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This work is concerned with the spatio-temporal structures that emerge when non-identical, diffusively coupled oscillators synchronize. It contains analytical results and their confirmation through extensive computer simulations. We use the Kuramoto model which reduces general oscillatory systems to phase dynamics. The symmetry of the coupling plays an important role for the formation of patterns. We have studied the ordering influence of an asymmetry (non-isochronicity) in the phase coupling function on the phase profile in synchronization and the intricate interplay between this asymmetry and the frequency heterogeneity in the system. The thesis is divided into three main parts. Chapter 2 and 3 introduce the basic model of Kuramoto and conditions for stable synchronization. In Chapter 4 we characterize the phase profiles in synchronization for various special cases and in an exponential approximation of the phase coupling function, which allows for an analytical treatment. Finally, in the third part (Chapter 5) we study the influence of non-isochronicity on the synchronization frequency in continuous, reaction diffusion systems and discrete networks of oscillators. / Die vorliegende Arbeit beschäftigt sich in Theorie und Simulation mit den raum-zeitlichen Strukturen, die entstehen, wenn nicht-identische, diffusiv gekoppelte Oszillatoren synchronisieren. Wir greifen dabei auf die von Kuramoto hergeleiteten Phasengleichungen zurück. Eine entscheidene Rolle für die Musterbildung spielt die Symmetrie der Kopplung. Wir untersuchen den ordnenden Einfluss von Asymmetrie (Nichtisochronizität) in der Phasenkopplungsfunktion auf das Phasenprofil in Synchronisation und das Zusammenspiel zwischen dieser Asymmetrie und der Frequenzheterogenität im System. Die Arbeit gliedert sich in drei Hauptteile. Kapitel 2 und 3 beschäftigen sich mit den grundlegenden Gleichungen und den Bedingungen für stabile Synchronisation. Im Kapitel 4 charakterisieren wir die Phasenprofile in Synchronisation für verschiedene Spezialfälle sowie in der von uns eingeführten exponentiellen Approximation der Phasenkopplungsfunktion. Schliesslich untersuchen wir im dritten Teil (Kap.5) den Einfluss von Nichtisochronizität auf die Synchronisationsfrequenz in kontinuierlichen, oszillatorischen Reaktions-Diffusionssystemen und diskreten Netzwerken von Oszillatoren.

Turbulent gas-particle suspensions are of great practical importance in many naturally phenomena, such as dust storms and snow avalanches, as well as in industrial applications such as fluidised, circulating bed reactors and pneumatic transport. Due to the difference in mass density of about three orders of magnitude between solids and gases, the mass loading is large, but the volume fraction of the particles is usually small. Since the length scale of these flows ranges from tens of centimeters to hundreds of meters, the Reynolds number based on the flow dimension and velocity is usually large. Due to this, these flows are almost always in the turbulent regime, and the fluid velocity fluctuations are significant. The particle sizes are typically small in these applications, of the order of 100 m or less. Due to this, the Reynolds number (based on the particle size and velocity) is usually low. This implies that the fluid inertia is not important, and the flow dynamics is dominated by fluid viscosity at the particle scale. At the same time, due to the density contrast between the particles and fluid, the Stokes number (ratio of particle inertia and fluid viscosity) is large. The inertia is sufficiently large that the particles cross the fluid streamlines. In this situation, there is a two-way coupling between the fluid turbulence and the particle dynamics. The turbulent fluid velocity fluctuations result in particle velocity fluctuations due to the drag force exerted by the particles on the fluid. In turbulent gas-particle suspensions, the fluctuating velocity of the particles results in a force on the fluid, which could either enhance or dampen the turbulent velocity fluctuations. The finite size of the particles could also result in fluid velocity effects which can not be captured by considering the particles as point masses. The dynamics of turbulent particle suspensions is analysed in the limit of low particle Reynolds number and high particle Stokes number, where there is a balance between particle inertia and fluid viscosity. The turbulent gas flow in a channel is considered for definiteness, in order to analyse the effect of turbulent fluctuations, as well as the effect of cross-stream variations in the turbulent statistics. The particle size is considered to be comparable to the Kolmogorov scales, which are the smallest scales in the turbulent flow. In addition, the fluid inertia at the particle scale is neglected, and the particles are dynamics is modeled using the Stokes equations. However, inertial effects are included at the macroscopic scale, where the Navier-Stokes equations are solved by Direct Numerical Simulations (DNS) using Chebyshev-Fourier spectral techniques. There are three important objectives in the present analysis. 1. The first is to examine the turbulence modification due to the reverse force of the particles. There are two models used for the reverse force of the particles on the fluid. The first is a point force, which is modeled as a delta function in real space. Instead of using smoothing functions for the delta function, we prefer to incorporate the point force in the momentum conservation equation in spectral space. A more complicated representation proposed here involves the inclusion of the symmetric and anti-symmetric force moments, calculated from the solution of Stokes equations for the flow around the sphere. These are represented as gradients of delta functions, and are also included in the momentum conservation equations in the spectral co-ordinates. 2. The second objective is to examine the effect of particle rotation and collisions on the flow dynamics. While particle rotation is usually included in the analysis of granular flows, this is not normally included in the treatment of particle collisions. 3. The third objective is to examine whether the effect of the fluid turbulence can be modeled as a fluctuating force. When the viscous relaxation time of the particles is larger than the integral time for the fluid velocity fluctuations, the fluid velocity fluctuations can be considered as delta function correlated in time, and the effect of these fluctuations can be incorporated using a Langevin description. In this case, the diffusion coeffcients in the Langevin equation for the particles is calculated from the correlation in the fluid velocity fluctuations. The new objective here is to include both the drag force and the torque exerted on the particles in the presence of particle rotation, and to examine whether these are sufficient to capture the effect of ff fluid turbulence on the particle phase. The Direct Numerical Simulations show that there is a significant attenuation of the turbulent velocity fluctuations when the reverse force exerted by the particles is added in the fluid momentum equations, and the particles are considered to be smooth. This turbulence attenuation is greater when the particle volume fraction increases, and when the particle mass density increases. However, when particle rotation is included, the turbulent velocity fluctuations are significantly larger than those without rotation, and in come cases are close the fluctuation levels when the reverse force is included. Thus, the particle rotation has a significant enhancement on the turbulent velocity fluctuations. The attenuation in the fluid turbulence is also reflected in the magnitude of the particle fluctuating velocities. The particle fluctuating velocities are higher when the effect of particle rotation is included. The reason for this is that there is particle rotation induced due to mean fluid shear, and this rotational energy gets transformed into translational energy in inter-particle collisions. The effect of inclusion of the symmetric and anti-symmetric force moments does not result in a significant change in the turbulence intensities for the range of volume fractions and mass densities considered here. There is a small but discernible increase in the turbulence for the largest volume fraction and mass density considered here, but this increase is much smaller than the significant turbulence attenuation due to the inclusion of particle rotation. Systematic trends are also observed in the particle linear and angular velocity distributions. The particle stream-wise linear velocity distribution, and the span-wise angular velocity distribution are broader than a Gaussian distribution near the zero, and exhibit steep decrease at larger velocity. They are also asymmetric, and the distribution depends on the location across the channel. The distribution of the cross-stream and span-wise linear velocity and the stream-wise and cross-stream angular velocity, is narrower than a Gaussian distribution at the center, and exhibits long tails for high velocities. Thus, there are systematic variations in the distribution functions for both the linear and angular velocities, which need to be included in kinetic theory descriptions for the particle phase. The fluctuating force model has also been simulated, where particle dynamics is explicitly simulated, the fluid velocity fields are not simulated, but are modeled as fluctuating forces and torques acting on the particles. The variance in the fluctuating force and torque are determined from the correlations in the fluid velocity and the vorticity fields, and these are modified to include the turbulence attenuation due to the reverse force exerted by the particles. The fluctuating force simulations do accurately capture the trends observed in the mean and fluctuating velocities. They are also able to capture the non-Gaussian nature of the linear and angular velocity distributions of the particles, even though the random forcing is considered to be a Gaussian function. Thus, the fluctuating force formulation can be used to accurately capture the effect of the fluid on the particles, only if the forces are modified to include the effect of turbulence attenuation due to the reverse force exerted by the particles.