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NASA NTRS1985

Critical evaluation of the unsteady aerodynamics approach to dynamic stability at high angles of attack

Bifurcation theory is used to analyze the nonlinear dynamic stability characteristics of an aircraft subject to single-degree-of-freedom. The requisite moment of the aerodynamic forces in the equations of motion is shown to be representable in a form equivalent to the response to finite amplitude oscillations. It is shown how this information can be deduced from the case of infinitesimal-amplitude oscillations. The bifurcation theory analysis reveals that when the bifurcation parameter is increased beyond a critical value at which the aerodynamic damping vanishes, new solutions representing finite amplitude periodic motions bifurcate from the previously stable steady motion. The sign of a simple criterion, cast in terms of aerodynamic properties, determines whether the bifurcating solutions are stable or unstable. For the pitching motion of flat-plate airfoils flying at supersonic/hypersonic speed and for oscillation of flaps at transonic speed, the bifurcation is subcritical, implying either the exchanges of stability between steady and periodic motion are accompanied by hysteresis phenomena, or that potentially large aperiodic departures from steady motion may develop.

Hui, W. H.↗

NASA NTRS1986

Transitions to chaos in two-dimensional double-diffusive convection

The partial differential equations governing two-dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically in a regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences of period-doubling bifurcations. Overstability occurs if the ratio of the solutal to the thermal diffusivity tau is less than 1 and the solutal Rayleigh number Rs is sufficiently large. Solutions have been obtained for two representative values of tau. For tau = 0.316, R(s) = 10,000, symmetrical oscillations undergo a bifurcation to asymmetry, followed by a cascade of period-doubling bifurcations leading to aperiodicity, as the thermal Rayleigh number R(T) is increased. At higher values of R(T), the bifurcation sequence is repeated in reverse, restoring simple periodic solutions. As R(T) is further increased more period-doubling cascades, followed by chaos, can be identified. Within the chaotic regions there are narrow periodic windows, and multiple branches of oscillatory solutions coexist. Eventually the oscillatory branch ends and only steady solutions can be found. The development of chaos has been investigated for tau = 0.1 by varying R(T) for several different values of R(s). When R(s) is sufficiently small there are periodic solutions whose period becomes infinite at the end of the oscillatory branch. As R(s) is increased, chaos appears in the neighborhood of these heteroclinic orbits. At higher values of R(s), chaos is found for a broader range in R(T). A truncated fifth-order model suggest that the appearance of chaos is associated with heteroclinic bifurcations.

Knobloch, E.↗

NASA NTRS2005

Two Stage Supersonic Inlet (TSSI):10-inch Model Calculations

The bifurcated inlet examined in this study (which is one of several being considered in the High Speed Research (HSR) Program) was chosen based upon paper trade studies of axisymmetric, single sided, and bifurcated inlets. For a given compression ratio and mass flow a bifurcated inlet weighs less than a single sided inlet. An axisymmetric inlet has less bleed requirements than 2D inlets but has trouble matching transonic airflow requirements without going to a variable diameter centerbody. The bifurcated inlet was selected as one of the candidates because of its ability to match airflow schedules. The inlet examined in this study, the Two Stage Supersonic Inlet (TSSI), was a candidate mixed compression bifurcated inlet. It has a novel concept to aid in inlet stability. This concept was tested in the 10x10 wind tunnel at NASA Glenn. CFD tools were used to predict and interpret the experimental results.

Chapman, Dave↗

NASA NTRS1974

Plastic buckling

The present article is concerned mainly with the post-bifurcation and imperfection-sensitivity aspects of plastic buckling. A simple two-degree-of-freedom model is used to introduce post-bifurcation behavior and a second model illustrates features of the behavior of continuous solids and structures. Hill's bifurcation criterion for a class of three-dimensional solids is applied to the Donnell-Mushtari-Vlasov (DMV) theory of plates and shells. A general treatment of the initial post-bifurcation behavior of plates and shells is given within the context of the DMV theory. This is illustrated by problems involving columns and circular plates under radial compression. Numerical results are given for a column under axial compression, a circular plate under radial compression, and spherical and cylindrical shells.

Hutchinson, J. W.↗

NASA NTRS1983

Multiple buoyancy driven flows in a vertical cylinder heated from below

The structure of axisymmetric buoyancy-driven convection in a vertical cylinder heated from below is probed by finite element solution of the Boussinesq equations coupled with computed-implemented perturbation techniques for detecting and tracking multiple flows and for determining flow stability. Results are reported for fluids with Prandtl number of one and for cylinders with aspect ratio (Lambda) (defined as the height to radius of the cylinder) between 0.5 and 2.25. Extensive calculations of the neutral stability curve for the static solution and of the nonlinear motions along the bifurcating flow families show a continuous evolution of the primary cellular motion from a single toroidal cell to two and three cells nested radially in the cylinder, instead of the sharp transitions found for a cylinder with shear-free sidewalls. The smooth transitions in flow structure with Rayleigh number and lambda are explained by nonlinear connectivity between the first two bifurcating flow families formed either by a secondary bifurcation point for Lambda or = Lambda * approximately 0.80 or by a limit point for Lambda Lambda *. The transition between these two modes may be described by the theory of multiple limit point bifurcation.

Yamaguchi, Y.↗

NASA NTRS1985

Nonlinear global stability analysis of compressor stall phenomena

Compressor stall phenomena are analyzed from the point of view of nonlinear control theory, based on bifurcation-catastrophe techniques. This new approach appears promising and offers insight into such well-known compressor instability problems as surge and rotating stall and suggests strategies for recovery. Three interlocking dynamic nonlinear state space models are developed. It is shown that the problem of rotating stall can be viewed as an induced bifurcation of solution of the unstalled model. Hysteresis effects are shown to exist in the stall/recovery process. Surge cycles are observed for some critical parameter values. The oscillatory behavior is seen to be due to development of limit cycles, generated by Hopf bifurcation of solutions. More specifically, it is observed that at certain critical values of parameters, a family of stable limit cycles with growning and then diminishing amplitudes is generated, then giving rise to an unstable family of limit cycles. This unstable family in turn bifurcates into other unstable families. To further illustrate the utility of the methodology, some partial computation of domains is carried out, and parameter sensitivity analysis is performed.

Razavi, H.↗

NASA NTRS1991

Nonlinear response of infinitely long circular cylindrical shells to subharmonic radial loads

The nonlinear response of infinitely long circular cylindrical shells (thin circular rings) in the presence of a two-to-one internal (autoparametric) resonance to a subharmonic excitation of order one-half of the higher mode is analyzed with the multiple-scale method. Four autonomous first-order ordinary differential equations are derived for the modulation of the amplitudes and phases of the interacting models. These modulation equations are used to determine the fixed points and their stability. The fixed points correspond to periodic oscillations of the shell, whereas the limit-cycle solutions of the modulation equations correspond to amplitude and phase-modulated oscillations of the shell. The force response curves exhibit saturation, jumps, and Hopf bifurcation. As excitation frequency changes, all limit cycles deform and lose stability through either pitchfork or cyclic-fold (saddle-node) bifurcations. Some of these saddle-node bifurcations cause a transition to chaos. The pitchfork bifurcations break the symmetry of the limit cycles.

Nayfeh, Ali H.↗

NASA NTRS1992

On the structure of cellular solutions in Rayleigh-Benard-Marangoni flows in small-aspect-ratio containers

Multiple steady flow patterns occur in surface-tension/buoyancy-driven convection in a liquid layer heated from below (Rayleigh-Benard-Marangoni flows). Techniques of numerical bifurcation theory are used to study the multiplicity and stability of two-dimensional steady flow patterns (rolls) in rectangular small-aspect-ratio containers as the aspect ratio is varied. For pure Marangoni flows at moderate Biot and Prandtl number, the transitions occurring when paths of codimension 1 singularities intersect determine to a large extent the multiplicity of stable patterns. These transitions also lead, for example, to Hopf bifurcations and stable periodic flows for a small range in aspect ratio. The influence of the type of lateral walls on the multiplicity of steady states is considered. 'No-slip' lateral walls lead to hysteresis effects and typically restrict the number of stable flow patterns (with respect to 'slippery' sidewalls) through the occurrence of saddle node bifurcations. In this way 'no-slip' sidewalls induce a selection of certain patterns, which typically have the largest Nusselt number, through secondary bifurcation.

Dijkstra, Henk A.↗

NASA NTRS2004

Flow field topology of transient mixing driven by buoyancy

Transient mixing driven by buoyancy occurs through the birth of a symmetric Rayleigh-Taylor morphology (RTM) structure for large length scales. Beyond its critical bifurcation the RTM structure exhibits self-similarity and occurs on smaller and smaller length scales. The dynamics of the RTM structure, its nonlinear growth and internal collision, show that its genesis occurs from an explosive bifurcation which leads to the overlap of resonance regions in phase space. This event shows the coexistence of regular and chaotic regions in phase space which is corroborated with the existence of horseshoe maps. A measure of local chaos given by the topological entropy indicates that as the system evolves there is growth of uncertainty. Breakdown of the dissipative RTM structure occurs during the transition from explosive to catastrophic bifurcation; this event gives rise to annihilation of the separatrices which drives overlap of resonance regions. The global bifurcation of explosive and catastrophic events in phase space for the large length scale of the RTM structure serves as a template for which mixing occurs on smaller and smaller length scales. Copyright 2004 American Institute of Physics.

Nonlinear Dynamics↗

NASA NTRS2005

Correlation Between Low Frequency Auroral Kilometric Radiation (AKR) and Auroral Structures

Auroral Kilometric Radiation (AKR) is a radio wave emission that has long been associated with auroral activity. AKR is normally observed in the frequency range from -60 - 600 kHz. Low frequency AKR (or LF-AKR) events are characterized as a rapid extension of AKR related emissions to 30 kHz or lower in frequency for typically much less than 10 minutes. LF-AKR emissions predominantly occur within a frequency range of 20 kHz - 30 kHz, but there are LF-AKR related emissions that reach to a frequency of 5 kHz. This study correlates all instances of LF-AKR events during the first four years of observations from the IMAGE spacecraft's Radio Plasma Imager (WI) instrument with auroral observations from the wideband imaging camera (WIC) onboard IMAGE. The correlation between LF-AKR occurrence and WIC auroral observations shows that in the 295 confirmed cases of LF-AKR emissions, bifurcation of the aurora is seen in 74% of the cases. The bifurcation is seen in the dusk and midnight sectors of the auroral oval, where AKR is believed to be generated. The polarization of these LF-AKR emissions has yet to be identified. Although LF-AKR may not be the only phenomena correlated with bifurcated auroral structures, bifurcation will occur in most instances when LF-AKR is observed. The LF-AKR emissions may be an indicator of specific auroral processes sometimes occurring during storm-time conditions in which field-aligned density cavities extend a distance of perhaps 5-6 RE tailward from the Earth for a period of 10 minutes or less.

Paxamickas, Katherine A.↗

NASA NTRS1973

Experiments on stiffened conical shell structures using cast epoxy models

Description of a casting technique for fabricating high-quality plastic structural models, and review of results regarding the use of such specimens to parametrically study the effect of base ring stiffness on the critical buckling pressure of a ring-stiffened conical shell. The fabrication technique involves machining a metal mold to the desired configuration and vacuum-drawing the plastic material into the mold. A room-temperature curing translucent thermoset epoxy was the casting material selected. A shell of revolution computer program which employs a nonlinear axisymmetric prebuckling strain field to obtain a bifurcation buckling solution was used to guide the selection of congifurations tested. The shell experimentally exhibited asymmetric collapse behavior, and the ultimate load was considerably higher than the analytical bifurcation prediction. The asymmetric buckling mode shape, however, initially appeared at a pressure near the analysis bifurcation solution.

Williams, J. G.↗

NASA NTRS1975

Sustained small oscillations in nonlinear control systems

Some results of bifurcation theory were used to study the existence of small-amplitude periodic behavior in launch vehicle dynamics, assuming that nonlinearity exists as a cubic term in the rudder response. The analysis follows closely Sattinger's (1973) approach to the theory of periodic bifurcations. The conditions under which a bifurcating branch of orbitally stable periodic solutions will exist are determined. It is shown that in more complicated cases, the conditions under which the system matrix has a pair of simple purely imaginary eigenvalues can be determined with the aid of linear stability techniques.

George, J. H.↗

NASA NTRS1979

Some potential blood flow experiments for space

Blood is a colloidal suspension of cells, predominantly erythrocytes, (red cells) in an aqueous solution called plasma. Because the red cells are more dense than the plasma, and because they tend to aggregate, erythrocyte sedimentation can be significant when the shear stresses in flowing blood are small. This behavior, coupled with equipment restrictions, has prevented certain definitive fluid mechanical studies from being performed with blood in ground-based experiments. Among such experiments, which could be satisfactorily performed in a microgravity environment, are the following: (1) studies of blood flow in small tubes, to obtain pressure-flow rate relationships, to determine if increased red cell aggregation can be an aid to blood circulation, and to determine vessel entrance lengths, and (2) studies of blood flow through vessel junctions (bifurcations), to obtain information on cell distribution in downstream vessels of (arterial) bifurcations, and to test flow models of stratified convergent blood flows downstream from (venous) bifurcations.

Cokelet, G. R.↗

NASA NTRS1981

Chaotic /strange/ and periodic behavior in instability saturation by the oscillating two-stream instability

The nonlinear Schroedinger equation with linear growth and damping is truncated to three waves. The resulting system of nonlinear ordinary differential equations describes the excitation of linearly damped waves by the oscillating two-stream instability driven by a linearly unstable pump wave. This system represents a simple model for the nonlinear saturation of a linearly unstable wave. The model is examined analytically and numerically as a function of the dimensionless parameters of the system. It is found that the model can exhibit a wealth of characteristic dynamical behavior including stationary equilibria, Hopf bifurcations to periodic orbits, period doubling bifurcations, chaotic solutions characteristic of a strange attractor, tangent bifurcations from chaotic to periodic solutions, transient chaos, and hysteresis. Many of these features are shown to be explainable on the basis of one-dimensional maps. In the case of chaotic solutions, evidence for the presence of a strange attractor is provided by demonstrating Cantor set-like structure (i.e., scale invariance) in the surface of section.

Russell, D. A.↗

NASA NTRS1988

Modifying PASVART to solve singular nonlinear 2-point boundary problems

To study the buckling and post-buckling behavior of shells and various other structures, one must solve a nonlinear 2-point boundary problem. Since closed-form analytic solutions for such problems are virtually nonexistent, numerical approximations are inevitable. This makes the availability of accurate and reliable software indispensable. In a series of papers Lentini and Pereyra, expanding on the work of Keller, developed PASVART: an adaptive finite difference solver for nonlinear 2-point boundary problems. While the program does produce extremely accurate solutions with great efficiency, it is hindered by a major limitation. PASVART will only locate isolated solutions of the problem. In buckling problems, the solution set is not unique. It will contain singular or bifurcation points, where different branches of the solution set may intersect. Thus, PASVART is useless precisely when the problem becomes interesting. To resolve this deficiency we propose a modification of PASVART that will enable the user to perform a more complete bifurcation analysis. PASVART would be combined with the Thurston bifurcation solution: as adaptation of Newton's method that was motivated by the work of Koiter 3 are reinterpreted in terms of an iterative computational method by Thurston.

Fulton, James P.↗

NASA NTRS1988

A new look at equatorial quasi-biennial oscillation models

Simplified quasi-biennial oscillation models are studied, taking bifurcation theory into account. It is found that the model has a trivial steady solution of no mean zonal flow when the two components of the wave forcing are symmetric. The steady solution becomes unstable with respect to an oscillatory eigenmode when the amplitude of the wave forcing exceeds a critical value. Periodic solutions branch off from the steady solution at this point because of Hopf bifurcation. If the two components are not symmetric, the model has a nontrivial steady solution with nonzero mean zonal flow. Hopf bifurcation takes place and periodic solutions which are not symmetric with respect to time appear. A two-level model is developed to analyze the quasi-biennial oscillation mechanism. It is shown that both vertical diffusion and the shielding effect are needed to obtain periodic solutions.

Yoden, Shigeo↗

NASA NTRS1990

Long-wave morphological instabilities in the directional solidification of a dilute binary mixture

Long-wave instabilities in a directionally-solidified binary mixture may occur in several limits. Sivashinsky (1983) identified a small-segregation-coefficient limit and obtained a weakly nonlinear evolution equation governing subcritical two-dimensional bifurcation. Brattkus and Davis (1988) identified a near-absolute-stability limit and obtained a strongly nonlinear evolution equation governing supercritical two-dimensional bifurcation. The present investigation identifies a third strongly nonlinear evolution equation, arising in the small-segregation-coefficient, large-surface-energy limit. This equation links both of the former and describes the change from the sub- to super-critical bifurcations. This study sets the previous long-wave analyses into a logical framework.

Riley, D. S.↗

NASA NTRS1993

Transition to chaos in an open unforced 2D flow

The present numerical study of unsteady, low Reynolds number flow past a 2D airfoil attempts to ascertain the bifurcation sequence leading from simple periodic to complex aperiodic flow with rising Reynolds number, as well as to characterize the degree of chaos present in the aperiodic flow and assess the role of numerics in the modification and control of the observed bifurcation scenario. The ARC2D Navier-Stokes code is used in an unsteady time-accurate mode for most of these computations. The system undergoes a period-doubling bifurcation to chaos as the Reynolds number is increased from 800 to 1600; its chaotic attractors are characterized by estimates of the fractal dimension and partial Liapunov exponent spectra.

Pulliam, Thomas H.↗