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NASA NTRS1993

Coronal heating by the resonant absorption of Alfven waves - Importance of the global mode and scaling laws

Numerical simulations of the MHD equations for a fully compressible, low-beta, resistive plasma are used to study the resonance absorption process for the heating of coronal active region loops. Comparisons with more approximate analytic models show that the major predictions of the analytic theories are, to a large extent, confirmed by the numerical computations. The simulations demonstrate that the dissipation occurs primarily in a thin resonance layer. Some of the analytically predicted features verified by the simulations are (a) the position of the resonance layer within the initial inhomogeneity; (b) the importance of the global mode for a large range of loop densities; (c) the dependence of the resonance layer thickness and the steady-state heating rate on the dissipation coefficient; and (d) the time required for the resonance layer to form. In contrast with some previous analytic and simulation results, the time for the loop to reach a steady state is found to be the phase-mixing time rather than a dissipation time. This disagreement is shown to result from neglect of the existence of the global mode in some of the earlier analyses. The resonant absorption process is also shown to behave similar to a classical driven harmonic oscillator.

Steinolfson, Richard S.↗

NASA NTRS1950

Linearized compressible-flow theory for sonic flight speeds

The partial differential equation for the perturbation velocity potential is examined for free-stream Mach numbers close to and equal to one. It is found that, under the assumptions of linearized theory, solutions can be found consistent with the theory for lifting-surface problems both in stationary three-dimensional flow and in unsteady two-dimensional flow. Several examples are solved including a three dimensional swept-back wing and two dimensional harmonically-oscillating wing, both for a free stream Mach number equal to one. Momentum relations for the evaluation of wave and vortex drag are also discussed. (author)

Heaslet, Max A↗

NASA NTRS1952

A Theoretical Analysis of the Effects of Fuel Motion on Airplane Dynamics

The general equations of motion for an airplane with a number of spherical fuel tanks are derived. The motion of the fuel is approximated by the motion of solid pendulums. The same type of derivation and equations are shown to apply to any type of fuel tank where the motion of the fuel may be represented in terms of undamped harmonic oscillators. Motions are calculated for a present-day high-speed airplane and a free-flying airplane model with two spherical tanks in the symmetry plane.

Schy, Albert A↗

NASA NTRS1952

Experimental Aerodynamic Derivatives of a Sinusoidally Oscillating Airfoil in Two-Dimensional Flow

Experimental measurements of the aerodynamic reactions on a symmetrical airfoil oscillating harmonically in a two-dimensional flow are presented and analyzed. Harmonic motions include pure pitch and pure translation, for several amplitudes and superimposed on an initial angle of attack, as well as combined pitch and translation. The apparatus and testing program are described briefly and the necessary theoretical background is presented. In general, the experimental results agree remarkably well with the theory, especially in the case of the pure motions. The net work per cycle for a motion corresponding to flutter is experimentally determined to be zero. Considerable consistent data for pure pitch were obtained from a search of available reference material, and several definite Reynolds number effects are evident.

Halfman, Robert L↗

NASA NTRS1956

Finite Span Wings in Compressible Flow

Equations are developed using the source distribution method for the velocity potential function and pressure on thin wings in steady and unsteady motion. Closed form solutions are given for harmonically oscillating wings of general plan form including the effect of the wing wake. Some useful examples are presented in an appendix for arrow, semielliptical, and hexagonal plan form wings.

WINGS, ELLIPTICAL↗

NASA NTRS1993

Smoothly deformed light

A single mode cavity is deformed smoothly to change its electromagnetic eigenfrequency. The system is modeled as a simple harmonic oscillator with a varying period. The Wigner function of the problem is obtained exactly by starting with a squeezed initial state. The result is evaluated for a linear change of the cavity length. The approach to the adiabatic limit is investigated. The maximum squeezing is found to occur for smooth change lasting only a fraction of the oscillational period. However, only a factor of two improvement over the adiabatic result proves to be possible. The sudden limit cannot be investigated meaningfully within the model.

Stenholm, Stig↗

NASA NTRS1993

One dimensional representations in quantum optics

The possibility of representing the quantum states of a harmonic oscillator not on the whole alpha-plane but on its one dimensional manifolds is considered. It is shown that a simple Gaussian distribution along a straight line describes a quadrature squeezed state while a similar Gaussian distribution along a circle leads to the amplitude squeezed state. The connection between the one dimensional representations and the usual Glauber representation is discussed.

Janszky, J.↗

NASA NTRS1993

On a Lagrange-Hamilton formalism describing position and momentum uncertainties

According to Heisenberg's uncertainty relation, in quantum mechanics it is not possible to determine, simultaneously, exact values for the position and the momentum of a material system. Calculating the mean value of the Hamiltonian operator with the aid of exact analytic Gaussian wave packet solutions, these uncertainties cause an energy contribution additional to the classical energy of the system. For the harmonic oscillator, e.g., this nonclassical energy represents the ground state energy. It will be shown that this additional energy contribution can be considered as a Hamiltonian function, if it is written in appropriate variables. With the help of the usual Lagrange-Hamilton formalism known from classical particle mechanics, but now considering this new Hamiltonian function, it is possible to obtain the equations of motion for position and momentum uncertainties.

Schuch, Dieter↗

NASA NTRS1993

Supersqueezed states from squeezed states

Using super-Baker-Campbell-Hausdorff relations on the elements of the supergroup OSP(2/2), we derive the supersqueeze operator and the supersqueezed states, which are the supersymmetric generalization of the squeezed states of the harmonic oscillator.

Nieto, Michael Martin↗

NASA NTRS1992

Flutter analysis of supersonic axial flow cascades using a high resolution Euler solver. Part 1: Formulation and validation

This report presents, in two parts, a dynamic aeroelastic stability (flutter) analysis of a cascade of blades in supersonic axial flow. Each blade of the cascade is modeled as a typical section having pitching and plunging degrees of freedom. Aerodynamic forces are obtained from a time accurate, unsteady, two-dimensional cascade solver based on the Euler equations. The solver uses a time marching flux-difference splitting (FDS) scheme. Flutter stability is analyzed in the frequency domain. The unsteady force coefficients required in the analysis are obtained by harmonically oscillating (HO) the blades for a given flow condition, oscillation frequency, and interblade phase angle. The calculated time history of the forces is then Fourier decomposed to give the required unsteady force coefficients. An influence coefficient (IC) method and a pulse response (PR) method are also implemented to reduce the computational time for the calculation of the unsteady force coefficients for any phase angle and oscillation frequency. Part 1, this report, presents these analysis methods and their validation by comparison with results obtained from linear theory for a selected flat plate cascade geometry. A typical calculation for a rotor airfoil is also included to show the applicability of the present solver for airfoil configurations. The predicted unsteady aerodynamic forces for a selected flat plate cascade geometry and flow conditions correlated well with those obtained from linear theory for different interblade phase angles and oscillation frequencies. All the three methods of predicting unsteady force coefficients, namely, HO, IC, and PR, showed good correlations with each other. It was established that only a single calculation with four blade passages is required to calculate the aerodynamic forces for any phase angle for a cascade consisting of any number of blades, for any value of the oscillation frequency. Flutter results, including mistuning effects, for a cascade of stator airfoils are presented in Part 2 of the report.

Reddy, T. S. R.↗

NASA NTRS1994

Nonlinear stability of oscillatory core-annular flow: A generalized Kuramoto-Sivashinsky equation with time periodic coefficients

In this paper the nonlinear stability of two-phase core-annular flow in a pipe is examined when the acting pressure gradient is modulated by time harmonic oscillations and viscosity stratification and interfacial tension is present. An exact solution of the Navier-Stokes equations is used as the background state to develop an asymptotic theory valid for thin annular layers, which leads to a novel nonlinear evolution describing the spatio-temporal evolution of the interface. The evolution equation is an extension of the equation found for constant pressure gradients and generalizes the Kuramoto-Sivashinsky equation with dispersive effects found by Papageorgiou, Maldarelli & Rumschitzki, Phys. Fluids A 2(3), 1990, pp. 340-352, to a similar system with time periodic coefficients. The distinct regimes of slow and moderate flow are considered and the corresponding evolution is derived. Certain solutions are described analytically in the neighborhood of the first bifurcation point by use of multiple scales asymptotics. Extensive numerical experiments, using dynamical systems ideas, are carried out in order to evaluate the effect of the oscillatory pressure gradient on the solutions in the presence of a constant pressure gradient.

Coward, Adrian V.↗

NASA NTRS1994

Multivariable Hermite polynomials and phase-space dynamics

The phase-space approach to classical and quantum systems demands for advanced analytical tools. Such an approach characterizes the evolution of a physical system through a set of variables, reducing to the canonically conjugate variables in the classical limit. It often happens that phase-space distributions can be written in terms of quadratic forms involving the above quoted variables. A significant analytical tool to treat these problems may come from the generalized many-variables Hermite polynomials, defined on quadratic forms in R(exp n). They form an orthonormal system in many dimensions and seem the natural tool to treat the harmonic oscillator dynamics in phase-space. In this contribution we discuss the properties of these polynomials and present some applications to physical problems.

Dattoli, G.↗

NASA NTRS1995

Molecular vibrational states during a collision

Alternative algebraic techniques to approximate a given Hamiltonian by a harmonic oscillator are described both for time-independent and time-dependent systems. We apply them to the description of a one dimensional atom-diatom collision. From the resulting evolution operator, we evaluate vibrational transition probabilities as well as other time-dependent properties. As expected, the ground vibrational state becomes a squeezed state during the collision.

Recamier, Jose A.↗

NASA NTRS1995

Noisy bases in Hilbert space: A new class of thermal coherent states and their properties

Coherent mixed states (or thermal coherent states) associated with the displaced harmonic oscillator at finite temperature, are introduced as a 'random' (or 'thermal' or 'noisy') basis in Hilbert space. A resolution of the identity for these states is proved and used to generalize the usual coherent state formalism for the finite temperature case. The Bargmann representation of an operator is introduced and its relation to the P and Q representations is studied. Generalized P and Q representations for the finite temperature case are also considered and several interesting relations among them are derived.

Vourdas, A.↗

NASA NTRS1994

Adiabatic invariants in stellar dynamics. 1: Basic concepts

The adiabatic criterion, widely used in astronomical dynamics, is based on the harmonic oscillator. It asserts that the change in action under a slowly varying perturbation is exponentially small. Recent mathematical results that precisely define the conditions for invariance show that this model does not apply in general. In particular, a slowly varying perturbation may cause significant evolution stellar dynamical systems even if its time scale is longer than any internal orbital time scale. This additional 'heating' may have serious implications for the evolution of star clusters and dwarf galaxies which are subject to long-term environmental forces. The mathematical developments leading to these results are reviewed, and the conditions for applicability to and further implications for stellar systems are discussed. Companion papers present a computational method for a general time-dependent disturbance and detailed example.

Weinberg, Martin D.↗

NASA NTRS1995

On the uniqueness of linear moving-average filters for the solar wind-auroral geomagnetic activity coupling

The relation between the solar wind input to the magetosphere, VB(sub South), and the auroral geomagnetic index AL is modeled with two linear moving-average filtering methods: linear prediction filters and a driven harmonic oscillator in the form of an electric circuit. Although the response of the three-parameter oscillator is simpler than the filter's, the methods yield similar linear timescales and values of the prediction-observation correlation and the prediction Chi(exp 2). Further the filter responses obtained by the two methods are similar in their long-term features. In these aspects the circuit model is equivalent to linear prediction filtering. This poses the question of uniqueness and proper interpretation of detailed features of the filters such as response peaks. Finally, the variation of timescales and filter responses with the AL activity level is discussed.

Vassiliadis, D.↗

NASA NTRS1994

Kinetic and thermodynamic properties of a convecting plasma in a two-dimensional dipole field

Charged particle guiding center motion is considered in the magnetic field of a two-dimensional ('line') dipole on which is superimposed a small, static, perpendicular electric field. The parallel equation of motion is that of a simple harmonic oscillator for cos theta, the cosine of magnetic colatitude theta. Equations for the perpendicular electric and magnetic drifts are derived as well as their bounce-averaged forms. The latter are solved to yield a bounce-averaged guiding center trajectory, which is the same as that obtained from conversation of magnetic moment mu, longitudinal invariant J, and total (kinetic plus electrostatic) energy K. The algebraic simplicity of the trajectory equations is also manifest in the forms of the invariants. An interesting result is that guiding centers drift in such a way that they preserve the values of their equatorial pitch angles and (equivalently) mirror latitudes. The most general Maxwellian form of the equilibrium one-particle distribution function f is constructed from the invariants, and spatially varying density and pressure moments, parallel and perpendicular to the magnetic field, are identified. Much of the paper deals with the more restricted problem in which f is specified as a bi-Maxwellian over a straight line of finite length in the equatorial plane of the dipole and perpendicular to field lines. This might be thought of as specifying a cross-tail ion injection source; our formalism then describes the subsequent spatial development. The distribution away from the source is a scaled bi-Maxwellian but one that is cut off at large and small kinetic energies, which depend on position. Density and pressure components are reduced from the values they would have if the total content of individual flux tubes convected intact. The equatorial and meridional variations of density and pressure components are examined and compared systematically for the isotropic and highly anisotropic situations. There appears to be little qualitative difference due to anisotropy. An anisotropy measure is defined, and its spatial variation determined as a signature of possible MHD instability. Extreme values are found, larger than at the source, but the plasma beta in such regions is probably so low as to render the effect inconsequential energetically. Finally, the possible consequence of 'nonadia- batic' pressure profiles on electrostatic interchanges is considered, and a boundary delineating stabilizing and destabilizing regions determined.

Huang, T. S.↗

NASA NTRS1996

The Squeezing Operator and the Squeezing States of Superspace

In this paper ,the unitary squeezing operator of 'superspace' is introduced and by making this operator act on the supercoherent state, the squeezing supercoherent states are obtained, then come out the four orthonormalization eigenstates of the square of annihilation operator A of the supersymmetry harmonic oscillator, and their squeezing character is also studied.

Aiqun, Ma↗