We study the dynamics of inertial particles in turbulence using datasets obtained from both direct numerical simulations and laboratory experiments of turbulent swirling flows. By analysing time series of particle velocity increments at different scales, we show that their evolution is consistent with a Markov process across the inertial range. This Markovian character enables a coarse-grained description of particle dynamics through a Fokker–Planck equation, from which we can extract drift and diffusion coefficients directly from the data. The inferred coefficients reveal scale-dependent relaxation and noise amplitudes, indicative of inertial filtering and intermittency effects. Beyond the kinematic description, we analyse the thermodynamic properties of particle trajectories by computing the trajectory-dependent entropy production. We show that the statistics of entropy fluctuations satisfy both the integral fluctuation theorem and, under certain conditions, the detailed fluctuation theorem. These results establish a quantitative bridge between stochastic thermodynamics and particle-laden flows, and open the door to modelling turbulent transport using effective stochastic theories constrained by data and physical consistency.

This study reveals the presence of frequency components lower than the wake vortex-shedding frequency within the classical Mode A. Primaries are one-third of the vortex-shedding frequency and frequencies corresponding to recirculation bubble pumping, as previously studied. However, when the spanwise domain size $L_z$ in numerical simulations is sufficiently large, their interaction relationship becomes obscured. To clarify the interaction relationship, we introduce a process in which distinct frequency components gradually emerge by starting with a small spanwise domain of $3.3D$ and then increasing it to $4.7D$, where $D$ represents the diameter of the cylinder. At $L_z \leqslant 3.5D$, only the vortex-shedding frequency harmonics are present. One-third of the vortex-shedding frequency component appeared in $L_z \geqslant 3.6D$. Bispectral mode decomposition and energy transfer analysis reveal that the difference interaction between the one-third-shedding frequency and the vortex-shedding frequency component transfers energy to another low-frequency component. The recirculation bubble pumping is evident in the flow fields $L_z \geqslant 3.8D$. The frequency components after this emergence are not only the harmonics of the lowest-frequency component, and the periodic nature is disrupted, which is marked as a quasi-periodic state. Nonlinear interactions between the lowest-frequency component corresponding to recirculation bubble pumping, primary frequency components such as wake vortex shedding, and approximately one-third of the vortex-shedding frequency complicate the temporal behaviour of the flow field. Utilising the constraint of the spanwise domain size, our approach effectively reveals the interaction relationship among frequency components inherent in a flow field with several coherent spectral components.

We present a theoretical framework for modelling a plane-strain hydraulic fracture propagating in a poroelastic rock in the toughness-dominated regime. The formulation explicitly incorporates two-dimensional (2-D) pore-pressure diffusion, thereby generalising the classical Carter leak-off model, which can be interpreted as the limiting case of one-dimensional (1-D) diffusion. The poroelastic response is captured by superposing pore pressure and backstress contributions from a spatial and temporal distribution of instantaneous point sources along the extending fracture. A scaling analysis reveals the existence of a class of large-time, self-similar solutions for which the fracture length grows as $\ell \sim t^{1/2}$, with a prefactor function of a dimensionless injection rate $\mathcal{I}$ and a poroelastic stress coefficient $\eta$. The injection rate $\mathcal{I}$ emerges as the dominant controlling parameter. Asymptotic analysis provides large-time closed-form solutions in the limits of both large and small $\mathcal{I}$, which show excellent agreement with full numerical simulations. For large $\mathcal{I}$, diffusion reduces to 1-D and the solution converges to the classical toughness- and leak-off-dominated solution governed by Carter’s law. For small $\mathcal{I}$, fracture growth is instead controlled by pseudo-steady (2-D) diffusion. The transition from 2-D to 1-D diffusion is characterised by an increase in the fracture length prefactor and a reduction in leak-off. The poroelastic coefficient $\eta$ acts to shorten and narrow the fracture while increasing both leak-off and driving pressure. This framework delineates the transition between 2-D and 1-D diffusion and establishes quantitative conditions under which Carter’s law remains valid in the large-time limit.

We propose a novel stability criterion for incompressible shear flows by combining input–output analysis and the small gain theorem. The criterion yields an explicit threshold on the magnitude of velocity perturbations about a given base flow that guarantees stability. If this threshold is crossed – either due to non-modal growth, exponential growth or a bypass transition scenario – our analysis predicts a loss of stability that may lead to transition to turbulence. We consider three approximated models for nonlinearity: unstructured, structured with non-repeated blocks and structured with repeated blocks. We show that the imposed threshold obtained by these three methods complies with a hierarchical relationship, where the unstructured case is the most conservative, imposing the lowest bound on disturbance magnitude. We apply this approach to three canonical and well-studied base flows: Couette, plane Poiseuille and Blasius. For these three base flows, we compare our results with experiments, direct numerical simulation results, non-modal nonlinear stability results and linear stability theory (LST). In the limit of infinitesimally small perturbation magnitude, our stability criterion for the unstructured case recovers the results of LST. For finite perturbations, the structured cases that account for nonlinear interactions provided stability thresholds that are consistent with experimental observations and simulation results of transition at both subcritical and post-critical Reynolds numbers for the considered base flows in our study. In particular, we utilise our stability criterion to demonstrate that Couette flow can become unstable and transition can be triggered at different Reynolds numbers, which is consistent with past experimental observations.