The motion of weakly inertial Brownian particles, transported by steady two-dimensional fluid flows, is investigated by means of asymptotic methods. This study focuses on the phenomenon of noise-induced separatrix crossing, which can force particles to enter or exit recirculation cells in an unpredictable manner. An analytical expression for the probability of separatrix crossing is obtained. It can be applied to a wide variety of flows, provided some elementary kinematical quantities of the fluid flow are known. It does not require solving particle trajectories.

Recent numerical simulations predict the formation of a thin circulation layer (TCL) near the interface in a steady axisymmetric swirling flow of two immiscible fluids. This phenomenon, of fundamental and practical interest, has not been detected in prior experiments. The current experimental study reveals that the TCL does develop, but in a way which has not been predicted. The flow is driven by the rotating lid in a vertical cylindrical container—a model of a vortex bioreactor. The centrifugal force pushes the upper fluid from the axis to the sidewall near the lid and the fluid goes back to the axis near the interface. This centrifugal circulation (CC) drives the anticentrifugal circulation (AC) of the lower fluid at a slow rotation. As the rotation speeds up, a CC cell emerges below the interface-axis intersection. It expands downward and occupies almost the entire lower-fluid domain while the AC shrinks into a thin ring adjacent to the interface-sidewall intersection. As the interface starts to visibly rise near the axis, a new AC cell emerges below the interface-axis intersection. This cell and the AC ring merge and form the TCL, attached to the entire interface from below. The TCL is observed in a wide range of the lid angular velocity, until the flow becomes unsteady and three-dimensional. The resulting multicellular flow pattern is favorable for efficient mixing in bioreactors.

The numerical simulation of supersonic flow over a cone is carried out to investigate oblique detonation waves. A three-dimensional (3D) conical oblique detonation wave is studied by changing the heat release. It is found that the formation of a conical oblique detonation wave shifts from a moderate transition to an abrupt transition and the frontal structure also changes from smooth to cellular features. Moreover, the conical oblique detonation wave approaches detachment as heat release increases. A comparison of oblique detonation waves attached to a 2D wedge and a 3D cone demonstrates that, for a fixed heat release, a cone is able to moderate the transition significantly, and that detaching behavior is also delayed significantly due to curvature as heat release changes. The critical heat release for detachment of the conical oblique detonation wave is much larger than that of the wedge-induced oblique detonation wave. Moreover, we assess the difference in angles of oblique detonation waves produced by wedges and cones and find that the angle is much smaller than that of wedge-induced oblique detonation wave because of flow divergence caused by the curvature (curved front in the circumference direction of the cone).

An object withdrawn from a liquid bath is coated with a thin layer of liquid. Along with the liquid, impurities such as particles present in the bath can be transferred to the withdrawn substrate. Entrained particles locally modify the thickness of the film, hence altering the quality and properties of the coating. In this study, we show that it is possible to entrain the liquid alone and avoid contamination of the substrate, at sufficiently low withdrawal velocity in diluted suspensions. Using a model system consisting of a plate exiting a liquid bath, we observe that particles can remain trapped in the meniscus which exerts a resistive capillary force to the entrainment. We characterize different entrainment regimes as the withdrawal velocity increases: from a pure liquid film, to a liquid film containing clusters of particles, and eventually individual particles. This capillary filtration is an effective barrier against the contamination of substrates withdrawn from a polluted bath and finds application against biocontamination.

In experiments and numerical simulations we measure angles between the symmetry axes of small spheroids advected in turbulence (passive directors). Since turbulent strains tend to align nearby spheroids, one might think that their relative angles are quite small. We show that this intuition fails in general because angles between the symmetry axes of nearby particles are anomalously large. We identify two mechanisms that cause this phenomenon. First, the dynamics evolves to a fractal attractor despite the fact that the fluid velocity is spatially smooth at small scales. Second, this fractal forms steps akin to scar lines observed in the director patterns for random or chaotic two-dimensional maps.

In the present work, we assess the capabilities of neural networks to predict temporally evolving turbulent flows. In particular, we use the nine-equation shear flow model by Moehlis et al. [New J. Phys. 6, 56 (2004)] to generate training data for two types of neural networks: the multilayer perceptron (MLP) and the long short-term memory (LSTM) networks. We tested a number of neural network architectures by varying the number of layers, number of units per layer, dimension of the input, and weight initialization and activation functions in order to obtain the best configurations for flow prediction. Because of its ability to exploit the sequential nature of the data, the LSTM network outperformed the MLP. The LSTM led to excellent predictions of turbulence statistics (with relative errors of $0.45\%$ and $2.49\%$ in mean and fluctuating quantities, respectively) and of the dynamical behavior of the system (characterized by Poincaré maps and Lyapunov exponents). This is an exploratory study where we consider a low-order representation of near-wall turbulence. Based on the present results, the proposed machine-learning framework may underpin future applications aimed at developing accurate and efficient data-driven subgrid-scale models for large-eddy simulations of more complex wall-bounded turbulent flows, including channels and developing boundary layers.

This paper presents the experimental results from an air bubble-in-chain induced flow at different bubble release frequencies, resulting in a range of 1510–2318 for the Reynolds number based on bubble parameters. Mean flow and turbulent statistics were measured using planar particle image velocimetry. The bubble rise velocities were validated to be the superposition of the mean vertical velocity of water and the terminal velocity of the isolated bubbles. We tested the scaling parameters for the bubble-in-chain induced flow using those typically for bubble plumes, which demonstrates the intrinsic similarities of these two bubbly flows. We found that the radial profiles of normalized mean vertical velocity for bubble-in-chain induced flows and bubble plumes collapse well using the center-line velocity and the half-width of the velocity profile. We found the half-width correlated with the characteristic length scales of the bubble wakes. The normalized Reynolds stresses using the local velocity square show the same shape of profiles for both bubbly flows, but they do not collapse onto universal curves. The normalized turbulent kinetic energy is a function of normalized gas flow rate and approaches an asymptotic value of 0.05. The dynamic length scale, a function of initial kinematic buoyancy flux, was found to govern turbulent dissipation rates in both bubbly flows. The normalized one-dimensional velocity energy spectra collapse in the regime of $k_v\eta >0.1$ where $k_v$ is the wave number and $\eta$ is the Kolmogorov length scale, showing a $-3$ slope as the result of energy balance between production from bubble motion and dissipation in the bubble wakes. The wake length scale was found to define the lower-end wave number of the $-3$ slope.

We investigate a small vortex-lattice system of four corotating vortices in an atomic Bose-Einstein condensate and find that the vortex dynamics display chaotic behavior after a system quench introduced by reversing the direction of circulation of a single vortex through a phase-imprinting process. By tracking the vortex trajectories and Lyapunov exponent, we show the onset of chaotic dynamics is not immediate, but occurs at later times and is accelerated by the close approach and separation of all vortices in a scattering event. The techniques we develop could potentially be applied to create locally induced chaotic dynamics in larger lattice systems as a stepping stone to study the role of chaotic events in turbulent vortex dynamics.

Raindrop impacts on plant leaves are responsible for the dispersal of several crop diseases. In this experimental study, we propose a generic impact configuration that shares several qualitative features with more complex impacts on leaves. We investigate the formation and breakup of a liquid sheet upon impact of a drop close to the horizontal edge of an inclined substrate. The impact Weber number, the distance from the impact point to the edge, and the substrate inclination are systematically varied. We analyze their influence on the kinematics of both the liquid spreading on substrate and the liquid sheet beyond the edge, as well as on the subsequent statistics of droplet ejection (radius, speed, and time of ejection).

We study the deformation and breakup of an axisymmetric electrolyte drop which is freely suspended in an infinite dielectric medium and subjected to an imposed electric field. The electric potential in the drop phase is assumed to be small, so that its governing equation is approximated by a linearized Poisson-Boltzmann or modified Helmholtz equation (the Debye-Hückel regime). An accurate and efficient boundary integral method is developed to solve the low-Reynolds-number flow problem for the time-dependent drop deformation, in the case of arbitrary Debye layer thickness. Extensive numerical results are presented for the case when the viscosity of the drop and surrounding medium are comparable. Qualitative similarities are found between the evolution of a drop with a thick Debye layer (characterized by the parameter $\chi \ll 1$, which is an inverse dimensionless Debye layer thickness) and a perfect dielectric drop in an insulating medium. In this limit, a highly elongated steady state is obtained for sufficiently large imposed electric field, and the field inside the drop is found to be well approximated using slender-body theory. In the opposite limit $\chi \gg 1$, when the Debye layer is thin, the drop behaves as a highly conducting drop, even for moderate permittivity ratio $Q={\epsilon }_1/{\epsilon }_2$, where ${\epsilon }_1,{\epsilon }_2$ is the dielectric permittivity of drop interior and exterior, respectively. For parameter values at which steady solutions no longer exist, we find three distinct types of unsteady solution or breakup modes. These are termed conical end formation, end splashing, and open end stretching. The second breakup mode, end splashing, resembles the breakup solution presented in a recent paper [R. B. Karyappa et al., J. Fluid Mech. 754, 550 (2014)]. We compute a phase diagram which illustrates the regions in parameter space in which the different breakup modes occur.

Using direct numerical simulations, we examine the effects of the Taylor Reynolds number, $R_{\lambda }\equiv u^{\prime }\lambda /\nu$ (where $u^{\prime }$, $\lambda$, and $\nu$ denote the fluid root-mean-squared velocity, the Taylor microscale and the fluid kinematic viscosity, respectively), and Froude number, $\text{Fr}\equiv a_{\eta }/g$ (where $a_{\eta }$ is the Kolmogorov acceleration, and $g$ is the acceleration due to gravity), on the motion of small, spherical, settling, bidisperse inertial particles (characterized by their Stokes number $\text{St}\equiv {\tau }_p/{\tau }_{\eta }$, which is the ratio of the particle response time to the Kolmogorov timescale) in isotropic turbulence. Particle accelerations play a key role in the relative motion of bidisperse particles, and we find that reducing Fr leads to an enhancement of the accelerations but a suppression of their intermittency. For Stokes numbers $\text{St}>1$, the effect of $R_{\lambda }$ on the accelerations is enhanced by gravity, since settling causes the particle accelerations to be affected by a larger range of flow scales. The results for the probability density function (PDF) of the particle relative velocities show that even when the particles are settling very fast, turbulence continues to play a key role in their vertical relative velocities, and increasingly so as $R_{\lambda }$ is increased. This occurs because although the settling velocity may be much larger than typical velocities of the turbulence, due to intermittency, there are significant regions of the flow where the turbulence contribution to the particle motion is of the same order as that from gravitational settling. Increasing $R_{\lambda }$ enhances the non-Gaussianity of the relative velocity PDFs, while reducing Fr has the opposite effect, and for fast settling particles, the PDFs become approximately Gaussian. Finally, we observe that low-order statistics such as the radial distribution function and the particle collision kernel are strongly affected by Fr and St, and especially by the degree of bidispersity of the particles. However, we also find that these low-order statistics are very weakly affected by $R_{\lambda }$ when $\text{St}\le O\left(1\right)$, irrespective of the degree of bidispersity. Therefore, although the mechanisms controlling the collision rates of monodisperse and bidisperse particles are different, they share the property of a weak sensitivity to $R_{\lambda }$ when $\text{St}\le O\left(1\right)$.

Two-dimensional (2D) flows can be controlled efficiently using spanwise “waviness,” i.e., a control (e.g., wall blowing and suction or wall deformation) that is periodic in the spanwise direction. This study tackles the global linear stability of 2D flows subject to small-amplitude three-dimensional (3D) spanwise-periodic control. Building on previous work for parallel flows, an adjoint method is proposed for computing the second-order sensitivity of eigenvalues. Since such control has indeed a zero net first-order (linear) effect, the second-order (quadratic) effect prevails. The sensitivity operator allows one (i) to predict the effect of any control without actually computing the controlled flow and (ii) to compute the optimal control (and an orthogonal set of suboptimal controls) for stabilization and destabilization or frequency modification. The proposed method takes advantage of the very spanwise-periodic nature of the control to reduce computational complexity (from a fully 3D problem to a 2D problem). The approach is applied to the leading eigenvalue of the laminar flow around a circular cylinder, and two kinds of spanwise-harmonic control are explored: (i) wall actuation via blowing and/or suction and (ii) wall deformation. By decomposing the eigenvalue variation, we found that the 3D contribution (from the spanwise-periodic first-order flow modification) is generally larger than the 2D contribution from the mean flow correction (spanwise-invariant second-order flow modification). Over a wide range of control spanwise wave numbers, the optimal control for flow stabilization is symmetric about the wake centerline, leading to varicose streaks in the cylinder wake. Analyzing the competition between amplification and stabilization shows that optimal varicose streaks are not significantly more amplified than sinuous streaks but have a stronger stabilizing effect. The optimal wall deformation induces a flow modification very similar to that induced by the optimal wall actuation. In general, spanwise and tangential actuation have a small contribution to the optimal control, so normal-only actuation is a good trade-off between simplicity and effectiveness. Our method opens the way to the systematic design of optimal spanwise-periodic control for a variety of control objectives other than linear stability properties.

Turbulent flows over streamwise traveling wavy boundaries are investigated by large eddy simulations at a friction Reynolds number of ${\mathrm{Re}}_{\tau }=1000$. Four wave ages (i.e., the ratio between wave phase speed and bulk mean velocity) are considered, sequentially corresponding to the wave moving against the wind, stationary wavy wall, and intermediate and fast waves. A triple decomposition is performed to extract the mean, wave-induced, and turbulent components of the flow field. Very large-scale motions (VLSMs) of turbulent flow are identified by using the one-dimensional premultiplied energy spectra, instantaneous flow fields and conditionally averaged results. Compared with the flat-wall case, VLSMs in the negative wave age and stationary wavy wall cases are stronger, with larger length scales in both the streamwise and spanwise directions. The length scale and intensity of these motions decrease as the wave age increases. The conditionally averaged VLSMs involve the elongated low- and high-speed momentum regions and the roll cells in the streamwise direction. The transport equation of the two-point velocity correlation is investigated in different length scales by applying a spectral analysis. The wave-induced production that represents the interaction between the wave-induced and turbulent components of flow velocities provides extra input for the large-scale energy at low wave ages but play an opposite role at high wave ages.

Numerical simulations are used to explore the use of flow-induced flutter of a membrane as a mechanism for effective mixing at inertial microscales. A simple two-dimensional model of mixing in a channel is employed and fluid-structure interaction modeling with an immersed boundary method-based solver is used to explore the flow physics, flutter dynamics, and scalar mixing of these flutter mixers. The performance of the mixers is characterized in terms of a mixing index and a nondimensional head loss, and the effect of Reynolds and Schmidt numbers, as well as the membrane length on these performance indices, is examined. Simulations indicate that flow-induced flutter is capable of significantly enhancing mixing for channel Reynolds numbers as low as 15 and the relatively low associated pressure loss might make these flutter mixers a viable alternative for such applications.

We use the feedback formulation of McKeon and Sharma [J. Fluid Mech. 658, 336 (2010)], where the nonlinear term in the Navier-Stokes equations is treated as an intrinsic forcing of the linear resolvent operator, to educe the structure of fluctuations in the range of scales (wave numbers) where linear mechanisms are not active. In this region, the absence of dominant linear mechanisms is reflected in the lack of low-rank characteristics of the resolvent and in the disagreement between the structure of resolvent modes and actual flow features. To demonstrate the procedure, we choose low Reynolds number cylinder flow and the Couette equilibrium solution EQ1, which are representative of very low-rank flows dominated by one linear mechanism. The former is evolving in time, allowing us to compare resolvent modes with dynamic mode decomposition (DMD) modes at the first and second harmonics of the shedding frequency. There is a match between the modes at the first harmonic but not at the second harmonic where there is no separation of the resolvent operator's singular values. We compute the self-interaction of the resolvent mode at the shedding frequency and illustrate its similarity to the nonlinear forcing of the second harmonic. When it is run through the resolvent operator, the “forced” resolvent mode shows better agreement with the DMD mode. A similar phenomenon is observed for the fundamental streamwise wave number of the EQ1 solution and its second harmonic. The importance of parasitic modes, labeled as such since they are driven by the amplified frequencies, is their contribution to the nonlinear forcing of the main amplification mechanisms as shown for the shedding mode, which has subtle discrepancies with its DMD counterpart.

The large-scale hydrodynamics of the solar tachocline have been modeled by a version of the well-known shallow water equations augmented to include the effects of magnetism. Due to the conservation of magnetic flux, these shallow water magnetohydrodynamic equations have an additional constraint that the classical shallow water equations (without magnetism) do not. This restricts the set of possible initial conditions, but in itself does not provide information about the dynamics. The linear theory of wave motion on both the f plane and $\beta$ plane is relatively well documented. We thus seek to explore the dynamics of nonlinear waves on the magnetohydrodynamic f plane. By initializing with a circular vortex and no free surface displacement, we make contact with a substantial literature in geophysical fluid dynamics, satisfy the magnetohydrodynamic constraint, and provide a means to generate all possible wave species through the adjustment process. We find that for Cowling numbers typical of the solar tachocline, the response is dominated by zonally propagating Alfvén waves, which take the form of vortices. These vortices are expressed in the velocity, free surface, and magnetic field variables. We explore how nonlinearity and variation in the Cowling number affect these dynamical structures, finding that nonlinearity warps the axially symmetric Alfvén wave modes of linear theory. Moreover, while the waves maintain a fixed shape during pure propagation, we find that wave shape is significantly modified during collisions. The extent of warping during collisions is amplitude dependent, and hence a nonlinear effect, albeit one that is fundamentally different from soliton models of nonlinear waves in shallow water. We explore parameter ranges outside those relevant to our Sun's tachocline and find that our results are robust over significant portions of parameter space. In the limit of rapidly rotating stars (i.e., rates 100 times larger than those associated with our Sun) we find that the Alfvén waves destabilize over time.

We report on a series of numerical results about a pair of bubbles rising side by side in various silicone oils, with the terminal Reynolds number to vary between 10 and 700 and the aspect ratio of the bubble shapes to vary between 1.0 and 3.2. Through fully solving the three-dimensional Navier-Stokes equations, different interactive behaviors between the two bubbles are reproduced, such as the repelling, the attracting coalescence, the attracting bounce, and the repeated bounce, which are also observed in the experiments. First, we determine that, if there are no toroidal vortex rings attached at the rear end of the bubbles, the moderate- to high-Reynolds-number bubbles tend to attract with one another after being released, and, subsequently, whether they will coalesce or bounce after contact depends on the development of the vortex structures. In contrast, if toroidal vortex rings are generated in the bubble wake in other parameter spaces, a pair of bubbles will repel one another after being released, and a subsequent repeated bounce will be even observed. Lastly, an available force calculation model enables us to identify the relation between the forces and the vorticity produced on the bubbles when the pair of bubbles collide.

We study contact angle hysteresis in a chemically heterogeneous microchannel by tracking static meniscus configurations in the microchannel upon varying the volume of liquid. We first construct a graphical force balance similar to a previous approach by Joanny and de Gennes for this system, though here with a straight contact line. It is shown that hysteresis is induced by wettability gradients above a finite threshold value. This is also visualized in a phase-plane plot enabling to easily predict stick-slip events of the contact line and the occurrence of hysteresis. Above the threshold and for nonoverlapping Gaussian defects, we find good agreement with the expressions by Joanny and de Gennes for the hysteresis amplitude induced by a dilute system of defects. In particular, the hysteresis amplitude is found to be proportional to the square of the defect force and to the defect concentration. For a model sinusoidal heterogeneity, decreasing the ratio between the heterogeneity wavelength and the microchannel gap size brings the system from a subthreshold regime, to a stick-slip dominated regime, and finally to a regime with a quasiconstant advancing and receding angle. In the latter case, the hysteresis amplitude is found to be proportional to the defect force. We also consider an unusual heterogeneity for which the gradients of increasing and decreasing wettability are different. In such a situation breaking the left/right symmetry, whether or not hysteresis is observed will depend on the side the liquid enters the microchannel.

Evaporating salty droplets are ubiquitous in nature, in our home and in the laboratory. Interestingly, the transport processes in such apparently simple systems differ strongly from evaporating “freshwater” droplets since convection is partly inverted due to Marangoni stresses. Such an effect has crucial consequences to the salt crystallization process and to the deposits left behind. In this work we show unprecedented measurements that, not only confirm clearly the flow inversion, but also elucidate their impact on the distribution of nonvolatile solutes. Contrary to what has been often reported in the literature, such a flow reversal does not prevent the formation of ring-shaped stains: particles accumulate at the contact line driven solely by the interfacial flow. We can therefore conclude that the classical “coffee-stain effect” is not the only mechanism that can generate ring-shaped stains in evaporating droplets.

Dynamics of a bubble impacting and sliding on a tilted surface has been investigated through experimental and computational methods. In experiments, bubble-wall interaction has been characterized with an air bubble of about 550 $\mu \mathrm{m}$ in radius and a rising speed of about 30 cm/s as it impacts a solid substrate of a wall angle between $0^{\circ }$ and ${40}^{\circ }$. Specifically, shear stress generated on the wall has been calculated and compared with bacterium adhesion force in order to evaluate a potential sanitization function. We numerically solved a force balance including buoyancy, hydrodynamic inertia and drag, and lift and thin film force to determine the bubble motion. Results showed that the shear stress increases with the wall inclination. The maximum shear stress goes up to more than 300 Pa as a single bubble impacts and scrubs a tilted wall. We found that such a high shear stress is attributed to a rapid change in thin film curvature (flipping bubble-water interface) during the bouncing stage. Later, during the sliding stage, a smaller shear stress up to around 45 Pa is generated for a longer period. We also showed that the shear stress generated during the bouncing and sliding stages is high enough to remove bacteria from a surface as a potential method for removing bacteria from tilted surfaces.