-dependent, and hereditary in the sense that current behavior depends not only on current conditions, but also on thermo-mechanical history. The objective of this work is to present the formulation of a macroscopic continuum theory that captures these time-dependent phenomena. Specifically, the overview contained in this paper focuses on the multiaxial derivation of the constitutive model, and examines the scalar threshold function and its attending geometrical implications.

Janosik, Lesley A.↗

A modification to linearized theory for prediction of pressure loadings on lifting surfaces at high supersonic Mach numbers and large angles of attack

A new linearized-theory pressure-coefficient formulation was studied. The new formulation is intended to provide more accurate estimates of detailed pressure loadings for improved stability analysis and for analysis of critical structural design conditions. The approach is based on the use of oblique-shock and Prandtl-Meyer expansion relationships for accurate representation of the variation of pressures with surface slopes in two-dimensional flow and linearized-theory perturbation velocities for evaluation of local three-dimensional aerodynamic interference effects. The applicability and limitations of the modification to linearized theory are illustrated through comparisons with experimental pressure distributions for delta wings covering a Mach number range from 1.45 to 4.60 and angles of attack from 0 to 25 degrees.

Carlson, H. W.↗

Causality, unitarity, and the weak gravity conjecture

We consider the shift of charge-to-mass ratio for extremal black holes in the context of effective field theory, motivated by the Weak Gravity Conjecture. We constrain extremality corrections in different regimes subject to unitarity and causality constraints. In the asymptotic IR, we demonstrate that for any supersymmetric theory in flat space, and for all minimally coupled theories, logarithmic running at one loop pushes the Wilson coefficient of certain four-derivative operators to be larger at lower energies, guaranteeing the existence of sufficiently large black holes with Q > M. We identify two exceptional cases of nonsupersymmetric theories involving large numbers of light states and Planck-scale nonminimal couplings, in which the sign of the running is reversed, leading to black holes with negative corrections to Q/M in the deep IR, but argue that these do not rule out extremal black holes as the requisite charged states for the WGC. We separately show that causality and unitarity imply that the leading threshold corrections to the effective action from integrating out massive states, in any weakly coupled theory, can be written as a sum of squares and is manifestly positive for black hole backgrounds. Quite beautifully, the shift in the extremal Q/M ratio is directly proportional to the shift in the on-shell action, guaranteeing that these threshold corrections push Q > M in compliance with the WGC. Our results apply for black holes with or without dilatonic coupling and charged under any number of U(1)s.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Experimentally determined stiffness and damping of an inherently compensated air squeeze-film damper

Values of damping and stiffness were determined experimentally for an externally pressurized, inherently compensated, compressible squeeze-film damper up to excitation frequencies of 36,000 cycles per minute. Experimental damping values were higher than theory predicted at low squeeze numbers and less than predicted at high squeeze numbers. Experimental values of air film stiffness were less than theory predicted at low squeeze numbers and much greater at higher squeeze numbers. Results also indicate sufficient damping to attenuate amplitudes and forces at the critical speed when using three dampers in the flexible support system of a small, lightweight turborotor.

Cunningham, R. E.↗

Investigation at Mach Numbers of 0.20 to 3.50 of Blended Wing-Body Combinations of Sonic Design with Diamond, Delta, and Arrow Plan Forms

The models had aspect-ratio-2 diamond, delta, and arrow wings with the leading edges swept 45.00 deg, 59.04 deg, and 70.82 deg, respectively. The wing sections were computed by varying the section shape along with the body radii (blending process) to match the prescribed area distribution and wing plan form. The wing sections had an average value of maximum thickness ratio of about 4 percent of the local chords in a streamwise direction. The models were tested with transition fixed at Reynolds numbers of about 4,000,000 to 9,000,0000, based on the mean aerodynamic chord of the wings. The effect of varying Reynolds number was checked at both subsonic and supersonic speeds. The diamond model was superior to the other plan forms at transonic speeds ((L/D)max = 11.00 to 9.52) because of its higher lift-curve slope and near optimum wave drag due to the blending process. For the wing thickness tested with the diamond model, the marked body and wing contouring required for transonic conditions resulted in a large wave-drag penalty at the higher supersonic Mach numbers where the leading and trailing edges of the wing were supersonic. Because of the low sweep of the trailing edge of the delta model, this configuration was less adaptable to the blending process. Removing a body bump prescribed by the Mach number 1.00 design resulted in a good supersonic design. This delta model with 10 percent less volume was superior to the other plan forms at Mach numbers of 1.55 to 2.35 ((L/D)max = 8.65 to 7.24), but it and the arrow model were equally good at Mach numbers of 2.50 to 3.50 ((L/D)max - 6.85 to O.39). At transonic speeds the arrow model was inferior because of the reduced lift-curve slope associated with its increased sweep and also because of the wing base drag. The wing base-drag coefficients of the arrow model based on the wing planform area decreased from a peak value of 0.0029 at Mach number 1.55 to 0.0003 at Mach number 3.50. Linear supersonic theory was satisfactory for predicting the aerodynamic trends at Mach numbers from 1.55 to 3.50 of lift-curve slope, wave drag, drag due to lift, aerodynamic-center location, and maximum lift-drag ratios for each of the models.

Holdaway, George H.↗

Baryonic Higgs and dark matter

We discuss the correlation between dark matter and Higgs decays in gauge theories where the dark matter is predicted from anomaly cancellation. In these theories, the Higgs responsible for the breaking of the gauge symmetry generates the mass for the dark matter candidate. We investigate the Higgs decays in the minimal gauge theory for Baryon number. After imposing the dark matter density and direct detection constraints, we find that the new Higgs can have a large branching ratio into two photons or into dark matter. Furthermore, we discuss the production channels and the unique signatures at the Large Hadron Collider.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Holomorphic modular bootstrap revisited

In this work we revisit the “holomorphic modular bootstrap”, i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters. By making use of the representation theory of PSL(2, $\mathbb{Z}_n$), we describe a method to classify allowed central charges and weights (c, h i ) for theories with any number of characters d. This allows us to avoid various bottlenecks encountered previously in the literature, and leads to a classification of consistent characters up to d = 5 whose modular differential equations are uniquely fixed in terms of (c, h i ). In the process, we identify the full set of constraints on the allowed values of the Wronskian index for fixed d ≤ 5.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Wake curvature and trailing edge interaction effects in viscous flow over airfoils

A theory developed for analyzing viscous flows over airfoils at high Reynolds numbers is described. The theory includes a complete treatment of viscous interaction effects induced by the curved wake behind the airfoil and accounts for normal pressure gradients across the boundary layer in the trailing edge region. A brief description of a computer code that was developed to solve the extended viscous interaction equations is given. Comparisons of the theoretical results with wind tunnel data for two rear loaded airfoils at supercritical conditions are presented.

Melnik, R. E.↗

Constitutive Theory Developed for Monolithic Ceramic Materials

With the increasing use of advanced ceramic materials in high-temperature structural applications such as advanced heat engine components, the need arises to accurately predict thermomechanical behavior that is inherently time-dependent and that is hereditary in the sense that the current behavior depends not only on current conditions but also on the material's thermomechanical history. Most current analytical life prediction methods for both subcritical crack growth and creep models use elastic stress fields to predict the time-dependent reliability response of components subjected to elevated service temperatures. Inelastic response at high temperatures has been well documented in the materials science literature for these material systems, but this issue has been ignored by the engineering design community. From a design engineer's perspective, it is imperative to emphasize that accurate predictions of time-dependent reliability demand accurate stress field information. Ceramic materials exhibit different time-dependent behavior in tension and compression. Thus, inelastic deformation models for ceramics must be constructed in a fashion that admits both sensitivity to hydrostatic stress and differing behavior in tension and compression. A number of constitutive theories for materials that exhibit sensitivity to the hydrostatic component of stress have been proposed that characterize deformation using time-independent classical plasticity as a foundation. However, none of these theories allow different behavior in tension and compression. In addition, these theories are somewhat lacking in that they are unable to capture the creep, relaxation, and rate-sensitive phenomena exhibited by ceramic materials at high temperatures. The objective of this effort at the NASA Lewis Research Center has been to formulate a macroscopic continuum theory that captures these time-dependent phenomena. Specifically, the effort has focused on inelastic deformation behavior associated with these service conditions by developing a multiaxial viscoplastic constitutive model that accounts for time-dependent hereditary material deformation (such as creep and stress relaxation) in monolithic structural ceramics. Using continuum principles of engineering mechanics, we derived the complete viscoplastic theory from a scalar dissipative potential function.

Janosik, Lesley A.↗