Liquid metal flows, generated in a cylindrical cell by a strong electric current flowing from the central bottom electrode to the sidewall, are studied experimentally and numerically. In the absence of an external magnetic field, the flow is caused by the interaction of the electric current with its own magnetic field and is known as electro-vortex flows (EVF), which has a pure poloidal configuration. It is also well known that even a relatively weak azimuthal force (which immediately arises if a vertical external magnetic field is applied) induces a swirl that completely changes the structure of the flow, suppressing the poloidal motion (Davidson et al., Eur. J. Mech. B Fluids, vol. 18, 1999, pp. 693–711). We study the steady-state modes and analyse the dynamics of transient modes, following the evolution of the flow structure at different ratios of azimuthal and poloidal Lorentz forces, governed by the applied current and vertical magnetic field. In the considered problem, the peculiarities of transient modes, leading to the suppression of the poloidal flow, are associated with the fact that the domain of acting forces is localized in the vicinity of the bottom electrode. The scenario of flow evolution strongly depends on the ratio of electro-vortex and rotational forces (whether the EVF has time to form before a noticeable swirl of the metal, or the swirl dominates from the beginning of flow evolution).

We study the endothermic reaction and flow of a granular solid reactant, where energy for the reaction is provided by a counter-current flow of hot gases through the porous reactant bed. Research into reacting flows typically focusses on exothermic combustion processes. However, endothermic processes are common in the metallurgy industry, including the production of cement, silicon and rutile titanium dioxide. Several common features are observed in experimental and numerical studies of these processes, including critical temperatures of the reactant at which the chemical reaction begins, and regions of the reactor with uniform reactant temperature. Motivated specifically by the processes in a silicon furnace, we analyse a model of endothermic, reacting counter-current flow using the method of matched asymptotic expansions. Assuming the Péclet number in the solid is large, we explore the full range of values for the dimensionless inter-phase heat-transfer rate, finding six distinguished limits. In all limits, we find a diffusive boundary layer in which there is a fast chemical reaction rate due to the high temperatures, analogous to exothermic flame fronts. Outside this region, the counter-current flow is crucial to the chemical processes. For intermediate values of the heat-transfer rate, we find the same qualitative properties as those observed across the metallurgy industry, and we quantify the dependence of these properties on the flow rate and heat-transfer rate. In the limit of large heat-transfer coefficient, we derive the single-temperature limit, in which the solution structure is dependent on the direction of net heat flux through the domain.

This paper examines the shape dynamics of deformable elastic and viscoelastic particles in an ambient Newtonian fluid subjected to simple shear. The particles are allowed to undergo large deformation, with the elastic stress determined using the neo-Hookean constitutive relation. We first present a method to determine the shape dynamics of initially ellipsoidal particles that is an extension of the method of Roscoe (J. Fluid Mech., vol. 28, issue 2, 1967, pp. 273–293), originally used to determine the shape at steady state of an initially spherical particle. We show that our method recovers earlier results for the in-plane trembling and tumbling dynamics of initially prolate spheroids in simple shear flow, obtained by a different approach. We then examine the in-plane dynamics of oblate spheroids and triaxial ellipsoids in simple shear flow, and show that they too, like prolate spheroids, exhibit time-periodic tumbling or trembling dynamics, depending on the initial aspect ratios of the particle and the elastic capillary number $G \equiv \mu \dot {\gamma }/\eta$, where $\mu$ is the viscosity of the fluid, $\eta$ is the elastic shear modulus of the particle and $\dot {\gamma }$ is the shear rate. In addition, we find a novel state wherein the particle extends indefinitely in time and asymptotically aligns with the flow axis. We demarcate all the dynamical regimes in the parameter space comprising $G$ and the initial particle aspect ratios. When the particles are viscoelastic, damped oscillatory dynamics is observed for initially spherical particles, and the tumbling–trembling boundary is altered for initially prolate spheroids so as to favour tumbling.

When the top of a sessile droplet is contacted by an opposing solid surface, the droplet can transfer depending on the wettabilities and relative velocity of the surfaces. What if the surface receiving the liquid was porous? High-speed imaging was used to capture the transfer of a droplet from a solid substrate to an opposing porous surface. The parameters that were varied include the wettability of the donor substrate, the pore size of the receiving surface and the droplet's volume and working fluid. Generally, the transfer process is split into two sequential regimes, wetting and wicking, with wicking being three orders of magnitude longer than wetting on average. The wetting regime is split into two sub-regimes, the donor-independent and donor-dependent regimes. The donor-independent regime follows the dynamics of droplet coalescence, starting in a mass-limited viscous regime followed by a capillary–inertial regime. The donor-dependent regime is driven by a global change in Laplace pressure across the liquid bridge, with the viscous wedge of the receding contact line being the rate-limiting factor. The wicking regime is governed by Darcy's law, completing the transfer process of the droplet.

Vortical impulse theory is used to investigate the relationship between turbine thrust and the near-wake velocity and vorticity fields. Three different hypotheses regarding the near-wake structure allow the derivation of novel expressions for the thrust on a steadily rotating wind turbine, and these are tested using stereoscopic particle-image velocimetry (PIV) data acquired just behind a rotor in a water channel. When one assumes that vortex lines and streamlines are aligned in a rotor-fixed frame of reference, one obtains a PIV-based thrust estimate that fails even to capture the trend of the directly measured thrust, and this failure is attributed to an implicit assumption that most of the generated thrust does useful work. When one neglects the axial gradients of radial velocity, the PIV-based thrust estimate captures the measured thrust trend, but underpredicts its magnitude by approximately $33\,\%$. The third and most promising physical proposition treats the trailing vortices as purely ‘rolling’ structures that exhibit zero-strain rate in their cores, with the corresponding thrust estimates in close agreement with direct thrust measurements. This best-performing expression appears as a correction to the classical thrust expression from momentum theory, possessing additional squared-velocity terms that can account for the high-thrust regime of turbine operation that is typically addressed empirically.

In this paper, we train turbulence models based on convolutional neural networks. These learned turbulence models improve under-resolved low-resolution solutions to the incompressible Navier–Stokes equations at simulation time. Our study involves the development of a differentiable numerical solver that supports the propagation of optimisation gradients through multiple solver steps. The significance of this property is demonstrated by the superior stability and accuracy of those models that unroll more solver steps during training. Furthermore, we introduce loss terms based on turbulence physics that further improve the model accuracy. This approach is applied to three two-dimensional turbulence flow scenarios, a homogeneous decaying turbulence case, a temporally evolving mixing layer and a spatially evolving mixing layer. Our models achieve significant improvements of long-term a posteriori statistics when compared with no-model simulations, without requiring these statistics to be directly included in the learning targets. At inference time, our proposed method also gains substantial performance improvements over similarly accurate, purely numerical methods.

In this work, we propose using an ensemble Kalman method to learn a nonlinear eddy viscosity model, represented as a tensor basis neural network, from velocity data. Data-driven turbulence models have emerged as a promising alternative to traditional models for providing closure mapping from the mean velocities to Reynolds stresses. Most data-driven models in this category need full-field Reynolds stress data for training, which not only places stringent demand on the data generation but also makes the trained model ill-conditioned and lacks robustness. This difficulty can be alleviated by incorporating the Reynolds-averaged Navier–Stokes (RANS) solver in the training process. However, this would necessitate developing adjoint solvers of the RANS model, which requires extra effort in code development and maintenance. Given this difficulty, we present an ensemble Kalman method with an adaptive step size to train a neural-network-based turbulence model by using indirect observation data. To our knowledge, this is the first such attempt in turbulence modelling. The ensemble method is first verified on the flow in a square duct, where it correctly learns the underlying turbulence models from velocity data. Then the generalizability of the learned model is evaluated on a family of separated flows over periodic hills. It is demonstrated that the turbulence model learned in one flow can predict flows in similar configurations with varying slopes.

Droplets impinging on solid surfaces, as well as the countless varieties of the resulting possible dynamics, are found frequently in both natural and industrial environments. Among such dynamics, the contact time with the surface, the amount of deformation and the occurrence of breaking induced by geometrical singularities of the solid surface are key aspects in a wide range of applications. We report the results of an extensive experimental activity investigating the capability of liquid droplets to jump over a gap while sliding/rolling over a hydrophobic solid plane. These drops impact on the downstream sharp edge of the gap and undergo the amount of deformation that allows them to climb the edge. We ascribe this unique behaviour to the transformation of rotational momentum into linear momentum. Such conversion can take place only if the right amount of deformation occurs upon impact. Indeed, within the explored range of Weber number $\left (0.5\lesssim {We}\lesssim 40 \right )$, we show the existence of a sub-range for which the drops show a significantly higher probability of jumping over the gap if compared to solid spheres, whose behaviour is predicted accurately by a purely ballistic and elastic impact model. We formulate a minimal energy balance in order to show that such peculiar drops are indeed the ones featuring high rotational energy. The results of this study also contribute to shed light on the debate about the amount of rotational speed characterizing liquid droplets running over hydrophobic surfaces.

The effect of surface vibrations on the propulsion augmentation and resistance in the relative movement of parallel plates has been studied. The analysis was focused on monochromatic waves and laminar flows. The effectiveness of the vibrations was gauged by determining the external force required to maintain the movement of one of the plates at a prescribed velocity. It is shown that waves propagating upstream always increase the resistance but the flow response to waves propagating downstream is more intricate and is a function of the flow Reynolds number. In general, waves must be sufficiently fast to reduce the flow resistance. This leads to a natural division between slow and fast waves; a characterization that is helpful for flows at a sufficiently small Reynolds number Re. An increase in Re brings into play the complication of possible resonances with the natural flow frequencies. Resonances are not possible with waves faster than the plate velocity and these supercritical waves generally decrease the flow resistance. More complex flow responses can occur with slower (subcritical) waves which tend to increase the flow resistance. A complete elimination of the resistance is possible if the waves are of sufficiently short wavelength and travel quickly. This suggests that our mechanism has great potential in the development of propulsion augmentation systems. None of the waves produced net energy savings.

We examine the translation of a sphere in a stratified ambient in the limit of small Reynolds numbers ($Re \ll 1$) and viscous Richardson numbers ($Ri_v \ll 1$); here, $Re = {\rho Ua}/{\mu }$ and $Ri_v = {\gamma a^3 g}/{\mu U}$, with $a$ being the sphere radius, $U$ the translation speed, $\rho$ and $\mu$ the density and viscosity of the stratified ambient, $g$ the acceleration due to gravity, and $\gamma$ the density gradient characterizing the ambient stratification. In contrast to most earlier efforts, our study considers the convection-dominant limit corresponding to $Pe = {Ua}/{D} \gg 1$, $D$ being the diffusivity of the stratifying agent. We characterize in detail the velocity and density fields around the particle in what we term the Stokes stratification regime, defined by $Re \ll Ri_v^{{1}/{3}} \ll 1$, and corresponding to the dominance of buoyancy over inertial forces. Buoyancy forces associated with the perturbed stratification fundamentally alter the viscously dominated fluid motion at large distances of order the stratification screening length that scales as $a\,Ri_v^{-{1}/{3}}$. The motion at these distances transforms from the familiar fore–aft symmetric Stokesian form to a fore–aft asymmetric pattern of recirculating cells with primarily horizontal motion within, except in the vicinity of the rear stagnation streamline. At larger distances, the motion is vanishingly small except within (a) an axisymmetric horizontal wake whose vertical extent grows as $O(r_t^{{2}/{5}})$, $r_t$ being the distance in the plane perpendicular to translation, and (b) a buoyant reverse jet behind the particle that narrows as the inverse square root of distance downstream. As a result, for $Pe = \infty$, the motion close to the rear stagnation streamline starts off pointing in the direction of translation, in the inner region, and decaying as the inverse of the downstream distance; the motion reverses beyond distance $1.15a\,Ri_v^{-{1}/{3}}$, with the eventual reverse flow in the far-field buoyant jet again decaying as the inverse of the distance downstream. For large but finite $Pe$, the narrowing jet is smeared out beyond a distance of $O(a\,Ri_v^{-{1}/{2}}\, Pe^{{1}/{2}})$, leading to an exponential decay of the aforementioned reverse flow.

Despite decades of investigations, there is still no consensus on whether inertial particles augment or dampen turbulence. Here, we perform the first experimental study in which the particle concentration is varied systematically across a broad range of volume fractions $\varPhi _{v}$, from nominally one-way coupled to heavily two-way coupled regimes, keeping all other parameters constant. We utilize a zero-mean flow chamber where steady, homogeneous and approximately isotropic air turbulence is realized, with a Taylor-microscale Reynolds number $Re_{\lambda } = 150\unicode{x2013}300$. We consider spherical solid particles of two sizes, both much smaller than the Kolmogorov length, and yielding Stokes numbers $St_{\eta } = 0.3$ and 2.6 based on the Kolmogorov time scale. By adjusting the turbulent intensity, the settling velocity parameter is kept constant for both cases, $Sv_{\eta } = V_{t}/u_{\eta }\approx 3$ (where $V_{t}$ is the still-air terminal velocity, and $u_{\eta }$ is the Kolmogorov velocity scale). Unlike previous studies focused on massively inertial particles, we find that the turbulent kinetic energy increases with particle loading, being more than doubled at $\varPhi _{v} =5\times 10^{-5}$. This is attributed to the energy input associated with gravitational settling: the particles release their potential energy into the fluid and increase its dissipation rate, while the time scale associated with the inter-scale energy transfer is not strongly changed. Two-point statistics indicate that the energy-containing eddies become vertically elongated in the presence of falling particles, and that the latter redistribute the energy more homogeneously across the scales compared to unladen turbulence. This is rooted in an enhanced cascade, as shown by the nonlinear inter-scale energy transfer rate.

In Wang & Pan (J. Fluid Mech., vol. 918, 2021, A19), the authors developed the first ensemble-based data assimilation (DA) capability for the reconstruction and forecast of ocean surface waves, namely the EnKF-HOS method coupling an ensemble Kalman filter (EnKF) and the high-order spectral (HOS) method. In this work, we continue to enrich the method by allowing it to simultaneously estimate the ocean current field, which is in general not known a priori and can (slowly) vary in both space and time. To achieve this goal, we incorporate the effect of ocean current (as unknown parameters) on waves to build the HOS-C method as the forward prediction model, and obtain a simultaneous estimation of (current) parameters and (wave) states via an iterative EnKF (IEnKF) method that is necessary to handle the complexity in this DA problem. The new algorithm, named the IEnKF-HOS-C method, is first tested in synthetic problems with various forms (steady/unsteady, uniform/non-uniform) of current. It is shown that the IEnKF-HOS-C method is able to not only estimate the current field accurately, but also boost the prediction accuracy of the wave field (even) relative to the state-of-the-art EnKF-HOS method. Finally, using real data from a shipborne radar, we show that the IEnKF-HOS-C method successfully recovers the current speed that matches the in situ measurement by a floating buoy.

In this work, we study the effect of insoluble surfactants on the three-dimensional rim-driven retraction dynamics of thin water sheets in air. We employ an interface-tracking/level-set method to ensure the full coupling between the surfactant-induced Marangoni stresses, interfacial diffusion and inertia. Our findings are contrasted with the (Newtonian) dynamics of a liquid sheet edge, finding that the surfactant concentration can delay, or effectively prevent, the breakup of the rim. Our simulations use the fastest growing Rayleigh–Plateau instability to drive droplet detachment from the fluid sheet (rim). The results of this work unravel the significant role of Marangoni stresses in the retracting sheet dynamics at large elasticity numbers. We study the sensitivity of the dynamics to the elasticity number and the rigidification of the interface.

The present paper analyses the vortical structures in the wake of a naval propeller operating underneath a free surface using detached-eddy simulation. We investigate the flow topology for several loading conditions and compare it with analogous observations behind a propeller operating in open water. We show that the wake topology is similar to that observed in open water only for low-loading conditions. For mild blade loading, the free surface's presence seems to stabilize the flow. On the contrary, for high blade loading, the mutual interaction between the vortex system and the free surface leads to vortex breakdown that overshadows the multiple pairing mechanisms observed in open-water conditions.

We study shear-driven liquid flow over a planar surface with an embedded gas-filled groove, with the gas–liquid interface protruding slightly above or below the planar surface. The flow direction is along the groove, which is taken to be much longer than it is wide, and the gas–liquid interface is assumed to be covered by an incompressible surface fluid, representing a surfactant phase. Using the incompressibility condition for the surface fluid, the equations of motion and corresponding boundary conditions for the liquid phase are obtained by minimizing the dissipation rate. Assuming a moderate deformation of the interface, a domain perturbation technique with the maximal deformation as the small parameter is employed. The Stokes equation in the liquid phase under corresponding boundary conditions is solved to second order in the deformation using the Keldysh–Sedov formalism. The obtained analytical results are compared with numerical calculations of the same problem, allowing an assessment of the limits of validity of the expansion. While on a planar gas–liquid interface no flow is induced, a recirculating flow is observed on an interface protruding slightly above or below the planar surface. The study sheds light onto the mobility of curved gas–liquid interfaces in the presence of surfactants acting as an incompressible surface fluid.

A numerical investigation is undertaken on the development of linear disturbances in the rotating disk boundary layer, in which a time-periodic modulation is applied to the disk rotation rate. The model gives a prototypical example of a three-dimensional oscillatory boundary layer, by adding a Stokes layer to the von Kármán flow that develops on a steady disk. The study extends the Floquet analysis of Morgan et al. (J. Fluid Mech., vol. 925, 2021, A20), who showed that disk modulation stabilises the stationary convective instabilities found on the steady rotating disk. Using a radial homogeneous flow approximation, whereby the radial dependence of the basic state is ignored, disturbance development is simulated for several modulation settings, with flow conditions matched to both convective and absolute forms of linear instability. Disturbances excited via a stationary periodic wall forcing display behaviour consistent with that found using Floquet theory; time-periodic modulation stabilises the cross-flow instability by reducing the radial growth rate. In addition, convective and absolute instabilities, generated by an impulsive wall forcing, are both stabilised by the introduction of modulation to the disk rotation rate. Modulation establishes a significant reduction in both the temporal growth rate and the disturbance amplitude as it propagates away from the impulse origin. Moreover, greater stabilising control benefits are realised as the modulation amplitude increases.

Aerosol particles are inertial particles. They cannot follow the surrounding fluid in a region of high vorticity. These encounters render the particle velocity field ${{\boldsymbol {v}}}$ locally compressible. Caustics occur if the trace of the gradient matrix $\textsf{$\boldsymbol{\sigma}$}$ of the field ${{\boldsymbol {v}}}$ locally diverges. For three-dimensional isotropic and homogeneous turbulence, the dynamics of the gradient matrix can be expressed in terms of three geometric invariants. In the present paper we establish a parametrisation of this problem where the dynamics takes the form of an excitable stochastic dynamical system with a three-dimensional phase space. The deterministic part of the dynamics is solved analytically. We show that the deterministic system has a globally attractive stable fixed point. Small noise induces excursions from the fixed point that typically relax straight back towards the fixed point. Caustics emerge as non-trivial return to a global fixed point when noise excites a trajectory across a stability threshold. The relaxation to the global fixed point will then involve at least one, and it may involve two or even three divergences of $\boldsymbol {Tr}\,\textsf{$\boldsymbol{\sigma}$}$. Based on a combination of analytical insights and numerical analysis, we determine the rate of occurrence, duration and relative observation probability of caustic events in turbulent aerosols. Our analysis reveals that each approach towards a divergence proceeds along a straight line in the phase space of the dynamical system, which can help identify caustics. Moreover, there are infinite ways in which caustics can arise, namely whenever $\boldsymbol {Tr}\,\textsf{$\boldsymbol{\sigma}$}$ tends to $-\infty$, so that no two caustics look the same.

Experiments are performed in a Mach-6 shock tunnel to examine the laminar-to-turbulent transition process associated with a sudden increase in surface angle on a slender body. A cone/flare geometry with a 5$^\circ$ frustum and compression angles ranging from 5$^\circ$ to 15$^\circ$ allow a range of mean flow configurations, spanning an attached shock-wave/boundary-layer interaction to a fully separated one; the unit Reynolds number of the flow is also varied to modify the state of incoming second-mode boundary-layer disturbances. Ultra-high-speed schlieren visualizations provide a global picture of the flow development, supplemented by high-frequency surface pressure measurements. For the 5$^\circ$ compression, the unsteady flow field is dominated by the second-mode waves, whose breakdown to turbulence is generally accelerated (compared with the straight-cone configuration) by encountering the angle change. As the compression angle is increased to induce separation, lower-frequency disturbances appear along the separated shear layer that exhibit much larger amplification rates than the incoming second-mode waves; the latter effectively freeze in amplitude downstream of the separation point before rapidly breaking down upon reattachment. The shear-layer disturbances become dominant at the largest compression angle tested. Radiation of disturbance energy to the external flow is consistently observed: this generally occurs along mean flow features (flare, separation or reattachment shocks) for the second-mode disturbances and spontaneously for the shear-layer waves. The combined application of spectral proper orthogonal decomposition and a global bispectral analysis allows the identification of important unsteady flow structures and the association of these with prominent nonlinear interactions in the various configurations.

Drainage and spreading processes in thin liquid films have received considerable attention in the past decades. Yet, our understanding of three-dimensional cases remains sparse, with only a few studies focusing on flat and axisymmetric substrates. Here, we exploit differential geometry to understand the drainage and spreading of thin films on curved substrates, under the assumption of negligible surface tension and hydrostatic gravity effects. We develop a solution for the drainage on a local maximum of a generic substrate. We then investigate the role of geometry in defining the spatial thickness distribution via an asymptotic expansion in the vicinity of the maximum. Spheroids with a much larger (respectively smaller) height than the equatorial radius are characterized by an increasing (respectively decreasing) coating thickness when moving away from the pole. These thickness variations result from a competition between the variations of the substrate's slope and mean curvature. The coating of a torus presents larger thicknesses and a faster spreading on the inner region than on the outer region, owing to the different curvatures in these two regions. In the case of an ellipsoid with three different axes, spatial modulations in the drainage solution are observed as a consequence of a faster drainage along the short principal axis, faithfully reproduced by a three-dimensional asymptotic solution. Leveraging the conservation of mass, an analytical solution for the average spreading front is obtained. The solutions are in agreement with numerical simulations and experimental measurements obtained from the coating of a curing polymer on diverse substrates.

In the present study, the flow–structure interaction of a starting jet through a flexible nozzle is experimentally investigated, with a focus on the optimal flexibility for thrust generation. Water slug is impulsively accelerated through a cylindrical nozzle, fabricated with silicone rubber of varying flexibility. In general, the flexible nozzle modifies the vortical structure of the jet and augments the thrust of the starting jet. The measurement of nozzle surface deformation revealed that a back-and-forth wave propagation on the nozzle surface is responsible for the jet-vortex evolution augmenting the thrust generation. Combining the hydrodynamic conservation equations and the linearized shell theory, we also formulated the governing equations, dominated by two relevant dimensionless parameters: the effective acceleration time of the jet ($\varPi _0$) and the effective nozzle stiffness ($\varPi _1$). Asymptotic analysis of the equation showed that the dimensionless wave speed ($\hat {c}$) is expressed as $\hat {c}=(\varPi _{0}^{2}\varPi _{1}/2)^{0.5}$, and the jet momentum is maximized at $\hat {c}=\hat {c}_{crit}$ ($\simeq 3.0$), the condition at which the release of elastic energy stored during nozzle contraction to the jet is synchronized with the instant of termination of jet acceleration. While $\hat {c}=\hat {c}_{crit}$, the achievable maximum jet velocity decreases with the effective acceleration time of the jet ($\varPi _0$), which is attributed to the reduced speed of the surface wave by the flow inside the nozzle.