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At least 199 records · Page 11
All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

Hopf bifurcation in the driven cavity

The algorithm employed in the present incompressible two-dimensional calculations of an impulsively-started lid-driven cavity has its basis in the time-dependent stream-function equation. While a Crank-Nicholson differencing scheme is used for the diffusion terms, the Adams-Bashforth scheme is used for the convection terms. The periodic asymptotic solutions obtained for Reynolds numbers of 5000 and 10,000 are found to be precisely periodic; it is demonstrated that they have reached asymptotic states. The indicators of that achievement are discussed.

Goodrich, John W.↗
NASA NTRS1998

Buckling and Damage Resistance of Transversely-Loaded Composite Shells

Experimental and numerical work was conducted to better understand composite shell response to transverse loadings which simulate damage-causing impact events. The quasi-static, centered, transverse loading response of laminated graphite/epoxy shells in a [+/-45(sub n)/O(sub n)](sub s) layup having geometric characteristics of a commercial fuselage are studied. The singly-curved composite shell structures are hinged along the straight circumferential edges and are either free or simply supported along the curved axial edges. Key components of the shell response are response instabilities due to limit-point and/or bifurcation buckling. Experimentally, deflection-controlled shell response is characterized via load-deflection data, deformation-shape evolutions, and the resulting damage state. Finite element models are used to study the kinematically nonlinear shell response, including bifurcation, limit-points, and postbuckling. A novel technique is developed for evaluating bifurcation from nonlinear prebuckling states utilizing asymmetric spatial discretization to introduce numerical perturbations. Advantages of the asymmetric meshing technique (AMT) over traditional techniques include efficiency, robustness, ease of application, and solution of the actual (not modified) problems. The AMT is validated by comparison to traditional numerical analysis of a benchmark problem and verified by comparison to experimental data. Applying the technique, bifurcation in a benchmark shell-buckling problem is correctly identified. Excellent agreement between the numerical and experimental results are obtained for a number of composite shells although predictive capability decreases for stiffer (thicker) specimens which is attributed to compliance of the test fixture. Restraining the axial edge (simple support) has the effect of creating a more complex response which involves unstable bifurcation, limit-point buckling, and dynamic collapse. Such shells were noted to bifurcate into asymmetric deformation modes but were undamaged during testing. Shells in this study which were damaged were not observed to bifurcate. Thus, a direct link between bifurcation and atypical damage could not be established although the mechanism (bifurcation) was identified. Recommendations for further work in these related areas are provided and include extensions of the AMT to other shell geometries and structural problems.

Wardle, Brian L.↗
NASA NTRS1973

Buckling of unstiffened and ring stiffened cylindrical shells under axial compression.

Bifurcation stresses and initial postbuckling behavior of both unstiffened and outside ring stiffened circular cylindrical shells under axial compression are analyzed. The shells are assumed to have axisymmetric sinusoidal imperfections with arbitrary wavelengths and amplitudes. It is found that for large imperfection amplitudes and wavelengths both the unstiffened and the stiffened shells have extremely small bifurcation loads. The postbuckling analysis shows that for small imperfection amplitudes the bifurcations from the axisymmetric state are initially unstable and collapse is associated with the bifurcation points. However, for larger values of the imperfection amplitudes the bifurcations are stable. For unstiffened shells the transition from unstable to stable bifurcations can take place at very small values of the load. On the other hand, for stiffened shells it is found that bifurcations at load levels less than about 40% of the classical buckling load have stable initial postbuckling behavior.

Pedersen, P. T.↗
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↗
DOE OSTI2022

Characterization and interaction of geometric and contact/impact nonlinearities in dynamical systems

Here, we study how a contact/impact nonlinearity interacts with a geometric cubic nonlinearity in an oscillator system. Specific focus is shown to the effects on bifurcation behavior and secondary resonances (i.e., super- and sub-harmonic resonances). The effects of the individual nonlinearities are first explored for comparison, and then the influences of the combined nonlinearities, varying one parameter at a time, are analyzed and discussed. Nonlinear characterization is then performed on an arbitrary system configuration to study super- and sub-harmonic resonances and grazing contacts or bifurcations. Both the cubic and contact nonlinearities cause a drop in amplitude and shift up in frequency for the primary resonance, and they activate high-amplitude subharmonic resonance regions. The nonlinearities seem to never destructively interfere. The contact nonlinearity generally affects the system’s superharmonic resonance behavior more, particularly with regard to the occurrence of grazing contacts and the activation of many bifurcations in the system’s response. The subharmonic resonance behavior is more strongly affected by the cubic nonlinearity and is prone to multistable behavior. Perturbation theory proved useful for determining when the cubic nonlinearity would be dominant compared to the contact nonlinearity. The limiting behaviors of the contact stiffness and freeplay gap size indicate the cubic nonlinearity is dominant overall. It is demonstrated that the presence of contact may result in the activation of several bifurcations. In addition, it is proved that the system’s subharmonic resonance region is prone to multistable dynamical responses having distinct magnitudes.

42 ENGINEERING↗
DOE OSTI2020

Simulations of divertor plasmas with inverse sheaths

The effect of strong electron emission from material surfaces has been proposed to form an “inverse sheath”: a region with a positive potential relative to the near-wall plasma which prevents the flow of ions to the wall [M. D. Campanell, “Negative plasma potential relative to electronemitting surfaces,” Phys. Rev. E. 88, 033103 (2013); M. D. Campanell and M. V. Umansky, “Strongly emitting surfaces unable to float below plasma potential,” Phys. Rev. Lett. 116, 1–5 (2016); M. D. Campanell and G. R. Johnson, “Thermionic cooling of the target plasma to a sub-ev temperature,” Phys. Rev. Lett. 122, 1–5 (2019)]. We assess the viability of this regime in a tokamak device using the 2D edge plasma transport code UEDGE [T. Rognlien et al ., “A fully implicit, time dependent 2-D fluid code for modeling tokamak edge plasmas,” J. Nucl. Mater. 196–198, 347–351 (1992)]. Since the UEDGE code does not consider the sheath region directly, we apply boundary conditions at the divertor targets which emulate the physics of both “standard” and “inverse” sheath regimes [R. Masline et al ., “Influence of the inverse sheath on divertor plasma performance in tokamak edge plasma simulations,” Contrib. Plasma Phys. 60, e201900097 (2020)]. Using these boundary conditions, we perform scoping studies to assess plasma parameters near the target by varying the density at the core-edge interface. We observe a smooth transition in the resultant profiles of plasma parameters for the standard sheath, and a bifurcation across the simulation set for plasmas with an inverse sheath. The cause of this bifurcation is assessed by performing the parameter scan both with and without impurity radiation; we observe that the bifurcation persists in both cases, indicating that this bifurcation is caused by plasma recombination.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Multi-harmonic Rutherford island theory

Rutherford island theory, which governs the nonlinear evolution of tearing modes in tokamak plasmas, is generalized to take into account situations in which the conventional one-harmonic approximation is not valid. The analysis incorporates non-inductive currents driven by radio frequency (RF) electromagnetic waves injected into the plasma. A multi-harmonic tearing mode dispersion relation is derived that takes the form of a nonlinear inhomogeneous matrix eigenvalue problem. The dispersion relation is solved in the so-called two-harmonic approximation, in which only the principal Fourier harmonic of the perturbed magnetic flux and its first overtone are included in the calculation. In the absence of RF current drive, the nonlinear behavior of a tearing mode predicted in the two-harmonic approximation does not differ substantially from that predicted in the one-harmonic approximation. On the other hand, RF current drive that is sufficiently localized in the vicinity of the O-points of the mode's magnetic island chain is capable of triggering bifurcations of the O-points (which is impossible in the one-harmonic approximation). However, the current drive is incapable of triggering bifurcations of the island X-points. This finding is significant because Bardóczi and Evans recently observed bifurcations of magnetic island chain O-points in the presence of RF current drive in the DIII-D tokamak but did not observe bifurcations of the X-points. Finally, the changes in the topology of the magnetic island flux-surfaces induced by RF current drive are found to facilitate the stabilization of the tearing mode.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗