We analyse the long-time dynamics of trajectories within the stability boundary between laminar and turbulent square duct flow. If not constrained to a symmetric subspace, the edge trajectories exhibit a chaotic dynamics characterised by a sequence of alternating quiescent phases and intense bursting episodes. The dynamics reflects the different stages of the well-known near-wall streak–vortex interaction. Most of the time, the edge states feature a single streak with a number of flanking vortices attached to one of the four surrounding walls. The initially straight streak undergoes a linear instability and eventually breaks in an intense bursting event. At the same time, the downstream vortices give rise to a new low-speed streak at one of the neighbouring walls, thereby causing the turbulent activity to ‘switch’ from one wall to the other. If the edge dynamics is restricted to a single or twofold mirror-symmetric subspace, the bursting and wall-switching episodes become self-recurrent in time, representing the first periodic orbits found in square duct flow. In contrast to the chaotic edge states in the non-symmetric case, the imposed symmetries enforce analogue bursting cycles to simultaneously appear at two parallel opposing walls in a mirror-symmetric configuration. Both the localisation of turbulent activity to one or two walls and the wall-switching dynamics are shown to be common phenomena in marginally turbulent duct flows. We argue that such episodes represent transient visits of marginally turbulent trajectories to some of the edge states detected here.

Symmetry-based analyses of multiscale velocity gradients highlight that strain self-amplification (SS) and vortex stretching (VS) drive forward energy transfer in turbulent flows. By contrast, a strain–vorticity covariance mechanism produces backscatter that contributes to the bottleneck effect in the subinertial range of the energy cascade. We extend these analyses by using a normality-based decomposition of filtered velocity gradients in forced isotropic turbulence to distinguish contributions from normal straining, pure shearing and rigid rotation at a given scale. Our analysis of direct numerical simulation (DNS) data illuminates the importance of shear layers in the inertial range and (especially) the subinertial range of the cascade. Shear layers contribute significantly to SS and VS and play a dominant role in the backscatter mechanism responsible for the bottleneck effect. Our concurrent analysis of large-eddy simulation (LES) data characterizes how different closure models affect the flow structure and energy transfer throughout the resolved scales. We thoroughly demonstrate that the multiscale flow features produced by a mixed model closely resemble those in a filtered DNS, whereas the features produced by an eddy viscosity model resemble those in an unfiltered DNS at a lower Reynolds number. This analysis helps explain how small-scale shear layers, whose imprint is mitigated upon filtering, amplify the artificial bottleneck effect produced by the eddy viscosity model in the inertial range of the cascade. Altogether, the present results provide a refined interpretation of the flow structures and mechanisms underlying the energy cascade and insight for designing and evaluating LES closure models.

Transonic buffet presents time-dependent aerodynamic characteristics associated with shock, turbulent boundary layer and their interactions. Despite strong nonlinearities and a large degree of freedom, there exists a dominant dynamic pattern of a buffet cycle, suggesting the low dimensionality of transonic buffet phenomena. This study seeks a low-dimensional representation of transonic airfoil buffet at a high Reynolds number with machine learning. Wall-modelled large-eddy simulations of flow over the OAT15A supercritical airfoil at two Mach numbers, $M_\infty = 0.715$ and 0.730, respectively producing non-buffet and buffet conditions, at a chord-based Reynolds number of ${Re} = 3\times 10^6$ are performed to generate the present datasets. We find that the low-dimensional nature of transonic airfoil buffet can be extracted as a sole three-dimensional latent representation through lift-augmented autoencoder compression. The current low-order representation not only describes the shock movement but also captures the moment when the separation occurs near the trailing edge in a low-order manner. We further show that it is possible to perform sensor-based reconstruction through the present low-dimensional expression while identifying the sensitivity with respect to aerodynamic responses. The present model trained at ${Re} = 3\times 10^6$ is lastly evaluated at the level of a real aircraft operation of ${Re} = 3\times 10^7$, exhibiting that the phase dynamics of lift is reasonably estimated from sparse sensors. The current study may provide a foundation towards data-driven real-time analysis of transonic buffet conditions under aircraft operation.

We explore the mechanisms and regimes of mixing in yield-stress fluids by simulating the stirring of an infinite, two-dimensional domain filled with a Bingham fluid. A cylindrical stirrer moves along a circular path at constant speed, with the path radius fixed at twice the stirrer diameter; the domain is initially quiescent and marked by a passive dye in the lower half. We first examine the mixing process in Newtonian fluids, identifying three key mechanisms: interface stretching and folding around the stirrer’s path, diffusion across streamlines and dye advection and interface stretching due to vortex shedding. Introducing yield stress leads to notable mixing localisation, manifesting through three mechanisms: advection of vortices within a finite distance of the stirrer, vortex entrapment near the stirrer and complete suppression of vortex shedding at high yield stresses. Based on these mechanisms, we classify three distinct mixing regimes: (i) regime SE, where shed vortices escape the central region, (ii) regime ST, where shed vortices remain trapped near the stirrer and (iii) regime NS, where no vortex shedding occurs. These regimes are quantitatively distinguished through spectral analysis of energy oscillations, revealing transitions and the critical Bingham and Reynolds numbers. The transitions are captured through effective Reynolds numbers, supporting the hypothesis that mixing regime transitions in yield-stress fluids share fundamental characteristics with bluff-body flow dynamics. The findings provide a mechanistic framework for understanding and predicting mixing behaviours in yield-stress fluids, suggesting that the localisation mechanisms and mixing regimes observed here are archetypal for stirred-tank applications.

We analyse the dynamics of a weakly elastic spherical particle translating parallel to a rigid wall in a quiescent Newtonian fluid in the Stokes limit. The particle motion is constrained parallel to the wall by applying a point force and a point torque at the centre of its undeformed shape. The particle is modelled using the Navier elasticity equations. The series solutions to the Navier and the Stokes equations are used to obtain the displacement and velocity fields in the solid and fluid, respectively. The point force and the point torque are calculated as series in small parameters $\alpha$ and $1/H$, using the domain perturbation method and the method of reflections. Here, $\alpha$ is the measure of elastic strain induced in the particle resulting from the fluid’s viscous stress and $H$ is the non-dimensional gap width, defined as the ratio of the distance of the particle centre from the wall to its radius. The results are presented up to $\textit {O}(1/H^3)$ and $\textit {O}(1/H^2)$, assuming $\alpha \sim 1/H$, for cases where gravity is aligned and non-aligned with the particle velocity, respectively. The deformed shape of the particle is determined by the force distribution acting on it. The hydrodynamic lift due to elastic effects (acting away from the wall) appears at $\textit {O}(\alpha /H^2)$ in the former case. In an unbounded domain, the elastic effects in the latter case generate a hydrodynamic torque at O($\alpha$) and a drag at O($\alpha ^2$). Conversely, in the former case, the torque is zero, while the drag still appears at O($\alpha ^2$).