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NASA NTRS1980

Asymptotic eigenstructures

The behavior of the closed loop eigenstructure of a linear system with output feedback is analyzed as a single parameter multiplying the feedback gain is varied. An algorithm is presented that computes the asymptotically infinite eigenstructure, and it is shown how a system with high gain feedback decouples into single input single ouput systems. Then a synthesis algorithm is presented which uses full state feedback to achieve a desired asymptotic eigenstructure.

Thompson, P. M.↗

NASA NTRS1981

Multigrid simulation of asymptotic curved-duct flows using a semi-implicit numerical technique

Asymptotic flows inside curved ducts of rectangular as well as polar cross section are analyzed using the Navier-Stokes equations in terms of the axial velocity and vorticity and the cross-flow stream function. Numerical solutions of the three second-order coupled elliptic partial differential equations governing this flow are obtained efficiently using the coupled alternating-direction implicit (ADI) method as well as the multigrid strongly-implicit (SI) scheme. For the flow configuration studied, the ADI method is found to be more sensitive to the time steps used than is the SI scheme. Use of the multigrid-coupled-strongly-implicit (MG-SI) scheme makes it possible to efficiently obtain fine-grid solutions for configurations having strong secondary flow. It is shown that, for this asymptotic curved-duct flow, the similarity parameter of significance is the Dean's number K rather than the Reynolds number Re. Results are obtained for curved ducts with square cross sections for K up to 900, which here corresponds to Re = 9,000 for this internal flow configuration.

Ghia, K. N.↗

NASA NTRS1983

Helicopter rotor loads using discretized matched asymptotic expansions

The numerical practicality of a matched asymptotic expansion approach for the computation of unsteady three dimensional airloads on a helicopter rotor was improved. This effort utilizes a discretized repesentation of the doublet strength distribution and helical streamlines to decrease the computational requirements of the original analysis. The continuous variation of the doublet strength was approximated by piecewise constant or piecewise quadratic distributions, and the helical trajectory of a fluid particle was approximated by connected straight line segments. As a direct result of these simplified representations the computational time required for the execution of a typical flight condition was reduced by an order of magnitude with respect to the requirements of the original analysis. Airloads which were computed using the discretized method for a two bladed model rotor and a full scale four bladed rotor are in close agreement with measured results and airloads from the original asymptotic analysis. For conditions characterized by significant rotor/wake interaction the piecewise constant representation requires a reduced azimuth spacing to maintain acceptable accuracy.

Pierce, G. A.↗

NASA NTRS1982

Atomic collisions in the presence of laser radiation - Time dependence and the asymptotic wave function

A time-dependent, wave-packet description of atomic collisions in the presence of laser radiation is extracted from the more conventional time-independent, stationary-state description. This approach resolves certain difficulties of interpretation in the time-independent approach which arise in the case of asymptotic near resonance. In the two-state model investigated, the approach predicts the existence of three spherically scattered waves in this asymptotically near-resonant case.

Devries, P. L.↗

NASA NTRS1983

Pulse asymptotics of three-dimensional baroclinic waves

The asymptotic development at large time of waves arising from localized disturbances in a baroclinic flow is examined. Vertical structures unlike those associated with the more commonly examined temporal normal modes are found both for the pulse confined to a channel as previously examined, and for the unconfined pulse on an infinite beta-plane. These structures and their implied transports are compared to observations in the regions of storm tracks. It is also found that the meridional extent of the asymptotic solution becomes large compared to observed cyclone wavetrains, emphasizing the importance of flow inhomogeneity and sphericity effects in determining the latitudinal structure of eddies.

Farrell, B. F.↗

NASA NTRS1985

An asymptotic theory for the interference of large aspect ratio swept wings and multiple propeller slipstreams

This paper presents an asymptotic method for the analysis of the interference of multiple tractor propeller slipstream with large aspect ratio swept wings. It is assumed that the height of the slipstream is of the order of the wing chord and its spanwise extent is of the order of the wing span. Three different flow regions are identified by employing different stretching transformations. Asymptotic expansions are made in each of the three regions, using the chord to span ratio as the small expansion parameter. The details of the nonuniform flow in the slipstream enter into the wing sectional analysis. In the outer limit, the wing shrinks to a swept lifting line, and the slipstream reduces to a thin sheet of jet carrying the momentum gain from the propeller. The curvature of this jet sheet results in a pressure difference which is represented by a vortex sheet. The governing equations are solved by discretization. Several examples are considered for which experimental data are available. Comparison of the present results with the experimental data as well as other numerical solutions showed generally good agreement.

Prabhu, R. K.↗

NASA NTRS1987

An asymptotic analysis of supersonic reacting mixing layers

The purpose of this paper is to present an asymptotic analysis of the laminar mixing of the simultaneous chemical reaction between parallel supersonic streams of two reacting species. The study is based on a one-step irreversible Arrhenius reaction and on large activation energy asymptotics. Essentially it extends the work of Linan and Crespo to include the effect of free shear and Mach number on the ignition regime, the deflagration regime and the diffusion flame regime. It is found that the effective parameter is the product of the characteristic Mach number and a shear parameter.

Jackson, T. L.↗

NASA NTRS1989

Asymptotic behavior of a flat plate wake

An experimental study has been conducted to investigate the far-field, self-similar properties of a flat plate wake. A plane turbulent wake was generated at the trailing edge of a smooth splitter plate separating two legs of a Mixing Layer Wind Tunnel, with both initial boundary layers tripped. For the present study, both legs were operated at a free-steam velocity in the test section of 15 m/s, giving a Reynolds number based on wake momentum thickness of about 1750. Single profile measurements were obtained at five streamwise locations using a Pitot probe for the mean velocity measurements and a single cross-wire probe for the turbulence data, which included statistics up to third order. The mean flow data indicated a self-similar behavior beyond a streamwise distance equivalent to about 350 wake momentum thicknesses. However, the turbulence data show better collapse beyond a distance equivalent to about 500 momentum thicknesses, with all the measured peak Reynolds stresses achieving constant, asymptotic levels. The asymptotic mean flow behavior and peak primary stress levels agree well with theoretical predictions based on a constant eddy viscosity model. The present data also agree reasonably well with previous measurements, of which only one set extends into the self-similar region. Detailed comparisons with previous data are presented and discussed in this report.

Weygandt, James H.↗

NASA NTRS1990

Asymptotic boundary conditions for dissipative waves: General theory

An outstanding issue in the computational analysis of time dependent problems is the imposition of appropriate radiation boundary conditions at artificial boundaries. Accurate conditions are developed which are based on the asymptotic analysis of wave propagation over long ranges. Employing the method of steepest descents, dominant wave groups are identified and simple approximations to the dispersion relation are considered in order to derive local boundary operators. The existence of a small number of dominant wave groups may be expected for systems with dissipation. Estimates of the error as a function of domain size are derived under general hypotheses, leading to convergence results. Some practical aspects of the numerical construction of the asymptotic boundary operators are also discussed.

Hagstrom, Thomas↗

NASA NTRS1990

An unsteady time asymptotic flow in the square driven cavity

Summary details of an aperiodic time asymptotic numerical solution for the square drive cavity at Re = 10000 are presented. The data presented is for 6100 less than t less than or equal to 7100, and is representative of the data that characterizes the aperiodic asymptotic state.

Goodrich, John W.↗

NASA NTRS1990

Asymptotic modal analysis and statistical energy analysis

The sound field of a structural-acoustic enclosure was subject to experimental analysis and theoretical description in order to develop an efficient and accurate method for predicting sound pressure levels in enclosures such as aircraft fuselages. Asymptotic Modal Analysis (AMA) is the method under investigation. AMA is derived from classical modal analysis (CMA) by considering the asymptotic limit of the sound pressure level as the number of acoustic and/or structural modes approaches infinity. Using AMA, results identical to those of Statistical Energy Analysis (SEA) were obtained for the spatially-averaged sound pressure levels in the interior. AMA is systematically derived from CMA and therefore the degree of generality of the end result can be adjusted through the choice of appropriate simplifying assumptions. For example, AMA can be used to obtain local sound pressure levels at particular points inside the enclosure, or to include the effects of varying the size and/or location of the sound source. AMA theoretical results were compared with CMA theory and also with experiment for the case where the structural-acoustic enclosure is a rectangular cavity with part of one wall flexible and vibrating, while the rest of the cavity is rigid.

Dowell, Earl H.↗

NASA NTRS1989

Viscous flow in simple curved gaps. I - An asymptotic theory. II - Viscous stress and shape function

The present asymptotic theory for generalized incompressible two-dimensional steady flow in curved channels has been constructed in the limit when gas thickness approaches zero with its lateral dimensions fixed; successive asymptotic solution terms are analytically generated by quadratures. In the second part of this work, the curvature of the gap treated is arbitrary. It is established that each term in the series solution of velocity and pressure is the product of a scale factor and a universal shape functions. Various interaction modes between the volume rate-of-flow, curvature, and its variations, are identified and quantitatively characterized.

Fan, D.-N.↗

NASA NTRS1990

Asymptotic-induced numerical methods for conservation laws

Asymptotic-induced methods are presented for the numerical solution of hyperbolic conservation laws with or without viscosity. The methods consist of multiple stages. The first stage is to obtain a first approximation by using a first-order method, such as the Godunov scheme. Subsequent stages of the method involve solving internal-layer problems identified by using techniques derived via asymptotics. Finally, a residual correction increases the accuracy of the scheme. The method is derived and justified with singular perturbation techniques.

Garbey, Marc↗

NASA NTRS1991

Asymptotic expansions of the kernel functions for line formation with continuous absorption

Asymptotic expressions are obtained for the kernel functions M2(tau, alpha, beta) and K2(tau, alpha, beta) appearing in the theory of line formation with complete redistribution over a Voigt profile with damping parameter a, in the presence of a source of continuous opacity parameterized by beta. For a greater than 0, each coefficient in the asymptotic series is expressed as the product of analytic functions of a and eta. For Doppler broadening, only the leading term can be evaluated analytically.

Hummer, D. G.↗

NASA NTRS1991

Asymptotic boundary conditions for dissipative waves - General theory

An outstanding issue in computational analysis of time dependent problems is the imposition of appropriate radiation boundary conditions at artificial boundaries. Accurate conditions are developed which are based on the asymptotic analysis of wave propagation over long ranges. Employing the method of steepest descents, dominant wave groups are identified and simple approximations to the dispersion relation are considered in order to derive local boundary operators. The existence of a small number of dominant wave groups may be expected for systems with dissipation. Estimates of the error as a function of domain size are derived under general hypotheses, leading to convergence results. Some practical aspects of the numerical construction of the asymptotic boundary operators are also discussed.

Hagstrom, Thomas↗

NASA NTRS1992

Application of the variational-asymptotical method to composite plates

A method is developed for the 3D analysis of laminated plate deformation which is an extension of a variational-asymptotical method by Atilgan and Hodges (1991). Both methods are based on the treatment of plate deformation by splitting the 3D analysis into linear through-the-thickness analysis and 2D plate analysis. Whereas the first technique tackles transverse shear deformation in the second asymptotical approximation, the present method simplifies its treatment and restricts it to the first approximation. Both analytical techniques are applied to the linear cylindrical bending problem, and the strain and stress distributions are derived and compared with those of the exact solution. The present theory provides more accurate results than those of the classical laminated-plate theory for the transverse displacement of 2-, 3-, and 4-layer cross-ply laminated plates. The method can give reliable estimates of the in-plane strain and displacement distributions.

Hodges, Dewey H.↗

NASA NTRS1991

The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices

Toeplitz matrices occur in many mathematical, as well as, scientific and engineering investigations. This paper considers the spectra of banded Toeplitz and quasi-Toeplitz matrices with emphasis on non-normal matrices of arbitrarily large order and relatively small bandwidth. These are the type of matrices that appear in the investigation of stability and convergence of difference approximations to partial differential equations. Quasi-Toeplitz matrices are the result of non-Dirichlet boundary conditions for the difference approximations. The eigenvalue problem for a banded Toeplitz or quasi-Toeplitz matrix of large order is, in general, analytically intractable and (for non-normal matrices) numerically unreliable. An asymptotic (matrix order approaches infinity) approach partitions the eigenvalue analysis of a quasi-Toeplitz matrix into two parts, namely the analysis for the boundary condition independent spectrum and the analysis for the boundary condition dependent spectrum. The boundary condition independent spectrum is the same as the pure Toeplitz matrix spectrum. Algorithms for computing both parts of the spectrum are presented. Examples are used to demonstrate the utility of the algorithms, to present some interesting spectra, and to point out some of the numerical difficulties encountered when conventional matrix eigenvalue routines are employed for non-normal matrices of large order. The analysis for the Toeplitz spectrum also leads to a diagonal similarity transformation that improves conventional numerical eigenvalue computations. Finally, the algorithm for the asymptotic spectrum is extended to the Toeplitz generalized eigenvalue problem which occurs, for example, in the stability of Pade type difference approximations to differential equations.

Beam, Richard M.↗

NASA NTRS1992

Asymptotic analysis with reduced chemistry for the burning of n-heptane droplets

The method of rate-ratio asymptotics is used with reduced chemistry to analyze the flame structure and extinction of an isolated n-heptane droplet burning under quasisteady, spherically symmetrical conditions. The outer transport zones are described by the classical Burke-Schumann solution. The inner reaction zone consists of a thin layer, on the rich side of the flame, where the fuel is consumed, and on the lean side, a broader but still asymptotically thin oxidation layer, where H2 and CO are consumed. Special attention is given to differences in predictions of extinction conditions, caused by different chemical-kinetic approximations in the reduced chemistry, including fuel-chemistry effects through molecules containing more than one carbon atom. From the analysis, extinction diameters for n-heptane droplets are estimated for different pressures and ambient oxygen concentrations. The results show that extinction diameters are extremely sensitive to the number of radicals consumed in breaking down each fuel molecule.

Card, J. M.↗