We study the statistical properties of scalar fields undergoing reversible chemical reactions in a turbulent environment by means of numerical simulations. To produce strong chemical fluctuations in a wide region of the domain, an original flow configuration has been proposed, where the species are supplied from buffer boundaries with adjustable thickness, while the flow is developed homogeneous and isotropic turbulence in a periodic domain. With the presence of the mean scalar gradient in the bulk region, the strength of turbulent advection is comparable with the chemical source, which is quantified by the Damköhler number. Our analysis focuses on the global and spatial properties of the reactive scalars in their statistically steady regime. We show how for the case of a second-order reaction such features can be connected to the properties of a non-reactive scalar field advected in the same system. Analytical predictions of the scalar moments in the fast reaction regime agree satisfactorily with the direct numerical simulation results. In comparison with the existing results of the isotropic turbulence case, we conclude that the scalar correlation is jointly determined by both the chemical source and the flow configuration. Moreover, the chemical reaction also plays an important role in determining the scalar energy spectra.

Modelling of fluid–particle interactions is a major area of research in many fields of science and engineering. There are several techniques that allow modelling of such interactions, among which the coupling of computational fluid dynamics (CFD) and the discrete element method (DEM) is one of the most convenient solutions due to the balance between accuracy and computational costs. However, the accuracy of this method is largely dependent upon mesh size, where obtaining realistic results always comes with the necessity of using a small mesh and thereby increasing computational intensity. To compensate for the inaccuracies of using a large mesh in such modelling, and still take advantage of rapid computations, we extended the classical modelling by combining it with a machine learning model. We have conducted seven simulations where the first one is a numerical model with a fine mesh (i.e. ground truth) with a very high computational time and accuracy, the next three models are constructed on coarse meshes with considerably less accuracy and computational burden and the last three models are assisted by machine learning, where we can obtain large improvements in terms of observing fine-scale features yet based on a coarse mesh. The results of this study show that there is a great opportunity in machine learning towards improving classical fluid–particle modelling approaches by producing highly accurate models for large-scale systems in a reasonable time.

Viscous contact problems describe the time evolution of fluid flows in contact with a surface from which they can detach and reattach. These problems are of particular importance in glaciology, where they arise in the study of grounding lines and subglacial cavities. In this work, we propose a novel numerical method for solving viscous contact problems based on a mixed formulation with Lagrange multipliers of a variational inequality involving the Stokes equations. The advection equation for evolving the geometry of the domain occupied by the fluid is then solved via a specially-built upwinding scheme, leading to a robust and accurate algorithm for viscous contact problems. We first verify the method by comparing the numerical results to analytical results obtained by a linearised method. Then we use this numerical scheme to reconstruct friction laws for glacial sliding with cavitation. Finally, we compute the evolution of cavities from a steady state under oscillating water pressures. The results depend strongly on the location of the initial steady state along the friction law. In particular, we find that if the steady state is located on the downsloping or rate-weakening part of the friction law, then the cavity evolves towards the upsloping section, indicating that the downsloping part is unstable.

Capillary flow of liquids plays a key role in many applications including lab-on-a-chip devices, heat pipes and printed electronics manufacturing. Open rectangular microchannels often appear in these applications, with the lack of a top resulting in a complex free-surface morphology and evaporation. In this work we develop a theoretical model based on lubrication theory and kinetically limited evaporation to examine capillary flow of evaporating liquid solutions in open rectangular microchannels connected to circular reservoirs. The model accounts for the complex free-surface morphology, solvent evaporation, Marangoni flows due to gradients in solute concentration and temperature and finite-size reservoir effects. Significant differences are predicted in flow behaviour between pure liquids and liquid solutions due to solvent evaporation and solute transport. Marangoni flows are found to promote more uniform solute deposition patterns after solvent evaporation. Model predictions of meniscus position evolution are in good agreement with prior capillary-flow experiments of aqueous poly(vinyl alcohol) solutions in the presence of evaporation. The model reveals that the principal mechanism through which evaporation influences the meniscus position in the experiments is the increase in viscosity with solute concentration.

Triggering is the process by which a linearly stable thermoacoustic system can reach self-sustained oscillations. This nonlinear phenomenon is activated only for sufficiently large amplitudes of perturbations to the equilibrium state. In this work, using a nonlinear variational optimisation method coupled with energy bisection, we compute the minimal thresholds for triggering in the Rijke tube. In particular, extending previous works, we take into account the effect of the time delay by optimising not only the perturbations at initial time, but also the velocity at the hot-wire position in the time-delay interval. We found that, for sufficiently large time delays, the nonlinearity linked to the delayed flow velocity bears a strong potential for energy growth, leading to transient amplifications of the energy reaching ${O}(10^{2})$, two orders of magnitude larger than those reported in previous studies. Notably, the gain increases with the time delay, but decreases with the initial energy of the perturbation, thus reaching very high values close to the triggering threshold of the system. The minimal energy for triggering self-sustained oscillations achieves energy values as low as ${O}(10^{-4})$, two orders of magnitude smaller than previous estimates. This indicates that, for thermoacoustic systems characterised by a non-negligible time delay, taking into account the effect of the time-delayed variables, as well as the system nonlinearity, is crucial for correctly evaluating the triggering energy thresholds.

The onset and evolution of the dynamic stall vortex (DSV) are analysed by means of large eddy simulations of an SD7003 aerofoil undergoing periodic plunging motion in a transitional Reynolds number flow ($Re =6\times 10^{4}$). Interactions between upstream propagating Kelvin–Helmholtz instabilities and a shear layer formed at the leading edge trigger flow separation. The former appear to be related to acoustic waves scattered at the trailing edge due to initial vortex shedding. Two freestream Mach numbers ($M_{\infty }=0.1$ and $0.4$) are employed to examine the flow differences due to compressibility variations. The existence of a common timing for the acoustic perturbations in both flows suggests a possible Mach number invariance for the birth of the Kelvin–Helmholtz instability. Increasing compressibility, however, induces earlier spanwise fluctuations, higher flow three-dimensionality and a weaker and more diffuse DSV, which is formed further downstream of the leading edge and has lower residency time. In order to better characterize the onset of the DSV, two empirical criteria are assessed: the leading edge suction parameter and the chord-normal shear layer height. Results demonstrate a higher robustness of the latter with respect to Mach number variations. Modal decomposition, performed with both the classical dynamic mode decomposition (DMD) and its multi-resolution variant (mrDMD), highlights key trends and demonstrates the capacity of the mrDMD to extract physically meaningful flow structures related to the stall onset. Such detailed characterization of the shear layer can be used for a systematic exploration of flow control strategies for unsteady aerofoils.

Recently, many authors have investigated the origin and growth of turbulent bands in shear flows, highlighting the role of streaks and their inflectional instability in the process of band generation and sustainment. Recalling that streaks are created by an optimal transient growth mechanism, and motivated by the observation of a strong increase of the disturbance kinetic energy corresponding to the creation of turbulent bands, we use linear and nonlinear energy optimisations in a tilted domain to unveil the main mechanisms allowing the creation of a turbulent band in a channel flow. Linear transient growth analysis shows an optimal growth for wavenumbers having an angle of approximately $35^\circ$, close to the peak values of the premultiplied energy spectra of direct numerical simulations. This linear optimal perturbation generates oblique streaks, which, for a sufficiently large initial energy, induce turbulence in the whole domain, due to the lack of spatial localisation. However, spatially localised perturbations obtained by adding nonlinear effects to the optimisation or by artificially confining the linear optimal to a localised region in the transverse direction are characterised by a large-scale flow and lead to the generation of a localised turbulent band. These results suggest that two main elements are needed for inducing turbulent bands in a tilted domain: (i) a linear energy growth mechanism, such as the lift-up, for generating large-amplitude flow structures, which produce inflection points; (ii) spatial localisation, linked to the presence or generation of large-scale vortices. We show that these elements alone generate isolated turbulent bands also in large non-tilted domains.

We obtain an analytical bound on the non-dimensional mean vertical convective heat flux $\langle w T \rangle$ between two parallel boundaries driven by uniform internal heating. We consider two configurations. In the first, both boundaries are held at the same constant temperature and $\langle wT \rangle$ measures the asymmetry of the heat fluxes escaping the layer through the top and bottom boundaries. In the second configuration, the top boundary is held at constant temperature, the bottom one is perfectly insulating, and $\langle wT \rangle$ is related to the difference between the horizontally-averaged temperatures of the two boundaries. For the first configuration, Arslan et al. (J. Fluid Mech., vol. 919, 2021, p. A15) recently provided numerical evidence that Rayleigh-number-dependent corrections to the only known rigorous bound $\langle w T \rangle \leq 1/2$ may be provable if the classical background method is augmented with a minimum principle stating that the fluid's temperature is no smaller than that of the top boundary. Here, we confirm this fact rigorously for both configurations by proving bounds on $\langle wT \rangle$ that approach $1/2$ exponentially from below as the Rayleigh number is increased. The key to obtaining these bounds is inner boundary layers in the background fields with a particular inverse-power scaling, which can be controlled in the spectral constraint using Hardy and Rellich inequalities. These allow for qualitative improvements in the analysis that are not available to standard constructions.

A lubrication analysis is presented for the resistances between permeable spherical particles in near contact, $h_0/a\ll 1$, where $h_0$ is the minimum separation between the particles, and $a=a_1 a_2/(a_1+a_2)$ is the reduced radius. Darcy's law is used to describe the flow inside the permeable particles and no-slip boundary conditions are applied at the particle surfaces. The weak permeability regime $K=k/a^{2} \ll 1$ is considered, where $k=\frac {1}{2}(k_1+k_2)$ is the mean permeability. Particle permeability enters the lubrication resistances through two functions of $q=K^{-2/5}h_0/a$, one describing axisymmetric motions, the other transverse. These functions are obtained by solving an integral equation for the pressure in the near-contact region. The set of resistance functions thus obtained provide the complete set of near-contact resistance functions for permeable spheres and match asymptotically to the standard hard-sphere resistances that describe pairwise hydrodynamic interactions away from the near-contact region. The results show that permeability removes the contact singularity for non-shearing particle motions, allowing rolling without slip and finite separation velocities between touching particles. Axisymmetric and transverse mobility functions are presented that describe relative particle motion under the action of prescribed forces and in linear flows. At contact, the axisymmetric mobility under the action of oppositely directed forces is $U/U_0=d_0K^{2/5}$, where $U$ is the relative velocity, $U_0$ is the velocity in the absence of hydrodynamic interactions and $d_0=1.332$. Under the action of a constant tangential force, a particle in contact with a permeable half-space rolls without slipping with velocity $U/U_0=d_1(d_2+\log K^{-1})^{-1}$, where $d_1=3.125$ and $d_2=6.666$; in shear flow, the same expression holds with $d_1=7.280$.

A physical description of the flow mechanisms that govern the distribution of the wall-pressure fluctuations over the surface of a serrated trailing edge is proposed. Three main mechanisms that define the variation of turbulent pressure fluctuations across the serrated edge are discussed and semi-empirical models are formulated accordingly. It is shown that the intensity of the wall-pressure fluctuations increases at the tips under the effect of an increased convective velocity as a result of sidewise momentum diffusion. Furthermore, the change of impedance across the edge causes a local reduction of the pressure fluctuations in the vicinity of the trailing edge. Finally, aerodynamic loading over the serrations due to the non-symmetric flow created at different angles of attack establishes secondary flow patterns that induce higher wall-pressure fluctuations over the serration edges. The latter effect is present only for serrations under high aerodynamic loading, while the former ones are observed under any conditions. Semi-empirical models are formulated for predicting the variation of the wall-pressure fluctuations over the serration surface based on the three physical mechanisms described. These models are calibrated and compared against experiments conducted on a symmetric airfoil model at high Reynolds numbers.

In this paper, we study the role of shear-induced migration and particle-induced normal stresses in the formation and stability of a particle-laden, gravity-driven shallow flow. We first examine the modification of the base-state Nusselt flow due to the underlying microstructure, how shear-induced migration leads to viscosity stratification. We inspect the development of the base state via the boundary layer formation in the ‘shallow’ limit and find a reduction in entrance length with increasing bulk particle concentration and an increase in entrance length with increasing Péclet number ($Pe_p = \dot {\gamma } a^2 / D_0$, where $\dot{\gamma}$ is the average shear rate, a is the particle size and $D_0$ is the single particle diffusivity). A linear stability analysis is then performed on the fully developed state to identify two modes of instability typically found in gravity-driven falling films – the long-wave surface and the short-wave shear modes. We find that when the associated Péclet number is $Pe_p \ll 1$, increasing bulk particle volume fraction delays the onset of instability for both the surface mode and shear mode. However, with $Pe_p = {O}(1)$, we find an enhancement in both modes of instability. We also find that, beyond a critical Péclet number, for a fixed particle volume fraction, the surface mode is unstable even in the absence of fluid inertia. The enhanced destabilisation is attributed to the combined effects of base-state viscosity stratification and momentum forcing via particle concentration perturbations. We also show that the physics behind the enhancement of instability is independent of the choice of the constitutive model used to describe the dynamics of the particle phase, provided the chosen model has elements of shear-induced migration.

The motion of finite-length cylindrical rods moving near a planar rigid surface is a scenario common across many engineering and natural settings. We study the low-Reynolds-number flow around finite rods that are allowed to rotate or translate in directions perpendicular or parallel to the plane. We develop a three-dimensional lubrication theory to characterize the pressure and hydrodynamic resistances of the cylinders through a special consideration of the cylinder's end effects. In addition, we use three-dimensional numerical simulations to solve these Stokes flows for cylinders of varying lengths and with varying gap sizes between the cylinder and plane, and the numerical results support the developed analytical descriptions. We also use visualizations of the flow to provide qualitative insights and rationalize the effect of the ends on the dynamics of the cylinders. The numerical simulations and theoretical predictions show good agreement in the long (isolated ends) and short (disk-like) limits.

We numerically investigate turbulent Rayleigh–Bénard convection through two immiscible fluid layers, aiming to understand how the layer thickness and fluid properties affect the heat transfer (characterized by the Nusselt number $\mbox {Nu}$) in two-layer systems. Both two- and three-dimensional simulations are performed at fixed global Rayleigh number $\mbox {Ra}=10^8$, Prandtl number $\mbox {Pr}=4.38$ and Weber number $\mbox {We}=5$. We vary the relative thickness of the upper layer between $0.01 \le \alpha \le 0.99$ and the thermal conductivity coefficient ratio of the two liquids between $0.1 \le \lambda _k \le 10$. Two flow regimes are observed. In the first regime at $0.04\le \alpha \le 0.96$, convective flows appear in both layers and $\mbox {Nu}$ is not sensitive to $\alpha$. In the second regime at $\alpha \le 0.02$ or $\alpha \ge 0.98$, convective flow only exists in the thicker layer, while the thinner one is dominated by pure conduction. In this regime, $\mbox {Nu}$ is sensitive to $\alpha$. To predict $\mbox {Nu}$ in the system in which the two layers are separated by a unique interface, we apply the Grossmann–Lohse theory for both individual layers and impose heat flux conservation at the interface. Without introducing any free parameter, the predictions for $\mbox {Nu}$ and for the temperature at the interface agree well with our numerical results and previous experimental data.

This study validated a two-phase compressible flow model considering elasto-plastic porous foams. The numerical data were compared with the previous experimental results in terms of the interaction between the planar shock wave and the porous foams and the mitigation effect of a porous foam filling a straight tube on a blast wave. The porous foams in the shock tube interacted with a planar shock wave. The drag between the shocked air and the foams reduced the shock wave strength. Moreover, the flexible foam was significantly deformed by the shock wave. The validation results confirmed good agreement and consistency between the numerical and experimental data. The mitigation effect on the blast wave caused by a high explosive, where the main parameter for comparison was the location of a rigid porous foam layer inside the straight tube, was investigated. In the first case, the porous foam plate was placed on the floor, whereas in the second case, the porous foam plates were placed on the floor, sidewalls and ceiling. The total energy transferred between the porous foam and the shocked air was computed to quantitatively understand the mitigation mechanism of the porous foam on the blast wave. The heat transfer was a dominant factor for the energy transfer from the shocked air to the porous foams. The second case further mitigated the blast wave outside, and the increment of the interface area of the air/porous foam greatly affected the blast wave mitigation.

The effect of electrostatic forces on the dispersion of small like-charged inertial particles transported by a homogeneous isotropic turbulent flow is analysed by means of direct numerical simulations coupled with the Lagrangian tracking of charged particles. The results show that particle dispersion decreases for increasing charge. Since turbulent particle dispersion is related to the intensity of particle agitation and to the Lagrangian particle integral time scale, we analyse the effects of electrostatic forces on these two quantities. Particle agitation decreases for increasing particle charge. In fact, particle kinetic energy is not directly modified by electrostatic forces, which are conservative, it is rather the particle entrainment by fluid turbulence that is modified. To support this claim, one shows that the fluid–particle velocity covariance is destructed by electrostatic forces, a destruction can be predicted by drawing an analogy between inter-particle collisions and Coulomb collisions. As expected, electrostatic forces decorrelate particle velocities leading to a decrease of the Lagrangian particle integral time scale. Finally, one shows that the analogy with inter-particle collisions allows us to predict the reduction of particle dispersion.

An unsteady analytical solution is proposed to predict the spreading rate of the crown generated by an impacting droplet onto wetted walls. The modelling strategy is based on the direct integration of the boundary layer correction into the potential flow solution that leads to the well-established square-root time dependence. The original potential flow has the structure of an unsteady, stagnation point flow with decaying strength. For initial strengths of the potential flow $a_0 \geqslant 100\ {\rm s}^{-1}$, we find that a self-similar solution can also be obtained for the boundary layer in the variables $\left (r \sqrt {a(t)/(\nu t)}, z \sqrt {a(t)/(\nu t)} \right )$. The self-similarity of the solution enables a straightforward estimation of momentum losses during the spreading of the liquid layer along the wall. The proposed modelling approach yields an excellent agreement with experiments during the entire spreading phase. Moreover, it enables a smooth transition from the inertia-driven to the shear-controlled regime of crown propagation. In general, the analysis shows that momentum losses arising from viscous effects cannot be neglected during a significant portion of crown propagation, particularly for thin wall films.

In this paper, slip at liquid–liquid interfaces is studied focusing on the ubiquitous case in which a third species (e.g. a gas) is present. Non-equilibrium molecular dynamics simulations demonstrate that the contaminant species accumulate at the liquid–liquid interface, enriching it and affecting momentum transfer in a non-trivial fashion. The Navier boundary condition is seen to apply at this interface, accounting for slip between the liquids. Opposite trends are observed for soluble and poorly soluble species, with the slip length decreasing with concentration in the first case and significantly increasing in the latter. Two regimes are found, one in which the liquid–liquid interface is altered by the third species but changes in slip length remain limited to molecular sizes (intrinsic slip). In the second regime, further accumulation of non-soluble gas at the interface gives rise to a gaseous layer replacing the liquid–liquid interface; in this case, the apparent slip lengths are one order of magnitude larger and grow linearly with the layer width as captured quantitatively by a simple three-fluids model. Overall, results show that the presence of a third species considerably enriches the slip phenomenology both calling for new experiments and opening the door to novel strategies to control liquid–liquid slip, e.g. in liquid infused surfaces.

An analytical solution is developed for studying transient water wave-induced responses inside an unsaturated poroelastic seabed of finite thickness. The soil skeleton and the pore fluid are compressible and the constitutive relationship of the soil skeleton is described by Hooke's law. Assuming that the horizontal length scale of wave motion is much larger than the seabed thickness, the leading-order analytical solutions for the seabed responses, including pore fluid pressure and soil skeleton motion, are obtained. The present solutions are suitable for general transient wave loading and for the shear modulus of the soil skeleton being of the same order of magnitude as the effective bulk modulus of elasticity of the pore fluid. The present theory is first validated by checking the solutions with the experimental data for the pore pressure induced by periodic-wave loading. The present analytical solutions are then used to investigate the seabed responses under transient waves, including linear periodic wave, a solitary wave and a bore. The effects of the wave-induced effective stresses on the bed failure potential are further analysed. The results show that the shear failure potential and its duration are highly dependent on the soil properties, such as saturation degree, shear modulus and permeability. Sensitivity analyses are presented.