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NASA NTRS1995

Modeling of nonequilibrium space plasma flows

Godunov-type numerical solution of the 20 moment plasma transport equations. One of the centerpieces of our proposal was the development of a higher order Godunov-type numerical scheme to solve the gyration dominated 20 moment transport equations. In the first step we explored some fundamental analytic properties of the 20 moment transport equations for a low b plasma, including the eigenvectors and eigenvalues of propagating disturbances. The eigenvalues correspond to wave speeds, while the eigenvectors characterize the transported physical quantities. In this paper we also explored the physically meaningful parameter range of the normalized heat flow components. In the second step a new Godunov scheme type numerical method was developed to solve the coupled set of 20 moment transport equations for a quasineutral single-ion plasma. The numerical method and the first results were presented at several national and international meetings and a paper describing the method has been published in the Journal of Computational Physics. To our knowledge this is the first numerical method which is capable of producing stable time-dependent solutions to the full 20 (or 16) moment set of transport equations, including the full heat flow equation. Previous attempts resulted in unstable (oscillating) solutions of the heat flow equations. Our group invested over two man-years into the development and implementation of the new method. The present model solves the 20 moment transport equations for an ion species and thermal electrons in 8 domain extending from a collision dominated to a collisionless region (200 km to 12,000 km). This model has been applied to study O+ acceleration due to Joule heating in the lower ionosphere.

Gombosi, Tamas↗

NASA NTRS1994

Lifetimes and eigenstates in atmospheric chemistry

The time scales and mode of the atmosphere's response to chemical perturbations are defined by the eigenvalues and eigenvectors of the system. The eigenstates of a simplified one-box CH4-CO-OH system are analyzed. The longest time constant (smallest eigenvalue) always exceeds the lifetime defined by the steady-state loss frequency for CH4, the longest lived gas. Thus, the extent of a CH4 perturbation -- the methane response time -- is always longer than predicted by the steady-state lifetime and is independent of size of the perturbation in the linear limit. This lengthening of the atmospheric recovery time can be diagnosed by how close we are to a chemically unstable troposphere, i.e., how much OH production exceeds that minimum needed to oxidize just the global emissions of CH4, CO, and other hydrocarbons and species.

Prather, Michael J.↗

NASA NTRS1994

Coherent pulses in the diffusive transport of charged particles`

We present exact solutions to the diffusive transport of charged particles following impulsive injection for a simple model of scattering. A modified, two-parameter relaxation-time model is considered that simulates the low rate of scattering through perpendicular pitch-angle. Scattering is taken to be isotropic within each of the foward- and backward-pointing hemispheres, respectively, but, at the same time, a reduced rate of sccattering is assumed from one hemisphere to the other one. By applying a technique of Fourier- and Laplace-transform, the inverse transformation can be performed and exact solutions can be reached. By contrast with the first, and so far only exact solutions of Federov and Shakov, this wider class of solutions gives rise to coherent pulses to appear. The present work addresses omnidirectional densities for isotropic injection from an instantaneous and localized source. The dispersion relations are briefly discussed. We find, for this particular model, two diffusive models to exist up to a certain limiting wavenumber. The corresponding eigenvalues are real at the lowest wavenumbers. Complex eigenvalues, which are responsible for coherent pulses, appear at higher wavenumbers.

Kota, J.↗

NASA NTRS1996

Predicting the stability of a compressible periodic parallel jet flow

It is known that mixing enhancement in compressible free shear layer flows with high convective Mach numbers is difficult. One design strategy to get around this is to use multiple nozzles. Extrapolating this design concept in a one dimensional manner, one arrives at an array of parallel rectangular nozzles where the smaller dimension is omega and the longer dimension, b, is taken to be infinite. In this paper, the feasibility of predicting the stability of this type of compressible periodic parallel jet flow is discussed. The problem is treated using Floquet-Bloch theory. Numerical solutions to this eigenvalue problem are presented. For the case presented, the interjet spacing, s, was selected so that s/omega =2.23. Typical plots of the eigenvalue and stability curves are presented. Results obtained for a range of convective Mach numbers from 3 to 5 show growth rates omega(sub i)=kc(sub i)/2 range from 0.25 to 0.29. These results indicate that coherent two-dimensional structures can occur without difficulty in multiple parallel periodic jet nozzles and that shear layer mixing should occur with this type of nozzle design.

Miles, Jeffrey H.↗

NASA NTRS1996

ASTROP2 Users Manual: A Program for Aeroelastic Stability Analysis of Propfans

This manual describes the input data required for using the second version of the ASTROP2 (Aeroelastic STability and Response Of Propulsion systems - 2 dimensional analysis) computer code. In ASTROP2, version 2.0, the program is divided into two modules: 2DSTRIP, which calculates the structural dynamic information; and 2DASTROP, which calculates the unsteady aerodynamic force coefficients from which the aeroelastic stability can be determined. In the original version of ASTROP2, these two aspects were performed in a single program. The improvements to version 2.0 include an option to account for counter rotation, improved numerical integration, accommodation for non-uniform inflow distribution, and an iterative scheme to flutter frequency convergence. ASTROP2 can be used for flutter analysis of multi-bladed structures such as those found in compressors, turbines, counter rotating propellers or propfans. The analysis combines a two-dimensional, unsteady cascade aerodynamics model and a three dimensional, normal mode structural model using strip theory. The flutter analysis is formulated in the frequency domain resulting in an eigenvalue determinant. The flutter frequency and damping can be inferred from the eigenvalues.

Reddy, T. S. R.↗

NASA NTRS1996

Expendable Launch Vehicle Studies

Analytical support studies of expendable launch vehicles concentrates on the stability of the dynamics during launch especially during or near the region of maximum dynamic pressure. The in-plane linearized dynamic equations of a generic launch vehicle with multiple flexible bending and fuel sloshing modes are developed. The design of a robust LQR controller based on the reduced order system is accomplished using the parameter perturbation technique. The ELV modeling and analysis team has spent the past three years working on the theoretical development and application of sensitivity analysis to solve eigenvalue problems associated with structural dynamics. Specific application areas include stochastic vibrations, viscously damped vibrations and nonlinear dynamics. In stochastic linear vibrations, sensitivity analysis methods have been developed which determined the expectation and variance of the response for given probability density functions of various stochastic parameters. Examples worked include beam vibrations with up to six stochastic parameters; and the eigensensitivity results differ little from those obtained, with much effect, from Monte Carlo techniques. For viscously damped vibrations, eigensensitivity analysis has given excellent results for homogeneous beams, modeled by either quadratic or quartic eigenvalue equations. Specific applications include Kevin and Maxwell-type viscoelastic beams.

Bainum, P. M.↗

NASA NTRS1996

Thermal and Mechanical Buckling Analysis of Hypersonic Aircraft Hat-Stiffened Panels With Varying Face Sheet Geometry and Fiber Orientation

Mechanical and thermal buckling behavior of monolithic and metal-matrix composite hat-stiffened panels were investigated. The panels have three types of face-sheet geometry: Flat face sheet, microdented face sheet, and microbulged face sheet. The metal-matrix composite panels have three types of face-sheet layups, each of which is combined with various types of hat composite layups. Finite-element method was used in the eigenvalue extractions for both mechanical and thermal buckling. The thermal buckling analysis required both eigenvalue and material property iterations. Graphical methods of the dual iterations are shown. The mechanical and thermal buckling strengths of the hat-stiffened panels with different face-sheet geometry are compared. It was found that by just microdenting or microbulging of the face sheet, the axial, shear, and thermal buckling strengths of both types of hat-stiffened panels could be enhanced considerably. This effect is more conspicuous for the monolithic panels. For the metal-matrix composite panels, the effect of fiber orientations on the panel buckling strengths was investigated in great detail, and various composite layup combinations offering, high panel buckling strengths are presented. The axial buckling strength of the metal-matrix panel was sensitive to the change of hat fiber orientation. However, the lateral, shear, and thermal buckling strengths were insensitive to the change of hat fiber orientation.

Ko, William L.↗

NASA NTRS1998

Nonlinear Analysis of the Space Shuttle Superlightweight LO2 Tank: Behavior Under 3g End-of-Flight Loads - Part 2

Results of linear bifurcation and nonlinear analyses of the Space Shuttle super lightweight (SLWT) external liquid-oxygen (LO2) tank are presented for an important end-of-flight loading condition. These results illustrate an important type of response mode for thin-walled shells, that are subjected to combined mechanical and thermal loads, that may be encountered in the design of other liquid-fuel launch vehicles. Linear bifurcation analyses are presented that predict several nearly equal eigenvalues that correspond to local buckling modes in the aft dome of the LO2 tank. In contrast, the nonlinear response phenomenon is shown to consist of a short-wavelength bending deformation in the aft elliptical dome of the LO2 tank that grows in amplitude in a stable manner with increasing load. Imperfection sensitivity analyses are presented that show that the presence of several nearly equal eigenvalues does not lead to a premature general instability mode for the aft dome. For the linear bifurcation and nonlinear analyses, the results show that accurate predictions of the response of the shell generally require a large-scale, high fidelity finite-element model. Results are also presented that show that the SLWT LO2 tank can support loads in excess of approximately 1.9 times the values of the operational loads considered.

Nemeth, Michael P.↗

NASA NTRS1998

Nonlinear Analysis of the Space Shuttle Superlightweight LO2 Tank: Bahavior Under Booster Ascent Loads - Part 1

Results of linear bifurcation and nonlinear analyses of the Space Shuttle superlightweight (SLWT) external liquid-oxygen (LO2) tank for an important early booster ascent loading condition are presented. These results for thin-walled linear elastic shells that are subjected to combined mechanical and thermal loads illustrate an important type of response mode that may be encountered in the design of other liquid-fuel launch vehicles. Linear bifurcation analyses are presented that predict several nearly equal eigenvalues that correspond to local buckling modes in the forward ogive section of the LO2 tank. In contrast, the nonlinear response phenomenon is shown to consist of short-wavelength bending deformations in the forward ogive and barrel sections of the LO2 tank that growing amplitude in a stable manner increasing load. Imperfection sensitivity analyses are presented that show that the presence of several nearly equal eigenvalues does not lead to a premature general instability mode for the forward ogive section. For the linear bifurcation and nonlinear analyses, the results show that accurate predictions of the response of the shield generally require a large-scale, high-fidelity finite-element model. Results are also presented that show that the SLWT LO2 tank can support loads in excess of approximately 2.6 times the values of the operational loads considered.

Young, Richard D.↗

NASA NTRS1998

Self-Similar Compressible Free Vortices

Lie group methods are used to find both exact and numerical similarity solutions for compressible perturbations to all incompressible, two-dimensional, axisymmetric vortex reference flow. The reference flow vorticity satisfies an eigenvalue problem for which the solutions are a set of two-dimensional, self-similar, incompressible vortices. These solutions are augmented by deriving a conserved quantity for each eigenvalue, and identifying a Lie group which leaves the reference flow equations invariant. The partial differential equations governing the compressible perturbations to these reference flows are also invariant under the action of the same group. The similarity variables found with this group are used to determine the decay rates of the velocities and thermodynamic variables in the self-similar flows, and to reduce the governing partial differential equations to a set of ordinary differential equations. The ODE's are solved analytically and numerically for a Taylor vortex reference flow, and numerically for an Oseen vortex reference flow. The solutions are used to examine the dependencies of the temperature, density, entropy, dissipation and radial velocity on the Prandtl number. Also, experimental data on compressible free vortex flow are compared to the analytical results, the evolution of vortices from initial states which are not self-similar is discussed, and the energy transfer in a slightly-compressible vortex is considered.

vonEllenrieder, Karl↗

NASA NTRS1999

Eigenmodes of Ducted Flows With Radially-Dependent Axial and Swirl Velocity Components

This report characterizes the sets of small disturbances possible in cylindrical and annular ducts with mean flow whose axial and tangential components vary arbitrarily with radius. The linearized equations of motion are presented and discussed, and then exponential forms for the axial, circumferential, and time dependencies of any unsteady disturbances are assumed. The resultant equations form a generalized eigenvalue problem, the solution of which yields the axial wavenumbers and radial mode shapes of the unsteady disturbances. Two numerical discretizations are applied to the system of equations: (1) a spectral collocation technique based on Chebyshev polynomial expansions on the Gauss-Lobatto points, and (2) second and fourth order finite differences on uniform grids. The discretized equations are solved using a standard eigensystem package employing the QR algorithm. The eigenvalues fall into two primary categories: a discrete set (analogous to the acoustic modes found in uniform mean flows) and a continuous band (analogous to convected disturbances in uniform mean flows) where the phase velocities of the disturbances correspond to the local mean flow velocities. Sample mode shapes and eigensystem distributions are presented for both sheared axial and swirling flows. The physics of swirling flows is examined with reference to hydrodynamic stability and completeness of the eigensystem expansions. The effect of assuming exponential dependence in the axial direction is discussed.

Kousen, Kenneth A.↗

NASA NTRS1967

Integrating Matrix Method for Determining the Natural Vibrations of a Rotating, Unsymmetrical Beam with Application to Twisted Propeller Blades

A numerical method is Presented for determining the natural vibration frequencies, and the corresponding mode shapes, of a rotating cantilever beam which has a nonuniform, unsymmetrical cross section. Two coupled fourth-order differential equations of motion with variable coefficients are derived which govern the motion of such a beam having deformations in two directions. Through the development and utilization of the integrating matrix, the solution of the differential equations is obtained in the form of an eigenvalue problem. The solutions to the eigenvalue problem are determined by an iteration method based upon a special orthogonality relationship which is derived. Numerical examples, including an application to a twisted propeller blade, are presented with the results of the integrating matrix solutions being compared to exact solutions and experimental data.

Hunter, William F.↗

NASA NTRS1999

On the Stability of Three-Dimensional Boundary Layers: Secondary Instability - Pt. 2

The secondary instability of three-dimensional incompressible boundary layers is studied using Floquet theory. Starting from the equilibrium solutions that we obtained from the PSE computations documented in Part 1, we investigate the region where a purely stationary crossflow disturbance saturates for its secondary instability characteristics utilizing developed global and local eigenvalue solvers that are based on the Implicitly Restarted Arnoldi Method, and a Newton-Raphson technique, respectively. The main focuses of this study are on the existence of multiple roots in the eigenvalue spectrum that could explain experimental observations of time-dependent occurrences of an explosive growth of traveling disturbances, on the routes by which high-frequency disturbances enter the boundary layer, as well as on gaining more information about threshold amplitudes for the growth of secondary disturbances.

Janke, Erik↗

NASA NTRS1997

The Topology of Symmetric Tensor Fields

Combinatorial topology, also known as "rubber sheet geometry", has extensive applications in geometry and analysis, many of which result from connections with the theory of differential equations. A link between topology and differential equations is vector fields. Recent developments in scientific visualization have shown that vector fields also play an important role in the analysis of second-order tensor fields. A second-order tensor field can be transformed into its eigensystem, namely, eigenvalues and their associated eigenvectors without loss of information content. Eigenvectors behave in a similar fashion to ordinary vectors with even simpler topological structures due to their sign indeterminacy. Incorporating information about eigenvectors and eigenvalues in a display technique known as hyperstreamlines reveals the structure of a tensor field. The simplify and often complex tensor field and to capture its important features, the tensor is decomposed into an isotopic tensor and a deviator. A tensor field and its deviator share the same set of eigenvectors, and therefore they have a similar topological structure. A a deviator determines the properties of a tensor field, while the isotopic part provides a uniform bias. Degenerate points are basic constituents of tensor fields. In 2-D tensor fields, there are only two types of degenerate points; while in 3-D, the degenerate points can be characterized in a Q'-R' plane. Compressible and incompressible flows share similar topological feature due to the similarity of their deviators. In the case of the deformation tensor, the singularities of its deviator represent the area of vortex core in the field. In turbulent flows, the similarities and differences of the topology of the deformation and the Reynolds stress tensors reveal that the basic addie-viscosity assuptions have their validity in turbulence modeling under certain conditions.

Levin, Yingmei↗

NASA NTRS2001

Characterizing Aeroelastic Systems Using Eigenanalysis, Explicitly Retaining The Aerodynamic Degrees of Freedom

Discrete time aeroelastic models with explicitly retained aerodynamic modes have been generated employing a time marching vortex lattice aerodynamic model. This paper presents analytical results from eigenanalysis of these models. The potential of these models to calculate the behavior of modes that represent damped system motion (noncritical modes) in addition to the simple harmonic modes is explored. A typical section with only structural freedom in pitch is examined. The eigenvalues are examined and compared to experimental data. Issues regarding the convergence of the solution with regard to refining the aerodynamic discretization are investigated. Eigenvector behavior is examined; the eigenvector associated with a particular eigenvalue can be viewed as the set of modal participation factors for that particular mode. For the present formulation of the equations of motion, the vorticity for each aerodynamic element appears explicitly as an element of each eigenvector in addition to the structural dynamic generalized coordinates. Thus, modal participation of the aerodynamic degrees of freedom can be assessed in M addition to participation of structural degrees of freedom.

Heeg, Jennifer↗

NASA NTRS2001

Analysis of Preconditioning and Relaxation Operators for the Discontinuous Galerkin Method Applied to Diffusion

The explicit stability constraint of the discontinuous Galerkin method applied to the diffusion operator decreases dramatically as the order of the method is increased. Block Jacobi and block Gauss-Seidel preconditioner operators are examined for their effectiveness at accelerating convergence. A Fourier analysis for methods of order 2 through 6 reveals that both preconditioner operators bound the eigenvalues of the discrete spatial operator. Additionally, in one dimension, the eigenvalues are grouped into two or three regions that are invariant with order of the method. Local relaxation methods are constructed that rapidly damp high frequencies for arbitrarily large time step.

Atkins, H. L.↗

NASA NTRS2001

Discontinuous Galerkin Methods for NonLinear Differential Systems

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

Barth, Timothy↗

NASA NTRS2001

The Effect of Stochastic Perturbation of Fuel Distribution on the Criticality of a One Speed Reactor and the Development of Multi-Material Multinomial Line Statistics

The effect of random fuel redistribution on the eigenvalue of a one-speed reactor is investigated. An ensemble of such reactors that are identical to a homogeneous reference critical reactor except for the fissile isotope density distribution is constructed such that it meets a set of well-posed redistribution requirements. The average eigenvalue, , is evaluated when the total fissile loading per ensemble element, or realization, is conserved. The perturbation is proven to increase the reactor criticality on average when it is uniformly distributed. The various causes of the change in reactivity, and their relative effects are identified and ranked. From this, a path towards identifying the causes. and relative effects of reactivity fluctuations for the energy dependent problem is pointed to. The perturbation method of using multinomial distributions for representing the perturbed reactor is developed. This method has some advantages that can be of use in other stochastic problems. Finally, some of the features of this perturbation problem are related to other techniques that have been used for addressing similar problems.

Jahshan, S. N.↗