Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Nonlinear equations”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,153 records · Page 64
All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

On the Nonlinear Evolution of a Stationary Cross-Flow Vortex in a Fully Three-Dimensional Boundary Layer Flow

We consider the nonlinear stability of a fully three-dimensional boundary layer flow in an incompressible fluid and derive an equation governing the nonlinear development of a stationary cross-flow vortex. The amplitude equation is a novel integro-differential equation which has spatial derivatives of the amplitude occurring in the kernal function. It is shown that the evolution of the cross-flow vortex is strongly coupled to the properties of an unsteady wall layer which is in fact driven by an unknown slip velocity, proportional to the amplitude of the cross-flow vortex. The work is extended to obtain the corresponding equation for rotating disk flow. A number of special cases are examined and the numerical solution for one of cases, and further analysis, demonstrates the existence of finite-distance as well as focussing type singularities. The numerical solutions also indicate the presence of a new type of nonlinear wave solution for a certain set of parameter values.

Gajjar, J. S. B.↗
NASA NTRS1972

A numerical investigation of film cooling as an alternate thermal protection system for spacecraft during atmospheric reentry

Research was undertaken to determine the mass required to protect a surface during atmospheric reentry by injecting a cold gas tangentially along the surface. A consideration of this problem revealed that the boundary layer equations could be employed to approximate the most important flow configurations which occur in film cooling. However, even with this simplification, the equations to be solved are complex, nonlinear, coupled partial differential equations. Due to the relative weakness of classical methods of boundary layer analysis, finite difference methods were chosen as the best approach to the formidable problem under consideration. Instead of blindly adopting a proposed technique and trying to force it to the problem, a more systematic study was deemed to be not only desirable but necessary. A relatively basic study of the application of finite difference methods was undertaken with the objective of developing an algorithm capable of obtaining accurate solutions to film cooling problems.

Clausing, A. M.↗
NASA NTRS1986

Efficient numerical techniques for complex fluid flows

The central feature in any flow prediction method is the treatment of the coupling between the momentum and continuity equations. In natural-convection flows, the energy equation also becomes strongly coupled with the momentum equations. Because of the nonlinear nature of the coupling, these equations are solved iteratively. Iterative methods are often prone to slow convergence, divergence, and extreme sensitivity to underrelaxation factors. The aim of the present research is to develop more efficient and reliable solution schemes for the coupled flow equations. Such schemes will significantly reduce the expense of computing complex flows encountered in combustion chambers, gas turbines, heat exchangers, and other practical equipment. In the work completed so far, a technique employing norm reduction in conjunction with the successive-substitution and Newton-Raphson techniques was developed. Also, a block-correction procedure for the flow equations is currently being formulated and tested.

Patankar, Suhas V.↗
NASA NTRS1993

Burnett solutions along the stagnation line of a cooled cylinder in low-density hypersonic flows

The Burnett equations in the cylindrical coordinate system have been applied to the hypersonic flow over a cylinder in low density environments. Through a local similarity transformation, the Burnett equations are degenerated into nonlinear third-order ordinary differential equations which are only valid along the stagnation line. These equations are solved by the multi-staged Runge-Kutta integration technique. Steady state solutions are asymptotically reached by the time-marching procedure. A newly-developed computer code is used to solve both the Navier-Stokes and the Burnett equations within the shock layer. The computed heat transfer coefficients are consistent with wind-tunnel and STS flight data. Comparisons with the DSMC solutions and the Boltzmann solutions show that the Burnett equations are more appropriate as the governing equations than the Navier-Stokes equations in the transitional flow regime.

Liaw, G. S.↗
NASA NTRS1985

Interaction of finite-amplitude sound with air-filled porous materials

The propagation of high intensity sound waves through an air-filled porus material was studied. The material is assumed: (1) to be rigid, incompressible, and homogeneous, and (2) to be adequately described by two properties: resistivity r and porosity. The resulting wave equation is still nonlinear, however, because of the u sgn(u) term in the resistivity. The equation is solved in the frequency domain as an infinite set of coupled inhomogeneous Helmholtz equations, one for each harmonic. An approximate but analytical solution leads to predictions of excess attenuation, saturation, and phase speed reduction for the fundamental component. A more general numerical solution is used to calculate the propagation curves for the higher harmonics. The u sgn(u) nonlinearity produces a cubic distortion pattern; when the input signal is a pure tone, only odd harmonic distortion products are generated.

Nelson, D. A.↗
NASA NTRS1989

Integrated approach for active coupling of structures and fluids

Strong coupling of structure and fluids is common in many engineering environments, particularly when the flow is nonlinear and very sensitive to structural motions. Such coupling can give rise to physically important phenomena, such as a dip in the transonic flutter boundary of a wing. The coupled phenomenon can be analyzed in closed form for simple cases that are defined by linear structural and fluid equations of motion. However, complex cases defined by nonlinear equations pose a more difficult task for solution. It is important to understand these nonlinear coupled problems, since they may lead to physically important new phenomena. Flow discontinuities, such as a shock wave, and structural discontinuities, such as a hinge line of a control surface of a wing, can magnify the coupled effects and give rise to new phenomena. To study such a strongly coupled phenomenon, an integrated approach is presented in this paper. The aerodynamic and structural equations of motion are simultaneously integrated by a time-accurate numerical scheme. The theoretical simulation is done using the time-accurate unsteady transonic aerodynamic equations coupled with modal structural equations of motion. As an example, the coupled effect of shock waves and hinge-line discontinuities are studied for aeroelastically flexible wings with active control surfaces. The simulation in this study is modeled in the time domain and can be extended to simulate accurately other systems where fluids and structures are strongly coupled.

Guruswamy, Guru P.↗
NASA NTRS1988

Linear and nonlinear interpretation of CV-580 lightning data

Numerical models developed for the study of lightning strike data acquired by in-flight aircraft are applied to the data measured on the CV-580. The basic technique used is the three dimensional time domain finite difference solution of Maxwell's equations. Both linear and nonlinear models are used in the analysis. In the linear model, the lightning channel and the aircraft are assumed to form a linear time invariant system. A transfer function technique can then be used to study the response of the aircraft to a given lightning strike current. Conversely, the lightning current can be inferred from the measured response. In the nonlinear model, the conductivity of air in the vicinity of the aircraft is calculated and incorporated into the solution of the Maxwell's equations. The nonlinear model thus simulates corona formation and air breakdown. Results obtained from the models are in reasonable agreement with the measured data. This study provides another validation of the models and increases confidence that the models may be used to predict aircraft response to any general lightning strike.

Ng, Poh H.↗
NASA NTRS2005

Magneto-thermo-elastokinetics of geometrically nonlinear laminated composite plates. Part 2: vibration and wave propagation

In Part 1 of this paper, the governing equations of geometrically nonlinear, anisotropic composite plates incorporating magneto-thermo-elastic effects have been derived. In order to gain insight into the implications of a number of geometrical and physical features of the system. three special cases are investigated: (i) free vibration of a plate strip immersed in a transversal magnetic field; (ii) free vibration of the plate strip immersed in an axial magnetic field; (iii) magneto-elastic wave propagations of an infinite plate. Within each of these cases, a prescribed uniform thermal field is considered. Special coupling characteristics between the magnetic and elastic fields are put into evidence. Extensive numerical investigations are conducted and pertinent conclusions which highlight the various effects induced by the magneto-elastic couplings and the finite electroconductivity, are outlined.

Qin, Zhanming↗
NASA NTRS1977

Updating prediction models by dynamical relaxation - An examination of the technique

A dynamical relaxation technique for updating prediction models is analyzed with the help of the linear and nonlinear barotropic primitive equations. It is assumed that a complete four-dimensional time history of some prescribed subset of the meteorological variables is known. The rate of adaptation of the flow variables toward the true state is determined for a linearized f-model, and for mid-latitude and equatorial beta-plane models. The results of the analysis are corroborated by numerical experiments with the nonlinear shallow-water equations.

Davies, H. C.↗
NASA NTRSDate not supplied

Recent Enhancements to Modeling Sonic Boom Propagation using Augmented Burgers’ Equation

Sonic boom propagation through the atmosphere is modeled with an augmented Burgers’ equation which includes nonlinearity and loss mechanisms. This work details an updated discretization of the governing equations which is fully conservative and duality preserving. Adjoint equations, for all the mechanisms involved, are re-derived and implemented using adjoint consistent discretizations. Computation of loudness metrics is performed using digital filters. The updated implementation is demonstrated and compared against the previous formulation for selected cases, and the differences are documented and discussed. The improved discretization results in faster mesh convergence of the loudness metrics and substantially de-creases runtime. In addition, the adjoint solutions provide mesh-converged gradients which are free from spurious oscillations.

Sonic Boom↗
NASA NTRS2000

Viscoelastic flow modeling in the extrusion of a dough-like fluid

This work attempts to investigate the effect of viscoelasticity and three-dimensional geometry in screw channels. The Phan-Thien Tanner (PTT) constitutive equation with simplified model parameters was solved in conjunction with the flow equations. Polyflow, a commercially available finite element code was used to solve the resulting nonlinear partial differential equations. The PTT model predicted one log scale lower pressure buildup compared to the equivalent Newtonian results. However, the velocity profile did not show significant changes for the chosen PTT model parameters. Past Researchers neglected viscoelastic effects and also the three dimensional nature of the flow in extruder channels. The results of this paper provide a starting point for further simulations using more realistic model parameters, which may enable the food engineer to more accurately scale-up and design extrusion processes.

NASA Discipline Life Support Systems↗
NASA NTRS1967

Hidden Variables and Magnetic Relaxation

Nonlinear generalization of Schroedinger equation describing hidden variables effect solved for spin-1/2 magnetic moment relaxing in constant magnetic field

MAGNETIC RELAXATION↗
NASA NTRS1984

Differential eigenvalue problems in which the parameter appears nonlinearly

Several methods are examined for determining the eigenvalues of a system of equations in which the parameter appears nonlinearly. The equations are the result of the discretization of differential eigenvalue problems using a finite Chebyshev series. Two global methods are considered which determine the spectrum of eigenvalues without an initial estimate. A local iteration scheme with cubic convergence is presented. Calculations are performed for a model second order differential problem and the Orr-Sommerfeld problem for plane Poiseuille flow.

Bridges, T. J.↗