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At least 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
NASA NTRS1996

The Evolution of Finite Amplitude Wavetrains in Plane Channel Flow

We consider a viscous incompressible fluid flow driven between two parallel plates by a constant pressure gradient. The flow is at a finite Reynolds number, with an 0(l) disturbance in the form of a traveling wave. A phase equation approach is used to discuss the evolution of slowly varying fully nonlinear two dimensional wavetrains. We consider uniform wavetrains in detail, showing that the development of a wavenumber perturbation is governed by Burgers equation in most cases. The wavenumber perturbation theory, constructed using the phase equation approach for a uniform wavetrain, is shown to be distinct from an amplitude perturbation expansion about the periodic flow. In fact we show that the amplitude equation contains only linear terms and is simply the heat equation. We review, briefly, the well known dynamics of Burgers equation, which imply that both shock structures and finite time singularities of the wavenumber perturbation can occur with respect to the slow scales. Numerical computations have been performed to identify areas of the (wavenumber, Reynolds number, energy) neutral surface for which each of these possibilities can occur. We note that the evolution equations will breakdown under certain circumstances, in particular for a weakly nonlinear secondary flow. Finally we extend the theory to three dimensions and discuss the limit of a weak spanwise dependence for uniform wavetrains, showing that two functions are required to describe the evolution. These unknowns are a phase and a pressure function which satisfy a pair of linearly coupled partial differential equations. The results obtained from applying the same analysis to the fully three dimensional problem are included as an appendix.

Hewitt, R. E.↗
NASA NTRS1988

An asymptotic induced numerical method for the convection-diffusion-reaction equation

A parallel algorithm for the efficient solution of a time dependent reaction convection diffusion equation with small parameter on the diffusion term is presented. The method is based on a domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. Parallelism is evident at two levels. Domain decomposition provides parallelism at the highest level, and within each domain there is ample opportunity to exploit parallelism. Run time results demonstrate the viability of the method.

Scroggs, Jeffrey S.↗
NASA NTRS2011

Toward Realistic Dynamics of Rotating Orbital Debris, and Implications for Lightcurve Interpretation

Optical observations of rotating space debris near GEO contain important information on size, shape, composition, and rotational states, but these aspects are difficult to extract due to data limitations and the high number of degrees of freedom in the modeling process. For tri-axial rigid debris objects created by satellite fragmentations, the most likely initial rotation state has a large component of initial angular velocity directed along the intermediate axis of inertia, leading to large angular reorientations of the body on the timescale of the rotation period. This lends some support to the simplest possible interpretation of light curves -- that they represent sets of random orientations of the objects of study. However, effects of internal friction and solar radiation are likely to cause significant modification of rotation states within a time as short as a few orbital periods. In order to examine the rotational dynamics of debris objects under the influences of these effects, a set of seven first-order coupled equations of motion were assembled in state form: three are Euler equations describing the rates of change of the components of angular velocity in the body frame, and four describe the rates of change of the components of the unit quaternion. Quaternions are a four-dimensional extension of complex numbers that form a seamless, singularity-free representation of body orientation on S3. The Euler equations contain explicit terms describing torque from solar radiation in terms of spherical harmonics, and terms representing effects of a prescribed rate of internal friction. Numerical integrations of these equations of motion are being performed, and results will be presented. Initial tests show that internal friction without solar radiation torque leads to rotation about the maximum principal axis of inertia, as required, and solar radiation torque is expected to lead to spin-up of objects. Because the axis of maximum rotational inertia tends to be roughly coincident with the normal to the largest projected cross-sectional area, internal friction is expected to lead to reduced variation of light curve amplitudes at a given phase angle, but a large dependence of the same on phase angle. At a given phase angle, databases are generated which contain reflected intensities for comprehensive sets of equally-likely orientations, represented as unit quaternions. When projected onto three dimensions (S2) and color-coded by intensity, the set is depicted as points within a solid, semi-transparent unit sphere, within which all possible reflected intensities for an object at a given phase angle may be inspected simultaneously. Rotational sequences are represented by trajectories through the sphere. Databases are generated for each of a set of phase angles separately, forming a comprehensive dataset of reflected intensities spanning all object orientations and solar phase angles. Symmetries in the problem suggest that preferred rotation states are likely, defined relative to the object-sun direction in inertial space and relative to the maximum principal axis of inertia in the body coordinate system. Such rotation states may greatly simplify the problem of light curve interpretation by reducing the number of degrees of freedom in the problem.

Ojakangas, Gregory W.↗
NASA NTRS1972

Singular perturbation of absolute stability.

The influence of a small parameter at the higher derivatives in the differential equations describing nonlinear systems of the Lur'e-Postnikov class on absolute stability in the parameter space is investigated. The conditions leading to singular perturbations of absolute stability are examined.

Siljak, D. D.↗
NASA NTRS1990

A parallel algorithm for nonlinear convection-diffusion equations

A parallel algorithm for the efficient solution of nonlinear time-dependent convection-diffusion equations with small parameter on the diffusion term is presented. The method is based on a physically motivated domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. The method is suitable for the solution of problems arising in the simulation of fluid dynamics. Experimental results for a nonlinear equation in two-dimensions are presented.

Scroggs, Jeffrey S.↗
NASA NTRS1989

Optimized resolved rate control of seven-degree-of-freedom Laboratory Telerobotic Manipulator (LTM) with application to three-dimensional graphics simulation

The Laboratory Telerobotic Manipulator (LTM) is a seven-degree-of-freedom robot arm. Two of the arms were delivered to Langley Research Center for ground-based research to assess the use of redundant degree-of-freedom robot arms in space operations. Resolved-rate control equations for the LTM are derived. The equations are based on a scheme developed at the Oak Ridge National Laboratory for computing optimized joint angle rates in real time. The optimized joint angle rates actually represent a trade-off, as the hand moves, between small rates (least-squares solution) and those rates which work toward satisfying a specified performance criterion of joint angles. In singularities where the optimization scheme cannot be applied, alternate control equations are devised. The equations developed were evaluated using a real-time computer simulation to control a 3-D graphics model of the LTM.

Barker, L. Keith↗
NASA NTRS1992

A robust direct-integration method for rotorcraft maneuver and periodic response

The Newmark-Beta method and the Newton-Raphson iteration scheme are combined to develop a direct-integration method for evaluating the maneuver and periodic-response expressions for rotorcraft. The method requires the generation of Jacobians and includes higher derivatives in the formulation of the geometric stiffness matrix to enhance the convergence of the system. The method leads to effective convergence with nonlinear structural dynamics and aerodynamic terms. Singularities in the matrices can be addressed with the method as they arise from a Lagrange multiplier approach for coupling equations with nonlinear constraints. The method is also shown to be general enough to handle singularities from quasisteady control-system models. The method is shown to be more general and robust than the similar 2GCHAS method for analyzing rotorcraft dynamics.

Panda, Brahmananda↗
NASA NTRS1979

A computational fluid dynamic technique valid at the centerline for non-axisymmetric problems in cylindrical coordinates

A technique is described for the numerical solution of non-axisymmetric flow problems posed in cylindrical coordinates when the z-axis is included in the flowfield. The highlight of the technique is the manner in which the singularities at the centerline are handled. Specifically, the governing flowfield equations at r = 0 are put in a special form by applying L'Hospital's Rule. The required radial derivatives are evaluated using a one-sided, second-order accurate, first-difference. This leads to a smooth, convergent calculation of the flowfield at the centerline. This appears to be the first generally applicable numerical method for avoiding coordinate system singularities in the context of a finite-difference scheme, and could have application to many nonaxisymmetric flows. The technique is illustrated by specific results for the time-dependent flowfield inside an internal combustion engine.

Griffin, M. D.↗
NASA NTRS1978

Numerical solution of a class of integral equations arising in two-dimensional aerodynamics

We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This allows the problem to be studied in terms of an equivalent integral equation of the second kind. A convergent numerical algorithm for its solution is derived by using Galerkin's method. It is shown that this algorithm is numerically equivalent to Bland's collocation method, which is then used as the method of computation. Extensive numerical calculations are presented establishing the validity of the theory.

Fromme, J.↗
NASA NTRS1963

ORBITAL ELEMENT EQUATIONS FOR OPTIMUM LOW THRUST TRAJECTORIES

The three dimensional optimum trajectory relations developed by Messrs J. G. Cox and W. A. Shaw in Reference I, are transformed into- a form that appears more amenable to low thrust trajectory calculations. Orbital element coordinates, commonly used in Celestial Mechanics, are employed due to their slow variation in low thrust applications. Combinations of these elements and a generalized eccentric anomaly are utilized in arranging the resulting equations- in a form which does not contain circular singularities.

Spacecraft Guidance↗
NASA NTRS1975

Viscous interaction of flow redevelopment after flow reattachment with supersonic external streams

A flow model has been developed to study the flow development after reattachment with supersonic external streams. Special attention is given to the pressure difference across the viscous layer, and it is suggested that such a flow redevelopment can be treated as a relaxation of this pressure difference. Upon correlating the pressure difference with a slope parameter of the velocity profile, the system of equations governing the flow would produce a saddle point singularity corresponding to the fully rehabilitated asymptotic flow condition. A method of calculation for this flowfield, in conjunction with the matching of the upstream flow, has been derived and is discussed. Samples of calculations are also presented. Reasonably good agreement with experimental data has also been observed.

Chow, W. L.↗
NASA NTRS1985

Postbuckling analysis using a general purpose code

A new capability for solving postbuckling problems in shell structures is described. The matrix theory to adapt Newton's method to nonlinear finite element shell analysis is outlined first. The matrix theory is directed at writing consistent linear algebratic equations for problems where the tangent stiffness matrix is singular or nearly singular. The matrix theory suggests a change of variables as part of the usual iterative procedure in Newton's method. The change of variables is shown to be feasible for introduction into the algorithm programmed in general purpose codes for finite element analysis of structures. Numerical results from a new option that has been programmed in an existing general purpose code are presented. The analysis of shell structures for collapse and for branching at bifurcation loads is illustrated by the numerical examples.

Thurston, G. A.↗
NASA NTRS1987

Notes on implementation of Coulomb friction in coupled dynamical simulations

A coupled dynamical system is defined as an assembly of rigid/flexible bodies that may be coupled by kinematic connections. The interfaces between bodies are modeled using hinges having 0 to 6 degrees of freedom. The equations of motion are presented for a mechanical system of n flexible bodies in a topological tree configuration. The Lagrange form of the D'Alembert principle was employed to derive the equations. The equations of motion are augmented by the kinematic constraint equations. This augmentation is accomplished via the method of singular value decomposition.

Vandervoort, R. J.↗
NASA NTRS1987

The thermal stability of coronal loops by nonlinear diffusion asymptotics

A nonlinear reaction-diffusion equation and some additional constraints are derived which describe the time-dependent behavior of the temperature structure of the plasma in coronal loops. The equation is analyzed using nonlinear diffusion asymptotics, in particular singular perturbation techniques, and the results are interpreted in the context of the physical problem of the thermal stability and temporal behavior of the plasma. The results are consistent with the possibility of cyclic thermal behavior of the plasma, as suggested by Kuin and Martens (1982).

Pakkert, J. W.↗
NASA NTRS1991

Nonlinear evolution of subsonic and supersonic disturbances on a compressible free shear layer

The effects of a nonlinear-nonequilibrium-viscous critical layer on the spatial evolution of subsonic and supersonic instability modes on a compressible free shear layer is considered. It is shown that the instability wave amplitude is governed by an integrodifferential equation with cubic-type nonlinearity. Numerical and asymptotic solutions to this equation show that the amplitude either ends in a singularity at a finite downstream distance or reaches an equilibrium value, depending on the Prandtl number, viscosity law, viscous parameter and a real parameter which is determined by the linear inviscid stability theory. A necessary condition for the existence of the equilibrium solution is derived, and whether or not this condition is met is determined numerically for a wide range of physical parameters including both subsonic and supersonic disturbances. it is found that no equilibrium solution exists for the subsonic modes unless the temperature ratio of the low-to-high-speed streams exceeds a critical value, while equilibrium solutions for the most rapidly growing supersonic mode exist over most of the parameter range examined.

Leib, S. J.↗
NASA NTRS1992

Adaptive mesh strategies for the spectral element method

An adaptive spectral method was developed for the efficient solution of time dependent partial differential equations. Adaptive mesh strategies that include resolution refinement and coarsening by three different methods are illustrated on solutions to the 1-D viscous Burger equation and the 2-D Navier-Stokes equations for driven flow in a cavity. Sharp gradients, singularities, and regions of poor resolution are resolved optimally as they develop in time using error estimators which indicate the choice of refinement to be used. The adaptive formulation presents significant increases in efficiency, flexibility, and general capabilities for high order spectral methods.

Mavriplis, Catherine↗
NASA NTRS1992

A new time scale based k-epsilon model for near wall turbulence

A k-epsilon model is proposed for wall bonded turbulent flows. In this model, the eddy viscosity is characterized by a turbulent velocity scale and a turbulent time scale. The time scale is bounded from below by the Kolmogorov time scale. The dissipation equation is reformulated using this time scale and no singularity exists at the wall. The damping function used in the eddy viscosity is chosen to be a function of R(sub y) = (k(sup 1/2)y)/v instead of y(+). Hence, the model could be used for flows with separation. The model constants used are the same as in the high Reynolds number standard k-epsilon model. Thus, the proposed model will be also suitable for flows far from the wall. Turbulent channel flows at different Reynolds numbers and turbulent boundary layer flows with and without pressure gradient are calculated. Results show that the model predictions are in good agreement with direct numerical simulation and experimental data.

Yang, Z.↗