A systematic formula for the asymptotic expansion of singular integrals

Asymptotic theories like the lifting-line, the slender body or the slender ship lead to lineintegrals with singular kernels. Sometimes these integrals are improper, that is to say that they are defined only by their Finite Part. To find asymptotic expansions of these integrals, the Matched Asymptotic Expansion Method is widely used along with other more specific methods depending on the kernel type. The first method is laborious and not systematic, and the other methods are sometimes too much specific to treat general cases. Moreover, all of them are not well adapted to deal with Finite Part integrals.Here, a new method is proposed to avoid the previous difficulties. This method is systematic for homogeneous kernels and gives approximations up to any order, as long as the derivative of the weight function exists at this given order. Moreover the occurrence of logarithmic terms in the expansion is explained and easily predictable. An elliptic integral and the classical lifting-line theory are treated to illustrate the ease of this method.

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Résumé Les théories asymptotiques telles que la ligne portante, le corps élancé ou le navire de grand allongement conduisent à des intégrales curvilignes à noyaux singuliers. Parfois, ces intégrales sont impropres c'est à dire qu'elles sont définies en Parties Finies. Différentes méthodes ont été mises au point pour trouver les développements asymptotiques de ces intégrales. Généralement elles dépendent fortement de la nature du noyau, et c'est finalement la méthode des développements raccordés qui est utilisées quand le noyau est trop compliqué. Cependant, cette méthode est laborieuse et comme les précèdentes non adaptée aux intégrales défines par leur Partie Finie.Une nouvelle méthode est proposée pour surmonter ces difficultés. Cette méthode est systématique pour les noyaux homogènes et donne les approximations à tout ordre pourvu que les dérivées de la fonction poids existent jusqu'à cet ordre. De plus la présence de termes logarithmiques dans le développement est expliquée et aisément prédictible.Une intégrale elliptique, ainsi que la fameuse théorie de la ligne portante sont traités pour illustrer les possibilités de la méthode.

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Nomenclature D domain of integration - f(x) weight function - FP Finite Part - h(x) weight function - I ,I _o bounded intervals - j, J integers - K(x, ) singular kernel - L, L integers - M integer defining the approximation order - P _k (x) Legendre polynomial - R set of real numbers - R(ß) equals 1 if is an integer and 0 if not - R _{f, J} ,R _{K, L} remainders of Taylor developments - S () equals either 1 or the sign function:sgn() - t, u, v, x variable of integration - , real numbers - homogeneity order of the kernel - F () Euler's integral (gamma function) - small parameter - [.] integer part of     

Random Version of the Theorems of the Alternative

Three theorems of the alternative for abstract linear inequality systems which depend on random parameter are produced in infinite dimensions.     

Accelerating flames in tubes—an analysis

Flame acceleration in tubes is studied. A tube filled with flammable mixture is closed at one end and open to the atmosphere at its second end. When ignition takes place near the closed end, it is well-known from experiments that the flame may accelerate, oscillate and eventually reach considerable speeds. A one-dimensional analysis is presented, based upon the assumption that the flame front propagates at a speed that is small compared to the speed of sound. The analysis leads to a construction of the complete unsteady solution. Results from the analysis and from a numerical simulation are compared. They are similar enough to validate the analysis. The tube acoustics are set in motion by the expansion of the fluid due to ignition at the closed end. Subsequently, both spectrum and amplitude evolve because of the motion of the temperature interface, and because of forcing by the flame front, which the analysis precisely quantifies. Oscillations in the front position are strong enough to result in flow reversal. In addition, the induced periodic acoustic acceleration of the temperature and density interface will periodically make the flame front Rayleigh–Taylor unstable, which should result in the dramatic increase in the propagation speed seen in experiments.     

Micro–macro analysis of granular material behavior along proportional strain paths

When granular materials are subjected to proportional strain loading paths, they manifest a variety of behaviors depending on the initial void ratio of the specimen as well as the imposed dilatancy/contractancy rate. In some cases, the stress components may vanish over the duration of the test, and the specimen may progressively liquefy. To investigate this behavior, the authors have developed a kinematic approach to be deployed in two parts. First, numerical simulations are performed by means of a discrete element method. Secondly, two micromechanical models have corroborated the DEM results. The performance of these models may explain a number of microstructural mechanisms responsible for the macroscopic constitutive behavior.     

Synthesis of Symmetrical and Nonsymmetrical Bisthienylcyclopentenes

Diarylethenes possess unique structural properties, which enabled them to find widespread applications in the field of photochromism. Nowadays, bisthienylcyclopentenes (BTCs) present the most popular subfamily of these compounds, which are widely used as P‐type chromophores. This minireview summarises the main strategies for the synthesis of symmetrical and nonsymmetrical BTCs. In addition, attention is drawn to desymmetrisations achieved by monosubstitutions, which is not frequently utilised, although it can be highly advantageous. This is supported with some of the authors’ latest results.     

The Intramolecular Diels-Alder Furan Approach in Synthesis : 11-oxatricyclo|6.2.1.01,6|undec-9-en-5-one

Although the synthetic potential of the intramolecular variant of the Diels-Alder reaction has now been fully recognized³, little attention has so far been paid to this type of addition using the furan nucleus as the diene partner4–8. The presence of an oxygenated cyclohexene system in several complex natural products (e.g. gibberellic acid, morphine), the diverse possibilities of the substituted 7-oxabicyclo |2.2.1|-2-heptene system for further transformation⁹, the expected enhanced reactivity and stereoselectivity of the intramolecular cycloaddition and the availability of a wide variety of commercial furan derivatives make the sequence a → b → c particularly attractive in total synthesis.     

Eulerian Scalar Projection in Lagrangian Point Source Context: An Approximate Inverse Filtering Approach

In many simulations of flows transporting a discrete phase (droplets or particles), point sources are followed numerically in the Lagrangian context over an Eulerian mesh storing the continuous phase variables. The projection of the properties of the moving sources over the Eulerian nodes requires interpolations, which may drastically reduce the overall accuracy of the algorithm and alter the flow physics. From the analytical solution of a scalar field downstream of a source, it is shown that the widely used Lagrange/Euler projection, constructed from the regressive normalized distance between the point source position and the nodes cell, provides results very close to an approximate Gaussian filtering of the exact solution. Then, an approximate deconvolution is discussed to allow for better estimating the Eulerian scalar field.     

DNS Analysis of Wall Heat Transfer and Combustion Regimes in a Turbulent Non-premixed Wall-jet Flame

Understanding the heat-release effects on the wall heat transfer in turbulent reacting flows, i.e. heat transfer with or without significant density variation, is essential for a wide variety of industrial flows, especially combustion problems. The present study focuses on the wall heat transfer and the near-wall reaction characteristics. The heat-release effects on the wall heat transfer and skin-friction coefficients are investigated using three-dimensional direct numerical simulations of a turbulent reacting wall-jet flow with and without heat release. Reductions in the skin-friction coefficient are observed in the exothermic case, compared to the isothermal one, and the underlying mechanism is explained. The absolute wall heat flux also increases, while the corresponding Nusselt number decreases with increasing heat release. Furthermore, the wall effects on the near-wall average burning rate are assessed. It is found that the isothermal cold wall results in an appreciable decrease of the burning rate in the exothermic cases. We observed indications that the wall increases the chances for the development of the premixed mode and its occurrence is very fast in the wall-normal direction.     

Influence of capillary effects on strength of non-saturated porous mediaInfluence des effets capillaires sur la résistance d'un milieu poreux non saturé

The mechanical behavior of a partially saturated porous medium is addressed by means of a micro-to-macro reasoning. First, an estimate of the quadratic average over the solid phase of the equivalent shear strain is proposed. The latter is used in the framework of a nonlinear homogenization technique (‘modified secant’ method) in order to model the nonlinear poroelastic behavior in partially saturated conditions. The determination of the macroscopic strength criterion is then considered. Finally, the influence of membrane tension effects on strength is investigated. To cite this article: L. Dormieux et al., C. R. Mecanique 334 (2006).