Particle laden turbulent flows find applications in many industrial processes such as energy conversion, air pollution control etc. In these types of flows, there are strong coupling between the turbulent fluctuations in the fluid velocity fields, and the fluctuating velocities of the particles. In order to analyze the stresses and the heat and mass transfer properties in turbulent suspensions, it is necessary to have a good understanding of not just the mean flow of the gas and particles, but also of the fluctuations in the two phases. The coupling is a two-way coupling; the fluid turbulence contributes to the velocity fluctuations in the particles, and conversely, the particle velocity fluctuations generate fluctuations in the fluid. Two-phase flow models capture these interactions only in an indirect way, usually through a ‘particle pressure’ term for the particle phase. In the present work the effect of fluid velocity fluctuations on the dynamics of the particles in a turbulent gas-solid suspension is analyzed in the low Reynolds number and high Stokes number limit, where the particle relaxation time is long compared to the correlation time for the fluid velocity fluctuations. The direct numerical simulation (DNS) is used for solving the Navier-Stokes equations for the fluid, the particles are modeled as hard spheres which undergo elastic collisions. A one-way coupling algorithm is used where the force exerted by the fluid on the particles is incorporated, but not the reverse force exerted by the particles on the fluid. This is because the main focus of our study is to examine the effect of the fluid turbulence on the particle fluctuations, and we are interested in examining whether a Langevin model with random forcing can accurately capture the effect of fluid turbulence on the particle phase. First, the turbulent flow in a plane Couette is analyzed. Though this is a model flow which is not encountered often in applications, it is easier to analyze because the turbulent velocity fluctuations are maximum at the center of the channel, in contrast to the Poiseuille flow, where the velocity fluctuations are maximum at a location between the center and the wall. Also, in a Couette flow, the wall-normal and the spanwise root mean square velocities are nearly a constant in the central region in the channel, and the percentage variation in the stream-wise velocity fluctuations is also less than that in a pressure driven Poiseuille flow. Therefore, it is possible to treat the central region as a region with homogeneous, but anisotropic, fluid velocity fluctuations and with a linear mean velocity variation. The particle mean and root mean square fluctuating velocities, as well as the probability distribution function for the fluid velocity fluctuations and the distribution of acceleration of the particles in the central region of the Couette, which comprises about 20% of the entire channel have been studied. It is found that the distribution of particle velocities is very different from a Gaussian, especially in the span-wise and wall-normal directions. However, the distribution of the acceleration fluctuation on the particles is found to be close to a Gaussian, though the distribution is highly anisotropic and there is a correlation between the fluctuations in the flow and gradient directions. The non-Gaussian nature of the fluid velocity fluctuations is found to be due to inter-particle collisions induced by the large particle velocity fluctuations in the flow direction. Another interesting result is a comparison of the distribution of the acceleration on a particle due to the fluid velocity fluctuation at the particle position, and the distribution of the ratio of fluid velocity fluctuation to the viscous relaxation time in the fluid. The comparison shows that these two distributions are almost identical, indicating that the fluid velocity fluctuations are not correlated over time scales comparable to the relaxation time of a particle. This result is important because it indicates that in order to model the fluctuating force on the particle, it is sufficient to obtain the variance of the force distribution from the variance of the fluid velocity distribution function. Finally, the correlation time for the acceleration correlations is calculated along the trajectory of a particle. The correlation time is found to be of the same magnitude as the correlation time for the fluid velocity in an Eulerian reference frame, and much smaller than the viscous relaxation time and the time between collisions of the particles. All of these results indicate that the effect of the turbulent fluid velocity fluctuations can be accurately represented by an anisotropic Gaussian white noise. The above results are used to formulate a ‘fluctuating force’ model for the particle phase alone, where the force exerted by the fluid turbulent velocity fluctuations is modeled as random Gaussian white noise, which is incorporated into the equation of motion for the particles. The variance of the distribution function for the fluctuating force distribution is obtained from the variance of the local turbulent fluid velocity fluctuations, assuming linear Stokes drag law. The force distribution is anisotropic, and it has a non-zero correlation between the flow and gradient directions. It is found that the results of the fluctuating force simulations are in quantitative agreement with the results of the complete DNS, both for the particle concentration and variances of the particle velocity fluctuations, at relatively low volume fractions where the viscous relaxation time is small compared to the time between collisions, as well as at higher volume fractions where the time between collisions is small compared to the viscous relaxation time. The simulations are also able to predict the velocity distributions in the center of the Couette, even in cases where the velocity distribution is very different from a Gaussian distribution. The fluctuating force model is applied to the turbulent flow of a gas-particle suspension in a vertical channel in the limit of high Stokes number. In contrast to the Couette flow analyzed the fluid velocity variances in the different directions in the channel are highly non-homogeneous, and they exhibit a significant variation across the channel. First, we analyze the fluctuating particle velocity and acceleration distributions at different locations across the channel using direct numerical simulation. The distributions are found to be non-Gaussian near the center of the channel, and they exhibit significant skewness. The time correlations of the fluid velocity fluctuations and the acceleration fluctuations on the particles are evaluated and compared. Unlike the case of Couette flow it is found that the time correlation functions for the fluid in the fixed Eulerian frame are not in agreement with the time correlation of the acceleration on the particles. However, the time correlations of the particle acceleration are in good agreement with the velocity time correlations in the fluid in a ‘moving Eulerian’ reference frame, moving with the mean velocity of the fluid. The fluctuating force simulations are used to model the particle phase, where the force on the particles due to the fluid velocity fluctuations are substituted by random white noise in the equations for the particle motion. The random noise is assumed to be Gaussian and anisotropic. The variances of the fluctuating force are calculated form the fluid velocity fluctuations in a moving Eulerian reference frame using DNS. The results from the fluctuating force simulations are then compared with the results obtained from DNS. Quantitative agreement between the two simulations are obtained provided the particle viscous relaxation time is at least five times larger than the fluid integral time. The interactions between the solid particles and the fluid turbulence have been investigated experimentally in a vertical fully developed channel flow of air and solid particles. Experiments are conducted at low volume fraction for which viscous relaxation time of the particle is expected to be lower than the particle particle collision time, as well as at moderately high volume fraction where the particle particle collision time is expected to be lower than the particle relaxation time. Velocity statistics of both the particle and gas phases are obtained using high spatial resolution Particle Image Velocimetry (PIV) system. It is observed that at low solid volume fraction, the particle root mean square velocities and the velocity distribution are in good agreement with those predicted by the fluctuating force simulation, provided the polydispersity in the particle size distribution is incorporated in the fluctuating force simulations. In this case, the modification of turbulence in the center of the channel due to the particles is small. At much higher volume fraction, the mean gas flow is significantly affected by the presence of particles, and the mean flow is no longer symmetric about the center line of the channel. Simultaneously, there is also a significant change in the volume fraction across the channel, and the volume fraction is also not symmetric about the center line. This seems to indicate that there is a spontaneous instability of the symmetric volume fraction and velocity profiles, giving rise to a region of high fluid velocity and high particle volume fraction coexisting with a region of low gas velocity and low particle volume fraction. There is some recirculation of the gas within the channel, and the gas phase turbulence intensity is significantly enhanced when the velocity and volume fraction profiles become asymmetric. As we have considered only one way coupling in the computation of the particle laden flow it is expected that the particle statistics obtained for this condition can not be predicted by our fluctuating force model due to modification of the gas phase statistics.

The objective is to contribute to several issues regarding the traditional view of the local structure of passive scalar fields: (1) probability density function (PDF) of the scalar concentration and scalar gradient, (2) the scalar power spectrum, (3) the structure functions, and (4) correlation functions and multi-point correlators. In addition, the research provides a geometric description of two-dimensional transects of the passive scalar iso-surfaces using the tools of fractal geometry. The local structure is analyzed as a function of large-scale anisotropy, intermittency factor, Reynolds number, and initial condition of the scalar injection. Experiments were performed in the bed boundary layer produced by a uniform depth open channel flow of water in a tilting flume for Re_lamda = 63, 94, and 120. A small nozzle iso-kinetically delivers a passive scalar of high Schmidt number ( Sc = 1000) at mid-depth to generate the turbulent scalar field. Three nozzle diameters are used to study the effects of the injection length scale. High-resolution planar laser induced fluorescence (PLIF) technique is used to measure the scalar field. The local structure far from isotropic and is influenced even at the smallest scales by large-scale anisotropy, initial injection length scale and the Reynolds number of the flow. The PDF of the scalar fluctuations is non-Gaussian and dependent on large-scale anisotropy. The PDF of scalar gradients show the influence of large-scale anisotropy on the structure at the smallest scales. The spectrum of the scalar field deviates from the in the inertial convection regime and is dependent on large-scale anisotropy, external intermittency, and low Reynolds number. There is no evidence of Batchelors k^-1 scaling law. The scaling exponents of the even-ordered structure functions appear to be inversely correlated with the kurtosis of the scalar fluctuations. The fractal geometry of the two dimensional transects of passive scalar iso-surfaces is scale dependent. The fractal dimension is 1.0 at the smallest length scale and increases in a universal manner in the viscous-convective regime. The coverage length underestimate reflects this universal behavior with practical significance. The lacunarity function shows that the instantaneous scalar field is most in-homogenous around the Kolmogorov scale.

The paper considers systems of linear first-order ODEs with a randomly perturbed system matrix and stationary additive noise. For the description of the long-term behavior of such systems it is necessary to study their stationary solutions. We deal with conditions for the existence of stationary solutions as well as with their representations and the computation of their moment functions. Assuming small perturbations of the system matrix we apply perturbation techniques to find series representations of the stationary solutions and give asymptotic expansions for their first- and second-order moment functions. We illustrate the findings with a numerical example of a scalar ODE, for which the moment functions of the stationary solution still can be computed explicitly. This allows the assessment of the goodness of the approximations found from the derived asymptotic expansions.

<p>We investigate the clustering of HI-selected galaxies in the ALFALFA survey and compare results with those obtained for HIPASS. Measurements of the angular correlation function and the inferred 3D-clustering are compared with results from direct spatial-correlation measurements. We are able to measure clustering on smaller angular scales and for galaxies with lower HI masses than was previously possible. We calculate the expected clustering of dark matter using the redshift distributions of HIPASS and ALFALFA and show that the ALFALFA sample is somewhat more anti-biased with respect to dark matter than the HIPASS sample. We are able to conform the validity of the dark matter correlation predictions by performing simulations of the non-linear structure formation. Further we examine how the bias evolves with redshift for radio galaxies detected in the the first survey.</p>

We review properties of the Beurling correlation function related to differentiability and functional equations. The relevance of this function is due to the fact that some properties of the Riemann zeta function can be expressed interms of it. / Se repasan algunas propiedades de la función de correlación de Beurling, que sirven para expresar ciertas propiedades de la función zeta de Riemann.

Philosophiae Doctor - PhD / We investigate the clustering of HI-selected galaxies in the ALFALFA survey and compare results with those obtained for HIPASS. Measurements of the angular correlation function and the inferred 3D-clustering are compared with results from direct spatial-correlation measurements. We are able to measure clustering on smaller angular scales and for galaxies with lower HI masses than was previously possible. We calculate the expected clustering of dark matter using the redshift distributions of HIPASS and ALFALFA and show that the ALFALFA sample is somewhat more anti-biased with respect to dark matter than the HIPASS sample. We are able to conform the validity of the dark matter correlation predictions by performing simulations of the non-linear structure formation. Further we examine how the bias evolves with redshift for radio galaxies detected in the the first survey. / South Africa

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Parameters of Heine and Abarenkov potential has been computed in this paper for twenty two hexagonal closed pack (hcp) crystals. From the minimization of structure dependent energy of the pure crystal the inter-relation between the two parameters of the potential is first determined. Calculation uses pseudopotential technique with nine different exchange and correlation functions and either only available experimental value of vacancy formation energy (E1vF) or that obtained from an empirical relation based on other experimental parameters (Melting temperature, cohesive energy or activation energy) as tool. The variation of E1vF with parameter A of HAP and different exchange and correlation functions (ECF) show sharp fall in E1vF near very small value of A after which it shows constancy for all hcp crystals. Comparison is made with parameter of Ashcroft model also. For Aschroft this variation is almost flat showing averageness while for Heine and Abarenkov sharp variations are there from one hcp crystal to other.

La synthèse d'ouverture est une technique interférométrique similaire à la synthèse par rotation de la terre utilisée en radioastronomie où les signaux reçus par une paire de petites antennes sont traités de telle manière à synthétiser une seule grande antenne. Le concept de synthèse d'ouverture a été réadapté pour l'observation de la terre dans le cas de la télédétection de sources étendues de température. L'utilisation de cette technique pour l'observation de la terre a permis de contourner les limitations sur la taille d'antenne en télédétection passive. La fonction de corrélation, ou de visibilité, obtenue en inter-corrélant les signaux reçus par les an- tennes d'un système interférométrique employant une synthèse d'ouverture est définie comme étant la transformée de Fourier de la carte des températures de bril lance de la scène observée. Cette relation est connue sous le nom du théorème de Van Cittert-Zernike pour des observateurs en repos par rapport aux sources de température. La forme classique de ce théorème a été dérivée en inter-corrélant les échantillons temporels instantanés du champ électrique mesurés par différentes antennes. Un nouveau concept basé une interférométrie spatio-temporelle passive a été proposé comme étant la nouvelle génération qui succédera à la mission SMOS (Soil Moisture and Ocean Salinity) opérant dans l'espace depuis Novembre 2009. Celui-ci a pour objectif principal l'amélioration de la résolution spatiale à des ordres pouvant répondre aux applications hydrologiques à l'échelle locale où des résolutions kilométriques sont exigées. Ce concept interférométrique se base sur l'idée d'intégrer le déplacement de l'observateur (l'antenne) et ainsi la variable temps dans le calcul de la fonction de corrélation. Ceci engendre la création de nouvelles lignes de base virtuelles entre les positions des antennes à des instants différents, en plus des lignes de base physiques formées entres les positions des antennes instantanées. L'étude de ce concept de corrélation a malheureusement démontré la suppression exacte de l'information additionnelle due aux lignes de base virtuelles par le décalage Doppler induit par le déplacement. Une seconde étude du concept d'interférométrie spatio-temporelle combinée à une nouvelle procé- dure d'imagerie par corrélation dans l'espace fréquentiel, accomplie en inter-corrélant les spectres fréquentiels des champs électriques mesurés par une paire d'antennes séparées d'une distance Δr à bord d'un satellite à une hauteur h, a démontré l'obtention d'une information en 2D en températures de brillance de la scène observée. En plus, le développement théorique de la fonction de corrélation a mis en évidence une relation liant les visibilités aux températures de brillance par l'intermédiaire d'un noyau hautement oscillatoire. L'élément nouveau apporté par la corrélation dans l'espace fréquentiel consiste à exploiter l'informati- on de corrélation acquise par les antennes du satellite pour des fréquences présentant de petites dif- férences et pas seulement l'auto-corrélation. Cette propriété permet une reconstruction en 2D des températures de brillance avec seulement deux antennes / Aperture synthesis is an interferometric technique similar to Earth rotation synthesis employed in radio astronomy in which the signals received by a pair of small antennas are processed in a way to synthesize a single large antenna. The aperture synthetic concept used in radioastronomy was readapted to Earth remote sensing for large thermal sources. Thanks to this technique, limitations on antenna size in passive microwave remote sensing have been overcome. The correlation, or visibility, function obtained by cross-correlating the signals received by the antennas of an interferometric system using aperture synthesis is linked to the brightness temperature map of the observed scene by means of a Fourier-transform law. This is know as the standard form of the Van Cittert-Zernike theorem for fixed observers with respect to sources of temperature. This stan- dard formulation is derived by cross-correlating the instantaneous temporal components of the measured electric fields by different antennas. A new concept based on a passive spatio-temporal interferometry was proposed as the new generation to follow the well-known SMOS (Soil Moisture and Ocean Salinity) mission successfully operating since November 2, 2009. The aim of the proposed concept is a jump in the current achieved geometric resolution to orders capable of meeting the stringent users' needs for the study of hydrological applications in the local scale where sub-kilometric resolutions are required. This interferometric concept is based on the idea of integrating the displacement of the observer (satellite's antenna), and hence the time variable, in the calculation of the correlation function, which yields the creation of virtual baselines between the positions of antennas at different instants, in addition to the physical ones formed between the instantaneous antennas' spatial positions. Sadly, the additional information due to the virtual baseline was shown to be exactly canceled by the induced Doppler shift due to the observer's motion. We show furthermore that when using the aforementioned spatio-temporal interferometric system combined with a revolutionary Fourier Correlation Imaging (FouCoIm) procedure, consisting in cross-correlating, at slightly different frequencies, the Fourier components of the fluctuations of the re- ceived electric fields by a pair of antennas separated by a distance Δr on board of a satellite flying at height h, the 2D position-dependent brightness temperature can be reconstructed. Besides, the analytical derivation of the correlation function gives rise to a relationship linking the measured cor- relations to the position-dependent brightness temperatures by means of a Highly Oscillatory Integral (HOI) kernel. Interestingly, the analytical study of the HOI kernel showed the remarkable property that a corre- lation between both antenna-signals remains within a small frequency interval (different frequencies) outside the simple auto-correlation (same frequency). As a matter of fact, while existing systems had, until now, only considered the simple 1D information contained in the auto-correlation, it appears that the resulting correlation function from this concept bears a 2D information for the measurement of the position-dependent brightness temperature. Based on this, one is capable of reconstructing 2D bright- ness temperatures starting from a simple 1D geometry (two antennas arranged perpendicularly to the flight direction)

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