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An analytic solution of the time-dependent radiation transport problem in a one-dimensional, stationary and homogeneous medium of finite thickness is presented. The solution is found by the method of images, and is compared with an eigenfunction expansion. Previous conjectures about the structure of such an expansion are clarified. The Green's function of this problem is also expanded in scattering orders.
A general relationship is established between wave-packet evolution and quantization in classically integrable systems. Because of wave-packet spreading, one cannot simply take the Fourier transform of the time evolution of a wave-packet. Instead, one must propagate the wave-packet using the actions as Hamiltonians. The energy eigenvalues which result are the Einstein-Brillouin-Keller values, and new forms for the eigenfunctions appear. These are free of caustic singularities, and represent averages of wave packets over the invariant torus.
This paper presents a detailed account of the hydrodynamic stability characteristics of the initial mixing region of an elliptic jet. A realistic mean velocity profile is used. Calculations of growth rates, phase velocities and eigenfunctions are presented. The growth rates of all modes in the initial mixing region scale with the minimum momentum thickness. Pressure fluctuations are found to be greatest for all modes close to the major axis. An irregular normal mode is found at larger eccentricities. The flapping mode about the major axis becomes the least stable as the momentum thickness of the mixing layer increases.
A finite element Galerkin formulation has been developed to study electromagnetic propagation in complex two-dimensional absorbing ducts. The reflection and transmission at entrance and exit boundaries are determined by coupling the finite element solutions at the entrance and exit to the eigenfunctions of an infinite uniform perfect conducting duct. Example solutions are presented for electromagnetic propagation with absorbing duct walls and propagating through dielectric-metallic matrix materials.
The effect of a planetary-scale, wavenumber 2 topography on baroclinically active disturbances is investigated for a channel domain in a two-layer, quasi-geostrophic context. When the lower-layer zonal velocity is nonzero, the topography influences the disturbances by forcing a stationary wave, and the topography and the forced wave influence the growth rates and the spatial structures of the time-dependent solutions. The case of zero zonal velocity in the lower layer was also investigated, for which no forced wave exists. Asymptotic forms of the equations, valid when the topographic effect (governed by the ratio of the nondimensional topographic height to the rotational Froude number) is small, are used to obtain both the stationary and time-dependent solutions. The latter are also obtained using a numerical approach, in which is determined the eigenvalues and eigenfunctions of a matrix representing the dynamical equations. Agreement is good between the two approaches. Recent laboratory experiments with a baroclinic annulus in which there is a false bottom with wavenumber 2 topography, are used to select governing parameters. The simultaneous presence of a stationary forced wave of wavenumber 2 and a time-dependent baroclinic wave of wavenumber 4, which has wavenumber 2 and 6 sidebands due to the topography, yields a flow field that exhibits some principal features of the laboratory experiments. The position of the forced wave and the location of an excursion in latitude of the storm track show qualitative resemblance to those features observed in the atmosphere.
An estimation technique was developed to extrapolate tidal amplitudes and phases over entire ocean basins using existing gauge data and the altimetric measurements provided by satellite oceanography. The technique was previously tested. Some results obtained by using a 3 deg by 3 deg grid are presented. The functions used in the interpolation are the eigenfunctions of the velocity (Proudman functions) which are computed numerically from a knowledge of the basin's bottom topography, the horizontal plan form and the necessary boundary conditions. These functions are characteristic of the particular basin. The gravitational normal modes of the basin are computed as part of the investigation; they are used to obtain the theoretical forced solutions for the tidal constituents. The latter can provide the simulated data for the testing of the method and serve as a guide in choosing the most energetic functions for the interpolation.
Using the lifting surface theory and the acceleration potential method for the flow field of an axial turbocompressor stage, a recursive and a direct method are presented that make use of the eigenfunction solutions of the isolated rotor and stator to solve for the rotor-stator interaction problem. The net pressure distribution on the rotor and stator blades is represented by modified Birnbaum series, whose coefficients are determined using a matrix procedure and satisfying the boundary conditions on the surface of the blades. The relation between the matrix operators of the recursive and the direct methods is also shown. Expressions have been given for the blade circulation, the axial and tangential forces on the blade, the rotor power required, and the induced upwash velocity of the stage.
Effects of boundaries (vertical sides) on the Marangoni-Rayleigh problem are reviewed. Spectral crowding and a dimensional singularity when Rayleigh effects are neglected are discussed. In the unbounded case, there is a unique Ma with unique eigenfunction for fixed Ra but infinite Ra for fixed Ma. These infinite values of Ra correspond to vertical cellular modes. In the bounded case, there is infinite Ma for a fixed Ra. The stress free side wall case gives double infinite of Ra for fixed Ma and wave number. Clearly, a dimensional singularity occurs for the Marangoni problem when the side walls go to infinity and a fixed Ra is considered. There is no direct analogy regarding this singularity in the morphological instability problem. Results on the behavior of Mac, Rac with boundary conditions also show only marginal connection to the morphological stability problem.
This paper deals with nearly singular, possibly indefinite problems for which the usual multigrid solvers converge very slowly or even diverge. The main difficulty is related to some badly approximated smooth functions which correspond to eigenfunctions with nearly zero eigenvalues. A correction to the usual coarse-grid equations is derived, both in the correction scheme and in the full approximation scheme. The performance of the new algorithm using this correction is essentially as that of usual multigrid for definite problems.
The characteristic Rossby frequency is defined for a fixed zonal wavenumber perturbation as the variational integral of the Rayleigh-Ritz method. It is a measure of the time scale of the disturbance. For a disturbance which locally has the shape of an eigenfunction but is not global in extent, the characteristic Rossby frequency is very close to the true eigenvalue, and additionally remains unchanged under linear inviscid dynamics. Results are presented for the shallow water equations, both with and without a mean zonal wind. The characteristic Rossby frequency of a wavenumber 1 perturbation having the shape of the second symmetric Rossby mode but confined to the Northern Hemisphere is close to the corresponding Rossby frequency. This finding is helpful in understanding the behavior of the observed wavenumber 1 pattern of January 1979, which propagated westward with nearly the pure Rossby frequency but was discernible only in the Northern Hemisphere (as discussed by Daley and Williamson).
A compressible stability analysis computer code is developed. The code uses a matrix finite-difference method for local eigenvale solution when a good guess for the eigenvalue is available and is significantly more computationally efficient than the commonly used inital-value approach. The local eigenvalue search procedure also results in eigenfunctions and, at little extra work, group velocities. A globally convergent eigenvalue procedure is also developed that may be used when no guess for the eigenvalue is available. The global problem is formulated in such a way that no unstable spurious modes appear so that the method is suitable for use in a black-box stability code. Sample stability calculations are presented for the boundary layer profiles of an LFC swept wing.
The diffusive acceleration of highly relativistic particles at a shock is reconsidered. Using the same physical assumptions as Blandford and Ostriker (1978), but dropping the restriction to nonrelativistic shock velocities, the authors find approximate solutions of the particle kinetic equation by generalizing the diffusion approximation to higher order terms in the anisotropy of the particle distribution. The general solution of the transport equation on either side of the shock is constructed, which involves the solution of an eigenvalue problem. By matching the two solutions at the shock, the spectral index of the resulting power law is found by taking into account a sufficiently large number of eigenfunctions. Low-order truncation corresponds to the standard diffusion approximation and to a somewhat more general method described by Peacock (1981). In addition to the energy spectrum, the method yields the angular distribution of the particles and its spatial dependence.
Using the lifting surface theory and the acceleration potential method for the flow field of an axial turbocompressor stage, a recursive and a direct method are presented that make use of the eigenfunction solutions of the isolated rotor and stator to solve for the rotor-stator interaction problem. The net pressure distribution on the rotor and stator blades is represented by modified Birnbaum series, whose coefficients are determined using a matrix procedure and satisfying the boundary conditions on the surface of the blades. The relation between the matrix operators of the recursive and the direct methods is also shown. Expressions have been given for the blade circulation, the axial and tangential forces on the blade, the rotor power required, and the induced upwash velocity of the stage.
The instability of large amplitude Goertler vortices in a growing boundary layer is discussed in the fully nonlinear regime. It is shown that a three-dimensional breakdown to a flow with wavy vortex boundaries similar to that which occurs in the Taylor vortex problem takes place. However, the instability is confined to the thin shear layers which have been shown to trap the region of vortex activity. The disturbance eigenfunctions decay exponentially away from the center of these layers so that the upper and lower shear layers can support independent modes of instability. The structure of the instability, in particular its location and speed of downstream propagation, is found to be entirely consistent with recent experimental results. Furthermore, it is shown that the upper and lower layers support wavy vortex instabilities with quite different frequencies. This result is again consistent with the available experimental observations.
Thermal waves generated by either turbine rotor blades cutting through nonuniform combustor temperature fields or unsteady burning could lead to thermal fatigue cracking in the blades. To determine the magnitude of the thermal oscillation in blades with complex shapes and material compositions, a finite element Galerkin formulation has been developed to study combustor generated thermal wave propagation in a model two-dimensional duct with a uniform plug flow profile. The reflection and transmission of the thermal waves at the entrance and exit boundaries are determined by coupling the finite element solutions at the entrance and exit to the eigenfunctions of an infinitely long adiabatic duct. Example solutions are presented. In general, thermal wave propagation from an air passage into a metallic blade wall is small and not a problem. However, if a thermal barrier coating is applied to a metallic surface under conditions of a high heat transfer, a good impedance match is obtained and a significant portion of the thermal wave can pass into the blade material.
A finite-element Galerkin formulation has been developed to study transverse magnetic (TM) wave propagation in 2-D S-curved ducts with both perfectly conducting and absorbing walls. The reflection and transmission at the entrances and the exits of the curved ducts are determined by coupling the finite-element solutions in the curved ducts to the eigenfunctions of an infinite, uniform, perfectly conducting duct. Example solutions are presented for a double mitred and S-ducts of various lengths. The length of the S-duct is found to significantly effect the reflective characteristics of the duct. Also, the effect of curvature on an absorbing duct is illustrated.
The techniques for computing the eigenfunctions of the velocity potential (Proudman functions) set out in Sanchez, et al. (1986) in relation to the Atlantic-Indian Oceans are here applied to the Pacific Ocean, using a 6x6 degree grid of 510 points (455 points for the associated stream functions). Normal modes are computed from the first Proudman functions and have natural periods from 43.9h downward. Tidal syntheses are derived from these modes by direct application of the (frictionless) dynamic equations and by least-squares fitting of Proudman functions to the dynamically interpolated tide-gauge data of Schwiderski (1983). The modes contributing the most energy to the principal harmonic tidal constituents are different in the two computations: their natural periods are typically in the range of 9 to 16h for semidiurnal, and 14 to 43h for diurnal tides. The rms of fit for the Proudman functions is, in all cases, better than the corresponding value for the same number of spherical harmonics.
Unsteady two- and three-dimensional flow structure at leading and trailing edges of bodies can be characterized effectively using recently developed techniques for acquisition and interpretation of flow visualization. The techniques addressed here include: flow image/surface pressure correlations; 3-D reconstruction of flow structure from flow images; and interactive interpretation of flow images with theoretical simulations. These techniques can be employed in conjunction with: visual correlation and ensemble-averaging, both within a given image and between images; recognition of patterns from images; and estimates of velocity eigenfunctions from images.