Experimental study of transition to turbulence of a round thermal plume by ultrasound scattering

In this study, we carried out the characterization of the transition to turbulence of a thermal pure plume by using ultrasound scattering. For this, the position, amplitude and broadening of the scattering peak are analyzed. The technique is based upon the scattering of an ultrasound wave coupling with an unstable flow. The coupling between the acoustic mode with both vorticity and entropy modes is derived from non-linear terms of Navier–Stokes and energy equations. When the scattering mechanism occurs, the characteristic length scale of the flow structure under observation is comparable with the wavelength of incoming sound. Thus, the flow can be probed at different length scales by only changing the frequency of incoming sound. The thermal plume rises from a heated disk immersed into a quiescent medium and can reach transition and fully turbulent regimes. Criteria allowing the identification of both the beginning and the end of transition are derived from the results. The characteristics of the scattering process show evidence that allows us to discern the beginning of transition. The analysis of the amplitude of the scattering peak revealed a homogeneous behavior and led us to think of a possible principle of similarity. The evolution of both thermal and velocity fluctuations has made it possible to establish the limits of both the beginning and the end of transition, in terms of local Grashof number Gr_z and position of the measurement zone z/D. The limits for transition reported in this work are comparable in its magnitude order with those of the literature. It was verified that thermal and velocity transition are phenomena that begin and finish almost simultaneously.     

The scattering of electroelastic waves by an ellipsoidal inclusion in piezoelectric medium

The scattering problem for a single ellipsoidal piezoelectric inclusion embedded in piezoelectric medium is investigated. Based on the polarization method, the extended displacements are expressed in terms of integral equations, whose kernels are obtained from the Green’s functions of homogenous matrix. In this paper, the 3D dynamic Green’s functions are derived by means of the Radon transform technique. To illustrate the use of the equations, scattering by a piezoelectric, ellipsoidal inhomogeneity in a piezoelectric medium is considered in the low frequency and an asymptotic formula for this scattering cross-section is obtained. Numerical results of the scattering cross-sections are carried out for a spheroidal BaTiO₃-inclusion in a PZT-5H-matrix.     

Transition Matrix Method for Determining Stress Distribution of an Inhomogeneity in Elastostatic Medium

The transition matrix method has been extensively utilized to solve scattering in elastodynamic media. It is based on the reciprocity theorem, continuity of the interface boundary conditions, and applicable to arbitrary shape of inhomogeneity in systematic matrix multiplication. However, the transition matrix method has never been applied to determine stress distribution in elastostatic media. One important reason is the problem of the shortage of the basis functions of the elastostatic media that must be used to develop the transition matrix. This study investigates the required basis functions, and finds a set thereof that include Love??s special solutions of three dimensional elastostatics and three vector functions that are applicable to elastic waves. The proposed basis functions also can be adopted to derive the three significant orthogonality conditions for reciprocity at the surface of the inhomogeneity, which are useful in developing the transition matrix. The novel basis functions make the process of derivation of the T-matrix in elastostatics similar to that in elastodynamics. This process is illustrated for a spherical inhomogeneity that embedded in an elastic medium and stress patterns are compared with Goodier??s solutions, demonstrating high accuracy.     

覆水饱和双相介质圆弧场地P波激励下散射问题的理论解

采用波函数展开法对平面P波入射复杂水域地形的空间变异性地震动场进行研究,该水域地形具有覆水层、饱和双相介质、场地非平坦以及第二类分层(场地跨越分层界面)等属性.首先,依据地震波反射和透射特性推导直角坐标系下的自由波场分布;然后,根据场地属性并引入大圆弧法分析极坐标系下的含有待定系数的散射波场;进而,结合土-水分界面和饱和土层分界面边界条件,求解散射波场中的待定系数;最后,通过自由波场和散射波场得到覆水饱和双相介质圆弧场地波函数理论解.基于理论解,通过算例验证了理论推导的合理性及可靠性,分析了地表位移在不同入射条件下的差异性.结果 表明,相对于均匀介质,饱和双相介质会显著影响地表位移分布.此外,入射波频率和角度对地震地面运动特性也有较大的影响.     

Flow and geometry induced scattering of high frequency acoustic duct modes

Cut-on cut-off transition of acoustic modes in hard-walled ducts with irrotational mean flow is well understood for Helmholtz numbers of order unity. Previous finite-element simulations of this phenomenon, however, appear to indicate the possibility of energy scattering into neighbouring modes at moderately large Helmholtz numbers. In this paper, such scattering phenomena are explained and predicted in slowly varying aeroengine ducts using a multiple-scales approach. It is found that, for sufficiently high frequencies, two mechanisms exist whereby energy can be scattered into neighbouring modes by an incident propagating mode. One mechanism occurs only when there is a mean flow inside the duct and induces scattering at significantly lower frequencies than the other mechanism which remains present without mean flow. A coupled system of ordinary differential equations is derived and then solved numerically for a number of example cases to obtain the corresponding transmitted and reflected amplitudes of the scattered modes as well as the overall acoustic pressure field. The theory appears to demonstrate that some exchange of energy between the acoustic and mean flow fields occurs during scattering.     

Scattering of antiplane shear waves by a submerged crack

The scattering of time harmonic antiplane shear waves by a crack situated parallel to the surface of a homogenious elastic half-space is considered. The problem is investigated for both clamped and stress-free conditions at the surface of the half-space. Singular integral equations are derived for each case and treated numerically by the Gauss-Chebyshev technique. Harmonic stress intensity factors and crack opening displacements are computed as functions of dimensionless frequency, submerged depth-to-crack width ratio, and angle of incidence. Results are obtained for a moderately wide range of frequencies and are set out in graphical form.     

Finite Element Analysis of Elasto-plastic Plate Bending Problems using Transition Rectangular Plate Elements

In this work, the finite element analysis of the elasto-plastic plate bending problems is carried out using transition rectangular plate elements. The shape functions of the transition plate elements are derived based on a practical rule. The transition plate elements are all quadrilateral and can be used to obtain efficient finite element models using minimum number of elements. The mesh convergence rates of the models including the transition elements are compared with the regular element models. To verify the developed elements, simple tests are demonstrated and various elasto-plastic problems are solved. Their results are compared with ANSYS results.The English text was polished by Keren Wang.     

Theory of scattering from multilayered bodies of arbitrary shape

Green 's functions and boundary integral equation methods are used to derive a matrix set of equations for scattering from a multilayered homogeneous elastic body embedded in an infinite elastic material. The surfaces separating the layers have arbitrary shape. The formalism for the three-layer material is derived in detail and generalized to N-layers. A matrix factorization method (MFM) is shown to considerably simplify the computational problem. The relation to the problems of acoustic waves in fluids and electromagnetic waves in a dielectric material is briefly indicated.     

On Landau theory and symmetric energy landscapes for phase transitions

The aim of this presentation is the development of a general approach for modelling the global complex energy landscapes of phase transitions. For the sake of clarity and brevity the exposition is restricted to martensitic phase transition (i.e., diffusionless phase transitions in crystalline solids). The methods, however, are more broadly applicable. Explicit energy functions are derived for the cubic-to-tetragonal phase transition, where data are fitted for InTl. Another example is given for the cubic-to-monoclinic transition in CuZnAl. The resulting energies are defined globally, in a piecewise manner. We use splines that are twice continuously differentiable to ensure sufficient smoothness. The modular (piecewise) technique advocated here allows for modelling elastic moduli, energy barriers and other characteristics independent of each other.     

局部散射理论在高超声速边界层转捩预测中应用的检验

eN方法是物理意义明确的转捩预测方法之一, 但它无法考虑边界层中的局部突变(如粗糙元、缝隙、台阶等)对转捩的影响. 而后者在飞行器表面经常出现. 近期发展的局部散射理论框架提供了该问题的有效解决途径. 该理论框架从转捩的物理机理出发, 定量刻画局部感受性和线性模态的局部散射两个机制, 并用参数化的感受性系数和透射系数修正转捩判据. 为了验证该理论框架的有效性, 设计了一套高超声速边界层的直接数值模拟方案: 分别在光滑壁与粗糙壁两种工况下引入相同的初始失稳模态, 计算它们触发转捩的过程, 并定量考察粗糙元对转捩的影响. 数值模拟结果与描述线性模态局部散射机制的理论预测吻合很好.      

Generalized Born scattering in anisotropic media

The Born scattering approximation has been widely used in seismology to study scattered waves, and to linearize the propagation problem for inversion. The standard Born theory requires the model be separated into a smooth, reference model and a perturbation. Scattering occurs from the pertubation. In the distorted Born approximation, when the reference model is inhomogeneous, the reference Green's functions are normally not known exactly, but the error in these Green's functions is rarely quantified. In this paper, we generalize Born scattering theory to include the errors in the Green's functions explicitly, and obtain scattering integrals from these errors. For forward modelling, there is no need to separate the model into a reference and perturbation part - approximate Green's functions in the true model can be used to calculate the scattered signals.

The theory is developed for inhomogeneous, anisotropic media. Asymptotic ray theory results are suitable approximate Green's functions for the generalized Born scattering theory. The error terms are simple, easily calculated and included in the scattering integrals. Various applications of generalized Born scattering theory have already appeared in the literature, e.g. quasi-shear ray coupling, and this paper is restricted to an improved and more complete theoretical development. Further applications will appear elsewhere.     

Elastic wave scattering and dynamic stress concentrations in exponential graded materials with two elliptic holes

Based on the elastodynamics, employing complex functions and conformal mapping methods, and local coordinates, the scattering of elastic waves and dynamic stress concentrations in infinite exponential graded materials with two holes are investigated. A general solution of the problem and expression satisfying the given boundary conditions are derived. The problem can be reduced to the solution of an infinite system of algebraic equations. As an example, numerical results of dynamic stress concentration factors for two elliptic holes in exponential graded materials are presented, and the influence of incident wave number and holes spacing on dynamic stress distributions is analyzed.     

A variational principle for waves in discrete random media

A new variational principle is derived for the Green's function of the linear harmonic response of a scalar wave field in a discretely heterogeneous medium. The variational principle is derived directly from exact multiple scattering integral equations and is stated in terms of a certain functional of trial configuration dependent fields. The functional is found to be stationary with respect to small variations in the fields when those fields have their correct configuration dependence. A certain trial dependence in the fields is shown to generate the Lax quasicrystalline multiple scattering equations. It is furthermore clear that these equations are optimal in the sense that any more realistic form for the trial fields unavoidably generates approximate equations entailing the generally unknown high order correlation functions. At its stationary point the proposed functional takes on a physically significant value. It becomes the change in the medium admittance due to the introduction of the scatterers. In a nonlossy medium this is related to the spectral density of modes. As errors in the trial fields cancel to first order, the final evaluation of the functional at its stationary point is especially accurate. Other related functionals are proposed and discussed as well.     

Diffraction of a plane electromagnetic wave by a perfectly conducting elliptic cylinder

The scattering of a plane electromagnetic wave by a perfectly conducting elliptic cylinder is investigated theoretically. The calculations are based upon the expansion of the scattered wave functions in terms of Mathieu functions. Both E- and H-polarized waves are considered. Numerical results, in particular for the scattering cross-section, are presented for cylinders the cross-sectional dimensions of which are up to many wavelengths (e.g. distance between the focal lines up to 20 wavelengths).     

The singularity expansion method as applied to the elastodynamic scattering problem

A basic problem in non-destructive testing of materials is the classification of defects with respect to size and geometry. Utilizing elastrodynamic scattering, ultrasonic test methods reduce this problem either to the application of reconstruction algorithms or to the parametrization of experimental data in terms of quantities being related to the physical scattering mechanisms. The so-called Singularity Expansion Method (SEM), originally developed for broadband electromagnetic scattering by arbitrarily shaped targets proves essentially useful to predict and understand impulsive scattering of ultrasound; in addition, even though not yet fully solved in practice, it seems possible to parametrize experimentally obtained time records in a sense which is physically intuitive.

SEM starts either with the eigenfunction expansion in the complex Laplace domain for canonical objects such as a sphere or spheroids, or with a corresponding integral equation formulation for more arbitrarily shaped defects. The essential point is then an expansion in terms of the singularities of the scattered field in the Laplace domain similar to the expansion of transfer functions in linear system theory. It turns out that the location of these singularities is characteristic for the geometry of the scattering body; therefore, it might be a useful tool to parametrize size and shape of the defects.

Several theoretically derived singularity patterns are presented for various body shapes and material compositions, which yield a thorough and physically intuitive interpretation in terms of distinct creeping wave modes. They are compared with first experimental results for a spherical void in a steel test specimen.     

AN ENTROPY VARIABLE FORMULATION AND APPLICATIONS FOR THE TWO-DIMENSIONAL SHALLOW WATER EQUATIONS

A new symmetric formulation of the two-dimensional shallow water equations and a streamline upwind Petrov–Galerkin (SUPG) scheme are developed and tested. The symmetric formulation is constructed by means of a transformation of dependent variables derived from the relation for the total energy of the water column. This symmetric form is well suited to the SUPG approach as seen in analogous treatments of gas dynamics problems based on entropy variables. Particulars related to the construction of the upwind test functions and an appropriate discontinuity-capturing operator are included. A formal extension to the viscous, dissipative problem and a stability analysis are also presented. Numerical results for shallow water flow in a channel with (a) a step transition, (b) a curved wall transition and (c) a straight wall transition are compared with experimental and other computational results from the literature.     

Determination of the similarity criteria for thermal gravitational convection in a supercritical fluid

Natural-convection flows in continua are studied at the parameter values beyond the critical point of the liquid-gas phase transition. The theory of similarity of supercritical convective flows is generalized by taking account for the dependence of the thermodynamic properties of a medium on the temperature and the density. In the case of an arbitrary two-parameter medium calibration relations, which serve for determining the similarity criteria of convection, are derived. In the fluid with the Van der Waals equation of state the calibration relations are reduced to the form of functions of the dimensionless temperature and density.     

Thermodynamic aspects of the glass-rubber transition

Summary In 1933 Ehren/est defined transitions in which not only the thermodynamical potential but also the specific volume and entropy of the two states are equal. For these transitions he derived three relations, between the differences of the coefficients of dilatation and of compressibility and the specific heat on one hand and the slope of the transition curve in /)-T-space on the other hand.Although it is beyond any doubt that the glass-rubber transition in polymers is not a second order transition as defined by Ehren]est, yet the term second order transition is often applied to it.The confusion arising from this abuse of a physically well defined eonce]?t is enhanced by the fact that some of Ehren/est's relations are under certain circumstances valid for the glass rubber transition, although in those cases the physical background of these relations is quite different from that of a second order transition in Ehren/est's sense. In fact, the essence of the glass transition has been formulated clearly and correctly by Simon in 1931 and consists in the assumption that in a glass one or more internal parameters are not in equilibrium but are frozen in.The conditions required for the validity of each of EhrenJest's relations are inspected. The equality of two functions of the dilatation and the compressibility coefficients and the specific heats of the two states depends on the condition that there is only one frozen parameter in the glass state. I f there is more than one fl'ozen parameter, then the equality turns into an inequality and the physical meaning of the difference between the two sides is derived.Similar relations as those derived between the quantities mentioned, can be derived between the specific heat, the modulus of elasticity and the temperature coefficient of that modulus. These new relations are more interesting for practical purposes and less subject to experimental errors than the older ones.Finally the expressions for the slope of the transition curve are investigated. The validity of any one of the expressions depends on the physical condition required for a material to become a glass. Reversely from the validity of any of the possible expressions insight in the glass transformation can be derived. The scarce available data give the impression that a material turns into a glass whenever its excess entropy decreases below a critical value.     

Ultrasonic scattering in polycrystals with orientation clusters of orthorhombic crystallites

Scattering-induced ultrasonic attenuation and backscattering in a polycrystalline medium with orientation clusters composed of orthotropic crystallites are studied theoretically, aiming to improve understanding of ultrasonic wave interaction with such clustered microstructures for application to the modeling of titanium alloys. Both orthorhombic crystallites and their arrangements in orientation clusters (also termed microtexture regions, MTRs) are of general ellipsoidal shape. The preferred orientation of orthotropic crystallites in the clusters is represented by a generalized Gaussian orientation distribution function with three independent texture parameters. The effective elastic properties of the clusters, which have in general orthorhombic symmetry, are determined by a volume average of elastic constants weighted by orientation distribution functions and then used to obtain the cluster-scattering-induced attenuation and backscattering in the polycrystalline medium. In the model the wave propagation direction is arbitrary relative to the ellipsoidal axes of the clusters. The contribution of crystallite-scattering-induced attenuation is estimated by the untextured attenuation coefficient factored by a texture transition function. The total attenuation and backscattering are determined by combining scattering by the clusters and crystallites. Drastic effects of clustering and the transition to unclustered polycrystals are demonstrated. Reasonable agreement is observed between the model’s prediction and measurements on two Ti-alloy samples with different crystallite clustering.