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At least 1,441 records · Page 80
All sourcesNASA NTRSDOE OSTI
NASA NTRS2017

New Mathematical Functions for Vacuum System Analysis

A new bivariate function has been found that provides solutions of integrals having the form u (sup minus eta) e (sup u) du which arise when developing predictions for the behavior of pressure within a rigid volume under high vacuum conditions in the presence of venting as well as sources characterized by power law transient decay over the range [0,1] for eta and for u greater than or equal to 0. A few properties of the new function are explored in this work. For instance the eta equals 1/2 case reproduces the Dawson function. In addition, a slight variation of the solution technique reproduces the exponential integral for eta equals 1. The technique used to generate these functions leads to an approach for solving a more general class of nonlinear ordinary differential equations, with the potential for identifying other new functions that solve other integrals.

Dawson Function↗
NASA NTRS2018

Parameterization of Mixed Layer and Deep-Ocean Mesoscales Including Nonlinearity

In 2011, Chelton et al. carried out a comprehensive census of mesoscales using altimetry data and reached the following conclusions: "essentially all of the observed mesoscale features are nonlinear" and "mesoscales do not move with the mean velocity but with their own drift velocity," which is "the most germane of all the nonlinear metrics." Accounting for these results in a mesoscale parameterization presents conceptual and practical challenges since linear analysis is no longer usable and one needs a model of nonlinearity. A mesoscale parameterization is presented that has the following features: 1) it is based on the solutions of the nonlinear mesoscale dynamical equations, 2) it describes arbitrary tracers, 3) it includes adiabatic (A) and diabatic (D) regimes, 4) the eddy-induced velocity is the sum of a Gent and McWilliams (GM) term plus a new term representing the difference between drift and mean velocities, 5) the new term lowers the transfer of mean potential energy to mesoscales, 6) the isopycnal slopes are not as flat as in the GM case, 7) deep-ocean stratification is enhanced compared to previous parameterizations where being more weakly stratified allowed a large heat uptake that is not observed, 8) the strength of the Deacon cell is reduced. The numerical results are from a stand-alone ocean code with Coordinated Ocean-Ice Reference Experiment I (CORE-I) normal-year forcing.

Baroclinic flows; Mesoscale processes; Mixing; Sub↗
NASA NTRSDate not supplied

Effects of Random Micron-Sized Roughness on Swept-Wing Transition

This study examines the effect of random micron-sized distributed roughness on stationary crossflow instabilities. The roughness parameters are varied by creating nanoparticle coatings of various formulations and applying them to inserts that cover approximately the first 14% of the model. In addition to the baseline configuration (no added roughness, root-mean-square (RMS) ≈ 0.42 𝜇m), panels with RMS roughness values of 4.8 and 8.6 𝜇m were tested, with correlation lengths of 1029 and 385 𝜇m, respectively. Despite the significant roughness levels tested, the transition location was found to be only mildly impacted by the additional roughness, and the roughness panel with lower RMS amplitude caused a larger upstream movement of transition, on average. However, the stationary crossflow amplitudes and wavelength content were found to vary substantially depending on the roughness input. In particular, the panel with higher RMS roughness amplitude resulted in significantly larger amplitudes in the 7.5-9 mm wavelength range at the farthest upstream measurement station, while the lower roughness panel resulted in mildly larger amplitudes at 10 mm and wavelengths larger than 15 mm. Nonlinear Parabolized Stability Equations (PSE) computations were performed to attempt to estimate the initial amplitudes of the stationary crossflow instabilities. Wavelength spectra were matched at the most upstream measurement location, but large discrepancies exist between the predicted and measured growth behavior farther downstream, thus, more work is required to improve confidence in initial amplitude estimates.

boundary-layer transition↗
NASA NTRSDate not supplied

Effects of Random Micron-Sized Roughness on Swept-Wing Transition

This study examines the effect of random micron-sized distributed roughness on stationary crossflow instabilities. The roughness parameters are varied by creating nanoparticle coatings of various formulations and applying them to inserts that cover approximately the first 14% of the model. In addition to the baseline configuration (no added roughness, root-mean-square (RMS) ≈ 0.42 𝜇m), panels with RMS roughness values of 4.8 and 8.6 𝜇m were tested, with correlation lengths of 1029 and 385 𝜇m, respectively. Despite the significant roughness levels tested, the transition location was found to be only mildly impacted by the additional roughness, and the roughness panel with lower RMS amplitude caused a larger upstream movement of transition, on average. However, the stationary crossflow amplitudes and wavelength content were found to vary substantially depending on the roughness input. In particular, the panel with higher RMS roughness amplitude resulted in significantly larger amplitudes in the 7.5-9 mm wavelength range at the farthest upstream measurement station, while the lower roughness panel resulted in mildly larger amplitudes at 10 mm and wavelengths larger than 15 mm. Nonlinear Parabolized Stability Equations (PSE) computations were performed to attempt to estimate the initial amplitudes of the stationary crossflow instabilities. Wavelength spectra were matched at the most upstream measurement location, but large discrepancies exist between the predicted and measured growth behavior farther downstream, thus, more work is required to improve confidence in initial amplitude estimates.

boundary-layer transition↗
NASA NTRSDate not supplied

Effects of Random Micron-Sized Roughness on Swept-Wing Transition

This study examines the effect of random micron-sized distributed roughness on stationary crossflow instabilities. The roughness parameters are varied by creating nanoparticle coatings of various formulations and applying them to inserts that cover approximately the first 14% of the model. In addition to the baseline configuration (no added roughness, root-mean-square (RMS) ≈ 0.42 𝜇m), panels with RMS roughness values of 4.8 and 8.6 𝜇m were tested, with correlation lengths of 1029 and 385 𝜇m, respectively. Despite the significant roughness levels tested, the transition location was found to be only mildly impacted by the additional roughness, and the roughness panel with lower RMS amplitude caused a larger upstream movement of transition, on average. However, the stationary crossflow amplitudes and wavelength content were found to vary substantially depending on the roughness input. In particular, the panel with higher RMS roughness amplitude resulted in significantly larger amplitudes in the 7.5-9 mm wavelength range at the farthest upstream measurement station, while the lower roughness panel resulted in mildly larger amplitudes at 10 mm and wavelengths larger than 15 mm. Nonlinear Parabolized Stability Equations (PSE) computations were performed to attempt to estimate the initial amplitudes of the stationary crossflow instabilities. Wavelength spectra were matched at the most upstream measurement location, but large discrepancies exist between the predicted and measured growth behavior farther downstream, thus, more work is required to improve confidence in initial amplitude estimates.

boundary-layer transition↗
NASA NTRS1985

The general Lie group and similarity solutions for the one-dimensional Vlasov-Maxwell equations

The general Lie point transformation group and the associated reduced differential equations and similarity forms for the solutions are derived here for the coupled (nonlinear) Vlasov-Maxwell equations in one spatial dimension. The case of one species in a background is shown to admit a larger group than the multispecies case. Previous exact solutions are shown to be special cases of the above solutions, and many of the new solutions are found to constrain the form of the distribution function much more than, for example, the BGK solutions do. The individual generators of the Lie group are used to find the possible subgroups. Finally, a simple physical argument is given to show that the asymptotic solution for a one-species, one-dimensional plasma is one of the general similarity solutions.

Roberts, D.↗
NASA NTRS1998

Nonlinear Comparison of High-Order and Optimized Finite-Difference Schemes

The effect of reducing the formal order of accuracy of a finite-difference scheme in order to optimize its high-frequency performance is investigated using the I-D nonlinear unsteady inviscid Burgers'equation. It is found that the benefits of optimization do carry over into nonlinear applications. Both explicit and compact schemes are compared to Tam and Webb's explicit 7-point Dispersion Relation Preserving scheme as well as a Spectral-like compact scheme derived following Lele's work. Results are given for the absolute and L2 errors as a function of time.

Hixon, R.↗
NASA NTRS1977

Comparison of some optimal control methods for the design of turbine blades

This paper attempts a comparative study of some numerical methods for the optimal control design of turbine blades whose vibration characteristics are approximated by Timoshenko beam idealizations with shear and incorporating simple boundary conditions. The blade was synthesized using the following methods: (1) conjugate gradient minimization of the system Hamiltonian in function space incorporating penalty function transformations, (2) projection operator methods in a function space which includes the frequencies of vibration and the control function, (3) epsilon-technique penalty function transformation resulting in a highly nonlinear programming problem, (4) finite difference discretization of the state equations again resulting in a nonlinear program, (5) second variation methods with complex state differential equations to include damping effects resulting in systems of inhomogeneous matrix Riccatti equations some of which are stiff, (6) quasi-linear methods based on iterative linearization of the state and adjoint equation. The paper includes a discussion of some substantial computational difficulties encountered in the implementation of these techniques together with a resume of work presently in progress using a differential dynamic programming approach.

Desilva, B. M. E.↗
NASA NTRS1973

A general conclusion regarding the large amplitude flexural vibration of beams and plates.

Use is made of the two coupled partial differential equations that govern the large amplitude flexural vibration of flat plates. On the basis of an assumed vibration mode taken to be separable in the space coordinates and time, the modal equation is obtained. In the case of a one-term solution, it is shown that the modal equation is the same as the dynamic equation of a mass-spring combination where the restoring force of the nonlinear spring is a cubic and an odd function of the displacement. Since the modal equation is a nonlinear, ordinary differential equation with positive coefficients, it is shown that the nonlinearity associated with the large amplitude flexural vibration of beams and plates is of the hardening type, that is, the frequency increases with amplitude.

Pandalai, K. A. V.↗
NASA NTRS1982

Spline approximations for nonlinear hereditary control systems

A sline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.

Daniel, P. L.↗
NASA NTRS1996

Generation of Squeezed Light Using Photorefractive Degenerate Two-Wave Mixing

We present a quantum nonlinear model of two-wave mixing in a lossless photorefractive medium. A set of equations describing the quantum nonlinear coupling for the field operators is obtained. It is found that, to the second power term, the commutation relationship is maintained. The expectation values for the photon number concur with those of the classical electromagnetic theory when the initial intensities of the two beams are strong. We also calculate the quantum fluctuations of the two beams initially in the coherent state. With an appropriate choice of phase, quadrature squeezing or number state squeezing can be produced.

Lu, Yajun↗
NASA NTRS2015

Toward a Nonlinear Acoustic Analogy: Turbulence as a Source of Sound and Nonlinear Propagation

An acoustic analogy is proposed that directly includes nonlinear propagation effects. We examine the Lighthill acoustic analogy and replace the Green's function of the wave equation with numerical solutions of the generalized Burgers' equation. This is justified mathematically by using similar arguments that are the basis of the solution of the Lighthill acoustic analogy. This approach is superior to alternatives because propagation is accounted for directly from the source to the far-field observer instead of from an arbitrary intermediate point. Validation of a numerical solver for the generalized Burgers' equation is performed by comparing solutions with the Blackstock bridging function and measurement data. Most importantly, the mathematical relationship between the Navier- Stokes equations, the acoustic analogy that describes the source, and canonical nonlinear propagation equations is shown. Example predictions are presented for nonlinear propagation of jet mixing noise at the sideline angle

Miller, Steven A. E.↗
NASA NTRS2015

Toward a Nonlinear Acoustic Analogy: Turbulence as a Source of Sound and Nonlinear Propagation

An acoustic analogy is proposed that directly includes nonlinear propagation effects. We examine the Lighthill acoustic analogy and replace the Green's function of the wave equation with numerical solutions of the generalized Burgers' equation. This is justified mathematically by using similar arguments that are the basis of the solution of the Lighthill acoustic analogy. This approach is superior to alternatives because propagation is accounted for directly from the source to the far-field observer instead of from an arbitrary intermediate point. Validation of a numerical solver for the generalized Burgers' equation is performed by comparing solutions with the Blackstock bridging function and measurement data. Most importantly, the mathematical relationship between the Navier-Stokes equations, the acoustic analogy that describes the source, and canonical nonlinear propagation equations is shown. Example predictions are presented for nonlinear propagation of jet mixing noise at the sideline angle.

Miller, Steven A. E.↗
NASA NTRS1979

Lyapunov stability and its application to systems of ordinary differential equations

An outline and a brief introduction to some of the concepts and implications of Lyapunov stability theory are presented. Various aspects of the theory are illustrated by the inclusion of eight examples, including the Cartesian coordinate equations of the two-body problem, linear and nonlinear (Van der Pol's equation) oscillatory systems, and the linearized Kustaanheimo-Stiefel element equations for the unperturbed two-body problem.

Kennedy, E. W.↗
NASA NTRS1999

Application of the Space-Time Conservation Element and Solution Element Method to One-Dimensional Advection-Diffusion Problems

Test problems are used to examine the performance of several one-dimensional numerical schemes based on the space-time conservation and solution element (CE/SE) method. Investigated in this paper are the CE/SE schemes constructed previously for solving the linear unsteady advection-diffusion equation and the schemes derived here for solving the nonlinear viscous and inviscid Burgers equations. In comparison with the numerical solutions obtained using several traditional finite-difference schemes with similar accuracy, the CE/SE solutions display much lower numerical dissipation and dispersion errors.

Wang, Xiao-Yen↗
NASA NTRS1987

Parallels between control PDE's (Partial Differential Equations) and systems of ODE's (Ordinary Differential Equations)

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

Hunt, L. R.↗