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NASA NTRS1976

Finite difference procedure for boundary layers including effects of longitudinal and transverse curvatures

A second order viscous layer solution procedure has been developed that does in a consistent way include curvature effects and the corresponding normal pressure gradients. In the present system, the normal momentum equation is retained. The parabolic system of nonlinear partial differential equations is converted by linear finite differencing procedures to a system of linear algebraic equations and solved in primitive coordinates. The solutions have been shown to give smooth stable distributions for all the variables, most particularly the normal velocity which plays an important role in the interaction procedure. An algorithm for matching the viscous layer solution with a rotational characteristics outer solution has been developed.

Tassa, Y.↗

NASA NTRS1977

Derivation of an integral equation for transonic flows

It is noted that the nonlinear partial differential equation for the perturbation velocity potential and boundary conditions describing steady inviscid compressible transonic flow past a thin two-dimensional airfoil can be transformed into a singular integrodifferential equation and that differentiation of the latter yields an integral equation. Two forms of this integral equation currently exist: one for the singularity that is enclosed in an infinitely long strip of vanishing thickness and the other for the singularity that is enclosed in a vanishing circle. In the present article, a more general integral equation is derived by enclosing the singularity in a vanishing rectangular cavity of arbitrary aspect ratio. The two existing forms of this equation are deduced as special cases distinguished by the respective values for the aspect ratio (infinity for the first form and unity for the second).

Ogana, W.↗

NASA NTRS1976

Stability of elastic bending and torsion of uniform cantilever rotor blades in hover with variable structural coupling

The stability of elastic flap bending, lead-lag bending, and torsion of uniform, untwisted, cantilever rotor blades without chordwise offsets between the elastic, mass, tension, and areodynamic center axes is investigated for the hovering flight condition. The equations of motion are obtained by simplifying the general, nonlinear, partial differential equations of motion of an elastic rotating cantilever blade. The equations are adapted for a linearized stability analysis in the hovering flight condition by prescribing aerodynamic forces, applying Galerkin's method, and linearizing the resulting ordinary differential equations about the equilibrium operating condition. The aerodynamic forces are obtained from strip theory based on a quasi-steady approximation of two-dimensional unsteady airfoil theory. Six coupled mode shapes, calculated from free vibration about the equilibrium operating condition, are used in the linearized stability analysis. The study emphasizes the effects of two types of structural coupling that strongly influence the stability of hingeless rotor blades. The first structural coupling is the linear coupling between flap and lead-lag bending of the rotor blade. The second structural coupling is a nonlinear coupling between flap bending, lead-lag bending, and torsion deflections. Results are obtained for a wide variety of hingeless rotor configurations and operating conditions in order to provide a reasonably complete picture of hingeless rotor blade stability characteristics.

Hodges, D. H., Roberta.↗

NASA NTRS1986

An efficient method for solving the steady Euler equations

An efficient numerical procedure for solving a set of nonlinear partial differential equations, the steady Euler equations, using Newton's linearization procedure is presented. A theorem indicating quadratic convergence for the case of differential equations is demonstrated. A condition for the domain of quadratic convergence Omega(2) is obtained which indicates that whether an approximation lies in Omega(2) depends on the rate of change and the smoothness of the flow vectors, and hence is problem-dependent. The choice of spatial differencing, of particular importance for the present method, is discussed. The treatment of boundary conditions is addressed, and the system of equations resulting from the foregoing analysis is summarized and solution strategies are discussed. The convergence of calculated solutions is demonstrated by comparing them with exact solutions to one and two-dimensional problems.

Liou, M.-S.↗

NASA NTRS1974

Farfield viscous effects in nonlinear noise propagation

Discussion of the method of parametric differentiation in application to predictions of farfield noise propagation in both lossless and dissipative media. It is shown that, in the lossless medium, the governing equation, transformed to parameter space, reduces to a wave equation in the farfield. In the dissipative medium, the system of nonlinear partial differential equations, transformed to parameter space, reduces to a linear partial differential equation of the propagating type which contains a third-order derivative as well as the wave operator.

Harris, W. L., Sr.↗

NASA NTRS1988

Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

Mostrel, M. M.↗

NASA NTRS2019

On the Calculation of Exact Cumulative Distribution Statistics for Burgers Equation

A mathematical procedure is presented for the calculation of exact cumulative distribution statistics for a viscosity-free variant of Burgers nonlinear partial differential equation (PDE) in one space dimension and time subject to sinusoidal initial data with uncertain (random variable) amplitude or phase shift. Analytical solutions of nonlinear PDEs with uncertain initial and/or boundary data are invaluable benchmarks in assessing approximate uncertainty quantification techniques. The Burgers equation solution with uncertain initial data results in nonsmooth solution behavior in both physical and random variable dimensions which provides a severe test for approximate uncertainty quantification techniques. Mathematical proofs are provided to verify that exact cumulative distribution statistics can be systematically and robustly obtained for all forward time.

Burgers↗

NASA NTRS1984

Analytical results for postbuckling behavior of plates in compression and shear

The postbuckling behavior of long rectangular isotropic and orthotropic plates is determined. By assuming trigonometric functions in one direction, the nonlinear partial differential equations of von Karman large deflection plate theory are converted into nonlinear ordinary differential equations. The ordinary differential equations are solved numerically using an available boundary value problem solver which makes use of Newton's method. Results for longitudinal compression show different postbuckling behavior between isotropic and orthotropic plates. Results for shear show that change in inplane edge constraints can cause large change in postbuckling stiffness.

Stein, M.↗

NASA NTRS1985

Analytical results for post-buckling behaviour of plates in compression and in shear

Stein, M.↗

NASA NTRS1987

An Exponential Finite Difference Technique for Solving Partial Differential Equations

An exponential finite difference algorithm, as first presented by Bhattacharya for one-dimensianal steady-state, heat conduction in Cartesian coordinates, has been extended. The finite difference algorithm developed was used to solve the diffusion equation in one-dimensional cylindrical coordinates and applied to two- and three-dimensional problems in Cartesian coordinates. The method was also used to solve nonlinear partial differential equations in one (Burger's equation) and two (Boundary Layer equations) dimensional Cartesian coordinates. Predicted results were compared to exact solutions where available, or to results obtained by other numerical methods. It was found that the exponential finite difference method produced results that were more accurate than those obtained by other numerical methods, especially during the initial transient portion of the solution. Other applications made using the exponential finite difference technique included unsteady one-dimensional heat transfer with temperature varying thermal conductivity and the development of the temperature field in a laminar Couette flow.

Handschuh, Robert F.↗

NASA NTRS1996

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.

Molenaar, J.↗

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 NTRS1969

Application of the method of weighted residuals to the turbulent boundary-layer equations. I-The method of weighted residuals as a solution technique

Method of weighted residuals /MWR/ for nonlinear partial differential equations applied to incompressible two dimensional turbulent boundary layer momentum and continuity equations

Abbott, D. E.↗

NASA NTRS1973

Nonlinear Equations for Bending of Rotating Beams with Application to Linear Flap-Lag Stability of Hingeless Rotors

The nonlinear partial differential equations for the flapping and lead-lag degrees of freedom of a torisonally rigid, rotating cantilevered beam are derived. These equations are linearized about an equilibrium condition to study the flap-lag stability characteristics of hingeless helicopter rotor blades with zero twist and uniform mass and stiffness in the hovering flight condition. The results indicate that these configurations are stable because the effect of elastic coupling more than compensates for the destabilizing flap-lag Coriolis and aerodynamic coupling. The effect of higher bending modes on the lead-lag damping was found to be small and the common, centrally hinged, spring restrained, rigid blade approximation for elastic rotor blades was shown to be resonably satisfactory for determining flap-lag stability. The effect of pre-cone was generally stabilizing and the effects of rotary inertia were negligible.

Hodges, D. H.↗

NASA NTRS1975

Prolongation structures of nonlinear evolution equations

A technique is developed for systematically deriving a 'prolongation structure' - a set of interrelated potentials and pseudopotentials - for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg-de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear 'inverse scattering' equations for auxiliary functions, Backlund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.

Wahlquist, H. D.↗

NASA NTRS1972

Large amplitude flexural vibration of thin elastic flat plates and shells

The general equations governing the large amplitude flexural vibration of any thin elastic shell using curvilinear orthogonal coordinates are derived and consist of two coupled, nonlinear, partial differential equations in the normal displacement w and the stress function F. From these equations, the governing equations for the case of shells of revolution or flat plates can be readily obtained as special cases. The material of the shell or plate is isotropic and homogeneous and Hooke's law for the two-dimensional case is valid. It is suggested that the difference between the hardening type of nonlinearity in the case of flat plates and straight beams and the softening type of nonlinearity in the case of shells and rings can, in general, be traced to the amount of curvature present in the underformed median surface of the structure concerned.

Pandalia, K. A. V.↗

NASA NTRS1979

Time-varying linear systems and the theory of non-linear waves

The isospectral deformation of a Sturm-Liouville equation is extended to general linear time-varying systems and a method is described for determining the resulting nonlinear partial differential equations. Consideration is given to (1) isospectral deformation of I/O systems with boundary value conditions and (2) the spectral vector bundles attached to linear time-varying systems.

Hermann, R.↗

NASA NTRS1992

Wave equations and computational models for sonic boom propagation through a turbulent atmosphere

The improved simulation of sonic boom propagation through the real atmosphere requires greater understanding of how the transient acoustic pulses popularly termed sonic booms are affected by atmospheric turbulence. A nonlinear partial differential equation that can be used to simulate the effects of smaller-scale atmospheric turbulence on sonic boom waveforms is described. The equation is first order in the time derivative and involves an extension of geometrical acoustics to include diffraction phenomena. Various terms in the equation are explained in physical terms. Such terms include those representing convection at the wave speed, diffraction, molecular relaxation, classical dissipation, and nonlinear steepening. The atmospheric turbulence enters through an effective sound speed, which varies with all three spatial coordinates, and which is the sum of the local sound speed and the component of the turbulent flow velocity projected along a central ray that connects the aircraft trajectory with the listener.

Pierce, Allan D.↗