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Results for “Functional equations”

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 307 records · Page 17
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

The stability functions in the bulk similarity formulation for the unstable boundary layer

An update is presented for the function C in the heat transfer equation and for the function Bw in the momentum transfer equation of the bulk similarity approach for the atmospheric boundary layer (ABL). Motives for this update are recent developments in the formulation of Monin-Obukhov functions for the surface layer, and the availability of the new data set of FIFE-89, the 1989 phase of the First ISLSCP Field Experiment, which took place over the same hilly prairie terrain in northeastern Kansas as the 1987 phase, i.e., FIFE-87. Functional forms developed in earlier studies are considered. In addition, a new form is derived based on a simple dual structure of the ABL. The functions are calibrated with the data set obtained during FIFE-87; the results are then verified with the independent data set acquired during FIFE-89.

Sugita, Michiaki↗
NASA NTRS1976

Construction of analytically solvable models for interacting species

The basic form of a model representation for systems of n interacting biological species is a set of essentially nonlinear autonomous ordinary differential equations. A generic canonical expression for the rate functions in the equations is reported which permits the analytical general solution to be obtained by elementary computation. It is shown that a general analytical solution is directly obtainable for models where the rate functions are prescribed by the generic canonical expression from the outset. Some illustrative examples are given which demonstrate that the generic canonical expression can be used to construct analytically solvable models for two interacting species with limit-cycle dynamics as well as for a three-species interdependence.

Rosen, G.↗
NASA NTRS1980

Collisional redistribution of radiation. II - The effects of degeneracy on the equations of motion for the density matrix. III - The equation of motion for the correlation function and the scattered spectrum

The effect of correlations between an absorber atom and perturbers in the binary-collision approximation are applied to degenerate atomic systems. A generalized absorption profile which specifies the final state of the atom after an absorption event is related to the total intensities of Rayleigh scattering and fluorescence from the atom. It is suggested that additional dynamical information to that obtainable from ordinary absorption experiments is required in order to describe redistributed atomic radiation. The scattering of monochromatic radiation by a degenerate atom is computed in a binary-collision approximation; an equation of motion is derived for the correlation function which is valid outside the quantum-regression regime. Solutions are given for the weak-field conditions in terms of generalized absorption and emission profiles that depend on the indices of the atomic multipoles.

Burnett, K.↗
NASA NTRS1977

Hot corrosion resistance of nickel-chromium-aluminum alloys

The hot corrosion resistance of nickel-chromium-aluminum alloys was examined by cyclically oxidizing sodium sulfate-coated specimens in still air at 900, 1000, and 1100 C. The compositions tested were within the ternary region: Ni, Ni-50 at.% Cr, and Ni-50 at.% Al. At each temperature the corrosion data were statistically fitted to a third order regression equation as a function of chromium and aluminum contents. From these equations corrosion isopleths were prepared. Compositional regions with the best hot corrosion resistance were identified.

Santoro, G. J.↗
NASA NTRS1979

A simplified computer program for the prediction of the linear stability behavior of liquid propellant combustors

A program for predicting the linear stability of liquid propellant rocket engines is presented. The underlying model assumptions and analytical steps necessary for understanding the program and its input and output are also given. The rocket engine is modeled as a right circular cylinder with an injector with a concentrated combustion zone, a nozzle, finite mean flow, and an acoustic admittance, or the sensitive time lag theory. The resulting partial differential equations are combined into two governing integral equations by the use of the Green's function method. These equations are solved using a successive approximation technique for the small amplitude (linear) case. The computational method used as well as the various user options available are discussed. Finally, a flow diagram, sample input and output for a typical application and a complete program listing for program MODULE are presented.

Mitchell, C. E.↗
NASA NTRS1984

The influence of a wind tunnel on helicopter rotational noise: Formulation of analysis

An analytical model is discussed that can be used to examine the effects of wind tunnel walls on helicopter rotational noise. A complete physical model of an acoustic source in a wind tunnel is described and a simplified version is then developed. This simplified model retains the important physical processes involved, yet it is more amenable to analysis. The simplified physical model is then modeled as a mathematical problem. An inhomogeneous partial differential equation with mixed boundary conditions is set up and then transformed into an integral equation. Details of generating a suitable Green's function and integral equation are included and the equation is discussed and also given for a two-dimensional case.

Mosher, M.↗
NASA NTRS1988

Naturally occurring continued fractions in the variation of Kepler's equation

A family of functions involving integrals of universal functions is introduced. These functions have some interesting mathematical properties including the fact that they may be expressed as Gaussian continued fractions. A unique method of performing the integration is demonstrated which indicates why these functions may be important in the variation of Kepler's equation.

Shepperd, Stanley W.↗
NASA NTRS1972

Relation of the diffuse reflectance remission function to the fundamental optical parameters.

The Kubelka-Munk equations describing the diffuse reflectance of a powdered sample were compared to equations obtained using a uniformly-sized rough-surfaced spherical particle model. The comparison resulted in equations relating the remission function and the Kubelka-Munk constants to the index of refraction, the absorption coefficient, and the average particle diameter of a powdered sample. Published experimental results were used to test the equation relating to the remission function to the fundamental optical parameters.

Simmons, E. L.↗
DOE OSTI2024

Nonlinear dynamics of multiple Alfvén modes driven by trapped energetic ions in tokamaks: A triplet paradigm

Non-linear dynamics of multiple infernal Alfvén eigenmodes—a subset of global Alfvén eigenmodes in tokamak plasmas with extended low-shear central core [Marchenko et al., Phys. Plasmas 16, 092502 (2009)]—is studied. The analysis is carried out for a mode triplet with toroidal mode-numbers n=1, 2, 3. It was assumed that the n = 1 mode was linearly unstable due to precession resonance with trapped fast ions, whereas the other modes were linearly damped. The modes were coupled due to a non-linearity in a bounce-averaged drift kinetic equation for the distribution function of fast ions. Nonlinear equations for the mode amplitudes and phases are derived and solved numerically. It is found that the temporal evolution of the amplitudes and the phase (responsible for the frequency chirping) of the modes exhibit Hopf bifurcations to stable limit cycles. This can explain a synchronous cyclic destabilization of multiple modes in Alfvén avalanches (sudden growth of amplitudes of the mode cluster with different n and approximately equal frequency spacing) in NSTX and bursting modes in MAST—events, which resulted in enhanced loss of fast ions.

Marchenko, V. S. (ORCID:0000000261480533)↗
NASA NTRS1983

Postbuckling of long orthotropic plates in combined shear and compression

The nonlinear large-deflection partial differential equations of von karman for orthotropic plates loaded in combined shear and compression are converted into a set of first-order nonlinear ordinary differential equations by assuming trigonometric functions in one direction. These equations are solved numerically using a two point boundary problem solver which makes use of Newton's method. Results are obtained which determine the postbuckling behavior of rectangular plates with loading up to about three times the buckling load. Both isotropic and orthotropic composite plates are considered. Results show that orthotropic plates may behave quite differently than isotropic plates and that in-plane boundary conditions are important for plates loaded in shear.

Stein, M.↗
NASA NTRS1985

Stable boundary conditions and difference schemes for Navier-Stokes equations

The Navier-Stokes equations can be viewed as an incompletely elliptic perturbation of the Euler equations. By using the entropy function for the Euler equations as a measure of energy for the Navier-Stokes equations, it was possible to obtain nonlinear energy estimates for the mixed initial boundary value problem. These estimates are used to derive boundary conditions which guarantee L2 boundedness even when the Reynolds number tends to infinity. Finally, a new difference scheme for modelling the Navier-Stokes equations in multidimensions for which it is possible to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation was proposed.

Dutt, P.↗
NASA NTRS1948

The Problem of Torsion in Prismatic Members of Circular Segmental Cross Section

The problem is solved by approximation, by setting up a function complying with the differential equation of the stress function, and determining the coefficients appearing in it in such a way that the boundary condition is fulfilled as nearly as possible. For the semicircle, for which the solution is known, the method yields very accurate values; the approximated stress distribution is in good agreement with the accurately computed distribution. Stress and strain measurements indicate that the approximate solution is in sufficiently exact agreement with reality for segmental cross sections.

Weigand, A.↗
NASA NTRS1975

Solutions to Kuessner's integral equation in unsteady flow using local basis functions

The computational procedure and numerical results are presented for a new method to solve Kuessner's integral equation in the case of subsonic compressible flow about harmonically oscillating planar surfaces with controls. Kuessner's equation is a linear transformation from pressure to normalwash. The unknown pressure is expanded in terms of prescribed basis functions and the unknown basis function coefficients are determined in the usual manner by satisfying the given normalwash distribution either collocationally or in the complex least squares sense. The present method of solution differs from previous ones in that the basis functions are defined in a continuous fashion over a relatively small portion of the aerodynamic surface and are zero elsewhere. This method, termed the local basis function method, combines the smoothness and accuracy of distribution methods with the simplicity and versatility of panel methods. Predictions by the local basis function method for unsteady flow are shown to be in excellent agreement with other methods. Also, potential improvements to the present method and extensions to more general classes of solutions are discussed.

Fromme, J. A.↗
NASA NTRS1983

On the symmetric form of systems of conservation laws with entropy

The present investigation has the objective to review the general structure of system of conservation laws with entropy, giving particular attention to a presentation of symmetric formulations of the equations of gas dynamics. Systems of hyperbolic conservation laws are considered, taking into account the case in which a system of equations is symmetric hyperbolic, questions of symmetrization, and entropy functions. The Euler equations for a polytropic gas in conservation form are considered. The symmetric form retains the conservation properties of the equations. Weak solutions remain, therefore, unchanged. Of particular computational interest is the possibility of using the symmetric form.

Harten, A.↗
NASA NTRS2014

A Walsh Function Module Users' Manual

The solution of partial differential equations (PDEs) with Walsh functions offers new opportunities to simulate many challenging problems in mathematical physics. The approach was developed to better simulate hypersonic flows with shocks on unstructured grids. It is unique in that integrals and derivatives are computed using simple matrix multiplication of series representations of functions without the need for divided differences. The product of any two Walsh functions is another Walsh function - a feature that radically changes an algorithm for solving PDEs. A FORTRAN module for supporting Walsh function simulations is documented. A FORTRAN code is also documented with options for solving time-dependent problems: an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the usage of the Walsh function module including such features as operator overloading, Fast Walsh Transforms in multi-dimensions, and a Fast Walsh reciprocal.

Gnoffo, Peter A.↗