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Results for “parabolic problems”

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.

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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
NASA NTRS1985

Implicit/explicit analysis of interactive phenomena in: supersonic, chemically-reacting, mixing and boundary layer problems

An efficient implicit/explicit procedure for solving the parabolized Navier-Stokes (PNS) equations to analyze supersonic, viscous, chemically-reacting flows is presented. The approach combines a fully-implicit algorithm for solving the parabolic mixing/boundary layer equations with an explicit algorithm for determining the hyperbolic pressure field. The unified model, SPLITP, contains provisions for treating bounding/embedded subsonic regions using a pressure-splitting procedure which suppresses elliptic (upstream influence) effects without recourse to sublayer approximations. Several two-equation turbulence models with compressibility and curvature-correction terms are incorporated into SPLITP. Applications to a variety of flow problems are presented which include: ducted supersonic mixing and combustion flowfields, free jets and plumes, and, curved boundary layers and wall jets.

Dash, S. M.↗
NASA NTRS1988

Reliable use of determinants to solve nonlinear structural eigenvalue problems efficiently

The analytical derivation, numerical implementation, and performance of a multiple-determinant parabolic interpolation method (MDPIM) for use in solving transcendental eigenvalue (critical buckling or undamped free vibration) problems in structural mechanics are presented. The overall bounding, eigenvalue-separation, qualified parabolic interpolation, accuracy-confirmation, and convergence-recovery stages of the MDPIM are described in detail, and the numbers of iterations required to solve sample plane-frame problems using the MDPIM are compared with those for a conventional bisection method and for the Newtonian method of Simpson (1984) in extensive tables. The MDPIM is shown to use 31 percent less computation time than bisection when accuracy of 0.0001 is required, but 62 percent less when accuracy of 10 to the -8th is required; the time savings over the Newtonian method are about 10 percent.

Williams, F. W.↗
NASA NTRS1992

Analytic solution of the thickness problem of a rectangular wing in steady subsonic flow

An analytic solution of the thickness problem of a rectangular wing with parabolic airfoil section in three-dimensional flow is presented. The free-air solution is obtained by integrating the equation of the axial perturbation velocity. The Prandtl-Glauert rule can be used to derive the subsonic solution. Parts of the free-air solution are verified by taking the limit of the axial peturbation velocity on the model surface as the wing span goes to infinity. Pressure coefficients are studied for a selected wing geometry in the flow field. The solution of the thickness problem of a rectangular wing in a rectangular wind tunnel is derived using the free-air solution and the method of images.

Ulbrich, N.↗
NASA NTRS1985

Kalman-like estimation for static distributed systems Antenna shape from radiation measurements

This paper advances an approach to the determination of shape of static distributed systems. It also illustrates the application of the approach to the problems of surface diagnosis of large parabolic reflectors. The estimation methods developed combine in an optimal sense the information from an elliptic model of the structure and from measurements of the structural deflection and of the far-field pattern changes due to the structural deformation. The estimators have a predictor-corrector structure, quite similar to that of a Kalman filter. The system model is first used to obtain a predicted estimate. A correction term is then added to the prediction to obtain the final state estimate. The relative weighting between prediction and correction terms is determined by an estimator gain. As in a Kalman filter, the estimator gain can be expressed in terms of the state estimation error covariance.

Rodriguez, G.↗
DOE OSTI2018

Classical Solvability of the Radial Viscous Fingering Problem in a Hele–Shaw Cell with Surface Tension

We consider the one-phase problem on radial viscous fingering structures in a Hele–Shaw cell with surface tension. This problem is a nonlinear free-boundary problem for elliptic equations. Unlike the Stefan problem for the heat equation, we deal with a problem of hydrodynamic type. We establish the classical solvability of the one-phase Hele–Shaw problem with radial geometry by using the same method as that used for the Stefan problem and justifying the vanishing coefficient of the time-derivative in the parabolic equation.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1988

A comparative study of theoretical methods on Goertler instability

Goertler vortices arise in boundary layers along concave surfaces due to centrifugal effects and these vortices in combination with Tollmien-Schlichting (TS) waves and crossflow vortices may play an important role in triggering early transition. There have been two distinct theoretical approaches to solve the problem. In the classical, normal-mode approach (NMA) the linear development of Goertler vortices is reduced to an eigenvalue problem. In the other method, initiated by Hall, the governing partial differential equations for the perturbations which are parabolic in the streamwise direction, are solved as an initial value problem. This method predicts multiple neutral curves depending on how and where the mean flow is perturbed, unlike the NMA where a unique but different neutral curve is predicted by each investigator. The present paper attempts to compare the two techniques and shows that mutually compatible results can be obtained when physically consistent assumptions are made.

Kalburgi, Vijay↗
DOE OSTI2018

Some Problems of Distributed and Boundary Control for Systems with Integral Aftereffect

We consider the problem of exact control for a system described by an equation with integral “memory.” It is shown that, under certain conditions, this system can be brought to rest in finite time by distributed control bounded in absolute value, and, in a special one-dimensional case, by control applied to an end-point of the interval. We consider different types of kernels in the integral term of the equation and describe some relationships between problems of controllability of some hyperbolic and parabolic systems.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1979

On the efficient computation of Fraunhofer and Fresnel region fields radiated by reflector and planar-aperture antennas

Computation of fields radiated by parabolic reflectors and planar apertures is a problem of prime importance, and conventional approaches to performing these computations are often very time-consuming. The paper demonstrates the usefulness of a series expansion technique for efficient computation of radiated fields in both the Fraunhofer and Fresnel regions. The series approach is based on the use of the Jacobi polynomials. The application of the series method to determining the far-field pattern of the antenna from the near-field measurements for an arbitrary aperture antenna is also discussed.

Mittra, R.↗
NASA NTRS1986

Multi-dimensional initial-boundary value problems with strong nonlinearities

Initial boundary value problems for the general scalar singularly perturbed parabolic equation in a cylindrical domain are studied. The one-dimensional reaction-diffusion equation is considered in the absence of convection. Then, the general one-dimensional and N-dimensional reaction-convection-diffusion equations are analyzed.

Howes, F. A.↗
NASA NTRS1990

Wave propagation through random media: A local method of small perturbations based on the Helmholtz equation

Propagation of sound through the turbulent atmosphere is a statistical problem. The randomness of the refractive index field causes sound pressure fluctuations. Although no general theory to predict sound pressure statistics from given refractive index statistics exists, there are several approximate solutions to the problem. The most common approximation is the parabolic equation method. Results obtained by this method are restricted to small refractive index fluctuations and to small wave lengths. While the first condition is generally met in the atmosphere, it is desirable to overcome the second. A generalization of the parabolic equation method with respect to the small wave length restriction is presented.

Grosse, Ralf↗
NASA NTRS1981

Secondary concentrators for parabolic dish solar thermal power systems

A variety of different concepts are currently being studied with the objective to lower the cost of parabolic mirrors and to provide alternatives. One of the considered approaches involves the use of compound concentrators. A compound solar concentrator is a concentrator in which the sunlight is reflected or refracted more than once. It consists of a primary mirror or lens, whose aperture determines the amount of sunlight gathered, and a smaller secondary mirror or lens. Additional small optical elements may also be incorporated. The possibilities and problems regarding a use of compound concentrators in parabolic dish systems are discussed. Attention is given to concentrating secondary lenses, secondary imaging and concentrating mirrors, conical secondary mirrors, compound elliptic secondary concentrating mirrors, and hyperbolic trumpet secondary concentrating mirrors.

Jaffe, L. D.↗
DOE OSTI2021

Exponential Time Differencing Schemes for Fuel Depletion and Transport in Molten Salt Reactors: Theory and Implementation

A numerical framework for modeling depletion and mass transport in liquid-fueled molten salt reactions is presented based on exponential time differencing. The solution method involves using the finite volume method to transform the system of partial differential equations (PDEs) into a much larger system of ordinary differential equations. The key part of this method involves solving for the exponential of a matrix. We explore six different algorithms to compute the exponential in a series of progression problems that explore physical transport phenomena in molten salt reactors. This framework shows good results for solving linear parabolic PDEs with each of the six matrix exponential algorithms. For large problems, the series solvers such as Padé and Taylor have large run times, which can be mitigated by using the Krylov subspace.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2018

Singularly Perturbed System of Parabolic Equations in the Critical Case

We examine a system of singularly perturbed parabolic equations in the case where the small parameter is involved as a coefficient of both time and spatial derivatives and the spectrum of the limit operator has a multiple zero point. In such problems, corner boundary layers appear, which can be described by products of exponential and parabolic boundary-layer functions. Under the assumption that the limit operator is a simple-structure operator, we construct a regularized asymptotics of a solution, which, in addition to corner boundary-layer functions, contains exponential and parabolic boudary-layer functions. The construction of the asymptotics is based on the regularization method for singularly perturbed problems developed by S. A. Lomov and adapted to singularly perturbed parabolic equations with two viscous boundaries by A. S. Omuraliev.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1983

Spectral methods for time dependent partial differential equations

The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.

Gottlieb, D.↗
NASA NTRS1979

A fast semi-implicit algorithm for problems of mixed type

Certain physical processes are modeled by partial differential equations which are parabolic over part of the domain and elliptic over the remainder. A family of semi-implicit algorithms which are well suited to initial-boundary value problems of this mixed type is discussed. One important feature of these algorithms is the use of an approximate inverse for the solution of the implicit linear system. A strong error analysis results in an estimate of the total error as a function of approximate inverse error e and time step h.

Frederickson, P. O.↗