Intermittency observed prior to thermoacoustic instability is characterized by the occurrence of bursts of high-amplitude periodic oscillations (active state) amidst epochs of low-amplitude aperiodic fluctuations (rest state). Several model-based studies conjectured that bursting arises due to the underlying turbulence in the system. However, such intermittent bursts occur even in laminar and low-turbulence combustors, which cannot be explained by models based on turbulence. We assert that bursting in such combustors may arise due to the existence of subsystems with varying timescales of oscillations, thus forming slow–fast systems. Experiments were performed on a horizontal Rijke tube and the effect of slow–fast oscillations was studied by externally introducing low-frequency sinusoidal modulations in the control parameter. The induced bursts display an abrupt transition between the rest and the active states. The growth and decay patterns of such bursts show asymmetry due to delayed bifurcation caused by slow oscillations of the control parameter about the Hopf bifurcation point. Further, we develop a phenomenological model for the interaction between different subsystems of a thermoacoustic system by either coupling the slow and fast subsystems or by introducing noise in the absence of slow oscillations of the control parameter. We show that interaction between subsystems with different timescales leads to regular amplitude modulated bursting, while the presence of noise induces irregular amplitude modulations in the bursts. Thus, we speculate that bursting in laminar and low-turbulence systems occurs predominantly due to the interdependence between slow and fast oscillations, while bursting in high-turbulence systems is predominantly influenced by the underlying turbulence.

The emergence of oscillatory dynamics (order) from chaotic fluctuations is a well-known phenomenon in turbulent thermoacoustic, aero-acoustic, and aeroelastic systems and is often detrimental to the system. We study the dynamics of two distinct turbulent thermoacoustic systems, bluff-body and swirl-stabilized combustors, where the transition occurs from the state of combustion noise (chaos) to thermoacoustic instability (order) via the route of intermittency. Using unweighted complex networks built from phase space cycles of the acoustic pressure oscillations, we characterize the topology of the phase space during various dynamical states in these combustors. We propose the use of network centrality measures derived from cycle networks as a novel means to characterize the number and stability of periodic orbits in the phase space and to study the topological transformations in the phase space during the emergence of order from chaos in the combustors. During the state of combustion noise, we show that the phase space consists of several unstable periodic orbits, which influence the phase space trajectory. As order emerges in the system dynamics, the number of periodic orbits decreases and their stability increases. At the onset of oscillatory dynamics, the phase space consists of a stable periodic orbit. We also use network centrality measures to identify the onset of thermoacoustic instability in both the combustors. Finally, we propose that the onset of oscillatory instabilities in turbulent systems is analogous to Bose–Einstein condensation transition observed for bosons, if we define phase space cycles as particles and the periodic orbits as energy levels.

Very few problems are analytically solvable in quantum mechanics. We present an analytical approximation to the expression for quantized energies of a semiconductor quantum well placed in a constant electric field. The system is studied under the influence of the BenDaniel–Duke boundary condition. We obtain approximated scaling laws to understand the exact numerical results obtained. We study the size dependence, field dependence and charge densities on the mass ratio of electron outside and inside the well. We relate the obtained results to quantum confinement Stark effect. The approach is suitable to discuss in an undergraduate classroom.

Studying particle-laden flows is essential for understanding diverse physical processes such as rain formation in clouds, pathogen transmission, and pollutant dispersal. This work introduces a framework of complex networks to analyze the particle dynamics through a Lagrangian perspective. To illustrate this method, we study the clustering of inertial particles (small heavy particles) in Taylor–Green flow, where the dynamics depend on the particle Stokes number ( St). Using complex networks, we can obtain the instantaneous local and global clustering characteristics simultaneously. Furthermore, from the complex networks derived from the particle locations, we observe an emergence of a giant component through a continuous phase transition as particles cluster in the flow field, thus providing novel insight into the spatiotemporal dynamics of particles such as the rate of clustering. Finally, we believe that complex networks have a great potential for analyzing the spatiotemporal dynamics of particle-laden flows.

Thermoacoustic systems are complex systems where the interactions between the hydrodynamic, acoustic and heat release rate fluctuations lead to diverse dynamics such as chaos, intermittency and ordered dynamics. Such complex interactions cause catastrophically high-amplitude acoustic pressure oscillations and the emergence of order in the spatiotemporal dynamics, referred to as thermoacoustic instability. In this work, we use multilayer networks to study the spatial pattern of inter-subsystem interactions between the vorticity dynamics and thermoacoustic power generated due to acoustically coupled combustion in a bluff-body-stabilised turbulent dump combustor. We construct a two-layered network where the layers represent the thermoacoustic power and vorticity fields. The inter-layer links are determined using cross-variable short-window correlations between vorticity and thermoacoustic power fluctuations at any two locations in the flow field. Analysing the topology of inter-layer networks, using network properties such as degree correlations and link-rank distributions, helps us infer the spatial inhomogeneities in inter-subsystem interactions and unravel the fluid mechanical processes involved during different dynamical states. We show that, during chaotic dynamics, interactions between subsystems are non-localised and spread throughout the flow field of the combustor. During the state of thermoacoustic instability (order), we find that intense interactions occur in between regions of coherent vortex shedding and thermoacoustic power generation and we understand that these processes are strongly and locally coupled. Moreover, we discover that such dense inter-layer connections emerge in spatial pockets in the dump plane of the combustor during the state of intermittency much prior to the onset of order.