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NASA NTRS1984

The linear two-dimensional stability of inviscid vortex streets of finite-cored vortices

The stability of two-dimensional infinitesimal disturbances of the steady inviscid incompressible flow produced by infinite rows of finite-cored particles is studied. Numerical results are obtained for the growth rate and oscillation frequencies of disturbances of arbitrary subharmonic wavenumber, and the stability boundaries are calculated. The stabilization of the pairing instability by finite area demonstrated by Saffman and Schatzman (1982) is confirmed, as is Kida's (1982) result that this is not the most unstable disturbance when the area is finite. However, the finite area does not stabilize the inviscid Karman vortex street of finite-area vortices to infinitesimal two-dimensional disturbances of arbitrary wavelength. The finite area is always unstable except for one isolated value of the aspect ratio which depends upon the size of the vortices.

Meiron, D. I.↗

NASA NTRS2000

Solving ODE Initial Value Problems With Implicit Taylor Series Methods

In this paper we introduce a new class of numerical methods for integrating ODE initial value problems. Specifically, we propose an extension of the Taylor series method which significantly improves its accuracy and stability while also increasing its range of applicability. To advance the solution from t (sub n) to t (sub n+1), we expand a series about the intermediate point t (sub n+mu):=t (sub n) + mu h, where h is the stepsize and mu is an arbitrary parameter called an expansion coefficient. We show that, in general, a Taylor series of degree k has exactly k expansion coefficients which raise its order of accuracy. The accuracy is raised by one order if k is odd, and by two orders if k is even. In addition, if k is three or greater, local extrapolation can be used to raise the accuracy two additional orders. We also examine stability for the problem y'= lambda y, Re (lambda) less than 0, and identify several A-stable schemes. Numerical results are presented for both fixed and variable stepsizes. It is shown that implicit Taylor series methods provide an effective integration tool for most problems, including stiff systems and ODE's with a singular point.

Scott, James R.↗

NASA NTRS2015

Evolving Design Criteria for Very Large Aperture Space Based Telescopes and Their Influence on the Need for Integrated Tools in the Optimization Process

NASA's Advanced Mirror Technology Development (AMTD) program has been developing the means to design and build the future generations of space based telescopes. With the nearing completion of the James Webb Space Telescope (JWST), the astrophysics community is already starting to define the requirements for follow-on observatories. The restrictions of available launch vehicles and the possibilities of planned future vehicles have fueled the competition between monolithic primaries (with better optical quality) and segmented primaries (with larger apertures, but with diffraction, costs and figure control issues). Regardless of the current shroud sizes and lift capacities, these competing architectures share the need for rapid design tools. As part of the AMTD program a number of tools have been developed and tested to speed up the design process. Starting with the Arnold Mirror Modeler (which creates Finite Element Models (FEM) for structural analysis) and now also feeds these models into thermal stability analyses. They share common file formats and interchangeable results. During the development of the program, numerous trade studies were created for 4-meter and 8-meter monolithic primaries, complete with support systems. Evaluation of these results has led to a better understanding of how the specification drives the results. This paper will show some of the early trade studies for typical specification requirements such as lowest mirror bending frequency and suspension system lowest frequency. The results use representative allowable stress values for each mirror substrate material and construction method and generic material properties. These studies lead to some interesting relationships between feasible designs and the realities of actually trying to build these mirrors. Much of the traditional specifications were developed for much smaller systems, where the mass and volume of the primary where a small portion of the overall satellite. JWST shows us that as the aperture grows, the primary takes up the majority of the mass and volume and the established rules need to be adjusted. For example, a small change in lowest frequency requirement can change the cost by millions of dollars. The paper uses numerous trade studies created during the software development phase of the Arnold Mirror Modeler to illustrate the influences of system specifications on the design space. The future telescopes will require better performance, stability and documented feasibility to meet the hurdles of today's budget and schedules realities. AMTD is developing the tools, but the basic system planning mentality also has to adopt to the requirements of these very large and complex physical structures.

Arnold, William R., Sr.↗

NASA NTRS2008

3D RMHD Simulations of Magnetized Spine-Sheath Relativistic Jets

We have performed numerical simulations of weakly and strongly magnetized relativistic jets embedded in a weakly and strongly magnetized stationary or mildly relativistic (0.5c) sheath flow using the RAISHIN code. In the numerical simulations a jet with Lorentz factor gamma=2.5 is processed to break the initial equilibrium configuration. Results of the numerical simulations are compared to theoretical predictions from a normal mode analysis of the linearized RMHD equations describing a uniform axially magnetized cylindrical relativistic jet embedded in a uniform axially magnetized sheath flow. The prediction of increased stability of a weakly-magnetized system with mildly relativistic sheath flow to Kelvin-Helmholtz instabilities and the stabilization of a strongly magnetized system with mildly relativistic sheath flow is confirmed by the numerical simulations.

Mizuno, Yosuke↗

NASA NTRS2021

Detection of Transient Instabilities in Multi-Agent Systems and Swarms

In this paper, we introduce the novel concept of transient instabilities in multi-agent systems and swarms, i.e., a small disturbance leads to increasing-amplitude oscillations throughout the swarm, which results in a large number of interagent collisions. This instability is dominant in the transient phase of the system and it does not appear in the steady-state behavior of the system, as each agent uses Lyapunov-stable feedback control laws. We present a rigorous definition of transient instability in swarms, and discuss its key properties. We also present a sufficient condition to check if a swarm will be transient stable. Using both theoretical techniques and numerical results, we show that a simple Proportional-Integral-Derivative (PID) feedback-tracking control law is not suitable for these applications. We also present a robust control law for Euler- Lagrangian systems that maintains transient stability across static swarms or time-varying reconfiguration in swarms. Both theoretical results and numerical simulations are presented to demonstrate the effectiveness of our proposed approach.

Quadrelli, Marco B.↗

NASA NTRS1979

Transient unbalance response of four multilobe journal bearings

This work carries out an analysis of the transient response of four multilobe journal bearings (elliptical, offset, three-lobe, and four-lobe) subject to unbalance both below and above the linearized stability thresholds for the bearings. It extends the work of a previous paper on a balanced rotor in the same four bearing types. Transient orbits, bearing forces, and a numerical fast Fourier transform analysis of the orbits are presented. A comparison of bearing forces above the stability threshold for each bearing indicates that the elliptical bearing has the most violent whirl vibration amplitudes, while the offset bearing exhibits the least amount of subsynchronous vibration.

Allaire, P. E.↗

NASA NTRS1992

The stability of a trailing-line vortex in compressible flow

We consider the inviscid stability of the Batchelor (1964) vortex in a compressible flow. The problem is tackled numerically and also asymptotically, in the limit of large (aximuthal and streamwise) wavenumbers, together with large Mach numbers. The nature of the solution passes through different regimes as the Mach number increases, relative to the wavenumbers. At very high wavenumbers and Mach numbers, the mode which is present in the incompressible case ceases to be unstable, while new 'center mode' forms, whose stability characteristics, are determined primarily by conditions close to the vortex axis. We find that generally the flow becomes less unstable as the Mach number increases, and that the regime of instability appears generally confined to disturbances in a direction counter to the direction of the rotation of the swirl of the vortex. Throughout the paper, comparison is made between our numerical results and results obtained from the various asymptotic theories.

Stott, Jillian A. K.↗

NASA NTRS1988

Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: Stationary modes

Spatially discrete difference approximations for hyperbolic initial-boundary-value problems (IBVPs) require numerical boundary conditions in addition to the analytical boundary conditions specified for the differential equations. Improper treatment of a numerical boundary condition can cause instability of the discrete IBVP even though the approximation is stable for the pure initial-value or Cauchy problem. In the discrete IBVP stability literature there exists a small class of discrete approximations called borderline cases. For nondissipative approximations, borderline cases are unstable according to the theory of the Gustafsson, Kreiss, and Sundstrom (GKS) but they may be Lax-Richtmyer stable or unstable in the L sub 2 norm on a finite domain. It is shown that borderline approximation can be characterized by the presence of a stationary mode for the finite-domain problem. A stationary mode has the property that it does not decay with time and a nontrivial stationary mode leads to algebraic growth of the solution norm with mesh refinement. An analytical condition is given which makes it easy to detect a stationary mode; several examples of numerical boundary conditions are investigated corresponding to borderline cases.

Warming, Robert F.↗

NASA NTRS2006

Stability of Supersonic Boundary Layers Over Blunt Wedges

Receptivity and stability of supersonic boundary layers over blunt flat plates and wedges are numerically investigated at a free stream Mach number of 3.5 and at a high Reynolds number of 10(exp 6)/inch. Both the steady and unsteady solutions are obtained by solving the full Navier-Stokes equations using the 5th-order accurate weighted essentially non-oscillatory (WENO) scheme for space discretization and using third-order total-variation-diminishing (TVD) Runge-Kutta scheme for time integration. Computations are performed for a flat plate with leading edge thicknesses of 0.0001, 0.001, 0.005 and 0.01 inches that give Reynolds numbers based on the leading edge thickness ranging from 1000 to 10000. Calculations are also performed for a wedge of 10 degrees half angle with different leading edge radii 0.001 and 0.01 inches. The linear stability results showed that the bluntness has a strong stabilizing effect on the stability of two-dimensional boundary layers. The transition Reynolds number for a flat plate with a leading edge thickness of 0.01 inches is about 3.5 times larger than it is for the Blasius boundary layer. It was also revealed that boundary layers on blunt wedges are far more stable than on blunt flat plates.

Balakumar, Ponnampalam↗

NASA NTRS1993

Linear instability of curved free shear layers

The linear inviscid hydrodynamic stability of slightly curved free mixing layers is studied in this paper. The disturbance equation is solved numerically using a shooting technique. Two mean velocity profiles that represent stably and unstably curved free mixing layers are considered. Results are shown for cases of five curvature Richardson numbers. The stability characteristics of the shear layer are found to vary significantly with the introduction of the curvature effects. The results also indicate that, in a manner similar to the Goertler vortices observed in a boundary layer along a concave wall, instability modes of spatially developing streamwise vortex pairs may appear in unstable curved mixing layers.

Liou, William W.↗

NASA NTRS1993

Linear instability of curved free shear layers

Liou, William W.↗

NASA NTRS1966

A study of pulsation in RR Lyrae models.

Nonlinear calculations of intermediate amplitude to analyze stability of pulsating stars of varying composition, mass, luminosity, etc, using numerical integration of motion equations

STELLAR MASS↗

NASA NTRS1972

An implicit finite-difference solution to the viscous shock layer, including the effects of radiation and strong blowing

An implicit finite-difference scheme is developed for the fully coupled solution of the viscous, radiating stagnation-streamline equations, including strong blowing. Solutions are presented for both air injection and injection of carbon-phenolic ablation products into air at conditions near the peak radiative heating point in an earth entry trajectory from interplanetary return missions. A detailed radiative-transport code that accounts for the important radiative exchange processes for gaseous mixtures in local thermodynamic and chemical equilibrium is utilized in the study. With minimum number of assumptions for the initially unknown parameters and profile distributions, convergent solutions to the full stagnation-line equations are rapidly obtained by a method of successive approximations. Damping of selected profiles is required to aid convergence of the solutions for massive blowing. It is shown that certain finite-difference approximations to the governing differential equations stabilize and improve the solutions. Detailed comparisons are made with the numerical results of previous investigations. Results of the present study indicate lower radiative heat fluxes at the wall for carbonphenolic ablation than previously predicted.

Garrett, L. B.↗

NASA NTRS1973

Deformation and burst of a liquid droplet freely suspended in a linear shear field

A theoretical method is presented for predicting the deformation and the conditions for breakup of a liquid droplet freely suspended in a general linear shear field. This is achieved by expanding the solution to the creeping-flow equations in powers of the deformation parameter epsilon and using linear stability theory to determine the onset of bursting. When compared with numerical solutions and with the available experimental data, the theoretical results are generally found to be of acceptable accuracy although, in some cases, the agreement is only qualitative.

Barthes-Biesel, D.↗

NASA NTRS1975

The dynamics of spin stabilized spacecraft with movable appendages, part 1

The motion and stability of spin stabilized spacecraft with movable external appendages are treated both analytically and numerically. The two basic types of appendages considered are: (1) a telescoping type of varying length and (2) a hinged type of fixed length whose orientation with respect to the main part of the spacecraft can vary. Two classes of telescoping appendages are considered: (a) where an end mass is mounted at the end of an (assumed) massless boom; and (b) where the appendage is assumed to consist of a uniformly distributed homogeneous mass throughout its length. For the telescoping system Eulerian equations of motion are developed. During all deployment sequences it is assumed that the transverse component of angular momentum is much smaller than the component along the major spin axis. Closed form analytical solutions for the time response of the transverse components of angular velocities are obtained when the spacecraft hub has a nearly spherical mass distribution.

Bainum, P. M.↗

NASA NTRS1985

The dynamics and control of large flexible space structures, 8

A development of the in plane open loop rotational equations of motion for the proposed Spacecraft Control Laboratory Experiment (SCOLE) in orbit configuration is presented based on an Eulerian formulation. The mast is considered to be a flexible beam connected to the (rigid) shuttle and the reflector. Frequencies and mode shapes are obtained for the mast vibrational appendage modes (assumed to be decoupled) for different boundary conditions based on continuum approaches and also preliminary results are obtained using a finite element representation of the mast reflector system. The linearized rotational in plane equation is characterized by periodic coefficients and open loop system stability can be examined with an application of the Floquet theorem. Numerical results are presented to illustrate the potential instability associated with actuator time delays even for delays which represent only a small fraction of the natural period of oscillation of the modes contained in the open loop model of the system. When plant and measurement noise effects are added to the previously designed deterministic model of the hoop column system, it is seen that both the system transient and steady state performance are degraded. Mission requirements can be satisfied by appropriate assignment of cost function weighting elements and changes in the ratio of plant noise to measurement noise.

Bainum, P. M.↗

NASA NTRS1987

Removal of spurious modes encountered in solving stability problems by spectral methods

A technique based on the Galerkin approximation is developed to remove spurious roots arising when Chebyshev spectral methods are used to solve eigenvalue problems in hydrodynamic stability. The derivation of Galerkin-Chebyshev approximations is explained, and numerical results for the Orr-Sommerfeld equations of plane Poiseuille flow and a Blasius profile are presented in tables and compared with those obtained by the method of Zebib (1984). It is pointed out that the present method does not increase the size of the algebraic system to be solved.

Zebib, Abdelfattah↗

NASA NTRS1989

Secular resonances

Theories and numerical experiments regarding secular resonances are reviewed. The basic dynamics and the positions of secular resonances are discussed, and secular perturbation theories for the nu16 resonance case, the nu6 resonance, and the nu5 resonance are addressed. What numerical experiments have revealed about asteroids located in secular resonances, the stability of secular resonances, variations of eccentricities and inclinations, and chaotic orbits is considered. Resonant transport of meteorites is discussed.

Scholl, H.↗