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It is shown that the problem of computing astronomical refraction for any value of the zenith angle may be reduced to a simple, nonsingular, numerical quadrature when the proper choice is made for the independent variable of integration.
An approximate method to determine the index of refraction of a dielectric obstacle is presented. For simplicity one dimensional models of electromagnetic scattering are treated. The governing equations yield a second order boundary value problem, in which the index of refraction appears as a functional parameter. The availability of reflection coefficients yield two additional boundary conditions. The index of refraction by a k-th order spline which can be written as a linear combination of B-splines is approximated. For N distinct reflection coefficients, the resulting N boundary value problems yield a system of N nonlinear equations in N unknowns which are the coefficients of the B-splines.
An approximate method to determine the index of refraction of a dielectric obstacle is presented. For simplicity one dimensional models of electromagnetic scattering are treated. The governing equations yield a second order boundary value problem, in which the index of refraction appears as a functional parameter. The availability of reflection coefficients yield two additional boundary conditions. The index of refraction by a k-th order spline which can be written as a linear combination of B-splines is approximated. For N distinct reflection coefficients, the resulting N boundary value problems yield a system of N nonlinear equations in N unknowns which are the coefficients of the B-splines.
An analysis of tone noise propagation through a boundary layer and fuselage scattering effects was derived. This analysis is a three dimensional and the complete wave field is solved by matching analytical expressions for the incident and scattered waves in the outer flow to a numerical solution in the boundary layer flow. The outer wave field is constructed analytically from an incident wave appropriate to the source and a scattered wave in the standard Hankel function form. For the incident wave, an existing function - domain propeller noise radiation theory is used. In the boundary layer region, the wave equation is solved by numerical methods. The theoretical analysis is embodied in a computer program which allows the calculation of correction factors for the fuselage scattering and boundary layer refraction effects. The effects are dependent on boundary layer profile, flight speed, and frequency. Corrections can be derived for any point on the fuselage, including those on the opposite side from the source. The theory was verified using limited cases and by comparing calculations with available measurements from JetStar tests of model prop-fans. For the JetStar model scale, the boundary layer refraction effects produce moderate fuselage pressure reinforcements aft of and near the plane of rotation and significant attenuation forward of the plane of rotation at high flight speeds. At lower flight speeds, the calculated boundary layer effects result in moderate amplification over the fuselage area of interest. Apparent amplification forward of the plane of rotation is a result of effective changes in the source directivity due to boundary layer refraction effects. Full scale effects are calculated to be moderate, providing fuselage pressure amplification of about 5 dB at the peak noise location. Evaluation using available noise measurements was made under high-speed, high-altitude flight conditions. Comparisons of calculations made of free field noise, using a current frequency-domain propeller noise prediction method, and fuselage effects using this new procedure show good agreement with fuselage measurements over a wide range of flight speeds and frequencies. Correction factors for the JetStar measurements made on the fuselage are provided in an Appendix.
The numerically exact superposition T-matrix method is used to compute, for the first time to our knowledge, electromagnetic scattering by finite spherical volumes composed of polydisperse mixtures of spherical particles with different size parameters or different refractive indices. The backscattering patterns calculated in the far-field zone of the polydisperse multiparticle volumes reveal unequivocally the classical manifestations of the effect of weak localization of electromagnetic waves in discrete random media, thereby corroborating the universal interference nature of coherent backscattering. The polarization opposition effect is shown to be the least robust manifestation of weak localization fading away with increasing particle size parameter.
The Helmholtz equation (Delta + K(2)n(2))u = f with a variable index of refraction n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. Such problems can be solved numerically by first truncating the given unbounded domain and imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization (for example, using a finite element method). In practical applications, the mesh size h and the wave number K, are not independent but are constrained by the accuracy of the desired computation. It will be shown that the number of points per wavelength, measured by (Kh)(-1), is not sufficient to determine the accuracy of a given discretization. For example, the quantity K(3)h(2) is shown to determine the accuracy in the L(2) norm for a second-order discretization method applied to several propagation models.
The Helmholtz equation (Delta + K(2)n(2))u = f with a variable index of refraction n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. Such problems can be solved numerically by first truncating the given unbounded domain and imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization (for example, using a finite element method). In practical applications, the mesh size h and the wave number K, are not independent but are constrained by the accuracy of the desired computation. It will be shown that the number of points per wavelength, measured by (Kh)(-1), is not sufficient to determine the accuracy of a given discretization. For example, the quantity K(3)h(2) is shown to determine the accuracy in the L(2) norm for a second-order discretization method applied to several propagation models.
The T-matrix method pioneered by Peter C. Waterman is one of the most versatile and efficient numerically exact computer solvers of the time-harmonic macroscopic Maxwell equations. It is widely used for the computation of electromagnetic scattering by single and composite particles, discrete random media, periodic structures (including metamaterials), and particles in the vicinity of plane or rough interfaces separating media with different refractive indices. This paper is the eighth update to the comprehensive thematic database of peer-reviewed T-matrix publications initiated in 2004 and lists relevant publications that have appeared since 2015. It also references a small number of earlier publications overlooked previously.
The T-matrix method is among the most versatile, efficient, and widely used theoretical techniques for the numerically exact computation of electromagnetic scattering by homogeneous and composite particles, clusters of particles, discrete random media, and particles in the vicinity of an interface separating two half-spaces with different refractive indices. This paper presents an update to the comprehensive database of T-matrix publications compiled by us previously and includes the publications that appeared since 2007. It also lists several earlier publications not included in the original database.
The properties of a pencil of light as defined approximately in the geometric optics ray tracing method are investigated. The vector Kirchhoff integral is utilized to accurately compute the electromagnetic near field in and around the pencil of light with various beam base sizes, shapes, propagation directions and medium refractive indices. If a pencil of light has geometric mean cross section size of the order p times the wavelength, it can propagate independently to a distance p2 times the wavelength, where most of the beam energy diffuses out of the beam region. This is consistent with a statement that van de Hulst made in a classical text on light scattering. The electromagnetic near fields in the pencil of light are not uniform, have complicated patterns within short distances from the beam base, and the fields tend to converge to Fraunhofer diffraction fields far away from the base.
The Helmholtz Equation (-delta-K(2)n(2))u=0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm was developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized using the finite element method, thus allowing for the modeling of complicated geometrices (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes. The resulting large, non-selfadjoint system of linear equations with indefinite symmetric part is solved using the preconditioned conjugate gradient method applied to the normal equations. A new preconditioner is developed based on the multigrid method. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner.
According to ray theory, regions exist in an upward refracting atmosphere where no sound should be present. Experiments show, however, that appreciable sound levels penetrate these so-called shadow zones. Two mechanisms contribute to sound in the shadow zone: diffraction and turbulent scattering of sound. Diffractive effects can be pronounced at lower frequencies but are small at high frequencies. In the short wavelength limit, then, scattering due to turbulence should be the predominant mechanism involved in producing the sound levels measured in shadow zones. No existing analytical method includes turbulence effects in the prediction of sound pressure levels in upward refractive shadow zones. In order to obtain quantitative average sound pressure level predictions, a numerical simulation of the effect of atmospheric turbulence on sound propagation is performed. The simulation is based on scattering from randomly distributed scattering centers ('turbules'). Sound pressure levels are computed for many realizations of a turbulent atmosphere. Predictions from the numerical simulation are compared with existing theories and experimental data.
Since its inception in the mid-1960s, the T-matrix method has been one of the most versatile and efficient numerically exact computer solvers of the time-harmonic macroscopic Maxwell equations. It has been widely used for the computation of electromagnetic scattering by single and composite particles, discrete random media, periodic structures (including metamaterials), and particles in the vicinity of plane or rough interfaces separating media with different refractive indices. This compilation is the ninth update to the comprehensive thematic database of peer-reviewed T-matrix publications initiated in 2004 and lists relevant publications that have appeared since 2017. It also includes a few earlier publications that have so far been overlooked.
Ceramic parts and coatings are needed to withstand high temperatures in advanced aircraft engines. In hot environments, such as in the combustion chambers of these engines, infrared and visible radiation can penetrate into some ceramics and heat them internally. The internal temperatures depend on radiative effects combined with heat conduction, and on convection and radiation at the material boundaries. Since engine temperatures are high, radiant emission can be large from within translucent parts and coatings, and this must be included in the analysis. Transient and steady-state behavior are both important. During a transient, radiant penetration provides more rapid internal heating than conduction alone, and the temperature distributions are usually considerably different than for steady-state conditions; this can produce transient thermal stresses. Analytical and numerical methods are being used at the NASA Lewis Research Center to predict transient temperatures and heat flows in translucent materials. A transient analysis was done for a composite of two translucent layers. The layer refractive indices were larger than 1, producing internal reflections at the boundaries and at the internal interface. In addition, steady-state results were computed for a two-layer composite with one layer opaque. The results were used to assess the importance of internal radiation in a zirconia thermal barrier coating on a cooled metal wall. Since the radiative transfer equations are rather complex, especially when internal scattering is included, approximate methods are being investigated that might be convenient for computer design programs. An approximate two-flux method was used and verified by comparisons with solutions that used exact radiative transfer relations. The two-flux equations include scattering without increasing solution difficulty.
Atmospheric radiation in the infrared (IR) 8-13 microns spectral region contains a wealth of information that is very useful for the retrieval of ice cloud properties from aircraft or space-borne measurements. To provide the scattering and absorption properties of nonspherical ice crystals that are fundamental to the IR retrieval implementation, we use the finite-difference time domain (FDTD) method to solve for the extinction efficiency, single-scattering albedo, and the asymmetry parameter of the phase function for ice crystals smaller than 40 microns. For particles larger than this size, the improved geometric optics method (IGOM) can be employed to calculate the asymmetry parameter with an acceptable accuracy, provided that we properly account for the inhomogeneity of the refracted wave due to strong absorption inside the ice particle. A combination of the results computed from the two methods provides the asymmetry parameter for the entire practical range of particle sizes between 1 micron and 10000 microns over wavelengths ranging from 8 microns to 13 microns. For the extinction and absorption efficiency calculations, several methods including the IGOM, Mie solution for equivalent spheres (MSFES), and the anomalous diffraction theory (ADT) can lead to a substantial discontinuity in comparison with the FDTD solutions for particle sizes on the order of 40 microns. To overcome this difficulty, we have developed a novel approach called the stretched scattering potential method (SSPM). For the IR 8-13 microns spectral region, we show that SSPM is a more accurate approximation than ADT, MSFES, and IGOM. The SSPM solution can be further refined numerically. Through a combination of the FDTD and SSPM, we have computed the extinction and absorption efficiency for hexagonal ice crystals with sizes ranging from 1 to 10000 microns at 12 wavelengths between 8 and 13 microns Calculations of the cirrus bulk scattering and absorption properties are performed for 30 size distributions obtained from various field campaigns for midlatitude and tropical cirrus cloud systems. Parameterization of these bulk scattering properties is carried out by using second-order polynomial functions for the extinction efficiency and the single-scattering albedo and the power law expression for the asymmetry parameter. We note that the volume-normalized extinction coefficient can be separated into two parts: one is inversely proportional to effective size and is independent of wavelength, and the other is the wavelength-dependent effective extinction efficiency. Unlike conventional parameterization efforts, the present parameterization scheme is more accurate because only the latter part of the volume-normalized extinction coefficient is approximated in terms of an analytical expression. After averaging over size distribution, the single-scattering albedo is shown to decrease with an increase in effective size for wavelengths shorter than 10.0 microns whereas the opposite behavior is observed for longer wavelengths. The variation of the asymmetry parameter as a function of effective size is substantial when the effective size is smaller than 50 microns. For effective sizes larger than 100 microns, the asymmetry parameter approaches its asymptotic value. The results derived in this study can be useful to remote sensing applications involving IR window bands under cirrus cloud conditions.
The objective of this work was to use direct numerical simulation (DNS) techniques to study the physics of noise generation by a high-speed turbulent jet. A Mach 0.9, Reynolds number 3,600 jet was selected because of available experimental data. New numerical methods for generating disturbances at the nozzle and computing far-field sound were developed and reported in the course of this work. Over 25 million mesh points were used in the simulations which ran for over 50,000 timesteps and required over 50,000 processor hours on state-of-the-art parallel computer systems to complete. Figures show a visualization of the jet and sound field, a comparison of the mean flow development with the experiment, a directivity comparison with the experiment, and time spectrum comparison with the experiment. Agreement is seen to be excellent. These are fully document in the attached references. Full details of the work, detailed achievements and conclusions are discussed in appendices, which are copies of publications that resulted from this work. We have studied noise mechanisms in supersonic jets, the refraction of sound by turbulence in subsonic jets, and noise sources in conjunction with a DNS of a Mach 0.9 jet.
Optical communication links have the potential to solve many of the problems of current radio and microwave links to satellites and high-altitude aircraft. The higher frequency involved in optical systems allows for significantly greater signal bandwidth, and thus information transfer rate, in excess of 10 Gbps, and the highly directional nature of laser-based signals eliminates the need for frequency-division multiplexing seen in radio and microwave links today. The atmosphere, however, distorts an optical signal differently than a microwave signal. While the ionosphere is one of the most significant sources of noise and distortion in a microwave or radio signal, the lower atmosphere affects an optical signal more significantly. Refractive index fluctuations, primarily caused by changes in atmospheric temperature and density, distort the incoming signal in both deterministic and nondeterministic ways. Additionally, suspended particles, such as those in haze or rain, further corrupt the transmitted signal. To model many of the atmospheric effects on the propagating beam, we use simulations based on the beam-propagation method. This method, developed both for simulation of signals in waveguides and propagation in atmospheric turbulence, separates the propagation into a diffraction and refraction problem. The diffraction step is an exact solution, within the limits of numerical precision, to the problem of propagation in free space, and the refraction step models the refractive index variances over a segment of the propagation path. By applying refraction for a segment of the propagation path, then diffracting over that same segment, this method forms a good approximation to true propagation through the atmospheric medium. Iterating over small segments of the total propagation path gives a good approximation to the problem of propagation over the entire path. Parameters in this model, such as initial beam profile and atmospheric constants, are easily modified in a simulation such as this, which allows for the rapid analysis of different propagation scenarios. Therefore, this method allows the development of a near-optimal system design for a wide range of situations, typical of what would be seen in different atmospheric conditions over a receiving ground station. A simulation framework based upon this model was developed in FORTRAN, and for moderate grid sizes and propagation distances these simulations are computable in reasonable time on a standard workstation. This presentation will discuss results thus far.
A hybrid numerical procedure is presented for the prediction of the aerodynamic and acoustic performance of advanced turboprops. A hybrid scheme is proposed which in principle leads to a consistent simultaneous prediction of both fields. In the inner flow a finite difference method, the Approximate-Factorization Alternating-Direction-Implicit (ADI) scheme, is used to solve the nonlinear Euler equations. In the outer flow the linearized acoustic equations are solved via a Boundary-Integral Equation (BIE) method. The two solutions are iteratively matched across a fictitious interface in the flow so as to maintain continuity. At convergence the resulting aerodynamic load prediction will automatically satisfy the appropriate free-field boundary conditions at the edge of the finite difference grid, while the acoustic predictions will reflect the back-reaction of the radiated field on the magnitude of the loading source terms, as well as refractive effects in the inner flow. The equations and logic needed to match the two solutions are developed and the computer program implementing the procedure is described. Unfortunately, no converged solutions were obtained, due to unexpectedly large running times. The reasons for this are discussed and several means to alleviate the situation are suggested.