Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments

Angular structures in the far-field scattering from bubbles are observed and modeled. Mie theory supports a model of diffraction and interference near the critical scattering angle. A new expression for the angular spacing of fine structure is derived. Photographs of scattering show some of the predicted features. Application of these structures to bubble sizing and detection are summarized and the theoretical extinction coefficient in water is plotted.Mie computations for bubbles in water also reveal backward and forward glory effects. These are partially manifested as cross-polarized scattering. Observed scattering from bubbles in the near backward direction is found to have a strong cross-polarized component.     

测量两相流中颗粒特性的相位多普勒方法理论模型

本文在几何光学近似Ｍｉｅ散射理论的基础上建立了一种计算激光多普勒信号中相左与粒径关系的数学模型。对几种常用的相对折射率情况下的相差一粒径特性进行了计算,分析并给出了光学系统的最佳配置。计算结果与Ｍｉｅ散射理论的计算结果进行了对比,有限好的一致性。     

A hybrid numerical–asymptotic boundary element method for high frequency scattering by penetrable convex polygons

We present a novel hybrid numerical–asymptotic boundary element method for high frequency acoustic and electromagnetic scattering by penetrable (dielectric) convex polygons. Our method is based on a standard reformulation of the associated transmission boundary value problem as a direct boundary integral equation for the unknown Cauchy data, but with a nonstandard numerical discretization which efficiently captures the high frequency oscillatory behaviour. The Cauchy data is represented as a sum of the classical geometrical optics approximation, computed by a beam tracing algorithm, plus a contribution due to diffraction, computed by a Galerkin boundary element method using oscillatory basis functions chosen according to the principles of the Geometrical Theory of Diffraction. We demonstrate with a range of numerical experiments that our boundary element method can achieve a fixed accuracy of approximation using only a relatively small, frequency-independent number of degrees of freedom. Moreover, for the scattering scenarios we consider, the inclusion of the diffraction term provides an order of magnitude improvement in accuracy over the geometrical optics approximation alone.     

A numerical study of super-resolution through fast 3D wideband algorithm for scattering in highly-heterogeneous media

We present a wideband fast algorithm capable of accurately computing the full numerical solution of the problem of acoustic scattering of waves by multiple finite-sized bodies such as spherical scatterers in three dimensions. By full solution, we mean that no assumption (e.g. Rayleigh scattering, geometrical optics, weak scattering, Born single scattering, etc.) is necessary regarding the properties of the scatterers, their distribution or the background medium. The algorithm is also fast in the sense that it scales linearly with the number of unknowns. We use this algorithm to study the phenomenon of super-resolution in time-reversal refocusing in highly-scattering media recently observed experimentally (Lemoult et al., 2011), and provide numerical arguments towards the fact that such a phenomenon can be explained through a homogenization theory.     

A NEW ALGORITHM OF RELAXATION METHOD FOR PARTICLE ANALYSIS FROM FORWARD SCATTERED LIGHT

1. Introduction In many cases, particle dimension and size distribution play an important role in application. The analysis of parti- cle size is therefore necessary for research and develop- ment and also for process quality control in powder tech- nology and for many particle systems (Wang et al., 2000). Many instruments have been developed and constructed for this purpose. With the rapid development and availabil- ity of laser techniques, optoelectronics and microcomputer science, light sc…     

Asymptotic inverse scattering

Several inverse techniques are developed for determining the shape of an unknown scattering surface by analyzing backscattered acoustic or electromagnetic waves. These techniques are based on asymptotic high frequency representations of the fields and may be divided into three categories. The first one is the geometrical imaging method where the surface is reconstructed by means of a travel-time analysis which is here specified to the far field by utilizing Minkowski's support function. Furthermore, a geometrical method is developed for localizing edges from mid field data measured along a curve. The second category is called quasigeometrical imaging and uses geometric optics or higher order amplitudes for the reconstruction. It is shown that cross-polarized electromagnetic far field amplitude measurements permit one to deduce the complete quadratic approximation of the surface at the specular points from which the surface can be reconstructed pointwise. The third category may be subsumed under ‘asymptotic inverse scattering identities’. Here, asymptotic relations between scattered fields and distributions associated with the geometry of the scatterer are established. It is shown that the physical optics far field inverse scattering identity is only a leading order asymptotic relation but as such is also valid for non-convex scatterers. Furthermore, asymptotic inverse scattering identities are deduced which relate the singular function of a closed surface to the backscattered field data measured on a sphere enclosing the scatterer. This generalizes far field results of Cohen and Bleistein (Wave Motion 1 (1979), p. 153) to the mid field.     

可压缩流场中气泡脉动数值模拟

在应用边界元方法对气泡动力学的研究中, 绝大多数模型是建立在不压缩势流理论基础之上, 针对可压缩流场中气泡运动特性的研究很少. 从波动方程出发, 分别在气泡运动前期和后期对波动方程进行简化, 得到气泡运动局部和全局简化方程, 采用双渐进方法对简化方程进行匹配, 提出了考虑流场可压缩性的非球状气泡运动模型. 该模型的计算结果与Prospertti 等的解析结果吻合很好, 气泡脉动最大半径和内部最大压力随气泡脉动逐渐减小. 基于该模型对比了自由场中药包爆炸考虑可压缩性与不考虑可压缩性的计算结果, 发现考虑可压缩性气泡射流速度较小, 随后基于该模型计算了刚性边界下气泡的运动特性.     

The impact of high order refraction on optical microbubble sizing in multiphase flows

High order refractions have been found to have a great impact on optical particle sizing using phase-Doppler anemometry (PDA) especially when the relative refractive index of the media is less than one. The high order refractions cannot be neglected because of the differences in spatial frequencies and motion directions of each order. A model of phase-size correlation is proposed that considers these high order refractions. By using the conversion factors of the high order refractions, the particle diameter can be determined. The capability of the newly developed method was evaluated by using generalized Lorenz Mie theory (GLMT), which was validated by experiments. The results of the simulation and experiments are compared with those based on the conventional method. An optimization method for accurately sizing air bubbles in water has been suggested.     

Experimental investigation of the buckling of shallow spherical shells

In the present paper, the buckling behavior of clamped thin shallow spherical shells under external pressure is studied. Seventy-nine plastic shells formed by thermovacuum process were tested. The distributions of initial geometrical imperfections and vertical displacements were minutely measured with a differential transformer. It was possible to control the symmetrical initial geometrical imperfection of each specimen.Results indicate that the buckling phenomena of shallow spherical shells vary greatly with the symmetrical initial imperfection parameter η. In the case of the geometrical parameter λ larger than 5.5, the amplitude of the asymmetrical displacement component with the bifurcation buckling wave calculated by Huang becomes large immediately before buckling. The validity of Huang's theory for an initially perfect shell is experimentally demonstrated.     

Egocentric physics: Summing up Mie

We consider sum rules for the electric type (TM) multipole coefficients in the Mie theory of the scattering coefficients for electromagnetic waves incident upon spherical particles. These sum rules are derived from infinite product representations for the scattering coefficients and involve an analytically-determined multiplying factor in addition to the resonant eigenstate values. The product expansions converge rapidly to the scattering coefficients with increasing numbers of resonant state values only if the analytic multiplying factor is included in expansions, and the use of these sum rules further accelerates the convergence of scattering coefficient expansions. We present analytic asymptotic estimates for the resonant state eigenvalues in the dipole, quadrupole and hexapole cases, give the corresponding sum rules, and numerically illustrate their convergence.     

Sizing of bubbles by incoherent imaging: defocus bias

 Digitized images of millimeter bubbles are obtained with backlighting from a diffuse source. In order to investigate how the measured and real sizes of a bubble in or out of focus relate, a Ray Tracing Model of image formation, based on geometrical optics and photometric laws is developed. It emphasizes the contribution of direct, reflected and refracted rays. The model shows that the size of the source has a major role in the image formation. Moreover no reliable information on the defocus sign can be found in the image. A method for sizing blurred bubbles is suggested. Received: 21 June 1996 / Accepted: 4 February 1997     

Evolution of weak shocks in one dimensional planar and non-planar gasdynamic flows

Asymptotic decay laws for planar and non-planar shock waves and the first order associated discontinuities that catch up with the shock from behind are obtained using four different approximation methods. The singular surface theory is used to derive a pair of transport equations for the shock strength and the associated first order discontinuity, which represents the effect of precursor disturbances that overtake the shock from behind. The asymptotic behaviour of both the discontinuities is completely analysed. It is noticed that the decay of a first order discontinuity is much faster than the decay of the shock; indeed, if the amplitude of the accompanying discontinuity is small then the shock decays faster as compared to the case when the amplitude of the first order discontinuity is finite (not necessarily small). It is shown that for a weak shock, the precursor disturbance evolves like an acceleration wave at the leading order. We show that the asymptotic decay laws for weak shocks and the accompanying first order discontinuity are exactly the ones obtained by using the theory of non-linear geometrical optics, the theory of simple waves using Riemann invariants, and the theory of relatively undistorted waves. It follows that the relatively undistorted wave approximation is a consequence of the simple wave formalism using Riemann invariants.     

Non-intrusive measurements of bubble size and velocity

A non-intrusive measuring technique based on video-imaging has been developed for the measurement of bubble size, velocity and frequency. Measurements carried out with this method have been compared to those obtained by an optimized phase-Doppler system in standard configuration, for a wide range of bubble sizes produced from single injectors in a quiescent environment. The two measuring techniques have yielded velocities and frequencies that are in very good agreement while the size of spherical bubbles was consistently measured by both methods. The phase-Doppler system was also used to size oblate-spheroidal bubbles moving with their equatorial plane parallel to the scattering plane, yielding measurements reasonably close to the average radius of curvature of the bubbles in the neighborhood of the equatorial plane, as calculated from the video-imaging data. Both methods were used for detailed velocity measurements of the bubble-stream in the neighborhood of the injector tip. The observed bubble-velocity variation with the distance from the injector tip does not always display the usual increasing trend leading into the terminal velocity. When injection conditions are near the transition from discrete to jet injection mode and the bubbles are small, the latter decelerate into a terminal velocity due to direct interaction of successive bubbles at the injector tip. The measured terminal velocities of bubble-chains for a variety of bubble sizes and injection frequencies, are successfully predicted by using a far-field wake approximation to account for the drafting effect which is responsible for bubble-chain velocities higher than those of single bubbles.The authors gratefully acknowledge the support from the Louisiana Board of Reagents through grants LEQSF (1991-1992)-ENH-31 and LEQSF (1990-1992)-RD-A-07, ARCO Chemical Company through it's University Grants Program and the National Science Foundation through its Graduate Fellowship Program.     

The singular function of a surface and physical optics inverse scattering

It is shown how to recover both the location and the reflection coefficient of a scatterer using only high frequency backscattered data. The result is obtained without use of the far field approximation although a separate identity is derived when this approximation is introduced. This latter result improves upon previously derived physical optics far field inverse scattering identities.     

Ray method for surface waves on a ferromagnetic fluid

The ray method, or geometrical optics method, is used to study surface waves on a ferromagnetic fluid of variable depth. Both time-dependent and time-reduced cases are considered on the basis of the shallow-water approximation. An application of the time-reduced ray method leads to the discovery of edge waves on a ferromagnetic fluid over a uniformly sloping bottom.     

The method of geometrical optics for differential equations of the fourth order as applied to low-frequency plasma oscillations

The theory of osculations of a spatially inhomogeneous plasma [1] draws substantially on the theory of geometrical optics as applied to differential equations of the second order. The theory of asymptotic solutions for equations of the second order has now been thoroughly developed [2]. The quasi-classical quantization rules determining the spectrum of eigenvalues of such equations are written in the form of the well-known Bohr -Sommerfeld integrals [3]. However, in analyzing the spectrum of oscillations of an inhomogeneous plasma it is insufficient in many cases to confine oneself to equations of the second order. For example, in an inhomogeneous magnetoactive plasma, even when the thermal motion of the particles is neglected, the field equations, generally speaking, reduce to a differential equation of the fourth order. Equations of the fourth order also arise in investigating the stability of the hydrodynamical flow of a viscous fluid [4].Certain special forms of fourth-order equations were studied in [4–6]. The authors of [6] obtained a quasi-classical quantization rule for equations of the fourth order with a small parameter associated with the leading derivative. The present paper investigates the general fourth-order equation with real coefficients. Asymptotic solutions of such an equation are obtained with an accuracy to terms of the first order in the approximation of geometrical optics, and quasi-classical quantization rules are established for various concrete cases. Using the theory thus developed, a new spectrum of oscillations is determined, characteristic only for an inhomogeneous plasma in a magnetic field.In conclusion, the authors express their gratitude to V. P. Silin who suggested the idea of matching the quasi-classical solutions, and also to Yu. N. Dnestrovskii and D. P. Kostomarova for discussing the paper and offering valuable criticism.     

On the location of flame edge in Shadowgraph pictures of spherical flames: a theoretical and experimental study

In this paper, a theoretical model based on geometrical optics has been developed to analyze the light intensity pattern of Shadowgraph pictures of spherical flames. The theoretical results have been compared with experimental measurement of light intensity profiles across the flame front using commercially available image processing software. These results are in good agreement. The theory predicts that the sudden change of light intensity from dark to bright does not coincide with the flame edge unless the flame thickness is negligible. Experimental results agree very well with the theoretical predictions.     

用二维逆Born近似法对杆中圆形缺陷的反演

在三维弹性波散射问题的Born近似解基础上,进一步分析了在纵波入射条件下二维散射问题的Born近似解,对铝质长杆中的椭圆形空穴缺陷的散射场情况进行了对比分析;最后,提出了在低频下识别缺陷几何特征的二维逆Born近似法,并用此法对铝质长杆中的圆形空穴缺陷作了计算机模拟。     

光检测器参数对相位—多普勒粒子分析系统特性的影响

应用几何光学近似理论模型计算了激光相位多普勒系统中微粒的光散射特性,得到了散射光强与粒径关系、检测器表现的散射光强分布、相位差与粒径特性中的非线性跃变以及检测器形状对特性的影响。