Natural flyers are capable of producing excessive lift via a stabilized leading-edge vortex (LEV), which appears to linger above the wing for a longer duration than it could in an equivalent two-dimensional flow. Previous studies found this stabilization behaviour closely related to a spanwise flow along the LEV axis; however, it is still debatable how the spanwise flow influences the LEV stability. In this work, potential flow theory is adopted to model an LEV attached to a flat-plate wing. To account for the spanwise flow effect, we propose a finite-area sink (FAS) model which allows the dynamical interaction between co-located LEV and spanwise flow. Through linear stability analysis of the dynamical system associated with the LEV movement, we arrive at a stable spiral-sink type of equilibrium, which is the first mathematical evidence supporting LEV stabilization by spanwise flow. It is further concluded that the LEV stability can be enhanced by either increasing the strength or decreasing the cross-section area of the spanwise flow.

Convection from a buoyancy source distributed over a vertical wall has diverse applications, from the natural ventilation of buildings to the melting of marine-terminating glaciers which impacts on future sea level. A key challenge involves determining how the rate and mechanisms of turbulent heat transfer should be extrapolated across a range of scales. Ke et al. (J. Fluid Mech., vol. 964, 2023, A24) explore transitions in the turbulent flow dynamics using direct numerical simulation of a convective boundary layer at a heated vertical wall. A classical regime of heat transfer, consistent with previous laboratory experiments, gives way with increasing accumulation of buoyancy to an ultimate regime with enhanced heat transfer. The key to this transition lies in a near-wall sublayer, with a switch from laminar buoyancy-driven dynamics to a sublayer dominated by turbulence and shear instability from the mean flow.

A mathematical model for the effect of the spatial variation of the local evaporative flux on the evaporation of and deposition from a thin pinned particle-laden sessile droplet is formulated and solved. We then analyse the behaviour for a one-parameter family of local evaporative fluxes with the free parameter $n \, (>-1)$ that exhibits qualitatively different behaviours mimicking those that can be obtained by, for example, surrounding the droplet with a bath of fluid or using a mask with one or more holes in it to achieve a desired pattern of evaporation enhancement and/or suppression. We show that when $-1< n<1$ (including the special cases $n=-1/2$ of diffusion-limited evaporation into an unbounded atmosphere and $n=0$ of spatially uniform evaporation), all of the particles are eventually advected to the contact line, and so the final deposit is a ring deposit at the contact line, whereas when $n>1$ all of the particles are eventually advected to the centre of the droplet, and so the final deposit is at the centre of the droplet. In particular, the present work demonstrates that a singular (or even a non-zero) evaporative flux at the contact line is not an essential requirement for the formation of a ring deposit. In addition, we calculate the paths of the particles when diffusion is slower than both axial and radial advection, and show that in this regime all of the particles are captured by the descending free surface before eventually being deposited onto the substrate.

This paper proposes a low-divergence method to generate inhomogeneous and anisotropic turbulence. Based on the idea of correlation reconstruction, the proposed method uses the Cholesky decomposition matrix to re-establish turbulence correlation functions. This is in contrast with the time-consuming procedure used in conventional methods to solve eigenvalues and eigenvectors in each location required by coordinate transformation, thereby reducing the computational complexity and improving the efficiency of synthetic turbulence generation. The proposed method is based on the classical spectrum-based method that is widely used to generate homogeneous and isotropic turbulence. By adjusting the generation strategy of specific random vectors, inhomogeneous and anisotropic turbulence with a relatively low divergence level can be obtained in practice, with almost no additional computational burden. Two versions of the proposed method are presented: the inverter and shifter versions. Both the versions are highly efficient, easy to implement, and compatible with high-performance computing. This method is also suitable for providing high-quality initial or boundary conditions for scale-resolving turbulence simulations with large grid numbers (such as direct numerical simulations or large eddy simulations); it can be rapidly implemented using various open-source computational fluid dynamics codes or common commercial software.

The logarithmic law of the wall does not capture the mean flow when a boundary layer is subjected to a strong pressure gradient. In such a boundary layer, the mean flow is affected by the spatio-temporal history of the imposed pressure gradient; and accounting for history effects remains a challenge. This work aims to develop a universal mean flow scaling for boundary layers subjected to arbitrary adverse or/and favourable pressure gradients. We derive from the Navier–Stokes equation a velocity transformation that accounts for the history effects and maps the mean flow to the canonical law of the wall. The transformation is tested against channel flows with a suddenly imposed adverse or favourable pressure gradient, boundary layer flows subjected to an adverse pressure gradient, and Couette–Poiseuille flows with a streamwise pressure gradient. It is found that the transformed velocity profiles follow closely the equilibrium law of the wall.

A novel mild-slope equation is derived based on a manipulation of the cylindrical and Cartesian coordinate reference systems. The vertical profile of the velocity field is constructed by solving an approximate problem in cylindrical coordinates. This allows us to address the local derivatives on the bottom profile along a constant-slope line. This formulation is as opposed to the Cartesian-based mild-slope equations in terms of which the profile is constructed by assuming a constant depth. An angular profile is derived for the three-dimensional case on a sloping plane beach. For the two-dimensional case, a mild-slope polar-Cartesian equation is derived, for which an improved linear dispersion relation is reconstructed. This is accomplished due to the inclusion of first-order derivatives of the local bottom profile. The coefficients of the polar-Cartesian mild-slope equation contain the derivatives of the bottom profile up to third order as opposed to second-order derivatives in the Cartesian-based equations. The equation is derived by applying the variational principle to the Cartesian Lagrangian when formulated as a function of the profile in polar coordinates. It is then compared with existing models of the mild-slope equation for simulations of two-dimensional test cases and a quasi-three-dimensional case, which have known analytical solutions. Our modified equation exhibits better matching to the exact solutions for a majority of the investigated cases.

Axisymmetric numerical simulations of the hydrodynamics around rising bubbles are performed in order to investigate the impact of surfactants on the bubble dynamics. Surfactants are assumed to be insoluble. The transport of the adsorbed surfactants is computed along the deforming surface at large surface Péclet number, and Marangoni stresses are taken into account. This simulation model leads to the stagnant-cap regime, with partially immobile interfaces. A parametric study is performed on cases at given Archimedes number, by varying the degree of contamination (Marangoni number) but maintaining a nearly constant Eötvös number. The presence of surfactants affects the rise velocity for oblate bubbles less than for spherical bubbles: the increase of the drag coefficient, due to interface contamination, is mitigated by a lower bubble deformation. When the cap angle $\theta _{cap}$ belongs to the southern hemisphere, the aspect ratio $\chi$ is found to decrease with contamination: the dynamic pressure responsible for the bubble distortion is lowered, related to the decline of kinetic energy. As soon as $\theta _{cap}$ lies in the northern hemisphere, the pressure stress causing distortion becomes independent on $\theta _{cap}: \chi$ no longer evolves with contamination, and already matches the prediction for fully immobile interfaces. Mass transfer of a passive scalar across the contaminated interface is also analysed. Surprisingly, the Sherwood number $Sh$ is found to follow the same law as for spherical shapes (Kentheswaran et al., Intl J. Heat Mass Transfer, vol. 198, 2022, 123325), allowing us to predict the decrease in $Sh$ due to contamination. These results reveal the couplings between interface immobilisation, bubble deformation, rise velocity and interfacial mass transfer.

This work aims at studying the mechanisms behind the occurrence of extreme dissipation events in a channel flow, identifying nonlinear optimal perturbations as potential precursors of these events. Nonlinear optimal perturbations with respect to a generic turbulent instantaneous snapshot are computed for the first time using a direct-adjoint algorithm in the channel flow at $Re_{\tau }\approx 180$. The resulting initial perturbation displays the upstream tilting characteristic of Orr's mechanism and is positioned along the interfaces between two opposite-sign velocity streaks of the pre-existing turbulent field. Such a perturbation induces a sudden breakdown of the pre-existing structures and a heavier tail in the dissipation probability density function distribution. Different mechanisms are at play during this process: the high shear present at the interface between coherent low- and high-momentum regions is exploited to break down the larger structures and drive energy to small scales. This energy cascade is fed by an enhanced lift-up effect that produces intense streaks near the wall. It is found that the optimal perturbation grows exponentially during the first phase of its evolution reflecting the existence of a secondary modal instability of the streaks. To corroborate the results, the conditional spatiotemporal proper orthogonal decomposition (POD) analysis of Hack & Schimdt (J. Fluid Mech., vol. 907, 2021, A9) is performed both in the perturbed and in the unperturbed flow, showing a clear agreement between the two cases and with the reference study. Thus, the optimal perturbation at initial time can be considered as a precursor of extreme events.

Turbulent plane-Couette flow suspended with finite-size spheroidal particles is studied using fully particle-resolved direct numerical simulations. The effects of particle aspect ratio on turbulent arguments and particle statistics are explored, leading to the same conclusions as the previous experimental findings (Wang et al., J. Fluid Mech., vol. 937, 2022, A15). By performing stress analysis, we find that the presence of particles introduces extra stresses to the system and accounts for the global drag increases. The particle-laden flow cases exhibit spectra that are consistent with the scalings $k^{-5/3}$ and $k^{-3}$ in the large and small scales, respectively. While the $k^{-3}$ scaling observed in the particle-laden flow is reminiscent of bubbly flow, an examination of the particle Reynolds number suggests that the mechanism responsible may not be attributable to the pseudo-turbulence induced by particles as in the case of bubbles. In the view of particle statistics, we observe that spherical and non-spherical particles cluster preferentially in the near-wall and the bulk region, respectively, and that the orientations of non-spherical particles are affected by their aspect ratios, especially in the near-wall region. The present numerical results, combined with previous experimental findings in Wang et al. (J. Fluid Mech., vol. 937, 2022, A15), provide in-depth information on both the fluid and the particle phase, contributing to a better understanding of particle suspension in shear flows.

Green hydrogen is expected to have a key role in the transition towards a carbon-neutral society, particularly in the transportation sector. Exploration of novel solutions for the direct injection of hydrogen in internal combustion engines (ICEs) is a main topic for both academia and industry. Here, the authors explore the gas-dynamic and mixing features of hydrogen under-expanded jets exiting from non-axisymmetric cross-sections, with the aim to provide guidelines for designing novel generations of ICE hydrogen injectors. Triangular and star shapes have been compared with elliptical and rectangular sections with different aspect ratios. Differences in the shock wave systems are reported, and explanations of the gas-dynamic mechanisms developed by each section are proposed. Non-uniformity in the radial expansion of the jet boundary has been noticed in all of the non-axisymmetric sections, leading to an axis switching of the jet in some cases. Differences in the radial mixing and axial jet penetration have been reported too, and a robust correlation with the vorticity distribution along the jet boundary has been observed.

We present novel numerical simulations investigating the bag breakup of liquid droplets. We first examine the viscous effect on the early-time drop deformation, comparing with theory and experiment. Next, a bag film forms at late time and is susceptible to spurious mesh-induced breakup in numerical simulations, which has prevented previous studies from reaching grid convergence of fragment statistics. We therefore adopt the manifold death (MD) algorithm which artificially perforates thin films once they reach a prescribed critical thickness independent of the grid size, controlled by a numerical parameter $L_{sig}$. We show grid convergence of fragment statistics when utilising the MD algorithm, and analyse the fragment behaviour and bag film disintegration mechanisms including ligament breakup, node detachment and rim destabilisation. Our choice of the critical thickness parameter $L_{sig}$ is limited by numerical constraints and thus has not been matched to experiment or theory; consequently, the current simulations yield critical bag film perforation thicknesses larger than experimentally observed. The influence of the MD algorithm configuration on the bag breakup phenomena and statistics will be investigated in future work. We also study the effects of moderate liquid Ohnesorge number ($0.005 \leqslant Oh \leqslant 0.05$) on the bag breakup process and fragment statistics, where a non-monotonic dependency of the average diameter of bag film fragments on $Oh$ is found. These results highlight the utility of the MD algorithm in multiphase simulations involving topological changes, and pave the way for physics-based numerical investigations into spume generation at the air–sea interface.

This work discusses modons, or dipolar vortices, propagating along sloping topography. Two different regimes exist, which are studied separately using the surface quasi-geostrophic equations. First, when the modon propagates in the direction opposite to topographic Rossby waves, steady solutions exist and a semi-analytical method is presented for calculating these solutions. Second, when the modon propagates in the same direction as the Rossby waves, a wave wake is generated. This wake removes energy from the modon, causing it to decay slowly. Asymptotic predictions are presented for this decay and found to agree closely with numerical simulations. Over long times, decaying vortices are found to break down due to an asymmetry resulting from the generation of waves inside the vortex. A monopolar vortex moving along a wall is shown to behave in a similar way to a dipole, though the presence of the wall is found to stabilise the vortex and prevent the long-time breakdown. The problem is equivalent mathematically to a dipolar vortex moving along a density front, hence our results apply directly to this case.

This study presents the results of a numerical simulation of two horizontal flows of miscible magnetic and non-magnetic fluids at low Reynolds numbers in a vertical uniform magnetic field. The problem is solved by taking into account the dependence of the viscosity and magnetization of the fluid on the concentration of the magnetic phase, and the dependence of the magnetic field on the concentration. Four flow modes are found: the diffusion mixing mode with a flat diffusion front, the wave mode and two different plug flow modes. In the first of them, the growing wave instability forms the plugs, whereas in the second, the growing magnetostatic instability does. A combination of dimensionless criteria is found that determines the transition from one mode to another. The dependences of the phase velocity of the waves on the diffusion front and the period of the oscillations of the front near the point of the confluence of the two flows on dimensionless criteria are found.

Fluid-inertia torque remarkably affects the orientation of non-spherical particles in Newtonian flows whereas this torque induced by convective fluid inertia in particle-laden pseudo-plastic flows is still unknown. In the present study we numerically investigate the fluid-inertia torque on a neutrally buoyant spheroid in the Carreau-type pseudo-plastic fluid flows at finite Reynolds numbers with the immersed boundary method. The results show that compared with the fluid-inertia torque in Newtonian flows, the magnitude of the fluid-inertia torque on spheroids is remarkably attenuated by the shear-thinning rheology in pseudo-plastic fluid flows. The deviation of fluid-inertia torque between pseudo-plastic and Newtonian flows is more significant with decreasing Reynolds numbers, indicating the importance of the effect of shear-thinning rheology at small Reynolds numbers. Moreover, the spheroid rotation rate is reduced in pseudo-plastic fluids, and the equilibrium orientation of oblate spheroids changes non-monotonically with the shear-thinning effect in the linear shear flow of pseudo-plastic fluids. The present findings imply the importance of the effect of shear-thinning rheology on the torques of spheroids, which could be potentially applied for the control of particle orientations in pseudo-plastic fluids in the future.

We use experiments to explore the effect of surfactants on bubble-induced turbulence (BIT) at different scales, considering how the bubbles affect the flow kinetic energy, anisotropy and extreme events. To this end, high-resolution particle shadow velocimetry measurements are carried out in a bubble column in which the flow is generated by bubble swarms rising in water for two different bubble diameters (3 and 4 mm) and moderate gas volume fractions (0.5 %–1.3 %). We use tap water as the base liquid and add 1-Pentanol as an additional surfactant with varying bulk concentration, leading to different bubble shapes and surface boundary conditions. The results reveal that with increasing surfactant concentration, the BIT generated increases in strength, even though bubbles of a given size rise more slowly with surfactants. We also find that the level of anisotropy in the flow is enhanced with increasing surfactant concentration for bubbles of the same size, and that for the same surfactant concentration, smaller bubbles generate stronger anisotropy in the flow. Concerning the intermittency quantified by the normalized probability density functions of the fluid velocity increments, our results indicate that extreme values in the velocity increments become more probable with decreasing surfactant concentration for cases with smaller bubbles and low gas void fraction, while the effect of the surfactant is much weaker for cases with larger bubble and higher void fractions.

The eigenmodes and eigenfrequencies of two-dimensional elastic structures in contact with a liquid are investigated within the linear theory of hydroelasticity. The shapes of the structural vibrations and the hydrodynamic loads acting on the structure are calculated at the same time. The wet modes are obtained as superpositions of the dry modes of the structure, where the coefficients are solutions of a matrix equation with the hydrodynamic loads being represented by an added-mass matrix. The added-mass matrix of a homogenous elastic plate is calculated analytically through Bessel functions. Added-mass matrices of complex structures are obtained using representations of the dry modes through dry modes of the homogeneous plate of the same length. Relations between wet and dry modes and their frequencies depending on parameters of the problems are studied. The structure could be made of several plates, connected or not, completely or partially wetted, and of any variable thickness and rigidity. The main contribution to a wet mode comes from the corresponding dry mode. The wet frequency is below the corresponding dry frequency with their ratios being weakly dependent on the properties of the structure. This finding is new. The obtained added-mass matrices are suggested to use in problems of hydroelastic slamming for any geometry of impacting elastic body and any distributions of its elastic characteristics. The matrices can be also used to design an interface between hydrodynamic and structural solvers in numerical analysis of hydroelastic slamming.

The mean flow behaviour of a turbulent boundary layer over rough walls is expected to exhibit symmetries that govern the flow dynamics. In particular, when roughness elements are arranged in a spanwise symmetric manner, the mean flow above them should also exhibit spanwise symmetry. This symmetrical consideration has garnered substantial empirical support. We conduct direct numerical simulations (DNS) of flow over aligned cube arrays to test such symmetry considerations further. We vary the surface coverage density from 0.25 % to 6.25 %, and employ an averaging time of about 100 large-eddy turnover times, which is longer than the typical averaging time in prior DNS studies of rough-wall boundary layers. The results suggest the presence of spanwise asymmetry in the mean flow. Specifically, we observe the development of a prominent secondary vortex on one side of the cubical roughness, accompanied by a relatively smaller secondary vortex on the other side. This asymmetry becomes most pronounced when the surface coverage density is approximately 0.59 %, and diminishes as the coverage density approaches either a low or a high value. We also establish that this mean flow asymmetry is robust across variations in the domain size, the initial condition, and the placement of the cubes in the spanwise direction.

The global linear stability analysis for the magnetohydrodynamic liquid metal flow past an insulated sphere subjected to a constant streamwise magnetic field is investigated in the range of the Reynolds number $Re\leq 400$ and the interaction number $N\leq 40$ coupled with direct numerical simulations, where $N$ stands for strength of the electromagnetic force. The stability of the steady axisymmetric base flow to independent time-azimuthal modes is discussed. Five critical curves associated with various wake transitions are obtained in the $\{Re, N\}$ phase diagram. These critical curves reveal the stabilising effect of a weak magnetic field, the destabilising effect of a strong magnetic field and re-stabilising effect of a much stronger magnetic field. To explore the impact of the magnetic field on flow instability, a sensitivity analysis utilizing an adjoint method is performed for the first regular bifurcation. Sensitivity functions of growth rate to base-flow modifications and Lorentz force are defined to identify the region that has the most significant influence on flow instability, such as the recirculation region responsible for the stabilising effect at a weak magnetic field and the shear layer region responsible for the destabilising effect at a strong magnetic field. Furthermore, a competition between the stabilising and shear destabilising effects of the magnetic field is discussed. This analysis provides valuable insights into the non-monotonic effect of the magnetic field on flow instability.

The wetting effect has attracted great scientific interest because of its natural significance as well as technical applications. Previous models mostly focus on one-component fluids or binary immiscible liquid mixtures. Modelling of the wetting phenomenon for multicomponent and multiphase fluids is a knotty issue. In this work, we present a thermodynamically consistent diffuse interface model to describe the wetting effect for ternary fluids, as an extension of Cahn's theory for binary fluids. In particular, we consider both immiscible and miscible ternary fluids. For miscible fluids, we validate the equilibrium contact angle and the thermodynamic pressure with Young's law and the Young–Laplace equation, respectively. Distinct flow patterns for dynamic wetting are presented when the surface tension and the viscous force dominate the wetting effect. For immiscible ternary fluids, we manipulate the wettability of two contact droplets deposited on a solid substrate according to three scenarios: (I) both droplets are hydrophilic; (II) a hydrophilic droplet in contact with a hydrophobic one; (III) both droplets are hydrophobic. The contact angles at each triple junction from the simulations are compared with Young's contact angle and Neumann's triangle rule. Simulations for the validation of our work are performed in two and three dimensions. In addition, we model the evaporation process of a ternary droplet and obtain the same power law as that of previous experiments. Our model allows one to relate the interfacial energies with surface composition, enabling the modelling of the coffee-ring phenomenon in further perspective.

This paper reports several new classes of unstable recurrent solutions of the two-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions are in many ways analogous to recurrent solutions of the Navier–Stokes equation which are often referred to as exact coherent structures. In particular, we find that recurrent solutions of the Euler equation are dynamically relevant: they faithfully reproduce large-scale flows in simulations of turbulence at very high Reynolds numbers. On the other hand, these solutions have a number of properties which distinguish them from their Navier–Stokes counterparts. First of all, recurrent solutions of the Euler equation come in infinite-dimensional continuous families. Second, solutions of different types are connected, e.g. an equilibrium can be smoothly continued to a travelling wave or a time-periodic state. Third, and most important, they are only weakly unstable and, as a result, fully developed turbulence mimics some of these solutions remarkably frequently and over unexpectedly long temporal intervals.