Radiation and scattering from loaded bodies of revolution

The problem of electromagnetic radiation and scattering from loaded bodies of revolution of arbitrary shape is considered. The analysis assumes the existence of an impedance function relating the tangential electric field to the surface current on the body. A solution is obtained by the method of moments applied to the potential integral formulation of the problem. The results are expressed in terms of generalized network parameters, using formulas previously obtained for unloaded bodies. Representative computations are given for plane-wave scattering and radiation from apertures in loaded cylinders and hemispheres. A general computer program for arbitrary bodies of revolution is available.     

Optimization of the shape of a body with respect to two criteria: Radiation heat flux and wave drag

The numerical solution of the two-criteria variational problem of the body contour with minimum radiation heat flux and wave drag is obtained in the class of axisymmetric and plane slender bodies in hypersonic flow. Solutions obtained using the Pareto, ideal point and minimax methods are compared. It is shown that in the class of axisymmetric slender bodies the optimum body gives a decrease in the radiation heat flux as compared with a cone of up to 15% for the Pareto method, up to 13% for the ideal point method, and up to 5% for the minimax method. A solution is also obtained in the subclass of power-law slender bodies and it is shown that the optimum power-law bodies are inferior, as compared with the optimum bodies from the general class of such bodies, in reducing both radiation heating and resistance.     

New expressions for the radiation force function of spherical targets in stationary and quasi-stationary waves

A new expression for the radiation force function – which is the radiation force per unit energy density and unit cross-sectional surface area – for spheres in a stationary (or standing) and quasi-stationary wave is obtained based on the far-field acoustic scattering field. The radiation force function formulation has been simplified mathematically and improved into a more general form. Numerical results are presented for rigid and elastic spheres, air bubbles in water as well as liquid drops in air to illustrate the theory. It is demonstrated that expressions for the radiation force functions obtained from the far-field derivation approach are equivalent to those obtained from the near-field-based derivation.     

Stress singularities in an anisotropic body of revolution

To fill the gap in the literature on the application of three-dimensional elasticity theory to geometrically induced stress singularities, this work develops asymptotic solutions for Williams-type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials. The Cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution, so the problems under consideration are very complicated. The eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three-dimensional equilibrium equations in terms of the displacement components. The correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed-form characteristic equations for an isotropic body of revolution and using the commercial finite element program ABAQUS for orthotropic bodies of revolution. Thereafter, the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries, made of a single material or bi-materials, under different boundary conditions.     

Solution of the axisymmetric inverse problem by higher-order line doublets

The axial singularity inverse method for designing bodies of revolution has been improved by using higher-order doublet elements. The performance of the method for various element orders and other solution parameters is presented in detail. The results indicate that the method is generally more robust, less sensitive to insets and has a better-conditioned coefficient matrix compared with the source method of the same order. The condition number of the matrix is shown to increase with the thickness of the body, the order of the method, the number of elements and the degree of stretching of the node distribution. In general, good performance is attained for most bodies even with ?_r as low as 2 by using 10–12 second-order doublet elements with insets greater than 0.02L from rounded ends. Increasing the insets to 0.06L appears to improve the accuracy of the method for most bodies but slows its convergence.     

多个椭圆柱波浪力的一种解析解

波浪在大尺寸结构表面产生不可忽略的散射波, 该散射波在多柱体体系中继续传播, 并在同体系中的其他柱体上产生高次散射波. 本文基于椭圆坐标系和绕射波理论首先推导了波浪作用下椭圆单柱体产生的散射波压力公式, 随后考虑该散射波在多柱体系中的传播, 将其视为第二次入射波, 推导出柱体上第二次散射波压力公式, 同理可以推导出高次散射波压力公式, 最后得到椭圆多柱体波浪力解析解, 并用数值解验证了本文解析方法的正确性. 本文以双柱体和四柱体体系为例, 分析了不同参数(波数、净距、波浪入射角度等)下, 高次散射波对柱体上波浪作用的影响. 结果表明: 波数较大的情况下, 高次散射波引起柱体上的波浪力不能忽略; 结构间距较大的情况下, 虽然高次波的作用有减小的趋势但仍然明显; 高次散射波来自多个柱体对入射波的散射, 柱体数目的增加后, 高次波的影响会增加, 结构所受的高次波作用因参数变化而起的波动会变剧烈; 高次波对上游柱体波浪力的贡献较对下游柱体的贡献大.      

Computer extended series for a thermally driven gas centrifuge

Linearized, multidimensional, thermally driven flow in a gas centrifuge can be approximately described in regions away from the ends by Onsager's homogeneous pancake equation.¹ Upon reformulation of the general problem, we find a new, simple and rigorous closed form, analytical solution by assuming a special separable solution and replacing the usual Ekman end cap boundary conditions with idealized impermeable, free slip boundary conditions. Then the flow may be described by an ordinary differential equation with solutions in terms of simple, classical functions. By identifying a small parameter, say ?, defining the semi-long bowl approximation, and assuming a power series expansion in ?, a sequence of asymptotic approximations to the master potential is obtained. Not surprisingly, the leading order term involves the well known ‘long bowl’ solution. Using the so-called ‘solving’ property of the 1-D pancake Green's function,² we determine the next higher order solution. This recursive process is carried out on the computer to find all the terms up to O(?⁴). Consequently, the solution of some complex rotating, viscous, heat conducting flow problems that normally require large mainframe computers can be better understood.     

Computational and experimental study of supersonic flow over axisymmetric cavities

Laminar and turbulent computations are presented for annular rectangular-section cavities, on a body of revolution, in a Mach 2.2 flow. Unsteady ‘open cavity flows’ result for all laminar computations for all cavity length-to-depth ratios, L/D (1.33, 10.33, 11.33 and 12.33). The turbulent computations produce ‘closed cavity flows’ for L/D of 11.33 and 12.33. Surface pressure fluctuations at the front corner of the L/D = 1.33 cavity are periodic in some cases depending on the cavity length and depth, the boundary layer at the cavity front lip and the cavity scale. The turbulent computations are supported by experimental schlieren images, obtained using a spark light source, and time-averaged surface pressure data.     

Discrete-to-Continuum Limit of Magnetic Forces: Dependence on the Distance Between Bodies

We investigate force formulae for two rigid magnetic bodies in dependence on their mutual distance. These formulae are derived as continuum limits of atomistic dipole–dipole interactions. For bodies that are far apart in terms of the typical lattice spacing we recover a classical formula for magnetic forces. For bodies whose distance is comparable to the atomistic lattice spacing, however, we discover a new term that explicitly depends on the distance, measured in atomic units, and the underlying crystal lattice structure. This new term links the classical force formula and a limiting force formula obtained earlier in the case of two bodies being in contact on the atomistic scale.     

Two-Dimensional Elastic Herglotz Functions and Their Applications in Inverse Scattering

In this work solutions of the spectral Navier equation that satisfy the Herglotz boundedness condition in two-dimensional linear elasticity are presented. Navier eigenvectors in polar coordinates are introduced and it is established that they form a linearly independent and complete set in the L ²-sense on every smooth curve. It is also proved that the classical solutions of the spectral Navier equation are expressed via Navier eigenvectors, and this expansion converges uniformly over compact subsets of R ². Two far-field patterns, the longitudinal and the transverse one corresponding to the displacement field are introduced, and the Herglotz norm is expressed as the sum of the L ²-norms of these patterns over the unit circle. It is also established that the space of elastic Herglotz functions is dense in the space of the classical solutions of the spectral Navier equation. Finally, connection to inverse elasticity scattering is established and reconstructions of rigid bodies are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Fundamental Solution of the Linearized Navier–Stokes Equations for Spinning Bodies in Three Spatial Dimensions – Time Dependent Case

Explicit formulae for the fundamental solution of the linearized time dependent Navier–Stokes equations in three spatial dimensions are obtained. The linear equations considered in this paper include those used to model rigid bodies that are translating and rotating at a constant velocity. Estimates extending those obtained by Solonnikov in [23] for the fundamental solution of the time dependent Stokes equations, corresponding to zero translational and angular velocity, are established. Existence and uniqueness of solutions of these linearized problems is obtained for a class of functions that includes the classical Lebesgue spaces L^p(R³), 1 < p < ∞. Finally, the asymptotic behavior and semigroup properties of the fundamental solution are established.     

Instability of stationary and uniformly moving cylindrical fluid bodies—I. Newtonian systems

The early work of Tomotika provides a basis for analyzing the instability of stationary and uniformly moving cylindrical fluid bodies. The classical results of Rayleigh, Christiansen and Weber for critical growth rates and wavenumbers are obtained as zero order limits of Tomotika's general solution expanded in terms of the key characteristic parameters: the viscosity ratio, the density ratio and the dispersed and continuous phase Ohnesorge numbers. By employing more than one characteristic time, these limits, as well as others, are obtained in a general solution framework. Numerical results provide insights into the effects of the physical forces, as well as criteria and bounds for the application of the limiting cases.     

BEM solution to magnetohydrodynamic flow in a semi‐infinite duct

We consider the magnetohydrodynamic flow that is laminar and steady of a viscous, incompressible, and electrically conducting fluid in a semi‐infinite duct under an externally applied magnetic field. The flow is driven by the current produced by a pressure gradient. The applied magnetic field is perpendicular to the semi‐infinite walls that are kept at the same magnetic field value in magnitude but opposite in sign. The wall that connects the two semi‐infinite walls is partly non‐conducting and partly conducting (in the middle). A BEM solution was obtained using a fundamental solution that enables to treat the magnetohydrodynamic equations in coupled form with general wall conductivities. The inhomogeneity in the equations due to the pressure gradient was tackled, obtaining a particular solution, and the BEM was applied with a fundamental solution of coupled homogeneous convection–diffusion type partial differential equations. Constant elements were used for the discretization of the boundaries (y = 0, ?a ? x ? a) and semi‐infinite walls at x = ±a, by keeping them as finite since the boundary integral equations are restricted to these boundaries due to the regularity conditions as y → ∞ . The solution is presented in terms of equivelocity and induced magnetic field contours for several values of Hartmann number (M), conducting length (l), and non‐conducting wall conditions (k). The effect of the parameters on the solution is studied. Flow rates are also calculated for these values of parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

Characteristic relations of flow birefringence

The general problems of the flow-induced birefringence in liquids have been presented and discussed in Part 1 of this paper, in which some particular responses of an aqueous solution of NGS 1828 were presented, namely, the pertinent mechanical responses and the birefringence responses in transmitted radiation. This paper presents the optical responses of NGS 1828 observed in the scattered radiation. It is shown that the scattered-light techniques can become a powerful and indispensable tool of flow-birefringence techniques, if the actual patterns of light scattering are taken into account. Proof is given that the light scattering and the related birefringence cannot be described by Rayleigh's mathematical model of scattering (Part 1, Ref. 51). The optical effects of the modulation of the primary beam and the scattered beam, called scattered primary isochromatics and scattered secondary isochromatics, are described in terms of the parameters of the system. A practical example is presented. Samples of typical recordings of light-intensity modulation by typical flow patterns are given in the form of scattered primary and secondary isochromatics and integrated isochromatics. One type of intensity modulation enables the determination of shear-strain rate at an interior point in the flow field. Using the determined shear-strain rates at the interior points, a velocity profile along the axis of symmetry of a rectangular-conduit flow was obtained. The results so obtained were satisfactory when compared with the results of direct volumetric measurement. The other type of intensity modulation reveals the nonuniform distribution of birefringence along the path of the transmitted light in the rectangular-conduit flow. The results presented in Part 1 of this paper, are complementary to the results presented below.     

Magnetohydrodynamic flow in electrodynamically coupled rectangular ducts

In Sezgin^1,2 the problems considered are the magnetohydrodynamic (MHD) flows in an electrodynamically conducting infinite channel and in a rectangular duct respectively, in the presence of an applied magnetic field. In the present paper we extend the solution procedure of these papers to two rectangular channels connected by a barrier which is partially conductor and partially insulator. The problem has been reduced to the solution of a pair of dual series equations and then to the solution of a Fredholm's integral equation of the second kind. The infinite series obtained were transformed to finite integrals containing Bessel Junctions of the second kind to avoid the computations of slowly converging infinite series and infinite integrals with oscillating integrands. The results obtained compared well with those of Butsenieks and Shcherbinin3 which were obtained for the perfectly conducting barrier separating the flows.     

On the general expression of Fredholm Integral Equations Method in elasticity

In this paper, the concept of covering domain is introduced to develop a general expression for the Fredholm Integral Equations Method, by which elasticity problems of arbitrarily shaped bodies loaded by external forces can be solved. Some special expressions are given for a body with non-zero remote stresses, or subjected to some concentrated forces on its boundary. The relationship between the loading forces and solutions are also discussed. Some analytical solutions can be obtained for simple cases. When numerical computations are needed for the solution, the method proves to have high precision and fast convergency.     

Improving the differentiability of generalized fourier series solutions of boundary value problems of mechanics by using boundary functions

We prove a theorem on conditions for the differentiation of generalized Fourier series. We show that Fourier series solutions of boundary value problems can in general be differentiated term by term only once. To improve the differentiability properties of such series, we suggest to use pth-order boundary functions. We suggest an algorithm for constructing boundary functions for classical domains. This approach is illustrated by a new solution, with improved differentiability properties, of the problem on the torsion of an elastic rod of rectangular cross-section.     

On the stress-strain state of a transversally isotropic body with a paraboloidal inclusion

A closed solution is presented for the three-dimensional problem of the stress-strain state of an unbounded elastic body with a soldered-in transversally isotropic inclusion in the form of a paraboloid of revolution. Here, it is assumed that the body is under axisymmetric tension (compression). A solution of the corresponding problem for a paraboloidal recess is obtained as a special case. Podil’chuk [2, 3] has investigated similar problems for isotropic bodies with an inclusion assuming the form of a paraboloid of revolution or an elliptical paraboloid. Translated from Prikladnaya Mekhanika, Vol. 34, No. 11, pp. 16–22, November, 1998.     

Axisymmetric Collision Problem for Two Identical Elastic Solids of Revolution

A direct central collision of two identical bodies of revolution is studied. A nonstationary mixed boundary-value problem with an unknown moving boundary is formulated. Its solution is represented by a series in term of Bessel functions. An infinite system of Volterra equations of the second kind for the unknown expansion coefficients is derived by satisfying the boundary conditions. The basic characteristics of the collision process are determined depending on the curvature of the frontal surface of the bodies     

Opposing mixed convection and its interaction with radiation inside eccentric horizontal cylindrical annulus

In the present investigation, the coupled phenomenon of opposing mixed convection and radiation within differentially heated eccentric horizontal cylindrical annulus has been numerically simulated. The radiation transfer contributed from the participating medium is obtained by solving the nonlinear integro‐differential radiative transfer equation using discrete ordinate method. The participating gray medium is considered to be emitting, absorbing and isotropically scattering. The walls of the annulus are considered to be opaque, diffuse and gray. In the study it has been observed that the Richardson number ‘Ri’ has a small effect on the total Nusselt number ‘Nu’ in mixed convection heat transfer with or without radiation. From the present investigation it is found that substantial changes occur in isotherms as well as in flow patterns, when the Richardson number is allowed to vary in the range of 0.01–1. The influence of radiative parameters on the interaction phenomenon has been delineated through isotherm and streamline pattern. Copyright © 2008 John Wiley & Sons, Ltd.