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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.

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At least 271 records · Page 15
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NASA NTRS1982

Burgers approximation for two-dimensional flow past an ellipse

A linearization of the Navier-Stokes equation due to Burgers in which vorticity is transported by the velocity field corresponding to continuous potential flow is examined. The governing equations are solved exactly for the two dimensional steady flow past an ellipse of arbitrary aspect ratio. The requirement of no slip along the surface of the ellipse results in an infinite algebraic system of linear equations for coefficients appearing in the solution. The system is truncated at a point which gives reliable results for Reynolds numbers R in the range 0 R 5. Predictions of the Burgers approximation regarding separation, drag and boundary layer behavior are investigated. In particular, Burgers linearization gives drag coefficients which are closer to observed experimental values than those obtained from Oseen's approximation. In the special case of flow past a circular cylinder, Burgers approximation predicts a boundary layer whose thickness is roughly proportional to R-1/2.

Dorrepaal, J. M.↗
NASA NTRS1981

Analytic approximations to the modon dispersion relation

Three explicit analytic approximations are given to the modon dispersion relation developed by Flierl et al. (1980) to describe Gulf Stream rings and related phenomena in the oceans and atmosphere. The solutions are in the form of k(q), and are developed in the form of a power series in q for small q, an inverse power series in 1/q for large q, and a two-point Pade approximant. The low order Pade approximant is shown to yield a solution for the dispersion relation with a maximum relative error for the lowest branch of the function equal to one in 700 in the q interval zero to infinity.

Boyd, J. P.↗
NASA NTRS1982

An approximation theory for nonlinear partial differential equations with applications to identification and control

Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.

Banks, H. T.↗
NASA NTRS1984

The uniform asymptotic swallowtail approximation - Practical methods for oscillating integrals with four coalescing saddle points

Methods that can be used in the numerical implementation of the uniform swallowtail approximation are described. An explicit expression for that approximation is presented to the lowest order, showing that there are three problems which must be overcome in practice before the approximation can be applied to any given problem. It is shown that a recently developed quadrature method can be used for the accurate numerical evaluation of the swallowtail canonical integral and its partial derivatives. Isometric plots of these are presented to illustrate some of their properties. The problem of obtaining the arguments of the swallowtail integral from an analytical function of its argument is considered, describing two methods of solving this problem. The asymptotic evaluation of the butterfly canonical integral is addressed.

Connor, J. N. L.↗
NASA NTRS1984

Computing approximate random Delta v magnitude probability densities

This paper describes the development and use of an algorithm to compute approximate statistics of the magnitude of a single random trajectory correction maneuver (TCM) Delta v vector. The TCM Delta v vector is modeled as a three component Cartesian vector each of whose components is a random variable having a normal (Gaussian) distribution with zero mean and possibly unequal standard deviations. The algorithm uses these standard deviations as input to produce approximations to (1) the mean and standard deviation of the magnitude of Delta v, (2) points of the probability density function of the magnitude of Delta v, and (3) points of the cumulative and inverse cumulative distribution functions of Delta v. The approximates are based on Monte Carlo techniques developed in a previous paper by the author and extended here. The algorithm described is expected to be useful in both pre-flight planning and in-flight analysis of maneuver propellant requirements for space missions.

Chadwick, C.↗
NASA NTRS1984

Approximation methods for inverse problems involving the vibration of beams with tip bodies

Two cubic spline based approximation schemes for the estimation of structural parameters associated with the transverse vibration of flexible beams with tip appendages are outlined. The identification problem is formulated as a least squares fit to data subject to the system dynamics which are given by a hybrid system of coupled ordinary and partial differential equations. The first approximation scheme is based upon an abstract semigroup formulation of the state equation while a weak/variational form is the basis for the second. Cubic spline based subspaces together with a Rayleigh-Ritz-Galerkin approach were used to construct sequences of easily solved finite dimensional approximating identification problems. Convergence results are briefly discussed and a numerical example demonstrating the feasibility of the schemes and exhibiting their relative performance for purposes of comparison is provided.

Rosen, I. G.↗
NASA NTRS1985

Approximation of Optimal Infinite Dimensional Compensators for Flexible Structures

The infinite dimensional compensator for a large class of flexible structures, modeled as distributed systems are discussed, as well as an approximation scheme for designing finite dimensional compensators to approximate the infinite dimensional compensator. The approximation scheme is applied to develop a compensator for a space antenna model based on wrap-rib antennas being built currently. While the present model has been simplified, it retains the salient features of rigid body modes and several distributed components of different characteristics. The control and estimator gains are represented by functional gains, which provide graphical representations of the control and estimator laws. These functional gains also indicate the convergence of the finite dimensional compensators and show which modes the optimal compensator ignores.

Gibson, J. S.↗
NASA NTRS1985

Alternative approximation concepts for space frame synthesis

A method for space frame synthesis based on the application of a full gamut of approximation concepts is presented. It is found that with the thoughtful selection of design space, objective function approximation, constraint approximation and mathematical programming problem formulation options it is possible to obtain near minimum mass designs for a significant class of space frame structural systems while requiring fewer than 10 structural analyses. Example problems are presented which demonstrate the effectiveness of the method for frame structures subjected to multiple static loading conditions with limits on structural stiffness and strength.

Lust, R. V.↗
NASA NTRS1983

The application of Legendre-tau approximation to parameter identification for delay and partial differential equations

Approximation schemes based on Legendre-tau approximation are developed for application to parameter identification problem for delay and partial differential equations. The tau method is based on representing the approximate solution as a truncated series of orthonormal functions. The characteristic feature of the Legendre-tau approach is that when the solution to a problem is infinitely differentiable, the rate of convergence is faster than any finite power of 1/N; higher accuracy is thus achieved, making the approach suitable for small N.

Ito, K.↗
NASA NTRS1986

An approximate buckling analysis for rectangular orthotropic plates with centrally located cutouts

An approximate analysis for predicting buckling of rectangular orthotropic composite plates with centrally located cutouts is presented. In this analysis, prebuckling and buckling problems are converted from a two-dimensional to a one-dimensional system of linear differential equations with variable coefficients. The conversion is accomplished by expressing the displacements as series with each element containing a trigonometric function of one coordinate and a coefficient that is an arbitrary function of the other coordinate. Ordinary differential equations are then obtained from a variational principle. Analytical results obtained from the approximate analysis are compared with finite element analyses for isotropic plates and for specially orthotropic plates with central circular cutouts of various sizes. Experimental results for the specially orthotropic plates are also presented. In nearly all cases, the approximate analysis predicts the buckling mode shapes correctly and predicts the buckling loads to within a few percent of the finite element and experimental results.

Nemeth, M. P.↗
NASA NTRS1986

Comparative accuracy of the Albedo, transmission and absorption for selected radiative transfer approximations

Illustrations of both the relative and absolute accuracy of eight different radiative transfer approximations as a function of optical thickness, solar zenith angle and single scattering albedo are given. Computational results for the plane albedo, total transmission and fractional absorption were obtained for plane-parallel atmospheres composed of cloud particles. These computations, which were obtained using the doubling method, are compared with comparable results obtained using selected radiative transfer approximations. Comparisons were made between asymptotic theory for thick layers and the following widely used two stream approximations: Coakley-Chylek's models 1 and 2, Meador-Weaver, Eddington, delta-Eddington, PIFM and delta-discrete ordinates.

King, M. D.↗
NASA NTRS1986

Approximating linearizations for nonlinear systems

The following problem is examined: given a nonlinear control system dot-x(t) = f(x(t)) + the sum to m terms(i=1) u sub i (t)g sub i (x(t)) on R(n) and a point x(0) in R(n), approximate the system near x(0) by a linear system. One approach is to use the usual Taylor series linearization. However, the controllability properties of both the nonlinear and linear systems depend on certain Lie brackets of the vector field under consideration. This suggests that a linear approximation based on Lie bracket matching should be constructed at x(0). In general, the linearizations based on the Taylor method and the Lie bracket approach are different. However, under certain mild assumptions, it is shown that there is a coordinate system for R(n) near x(0) in which these two types of linearizations agree. The importance of this agreement is indicated by examining the time responses of the nonlinear system and its linear approximation and comparing the lower order kernels in Volterra expansions of each.

Hunt, L. R.↗
NASA NTRS1985

Approximate-factorization schemes for solving the transonic full-potential equation

The present paper provides a general discussion of approximate-factorization techniques applied to the transonic full-potential equation. Giving particular attention to the AF2 approximate-factorization scheme. This scheme was first introduced by Ballhaus and Steger (1975) for solving the low-frequency (unsteady), transonic small-disturbance equation. The full-potential equation algorithm is examined, taking into account the governing equations, grid generation, the artificial density scheme (spatial differencing), the alternating direction implicit scheme, the AF2 iteration scheme, temporal damping, and boundary conditions. Computed results are also presented. It is shown that fast, fully-implicit algorithms of the approximate-factorization variety are both efficient and reliable for solving the conservative full-potential equation.

Holst, T. L.↗
NASA NTRS1986

Higher order parabolic approximations of the reduced wave equation

Asymptotic solutions of order k to the nth are developed for the reduced wave equation. Here k is a dimensionless wave number and n is the arbitrary order of the approximation. These approximations are an extension of geometric acoustics theory, and provide corrections to that theory in the form of multiplicative functions which satisfy parabolic partial differential equations. These corrections account for the diffraction effects caused by variation of the field normal to the ray path and the interaction of these transverse variations with the variation of the field along the ray. The theory is applied to the example of radiation from a piston, and it is demonstrated that the higher order approximations are more accurate for decreasing values of k.

Mcaninch, G. L.↗
NASA NTRS1986

Convection in the vertical midplane of a horizontal cylinder Comparison of two-dimensional approximations with three-dimensional results

Experimental results and numerical data on the flow characteristics of core-driven and boundary-layer-driven regimes (CDRs) and (BLDRs) are compared to two-dimensional solutions for heated convective flow in the vertical midplane of a horizontal cylinder. The data were obtained from laser Doppler velocimetry (LDA) measurements and from three-dimensional numerical solutions. It is shown that the approximate solutions gave the correct Rayleigh-number-dependences and distributions for the velocities in the CDR and BLDR regimes. However, the transition between the CDR and the BLDR, as well as the magnitude of the distribution of velocity components, were found to be dependent on the type of two-dimensional approximation selected. In the case of CDR, core velocities were overestimated by as much as 30 percent by the two-dimensional approximations; and in the case of BLDR, the core velocities were overestimated by 40-50 percent as compared to the measured velocities. A schematic illustration of the flow patterns in the cylinder is provided.

Bontoux, P.↗
NASA NTRS1986

Approximation methods in gravitational-radiation theory

The observation of gravitational-radiation damping in the binary pulsar PSR 1913 + 16 and the ongoing experimental search for gravitational waves of extraterrestrial origin have made the theory of gravitational radiation an active branch of classical general relativity. In calculations of gravitational radiation, approximation methods play a crucial role. Recent developments are summarized in two areas in which approximations are important: (a) the quadrupole approxiamtion, which determines the energy flux and the radiation reaction forces in weak-field, slow-motion, source-within-the-near-zone systems such as the binary pulsar; and (b) the normal modes of oscillation of black holes, where the Wentzel-Kramers-Brillouin approximation gives accurate estimates of the complex frequencies of the modes.

Will, C. M.↗
NASA NTRS1986

Comparative accuracy of selected multiple scattering approximations

Doubling method computational results which have yielded plane albedo, total transmission and fractional absorption for plane-parallel atmospheres composed of cloud droplets are presently compared with data obtained with selected radiative transfer approximations. The relative and absolute accuracies of asymptotic theory for thick layers and delta-Eddington, Meador-Weaver (1980) and Coakley-Chylek (1975) approximations are compared as a function of optical thickness, solar zenith angle, and single scattering albedo. The delta-Eddington approximation is the most accurate for conservative scattering when the solar zenith angle is small, while Meador-Weaver is the most accurate for nonconservative scattering.

King, M. D.↗
NASA NTRS1986

A reformulation of the parabolic approximation for waves in stratified moving media

An asymptotic, large wave number approximation for the equations governing the propagation of acoustic disturbances through a stratified moving medium is developed. The theory is an extension of the geometric acoustics approximation and provides corrections to that approximation in the form of multiplicative functions which satisfy parabolic differential equations of second order. By properly accounting for variations in the acoustic field in directions normal to the rays both caustic surfaces and the secularity of the geometric theory may be avoided.

Mcaninch, G. L.↗