Vector Eigenfunction Expansion Research Articles

Vector eigenfunction expansion in the integral transform solution of transient natural convection

Purpose The purpose of this work is to revisit the integral transform solution of transient natural convection in differentially heated cavities considering a novel vector eigenfunction expansion for handling the Navier-Stokes equations on the primitive variables formulation. Design/methodology/approach The proposed expansion base automatically satisfies the continuity equation and, upon integral transformation, eliminates the pressure field and reduces the momentum conservation equations to a single set of ordinary differential equations for the transformed time-variable potentials. The resulting eigenvalue problem for the velocity field expansion is readily solved by the integral transform method itself, while a traditional Sturm–Liouville base is chosen for expanding the temperature field. The coupled transformed initial value problem is numerically solved with a well-established solver based on a backward differentiation scheme. Findings A thorough convergence analysis is undertaken, in terms of truncation orders of the expansions for the vector eigenfunction and for the velocity and temperature fields. Finally, numerical results for selected quantities are critically compared to available benchmarks in both steady and transient states, and the overall physical behavior of the transient solution is examined for further verification. Originality/value A novel vector eigenfunction expansion is proposed for the integral transform solution of the Navier–Stokes equations in transient regime. The new physically inspired eigenvalue problem with the associated integmaral transformation fully shares the advantages of the previously obtained integral transform solutions based on the streamfunction-only formulation of the Navier–Stokes equations, while offering a direct and formal extension to three-dimensional flows.

混响室的混沌特性及其场统计分布

Quantum chaos theory is used to study a reverberation chamber (RC). We evaluate the chaotic characteristics of a numerical RC by analyzing the eigenfrequency spacing distribution with the finite element method. Based on the electromagnetic chaotic features and the hypothesis, the electric field vector eigenfunction in the RC could be expressed as linear superposition of random plane waves. Then the electric field in the RC can be expressed as the vector eigenfunction expansion. The Rayleigh distribution model of probability density function of electric field in the RC is derived by statistical approach. The electric field intensity measurement in an actual mechanical stirring RC is completed with ETS HI-6105 prober. The electric field distribution in the chamber is in close agreement with the result from the statistical model, which proves the validity of the theoretical model.

Simulating the Response of Induction Logging‐While‐Drilling Tools by Vector Eigenfunction Expansion Formulae of Dyadic Green's Functions

AbstractThe vector eigenfunction expansion formulae of magnetic‐current‐source dyadic Green's functions for a perfect electric conduction wedge model in cylindrical coordinate system are adopted to compute the response of an induction logging‐while‐drilling tool within the collar's V‐shaped channel. It has shown from computation that the influence of the change of the V‐shaped channel's angle on different components of electromagnetic fields is different. If only a conventional induction logging tool is used, the channel's angle should be as small as possible in order to increase the signal's intensity fully. If a multi‐component induction logging tool is used, the value of the channel's angle should be selected compromisingly. Since the received signal's intensity will change when the tool is in the V‐shaped channel with different angle, the criterion should be fixed on the basis of the V‐shaped channel's angle when the formation's apparent conductivity is calibrated. It has also shown that different components of electromagnetic fields rely differently on the azimuth angle, and that the influence of the invasion zone's conductivity on different components of the received signal is also different, thus the radial depth of exploration is different for different components of the received signal.

Nonstationary axisymmetric electroelasticity problem for an anisotropic piezoceramic radially polarized cylinder

We consider an axisymmetric nonstationary electroelasticity problem for an anisotropic piezoceramic radially polarized cylinder of finite size whose lateral surface is subjected to an electric voltage that is an arbitrary function of the axial coordinate and time. A new closed-form solution is constructed by the vector eigenfunction expansion method in the form of a structural finite transform algorithm. This solution permits determining the natural vibration frequencies, the stress-strain state of an element, and the electric field potential and intensity. The results permit analyzing and optimizing the operation of inverse piezoelectric effect devices with cylindrical transducers.

Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells

We describe a novel and rigorous vector eigenfunction expansion of electric-type Green's dyadics for radially multi- layered uniaxial anisotropic media in terms of the modified spherical vector wave functions, which can take into account the effects of anisotropy ratio systematically. In each layer, the material constitutions e and epsiv macrmu macr are tensors and distribution of sources is arbitrary. Both the unbounded and scattering dyadic Green's functions (DGFs) for rotationally uniaxial anisotropic media are derived in spherical coordinates (r, thetas, phi). The coefficients of scattering DGFs, based on the coupling recursive algorithm satisfied by the coefficient matrix, are derived and expressed in a compact form. With these DGFs obtained, the electromagnetic fields in each layer are straightforward once the current source is known. A specific model is proposed for the scattering and absorption characteristics of multilayered uniaxial anisotropic spheres, and some novel performance regarding anisotropy effects is revealed.

Three-dimensional flow in a cylinder with a stress-free sidewall

Stokes flow in a cylindrical column of fluid, with a stress-free cylindrical sidewall, is considered. The motion is assumed to be generated by the linear, uniform motion of either or both of the flat endwalls. The field is obtained by a vector eigenfunction expansion procedure. If the field is assumed to have a θ - and z-dependence of the type exp ( i θ + kz ) , k has to satisfy the equation ( 1 - k 2 ) J 1 3 + J ^ 1 [ 2 J 1 2 - J ^ 1 { 2 J ^ 1 + ( 1 + k 2 ) J 1 } ] = 0 , where J ^ 1 = kJ 0 - J 1 and the argument of each Bessel function is k. This equation admits, unlike in the plane case and with important consequences, not just a real sequence { λ ˜ n } of eigenvalues but also a complex one { μ ˜ n } . Using a least squares procedure to satisfy the boundary conditions on the top and bottom walls, the three-dimensional velocity field in the column is determined for various values of column height h and wall speed ratio S. Detailed computations show that there are strong effects of both the stress-free boundary and three-dimensionality. The principal effect of the former is to permit motion on that boundary leading to large azimuthal motions and of the latter, unlike in the plane flow, to multiple primary eddies when h is sufficiently large. A number of new eddy structures are also found, which demonstrate that three-dimensionality often leads to the elimination of compactness found in plane flows. It is finally shown that the flow fields exhibit interesting bifurcations as S and h are varied.

Cylindrical Vector Eigenfunction Expansion of Green Dyadics for Multilayered Anisotropic Media and Its Application to Four-Layered Forest

A complete eigenfunction expansion of the dyadic Green's functions (DGFs) for planar, arbitrary multilayered anisotropic media using cylindrical vector wave functions is presented. These formulations are constructed based on the principle of scattering superposition. For the scattering dyadic Green's function in each layer, the scattering coefficients of TE and TM modes are determined from the boundary conditions matched at the planar interfaces. The explicit representation of the DGFs after reduction to the isotropic case agrees well with the existing results corresponding to the isotropic media. The general DGFs for multilayered anisotropic media are then reduced to those for a four-layered forest where the trunk layer is modeled as anisotropic medium. Application is further made for radio-wave propagation through forests of a four-layered geometry, whereas it is shown how these Green dyadic formulations are used in a practical way and how the field distributions due to a dipole can be obtained.

An efficient calculational approach to evaluation of microwave specific attenuation

An efficient calculational approach using the scattering-radiation conversion is developed in this paper to evaluate the microwave attenuation by arbitrarily distorted raindrops. For this modified first-order approach, the perturbation technique and the spherical vector eigenfunction expansion method are employed. A method of obtaining the volumetric current distribution of the assumed source that generates the plane waves is developed in the paper and the current distribution of such a source is derived. The electromagnetic fields outside the distorted raindrop scatterers are formulated in terms of integrals consisting of a volumetric current distribution located at infinity and the dyadic Green's functions. To illustrate the validity of this approach, the spheroidal raindrop and the Pruppacher and Pitter (1971) raindrop model of varying shapes are specifically investigated. Numerical results of the extinction cross sections and the specific attenuation due to the two models are obtained. While the former agrees well with the published results, the latter is in good agreement with the experimental specific attenuation data collected at 21.225 GHz in Singapore.

Three-dimensional Stokes flow in a cylindrical container

Consider a cylindrical container of circular section filled with a viscous fluid. We consider flow in such a cylinder driven by the motion of flat belts across the partially open end walls of the container. The flow field is determined by the use of a vector eigenfunction expansion. The critical points that are exhibited in the plane of symmetry include elliptic points, foci, and saddles. As the parameters are varied one can have bifurcations in which one type of critical point bifurcates to a collection of others. An example of a limiting surface is also demonstrated. Since the flow fields considered have little symmetry, the three-dimensional streamlines are for the most part not closed. As a consequence the flow fields tend to globalize structures that would otherwise have been isolated. This feature can have important consequences for mixing.

Equilibrium Vortex Motion in Circular Couette Flow

A possibility of equilibrium between secondary vortices of small but finite intensity in circular Couette flow is studied. Equations which describe the time development of the amplitudes of three different toroidal vortices are derived on a vector eigenfunction expansion procedure, nonlinear terms higher than the third order smallness being truncated. The equations determine the intensities of the vortices, if they exist at the same time, which are assumed to be of the same order. Analysis is limited to the case where the inner cylinder is rotating and the outer one at rest, and the gap between the cylinders is small compared with their radii. The calculation seems to suggest that the equilibrium vortex motion is rather periodic if we confine ourselves to these assumptions.