M-curves of degree 9 with deep nests

The first part of Hilbert's sixteenth problem deals with the classification of the isotopy types realizable by real plane algebraic curves of given degree $m$. For $m \geq 8$, one restricts the study to the case of the $M$-curves. For $m=9$, the classification is still wide open. We say that an $M$-curve of degree 9 has a deep nest if it has a nest of depth 3. In the present paper, we prohibit 10 isotopy types with deep nests and no outer ovals.Comment: 16 pages, 11 figures v.4 minimal correction

A Fully Spectral 3D Time-Domain Model for Second-Order Simulation of Wavetank Experiments. Part B: Validation; Calibration versus experiments and Sample Applications

International audienceA 3D second-order numerical wavetank (NWT) model, SWEET, is presented. In the first part (A) of the paper [Bonnefoy F, Le Touzé D, Ferrant P. A fully-spectral 3d time-domain model for second-order simultion of wavetank experiments. Part A: Formulation, implementation and numerical properties. Appl Ocean Res 2005. doi:10.1016/j.apor.2006.05.004], the fully-spectral formulation we employ has been detailed, and the numerical properties of the model analyzed. In the present part (B), careful validation by comparison to analytical and experimental results is first reported. Thanks to the efficiency of the proposed spectral method, the shortest wavelengths in the wavetank can be accounted for with moderate computational times. The consequent possibilities are illustrated here for the following 2D and 3D complex wave-pattern simulations, with experimental comparisons: wave-packet and geometric focusing cases, directional wavefields, long-time evolutions of irregular waves. The numerical model features all the physical characteristics of a wavetank (snake wavemaker, experimentally-calibrated absorbing zone, etc.). Its usefulness to help preparing and analyzing experiments is shown in relation to some key practical requirements: e.g. quality and evolution of the usable test zone and usability of enhanced wavemaker motions

Représentations Temps-Fréquence de la classe de puissance basées sur l'invariant océanique

National audienceEn acoustique sous marine Ultra Basse Fréquence (UBF < 200 Hz), un environnement océanique petit fond (D < 200 m) se comporte comme un guide d'onde dispersif, dans lequel de nombreuses applications (inversion géoacoustique, localisation de source) considèrent la propagation de sources impulsionnelles (canons à air, explosions, baleines, ...). Comment obtenir une représentation temps-fréquence correcte des signaux correspondants? Ces signaux sont multicomposantes, chaque composante possède un retard de groupe non-linéaire et il n'existe pas de lien direct entre les différentes composantes. Cependant, le retard de groupe de chaque composante peut être approximé par une fonction puissance en f puissance (−1/Beta) , où "Beta"est l'invariant océanique (un scalaire qui résume la dispersion globale dans le guide d'onde). On peut alors définir une Beta-classe de représentations temps-fréquence adaptées au signal reçu. La méthode est appliquée avec succès sur des données réelles marines et permet une bonne représentation temps-fréquence des signaux considérés. Elle est robuste au bruit naturel de l'environnement marin, et à une erreur dans le choix de "Beta

3-D HOS simulations of extreme waves in open seas

In the present paper we propose a method for studying extreme-wave appearance based on the Higher-Order Spectral (HOS) technique proposed by West et al. (1987) and Dommermuth and Yue (1987). The enhanced HOS model we use is presented and validated on test cases. Investigations of freak-wave events appearing within long-time evolutions of 2-D and 3-D wavefields in open seas are then realized, and the results are discussed. Such events are obtained in our periodic-domain HOS model by using different kinds of configurations: either i) we impose an initial 3-D directional spectrum with the phases adjusted so as to form a focused <i>forced</i> event after a while, or ii) we let 2-D and 3-D wavefields defined by a directional wave spectrum evolve up to the <i>natural</i> appearance of freak waves. Finally, we investigate the influence of directionality on extreme wave events with an original study of the 3-D shape of the detected freak waves

On computing the jump condition of the dissipation rate in the two-equation turbulence models for two-phase flow and application to air-water waves

Traditional turbulence models are derived for single-phase flow. Extension of the family of two-equation turbulence models for two-phase flow is obtained via scaling the transport equations by the density. In the special case of two-phase flow with a sharp interface, jump conditions exist. Two types of jump conditions are found: (1) jump in the partial differential equation (PDE) physical quantities such as density and viscosity and (2) jump in the turbulence frequency. We first derive and clarify the jump in the equations. The jump in the turbulence frequency is proportional to the kinematic viscosity ratio, which is approximately $10$ in the case of air-water. Then a new field, the inverse turbulence area, is considered to model the turbulence effects instead of the turbulence frequency. For the system of air and water, the effect of the jump of the kinematic viscosity is always greater than the effect arising from the jump of velocity gradient. This approximation leads to the assumption of a continuous inverse turbulence area scale. Validation versus experimental measurements from the literature is then presented to demonstrate the improvement of the model. In particular, the wave breaking phenomenon is simulated in two conditions: spilling and plunging wave breakers. The proposed model shows its ability to predict the turbulence in the surf zone accurately. Finally, it explains the low values of the time-averaged turbulent kinetic energy in the surf zone which is caused by the increase of the turbulence frequency in the air

Theoretical Analysis of SPH in Simulating Free-surface Viscous flows

A theoretical analysis on the performance, close to a free surface, of the most used SPH formulations for Newtonian viscous terms is carried out in this paper. After an introduction of the SPH formalism, the SPH expressions for the viscous term in the momentum equation are analyzed in their continuous form. Using a Taylor expansion, a reformulation of those expressions is undertaken which allows to characterize the behavior of the viscous term close to the free surface. Under specific flow conditions, we show that the viscous term close to the free surface is singular when the spatial resolution is increased. This problem is in essence related to the incompleteness of the kernel function close to the free surface and appears for all the formulations considered. In order to assess the impact of such singular behavior, an analysis of the global energy dissipation is carried out, which shows that such a free-surface singularity vanishes when the integral quantities are considered. Not with standing that, not all the SPH viscous formulas allow the correct evaluation of the energy dissipation rate and, consequently, they may lead to an inaccurate modelling of viscous free-surface flows