One of the cornerstones of turbulent dispersion is the celebrated Taylor's formula. This formula expresses the rate of transport (i.e., the eddy diffusivity) of a tracer as a time integral of the fluid velocity autocorrelation function evaluated along the fluid particle trajectories. Here, we review the hypotheses which permit us to extend Taylor's formula to particles of any inertia. The hypotheses are independent of the details of the inertial particle model. We also show by explicit calculation that the hypotheses encompass cases when memory terms such as Basset's and Faxén's corrections are taken into account in the modeling of inertial particle dynamics.

During the transport of colloidal suspensions in microchannels, the deposition of particles can lead to the formation of clogs, typically at constrictions. Once a clog is formed in a microchannel, advected particles form an aggregate upstream from the site of the blockage. This aggregate grows over time, which leads to a dramatic reduction of the flow rate. In this paper, we present a model that predicts the growth of the aggregate formed upon clogging of a microchannel. We develop an analytical description that captures the time evolution of the volume of the aggregate, as confirmed by experiments performed using a pressure-driven suspension flow in a microfluidic device. We show that the growth of the aggregate increases the hydraulic resistance in the channel and leads to a drop in the flow rate of the suspensions. We then derive a model for the growth of aggregates in multiple parallel microchannels where the clogging events are described using a stochastic approach. The aggregate growths in the different channels are coupled. Our work illustrates the critical influence of clogging events on the evolution of the flow rate in microchannels. The coupled dynamics of the aggregates described here for parallel channels is key to bridge clogging at the pore scale with macroscopic observations of the flow rate evolution at the filter scale.

The turbulent pipe flow of inelastic shear-thinning fluids has many practical applications; however, there is a deficit in understanding of how shear-thinning rheology modifies turbulence structure in the near-wall boundary layer (affecting shear stress and pressure drop) and in the core (affecting mixing). While previous direct numerical simulation studies have examined the effect of shear-thinning rheology at low Reynolds number (${\text{Re}}_{\tau ,\text{max}}=323$), the way in which these effects vary with ${\text{Re}}_{\tau }$ was unknown. In particular, from earlier work it was unclear if inner-scaled mean axial velocity profiles for Newtonian and shear-thinning fluids could collapse to a common curve with increasing Reynolds number. Via direct numerical simulations of Newtonian and one shear-thinning rheology for friction Reynolds number ${\text{Re}}_{\tau }=323\text{–}750$ (${\text{Re}}_G=10000\text{–}28000$), the present study investigates how increasing Reynolds number modifies turbulent pipe flow of a power-law fluid with particular focus on the boundary layer profiles. The results show that the inner-scaled mean axial velocity profiles for Newtonian and shear-thinning fluids cannot collapse to a common curve with increasing Reynolds number, which is consistent with predictions from the Dodge-Metzner correlation [Dodge and Metzner, Turbulent flow of non-Newtonian systems, AIChE J. 5, 189 (1959)]. In inner-scaled coordinates, mean viscosity profiles are shown to become independent of Reynolds number except close to the pipe center. The contribution of viscosity fluctuations in the mean shear budget and in the mean flow and turbulence kinetic energy budget remains small at all $\text{Re}$. Both increasing Reynolds number and shear thinning influence the turbulence kinetic energy budget near the wall; however, the region where shear thinning is important is much wider than the region where increasing Reynolds number influences the results. The persistence of shear-thinning effects on turbulence modification in pipe flow requires consideration in the development of suitable turbulence models for such fluids. The current results suggest that the effect of shear-thinning rheology in turbulence models can be captured via a Reynolds-number-independent mean viscosity model in the inner region.

In this work we investigate properties of the mean thermal energy balance equation using a multiscaling analysis approach. The analysis of the mean thermal energy balance (MHB) equation and the mean momentum balance (MMB) equation are presented side by side to better demonstrate the similarities and differences between the two. The main findings of this work include, first, the multiscaling for the MHB equation are similar to those for the MMB equation in the outer layer, mesolayer, and log layer. Péclet number ${\text{Pe}}_{\tau }=\text{Pr}{\text{Re}}_{\tau }$ in the MHB equation is the counterpart of ${\text{Re}}_{\tau }$ in the MMB equation. Here $\text{Pr}\equiv \nu /\alpha$ denotes the Prandtl number of the fluid, which is the ratio between the kinematic viscosity $\nu$ and the molecular thermal diffusivity $\alpha$. ${\text{Re}}_{\tau }=\delta u_{\tau }/\nu$ is the Reynolds number of the flow defined with the channel half-height $\delta$ and the friction velocity $u_{\tau }$. In the outer layer, mesolayer, and log layer, the shapes of the mean temperature and the mean velocity are very similar, and the shapes of the wall-normal turbulent transport of heat and momentum are also similar. Second, the molecular thermal diffusion sublayer is strongly influenced by the Prandtl number. At low Prandtl number ($\text{Pr}<1$), the thickness of the molecular thermal diffusion sublayer is $y_{{}_{I_t}}=O\left(\frac{1}{{\mathrm{Pr}}^{1/2}}\frac{\nu }{u_{\tau }}\right)$ or $y_{{}_{I_t}}^{+}=O\left(\frac{1}{{\mathrm{Pr}}^{1/2}}\right)$, a proper scaling for the wall-normal distance is ${\text{Pr}}^{1/2}{y}^{+}$ where $y^{+}=y/(\nu /u_{\tau })$ is the inner-scaled wall-normal distance, and a relevant Péclet number is ${\text{Pr}}^{1/2}{\text{Re}}_{\tau }$. At very low Prandtl number ($\text{Pr}\ll 1$), the molecular thermal diffusion sublayer becomes much thicker than the viscous sublayer (molecular momentum diffusion sublayer). At large Prandtl number ($\text{Pr}>1$), the thickness of the molecular thermal diffusion sublayer is $y_{{}_{I_t}}^{+}=O\left(\frac{1}{{\mathrm{Pr}}^{1/3}}\right)$, a proper scaling for the wall-normal distance is ${\text{Pr}}^{1/3}{y}^{+}$, and a relevant Péclet number is ${\text{Pr}}^{1/3}{\text{Re}}_{\tau }$. At very high Prandtl number ($\text{Pr}\gg 1$), the molecular thermal diffusion sublayer becomes much thinner than the viscous sublayer. Third, to be consistent with the scaled MHB equation, the “log law” for the mean temperature is presented using an inner-scaled wall-normal distance $y/(\alpha /u_{\tau })=\text{Pr}{y}^{+}$. The main effect of the Prandtl number is the shifting of the additive constant in the log law, due to the thickness of the molecular thermal diffusion sublayer and buffer layer. Fourth and finally, Zagarola-Smits-style scaling is applied to the mean velocity and the mean temperature deficit in the outer layer. An interpretation of the Zagarola-Smits-style scaling is provided. At sufficiently high Reynolds number and Péclet number, the Zagarola-Smits-style scaling is shown to be equivalent to the traditional scaling.

Several recent experiments investigate the orientational and transport behavior of self-driven bacteria and colloidal particles in nematic liquid crystals. Correspondingly, we study theoretically the dynamics of a minimal model microswimmer in a uniaxially anisotropic fluid. As a first step, the hydrodynamic Green's function providing the resulting fluid flow in response to a localized force acting on the anisotropic fluid is derived analytically. On this basis, the behavior of both puller- and pusher-type microswimmers in the anisotropic fluid is analyzed. Depending on the propulsion mechanism and the relative magnitude of different involved viscosities, we find alignment of the swimmers parallel or perpendicular to the anisotropy axis. Particularly, also an oblique alignment is identified under certain circumstances. The observed swimmer reorientation results from the hydrodynamic coupling between the self-induced fluid flow and the anisotropy of the surrounding fluid, which distorts the self-generated flow field. We support parts of our results by a simplified linear stability analysis. Our theoretical predictions are in qualitative agreement with recent experimental observations on swimming bacteria in nematic liquid crystals. They support the objective of utilizing the possibly switchable anisotropy of a host fluid to guide individual microswimmers and active particles along a requested path, enabling controlled active transport.

Vertical-axis wind turbines (VAWTs) are the subject of renewed interest due to the potential for higher power generation per unit land used, as well as their lower center of mass (the generator is at the bottom of tower), which renders them favorable for offshore deployment. However, VAWT farms have hardly been studied. In this paper, using a previously tested actuator line model in a large eddy simulation code, we investigate the transport of the mean kinetic energy (MKE) that replenishes the power in the farm. The primary sources of MKE are (1) the initial advective streamwise influx through the frontal area and (2) the vertical planform influx through the top and bottom interfaces of the farm. The results show that, for realistic finite-size farms, the planform MKE transport is a loss term over the first six rows: in this initial zone the mean flow adjusts by slowing down, and an upward mean advection develops that results in an efflux loss of MKE from the farm volume. The power extracted from farms is thus mainly from the frontal advection over the first few rows. When the initial streamwise advective flux is exhausted, the planform regeneration of MKE from above the wind farm becomes the dominant source; it is primarily affected by turbulent-mean interaction. This regeneration continues to adjust until rows 8 to 10 in our setups, beyond which a fully developed flow (similar to an infinite wind farm) can be observed. In the fully developed region, actual mechanical power generation by the turbines is about one third of replenishment. A primary conclusion is that more irregular farms designs should be studied, while the current literature continues to focus on the very classic layouts.

This paper investigates the formation number of vortex rings generated by a piston-cylinder mechanism in a confined tube. We use direct numerical simulations of axisymmetric confined vortex rings to study the influence of different parameters on the separation (or pinch-off) of the vortex ring from the trailing jet. It is shown that the structure of the vortex ring at pinch-off depends on the type of injection program (pulse dominated by either positive or negative acceleration ramps) and the confinement ratio $D_w/D$, where $D_w$ is the inner diameter of the tube and $D$ the diameter of the cylinder. For low confinement ratios ($D_w/D\le 2$), a vortex of opposite sign generated at the lateral wall strongly interacts with the vortex ring and the pinch-off is not clearly observed. The pinch-off is observed and analyzed for confinement ratios $D_w/D\ge 2.5$. Direct numerical simulation data are used to estimate the value of the formation time, which is the time necessary for the vortex generator to inject the same amount of circulation as carried by the detached vortex ring. The confined vortex ring at pinch-off is described by the model suggested by Danaila et al. [I. Danaila et al., A model for confined vortex rings with elliptical core vorticity distribution, J. Fluid Mech. 811, 67 (2017)]. This model allows us to take into account the influence of the lateral wall and the elliptical shape of the vortex core. The value of the formation time is predicted using this model and the slug-flow model.

A large variety of industrial and natural systems involve the adsorption of solid particles to the fluidic interface of droplets in motion. A diffuse interface model is here suggested to directly simulate the three-dimensional dynamics of a fluid droplet rising across a cloud of large particles. In this three-phase model the two solid-fluid boundaries and the fluidic boundary are replaced with smoothly spreading interfaces. The capillary effects and the three-phase flow hydrodynamics are fully resolved. A special treatment is adopted for the interparticle collisions. The effect of the particle concentration on the terminal velocity of a rising fluid droplet is then investigated. It is found that, at low Reynolds number, the terminal velocity of a rising fluid droplet decreases exponentially with the particle concentration. This exponential decay is confirmed by a simple rheological model.

The settling efficiency, and stability with respect to settling, of a dilute suspension of infinite circular cylinders in a quiescent viscoplastic fluid is examined by means of direct numerical simulations with varying solid volume fraction, $\varphi$, and yield number, $Y$. For $Y$ sufficiently large we find higher settling efficiency for increasing $\varphi$, similar to what is found in shear-thinning fluids and opposite to what is found in Newtonian fluids. The critical yield number at which the suspension is held stationary in the carrier fluid is found to increase monotonically with $\varphi$, while the transition to settling is found to be diffuse: in the same suspension, particle clusters may settle while more isolated particles remain arrested. In this regime, complex flow features are observed in the sedimenting suspension, including the mobilization of lone particles by nearby sedimentation clusters. Understanding this regime, and the transition to a fully arrested state, is relevant to many industrial and natural problems involving the sedimentation of viscoplastic suspensions under quiescent flow conditions.

The liquid drainage is a commonplace process that affects a wide array of industrial applications ranging from medical procedures and manufacturing processes to food processing. While many drainage-related applications involve the complex fluids comprising solid particles, the effects of suspended particles on the liquid drainage have not been considered, leaving simple fundamental questions on the suspension drainage unanswered. In this paper, we experimentally investigate the effects of particles on drainage by withdrawing suspensions from an air-filled Hele-Shaw cell in a radial sink flow. As expected, viscous fingering arises as air invades a draining viscous suspension. Despite numerous studies on viscous fingering, only a few have studied this “inward” viscous fingering for pure liquids and none for suspensions. We find that, while the overall behavior of fingering remains unchanged from the pure liquid case, suspended particles are shown to delay the onset of fingering but also to accelerate its growth rate. This surprising dual effect of particles results in the increase of the total drainage time and in the amount of drained suspension as a function of particle concentrations. In addition, the particle entrainment into the thin film of wetting oil causes particles with select sizes to remain on the channel walls instead of draining, which closely follows our simple theoretical prediction.

The statistical properties of a large number of weakly nonlinear waves can be described in the framework of the weak turbulence theory. The theory is based on the hypothesis of an asymptotically large system. In experiments, the systems have a finite size and the predictions of the theory may not apply because of the presence of discrete modes rather than a continuum of free waves. Our study focuses on the case of waves at the surface of water at scales close to the gravity-capillarity crossover (of order 1 cm). Wave turbulence has peculiar properties in this regime because one-dimensional resonant interactions can occur, as shown by Aubourg and Mordant [Phys. Rev. Lett. 114, 144501 (2015)]. Here we investigate the influence of the confinement on the properties of wave turbulence by reducing gradually the size of our wave tank along one of its axes, the size in the other direction being unchanged. We use space-time resolved profilometry to reconstruct the deformed surface of water. We observe an original regime of coexistence of weak wave turbulence along the length of the vessel and discrete turbulence in the confined direction.

Streamwise boundary undulations induce spanwise vorticity in flows. In Newtonian flow, vorticity decays exponentially with distance from the undulation. However, recent theory for inertioelastic polymeric flow predicts vorticity amplification in a “critical layer” far from the site of disturbance. Here we present the first experimental evidence for the existence of such critical layers, demonstrating their measurable role in real polymeric flows with inertia. These vorticity amplification effects should be considered in all evaluations of vorticity dynamics in inertioelastic flows with streamline curvature.

Membrane filters are in widespread industrial use, and mathematical models to predict their efficacy are potentially very useful, as such models can suggest design modifications to improve filter performance and lifetime. Many models have been proposed to describe particle capture by membrane filters and the associated fluid dynamics, but most such models are based on a very simple structure in which the pores of the membrane are assumed to be simple circularly cylindrical tubes spanning the depth of the membrane. Real membranes used in applications usually have much more complex geometry, with interconnected pores that may branch and bifurcate. Pores are also typically larger on the upstream side of the membrane than on the downstream side. We present an idealized mathematical model, in which a membrane consists of a series of bifurcating pores, which decrease in size as the membrane is traversed. Feed solution is forced through the membrane by applied pressure and particles are removed from the feed by adsorption within pores (which shrinks them). Thus, the membrane's permeability decreases as the filtration progresses. We discuss how filtration efficiency depends on the characteristics of the idealized branching structure.

Jet droplets from bubble bursting are determined by a limited parametrical space: the liquid properties (surface tension, viscosity, and density), mother bubble size, and acceleration of gravity. Thus, the two resulting parameters from dimensional analysis (usually the Ohnesorge and Bond numbers Oh and Bo) completely define this phenomenon when both the trapped gas in the bubble and the environment gas have negligible density. A detailed physical description of the ejection process to model both the ejected droplet radius and its initial launch speed is provided, leading to a scaling law including both Oh and Bo. Two critical values of Oh determine two limiting situations: One (${\text{Oh}}_1=0.038$) is the critical value for which the ejected droplet size is minimum and the ejection speed maximum, and the other (${\text{Oh}}_2=0.0045$) is a critical value which signals when viscous effects vanish. Gravity effects (Bo) are consistently introduced from energy conservation principles. The proposed scaling laws produce a remarkable collapse of published experimental measurements collected for both the ejected droplet radius and ejection speed.

We investigate the onset of three-dimensionality in a Mach 5 slender-double-wedge flow as the turn angle at the compression corner increases. Beyond a critical angle, the two-dimensional flow destabilizes to three-dimensional perturbations and results in growth of spanwise periodic flow structures. We carry out global linear stability analysis to identify the critical turn angle and the nature of the associated three-dimensional instability. At the critical angle, an unstable mode is present in the separation bubble and the reattached boundary layer. The mode is associated with streamwise streaks in wall temperature downstream of the corner. The existence of the unstable mode and its growth rate are confirmed with direct numerical simulations. It is found from budget analysis that streamwise deceleration of the recirculating flow plays a dominant role in the three-dimensional instability. Wave-maker analysis suggests that the instability does not have a centrifugal origin. Linear stability analysis of the steady-state flow at an angle beyond bifurcation is also carried out. The spanwise wavelength of the most unstable mode obtained at this turn angle compares well with experimental observations.

Turbulent/nonturbulent interfaces (TNTIs) are studied in the direct numerical simulation of temporally evolving turbulent boundary layers at Mach numbers 0.8 and 1.6 with Reynolds number based on the momentum thickness of about 2200. The computational grid size determined based solely on the wall unit results in insufficient resolutions near the TNTI even though it yields the well-known profiles of global statistics such as mean velocity and rms velocity fluctuations. The insufficient resolution near the TNTI layer causes the spiky patterns of the enstrophy isosurface used for detecting the outer edge of the TNTI layer and the thicker TNTI layer thickness. With the higher-resolution direct numerical simulation, where the resolution is determined based on both the wall unit and the smallest length scale of turbulence underneath the TNTI layer, we investigate the structures of the TNTI layer and the entrainment process in the compressible turbulent boundary layers. The mean vorticity profile and enstrophy evolutions near the TNTI layer show that the structure of the TNTI layer is similar to incompressible free shear flows: The thickness of the layer is about 15 times the Kolmogorov scale ${\eta }_I$ in turbulence near the TNTI layer; the turbulent sublayer (TSL) and viscous superlayer (VSL) are found based on the analysis of enstrophy transport equation, where the thicknesses of the TSL and VSL are $11{\eta }_I–12{\eta }_I$ and $4{\eta }_I$, respectively. The entrainment process across the TNTI layer is also studied based on the propagation velocity of the enstrophy isosurface and the mass transport equation in the local coordinate moving with the TNTI. The entrainment mechanism across the TNTI layer in compressible turbulent boundary layers is very similar to incompressible free shear flows until Mach number 1.6, where the mass transport within the TNTI layer is well predicted by an entrainment model based on a single vortex originally developed for incompressible flows. Furthermore, the mass entrainment rate per unit horizontal area of the temporally evolving turbulent boundary layers is consistent with the theoretical prediction for spatially evolving compressible turbulent boundary layers for both Mach numbers.

The wake behind a quarter-of-ring concave curved cylinder is investigated in this paper by means of direct numerical simulations. The plane of curvature is aligned with the uniform incoming flow. We have appended straight extensions to both ends of the curved part of the cylinder, such that free ends are eliminated from the simulations. The effect of the vertical extension, i.e., the straight extension with its axis normal to the inflow, is carefully studied and turns out to be significant. The results from several different Reynolds numbers ($\mathrm{Re}=100–500$) are presented, from which a clear picture of the wake transition behind this configuration could be sketched. The concave curved cylinder wake consists of different flow regimes along the span. Oblique shedding, vortex dislocations, and various shedding frequencies are captured in different flow regimes. At $\mathrm{Re}\le 300$, the flow regimes change abruptly, but at $\mathrm{Re}=400$ and 500, the changes are continuous, so the boundaries between them are difficult to observe. A frequency band, instead of one single dominating frequency, manifests itself in the three-dimensional (3D) energy spectrum.

Microfluidics have been used extensively for the study of flows of immiscible fluids, with a specific focus on the effects of interfacial forces on flow behavior. In comparison, inertia-driven flow of confined coflowing fluids has received scant attention at the microscale, despite the fact that the effects of microscale confinement are expected to influence inertia-driven flow behavior as observed in free jets. Herein, we report three distinct modes for breakup of coflowing, confined, microscale jets: the conventional Rayleigh mode and two additional inertia-driven modes occurring at higher Reynolds number flows, namely, a sinuous wave breakup and an atomizationlike mode. Each of the three modes is differentiated by a characteristic droplet size, size distribution, and dependence of the jet length as a function of the external fluid velocity $\left(v_{\text{ext}}\right)$. A unified phase diagram is proposed to categorize the jet breakup mechanisms and their transitions using, as a scale-up factor, the ratio of the jet inertial forces to the sum of the viscous and interfacial forces for both the inner and outer fluids. These results provide fundamental insights into the flow behavior of microscale-confined coflowing jets.

A normal compressive shock wave with supercritical upstream thermodynamic conditions is analyzed using the Soave-Redlich-Kwong equation of state (EoS) relations for real-gas density, enthalpy, and entropy for argon, nitrogen, oxygen, and carbon dioxide. Upstream pressure and temperature vary from 10 to 500 bar and 160 to 800 K. At high pressures, the flow does not follow the calorically perfect-gas behavior. For the perfect gas, the enthalpy and ratio of pressure-to-density are directly proportional to the square of the sound speed, allowing direct substitution of the sound speed in the conservation equations. A thermodynamic function is identified for the real-gas sound speed which is shown to remain as the proper characteristic speed. Although the sound speed does not emerge directly from the conservation equations as it does for a perfect gas, the shock speed goes to this limiting value as shock strength goes to zero. For the real gas, modifications are obtained for Prandtl's relation and the Rankine-Hugoniot relation. The modified real-gas Riemann invariants are constructed and discussed for application to weak shocks. A foundation is presented for use with other cubic EoS, multicomponent flows, and/or for more complex flow configurations. Near-similar solutions are developed by normalization of the variables using critical values for pressure and temperature. These exact solutions are compared with approximate solutions obtained via a linearization of the cubic EoS for deviation from ideal-gas behavior.

We perform direct numerical simulations of three-dimensional Rayleigh-Taylor turbulence with a nonuniform singular initial temperature background. In such conditions, the mixing layer evolves under the driving of a varying effective Atwood number; the long-time growth is still self-similar, but no longer proportional to $t^2$ and depends on the singularity exponent $c$ of the initial profile $\mathrm{\Delta }T\propto z^c$. We show that universality is recovered when looking at the efficiency, defined as the ratio of the variation rates of the kinetic energy over the heat flux. A closure model is proposed that is able to reproduce analytically the time evolution of the mean temperature profiles, in excellent agreement with the numerical results. Finally, we reinterpret our findings in the light of spontaneous stochasticity where the growth of the mixing layer is mapped into the propagation of a wave of turbulent fluctuations on a rough background.