Bounds for the non-local effective properties of random media—II

The paper deals with a random medium subjected to a static field with inhomogeneous mean values. Then, effective linear material parameters show dispersion, i.e. they depend on the ‘wave vector’ k of the mean field. Starting from a variational method previously developed by the authors, upper and lower bounds for k-dependent scalar effective parameters are derived in terms of two-point and three-point correlation functions of the stochastic material parameters. Taking into consideration the three-point correlation function gives a substantial improvement of the generalized Hashin-Shtrikman bounds obtained previously. In particular, composites with cell structure and arbitrary binary systems are considered. In order to illustrate the general results, numerical evaluations are carried out for effective permittivity of a binary cell material composed of nearly spherical grains of equal size.     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.     

Simulations of Three-Dimensional Turbulent Mixing for Schmidt Numbers of the Order 1000

We report basic results from new numerical simulations of passive scalar mixing at Schmidt numbers (Sc) of the order of 1000 in isotropic turbulence. The required high grid-resolution is made possible by simulating turbulence at very low Reynolds numbers, which nevertheless possesses universality in dissipative scales of motion. The results obtained are qualitatively consistent with those based on another study (Yeung et al., Phys. Fluids 14 (2002) 4178-4191) with a less extended Schmidt number range and a higher Reynolds number. In the stationary state maintained by a uniform mean scalar gradient, the scalar variance increases slightly with Sc but scalar dissipation is nearly constant. As the Schmidt number increases, there is an increasing trend towards k ^?1 scaling predicted by Batchelor (Batchelor, J. Fluid Mech. 5 (1959) 113-133) for the viscous-convective range of the scalar spectrum; the scalar gradient skewness approaches zero; and the intermittency measured by the scalar gradient flatness approaches its asymptotic state. However, the value of Sc needed for the asymptotic behavior to emerge appears to increase with decreasing Reynolds number of the turbulence. In the viscous-diffusive range, the scalar spectrum is in better agreement with Kraichnan's (Kraichnan., Phys. Fluids 11 (1968) 945-953) result than with Batchelor's.     

Reconstruction of the three mechanical material constants of a lossy fluid-like cylinder from low-frequency scattered acoustic fieldsReconstruction des trois constants mécaniques matériels d'un cylindre fluide dissipatif à partir de champs acoustiques basses fréquences

The inverse medium problem for a circular cylindrical domain is studied using low-frequency acoustic waves as the probe radiation. To second order in k₀a (k₀ the wavenumber in the host medium, a the radius of the cylinder), only the first three terms (i.e., of orders 0, ?1 and +1) in the partial wave representation of the scattered field are non-vanishing. This enables the scattered field to be expressed algebraically in terms of the unknown material constants, i.e., the density ρ₁, and the real and imaginary parts of complex compressibility κ₁ of the cylinder. It is shown that these relations can be inverted to yield explicit, decoupled expressions for ρ₁ and κ₁ in terms of the totality of the far-zone scattered field. These expressions furnish accurate estimations of the material parameters provided the probe frequency is low and the radius of the cylinder is known very precisely. To cite this article: T. Scotti, A. Wirgin, C. R. Mecanique 332 (2004).     

Numerical study of dispersing pollutant clouds in a built-up environment

In this paper, we study numerically the dispersion of a passive scalar released from an instantaneous point source in a built-up (urban) environment using a Reynolds-averaged Navier–Stokes method. A nonlinear k–? turbulence model [Speziale, C.G., 1987. On nonlinear k–l and k–? models of turbulence. J. Fluid Mech., 178, 459–475] was used for the closure of the mean momentum equations. A tensor diffusivity model [Yoshizawa, A., 1985. Statistical analysis of the anisotropy of scalar diffusion in turbulent shear flows. Phys. Fluids, 28, 3226–3231] was used for closure of the scalar transport equations. The concentration variance was also calculated from its transport equation, for which new values of Yoshizawa’s closure coefficients are used, in order to account for the instantaneous tracer release and the complex geometry. A new dissipation length-scale model, required for the modelling of the dissipation rate of concentration variance, is also proposed. The numerical results for the flow, the pollutant concentration and the concentration variance, are compared with experimental data. This data was obtained from a water-channel simulation of a full-scale field experiment of tracer dispersion through a large array of building-like obstacles known as the Mock Urban Setting Trial (MUST).     

On bounds and limit theorems for secondary creep in symmetric pressure vessels

The field equations governing creep in spherical and incompressible cylindrical pressure vessels subject to a nondecreasing internal pressure are reduced to a single equation in the effective stress. Using this equation, bounds are obtained for the effective stress and the displacement at any point in the body at any time. Also, in the case where the pressure tends to a limit as t → ∞. limit theorems are obtained which describe the long term behavior of the effective stress and the displacement.     

The calculation of free electron density in Cassandra

Cassandra is an AWE opacity code used to model plasmas in local thermal equilibrium: there is a desire to expand its use to calculating plasma equations of state. Cassandra's self-consistent field calculation (scf) uses the local density approximation for bounds states and has a free electron contribution based upon the Thomas-Fermi model [B.J.B. Crowley et al., J. Quant. Spectro. Radiat. Trans. 71, 257(2001)]. Whilst this is applicable for very high temperature or low density plasmas; in hot and dense matter the effect of ionization will lead to discontinuities in the effective ionisation, Z^?. The electron contribution to hydrostatic pressure is associated with Z^?, thus these discontinuities produce unphysical jumps in the resulting calculated material pressure.We describe a procedure to mitigate the effect by calculating the free electron wave functions within the generalized ion-cell model [B.J.B. Crowley et al., Phys. Rev. A 41, 2179(1990)], and thus explicitly calculate free-electron resonances.     

On the effect of small fluctuations in the volume fraction of constituents on the effective properties of composites

This study deals with three-scale composite materials comprised of nonlinear constituents. At the meso scale the composite can be considered as locally homogeneous with a macroscopic spatial variation of the constituents volume fraction. When these variations about a mean value are small, a Taylor expansion to second-order of the effective properties of the composite with respect to the fluctuations is given. This expansion can be used to discuss the beneficial or deleterious effects of clusters of inhomogeneities. It can also be used to derive new upper and lower bounds for the effective properties of nonlinear composites from dilute results. To cite this article: P. Suquet, C. R. Mecanique 333 (2005).     

Strain,Rotation and Curvature of Non-material Propagating Iso-scalar Surfaces in Homogeneous Turbulence

This research aims at gaining some physical insight into the problem of scalar mixing, following the time evolution of propagating iso-surfaces, Y (x, t) = constant, where Y (x, t) stands for any scalar field (e.g., species mass fraction or temperature). First, a rigorous kinematic analysis of non-material line, surface and volume elements, related to propagating iso-scalar surfaces, is presented; this formalism is valid for both constant and variable density flows. Time rates of change of the normal distance and volume between two adjacent iso-surfaces and of area elements, rotation rates of lines and surface elements and an evolution equation for the local mean curvature are obtained. Line and area stretch rates, which encompass additive contributions from the flow and the displacement speed (due to diffusion and reaction), are identified as total strain rates, normal and tangential to the iso-surfaces. Volumetric dilatation rates, addition of line plus area stretch rates, include the mass entrainment rate per unit mass into the non-material volume. Flow and added vorticities, the latter due to gradients of the displacement speed, yield the total vorticity, which provides the real angular velocity of lines and surface elements. A 512³ DNS database for the mixing of inert and reactive scalars in a box of forced statistically stationary and homogeneous turbulence of a constant-density fluid is then examined. A strongly segregated scalar field is prescribed as initial condition. A one-step reaction rate with a characteristic chemical time one order of magnitude greater than the Kolmogorov time micro-scale is used. Data are analyzed at 1.051 large-eddy turnover times after initialization of velocity and scalar fields. Mean negative normal (contractive) and positive tangential (stretching) flow strain rates occur over all mass fractions and scalar-gradient magnitudes. However, means of the total normal strain rate, conditional upon mass fraction, scalar-gradient and mean curvature, are positive everywhere and tend to destroy scalar-gradients for small times. Negative conditioned mean total tangential strain rates (area stretch factor) contract local areas, except for large values of scalar-gradients. Conditional averages of total and added enstrophies are almost identical, which implies a negligible contribution of the flow vorticity to the observed rotation of non-material line and surface elements. The added vorticity is exactly tangential to the iso-surfaces, whereas the flow and total ones are predominantly tangential. Flow sources/sinks of the mean curvature transport equation are much smaller than the added contributions; for this particular DNS database, the local mean curvature development is self-induced by spatial changes of the displacement speed.     

Bounds on the elastic moduli of statistically isotropic multicomponent materials and random cell polycrystals

Minimum energy and complementary energy principles are used to derive the upper and lower bounds on the effective elastic moduli of statistically isotropic multicomponent materials in d ($d=2$ or 3) dimensions. The trial fields, involving harmonic and biharmonic potentials, and free parameters to be optimized, lead to the bounds containing, in addition to the properties and volume proportions of the material components, the three-point correlation information about the microgeometries of the composites. The relations and restrictions among the three-point correlation parameters are explored. The upper and lower bounds are specialized to symmetric cell materials and asymmetric multi-coated spheres, which are optimal or even converge in certain cases. New bounds for random cell polycrystals are constructed with particular results for random aggregates of cubic crystals.     

Calcul à la rupture en présence d'un écoulement à surface libre : construction de champs de pression approchésYield design for porous media subjected to unconfined flow: construction of approximate pressure fields

We consider the stability of a porous medium submitted to a steady-state flow with free-boundary. Assuming some hypotheses, it is possible to implement the kinematic method by using an approximate pressure field bounding the true pressure field from below. We are interested in finding such approximate pressure fields and in proving that they bound the true pressure field from below without knowing the true pressure field. We use fields which are solutions of a problem with relaxed conditions with regard to the real problem. Under a uniqueness condition of the solution of a weak formulation of the problem, such fields are lower bounds for the true pressure field. Finally, we give the example of a vertical dam. To cite this article: A. Corfdir, C. R. Mecanique 334 (2006).     

Bounds on the effective conductivity of statistically isotropic multicomponent materials and random cell polycrystals

Upper and lower bounds on the effective conductivity of statistically isotropic multicomponent materials in d dimensions (d=2 or 3) are constructed from the minimum energy principles and appropriate trial fields. The trial fields, involving harmonic potentials and free parameters to be optimized, lead to the bounds containing up to three-point correlation information about the microgeometry of a composite. The bounds are applied to give estimates for the symmetric cell materials, which are optimal over some ranges of parameters, and asymmetric multicoated spheres, which yield the exact effective conductivity in certain cases. The results also agree with many known ones. New bounds for random cell polycrystals are obtained and illustrated on a number of polycrystalline aggregates.     

Étude numérique du champ de vitesse dans un échangeur à vortexNumerical study of the velocity field in a vortex heat exchanger

Flow through a cylindrical flat chamber, a model of some particular heat exchanger, is investigated numerically using the CFD-ACE code. Turbulence is modelled using the classical k–ε model. A better understanding of the secondary flow is then obtained: the k–ε model shows a strong dependence of the secondary flow velocity field with Reynolds number as was pointed out with precedent experimental results. Variations of the number of vortexes composing the secondary flow, giving a symmetrical or asymmetrical aspect, will influence the fluid particle trajectories and time residence. To cite this article: S. Petitot et al., C. R. Mecanique 330 (2002) 749–756.     

A method of plasticity for general aligned spheroidal void or fiber-reinforced composites

Based on the general concept of the secant moduli method, together with a new way of evaluating the average matrix effective stress originally proposed by Qiu and Weng (“A Theory of Plasticity for Porous Materials and Particle-Reinforced Composites”, ASME J. Appl. Mech. (1992), 59, 261.), a method for nonlinear effective properties of general aligned fiber or void composites is proposed. The method is capable of predicting composite (especially for porous materials) yielding under a hydrostatic load. Compared to the Tandon and Weng (“A Theory of Particle-Reinforced Plasticity,” ASME J. Appl. Mech. (1988), 55, 126.), model the proposed method always gives softer prediction in the uniaxial tension. The proposed method will predict the same nonlinear stress and strain relation as the Ponte Castaneda (“The Effective Mechanical Properties of Nonlinear Isotropic Composite,” J. Mech. Phys. Solids (1991), 39, 45.) variational model if the same estimates or bounds for the linear comparison composite are adopted.     

Validation of a numerical model for the simulation of an electrostatic powder coating process

Numerical modeling of a complete powder coating process is carried out to understand the gas-particle two-phase flow field inside a powder coating booth and results of the numerical simulations are compared with experimental data to validate the numerical results. The flow inside the coating booth is modeled as a three-dimensional turbulent continuous gas flow with solid powder particles as a discrete phase. The continuous gas flow is predicted by solving Navier–Stokes equations using a standard k–ε turbulence model with non-equilibrium wall functions. The discrete phase is modeled based on a Lagrangian approach. In the calculation of particle propagation, a particle size distribution obtained through experiments is applied. The electrostatic field, including the effect of space charge due to free ions, is calculated with the use of the user defined scalar transport equations and user defined scalar functions in the software package, FLUENT, for the electrostatic potential and charge density.     

Optimal lower bounds on the hydrostatic stress amplification inside random two-phase elastic composites

Composites made from two linear isotropic elastic materials are subjected to a uniform hydrostatic stress. It is assumed that only the volume fraction of each elastic material is known. Lower bounds on all rth moments of the hydrostatic stress field inside each phase are obtained for r?2. A lower bound on the maximum value of the hydrostatic stress field is also obtained. These bounds are given by explicit formulas depending on the volume fractions of the constituent materials and their elastic moduli. All of these bounds are shown to be the best possible as they are attained by the hydrostatic stress field inside the Hashin-Shtrikman coated sphere assemblage. The bounds provide a new opportunity for the assessment of load transfer between macroscopic and microscopic scales for statistically defined microstructures.     

New bounds and estimates for porous media with rigid perfectly plastic matrixNouvelles bornes et estimations pour les milieux poreux à matrice rigide parfaitement plastique

We derive new rigorous bounds and self-consistent estimates for the effective yield surface of porous media with a rigid perfectly plastic matrix and a microstructure similar to Hashin's composite spheres assemblage. These results arise from a homogenisation technique that combines a pattern-based modelling for linear composite materials and a variational formulation for nonlinear media. To cite this article: N. Bilger et al., C. R. Mecanique 330 (2002) 127–132.     

Homogenization of an elastic medium reinforced by anisotropic fibersHomogénéisation d'un milieu élastique renforcé par des fibres anisotropes

We present results in this Note concerning a vector version in the framework of linearized elasticity (see A. Sili, Homogenization of an elastic medium reinforced by anisotropic fibers, in press), of our previous work in which we have studied the homogenization of a scalar nonlinear monotone problem posed on a fibered medium (see A. Sili, Homogenization of a nonlinear monotone problem in an anisotropic medium, in press). Here, we assume that parallel elastic anisotropic fibers, periodically distributed with a period of size ε in a cube $\text{Ω}$, are surrounded by a soft elastic material, the elasticity coefficients of this material being in the ratio ε² with those of the fibers. We prove that the homogenized problem is nonlocal and involves variables linked together with the anisotropy of the fibers. To cite this article: A. Sili, C. R. Mecanique 331 (2003).     

Magnetic field and magneto elastic stress in an infinite plate containing an elliptical hole with an edge crack under uniform electric current

Two-dimensional solutions of the electric current, magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack under uniform electric current. Using a rational mapping function, the each solution is obtained as a closed form. The linear constitutive equation is used for the magnetic field and the stress analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate which raises a plane stress state for a thin plate and the deformation of the plate thickness. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, electric current, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that the stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solving the present magneto elastic stress problem, dislocation and rotation terms appear, which makes the present problem complicate. Solutions of the magneto elastic stress are nonlinear for the direction of electric current. Stresses in the direction of the plate thickness are caused and the solution is also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length and the electric current direction.     

The Effect of Reynolds Number on the Passive Scalar Field in the Turbulent Wake of a Circular Cylinder

In this paper we investigate by experiments the effect of Reynolds number on a passive scalar (temperature) field in the turbulent wake of a slightly heated circular cylinder. The Reynolds number defined by Re≡U _∞ d/ν (see Nomenclature) is varied from Re= 1200 to Re= 8600. Temperature differential above ambient is chosen to be the passive scalar quantity. Present measurements are conducted using a cold wire (0.63 μm) probe. Results obtained suggest that Reynolds number in general has significant influence on the scalar mixing characteristics in the entire wake flow. Specifically, as Re increases, the mean scalar spreads out more rapidly, the scalar fluctuation intensity increases; however, its variance decays at a lower rate with downstream distance. It is also found that an increase of Re accelerates the streamwise evolution of the scalar probability density function from highly non-Gaussianity to near Gaussianity along the wake centreline. This reflects the reduction in length of the Karman-vortex street caused by an increase of Re.