This study offers the first comprehensive analysis of convective onset in a uniformly internally heated, rotating porous sphere. By incorporating the Coriolis force into Darcy’s law, we formulate the linear stability problem using a vector potential expansion combined with spherical harmonic decomposition. The resulting equations are solved numerically via a spectral method based on Worland polynomials. In the absence of rotation, the most unstable mode corresponds to a spherical harmonic degree $l = 2$, with a critical Rayleigh number of 91.95. When rotation is introduced, the Coriolis force further stabilises the flow, causing the critical Rayleigh number to increase monotonically with the Taylor number. Notably, the system consistently favours an azimuthal wavenumber $m = 2$ across nearly the entire parameter range. This behaviour contrasts sharply with rotating fluid convection, where the preferred wavenumber increases indefinitely with the rotation rate. This fundamental difference arises from the disruption of the quasi-geostrophic balance by the strong Darcy resistance, leading to the failure of the Proudman–Taylor theorem. Under strong rotation, the system exhibits power-law scaling and develops an Ekman-like boundary layer localised near the polar axis, which channels global heat and mass transport. These findings provide novel theoretical insights into the thermal structure and evolution of early evolutionary bodies, such as rapidly rotating planetesimals or undifferentiated asteroids with high permeability, and lay the groundwork for future nonlinear studies of rotating porous convection.

This paper is devoted to revealing the effects of swirling flows with radially dependent velocity profiles on thermoacoustic instability. By reformulating the three-dimensional linearized Euler equations in cylindrical coordinates under standing-wave assumptions according to inlet/outlet boundary reflections, and integrating flame response dynamics, we develop a dispersion relation framework for thermoacoustic instability analysis. In so doing, the model generalizes prior network-based approaches for thermoacoustic instabilities (with axial and circumferential flow) to incorporate radial velocity gradients. Validation against analytical and numerical solutions for benchmark cases spanning thermoacoustic instabilities and aeroacoustic phenomena demonstrates quantitative agreement across configurations. Subsequent parametric studies further indicate that both radially dependent azimuthal and axial velocity components exert a considerable influence on the prediction of azimuthal thermoacoustic instability in an annular duct. This finding suggests that the assumption of uniform velocities in existing models may lead to inaccurate estimations of instability. Additionally, acoustic energy analysis indicates that axial flow components (perpendicular to inlet/outlet cross-sections) exhibit higher acoustic energy efflux at the combustor outlet than the circumferential components of swirling flows. Meanwhile, in comparison with burner-induced acoustic energy variations at the combustor inlet, an increase in the circumferential flow component reduces acoustic energy transmission at the combustor outlet for counter-propagating waves while amplifying it for copropagating waves. These effects are further enhanced by a stronger axial component of swirling flows. Overall, the proposed framework provides a foundational tool for elucidating thermoacoustic mechanisms and advancing instability mitigation strategies in a swirling flow configuration within confined spaces.

Surfactants at the air–sea interface are known to alter surface wave dynamics by modifying surface tension and Marangoni stresses. In this study, we perform two-dimensional direct numerical simulations of gravity-capillary waves with insoluble surfactants using a coupled phase field and volume-of-fluid method. We consider a nonlinear equation of state for surface tension and resolve Marangoni stresses induced by surfactant concentration gradients. We explore a broad parameter space characterised by initial wave steepness $ak$, Bond number $\textit{Bo}$ (comparing gravity and surface tension), Reynolds number $\textit{Re}$ (comparing inertia and viscosity), and the importance of surfactant concentration and strength of the gradient, characterised by a surfactant parameter $\beta$. We analyse the impact of surfactants on wave patterns, surface roughness, wave breaking, energy dissipation and surface vorticity. Our results reveal a non-monotonic dependence of wave shape, roughness, vorticity and energy dissipation on $\beta$, which is found to be governed by Marangoni effects that peak at intermediate surfactant concentrations. Wave regime transition at high $\textit{Bo}$ is governed by an effective $\textit{Bo}$, which accounts for the reduction in surface tension induced by surfactants. We further introduce a rescaled parameter $\textit{Bo}\,\textit{Re}^{-1/2}\,(ak)^{-1}$ based on force balance, which collapses the transition boundaries across different $\textit{Re}$. These findings provide a systematic understanding of surfactant-modulated wave dynamics for both laboratory and geophysical applications.

We explore the bifurcation structure of mode-1 solitary waves in a three-layer fluid confined between two rigid boundaries. A recent study Lamb (2023 J. Fluid Mech., vol. 962, A17) proposed a method to predict the coexistence of solitary waves with opposite polarity in a continuously stratified fluid with a double pycnocline by examining the conjugate states for the Euler equations. We extend this line of inquiry to a piecewise-constant three-layer stratification, taking advantage of the fact that the conjugate states for the Euler equations are exactly preserved by the strongly nonlinear model that we will refer to as the three-layer Miyata-Maltseva-Choi-Camassa (MMCC3) equations. In this reduced setting, solitary waves are governed by a Hamiltonian system with two degrees of freedom, whose critical points are used to explain the bifurcation structure. Through this analysis, we also discover families of solutions that have not been previously reported for a three-fluid system. Using the shared conjugate state structure between the MMCC3 model and the full Euler equations, we propose criteria for distinguishing the full range of solution behaviours. This alignment between the reduced and full models provides strong evidence that partitioning the parameter space into regions associated with distinct solution types is valid within both theories. This classification is further substantiated by numerical solutions to both models, which show excellent agreement.

Active flow control often exploits disturbance amplification mechanisms to achieve desired flow properties. Recently, theoretical predictions of optimal control based on stability analysis have gained traction. However, these methods are limited in their ability to predict nonlinear control strategies, such as burst-mode actuation for separated flows, which involve intermittent and high-amplitude forcing. To address this limitation, we developed a nonlinear optimal forcing analysis based on optimal perturbation theory. This method is specifically designed to capture non-harmonic forcing patterns and the nonlinear temporal evolution of the disturbance field. We applied this method to the two-dimensional high-subsonic, low-Reynolds number flow around a NACA0012 airfoil to reattach the separated flow and investigate the onset mechanism of low-frequency oscillation. The analysis identified an optimal temporal forcing pattern characterized by damped oscillation. This forcing exploits flow amplification mechanisms over the separation bubbles, promoting the formation of spanwise vortices in the shear layer. When implemented as a periodic forcing concentrated at the separated point, these vortices were stably generated, resulting in a significant lift increase via momentum exchange. A key finding is that the application of this optimal forcing induced long-term changes in the flow field, driven by the transient emergence of low-frequency oscillations. Furthermore, we explored the intermittent application of this forcing and found that an appropriate duty cycle can enhance the lift coefficient while reducing energy consumption.

We present a method to simulate non-coalescing impacts and rebounds of droplets onto the free surface of a liquid bath, together with new experimental data, focused on the low-speed impact of droplets. The method is derived from first principles and imposes only natural geometric and kinematic constraints on the motion of the impacting interfaces, yielding predictions for the evolution of the contact area, pressure distribution and wave field generated on both impacting masses. This work generalises an existing kinematic-match method whose prior applications dealt with deformation of the surface of the bath only; i.e. neglecting that of the droplet. The method’s extension to include droplet deformation gives predictions that compare favourably with existing experimental results and our new experiments conducted in the low-Weber-number regime.

The concept of inverse energy cascades has played a central role in the development of turbulence theory, with applications in two-dimensional and quasi-two-dimensional flows. We examine the presence or absence of inverse energy cascades in rotating stably stratified flows constrained to anisotropic yet fully three-dimensional domains, in a range of parameters that are relevant for planetary atmospheres. Our results show that inverse energy cascades can indeed emerge when rotation overcomes a certain threshold that depends on the stratification. Implications for the self-organisation processes of planetary atmospheres are discussed.