Showing papers on "Spherical shell published in 2006"

Experimental study of super-rotation in a magnetostrophic spherical Couette flow

Abstract: We report measurements of electric potentials at the surface of a spherical container of liquid sodium in which a magnetized inner core is differentially rotating. The azimuthal angular velocities inferred from these potentials reveal a strong super-rotation of the liquid sodium in the equatorial region, for small differential rotation. Super-rotation was observed in numerical simulations by Dormy et al. (Dormy, E., Cardin, P. and Jault, D., MHD flow in a slightly differentially rotating spherical shell, with conducting inner core, in a dipolar magnetic field, Earth Planet. Sci. Lett., 1998, 160, 15--30). We find that the latitudinal variation of the electric potentials in our experiments differs markedly from the predictions of a similar numerical model, suggesting that some of the assumptions used in the model – steadiness, equatorial symmetry, and linear treatment for the evolution of both the magnetic and velocity fields – are violated in the experiments. In addition, radial velocity measurements, usi...

Dipolar and non-dipolar dynamos in a thin shell geometry with implications for the magnetic field of Mercury

Abstract: [1] Dynamo action possibly working in the fluid core of Mercury is examined using numerical models in a thin spherical shell Dipolar (DP) dynamos are obtained in the regime of columnar flows outside the tangent cylinder (TC), whereas non-dipolar (NDP) dynamos dominated by the multipole components are found in the regime of flows both inside and outside the TC It turns out that columnar-like convective motions not only outside but also inside the TC are responsible for the NDP dynamo The electrically conducting inner core enhances the strength of large-scale magnetic field, but predominance of the NDP components remains due to the thin shell geometry These results suggest that Mercury may have more complicated magnetic field than has been considered

Low‐degree mantle convection with strongly temperature‐ and depth‐dependent viscosity in a three‐dimensional spherical shell

Abstract: [1] A series of numerical simulations of thermal convection of Boussinesq fluid with infinite Prandtl number, with Rayleigh number 10 7 , and with the strongly temperature-and depth-dependent viscosity in a three-dimensional spherical shell is carried out to study the mantle convection of single-plate terrestrial planets like Venus or Mars without an Earth-like plate tectonics. The strongly temperature-dependent viscosity (the viscosity contrast across the shell is > 10 5 ) makes the convection under stagnant lid short-wavelength structures. Numerous, cylindrical upwelling plumes are developed because of the secondary downwelling plumes arising from the bottom of lid. This convection pattern is inconsistent with that inferred from the geodesic observation of the Venus or Mars. Additional effect of the stratified viscosity at the upper/lower mantle (the viscosity contrast is varied from 30 to 300) are investigated. It is found that the combination of the strongly temperature- and depth-dependent viscosity causes long-wavelength structures of convection in which the spherical harmonic degree l is dominant at 1-4. The geoid anomaly calculated by the simulated convections shows a long-wavelength structure, which is compared with observations. The degree-one (l = 1) convection like the Martian mantle is realized in the wide range of viscosity contrast from 30 to 100 when the viscosity is continuously increased with depth at the lower mantle.

Analysis of a rotation‐free 4‐node shell element

Abstract: In this paper, a shell element for small and large deformations is presented based on the extension of the methodology to derive triangular shell element without rotational degrees of freedom (so-called rotation-free). As in our original triangular S3 element, the curvatures are computed resorting to the surrounding elements. However, the extension to a quadrilateral element requires internal curvatures in order to avoid singular bending stiffness. The quadrilateral area co-ordinates interpolation is used to establish the required expressions between the rigid-body modes of normal nodal translations and the normal through thickness bending strains at mid-side. In order to propose an attractive low-cost shell element, the one-point quadrature is achieved at the centre for the membrane strains, which are superposed to the bending strains in the centred co-rotational local frame. The membrane hourglass control is obtained by the perturbation stabilization procedure. Free, simply supported and clamped edges are considered without introducing virtual nodes or elements. Several numerical examples with regular and irregular meshes are performed to show the convergence, accuracy and the reasonable little sensitivity to geometric distortion. Based on an updated Lagrangian formulation and Newton iterations, the large displacements of the pinched hemispherical shell show the effectiveness of the proposed simplified element (S4). Finally, the deep drawing of a square box including large plastic strains with contact and friction completes the ability of the rotation-free quadrilateral element for sheet-metal-forming simulations. Copyright © 2005 John Wiley & Sons, Ltd.

Integrating cavities: temporal response.

Abstract: The temporal response of an integrating cavity is examined and compared with the results of a Monte Carlo analysis. An important parameter in the temporal response is the average distance d¯ between successive reflections at the cavity wall; d¯ was calculated for several specific cavity designs--spherical shell, cube, right circular cylinder, irregular tetrahedron, and prism; however, only the calculation for the spherical shell and the right circular cylinder will be presented. A completely general formulation of d¯ for arbitrary cavity shapes is then derived, d¯=4V/S where V is the volume of the cavity, and S is the surface area of the cavity. Finally, we consider an arbitrary cavity shape for which each flat face is tangent to a single inscribed sphere of diameter D (a curved surface is considered to be an infinite number of flat surfaces). We will prove that for such a cavity d¯=2D/3, exactly the same as d¯ for the inscribed sphere.

Thermal Instability of Functionally Graded Shallow Spherical Shell

Abstract: In this paper, thermal instability of shallow spherical shells made of functionally graded material (FGM) is considered. The governing equations for a thin spherical shell based on the Donnell–Mushtari–Vlasov theory are obtained. The equations are derived using the Sanders simplified kinematic relations and variational method. It is assumed that the mechanical properties vary linearly through the shell thickness. The constituent material of the functionally graded shell is assumed to be a mixture of ceramic and metal. Analytical solutions are obtained for three types of thermal loading including Uniform Temperature Rise (UTR), Linear Radial Temperature (LRT), and Nonlinear Radial Temperature (NRT). The results are validated with the known data in the literature. Communicated by Theodore R. Tauschert on September 1, 2005.

Implications of deformation and shape coexistence for the nuclear shell model.

Abstract: The successes of the nuclear shell model in explaining the stability properties of magic nuclei are challenged by the observation of rotational bands for which the sequential filling of single-particle energy levels of the spherical shell model are not respected. This Letter proposes criteria for identifying the shell-model configurations appropriate for describing such bands of states.

Free Vibration Analysis of Spherical Caps Using a G.D.Q. Numerical Solution

Abstract: In this paper we present the frequency evaluation of spherical shells by means of the generalized differential quadrature method (G.D.Q.M.), an effective numerical procedure which pertains to the class of generalized collocation methods. The shell theory used in this study is a first-order shear deformation theory with transverse shearing deformations and rotatory inertia included. The shell governing equations in terms of mid-surface displacements are obtained and, after expansion in partial Fourier series of the circumferential coordinate, solved with the G.D.Q.M. Several comparisons are made with available results, showing the reliability and modeling capability of the numerical scheme in argument.

Nonlinear Transient Analysis of Laminated Composite Shells

Abstract: In this study the geometrically nonlinear transient response of the laminated composite doubly curved shells is investigated using the finite element method. The finite element model includes nonlinearity due to large deflection. A nine-noded isoparametric composite shell element including the first-order shear deformation is developed in curvilinear coordinates for the nonlinear transient analysis. The present formulation is based on the total Lagrangian approach and the material behavior is assumed to be linear and elastic. The governing equations have been solved by the Newton–Raphson iteration technique, wherein the Newmark method has been used for the time integration. The present results are found to compare well with those available in the literature. The nonlinear transient response of the cross-ply and three specially laminated spherical, cylindrical, and hyperbolic paraboloidal shell panels are also analyzed and the results are discussed.

Multiple jets and bursting in the rapidly rotating convecting two-dimensional annulus model with nearly plane-parallel boundaries

Abstract: We analyse numerical solutions in the annulus model of rotating convection outside the tangent cylinder in a spherical shell. This model is capable of producing zonal flows with multiple jets. We investigate the conditions under which multi-jet solutions can be found. Although boundary friction reduces the strength of the zonal flow, it enhances the formation of multi-jets. More general models have a well-defined Ekman-layer term. In the annulus model, the Ekman-layer term has a similar form, but with variable strength. We have explored how the strength of the Ekman-layer term affects the form and strength of the zonal flows. We find that strong multi-jet zonal flows can be found for realistic values of the boundary friction, and hence have implications for convection in experiments and enclosed planetary cores. In addition, at higher Rayleigh numbers the importance of boundary friction is enhanced relative to bulk viscosity. Convection in the annulus model often occurs in the form of shortlived bursts as opposed to quasi-steady equilibriums. We have investigated when these events occur and their characteristics. In particular, we find precursors and afterglows of the convective bursts. We have obtained the β-scaling for a range of quantities when the thermal forcing is moderate. An examination of the components of the energy rate of change shows that the total Ekman-layer dissipation is of second order in the large β limit. However, the β-scaling of the forces driving the zonal flow seems to suggest that the zonal Ekman-layer dissipation remains important. We have introduced the concept of flow Taylorization, an analogue to the Taylorization used in magnetohydrodynamics studies and find a β-scaling of this quantity compatible with the moderate strength of the zonal flow. We also determine the typical length scale on which convection operates and compare this to the numerically determined length scale.

Local Casimir energies for a thin spherical shell

Abstract: The local Casimir energy density for a massless scalar field associated with step-function potentials in a 3+1 dimensional spherical geometry is considered. The potential is chosen to be zero except in a shell of thickness $\delta$, where it has height $h$, with the constraint $h\delta=1$. In the limit of zero thickness, an ideal $\delta$-function shell is recovered. The behavior of the energy density as the surface of the shell is approached is studied in both the strong and weak coupling regimes. The former case corresponds to the well-known Dirichlet shell limit. New results, which shed light on the nature of surface divergences and on the energy contained within the shell, are obtained in the weak coupling limit, and for a shell of finite thickness. In the case of zero thickness, the energy has a contribution not only from the local energy density, but from an energy term residing entirely on the surface. It is shown that the latter coincides with the integrated local energy density within the shell. We also study the dependence of local and global quantities on the conformal parameter. In particular new insight is provided on the reason for the divergence in the global Casimir energy in third order in the coupling.

McSCIA: application of the Equivalence Theorem in a Monte Carlo radiative transfer model for spherical shell atmospheres

Abstract: A new multiple-scattering Monte Carlo 3-D radiative transfer model named McSCIA (Monte Carlo for SCIAmachy) is presented. The backward technique is used to efficiently simulate narrow field of view instruments. The McSCIA algorithm has been formulated as a function of the Earth's radius, and can thus perform simulations for both plane-parallel and spherical atmospheres. The latter geometry is essential for the interpretation of limb satellite measurements, as performed by SCIAMACHY on board of ESA's Envisat. The model can simulate UV-vis-NIR radiation. First the ray-tracing algorithm is presented in detail, and then successfully validated against literature references, both in plane-parallel and in spherical geometry. A simple 1-D model is used to explain two different ways of treating absorption. One method uses the single scattering albedo while the other uses the equivalence theorem. The equivalence theorem is based on a separation of absorption and scattering. It is shown that both methods give, in a statistical way, identical results for a wide variety of scenarios. Both absorption methods are included in McSCIA, and it is shown that also for a 3-D case both formulations give identical results. McSCIA limb profiles for atmospheres with and without absorption compare well with the one of the state of the art Monte Carlo radiative transfer model MCC++. A simplification of the photon statistics may lead to very fast calculations of absorption features in the atmosphere. However, these simplifications potentially introduce biases in the results. McSCIA does not use simplifications and is therefore a relatively slow implementation of the equivalence theorem.

McSCIA: application of the equivalence theorem in a Monte Carlo radiative transfer model for spherical shell

Abstract: A new multiple-scattering Monte Carlo 3-D radiative transfer model named McSCIA (Monte Carlo for SCIAmachy) is presented. The backward technique is used to efficiently simulate narrow field of view instruments. The McSCIA algorithm has been formulated as a function of the Earth’s radius, and can thus perform simulations for both plane-parallel and spherical atmospheres. The latter geometry is essential for the interpretation of limb satellite measurements, as performed by SCIAMACHY on board of ESA’s Envisat. The model can simulate UV-vis-NIR radiation. First the ray-tracing algorithm is presented in detail, and then successfully validated against literature references, both in plane-parallel and in spherical geometry. A simple 1-D model is used to explain two different ways of treating absorption. One method uses the single scattering albedo while the other uses the equivalence theorem. The equivalence theorem is based on a separation of absorption and scattering. It is shown that both methods give, in a statistical way, identical results for a wide variety of sce narios. Both absorption methods are included in McSCIA, and it is shown that also for a 3-D case both formulations give identical results. McSCIA limb profiles for atmospheres with and without absorption compare well with the one of the state of the art Monte Carlo radiative transfer model MCC++. A simplification of the photon statistics may lead to very fast calculations of absorp tion features in the atmosphere. However, these simplifications potentially introduce biases in the results. McSCIA does not use simplifications and is therefore a relatively slow implementation of the equivalence theorem. For the first time, however, the validity of the equivalence theorem is demonstrated in a spherical 3-D radiative transfer model.

Elasto‐plastic finite element analysis of shells with damage due to microvoids

Abstract: This paper presents a non-linear finite element analysis for the elasto-plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41:3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C0 quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto-plastic model for shells presented by Voyiadjis and Woelke (General non-linear finite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek-Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD-Vol. 183/MD-50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage effects, the influence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34:1089–1104) is used to derive the large rotation, elasto-plastic-damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted). In addition, a non-layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto-plastic-damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright © 2006 John Wiley & Sons, Ltd.

Depressions at the surface of an elastic spherical shell submitted to external pressure.

Abstract: Elasticity theory calculations predict the number N of depressions that appear at the surface of a spherical thin shell submitted to an external isotropic pressure. Using a model that mainly considers curvature deformations, we show that N depends on the relative volume variation and on an adimensional parameter that takes into account both the relative spontaneous curvature and the relative thickness of the shell. Equilibrium configurations show single depression (N = 1) for small volume variations, then N increases, at maximum up to 6, before decreasing more abruptly due to steric constraints, down to N = 1 again for maximal volume variations. These static predictions are consistent with previously published experimental observations.

The dynamical Casimir effect for different geometries

Abstract: We consider the problem of motion-induced photon creation from quantum vacuum inside closed, perfectly conducting cavities with time-dependent geometries. These include one-dimensional Fabry–Perot resonators with Dirichlet or Neumann boundary conditions, three-dimensional cylindrical waveguides, and a spherical shell. The number of Casimir TE, TM and TEM photons is computed. We also present a classical mechanical analogue of the one-dimensional dynamical Casimir effect.

Axisymmetric vibrations of closed spherical shells: equations of motion and bifurcation analysis

Abstract: We investigate the torsionless axisymmetric dynamics of a forced closed spherical shell. The non-linear equations of motion are formulated using a variational approach and surface analysis. First, we revisit the linear eigenvalue problem. Then, using the method of multiple scales, we assess the possibility of the activation of two-to-one internal resonances between the different types of modes. Lastly, we examine the shell's non-linear responses to an axisymmetric primary-resonance excitation and analyse their bifurcations. Copyright © 2005 John Wiley & Sons, Ltd.

A Non-axisymmetric Spherical α 2 -Dynamo

Abstract: Using the Chebyshev-tau method, the generation of oscillatory non-axisymmetric stellar magnetic fields by the α 2 -dynamo is studied in spherical geometry. Following the boundary conditions given by Schubert & Zhang, the spherical α 2 -dynamo consists of a fully convective spherical shell with inner radius r i and outer radius r o . A comparison of the critical dynamo numbers of axisymmetric and -dependent modes for different thicknesses of the convective shell and different α -profiles leads to the following qualitative results: (i) when the angular factor of α -profile is sin n θ cos θ ( n = 1,2,4) the solutions of the α 2 -dynamo are oscillatory and non-axisymmetric, (ii) the thinner the convective shell, the more easily is the non-axisymmetric mode excited and the higher is the latitudinal wave number, (iii) the thickness of the outer convective shell has an effect on the symmetries of the magnetic fields.

Snap-through of the system of open shallow axi-symmetric bimetallic shell by non-linear theory

Abstract: The paper deals with stresses, strains and buckling conditions in the thin axi-symmetric shallow bimetallic shells with circular opening at the top of the shell. According to the third order theory by the Czech researcher E. Chawalla, which takes into account the equilibrium state of forces and moments acting on the deformed system, the paper presents a model for mathematical description of the system's geometry, stresses, thermo-elastic strains and displacements. The mathematical formulation is based on the theory of large displacements. As an example, the results for spherical shallow shells are shown, approximated by a parabolic function. The shells are loaded with temperature and/or with load acting at the top of the shell or at the shell's inner edge. The displacement state and the snap-through temperature are calculated numerically by the Runge Kutta algorithm of the fourth order, by non-linear shooting method.

Dynamical Casimir effect for different geometries

Abstract: We consider the problem of motion-induced photon creation from quantum vacuum inside closed, perfectly conducting cavities with time-dependent geometries. These include one dimensional Fabry-Perrot resonators with Dirichlet or Neumann boundary conditions, three dimensional cylindrical waveguides, and a spherical shell. The number of Casimir TE, TM and TEM photons is computed. We also present a classical mechanical analogue of the one dimensional dynamical Casimir effect.

Modal analysis of conical shell filled with fluid

Abstract: As a basic study on the fluid-structure interaction of the shell structure, a theoretical formulation has been suggested on the free vibration of a thin-walled conical frustum shell filled with an ideal fluid, where the shell is assumed to be fixed at both ends. The motion of fluid coupled with the shell is determined by means of the velocity potential flow theory. In order to calculate the normalized natural frequencies that represent the fluid effect on a fluid-coupled system, finite element analyses for a fluid-filled conical frustum shell are carried out. Also, the effect of apex angle on the frequencies is investigated.

Analytical modeling of natural convection in concentric spherical enclosures

Abstract: A modeling procedure is developed for natural convection heat transfer from an isothermal heated sphere located at the center of an isothermal, cooled, spherical-shaped enclosure. The model is based on the linear superposition of conduction and convection solutions, where the convective component is determined based on a combination of two limiting cases, laminar boundary-layer convection and transition flow convection. Validation of the model is performed using experimental and numerical data from the literature. The model and data are shown to be in good agreeement, with an rms difference of 2-4%.