Arbitrary Cross-sectional Shape Research Articles

Use of surface-integral equations for the analysis of the TE-induction problem

A surface-integral-equations approach for the analysis of the linear transverse-electric (TE) induction problem at an arbitrary frequency is presented. It parallels an earlier analysis of the TM induction problem, and uses Helmholtz potentials to derive integral equations correlating the surface values of the tangential current density and axial magnetic field. Two Fredholm integral equations with weakly singular kernels reduce to a single equation for a quasistatic magnetic field. The present approach reduces the dimensions of the numerical problem by one, and is valid for arbitrary cross-sectional shapes, arbitrary material properties and arbitrary frequencies. It is free of the limitations of the separation of variables method used in the low-frequency range and perfect-conductor analysis used in the high-frequency range. It is an exact dual of the TM analysis. This approach is used to compute the power loss over the whole frequency range for a rectangular conducting cylinder placed normal to an H-wave. It is shown that, in the low-frequency range, the surface charges have no appreciable effect on the power loss.

Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides

Using the finite element technique, a numerical method is devloped so that one may obtain the propagation characteristics of optical waves along guiding structures whose cores may be of arbitrary cross-sectional shape and whose material media may be inhomogeneous in more than one transverse direction. Several specific examples are given and the results are compared with those obtained by other exact or approximate methods. Very close agreement was found. The method developed here can easily be applied to many important problems dealing with practical optical fiber or integrated optical waveguides whose cross-sectional index of refraction distribution may be quite arbitrary.

Open-Channel Flow Equations Revisited

Equations of continuity, momentum, and energy in integral form are formulated herein by integrating over the cross section the corresponding equations for a point and then transformed into one-dimensional form by introducing the necessary correction factors. These one-dimensional equations can be regarded as the unified general open-channel flow equations. The channel can have arbitrary cross sectional shape and alignment with fixed or erodible and impervious or pervious bed. The fluid can be homogeneous or nonhomogeneous, compressible or incompressible, and viscous or inviscid. The flow can be turbulent or laminar, rotational or irrotational, steady or unsteady, uniform or nonuniform supercritical or subcritical, gradually or rapidly varied, and with or without lateral discharge. Based on these equations, the assumptions involved in conventionally used open-channel flow equations such as backwater flow equation and St. Venant equations are examined. The inherent differences between the flow equations derived based on the momentum and energy concepts are compared.

Ordinary size effects and deviations from Matthiessen's rule in the resistance of fine wires

Contrary to previous statements in the literature, large deviations from Matthiessen's rule in fine wiresare to be expected on the basis of a straight-forward solution of the ordinary transport equation, assuming the relaxation-time approximation and imposing the idealized condition of “diffuse” scattering of electrons at the boundaries. Using Chambers' path-integral method to evaluate the current density in a wire of arbitrary cross-sectional shape, the effects of boundary scattering on the resistivity in the regimed ≲0.1λ have been calculated for two model Fermi surface geometries. For the temperature-dependent part of the resistivity, Δρ d (T)≡ρ d (T)−ρ d (0), two distinct types of behavior are found in the alternative cases: (1) for a spherical Fermi surface, Δρ d(T) increases logarithmically with ρd(0); (2) for a cylindrical Fermi surface, Δρ d (T) increases essentially linearly with ρ d (0). [In each case the qualitative dependence of ρd(0) on λ/d is, for practical purposes, “linear.” However, the correct value of the product ρ∞λ in the cylindrical case is not simply given in the ordinary way by the slope of an empirical plot of ρ d (0) vs.d−1.] A comparison of theoretical results for the two simple models with the published data for indium and gallium shows that the actual temperature-dependent size effects are consistent, both qualitatively and, by a rough estimation, quantitatively, with the expected behavior.

Edge and corner contributions in Weyl's problem

The smoothed asymptotic eigenvalue distributions for scalar and electromagnetic waves are calculated for a finite cylindrical domain with arbitrary cross section shape. Four terms of the expansion are given including the edge and corner contributions unknown hitherto.

Seepage from Trenches through Nonhomogeneous Soils

Infiltration from shallow trenches of arbitrary cross-sectional shape and through a layer of nonhomogeneous soil was studied analytically. The soil layer is underlain by a bed of gravel that is represented mathematically by a line of constant pressure. The basic flow equations were derived and the corresponding boundary value problems constructed by use of the perturbation theory. The effect of an exponential variation of the hydraulic conductivity on the quantity of seepage and the free surface is determined. A comparison with the case of constant permeability is made for different layer thicknesses. A triangular trench was chosen to indicate the effect of trench dimensions on the amount of seepage through layers of nonuniform permeability. Curves are presented to illustrate this effect.

Hall effect in a long cylindrical sample having a cross section of arbitrary shape

Hall effect in a long cylindrical sample having a cross section of arbitrary shape

Ferrite-Filled Elliptical Waveguides. I. Propagation Characteristics

This paper deals with electromagnetic wave propagation in a longitudinally magnetized, ferrite-filled elliptical waveguide. Using the wave equations and boundary conditions for a ferrite-filled waveguide of arbitrary cross-sectional shape, it is shown that for an elliptical cross section the characteristic equations for the propagation constant take the form of even and odd infinite determinants. Solutions for the characteristic equations are obtained for various values of applied field strength, saturation magnetization, and waveguide ellipticity. The existence of three types of cutoff conditions is demonstrated.

Special Relationships between the Farfield and the Radiation Impedance

Some new relationships between the sound pressure at certain points in the farfield and the radiation impedance for certain acoustic sources have been derived from the Helmholtz integral formula. Under special conditions these relationships reduce to exact analytical solutions of problems that can otherwise only be solved numerically. For example, for a cylinder of arbitrary cross sectional shape with the ends vibrating uniformly and the sides motionless the farfield pressure in the direction of the cylinder axis is simply related to the radiation impedance. Such relationships provide helpful checkpoints for evaluating the success of some of the elaborate numerical methods which are currently being developed for attacking previously unsolved acoustic radiation problems with large digital computers. In certain cases, they can also provide new methods for determining radiation impedance from farfield calculations or possibly from farfield measurements. [Work supported by U. S. Navy Underwater Sound Laboratory.]

Non-Newtonian flow through packed beds and porous media

The general problem of viscous flow of an arbitrary time-independent non-Newtonian fluid through packed beds and porous media is examined. The Blake-Kozeny capillary model, utilizing general relationships describing flow in straight ducts of arbitrary cross-sectional shape, has yielded a general relationship involving the non-Newtonian or apparent viscosity of the fluid and the impermeability factor plus an additional aspect factor characteristic of the bed, which satisfactorily correlates the available experimental data. The authors' experimental data and data obtained in an independent study by Sadowski with high molecular weight Natrosol-250 H aqueous solutions was successfully correlated by allowing for the existence of an anisotropic anomalous layer on the surface of the particles. The negative effective velocity on the surface computed from the data suggests polymer adsorption-gel formation at the surface, also observed physically by Sadowski. Estimated thickness of the anomalous layer are presented.

Non-Newtonian flow in ducts of arbitrary cross-sectional shape

Two general equations are proposed for prediction of the flow rate and maximum velocity versus pressure drop relationships in the isotherma,l steady, uniform, laminar flow of incompressible, time-independent non-Newtonian fluids in ducts of arbitrary cross section. The equations are expressed in terms of two parametric constants characteristic of the shape of the flow cross section and a function of shear stress characterizing the fluid. Numerical values of the geometric parameters have been determined for circular, slit, concentrically annular, rectangular, elliptical and isosceles triangular ducts. Comparisons with available experimental and analytical results for Newtonian, Ostwald-de Waele, Bingham and Rabinowitsch fluids indicate good agreement. A generalization of the Fanning friction factor-Reynolds number considerations is presented yielding the relationship f = 16/Re * applicable to laminar flow through ducts of arbitrary cross section. Methods for prediction of the transition point from laminar flow in the general case are suggested. The effect of the existence of a slip velocity at the wall is considered. Applications are illustrated by examples.

TE-wave scattering by a dielectric cylinder of arbitrary cross-section shape

The theory and equations are developed for the scattering pattern of a dielectric cylinder of infinite length and arbitrary cross-section shape. The harmonic incident wave is assumed to have its electric vector perpendicular to the axis of the cylinder, and the fields are assumed to have no variations along this axis. Although some investigators have approximated the field within the dielectric body by the incident field, a more accurate solution is obtained here by treating the field as an unknown function which is determined by solving a system of linear equations. Scattering patterns obtained by this method are presented for dielectric shells of circular and semicircular cross section, and for a thin plane dielectric slab of finite width. The results for the circular shell agree accurately with the exact classical solution. The effects of surface-wave excitation and mutual interaction among the various portions of the shell are included automatically in this solution.

Modal network theory of skin effect in flat conductors

A new theory of skin effect in linear conductors of arbitrary cross-sectional shape is proposed and illustrated by calculating the ac resistance of a flat conductor. The current flowing in any conductor may be decomposed into an infinite set of normal mode currents, all of which exist independently of each other. Each modal current may be calculated from the conductor terminal conditions by simple circuit theory. Use of this method presupposes knowledge of the normal modes, which are different for every cross-sectional shape. A general method of finding the normal modes is given. Excellent agreement with experimental data is reported.

Elastic Torsional Analysis of Irregular Shapes

A direct stiffness (finite element) solution method for the torsional analysis of shafts is presented. The technique is applicable to prismatic shafts (linear elastic material) of arbitrary cross-sectional shapes (single or multiple connected bodies) that are free to warp. The analysis yields the torsional stiffness, shear stress distribution, and warping function distribution for the shaft. The accuracy of the analysis is demonstrated by the consideration of problems with known solutions.

Scattering by a dielectric cylinder of arbitrary cross section shape

The theory and equations are developed for the scattering pattern of a dielectric cylinder of arbitrary cross section shape. The harmonic incident wave is assumed to have its electric vector parallel with the axis of the cylinder, and the field intensities are assumed to be independent of distance along the axis. Solutions are readily obtained for inhomogeneous cylinders when the permittivity is independent of distance along the cylinder axis. Although other investigators have approximated the field within the dielectric body by the incident field, we treat the total field as an unknown function which is determined by solving a system of linear equations. In the case of the dielectric cylindrical shell of circular cross section, this technique yields results which agree accurately with the exact classical solution. Scattering patterns are also presented in graphical form for a dielectric shell of semicircular cross section, a thin homogeneous plane dielectric sheet of finite width, and an inhomogeneous plane sheet. The effects of surface-wave excitation and mutual interaction among the various portions of the dielectric shell are included automatically in this solutiom

Scattering from cylinders with arbitrary surface impedance

This paper presents an integral equation method for the solution of the field scattered by a set of cylinders with arbitrary cross-sectional shape, and arbitrarily varying anisotropic surface impedance. The integral equations are given for an arbitrary source with arbitrary harmonic variation along the cylinder axis. The scattering problem can be solved for arbitrary three-dimensional sources by expansion of the source in a Fourier integral over the axial propagation constant. The integral equations have been programmed for a CDC 1604A computer. The program developed has been used to solve a great variety of scattering, antenna, and propagation problems, and, depending upon accuracy desired, will handle cylinders up to about 150-wavelengths total perimeter. Numerical results on scattering from cylinders with specific cross sections are presented to illustrate the utility of the program developed.

Sound Propagation in a Tube of Arbitrary Cross-Sectional Shape

Sound Propagation in a Tube of Arbitrary Cross-Sectional Shape

Relaxation effects on the flow over slender bodies

The effects of heat-capacity lag on the flow over slender bodies are examined by means of an extension of Ward's (1949) generalized treatment of the slender-body problem. The results are valid for smooth bodies of arbitrary cross-sectional shape and attitude in the complete Mach number range up to, but not including, hypersonic conditions. Transonic flow can be treated owing to the presence of a dissipative mechanism in the basic differential equation, but the results in this Mach number range are probably of limited practical value.The results show that cross-wind forces are unaffected to a first approximation, but that drag forces comparable with laminar skin-friction values can arise as a result of the relaxation of the internal degrees of freedom. The magnitude and sign of these effects depend strongly on body shape and free-stream Mach number.Results are given for the surface pressure coefficient, and the variations of translational and internal mode temperature on and near the body are also found. The influence of these latter effects on heat transfer to the body is discussed.

On the Plastic Buckling of Stiffened Panels under Axial Compression

In the previous paper the buckling of a stiffened panel with a symmetric type of stiffener was treated as an eigenvalue problem of an orthotropic plate with proper moduli of materials in the strainhardening range. The equation of echuilibrium of a stiffener can be generalized for an arbitrary shape of thin-walled open cross-section attached to a plate if proper consideration is given to the mutual interaction between plate and stiffener.V if a beam with an arbitrary cross-section is subjected to transverse load, the beam is bent and twisted simultaneously unless the line of loading passes through the shear center. When a beam is attached to a plate, the enforced axis causes twisting of the stiffener if it is bent or the other way around, bending of the stiffener if it is twisted. Therefore the bending and the twisting of the stiffener are no longer separable for such a stiffener with an unsymmetric cross-section.When the plate starts to buckle, the distributed resistance due to the longitudinal and or transverse stiffener can be obtained from the equation of equilibrium of a stiffener attached to the plate. The integral equation for the buckling of a longitudinally and transversely stiffened plate can be formulated as in the previous paper. The buckling strength of a stiffened plate is obtained as an eigenvalue of the integral equation. From the computation given in this paper it may be concluded that a longitudinal stiffener is tolerably effective to prevent the buckling of a plate compared to a transverse stiffener, and the efficiency of an inverted angle stiffener is considerably lower than that of a Tee stiffener when the stiffener is subjected to axial loads as in a longitudinal stiffener. In other words, the existence of axial load weakens the torsional resistance of a stiffener with an unsymmetric open cross-section as well as the bending resistance of the stiffener, and its influence for an unsymmetric cross-section is considerably greater that for a symmetric one. The results of this paper, together with the previous one, are used to specify the proper geometric conditions of the siffened plate such that each panel may develop large plastic deformation without buckling and a consequent fall-off in load. These requirements are essential for a successful application of Plastic Design Methods to plate structures.

Note on the Numerical Evaluation of the Wave Drag of Smooth Slender Bodies Using Optimum Area Distributions for Minimum Wave Drag

The linearised theory value of the wave drag D at zero lift of a smooth slender body of arbitrary cross-sectional shape was shown by Ward to be given bywhere S (x), 0 ⩽ x ⩽ 1, is the cross-sectional area distribution of the body and q is the kinetic pressure. The development of this result has aroused interest in two problems: the derivation of the optimum area distribution for minimum wave drag under certain specified conditions and the numerical evaluation of the wave drag of a specified area distribution. These apparently distinct problems have hitherto been treated separately, but it is shown here how an attempt to solve the first problem has led to a practical method of solving the second.