Applying a focused ultrasonic field on a free liquid surface results in its growth eventually leading to the so-called acoustic fountain. In this work, a numerical approach is presented to further increase the understanding of the acoustic fountain phenomenon. The developed simulation method enables the prediction of the free surface motion and the dynamic acoustic field in the moving liquid. The dynamic system is a balance between inertia, surface tension and the acoustic radiation force, and its nonlinearity is demonstrated by studying the relation between the ultrasonic excitation amplitude and corresponding liquid deformation. We show that dynamic resonance is the main mechanism causing the specific acoustic fountain shapes, and the analysis of the dynamic acoustic pressure allows us to predict Faraday-instability atomisation. We show that strong resonance peaks cause atomisation bursts and strong transient deformations corresponding to previously reported experimental observations. The quantitative prediction of the dynamic acoustic pressure enables us to assess the potential of cavitation generation in acoustic fountains. The observed local high acoustic pressures above both the cavitation and the atomisation threshold hint at the coexistence of these two phenomena in acoustic fountains.

The inverse problem of steady two-dimensional open channel free-surface flow is considered, with the focus on determining two types of disturbances: a surface pressure distribution and solid channel bottom topography. A closed-form expression for the inverse surface pressure is derived, and a linear Fredholm equation of the first kind is shown to describe the inverse topography problem, which then needs to be descretised and solved numerically. However, the equation for the channel bottom is prone to instability, so the truncated singular value decomposition (TSVD) method is proposed as a way to stabilise the associated discrete solution. The effectiveness of the TSVD method is demonstrated through several numerical examples, and its performance in the presence of error-contaminated input data is also examined. The results show that the TSVD method can recover the topography accurately from the forward free-surface problem, and provide good approximations even with noisy input data.

The unsteady wake interference of unequal-height tandem finite wall-mounted cylinders (FWMCs) fully submerged in a turbulent boundary layer (TBL) was investigated using time-resolved particle image velocimetry. The aspect ratios of the cylinders were fixed at $h/d = 5.3$ for the upstream cylinder (UC) and $H/d = 7.0$ for the downstream cylinder (DC) to achieve a height ratio of $h/H = 0.75$, where d is the diameter of the cylinders. The Reynolds number based on the cylinder diameter was $Re = 5540$ and the submergence ratio was $\delta /H = 1.2$, where $\delta $ is the TBL thickness. Three main flow regimes of tandem FWMCs were examined by varying the centre-to-centre spacing ($s$) between the cylinders: extended-body ($s/d = 2$), reattachment ($s/d = 4$) and co-shedding ($s/d = 6$) regimes. These test cases denoted as SR2, SR4 and SR6, respectively, were compared with a reference isolated cylinder (SC) with an aspect ratio similar to that of the DC. Spatio-temporal analysis of the flow field showed that the gap region of SR2 is characterized by a strong downwash of alternating low- and high-momentum fluid induced by the approach flow that is deflected from the unsheltered portion of the DC. In contrast, the gap region of SR4 and SR6 exhibited both downwash and upwash flow with a saddle point that moves closer to the mid-height of the UC as the spacing ratio increases. The upwash and downwash shear layers were associated with small-scale vortices with Strouhal numbers larger than that of the Kármán vortex shedding in the spanwise shear layers. The wake structure behind the DC was significantly altered compared with the SC due to sheltering effects, and the spacing ratio had a significant impact on the spatio-temporal evolution of the vortices.

A rigid object moving in a viscous fluid and in close proximity to an elastic wall experiences self-generated elastohydrodynamic interactions. This has been the subject of intense research activity, with recent and growing attention given to the particular case of elastomeric and gel-like substrates. Here, we address the situation where the elastic wall is replaced by a capillary surface. Specifically, we analyse the lubrication flow generated by the prescribed normal motion of a rigid infinite cylinder near the deformable interface separating two immiscible and incompressible viscous fluids. Using a combination of analytical and numerical treatments, we compute the emergent capillary-lubrication force at leading order in capillary compliance, and characterize its dependencies with the interfacial tension, viscosities of the fluids, and length scales of the problem. Interestingly, we identify two main contributions: (i) a velocity-dependent adhesive-like force; (ii) an acceleration-dependant inertia-like force. Our results may have implications for the mobility of colloids near complex interfaces and for the motility of confined microbiological entities.

In this study, mean velocity and temperature profiles for turbulent vertical convection (VC) confined in an infinite channel are investigated theoretically. The analysis starts from the governing equations of the thermal flow, with Reynolds shear stress and turbulent heat flux closed by the mixing length theory. Employing a three-sublayer description of the mean fields, the mean velocity and temperature profiles are found to be linear laws near the channel wall (viscosity-dominated sublayer), and they follow power laws close to the channel centre (turbulence-dominated sublayer). The characteristic scales of velocity, temperature and length in the present profiles arise naturally from the system normalisation, rather than from scaling analyses, thus ensuring a sound mathematical description. The derived profiles are verified fully via various literature data available in the classical regime; further, they are compared with the reported profiles, and the results indicate that the present profiles are the only ones with the ability to interpret data accurately from different sources, demonstrating much better versatility. Meanwhile, we provide analytical arguments showing that in the ultimate regime, the mean profiles in VC may remain in power laws, rather than the log laws inferred by analogy with Rayleigh–Bénard convection (RBC) systems. The power profiles recognised in this study are induced by the effect of buoyancy, which is in parallel with the mean flow in VC and contributes to the streamwise momentum transport, whereas in RBC systems, buoyancy is perpendicular to the mean flow, and does not influence the streamwise momentum transport, resulting in log profiles, being similar to the case of wall shear flows.

The scaling and mechanism of the propagation speed of turbulent fronts in pipe flow with the Reynolds number has been a long-standing problem in the past decades. Here, we derive an explicit scaling law for the upstream front speed, which approaches a power-law scaling at high Reynolds numbers, and we explain the underlying mechanism. Our data show that the average wall distance of low-speed streaks at the tip of the upstream front, where transition occurs, appears to be constant in local wall units in the wide bulk-Reynolds-number range investigated, between 5000 and 60 000. By further assuming that the axial propagation of velocity fluctuations at the front tip, resulting from streak instabilities, is dominated by the advection of the local mean flow, the front speed can be derived as an explicit function of the Reynolds number. The derived formula agrees well with the speed measured by front tracking. Our finding reveals a relationship between the structure and speed of a front, which enables a close approximation to be obtained of the front speed based on a single velocity field without having to track the front over time.

To explore the impact of surface viscosity on coexisting fluid domains in biomembranes we consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modelled by incompressible surface Navier–Stokes–Cahn–Hilliard-like equations with bending forces. We derive this model using the Lagrange–d’Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor–Hood element for the surface Navier–Stokes part, a semi-implicit approach for the Cahn–Hilliard part, higher-order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified subproblems.

The effect of magnetic as well as electromagnetic fields on the stability of an electrically conducting viscous liquid film flowing down an inclined plane has been investigated for the full range of inclination angles $\theta$ ($0 < \theta \le 90^{\circ }$) in association with a given value of the Reynolds number $Re$ ($0 < Re \le 100$), and vice versa. A nonlinear evolution equation is derived by using the momentum-integral method, which is valid for both small and large values of $Re$. Use of the normal mode approach on the linearized surface evolution equation gives the stability criterion and the critical value of the wavenumber $k_c$ (for which the imaginary part of the complex frequency $\omega _i^+$ is zero) which conceive the electric parameter $E$, magnetic parameter $M$, Reynolds number $Re$, Weber number $We$ and inclination angle $\theta$. The nonlinear stability analysis based on the second Landau constant $J_2$ helps to demarcate all four possible distinct flow zones (explosive, supercritical, unconditional and subcritical) of this problem. A novel result of this analysis is a simple relationship between the critical values of $k_c$ and $k_j$ (for which $J_2$ is zero) that basically gives the necessary conditions for the existence of the range of $k$ for an explosive unstable zone, which is either one or two accordingly as $k_j >k_c$ or $k_j< k_c$, and the non-existence of an unconditional stable zone is $k_j \le k_c$ depending upon the values of $M$. The analysis confirms the existence of two critical values of $M$, namely, $M_c$ (for which $k_c$ is zero) and $M_j$ (for which $k_j$ is zero). Here, $M_j > M_c$ except for $\theta = 90^{\circ }$; and we have found the existence of all four or two (unconditional and subcritical) or one (subcritical) zone(s) of this flow problem accordingly, as $0 \le M < M_c$ or $M_c \le M < M_j$ and $M > M_j$ or $M = M_j$.

In this work, the relationship between the velocity of an elongated bubble and its shape is investigated, in the case where the elongated bubble flows in a viscous liquid initially at rest in a pipe. The velocity, expressed as a Froude number, depends on the angle of the inclined pipe, the Eötvös number and the buoyancy Reynolds number. The diameter of the pipe and the surface tension being fixed, the Eötvös number remains constant; this study focuses on the dependence of the velocity on the pipe inclination angle and the viscosity of the liquid. The velocity of the elongated bubble was measured for different angles between 0 and 15 degrees and for liquid viscosities 10 to 200 times that of water. As the velocity of elongated bubbles depends closely on their shape, shadowgraphy coupled with particle image velocimetry was used. The results show that the velocity of the elongated bubbles is highly sensitive to the inclination angle of the pipe and to the viscosity of the liquid, particularly for low pipe inclinations and large viscosities. In the layer of liquid located downstream of the elongated bubble, laminar flow develops rapidly in the liquid, resulting from a balance between gravity and friction at the wall. The identification of the position of the stagnation point close to the nose of the elongated bubble and the curvature of the interface at this point helps to explain why the velocity of the elongated bubble decreases for low angles and high viscosities.

We introduce a geometric analysis of turbulent mixing in density-stratified flows based on the alignment of the density gradient in two orthogonal bases that are locally constructed from the velocity gradient tensor. The first basis connects diapycnal mixing to rotation and shearing motions, building on the recent ‘rortex–shear decomposition’ in stratified shear layers (Jiang et al., J. Fluid Mech., vol. 947, 2022, A30), while the second basis connects mixing to the principal axes of the viscous dissipation tensor. Applying this framework to datasets taken in the stratified inclined duct laboratory experiment reveals that density gradients in locations of high shear tend to align preferentially (i) along the direction of minimum dissipation and (ii) normal to the plane spanned by the rortex and shear vectors. The analysis of the local alignment across increasingly turbulent flows offers new insights into the intricate relationship between the density gradient and dissipation, and thus diapycnal mixing.

The strong coupling interactions of non-equilibrium flow, microscopic particle collisions and radiative transitions within the shock layer of hypersonic atmospheric re-entry vehicles makes accurate prediction of the aerothermodynamics challenging. Therefore, in this study a self-consistent non-equilibrium flow, collisional–radiative reactions and radiative transfer fully coupled model are established to study the non-equilibrium characteristics of the flow field and radiation of vehicle atmospheric re-entry. The comparison of the present calculation results with flight data of FIRE II and previous results in the literature shows a reasonable agreement. The thermal, chemical and excited energy level non-equilibrium phenomena are obtained and analysed for the different FIRE II trajectory points, which form the critical basis for studying the heat transfer and radiation. The non-equilibrium distribution of excited energy levels significantly exists in the post-shock and near-wall regions due to the rapid vibrational dissociation and electronic under-excitation, as well as the wall catalytic reactions. The analysis of stagnation-point heating of FIRE II illustrates that the translational–rotational convection and the dissociation component diffusion play key roles in the aerodynamic heating of the wall region. The spectrally resolved radiative intensity in the entire flow field indicates that the vacuum ultraviolet radiation caused by the high-energy nitrogen atomic spectral lines makes the main contribution to the radiative transfer. Finally, it is found that the non-equilibrium flow–radiation coupling effect can exacerbate the excited energy level non-equilibrium, and further affect the gas radiative properties and radiative transfer. This fully coupled study provides an effective method for reasonable prediction of atmospheric re-entry flow and radiation fields.

We study the large-scale dynamics and prediction of hydrodynamic transport in random fracture networks. The flow and transport behaviour is characterized by first passage times and displacement statistics, which show heavy tails and anomalous dispersion with a strong dependence on the injection condition. The origin of these behaviours is investigated in terms of Lagrangian velocities sampled equidistantly along particle trajectories, unlike classical sampling strategies at a constant rate. The velocity series are analysed by their copula density, the joint distribution of the velocity unit scores, which reveals a simple, albeit hidden, correlation structure that can be described by a Gaussian copula. Based on this insight, we derive a Langevin equation for the evolution of equidistant particle speeds. In this framework, particle motion is quantified by a stochastic time-domain random walk, the joint density of particle position, and speed satisfies a Klein–Kramers equation. The upscaled theory quantifies particle motion in terms of the characteristic fracture length scale and the distribution of Eulerian flow velocities. That is, it is predictive in the sense that it does not require the a priori knowledge of transport attributes. The upscaled model captures non-Fickian transport features, and their dependence on the injection conditions in terms of the velocity point statistics and average fracture length. It shows that the first passage times and displacement moments are dominated by extremes occurring at the first step. The presented approach integrates the interaction of flow and structure into a predictive model for large-scale transport in random fracture networks.

The present paper concerns the linear fate of transverse perturbations in a gravity-driven, thin-film flow over a soluble substrate. We propose a reduced-order model, based on a boundary-layer treatment of the solute transport and a depth-integration of the Stokes equations, using two extended lubrication methodologies found in the literature. We obtain a closed-form dispersion relation, which we compare to a previous, fully resolved analytical investigation (Bertagni and Camporeale, J. Fluid Mech., vol. 913, 2021, A34). The results allow us to distil the essential physical mechanisms behind the instability.

‘Storm oil’ – nearly water-insoluble oil poured into the ocean and acting as a surfactant – has been used since ancient times to smooth the waves on the ocean. It was first scientifically described by Benjamin Franklin (Phil. Trans. R. Soc. Lond., vol. 64, 1774, pp. 445–460). In a recent paper, by combining highly controlled experiments in a wave tank and direct numerical simulations, Erinin et al. (J. Fluid Mech., vol. 972, 2023, R5) have now beautifully revealed the strong effect of soluble surfactants on the dynamics of plunging breakers. Remarkably, it is not the change in surface tension which mainly matters, but the surface tension gradient which emerges through compression and dilation of the plunging breaker surface.