Rigid Spheroids Research Articles

Near-field calculations for a rigid spheroid with an arbitrary incident acoustic field.

A general spheroidal coordinate separation-of-variables solution is developed for the determination of the acoustic pressure distribution near the surface of a rigid spheroid for a monofrequency incident acoustic field of arbitrary character. Calculations are presented, for both the prolate and oblate geometries, demonstrating the effects of incident field orientation and character (plane-wave, spherical wave, cylindrical wave, and focused beam) on the resultant acoustic pressure distribution.

A numerical study of the interfacial transport characteristics outside spheroidal bubbles and solids

In this paper, the interfacial transport characteristics to or from spheroidal bubbles are studied numerically and the results of numerical simulations are reported. The numerical modelling uses the SIMPLE method with a non-orthogonal, boundary fitted staggered grid. The aim of this paper is to provide an understanding of the interfacial transport characteristics of inviscid spheroidal bubbles, of different geometric parameters, rising in a stagnant hot or bi-solution liquid. The flow and concentration (or temperature) fields around bubbles and similarly sized rigid spheroids are computed. Detailed analyses of the pressure and vorticity distributions at surfaces of the inviscid spheroidal bubbles are made and compared with those of similar rigid spheroids. Drag coefficients for and wake lengths behind rigid spheroids and inviscid spheroidal bubbles are also presented. Local and mean Sherwood numbers and Nusselt numbers at bubble and rigid spheroid surfaces are compared, as well as their change with the Reynolds and Schmidt numbers and geometric parameters.

水中生物の音響散乱への境界要素法の応用

The Boundary Element Method (BEM) is applied to calculate the target strength (TS) and the total cross section of a scatterer. The scattering amplitudes of the prolate spheroids of four types, namely, the vacant, rigid, liquid-filled, and gas-filled prolate spheroids are calculated by the BEM. The comparison of the results of the BEM and those of Prolate Spheroid Model (PSM), which is a numerical calculation using the spheroidal wave functions, shows excellent agreement for the vacant, rigid, and liquid-filled prolate spheroids. The TS of a gas-filled prolate spheroid calculated by the BEM shows resonance at a low frequency.

Acoustic scattering by an arbitrarily shaped body: an application of the boundary-element method

Abstract The boundary-element method (BEM) is applied to calculate target strength (TS) and extinction cross-section of a scatterer. The scattering amplitudes of four types of prolate spheroid, namely, the vacant, rigid, liquid-filled, and gas-filled, are calculated using the BEM. Comparison between the results of the BEM and the prolate-spheroid model (PSM), which is a numerical calculation using spheroidal-wave functions, shows excellent agreement for the vacant, rigid, and liquid-filled spheroids. The TS of a gas-filled prolate spheroid, calculated by the BEM, shows resonance at a low frequency. The lowest resonant frequency, the peak ratio, and the quality factor (Q) of this resonance are compared with those given by the T-matrix and an analytical solution. Again the results show excellent agreement.

Motion of diatom chains in steady shear flow

Information on the motion of phytoplankton in shear flows is essential for predicting or understanding effects of turbulence on processes such as nutrient uptake, aggregate formation, and phytoplankter‐herbivore interactions. Of particular interest is the motion of diatom chains because they are expected to benefit most from enhancement of nutrient flux due to turbulence and are often the most abundant components in aggregates. We studied the motion of two chain‐forming diatoms, Skeletonema costatum and Thalassiosira nordenskiiildii, in steady shear flow and in the light of available theory for rigid, elongated spheroids. Both species underwent periodic rotation upon exposure to a simple shear flow, as predicted by theory. Whereas the rotational orbits of S. costutum resembled those predicted by theory for rigid spheroids, the rotational motion of T. nordenskiiildii was more like the motion of flexible fibers observed in engineering applications. Periods of rotation of S. costutum increased linearly with increasing axis ratio, whereas no clear relationship was observed between periods of rotation and axis ratios for T. nordenskiiildii. Measured periods of rotation of both species were smaller than predicted by theory for rigid spheroids of similar axis ratios. The diverse behaviors observed imply that fluxes of nutrients and collision frequencies experienced vary greatly with detailed shapes and mechanical properties of chains and their unit cells.

Dynamic simulation of flexible fibers composed of linked rigid bodies

A particle-level simulation method is employed to study the dynamics of flowing suspensions of rigid and flexible fibers. Fibers are modeled as chains of prolate spheroids connected through ball and socket joints. By varying the resistance in the joints, both flexible and rigid fibers can be modeled. Repulsive interactions between fibers are included, but hydrodynamic interactions and particle inertia are neglected in this implementation. The motion of a fiber is determined by solving the translational and rotational equations of motion for each spheroid. Simulations of isolated fibers in shear flow demonstrate that the method can reproduce known dynamical behavior of both rigid and flexible fibers. The transient behavior in the suspension relative viscosity under simple shear flow was also investigated. An oscillatory response similar to the experimental observations of Ivanov et al. was obtained for rigid fibers. Fiber flexibility reduced the period of oscillation, but had little effect on steady-state viscosities. These results verify that rigid and flexible fibers can be modeled with linked rigid prolate spheroids. Modeling fibers with fewer elongated bodies, as opposed to many spheres, significantly reduces computation time and facilitates the study of suspensions of many interacting, long fibers.

Determination of Far-Field Pattern of Rigid Scatterers Using Independent Finite Element Method and Eigenfunction Expansion, Part 2: Nonaxisymmetric Scattering

The near- and far-field steady state scattered potentials around a rigid arbitrary obstacle subjected to plane incident wave are computed by FE and eigenfunction expansion approach. The near-field is bounded by a truncation surface which is spherical. The nonaxisymmetric damper equation developed by Bayliss et al. (SIAM J. Appl. Math. Vol. 42, pp. 430–451, 1982) is employed on this surface. The computed FE near-field potential on the truncation boundary is employed as a Dirichlet boundary condition to obtain the expansion coefficients which eventually help in obtaining the complete solution in the entire outer domain including the far-field. The numerical technique is applied to problems of rigid scattering of beam-on- and oblique-incident waves by rigid prolate spheroid and hemispherically capped cylinder and contour plots of far-field scattered potential are presented.

Transport properties of a binary gas mixture of molecules with internal energy. II. Thermal conductivity

General expressions in terms of various square-bracket integrals for the thermal conductivity λ and its first-order approximation [λ]1 of a classical dilute binary gas mixture of nonspherical molecules have been obtained by extending the Taxman classical kinetic theory of transport phenomena within the framework of the Chapman–Enskog second approximation method for the mixture of spherical molecules. In the limit, [λ]1 formula tends to the corresponding formula for a dilute simple gas of nonspherical molecules or a dilute binary gas mixture of spherical molecules. The dimensionalities of the square-bracket integrals belonging to [λ]1 for the mixture of rigid spheroids having C∞v symmetry have been reduced by integrating analytically the velocity and angular velocity parts of the integrals. The resulting reduced quadratures are over the orientational coordinates of two colliding like or unlike spheroids. Two limiting checks are discussed which confirm that the quadrature formulas are correct.

The surface structure of cake formed by uniform-sized rigid spheroids in cake filtration

The cake formed by uniform-sized spheroidal particles during cake filtration is simulated with the view of force and torque analyses on the particles. Various effects on the structure of the filter cake, such as the filtration rate, concentration of slurry, particle shape, and friction coefficient between particles, are discussed in depth. The simulated results show that the particle shape plays the major role in the packing porosity. As the shape of particles departs from spherical, the structure of the filter cake becomes looser. The simulation results of cake porosity are compared with the available experimental data of random packing under gravity, and good agreement between them exists when the relative sphericity of the particles is larger than 0.7.

Computational analysis of convection heat transfer to non-spherical particles

Steady laminar axisymmetric thermal flow past solitary oblate spheroids and non-spherical droplets at constant temperature has been numerically analysed. Mean Nusselt number correlations as a function of Reynolds number (i.e. 40 ≤ Re ≤ 120), aspect ratio (i.e. 0.2 ≤ E ≤ 1.0) and Prandtl number (i.e. Pr = 0.7 and 7.0) are presented for rigid spheroids. Non-sphericity effects of water and n-hexane fuel droplets on heat transfer mechanisms are also discussed. Thus, the analysis may form a base case for studying spheroidal particle effects in convection heat transfer systems.

Signal rétrodiffusé par un corps élancé de révolution

The echoes backscattered in the far field by a rigid prolate spheroid at low and medium frequencies are computed and commented. The so-called slender-body aproximation is applied to express the pressure backscattered by the body when insonified by a harmonic plane wave. The impulse response is then calculated by Fast Fourier Transform for different angles of incidence. The computed echoes exhibit two main peaks, one narrow and one spread out, corresponding respectively to the reflected echo and to the diffracted waves that encircle the body once («creeping waves»). The time of arrival of the specular echo corresponds to that provided by Geometrical Optics. An explanation, resulting from the analysis of the computed echos, is proposed for the times of arrival of the diffracted waves at oblique incidence. However, a theoretical study is required to check the validity of that explanation

Diffusiophoresis and Electrophoresis of Colloidal Spheroids

An analytical study is presented for the diffusiophoresis and electrophoresis of a rigid nonconducting spheroid in a uniform applied field that is oriented arbitrarily with respect to its axis of revolution. The range of the interaction between the solute species and the particle surface is assumed to be small relative to the dimension of the particle, but the effect of polarization of the diffuse species in the thin solute-particle interaction layer is incorporated. To solve the conservative equations governing the system, a slip velocity of fluid and normal fluxes of solute species at the outer edge of the thin diffuse layer which balance convection and diffusion of the solute species along the particle surface are used as the boundary conditions for the fluid domain outside the diffuse layer. Explicit expressions for the migration velocity of the spheroidal particle are obtained for the cases of diffusiophoresis in a nonionic solute concentration gradient, diffusiophoresis in a concentration gradient of symmetric electrolyte, and electrophoresis in an external electric field. An interesting feature is found that the diffusiophoretic or electrophoretic velocity of the particle decreases with the reduction of the maximum length of the particle in the direction of migration. Also, the average migration velocity for an ensemble of identical, non-interacting spheroids with random orientation distribution is obtained for each case considered.

Acoustic image analysis of acoustic scattering from rigid spheroids and cylinders

Acoustic imaging can be used to display the spatial, frequency, and temporal effects of acoustic scattering. The quantitative images can be coupled with simple scattering models to elucidate scattering mechanisms. Acoustic scattering results are presented from rigid spheroids and spheroidally endcapped cylinders with similar length-to-width ratio. The bistatic patterns produced from end-ensonification differ due to the reflectivity of the ends and the amount of reradiation along the length of the scatterer. The ends of prolate spheroids possess higher curvature than endcapped cylinders and therefore reflect less into the back direction. The cylinders possess a straight section along which the creeping wave does not radiate. These scattering phenomena are shown in acoustic images formed from the far-field scattering patterns.

Acoustic scattering from a rigid prolate spheroid

Acoustic scattering from a rigid prolate spheroid

A comparison between the boundary element method and the wave superposition approach for the analysis of the scattered fields from rigid bodies and elastic shells

The steady state analysis of the scattering of plane acoustic waves from submerged rigid and elastic bodies using two approaches is presented. The first approach uses a combined finite element/boundary element (FE/BE) methodology. The NASA structural analysis (NASTRAN) program is used to formulate the structural matrices based on the finite element method (FEM). The surface integral equation radiation and scattering (SIERRAS) program creates the fluid matrices based on the boundary element method (BEM) and solves the coupled fluid-structure interaction problem. A superparametric boundary element (BE) with nine nodes is employed. The combined Helmholtz integral equation formulation (CHIEF) is employed to provide a unique solution for all frequencies. In the second approach, the superposition method (SUP) is used for modeling the fluid. The SUP method is an off-boundary approach that employs a number of point sources moved inside the body to represent the fluid response at the surface. This allows the fluid matrices to be formed without surface integration. Formulations for the SUP method are introduced for both the radiation and scattering problems. The program SUPER reads the NASTRAN structural matrices and solves the combined FE/SUP fluid-structure equations. The FE/BEM and FE/SUP methods are applied to the scattering of an infinite set of plane waves from a submerged rigid sphere, rigid prolate spheroid, right cylinder, and from an elastic cylindrical shell with hemispherical endcaps. The SUP method is found to be easier to implement and to provide an accurate result at internal resonance frequencies where the BEM requires additional equations to ensure an accurate solution.

Flexible and rigid macromolecules in shear and electric fields

Models for rigid and flexible polymers dissolved at low concentration in a dielectric Newtonian fluid and subjected to shear and electric (or magnetic) fields are developed. The rigid polymers are taken to be a rigid spheroid with high aspect ratio and the flexible chains are considered as an elastic dumb-bell with a nonlinear spring constanl. Specific calculation schemes are developed for transient shear and/or electric fields. Rheo-optical properties such as birefringence and extinction angle are calculated and also interesting components for stress tensor are predicted.

Electrohydrodynamics of rigid macromolecules with permanent and induced dipole moments. II

The orientation distribution of rigid macromolecules dissolved at low concentration in a dielectric Newtonian fluid is solved for the case in which simultaneous hydrodynamic and electric fields are applied. Macromolecules are taken to be Brownian rigid spheroids of arbitrary aspect ratio and both permanent and induced dipole moments are included. To solve the problem with arbitrary strength of fields, the Galerkin method based on spherical harmonics is adopted. The converged solutions are used in the prediction of the steady state birefringence and extinction angle over very wide ranges of field strengths. Due to the coupled effects of flow and external field, somewhat complicated responses are predicted and saturation features are revealed at the strong flow limit. External field dependence due to the induced dipole is similar to that of the permanent case for most field conditions. Finally, it is shown, taking poly‐γ‐benzyl‐L‐glutamate solutions with various molecular weights as an example, how results in this article are applicable to an analysis of a real system.

Electrohydrodynamics of Rigid Macromolecules with Permanent and Induced Dipole Moments

A model of rigid macromolecules dissolved at low concentration in a dielectric Newtonian fluid and subjected to shear and electric fields is developed. Macromolecules are taken to be rigid spheroids of arbitrary aspect ratio and with both permanent and induced dipole moments. Specific calculations are reported for the steady‐state birefringence, extinction angle, and rheological material functions. The procedure developed here is general, so that it may be applied both to transient properties and to strong field problems since terms of higher order in the shear and electric fields may be calculated readily.

Comparison of asymptotic and exact results for scattering from rigid spheroids

Results generated with the Kirchhoff approximation, which is effective for backscattering, and the diffraction integral method, which is effective for forward scattering, will be compared with the exact solution obtained with Waterman's extended boundary condition method. Calculations will be presented for kL/2 ranging from 5 to 300 and for aspect ratios ranging from 1 to 15. An additional asymptotic approximation, the method of stationary phase, is used for higher frequencies. For higher aspect ratio targets, one would expect the accuracy of high-frequency methods to be difficult to predict and to depend not only on kL/2 but also on bistatic configurations due to the large variation of Gaussian curvature on these targets. Hence, the focus of the presentation will be to illustrate how accuracy depends upon kL/2, angle of incidence, and angle of observation for high aspect ratio targets. [Work supported by ONR and NORDA.]

The effect of an object in a waveguide on the field produced by a point source

When the acoustic wave produced by a point source in a shallow water waveguide ensonifies an object, the object acts as a secondary source and in turn generates a guided wave. The effect that the object has on the initial incident field is estimated. In particular, comparisons are made with transmission loss versus range for the point source by itself and with transmission loss versus range for the object as ensonified by the point source. To obtain the scattered field from the object, the scattered field is first generated in the vicinity of the object using a transition matrix that relates the incident field to the scattered field in a waveguide. The transition matrix is obtained from the extended boundary condition (EBC) method. This solution is coupled with the waveguide solution [G. V. Norton and M. F. Werby, “Some Numerical Approaches to Describe Acoustical Scattering from Objects in a Waveguide,” in Proc. Sixth Int. Conf. Math. Modeling, St. Louis, MO, 4-7 Aug. 1987]. This method satisfies all appropriate boundary conditions and yields a continuous solution throughout the waveguide. The object used is a rigid spheroid of aspect ratio 5:1. The frequencies are 100 and 450 Hz. The source, receiver, and object depths are different combinations of 50, 75, and 200 m, and two waveguide water depths at 150 and 400 m are considered. The nature of the effect on the initial incident field due to the object and as influenced by the relative placement of the source and receiver relative to it is demonstrated. [Work supported by NORDA.]