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NASA NTRS1985

Distributed reacceleration of cosmic rays

A model is developed in which cosmic rays, in addition to their initial acceleration by a strong shock, are continuously reaccelerated while propagating through the Galaxy. The equations describing this acceleration scheme are solved analytically and numerically. Solutions for the spectra of primary and secondary cosmic rays are given in a closed analytic form, allowing a rapid search in parameter space for viable propagation models with distributed reeacceleration included. The observed boron-to-carbon ratio can be reproduced by the reacceleration theory over a range of escape parameters, some of them quite different from the standard leaky-box model. It is also shown that even a very modest amount of reacceleration by strong shocks causes the boron-to-carbon ratio to level off at sufficiently high energies.

Wandel, Amri↗

NASA NTRS1988

Total Variation Diminishing (TVD) schemes of uniform accuracy

Explicit second-order accurate finite-difference schemes for the approximation of hyperbolic conservation laws are presented. These schemes are nonlinear even for the constant coefficient case. They are based on first-order upwind schemes. Their accuracy is enhanced by locally replacing the first-order one-sided differences with either second-order one-sided differences or central differences or a blend thereof. The appropriate local difference stencils are selected such that they give TVD schemes of uniform second-order accuracy in the scalar, or linear systems, case. Like conventional TVD schemes, the new schemes avoid a Gibbs phenomenon at discontinuities of the solution, but they do not switch back to first-order accuracy, in the sense of truncation error, at extrema of the solution. The performance of the new schemes is demonstrated in several numerical tests.

Hartwich, PETER-M.↗

NASA NTRS1988

Neural networks with hierarchically correlated patterns

The Hopfield model of neural networks is extended to allow for the storage and retrieval of hierarchically correlated patterns. The overlaps between these patterns form an ultrametric tree. Intermediate states, which serve as ancestors for the following levels, are generated at each level of the tree. The states belonging to each level are stored, by a modified learning rule, in a series of identical networks, one for each level. The retrieval of a particular pattern is preceded and assisted by the successive retrieval of its ancestors. The performance of this scheme is studied analytically and numerically.

Gutfreund, H.↗

NASA NTRS1988

A three dimensional calculation of elastic equilibrium for composite materials

A compact scheme is applied to three-dimensional elasticity problems for composite materials, involving simple geometries. The mathematical aspects of this approach are discussed, in particular the iteration method. A vector processor code implementing the compact scheme is presented, and several numerical experiments are summarized.

Lustman, Liviu R.↗

NASA NTRS1987

Flow solution on a dual-block grid around an airplane

The compressible flow around a complex fighter-aircraft configuration (fuselage, cranked delta wing, canard, and inlet) is simulated numerically using a novel grid scheme and a finite-volume Euler solver. The patched dual-block grid is generated by an algebraic procedure based on transfinite interpolation, and the explicit Runge-Kutta time-stepping Euler solver is implemented with a high degree of vectorization on a Cyber 205 processor. Results are presented in extensive graphs and diagrams and characterized in detail. The concentration of grid points near the wing apex in the present scheme is shown to facilitate capture of the vortex generated by the leading edge at high angles of attack and modeling of its interaction with the canard wake.

Eriksson, Lars-Erik↗

NASA NTRS1990

Combination Of Techniques For Computing Incompressible Flow

Pseudocompressibility, upwind differencing, and other techniques used to solve Navier-Stokes equations. Scheme for finite-difference numerical solution of two-dimensional Navier-Stokes equations of incompressible flow combines several recently developed methods, each developed to increase speed and/or accuracy of computations of this kind.

Kwak, D.↗

NASA NTRS1989

Spectral element methods for the incompressible Navier-Stokes equations

Spectral element methods are high-order weighted-residual techniques for partial differential equations that combine the geometric flexibility of finite element techniques with the rapid convergence rate of spectral schemes. The theoretical foundations and numerical implementation of spectral element methods for the incompressible Navier-Stokes equations are presented, considering the construction and analysis of optimal-order spectral element discretizations for elliptic and saddle (Stokes) problems, as well as the efficient solution of the resulting discrete equations by rapidly convergent tensor-product-based iterative procedures. Several examples of spectral element simulation of moderate Reynolds number unsteady flow in complex geometry are presented.

Maday, Yvon↗

NASA NTRS1990

Collisionless drift-tearing modes in the magnetopause

The linear stability properties of collisionless drift-tearing modes are analyzed in a modified Harris equilibrium model of the magnetopause. Particular attention is paid to the relevance of the parametric behavior of growth rates to the 'magnetic percolation' theory of flux transfer event formation (Galeev et al., 1986). Numerical methods are used to solve the drift-tearing eigenmode equations and the results are compared with those previously obtained by analytical methods. The analytical results are found to correctly model important parametric dependencies but to typically overestimate the rate of growth. The eigenmode equations are numerically difficult, and an integration scheme utilizing Ricatti transforms is developed to affect their solution.

Gladd, N. T.↗

NASA NTRS1992

Tables Of Gaussian-Type Orbital Basis Functions

NASA technical memorandum contains tables of estimated Hartree-Fock wave functions for atoms lithium through neon and potassium through krypton. Sets contain optimized Gaussian-type orbital exponents and coefficients, and near Hartree-Fock quality. Orbital exponents optimized by minimizing restricted Hartree-Fock energy via scaled Newton-Raphson scheme in which Hessian evaluated numerically by use of analytically determined gradients.

Partridge, Harry↗

NASA NTRS1993

Numerical investigation of receptivity to freestream vorticity

The receptivity to freestream vorticity of the boundary layer over a flat plate with an elliptic leading edge is studied numerically. A finite-difference scheme that is second-order accurate in both time and space is utilized to solve the incompressible Navier-Stokes system in general curvilinear coordinates with the vorticity and stream function as dependent variables. Simulations show, for the conditions considered, a linear variation in the Tollmien-Schlichting response with forcing amplitude for perturbations of freestream velocity that are either symmetrical or asymmetrical with respect to the basic state stagnation streamline.

Buter, Thomas A.↗

NASA NTRS1992

Prediction of asymmetric vortical flows around slender bodies using Navier-Stokes equations

Steady and unsteady asymmetric vortical flows around slender bodies at high angles of attack are solved using the unsteady, compressible, thin-layer Navier-Stokes equations. An implicit, upwind-biased, flux-difference splitting, finite-volume scheme is used for the numerical computations. For supersonic flows past point cones, the locally conical flow assumption have been used for efficient computational studies of this phenomenon. Asymmetric flows past a 5-deg semiapex-angle circular cone at different angles of attack, free-stream Mach numbers, and Reynolds numbers have been studied in responses to different sources of disturbances. The effects of grid fineness and computational domain size have also been investigated. Next, the responses of three-dimensional supersonic asymmetric flow around a 5-deg circular cone at different angles of attack and Reynolds numbers to short-duration sideslip disturbances are presented. The results show that flow asymmetry becomes stronger as the Reynolds number and angles of attack are increased. One of the cases of flow over a cone-cylinder configuration is validated fairly well by experimental data.

Liu, C. H.↗

NASA NTRS1992

Fast neural solution of a nonlinear wave equation

A neural algorithm for rapidly simulating a certain class of nonlinear wave phenomena using analog VLSI neural hardware is presented and applied to the Korteweg-de Vries partial differential equation. The corresponding neural architecture is obtained from a pseudospectral representation of the spatial dependence, along with a leap-frog scheme for the temporal evolution. Numerical simulations demonstrated the robustness of the proposed approach.

Toomarian, Nikzad↗

NASA NTRS1993

Drag reduction in channel flow using nonlinear control

Two nonlinear control schemes have been applied to the problem of drag reduction in channel flow. Both schemes have been tested using numerical simulations at a mass flux Reynolds numbers of 4408, utilizing 2D nonlinear neutral modes for goal dynamics. The OGY-method, which requires feedback, reduces drag to 60-80 percent of the turbulent value at the same Reynolds number, and employs forcing only within a thin region near the wall. The H-method, or model-based control, fails to achieve any drag reduction when starting from a fully turbulent initial condition, but shows potential for suppressing or retarding laminar-to-turbulent transition by imposing instead a transition to a low drag, nonlinear traveling wave solution to the Navier-Stokes equation. The drag in this state corresponds to that achieved by the OGY-method. Model-based control requires no feedback, but in experiments to date has required the forcing be imposed within a thicker layer than the OGY-method. Control energy expenditures in both methods are small, representing less than 0.1 percent of the uncontrolled flow's energy.

Keefe, Laurence R.↗

NASA NTRS1994

A sensitivity equation approach to shape optimization in fluid flows

A sensitivity equation method to shape optimization problems is applied. An algorithm is developed and tested on a problem of designing optimal forebody simulators for a 2D, inviscid supersonic flow. The algorithm uses a BFGS/Trust Region optimization scheme with sensitivities computed by numerically approximating the linear partial differential equations that determine the flow sensitivities. Numerical examples are presented to illustrate the method.

Borggaard, Jeff↗

NASA NTRS1994

A three-dimensional structured/unstructured hybrid Navier-Stokes method for turbine blade rows

A three-dimensional viscous structured/unstructured hybrid scheme has been developed for numerical computation of high Reynolds number turbomachinery flows. The procedure allows an efficient structured solver to be employed in the densely clustered, high aspect-ratio grid around the viscous regions near solid surfaces, while employing an unstructured solver elsewhere in the flow domain to add flexibility in mesh generation. Test results for an inviscid flow over an external transonic wing and a Navier-Stokes flow for an internal annular cascade are presented.

Tsung, F.-L.↗

NASA NTRS1994

Eulerian-Lagrangian Simulations of Transonic Flutter Instabilities

This paper presents an overview of recent applications of Eulerian-Lagrangian computational schemes in simulating transonic flutter instabilities. This approach, the fluid-structure system is treated as a single continuum dynamics problem, by switching from an Eulerian to a Lagrangian formulation at the fluid-structure boundary. This computational approach effectively eliminates the phase integration errors associated with previous methods, where the fluid and structure are integrated sequentially using different schemes. The formulation is based on Hamilton's Principle in mixed coordinates, and both finite volume and finite element discretization schemes are considered. Results from numerical simulations of transonic flutter instabilities are presented for isolated wings, thin panels, and turbomachinery blades. The results suggest that the method is capable of reproducing the energy exchange between the fluid and the structure with significantly less error than existing methods. Localized flutter modes and panel flutter modes involving traveling waves can also be simulated effectively with no a priori knowledge of the type of instability involved.

Bendiksen, Oddvar O.↗

NASA NTRS2001

High Order Difference Method for Low Mach Number Aeroacoustics

A high order finite difference method with improved accuracy and stability properties for computational aeroacoustics (CAA) at low Mach numbers is proposed. The Euler equations are split into a conservative and a symmetric non- conservative portion to allow the derivation of a generalized energy estimate. Since the symmetrization is based on entropy variables, that splitting of the flux derivatives is referred to as entropy splitting. Its discretization by high order central differences was found to need less numerical dissipation than conventional conservative schemes. Owing to the large disparity of acoustic and stagnation quantities in low Mach number aeroacoustics, the split Euler equations are formulated in perturbation form. The unknowns are the small changes of the conservative variables with respect to their large stagnation values. All nonlinearities and the conservation form of the conservative portion of the split flux derivatives can be retained, while cancellation errors are avoided with its discretization opposed to the conventional conservative form. The finite difference method is third-order accurate at the boundary and the conventional central sixth-order accurate stencil in the interior. The difference operator satisfies the summation by parts property analogous to the integration by parts in the continuous energy estimate. Thus, strict stability of the difference method follows automatically. Spurious high frequency oscillations are suppressed by a characteristic-based filter similar to but without limiter. The time derivative is approximated by a 4-stage low-storage second-order explicit Runge-Kutta method. The method has been applied to simulate vortex sound at low Mach numbers. We consider the Kirchhoff vortex, which is an elliptical patch of constant vorticity rotating with constant angular frequency in irrotational flow. The acoustic pressure generated by the Kirchhoff vortex is governed by the 2D Helmholtz equation, which can be solved analytically using separation of variables.

Mueller, B.↗

NASA NTRS1999

An O(Nm(sup 2)) Plane Solver for the Compressible Navier-Stokes Equations

A hierarchical multigrid algorithm for efficient steady solutions to the two-dimensional compressible Navier-Stokes equations is developed and demonstrated. The algorithm applies multigrid in two ways: a Full Approximation Scheme (FAS) for a nonlinear residual equation and a Correction Scheme (CS) for a linearized defect correction implicit equation. Multigrid analyses which include the effect of boundary conditions in one direction are used to estimate the convergence rate of the algorithm for a model convection equation. Three alternating-line- implicit algorithms are compared in terms of efficiency. The analyses indicate that full multigrid efficiency is not attained in the general case; the number of cycles to attain convergence is dependent on the mesh density for high-frequency cross-stream variations. However, the dependence is reasonably small and fast convergence is eventually attained for any given frequency with either the FAS or the CS scheme alone. The paper summarizes numerical computations for which convergence has been attained to within truncation error in a few multigrid cycles for both inviscid and viscous ow simulations on highly stretched meshes.

Thomas, J. L.↗