In our everyday life, turbulence is an omnipresent phenomenon and yet remains poorly understood. Its random and chaotic nature makes it a subject almost impossible to treat from the mathematical point of view and, at present, there is no real prospect of a simple analytic theory. Scientists have therefore regarded the numerical simulation as an alternative to compute the relevant properties of turbulent flows. In this context, our thesis aims at developing and using accurate computational methods, namely pseudo-spectral methods, for studying hydrodynamic (1st part) and magnetohydrodynamic (2nd part) turbulence. In the hydrodynamic part, Chapter I introduces the governing equations of fluid mechanics as well as the main issues related to the numerical study of turbulent flows. In particular, the Direct Numerical Simulations (DNS) of turbulence, in which accurate numerical solutions of the Navier-Stokes equations are obtained, are shown to be limited to moderately turbulent flows. Chapter II introduces the Large Eddy Simulation (LES) technique which aims at simulating highly turbulent flows and which is based on a separation of scales. In practice, it consists of simulating the large - resolved - scales of the flow explicitly while modelling the small - unresolved - scales. Two different approaches for modelling the kinetic energy of the unresolved scales are proposed and their respective advantages and drawbacks are discussed. Chapter III is devoted the study of the mixing-layer using both DNS and LES. It consists of an inhomogeneous turbulent flow which has been studied experimentally and for which well-documented measurements are available. A highly accurate DNS mimicking the same experiment has been produced. It allows to study the inhomogeneity and anisotropy properties of this flow. Also, LES of the same flow, using different models, have been evaluated. In Chapter IV, we explore a pseudo-spectral method to investigate turbulence in a pipe. In this case, the method has to take into account two additional difficulties: i) the presence of the boundary and ii) the axis singularity. We detail how to circumvent these issues. The second part of the thesis is devoted to magnetohydrodynamic (MHD) turbulence. It concerns phenomena where electrically conducting flows interact with electromagnetism and for which governing equations are derived in Chapter V. In Chapter VI, a detailed analysis of the energy transfers between the magnetic and velocity fields is performed thanks to a high resolution database of homogeneous MHD turbulence. It provides some insights to understand the physics of the nonlinear interactions and is also a valuable diagnostic in the framework of LES modelling. Finally, the inhomogeneous configuration studied in Chapter III has been extended to MHD. Several statistics related to the kinetic and magnetic energies are measured and LES of this flow are performed and presented in Chapter VII.

A number of different ways of using galaxy clusters to provide information concerning fundamental cosmological parameters are considered. Using the observed local cluster X-ray temperature function in conjunction with the Press-Schechter formalism, the normalisation of a CDM power spectrum is found to be σ(_8) = (0.52 ± 0.04)Ω(_o)(^-0.46+0.10Ωo) if Ʌ(_o) = 0 or σ(_8) = (0.52 ± 0.04)Ω(_o)(^-0.52+0.13Ωo) if Ʌ(_o) = 1 — Ω(_0). This result is employed to provide detailed predictions for the abundance of clusters at high redshift, and the differences between predictions for various cosmologies are emphasised. New tests using available high-redshift cluster data are presented. For the adopted power spectrum normalisation, it is found that an Ω(_o) = 0.3, Ʌ(_o) = 0 cosmology vastly overpredicts the number of clusters that were actually found with 0.4 < z < 0.6 in the Extended Medium Sensitivity Survey. The rapid variation in the expected abundance with both σ(_8) and the assumed scatter in the L(_x) – T_x) relation limits the significance of this result, but this model is still ruled out at the ~ 95% confidence level. Order statistics are utilised to calculate the probability of finding extremely massive clusters at high redshifts. With presently available observations, no interesting upper limit can yet be placed on Ω(_o). Systematic variations in the cluster-cluster correlation length calculated using numerical simulations and resulting from the definition of clusters, the chosen σ(_8), the mean intercluster separation and whether or not redshift space distortions are included, are found to exceed the statistical errors on the measurements. Although the uncertainty in ε(_cc) derived from an ensemble of 10 Standard CDM simulations is not sufficient at large separations to remove the discrepancy between this model and results from the APM Cluster Survey, this does suggest that the level at which such a scenario has previously been rejected using ε(_cc) should be significantly reduced. Details and a few tests of a procedure for improving mass and spatial resolution in cosmological simulations are presented. After showing that a coarse-sampling technique can be used to represent the large-scale forces sufficiently accurately, the method is then used to perform ten simulations of clusters forming in an Ω(_o) = 0.3, Ʌ(_o) = 0.7 CDM cosmology. To incorporate non-radiative gas, an SPH code adapted to work on a GRAPEsupercomputer is used. The resulting clusters are found to have virial radii in good agreement with the predictions of the spherical collapse model, dark matter density profiles well described by the 'NFW formula and isothermal central gas components, with temperatures dropping by a factor of ~ 2 near the virial radius. The evolution of these properties is studied as well as that of the bulk quantities describing the clusters, with particular reference to the β parameters relating cluster gas temperatures with virial mass or velocity dispersion. Slightly greater evolution in the luminosity is seen than in previous Ω(_o) = 1 simulations, suggesting that the improved resolution is important. The β parameter relevant to the normalisation of the mass fluctuation spectrum is found to be 0.98 ± 0.07.

A large direct numerical simulation database spanning a wide range of Reynolds and Schmidt number is used to examine fundamental laws governing passive scalar mixing and turbulence structure. Efficient parallel algorithms have been developed to calculate quantities useful in examining the Kolmogorov small-scale phenomenology. These new algorithms are used to analyze data sets with Taylor scale Reynolds numbers as high as 650 with grid-spacing as small as the Kolmogrov length scale. Direct numerical simulation codes using pseudo-spectral methods typically use transpose based three-dimensional (3D) Fast Fourier Transforms (FFT). The ALLTOALL type routines to perform global transposes have a quadratic dependence on message size and typically show limited scaling at very large problem sizes. A hybrid MPI/OpenMP 3D FFT kernel has been developed that divides the work among the threads and schedules them in a pipelined fashion. All threads perform the communication, although not concurrently, with the aim of minimizing thread-idling time and increasing the overlap between communication and computation. The new algorithm is seen to reduce the communication time by as much as 30% at large core-counts, as compared to pure-MPI communication. Turbulent mixing is important in a wide range of fields ranging from combustion to cosmology. Schmidt numbers range from O(1) to O(0.01) in these applications. The Schmidt number dependence of the second-order scalar structure function and the applicability of the so-called Yaglomﾒs relation is examined in isotropic turbulence with a uniform mean scalar gradient. At the moderate Reynolds numbers currently achievable, the dynamics of strongly diffusive scalars is inherently different from moderately diffusive Schmidt numbers. Results at Schmidt number as low as 1/2048 show that the range of scales in the scalar field become quite narrow with the distribution of the small-scales approaching a Gaussian shape. A much weaker alignment between velocity gradients and principal strain rates and a strong departure from Yaglomﾒs relation have also been observed. Evaluation of different terms in the scalar structure function budget equation assuming statistical stationarity in time shows that with decreasing Schmidt number, the production and diffusion terms dominate at the intermediate scales possibly leading to non-universal behavior for the low-to-moderate Peclet number regime considered in this study. One of the few exact, non-trivial results in hydrodynamic theory is the so-called Kolmogorov 4/5th law. Agreement for the third order longitudinal structure function with the 4/5 plateau is used to measure the extent of the inertial range, both in experiments and simulations. Direct numerical simulation techniques to obtain the third order structure structure functions typically use component averaging, combined with time averaging over multiple eddy-turnover times. However, anisotropic large scale effects tend to limit the inertial range with significant variance in the components of the structure functions in the intermediate scale ranges along the Cartesian directions. The net result is that the asymptotic 4/5 plateau is not attained. Motivated by recent theoretical developments we present an efficient parallel algorithm to compute spherical averages in a periodic domain. The spherically averaged third-order structure function is shown to attain the K41 plateau in time-local fashion, which decreases the need for running direct numerical simulations for multiple eddy-turnover times. It is well known that the intermittent character of the energy dissipation rate leads to discrepancies between experiments and theory in calculating higher order moments of velocity increments. As a correction, the use of three-dimensional local averages has been proposed in the literature. Kolmogorov used the local 3D averaged dissipation rate to propose a refined similarity theory. An algorithm to calculate 3D local averages has been developed which is shown to scale well up to 32k cores. The algorithm, computes local averages over overlapping regions in space for a range of separation distances, resulting in N^3 samples of the locally averaged dissipation for each averaging length. In light of this new calculation, the refined similarity theory of Kolmogorov is examined using the 3D local averages at high Reynolds number and/or high resolution.

An understanding of the shape and nature of galaxy density profiles remains a major challenge to galaxy structure studies. The physical mechanisms thought to control these profiles include star formation rates and dynamical interactions, but we focus in this thesis on the contribution of dynamical parameters associated with the dark and baryonic matter. We follow the evolution of mass density profiles, and investigate the development of a truncation radius. Using GADGET-2, an N-body/SPH code with a prescription for star formation and feedback, and the SHARCNET computational facilities, we have generated over 200 galaxy models covering a full range of structural parameters. The galaxy models have a minimum of 1.4 million particles and most are evolved over a period of 10 Gyr. We find that the evolution of the galaxy mass density profile is controlled by the ratio of the disk mass fraction, $m_{d}$, to the halo spin parameter, $\lambda$. The strength of the two-component structure in disk profiles and speed at which this structure develops, is directly proportional to $m_{d}/\lambda$. While the development of a two-component profile is coupled to bar formation, not all barred galaxies develop a two-component profile. We also show that the slope of the outer profile is in close agreement with that of the initial profile and remains stable over time, whereas the inner profile slope evolves considerably. This result will greatly improve comparisons of observed with predicted measures of galaxy density profiles. Our galaxy database is the largest of its kind and a valuable resource for many potential galaxy structural studies. We conclude with a list of future investigations based on our study and new database. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2007-07-30 14:46:24.568

Dust grains are a crucial component of disks around young stellar systems where current observations and theory show that planets form. Dust grains must grow 10 orders of magnitude in size to become planets. However, one of the early steps in this growth phase faces stringent theoretical constraints. The metre barrier relates to two well-studied physical mechanisms which inhibit grain growth beyond centimetre sizes. We report on numerical studies which probe these early stages of planet formation including instabilities that promote dust concentration such as the streaming instability (SI). We explore several different SPH models for dusty gas evolution. We find the linear SI is difficult for SPH to capture because it begins with perturbations below the 1% level. We also employ the Athena 3rd order Eulerian code which has been used to study the SI in the linear phase and the non-linear or saturated phase. We present numerical confirmations of recent analytical predictions of enhancements to the SI growth rates caused by the dust settling to the disk midplane in the earliest stages of the protoplanetary disk evolution. Symmetric analytical predictions for SI growth are not directly relevant to the non-axisymetric, planar geometry of the saturated, non-linear phase. We lay the ground work to explore this in future work. / Thesis / Master of Science (MSc)

Parametric instability processes are thought to produce Stimulated Electromagnetic Emissions (SEE) during ionospheric heating experiments. The phenomenon is primarily attributed to plasma turbulence excited by the high frequency HF heater in the altitude region where the pump frequency <i>ω</i>₀ is near the plasma upper hybrid frequency <i>ω_uh</i>. In this study, parametric instability processes thought to produce SEE are studied using theoretical and electrostatic Particle-In-Cell PIC simulation models. The simulation plasma is driven with a uniform oscillating electric field directed nearly perpendicular to the background geomagnetic field {B} to consider interactions when <i>ω_uh</i> is near electron cyclotron harmonics <i>nΩ_ce</i>. The pump frequency and amplitude are varied to consider the effects on the simulation electric field power spectrum. In this study, theoretical predictions and numerical simulations are used to study the three-wave decay instability process thought to be responsible for the generation of the down-shifted sidebands, the downshifted peak DP and the downshifted maximum DM. In particular, the lower hybrid decay instability LHDI and the ion cyclotron decay instability ICDI are studied in detail. The theory is used to provide the angular regime, with respect to the direction perpendicular to the magnetic field, at which the sidebands develop as well as the frequency and wavenumber regimes of both the LHDI and the ICDI. The effect of the temperature ratio <i>T_e/T_i</i> for both instabilities is discussed. A comparison between the theoretical predictions, the simulation electric field power spectrum and the experimental observations are presented in this study. Time evolution of both the LHDI and the ICDI is also investigated. The theoretical predictions are also used to investigate the cascading of the LHDI and the ICDI. The spectra show consistencies with the experimental observations. A four-wave parametric decay instability process thought to be responsible for SEE broad up-shifted sideband spectral features is discussed as well. Many theoretical results are presented, in which the effect of stepping the heater frequency closer to the upper hybrid frequency on the angle of maximum growth <i>θ_max</i>, the growth rate γ and on both the frequency and wavenumber regimes of the four-wave process is investigated. The simulation electric field power spectrum showed a large amplitude up-shifted sideband and a much smaller amplitude down-shifted sideband, consistent with the experimental observations. Comparisons between the theoretical predictions, the simulation electric field power spectrum and the experimental observations are discussed in detail. The time evolution of the four-wave process is one important aspect that is also presented in this study. The development of density irregularities, cavities and particle heating is also analyzed and investigated in this study. / Ph. D.

Baroclinic tides play a significant role in driving deep-ocean mixing, which not only influences the transport of nutrients and other biological tracers, but also affects the large scale circulations. This thesis combines advanced numerical modeling techniques and available observations to investigate the characteristics of baroclinic tides in the Red Sea, and understand their formation and fate, and their influence on the circulation and ecosystem. Based on a validated 3D, non-hydrostatic and high-resolution MIT General Circulation Model (MITgcm), we first established four potential areas for the generation of baroclinic tide in the Red Sea: the Strait of Bab-el-Mandeb (BAM), the southern Red Sea, the Gulf of Suez, and the Strait of Tiran. These areas are consistently suggested by the spatial distribution of baroclinic tidal kinetic energy and energy fluxes. The majority of the baroclinic energy disappears within the basin; either dissipates due to friction and bottom drag or converts back into barotropic energy due to pressure. We next conducted 2D numerical simulations to investigate the generation mechanism behind the formation of the observed internal solitary waves(ISWs) in the southern Red Sea, revealing the dominant role of tide-topography interactions. Instead of evolving from the disturbed thermocline due to a locally impinging tidal beam ISWs are generated from the depression formed near the shelf. As this depression propagates out of the shelf, it gradually steepens and ultimately breaks into a group of ISWs. To further study the breaking and dissipation processes of the ISWs, we implemented a regional high-resolution MITgcm for the southern Red Sea with a realistic topography revealing that the breaking process is triggered by the fission mechanism. The associated particles transport is further quantitatively analyzed, revealing a significant transport, both vertically and horizontally. The thesis, finally examined the tidal influences on the basin-scale circulation of the Red Sea. Comparison experiments with and without tides show that, the intrusion of the Gulf of Aden Intermediate Water (GAIW) in summer is strongly influenced by the tides. It is suggested that tides enforce more turbulence and mixing at the strait, which significantly increase the diffusivity of heat and salinity.

The aim of this thesis is to implement source terms to numerical model for curent sheet in solar atmosphere. The chapters are structured in the way that can present us with basic knowledge of the Sun and processes in its atmosphere. Than we present important equations for numerical solution and for the initial equilibrium of our simulation. One of the last chapters is dedicated to software we use for our numerical simulations called FLASH. In the end of the thesis we present results of our numerical simulations.

In this work, the peroxide-initiated degradation of polypropylene (PP) in co-rotating intermeshing twin-screw extruders (COITSEs) is analyzed by means of numerical simulations. This reactive extrusion (REX) operation is simulated by implementing (i) a one-dimensional and (ii) a three-dimensional (3D) modeling approach. In the case of the 1D modeling, a REX mathematical model previously developed and implemented as a computer code is used for the evaluation of two scale-up rules for COITSEs of various sizes. The first scale-up rule which is proposed in this work is based on the concept of thermal time introduced by Nauman (1977), and the second one is based on specific energy consumption (SEC) requirements. The processing parameters used in testing the previously referred to scale-up approaches are the mass throughput, the screw rotating speed, and the peroxide concentration, whereas the extruder screw configuration and the barrel temperature profiles are kept constant. The results for the simulated operating conditions show that when the REX operation is scaled-up under constant thermal time, very good agreement is obtained between the weight-average molecular weight (Mw) and poly-dispersity index (PDI) from the larger extruders and the values of these parameters corresponding to the reference extruder. For the constant SEC approach, on the other hand, more significant variations are observed for both of the aforementioned parameters. In the case of the implemented constant thermal time procedure, a further analysis of the effect of the mass throughput and screw speed of the reference device on the scaled-up operation is performed. It is observed that when the lower mass throughput is implemented for the smaller extruder keeping a constant screw speed, the predicted residence times of extrusion for the larger extruders are lower, in general terms, than those corresponding to the reference device, and a converse situation occurs for the higher implemented value of the mass throughput. Also, in general terms, the higher increase of the reaction temperature on the scaled-up operation corresponds to the lower mass throughputs and higher screw speeds specified for the reference extruder. For the 3D modeling approach, two different case studies are analyzed by means of a commercial FEM software package. The REX simulations are performed under the assumption of steady-state conditions using the concept of a moving relative system (MRS). To complement the information obtained from the MRS calculations, simulations for selected conditions (for non-reactive cases) are performed considering the more realistic transient-state (TS) flow conditions. The TS flow conditions are associated to the time periodicity of the flow field inside the conveying elements of COITSEs. In the first case study, the peroxide-initiated degradation of PP is simulated in fully-filled screw elements of two different size COITSEs in order to evaluate scale-up implications of the REX operation. In the second case, the reacting flow is simulated for a conventional conveying screw element and a conveying screw element having a special design and corresponding to the same extruder size. For both of the analyzed cases, the effects of the initial peroxide concentration and mass throughput on the final Mw and PDI of the degraded resin are studied. The effect of the processing conditions is discussed in terms of the residence time distribution (RTD), the temperature of reaction, and the distributive mixing capabilities of the REX system. When analyzing the scale-up case, it is found that for the implemented processing conditions, the final Mws and PDIs are very close to each other in both of the analyzed flow geometries when the specified flow is close to that corresponding to the maximum conveying capabilities of the screw elements. For more restrictive flow conditions, the final Mws and PDIs are lower in the case of the screw element of the larger extruder. It is found that the distributive mixing ability of the reactive flow is mainly related to the specified mass throughput and almost independent of the specified peroxide concentration for a particular extruder size. For the analyzed screw elements, the conveying element corresponding to the small size extruder shows a slightly better distributive mixing performance. For this same case study, a further evaluation of the proposed scale-up criterion under constant thermal time confirms the trend of the results observed for the 1D simulations. In the second case study, the special type of screw element consists of screws rotating at different speeds which have different cross sections. In this case, the outer and inner diameters of both the special and the conventional type of screw elements are specified to be the same. As in the previous case study, the distributive mixing capabilities appear to be independent of the specified peroxide concentrations but dependent on the mass flow rate. It is speculated from the simulation results, from both the transient- as well as the steady-state flow conditions, that the screw element with the special design would yield lower final values of the PDI and Mw. Also, this screw element appears to have improved distributive mixing capabilities as well as a wider RTD.

In this work, the peroxide-initiated degradation of polypropylene (PP) in co-rotating intermeshing twin-screw extruders (COITSEs) is analyzed by means of numerical simulations. This reactive extrusion (REX) operation is simulated by implementing (i) a one-dimensional and (ii) a three-dimensional (3D) modeling approach. In the case of the 1D modeling, a REX mathematical model previously developed and implemented as a computer code is used for the evaluation of two scale-up rules for COITSEs of various sizes. The first scale-up rule which is proposed in this work is based on the concept of thermal time introduced by Nauman (1977), and the second one is based on specific energy consumption (SEC) requirements. The processing parameters used in testing the previously referred to scale-up approaches are the mass throughput, the screw rotating speed, and the peroxide concentration, whereas the extruder screw configuration and the barrel temperature profiles are kept constant. The results for the simulated operating conditions show that when the REX operation is scaled-up under constant thermal time, very good agreement is obtained between the weight-average molecular weight (Mw) and poly-dispersity index (PDI) from the larger extruders and the values of these parameters corresponding to the reference extruder. For the constant SEC approach, on the other hand, more significant variations are observed for both of the aforementioned parameters. In the case of the implemented constant thermal time procedure, a further analysis of the effect of the mass throughput and screw speed of the reference device on the scaled-up operation is performed. It is observed that when the lower mass throughput is implemented for the smaller extruder keeping a constant screw speed, the predicted residence times of extrusion for the larger extruders are lower, in general terms, than those corresponding to the reference device, and a converse situation occurs for the higher implemented value of the mass throughput. Also, in general terms, the higher increase of the reaction temperature on the scaled-up operation corresponds to the lower mass throughputs and higher screw speeds specified for the reference extruder. For the 3D modeling approach, two different case studies are analyzed by means of a commercial FEM software package. The REX simulations are performed under the assumption of steady-state conditions using the concept of a moving relative system (MRS). To complement the information obtained from the MRS calculations, simulations for selected conditions (for non-reactive cases) are performed considering the more realistic transient-state (TS) flow conditions. The TS flow conditions are associated to the time periodicity of the flow field inside the conveying elements of COITSEs. In the first case study, the peroxide-initiated degradation of PP is simulated in fully-filled screw elements of two different size COITSEs in order to evaluate scale-up implications of the REX operation. In the second case, the reacting flow is simulated for a conventional conveying screw element and a conveying screw element having a special design and corresponding to the same extruder size. For both of the analyzed cases, the effects of the initial peroxide concentration and mass throughput on the final Mw and PDI of the degraded resin are studied. The effect of the processing conditions is discussed in terms of the residence time distribution (RTD), the temperature of reaction, and the distributive mixing capabilities of the REX system. When analyzing the scale-up case, it is found that for the implemented processing conditions, the final Mws and PDIs are very close to each other in both of the analyzed flow geometries when the specified flow is close to that corresponding to the maximum conveying capabilities of the screw elements. For more restrictive flow conditions, the final Mws and PDIs are lower in the case of the screw element of the larger extruder. It is found that the distributive mixing ability of the reactive flow is mainly related to the specified mass throughput and almost independent of the specified peroxide concentration for a particular extruder size. For the analyzed screw elements, the conveying element corresponding to the small size extruder shows a slightly better distributive mixing performance. For this same case study, a further evaluation of the proposed scale-up criterion under constant thermal time confirms the trend of the results observed for the 1D simulations. In the second case study, the special type of screw element consists of screws rotating at different speeds which have different cross sections. In this case, the outer and inner diameters of both the special and the conventional type of screw elements are specified to be the same. As in the previous case study, the distributive mixing capabilities appear to be independent of the specified peroxide concentrations but dependent on the mass flow rate. It is speculated from the simulation results, from both the transient- as well as the steady-state flow conditions, that the screw element with the special design would yield lower final values of the PDI and Mw. Also, this screw element appears to have improved distributive mixing capabilities as well as a wider RTD.