This article presents a review and perspectives on the models for space-time energy spectra in turbulent shear flows. The Taylor, Kraichnan-Tennekes, and elliptic approximation (EA) models are re-examined in terms of the picture of turbulent passage, which is proposed by Taylor's frozen-flow hypothesis and the Kraichnan-Tennekes random sweeping hypothesis; the stochastic dynamic models for reproduction of space-time energy spectra, such as dynamic autoregression model, are discussed; and the statistical models for reconstruction of space-time energy spectra from incomplete data sets in experimental measurements are revisited. We present three distinct approaches of successive approximation for developing the models of space-time energy spectra and use the conditional moments of energy distribution to characterize space-time energy spectra, such as propagation velocities and spectral bandwidths.

In this direct numerical simulation (DNS) study, a fully developed turbulent plane Couette flow (pCf), where the bottom wall is roughened and the top smooth wall moves with a constant velocity $U_w$, is reported. For roughening, roughness elements in the form of square ribs of height $r=0.2h$ (where $h$ is channel half-height) are mounted only on the bottom wall with streamwise pitch separations of $5r$ and $10r$. The Reynolds number, based on half the velocity difference between the walls and $h$, is set to 1300. Upon roughening a wall, roughness would generate vortical structures near the vicinity of the roughness elements and enhance turbulence locally. We report that in the current problem, turbulence kinetic energy (TKE) is enhanced near both the rough and the smooth walls. The major factor aiding this process is the blockage effect offered by the rib roughness, which enhances the skin friction on both walls, and therefore the average friction Reynolds number is increased by 25% and 33% in the cases with pitches $5r$ and $10r$, respectively. In addition, in each of the rough pCf cases, streamwise variations in the mean flow, Reynolds stresses and the TKE budget terms are confined within the channel centerline, thereby indicating streamwise homogeneity above the channel centerline.

The head-on collision of two vortex rings can produce diverse phenomena: A tiara of secondary rings, vortex sheets which flatten and interact iteratively, or the violent disintegration of the rings into a turbulent cloud. The outcome of the interaction is determined by the nature of the instability affecting two impinging vortex rings. Here we carry out a systematic study to determine the dominant instability as a function of the parameters of the problem. To this end, we numerically simulate the head-on collision of vortex rings with circulation Reynolds numbers between 1000 and 3500 and varying slenderness ratios $\mathrm{\Lambda }=a/R$ ranging from $\mathrm{\Lambda }=0.1$ to 0.35, with $a$ the core radius and $R$ the ring radius. By studying the temporal evolution of the energy and viscous dissipation, we elucidate the role azimuthal instabilities play in determining what the outcomes of the collision are. We then compare these collisions to the head-on impact of a vortex ring on a free-slip and a no-slip wall. The free-slip wall imposes a mirror symmetry, which impedes certain instabilities and at sufficiently large Reynolds numbers leads to the formation of a half-tiara of vortices. Impact against a no-slip wall results in the process where a secondary vortex ring is formed after the ejection of the resulting boundary layer. When the Reynolds number is above a certain threshold, which increases with $\mathrm{\Lambda }$, the vortices disintegrate through azimuthal instabilities, resulting in a turbulent cloud.

Using a slender-jet approach, we numerically investigate the control of jet breakup using temperature modulation at the nozzle with a specified frequency and amplitude. Our results show that temperature modulation does lead to instability through capillary and Marangoni stresses, providing control of the droplet formation in terms of intact length and resultant drop size distribution, which is otherwise irregular due to inevitable presence of background noise. For understanding the mechanisms underpinning the breakup of a thermally modulated jet in the presence of noise, it is useful to decompose the surface tension forces into a contribution from curvature-gradient forces and a contribution from surface tension-gradient forces, associated with axial variations in the jet curvature and the temperature-dependent surface tension coefficient, respectively. We show that in the limit of slow axial heat diffusion and slow cooling to the ambient, as considered here, the breakup of a thermally modulated jet is governed by the ratio of the surface tension-gradient force to the imposed random perturbation force at nozzle exit. This so-called “thermal modulation strength number” depends on the amplitude and frequency of the thermal modulation, the sensitivity of the surface tension coefficient to variations in temperature, the Weber number, and the strength of the Gaussian white noise added to the nozzle exit velocity. We show that the thermal modulation strength number governs the shift in breakup characteristics from forward to rear pinchoff for increasing modulation strength as well as the nature of the instability. When thermal modulation is weak, the surface tension-gradient forces act only as a trigger, and curvature-gradient forces soon take over and grow exponentially downstream from the jet due to inertio-capillary growth. When thermal modulation is strong, the surface tension-gradient forces not only act as a trigger, but remain significant until breakup. The thermal modulation strength number is thus useful to the design of thermal modulation in practical applications as a possible alternative to often-used mechanical excitation mechanisms to control jet breakup.

Wind- and water-driven migrating sand dunes frequently interact with elevated natural and artificial topographical features. The dune-obstacle interaction can alter the migrating behavior of the dune and, depending on the nature of the obstacle, it may generate various societal and technological risks. Here we study the problem of dune-obstacle interaction in a paradigmatic quasi-two-dimensional domain realized in a subaqueous laboratory experiment. Generically, dunes interact with obstacles either by crossing over the obstacle or by being trapped. We describe how the selection of these two distinct dynamical behaviors depends on the size and shape of the obstacle, focusing in particular on the fluid flow in the immediate vicinity of the obstacle. Specifically, we perform a modal decomposition of the measured flow field and we discover that the outcome of the dune-obstacle interaction is closely related to the flow structure above the obstacle.

The local contact angle distribution within the pore space of a porous medium of mixed wettability is identified by combining pore network simulations and an optimization method. The latter is used to solve the inverse problem consisting of determining the local contact angle distribution from the capillary pressure and relative permeability curves. The inverse optimization method combines a genetic algorithm and the hill-climbing algorithm. The method is illustrated considering a Boolean distribution of the local contact angle in which the pore bodies and throats in the pore space are either hydrophilic or hydrophobic.

The higher order corrections to the equations that describe nonlinear wave motion in shallow water are derived from the water wave equations. In particular, the extended cylindrical Korteweg–de Vries and Kadomtsev-Petviashvili equations—which include higher order nonlinear, dispersive, and nonlocal terms—are derived from the Euler system in (2+1) dimensions, using asymptotic expansions. It is thus found that the nonlocal terms are inherent only to the higher dimensional problem, both in cylindrical and Cartesian geometry. Asymptotic theory is used to study the resonant radiation generated by solitary waves governed by the extended equations, with an excellent comparison obtained between the theoretical predictions for the resonant radiation amplitude and the numerical solutions. In addition, resonant dispersive shock waves (undular bores) governed by the extended equations are studied. It is shown that the asymptotic theory, applicable for solitary waves, also provides an accurate estimate of the resonant radiation amplitude of the undular bore.

The present study uses direct numerical simulations to calculate two different transitional flows from laminar flow to turbulence, that is, the two-scale wake flow and the two-dimensional Reyleigh-Taylor unstable flow, respectively. Both results show that the skewness of the longitudinal velocity gradient, $S_k$, can become positive in the early transition stage, which is beyond our expectation since the turbulent equilibrium state always implies negative values of $S_k$. These phenomena are explained analytically by considering only two dominant Fourier modes with harmonic relations. It is illustrated that the sign of $S_k$ is not only affected by the amplitudes of the perturbation velocities, but also related to their phases. We expect that the present results will be helpful for understanding the formation of turbulent equilibrium state and for constructing a new measurable criterion of the transition onset.

Based on the nearest particle statistics [Zhang, J. Fluid Mech. 910, A16 (2021)], the phase interaction force in a multiphase flow is decomposed into a particle–mean-field force and the divergence of the particle–fluid–particle (PFP) stress. The PFP stress is proportional to the correlation product of the distance from a particle to its nearest neighbor and the force on the particle conditionally averaged on the nearest-neighbor location. In this work, a functional form of the stress is obtained corrected to the first order of the ratio between the interparticle distance and the macroscopic length scale of the flow. Particle-resolved numerical simulations are used to calculate the PFP stress in random arrays of fixed particles and to explore the physics represented by the stress. The numerical results show that the PFP stress is attractive along the direction of the flow and is repulsive in the directions perpendicular to the flow. In the flow regime simulated, this PFP stress can be considered as a macroscopic representation of the drafting–kissing–tumbling mechanism. The Reynolds stress for the fluid phase is also calculated and compared with the PFP stress.

Dynamic jamming is a phenomenon whereby a dense suspension switches from a fluidlike to a solidlike state when subjected to sufficient stress and deformation. Large enough systems show that this transition is accompanied by a distinct jamming front. We present an experimental study where two jamming fronts are created simultaneously using two cylinders moving in parallel. We focus our observations on the collision of the jammed regions when the two fronts meet. Surprisingly, our measurements, combining surface texture visualization and time-resolved particle image velocimetry, show the formation of an unjammed region contained within the otherwise jammed suspension.

Interfacial swimmers are objects that self-propel at an interface by autonomously generating a gradient of surface tension, often through the continuous release of a surfactant. While the case of asymmetric swimmers has long been studied, experiments have shown that spontaneous motion is also possible for symmetric swimmers. The basic mechanism of symmetry-breaking is qualitatively well-established but one key aspect of the phenomenon that has proved particularly difficult to elucidate is the role of Marangoni effects in the self-propulsion. We address this question by numerical methods, which can fully handle the complex interplay between swimmer motion, fluid flow, surfactant distribution, and Marangoni stresses. Our swimmer is a disk releasing a soluble surfactant in a deep-layer fluid. We investigate how the swimming velocity, represented by a Péclet number ${\mathrm{Pe}}^{*}$ depends on its characteristics, as encapsulated in the Marangoni number $\mathrm{M}$. We analyze the properties of the swimming diagram ${\mathrm{Pe}}^{*}\left(\mathrm{M}\right)$ and compare with approximate models to understand their origin. We find that the low-${\mathrm{Pe}}^{*}$ regime exhibits a bistability region: spontaneous swimming involves a threshold Marangoni number, a discontinuity in velocity and possibly hysteresis. Those features are present only for a full description of the problem and reveal the subtle but key role of Marangoni flows. The large-${\mathrm{Pe}}^{*}$ regime features a robust asymptotic scaling law ${\mathrm{Pe}}^{*}\sim {\mathrm{M}}^{\alpha }$, whose exponent $\alpha \simeq 0.72$ is close to the $3/4$ value predicted by a simplified model, indicating a much weaker influence of Marangoni flows. While our results were obtained assuming a point-source swimmer in the Stokes flow regime, we show that the picture remains very similar when considering a spatially extended source size, finite Reynolds number, or a fixed concentration swimmer. We discuss our findings in relation to experiments.

A nonintrusive data assimilation methodology is developed to improve the statistical predictions of large-eddy simulations (LES). The ensemble-variational (EnVar) approach aims to minimize a cost function that is defined as the discrepancy between LES predictions and reference statistics from experiments or, in the present demonstration, independent direct numerical simulations (DNS). This methodology is applied to adjust the Smagorinsky subgrid model and obtain data assimilated LES (DA-LES) which accurately estimate the statistics of turbulent channel flow. To separately control the mean and fluctuations of the modeled subgrid tensor, and ultimately the first- and second-order flow statistics, two types of model corrections are considered. The first one optimizes the wall-normal profile of the Smagorinsky coefficient, while the second one introduces an adjustable steady forcing in the momentum equations to independently act on the mean flow. Using these two elements, the data assimilation procedure can satisfactorily modify the subgrid model and accurately recover reference flow statistics. In a posteriori testing, the retrieved subgrid model significantly outperforms more elaborate baseline models such as dynamic and mixed models. The robustness of the present data assimilation methodology is assessed by changing the Reynolds number and considering grid resolutions that are away from usual recommendations. Taking advantage of the stochastic formulation of EnVar, the developed framework also provides the uncertainty of the retrieved model.

In this work, we analyze the motion of an active particle, modeled as a spherical squirmer, in linearly varying viscosity fields. In general, the presence of a particle will disturb a background viscosity field and the disturbance generated depends on the boundary conditions imposed by the particle on the viscosity field. We find that, irrespective of the details of the disturbance, active squirmer-type particle tend to align down viscosity gradients (negative viscotaxis). However, the rate of rotation and the swimming speed along the gradient do depend on the details of the interaction of the particle and the background viscosity field. In addition, we explore the relative importance on the dynamics of the local viscosity changes on the surface of active particles versus the (nonlocal) changes in the flow field due to spatially varying viscosity (from that of a homogeneous fluid). We show that the relative importance of local versus nonlocal effects depends crucially on the boundary conditions imposed by the particle on the field. This work demonstrates the dangers in neglecting the disturbance of the background viscosity caused by the particle as well as in using the local effects alone to capture the particle motion.

The present paper aims to investigate the magnetohydrodynamic effects on binary liquid metal droplet collision in the presence of horizontal magnetic field ($0<B<1.5$  T) normal to collision velocity. Here we observe that for small magnetic interaction parameter $N$ ($<0.4$), one type of collision regimes, reflexive separation, could be obviously facilitated by magnetic field in comparison with no magnetic cases. For collision regimes except reflexive separation, we present a correlation of $N$ = $f(B/B_0,\text{We}/48)$ as well as a $N-\text{We}/48$ map that could help to interpret why little influence of magnetic field on them and predict possible remarkable influences when $B$ is larger than a critical value of 4 T. Furthermore, we draw the possible geometrical morphology of droplets after collision based on our understanding of the physical process of liquid metal droplets collision under the horizontal magnetic field.

Electrohydrodynamic instabilities of fluid-fluid interfaces can be exploited in various microfluidic applications to enhance mixing, replicate well-controlled patterns, or generate drops of a particular size. In this work, we study the stability and dynamics of a system of three superimposed layers of two immiscible fluids subject to a normal electric field. Following the Taylor-Melcher leaky dielectric model, the bulk remains electroneutral while a net charge accumulates on the interfaces. The interfacial charge dynamics is captured by a conservation equation accounting for Ohmic conduction, advection by the flow, and finite charge relaxation. Using this model, we perform a linear stability analysis and identify different modes of instability, and we characterize the behavior of the system as a function of the relevant dimensionless groups in each mode. Further, we perform numerical simulations using the boundary element method to study the effect of nonlinearities on long-time interfacial dynamics. We demonstrate how the coupling of flow and surface charge transport in different modes of instability can give rise to nonlinear phenomena such as tip streaming or pinching of the film into droplets.

In the present work we show how the subgrid-scale (SGS) energy transfer among resolved scales in large eddy simulations (LESs) and its wave number distribution can be obtained from the evolving LES velocity fields. This information, supplemented by the known asymptotic properties of energy flux in the inertial range, when cast in the form of a spectral eddy viscosity, allows self-contained simulations without use of extraneous SGS models. The method is tested in LESs of isotropic turbulence at high Reynolds number where the inertial range dynamics is expected and for lower-Reynolds-number decaying turbulence under conditions of the classical Comte-Bellot–Corrsin experiments.

This article elucidates how unsteady hydrodynamic interactions between two closely situated spheres in a viscous liquid affect their time-dependent motion. The system represents typical Brownian particles for which temporal inertia is always comparable to the viscous forces even though convective inertia is negligible. The analysis quantifies the transient mutual interactions in terms of frequency-dependent friction coefficients of both spheres as well as their temporally varying mobility response to an impulsive force. To this end, a generalization of Stokesian dynamics is formulated, where instead of Stokes equation, linearized unsteady Navier-Stokes is Fourier transformed in frequency space to describe flow fields. Accordingly, two complete sets of basis functions for the Brinkman equation instead of the Stokes equation are constructed in spherical coordinates centered around two particles. The mutual transformations between these two sets enable the enforcement of the no-slip boundary conditions on all solid-liquid interfaces. The resulting algebraic relations provide the frequency-dependent two-body frictions, whereas inverse Fourier transform of these after adding appropriate inertial contributions yields a time-dependent mobility response. The friction and mobility values are validated in limiting cases under short-time and long-time limits. The scaling laws of these quantities are also explored as functions of the separation distance between two solid bodies, revealing important physical insight into the complicated dynamics.

Turbulence in two-dimensional flows is expected to produce a dynamical state in which energy condenses into the largest scale allowed by the system when the scale at which energy is dissipated exceeds the domain size. We study this phenomenon in a laboratory quasi-two-dimensional turbulent flow in a thin layer of electromagnetically driven fluid where the energy is primarily dissipated by bottom friction. By inserting boundaries of different sizes, we fix the driving and damping and vary only the domain size. Although we observe flow patterns that are consistent with previous claims of spectral condensation, we see no signatures in the energy spectrum. An analysis of the scale-to-scale energy flux reveals that small domains weaken the turbulent cascade, even though the bulk forcing and frictional dissipation remain the same. Our results suggest that we lack a robust set of criteria for the existence of spectral condensation, and that claims of condensation in experimental flows with small scale separations must be supported by strong evidence.

Vortex rings are generated every time an impulsive jet occurs in an ambient homogeneous fluid, for instance in thermal or density plumes in environmental applications. Those vortical structures, often referenced as building blocks of turbulence, exist in a wide range of scales. The velocity field associated with these toroidal vortical structures makes them self-propagating. Such vortices are likely to carry mass and momentum along their path and are therefore good candidates for driving mixing to locations that are remote from where the ring is created. In numerous environmental applications, vortex rings interact with a density stratification due to variations of salinity and/or temperature. The path followed by a light homogeneous and axisymmetric vortex ring impinging a layer of fluid stably stratified in density with a given angle $\theta$ to the vertical, along with its dynamics and subsequent reorganization, are experimentally addressed in the present study. A three-dimensional time-resolved particle tracking velocimetry technique is set up to highlight the response of the stratified layer that is characterized by the generation of baroclinic vorticity and internal gravity waves. For the experimental parameters of the present study, the vorticity generation and reorganization, as well as the internal gravity waves in the stratified layer, are shown to remain axisymmetric for normal impacts. When the vortex ring propagation axis is not aligned with the density gradient, the expected symmetry breaking of the flow is observed. For inclined impact, the flow reorganizes in the form of a vertically flattened dipolar structure. In this case, internal gravity waves radiate away from two sources that match with the dipole cores.

A trailing vortex behind a wing moving through a stratified fluid will act to twist the density field into a pattern where the density profile continuously overturns along the axis of the vortex. This configuration is approximated here with a density field that is overturning periodically along the axis of the vortex. The matching velocity field is found approximately assuming weak stratification and constant axial vorticity at leading order and confining the flow to a fixed radius. This base flow is shown to be unstable to a wave triad consisting of two disturbance waves and a component of the base flow as the third “wave.” Three components of this base flow lead to instability: (1) the twirling component, (2) the streaming component, and (3) the rolling component. All three instabilities depend strongly on the axial length for the density field to overturn (the pitch). The twirling and rolling instabilities are important when the pitch is small and they also depend on the Froude number. The streaming instability is dominant when the pitch is large and is independent of the Froude number.