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NASA NTRS1976

Lagrangian density for collisional plasma

For the purpose of deriving appropriate Lagrangians for plasma equations that include effects of energy loss, the paper examines the inverse problem of the calculus of variations for systems of first- and second-order quasi-linear partial differential equations. This results in convenient forms of the sufficient conditions under which the given differential equations are Euler-Lagrange equations of a Lagrangian. These conditions are then applied to determine the necessary transformation that converts equations, apparently not already in it, into Euler-Lagrange form. The appropriate Lagrangian for a warm collisional plasma is obtained, and the Lagrangian is derived for a resistive transmission line.

Peng, Y.-K. M.↗

NASA NTRS1976

Diffraction of a shock wave by a compression corner. I - Regular reflection

The unsteady, two-dimensional flowfield resulting from the interaction of a moving planar shock wave with a compression corner is determined using a second-order, discontinuity-fitting, finite-difference approach. The time-dependent Euler equations are transformed to normalize the distance between the body and peripheral shock and to include the existing self-similar property of the flow. The resulting set of partial differential equations in conservation-law form is then solved in a time-dependent fashion using MacCormack's scheme. The vortical singularity, which lies on the body surface, and the single reflected shock are both treated as discontinuities in the numerical procedure. The results of the numerical simulation compare quite favorably with existing experimental interferograms and yield better flowfield resolution than previous first-order, shock-capturing, numerical solutions.

Kutler, P.↗

NASA NTRS1986

Preconditioned conjugate residual methods for the solution of spectral equations

Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.

Wong, Y. S.↗

NASA NTRS1993

A least-squares finite element method for 3D incompressible Navier-Stokes equations

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

Jiang, Bo-Nan↗

NASA NTRS1979

Analysis of turbulent free-jet hydrogen-air diffusion flames with finite chemical reaction rates

A numerical analysis is presented of the nonequilibrium flow field resulting from the turbulent mixing and combustion of an axisymmetric hydrogen jet in a supersonic parallel ambient air stream. The effective turbulent transport properties are determined by means of a two-equation model of turbulence. The finite-rate chemistry model considers eight elementary reactions among six chemical species: H, O, H2O, OH, O2 and H2. The governing set of nonlinear partial differential equations was solved by using an implicit finite-difference procedure. Radial distributions were obtained at two downstream locations for some important variables affecting the flow development, such as the turbulent kinetic energy and its dissipation rate. The results show that these variables attain their peak values on the axis of symmetry. The computed distribution of velocity, temperature, and mass fractions of the chemical species gives a complete description of the flow field. The numerical predictions were compared with two sets of experimental data. Good qualitative agreement was obtained.

Sislian, J. P.↗

NASA NTRS1988

Extensions of the CSCM methodology for nonequilibrium reacting gas flows

The paper describes the extension of the CSCM flux difference splitting to produce conservative unconditionally stable implicit upwind methods for the extended equation set. The CSCM-S single data level relaxation algorithm is argued to be particularly advantageous for efficiently solving the set of equations. Attention is given to issues of alternative state variable representations and to complementary thermodynamic relations that parameterize and need to be effectively coupled with the solution procedure for the partial differential equations. Within the framework of iteration, consideration is given to more efficient approximate solution procedures than the full block coupled LU decomposition. Results of early numerical experiments are given to illustrate the developments.

Lombard, C. K.↗

NASA NTRS1983

Planar dynamics of a uniform beam with rigid bodies affixed to the ends

The planar dynamics of a uniform elastic beam subject to a variety of geometric and natural boundary conditions and external excitations were analyzed. The beams are inextensible and capable of small transverse bending deformations only. Classical beam vibration eigenvalue problems for a cantilever with tip mass, a cantilever with tip body and an unconstrained beam with rigid bodies at each are examined. The characteristic equations, eigenfunctions and orthogonality relations for each are derived. The forced vibration of a cantilever with tip body subject to base acceleration is analyzed. The exact solution of the governing nonhomogeneous partial differential equation with time dependent boundary conditions is presented and compared with a Rayleigh-Ritz approximate solution. The arbitrary planar motion of an elastic beam with rigid bodies at the ends is addressed. Equations of motion are derived for two modal expansions of the beam deflection. The motion equations are cast in a first order form suitable for numerical integration. Selected FORTRAN programs are provided.

Storch, J.↗

NASA NTRS1992

Bleed-boundary conditions for numerically simulated mixed-compression supersonic inlet flow

A numerical study was performed to investigate boundary conditions for bleed openings that are needed in the numerical simulation of critical flow through turbojet inlets with bleed. Nine different boundary conditions (BCs) were devised and examined. Of the BCs examined, three were able to stabilize the terminal shock near the inlet's throat. This numerical study was based on the ensemble-averaged, unsteady, three-dimensional, compressible Navier-Stokes equations closed by the Baldwin-Lomax algebraic turbulence model. Solutions to the Navier-Stokes equations were obtained by using a hybrid implicit-explicit method (Beam and Warming/MacCormack). The multiblock grid system used in the solution procedure was generated by solving a set of elliptic partial differential equations. Solutions obtained were compared with available experimental data.

Chyu, W. J.↗

NASA NTRS1977

On Szebehely's equation for the potential of a prescribed family of orbits

In the present note we first give a simple proof of the Dainelli formulas for the force field generating a given family of orbits. We also show that the Szebehely partial differential equation for the potential can be derived from the Dainelli formulas if the energy integral is assumed. The Szebehely equation can be solved directly with the method of characteristics or indirectly with the Joukovsky formulas. Several examples are briefly described in the article. In particular we find some rather general potential functions corresponding to circular motion.

Broucke, R.↗

NASA NTRS1981

Analytical description of charged particle transport along arbitrary guiding field configurations

Aspects of charged particle transport theory have been studied by Earl (1973, 1974, 1976) in a number of investigations. The investigations were conducted under the assumption that particles propagate along a DC guiding field with negligible perpendicular transport and that pitch-angle scattering by random fields can be described by the Fokker-Planck formalism. The present study provides a generalized and simplified treatment of focused transport. The first step is taken with respect to a perturbation analysis by which the Boltzmann equation is solved in terms of successively improved approximations to its exact solution. The first approximation is the sum of isotropic pseudodiffusive and anisotropic supercoherent components whose coefficients must satisfy two coupled partial differential equations. The normalization constant K and the pseudodiffusive velocity, which appear in the first-order transport equations, are given by simple integrals over the pitch angle.

Earl, J. A.↗

NASA NTRS1982

A mathematical model of post-instability in fluid mechanics

Postinstability of fluids is eliminated in numerical models by introducing multivalued velocity fields after discarding the principle of impenetrability. Smooth functions are shown to be incapable of keeping the derivatives from going towards infinity when iterating solutions for the governing equations such as those defined by Navier-Stokes. Enlarging the class of functions is shown to be necessary to eliminate the appearance of imaginary characteristic roots in the systems of arbitrary partial differential equations, a condition which leads to physically impossible motions. The enlarging is demonstrated to be achievable by allowing several individual particles with different velocities to appear at the same point of space, and the subsequent multivaluedness of the solutions is purely a mathematical concern, rather than one of actual physical existence. Applications are provided for an inviscid fluid and for turbulence.

Zak, M. A.↗

NASA NTRS1986

A numerical algorithm for optimal feedback gains in high dimensional LQR problems

A hybrid method for computing the feedback gains in linear quadratic regulator problems is proposed. The method, which combines the use of a Chandrasekhar type system with an iteration of the Newton-Kleinman form with variable acceleration parameter Smith schemes, is formulated so as to efficiently compute directly the feedback gains rather than solutions of an associated Riccati equation. The hybrid method is particularly appropriate when used with large dimensional systems such as those arising in approximating infinite dimensional (distributed parameter) control systems (e.g., those governed by delay-differential and partial differential equations). Computational advantage of the proposed algorithm over the standard eigenvector (Potter, Laub-Schur) based techniques are discussed and numerical evidence of the efficacy of our ideas presented.

Banks, H. T.↗

NASA NTRS1990

Algebraic Generation Of Two-Dimensional Grids

Two-Boundary Grid Generation (TBGG) computer program applies interactive algebraic grid-generation technique in two dimensions. Incorporates algebraic equations that relate uniform rectangular computational domain to arbitrary physical domain. Hermite cubic interpolation performed between fixed, nonintersecting boundaries. Applied to variety of problems in which discrete techniques used to solve such partial differential equations as governing equations of fluid flow. Written in FORTRAN 77.

Smith, Robert E.↗

NASA NTRS1996

From Individuals to Epidemics

Heterogeneous mixing fundamentally changes the dynamics of infectious diseases; finding ways to incorporate it into models represents a critical challenge. Phenomenological approaches are deficient in their lack of attention to underlying processes; individual-based models, on the other hand, may obscure the essential interactions in a sea of detail. The challenge then is to find ways to bridge these levels of description, starting from individual-based models and deriving macroscopic descriptions from them that retain essential detail, and filter out the rest. In this paper, attempts to achieve this transformation are described for a class of models where non-random mixing arises from the spatial localization of interactions. In general, the epidemic threshold is found to be larger owing to spatial localization than for a homogeneous mixing population. An improved estimate of the dynamics is developed by the use of moment equations, and a simple estimate of the threshold in terms of a 'dyad heuristic'. For more general models in which local infection is not described by mass action, the connection with related partial differential equations is investigated.

Levin, Simon A.↗

NASA NTRS1990

Quadrature imposition of compatibility conditions in Chebyshev methods

Often, in solving an elliptic equation with Neumann boundary conditions, a compatibility condition has to be imposed for well-posedness. This condition involves integrals of the forcing function. When pseudospectral Chebyshev methods are used to discretize the partial differential equation, these integrals have to be approximated by an appropriate quadrature formula. The Gauss-Chebyshev (or any variant of it, like the Gauss-Lobatto) formula can not be used here since the integrals under consideration do not include the weight function. A natural candidate to be used in approximating the integrals is the Clenshaw-Curtis formula, however it is shown that this is the wrong choice and it may lead to divergence if time dependent methods are used to march the solution to steady state. The correct quadrature formula is developed for these problems. This formula takes into account the degree of the polynomials involved. It is shown that this formula leads to a well conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.

Gottlieb, D.↗

NASA NTRS1991

A numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems

A hybrid method for computing the feedback gains in linear quadratic regulator problem is proposed. The method, which combines use of a Chandrasekhar type system with an iteration of the Newton-Kleinman form with variable acceleration parameter Smith schemes, is formulated to efficiently compute directly the feedback gains rather than solutions of an associated Riccati equation. The hybrid method is particularly appropriate when used with large dimensional systems such as those arising in approximating infinite-dimensional (distributed parameter) control systems (e.g., those governed by delay-differential and partial differential equations). Computational advantages of the proposed algorithm over the standard eigenvector (Potter, Laub-Schur) based techniques are discussed, and numerical evidence of the efficacy of these ideas is presented.

Banks, H. T.↗

NASA NTRS1993

Approximate optimal guidance for the advanced launch system

A real-time guidance scheme for the problem of maximizing the payload into orbit subject to the equations of motion for a rocket over a spherical, non-rotating earth is presented. An approximate optimal launch guidance law is developed based upon an asymptotic expansion of the Hamilton - Jacobi - Bellman or dynamic programming equation. The expansion is performed in terms of a small parameter, which is used to separate the dynamics of the problem into primary and perturbation dynamics. For the zeroth-order problem the small parameter is set to zero and a closed-form solution to the zeroth-order expansion term of Hamilton - Jacobi - Bellman equation is obtained. Higher-order terms of the expansion include the effects of the neglected perturbation dynamics. These higher-order terms are determined from the solution of first-order linear partial differential equations requiring only the evaluation of quadratures. This technique is preferred as a real-time, on-line guidance scheme to alternative numerical iterative optimization schemes because of the unreliable convergence properties of these iterative guidance schemes and because the quadratures needed for the approximate optimal guidance law can be performed rapidly and by parallel processing. Even if the approximate solution is not nearly optimal, when using this technique the zeroth-order solution always provides a path which satisfies the terminal constraints. Results for two-degree-of-freedom simulations are presented for the simplified problem of flight in the equatorial plane and compared to the guidance scheme generated by the shooting method which is an iterative second-order technique.

Feeley, T. S.↗

NASA NTRS1995

Analysis of gas absorption to a thin liquid film in the presence of a zero-order chemical reaction

The paper presents a detailed theoretical analysis of the process of gas absorption to a thin liquid film adjacent to a horizontal rotating disk. The film is formed by the impingement of a controlled liquid jet at the center of the disk and subsequent radial spreading of liquid along the disk. The chemical reaction between the gas and the liquid film can be expressed as a zero-order homogeneous reaction. The process was modeled by establishing equations for the conservation of mass, momentum, and species concentration and solving them analytically. A scaling analysis was used to determine dominant transport processes. Appropriate boundary conditions were used to solve these equations to develop expressions for the local concentration of gas across the thickness of the film and distributions of film height, bulk concentration, and Sherwood number along the radius of the disk. The partial differential equation for species concentration was solved using the separation of variables technique along with the Duhamel's theorem and the final analytical solution was expressed using confluent hypergeometric functions. Tables for eigenvalues and eigenfunctions are presented for a number of reaction rate constants. A parametric study was performed using Reynolds number, Ekman number, and dimensionless reaction rate as parameters. At all radial locations, Sherwood number increased with Reynolds number (flow rate) as well as Ekman number (rate of rotation). The enhancement of mass transfer due to chemical reaction was found to be small when compared to the case of no reaction (pure absorption), but the enhancement factor was very significant when compared to pure absorption in a stagnant liquid film. The zero-order reaction processes considered in the present investigation included the absorption of oxygen in aqueous alkaline solutions of sodiumdithionite and rhodium complex catalyzed carbonylation of methanol. Present analytical results were compared to previous theoretical results for limiting conditions, and were found to have very good agreement.

Rajagopalan, S.↗