The classical Gill's problem, focusing on the stability of thermal buoyancy-driven convection in a vertical porous slab with impermeable isothermal boundaries, is studied from a different perspective by considering a triple-diffusive fluid system having different molecular diffusivities. The assessment of stability/instability of the basic flow entails a numerical solution of the governing equations for the disturbances as Gill's proof of linear stability falls short. The updated problem formulation is found to introduce instability in contrast to Gill's original set-up. A systematic examination of neutral stability curves is undertaken for KCl–NaCl–sucrose and heat–KCl–sucrose aqueous systems which are found to exhibit an anomalous behaviour on the stability of base flow. It is found that, in some cases, the KCl–NaCl–sucrose system necessitates the requirement of four critical values of the Darcy–Rayleigh number to specify the linear stability criteria ascribed to the existence of two isolated neutral curves positioned one below the other. Conversely, the heat–KCl–sucrose system demands only two critical values of the Darcy–Rayleigh number to decide the stability of the system. The stability boundaries are presented and the emergence of a travelling-wave mode supported back and forth with stationary modes is observed due to the introduction of a third diffusing component. In addition, some intriguing outcomes not recognized hitherto for double-diffusive fluid systems are manifested.

Symmetry-breaking bifurcations, where a flow state with a certain symmetry undergoes a transition to a state with a different symmetry, are ubiquitous in fluid mechanics. Much can be understood about the nature of these transitions from symmetry alone, using the theory of groups and their representations. Here, we show how the extensive databases on groups in crystallography can be exploited to yield insights into fluid dynamical problems. In particular, we demonstrate the application of the crystallographic layer groups to problems in fluid layers, using thermal convection as an example. Crystallographic notation provides a concise and unambiguous description of the symmetries involved, and we advocate its broader use by the fluid dynamics community.

The actuator line method (ALM) is a commonly used technique to simulate slender lifting and dragging bodies such as wings or blades. However, the accuracy of the method is significantly reduced near the tip. To quantify the loss of accuracy, translating wings with various aspect and taper ratios are simulated using several methods: wall-resolved Reynolds-averaged Navier–Stokes (RANS) simulations, an advanced ALM with two-dimensional (2-D) mollification of the force, a lifting line method, a mollified lifting line method and a vortex lattice method. Significant differences in the lift and drag distributions are found on the part of the wing where the distance to the tip is smaller than approximately 3 chords and are identified to arise from both the forces mollification and the uneven induced velocity along the chord. Correction functions acting on the lift coefficient and effective angle of attack near the wing tip are then derived for rectangular wings of various aspect ratios. They are then also applied to wings of various taper ratios using the ‘effective dimensionless distance to the tip’ as the main parameter. The application of the correction not only leads to a much improved lift distribution, but also to a more consistent drag distribution. The correction functions are also obtained for various mollification sizes, as well as for ALM with three-dimensional (3-D) mollification. These changes mostly impact the correction for the effective angle of attack. Finally, the correction is applied to simulations of the NREL Phase VI wind turbine, leading to an enhanced agreement with the experimental data.

A new wall-wake law is proposed for the streamwise turbulence in the outer region of a turbulent boundary layer. The formulation pairs the logarithmic part of the profile (with a slope $A_1$ and additive constant $B_1$) to an outer linear part, and it accurately describes over 95 % of the boundary layer profile at high Reynolds numbers. Once the slope $A_1$ is fixed, $B_1$ is the only free parameter determining the fit. Most importantly, $B_1$ is shown to follow the same trend with Reynolds number as the wake factor in the wall-wake law for the mean velocity, which is tied to changes in scaling of the mean flow and the turbulence that occur at low Reynolds number.

We investigate flow of liquid which is partially filled in a cylindrical container horizontally rotating about its axis of symmetry. Even if the rotation is slow enough to keep the liquid–gas interface almost undeformed, convection cells whose circulation axis is perpendicular to the container's rotational axis can be sustained. We conduct experiments by particle image velocimetry and direct numerical simulations with the S-CLSVOF and immersed boundary methods to reveal the condition of the Reynolds number, the aspect ratio of the container and the filling ratio of liquid for the onset of these convection cells. When the filling ratio is not too large, as the Reynolds number increases, convection cells appear through a pitchfork bifurcation in an infinitely long cylinder. This bifurcation becomes imperfect in the case of a finite-length cylinder. In contrast, when the filling ratio is large enough, convection cells appear through a subcritical bifurcation. Through these investigations, it becomes evident that the axial wavelength of sustained convection cells is an increasing function of the filling ratio in an infinitely long cylinder. In practice, to sustain intense convection cells, we should use a cylinder with the length equal to an integer multiple of the wavelength of the most unstable mode in the infinite-length cylinder. Although we focus on the liquid-pool regime with small Froude numbers, the critical Reynolds number for the pitchfork bifurcation weakly depends on the Froude number. This dependence is explained by considering the changes in the effective filling ratio and the convection velocity.

Low Stokes number particles at dilute concentrations in turbulent flows can reasonably be approximated as passive scalars. The added presence of a drift velocity due to buoyancy or gravity when considering the transport of such passive scalars can reduce the turbulent dispersion of the scalar via a diminution of the eddy diffusivity. In this work, we propose a model to describe this decay and use a recently developed technique to accurately and efficiently measure the eddy diffusivity using Eulerian fields and quantities. We then show a correspondence between this method and standard Lagrangian definitions of diffusivity and collect data across a range of drift velocities and Reynolds numbers. The proposed model agrees with data from these direct numerical simulations, offers some improvement to previous models in describing other computational and experimental data and satisfies theoretical constraints that are independent of Reynolds number.

Dense mixtures of particles of varying size tend to segregate based on size during flow. Granular size segregation impacts many industrial and geophysical processes, but the development of coupled, continuum models capable of predicting the evolution of segregation dynamics and flow fields in dense granular media across different geometries remains a challenge. One reason is because size segregation stems from two driving forces: pressure gradients and shear-strain-rate gradients. Another reason is the challenge of integrating segregation models with rheological constitutive equations for dense granular flow. In this paper we develop a continuum model that accounts for pressure-gradient-driven and shear-strain-rate-gradient-driven segregation, coupled to rheological modelling of a dense granular medium across the quasi-static and dense inertial flow regimes. To calibrate and test the continuum model, we perform discrete element method (DEM) simulations of dense flow of bidisperse granular systems in two flow geometries in which both segregation driving forces are present: inclined plane flow and planar shear flow with gravity. Steady-state DEM data from inclined plane flow is used to determine the dimensionless material parameters in the pressure-gradient-driven segregation model for both spheres and disks. Then, predictions of the continuum model are tested against DEM data across different cases of inclined plane flow and planar shear flow with gravity, while varying parameters such as the size of the flow geometry, the flow speed and the initial conditions. We find that it is crucial to account for both driving forces to capture segregation dynamics across both flow geometries with a single set of parameters.

This paper gives, in the limit of infinite Froude number, a closed-form, analytical solution for steady, two-dimensional, irrotational, infinite-depth, free-surface, attached flow over a submerged tandem cascade of hydrofoils for arbitrary angle of attack, depth of submergence and interfoil separation. The multiply connected flow domain is conformally mapped to a concentric annulus in an auxiliary plane. The complex flow potential and its derivative, the complex velocity, are obtained in the auxiliary plane by considering their form at known special points in the flow and the required conformal mapping is determined by explicit integration, allowing accurate evaluation of various flow quantities including the lift on each foil. The circulation around the foils causes the foil array to act as a row of point vortices, or a shear layer, and so, for positive angles of attack, the flow speed at the free surface can substantially exceed the speed at depth, with the speeds simply related through the lift coefficient. Decreasing the interfoil separation decreases the disturbance to the free surface and greatly increases the lift per hydrofoil, thus allowing for the shallower operation of a hydrofoil array than of an isolated foil for a given lift requirement. Further, the flow over a hydrofoil array approaches its infinite depth form significantly more rapidly than that over an isolated foil. In contrast to the infinite-submergence case where a through-array flow can be imposed, in the finite submergence case, periodicity and the presence of the free surface mean that there is no net flow between the foils.

Turbulent boundary layers (TBLs) over surface perturbations like bumps with roughness – notably altering heat and mass transfer, drag, etc. – are prevalent in nature (mountains, dunes, etc.) and technology. We study a channel flow with a transverse bump on one wall superimposed with small-scale longitudinal grooves via direct numerical simulation (DNS) of incompressible flow. Turbulence statistics and dynamics are compared between grooved wall (GW) and smooth wall (SW) bumps. Streamwise spinning jets emanating from the crests’ corners alter the flow structure within the separation bubble (SB), extending the SB length by 30 % over that for SW, and have lingering effects far downstream. Grooves decrease skin friction but increase the bump's form drag by 25 %. In GW, the peaks of turbulence intensity and production decrease by 20 % and shift downstream, compared with SW. Three regions of negative production, found upstream as well as downstream of the bump, are explained in terms of two separate mechanisms: normal and shear productions. Separation upstream of the bump occurs always for GW, but intermittently for SW. Within the downstream SB, counter-rotating minibubbles form intermittently for SW but always for GW. Interestingly, a minibubble causes streamwise vorticity reversal of the upstream moving secondary flow around each crest corner. The wall pressure in GW is invariant in the spanwise direction and is explained in terms of its non-local nature and its connection with outer structures. The grooved bump unearths rich TBL flow physics – upstream separation, dynamics of the downstream minibubble, altered reattachment dynamics and negative production.

In 2017, Brosseau & Vlahovska (Phys. Rev. Lett, vol. 119, no. 3, 2017, p. 034501) found that, in a strong electric field, a weakly conductive, low-viscosity droplet immersed in a highly conductive, high-viscosity medium formed a lens shape, and liquid rings continuously detached from its equatorial plane and subsequently broke up into satellite droplets. This fascinating multiphase electrohydrodynamic (EHD) phenomenon is known as droplet equatorial streaming. In this paper, based on the unified lattice Boltzmann method framework proposed by Luo et al. (Phil. Trans. R. Soc. A Math. Phys. Engng Sci, vol. 379, no. 2208, 2021, p. 20200397), a novel lattice Boltzmann (LB) model is constructed for multiphase EHD by coupling the Allen–Cahn type of multiphase LB model and two new LB equations to solve the Poisson equation of the electric field and the conservation equation of the surface charge. Using the proposed LB model, we successfully reproduced, for the first time, the complete process of droplet equatorial streaming, including the continuous ejection and breakup of liquid rings on the equatorial plane. In addition, it is found that, under conditions of high electric field strength or significant electrical conductivity contrast, droplets exhibit fingering equatorial streaming that was unknown before. A power-law relationship is discovered for droplet total charge evolution and a theoretical model is then proposed to describe the droplet radius and height over time. The breakup of liquid rings is found to be dominated by capillary instability, while the breakup of liquid fingers is governed by the end-pinching mechanism. Finally, a phase diagram is constructed for fingering equatorial streaming and ring equatorial streaming, and a criterion equation is established for the phase boundary.

Coherent small-amplitude unsteadiness of the shock wave and the separation region over a canonical double cone flow, termed in literature as oscillation-type unsteadiness, is experimentally studied at Mach 6. The double cone model is defined by three non-dimensional geometric parameters: fore- and aft-cone angles ($\theta _1$ and $\theta _2$), and ratio of the conical slant lengths ($\varLambda$). Previous studies of oscillations have been qualitative in nature, and mostly restricted to a special case of the cone model with fixed $\theta _1 = 0^\circ$ and $\theta _2 = 90^\circ$ (referred to as the spike-cylinder model), where $\varLambda$ becomes the sole governing parameter. In the present effort we investigate the self-sustained flow oscillations in the $\theta _1$-$\varLambda$ parameter space for fixed $\theta _2 = 90^\circ$ using high-speed schlieren visualisation. The experiments reveal two distinct subtypes of oscillations, characterised by the motion (or lack thereof) of the separation point on the fore-cone surface. The global time scale associated with flow oscillation is extracted using spectral proper orthogonal decomposition. The non-dimensional frequency (Strouhal number) of oscillation is seen to exhibit distinct scaling for the two oscillation subtypes. The relationship observed between the local flow properties, instability of the shear layer, and geometric constraints on the flow suggests that an aeroacoustic feedback mechanism sustains the oscillations. Based on this understanding, a simple model with no empiricism is developed for the Strouhal number. The model predictions are found to match well with experimental measurements. The model provides helpful physical insight into the nature of the self-sustained flow oscillations over a double cone at high speeds.

In particle-laden turbulent wall flows, lift forces can influence the near-wall turbulence. This has been observed recently in particle-resolved simulations, which, however, are too expensive to be used in upscaled models. Instead, point-particle simulations have been the method of choice to simulate the dynamics of these flows during the last decades. While this approach is simpler, cheaper and physically sound for small inertial particles in turbulence, some issues remain. In the present work, we address challenges associated with lift force modelling in turbulent wall flows and the impact of lift forces in the near-wall flow. We performed direct numerical simulations of small inertial point particles in turbulent channel flow for fixed Stokes number and mass loading while varying the particle size. Our results show that the particle dynamics in the buffer region, causing the apparent particle-to-fluid slip velocity to vanish, raises major challenges for modelling lift forces accurately. While our results confirm that lift forces have little influence on particle dynamics for sufficiently small particle sizes, for inner-scaled diameters of order one and beyond, lift forces become quite important near the wall. The different particle dynamics under lift forces results in the modulation of streamwise momentum transport in the near-wall region. We analyse this lift-induced turbulence modulation for different lift force models, and the results indicate that realistic models are critical for particle-modelled simulations to correctly predict turbulence modulation by particles in the near-wall region.

The orientational dynamics of a spherical magnetic particle in linear shear flow subjected to an oscillating magnetic field in the flow plane is analysed in the viscous limit. The shear is in the $X$–$Y$ plane, the magnetic field is in the $X$ direction and the vorticity is perpendicular to the flow in the $Z$ direction. The relevant dimensionless groups are $\omega ^\ast$, the ratio of the frequency of the magnetic field and the strain rate, and $\varSigma$, the ratio of the magnetic and hydrodynamic torques. As $\varSigma$ is decreased, there is a transition from in-plane rotation, where the rotation is in the flow ($X$–$Y$) plane, to out-of-plane rotation, where the orientation vector is not necessarily in the $X$–$Y$ plane and the dynamics depends on the initial orientation. The particle rotation is phase-locked for in-plane rotation with discrete odd rotation number (number of rotations in one period of magnetic field oscillation), while the orbits are quasi-periodic with non-integer rotation number for out-of-plane rotation. For $\varSigma \gg 1$, regions of odd rotation number $n_o$ are bound by the lines $8 (n_o-1) \varSigma \omega ^\ast = 1$ and $8 (n_o+1) \varSigma \omega ^\ast = 1$, and there are discontinuous changes in the rotation number and mean and root-mean-square torque at these lines. For $\varSigma \ll 1$, the domains of in-plane rotation of finite width in the $\omega ^\ast$–$\varSigma$ plane extend into downward cusps at $\omega ^\ast = {1}/{2 n_o}$. The orbits are quasi-periodic between these domains, where the rotation is out of plane.

Microbes play a primary role in wide-ranging biogeochemical and physiological processes, where ambient fluid flows are responsible for cell dispersal as well as mixing of dissolved resources, signalling molecules and biochemical products. Determining the simultaneous (and often coupled) transport properties of actively swimming cells together with passive scalars is key to understanding and ultimately predicting these complex processes. In recent work, Ran & Arratia (J. Fluid Mech., vol. 988, 2024, A25) present the striking observation that dilute concentrations of swimming bacteria severely hinder scalar transport through Lagrangian vortex boundaries in a chaotic flow. Analysis of rotation-dominated regions suggests that local accumulation of bacteria enhances the strength of transport barriers and highlights the role of understudied elliptical Lagrangian coherent structures in bacterial and multicomponent transport.

Seafloor roughness profoundly influences the pattern and dynamics of large-scale oceanic flows. However, these kilometre-scale topographic patterns are unresolved by global numerical Earth system models and will remain subgrid for the foreseeable future. To properly represent the effects of small-scale bathymetry in analytical and coarse-resolution numerical models, we develop the stratified ‘sandpaper’ theory of flow–topography interaction. This model, which is based on the multilayer shallow-water framework, extends its barotropic antecedent to stratified flows. The proposed theory is successfully tested on the configuration representing the interaction of a zonal current with a corrugated cross-flow ridge.

Plane turbulent wall jets are traditionally considered to be composed of a turbulent boundary layer (TBL) topped by a half-free jet. However, certain peculiar features, such as counter-gradient momentum flux occurring below velocity maximum in experiments and numerical simulations, suggest a different structure of turbulence therein. Here, we hypothesize that turbulence in wall jets has two distinct structural modes, wall mode scaling on wall variables and free-jet mode scaling on jet variables. To investigate this hypothesis, experimental data from our wall jet facility are acquired using single hot-wire anemometry and two-dimensional particle image velocimetry at three nozzle Reynolds numbers 10 244, 15 742 and 21 228. Particle image velocimetry measurements with four side-by-side cameras capture the longest field of view studied so far in wall jets. Direct spatial spectra of these fields reveal modal spectral contributions to variances of velocity fluctuations, Reynolds shear stress, shear force, turbulence production, velocity fluctuation triple products and turbulent transport. The free-jet mode has wavelengths scaling on the jet length scale ${z_{T}}$, and contains two dominant submodes with wavelengths $5{z_{T}}$ and $2.5{z_{T}}$. The region of flow above the velocity maximum shows the presence of the outer jet mode whereas the region below it shows robust bimodal behaviour attributed to both wall and inner jet modes. Counter-gradient momentum flux is effected by the outer jet mode intruding into the region below velocity maximum. These findings support the hypothesis of wall and free-jet structural modes, and indicate that the region below velocity maximum could be much complex than a conventional TBL.

In this study we propose a novel data-driven reduced-order model for complex dynamics, including nonlinear, multi-attractor, multi-frequency and multiscale behaviours. The starting point is a fully automatable cluster-based network model (CNM) (Li et al., J. Fluid Mech., vol. 906, 2021, A21) that kinematically coarse grains the state with clusters and dynamically predicts the transitions in a network model. In the proposed dynamics-augmented CNM (dCNM) the prediction error is reduced with trajectory-based clustering using the same number of centroids. The dCNM is first exemplified for the Lorenz system and then demonstrated for the three-dimensional sphere wake featuring periodic, quasi-periodic and chaotic flow regimes. For both plants, the dCNM significantly outperforms the CNM in resolving the multi-frequency and multiscale dynamics. This increased prediction accuracy is obtained by stratification of the state space aligned with the direction of the trajectories. Thus, the dCNM has numerous potential applications to a large spectrum of shear flows, even for complex dynamics.

The non-Oberbeck–Boussinesq effects on the stability of a vertical natural convection boundary layer are investigated using the linearised disturbance equations for air flows up to a temperature difference of $\Delta T=100\,{\rm K}$. Based on the linear stability results, the neutral curve is shown to be sensitive to the choice of reference temperature. When evaluated using the film temperature $T_f$, a lower film Grashof number is required to trigger the linear instability for larger $\Delta T$. The relative contributions of shear and buoyant production to the perturbation kinetic energy budget reveals that the marginally unstable modes are amplified based on different mechanisms: for lower wavenumbers at relatively small Grashof number, the instability is driven by buoyancy; whereas for higher wavenumbers and larger Grashof number, the flow becomes unstable due to a shear instability. The use of reference temperature is found to scale the shear- and buoyant-driven instabilities differently so that no single reference temperature definition would collapse the neutral curves. The linear stability result further demonstrates that at a given Grashof number a higher temperature difference would give a larger amplification rate of the perturbation, which then leads to an earlier onset of the nonlinearities when evaluated at $T_f$. Finally, by comparing the amplification rates obtained from direct numerical simulation and the linear stability results, the extent of the linear regime is determined for $\Delta T = 100\,{\rm K}$.

In this study, the asymptotic solutions of the pressure variations induced by two trains passing each other in a tunnel are theoretically investigated. The one-dimensional inviscid compressible airflow is analysed, and two methods to obtain numerically exact solutions and $M_{H}$ expansion formulas for approximate equations are presented, where $M_{H}$ is the Mach number of the high-speed train. The pressure coefficient, corresponding to the maximum value of the magnitude of the pressure, is expressed as $|c_{p}|_{max}=|c_{p,min}|=[({R}/({1-R}))$ $(1+\alpha )^{2}+({R(1-R)}/{(1-2R)^{2}})(1-\alpha )^{2}]+O[M_{H}]$, where $c_{p,min}<0$, $\alpha =U_{L}/U_{H}$ and $U_{L}$ and $U_{H}$ denote the speeds of the low- and high-speed trains, respectively, and $R$ is the cross-sectional area ratio of the train to the tunnel. The theoretical results indicate the dependence of the speeds of the two trains on the pressure distribution and that the maximum magnitude of the asymptotic pressure for a fixed value of $M_{H}$ is obtained for $\alpha =1$ and $\alpha =0$ when $R< R_{c}$ and $R>R_{c}$, respectively, where $R_{c}$ denotes the critical blockage ratio. Because the airflow along the side of the low-speed train, induced by the low-speed train, is along the running direction of the high-speed train and reduces the relative velocity of the high-speed train as the two trains pass each other, $|c_{p}|_{max}$ for $\alpha =0$ is larger than $|c_{p}|_{max}$ for $\alpha =1$ when $R>R_{c}$. It is theoretically demonstrated that, as conventional high-speed railway systems satisfy $R< R_{c}$, a conservative pressure estimation can be established assuming $\alpha =1$.

Vortex-induced vibrations and galloping of an elastically mounted square cylinder are investigated for cylinder mass ratio m* = 2–50, damping ratio ζ = 0–1.0, mass-damping ratio m*ζ = 0–50 and flow reduced velocity Ur = 1–80. We home in on the effects of m*, ζ, m*ζ, $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ and $({m^\ast } + m_{ae}^\ast )\zeta$ on the critical reduced velocity Urc marking the onset of galloping, where $m_{a\textrm{0}}^\ast $ is the quiescent-fluid added mass ratio and $m_{ae}^\ast $ is the effective added mass ratio. Vibration responses, forces, vibration frequencies and added mass ratios are studied and discussed. The different branches of vortex-induced vibrations have different dependencies of $m_{ae}^\ast $ on Ur. The $m_{ae}^\ast $ in the initial branch is positive and drops rapidly with Ur, but that in the lower branch is negative and declines gently. In the galloping regime, $m_{ae}^\ast $ jumps from negative to positive at the onset of galloping, declining slightly with increasing Ur. Our results and prediction equations show that when ζ = 0, Urc is independent of m* for m* ≥ 5, albeit slightly higher for m* = 3. The latter is ascribed to mode competition. When ζ > 0, Urc linearly increases with increasing ζ. Detailed analysis substantiates that m*ζ or $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ does not serve as the unique criterion to predict the galloping occurrence. Here, we propose a new combined mass-damping parameter $({m^\ast } + m_{ae}^\ast )\zeta$ in the relationship between galloping onsets and structural properties, which successfully scales all data of Urc at different m* and ζ values.