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All sourcesNASA NTRSDOE OSTI
NASA NTRS2004

Deriving Laws from Ordering Relations

The effect of Richard T. Cox's contribution to probability theory was to generalize Boolean implication among logical statements to degrees of implication, which are manipulated using rules derived from consistency with Boolean algebra. These rules are known as the sum rule, the product rule and Bayes Theorem, and the measure resulting from this generalization is probability. In this paper, I will describe how Cox s technique can be further generalized to include other algebras and hence other problems in science and mathematics. The result is a methodology that can be used to generalize an algebra to a calculus by relying on consistency with order theory to derive the laws of the calculus. My goals are to clear up the mysteries as to why the same basic structure found in probability theory appears in other contexts, to better understand the foundations of probability theory, and to extend these ideas to other areas by developing new mathematics and new physics. The relevance of this methodology will be demonstrated using examples from probability theory, number theory, geometry, information theory, and quantum mechanics.

Knuth, Kevin H.↗
NASA NTRS1990

On the instabilities of supersonic mixing layers - A high-Mach-number asymptotic theory

The stability of a family of tanh mixing layers is studied at large Mach numbers using perturbation methods. It is found that the eigenfunction develops a multilayered structure, and the eigenvalue is obtained by solving a simplified version of the Rayleigh equation (with homogeneous boundary conditions) in one of these layers which lies in either of the external streams. This analysis leads to a simple hypersonic similarity law which explains how spatial and temporal phase speeds and growth rates scale with Mach number and temperature ratio. Comparisons are made with numerical results, and it is found that this similarity law provides a good qualitative guide for the behavior of the instability at high Mach numbers. In addition to this asymptotic theory, some fully numerical results are also presented (with no limitation on the Mach number) in order to explain the origin of the hypersonic modes (through mode splitting) and to discuss the role of oblique modes over a very wide range of Mach number and temperature ratio.

Balsa, Thomas F.↗
NASA NTRS1980

Numerical analysis of natural convection in two-dimensional square and circular containers in low gravity

A numerical study of natural convection in circular cylinder and square enclosures shows that the analytic low Rayleigh number theory of previous investigators is valid for Rayleigh numbers up to 1000. For a Rayleigh number of 5000, steady state values of maximum fluid velocity differ by 20 percent. This deviation between analytic theory and numerical results increases for higher Rayleigh numbers. In addition, the low Rayleigh number theory is shown to be valid for higher Rayleigh numbers for a portion of the transient phase before significant deviation becomes apparent. It is also shown that square shaped experimental configurations may be analytically approximated with good accuracy by circular cylinders of equal cross sectional area for the prediction of convection velocities and flow patterns at low Rayleigh number.

Robertson, S. J.↗
NASA NTRS1986

Very high Mach number shocks - Theory

The theory and simulation of collisionless perpendicular supercritical shock structure is reviewed, with major emphasis on recent research results. The primary tool of investigation is the hybrid simulation method, in which the Newtonian orbits of a large number of ion macroparticles are followed numerically, and in which the electrons are treated as a charge neutralizing fluid. The principal results include the following: (1) electron resistivity is not required to explain the observed quasi-stationarity of the earth's bow shock, (2) the structure of the perpendicular shock at very high Mach numbers depends sensitively on the upstream value of beta (the ratio of the thermal to magnetic pressure) and electron resistivity, (3) two-dimensional turbulence will become increasingly important as the Mach number is increased, and (4) nonadiabatic bulk electron heating will result when a thermal electron cannot complete a gyrorbit while transiting the shock.

Quest, Kevin B.↗
NASA NTRS1972

The large Reynolds number - Asymptotic theory of turbulent boundary layers.

A self-consistent, asymptotic expansion of the one-point, mean turbulent equations of motion is obtained. Results such as the velocity defect law and the law of the wall evolve in a relatively rigorous manner, and a systematic ordering of the mean velocity boundary layer equations and their interaction with the main stream flow are obtained. The analysis is extended to the turbulent energy equation and to a treatment of the small scale equilibrium range of Kolmogoroff; in velocity correlation space the two-thirds power law is obtained. Thus, the two well-known 'laws' of turbulent flow are imbedded in an analysis which provides a great deal of other information.

Mellor, G. L.↗
NASA NTRS1975

Quantization and symmetry in periodic coverage patterns with applications to earth observation

Analytical approaches based on an idealized physical model and concepts from number theory show that in periodic coverage patterns, uniquely defined by their revolution numbers R (orbital) and N (rotational), the subnodal points are earth-fixed, and they divide the equator into R equal segments of length s. The ascending subsatellite trace crosses each point once (only) each period. The descending subnodal points coincide with the ascending points if the integers N and R have like parity, and bisect the intervals between them if opposite. The interval between consecutive unidirectional crossings is Ns. Symmetries extend the equatorial results to all parallels of latitude. Complete periodic patterns of traces exhibit an overall symmetry, with trace intersections confined to discrete coordinate values which are quantized in longitude (basic s-unit) and symmetric in latitude.

King, J. C.↗
NASA NTRS1975

Quantization and symmetry in periodic coverage patterns with applications to earth observation

The general orbit-coverage problem in a simplified physical model is investigated by application of numerical approaches derived from basic number theory. A system of basic and general properties is defined by which idealized periodic coverage patterns may be characterized, classified, and delineated. The principal common features of these coverage patterns are their longitudinal quantization, determined by the revolution number R, and their overall symmetry.

King, J. C.↗
NASA NTRS1979

NASF transposition network: A computing network for unscrambling p-ordered vectors

The viewpoints of design, programming, and application of the transportation network (TN) is presented. The TN is a programmable combinational logic network that connects 521 memory modules to 512 processors. The unscrambling of p-ordered vectors to 1-ordered vectors in one cycle is described. The TN design is based upon the concept of cyclic groups from abstract algebra and primitive roots and indices from number theory. The programming of the TN is very simple, requiring only 20 bits: 10 bits for offset control and 10 bits for barrel switch shift control. This simple control is executed by the control unit (CU), not the processors. Any memory access by a processor must be coordinated with the CU and wait for all other processors to come to a synchronization point. These wait and synchronization events can be a degradation in performance to a computation. The TN application is for multidimensional data manipulation, matrix processing, and data sorting, and can also perform a perfect shuffle. Unlike other more complicated and powerful permutation networks, the TN cannot, if possible at all, unscramble non-p-ordered vectors in one cycle.

Lim, R. S.↗
NASA NTRS1959

Hydrodynamic Characteristics of Two Low-Drag Supercavitating Hydrofoils

An experimental investigation has been conducted in Langley tank no. 2 to determine the hydrodynamic characteristics of two low-drag supercavitating hydrofoils operating in a range of cavitation numbers from 0 to approximately 6. The hydrofoils had aspect ratios of 1 and 3, and the sections were derived by assuming five terms in the vorticity-distribution expansion of the equivalent airfoil. The aspect-ratio-1 hydrofoil was also tested at zero cavitation number with two sets of end plates having depths of 3/8 and 1/4 chords. Zero cavitation number was established by operating the hydrofoils near the water surface so that complete ventilation of the upper surfaces could be obtained. For those depths of submersion where complete ventilation was not obtained through vortex ventilation, two probes were used to introduce air to the upper surfaces of the hydrofoils and to induce complete ventilation. Data were obtained for a range of speeds from 20 to 80 fps, angles of attack from 2 to 20 deg, and ratios of depth of submersion to chord from 0 to 0.85. The experimental results obtained from the aspect-ratio-1 and aspect-ratio-3, five-term hydrofoils were compared with a three-dimensional zero-cavitation-number theory. The theoretical and experimental values of lift and center of pressure for the aspect-ratio-1 hydrofoil were in agreement, within engineering accuracy, for the range of lift coefficients investigated. The theoretical drag coefficients were lower, by a constant amount, than the experimental drag coefficients. The theoretical expressions derived for the lift, drag, and center of pressure of the aspect-ratio-3 hydrofoil were in agreement, within engineering accuracy, with the experimental values. The theoretical and experimental drag coefficients of the aspect-ratio-3 five-term hydrofoil were lower than the theoretical drag coefficients computed for a comparable Tulin-Burkart hydrofoil.

McGehee, John R.↗
NASA NTRS1960

Experimental Investigation of Two Low-Drag Supercavitating Hydrofoils at Speeds up to 200 Feet per Second

An experimental investigation has been made in the Langley highspeed hydrodynamics facility to determine the force and moment characteristics of two hydrofoils (one having an aspect ratio of 1 and the other having an aspect ratio of 3) designed to have improved lift-drag ratios when operating under either supercavitating or ventilated conditions. Measurements were made of lift, drag, and pitching moment over a range of angles of attack from 40 to 200 for depths of submersion varying from 0 to approximately 1 chord. The range of speed for the investigation was from 110 to 200 feet per second. When the upper surface of the hydrofoils was completely unwetted, the experimental values of lift and drag forces were in good agreement with the theoretical values obtained from the zero-cavitation-number theory. The theoretical values for minimum angle of attack for operation with the upper surface of the hydrofoil unwetted define the lower limits of angle of attack for which the experimental values of lift coefficient are either in agreement with or slightly greater than those predicted by theory.

Christopher, Kenneth W.↗
NASA NTRS1979

Modal propagation angles in a cylindrical duct with flow and their relation to sound radiation

The main emphasis is upon the propagation angle with respect to the duct axis and its relation to the far-field acoustic radiation pattern. When the steady flow Mach number is accounted for in the duct, the propagation angle in the duct is shown to be coincident with the angle of the principal lobe of far-field radiation obtained using the Wiener-Hopf technique. Different Mach numbers are allowed within the duct and in the external field. For static tests with a steady flow in an inlet but with no external Mach number the far-field radiation pattern is shifted considerably toward the inlet axis when compared to zero Mach number radiation theory. As the external Mach number is increased the noise radiation pattern is shifted away from the inlet axis. The theory is developed using approximations for sound propagation in circular ducts. An exact analysis using Hankel function solutions for the zero Mach number case is given to provide a check of the simpler approximate theory.

Rice, E. J.↗
DOE OSTI2026

Excitation energy of fission fragments within nuclear time-dependent density functional theory

The number and properties of the neutrons and photons emitted in nuclear fission are directly related to the excitation energy of the fission fragments when they are formed at scission. Though not observable experimentally because of the extremely short timescales, the excitation energy of fission fragments can be predicted by microscopic theory based on time-dependent density functional theory (TDDFT). Initial results on the value of the total kinetic energy of fission reactions were very promising, but could not probe all possible fragmentations. Here, in this work, we perform large-scale TDDFT calculations in 240 Pu enabled by the development of a new TDDFT solver. We obtain TDDFT trajectories covering nearly all possible fragmentations. We find that the total kinetic energy is close to experimental values only for the most likely fission while it is severely underestimated at both small and large asymmetries. This conclusion seems rather independent of the parametrization of the energy functional, both in its particle-hole and particle-particle channels.

Physics - Nuclear physics and radiation physics↗
NASA NTRS1972

Assessment of a transitional boundary layer theory at low hypersonic Mach numbers

An investigation was carried out to assess the accuracy of a transitional boundary layer theory in the low hypersonic Mach number regime. The theory is based upon the simultaneous numerical solution of the boundary layer partial differential equations for the mean motion and an integral form of the turbulence kinetic energy equation which controls the magnitude and development of the Reynolds stress. Comparisions with experimental data show the theory is capable of accurately predicting heat transfer and velocity profiles through the transitional regime and correctly predicts the effects of Mach number and wall cooling on transition Reynolds number. The procedure shows promise of predicting the initiation of transition for given free stream disturbance levels. The effects on transition predictions of the pressure dilitation term and of direct absorption of acoustic energy by the boundary layer were evaluated.

Shamroth, S. J.↗
DOE OSTI2025

Foaming prediction in pure liquids from dimensionless numbers inspired by the theory of fluid behavior for drops

Foaming prediction is critical for selecting materials and designing processes in industries such as bioprocessing and gas processing. Existing models lack the generality needed for a wide range of materials and overlook the foaming behavior in pure liquids. Here, this work presents a novel method for predicting foaming in pure liquids based on their density, surface tension, and viscosity, using Reynolds ( Re ) and Ohnesorge ( Oh ) numbers. A foaming prediction map, leveraging the theory of fluid drop behavior, was developed by plotting these numbers. This map delineates distinct non-foaming and foaming regions, functioning as a binary classifier for foaming predictions. The map was fitted and validated through shake test experiments on 46 liquids, demonstrating reliable predictions, except for a specific region characterized by small Oh and large Re numbers. This region corresponded to relatively low foam stability and high turbulence, making foaming predictions challenging for liquids in this category.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗