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All sourcesNASA NTRSDOE OSTI
NASA NTRSDate not supplied

Bypass Duct Acoustic Liner Design With and Without Bifurcation Effects

The growth in air traffic and the commitment to sustainable aviation continue to provide new challenges to reducing aircraft noise levels. Acoustic liner design methodologies must therefore provide the capability to efficiently predict the acoustic benefits of novel liner configurations within complex aircraft nacelle geometries. With these observations in mind, a broadband acoustic liner optimization process has been developed and assessed through a series of design and experimental studies at increasing technology readiness levels. This work applies the design process to the aft-fan noise component and explores the effects of bypass duct bifurcations (e.g., the pylon and lower bifurcation). In addition to this new application, the design study is expanded to include the use of a commercially available duct propagation code. Despite the different general workflow for the two propagation codes, consistent optimized impedance spectra and in-duct attenuation predictions were obtained for several acoustic treatment options. The preliminary results are promising, and this work increases confidence in the enhanced broadband liner design methodology and lays the groundwork for complimentary use of the codes in future studies. The potential benefits of acoustic treatment on the upper and lower bifurcations are also demonstrated. The knowledge gained through this preliminary stage of the liner design process will be used to guide the identification of candidate liner designs for a proposed static engine test within the next year.

Acoustic Liner Design↗
NASA NTRSDate not supplied

Bypass Duct Acoustic Liner Design with and without Bifurcation Effects

The growth in air traffic and the commitment to sustainable aviation continue to provide new challenges to reducing aircraft noise levels. Acoustic liner design methodologies must therefore provide the capability to efficiently predict the acoustic benefits of novel liner configurations within complex aircraft nacelle geometries. With these observations in mind, a broadband acoustic liner optimization process has been developed and assessed through a series of design and experimental studies at increasing technology readiness levels. This work applies the design process to the aft-fan noise component and explores the effects of bypass duct bifurcations (e.g., the pylon and lower bifurcation). In addition to this new application, the design study is expanded to include the use of a commercially available duct propagation code. Despite the different general workflow for the two propagation codes, consistent optimized impedance spectra and in-duct attenuation predictions were obtained for several acoustic treatment options. The preliminary results are promising, and this work increases confidence in the enhanced broadband liner design methodology and lays the groundwork for complimentary use of the codes in future studies. The potential benefits of acoustic treatment on the upper and lower bifurcations are also demonstrated. The knowledge gained through this preliminary stage of the liner design process will be used to guide the identification of candidate liner designs for a proposed static engine test within the next year.

Acoustic Liner Design↗
NASA NTRS1974

Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension

The bifurcation problem governing the onset of axisymmetric necking in a circular cylindrical specimen in uniaxial tension is analysed. The specimen is made of an incompressible elastic/plastic material. One end is subject to a prescribed uniform axial displacement relative to the other and both ends are shear free. The true stress at bifurcation is greater than the stress at which the maximum load is attained by an amount which depends on (a) the radius to length ratio of the specimen, (b) the ratio of the elastic shear modulus to the tangent modulus, and (c) the derivative of the tangent modulus with respect to the stress. Bifurcation takes place immediately following attainment of the maximum load when the specimen is sufficiently slender.

Hutchinson, J. W.↗
NASA NTRS1983

An explicit example of Hopf bifurcation in fluid mechanics

It is observed that a complete and explicit example of Hopf bifurcation appears not to be known in fluid mechanics. Such an example is presented for the rotating Benard problem with free boundary conditions on the upper and lower faces, and horizontally periodic solutions. Normal modes are found for the linearization, and the Veronis computation of the wave numbers is modified to take into account the imposed horizontal periodicity. An invariant subspace of the phase space is found in which the hypotheses of the Joseph-Sattinger theorem are verified, thus demonstrating the Hopf bifurcation. The criticality calculations are carried through to demonstrate rigorously, that the bifurcation is subcritical for certain cases, and to demonstrate numerically that it is subcritical for all the cases in the paper.

Kloeden, P.↗
NASA NTRS1985

Nonlinear problems in flight dynamics involving aerodynamic bifurcations

Aerodynamic bifurcation is defined as the replacement of an unstable equilibrium flow by a new stable equilibrium flow at a critical value of a parameter. A mathematical model of the aerodynamic contribution to the aircraft's equations of motion is amended to accommodate aerodynamic bifurcations. Important bifurcations such as, the onset of large-scale vortex-shedding are defined. The amended mathematical model is capable of incorporating various forms of aerodynamic responses, including those associated with dynamic stall of airfoils.

Tobak, M.↗
NASA NTRS1985

Nonlinear problems in flight dynamics involving aerodynamic bifurcations

Aerodynamic bifurcation is defined as the replacement of an unstable equilibrium flow by a new stable equilibrium flow at a critical value of a parameter. A mathematical model of the aerodynamic contribution to the aircraft's equations of motion is amended to accommodate aerodynamic bifurcations. Important bifurcations such as, the onset of large-scale vortex shedding are defined. The amended mathematical model is capable of incorporating various forms of aerodynamic responses, including those associated with dynamic stall of airfoils.

Murray Tobak↗
NASA NTRS1985

Bifurcation techniques for nonlinear dynamic analysis of compressor stall phenomena

Compressor stall phenomena is analyzed from nonlinear control theory viewpoint, 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; furthermore it suggests strategies for recovery from stall. 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 effect is shown to exist in the stall/recovery process. Surge cycles are observed to develop for some critical parameter values. It is shown that the oscillatory behavior is due to development of limit cycles, generated by Hopf bifurcation of solutions. Both stable and unstable limit cycles are observed. To further illustrate the usefulness of the methodology some partial computation of domains of attraction of equilibria is carried out, and parameter sensitivity analysis is performed.

Razavi, H. C.↗
NASA NTRS1986

Bifurcations in unsteady aerodynamics

Nonlinear algebraic functional expansions are used to create a form for the unsteady aerodynamic response that is consistent with solutions of the time dependent Navier-Stokes equations. An enumeration of means of invalidating Frechet differentiability of the aerodynamic response, one of which is aerodynamic bifurcation, is proposed as a way of classifying steady and unsteady aerodynamic phenomena that are important in flight dynamics applications. Accomodating bifurcation phenomena involving time dependent equilibrium states within a mathematical model of the aerodynamic response raises an issue of memory effects that becomes more important with each successive bifurcation.

Tobak, M.↗
NASA NTRS1987

Bifurcations in unsteady aerodynamics

Nonlinear algebraic functional expansions are used to create a form for the unsteady aerodynamic response that is consistent with solutions of the time dependent Navier-Stokes equations. An enumeration of means of invalidating Frechet differentiability of the aerodynamic response, one of which is aerodynamic bifurcation, is proposed as a way of classifying steady and unsteady aerodynamic phenomena that are important in flight dynamics applications. Accommodating bifurcation phenomena involving time dependent equilibrium states within a mathematical model of the aerodynamic response raises an issue of memory effects that becomes more important with each successive bifurcation.

Tobak, M.↗
NASA NTRS1987

Bifurcation properties of a stratospheric vacillation model

Nonlinear properties of the stratospheric vacillation model of Holton and Mass (1976) are studied numerically using bifurcation theory. Severe truncation and vertical differencing are used to obtain 81 nonlinear ordinary differential equations with time-dependent variables. Three branches of the steady solutions are determined using Powell's hybrid method and the pseudoarclength continuation method, and a multiplicity of stable steady-state solutions with different vertical structures are found to exist in some range of the bifurcation parameter. Periodic solutions are found which branch off from a steady solution by a Hopf bifurcation. It is suggested that the interannual variability of the stratospheric circulation in the middle and high latitudes during winter may be explained by the multiplicity of steady and periodic stable solutions.

Yoden, Shigeo↗
NASA NTRS1988

Bifurcations in unsteady aerodynamics-implications for testing

The various forms of bifurcations that can occur between steady and unsteady aerodynamic flows are reviewed. Examples are provided to illustrate the various ways in which bifurcations may intervene to influence the outcome of dynamics tests involving unsteady aerodynamics. The presence of bifurcation phenomena in such tests must be taken into consideration to ensure the proper interpretation of results, and some recommendations are made to that end.

Chapman, Gary T.↗
NASA NTRS1989

Bifurcation structure and the Eckhaus instability

The bifurcation diagram corresponding to the Eckhaus stability curve has been constructed for the one-dimensional Swift-Hohenberg equation in a finite domain. Finite-amplitude solutions with particular spatial wavelength recover linear stability, as predicted by the Eckhaus curve, after a sequence of secondary bifurcations from the branch of solutions with this wavelength. No connectivity between the primary-solution branches is admissible if the stability predicted by this bifurcation diagram is to correspond to the prediction of the Eckhaus analysis. The Eckhaus curve does not exist if nonlinear couplings destroy this pattern. This is demonstrated by analysis of a coupled pair of Swift-Hohenberg equations.

Tsiveriotis, K.↗
NASA NTRS1990

A numerical and analytical study of nonlinear bifurcations associated with the morphological stability of two-dimensional single crystals

The nonlinear stability of a two-dimensional single crystal of pure material in an undercooled melt is studied both analytically and numerically. The quasi-steady state approximation is used for the thermal fields, and the effects of different solid and liquid thermal conductivities, isotropic interfacial growth kinetics, and isotropic surface tension are included. The bifurcation analysis is performed by calculating the instantaneous value of the fundamental component of the local normal growth speed for an interface perturbed by a single Fourier shape component. Numerically, the fundamental component of the interfacial growth speed is found by Fourier analysis of the solution to an integrodifferential equation obeyed at the interface. Analytically, an expansion technique is used to derive a solvability condition defining each of these bifurcation points. The analytical and numerical results are in very close agreement. Almost all of the bifurcations are subcritical, and the results are presented by giving values of the Landau coefficient as a function of the different dimensionless parameters used in the model.

Brush, L. N.↗
NASA NTRS1992

Bifurcations In Unsteady Flows

Report discusses various types of bifurcations occurring between steady and unsteady aerodynamic flows. Provides examples to illustrate ways bifurcations influence results of experiments. Recommends experimenters take bifurcation phenomena into account in interpretation of measurements.

Chapman, Gary T.↗
NASA NTRS1991

Evidence of chaotic pattern in solar flux through a reproducible sequence of period-doubling-type bifurcations

Presented here is a preliminary study of the limits to solar flux intensity prediction, and of whether the general lack of predictability in the solar flux arises from the nonlinear chaotic nature of the Sun's physical activity. Statistical analysis of a chaotic signal can extract only its most gross features, and detailed physical models fail, since even the simplest equations of motion for a nonlinear system can exhibit chaotic behavior. A recent theory by Feigenbaum suggests that nonlinear systems that can be led into chaotic behavior through a sequence of period-doubling bifurcations will exhibit a universal behavior. As the control parameter is increased, the bifurcation points occur in such a way that a proper ratio of these will approach the universal Feigenbaum number. Experimental evidence supporting the applicability of the Feigenbaum scenario to solar flux data is sparse. However, given the hypothesis that the Sun's convection zones are similar to a Rayleigh-Bernard mechanism, we can learn a great deal from the remarkable agreement observed between the prediction by theory (period doubling - a universal route to chaos) and the amplitude decrease of the signal's regular subharmonics. The authors show that period-doubling-type bifurcation is a possible route to a chaotic pattern of solar flux that is distinguishable from the logarithm of its power spectral density. This conclusion is the first positive step toward a reformulation of solar flux by a nonlinear chaotic approach. The ultimate goal of this research is to be able to predict an estimate of the upper and lower bounds for solar flux within its predictable zones. Naturally, it is an important task to identify the time horizons beyond which predictability becomes incompatible with computability.

Ashrafi, S.↗
NASA NTRS1991

Evidence of chaotic pattern in solar flux through a reproducible sequence of period-doubling-type bifurcations

A preliminary study of the limits to solar flux intensity prediction, and of whether the general lack of predictability in the solar flux arises from the nonlinear chaotic nature of the Sun's physical activity is presented. Statistical analysis of a chaotic signal can extract only its most gross features, and detailed physical models fail, since even the simplest equations of motion for a nonlinear system can exhibit chaotic behavior. A recent theory by Feigenbaum suggests that nonlinear systems that can be led into chaotic behavior through a sequence of period-doubling bifurcations will exhibit a universal behavior. As the control parameter is increased, the bifurcation points occur in such a way that a proper ratio of these will approach the universal Feigenbaum number. Experimental evidence supporting the applicability of the Feigenbaum scenario to solar flux data is sparse. However, given the hypothesis that the Sun's convection zones are similar to a Rayleigh-Bernard mechanism, we can learn a great deal from the remarkable agreement observed between the prediction by theory (period doubling - a universal route to chaos) and the amplitude decrease of the signal's regular subharmonics. It is shown that period-doubling-type bifurcation is a possible route to a chaotic pattern of solar flux that is distinguishable from the logarithm of its power spectral density. This conclusion is the first positive step toward a reformulation of solar flux by a nonlinear chaotic approach. The ultimate goal of this research is to be able to predict an estimate of the upper and lower bounds for solar flux within its predictable zones. Naturally, it is an important task to identify the time horizons beyond which predictability becomes incompatible with computability.

Ashrafi, S.↗
NASA NTRS2010

Nondimensional Parameters and Equations for Nonlinear and Bifurcation Analyses of Thin Anisotropic Quasi-Shallow Shells

A comprehensive development of nondimensional parameters and equations for nonlinear and bifurcations analyses of quasi-shallow shells, based on the Donnell-Mushtari-Vlasov theory for thin anisotropic shells, is presented. A complete set of field equations for geometrically imperfect shells is presented in terms general of lines-of-curvature coordinates. A systematic nondimensionalization of these equations is developed, several new nondimensional parameters are defined, and a comprehensive stress-function formulation is presented that includes variational principles for equilibrium and compatibility. Bifurcation analysis is applied to the nondimensional nonlinear field equations and a comprehensive set of bifurcation equations are presented. An extensive collection of tables and figures are presented that show the effects of lamina material properties and stacking sequence on the nondimensional parameters.

Nemeth, Michael P.↗
NASA NTRS1975

Bifurcation and collapse analysis of stringer and ring-stringer stiffened cylindrical shells with cutouts

Results for cylindrical configurations using the STAGS computer program were presented. Discontinuities were imposed upon the shell's skin by incorporating symmetrical cutout openings. In addition, the surface is stiffened with both stringer and ring-stringer arrangements. The cutout problem has been shown to be highly nonlinear for smooth surface shells, but it was found that bifurcation and collapse loads are close when one is considering stiffened skin configurations. In order to arrive at this conclusion, it was necessary to evaluate the following: (1) comparison between smeared and discrete stiffener theory for linear solutions, (2) numerical finite difference convergence as directed toward buckling determination, (3) collapse load results with the various skin stiffeners. A linear bifurcation study relating to stiffening effects around cutout areas present within stringer and ring-stringer shell surfaces was included. Comparisons were made between a variety of geometric positions considering cutout frame and thickened skin additions.

Palazotto, A. N.↗