Supersonic leading edge problem according to the ellipsoidal model
Supersonic leading edge problem solved by ellipsoidal model using discrete ordinate method compared with BGK model solution
Huang, A. B.↗
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Calculations of neutron flux spectra induced in the earth's atmosphere by galactic cosmic rays
Calculations have been carried out to determine the neutron flux induced in the earth's atmosphere by galactic protons and alpha particles at solar minimum for a geomagnetic latitude of 42 N. Neutron flux spectra were calculated using Monte Carlo and discrete ordinates methods, and various comparisons with experimental data are presented. The magnitude and shape of the calculated neutron-leakage spectrum at the particular latitude considered support the theory that the cosmic-ray-albedo-neutron-decay mechanism is the source of the protons and electrons trapped in the Van Allen belts.
Armstrong, T. W.↗
Calculations of neutron flux spectra induced in the earth's atmosphere by galactic cosmic rays.
Calculations have been carried out to determine the neutron flux induced in the earth's atmosphere by galactic protons and alpha particles at solar minimum for a geomagnetic latitude of 42 N. Neutron flux spectra in the energy range from about 10 to the minus 8th to about 100,000 MeV at various depths in the atmosphere were calculated by using Monte Carlo and discrete ordinates methods, and various comparisons with experimental data are presented. The magnitude and shape of the calculated neutron leakage spectrum at the particular latitude considered support the theory that the cosmic ray albedo neutron decay (Crand) mechanism is the source of the protons trapped in the inner radiation belt.
Armstrong, T. W.↗
High-speed leading edge problem.
The sharp leading edge problem has been studied for both monatomic and diatomic gases using the Boltzmann equation with the Bhatnagar-Gross-Krook type models as the governing equation and the discrete ordinate method with a closed-boundary value approach as a tool. Plate length relative to the freestream mean free path is taken to be 52. The gas-surface interaction law is assumed to be diffuse reflection. The local distribution functions of molecular velocities and internal energies (for the diatomic gas) for the entire flowfield have been calculated for a freestream Mach number of 6.1. Comparisons are made between the calculated results and experimental data.
Huang, A. B.↗
Shock-wave structure in a fully ionized gas.
The structure of a steady plane shock in a fully ionized gas has been investigated using the Boltzmann equation with the Gross-Krook type model as the governing equation and the discrete-ordinate method as a tool. The present results agree well with the results obtained by the continuum approach for weak shocks. For strong shocks the present approach gives results that are considerably different from those yielded by the continuum approach, particularly in the high pressure region. Also, the present method gives smooth and continuous shock profiles for high Mach numbers. On the other hand, the results from the continuum approach contain discontinuities in slope. The ion and electron distribution functions across the shock are found to be singly peaked for all cases considered and the effects of the induced electric field is small.
Lu, C. S.↗
Shock-wave structure in a partially ionized gas
The structure of a steady plane shock in a partially ionized gas has been investigated using the Boltzmann equation with a kinetic model as the governing equation and the discrete ordinate method as a tool. The effects of the electric field induced by the charge separation on the shock structure have also been studied. Although the three species of an ionized gas travel with approximately the same macroscopic velocity, the individual distribution functions are found to be very different. In a strong shock the atom distribution function may have double peaks, while the ion distribution function has only one peak. Electrons are heated up much earlier than ions and atoms in a partially ionized gas. Because the interactions of electrons with atoms and with ions are different, the ion temperature can be different from the atom temperature.
Lu, C. S.↗
Turbulent plane Couette flow using probability distribution functions
A numerical scheme employing a combination of the discrete ordinate method and finite differences is developed for solving the one-dimensional form of Lundgren's (1967) model equation for turbulent plane Couette flow. The approach used requires no a priori assumption about the form of the turbulent distribution function, and the numerical solution is obtained directly from the governing differential equations. Two different types of boundary conditions (zero-gradient and Chapman-Enskog) for the distribution function are evaluated by comparing the numerical results with experimental data. It is found that: (1) the present approach gives convergent and stable results over a wide range of Reynolds numbers; (2) Lundgren's equation yields results that compare well with experimental data for mean velocity and skin friction in the case of simple Couette flow; (3) the zero-gradient boundary condition leads to a logarithmic flow profile; and (4) the Chapman-Enskog boundary condition provides very good agreement with experimental data when applied within the near-wall region.
Srinivasan, R.↗
Kinetic theory analysis of rarefied gas flow through finite length slots
A kinetic-theory analysis is made of the flow of a rarefied monatomic gas through a two-dimensional slot connecting two reservoirs. Numerical solutions are obtained by the moment and discrete-ordinate methods. The former method portrays the transition-regime characteristics well but has limitations in the free-molecule regime. The latter method gives accurate results in the free-molecule and slip regimes and bolsters confidence in the accuracy of the transition-regime results. The numerical solution for the mass flux through the slot agrees well with an approximate analytical solution of the moment equations for length-to-width ratios from 6 to 0.5, pressure ratios from 0.8 to 0.1, and Knudsen numbers from 5 to 0.5.
Raghuraman, P.↗
The delta-four-stream approximation for radiative flux transfer
An approximate technique for solving the radiative transfer problem for fluxes reflected, transmitted, and absorbed by a homogeneous scattering layer is presented. The technique is a straightforward delta-function modification of a solution to the transfer equation in closed form via the discrete-ordinates method with four streams. By the use of four streams, significant improvement in accuracy is obtained for anisotropic phase functions over that obtained by similarly employed two-stream approximations with delta-functions. However, the computational effort is increased only slightly. Equations are developed for the general case of a layer underlaid by a reflecting Lambert surface. Graphical comparisons are given of fractional error resulting from the use of this method with that resulting from the use of typical four-stream and delta-two-stream techniques.
Cuzzi, J. N.↗
Introduction to the Theory of Atmospheric Radiative Transfer
The fundamental physical and mathematical principles governing the transmission of radiation through the atmosphere are presented, with emphasis on the scattering of visible and near-IR radiation. The classical two-stream, thin-atmosphere, and Eddington approximations, along with some of their offspring, are developed in detail, along with the discrete ordinates method of Chandrasekhar. The adding and doubling methods are discussed from basic principles, and references for further reading are suggested.
Buglia, J. J.↗
Angular dependent transport of auroral electrons in the upper atmosphere
The transport of auroral electrons through the upper atmosphere is analyzed. The transport equation is solved using a discrete-ordinate method, including elastic and inelastic scattering of electrons (resulting in changes of pitch angle) and degradation in energy as the electrons penetrate into the atmosphere. The transport equation is solved numerically for the electron intensity as a function of altitude, pitch angle, and energy. In situ measurements of the pitch angle and energy distribution of precipitating electrons over an auroral arc provide boundary conditions for the calculation. Model calculations were carried out with various different phase functions for elastic and inelastic collisions to attempt changing the angular scattering, but the observed pitch angle distributions remain unexplained. It is suggested that mechanisms other than collisional scattering influence the angular distribution of auroral electrons at or below 300 km altitude in the low-energy domain.
Lummerzheim, D.↗
Flow of rarefied gases over two-dimensional bodies
A kinetic-theory analysis is made of the flow of rarefied gases over two-dimensional bodies of arbitrary curvature. The Boltzmann equation simplified by a model collision integral is written in an arbitrary orthogonal curvilinear coordinate system, and solved by means of finite-difference approximation with the discrete ordinate method. A numerical code is developed which can be applied to any two-dimensional submerged body of arbitrary curvature for the flow regimes from free-molecular to slip at transonic Mach numbers. Predictions are made for the case of a right circular cylinder.
Jeng, Duen-Ren↗
Rarefied gas flow through two-dimensional nozzles
A kinetic theory analysis is made of the flow of a rarefied gas from one reservoir to another through two-dimensional nozzles with arbitrary curvature. The Boltzmann equation simplified by a model collision integral is solved by means of finite-difference approximations with the discrete ordinate method. The physical space is transformed by a general grid generation technique and the velocity space is transformed to a polar coordinate system. A numerical code is developed which can be applied to any two-dimensional passage of complicated geometry for the flow regimes from free-molecular to slip. Numerical values of flow quantities can be calculated for the entire physical space including both inside the nozzle and in the outside plume. Predictions are made for the case of parallel slots and compared with existing literature data. Also, results for the cases of convergent or divergent slots and two-dimensional nozzles with arbitrary curvature at arbitrary knudsen number are presented.
De Witt, Kenneth J.↗
A new spherical model for computing the radiation field available for photolysis and heating at twilight
Accurate computation of atmospheric photodissociation and heating rates is needed in photochemical models. These quantities are proportional to the mean intensity of the solar radiation penetrating to various levels in the atmosphere. For large solar zenith angles a solution of the radiative transfer equation valid for a spherical atmosphere is required in order to obtain accurate values of the mean intensity. Such a solution based on a perturbation technique combined with the discrete ordinate method is presented. Mean intensity calculations are carried out for various solar zenith angles. These results are compared with calculations from a plane parallel radiative transfer model in order to assess the importance of using correct geometry around sunrise and sunset. This comparison shows, in agreement with previous investigations, that for solar zenith angles less than 90 deg adequate solutions are obtained for plane parallel geometry as long as spherical geometry is used to compute the direct beam attenuation; but for solar zenith angles greater than 90 deg this pseudospherical plane parallel approximation overstimates the mean intensity.
Dahlback, Arne↗
How well is Martian ozone inferred with reflectance spectroscopy?
The Mariner 9 UV spectrometer scanned from 2100 to 3500 Angstroms in one of its two spectral channels every 3 seconds with a spectral resolution of 15 Angstroms and an effective field-of-view of approximately 300 km(exp 2). The only gaseous absorption in the 2000 to 3000 Angstrom region was assumed to come from the Hartley band system of ozone, and therefore the amount of ozone was inferred by fitting this absorption feature with laboratory data of ozone absorption. Mars O3 as inferred from these spectra shows strong seasonal and latitudinal variation, with column abundances ranging from 0.2 microns at equatorial latitudes to 60 microns over the northern winter polar latitudes. The detectability limit of the spectrometer was approximately 3 microns. I use a radiative transfer model based on the discrete ordinate method to calculate synthetic radiance spectra. When typical amounts of dust and cloud are present, significant underestimation of O3 occurs. A factor of 3 times as much O3 is needed to generate the same spectrum for cloudy, dusty atmospheres as for a clear atmosphere. If the scattering properties of Martian clouds and dust were well known, then their appearance would not be a problem, as a model would be capable of retrieving the O3 abundance. However, these properties are not well known, which raises doubts about the effectiveness of the current UV spectroscopy technique used to measure O3. Spatial and temporal variability in temperature and water vapor have been claimed to account for the scatter of the data points. However, water vapor is a small source of odd hydrogen in the winter polar atmosphere, and may not account for most of the variability. Masking by clouds and dust may also account for some of the observed O3 variability, because the nature and opacity of the clouds and dust in the polar hood change dramatically in latitude and even on a day-to-day basis. As the maximum O3 abundance resides near the surface, spacecraft must be able to observe through the entire cloud and dust abundance in order to actually see the total O3 column abundance. If reflectance spectroscopy is used, as on Mariner 9, then the cloud and the airborne dust must be traversed twice; first by the incoming solar flux down to the surface, and then once again upon reflection from the surface out to the spacecraft.
Lindner, Bernhard Lee↗
The effect of polar clouds and dust on the radiative budget of the Martian polar cap
How ozone, clouds, and airborne dust affect CO2 ice formation and sublimation is examined to determine if they help explain the asymmetry in the residual polar caps. The radiative flux striking the surface is calculated for 57 solar and 10 IR wavelength intervals from 0 to 100 microns including the scatering, absorption, and emission by O3, CO2, clouds, and dust. UV absorption by H2O, O2, HO2, and H2O2 is also included, as is Rayleigh scattering. The discrete ordinate method of Stamnes et al. (1988) treats the emission, scattering, and absorption of monochromatic radiation.
Lindner, Bernhard L.↗
Discrete ordinate solution of the radiative transfer equation in the 'polarization normal wave representation'
The transfer equations for normal waves in finite, inhomogeneous and plane-parallel magnetoactive media are solved using the discrete ordinate method. The physical process of absorption, emission, and multiple scattering are accounted for, and the medium may be forced both at the top and bottom boundary by anisotropic radiation as well as by internal anisotropic sources. The computational procedure is numerically stable for arbitrarily large optical depths, and the computer time is independent of optical thickness.
Kylling, A.↗
Radiative transfer solutions for cylindrical coordinates with emitting, absorbing, and anisotropic scattering medium
The radiative transfer equation is solved by the S-N discrete ordinates method in one-dimensional radial and two-dimensional r-z coordinates systems. The walls of the enclosures are diffuse, and the participating medium absorbs, emits and anisotropically scatters the radiative energy. Diffuse wall incidence, isothermal medium emission and collimated incidence problems are considered. Effects of the scattering phase functions on average incident radiation and net radiative heat fluxes are studied. In addition, the effects of scattering albedo, optical thickness and the wall emissivity are briefly discussed.
Jendoubi, Slah↗