Although the “log” law of the mean-velocity profile (MVP) and the “−5/3 or −1” power law of the energy spectrum have been well studied in wall-bounded turbulent flows over a long period of time, a possible connection between them is still worth exploring. In this paper, an eddy's inclination angle is introduced into a model of a “turbulence eddy” to refine the connection. An expression for the inclination angle depending on both wall-normal height and Reynolds number is proposed. We find that the predicted MVP can follow the logarithmic law excellently, which is also in good agreement with direct numerical simulation results even at much higher Reynolds numbers. Our refined model may provide a deeper relationship between the mean velocity and the turbulence intensity through the eddy's inclination angle. Moreover, through our refined model, the hardly measured inclination angle can be obtained according to the given MVP and streamwise turbulence intensity.

An extended lubrication model is proposed for improvement of the lubrication theory by taking into account a larger surface-to-surface distance than that for the Reynolds lubrication theory. The analysis shows that when considering the non-negligible pressure gradient in the surface-normal direction, the local pressure is separated into (i) a base component satisfying the Reynolds lubrication theory and (ii) an adjusting component varying in the surface-normal direction, which is found to take the form proportional to the longitudinal derivative of the local velocity of the Couette-Poiseuille flow. Comparison of the results obtained by analytical and numerical methods for the lubrication between a moving curved object and stationary object shows that the proposed lubrication model reproduces the pressure distribution in both wall-normal and longitudinal directions. In a problem of a spherical particle approaching to a plane wall, the hydrodynamic force calculated by the proposed model exhibits an inverse-proportional trend to the surface-to-surface distance. The results suggest extended applicability of the lubrication theory to a non-Reynolds regime.

We study the steady-state displacement of nonwetting liquid ganglia during immiscible two-phase flows in realistic, stochastically reconstructed porous domains, focusing primarily on the nonlinear Darcian regime that arises when capillary to viscous (or gravity) forces become comparable at the pore scale. During this process, the ganglia undergo a continuous cycle of dynamic coalescence and fragmentation, resulting in two populations (a mobile and a stranded one) with distinct structural and rheological features, that continuously exchange mass between them under “stationary” flow conditions. We use a lattice Boltzmann model for the explicit solution of flow and interfacial dynamics at the pore scale driven by a constant body force field (i.e., gravity), and a periodic clustering algorithm for the identification and classification of mobile and stranded ganglia. Our simulation results reveal that an increase in the applied Bond number (Bo) leads to a gradual mobilization of the initially stranded ganglia population, resulting in a power-law scaling with an exponent being a strong function of the nonwetting-phase saturation. The linear Darcian scaling for the nonwetting phase is progressively restored at high Bo, while the wetting phase appears to maintain a linear Darcian scaling over the entire range of Bo values. We show that the mobilization process is characterized by a critical Bo, which is independent of saturation, above which new flow paths are created, in a similar fashion as a yield-stress fluid flows in a porous medium. Our results also offer a unique insight on the distinct structural characteristics of the mobile and stranded populations (e.g., ganglia size and length), as well as on their velocity and orientation with respect to their size.

Biological systems are influenced by fluid mechanics at nearly all spatiotemporal scales. This broad relevance of fluid mechanics to biology has been increasingly appreciated by engineers and biologists alike, leading to continued expansion of research in the field of biological fluid dynamics. While this growth is exciting, it can present a barrier to researchers seeking a concise introduction to key challenges and opportunities for progress in the field. Rather than attempt a comprehensive review of the literature, this article highlights a limited selection of classic and recent work. In addition to motivating the study of biological fluid dynamics in general, the goal is to identify both longstanding and emerging conceptual questions that can guide future research. Answers to these fluid mechanics questions can lead to breakthroughs in our ability to predict, diagnose, and correct biological dysfunction, while also inspiring a host of new engineering technologies.

Singularities of the Navier-Stokes equations occur when some derivative of the velocity field is infinite at any point of a field of flow (or, in an evolving flow, becomes infinite at any point within a finite time). Such singularities can be mathematical (as, e.g., in two-dimensional flow near a sharp corner, or the collapse of a Möbius-strip soap film onto a wire boundary), in which case they can be “resolved” by refining the geometrical description; or they can be physical (as, e.g., in the case of cusp singularities at a fluid/fluid interface), in which case resolution of the singularity involves incorporation of additional physical effects; these examples will be briefly reviewed. The “finite-time singularity problem” for the Navier-Stokes equations will then be discussed and a recently developed analytical approach will be presented; here it will be shown that, even when viscous vortex reconnection is taken into account, there is indeed a physical singularity, in that, at sufficiently high Reynolds number, vorticity can be amplified by an arbitrarily large factor in an extremely small point-neighborhood within a finite time, and this behavior is not resolved by viscosity. Similarities with the soap-film-collapse and free-surface-cusping problems are noted in the concluding section and the implications for turbulence are considered.

There comes a time in each of our lives where we grab a thick section of the morning paper, roll it up and set off to do battle with one of nature's most accomplished aviators—the fly. If, however, instead of swatting we could magnify our view and experience the world in slow motion we would be privy to a world-class ballet full of graceful figure-eight wing strokes, effortless pirouettes, and astonishing acrobatics. After watching such a magnificent display, who among us could destroy this virtuoso? How do flies produce acrobatic maneuvers with such precision? What control mechanisms do they need to maneuver? More abstractly, what problem are they solving as they fly? In this article and the associated video presentation of my invited lecture from the ${71}^{\mathrm{st}}$ Annual Meeting of the American Physical Society's Division of Fluid Dynamics, I describe the challenges associated with answering these questions, our attempts to investigate them, and, through various demonstrations depicted in the video, qualitatively illustrate the mechanisms used by these insects to achieve these astonishing behaviors.

Detailed studies of modern large-scale wind turbines represent a significant challenge. The immense length scales characteristic of these machines, in combination with rotational effects, render numerical simulations and conventional wind tunnel tests unfeasible. Field experiments can give us important insight into the aerodynamics and operation, but they are always accompanied by large amounts of uncertainty, due to the changing nature of the inflow and the lack of accurate control of the test conditions. Here, a series of experiments is presented, using an alternative method that enables us to represent and study much of the physics governing the large-scale wind turbines in small-scale models. A specialized, compressed-air wind tunnel is used to achieve dynamic similarity with the field-scale, but under accurately controlled conditions of the laboratory. Power and thrust coefficients are investigated as a function of the Reynolds number up to ${\text{Re}}_D=14\times {10}^6$, at tip speed ratios representative of those typical in the field. A strong Reynolds number dependence is observed in the power coefficient, even at very high Reynolds numbers (well exceeding those occurring in most laboratory studies). We show that for an untripped rotor, the performance reaches a Reynolds number invariant state at ${\text{Re}}_c\ge 3.5\times {10}^6$, regardless of the tip speed ratio. The same model was also tested with scaled tripping devices, with a height of only $9\mu \mathrm{m}$, to study the effect of transition on the rotor performance. In the tripped case, the Reynolds number dependence was eliminated for all tested cases, suggesting that the state of the boundary layer is critical for correct predictions of rotor performance.

Coherent structures in the form of streamwise elongated streaks and vortices play a key role in energy growth, momentum transfer, and the self-sustaining processes underlying wall-bounded turbulent flow. A wide range of conceptual and physics-based models have been employed to analyze the role of these structures. This article focuses on the restricted nonlinear (RNL) modeling framework, a physics-based approach that simplifies the flow representation based on the dominance of streamwise coherent structures. This model is formed by decomposing the Navier-Stokes (NS) equations into a streamwise constant (averaged) mean flow and perturbations about that mean. Order reduction is then obtained through a dynamical restriction of the nonlinear interactions between the perturbations. We review the success of this model in reproducing statistical and spectral properties of wall-bounded turbulent flows at moderate Reynolds numbers and within a large-eddy simulation (LES) framework in the limit of infinite Reynolds number. An analysis of energy transfer in half-channel RNL flow highlights the critical nonlinearity and scale interactions necessary to sustain turbulence at moderate Reynolds numbers. Our results also indicate that the fundamental properties of wall-bounded turbulence such as skin friction drag are robust to dynamical restrictions of streamwise-varying interactions, which may lend to the difficulty in controlling these flows. The article concludes with a discussion of ongoing challenges for the RNL model and the need to unify existing approaches to meet the challenges of characterizing and controlling high-Reynolds number wall turbulence.

One of the ultimate goals of science is an understanding of biological cells so complete that one can construct a living cell from its constituent molecules, control its dynamics, and repair its machinery. Advances in experimental and computational biology techniques over the past 30 years have led to landmark progress toward this goal, from atomistic models of proteins to synthesis of entire bacterial genomes. However, the current frontier in operational mastery of cells arguably resides at the interface between biology and fluid physics: cellular processes that operate over colloidal length scales, where continuum fluid mechanics and Brownian motion underlie whole-cell-scale behavior. It is at the colloidal scale that much of cell machinery operates and where reconstitution and manipulation of cells is most challenging. This operational regime is centered between the two well-understood regimes of structural biology and systems biology, where the former focuses on atomistic-scale spatial resolution with little time evolution and the latter on kinetic models that abstract space away. Low-Reynolds-number colloidal hydrodynamics modeling bridges the divide between these regimes by unifying the disparate length scales and timescales of solvent molecule and colloidal dynamics and may hold the key to operational mastery of cells. Bridging the divide between the two disciplines of biology and fluid physics is as much a part of the way forward as are developing new tools and asking new questions. In this paper we highlight the central and nontrivial roles played by low-Reynolds-number hydrodynamics and colloidal-scale motion that appear in common across cell functions, types, and conditions.

Painting is a fluid mechanical process. The action of covering a solid surface with a layer of a viscous fluid is one of the most common human activities; virtually all manmade surfaces are painted to provide protection against the environment or simply for decoration. This process, in an industrial context, has been vastly studied and it is well understood. In the case of artistic painting the purpose is different. Painters learn how to manipulate the nonuniform deposition of paint onto a surface, through lengthy empirical testing of the action and modifying the physical properties of the fluids, to create textures and patterns of aesthetic value. In this paper, an analysis of some notable painting techniques is presented from the point of view of fluid mechanics. In particular, we discuss the so-called “accidental painting” technique, originally devised by David A. Siqueiros, which is the result of a Rayleigh-Taylor instability. An analysis of several techniques used by Jackson Pollock is also presented, showing how he learned to carefully control the motion of viscous filaments to create his famous abstract compositions. We also briefly discuss how pattern and textures are produced in decalcomania and watercolor painting. These investigations indicate that it is possible to establish concrete scientific discussions among modern fluid mechanics, art, art history, and conservation.

Flexible slender structures in flow are everywhere. While a great deal is known about individual flexible fibers interacting with fluids, considerably less work has been done on fiber ensembles, such as fur or hair, in flow. These hairy surfaces are abundant in nature and perform multiple functions from thermal regulation to water harvesting to sensing. Motivated by these systems, I consider two examples of hairy surfaces interacting with flow which were presented in the Corrsin lecture at the 2018 APS-DFD meeting (71st Annual Meeting of the APS Division of Fluid Dynamics, Atlanta, 2018). In the first example I consider a toy problem of angled hairs in Couette flow. Using this simple model, I explore asymmetry in the flow and anomalous drag scaling by exploiting various limits in the parameter space. In the second example I consider viscous dipping, a feeding method utilized by many nectar drinking animals. Previous studies have analyzed these drinking strategies through the Landau-Levich-Derjaguin framework; however, many viscous dippers have hairy structures on their tongues that enhance fluid uptake. Here I investigate the impact of mesoscale hairy structures on feeding efficiency and conclude with general comments on drainage through beds of hairs.

Suspensions of anisotropic Brownian particles are commonly encountered in a wide array of applications such as drug delivery and manufacturing of fiber-reinforced composites. Technological applications and fundamental studies of small anisotropic particles critically require precise control of particle orientation over defined trajectories and paths. In this work, we demonstrate robust control over the two-dimensional center-of-mass position and orientation of anisotropic Brownian particles using only fluid flow. We implement a path-following model predictive control scheme to manipulate colloidal particles over defined trajectories in position space, where the speed of movement along the path is a degree of freedom in the controller design. We further explore how the external flow field affects the orientation dynamics of anisotropic particles in steady and transient extensional flow using a combination of experiments and analytical modeling. Overall, this technique offers new avenues for fundamental studies of anisotropic colloidal particles using only fluid flow, without the need for external electric or optical fields.

The entrainment phenomena of the quiescent core region in a turbulent pipe flow are examined by characterizing the tall wall-attached structures of the streamwise velocity fluctuations $\left(u\right)$ extracted from direct numerical simulation data for $\mathrm{R}{\mathrm{e}}_{\tau }=926$. The quiescent core region is the uniform momentum zone with the highest streamwise velocity magnitude. The conditionally averaged turbulence statistics undergo an abrupt jump across the boundary of the core region, which indicates that there is a viscous shear layer at this interface, and that it is similar to the turbulent-nonturbulent interface. The two-point correlation of the interface height fluctuations extends along the streamwise direction in a manner similar to that of $u$ at the core boundary. The magnitudes of the local entrainment velocity were conditionally sampled by measuring the local curvatures of the core boundary, which are associated with large-scale structures. The local entrainment velocity $\left(v_{\mathrm{n}}\right)$ is enhanced at saddle concave and elliptic convex regions of the interface. This behavior mainly arises from small-scale motions due to viscous diffusion, which indicates that the large-scale features modulate the small scales associated with the entrainment velocity. In particular, the saddle concave shape formed by the bulgelike low-speed structures significantly strengthens the viscous component of $v_{\mathrm{n}}$. The turbulence statistics near the core boundary were obtained based on the projection area of the tall wall-attached structures that were extracted from the instantaneous flow field. Sharp changes in the turbulence statistics are evident; the magnitude of these changes increases with increases in the height $\left(l_y\right)$ of the identified structures. We reconstructed the turbulence statistics by using the tall structures with ${l_y}^{+}>100$ and found that the reconstructed profiles are remarkably similar to the turbulence statistics across the core boundary. These results confirm that the sudden changes in the averaged turbulence statistics and the modulation of $v_{\mathrm{n}}$ are associated with the instantaneous wall-attached structures that reach the core region.

The swimming characteristics achieved by flapping wings, translating motion, and shell pitching are studied from observations of shelled Antarctic pteropods (aquatic snails nicknamed “sea butterflies”). These pteropods (Limacina helicina antarctica) swim with a pair of parapodia (or “wings”) via a unique flapping propulsion mechanism that incorporates similar techniques as observed in small flying insects. The geometric scaling of the wing span ($L$), wing chord ($c$), and minor shell diameter ($d$) with respect to the major shell diameter ($D$) reveal geometric similitude. Thus, the major shell diameter ($D$) is the only length scale required to describe the size of the pteropods. The motion of swimming pteropods is characterized using flapping, translational, and rotational Reynolds numbers (i.e., ${\text{Re}}_f, {\text{Re}}_U$, and ${\text{Re}}_{\mathrm{\Omega }}$). A critical value of the flapping Reynolds number, ${\text{Re}}_f=35$, is found for the onset of translating and pitching locomotion. Finally, the relationship is obtained for the Strouhal number (${\text{St}}_A=fA/U$) for the pteropods using the geometric scalings and the translational and flapping Reynolds numbers. The Strouhal number is found to be between 0.2 and 0.4, which indicates general agreement with other oscillating organisms moving with high propulsion efficiency.

Channels are fundamental building blocks from biophysics to soft robotics, often used to transport or separate solutes. As solute particles inevitably transverse between streamlines along the channel by molecular diffusion, the effective diffusion of the solute along the channel is enhanced, an effect known as Taylor dispersion. Here we investigate how the Taylor dispersion effect can be suppressed or enhanced in different settings. Specifically, we study the impact of flow profile and active or pulsating channel walls on Taylor dispersion. We derive closed analytic expressions for the effective dispersion equation in all considered scenarios providing hands-on effective dispersion parameters for a multitude of applications. In particular, we find that active channel walls may lead to three regimes of dispersion: either dispersion decrease by entropic slow down at small Péclet number, or by dispersion increase at large Péclet number dominated either by shuttle dispersion, or by Taylor dispersion. This improves our understanding of solute transport, e.g., in biological active systems such as blood flow, and opens a number of possibilities to control solute transport in artificial systems such as soft robotics.

In almost all of the studies on the circular hydraulic jump (CHJ), gravity had been considered as a significant variable that affects the formation of the jump. Most recently, gravity was deprived of being important in the origin of the CHJ, which challenged researchers in this field of fluid mechanics. This study addresses in detail the physical concepts behind this intriguing phenomenon occurring in the radial outspreading of a vertically downward free-surface liquid impinging jet upon a horizontal plate. The aim is to find out whether gravity plays any role in the origin of the CHJ. Accordingly, the jump evolution is investigated in two cases: first, the initial formation of the CHJ in which the subcritical flow downstream from the jump is approaching the outlet boundary (developing jump). Second, the final evolution of the CHJ in which a steady-state flow is circumventing an obstacle at the edge of the impinged plate and falling uniformly down from the outlet boundary (developed jump). The results indicate the existence of two different flow regimes in the jump formation: gravity- and capillary-dominant flow regimes. In general, the role of gravity in the formation of developing or developed jumps cannot be eliminated; however, its importance lies in the fact of which regime dominates the flow. Intensification of gravitational effects is observed when capillary waves are dampened by increasing viscosity, density, or volume flow rate as well as by decreasing surface tension. Finally, a generalized scaling relation for the jump radius is obtained considering both capillary and gravitational effects in the critical flow condition. In contrast to the previous results, this generalized scaling relation predicts more accurately the radius of both a developing and a developed jump.

Stability analysis of stratified multiphase flow for a spanwise system of rotation plays a pivotal role in micromixing and micromachines. In several such systems, centrifugal actuation is the driving force, which creates a pressure gradient in a rotating channel and Coriolis force enhances mixing in a short span by destabilizing the flow. Here, we focus on the impact of the Coriolis force on a rotating two-fluid flow through a microchannel, which is miscible in nature, having small viscosity difference and thereby forming a thin diffusive interface between fluids due to viscosity stratification. Modal stability analysis is used to estimate the critical flow parameters which are, in turn, responsible for regulating the instability mechanism for different viscosity contrasts and mixed layer thicknesses. Usually, viscosity stratified flow with respect to streamwise disturbance becomes more unstable for a thinner mixed layer. On the contrary, our numerical computation confirms a completely discrepant scenario by considering Coriolis force-driven instability of a miscible flow system on account of spanwise disturbances. Possible physical mechanisms for the same are discussed in terms of base flow pattern and the energy fluctuation between the perturbed and base flow. Comparison of three-dimensional disturbances of the flow field, in both clockwise and anticlockwise directions (for two different viscosity ratios), is executed to provide an insight into the dynamics of the flow system. Distributions of the velocity perturbations display a critical bonding between the vortices near and away from the mixed layer. These vortices are, in turn, responsible for the variation in instability mechanism with respect to different viscosity ratios and rotational directions.

Large-eddy simulation (LES) of turbulence in complex geometries and domains is often conducted with high-aspect-ratio resolution cells of varying shapes and orientations. The effects of such anisotropic resolution are often simplified or neglected in subgrid model formulations. Here, we examine resolution-induced anisotropy and demonstrate that even for isotropic turbulence, anisotropic resolution induces mild resolved Reynolds stress anisotropy and significant anisotropy in second-order resolved velocity gradient statistics. In large-eddy simulations of homogeneous isotropic turbulence with anisotropic resolution, it is shown that commonly used subgrid models, including those that consider resolution anisotropy in their formulation, perform poorly. The one exception is the anisotropic minimum dissipation model proposed by Rozema et al. [Phys. Fluids 27, 085107 (2015)]. A simple model is presented here that is an anisotropic eddy diffusivity extension of the “Kolmogorov expression” eddy viscosity of Carati et al. [A family of dynamic models for large-eddy simulation, in Annual Research Briefs, Center for Turbulence Research (Stanford University and NASA AMES, 1995), pp. 35–40] that depends explicitly on the anisotropy of the resolution. It also performs well and is remarkable because, unlike other LES subgrid models, the eddy diffusivity only depends on statistical characteristics of the turbulence (in this case, the dissipation rate), not on fluctuating quantities. In other subgrid modeling formulations, such as the dynamic procedure, limiting flow dependence to statistical quantities in this way could have advantages.

The dynamics of geometrically nonlinear flexible filaments play an important role in a host of biological processes, from flagella-driven cell transport to the polymeric structure of complex fluids. Such problems have historically been computationally expensive due to numerical stiffness associated with the inextensibility constraint, as well as the often nontrivial boundary conditions on the governing high-order PDEs. Formulating the problem for the evolving shape of a filament via an integral equation in the tangent angle has recently been found to greatly alleviate this numerical stiffness. The contribution of the present manuscript is to enable the simulation of nonlocal interactions of multiple filaments in a computationally efficient manner using the method of regularized stokeslets within this framework. The proposed method is benchmarked against a nonlocal bead and link model, and recent code utilizing a local drag velocity law. Systems of multiple filaments (1) in a background fluid flow, (2) under a constant body force, and (3) undergoing active self-motility are modeled efficiently. Buckling instabilities are analyzed by examining the evolving filament curvature, as well as by coarse graining the body frame tangent angles using a Chebyshev approximation for various choices of the relevant nondimensional parameters. From these experiments, insight is gained into how filament-filament interactions can promote buckling, and further reveal the complex fluid dynamics resulting from arrays of these interacting fibers. By examining active moment-driven filaments, we investigate the speed of worm- and spermlike swimmers for different governing parameters. The matlab® implementation is made available as an open-source library, enabling flexible extension for alternate discretizations and different surrounding flows.

We investigate the development of flow-induced small-amplitude high-frequency oscillations occurring in a rigid fluid-conveying channel with a section of the upper wall replaced by a taut elastic sheet. We consider an axially nonuniform base state, caused by a negative transmural pressure (internal minus external), which results in the sheet bending inward, and then examine the evolution of oscillatory perturbations about this state. Due to the curvature of the base state, any normal displacement of the sheet will lead to a change in the axial stretching of the sheet, which is an order of magnitude higher than would be the case for perturbations to a uniform sheet. This stretching provides an additional restoring force that can be dominant. We derive a modified tube law to describe the wall mechanics with this additional effect and combine it with an existing fluid model to obtain a complete description of the system. At leading order, we obtain a one-dimensional eigenvalue problem for the frequencies and mode shapes of the oscillatory perturbations. The nonlinear interaction between the perturbations and the base-state curvature manifests itself as an additional integral term in the eigenvalue problem, corresponding to the total axial stretching in the wall. The normal modes are neutrally stable at leading order, and their slow growth or decay is determined by considering the global energy budget of the system. The stability of the system can be expressed in terms of a critical Reynolds number of the mean flow through the channel. We explain the behavior of the system as two key dimensionless parameters $\mathcal{F}$ and $\mathcal{K}$ are varied. These quantify the dimensionless axial tension and base-state curvature effects, respectively. Numerical simulations are used to identify distinct flow regimes in parameter space, which we explore further in detail using asymptotic analysis. This reveals that at large $\mathcal{K}$, i.e., strong base-state curvature effects, the leading-order axial profiles of the modes are forced to adjust in order to eliminate the need for significant stretching in the wall. This has the effect of stabilizing both the fundamental and, to a lesser extent, the first harmonic modes, which results in the first harmonic becoming the most unstable mode.