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NASA NTRS1979

Integral equations for flows in wind tunnels

This paper surveys recent work on the use of integral equations for the calculation of wind tunnel interference. Due to the large number of possible physical situations, the discussion is limited to two-dimensional subsonic and transonic flows. In the subsonic case, the governing boundary value problems are shown to reduce to a class of Cauchy singular equations generalizing the classical airfoil equation. The theory and numerical solution are developed in some detail. For transonic flows nonlinear singular equations result, and a brief discussion of the work of Kraft and Kraft and Lo on their numerical solution is given. Some typical numerical results are presented and directions for future research are indicated.

Fromme, J. A.↗

NASA NTRS1980

Stresses in a three-dimensional unidirectional composite containing broken fibers

An approximate solution is developed for the determination of the interlaminar normal and shear stresses in the vicinity of a crack in a three dimensional composite containing unidirectional linearly elastic fibers in an infinite linearly elastic matrix. In order to reduce the complexity of the formulation, certain assumptions are made as to the physically significant stresses to be retained. These simplifications reduce the partial differential equations of elasticity to differential-difference equations which are tractable using Fourier transform techniques. This 'material modeling' approach is in contrast with solutions developed by considering each lamina as a homogeneous, orthotropic layer. The resulting solution does not contain the classical singular stress field for the fibers and the influence of broken fibers on unbroken fibers is felt by a change in stress concentration factors. The matrix stresses however, are unbounded as the fiber spacing vanishes and an equivalent fiber-matrix geometry is proposed which gives the correct singular behavior. The numerical solution is considered in detail and several specific examples are presented. The potential for damaged or debonded zones to be generated by an embedded crack is discussed, and stress concentration factors for fibers near the crack are given. Detailed comparisons are made between the present solution, the analogous two-dimensional problem, and corresponding shear-lag models.

Goree, J. G.↗

NASA NTRS1980

A magnetohydrodynamic model for corotating interplanetary structures

A model is developed which treats interplanetary structures as a small-scale incompressible Alfvenic fluctuation superposed on a large-scale background flow, and uses the complete magnetohydrodynamic equations and the full energy equation to describe the interplanetary structures. For steady-state corotating structures the model assumes the background flow to be field aligned, and the nonlinear system is treated as being composed of two subsets: one dealing with the expansion of the solar wind in stream tubes, the other with the interaction between neighboring stream tubes. The expansion subset has a singularity at Parker's critical radius, and the interaction subset has a singularity at the Alfvenic point. Concerning the region where the flow velocity is super-Alfvenic, the characteristic equation is derived, and a general method of numerical solution for the three-dimensional problems based on the characteristics theory is discussed. In addition, a viscous WKB solution of the microscale system is presented.

Whang, Y. C.↗

NASA NTRS1981

Computation of unsteady turbulent boundary layers with flow reversal and evaluation of two separate turbulence models

A procedure which solves the governing boundary layer equations within Keller's box method was developed for calculating unsteady laminar flows with flow reversal. This method is extended to turbulent boundary layers with flow reversal. Test cases are used to investigate the proposition that unsteady turbulent boundary layers also remain free of singularities. Turbulent flow calculations are performed. The governing equations for both models are solved. As in laminar flows, the unsteady turbulent boundary layers are free from singularities, but there is a clear indication of rapid thickening of the boundary layer with increasing flow reversal. Predictions of both turbulence models are the same for all practical purposes.

Cebeci, T.↗

NASA NTRS1981

Topology of three-dimensional separated flows

Based on the hypothesis that patterns of skin-friction lines and external streamlines reflect the properties of continuous vector fields, topology rules define a small number of singular points (nodes, saddle points, and foci) that characterize the patterns on the surface and on particular projections of the flow (e.g., the crossflow plane). The restricted number of singular points and the rules that they obey are considered as an organizing principle whose finite number of elements can be combined in various ways to connect together the properties common to all steady three dimensional viscous flows. Introduction of a distinction between local and global properties of the flow resolves an ambiguity in the proper definition of a three dimensional separated flow. Adoption of the notions of topological structure, structural stability, and bifurcation provides a framework to describe how three dimensional separated flows originate and succeed each other as the relevant parameters of the problem are varied.

Tobak, M.↗

NASA NTRS1980

Hybrid-finite-element analysis of some nonlinear and 3-dimensional problems of engineering fracture mechanics

In this paper, efficient numerical methods for the analysis of crack-closure effects on fatigue-crack-growth-rates, in plane stress situations, and for the solution of stress-intensity factors for arbitrary shaped surface flaws in pressure vessels, are presented. For the former problem, an elastic-plastic finite element procedure valid for the case of finite deformation gradients is developed and crack growth is simulated by the translation of near-crack-tip elements with embedded plastic singularities. For the latter problem, an embedded-elastic-singularity hybrid finite element method, which leads to a direct evaluation of K-factors, is employed.

Atluri, S. N.↗

NASA NTRS1981

Optimal trajectories in atmospheric flight

Aspects of optimization theory and switching theory are discussed, taking into account the necessary conditions for extrema, a solution subject to constraints, the calculus of variations, the Pontryagin maximum principle, the canonical transformation, Contensou's domain of maneuverability, optimal switching, a junction with singular arc, and linearized singular control. Equations of motion are considered along with aerodynamic and propulsive forces, the general properties of optimal trajectories, flight in a horizontal plane, optimal coasting flight, supersonic cruise, the supersonic turn, supersonic maneuvers in a vertical plane, energy state approximation, a modified Chapman's formulation for optimal reentry trajectories, optimal planar reentry trajectories, and an optimal glide of reentry vehicles. Orbital aerodynamic maneuvers are examined, giving attention to aerodynamic capture, a change in the apogee, a change in the eccentricity, a change in the perigee, an orbital maneuver, an aerodynamic maneuver, and a combined maneuver.

Vinh, N. X.↗

NASA NTRS1982

Subsonic panel method for the efficient analysis of multiple geometry perturbations

An accurate and efficient method was developed for the aerodynamic analysis of a series of arbitrary small geometry perturbations to a given baseline configuration. The method is appropriate for wing-fuselage configurations in incompressible potential flow. Mathematical formulations are presented for three computer programs that are employed. The first program is a conventional surface panel method for calculating the baseline singularity distribution. The second program calculates a partial derivative matrix. Each element of the matrix is the rate of change of singularity strength at one point with respect to a surface coordinate of a different point. For each baseline configuration, the calculated quantities from the first two programs establish an input file for the third. The third program calculates the surface pressure distribution and forces and moments for a series of geometry perturbations.

Bristow, D. R.↗

NASA NTRS1981

Boundary layer thermal stresses in angle-ply composite laminates, part 1

Thermal boundary-layer stresses (near free edges) and displacements were determined by a an eigenfunction expansion technique and the establishment of an appropriate particular solution. Current solutions in the region away from the singular domain (free edge) are found to be excellent agreement with existing approximate numerical results. As the edge is approached, the singular term controls the near field behavior of the boundary layer. Results are presented for cases of various angle-ply graphite/epoxy laminates with (theta/-theta/theta/theta) configurations. These results show high interlaminar (through-the-thickness) stresses. Thermal boundary-layer thicknesses of different composite systems are determined by examining the strain energy density distribution in composites. It is shown that the boundary-layer thickness depends on the degree of anisotropy of each individual lamina, thermomechanical properties of each ply, and the relative thickness of adjacent layers. The interlaminar thermal stresses are compressive with increasing temperature. The corresponding residual stresses are tensile and may enhance interply delaminations.

Wang, S. S.↗

NASA NTRS1982

Creep crack-growth: A new path-independent integral (T sub c), and computational studies

The development of valid creep fracture criteria is considered. Two path-independent integral parameters which show some degree of promise are the C* and (Delta T)sub c integrals. The mathematical aspects of these parameters are reviewed by deriving generalized vector forms of the parameters using conservation laws which are valid for arbitrary, three dimensional, cracked bodies with crack surface tractions (or applied displacements), body forces, inertial effects, and large deformations. Two principal conclusions are that (Delta T)sub c has an energy rate interpretation whereas C* does not. The development and application of fracture criteria often involves the solution of boundary/initial value problems associated with deformation and stresses. The finite element method is used for this purpose. An efficient, small displacement, infinitesimal strain, displacement based finite element model is specialized to two dimensional plane stress and plane strain and to power law creep constitutive relations. A mesh shifting/remeshing procedure is used for simulating crack growth. The model is implemented with the quartz-point node technique and also with specially developed, conforming, crack-tip singularity elements which provide for the r to the n-(1+n) power strain singularity associated with the HRR crack-tip field. Comparisons are made with a variety of analytical solutions and alternate numerical solutions for a number of problems.

Stonesifer, R. B.↗

NASA NTRS1982

Solution of second-order linear system by matched asymptotic expansions

Matched asymptotic expansions (MAE) are used to obtain a first order approximation to the solution of a singularly perturbed second order system. A special case is considered in which the uniform asymptotic solution obtained by MAE is shown to converge to the exact solution. Ways in which the method can be used to sole higher-order linear systems, including those which are not singularly perturbed, are also discussed.

Ardema, M. D.↗

NASA NTRS1981

On-line optimization of aircraft altitude and flight path angle dynamics

The optimization of aircraft altitude and flight path angle dynamics is addressed in a form suitable for on-line computation and control. The approach here is a direct extension of the work reported by Calise (1979), where singular perturbation methods were used to optimize position, energy and heading dynamics; it thus represents an optimal control solution that models all of the primary trajectory related dynamics. It is pointed out that the resulting algorithm can be regarded as a nonlinear feedback control law. The minimum time intercept of a fixed terminal point is used in setting the framework in which the analytical results are developed. The main theoretical result is that the dynamics, while not completely separable, can be approximated by singular perturbation methods when the control model includes relative position dynamics.

Calise, A. J.↗

NASA NTRS1982

Numerical techniques in radiative heat transfer for general, scattering, plane-parallel media

The study of radiative heat transfer with scattering usually leads to the solution of singular Fredholm integral equations. The present paper presents an accurate and efficient numerical method to solve certain integral equations that govern radiative equilibrium problems in plane-parallel geometry for both grey and nongrey, anisotropically scattering media. In particular, the nongrey problem is represented by a spectral integral of a system of nonlinear integral equations in space, which has not been solved previously. The numerical technique is constructed to handle this unique nongrey governing equation as well as the difficulties caused by singular kernels. Example problems are solved and the method's accuracy and computational speed are analyzed.

Sharma, A.↗

NASA NTRS1982

Wall interference correction improvements for the ONERA main wind tunnels

This paper describes improved methods of calculating wall interference corrections for the ONERA large windtunnels. The mathematical description of the model and its sting support have become more sophisticated. An increasing number of singularities is used until an agreement between theoretical and experimental signatures of the model and sting on the walls of the closed test section is obtained. The singularity decentering effects are calculated when the model reaches large angles of attack. The porosity factor cartography on the perforated walls deduced from the measured signatures now replaces the reference tests previously carried out in larger tunnels. The porosity factors obtained from the blockage terms (signatures at zero lift) and from the lift terms are in good agreement. In each case (model + sting + test section), wall corrections are now determined, before the tests, as a function of the fundamental parameters M, CS, CZ. During the windtunnel tests, the corrections are quickly computed from these functions.

Vaucheret, X.↗

NASA NTRS1983

An implicit, transonic, full-potential code for cascade flow on H-grid topology

A transonic, full-potential code is developed for computing the flow through two-dimensional cascades using an H-type grid topology that employs an implicit approximate-factorization scheme. The body-conforming H-grid is generated numerically by solving Poisson's equation. The flow-solution algorithm at the coordinate mapping singularity associated with this grid is investigated using two different types of finite-difference schemes. The grid-geometry effect on these schemes is also studied by noting free-stream capturing properties. It is found that by implementing a consistent spatial differencing scheme, the mapping singularities can be resolved numerically, and the grid-geometry-induced error minimized. The code is verified by computing model cascade flow problems.

Kwak, D.↗

NASA NTRS1982

Modified SUMT for structural synthesis

The concepts of the singular perturbation theory are employed to modify the SUMT (Sequential Unconstrained Minimization Technique) algorithm of Fiacco-McCormick (1968) in order to make it more robust and reliable. The algorithm, which uses a sequence of penalty parameters to convert a constrained problem into a sequence of unconstrained problems, suffers from the need to minimize an ill-conditioned penalty function. By using the modified SUMT algorithm on two different structural optimization problems, it is shown that the singular perturbation SUMT easily converges to accurate solutions.

Kamat, M. P.↗

NASA NTRS1983

Derivation of the fundamental equation of sound generated by moving aerodynamic surfaces

Ffowcs Williams and Hawkings (1969) based their derivation of the fundamental equation of the sound generated by arbitrarily moving aerodynamic surfaces on the study of mass and momentum balance of a control volume imbedding a mathematical surface(s) exactly corresponding to real surface(s). These investigators also sketched an alternative method, employing generalized functions, for its derivation. This latter method, which was later developed by Farassat (1975), is purely mathematical and formal. Goldstein (1976) used the free-space Green function to produce an implicit derivation of the Ffowcs Williams and Hawkings equation. In the study presented here, Lowson's (1965) concept of moving point singularities is generalized to moving surface singularities, and a new derivation is given of the fundamental equation. The derivation is based on topological considerations of the underlying space, the fluid medium, and the integral properties of the Dirac delta function.

Aggarwal, H. R.↗

NASA NTRS1983

Post instability in continuous systems. II - Post-instability models of continua

New models simulating postinstability behavior of continuous systems are discussed. Multivalued velocity fields of the motion of a medium are analyzed, deriving concepts which are applied to a new model of turbulence. Postinstability models of solids are examined, emphasizing the multivaluedness of the parameters. The consequences are addressed of the simplifying assumption that in solids the rate of dissipation of energy of the pulsations is much higher than in fluids. The principal stress formulation of postinstability models is considered, including the geometry, kinematics, and dynamics of the principal stress coordinates. The fundamental properties of enlarged models of a solid are illustrated by the theory of wrinkling films. Results are presented for the statics of such films, the effects of shocks and shock waves, and the behavior of singular effects and singular thick wrinkles.

Zak, M.↗