The wetting and contact line dynamics of a leaky dielectric sessile droplet under an alternating (ac) electrostatic field applied in the vertical direction is investigated. A thin precursor film-based reduced-order model using the weighted residual integral boundary layer technique is developed. The limiting cases of perfect conducting and perfect dielectric droplets are also considered. It is shown that the droplet oscillates with a frequency twice that of the forcing potential due to the quadratic dependence of the Maxwell stress on the applied ac electric field. These oscillations take place about an equilibrium configuration, which can be achieved with a constant (dc) electric potential equivalent to the root-mean-square potential of the applied ac field. It is also shown that the contact line motion increases monotonically with the amplitude of the ac electric forcing. A significant increase in ac field leads to spiking and the interface ruptures at the top electrode. Depending on the static contact angle, the droplet deformation can become nonmonotonic as the applied frequency of the ac electric field increases. This behavior is attributed to the competition between the timescale of forcing and the timescale of the response as affected by the drop's wettability. The role of conductivity ratio, permittivity ratio, and different waveforms of ac forcing are also investigated.

The probability density function $p\left(k\right)$ of the turbulent kinetic energy $k$ is investigated for diabatic atmospheric surface layer (ASL) flows. When the velocity components are near-Gaussian and their squared amplitudes are nearly independent, the resulting $p\left(k\right)$ is shown to be $\gamma$-distributed with exponents that vary from 0.8 to 1.8. A nonlinear Langevin equation that preserves a $\gamma$-distributed $p\left(k\right)$, but allows linear relaxation of $k$ to its mean state, is proposed and tested using multiple ASL data sets. The three parameters needed to describe the drift and nonlinear diffusion terms can be determined from the ground shear stress and the mean velocity at height $z$. Using these model parameters, the Langevin equation reproduces the measured $p\left(k\right)$ with minimal Kullback-Leibler divergence.

A short focused laser pulse can generate a hot plasma, which expands then contracts and can eject a hot jet, the speed and direction of which is sensitive to the details of the plasma kernel. The coupling of thermal and chemical nonequilibrium plasma mechanisms with this hydrodynamic development is assessed with detailed simulations of a two-temperature, three-species plasma model. Time scales for electron recombination, thermal relaxation, and diffusion are compared to that of the plasma expansion to anticipate conditions in which these mechanisms might affect the vorticity generation that leads to the ultimate flow pattern. The effect of these mechanisms are analyzed through comparison with corresponding inert-gas and equilibrium models. Thermal-nonequilibrium effects are found to be weak due to rapid relaxation of the heavy-particle and electronic temperatures. In contrast, chemical equilibration occurs at a rate comparable to the expansion and thereby enhance both it and subsequent hydrodynamic mechanisms as the energy stored in ion formation during the preceding breakdown is released by electron recombination. Thermal conduction, enhanced by high-temperature free electrons, weakens the ejection.

In shock wave/boundary layer interactions, two mechanisms have been recognized to drive the low-frequency unsteadiness of the reflected shock: upstream boundary layer forcing and downstream feedback. The current work presents a quantitative analysis of the causal mechanisms underlying such flow unsteadiness. The analysis is based on a large-eddy simulation database covering approximately 300 cycles of the low-frequency shock fluctuations in a Mach 2 turbulent boundary layer. This time span enables the accurate application of frequency-domain system identification methods targeting such low frequencies. The evaluation of the spectrum in the interaction zone indicates that the broadband low-frequency unsteadiness is predominantly two-dimensional and can be isolated via spanwise averaging. Empirically derived transfer functions are computed using the averaged flow field and indicate the occurrence of a feedback between the locations downstream of the flow separation and the shock fluctuations. The results indicate that this mechanism dominates over the upstream forcing of the interaction region. Accordingly, the computed transfer functions are also used as an estimation tool to predict the shock motion accurately; for the largest streamwise separation between input and output signals, correlations above 0.6 are observed between predictions and raw data. Computation of spectral proper orthogonal decomposition modes reveals the existence of upstream traveling waves in the leading spectral mode at the main shock frequency; higher frequencies do not exhibit this trend. Furthermore, the spectral modes obtained using selected flow regions downstream of the shock enable the reconstruction of a significant portion of the energy in the interaction zone. Finally, a linear stability analysis is conducted using the mean turbulent flow, showing the existence of upstream traveling waves. Evaluation of a vortex sheet model indicates that these upstream traveling modes are of acoustic nature. The predicted modes from this local analysis present a compelling match against the spectral modes, both in terms of the shape and phase speed of the fluctuations. The combined analysis of the techniques indicates that downstream disturbances are the dominant cause of shock oscillations in the present configuration, leading to shock motion by upstream traveling acoustic modes.

Surfactants (amphiphilic surface active agents) are often used to control the dynamics of viscous drops and bubbles in microfluidics applications. Surfactant sorption kinetics has been shown to play a critical role in the deformation of drops in extensional and shear flows, yet to the best of our knowledge these kinetics effects on a viscous drop in an electric field have not been accounted for. In this paper we numerically investigate the effects of sorption kinetics on a surfactant-covered viscous drop in an electric field. Over a range of electric conductivity and permittivity ratios between the interior and exterior fluids, we focus on the dependence of deformation and flow on the transfer parameter $J$, and Biot number Bi that characterize the extent of surfactant exchange between the drop surface and the bulk. Our findings suggest solubility affects the electrohydrodynamics of a viscous drop in distinct ways as we identify parameter regions where (1) surfactant solubility alters both the drop deformation and circulation of fluid around a drop and (2) surfactant solubility affects mainly the flow and not the deformation.

We perform a numerical analysis of Rayleigh-Bénard convection during solid-liquid phase change in the presence of an electric field. The objective is to examine the effects of the melting boundary and the electric field on the onset of flow motion and pattern formation. We characterize the timescale separation between the dynamics of the flow and that of the interface, which is dependent on the Rayleigh number Ra, the Prandtl number Pr, the Stefan number St, and the Fourier number Fo. It is found that (i) the delay of onset is a notable feature of the systems exhibiting phase transitions and convective instability compared to the traditional Rayleigh-Bénard convection; (ii) in the absence of the electric field, the critical Rayleigh number decreases monotonically with the timescale separation; (iii) in the presence of the electric field, the critical Rayleigh number can be lower than the classical value of ${\mathrm{Ra}}_c=1707.76$. For moderate Rayleigh number ($\mathrm{Ra}={10}^5$), we predict the robust “locked-in” effect of the topography on convection. When the electric field is strong enough, this locked-in pattern will be broken and the system will evolve to a strong nonlinear regime in an inherently time-dependent fashion.

Advection-diffusion coupling can enhance particle and solute dispersion by orders of magnitude as compared to pure diffusion, with a steady state being reached for confined flow regions such as a nanopore or blood vessel. Here, by using evanescent wave microscopy, we measure for the first time the full dynamics of Taylor dispersion, highlighting the crucial role of the initial concentration profile. We make time-dependent, nanometrically resolved particle dispersion measurements varying nanoparticle size, velocity gradient, and viscosity in submicrometric near-surface flows. Such resolution permits a measure of the full dynamical approach and crossover into the steady state, revealing a family of master curves. Remarkably, our results show that the dynamics depend sensitively on the initial spatial distribution of the nanoparticles. These observations are in quantitative agreement with existing analytical models and numerical simulations performed herein. We anticipate that our study will be a first step toward observing and modelling more complex situations at the nanoscale, such as target finding and chemical reactions in nanoconfined flows, dynamical adsorption and capture problems, as well as nanoscale drug delivery systems.

Modeling of particle-size segregation in dense granular flows could benefit from a better understanding of the segregation behavior of individual large grains that are surrounded by smaller grains. In a previous study, the force in the direction of segregation experienced by such a particle was measured and decomposed into a modified Archimedean buoyancy force and a segregation lift force. Here we present a new micromechanical analysis of this granular buoyancy force that connects the microscale contact behavior to the macroscopic force, thereby lending further support to its validity. In the process of this validation we uncover evidence linking the average surface contact density on a grain to the scaling of the buoyancy force with its size. Our findings support the use of the existing Voronoi approximation for calculation of the granular buoyancy force and substantiate the decomposition of the net force into buoyancy and segregation lift. Ultimately, these insights will aid development of new models for size segregation by closely linking micro and macro behavior.

We consider the flow of Burgers's equation on an open set of (small) functions in $L^2\left([0,1]\right)$. We derive explicitly the Koopman decomposition of the Burgers flow. We identify the frequencies and the coefficients of this decomposition as eigenvalues and eigenfunctionals of the Koopman operator. We prove the convergence of the Koopman decomposition for $t>0$ for small Cauchy data and up to $t=0$ for regular Cauchy data. The convergence up to $t=0$ leads to a “completeness” property for the basis of Koopman modes. We construct all modes and eigenfunctionals, including the eigenspaces involved in geometric multiplicity. This goes beyond the summation formulas provided by Page and Kerswell [Phys. Rev. Fluids 3, 071901(R) (2018)], where only one term per eigenvalue was given. A numeric illustration of the Koopman decomposition is given and the Koopman eigenvalues compares to the eigenvalues of a dynamic mode decomposition.

Scalar fields which are subject to turbulent mixing typically feature a broad range of scales. When focusing on the large-scale dynamics, it remains a question how to effectively parametrize the small scales. Here, we address this question within the framework of a stochastic, one-dimensional passive scalar model. We show that small-scale averaging, i.e., an ensemble average over small-scale velocity fluctuations, results in an effective diffusivity reminiscent of phenomenological eddy viscosity models, while reducing the effective Reynolds number of the advecting velocity field. Based on that, we establish a filtering procedure that exactly maps second-order statistics of the fully resolved passive scalar field to the one obtained by small-scale averaging. Using fully resolved simulations, we show that small-scale averaging also captures higher-order large-scale statistics of passive scalar fields.

An investigation based on the application of spectral proper orthogonal decomposition (SPOD) of pressure measurements and Fourier analysis of velocity measurements is carried out to further elucidate the properties of separated flow behind a backward-facing ramp in uncontrolled and controlled conditions. Several flow control conditions are considered, varying the duty cycle and the frequency of an actuation system composed of 89 pulsed jets, issuing normally to the flow direction. The SPOD is applied to unsteady pressure measurements to characterize the main features of the flow and to extract the most energetically and dynamically relevant space-time coherent structures. Fourier analysis is applied to the velocity measurements and is complementary to the SPOD analysis. In uncontrolled conditions four major modes, together with their typical frequencies, have been identified and associated with (1) presence of large-scale longitudinal roll-up vortices which oscillate at the slanted ramp side edges, (2) flapping motion of the central recirculation region, (3) large-scale vortex emission from the shear layer at the slant edge, and (4) onset of high-frequency Kelvin-Helmholtz instability occurring in the shear layer near the slant edge. The control suppresses the recirculation mode and all associated mechanisms. As expected, the mode related to the lateral roll-up vortices persists. The modal analysis also shows that distinctive peaked bands emerge, which are ascribed to the presence of the control pulsed jets. Focusing on the SPOD mode associated with the actuation allows one to analyze the extension of the receptivity region of the flow to the action of control. Some peculiar aspects of SPOD field reconstruction are suggested by the specific physical characteristics of the flow.

Vortices play a fundamental role in the physics of two-dimensional (2D) fluids across a range of length scales, from quantum superfluids to geophysical flows. Despite a history dating back to Helmholtz, point vortices in a 2D fluid continue to pose interesting theoretical problems, owing to their unusual statistical mechanics. Here we show that the strongly interacting Helmholtz-Onsager vortex systems can form statistical edge modes at low energies, extending a previously identified analogy between vortex matter and quantum Hall systems. Through dynamical simulations, Monte-Carlo sampling, and mean-field theory, we demonstrate that these edge modes are associated with the formation of dipoles of real and image vortices at boundaries. The edge modes are robust, persisting in nonconvex domains, and are in quantitative agreement with the mean-field predictions.

The unsteady radial flow of a relatively dense fluid released from an axisymmetric lock is analyzed when it freely drains over an edge and when it generates a gravity current propagating over a horizontal surface. In both situations when modeled using the shallow water equations, despite initiation from rest within a lock, the flow thins, accelerates, and becomes supercritical in a region close to the symmetry axis. For free drainage, this alters the outflow, while for radial gravity currents, it leads to the formation of an internal jump connecting rapidly moving fluid to more tranquil flow at the front. Both scenarios share the same supercritical flow structure, which is related to their initiation from lock-release conditions and is a function of axisymmetry; the phenomenon is not found in analogous two-dimensional flows. Through analysis and numerical integration of the governing equations, the common onset and consequences of supercriticality are revealed in both flows, as well as, the progression of the fluid motions to self-similar states that develop at late times.

The viscous drag on a slender rod by a wall is important to many biological and industrial systems. This drag critically depends on the separation between the rod and the wall and can be approximated asymptotically in specific regimes, namely far from, or very close to, the wall, but is typically determined numerically for general separations. In this article we determine an asymptotic representation of the local drag for a slender rod parallel to a wall which is valid for all separations. This is possible through matching the behavior of a rod close to the wall and a rod far from the wall. We show that the leading order drag in both these regimes has been known since 1981 and that they can be used to produce a composite representation of the drag which is valid for all separations. This is in contrast to a sphere above a wall, where no simple uniformly valid representation exists. We estimate the error on this composite representation as the separation increases, discuss how the results could be used as resistive-force theory, and demonstrate their use on a two-hinged swimmer above a wall.

We perform numerical simulations on inertial migration of a non-neutrally buoyant particle with a density ratio of 0.98–1.02 in a linear shear flow dominated channel with a Reynolds number up to 500 in the presence of thermal convection using a double-population lattice Boltzmann method. It is found that under the isothermal condition, the particle with a larger density difference from the fluid will either settle to the bottom of the channel or float to the top of the channel, while the particle with a smaller particle-fluid density difference remains suspended in the channel due to the inertial lift force. The presence of thermal convection (characterized by the Grashof number $\mathrm{Gr}$) induces an additional downward lift force, which results in distinctive migration behaviors that depend on whether the particle density is larger or smaller than that of the fluid. For a particle heavier than the fluid, the settling is enhanced by thermal convection due to the synergistic effect of the thermal lift force and the gravitational force. The critical Reynolds number for lifting the particle increases compared with the isothermal case and is linearly correlated with the dimensionless density ratio ($\sigma$). On the other hand, for a particle lighter than the fluid, an empirical dimensionless number ${\mathrm{Gr}}^{*}$, defined as $\mathrm{Gr}/\left[\sigma \left(1.59\mathrm{Re}+9.31\right)\right]$, is introduced to characterize the particle migration. It is discovered that the particle's equilibrium position depends on whether it migrates to the top wall or remains suspended in the shear flow under the isothermal condition. For the former case, when thermal convection is introduced, the particle stays at the top wall when ${\mathrm{Gr}}^{*}<1$, and becomes suspended in the channel when ${\mathrm{Gr}}^{*}>1$.

Most studies of vortex shedding from a circular cylinder in a gas flow have explicitly or implicitly assumed that the no-slip condition applies on the cylinder surface. To investigate the effect of slip, vortex shedding is simulated using molecular gas dynamics (the direct simulation Monte Carlo method) and computational fluid dynamics (the incompressible Navier-Stokes equations with a slip boundary condition). A Reynolds number of 100, a Mach number of 0.3, and a corresponding Knudsen number of 0.0048 are examined. For these conditions, compressibility effects are small, and periodic laminar vortex shedding is obtained. Slip on the cylinder is varied using combinations of diffuse and specular molecular reflections with accommodation coefficients from zero (maximum slip) to unity (minimum slip). Although unrealistic, bounce-back molecular reflections are also examined because they approximate the no-slip boundary condition (zero slip). The results from both methods are in reasonable agreement. The shedding frequency increases slightly as the accommodation coefficient is decreased, and shedding ceases at low accommodation coefficients (large slip). The streamwise and transverse forces decrease as the accommodation coefficient is decreased. Based on the good agreement between the two methods, computational fluid dynamics is used to determine the critical accommodation coefficient below which vortex shedding ceases for Reynolds numbers of 60–100 at a Mach number of 0.3. Conditions to observe the effect of slip on vortex shedding appear to be experimentally realizable, although challenging.

Penetrative turbulence, which occurs in a convectively unstable fluid layer and penetrates into an adjacent, originally stably stratified layer, is numerically and theoretically analyzed. As example we pick the canonical Rayleigh-Bénard geometry, but now with the bottom plate temperature $T_b>4{}^{\circ }\mathrm{C}$, the top plate temperature $T_t\le 4{}^{\circ }\mathrm{C}$, and the density maximum around $T_m\approx 4{}^{\circ }\mathrm{C}$ in between, resulting in penetrative turbulence. Next to the Rayleigh number Ra, the crucial new control parameter as compared to standard Rayleigh-Bénard convection is the density inversion parameter ${\theta }_m\equiv (T_m-T_t)/(T_b-T_t)$. The crucial response parameters are the relative mean midheight temperature ${\theta }_c$ and the overall heat transfer (i.e., the Nusselt number Nu). We numerically show (for Ra up to ${10}^{10}$) and theoretically derive that ${\theta }_c\left({\theta }_m\right)$ and $\text{Nu}\left({\theta }_m\right)/\text{Nu}\left(0\right)$ are universally(i.e., independently of $\text{Ra}$) determined only by the density inversion parameter ${\theta }_m$ and succeed to derive these universal dependences. In particular, ${\theta }_c\left({\theta }_m\right)=(1+{\theta }_m^2)/2$, which holds for ${\theta }_m$ below a $\text{Ra}$-dependent critical value, beyond which ${\theta }_c\left({\theta }_m\right)$ sharply decreases and drops down to ${\theta }_c=1/2$ at ${\theta }_m={\theta }_{m,c}$. This critical density inversion parameter ${\theta }_{m,c}$ can be precisely predicted by a linear stability analysis. Finally, we numerically identify and discuss rare transitions between different turbulent flow states for large ${\theta }_m$.

The dynamics of a bi-dimensional solutal plume generated by the convective dissolution of an extended source of carbon dioxide (${\mathrm{CO}}_2$) is studied from both experimental and numerical standpoints. In the experiments, the ${\mathrm{CO}}_2$ is injected into a 1-mm-thick water filled Hele-Shaw cell at pressures ranging between 1 and 5 bars. The plume is visualized using a $p\mathrm{H}$-sensitive dye, and the velocity field is measured by means of standard particle image velocimetry (PIV). Concentration and velocity fields are scrutinized over one order of magnitude in the Rayleigh number (Ra), and for two different values of the Darcy number (Da). In order to extend the explored range of these dimensionless numbers, experiments are complemented by a comprehenvive set of numerical simulations. At low Darcy numbers ($\mathrm{Da}<1$), the morphology of the numerical plume is fairly close to that of the analytical solution previously derived by Wooding [J. Fluid Mech. 15, 527 (1963)] for the case of a line source in a porous medium. At larger Darcy numbers ($\mathrm{Da}>1$), the numerical solution departs on several aspects from this solution and exhibits closer similarity to the analytical solution derived by Spalding and Cruddace [Int. J. Heat Mass Transfer. 3, 55 (1961)] for the case of a line source in a viscous fluid. As the introduction of an additional length scale (the lateral size of the source) in the problem allows for the definition of a length-based Rayleigh number Ra, the respective amplitudes of the plume velocity $w$, plume width $\sigma$, and dimensionless total injection rate (i.e., the Nusselt number Nu) can be explicitly obtained as functions of Ra and Da. The scaling laws are reasonably close to those obtained from simple mass conservation considerations: (i) $w\sim \mathrm{Ra},\sigma \sim {\mathrm{Ra}}^{-1/2}$, and $\mathrm{Nu}\sim {\mathrm{Ra}}^{1/2}$ for $\mathrm{Ra}\mathrm{Da}<1$, and (ii) $w\sim {\left(\mathrm{Ra}/\mathrm{Da}\right)}^{1/2},\sigma \sim {\left(\mathrm{Ra}/\mathrm{Da}\right)}^{-1/4}$, and $\mathrm{Nu}\sim {\left(\mathrm{Ra}/\mathrm{Da}\right)}^{1/4}$ for $\mathrm{Ra}\mathrm{Da}>1$. Although the concentration field measurements are marred by a large uncertainty due to the logarithmic nature of the $p\mathrm{H}$-concentration relationship, the experimental data obtained from the PIV measurements show good agreement with the numerical results.

We study the effect of gravity on barrel-shaped droplets on fibers using theoretical analyses, numerical simulations, and experiments. By performing a perturbation analysis in the limit of small Bond numbers and small dimensionless droplet volumes, we formulate a nearly axisymmetric solution that describes the shape of the droplet. The leading-order solution yields the axisymmetric profile and the first-order correction incorporates the effect of gravity. Thus, we report the droplet shape as a function of dimensionless droplet volume, contact angle, and Bond number. We find that, due to gravity, the contact line position varies sinusoidally with the azimuthal angle around the fiber. We validate our solution by comparing droplet shape and contact line predictions with experiments and numerical simulations.

Solute transport in porous media is affected by several factors. The heterogeneous structure of the permeability field is a key factor controlling the spreading and mixing behaviors of a solute cloud. On the other hand, other factors such as the viscosity contrast between the dissolved solute and the ambient fluid can also play an important role. Although both these mixing mechanisms (field heterogeneity and viscosity contrast) have been acknowledged and studied, more investigations are needed in order to better characterize the effect of the variation of both the degree of viscous fingering and the level of disorder of the porous medium. This work aims to explore the impact of field heterogeneity and viscosity contrast on the transport behavior of an inert solute in a two-dimensional flow field. To achieve this, we performed high-resolution numerical simulations based on the spectral method to solve coupled flow and transport equations for a given range of viscosity contrast and log-permeability variance. We analyze the degree and rate of mixing, contour length of the solute cloud, spatial statistics of the concentration field, and arrival times at a control plane to characterize spreading and mixing in the domain. Through the use of numerical simulations, we provide a quantitative separation of the impacts of fingering and heterogeneity and we parametrize the concentration probability distribution function. We find that the interplay among viscous fingering, high-permeability channeling, and low-permeability stagnation at small scales create important features in the spreading and mixing characteristics. In particular, our results indicate that at early times viscosity contrast has a more significant impact on mixing than permeability heterogeneity, and the effect of viscosity contrast on early and late arrival times at a control plane is enhanced by increasing levels of permeability heterogeneity; on the other hand, heterogeneity reduces the peak concentration at a control plane and causes larger solute cloud spreading when the solute is more viscous than the ambient fluid compared to when the solute is less viscous. Moreover, we find that the concentration cumulative distribution function of the solute cloud can be described as a beta distribution for the range of viscosity contrast and permeability heterogeneity considered.