Direct numerical simulation (DNS) is performed to explore turbulent Rayleigh–Bénard convection in spherical shells. Our simulations cover six distinct values of radius ratio, $\eta = r_i/r_o = 0.2$, 0.3, 0.4, 0.5, 0.6 and 0.8, under the assumption of a centrally condensed mass with gravity profile $g \sim 1/r^{2}$; where $r_i$, $r_o$ and $r$ denote the inner shell radius, the outer shell radius and the local radial coordinate, respectively. The Prandtl number is kept constant at unity while the Rayleigh number ($Ra$) is varied from $3 \times 10^{3}$ to $5 \times 10^8$. Our primary aim is to analyze how the radius ratio influences the global transport properties and flow physics. To gain insights into the scaling behaviour of the Nusselt number ($Nu$) and the Reynolds number ($Re$) with respect to $Ra$ and $\eta$, we apply the Grossmann–Lohse (GL) theory (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56) to the system. It is observed that the scaling exponents for $Nu$ and $Re$ in relation to $Ra$ are more significant for smaller $\eta$ values, suggesting that the simulations with smaller $\eta$ reach the classical $Nu\sim Ra^{1/3}$ regime at a relatively lower $Ra$. This observation could also imply the systems with smaller $\eta$ might transition to the ultimate regime earlier at a smaller $Ra$. Based on our extensive DNS data, we establish that the thickness of the inner thermal boundary, $\lambda _{\vartheta }^{i}$, follows a scaling relationship of $\lambda _{\vartheta }^{i} \sim \eta ^{1/2}$. This relationship, in turn, leads to a scaling law for $Nu$ in the form of $Nu \sim f(\eta ) Ra^{\gamma }$, where the function $f(\eta )$ is defined as $f(\eta ) = {\eta ^{1/2}}/{(1+\eta ^{4/3})}$, and the exponent $\gamma$ depends on both $Ra$ and $\eta$. Additionally, we characterize and explain the asymmetry in the velocity field by introducing the separate Reynolds numbers for the inner and outer shells. The asymmetry of the kinetic and thermal energy dissipation rates in the inner and outer boundary layers (BLs) is also quantified.

By comparing the budget of a data-driven quasi-linear approximation (DQLA) (Holford, Lee & Hwang, J. Fluid Mech., vol. 980, 2024, A12) and direct numerical simulation (DNS) (Lee & Moser, J. Fluid Mech., vol. 860, 2019, pp. 886–938), the energetics of linear models for wall-bounded turbulence are assessed. The DQLA is implemented with the linearised Navier–Stokes equations with a stochastic forcing term and an eddy viscosity diffusion model. The self-consistent nature of the DQLA allows for a global comparison across all wavenumbers to assess the role of the various terms in the linear model in replicating the features present in DNS. Starting from the steady-state second-order statistics of a Fourier mode, a spectral budget equation is derived, connecting Lyapunov-like equations to the transport budget equations obtained from DNS. It is found that the DQLA and DNS are in good qualitative agreement for the streamwise-elongated structures present in DNS, comparing well for production, viscous transport and wall-normal turbulent transport. However, the DQLA does not have an energy-conservative nonlinear term. This results in no dissipation under molecular viscosity, with energy instead being dissipated locally through the eddy viscosity model, which models the energy removal by the nonlinear term at integral length scales. Comparison of the pressure–strain statistics also highlights the absence of the streak instability, with production and forcing mainly being retained in the streamwise and wall-normal components or shifted to the spanwise component. It is demonstrated that the eddy viscosity diffusion term locally enforces a self-similar budget, making the model for the nonlinear term self-consistent with a logarithmic mean profile. Implications and recommendations to improve the current eddy viscosity enhanced linear models are also discussed concerning the comparison with DNS, as well as considerations with regard to pressure statistics to mimic the role of the streak instability through colour of turbulence models.

We conduct a numerical study on the drag-reduction mechanism of an opposition- controlled turbulent channel flow from the viewpoint of a symbolic dynamics approach. The effect of the virtual wall formed by the opposition control is maximised at the location of the detection plane $y_d^+ \approx 10$. At this wall-normal location, the local link strength of the self-loop of network nodes representing the negative correlation pattern between the streamwise and wall-normal velocity fluctuations is maximised in the uncontrolled flow. In the controlled case, the multiscale complexity–entropy causality plane and the spatial permutation entropy at $y_d^+ \approx 10$ indicate that the drag-reduction effect is attributed to the reduction of the region where streaks actively coalesce and separate and the suppression of the regeneration cycle in the region near the wall.

We study the effect of geometrical confinement on thermal convection by laboratory experiments and direct numerical simulations using Hele-Shaw geometries (typically the gap-to-height aspect ratio $0.12$) for the Prandtl number $Pr \geq 40$ and the Rayleigh number $Ra \leq 6 \times 10^7$. Under such strong unidirectional confinement, the convective flows are forced to squeeze within the narrow gap and exhibit unique spatiotemporal signatures, which contrast those in unconfined systems. With the increase of $Ra$, we identify that the system experiences five convective regimes that can be classified from two aspects, time dependency and flow dimensionality: (I) quasi-two-dimensional (quasi-2-D) steady flow; (II) quasi-2-D flow with oscillatory corner rolls; (III) three-dimensional (3-D) flow with oscillatory corner rolls; (IV) 3-D steady flow; and (V) 3-D time-dependent motion of plumes around sidewalls. Notably, unsteadiness does not emerge globally, but is localised near the sidewalls as oscillatory corner rolls, resulting in the regime transitions happening in a quasi-steady manner. We confirm that these regime transitions show less dependence on both $Pr$ and the other (wider) horizontal scale of the geometry. Moreover, we find that a recently proposed criterion ‘degree of confinement’ (Noto et al., Proc. Natl Acad. Sci. USA, vol. 121, issue 28, 2024, e2403699121) successfully explains the emergence of 3-D structures, expanding its applicable range to smaller $Ra$. This study deepens the comprehension of the thermal convection emerging in tight geometries, impacting across disciplines, such as Earth and planetary science, and thermal engineering.

The flow near a moving contact line depends on the dynamic contact angle, viscosity ratio and capillary number. We report experiments involving immersing a plate into a liquid bath, concurrently measuring the interface shape, interfacial velocity and fluid flow using digital image processing and particle image velocimetry. All experiments were performed at low plate speeds to maintain small Reynolds and capillary numbers for comparison with viscous theories. The dynamic contact angle, measured in the viscous phase, was kept below $90^{\circ }$ and the viscosity ratio, $\lambda < 1$. This region of parameter space is largely unexplored for advancing contact lines. An important aim of the present study is to provide new experimental data against which new contact line models can be developed. The flow field is directly compared against the prediction from the viscous theory of Huh & Scriven (J. Colloid Interface Sci., vol. 35, issue 1, 1971, pp. 85–101) but with a slight modification involving the curved interface. Remarkable agreement is found between experiments and theory across a wide parameter range. The prediction for interfacial speed from Huh & Scriven is also in excellent agreement with experiments except in the vicinity of the contact line. Material points along the interface were found to rapidly slow down near the contact line, thus alleviating the singularity at the moving contact line. To the best of our knowledge, such a detailed test of theoretical models has not been performed before and we hope the present study will spur new modelling efforts in the field.

Porous membranes are thin solid structures that allow the flow to pass through their tiny openings, called pores. Flow inertia may play a significant role in several filtration flows of natural and engineering interest. Here, we develop a predictive macroscopic model to describe solvent and solute flows past thin membranes for non-negligible inertia. We leverage homogenization theory to link the solvent velocity and solute concentration to the jumps of solvent stress and solute flux across the membrane. Within this framework, the membrane acts as a boundary separating two distinct fluid regions. These jump conditions rely on several coefficients, stemming from closure problems at the microscopic pore scale. Two approximations for the advective terms of Navier–Stokes and advection–diffusion equations are introduced to include inertia in the microscopic problem. The approximate inertial terms couple the micro- and macroscopic fields. Here, this coupling is solved numerically using an iterative fixed-point procedure. We compare the resulting models against full-scale simulations, with a good agreement both in terms of averaged values across the membrane and far-field values. Eventually, we develop a strategy based on unsupervised machine learning to improve the computational efficiency of the iterative procedure. The extension of homogenization towards weak-inertia flow configurations as well as the performed data-driven approximation may find application in preliminary analyses as well as optimization procedures towards the design of filtration systems, where inertia effects can be instrumental in broadening the spectrum of permeability and selectivity properties of these filters.

Rotor-stator flows are known to exhibit instabilities in the form of circular and spiral rolls. While the spirals are known to emanate from a supercritical Hopf bifurcation, the origin of the circular rolls is still unclear. In the present work we suggest a quantitative scenario for the circular rolls as a response of the system to external forcing. We consider two types of axisymmetric forcing: bulk forcing (based on the resolvent analysis) and boundary forcing using direct numerical simulation. Using the singular value decomposition of the resolvent operator the optimal response is shown to take the form of circular rolls. The linear gain curve shows strong amplification at non-zero frequencies following a pseudo-resonance mechanism. The optimal energy gain is found to scale exponentially with the Reynolds number $Re$ (for $Re$ based on the rotation rate and interdisc spacing $H$). The results for both types of forcing are compared with former experimental works and previous numerical studies. Our findings suggest that the circular rolls observed experimentally are the effect of the high forcing gain together with the roll-like form of the leading response of the linearised operator. For high enough Reynolds number it is possible to delineate between linear and nonlinear responses. For sufficiently strong forcing amplitudes, the nonlinear response is consistent with the self-sustained states found recently for the unforced problem. The onset of such non-trivial dynamics is shown to correspond in state space to a deterministic leaky attractor, as in other subcritical wall-bounded shear flows.

Depth-limited overturning wave shape affects water turbulence and sediment suspension. Experiments have shown that wind affects shoaling and overturning wave shape, with uncertain mechanism. Here, we study wind effects (given by the wind Reynolds number) on solitary wave shoaling and overturning with the two-phase direct numerical simulations model Basilisk run in two dimensions on steep bathymetry for fixed wave Reynolds number and Bond number. For all wind, the propagating solitary wave sheds a two-dimensional turbulent air wake and has nearly uniform speed with minimal wave energy changes over the rapidly varying bathymetry. Wave-face slope is influenced by wind, and shoaling wave shape changes are consistent with previous studies. As overturning jet impacts, wind-dependent differences in overturn shape are quantified. The non-dimensional breakpoint location and overturn area have similar wind dependence as previous experience, whereas the overturn aspect ratio has opposite wind dependence. During shoaling, the surface viscous stresses are negligible relative to pressure. Surface tension effects are also small but grow rapidly near overturning. In a wave frame of reference, surface pressure is low in the lee and contributes 2–5 % to the velocity potential rate of change in the surface dynamic boundary condition, which, integrated over time, changes the wave shape. Reasons why the overturn aspect ratio is different than in experiment and why a stronger simulated wind is required are explored. The dramatic wind effects on overturning jet area, and thus to the available overturn potential energy, make concrete the implications of wind-induced changes to wave shape.

In this study we investigate the sedimentation of prolate spheroids in a quiescent fluid by means of the particle-resolved direct numerical simulation. With the increase of the particle volume fraction $\phi$ from $0.1\,\%$ to $10\,\%$, we observe a non-monotonic variation of the mean settling velocity of particles, $\langle V_s \rangle$. By virtue of the Voronoi analysis, we find that the degree of particle clustering is highest when $\langle V_s \rangle$ reaches the local maximum at $\phi =1\,\%$. Under the swarm effect, clustered particles are found to preferentially sample downward fluid flows in the wake regions, leading to the enhancement of the settling speed. As for lower or higher volume fractions, the tendency of particle clustering and the preferential sampling of downward flows are attenuated. The hindrance effect becomes predominant when the volume fraction exceeds 5 % and reduces $\langle V_s \rangle$ to less than the isolated settling velocity. Particle orientation plays a minor role in the mean settling velocity, although individual prolate particles still tend to settle faster in suspensions when they deviate more from the broad-side-on alignment. Moreover, we also demonstrate that particles are prone to form column-like microstructures in dilute suspensions under the effect of wake-induced hydrodynamic attractions. The radial distribution function is higher at a lower volume fraction. As a result, the collision rate scaled by the particle number density decreases with the increasing volume fraction. By contrast, as another contribution to the particle collision rate, the relative radial velocity for nearby particles shows a minor degree of variation due to the lubrication effect.

We introduce a continuum framework for the energetics of particle-size segregation in bidisperse granular flows. Building on continuum segregation equations and a recent segregation flux model, the proposed framework offers general analytical expressions to study the physics of granular flows from a mechanical energy perspective. We demonstrate the framework's applicability by examining the energetics of shear-driven granular flows. Numerical experiments with varying frictional coefficients and particle-size ratios reveal two distinct phases in the energetics, marked by the separate onset of particle segregation and diffusive remixing. Furthermore, our numerical simulations alongside previous experimental results show that the bulk Richardson number $Ri$, defined as the potential energy to kinetic energy ratio at steady state, follows the scaling relationship $Ri\equiv \hat {E}^{(s)}_{gp} / \hat {E}^{(s)}_{k} \propto Pe^{-1/2}_{sr}$ for $0.1 \leq Ri\leq ~10^{3}$ and $10^{-4} \leq Pe_{sr} \leq ~300$, the segregation–rheology Péclet number. Finally, we present a Péclet-number-dependent theoretical expression for the degree of mixing (or segregation), validated by the compiled numerical and experimental dataset. Our findings hint that the bulk segregation–mixing state can be predicted and controlled using the segregation Péclet number $Pe$ and $Pe_{sr}$, both determined from known system parameters, providing an instrumental tool for engineering and geophysical applications.

This study presents an approach to investigate the role of eddy viscosity in linearized mean-field analysis of broadband turbulent flows. The procedure is based on spectral proper orthogonal decomposition (SPOD), resolvent analysis and the energy budget of coherent structures and is demonstrated using the example of a turbulent jet. The focus is on the coherent component of the Reynolds stresses, the nonlinear interaction term of the fluctuating velocity component in frequency space, which appears as an unknown in the derivation of the linearized Navier–Stokes equations and which is the quantity modelled by the Boussinesq approach. For the considered jet the coherent Reynolds stresses are found to have a mostly dissipative effect on the energy budget of the dominant coherent structures. Comparison of the energy budgets of SPOD and resolvent modes demonstrates that dissipation caused by nonlinear energy transfer must be explicitly considered within the linear operator to achieve satisfactory results with resolvent analysis. Non-modelled dissipation distorts the energy balance of the resolvent modes and is not, as often assumed, compensated for by the resolvent forcing vector. A comprehensive analysis, considering different predictive and data-driven eddy viscosities, demonstrates that the Boussinesq model is highly suitable for modelling the dissipation caused by nonlinear energy transfer for the considered flow. Suitable eddy viscosities are analysed with regard to their frequency, azimuthal wavenumber and spatial dependence. In conclusion, the energetic considerations reveal that the role of eddy viscosity is to ensure that the energy the structures receive from the mean-field is dissipated.

The flow-induced oscillation of an S-shaped buckled flexible filament was explored using the penalty immersed boundary method. As the length and bending rigidity of the filament were varied, three distinct modes emerged: the equilibrium mode, streamwise oscillation (SO) mode and transverse oscillation (TO) mode. A transition region between the SO and TO modes was identified. Notably, the filament exhibited a 3P wake pattern under SO and a 2S wake pattern under TO. The former was induced by fluid–elastic instability, while the latter was attributed to vortex-induced oscillation. The interaction between the filament's motion and vortex shedding was examined for both modes. To elucidate the disparity between the TO of the S-shaped buckled filament and snap-through oscillation (STO), a ball-on-a-hill analogy was introduced. The performance of energy harvesting was evaluated using metrics including the elastic energy and power coefficient. The TO mode was found to show significantly higher energy harvesting performance than the SO and STO modes. The majority of the strain energy was concentrated at the upper and lower midpoints of the filament.

Energetics of mode-1 internal waves interacting with topographic ridges are investigated using high-resolution two-dimensional simulations at spatial scales of $O(100)$ m that span between classical laboratory-scale ($O(10)$ m) and field-scale simulations ($O(1000\unicode{x2013}10\,000)$ m). This paper focuses on the energetics of wave–topography interaction, with emphasis on systematically examining the partitioning of the incident wave energy as a function of wave forcing and topographic parameters. Partitioning of energy into the transmitted, reflected and dissipated components is quantified as a function of wave Froude number $Fr=U_0/c_{ph}$ ($U_0=$ velocity amplitude of forcing and $c_{ph}=$ internal wave celerity), slope criticality $=\gamma /s$, where $\gamma =$ topographic slope and $s=$ wave characteristic slope, and the ratio of topographic height $h_t$ to water depth $d$. As $Fr$ increases, dense fluid from the base of the stratified water column surges upslope with significant vertical inertia, leading to the formation of internal boluses that plunge over and onto the downstream side of the ridge, resulting in elevated dissipation. Results show that non-hydrostatic contributions to the total energy flux are significant (up to 50 %). Analysis of the energy flux budget shows that transmitted energy flux decreases monotonically as $\gamma /s$ increases for any given $Fr$ and $h_t/d$. At critical slopes ($\gamma /s=1$), the transmitted energy flux scales as a linear function of $h_t/d$, with a mild dependence on $Fr$, a key result that can be useful in energy flux parameterizations. Reflected energy flux exhibits a nonlinear dependence on the ridge height, increasing sharply when $h_t/d > 0.5$. Dissipation is enhanced at critical slopes, with a plateau evident for $\gamma /s \ge 1$ and $h_t/d = 0.5$ for all $Fr$.

This paper presents numerical simulations of the free fall of homogenous cylinders of length-to-diameter ratios $2$, $3$ and 5 and solid-to-fluid-density ratios $\rho _s/\rho$ going from 0 to 10 in transitional regimes. The path instabilities are shown to be due to two types of transitional states. The well-known fluttering state is a solid mode, characterised by significant oscillations of the cylinder axis due to a strong interaction between the vortex shedding in the wake and the solid degrees of freedom. Weakly oscillating, mostly irregular trajectories, are fluid modes, associated with purely fluid instabilities in the wake. The interplay of solid and fluid modes leads to a varying scenario in which the length-to-diameter and density ratios play an important role. The description is accompanied by the presentation of the identified transitional states in terms of path characteristics and vorticity structure of the wakes and by bifurcation diagrams showing the evolution of asymptotic states with increasing Galileo numbers. There appears to be a strong difference between the behaviour of cylinders of aspect ratio $L/d=3$ and 5. A similar contrast is stated between light cylinders of density ratios $\rho _s/\rho \le 2$ and dense cylinders of density ratios 5 and 10. Finally, the question of the scatter of values of the drag coefficient and of the frequency of oscillations raised in the literature is addressed. It is shown, that in addition to external parameters (Galileo number, density and aspect ratio) the amplitude of oscillations characterising the instability development is to be taken into account to explain this scatter. Fits of the simulation results to simple correlations are proposed. Namely that of the drag coefficient proves to be accurate (better than 1 % of accuracy) but also that of the Strouhal number (a few per cent of accuracy) may be of practical use.

Direct numerical simulations of the turbulence of a Herschel–Bulkley (HB) fluid in a rough channel are performed at a shear Reynolds number $Re_{\tau } \approx 300$ and a Bingham number ${Bn} \approx 0.9$. For the type of rough surface used in this study, the results indicate that Townsend's wall similarity hypothesis also holds for HB fluids. However, there are notable differences compared with the effect of roughness on Newtonian fluids. More specifically, the effect of roughness appears to be slightly stronger for HB fluids, in the sense that the bulk Reynolds number, based on the viscosity at the wall, is reduced further due to the increase in viscosity in the troughs of the roughness surface induced by the low shear. At the same time, for the simulated rough surface, the contribution of form drag to the total pressure drop is reduced from 1/4 to about 1/5 due to the persistence of viscous shear in the boundary layer, reducing its shielding effect. As for the friction factor, due to the nonlinearity of the HB constitutive relation, its use with the wall shear rate from the mean wall shear stress underpredicts the minimum viscosity at the wall by up to 18 %. This inevitably leads to uncertainties in the prediction of the friction factor. Finally, it is observed that the rough surface is unable to break the peculiar near-wall flow structure of HB fluids, which consists of long persistent low-speed streaks occupying the entire domain. This means that the small-scale energy is significantly reduced for HB fluids, even in rough channels, with the energy more concentrated in the lower wavenumber range, implying an increase in the slope of the power spectrum to $-7/2$ in the inertial range, as shown by Mitishita et al. (J. Non-Newtonian Fluid Mech., vol. 293, 2021, 104570).

The dominant mode instability in hypersonic boundary-layer transition is the so-called second-mode instability, which has a peculiar nature strongly coupled with thermoacoustic phenomena. In linear stability theory, the unstable wave is associated with one of the two eigenvalues that originate from the acoustic branches, referred to as slow and fast modes. Interestingly, the unstable mode (slow or fast) reaches its maximum amplification as the other mode (fast or slow) attains a minimum. The phase velocity of the two modes is then very close, and this phenomenon is called synchronization. The aim of the present study is to unravel the physical mechanism that explains the second-mode growth. To that aim, second-order nonlinear equations are written for the disturbances given by linear stability. In this framework, entropy, kinetic energy and temperature energy budgets are obtained up to second order. The budgets are scrutinized for various Mach numbers and for adiabatic and cold-wall thermal conditions. Perturbation entropy budgets clearly show the process is a reversible one. An energy exchange between kinetic energy and temperature energy of the weakly nonlinear modes is driven by pressure–dilatation terms. As underlined in previous studies, the unstable mode experiences an alternate heating and cooling near the wall, which is shown to be a rather nonlinear process. The change in fluctuating thermal energy in the form of a dilatational wave is sustained by pumping disturbance kinetic energy through the pressure–dilatation term, the direction of the conversion being driven by the relative phase between pressure and dilatation. This process is similar for the slow and fast modes, the unstable mode being amplified and the other being damped. No change in the process has been noted at the location of the synchronization, suggesting that the modes have the same nature but evolve independently.

The stability of liquid-film flows is essential in many industrial applications. In the dip-coating process, a liquid film forms over a substrate extracted at a constant speed from a bath. We studied the linear stability of this film considering different thicknesses $\hat {h}$ for four liquids, spanning an extensive range of Kapitza numbers ($Ka$). By solving the Orr–Sommerfeld eigenvalue problem with the Chebyshev–Tau spectral method, we calculated the threshold between growing and decaying perturbations, investigated the instability mechanism, and computed the absolute/convective threshold. The instability mechanism was studied by analysing the perturbations’ vorticity distribution and the kinetic energy balance. It was found that liquids with low $Ka$ (e.g. corn oil, $Ka = 4$) are stable for a smaller range of wavenumbers compared with liquid with high $Ka$ (e.g. liquid zinc, $Ka = 11\,525$). Surface tension has a stabilising and a destabilising effect. For long waves, it curves the vorticity lines near the substrate, reducing the flow under the crests. For short waves, it fosters vorticity production at the interface and creates a region of intense vorticity near the substrate. In addition, we discovered that the surface tension contributes to both the production and dissipation of perturbation's energy depending on the $Ka$ number. Regarding the absolute/convective threshold, we identified a window in the parameter space where unstable waves propagate throughout the entire domain (indicating absolute instability). Perturbations affecting Derjaguin's solution ($\hat {h}=1$) for $Ka<17$ and the Landau–Levich–Derjaguin solution ($\hat {h}=0.945 Re^{1/9}Ka^{-1/6}$), are advected by the flow (indicating convective instability).

We employ direct numerical simulations to investigate the heat transfer and flow structures in turbulent Rayleigh–Bénard convection in both cylindrical cells and laterally periodic domains, spanning an unprecedentedly wide range of aspect ratios $0.075 \leqslant \varGamma \leqslant 32$. We focus on Prandtl number ${Pr}=1$ and Rayleigh numbers ${{Ra}}=2\times 10^7$ and ${{Ra}}=10^8$. In both cases, with increasing aspect ratio, the heat transfer first increases, then reaches a maximum (which is more pronounced for the cylindrical case due to confinement effects), and then slightly goes down again before it finally saturates at the large aspect ratio limit, which is achieved already at $\varGamma \approx 4$. Already for $\varGamma \gtrsim 0.75$, the heat transfers in both cylindrical and laterally periodic domains become identical. The large-$\varGamma$ limit for the volume-integrated Reynolds number and the boundary layer thicknesses are also reached at $\varGamma \approx 4$. However, while the integral flow properties converge at $\varGamma \approx 4$, the confinement of a cylindrical domain impacts the temperature and velocity variance distributions up to $\varGamma \approx 16$, as thermal superstructures cannot form close to the sidewall.

We report an experimental study of Rayleigh–Bénard convection of liquid metal GaInSn in a cuboid cell with an aspect ratio of 0.5 under the effect of a horizontal magnetic field. The Rayleigh number spans a range of $3.8\times 10^5 \leqslant Ra \leqslant 1.1\times 10^7$, while the magnetic field strength reaches up to 0.5 T, corresponding to a maximum Hartmann number to 2041. By combining temperature and velocity measurements, we identify several flow morphologies, including a novel cellular pattern characterized by four stacked vortices that periodically squeeze and induce velocity reversals. Based on the identified flow morphologies, we partition the entire ($Ra, Ha$) parameter space into five distinct flow regimes and systematically investigate the flow characteristics within each regime. The temperature gradient and oscillation frequency exhibit scaling relationships with the combined parameters $Ra$ and $Ha$. Notably, we observe a coupling between flow regime and global transport efficiencies, particularly in a regime dominated by the double-roll structure, which experiences a maximum 36 % decrease in heat transfer efficiency compared with the single-roll structure. The dependencies of heat and momentum transport on $Ra$ and $Ha$ follow scaling laws as $Nu \sim (Ha^{-2/3}RaPr^{-1})^{3/5}$ and $Re \sim (Ha^{-1}RaPr^{-1})^{4/3}$, respectively.

Experiments and numerical simulations of inertial particles in underexpanded jets are performed. The structure of the jet is controlled by varying the nozzle pressure ratio, while the influence of particles on emerging shocks and rarefaction patterns is controlled by varying the particle size and mass loading. Ultra-high-speed schlieren and Lagrangian particle tracking are used to experimentally determine the two-phase flow quantities. Three-dimensional simulations are performed using a high-order, low-dissipative discretization of the gas phase while particles are tracked individually in a Lagrangian manner. A simple two-way coupling strategy is proposed to handle interphase exchange in the vicinity of shocks. Velocity statistics of each phase are reported for a wide range of pressure ratios, particle sizes and volume fractions. An upstream shift of the Mach disk in the presence of particles reveals significant two-way coupling even at low mass loading. A semi-analytic model that predicts the extent of the Mach disk shift is presented based on a one-dimensional Fanno flow that takes into account volume displacement by particles and interphase exchange due to drag and heat transfer. The per cent shift in Mach disk is found to scale with the mass loading, nozzle pressure ratio and interphase slip velocity and inversely with the particle diameter.