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We develop a simple semi-algebraic 2-level solver built on traditional multigrid ideas. It is designed to be easily incorporated into existing simulation software. It exhibits good convergence for many classes of challenging problems including discontinuous diffusion, convection- diffusion, and Helmholtz equations. It has built-in structure that makes it simple to generalize in several interesting directions.
New analytical tool is available to calculate the degree of foam heterogeneity based on the measurement of gas diffusivity values. Diffusion characteristics of plastic foam are described by a system of differential equations based on conventional diffusion theory. This approach saves research and computation time in studying mass or heat diffusion problems.
From March through May 1967, the high-altitude balloon satellites Pageos (1966 56A) and Dash 2 (1963 30D) were both in continuous sunlight orbits. Observed variations in the orbital accelerations of both satellites are found to be due primarily to a combination of the perturbing effects of earth albedo radiation and atmospheric drag. An equation based on diffuse Lambertian reflection of sunlight from the earth predicts quite well the observed trend of orbital accelerations for both satellites due to the force of earth albedo radiation. After correcting for this effect, the remaining orbital energy changes are found to be consistent with atmospheric drag perturbations, making it possible to deduce mean exospheric densities above 2300 km. The results indicate that hydrogen concentrations during the observation period were about three times greater than that given by the U.S. Standard Atmosphere Supplements (1966).
An acoustical theory of turbulence was developed to aid in the study of the generation of sound in turbulent flows. The statistical framework adopted is a quantum-like wave dynamical formulation in terms of complex distribution functions. This formulation results in nonlinear diffusion-type transport equations for the probability densities of the five modes of wave propagation: two vorticity modes, one entropy mode, and two acoustic modes. This system of nonlinear equations is closed and complete. The technique of analysis was chosen such that direct applications to practical problems can be obtained with relative ease.
Recently measured positron and negatron spectra are discussed with regard to the problem of solar modulation. At energies above 180 MeV, the spherically symmetric Fokker-Planck equation with a diffusion coefficient proportional to particle rigidity provides reasonable fits to both the positron and total electron data. At energies below 180 MeV, the data are consistent with a continuation of the same diffusion coefficient and a local source of negatrons or with a change in the diffusion coefficient to a constant value.
The flux density of cosmic-ray positrons and electrons was measured by a balloon-borne spark chamber magnetic spectrometer in two flights in an attempt to study the solar modulation of galactic cosmic rays. The total electron flux was found to be about 0.03/cu m/s/sr/MeV between 70 and 800 MeV and to increase toward lower energies, while the positron flux decreased sharply toward lower energies from about 0.008/cu m/s/sr/MeV at 650 MeV and only upper limits were obtained for positrons below 200 MeV. At energies above 180 MeV, the spherically symmetrical Fokker-Planck equation with a diffusion coefficient proportional to the particle rigidity provided a reasonable fit to both positron and total electron data, while at energies below 180 MeV, the data were consistent with a continuation of the same diffusion coefficient and local source of electrons, or a change in the diffusion coefficient to a constant value.
A computer coded Lagrangian marker particle in Eulerian finite difference cell solution to the three dimensional incompressible mass transport equation, Water Advective Particle in Cell Technique, WAPIC, was developed, verified against analytic solutions, and subsequently applied in the prediction of long term transport of a suspended sediment cloud resulting from an instantaneous dredge spoil release. Numerical results from WAPIC were verified against analytic solutions to the three dimensional incompressible mass transport equation for turbulent diffusion and advection of Gaussian dye releases in unbounded uniform and uniformly sheared uni-directional flow, and for steady-uniform plug channel flow. WAPIC was utilized to simulate an analytic solution for non-equilibrium sediment dropout from an initially vertically uniform particle distribution in one dimensional turbulent channel flow.
The atmospheric species concentration distribution-estimation problem that arises from the measurement configuration of remote sensing, which involves a field of view that moves across the region, is solved by means of the partial differential equation for atmospheric diffusion and the Wiener-Hopf theory. A numerical study is also undertaken for the estimation of concentration distribution downwind of a hypothetical, continuous ground-level source of pollutants.
The various steps that could control the vaporization rate of a material are discussed. These steps include the actual vaporization, flow rate of matrix gas, chemical reaction, gas diffusion, and solid state diffusion. The applicable equations have been collected from diverse appropriate sources, and their use is explained. Rate equations are derived for conditions where more than one step is rate controlling. Calculations are made for two model materials: rhenium which vaporizes congruently, and tantalum carbide which vaporizes incongruently. The case of vaporization under thermal gradient conditions is also treated. The existence of a thermal gradient in the resistojet means that the vaporization rate of a material may be only one thousandth of that predicted under isothermal conditions. Calculations show that rhenium might have a 100,000 hr lifetime at temperature in a 2500 C resistojet. Tantalum carbide would have a life of only 660 sec under similar conditions.
The diffusion-convection (modulation) equation is derived directly from the Boltzmann equation on the basis of a minimum number of assumptions concerning the scattering process, among which are: (1) that the scattered particles undergo no energy change, and (2) that isotropy is an equilibrium state. It is noted that, in the event that the background plasma contains a magnetic field and the flow speeds of the plasma and scattering centers are different, additional terms arise that will modify the equations. If, moreover, the scatterers have individual motions relative to their average flow, the second-order Fermi acceleration term will appear.
A new wall function approach is developed based on a wall law suitable for incompressible turbulent boundary layers under strong adverse pressure gradients. This wall law was derived from a one-dimensional analysis of the turbulent kinetic energy equation with gradient diffusion concept employed in modeling the near-wall shear stress gradient. Numerical testing cases for the present wall functions include turbulent separating flows around an airfoil and turbulent recirculating flows in several confined regions. Improvements on the predictions using the present wall functions are illustrated. For cases of internal recirculating flows, one modification factor for improving the performance of the k-epsilon turbulence model in the flow recirculation regions is also included.
The paper presents a brief account of various turbulence models employed in the computation of turbulent flows, and evaluates the application of these models to selected internal flow configurations. The main conclusions of this analysis are: (1) the k-epsilon model is used in a majority of all the two-dimensional flow calculations reported in the literature; (2) modified forms of the k-epsilon model improve the performance for flows with streamline curvature and heat transfer; (3) for flows with swirl, the k-epsilon model performs rather poorly (the algebraic stress model performs better in this case); and (4) for flows with regions of secondary flow (noncircular duct flows), the algebraic stress model performs fairly well. Two important factors in the numerical solution of the model equations, namely false diffusion and inlet boundary conditions, are discussed. The existence of countergradient transport and its implications in turbulence modeling are examined. Finally, some recommendations for improving the model performance are made. The need for detailed experimental data in flows with strong curvature is emphasized.
The November 22, 1977 and December 27, 1977 solar energetic particle (SEP) events observed simultaneously by the Helios and the Voyager space probes and the ISEE 1 satellite are described. The spacecraft instruments indicated flux increases for H, He, and (for November 22) for O and Fe, with the energy range of 0.5-20 MeV per nucleon. It was found that the data can be well fitted by a model based on the spherically symmetric Fokker-Planck equation including convection, diffusion, and adiabatic deceleration, assuming that the radial interplanetary diffusion coefficient increases in proportion to the heliocentric radius to the power of 0.6 and that there is an extended particle injection at the sun. This extended injection is of crucial importance for the event of November 22, where the injection functions are strongly dependent on times associated with a soft energy spectrum.
The application of Galerkin/least-squares methodology to computational fluid dynamics is reviewed. Applications to advective-diffusive systems of equations and incompressible and compressible Euler and Navier-Stokes equations are described. Space-time finite-element methodology for time-dependent problems is emphasized and the status and role of so-called 'discontinuity-capturing operators' is reviewed.
A family of secant methods based on general rank-1 updates was revisited in view of the construction of iterative solvers for large non-Hermitian linear systems. As it turns out, both Broyden's good and bad update techniques play a special role, but should be associated with two different line search principles. For Broyden's bad update technique, a minimum residual principle is natural, thus making it theoretically comparable with a series of well known algorithms like GMRES. Broyden's good update technique, however, is shown to be naturally linked with a minimum next correction principle, which asymptotically mimics a minimum error principle. The two minimization principles differ significantly for sufficiently large system dimension. Numerical experiments on discretized partial differential equations of convection diffusion type in 2-D with integral layers give a first impression of the possible power of the derived good Broyden variant.
Evidence is presented for focused transport of energetic particles along magnetic field lines draped around a coronal mass ejection. This evidence was obtained with the University of Maryland/Max-Planck-Institute experiment on the ISEE-3 spacecraft during the decay phase of the June 6, 1979, solar particle event. During the early portion of the decay phase of this event, interplanetary magnetic field lines were apparently draped around a coronal mass ejection, leading to a small focusing length on the western flank where ISEE 3 was located. A period of very slow decrease of particle intensity was observed, along with large sunward anisotropy in the solar wind frame, which is inconsistent with predictions of the standard Fokker-Planck equation models for diffusive transport. It was found possible to fit the observations, assuming that focused transport dominates and that the particle pitch angle scattering is isotropic.
Properties of supersaturated solutions such as the density, viscosity, and solute diffusivity are dependent on the solute concentration. The diffusion-boundary-layer equations are derived and solved for the natural convection case with the viscosity and density dependent on the solute concentration. The solution obtained demonstrates that, at the vicinity of the saturation concentration c(sub s), there is a non-trivial dependence of the solution viscosity eta on its density rho: eta(c(sub s)) = eta(rho(sub s)) varies as rho(sub s)(exp 1/2), where rho(sub s) = rho(c(sub s)). This result has been verified in experiments with aqueous solutions of inorganic and organic salts.
A zonal mean chemistry transport model (2-D CTM) coupled with a semi-spectral dynamical model is used to simulate the distributions of trace gases in the present day atmosphere. The zonal-mean and eddy equations for the velocity and the geopotential height are solved in the semi-spectral dynamical model. The residual mean circulation is derived from these dynamical variables and used to advect the chemical species in the 2- D CTM. Based on a linearized wave transport equation, the eddy diffusion coefficients for chemical tracers are expressed in terms of the amplitude, frequency and growth rate of dynamical waves; local chemical loss rates; and a time constant parameterizing small scale mixing. The contributions to eddy flux are from the time varying wave amplitude (transient eddy), chemical reactions (chemical eddy) and small scale mixing. In spite of the high truncation in the dynamical module (only three longest waves are resolved), the model has simulated many observed characteristics of stratospheric dynamics and distribution of chemical species including ozone. Compared with the values commonly used in 2-D CTMs, the eddy diffusion coefficients for chemical species calculated in this model are smaller, especially in the subtropics. It is also found that the chemical eddy diffusion has only a small effects in determining the distribution of most slow species, including ozone in the stratosphere.