Solution of the wave equation in a tridiagonal representation space

We use variable transformation from the real line to finite or semi-infinite spaces where we expand the regular solution of the 1D time-independent Schrodinger equation in terms of square integrable bases. We also require that the basis support an infinite tridiagonal matrix representation of the wave operator. By this requirement, we deduce a class of solvable potentials along with their corresponding bound states and stationary wavefunctions expressed as infinite series in terms of these bases. This approach allows for simultaneous treatment of the discrete (bound states) as well as the continuous (scattering states) spectrum on the same footing. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. These are written in terms of orthogonal polynomials, some of which are modified versions of known polynomials. The examples given, which are not exhaustive, illustrate the power of this approach in dealing with 1D quantum problems.Comment: 13 page

Validating CFD predictions of flow over an escarpment using ground-based and airborne measurement devices

Micrometeorological observations from a tower, an eddy-covariance (EC) station and an unmanned aircraft system (UAS) at the WINSENT test-site are used to validate a computational fluid dynamics (CFD) model, driven by a mesoscale model. The observation site is characterised by a forested escarpment in a complex terrain. A two-day measurement campaign with a flow almost perpendicular to the escarpment is analysed. The first day is dominated by high wind speeds, while, on the second one, calm wind conditions are present. Despite some minor differences, the flow structure, analysed in terms of horizontal wind speeds, wind direction and inclination angles shows similarities for both days. A real-time strategy is used for the CFD validation with the UAS measurement, where the model follows spatially and temporally the aircraft. This strategy has proved to be successful. Stability indices such as the potential temperature and the bulk Richardson number are calculated to diagnose atmospheric boundary layer (ABL) characteristics up to the highest flight level. The calculated bulk Richardson values indicate a dynamically unstable region behind the escarpment and near the ground for both days. At higher altitudes, the ABL is returning to a near neutral state. The same characteristics are found in the model but only for the first day. The second day, where shear instabilities are more dominant, is not well simulated. UAS proves its great value for sensing the flow over complex terrains at high altitudes and we demonstrate the usefulness of UAS for validating and improving models

Microscale CFD Simulations of a Wind Energy Test Site in the Swabian Alps with Mesoscale Based Inflow Data

The current study describes analyses of the WINSENT wind energy test site located in complex terrain in Southern Germany by highly resolved numerical simulations. The resolved atmospheric turbulence is simulated with Delayed Detached Eddy Simulations by the flow solver FLOWer without consideration of the research wind turbines. The mean inflow and wind direction of the analysed time period is provided by precursor simulations of project partners. The simulation model chain consists of three codes with different time scales and resolutions. The model chain provides a data transfer from mesoscale WRF simulations to OpenFOAM. As a next step OpenFOAM provides inflow data in the valley of the terrain site for the present FLOWer simulations, the code with the highest resolution in space and time. The mean velocity field provided by OpenFOAM is superimposed with fluctuations that are based on measurements to obtain the small turbulent scales within the FLOWer simulations, which the previous tools of the model chain can not resolve. Comparisons with the two already installed met masts clarify that the current FLOWer simulations provide an adequate agreement with measured data. The results are verified with the application of a second simulation, in which a homogeneous velocity profile is superimposed with turbulence. Thus, comparisons with measured data showed that the benefit of using the inflow data of this model chain is especially evident near the ground

Solution of One-dimensional Dirac Equation via Poincare Map

We solve the general one-dimensional Dirac equation using a "Poincare Map" approach which avoids any approximation to the spacial derivatives and reduces the problem to a simple recursive relation which is very practical from the numerical implementation point of view. To test the efficiency and rapid convergence of this approach we apply it to a vector coupling Woods--Saxon potential, which is exactly solvable. Comparison with available analytical results is impressive and hence validates the accuracy and efficiency of this method.Comment: 8 pages, 6 figures. Version to appear in EP

Transmission through Biased Graphene Strip

We solve the 2D Dirac equation describing graphene in the presence of a linear vector potential. The discretization of the transverse momentum due to the infinite mass boundary condition reduced our 2D Dirac equation to an effective massive 1D Dirac equation with an effective mass equal to the quantized transverse momentum. We use both a numerical Poincare Map approach, based on space discretization of the original Dirac equation, and direct analytical method. These two approaches have been used to study tunneling phenomena through a biased graphene strip. The numerical results generated by the Poincare Map are in complete agreement with the analytical results.Comment: 9 pages, 6 figure

Analyse multi-élémentaire par fluorescence X à dispersion d'énergie des apports fluviatiles et des sédiments superficiels de la plate-forme atlantique du Maroc

L'analyse multi-élémentaire d'une trentaine de sédiments d'oueds et de l'océan atlantique du Maroc a été réalisée en utilisant des spectromètres X à dispersion d'énergie et des sources radioactives d'excitation en Fer-55, Cd-109 et Am-241. Les échantillons ont été collectés de la plate-forme continentale atlantique, de Tanger à Agadir et des oueds situés entre ces deux villes. Les objectifs de cette analyse sont : * la caractérisation géochimique des sédiments en vue de l'approche du transfert de matière du continent vers l'océan en tenant compte des données morphologiques, climatiques et géologiques de l'environnement sédimentaire. Les échantillons ont été prélevés par benne de la plate-forme continentale atlantique et des lits des oueds suivants, du Nord vers le Sud : Tahaddart, Loukkos, Sebou, Bou Regreg, Oum er Rbia, Tensift et Souss. Les régions du Nord dont la plupart sont irriguées présentent un climat humide alors que celles du Sud ont un climat semi-aride à aride avec des oueds intermittents. * la détermination de pollutions possibles de la zone littorale en relation avec l'activité anthropique, principalement au large des zones urbanisées. la précision des résultats a été évaluée par l'analyse de deux matériaux standards de référence de 1'AIEA (Agence International de 1'Energie Atomique), le sédiment «SL-1» et le sol «soil-7». Cette précision est de 5% pour la plupart des éléments majeurs alors qu'elle n'est que de 10 à 20% pour les éléments traces.The XRF analysis of about 30 sediments from Moroccan oueds (rivers) and atlantic continental shelf for the content of major and trace elements has been performed, utilizing the energy dispersive X-ray spectrometer and radio-isotope excitation with Fe-55, Cd-109 and Am-241 annular sources. Samples have been collected from Moroccan atlantic continental shelf from Tanger to Agadir and oueds between these cities. The objectives of this analysis were : * to determine the geochemical areas on Moroccan atlantic continent, * to explain the mechanisms of sediment transport from the continent to the sea throw the oueds using morphological, climate and hiterland impacts. Thus, sample have been collected from atlantic continental shelf and the following oueds, from the north with a humid climate and irrigated areas to the south with a semi-arid to arid climate : Tahaddart, Loukkos, Sebou, Bou Regreg, Oum er Rbia, Tensift and Souss, * to assess the eventual pollution of the coast by human activities, mainly at regions which are very urbanized. Results were assessed for the accuracy by the analysis of standard reference materials from IAEA (International Atomic Energy Agency), soil-7 and SL-1 sediment. The results obtained for most of the major elements were accurate to about 5% while the accuracy of trace element determination was much worse, 10 to 20%

Fiabilité de la mesure de vitesse débitante à l'aval d'une singularité en réseau d'assainissement

When managing a sewer network, the water discharge has to be measured in real time at several locations in the network. In locations where a simple stage discharge relationship cannot be applied due to an influence of the backwater curve, this measurement is to be performed using sensors that measure the bulk velocity locally in order to estimate the discharge. The aim of this work is to quantify the errors made when such measurement is performed downstream from two types of singularities: a bend and a confluence, which are commonly encountered in sewer networks. Two types of sensors are tested: a time-of-flight flow meters and a acoustic Doppler flowmeter. Both technologies are first detailed along with their possible sources of errors when used in the vicinity of singularities. The second section then introduces the numerical calculation of all flow configurations tested downstream both kinds of singularities. The third section shows the results in terms of error of discharge estimation as a function of the location with regards to the singularities, the type of singularity and the flow conditions. This permits to finally make recommendations for the use of such sensors in a real sewer network.Les gestionnaires de réseau d'assainissement urbain sont amenés à équiper leurs réseaux de points de mesures débimétriques. Lorsqu'une relation permanente permettant de relier localement la hauteur d'eau et le débit n'existe pas, il est nécessaire de mesurer la vitesse moyenne de l'écoulement afin d'en déduire le débit localement. L'objectif de cet article est de quantifier les erreurs faites lorsqu'une mesure de vitesse est effectuée à l'aval d'une singularité pour des situations réalistes de collecteurs d'assainissement urbain. La première section de cet article présente les deux types de capteurs (cordes et Doppler) en détaillant leur mode d'échantillonnage spatial ainsi que les sources d'erreur associées. La deuxième section présente la démarche de modélisation numérique ainsi que les domaines de calcul associés à l'aval respectivement d'une confluence et d'un coude. Enfin la troisième partie présente les résultats en termes de réponse des capteurs à ces champs de vitesse et discute les erreurs pour les différentes configurations considérées