Turbulent flows are strongly chaotic and unpredictable, with a Lyapunov exponent that increases with the Reynolds number. Here, we study the chaoticity of the surface quasi-geostrophic system, a two-dimensional model for geophysical flows that displays a direct cascade similar to that of three-dimensional turbulence. Using high-resolution direct numerical simulations, we investigate the dependence of the Lyapunov exponent on the Reynolds number and find an anomalous scaling exponent larger than that predicted by dimensional arguments. We also study the finite-time fluctuation of the Lyapunov exponent by computing the Cramér function associated with its probability distribution. We find that the Cramér function attains a self-similar form at large $\textit{Re}$.

This study presents a numerical investigation of wall-mounted tandem flexible plates with unequal lengths in a laminar boundary layer flow, examining both two-dimensional (2-D) and three-dimensional (3-D) configurations. Key parameters influencing the system include the plate’s bending stiffness ($K$), Reynolds number (${Re}$) and length ratio ($L^*$). Five motion modes are identified: dual collapse (DC), flapping collapse (FC), dual flapping (DF), static flapping (SF) and dual static (DS). A phase diagram in the ($K,L^*$) space is constructed to illustrate their regimes. We focus on DF and SF modes, which significantly amplify oscillations in the downstream plate – critical for energy harvesting. These amplification mechanisms are classified into externally driven and self-induced modes, with the self-induced mechanism, which maximises the downstream plate’s amplitude, being the main focus of our study. A rigid–flexible (RF) configuration is introduced by setting the upstream plate as rigid, showing enhanced performance at high ${Re}$, with oscillation amplitudes up to 100 % larger than the isolated flexible (IF) plate configuration. A relation is developed to explain these results, relating oscillation amplitude to trailing-edge velocity, oscillation frequency and chord length. Force analysis reveals that the RF configuration outperforms both IF and flexible–flexible (FF) configurations. Unlike frequency lock-in, the RF configuration exhibits frequency unlocking, following a $-2/3$ scaling law between the Strouhal number ($St$) and ${Re}$. Results from the 3-D RF configuration confirm that the 2-D model remains applicable, with the self-induced amplification mechanism persisting in 3-D scenarios. These findings enhance understanding of fluid–structure interactions, and offer valuable insights for designing efficient energy harvesting systems.

This paper considers the propagation, arrest and recession of a planar hydraulic fracture in a porous elastic medium whose footprint is constrained to a growing or shrinking rectangular region with a constant height. Hydraulic fractures with large aspect ratio rectangular footprints are frequently referred to as PKN fractures in recognition of the original researchers (Perkins & Kern 1961 J. Petrol. Tech. 13, 937–949) and (Nordgren 1972 J. Petrol Technol. 1972, 306–314) who first analyzed models of such fracture geometries. We investigate the one-dimensional non-local PKN approximation to a fully planar rectangular hydraulic fracture model in a three-dimensional elastic medium. By analysing the tip behaviour of the non-local PKN model, a transformation procedure is established to render the asymptotic equations for the dynamics of the steady semi-infinite PKN and plane strain models formally identical, which implies that all the existing multiscale plane strain asymptotes can be converted directly to the PKN case by making use of this transformation. Using this transformation, it is shown that the appropriate PKN asymptotes for the average aperture $\bar {w}$ with distance $\hat {x}$ to the fracture front are $\bar {w}\sim \hat {x}^{1/2},\ \hat {x}^{5/8}\ {\textrm{and}}\, \ \hat {x}^{2/3}$ in the toughness, leak-off and viscous modes of propagation, respectively; as well as the linear elastic fracture mechanics tip asymptote $\bar {w}\sim \hat {x}^{1/2}$ for arrest, which transitions to the linear asymptote tip $\bar {w}\sim \hat {x}$ for a fracture driven to recede due to fluid leak-off. Both the arrest and recession tip asymptotes share the intermediate leak-off asymptote $\bar {w}\sim \hat {x}^{3/4}$. A scaling analysis yields the arrest time, length and aperture as functions of a dimensionless injection-cessation time $\omega$. An asymptotic analysis of the non-local PKN model is used to establish the fundamental decoupling between dynamics and kinematics, which leads to the emergence of a similarity solution – termed the sunset solution – close to the time of collapse of the fracture. The multiscale PKN numerical solutions agree well with those for a fully planar multiscale rectangular hydraulic fracture model in a three-dimensional elastic medium. The scaling laws and the emergence of the sunset solution are confirmed by the PKN numerical model. The sunset solution also emerges in the fully planar numerical model and persists beyond the collapse time of the PKN model, by which time its footprints have separated from the upper and lower constraining sedimentary layer boundaries and have assumed self-similar elliptic shapes that shrink as they approach collapse.

In the rapidly rotating limit, we derive a balanced set of reduced equations governing the strongly nonlinear development of the convective wall-mode instability in the interior of a general container. The model illustrates that wall-mode convection is a multiscale phenomenon where the dynamics of the bulk interior diagnostically determine the small-scale dynamics within Stewartson boundary layers at the sidewalls. The sidewall boundary layers feedback on the interior via a nonlinear lateral heat-flux boundary condition, providing a closed system. Outside the asymptotically thin boundary layer, the convective modes connect to a dynamical interior that maintains scales set by the domain geometry. In many ways, the final system of equations resembles boundary-forced planetary geostrophic baroclinic dynamics coupled with barotropic quasi-geostrophic vorticity. The reduced system contains the results from previous linear instability theory but captured in an elementary fashion, providing a new avenue for investigating wall-mode convection in the strongly nonlinear regime. We also derive the dominant Ekman-flux correction to the onset Rayleigh number for large Taylor number, ${\textit {Ra}} \approx 31.8 \,{\textit{Ta}}^{1/2} - 4.43 \,{\textit{Ta}}^{5/12} + {\mathcal{O}}({\textit{Ta}}^{1/3})$ for no-slip boundaries. However, we find that the linear onset in a finite cylinder differs noticeably compared with a Cartesian channel. We demonstrate some of the reduced model’s nonlinear dynamics with numerical simulations in a cylindrical container.

This study employs a direct numerical simulation method to investigate the wake pattern evolutions of flows past an insulated spheroid and provides expressions of force and torque coefficients influenced by a streamwise magnetic field in an incompressible, conducting, viscous fluid. A total of 1150 cases are examined covering a parameter range of Reynolds number $50 \leqslant \textit{Re} \leqslant 250$, aspect ratio $1.5 \leqslant \beta \leqslant 6$, inclination angle $0^\circ \leqslant \theta \leqslant 90^\circ$, and interaction parameter $0 \leqslant N \leqslant 10$, where $\beta$ and $N$, respectively, reflect the anisotropy of the spheroid and the strength of magnetic field. Nine wake patterns are classified based on wake structure features and summarised in three maps of regimes according to the inclination angle. The transition mechanisms among these wake patterns are also investigated under the influence of a streamwise magnetic field. Furthermore, expressions for drag, lift and torque coefficients are derived with the help of three fundamental physical criteria. Results indicate that the force and torque expressions give a good prediction within the present parameter space $\{\textit{Re}, \beta , \theta , N\}$.

In conventional hypersonic wind tunnels, tunnel noise is dominated by acoustic radiation from turbulent nozzle-wall boundary layers, which can directly influence the boundary-layer transition (BLT) over the model in the test section. To offer new insights into BLT in conventional ground facilities, direct numerical simulations (DNS) were performed to simulate the receptivity and transition processes of a Mach 8 boundary layer over a nearly sharp $7^\circ$ half-angle cone, with transition triggered by tunnel-like broadband free-stream acoustic disturbances radiated from the nozzle wall of the Sandia hypersonic wind tunnel at Mach 8 (Sandia HWT-8). The DNS captured all the stages of the transition to turbulence caused by tunnel noise, including the passage of broadband free-stream noise through the shock wave, the receptivity process leading to the generation of Mack’s second-mode waves, their nonlinear growth to saturation, the laminar breakdown to turbulence and the post-transitional, fully turbulent flow. The transition location predicted by DNS compared well with that of Pate’s theory and was also consistent with the locations of peak pressure fluctuations as measured in the Sandia HWT-8 facility. The computed skin friction and Stanton number distributions in the initial breakdown region showed an overshoot compared with the turbulent predictions by the van Driest II theory. The wall-pressure spectra in both the transitional and turbulent regions of the cone compared well with those measured in the Sandia HWT-8. The second-mode breakdown amplitude $A_{max}$ predicted by the DNS was also consistent with sharp-cone measurements from multiple conventional wind tunnels.

The influence of compressibility on shear flow turbulence is investigated within a self-preservation framework. This study focuses on the axisymmetric jet to examine compressibility effects in a slowly spatially evolving flow, unlike mixing layers, where the convective Mach number remains constant. Revisiting self-preservation, an a priori description of the compressible scaling for Reynolds stresses and higher-order velocity moments is developed. Turbulence moments are found to scale with powers of the spreading rate, suggesting Reynolds stress anisotropy results from compressibility effects consistent with self-preservation of the governing equations. Particle image velocimetry measurements for Mach 0.3 and perfectly expanded Mach 1.25 jets confirm the scaling predictions. The attenuation function, $\varPhi (M_c)$, describing the relationship between the convective Mach number, $M_c$, and the spreading rate, follows a similar trend in jets and mixing layers, where a higher $M_c$ results in reduced spreading rates. In the jet where $M_c$ decays, the relationship between the local $M_c$ and turbulence attenuation remains captured through $\varPhi (M_c)$, which scales proportionally with the spreading rate. A new scale is introduced, where the pressure in the mean momentum equation is substituted. The difference between the streamwise and radial-Reynolds-normal stresses was found to be a scale which is independent of Mach number and spreading rate. Further analysis of the Reynolds-stress-transport budget shows that internal redistribution of energy occurs within the Reynolds-normal stresses, and the role of pressure modification in turbulence attenuation supports previous observations. These findings confirm that the compressible axisymmetric jet exhibits self-preservation, with scaling extending into supersonic regimes.

Flame–flame interactions in continuous combustion systems can induce a range of nonlinear dynamical behaviours, particularly in the thermoacoustic context. This study examines the mutual coupling and synchronisation dynamics of two thermoacoustic oscillators in a model gas-turbine combustor operating within a stochastic environment and subjected to external sinusoidal forcing. Experimental observations from two flames in an annular combustor reveal the emergence of dissimilar limit cycles, indicating localised lock-in of thermoacoustic oscillators. To interpret these dynamics, we introduce a coupled stochastic oscillator model with sinusoidal forcing terms, which highlights the critical role of individual synchronisation in enabling local lock-in. Furthermore, through stochastic system identification using this phenomenological low-order model, we mathematically demonstrate that a transition towards self-sustained oscillations can be driven solely by enhanced mutual coupling under external forcing. This combined experimental and modelling effort offers a novel framework for characterising complex coupled flame dynamics in practical combustion systems.

A wall-modelled large eddy simulation approach is proposed in a discontinuous Galerkin (DG) setting, building on the slip-wall concept of Bae et al. (J. Fluid Mech., vol. 859, 2019, pp. 400–432) and the universal scaling relationship by Pradhan and Duraisamy (J. Fluid Mech., vol. 955, 2023, A6). The effect of the order of the DG approximation is introduced via the length scales in the formulation. The level of under-resolution is represented by a slip Reynolds number and the model attempts to incorporate the effects of the numerical discretization and the subgrid-scale model. The dynamic part of the new model is based on a modified form of the Germano identity -- performed on the universal scaling parameter -- and is coupled with the dynamic Smagorinsky model. A sharp modal cutoff filter is used as the test filter for the dynamic procedure, and the dynamic model can be easily integrated into any DG solver. Numerical experiments on channel flows show that grid independence of the statistics is achievable and predictions for the mean velocity and Reynolds stress profiles agree well with the direct numerical simulation, even with significant under-resolution. When applied to flows with separation and reattachment, the model also consistently predicts one-point statistics in the reverse flow and post-reattachment regions in good agreement with experiments. The performance of the model in accurately predicting equilibrium and separated flows using significantly under-resolved meshes can be attributed to several aspects that work synergistically: the optimal finite-element projection framework, the interplay of the scale separation and numerical discretization within the DG framework, and the consistent dynamic procedures for subgrid and wall modelling.