A study of the backflow events in the flow through a toroidal pipe at friction Reynolds number ${\text{Re}}_{\tau }\approx$ 650 is performed and compared with the results in a straight turbulent pipe flow at ${\text{Re}}_{\tau }\approx$ 500. The statistics and topological properties of the backflow events are analysed and discussed. Conditionally averaged flow fields in the vicinity of the backflow event are obtained, and the results for the torus show a similar streamwise wall-shear stress topology which varies considerably for the azimuthal wall-shear stress when compared to the pipe flow. In the region around the backflow events, critical points are observed. The comparison between the toroidal pipe and its straight counterpart also shows fewer backflow events and critical points in the torus. This is attributed to the secondary flow of Prandtl's first kind present in the toroidal pipe, which is responsible for the convection of momentum from the inner to the outer bend through the core of the pipe, and back from outer bend to the inner bend along the azimuthal direction. These results indicate that backflow events and critical points are genuine features of wall-bounded turbulence, and are not artefacts of specific boundary or inflow conditions in simulations and/or measurement uncertainties in experiments.

We present an experimental study of resonant generation of superharmonic internal waves as a result of interaction between horizontally propagating vertical internal wave modes $m$ and $n$ at frequency ${\omega }_0$ in a uniformly stratified finite-depth fluid. Thorpe [J. Fluid Mech. 24, 737 (1966)] has shown theoretically that modes $m$ and $n$ at frequency ${\omega }_0$ and mode $p=|m-n|$ at frequency $2{\omega }_0$ are in triadic resonance at specific values of ${\omega }_0$. We demonstrate the occurrence of this triadic resonance by forcing a primary wave field of modes $m$ and $n$ at various ${\omega }_0$ using a novel internal wave generator, and observing the spontaneous growth (or lack thereof) of the superharmonic mode $p=|m-n|$ at frequency $2{\omega }_0$. A superharmonic wave field with a predominantly mode-$p=|m-n|$ structure is observed over a finite range of frequency ($\mathrm{\Delta }{\omega }_0\simeq 0.03N$) around the resonant value, where $N$ is the uniform buoyancy frequency. The spatial growth of the superharmonic wave field is then quantitatively measured, to subsequently compare with the predictions from amplitude evolution equations at resonance at various forcing amplitudes, thereby validating this model. It is furthermore shown that a large-scale spatial evolution of the wave field is more suited to describe our experiments than the slow temporal evolution approach. The paper concludes with a brief discussion of viscous effects.

We investigate the effects of turbulent fluctuations on the Lagrangian statistics of absorption of a scalar field by tracer particles, as a model for nutrient uptake by suspended nonmotile microorganisms. By means of extensive direct numerical simulations of an Eulerian-Lagrangian model we quantify, in terms of the Sherwood number, the increase of the scalar uptake induced by turbulence and its dependence on the Peclet and Reynolds numbers. Numerical results are compared with classical predictions for a stationary shear flow extended here to take into account the presence of a restoring scalar flux. We find that mean-field predictions agree with numerical simulations at low Peclet numbers but are unable to describe the large fluctuations of local scalar uptake observed for large Peclet numbers. We also study the role of velocity fluctuations in the local uptake by looking at the temporal correlation between local shear and uptake rate and we find that the latter follows fluid velocity fluctuations with a delay given by Kolmogorov timescale. The relevance of our results for aquatic microorganisms is also discussed.

This paper studies the spontaneous radiant energy emitted by a trapped vortex flame, focusing on the intermittent occurrence of the flame blowout during thermoacoustic oscillation. We tackle this issue by a wavelet-based auto-conditioning technique, that on the grounds of an energy criterion, performs an objective detection and selection of those events whose energy content exceeds a proper threshold. The result is the time history of a signal containing extinction-reignition events only. As further outcomes, the detected intermittent events are statistically characterized in terms of shape, amplitude, time lag, and timescales. Moreover, intermittent events are found to be in phase with the tonal component of the radiant energy signal, whereas they occur randomly in time. Although the wavelet auto-conditioning technique is a well-established method in fluid dynamics and aeroacoustics, we apply it in the combustion field, where alternative methods, based on wavelet transforms, have already been successfully applied. Additional information about intermittency in the flame is inferred from the chaotic analysis: recurrence plot (RP) of radiant energy shows that the time of boundary crisis is much larger than the characteristic timescale of extinction-reignition event. Furthermore, cross-conditioning of RPs with intermittent events found by wavelet transform results in a characteristic “pike-like” pattern, which identifies an intermittent state different from the most common intermittency types already described in the literature.

The impact dynamics and internal mixing of a droplet onto a liquid-gas interface of lower surface tension was studied both experimentally and numerically, with both the Ohnesorge number (Oh) and the Bond number (Bo) being fixed. Compared to the droplet impact onto a pool of identical liquid, the interfacial Marangoni flow entrains abundant fluid upward and hence induces an additional jet breakup during crater formation (the first breakup), and it facilitates the emergence of the Rayleigh jet breakup (the second breakup) during crater restoration and enhances the vortical mixing beneath the liquid surface. Specifically, with the increase of the impact inertia, the first breakup manifests a nonmonotonic trend of “absence-presence-absence.” The former transition of “absence-presence” at a low droplet-based Weber number (${\mathrm{We}}_{\mathrm{d}}$) is caused by the shortened path of the Marangoni flow on the faster-growing liquid bridge, and the later transition of “presence-absence” at a high ${\mathrm{We}}_{\mathrm{d}}$ is resulted from the reduced displacement velocity of the pool fluid on the expanding crater surface. The second breakup corresponds to the Rayleigh jet breakup without surface tension difference and occurs monotonically beyond a certain ${\mathrm{We}}_{\mathrm{d}}$. Due to the relatively short displacement time of the Marangoni flow on the crater surface compared to the time for crater formation, the critical condition for the emergence of the second jet breakup could be described by the critical reservoir-fluid-based We number (${\mathrm{We}}_{\mathrm{r}}$). The critical ${\mathrm{We}}_{\mathrm{r}}$ contains two parts: the Bo-dependent critical ${\mathrm{We}}_{\mathrm{r}0}$ without surface tension difference, and the increased viscous dissipation from the wrap-up motion of the Marangoni flow. Furthermore, capillary waves are also induced by the Marangoni flow during crater restoration, and the accompanied vorticity generation causes the mixing pattern to exhibit multiple vortex rings and even a clawlike structure, which is substantially enhanced compared to the vortical mixing without surface tension difference.

Solutions to the Stokes equations written in terms of a small number of hydrodynamic image singularities have been a useful tool in theoretical and numerical computations for nearly 50 years. In this article, we extend the catalog of known solutions by deriving the flow expressions due to a general point torque and point source in the presence of a stationary sphere with either a no-slip or a stress-free (no-shear) boundary condition. For an axisymmetric point torque and a no-slip sphere the image system simplifies to a single image point torque, reminiscent of the solution for a point charge outside an equipotential sphere in electrostatics. By symmetry, this also gives a simple representation of the solution due to an axisymmetric point torque inside a rigid spherical shell. In all remaining cases, the solution can be described by a collection of physically intuitive point and line singularities. Our results will be useful for the theoretical modeling of the propulsion of microswimmers and efficient numerical implementation of far-field hydrodynamic interactions in this geometry.

The effect of a network of fixed rigid fibers on fluid flow is investigated by means of three-dimensional direct numerical simulations using an immersed boundary method for the fluid-structure coupling. Different flows are considered (i.e., cellular, parallel, and homogeneous isotropic turbulent flow) to identify the modification of the classic energy budget occurring within canopies or fibrous media, as well as particle-laden flows. First, we investigate the stabilizing effect of the network on the Arnold-Beltrami-Childress cellular flow, showing that, the steady configuration obtained for a sufficiently large fiber concentration mimics the single-phase stable solution at a lower Reynolds number. Focusing on the large-scale dynamics, the effect of the drag exerted by the network on the flow can be effectively modeled by means of a Darcy's friction term. For the latter, we propose a phenomenological expression that is corroborated when extending our analysis to the Kolmogorov parallel flow and homogeneous isotropic turbulence. Furthermore, we examine the overall energy distribution across the various scales of motion, highlighting the presence of small-scale activity with a peak in the energy spectra occurring at the wave number corresponding to the network spacing.

We study the equilibrium shape of a liquid drop resting on top of a liquid surface, i.e., a floating lens. We consider the surface tension forces in nonwetting situations (negative spreading factor), as well as the effects of gravity. We obtain analytical expressions for the drop shape when gravity can be neglected. Perhaps surprisingly, when including gravity in the analysis, we find two different families of equilibrium solutions for the same set of physical parameters. These solutions correspond to drops whose center of mass is above or below the level of the external liquid surface. By means of energetic considerations, we determine the family that has the smallest energy, and therefore is the most probable to be found in nature. A detailed explanation of the geometrical differences between the two types of solutions is provided.

Dynamic mode decomposition (DMD) provides a principled approach to extract physically interpretable spatial modes from time-resolved flow-field data, along with a linear model for how the amplitudes of these modes evolve in time. Recently, DMD has been extended to work with more realistic data that are underresolved either in time or space or with data collected in the same spatial domain over multiple independent time windows. In this work, we develop an extension to DMD to synthesize globally consistent modes from velocity fields collected independently in multiple partially overlapping spatial domains. We propose a tractable optimization to identify modes that span multiple windows and align their phases to be consistent in the overlapping regions. First, we demonstrate this approach on data from direct numerical simulation, where it is possible to split the data into overlapping domains and benchmark against ground-truth modes. We consider the laminar flow past a cylinder as an example with distinct frequencies, along with the spatially developing mixing layer, which exhibits a frequency spectrum that evolves continuously as the measurement window moves downstream. Next, we analyze experimental velocity fields from particle image velocimetry in six overlapping domains in the wake of a cross-flow turbine. On the numerical examples, we demonstrate the robustness of this approach to increasing measurement noise and decreasing size of the overlap regions. In all cases, it is possible to obtain a phase-aligned, composite reconstruction of the full time-resolved flow field from the phase-consistent modes over the entire domain.

The sedimentation of a suspension of rigid spherical particles in a polymeric fluid is studied employing experiments and particle-resolved numerical simulations. It is shown that the settling process in a viscoelastic Boger fluid is time-dependent and inhomogeneous; both experiments and simulations exhibit the formation of particle-rich, fast-settling regions and particle-depleted regions with backflow. The settling process in a Newtonian fluid is investigated for comparison and is shown to be homogeneous, with velocity fluctuations that decay following random mixing. In the viscoelastic Boger fluid, the mean settling rate is higher than that measured in a similar Newtonian suspending fluid, and in our experiments, higher than that of a single particle settling in isolation. When a cross shear flow is imposed, the mean settling rate in the viscoelastic Boger fluid is drastically reduced; the magnitude of the reduction is greater than that measured for a single particle, suggesting that both fluid elasticity and the suspension's particle volume fraction have important effects.

The forward enstrophy cascade in two-dimensional quantum turbulence in a superfluid film connected to a thermal bath is investigated using a Fokker-Planck equation based on Kosterlitz-Thouless renormalization. The steady-state cascade is formed by injecting vortex pairs of large initial separation at a constant rate. They diffuse with a constant flux to smaller scales, finally annihilating when reaching the core size. The energy spectrum varies as $k^{-3}$, similar to the spectrum known for two-dimensional (2D) classical-fluid enstrophy cascades. The dynamics of the cascade can also be studied, and for the case of a sharply peaked initial vortex-pair distribution, it takes about four eddy turnover times for the system to evolve to the decaying $k^{-3}$ cascade, in agreement with recent computer simulations. These insights into the nature of the cascade also allow a better understanding of the phase-ordering process of temperature-quenched 2D superfluids, where the decay of the vorticity is found to proceed via the turbulent cascade. This connection with turbulence may be a fundamental characteristic of phase ordering in general.

The ideal droplet impact, i.e., the orthogonal impact on a smooth surface, results in the formation of an axisymmetric lamella followed by various symmetry-breaking instabilities. Impacts found in nature are nonideal, i.e., are affected by symmetry-breaking factors, e.g., surface topography, surface elasticity, impact nonorthogonality, etc. This work is focused on oblique impacts. The lamella retains a nearly circular shape during the early stages of such impacts, showing a weak effect of the inclination angle. A strong effect is observed during splashing which occurs at different locations along the lamella edge. The variations of the location of the border between the splash and no-splash zones as a function of the relevant parameters were determined. New features of the splashing referred to as abnormal splashing were observed at reduced ambient pressures. It is shown that the abnormal splashing is a direct consequence of the nonmonotonic variations of the threshold pressure as a function of the Weber number for orthogonal impacts. A simple model being able to capture the basic features of the splash process was proposed and validated through comparisons with the experimental results.

This paper presents an unsteady discrete vortex model to simulate the unsteady aerodynamics of a wing encountering a large-amplitude transverse gust with a sine-squared velocity profile. The model is used to investigate wing-gust encounters and the impact of closed-loop control on mitigating the gust. A simple proportional closed-loop control law based on pitching input about the wing's midchord is selected to regulate lift during a gust encounter by evaluating its rejection of lift disturbances using Theodorsen's unsteady aerodynamic model. The control law is tested on the discrete vortex model and is shown to reduce the peak lift coefficient of the gust encounter by 92.3% for gust ratio $\text{GR}=V_{\text{max}}/U_{\infty }=0.50$ and by 91.8% for $\text{GR}=1.00$. The flow field and vorticity shedding behavior of the leading edge is studied and the effect of closed-loop control on the circulatory contribution to lift is discussed. The total lift force is decomposed into its individual contributions to study the various mechanisms involved in achieving the lift regulation objective.

Basic two-layer shock-joining theories for internal hydraulic jumps in flows with upstream shear do not have solutions when the shear becomes large. These theories conserve momentum flux and layer volume flow rate across the jump, and conserve energy in either of the two layers to close the set of equations. Alternatively, a closed set of equations conserving vorticity and layer volume flow rate can be used to predict jumps. As shear increases, the physics of the jumps changes, and entrainment becomes important. The theories can be modified to allow entrainment and to indirectly account for continuous velocity profiles, producing solutions where the basic theories failed. These jumps are investigated to illuminate the changing physics as shear increases. Simulations show that the amount of entrainment is related to the square of the upstream shear in both two-dimensional (2D) and three-dimensional (3D) simulations. However, they also show that the two-layer approximation becomes increasingly inaccurate, and 2D and 3D simulations disagree, at very high shear values. Furthermore, super- to super-critical transitions are also possible at higher shear, and while they exhibit a more gradual transition than a hydraulic jump, they can be analyzed using the same framework. The simulation results also illustrate the changing structure, vorticity balance, mixing, and energetic properties of the jumps as the upstream shear increases.

We investigate experimentally stratified turbulence forced by waves. Stratified turbulence is present in oceans and it is expected to be dominated by nonlinear interaction of internal gravity waves as described by the Garrett and Munk spectrum. To reach turbulent regimes dominated by stratification we use the Coriolis facility in Grenoble (France), where the large size enables us to reach regimes with both low Froude number and large Reynolds number. Stratification is obtained by using vertically linearly varying salt concentration and we force large-scale waves in a $6\times 6\times 1{\mathrm{m}}^3$ domain. We perform time-resolved particle image velocimetry to probe the space-time structure of the velocity field. We observe a wide band spectrum which is made of waves. Discrete modes are observed due to the square shape of the flow container as well as a continuum part which appears consistent with an axisymmetric superposition of random weakly nonlinear waves. Our observations support the interpretation of turbulence of a strongly stratified fluid as wave turbulence of internal waves although our spectrum is quite different from the Garrett and Munk spectrum. Weak turbulence proceeds down to a small cutoff length scale (the buoyancy wavelength) at which a transition to more strongly nonlinear turbulence is expected.

Dynamic behaviors of spherical microbubbles, within a one-way coupling frame, in stratified turbulence are investigated by direct numerical simulation. To simulate stratified turbulence, the Navier-Stokes equations and heat equation with a background linear temperature variation are solved in a periodic cube domain. Stratification creates a predominantly horizontal motion in a fluid, thereby inducing the well-defined “quasi” mean oscillatory horizontal flow. By solving the equation of motion for microbubbles, we found that the clustering of bubbles with a Stokes number below 0.1 becomes weaker as the stratification is increased. Consequently, the reduction of rise velocity of bubbles in turbulence compared to that in still fluid decreases from 7% to 2%. Owing to the alternating mean motion of the fluid, bubbles rise in a zigzag pattern, thereby resulting in an oscillatory Lagrangian correlation of the horizontal velocity component of bubbles. Despite this oscillating rising motion of the bubbles, the horizontal dispersion of a single bubble is suppressed. From the statistics of the horizontal separation of pair bubbles, we observe a power-law growth in the Batchelor and Richardson regimes for the separation.

Solutal buoyancy has a large impact on the flow of the alloy phase composing the positive electrode in liquid metal batteries. During discharge solutal buoyancy creates a stabilizing stratification, during charge it creates a vigorous solutal convection. In this article we provide new physical understandings of the role of solutal buoyancy during both charge and discharge. In particular we find that during discharge the electrovortex mechanism is in general not strong enough to counter the stabilizing effect of solutal buoyancy, and therefore this mechanism cannot be used to mix the alloy as is sometimes suggested in the literature. We show that the mixing capability of a generic flow in the alloy phase can be estimated by comparing the typical flow magnitude $U$ to two velocity scales: $U_p$ and $U_m$. Below $U_p$ the flow cannot mix the alloy, and above $U_m$ the flow significantly opposes solutal buoyancy. Although we focus on $\mathrm{Li}\left|\right|\mathrm{Pb}$-based batteries, these simple mixing criteria can be used during the discharging phase in other types of liquid batteries. We also present new, fully three-dimensional simulations of solutal convection during the charging cycle. These simulations suggest scaling laws for the magnitude of the convective flow, the time for the onset of solutal convection, and the typical inhomogeneity level in the alloy during charge. We propose physical arguments to explain these scaling laws.

The present work deals with the characteristics of the streamwise component of the fluctuating wall-shear stress ${\tau }_w^{\prime }$ in a smooth-wall turbulent boundary layer. A near-wall two-dimensional time-resolved particle image velocimetry measurement was performed in a low-to-moderate Reynolds number range (${\text{Re}}_{\tau }=400\sim 2900$). The algorithm of single-row cross correlation, together with a newly proposed iterative fitting method, was used to estimate ${\tau }_w^{\prime }$ from instantaneous near-wall velocity fields at single pixel row resolution. Additionally, existing direct numerical simulation datasets in the same ${\text{Re}}_{\tau }$ range were also analyzed for direct comparison. The fluctuating intensity of ${\tau }_w^{\prime }$ is found to follow the empirical log-law correlation, i.e., ${\sigma }_{{\tau }_w}^{+}\propto ln{\text{Re}}_{\tau }$. This indicates the existence of the influence of outer-layer large-scale or very-large-scale motions (LSMs or VLSMs) on the wall. The following scale decomposition analysis suggests an inner-scaling nature of the cutting-off boundary between small- and large-scale components of ${\tau }_w^{\prime }$, the magnitude of which is ${\lambda }_{xc}^{+}\approx 1000$, equivalent to the mean streamwise length of near-wall streaky structures. Combining with the weak Re-dependency of both the skewness of ${\tau }_w^{\prime }$ and its convection speed, i.e., $S_{{\tau }_w}\approx 1$ and ${\overline{u_c}}^{+}\approx 12$, it can be inferred that in the present studied Re range, LSMs or VSLMs only leave a mild footprint on the wall. Instead, the amplitude modulation effect is inferred to play a more prominent role in affecting both the statistics and the multiscale characteristics of ${\tau }_w^{\prime }$.

Jetting is a universal phenomenon frequently observed in nature and industries, for instance, in rain drop impact, inkjet printing, spray cooling, fuel atomization, etc. In drop impact on a superheated surface, we observe the formation of a vapor cavity beneath the dynamic Leidenfrost drop and a consecutive downward ejection of a jet into the cavity using ultrafast x-ray phase contrast imaging. We reveal that the cavity is induced mostly by the retraction of the drop and the jetting is caused by the convergence of capillary waves along the liquid-cavity interface. We find a jetting criterion based on the viscous damping of capillary waves: $\left[\mathrm{OhW}{\mathrm{e}}^2\right]\le 66\pm 10$. These results can provide important insight that leads to understanding and modeling of jets in nature.