The electrophoresis of charged colloidal particles in fluids exhibiting odd viscosity represents a fundamental challenge in understanding transport phenomena within charge-stabilised chiral active suspensions. Here, we consider a charged chiral active fluid, where electrokinetics is coupled to odd Stokes flow, to explore how classical results from electrophoresis in Newtonian fluids are modified in the presence of odd viscosity. In particular, we derive a general expression for the electrophoretic mobility for particles of any shape, under weak external electric fields, using the Lorentz reciprocal theorem for odd fluids. By applying this result to a charged sphere at low zeta potentials, we obtain an exact, closed-form analytical expression for the electrophoretic mobility, valid for arbitrary values of the Debye screening length and the odd-viscosity coefficient. Similar to Newtonian fluids, we find that the electrophoretic mobility is proportional to the translational mobility of an uncharged sphere, modulated by the Henry function. However, unlike in Newtonian fluids, odd viscosity leads to directional asymmetries in the electrophoretic mobility tensor that persist even for thin electric double layers. This case contrasts significantly with a charged anisotropic particle suspended in an isotropic Newtonian fluid, where anisotropic effects would vanish under the same electrostatic-screening conditions.

The pressure fluctuations in turbulent bubbly channel flows are investigated using incompressible large-eddy simulations under zero-gravity conditions. The volume-of-fluid method is adopted to capture the bubble–liquid interfaces. The gas volume fraction varies from 1.68 % to 11.31 % by adjusting the number and size of bubbles. A constant bulk mean velocity is imposed, yielding friction Reynolds numbers around 550, the highest achieved so far. To accurately reflect the contributions of the pressure fluctuations, the densities and viscosities are set to those of water and air under standard conditions. After careful validation of the numerical approach, the different source terms in the Poisson equation for pressure fluctuations are thoroughly examined based on the simulation results. It is found that (i) pressure fluctuations are primarily governed by spatio-temporal density variations induced by bubble motion, particularly in the channel core region; (ii) the surface-tension contribution is prominent throughout the bubble region, owing to the resulting Laplace pressure jump across the bubble interface; (iii) for the incompressible source terms, the linear component exceeds the nonlinear one beyond the buffer layer, in contrast to the behaviour in the single-phase channel flows. The inversion arises from the important contribution of spatial gradients of mean density and mean velocity due to bubble motions. Next, the wall-pressure fluctuations are analysed in terms of the wavenumber–frequency spectra. For a high gas volume fraction $\alpha _g \gtrsim 5.65\,\%$, a unique bubble-induced convection ridge, distinct from the turbulence-induced ridge, emerges and may even dominate. Owing to bubble-induced disturbances in the near-wall boundary layer, the convection velocity of energy-containing wall-pressure fluctuations exceeds that of single-phase case and increases with gas volume fraction. The bubbly flows, even with a bubble-deficient layer near the channel wall, modify the vortex structures in the boundary layer and modulate the wall-pressure fluctuations.

Two-fluid capillary flows in channels and pipes form a fundamental problem in fluid mechanics with widespread applications in physical, biological and microfluidic systems. A key phenomenon in such flows is the generation of long capsular bubbles, known as Taylor bubbles. A simple configuration that forms Taylor bubbles is coflow, where two fluids (one viscous, the other inviscid) are injected concurrently into a pipe or channel, producing a periodic pinch-off of inviscid bubbles at a regular frequency. By considering a lubrication theory for the necking region between the bubble injection nozzle and the developing Taylor bubble, we conduct a mathematical study of the formation of two-dimensional Taylor bubbles, yielding analytical insight into injection-driven pinch-off dynamics and the parametric control of its frequency. A key result is to demonstrate a quasi-static limiting theory of the necking dynamics for small capillary numbers. The theory is based on coupling an outer near-static regime of the pinching film with an apparent dynamic contact line that matches the necking zone to the film surrounding the Taylor bubble. This asymptotic structure, which becomes more refined in the limit of small capillary number, reveals a fundamental link between capillary pinch-off dynamics and the quasi-static regimes underlying spreading droplets. The theory yields a prediction for the dimensionless pinch-off time, $t_* \propto d^{7/3} / q_F^{4/3}$, where $q_F$ represents the flux of viscous fluid and $d$ the size of the capillary. The study thereby yields new mathematical understanding of flow regimes that underlie bubble generation in capillary systems from first principles, and the first step towards application of quasi-static analysis to more complex axisymmetric and three-dimensional examples of capillary pinch-off.

This study investigates the response of a Mach 4 flow over blunted plates to free stream disturbances using the linearised Navier–Stokes equations. Six nose radii, ranging from 0.1 to 2 mm, are considered. The free stream conditions match those of the experiments by Lysenko (J. Appl. Mech. Tech. Phys. vol. 31, 1990, pp. 868–873), in which the critical nose radius for transition reversal was found to be approximately 0.5 mm. The current study finds that increasing the nose radius strengthens low-frequency streamwise streaks while stabilising the oblique first mode. This leads to a shift in the dominant instability inside the boundary layer from the oblique first mode to streamwise streaks. For slow acoustic waves, the changeover nose radius is 0.7 mm. For entropy waves, vorticity waves and optimal free stream forcing obtained from resolvent analysis, the changeover nose radius is 0.2 mm. The results reproduce the transition reversal observed in the experiments and suggest that it can occur regardless of the type of free stream disturbances. The streak enhancement is attributed to the increasing magnitude of velocity perturbations near the stagnation line as the nose radius increases. Streamwise vortex-like disturbances of inviscid nature at the stagnation line generate streamwise streaks downstream via the lift-up mechanism. For the most amplified streaks, the ratio of the local boundary-layer thickness to the spanwise wavelength approaches a constant value downstream. This ratio is found to be independent of the nose radius and the type of free stream forcing. For slow acoustic disturbances, the streak strength is found to be insensitive to the angle of incidence.

We report experimental results on the low-frequency dynamics in the wake of a stationary sphere and the corresponding bifurcation. This low frequency is the second frequency appearing past a sphere when the Reynolds number is increased. We measure the velocity field using particle image velocimetry in transverse planes past a sphere placed in a water channel, and obtain the streamwise vorticity. We compute the position of the barycentre of the absolute value of the vorticity and projected the vorticity on azimuthal mode 1. We deduce from periodograms of these signals the Strouhal numbers $\mathit{St}_1$ and ${{\textit{St}}_2}$ associated with the first ($f_1$) and second ($f_2$) frequencies. The frequency ratio $f_2/f_1$ decreases with the Reynolds number, and is close to $2/7$ near the threshold. The two frequencies are generally incommensurate. In the periodograms, many combinations of $f_1$ and $f_2$ are present, in particular $f_1+f_2$ and $f_1-f_2$, consistent with a $T^2$ torus dynamics. We also obtain the squared magnitude associated with $f_1$ and $f_2$. The linear variation of this squared magnitude above the threshold ${\textit{Re}}_{1}$ and the variation of the frequency $f_1$ confirm that this first oscillatory bifurcation is a supercritical Hopf bifurcation. Similar variations for $f_2$ above the corresponding threshold ${\textit{Re}}_{2}$ show that the second oscillatory bifurcation is a supercritical secondary Hopf bifurcation (supercritical Neimark–Sacker bifurcation). The component at $f_2$ of the vorticity field remains significant even $9$ diameters downstream of the sphere.

This work presents high-fidelity simulations of the aerodynamic noise characteristics of a forward-flight propeller ingesting turbulent wakes of cylinders. Two cylinders with different diameters are employed to generate turbulent wakes with distinct turbulence intensity and length scales, namely small-scale turbulence (SST) and large-scale turbulence (LST). The lattice Boltzmann–very large-eddy simulation solver, PowerFLOW, coupled to an impermeable surface Ffowcs Williams–Hawkings formulation, is used for the forward-flight propeller configuration, matching that of an earlier experimental study. The aerodynamic results show that SST ingestion preserves a coherent tip–vortex path and confines elevated fluctuations to a limited portion of the blades. In contrast, LST ingestion disrupts the tip–vortex trajectory over a wide azimuth and spreads axial-velocity root-mean-square across the disk. Correspondingly, ingesting SST produces a series of discrete tonal sidebands at $m\mathrm{BPF}\pm nf_0$ (where $m, n \in \mathbb{N} = \{1,\, 2,\,3,\, \ldots \}$, $\textrm{BPF}$ is blade passing frequency and $f_0$ is the cylinder shedding frequency), while the LST case primarily yields notable noise increases in broadband components with clear haystacking around the blade passing frequency harmonics. Blade-level correlation and modulation intensity analyses confirm that the SST case gives rise to compact amplitude modulation, whereas the larger turbulence length scales in LST lead to redistribution of the turbulent energy via haystacking and broadband humps, linking inflow scale and coherence to the observed noise characteristics and generation mechanism. Hemispherical change in overall sound pressure level maps confirm axis-aligned rises in noise levels with turbulence ingestion. These results establish how the scale and coherence of ingested turbulence are crucial to both the near-wake dynamics and the radiated sound of propellers.

Convective mixing in porous media is crucial in both geophysical and industrial fields, spanning applications ranging from carbon dioxide sequestration to geothermal energy extraction. Key processes are affected by convective heat transport or diffusion of chemical species in porous formations. Intense convection flow and mixing create complex, dynamic patterns that are difficult to predict and measure. The present work focuses on the use of topological data analysis, in particular, the measures emerging from the growing field of persistent homology (PH), to quantify these patterns. These measures are objective and quantify structures across all temperature or concentration values simultaneously. These techniques, when applied to classical porous media set-ups, such as one-sided and Rayleigh–Bénard flow configurations, provide new insights into a system’s structure, flow patterns and macroscopic mixing properties. Using large datasets we make publicly available, comprising original simulations as well as those presented in previous works, we correlate the behaviour of the heat transport rate (quantified by the Nusselt number) with the evolution of the flow structures (quantified by the PH measures). Finally, we provide a detailed analysis of the flow evolution over a wide range of governing parameters, namely the Rayleigh–Darcy number and the domain size.

A key challenge in multiphase flow through porous media is to understand and predict the conditions under which trapped fluid clusters become mobilised. Here, we investigate the stability of such clusters in two-phase flow and present a simple, quasistatic model that accurately determines the critical Bond number (that is, the critical ratio between the average pressure gradient of the flow and the surface tension) for the onset of cluster mobilisation. The model is derived by combining elementary geometrical considerations with mass conservation and a mechanical equilibrium condition, resulting in a system of coupled differential equations. Our derivation sheds new light on the mechanisms that govern cluster stability. In addition, since the number of equations equals the number of cluster openings, our model is significantly faster than direct numerical simulations of the same problem and enables efficient exploration of the system’s parameter space. Using this approach, we highlight a discrepancy with the prediction of current mean-field theories, which predict that the largest stable cluster size scales in proportion to $r/{\textit{Bo}}^\alpha$, where $r$ is a typical pore size, ${\textit{Bo}}$ is the Bond number and $\alpha$ is a fixed exponent. We discuss the mechanisms that explain the breakdown of the mean field theories and we show that a scaling law of this form can only exist if $\alpha$ is allowed to depend on a broad set of flow and geometric parameters.

In laser-produced plasma (LPP) extreme ultraviolet (EUV) sources, deformation of a tin droplet into an optimal target shape is determined by its interaction with a pre-pulse laser-generated plasma. This interaction is mediated by a transient ablation pressure, whose complex spatio-temporal evolution remains experimentally inaccessible. Existing modelling approaches are limited: empirical pressure–impulse models neglect dynamic plasma feedback, while advanced radiation hydrodynamics codes often fail to resolve late-time droplet hydrodynamics. To bridge this gap, we propose a radiation two-phase flow model based on a diffuse interface approach. The model integrates radiation hydrodynamics for the plasma with the Euler equations for a weakly compressible liquid, extending a five-equation diffuse interface formulation to incorporate radiation transport, thermal conduction, ionisation and surface tension. This formulation enforces pressure and velocity equilibrium across the diffuse interface region, with closure models constructed to ensure correct jump conditions at interfaces and asymptotically recover the pure-phase equations in bulk regions. Then we apply the model to simulate a benchmark pre-pulse scenario, where a $50\ \unicode{x03BC} \mathrm{m}$ tin droplet is irradiated by a $10\ \mathrm{ns}$ laser pulse. Our axisymmetric simulations capture the rapid plasma expansion and subsequent inertial flattening of the droplet into a thin, curved sheet over microsecond time scales. Notably, the model reproduces experimentally observed features such as an axial jet – rarely replicated in prior simulations. Quantitative agreement with experimental data for sheet dimensions and velocity validates the approach. The proposed model self-consistently couples laser–plasma physics with compressible droplet dynamics, providing a powerful tool for studying plasma–liquid interactions in LPP-EUV source optimisation.

Building upon the theoretical investigation of fully developed porous-channel flow by Nield et al. (J. Fluid Mech. vol. 322, 1996, 201–214), the present study examines hydrodynamically developing flow in anisotropic porous channels to better characterise realistic transport behaviour. A variational formulation for the steady incompressible Darcy–Brinkman–Forchheimer (DBF) equations is derived, where the Darcy, Brinkman and Forchheimer terms represent porous drag, viscous shear effects and nonlinear inertial resistance, respectively. The existence and uniqueness of solution are established using the Browder–Minty theorem together with the inf-sup condition to ensure the uniqueness of the pressure field. The porous medium is represented through a spatially varying anisotropic permeability tensor to account for directional resistance effects. A physics-informed neural network (PINN) framework using feed-forward deep neural networks (DNNs) is employed to solve the coupled nonlinear DBF equations solely from the governing equations and boundary conditions, without the need for external simulation or experimental data. Results indicate that lower anisotropic permeability ratios enhance both axial and transverse velocity components, whereas increasing the Darcy number substantially reduces pressure drop due to weakened porous resistance. The proposed framework provides an effective approach for modelling anisotropic porous-medium flows relevant to subsurface transport, filtration, biomedical devices and energy-storage systems.

Several mechanisms have been identified by which turbulence may either enhance or reduce the settling velocity of heavy particles. However, the conditions under which some of them may dominate over others are not well known, thwarting our ability to predict the net outcome. We investigate this issue experimentally, focusing on spherical particles of size comparable to the Kolmogorov scale in homogeneous turbulence. We combine results obtained in two zero-mean-flow chambers working with water and air and various particle densities, leveraging high-resolution imaging of both phases and spanning a substantially wider range of parameters compared with previous studies. The change in settling velocity is found to be dictated by a single parameter: the ratio between the still-fluid fall speed and the characteristic velocity of the turbulent fluctuations, with settling enhancement and reduction occurring when this is smaller and larger than unity, respectively. This marks the transition from particles being preferentially swept by downward fluctuations to them loitering in upward ones. Counterintuitively, nonlinear drag effects are most significant in the regime of settling enhancement, while their contribution to settling reduction is modest.

We present a numerical study of three-dimensional gravity-capillary standing waves by using cubic and quintic truncated Hamiltonian formulations and the Craig–Sulem expansion of the Dirichlet–Neumann operator. The resulting models are treated as triply periodic boundary-value problems and solved via a spatio-temporal collocation method without executing initial-value calculations. This approach avoids the numerical stiffness associated with surface tension, and numerical instabilities arising from time integration. We reduce the number of unknowns significantly by exploiting the spatio-temporal symmetries for three types of symmetric standing waves. Comparisons with existing asymptotic and numerical results illustrate excellent agreement between the models and the full potential-flow formulation. We investigate typical bifurcations and standing waves that feature square, hexagonal and more complex flower-like patterns under the three-wave resonance. These solutions are generalisations of the classical Wilton ripples. Temporal simulations of the computed three-dimensional standing waves exhibit perfect periodicity and reveal an instability mechanism based on the previous reported oblique instability in two-dimensional standing waves.

The surface-gravity wave evolution, imitating tsunamis triggered by the ocean floor’s arbitrary temporal motion over a generic seafloor topography, is investigated using the linearised water-wave theory of a compressible ocean. The pressure wave bounce between the water surface and ocean floor, followed by the generation of acoustic-gravity wave (AGW) propagation away from the initial disturbance, is shown with unprecedented details. The computational novelty covers accurate pressure-wave-field computation using high-frequency AGW modes, unlike the dominant first AGW in surface waves. The mathematical problem is solved using the Fourier transformation and eigenfunction expansion techniques, following a multistep approximation of the arbitrarily deep sea. Test cases of two idealistic subsurface geometries (continental shelf and mountain range) using the ocean floor’s linear and parabolic temporal growth show elongation and shortening of the surface wavelength and changes in the propagation speed due to the variable seafloor topography through simulations. The mutual interaction of the pressure waves within the water column unravels depth-change-induced scattering. While high-frequency pressure at a shallower depth is dominant for the continental shelf, a significantly reduced amplitude across a mountain ridge is visualised with a corresponding lower impact of the acoustic-gravity wave on the surface wave. Substantial contrast in the pressure wave field triggered by a slight change in the rupture’s properties suggests its careful consideration when building tsunami prediction models. Since these AGWs are precursors to tsunamis, we believe their detection through an accurate pressure measurement will give very important information about the associated tsunami wave.

We investigate three-dimensional Kolmogorov flow as a model flow featuring turbulent superstructures, i.e. large-scale coherent structures within a turbulent flow. Previous works have shown that turbulent Kolmogorov flow features the emergence of various configurations of large-scale vortex pairs and dynamic switching between these flow states. Here, we devise a low-dimensional model and characterise the meandering of these large scales between different numbers of vortex pairs through two complex amplitudes. We identify these different large-scale states as remnants of instabilities far from the onset of turbulence. We obtain the statistics of the two complex amplitudes from direct numerical simulations. We then construct a set of stochastic amplitude equations whose statistical properties reproduce the ones observed in the three-dimensional Kolmogorov flow up to a fair agreement. Based on these results, we discuss how fast-evolving terms in the form of noise can be introduced into the averaged equations to capture the complex dynamics of the large scales. We thereby shed light on the interplay between the large scales and the small-scale fluctuations.

Fluid–structure interaction (FSI) poses a significant computational challenge due to the complex, multiscale nonlinearities of both fluid and structural dynamics. In this study, a novel strongly coupled FSI network is developed for accurate and efficient predictive modelling of FSI problems. Specifically, the framework architecture integrates a physics-constrained convolutional neural network autoencoder (CAE) with a strong coupling (SC) prediction module containing both fluid and structural dynamic prediction modules (DP) that perform recursive prediction simultaneously and interactively. First, the physics-constrained CAE learns low-dimensional nonlinear normal modes (NNMs) representations of the high-dimensional fluid field’s spatiotemporal dynamics. Subsequently, the fluid DP module in the SC module leverages these NNMs combined with structural states determined by embedding the motion equation into the structural DP, to predict the future-state flow fields efficiently. Such a strongly coupled FSI framework is achieved by integrating fluid NNMs and structural states within each time step to recursively correct the learned mapping of the trained CAE and fluid DP modules, thereby efficiently and accurately predicting future-state FSI dynamics simultaneously. The developed SC-CAE-NNM FSI framework is applied to the classic problem of vortex-induced vibrations of a circular cylinder, analysing both laminar and high-$ \textit{Re}$ flows. It is observed that the identified NNMs of fluid flows in association with the structural state achieve superior accuracy in the prediction of the flow fields and structural responses, indicating that the SC scheme effectively captures the dynamic flow characteristics and strong nonlinear interactions. Furthermore, the framework is found to be able to reconstruct small-scale flow structures in high-$ \textit{Re}$ flows accurately and predict structural responses efficiently. Additionally, the analysis of NNMs energy distributions reveals that the majority of the total energy of the flow field is captured by the first four NNMs, demonstrating significant advantages of nonlinear feature representation for efficient reduced-order modelling of complex flows. Overall, this novel framework shows strong capability for accurate and efficient predictive modelling of complex nonlinear dynamics of FSIs.

The stability of two bubbles rising initially in line through a viscous liquid is revisited using a global linear stability analysis formulated within an arbitrary Lagrangian–Eulerian framework, complemented by fully resolved embedded boundary method simulations. Whereas previous studies attributed the promoted in-line stability of oblate bubbles to a deformation-enhanced wake entrainment, the present analysis demonstrates that the dominant stabilizing mechanism arises instead from an inclination-induced rotational feedback generated as the trailing bubble (TB) experiences the asymmetric shear of the leading bubble’s wake. This inclination–shear coupling, rather than deformation itself, governs the recovery of stability with increasing aspect ratio. Furthermore, the results reveal that the unstable drafting–kissing–tumbling mode originates from short-range, two-way coupling between the bubbles, whereas the asymmetric side-escape mode corresponds to a long-range, one-way interaction dominated by the TB response. In addition, a previously unreported oscillatory global mode emerges from the unsteady recirculation linking the two bubbles, acting as a hydrodynamic spring whose effective stiffness and damping govern the oscillation frequency and growth rate. Together, these findings identify inclination-induced lift as the primary mechanism controlling the stability of rising bubble pairs and provide a unified framework for interpreting their stationary and oscillatory transitions across a broad range of inertial and deformable regimes.

In this paper, we use a two-dimensional depth-averaged model to explore the spreading of a ${\textrm {CO}}_{2}$ plume from a series of injection wells arranged around the crest of an axisymmetric anticline structure of finite vertical extent and which is connected to a laterally extensive aquifer. For a constant ${\textrm {CO}}_{2}$ injection rate, we calculate both the fraction of available pore space in the anticline filled by structurally trapped ${\textrm {CO}}_{2}$ at the point at which the plume spills from the lower boundary of the anticline into the aquifer and the maximum pressure within the anticline throughout the injection process. We find that the fraction of the anticline filled with ${\textrm {CO}}_{2}$ varies considerably between slow, buoyancy-controlled filling and fast, pressure-controlled filling, with the latter case leading to much smaller fractions of the anticline accessed by the ${\textrm {CO}}_{2}$ plume. We also find that there is a correspondence between the maximum pressure within the anticline produced by a single injection well at the centre and the maximum pressure produced by a series of equally spaced injection wells. The results of this modelling provide key insights into the tensions between the injection rate, total injected volume, and pressure in a ${\textrm {CO}}_{2}$ storage system, and we discuss some of the challenges this presents.

The current study presents a three-dimensional linear stability analysis of particle-laden Couette-Poiseuille flow (CPF) suspended in a Newtonian fluid between two parallel plates, with the lower plate coated by a porous medium. The influence of suspended particles is examined using a two-domain formulation in which particles are confined to the fluid layer and do not penetrate the porous substrate. The particle-laden suspension is modelled using the dusty gas framework, while the flow within the porous layer is described by the volume-averaged Navier–Stokes equations. In particle-laden flows over impermeable walls, particle inertia may either stabilise or destabilise the flow depending on the governing parameters. In contrast, the presence of a porous layer introduces an additional permeability-dependent destabilising mechanism that fundamentally modifies these classical trends. Consequently, particle loading can reduce the critical Reynolds number at sufficiently high permeability, even in parameter regimes where particles stabilise the corresponding rigid-wall flow. The coupled formulation also introduces additional disturbance branches associated with fluid–particle coupling near the permeable interface. Although these modes remain stable throughout the parameter space investigated, they modify the eigenspectrum and influence the dominant instability through altered coupling pathways. Furthermore, unlike impermeable-wall CPF, where increasing the Couette component generally stabilises the flow, the porous-wall configuration exhibits a monotonic decrease in the critical Reynolds number over the range examined. These results demonstrate that porous boundaries can fundamentally alter established stability behaviour in particle-laden shear flows through permeability-dependent coupling between the suspension and the porous substrate.

We study the navigation of a self-propelled inertial particle in two-dimensional Rayleigh–Bénard convection at Prandtl number $\textit{Pr}=0.71$ and cell aspect ratio $\varGamma =4$ for Rayleigh numbers $Ra$ ranging from $10^7$ to $10^{11}$. A reinforcement-learning (RL) controller selects the propulsive acceleration, subject to an upper bound $\mathcal{A}_{\textit{max}}$, to achieve a prescribed horizontal displacement. We find that the success rate increases abruptly with $\mathcal{A}_{\textit{max}}$ at moderate $Ra$, whereas at higher $Ra$ the transition becomes more gradual and shifts to larger $\mathcal{A}_{\textit{max}}$. Moreover, although the completion time increases with $Ra$, the propulsion energy required for successful traversal decreases. Proper orthogonal decomposition indicates that these performance differences are associated with reorganisation of the carrier flow. At moderate $Ra$, the dominant large-scale circulation partitions the domain through persistent transport barriers, requiring a finite thrust surplus to cross them; at higher $Ra$, energy is distributed across many modes, the barriers fragment and transient plume-assisted pathways emerge. Compared with a constant-heading baseline, the learned policy aligns with local currents and consumes significantly less energy. Lagrangian coherent structure analysis further suggests that the RL agent tends to cross repelling barriers and surf along attracting pathways. Finally, by mapping these behaviours onto the local Eulerian flow topology using Voronoi tessellation and the $Q$-criterion, we distil an interpretable, physics-based heuristic strategy that retains robust navigability. These results connect turbulent-flow organisation with autonomous navigation under bounded actuation.

We investigate the heat transport in porous media convection (PMC) over a wide Rayleigh–Darcy number range of $26.8\leqslant Ra\leqslant 2.62\times 10^5$, and a Darcy number range of $6.18\times 10^{-7}\leqslant Da\leqslant 1.21\times 10^{-5}$. In the experiments, we employ 3D-printed lattice structures as the solid porous matrix and water as the working fluid. Quantitative analyses of the porous medium Nusselt number $ \textit{Nu}_m$ and local temperature statistics reveal that the present system undergoes a transition through five distinct regimes: (I) conduction, (II) convection, (III) oscillation, (IV) transition, (V) classical Rayleigh–Bénard convection. This transitional process bridges the gap between Rayleigh–Darcy-like behaviour and Rayleigh–Bénard-like behaviour in PMC. By varying the permeability of the matrix, we further examine the role of the Darcy number $ \textit{Da}$, which turns out to have a profound impact on the transitional processes across different regimes. Flow field measurements reveal that the flow structures within regime IV and regime V evolve from several horizontally stacked convection rolls to a single-roll structure, and the pore-scale Reynolds numbers both exceed unity in these two regimes. Finally, we report the corresponding phase diagram in the $ \textit{Ra}$–$ \textit{Da}$ space.