A numerically efficient configuration to simulate turbulent flows is to use triply periodic domains, with numerical forcing techniques to sustain turbulence. Previous homogeneous shear turbulence simulations considered only idealized homogeneous shear flows and not the statistically stationary shear turbulence observed in practical free shear flows. In contrast, the current study mathematically derives the complete forcing technique from the large scales of the turbulent free shear flows. Different statistically stationary free shear flows are considered in this study, namely, a nearly homogeneous shear turbulent flow, turbulent mixing layer, a turbulent planar jet, and a turbulent round jet. The simulations are performed on triply periodic, statistically homogeneous cubic domains in the vicinity of the shear layer in the self-similar region. An a priori analysis is performed to calculate the effects of the different forcing terms and to predict the expected turbulence quantities. The forcing technique is then used to perform direct numerical simulations at different Reynolds numbers. Numerical results for the different cases are discussed and compared with results from experiments and other simulations of free shear turbulent flows. Anisotropy is observed both in the components of velocity and vorticity, with stronger Reynolds number dependence in the anisotropy of vorticity. Energy spectra obtained from the present homogeneous shear turbulence agree well with the spectra from temporally evolving shear layers. The results also highlight the effects of the additional forcing terms that were neglected in previous studies and the role of shear convection and the associated splitting errors in the unbounded evolution of previous numerical simulations.

In this paper we study the enstrophy transfers in helical turbulence using direct numerical simulation. We observe that the helicity injection does not have significant effects on the inertial-range energy and helicity spectra ($\sim k^{-5/3}$) and fluxes (constants). We also calculate the separate contributions to enstrophy transfers via velocity-to-vorticity and vorticity-to-vorticity channels. There are four different enstrophy fluxes associated with the former channel or vorticity stretching, and one flux associated with the latter channel or vorticity advection. In the inertial range, the fluxes due to vorticity stretching are larger than that due to advection. These transfers too are insensitive to helicity injection.

This paper is concerned with thermal counterflow in superfluid ${}^4\mathrm{He}$, particularly with the motion of the normal component at small heat fluxes when this motion is believed to be laminar. Recent experiments in which this motion is traced with micron-sized particles of solid deuterium show that the motion is not spatially uniform on a scale of order the spacing of the quantized vortex lines in the superfluid component. It is argued that this lack of uniformity has its origin in the fact that the force of mutual friction, which limits the thermal conductivity, is concentrated near the cores of the vortex lines. Possible effects of this concentration of force are discussed, and it is concluded that the experimental observations can be explained only if the vortex lines are arranged randomly in space, so that the observed lack of uniformity in the normal-fluid velocity can be regarded as being due partly to spatial variations in the vortex line density. However, problems remain, in that the form of the observed velocity correlation function has still to be understood.

Magnetic induction in magnetohydrodynamic fluids at magnetic Reynolds number (Rm) less than 1 has long been known to cause magnetic drag. Here we show that when $\mathrm{Rm}\gg 1$ and the fluid is in a hydrodynamic-dominated regime in which the magnetic energy is much smaller than the kinetic energy, induction due to a mean shear flow leads to a magnetic eddy viscosity. The magnetic viscosity is derived from simple physical arguments, where a coherent response due to shear flow builds up in the magnetic field until decorrelated by turbulent motion. The dynamic viscosity coefficient is approximately $(B_p^2/2{\mu }_0){\tau }_{\mathrm{corr}}$, the poloidal magnetic energy density multiplied by the correlation time. We confirm the magnetic eddy viscosity through numerical simulations of two-dimensional incompressible magnetohydrodynamics. We also consider the three-dimensional case, and in cylindrical or spherical geometry, theoretical considerations similarly point to a nonzero viscosity whenever there is differential rotation. Hence, these results serve as a dynamical generalization of Ferraro's law of isorotation. The magnetic eddy viscosity leads to transport of angular momentum and may be of importance to zonal flows in astrophysical domains such as the interior of some gas giants.

Since their development in 2001, regularized stokeslets have become a popular numerical tool for low-Reynolds-number flows since the replacement of a point force by a smoothed blob overcomes many computational difficulties associated with flow singularities [Cortez, SIAM J. Sci. Comput. 23, 1204 (2001)]. The physical changes to the flow resulting from this process are, however, unclear. In this paper, we analyze the flow induced by general regularized stokeslets. An explicit formula for the flow from any regularized stokeslet is first derived, which is shown to simplify for spherically symmetric blobs. Far from the center of any regularized stokeslet we show that the flow can be written in terms of an infinite number of singularity solutions provided the blob decays sufficiently rapidly. This infinite number of singularities reduces to a point force and source dipole for spherically symmetric blobs. Slowly decaying blobs induce additional flow resulting from the nonzero body forces acting on the fluid. We also show that near the center of spherically symmetric regularized stokeslets the flow becomes isotropic, which contrasts with the flow anisotropy fundamental to viscous systems. The concepts developed are used to identify blobs that reduce regularization errors. These blobs contain regions of negative force in order to counter the flows produced in the regularization process but still retain a form convenient for computations.

It is demonstrated that time-evolving coherent structures with features consistent with Townsend's attached eddies and the developments associated with the reconstruction of flow statistics using the attached eddy hypothesis can be obtained from analysis of the (linear) resolvent operator associated with the Navier-Stokes equations. A discrete number of members of a single self-similar hierarchy, chosen by assuming a constant ratio of critical layer heights between neighboring members with overlapping wall-normal footprints, produces a geometrically self-similar, fractal-like structure of the spatial distribution of the velocity field and an associated signature of uniform momentum zones. The range of convection velocities associated with the hierarchy gives rise to time evolution of the velocity. The scaling of the streamwise wave number on a hierarchy can be reconciled with Townsend's distance-from-the-wall scaling for self-similar, attached eddies, at least conceptually, by considering coherent spatial structures in the velocity field and thus enforcing an exact equivalence between compared features. It has been shown previously that both the linear and nonlinear terms in the resolvent framework have self-similar representations in the logarithmic overlap region of the turbulent mean velocity profile. The conditions under which self-sustaining self-similar behavior may be obtained from self-similar hierarchical members in this region are elaborated and some similarities with other results and theories associated with self-similarity in this region are identified.

Ferrofluid as a smart fluid has a wide range of applications. Although the spreading dynamics of water droplets have been well investigated, spreading dynamics of ferrofluid droplets under a magnetic field has rarely been studied. This paper reports our findings of the impact dynamics of a ferrofluid droplet onto a tempered glass surface in the presence of a vertical magnetic field. The effects of magnetic intensity, impact velocity, and $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$ nanoparticle concentration were investigated. It turned out that with the increased magnetic intensity, the height of the ferrofluid droplet would decrease owing to the energy dissipation increase of ferrofluids under a magnetic field and the additional stretching force in the vertical direction. Interestingly, we found that exertion of the magnetic field could significantly diminish the influence of velocity differences on the droplet spread dynamics in height direction. Satellite droplets were also observed in certain cases when the rebound kinetic energy could overcome the restraint of surface tension and adhesion of viscosity in the presence of the magnetic field. Our work will be a significant reference to with regard to various practical applications, especially when the impact dynamics of ferrofluid droplets need to be under precise control, for instance, as in three-dimensional printing or spray coating, etc.

We combine lubrication theory and solid linear elasticity to model the elastohydrodynamic deformation of a rigid, spherical particle impacting a stratified surface—a soft incompressible coating atop a rigid substrate. The model depends on three dimensionless parameters: Stokes number, elasticity parameter, and thickness parameter. These parameters collectively determine whether the particle will stick to or rebound against the stratified surface. We show that coating thickness moderates the degree to which elasticity affects the rebound criteria (and vice versa) in a fashion distinct from the effect of the elasticity or of the Stokes number. We also demonstrate the possibility of approximating a stratified surface as an elastic half space with an elasticity between the rigid substrate and soft coating. This “effective elasticity” of the half space couples both the elasticity parameter and thickness parameter of the stratified surface condition while maintaining the same sticking criteria. Lastly, we discuss the limitations of this effective elasticity, such as in thin or more-rigid coatings.

When wind blows at the surface of a liquid of sufficiently high viscosity, a wave packet of small amplitude is first generated, which sporadically forms large-amplitude fluid bumps that rapidly propagate downstream. These nonlinear structures, first observed by Francis [J. R. D. Francis, Philos. Mag. 42, 695 (1954)], have an almost vertical rear facing the wind and a weak slope at the front. We call them viscous solitons. We investigate their dynamics in a wind-tunnel experiment using silicon oil of kinematic viscosity $1000{\mathrm{mm}}^2{\mathrm{s}}^{-1}$ by means of laser sheet profilometry and particle image velocimetry. We give evidence of their subcritical nature: They are emitted in a region of large shear stress but, once formed, they are sustained by the wind and propagate in a region of lower stress. Their propagation velocity is given by the balance between aerodynamic drag in the air and viscous drag in the liquid. The stable soliton branch of the subcritical bifurcation diagram is reconstructed from the measured soliton amplitude at various wind velocities and distances along the channel. At large wind velocity, the emission frequency of solitons increases, resulting in a long-range sheltering of downstream mature solitons by newly formed upstream solitons, which limits their course.

In this work we revisit the dynamic Wilhelmy plate method for measuring dynamic contact angles with a specific focus on the viscous drag force which acts along the submerged surface of the plate. Particle image velocimetry shows that the viscous drag force can induce significant contact angle measurement errors if ignored. Preexisting models are found to underestimate the viscous drag force at moderate to high Capillary numbers, in finite-sized fluid domains, and for curved interfaces. Therefore, we propose a new model for the shear stress along a Wilhelmy plate by combining the dominant terms from a moving contact line flow, a Stokes flow past the leading edge of a flat plate, and a Couette-Poiseuille flow. This new model accounts for the shape of the fluid interface, the finite length of the plate, and the finite-size of the fluid container. Good agreement is obtained in comparisons with particle image velocimetry measurements over a Capillary number range from 0.001 to 0.3. This viscous drag model is applied to a dynamic Wilhelmy plate experiment and we obtain dynamic microscopic contact angle measurements and unbalanced Young's forces that are consistent with hydrodynamic theories of the moving contact line.

Capillary fingering is a displacement process that can occur when a nonwetting fluid displaces a wetting fluid from a homogeneous disordered porous medium. Here, we investigate how this process is influenced by a pore size gradient. Using microfluidic experiments and computational pore-network models, we show that the nonwetting fluid displacement behavior depends sensitively on the direction and the magnitude of the gradient. The fluid displacement depends on the competition between a pore size gradient and pore-scale disorder; indeed, a sufficiently large gradient can completely suppress capillary fingering. By analyzing capillary forces at the pore scale, we identify a nondimensional parameter that describes the physics underlying these diverse flow behaviors. Our results thus expand the understanding of flow in complex porous media and suggest a new way to control flow behavior via the introduction of pore size gradients.

Flow reversals in natural convection of particle-dispersed two-phase flow in a two-dimensional square box are studied by numerical simulation. The Rayleigh number based on the domain side length is set to ${10}^4$. The domain accommodates ${11}^2$ neutrally buoyant circular particles, and the thermal conductivity of the particle is set to ${10}^2$ times higher than the ambient fluid. The particle-dispersed flow driven by buoyancy develops into circulating flow, which transports particles into a diagonal pair of corner regions of the domain. The particles vertically aligned in the corner regions are a strong source of moment of buoyancy in the counterconvective direction, and flow reversals take place at the intervals of several hundred convective-time scales. The mechanism is different from that of the reversals or oscillation in single-phase or particle-dispersed natural convection reported in the literature. Thermal effect of the vertically aligned particles in a corner is modeled by nondimensionalized heat flux, and a cross-coupled sum of those of the four corners is found to be a precursor indicator of the reversal events. The investigation on the effects of three major parameters (Rayleigh number, conductivity ratio, and average interparticle spacing) suggests that the reversals occur in a small region in the parameter space.

Thinning dynamics in spin coating of viscous films is influenced by many physical processes. Temperature gradients are known to affect thin liquid films through their influence on the local fluid surface tension as Marangoni stresses. We show here experimentally and numerically that adding a static temperature gradient has a significant effect on the equilibrium film thickness and height profile reached in spin coating. Most notably, we find that the thickness of the resulting thin film in spin coating scales linearly with the strength of the thermal surface tension gradient. Once equilibrated, the thin film height profile is controlled by the temperature profile. For small but nonnegligible Marangoni number (Ma) the surface has a negative curvature at the center and reaching equilibrium takes progressively longer with smaller Ma. In this limit, the steady state reached is set by competition between Marangoni effects and the disjoining pressure.

The process of liquid drop impact onto a solid surface can be widely seen in both nature and industrial applications. Depending on the distance between the impact point and the surface edge ($W$), the impact can be categorized into three groups: (a) drop impact onto an infinite surface (surface is large, $W$ is larger than the maximum value of radius of the lamella in all directions); (b) finite surface (surface is small, $W$ is smaller than the maximum value of radius of the lamella in all directions); and (c) semi-infinite solid surface ($W$ is smaller than the maximum radius of the lamella in some directions, but not all). Different from most of the literature which typically focused on the drop impact onto an infinite and/or finite solid surface, an experimental study was performed to investigate the liquid drop behavior during an impact onto a semi-infinite solid surface. In this case, part of the lamella spreads out of the surface (free lamella), while the other part of the liquid remains on the solid surface. Depending on the normalized distance ($\overline{W}=W/D$; $D$ is the drop diameter), the free lamella can either recede back onto the surface ($\overline{W}>0.61$) or completely break off at the surface edge ($\overline{W}<0.61$). The distinctive behavior can be explained by the vertical velocity component of the free lamella and the Laplace pressure difference across the interface at the edge. The surface wettability and Weber numbers cannot determine whether the free lamella can break or recede back, but can significantly affect the breaking morphology of the free lamella. The liquid lamella on the surface behaves similarly to the drop impact on an infinite surface in the spreading phase. The value of the maximal spreading radius $\left(r_{\max }\right)$ on the surface is not affected by $\overline{W}$ when $\overline{W}>0.3$; $R_{\max }$ only starts to decrease when $\overline{W}$ is smaller than 0.3. In the receding phase, the liquid lamella on the solid surface evolves into a semicircle and a sausage shape before eventually evolving to a rounded shape.

Swirling flows are ubiquitous over a large range of length scales and applications including micron-scale microfluidic devices up to geophysical flows such as tornadoes. As the viscous dissipation, shear, and centrifugal stresses interact, such flows can often exhibit unexpected fluid dynamics. Here, we use microfluidic experiments and numerical simulations to study the flow in a vortex T-mixer: a T-shaped channel with staggered, offset inlets. The vortex T-mixer flow is characterized by a single dominant vortex, the stability of which is closely coupled to the appearance of vortex breakdown. Specifically, at a Reynolds number of $\mathrm{Re}\approx 90$, a first vortex breakdown region appears in the steady-state solution, rendering the vortex pulsatively unstable. A second vortex breakdown region appears at $\mathrm{Re}\approx 120$, which restabilizes the vortex. Finally, a third vortex breakdown region appears at $\mathrm{Re}\approx 180$, which renders the vortex helically unstable. Thus, a counterintuitive flow regime exists for the vortex T-mixer in which increasing the Reynolds number has a stabilizing effect on the steady-state flow. The pulsatively unstable vortex evolves into a periodically pulsating state with a Strouhal number of $\mathrm{St}\approx 0.5$, and the helically unstable vortex evolves into a helically oscillating state with $\mathrm{St}\approx 1.75$. These transitions can be explained within the framework of linear hydrodynamic stability. In addition, the vortex T-mixer flow exhibits multistability; multiple flow states are stable over various ranges of $\mathrm{Re}$, including a narrow range of tristability for $160\le \mathrm{Re}\le 170$, in which the steady state, the pulsatile oscillation, and the helical oscillation are all stable. This study provides experimental and numerical evidence of the close coupling between vortex breakdown and flow stability, including the restabilization of the flow with increasing Reynolds number due to the appearance of a vortex breakdown region, which will provide new insights into how vortex breakdown can affect the stability of a swirling flow.

Experimental and numerical results reveal a distinct signature of electroconvective instability in the transient response for a microchannel-nanoslot system. We study the development of the electroconvective instability from equilibrium during the transient chronopotentiometric (galvanostatic operation) and chronoamperometric (potentiostatic operation) responses at overlimiting current conditions and observe a distinct signature of nonmonotonic transient response resulting from the emergence of electroconvective instability. This stands in contrast to previously reported experimental nonmonotonic transient responses in heterogeneous membrane systems associated with linear electro-osmotic flow.

We experimentally and numerically investigate the angular momentum transport in turbulent Taylor-Couette flow for independently rotating cylinders at a small radius ratio of $\eta =0.357$ for various shear Reynolds numbers $\left(4.5\times {10}^3\le {\text{Re}}_S\le 1.2\times {10}^5\right)$ and ratios of angular velocities $\left(-0.5\le \mu \le 0.2\right)$. The momentum transport in terms of the pseudo-Nusselt number ${\text{Nu}}_{\omega }$ does not show a pure power law scaling with the forcing ${\text{Re}}_S$ and features nonconstant effective scaling between $1.3\times {10}^4\le {\text{Re}}_S\le 4\times {10}^4$. This transition lies in the classical turbulent regime and is caused by the curvature-dependent limited capacity of the outer cylinder to emit small-scale plumes at a sufficient rate to equalize the angular momentum in the bulk. For counter-rotating cylinders, a maximum in the torque occurs at ${\mu }_{max}=-0.123\pm 0.030$. The origin of this maximum can be attributed to a strengthening of turbulent Taylor vortices, which is revealed by the flow visualization technique. In addition, different flow states at ${\mu }_{max}$ concerning the wavelength of the large-scale vortices have been detected. The experimental and numerical results for the Nusselt number show a very good agreement.

The motion of helicoidal swimmers moving in a pool filled with a granular medium is studied experimentally. The horizontal displacement through granular beads is measured considering geometrical modifications of the swimmer, the size, and frictional properties of the media. We found three main parameters which affect the swimming performance: the diameter, the wavelength, and the angle of the helix. The swimming speed scales with the rotation speed, $\omega R$. The size of particles does not affect the swimming speed significantly; the swimming speed is reduced when the particle's angle of repose increases. It was found that a maximum swimming speed is achieved when the helix angle is close to ${55}^{\circ }$. The experimental data are compared with predictions of the granular resistive force theory, which was extended to be applicable for a swimmer with a rigid helical tail, leading to good agreement.

This paper investigates the scaling of turbulent kinetic energy (TKE) and temperature variance production in a differentially heated vertical channel (DHVC). In a DHVC, TKE is produced by two distinctively different mechanisms: buoyancy production and shear production. In the present work, identity equations are derived for the global integrals of shear-produced TKE and temperature variance production. The derived identity equations agree well direct numerical simulation (DNS) data. At sufficiently high Rayleigh number the global integral of the shear-produced TKE is found, based on the DNS data, to scale as ${\int }_0^{\delta }{R}_{\mathrm{wu}}\frac{dU}{dz}dz\approx 0.385u_{\tau }{U}_{\max }^2$. Here $z$ is the wall-normal direction, $\delta$ is the channel half-width, $U$ is the mean streamwise velocity in the $x$ direction, $U_{\max }$ is the maximum mean streamwise velocity, and $u_{\tau }$ is the friction velocity. $R_{\mathrm{wu}}=-\langle \mathrm{wu}\rangle$ is the Reynolds shear stress, where $\mathrm{w}$ is the velocity fluctuation in the $z$ direction, $\mathrm{u}$ is the velocity fluctuation in the $x$ direction, and angle brackets $\langle \rangle$ denote averaging operation. The global integral of the buoyancy-produced TKE at sufficiently high Grashof number is found to scale as ${\int }_0^{\delta }g\alpha R_{\mathrm{u}\theta }dz\approx u_{\tau }^2{U}_{\max }$ where $g$ is the gravitational acceleration, $\alpha$ is the thermal expansion coefficient, and $R_{\mathrm{u}\theta }=-\langle \mathrm{u}\theta \rangle$ is the covariance of the streamwise velocity fluctuation $u$ and the temperature fluctuation $\theta$. The global integrals of temperature variance production and temperature dissipation ${\epsilon }_{\theta }$ are found to grow with the Grashof number in a logarithmic-like fashion as ${\int }_0^{\delta }{R}_{\mathrm{w}\theta }\frac{d\mathrm{\Theta }}{dz}dz={\int }_0^{\delta }{\epsilon }_{{}_{\theta }}dz\approx \left[0.5\ln \left(\mathrm{Gr}\right)-1.8\right]u_{\tau }{\theta }_{\tau }^2$ where $\mathrm{\Theta }$ is the mean transformed temperature, $R_{\mathrm{w}\theta }=-\langle \mathrm{w}\theta \rangle$ is the wall-normal turbulent transport of heat, ${\theta }_{\tau }$ is the friction temperature, and $\mathrm{Gr}$ is the Grashof number. Based on the characteristics of the flux Richardson number, a four-layer structure is proposed for the TKE budget equation in a DHVC.

The steady laminar flow of two immiscible, uniformly rotating fluid layers is studied and exact similarity solutions of the axisymmetric Navier-Stokes equations in cylindrical polar coordinates are found. The similarity solutions occur with a flat interface at $z=0$ under the parameter restriction that ${\sigma }^2\rho =1$, where $\sigma$ is the ratio of the fluid angular velocities at $z=\pm \infty$ and $\rho$ is the density ratio of the two fluids. Under this restriction the problem reduces to one with two independent parameters $\sigma$ and $\mu$, the viscosity ratio of the fluids. Numerical results of the resulting system of ordinary differential equations are found for selected values of $\mu$ and $\sigma$, and it is shown that similarity solutions exist for ${\sigma }_c\left(\mu \right)\le \sigma \le 1$, where ${\sigma }_c\left(\mu \right)<0$ (i.e., counterrotating flows). For $\sigma <0$ the lower fluid can become divided into two distinct recirculation regions between which fluid cannot transfer.