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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 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS1984

Spectral methods for partial differential equations

Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid-dynamical applications are emphasized.

Hussaini, M. Y.↗
NASA NTRS1972

Vortex equations: Singularities, numerical solution, and axisymmetric vortex breakdown

A method of weighted residuals for the computation of rotationally symmetric quasi-cylindrical viscous incompressible vortex flow is presented and used to compute a wide variety of vortex flows. The method approximates the axial velocity and circulation profiles by series of exponentials having (N + 1) and N free parameters, respectively. Formal integration results in a set of (2N + 1) ordinary differential equations for the free parameters. The governing equations are shown to have an infinite number of discrete singularities corresponding to critical values of the swirl parameters. The computations point to the controlling influence of the inner core flow on vortex behavior. They also confirm the existence of two particular critical swirl parameter values: one separates vortex flow which decays smoothly from vortex flow which eventually breaks down, and the second is the first singularity of the quasi-cylindrical system, at which point physical vortex breakdown is thought to occur.

Bossel, H. H.↗
NASA NTRS1973

Swirling flows in streamtubes of variable cross section.

The behavior of vortex flow at high swirls is investigated. The problem was first formulated in the method of weighted residuals, using Bessel functions as approximating functions. Convergent solutions require a large number of steps in the axial direction and do not seem to offer an advantage over traditional methods. As the results do not differ substantially from those of the zeroth approximation (the solution for 'cylindrical' flow), this approximation is then explored and discussed in more detail. It is used to derive a relationship predicting a critical radius for which stagnation on the axis (vortex breakdown) can be expected for a given swirl. Flow patterns with open and closed vortex bubbles are explored. At high swirls, even minute variations in outer streamtube radius have profound effects on the flow. Vortex breakdown bubbles in divergences, tubes, and free flows can be explained.

Bossel, H. H.↗
NASA NTRS1978

Application of finite element techniques in predicting the acoustic properties of turbofan inlets

An analytical technique was developed for predicting the acoustic performance of turbofan inlets carrying a subsonic axisymmetric steady flow. The finite element method combined with the method of weighted residuals is used in predicting the acoustic properties of variable area, annular ducts with or without acoustic treatments along their walls. An approximate solution for the steady inviscid flow field is obtained using an integral method for calculating the incompressible potential flow field in the inlet with a correction to account for compressibility effects. The accuracy of the finite element technique was assessed by comparison with available analytical solutions for the problems of plane and spinning wave propagation through a hard walled annular cylinder with a constant mean flow.

Majjigi, R. K.↗
NASA NTRS1983

A lifting surface theory in rotational flow

The partial differential equation for small disturbance steady rotational flow in three dimensions is solved through an integral equation approach. The solution is obtained by using the method of weighted residuals. Specific applications are directed to wings in nonuniform subsonic parallel streams with velocity varying in vertical and spanwise directions and to airfoils in nonuniform freestream. Comparison with limited known results indicates that the present method is reasonably accurate. Numerical results for the lifting pressure of airfoil, lift, induced drag, and pitching moments of airfoil, lift, induced drag, and pitching moments of elliptic, rectangular, and delta wings in a jet, wake, or monotonic sheared stream are presented. It is shown that, in addition to the effect of local dynamic pressures, a positive velocity gradient tends to enhance the lift.

Shiau, M. J.↗
NASA NTRS1972

Adjoint variational methods in nonconservative stability problems.

A general nonself-adjoint eigenvalue problem is examined and it is shown that the commonly employed approximate methods, such as the Galerkin procedure, the method of weighted residuals and the least square technique lack variational descriptions. When used in their previously known forms they do not yield stationary eigenvalues and eigenfunctions. With the help of an adjoint system, however, several analogous variational descriptions may be developed and it is shown in the present study that by properly restating the method of least squares, stationary eigenvalues may be obtained. Several properties of the adjoint eigenvalue problem, known only for a restricted group, are shown to exist for the more general class selected for study.

Prasad, S. N.↗
DOE OSTI2025

A novel ignition model for low velocity impact of heterogeneous explosives based on interacting hot spots

While numerous studies have focused on the ignition of explosives occurring in high velocity impact and the associated shock-to-detonation transition, there has been growing interest in developing computational models focused on low-velocity impact regimes. A predictive low-velocity impact ignition model will be important for analyzing high explosive safety and potential accident scenarios. This work introduces a novel ignition model based on the concept of thermally interacting hot spots to simulate low velocity impacted heterogeneous explosives where observed ignition times are on the order of milliseconds. The model asserts that relevant hot spots are micron-sized, the typical separation between neighboring hot spots is on the order of a hundred microns, and that neighbors interact thermally through heat conduction across the interstitial region between them. To achieve tractable numerical solutions, hot spots are assumed to form a periodic array as opposed to the highly irregular positioning in an actual explosive. This idealization allows a single two hotspot system to characterize the ignition process. Consequently, the model is referred to as the two hot spot Frank-Kamenetskii ignition model. In the present study, hot spots are modeled as constant heat sources terms, but this can be extended to include grain-scale phenomena like frictional heating of micron-sized growing cracks that are confined under high pressure. Because the micron-sized features are below the scale that can be efficiently resolved at a systems level, an efficient subscale scheme based on the Method of Weighted Residuals (MWR) is used to efficiently solve the equations. In conclusion, we carry out numerical examples and analytic predictions illustrating the accuracy and the functioning of the model.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1973

The prediction of nonlinear three dimensional combustion instability in liquid rockets with conventional nozzles

An analytical technique is developed to solve nonlinear three-dimensional, transverse and axial combustion instability problems associated with liquid-propellant rocket motors. The Method of Weighted Residuals is used to determine the nonlinear stability characteristics of a cylindrical combustor with uniform injection of propellants at one end and a conventional DeLaval nozzle at the other end. Crocco's pressure sensitive time-lag model is used to describe the unsteady combustion process. The developed model predicts the transient behavior and nonlinear wave shapes as well as limit-cycle amplitudes and frequencies typical of unstable motor operation. The limit-cycle amplitude increases with increasing sensitivity of the combustion process to pressure oscillations. For transverse instabilities, calculated pressure waveforms exhibit sharp peaks and shallow minima, and the frequency of oscillation is within a few percent of the pure acoustic mode frequency. For axial instabilities, the theory predicts a steep-fronted wave moving back and forth along the combustor.

Powell, E. A.↗
NASA NTRS1974

Finite element solution algorithm for incompressible fluid dynamics

A finite element solution algorithm is established for the two-dimensional Navier-Stokes equations governing the transient motion of a viscous incompressible fluid, i.e., hydrodynamics. Dependent variable transformation renders the differential equation description uniformly elliptic. The finite element algorithm is established using the Galerkin criterion on a local basis within the Method of Weighted Residuals. It is unconstrained with respect to system linearity, computational mesh uniformity or solution domain closure regularity. The finite element matrices are established using a linear 'natural coordinate function' description. Computational solutions using the COMOC computer program illustrate the various features of the algorithm including recirculating flows.

Baker, A. J.↗
NASA NTRS1976

A wave-envelope technique for wave propagation in non-uniform ducts

The method of variation of parameters is used to analyze wave propagation in variable area, plane ducts with no mean flow. The method has an explicit representation of the fast axial variation of the acoustic modes, and numerical integration is required only for the slower axial variations of the mode amplitudes and phases. Results are presented which demonstrate the numerical advantages of the method. Comparison of the results with those of a small perturbation theory are given. The relationship between this method and the method of weighted residuals is discussed.

Kaiser, J. E.↗
NASA NTRS1978

Computational flow development for unsteady viscous flows: Foundation of the numerical method

A procedure is presented for effective consideration of viscous effects in computational development of high Reynolds number flows. The procedure is based on the interpretation of the Navier-Stokes equations as vorticity transport equations. The physics of the flow was represented in a form suitable for numerical analysis. Lighthill's concept for flow development for computational purposes was adapted. The vorticity transport equations were cast in a form convenient for computation. A statement for these equations was written using the method of weighted residuals and applying the Galerkin criterion. An integral representation of the induced velocity was applied on the basis of the Biot-Savart law. Distribution of new vorticity, produced at wing surfaces over small computational time intervals, was assumed to be confined to a thin region around the wing surfaces.

Bratanow, T.↗
NASA NTRS1977

A finite element analysis for acoustic transmission in nonuniform ducts

A multimodal finite-element analysis for two-dimensional and axisymmetric ducts with flow has been adopted to predict acoustic transmission in nonuniform ducts carrying a nonuniform flow (i.e., turbofan aircraft engine conditions). The analysis is based on a Galerkin method with a finite-element discretization, and multimodal matching at the duct ends. The element mesh and the number of modes required at each end are found to have a strong influence on the analysis. The numerical results compare well with results obtained from other techniques, including the method of weighted residuals.

Astley, R. J.↗
NASA NTRS1976

Calculation methods for compressible turbulent boundary layers

Calculation procedures for non-reacting compressible two- and three-dimensional turbulent boundary layers were reviewed. Integral, transformation and correlation methods, as well as finite difference solutions of the complete boundary layer equations summarized. Alternative numerical solution procedures were examined, and both mean field and mean turbulence field closure models were considered. Physics and related calculation problems peculiar to compressible turbulent boundary layers are described. A catalog of available solution procedures of the finite difference, finite element, and method of weighted residuals genre is included. Influence of compressibility, low Reynolds number, wall blowing, and pressure gradient upon mean field closure constants are reported.

Bushnell, D. M.↗
NASA NTRS1979

A split finite element algorithm for the compressible Navier-Stokes equations

An accurate and efficient numerical solution algorithm is established for solution of the high Reynolds number limit of the Navier-Stokes equations governing the multidimensional flow of a compressible essentially inviscid fluid. Finite element interpolation theory is used within a dissipative formulation established using Galerkin criteria within the Method of Weighted Residuals. An implicit iterative solution algorithm is developed, employing tensor product bases within a fractional steps integration procedure, that significantly enhances solution economy concurrent with sharply reduced computer hardware demands. The algorithm is evaluated for resolution of steep field gradients and coarse grid accuracy using both linear and quadratic tensor product interpolation bases. Numerical solutions for linear and nonlinear, one, two and three dimensional examples confirm and extend the linearized theoretical analyses, and results are compared to competitive finite difference derived algorithms.

Baker, A. J.↗
NASA NTRS1989

Axisymmetric shapes and stability of charged drops in an external electric field

A highly conducting charged drop that is surrounded by a fluid insulator of another density can be levitated by suitably applying a uniform electric field. Axisymmetric equilibrium shapes and stability of the levitated drop are found by solving simultaneously the augmented Young-Laplace equation for surface shape and the Laplace equation for the elecric field, together with constraints of fixed drop volume, charge, and center of mass. The means are a method of subdomains, finite element basis functions, and Galerkin's method of weighted residuals, all facilitated by a large-scale computer. Shape families of fixed charge are treated systematically by first-order continuation. Previous analyses by Abbas et al. in 1967 and Abbas and Latham in 1969, in which the shapes of levitated drops are approximated as spheroids, are corrected. The new analysis shows that drops charged to less than the Rayleigh limit lose shape stability at turning points, with respect to external field strength, and that the instability seen in experiments of Doyle et al. in 1964 and others is not a bifurcation to a family of two-lobed shapes, but rather is a related imperfect bifurcation.

Basaran, O. A.↗
NASA NTRS1989

Prediction of the thermal environment and thermal response of simple panels exposed to radiant heat

A method of predicting the radiant heat flux distribution produced by a bank of tubular quartz heaters was applied to a radiant system consisting of a single unreflected lamp irradiating a flat metallic incident surface. In this manner, the method was experimentally verified for various radiant system parameter settings and used as a source of input for a finite element thermal analysis. Two finite element thermal analyses were applied to a thermal system consisting of a thin metallic panel exposed to radiant surface heating. A two-dimensional steady-state finite element thermal analysis algorithm, based on Galerkin's Method of Weighted Residuals (GFE), was formulated specifically for this problem and was used in comparison to the thermal analyzers of the Engineering Analysis Language (EAL). Both analyses allow conduction, convection, and radiation boundary conditions. Differences in the respective finite element formulation are discussed in terms of their accuracy and resulting comparison discrepancies. The thermal analyses are shown to perform well for the comparisons presented here with some important precautions about the various boundary condition models. A description of the experiment, corresponding analytical modeling, and resulting comparisons are presented.

Turner, Travis L.↗
NASA NTRS2009

Ablative Thermal Response Analysis Using the Finite Element Method

A review of the classic techniques used to solve ablative thermal response problems is presented. The advantages and disadvantages of both the finite element and finite difference methods are described. As a first step in developing a three dimensional finite element based ablative thermal response capability, a one dimensional computer tool has been developed. The finite element method is used to discretize the governing differential equations and Galerkin's method of weighted residuals is used to derive the element equations. A code to code comparison between the current 1-D tool and the 1-D Fully Implicit Ablation and Thermal Response Program (FIAT) has been performed.

Dec John A.↗
NASA NTRS2013

Microscopic Simulation and Macroscopic Modeling for Thermal and Chemical Non-Equilibrium

This paper deals with the accurate microscopic simulation and macroscopic modeling of extreme non-equilibrium phenomena, such as encountered during hypersonic entry into a planetary atmosphere. The state-to-state microscopic equations involving internal excitation, de-excitation, dissociation, and recombination of nitrogen molecules due to collisions with nitrogen atoms are solved time-accurately. Strategies to increase the numerical efficiency are discussed. The problem is then modeled using a few macroscopic variables. The model is based on reconstructions of the state distribution function using the maximum entropy principle. The internal energy space is subdivided into multiple groups in order to better describe the non-equilibrium gases. The method of weighted residuals is applied to the microscopic equations to obtain macroscopic moment equations and rate coefficients. The modeling is completely physics-based, and its accuracy depends only on the assumed expression of the state distribution function and the number of groups used. The model makes no assumption at the microscopic level, and all possible collisional and radiative processes are allowed. The model is applicable to both atoms and molecules and their ions. Several limiting cases are presented to show that the model recovers the classical twotemperature models if all states are in one group and the model reduces to the microscopic equations if each group contains only one state. Numerical examples and model validations are carried out for both the uniform and linear distributions. Results show that the original over nine thousand microscopic equations can be reduced to 2 macroscopic equations using 1 to 5 groups with excellent agreement. The computer time is decreased from 18 hours to less than 1 second.

Modeling↗