Showing papers on "Spherical shell published in 1994"
TL;DR: In this article, the effects of rotation and a solid inner core on high Reynolds number buoyancy-driven flow in a fluid with the geometry of the Earth's liquid outer core were determined.
159 citations
TL;DR: In this paper, the authors investigate numerically the flow of an electrically conducting fluid in a differentially rotating spherical shell, in the presence of an imposed magnetic field, and show that the interior flow consists essentially of a solid-body rotation, with the precise rate determined by a torque balance between the inner and outer Ekman-Hartmann boundary layers.
Abstract: I investigate numerically the flow of an electrically conducting fluid in a differentially rotating spherical shell, in the presence of an imposed magnetic field. For a very weak field the flow is seen to consist of an Ekman layer on the inner and outer spherical boundaries, and a Stewartson layer on the cylinder circumscribing the inner sphere and parallel to the axis of rotation, in agreement with the classical non-magnetic analysis. As the field strength is increased, the non-magnetic Ekman layers merge smoothly into magnetic Ekman-Hartmann layers, and the Stewartson layer is suppressed. In the fully magnetic regime the interior flow consists essentially of a solid-body rotation, with the precise rate determined by a torque balance between the inner and outer Ekman-Hartmann boundary layers.
80 citations
TL;DR: In this article, the authors adapted an algorithm originally devised for calculating the diffusion-controlled reaction rate toward an arbitrarily shaped object to calculate the scalar translational hydrodynamic friction and the electrostatic capacitance of the object.
Abstract: An algorithm originally devised for calculating the diffusion‐controlled reaction rate toward an arbitrarily shaped object is adapted to calculate the scalar translational hydrodynamic friction and the electrostatic capacitance of the object. In this algorithm Brownian particles are launched from a spherical surface enclosing the object. Each particle is propagated until it either hits the enclosed object or crosses the starting surface. In the latter case the particle is allowed to escape to infinity with an analytically known probability. If the particle does not escape to infinity, it is put back on the starting surface with the correct distribution density and the process is repeated. The scalar friction or capacitance of the ‘‘probed’’ object is proportional to the fraction of particles that hit the object. This algorithm is illustrated on a dumbbell made of two equal‐size spheres, a cube, and a phantom spherical shell having random distributed beads embedded in its surface.
78 citations
TL;DR: A method for numerically solving the equation of radiative transfer in a spherical shell atmosphere using a conical boundary and a Gauss-Seidel iteration scheme to solve for all orders of scattering along a single radial line in the atmosphere is presented.
Abstract: A method for numerically solving the equation of radiative transfer in a spherical shell atmosphere is presented. The method uses a conical boundary and a Gauss-Seidel iteration scheme to solve for all orders of scattering along a single radial line in the atmosphere. Tests of the model indicate an accuracy better than 1% for most Earth-atmosphere situations. Results from this model are compared with flat-atmosphere model results for a scattering-only atmosphere. These comparisons indicate that excluding spherical effects for solar zenith angles greater than 85° leads to errors larger than 5% at optical depths as small as 0.10.
71 citations
50 citations
TL;DR: In this paper, a two-dimensional axisymmetric model of convection in a spherical shell was developed in order to investigate the effects of curvature on model predictions of heat flow and temperature in the Earth's mantle.
Abstract: A two-dimensional axisymmetric model of convection in a spherical shell has been developed in order to investigate the effects of curvature on model predictions of heat flow and temperature in the Earth's mantle. We restrict the solution domain to a belt centered midway between the poles of the axisymmetric coordinate system. This facilitates direct comparisons with models in cylindrical shells and plane layers, and avoids the physically unrealistic near-pole effects associated with the axial symmetry at the poles. A boundary layer argument is suggested which accounts for the observed variations in heat flow and mean temperature from one level of curvature to another. In particular, plane layer results may be scaled to any desired degree of curvature. Similarities between previous model results in cylindrical shells and our results in spherical shells are noted, and it is shown that results in one geometry may be mapped into the other provided the ratio of the surface areas of the upper and lower boundaries is the same.
40 citations
TL;DR: In this paper, the authors investigated the vibrational characteristics of a point-driven double shell in wave number-frequency space and found that the inner shell of the double shell exhibits two separate dispersion curves: a higher-frequency dispersion curve with respect to the inner and a second lower-frequency curve, out-of-phase vibration.
Abstract: The vibrational characteristics of a point‐driven ‘‘double shell’’ (two concentric submerged cylindrical shells coupled by the entrained fluid) are investigated theoretically and experimentally. Of particular interest are the shielding effects, if any, of the outer shell upon the inner shell. The theory on the double shell is based on Flugge’s infinite‐shell equations, the Helmholtz wave equation, and boundary conditions at the fluid–structure interfaces. This theory is used to model a finite double‐shell structure in wave number‐frequency space. Experiments are carried out in which generalized near‐field acoustical holography (GENAH) is employed to provide the experimental vibration characteristics in wave number‐frequency space of the finite double shell. It is confirmed theoretically and experimentally that the outer shell of the double shell exhibits two separate dispersion curves: A higher‐frequency dispersion curve exhibits in‐phase vibrations with respect to the inner shell, and a second lower‐frequency curve, out‐of‐phase vibration. The higher‐frequency dispersion curve of the double shell is very similar to the dispersion curve of the single shell (the inner shell without the outer shell), and thus is identified as a forced wave number response. The lower‐frequency curve seems to be dependent on the free wave number response of the outer shell alone of the double shell. A double‐shell structure can usually reduce its vibrational amplitudes by splitting the response of single‐shell’s forced vibration into the responses of inner‐shell’s forced vibration and outer‐shell’s induced vibration. However, it radiates low‐frequency underwater sounds inevitably according to the lower‐frequency dispersion curve. Furthermore, the appearance of the inner shell’s dispersion curve on the outer shell seems to indicate that the shielding influence of the outer shell is not completely effective.
36 citations
TL;DR: In this article, a model for the vibrations of a cage molecule in terms of the motions of an infinitely thin spherical shell was applied to icosahedral C60, and the model accounts for the general pattern, energy ordering, radial, and tangential characteristics of the vibrations, giving results comparable to those from a six-parameter conventional valence force field.
Abstract: A model for the vibrations of a cage molecule in terms of the motions of an infinitely thin spherical shell is applied to icosahedral C60. When appropriate values are given to the two free parameters, the model accounts for the general pattern, energy ordering, radial, and tangential characteristics of the vibrations, giving results comparable to those from a six‐parameter conventional valence force field. In the model, the lowest frequency of any spherical‐shell molecule is the ‘‘d’’ squashing deformation that describes tidal waves on a flooded planet.
34 citations
TL;DR: In this article, the authors considered the crushing which occurs in a spherical shell made of ductile material on striking a rigid wall and developed a static analysis which allows for strain hardening.
Abstract: This paper considers the crushing which occurs in a spherical shell made of ductile material on striking a rigid wall. A static analysis is developed which allows for strain hardening, whil...
34 citations
TL;DR: In this article, a comparison between a ray theory approximation and experiments in which tone bursts having carrier frequencies in the range 585
Abstract: A prominent feature predicted for the backscattering of tone bursts by thin spherical shells is an enhancement of a guided wave contribution near the first longitudinal resonance. This has been explained with a backward ray model of a leaky Lamb wave where energy is leaked off without having circumnavigated the far side of the shell [P. L. Marston et al., J. Acoust. Soc. Am. 90, 2341 (1991); D. H. Hughes, Ph.D. thesis, Washington State University (1992)]. The relevant s2b Lamb wave has opposing group and phase velocities giving rise to prompt radiation following the direct specular echo. The present research gives a comparison between a ray theory approximation and experiments in which tone bursts having carrier frequencies in the range 585
33 citations
TL;DR: In this article, the slow motion of a Newtonian fluid past a porous spherical shell has been examined and the streamlines for the flow outside of the inner core and around the spherical shell have been depicted graphically and compared with the corresponding streamlines around a non-porous sphere.
Abstract: Slow motion of a Newtonian fluid past a porous spherical shell has been examined. The flow in the free fluid region (inside the core and outside the shell) is governed by the Navier-Stokes equations whereas the flow in the porous region (shell region) is governed by the Brinkman model. The exact solution has been found under Stokes' approximation. The drag experienced by the shell has been discussed numerically for a range of values of governing parameters. The streamlines for the flow outside of the inner core and around the spherical shell have been depicted graphically and compared with the corresponding streamlines around a non-porous sphere.
TL;DR: In this article, an axisymmetric field is imposed across the viscous shear layer, and it is demonstrated that for a sufficiently strong field this adjustment does indeed occur, and that in the presence of a magnetic field the Lorentz force would adjust to satisfy this constraint.
Abstract: In the context of the generation of the Earth’s magnetic field, nonaxisymmetric solutions of the forced momentum equation in a rotating spherical shell are considered in the inertia‐less, inviscid limit. It has previously been pointed out that, in general, such solutions are singular, with all three flow components discontinuous across the cylinder tangent to the inner core and parallel to the axis of rotation. An integral constraint on the forcing was derived, which must be satisfied if the inviscid solution is to be nonsingular, and it was suggested that in the presence of a magnetic field the Lorentz force would adjust to satisfy this constraint, thereby eliminating the need for the viscous shear layer, which otherwise resolves the singularity. In this work an axisymmetric field is imposed across this shear layer, and it is demonstrated that for a sufficiently strong field this adjustment does indeed occur.
TL;DR: An exact elasto-plastic analytical solution for a finitely deformed, internal-pressurized, thick-walled spherical shell made of elastic linear-hardening material is derived in this article.
TL;DR: In this article, the LBB constant in terms of the wave number was evaluated for typical problems in elastic scattering, including the problem of scattering of a plane wave on an elastic spherical shell, for different wave numbers.
Abstract: The paper is a continuation of [1] and contains the evaluation of the (exact) LBB constant, in terms of the wave number, for typical problems (all with spherical geometry) in elastic scattering. Solutions to the problem of scattering of a plane wave on an elastic spherical shell, for different wave numbers, illustrate the dramatic effect of the magnitude of the LBB constant on the convergence.
TL;DR: The exact series solutions for the transient shell displacement and fluid pressure fields resulting from the axisymmetric acoustic loading of a submerged, thin elastic, spherical shell are well known.
Abstract: The exact series solutions for the transient shell displacement and fluid pressure fields resulting from the axisymmetric acoustic loading of a submerged, thin elastic, spherical shell are well known. Plots of the shell displacements, velocities, accelerations, strains, and strain rates have been published previously. Further investigation for the more general prolate spheroidal geometry has elucidated a complex exchange of energy between the structure and the fluid through the kinematic boundary conditions, which is presented herein for the simpler spherical geometry. The importance of the time evolution of the acoustic radiation coupling to the structure is discussed, including phase information. A comparison of the unloaded versus fluid‐loaded frequencies for an example problem demonstrates the ‘‘softening’’ effect of the fluid on the structural response and reveals it to be more complex than simple damping. The reloading of the shell due to radiation in the fluid introduces a proliferation of frequenc...
TL;DR: In this article, the influence of axisymmetric initial geometric imperfections and a nonlinear compressive stress state on the natural frequencies and nonlinear oscillations of a pressure loaded shallow spherical shell is investigated.
TL;DR: In this paper, numerical simulations of thermal convection in a rapidly rotating spherical fluid shell with and without inhomogeneous temperature anomalies on the top boundary have been carried out using a three-dimensional, time-dependent, spectral-transform code.
Abstract: Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell with and without inhomogeneous temperature anomalies on the top boundary have been carried out using a three-dimensional, time-dependent, spectral-transform code. The spherical shell of Boussinesq fluid has inner and outer radii the same as those of the Earth's liquid outer core. The Taylor number is 107, the Prandtl number is 1, and the Rayleigh number R is 5Rc (Rc is the critical value of R for the onset of convection when the top boundary is isothermal and R is based on the spherically averaged temperature difference across the shell). The shell is heated from below and cooled from above; there is no internal heating. The lower boundary of the shell is isothermal and both boundaries are rigid and impermeable. Three cases are considered. In one, the upper boundary is isothermal while in the others, temperature anomalies with (l,m) = (3,2) and (6,4) are imposed on the top boundary. The spherically averaged te...
TL;DR: In this paper, the radial component of the magnetic field is modeled as a homogeneous sphere and the expectation value for the product of magnetic fields simultaneously measured at two locations, the covariance, depends on only two parameters of the measurement locations: the angular distance and the products of the radii.
Abstract: Certain types of magnetic noise arising from physical as well as biological sources can be explained by the model of current dipoles randomly distributed in a volume conductor. The volume conductor investigated in the present study is the homogeneous sphere, which is the model commonly used in analyses of the magnetic field of the human brain (magnetoencephalogram). Uniform distributions of random dipoles in three different source spaces are considered: a spherical surface, a spherical shell, and a sphere. The main emphasis is put on the radial component of the magnetic field. It is shown that the expectation value for the product of magnetic fields simultaneously measured at two locations, the covariance, depends on only two parameters of the measurement locations: the angular distance and the product of the radii. The formulas derived for the covariance can be expressed in terms of elliptic integrals of the first and the second kind, so that a very efficient numerical calculation is possible. For the special case of the variance (two identical measurement locations) these formulas reduce to expressions composed of elementary functions. Numerical examples show that the noise produced by random dipoles in a sphere is similar to the noise generated by random dipoles on a spherical surface having a slightly smaller radius.
TL;DR: In this paper, the authors studied the scattering interaction of short electromagnetic pulses with a spherical target, where the target is assumed penetrable and they model it as an air-filled dielectric shell.
Abstract: The authors study the scattering interaction of short electromagnetic pulses with a spherical target. The target is assumed penetrable and they model it as an air-filled dielectric shell. The radar cross-section (RCS) of such a target is obtained and its resonance features are analyzed. A dielectric composition makes the resonance features become very prominent compared with the case of a perfectly conducting sphere. When the interrogating waveform is a pulse of short duration, the resonance features of the RCS can be extracted within the frequency band of the spectrum of the incident pulse. To verify their theoretical predictions they illuminate spherical targets with short, broadband pulses using an impulse radar system. The actual shape of the pulse that is incident on the targets is theoretically modeled using a digital filter design technique together with pulse returns from a reference target. They verify that the shape of the predicted, backscattered pulse that results from their design method agrees well with the experimental findings using three additional targets of different sizes and materials. They investigate in the combined time-frequency domain the development in time of the various frequency features of the spectra of backscattered pulses using time-windowed Fourier transforms. The methodology developed can handle broadband pulses of any sufficiently smooth spectrum, interacting with (lossy or lossless) dielectric scatterers, and can extract resonance features within the frequency band of the spectrum of the transmitted pulse. Accordingly, this method could be also used for assessing the performance of high-power impulse radar systems. >
TL;DR: In this paper, the Ritz method is used to obtain an eigenvalue equation for the free vibration of a class of solids, where each solid is modelled by means of a segment which is described in terms of Cartesian coordinates and is bounded by the yz, zx and xy orthogonal coordinate planes as well as by two curved surfaces which are defined by polynomial expressions in the coordinates x, y and z.
TL;DR: In this article, a combination of analytical and numerical techniques were used to analyse the onset of steady Marangoni convection in a spherical shell of fluid with an outer free surface surrounding a rigid sphere.
Abstract: In this paper we use a combination of analytical and numerical techniques to analyse the onset of steady Marangoni convection in a spherical shell of fluid with an outer free surface surrounding a rigid sphere. In so doing we correct the formulation of the problem and the results presented by Cloot & Lebon (Microgravity sci. technol.3 (1) 1990: 44–46). We find that if the free surface of the layer is non-deformable then the layer is always stable when heated from the outside and is unstable when heated from the inside if the magnitude of the (positive) non-dimensional Marangoni number is sufficiently large. If the free surface of the layer is deformable then the layer is always unstable when heated from the inside. It is stable when heated from the outside if Cr r2/4 then it is unstable if the magnitude of the (negative) Marangoni number is sufficiently large, where Cr is the non-dimensional Crispation number and r2 the non-dimensionalradius of the undisturbed outer free surface of the fluid.
TL;DR: In this paper, the linear polarizability and hyperpolarizability of a free electron gas confined to a spherical shell are calculated, and the magnitude of the calculated polarizabilities of C60 and C70 molecules were found to be in reasonable agreement with some of the experiments.
TL;DR: In this paper, a method is presented to evaluate the acoustic transient radiation from fluid-loaded shells of revolution which are excited by axisymmetric mechanical or acoustical excitations.
Abstract: A new method is presented to evaluate the acoustic transient radiation from fluid‐loaded shells of revolution which are excited by axisymmetric mechanical or acoustical excitations. This method is based on the use of an in‐vacuo modal vector (eigenvector) expansion with time‐dependent coefficients to describe the velocity field of the fluid‐loaded shell. Modal acoustic impulse responses which account for the coupling between the in‐vacuo modal vectors for the fluid‐loaded shell are introduced in the formulation of the general solution. The time‐dependent modal velocity coefficients are expressed as the solution of a set of coupled convolution integral equations which are readily solved by marching forward in time. The associated time‐dependent surface pressure is subsequently expressed as a modal vector expansion in which the time‐dependent coefficients are readily related to the modal velocity coefficients. Since the surface velocity and pressure are then known, the time‐dependent pressures in the field are simply obtained via quadrature methods from the space‐time Kirchhoff integral solution. The special case of a fluid‐loaded spherical shell is addressed to illustrate the application of the method to determine the general characteristics of the transient velocity response of the shell and the associated pressure field resulting from axisymmetric mechanical force excitations of spherical shells. Numerical results are presented to illustrate the effects on the velocity response and pressure field of time‐dependent fluid coupling between the modal vectors corresponding to the upper and lower branches of the in‐vacuo frequency spectrum of the spherical shell.
TL;DR: In this paper, the free vibration analysis of a thin spherical shell filled with a compressible fluid is investigated, where the interactions at the interface between the elastic structure and the compressible fluids are taken into account.
Abstract: Free vibration of a thin spherical shell filled with a compressible fluid is investigated. The interactions at the interface between the elastic structure and the compressible fluid are taken into account. The objective of this study is to develop a hybrid numerical technique for the free vibration analysis of sound–structure interaction problems. The boundary element method is employed for modeling the acoustic disturbances in the cavity, while the finite element method is used for modeling the structural dynamics of the shell. The formulations are then combined into a coupled numerical scheme for the total pressure‐displacement field. Natural frequencies and mode shapes are calculated by using the singular value decomposition algorithm. Physical insights into the resonance phenomena associated with sound–structure interactions are derived from the comparison between the results of the thin spherical shell, with and without the fluid loading effect.
TL;DR: In this article, the authors studied the nonlinear oscillations of an ideal gas contained in a cylindrical shell with a harmonically oscillating line source positioned on the cylinder axis and symmetric harmonic displacement of the cylinder wall of the same circular frequency.
Abstract: Resonant nonlinear oscillations of an ideal gas contained in a cylindrical shell are studied. Excitation is generated by a harmonically oscillating line source positioned on the cylinder axis and symmetric, harmonic displacement of the cylinder wall of the same circular frequency; also there is a constant phase shift between both excitations taken into account. The problem is nonlinear and qualitatively resembles the analogous one in a spherical shell in that there is a similiar response curve in both cases with the response amplitude being of the order of magnitude of the cubic root of the excitation amplitude in either situation, but there are quantitative differences.
TL;DR: In this paper, a new analytical expression for the impulse response of a transducer in the form of a curved strip is derived, where the radiator is defined as part of a concave spherical shell delimited by two horizontal parallel planes and two vertical parallel planes.
Abstract: A new analytical expression for the impulse response of a transducer in the form of a curved strip is derived. The radiator is defined as part of a concave spherical shell delimited by two horizontal parallel planes and two vertical parallel planes. The field of the transducer is obtained by means of the impulse response method proposed by F. Oberhettinger [J. Res. Natl. Bur. Stand. 65, 1–6 (1961)] and P. R. Stepanishen [J. Acoust. Soc. Am. 49, 841–849 (1971)] and the formalism developed by M. Arditti et al. [Ultrason. Imag. 3, 37–61 (1981)]. Exact analytical expressions of the impulse response and of the continuous pressure field were obtained in the three planes xOy, xOz, yOz, where Oz is the propagating axis of the ultrasound and O the focal point. The results of numerical modeling were compared with experimental results obtained using a 5‐MHz rectangular spherical strip (7×21 mm) focused at 75 mm. The agreement was excellent.
TL;DR: In this paper, the principle, technical procedure and features of the dieless hydroforming technology of double layer spherical vessels are introduced, as well as the technical problems associated with bulging of the shells, taking account of experiments on a few double-layer spherical shells.
TL;DR: In this paper, a three-dimensional finite element model of the integral hydro-bulge forming (IHBF) process used in manufacturing of spherical pressure vessels is developed using the ABAQUS finite element code.
TL;DR: In this article, the authors obtain analytic solutions for the surface universality class of extraordinary transitions in $d = 4$ for a spherical shell, which may serve as a starting point for a pertubative calculation.
Abstract: In the framework of mean-field theory the equation for the order-parameter profile in a spherically-symmetric geometry at the bulk critical point reduces to an Emden-Fowler problem. We obtain analytic solutions for the surface universality class of extraordinary transitions in $d=4$ for a spherical shell, which may serve as a starting point for a pertubative calculation. It is demonstrated that the solution correctly reproduces the Fisher-de Gennes effect in the limit of the parallel-plate geometry.
TL;DR: In this paper, the rise time of a strong shock wave propagating through a porous material is estimated by analyzing the finite deformation of an elastic/viscoplastic spherical shell under impulsive pressure loading.
Abstract: The rise time of a strong shock wave propagating through a porous material is estimated by analyzing the finite deformation of an elastic/viscoplastic spherical shell under impulsive pressure loading. The analysis examines explicitly the effects of dynamic loading rate and initial temperature and explains the relatively large shock rise times observed in porous materials. Results of the analysis provide a consistent and realistic interpretation of available experimental data.