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NASA NTRS1983

Fast Euler solver for steady, 1-dimensional flows

A numerical technique to solve the Euler equations for steady, one dimensional flows is presented. The technique is essentially implicit, but is structured as a sequence of explicit solutions for each Riemann variable separately. Each solution is obtained by integrating in the direction prescribed by the propagation of the Riemann variables. The technique is second-order accurate. It requires very few steps for convergence, and each step requires a minimal number of operations. Therefore, it is three orders of magnitude more efficient than a standard time-dependent technique. The technique works very well for transonic flows and provides shock fitting with errors as small as 0.001. Results are presented for subsonic problems. Errors are evaluated by comparison with exact solutions.

Moretti, G.↗

NASA NTRS1983

Initial-value problems for Rossby waves in a shear flow with critical level

The time-dependent evolution of sheared Rossby waves starting from an initial disturbance is studied for the simple case in which the shear is uniform. The uniform-shear assumption allows explicit solutions to be obtained which are useful in addressing the issue of the long-time asymptotic approach to normal modes and in assessing the relative importance of viscosity, nonlinearity and time-dependence in the evolution of Rossby waves in the presence of critical layers.

Tung, K. K.↗

NASA NTRS1983

Direct and implicit optical matrix-vector algorithms

New direct and implicit algorithms for optical matrix-vector and systolic array processors are considered. Direct rather than indirect algorithms to solve linear systems and implicit rather than explicit solutions to solve second-order partial differential equations are discussed. In many cases, such approaches more properly utilize the advantageous features of optical systolic array processors. The matrix-decomposition operation (rather than solution of the simplified matrix-vector equation that results) is recognized as the computationally burdensome aspect of such problems that should be computed on an optical system. The Householder QR matrix-decomposition algorithm is considered as a specific example of a direct solution. Extensions to eigenvalue computation and formation of matrices of special structure are also noted.

Casasent, D.↗

NASA NTRS1986

Vortex shedding from solid rocket propellant inhibitors

Vortex shedding frequency caused by the protrusion of inhibitors into the flow field of a solid rocket motor is investigated by experimental and mathematical models. The time dependent Navier-Stokes equations are solved using a finite difference technique assuming incompressible, two-dimensional flow under both laminar and turbulent flow conditions. For laminar flow, explicit solutions are obtained using a vorticity-transport equation in place of the Navier-Stokes equations. For turbulent flow, a two-equation (k-epsilon) model is used for turbulent modeling and the SIMPLE algorithm is employed as the computational scheme. Cold flow tests were conducted to confirm the basic flow structure and to determine the vortex shedding frequency under both laminar and turbulent flow conditions. The vortex shedding frequencies were determined using a stroboscope to measure the oscillating frequency of yarn tufts which were fastened to one inhibitor in the models. A hot-film anemometer established the velocity history behind the inhibitor. Good agreement between the theoretical results and measurements of the vortex shedding frequencies is demonstrated.

Shu, P. H.↗

NASA NTRS1986

Computer code for gas-liquid two-phase vortex motions: GLVM

A computer program aimed at the phase separation between gas and liquid at zero gravity, induced by vortex motion, is developed. It utilizes an explicit solution method for a set of equations describing rotating gas-liquid flows. The vortex motion is established by a tangential fluid injection. A Lax-Wendroff two-step (McCormack's) numerical scheme is used. The program can be used to study the fluid dynamical behavior of the rotational two-phase fluids in a cylindrical tank. It provides a quick/easy sensitivity test on various parameters and thus provides the guidance for the design and use of actual physical systems for handling two-phase fluids.

Yeh, T. T.↗

NASA NTRS1987

Development of response models for the Earth Radiation Budget Experiment (ERBE) sensors. Part 2: Analysis of the ERBE integrating sphere ground calibration

An explicit solution of the spectral radiance leaving an arbitrary point on the wall of a spherical cavity with diffuse reflectivity is obtained. The solution is applicable to spheres with an arbitrary number of openings of any size and shape, an arbitrary number of light sources with possible non-diffuse characteristics, a non-uniform sphere wall temperature distribution, non-uniform and non-diffuse sphere wall emissivity and non-uniform but diffuse sphere wall spectral reflectivity. A general measurement equation describing the output of a sensor with a given field of view, angular and spectral response measuring the sphere output is obtained. The results are applied to the Earth Radiation Budget Experiment (ERBE) integrating sphere. The sphere wall radiance uniformity, loading effects and non-uniform wall temperature effects are investigated. It is shown that using appropriate interpretation and processing, a high-accuracy short-wave calibration of the ERBE sensors can be achieved.

Halyo, Nesim↗

NASA NTRS1987

Investigation of the feasibility of an analytical method of accounting for the effects of atmospheric drag on satellite motion

An analytic technique for accounting for the joint effects of Earth oblateness and atmospheric drag on close-Earth satellites is investigated. The technique is analytic in the sense that explicit solutions to the Lagrange planetary equations are given; consequently, no numerical integrations are required in the solution process. The atmospheric density in the technique described is represented by a rotating spherical exponential model with superposed effects of the oblate atmosphere and the diurnal variations. A computer program implementing the process is discussed and sample output is compared with output from program NSEP (Numerical Satellite Ephemeris Program). NSEP uses a numerical integration technique to account for atmospheric drag effects.

Bozeman, Robert E.↗

NASA NTRS1990

The distortion of the magnetotail equilibrium structure by a net cross-tail magnetic field

Explicit solutions are presented for quiet, quasi-static magnetotail configurations in the presence of a net cross-tail magnetic field component in addition to field components associated with tail flaring. The solutions are derived using the general three-dimensional equilibrium theory of Birn (1987). It is found that the net By(N) field varies only weakly along field lines and across the tail. The presence of the net By(N) in the neutral sheet implies that a part of the cross-tail current becomes field-aligned, flowing from the southern to the northern hemisphere for By(N) larger than zero and from north to south for By(n) less than zero. The deformations of magnetic flux surfaces suggest that there may be a class of field lines extending from the earth into the magnetosheath region through the low-latitude flanks of the tail. A topological connection is found of the low-latitude region near the magnetopause with the plasma sheet boundary layer, separating closed plasma sheet and open lobe field lines.

Birn, J.↗

NASA NTRS1992

FNAS: The magnetic configuration leading to solar flares

We present a method for solving plasma magnetohydrodynamic (MHD) problems arising from the interaction of plasmas with magnetic fields in stellar atmospheres. Our approach, in contrast to previous methods, is not based on solving equations for the magnetic field and plasma velocity but rather studies the evolution of the electric current and density (and the related gas pressure). We have applied the method to several studies involving linearized departures from static, current-free equilibria. The applications show explicit solutions for cases found in astrophysics and to problems encountered with earlier studies where the gas pressure was neglected. The method is particularly well suited for studying situations which involve a transition between high and low plasma-beta regions. It shows precisely how electric currents, and magnetic free-energy, build up in the plasma as a result of the slow stressing of a potential magnetic field configuration. The method also demonstrates how transverse-current waves, a mix of Alfven and magneto-acoustic modes, propagate in a low-beta plasma for any density stratification and background field geometry.

Fontenla, Juan M.↗

NASA NTRS1992

Quasi-steady current sheet structures with field-aligned flow

The paper discusses the characteristics of quasi-steady plasma and field structures with field-aligned flow. Explicit solutions are developed for modeling the compressible flow around a plasmoid in the distant magnetotail. The expected and observed plasmoid signatures are found. Field signatures outside the plasmoid are typically those of encounters of traveling compression region: a north-south signature of Bz accompanied by an enhancement of Bx.

Birn, J.↗

NASA NTRS1991

Stationary turbulent closure via the Hopf functional equation

Explicit solutions of the stationary Hopf equation are discussed and their computational possibilities are explored. The motivation is to circumvent the infinite hierarchy of coupled equations for the velocity moments and obtain an exact closure of the steady-state 3D Navier-Stokes equations, without modeling assumptions or truncation. The Hopf formulation of the Navier-Stokes equation is reviewed. A stationary homogeneous solution for 2D flow is displayed and discussed. It is shown how depletion of nonlinearity may arise for 3D forced homogeneous flow. The general 3D forced case is considered and a method for closing the 3D unforced equations with arbitrary boundary conditions is derived.

Shen, Hubert H.↗

NASA NTRS1944

The Influence of the Aerodynamic Span Effect on the Magnitude of the Torsional-divergence Velocity and on the Shape of the Corresponding Deflection Mode

A procedure which takes into account the aerodynamic span effect is given for the determination of the torsional-divergence velocities of monoplanes. The explicit solutions obtained in several cases indicate that the aerodynamic span effect may increase the divergence velocities found by means of the section-force theory by as much as 17 to 40 percent. It is found that the magnitude of the effect increases with increasing degree of stiffness taper and decreases with increasing degree of chord taper. By a slight extension of the present method it is possible to analyze the elastic deformations of wings, and the resultant lift distributions, before torsional divergence occurs.

MONOPLANES↗

NASA NTRS1994

Three-dimensional fully spectral numerical method for mantle convection with depth-dependent properties

A semi-implicit fully spectral collocation method for the simulation of three-dimensional mantle convection with depth-dependent thermo-dynamic and transport properties is presented. The variable property Navier-Stokes equation expressed in terms of the primitive variable velocity and pressure is solved with the mass continuity and temperature equations. The periodic horizontal boundary conditions allow a Fourier expansion for the two horizontal directions. The stress-free, impermeable isothermal boundary conditions along with the depth dependent coefficients are handled with a Chebyshev expansion in the vertical direction. In the limit of an infinite Prandtl number appropriate to mantle convection, the inertial terms in the momentum equation are unimportant. In this case an explicit solution of a Poisson equation for pressure can be avoided; instead a fourth-order equation for vertical velocity can be solved. Simultaneous imposition of both impermeable and continuity boundary conditions during the vertical velocity evaluation is discussed. The pressure distributions on the top and bottom bounding planes were determined by means of an influence matrix technique. The numerical method employed here avoids time-splitting errors and enforces velocity boundary conditions and continuity over the entire domain, including the boundaries, to machine accuracy. Strongly time-dependent three-dimensional solutions up to a surface Rayleigh number of 1 x 10(exp 7) have been obtained. Strong upwellings, pulsating chaotically, are formed by the collective merging of cylindrical plumes.

Balachandar, S.↗

NASA NTRS1995

Vortex perturbation dynamics

An initial value approach is used to examine the dynamics of perturbations introduced into a vortex under strain. Both the basic vortex considered and the perturbations are taken as fully three-dimensional. An explicit solution for the time evolution of the vorticity perturbations is given for arbitrary initial vorticity. Analytical solutions for the resulting velocity components are found when the initial vorticity is assumed to be localized. For more general initial vorticity distributions, the velocity components are determined numerically. It is found that the variation in the radial direction of the initial vorticity disturbance is the most important factor influencing the qualitative behavior of the solutions. Transient growth in the magnitude of the velocity components is found to be directly attributable to the compactness of the initial vorticity.

Criminale, W. O.↗

NASA NTRS1997

Time Evolution of Modeled Reynolds Stresses in Planar Homogeneous Flows

The analytic expression of the time evolution of the Reynolds stress anisotropy tensor in all planar homogeneous flows is obtained by exact integration of the modeled differential Reynolds stress equations. The procedure is based on results of tensor representation theory, is applicable for general pressure-strain correlation tensors, and can account for any additional turbulence anisotropy effects included in the closure. An explicit solution of the resulting system of scalar ordinary differential equations is obtained for the case of a linear pressure-strain correlation tensor. The properties of this solution are discussed, and the dynamic behavior of the Reynolds stresses is studied, including limit cycles and sensitivity to initial anisotropies.

Jongen, T.↗

NASA NTRS2005

What Information Theory Says about Bounded Rational Best Response

Probability Collectives (PC) provides the information-theoretic extension of conventional full-rationality game theory to bounded rational games. Here an explicit solution to the equations giving the bounded rationality equilibrium of a game is presented. Then PC is used to investigate games in which the players use bounded rational best-response strategies. Next it is shown that in the continuum-time limit, bounded rational best response games result in a variant of the replicator dynamics of evolutionary game theory. It is then shown that for team (shared-payoff) games, this variant of replicator dynamics is identical to Newton-Raphson iterative optimization of the shared utility function.

Wolpert, David H.↗

NASA NTRS1996

A Derivation of the Long-Term Degradation of a Pulsed Atomic Frequency Standard from a Control-Loop Model

The phase of a frequency standard that uses periodic interrogation and control of a local oscillator (LO) is degraded by a long-term random-walk component induced by downconversion of LO noise into the loop passband. The Dick formula for the noise level of this degradation is derived from an explicit solution of an LO control-loop model.

Dick formula local oscillator frequency phase freq↗

NASA NTRS2010

Thermal-Acoustic Analysis of a Metallic Integrated Thermal Protection System Structure

A study is undertaken to investigate the response of a representative integrated thermal protection system structure under combined thermal, aerodynamic pressure, and acoustic loadings. A two-step procedure is offered and consists of a heat transfer analysis followed by a nonlinear dynamic analysis under a combined loading environment. Both analyses are carried out in physical degrees-of-freedom using implicit and explicit solution techniques available in the Abaqus commercial finite-element code. The initial study is conducted on a reduced-size structure to keep the computational effort contained while validating the procedure and exploring the effects of individual loadings. An analysis of a full size integrated thermal protection system structure, which is of ultimate interest, is subsequently presented. The procedure is demonstrated to be a viable approach for analysis of spacecraft and hypersonic vehicle structures under a typical mission cycle with combined loadings characterized by largely different time-scales.

Behnke, Marlana N.↗