The emergence of large-scale spatial modulations of turbulent channel flow, as the Reynolds number is decreased, is addressed numerically using the framework of linear stability analysis. Such modulations are known as the precursors of laminar–turbulent patterns found near the onset of relaminarisation. A synthetic two-dimensional base flow is constructed by adding finite-amplitude streaks to the turbulent mean flow. The streak mode is chosen as the leading resolvent mode from linear response theory. In addition, turbulent fluctuations can be taken into account or not by using a simple Cess eddy viscosity model. The linear stability of the base flow is considered by searching for unstable eigenmodes with wavelengths larger than the base flow streaks. As the streak amplitude is increased in the presence of the turbulent closure, the base flow loses its stability to a large-scale modulation below a critical Reynolds-number value. The structure of the corresponding eigenmode, its critical Reynolds number, its critical angle and its wavelengths are all fully consistent with the onset of turbulent modulations from the literature. The existence of a threshold value of the Reynolds number is directly related to the presence of an eddy viscosity, and is justified using an energy budget. The values of the critical streak amplitudes are discussed in relation with those relevant to turbulent flows.

The two-dimensional (2-D) evolution of perturbed long weakly nonlinear surface plane, ring and hybrid waves, consisting, to leading order, of a part of a ring and two tangent plane waves, is modelled numerically within the scope of the 2-D Boussinesq–Peregrine system. Numerical runs are initiated and interpreted using the reduced 2-D cylindrical Korteweg–de Vries (cKdV)-type and Kadomtsev–Petviashvili II (KPII) equations. The cKdV-type equation leads to two different models, the KdV$\theta$, where $\theta$ stands for a polar angle, and cKdV equations, depending on whether we use the general or singular (i.e. the envelope of the general) solution of the associated nonlinear first-order differential equation. The KdV$\theta$ equation is also derived directly from the 2-D Boussinesq–Peregrine system and used to analytically describe the intermediate 2-D asymptotics of line solitons subject to sufficiently long transverse perturbations of finite strength, while the cKdV equation is used to initiate outward- and inward-propagating ring waves with localised and periodic perturbations. Both of these equations, together with the KPII equation, are used to model the evolution of hybrid waves, where we show, in particular, that large localised waves (lumps) can appear as transient (emerging and then disappearing) states in the evolution of inward-propagating waves, contributing to the possible mechanisms for the generation of rogue waves. Detailed comparisons are made between the key features of the non-stationary 2-D modelling and relevant predictions of the reduced equations.

In this experimental study, we investigate, for the first time, the structure and evolution of the near wake of a circular cylinder in a flowing soap film at the onset of vortex shedding. The study primarily focuses on the changes occurring within the recirculation bubble, along with the evolution of vortex shedding. A significantly large recirculation bubble forms behind the cylinder in the soap film environment, characterized by small-scale vortices along its edges, an observation that starkly contrasts with its three-dimensional counterparts. These small-scale vortices driven by the Kelvin–Helmholtz instability, further induce a transverse deflection of the recirculation bubble, leading to an intermittent generation of the wake vortices. The instantaneous velocity field in the wake is examined, highlighting the clear evidence of intermittency in vortex formation. The frequency and wavelength of the chain of small-scale vortices on the recirculation bubble is evaluated, and a functional relationship with the flow Reynolds number is determined. We believe this observation to be novel, potentially revealing a new pathway for understanding the two-dimensional transition in bluff-body wakes.

We investigate the effect of inertial particles on Rayleigh-Bénard convection using weakly nonlinear stability analysis. An Euler–Euler/two-fluid formulation is used to describe the flow instabilities in particle-laden Rayleigh–Bénard convection. The weakly nonlinear results are presented near the critical point (bifurcation point) for water droplets in the dry air system. We show that supercritical bifurcation is the only type of bifurcation beyond the critical point in particle-laden Rayleigh–Bénard convection. Interaction of settling particles with the flow and the Reynolds stress or distortion terms emerges due to the nonlinear self-interaction of fundamental modes breaking down the top–bottom symmetry of the secondary flow structures. In addition to the distortion functions, the nonlinear interaction of fundamental modes generates higher harmonics, leading to the tendency of preferential concentration of uniformly distributed particles, which is completely absent in the linear stability analysis. Further, we show that in the presence of thermal energy coupling between the fluid and particles, the difference between the horizontally averaged heat flux at the hot and cold surfaces is equal to the net sensible heat flux advected by the particles. The difference between the heat fluxes at hot and cold surfaces increases with an increase in particle concentration.

Both experiments and direct numerical simulation (DNS) of hypersonic flow over a compression ramp show streamwise aligned streaks/vortices near the corner as the ramp angle is increased. The origin of this three-dimensional disturbance growth is not definitively known in the existing literature, but is typically connected to flow deceleration, centrifugal (Görtler) and/or baroclinic effects. In this work we consider the hypersonic problem with moderate wall cooling in the high Reynolds/Mach number, weak interaction limit. In the lower deck of the corresponding asymptotic triple-deck description we pose the linearised, three-dimensional, Görtler stability equations. This formulation allows computation of both receptivity and biglobal stability problems for linear spanwise-periodic disturbances with a spanwise wavelength of the same order as the lower-deck depth. In this framework the dominant response near the ramp surface is of constant density and temperature (at leading order) ruling out baroclinic mechanisms. Nevertheless, we show that there remains strong energy growth of upstream spanwise-varying perturbations and ultimately a bifurcation from two-dimensional to three-dimensional ramp flow. The unstable eigenmodes are localised to the separation region. The bifurcation points are obtained over a range of ramp angle, wall-cooling parameter and disturbance wavelength. Consistent with DNS results, the three-dimensional perturbations in this asymptotic formulation are streamwise aligned streaks/vortices, displaced above the separation region. In addition, the growth of upstream disturbances peaks near to the reattachment point, whilst the streaks persist beyond it, decaying relatively slowly downstream along the deflected ramp.

The Monin–Obukhov similarity theory (MOST) is a cornerstone of atmospheric science for describing turbulence in stable boundary layers. Extending MOST to stably stratified turbulent channel flows, however, is non-trivial due to confinement by solid walls. In this study, we investigate the applicability of MOST in closed channels and identify where and to what extent the theory remains valid. A key finding is that the ratio of the half-channel height to the Obukhov length serves as a governing parameter for identifying distinct flow regions and determining their corresponding mean velocity scaling. Hence, we propose a relation to estimate this ratio directly from the governing input parameters: the friction Reynolds and friction Richardson numbers ($\textit{Re}_{\tau }$ and $Ri_{\tau }$). The framework is tested against a series of direct numerical simulations across a range of $\textit{Re}_{\tau }$ and $Ri_{\tau }$. The reconstructed velocity profiles enable accurate prediction of the skin-friction coefficient crucial for quantifying pressure losses in stratified flows in engineering applications.

This paper investigates the flow past a flexible splitter plate attached to the rear of a fixed circular cylinder at low Reynolds number 150. A systematic exploration of the plate length ($L/D$), flexibility coefficient ($S^{*}$) and mass ratio ($m^{*}$) reveals new laws and phenomena. The large-amplitude vibration of the structure is attributed to a resonance phenomenon induced by fluid–structure interaction. The modal decomposition indicates that resonance arises from the coupling between the first and second structural modes, where the excitation of the second structural mode plays a critical role. Due to the combined effects of added mass and periodic stiffness variations, the two modes become synchronised, oscillating at the same frequency while maintaining fixed phase difference $\pi /2$. This further results in the resonant frequency being locked at half of the second natural frequency, which is approximately three times the first natural frequency. A reduction in plate length and an increase in mass ratio are both associated with a narrower resonant locking range, while a higher mass ratio also shifts this range towards lower frequencies. A symmetry-breaking bifurcation is observed for cases with $L/D\leqslant 3.5$, whereas for $L/D=4.0$, the flow remains in a steady state with a stationary splitter plate prior to the onset of resonance. For cases with a short flexible plate and a high mass ratio, the shortened resonance interval causes the plate to return to the symmetry-breaking stage after resonance, gradually approaching an equilibrium position determined by the flow field characteristics at high flexibility coefficients.

This study investigates the dynamics of free-surface turbulence (FST) using direct numerical simulations (DNS). We focus on the energy exchange between the deformed free-surface and underlying turbulence, examining the influence of high Reynolds (${\textit{Re}}$) and Weber (${\textit{We}}$) numbers at low to moderate Froude (${\textit{Fr}}$) numbers. The two-fluid DNS of FST at the simulated conditions is able to incorporate air entrainment effects in a statistical steady state. Results reveal that a high ${\textit{We}}$ number primarily affects entrained bubble shapes (sphericity), while ${\textit{Fr}}$ significantly alters free-surface deformation, two-dimensional compressibility and turbulent kinetic energy (TKE) modulation. Vortical structures are mainly oriented parallel to the interface. At lower ${\textit{Fr}}$, kinetic energy is redistributed between horizontal and vertical components, aligning with rapid distortion theory, whereas higher ${\textit{Fr}}$ preserves isotropy near the surface. Evidence of a reverse or dual energy cascade is verified through third-order structure functions, with upscale transfer near the integral length scale, and enhanced vertical kinetic energy in upwelling eddies. Phase-based discrete wavelet transforms of TKE show weaker decay at the smallest scales near the interface, suggesting contributions from gravitational energy conversion and reduced dissipation. The wavelet energy spectra also exhibits different scaling laws across the wavenumber range, with a $-3$ slope within the inertial subrange. These findings highlight scale- and proximity-dependent effects on two-phase TKE transport, with implications for subgrid modelling.

In this work, we study the reaction-controlled dual bubbles ripening on a heterogeneous substrate with high surface wettability hysteresis, where the bubbles evolve with constant contact radius but varied contact angle. We first theoretically derived the governing kinetic equation of bubble curvature radius $R_B$, based on which we surprisingly found three possible ripening processes under six different conditions, i.e. the classical Ostwald ripening (the bubble with the larger curvature radius $R_B$ exhibits an increase in $R_B$, while the bubble with the smaller curvature radius $R_B$ experiences a decrease in $R_B$), the reversed ripening (converse to Ostwald ripening), and the consistent ripening ($R_B$ of both bubbles increases or reduces consistently). Further analyses from the aspects of chemical potential and free energy lead to an interesting finding that the $R_B$ of two bubbles finally reach egalitarianism, independently of different ripening processes. Numerical results obtained from two-phase lattice Boltzmann modelling demonstrate excellent agreement with theoretical predictions, specifically concerning the kinetic equation, the various ripening processes, and the egalitarianism of bubble radii $R_B$ after ripening completion.

Three-dimensional laminar flow over an inclined spinning disk is investigated at a Reynolds number of ${\textit{Re}} = 500$ and an angle of attack of $\alpha = 25^\circ$, for tip-speed ratios up to 3. Numerical simulations are performed to investigate the effect of spin on the aerodynamics and characterise the instabilities that occur. Increasing tip-speed ratio significantly increases both lift and drag monotonically. Several distinct wake regimes are observed, including vortex shedding in the non-spinning case, vortex-shedding suppression at moderate tip-speed ratios and a distinct corkscrew-like short-wavelength instability in the advancing tip vortex at higher tip-speed ratios. Vorticity generated by the spinning disk strengthens the advancing tip vortex, inducing a spanwise stretching in the trailing-edge vortex sheet. This helps to dissipate the vorticity, which in turn prevents roll up and suppresses vortex shedding. The short-wavelength instability shows qualitative and quantitative matches to the $(-2,0,1)$ principal mode of the elliptic instabilities seen in pairs of counter-rotating Batchelor vortices. The addition of vorticity from the disk rotation significantly alters the circulation and axial velocity in the tip vortices, giving rise to elliptic instability despite its absence in the non-spinning case. In select cases, lock-in between the frequency of the elliptic instability and twice the spin frequency is observed, indicating that disk rotation acts as an additional forcing for the elliptic instability. Additional simulations at different Reynolds numbers and angle of attacks are considered to examine the robustness of observed phenomena across different parameter combinations.