Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Diffusion equation”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 865 records · Page 48
All sourcesNASA NTRSDOE OSTI
NASA NTRS1971

Transport of solar flare protons: Comparison of a new analytic model with spacecraft measurements

An analytic solution was obtained to the complete Fokker-Planck equation including the effects of convection, interplanetary deceleration and acceleration, corotation, and anisotropic diffusion. The solution yields a time to maximum for the particle flux of about 10 h and an exponential decay time of about 5 h. Several solar flare particle events were observed with the solar and galactic cosmic ray experiment on OGO-6. Comparisons of the calculated time dependence of the fluxes with these observations of 1 to 70 MeV protons show that the model adequately describes both the rise and decay times.

Lupton, J. E.↗
NASA NTRS1978

Cosmic-ray streaming perpendicular to the mean magnetic field. II - The gyrophase distribution function

The distribution function of cosmic rays streaming perpendicular to the mean magnetic field in a turbulent medium is reexamined. Urch's (1977) discovery that in quasi-linear theory, the flux is due to particles at 90 deg pitch angle is discussed and shown to be consistent with previous formulations of the theory. It is pointed out that this flux of particles at 90 deg cannot be arbitrarily set equal to zero, and hence the alternative theory which proceeds from this premise is dismissed. A further, basic inconsistency in Urch's transport equation is demonstrated, and the connection between quasi-linear theory and compound diffusion is discussed.

Forman, M. A.↗
NASA NTRS1978

A steady-state stratospheric aerosol model

This paper deals with the development of a one-dimensional steady-state stratospheric aerosol model and the subsequent perturbations caused by including the expected Space Shuttle particulate effluents in the model. Two approaches to the basic modeling effort have been made: in one, enough simplifying assumptions were introduced so that a more or less exact solution to the descriptive equations could be obtained; in the other, very few simplifications were made and a computer technique was used to solve the equations. The most complete form of the model contains the effects of sedimentation, diffusion, particle growth and coagulation. The results indicate that the model is capable of describing many aspects of the stratospheric aerosol layer, such as size distribution and the vertical profile of particles greater than 0.3 micron diameter.

Rosen, J. M.↗
NASA NTRS1977

A stratospheric aerosol model with perturbations induced by the space shuttle particulate effluents

A one dimensional steady state stratospheric aerosol model is developed that considers the subsequent perturbations caused by including the expected space shuttle particulate effluents. Two approaches to the basic modeling effort were made: in one, enough simplifying assumptions were introduced so that a more or less exact solution to the descriptive equations could be obtained; in the other approach very few simplifications were made and a computer technique was used to solve the equations. The most complex form of the model contains the effects of sedimentation, diffusion, particle growth and coagulation. Results of the perturbation calculations show that there will probably be an immeasurably small increase in the stratospheric aerosol concentration for particles larger than about 0.15 micrometer radius.

Rosen, J. M.↗
NASA NTRS1981

Generalized Dufort-Frankel spectral methods

An explicit time-advancing scheme for the spectral solution of parabolic equations is presented. Several two-dimensional examples are considered, including convection-diffusion and nonlinear problems, under various boundary conditions. Numerical evidence demonstrates the efficiency and accuracy of the spectral approach.

Lustman, L.↗
NASA NTRS1984

Chemical application of diffusion quantum Monte Carlo

The diffusion quantum Monte Carlo (QMC) method gives a stochastic solution to the Schroedinger equation. This approach is receiving increasing attention in chemical applications as a result of its high accuracy. However, reducing statistical uncertainty remains a priority because chemical effects are often obtained as small differences of large numbers. As an example, the single-triplet splitting of the energy of the methylene molecule CH sub 2 is given. The QMC algorithm was implemented on the CYBER 205, first as a direct transcription of the algorithm running on the VAX 11/780, and second by explicitly writing vector code for all loops longer than a crossover length C. The speed of the codes relative to one another as a function of C, and relative to the VAX, are discussed. The computational time dependence obtained versus the number of basis functions is discussed and this is compared with that obtained from traditional quantum chemistry codes and that obtained from traditional computer architectures.

Reynolds, P. J.↗
NASA NTRS1984

A spectral collocation method for the Navier-Stokes equations

A Fourier-Chebyshev spectral method for the incompressible Navier-Stokes equations is described. It is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time. In this fully spectral algorithm, a preconditioned iterative technique is used for solving the implicit equations arising from semi-implicit treatment of pressure, mean advection and vertical diffusion terms. The algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity.

Malik, M. R.↗
NASA NTRS1984

Turbulent compressible convection in a deep atmosphere. II - Two-dimensional results for main-sequence A5 and F0 type envelopes

In the present two-dimensional numerical study of turbulent compressible convection in the A5 and F0 main-sequence envelope types, ionization effects are included in the equation of state of the gas, and radiative transfer is modeled in a diffusive process with tabulated gas opacities. It is noted that the thermal effects of ionization significantly affect the dynamics of the flows and that an inversion of the mean density can be created and sustained in a dynamical situation. The substantial differences in the flows of A5 and F0 indicate development trends in the transition from the radiative to the convective mode of energy transport. As convection becomes more effective, the flow becomes more turbulent and the scaling effects of local scale heights become more significant.

Sofia, S.↗
NASA NTRS1985

The cosmic-ray shock structure problem for relativistic shocks

The time asymptotic behaviour of a relativistic (parallel) shock wave significantly modified by the diffusive acceleration of cosmic-rays is investigated by means of relativistic hydrodynamical equations for both the cosmic-rays and thermal gas. The form of the shock structure equation and the dispersion relation for both long and short wavelength waves in the system are obtained. The dependence of the shock acceleration efficiency on the upstream fluid spped, long wavelength Mach number and the ratio N = P sub co/cP sub co+P sub go)(Psub co and P sub go are the upstream cosmic-ray and thermal gas pressures respectively) are studied.

Webb, G. M.↗
NASA NTRS1986

On the modeling of low-Reynolds-number turbulence

A full Reynolds-stress closure that is capable of describing the flow all the way to the wall was formulated for turbulent flow through circular pipe. Since viscosity does not appear explicitly in the pressure redistribution terms, conventional high-number models for these terms are found to be applicable. However, the models for turbulent diffusion and viscous dissipation have to be modified to account for viscous diffusion near a wall. Two redistribution and two diffusion models are investigated for their effects on the model calculations. Wall correction to pressure redistribution modeling is also examined. Diffusion effects on calculated turbulent properties are further investigated by simplifying the transport equations to algebraic equations for Reynolds stress. Two approximations are explored. These are the equilibrium and nonequilibrium turbulence assumptions. Finally, the two-equation closure is also used to calculate the flow in question and the results compared with all the other model calculations. Fully developed pipe flows at two moderate Reynolds numbers are used to validate these model calculations.

So, R. M. C.↗
NASA NTRS1985

A spectral collocation method for the Navier-Stokes equations

A Fourier-Chebyshev spectral method for the incompressible Navier-Stokes equations is described. It is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time. In this fully spectral algorithm, a preconditioned iterative technique is used for solving the implicit equations arising from semi-implicit treatment of pressure, mean advection and vertical diffusion terms. The algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity.

Malik, M. R.↗
NASA NTRS1988

Model structure of a cosmic-ray mediated stellar or solar wind

An idealized hydrodynamic model is presented for the mediation of a free-streaming stellar wind by galactic cosmic rays or energetic particles accelerated at the stellar wind termination shock. The spherically-symmetric stellar wind is taken to be cold; the only body force is the cosmic ray pressure gradient. The cosmic rays are treated as a massless fluid with an effective mean diffusion coefficient k proportional to radial distance r. The structure of the governing equations is investigated both analytically and numerically. Solutions for a range of values of k are presented which describe the deceleration of the stellar wind and a transition to nearly incompressible flow and constant cosmic ray pressure at large r. In the limit of small k the transition steepens to a strong stellar wind termination shock. For large k the stellar wind is decelerated gradually with no shock transition. It is argued that the solutions provide a simple model for the mediation of the solar wind by interstellar ions as both pickup ions and the cosmic ray anomalous component which together dominate the pressure of the solar wind at large r.

Lee, M. A.↗
NASA NTRS1993

Software For Analyzing Performances Of Wind Tunnels

Subsonic Wind Tunnel Performance Analysis computer program developed as aid in design and analysis of subsonic wind tunnels. Brings together and refines previously scattered and oversimplified techniques used for design of, and prediction of losses in, components of subsonic wind tunnels. Implements system of equations for determining total pressure losses and provides general guidelines for design of diffusers, contractions, corners, and inlets and exits of nonreturn tunnels. Written in FORTRAN and BASIC.

Decker, C.↗
NASA NTRS1993

Numerical simulations of three-dimensional laminar flow over a backward facing step; flow near side walls

This paper reports the results of numerical simulations of steady, laminar flow over a backward-facing step. The governing equations used in the simulations are the full 'compressible' Navier-Stokes equations, solutions to which were computed by using a cell-centered, finite volume discretization. The convection terms of the governing equations were discretized by using the Advection Upwind Splitting Method (AUSM), whereas the diffusion terms were discretized using central differencing formulas. The validity and accuracy of the numerical solutions were verified by comparing the results to existing experimental data for flow at identical Reynolds numbers in the same back step geometry. The paper focuses attention on the details of the flow field near the side wall of the geometry.

Steinthorsson, Erlendur↗
NASA NTRS1993

Flux-difference split parabolized Navier-Stokes algorithm for non-equilibrium chemically reacting flows

A flux-difference split explicit finite-difference algorithm is presented for solving the parabolized form of the equations governing three-dimensional nonequilibrium chemically reacting flows. The algorithm is based on an explicit noniterative, upwind space-marching scheme developed by Korte, but differs in that the unsteady Riemann problem, rather than the steady Riemann problem, is solved. The algorithm allows either a second or an approximately third-order accurate upwind treatment of the convection terms by employing the unsteady approximate Riemann solver of Roe. The source terms of the species transport equations are treated in either an explicit or implicit manner, and the species diffusion terms are modeled with either a Fickian or a multicomponent model. A validation of the algorithm is performed by comparing computational results with the 2-D Mach 14, 15 degree compression-corner data of Holden. The three-dimensional capability of the algorithm is demonstrated by computing Mach 2.7 flow over a swept wedge scramjet fuel injector, and three-dimensional reacting flow capability is demonstrated by a computing a shock-jet interaction concept for mixing and combustion enhancement.

White, J. A.↗
NASA NTRS1993

Double-diffusive fingering convection in a porous medium

The characteristics of nonlinear two-dimensional horizontally periodic double-diffusive fingering convection in a saturated porous medium is investigated, using the Darcy equation including Brinkman and Forchheimer terms for the momentum equation. To solve the equations and the corresponding initial and boundary conditions, a Galerkin method is applied in the horizontal direction, and a hybrid finite difference method is used in the vertical direction. The developed computer code was used to compute the thermal convection case, and the results were found to be in good agreement with existing results.

Chen, Falin↗
NASA NTRS2005

Factorization of the Compressible Navier-Stokes Equations

The Navier-Stokes equations for a Newtonian ideal gas are examined to determine the factorizable form of the equations relevant to the construction of a factorizable relaxation scheme. The principal linearization of the equations is found by examining the relative magnitude of the terms for short-wavelength errors. The principal part of the operator is then found. Comparison of the factors of the Navier-Stokes and Euler equations differ qualitatively because of the coupling of entropy and pressure through thermal diffusion. Special cases of the factorization are considered.

Roberts, Thomas W.↗