Particles migrate in the transverse direction of the flow due to the existence of normal stress anisotropy in weakly viscoelastic liquids. We test the ability of theoretical predictions to predict the transverse velocity migration of particles in a confined Poiseuille flow according to the viscoelastic constitutive parameters of dilute polymer solutions. First, we carefully characterize the viscoelastic properties of two families of dilute polymer solutions at various concentrations using shear rheometry and capillary breakup experiments. Second, we develop a specific three-dimensional particle tracking velocimetry method to measure with a high accuracy the dynamics of particles focusing in flow for Weissenberg numbers Wi ranging from ${10}^{-2}$ to ${10}^{-1}$ and particle confinement $\beta$ of 0.1 and 0.2. The results show unambiguously that the migration velocity scales as $\text{Wi}{\beta }^2$, as expected theoretically for weakly elastic flows of an Oldroyd-B liquid. We conclude that classic constitutive viscoelastic laws are relevant to predict particle migration in dilute polymer solutions whereas detailed analysis of our results reveals that theoretical models overestimate by a few tenths the efficiency of particle focusing.

This work reports on the unstable region and the transition process of the boundary-layer flow induced by a rotating cone with a half apex angle of 60 degrees using the probability density function (PDF) contour map of the azimuthal velocity fluctuation, which was first used by Imayama et al. [Phys. Fluids 24, 031701 (2012)] for the similar boundary-layer flow induced by a rotating disk. The PDF shows that the transition behavior of the rotating-cone flow is similar to that on the rotating disk. The effects of roughness elements on the cone surface have been examined. For the cone with roughnesses, we reconstructed the most probable vortex structure within the boundary layer from the hot-wire anemometry time signals. The results show that the PDF clearly describes the overturning process of the high-momentum upwelling of the spiral vortices, which due to vortex meandering cannot be detected in the phase-averaged velocity field reconstructed from the point measurements. At a late stage of the overturning process, our hot-wire measurements captured high-frequency oscillations, which may be related to secondary instability.

For two-dimensional flow past a linearly sprung circular cylinder to which has been attached an internal “nonlinear energy sink” consisting of a mass allowed to rotate about the cylinder axis and a viscous damper that linearly retards the rotational motion of that mass, we show, for a given set of parameters, that as many as three distinct unsteady long-time solutions (two periodic and one chaotic), in addition to one or more steady solutions, can coexist. For other combinations of the parameters, two unsteady solutions (both periodic, one periodic and one quasiperiodic, one periodic and one chaotic, one quasiperiodic and one chaotic, or both chaotic) can coexist with one or more steady solutions. These multiple solutions, all of which appear to be linearly stable, are found in a range of Reynolds number ($15<\text{Re}<50$) in which the flow is expected to be two dimensional. The discovery of this unprecedented degree of solution multiplicity establishes the potential of this system to serve, at low Re, as a test bed for detailed investigation of basins of attraction of the Navier-Stokes equations, and in studies of noise- and disturbance-induced transitions between different long-time solutions.

Available direct numerical simulation of turbulent channel flow at moderately high Reynolds numbers data show that the logarithmic diagnostic function is a linearly decreasing function of the outer-normalized wall distance $\eta =y/\delta$ with a slope proportional to the von Kármán constant, $\kappa =0.4$. The validity of this result for turbulent pipe and boundary layer flows is assessed by comparison with the mean velocity profile from experimental data. The results suggest the existence of a flow-independent logarithmic law ${\overline{U}}^{+}=\overline{U}/u_{\tau }=(1/\kappa )ln(y^{*}/a)$, where $y^{*}=y{U}_S/\nu$ with $U_S=yS\left(y\right)$ being the local shear velocity and the two flow-independent constants $\kappa =0.4$ and $a=0.36$. The range of its validity extends from the inner-normalized wall distance $y^{+}=300$ up to half the outer-length scale $\eta =0.5$ for internal flows, and $\eta =0.2$ for zero-pressure-gradient turbulent boundary layers. Likewise, and within the same range, the mean velocity deficit follows a flow-dependent logarithmic law as a function of a local mean-shear-based coordinate. Furthermore, it is illustrated how the classical friction laws for smooth pipe and zero-pressure-gradient turbulent boundary layer are recovered from this scaling.

This Rapid Communication investigates the scaling of the Reynolds shear and normal stresses in a differentially heated vertical channel (DHVC). Analysis of the mean momentum balance equation, with an approximation of the mean temperature in the outer layer via a logarithmic function, reveals a relationship, $\frac{g\alpha {\theta }_{\tau }\delta }{|R_{\mathrm{wu}}{|}_{\mathrm{ctr}}}=O\left(1\right)$, among key parameters in a DHVC including the gravitational acceleration $g$, thermal expansion coefficient $\alpha$, channel half width $\delta$, friction temperature ${\theta }_{\tau }$, and the maximum magnitude of Reynolds shear stress $|R_{\mathrm{wu}}{|}_{\mathrm{ctr}}$. Direct numerical simulation (DNS) data indicate that at sufficiently high Grashof numbers, $\frac{g\alpha {\theta }_{\tau }\delta }{|R_{\mathrm{wu}}{|}_{\mathrm{ctr}}}\approx 0.95–1$, supporting the validity of the analysis. Based on analysis of the DNS data, a mixed scale $u_{\tau }{U}_{\max }$ is found to be a proper scale for the Reynolds shear stress in a DHVC. Here, $U_{\max }$ is the maximum mean velocity in the $x$ direction, and $u_{\tau }$ is the friction velocity. This mixed scale also captures the scaling of the Reynolds normal stresses better than previously used scales.

The macroscale structure and microscale fluctuation statistics of late-time asymptotic steady-state flows in cylindrical geometries is studied using the methods of equilibrium statistical mechanics. The axisymmetric assumption permits an effective two-dimensional (2D) description in terms of the (toroidal) flow field $\sigma$ about the cylinder axis and the vorticity field $\xi$ that generates mixing within the (poloidal) planes of fixed azimuth. As for a number of other 2D fluid systems, extending the classic 2D Euler equation, the flow is constrained by an infinite number of conservation laws, beyond the usual kinetic energy and angular momentum. All must be accounted for in a consistent equilibrium description. It is shown that the most directly observable impact of the conservation laws is on $\sigma$, which displays interesting large-scale radius-dependent flow structure. However, unlike in some previous treatments, we find that the thermodynamic temperature is always positive. As a consequence, except for an infinitesimal boundary layer that maintains the correct (conserved) value of the overall poloidal circulation, the impact on $\xi$ resides in the statistics of the strongly fluctuating, fine-scale mixing, where it is sensitive to “antiferromagnetic” microscale correlations that help maintain the analog of local charge neutrality. The poloidal flow is macroscopically featureless, displaying no large-scale circulating jet- or eddylike features (which typically emerge as negative temperature states in analogous Euler and quasigeostrophic equilibria).

Piezoelectric bending actuators have been used to drive flaps to manipulate vortex shedding from a nominally two-dimensional blunt trailing edge (BTE) using three different actuation methods. The experiments were conducted in a wind tunnel at a Reynolds number of $\text{Re}\left(h\right)=2600$ where $h$ is the height of the BTE. A combination of base-pressure measurements at the BTE and high-speed particle image velocimetry was used to characterize actuator performance over their operating range. Proper orthogonal decomposition was also applied to study the energy content of the primary vortex shedding structures in the wake during actuation. Cases of significant vortex shedding amplification and suppression were investigated. Vortex shedding amplification occurred when the frequency of actuation matched that of the natural wake process, resulting in the turbulent kinetic energy associated with two-dimensional vortex shedding increasing from 70% to 90%. This led to higher turbulence intensities, a shortened recirculation region, and more organized vortex shedding in general. Vortex shedding suppression was caused by the generation of small spanwise vortices at the tips of the actuators, which disrupted the interaction between the separating shear layers. The energy associated with vortex shedding was reduced from 70% to 20% in this case, resulting in a significant attenuation of the shedding pattern for at least six BTE thicknesses downstream. Vortex shedding suppression was accompanied by greatly reduced turbulence intensities, a narrowed wake, and a lengthened recirculation region. The actuators were also capable of forcing symmetry in the near-wake using a symmetric actuation method, but the vortex shedding resumed downstream, although with a reduced frequency. Finally, an adaptive slope-seeking controller was designed using the base-pressure measurements at the BTE in real time. The closed-loop controller was capable of seeking and maintaining an optimal input for vortex shedding suppression, even during slow variations in freestream velocity.

Viscous fingering (VF) is a hydrodynamic flow instability that occurs when a less-viscous liquid (LVL) displaces a more-viscous liquid (MVL) in porous media and takes place in either miscible or immiscible systems. In this study, we investigated the three-dimensional (3D) characteristics of VF for miscible non-Newtonian fluids with and without chemical reactions using a microfocus x-ray computed tomography scanner. When the viscosity of a system increased with a chemical reaction, the VF was suppressed, and pistonlike displacement occurred. The cylindrical domain of displacing LVL was surrounded by a high-viscosity filmlike layer because of the chemical reaction that impeded the mixing of the LVL and MVL. Consequently, the high LVL concentration inside this layer was retained without mixing with the MVL. A clear local peak in viscosity appeared in the viscosity profiles at the interface between the LVL and MVL. Behind the peak, an unfavorable viscosity profile of increasing viscosity along the flow direction appeared, but the VF was not induced because of the smooth viscosity gradients. After the peak, the viscosity profiles were in favorable conditions. In the nonreactive case, the tip of the fingers further advanced compared with that in the reactive case and went further with the decrease in Pe, which reflected the property of shear-thinning fluids. By contrast, in a decreasing-viscosity system with a chemical reaction, the tips of the fingers were located at a larger distance compared with that without reaction. The concentration in the finger intensively decreased from the point of injection because of the intensive mixing of the LVL and MVL. As a result, the area fraction of injected LVL in the reactive cases was lower than that in the nonreactive cases. The tip of the finger advanced with time, whereas LVL concentration in the finger hardly recovered above 0.4 in the reactive case, which reflected intensive fingering and mixing.

The incessant swimming motion of microbes in dense suspensions can give rise to striking collective motions and coherent structures. However, theoretical investigations of these structures typically utilize either computationally demanding numerical simulations or simplified continuum models. Here we analytically investigate the collective dynamics of a dense array of steady, spherical squirmers. We first calculate the forces and torques acting on two closely separated squirmers, through solving the Stokes equations to second order in the ratio of mean spacing to squirmer radius. This lubrication analysis is then used to assess the stability of a dense, vertical, planar array of identical three-dimensional squirmers. The system of vertically oriented squirmers is unstable if there is no short-range repulsive force between them, even when there is a strong gravitational torque on them because they are bottom-heavy. When there is a repulsive force the positions of the squirmers are stable, but the orientations are unstable unless the bottom-heaviness parameter $G_{bh}$ is sufficiently large. The predictions of instability and possible long time behavior are qualitatively the same for monolayers confined between two parallel rigid planes as for unconfined monolayers. The predictions compare favorably with published numerical simulations, and reveal the existence of additional dynamic structures not previously observed; puller-type squirmers show a greater range of structures than pushers. The use of pairwise lubrication interactions provides an efficient means of assessing stability of dense suspensions usually tackled using full numerical simulations.

Anti-phase and in-phase flickering modes of dual buoyant diffusion flames were numerically investigated and theoretically analyzed in this study. Inspired by the flickering mechanism of a single buoyant diffusion flame, for which the deformation, stretching, or even pinch-off of the flame surface result from the formation and evolution of the toroidal vortices, we attempted to understand the anti-phase and in-phase flickering of dual buoyant diffusion flames from the perspective of vortex dynamics. The interaction between the inner-side shear layers of the two flames was identified to be responsible for the different flickering modes. Specifically, the transition between anti-phase and in-phase flickering modes can be predicted by a unified regime nomogram of the normalized flickering frequency versus a characteristic Reynolds number, which accounts for the viscous effect on vorticity diffusion between the two inner-side shear layers. Physically, the transition of the vortical structures from symmetric (in-phase) to staggered (anti-phase) in a dual-flame system can be interpreted as being similar to the mechanism causing flow transition in the wake of a bluff body and forming the Karman vortex street.

We present a reduced system of seven ordinary differential equations that captures the time evolution of spatial gradients of the velocity and the temperature in fluid elements of stratified turbulent flows. We show the existence of invariant manifolds (further reducing the system dimensionality), and compare the results with data stemming from direct numerical simulations of the full incompressible Boussinesq equations in the stably stratified case. Numerical results accumulate over the invariant manifolds of the reduced system, indicating the system lives at the brink of an instability. Finally, we study the stability of the reduced system, and show that it is compatible with recent observations in stratified turbulence of nonmonotonic dependence of intermittency with stratification.

In dynamical systems theory, suppression of instabilities around a fixed point is generally achieved by controlling the linearized dynamics of infinitesimal perturbations, because considering small-amplitude disturbances allows for application of a range of celebrated techniques from linear control theory. In this paper, we consider the problem of design and implementation of a controller for fully nonlinear, high-dimensional, dynamical systems with the goal of steering trajectories to an unstable fixed point of the governing equations. Our control strategy is based on our previous work [A. Blanchard, S. Mowlavi, and T. P. Sapsis, Nonlinear Dynam. 95, 2745 (2019)] and takes advantage of the unique properties of the optimally time-dependent (OTD) modes, a set of global, time-evolving, orthonormal modes that track directions in phase space associated with transient growth and persistent instabilities. We show that the OTD control strategy introduced previously is robust with respect to perturbation amplitude even in cases in which the trajectory initially evolves on an attractor that lies far away from the target fixed point. In recognition of the fact that actuation capabilities are generally limited in practice, we also formulate a localized control strategy in which the OTD modes are computed in a spatially localized subdomain of the physical domain of interest. We suggest a strategy for selecting the optimal control domain based on a quantitative criterion derived from the OTD modes. We show that even when the range of the controller is reduced, OTD control is able to steer trajectories toward the target fixed point.

The effect of phase difference on the collective locomotion of two tandem flapping foils is numerically studied in this paper. The numerical results indicate that the collective locomotion is greatly affected by the phase difference. Two distinct collective modes are observed, i.e., the fast mode and the slow mode. The fast mode is only observed in part of the range of the phase difference ($\varphi$), i.e., $\varphi =0.0-0.1\pi$ and 1.5π–1.9π with appropriate initial distance, and the slow mode appears in the range of $\varphi =0.3\pi -1.4\pi$. Meanwhile, the follower of the two foils has hydrodynamic benefit in both fast and slow modes, and it can obtain the highest efficiency in the fast mode at $\varphi =1.6\pi$. However, the leader can only achieve hydrodynamic benefit in the fast mode, and the highest efficiency occurs at $\varphi =0.1\pi$. In addition, the stable distance between two foils in the slow mode can be quantized with the phase difference. Furthermore, the fluid-structure interactions between two foils are also analyzed. Two distinct vortex interactions are observed in the fast mode, i.e., merging interaction and broken interaction, which, respectively, result in the highest propulsive efficiency for the follower and the leader. In the merging interaction, the leading edge vortex of the leader is captured by the follower, which results in the weak trailing edge vortex of the leader but a strong trailing edge vortex of the follower. In the broken interaction, the leading edge vortex of the follower sheds into the wake together with the trailing edge vortex of the leader, and induces the trailing edge vortex of the follower to be broken into two parts. Which kind of vortex interaction occurs depends on the phase difference. The results obtained here may provide some light on understanding the coordinated behavior of biological collectives.

The spreading of liquid droplets impacting a surface at high speed is well understood by now. However, when a droplet impacts a surface at relatively low impact velocities ($<1$ m/s), the wetting properties of the fluid become important, and the entrapped air layer between the impacting drop and the solid surface prevents the immediate wetting of the surface. To determine the influence of both wetting and the entrapped air, we perform experiments by systematically varying the surface tension of the liquid and the air pressure. Drop impact measurements at reduced air pressures show that the spreading is independent of the pressure; dynamic contact-angle measurements indicate that this happens because the air film breaks rapidly. By varying the surface tension and surface wettability, we show that droplet spreading at low velocities can be predicted from the wetting properties of the surface and the known energy balance for the impact.

Ongoing development of fuel storage and delivery systems for space probes, interplanetary vehicles, satellites, and orbital platforms continues to drive interest in propellant management systems that utilize surface tension to retain, channel, and control flow in microgravity environments. Although it has been known for decades that capillary flows offer an ideal method of fuel management, there has been little research devoted to the general stability properties of such flows. In this work we demonstrate theoretically why capillary flows which channel wetting liquids in slender open triangular channels tend to be very stable against disturbances. By utilizing the gradient flow form of the governing fluid interface equation, we first prove that stationary interfaces in the presence of steady flow are asymptotically nonlinearly and exponentially stable in the Lyapunov sense. We then demonstrate that fluid interfaces exhibiting self-similar Washburn dynamics are transiently and asymptotically linearly stable to small perturbations. This second finding relies on a generalized nonmodal stability analysis due to the non-normality of the governing disturbance operator. Taken together, these findings reveal the robust nature of transient and steady capillary flows in open grooved channels and likely explain the prevalent use of capillary flow management systems in many emerging technologies ranging from CubeSats to point-of-care microfluidic diagnostic systems.

The hysteresis behavior of spanwise rotating plane Couette flow is investigated by direct numerical simulations, where the simulations are carried out at a computational box $8\pi h\times 2h\times 4\pi h$ (here $h$ is the half-channel height) along two opposite directions in rotation number ($\text{Ro}=2\mathrm{\Omega }h/U_w$, with $\mathrm{\Omega }$ the constant angular velocity in the spanwise direction and $U_w$ half of the wall velocity difference) space. When rotation speed increases, structures with two pairs of roll cells appear in $0.02\le \text{Ro}\le 0.3$. However, structures with three pairs of roll cells are observed for $0.03\le \text{Ro}\le 0.3$ when Ro decreases from 0.5. Turbulent statistics on the two branches, such as the friction Reynolds number, turbulent kinetic energy, and kinetic energy from the secondary flows, exhibit similar hysteresis behaviors during $0.03\le \text{Ro}\le 0.3$. Intuitive discussions are also made to explain the hysteresis behavior.

We report an experimental and analytical approach to characterize the pulsatile blood flow field based on Doppler ultrasound imaging of the carotid and brachial arteries. The diameter-averaged velocity, obtained from the instantaneous velocity histograms extracted from the Doppler waveform, was adapted to the solution of a pulsatile flow in a pipe, from which the instantaneous velocity profiles were predicted and compared to local velocity measurements in the carotid and brachial arteries of four healthy human subjects. Very good agreement as demonstrated by the regression slope of 0.97 and the near-zero intercept was observed between the spatiotemporal flow field predictions and local velocity measurements at specific distances from the vessel wall. Near-real-time in vivo measurements statistically demonstrate that the analytical and experimental approach presented herein precisely captures the pulsatile blood flow behavior in large blood vessels.

The vibrational excitation and dissociation of oxygen are modeled using different approaches with a range of fidelity, including the conventional two-temperature model, the state-specific method, and two variations of a model based on the maximum entropy principle. Comparison of the post-shock predictions with recent shock tube experimental data shows that the maximum entropy quadratic model predicts similar trends to the state-specific method and the experimental data. Although the maximum entropy quadratic model has significantly fewer equations than the state-specific method, no gain in computational efficiency is seen. Hence, the former model is further simplified by assuming that the vibrational relaxation can be described by the Landau-Teller formulation, with the corresponding relaxation times for ${\mathrm{O}}_2\text{−}{\mathrm{O}}_2$ and ${\mathrm{O}}_2\text{−}\mathrm{O}$ interactions determined from state-specific calculations of relaxation in a heat bath. The post-shock simulations indicate that the modified maximum entropy quadratic model maintains sufficient prediction accuracy while significantly improving computational efficiency. The proposed model could be used in computational fluid dynamics solvers for hypersonic nonequilibrium flow simulations.

We present here experimental measurements of the flow field around a swimming water droplet, using confocal particle image velocimetry in three dimensions. The droplet is denser than the continuous oil phase and swims close and parallel to the bottom wall. The measured flow field is first quantitatively characterized and compared to the flow field obtained for an axisymmetric swimmer in unbounded flow. Important qualitative differences are observed, in particular the emergence of a strong isotropic radial flow field in the planes parallel to the wall. These fundamental differences stress the critical impact of confinement on the flow field around the swimming droplet. We then propose an analytical formulation, based on a reduced-order description of the swimming droplet in terms of fundamental hydrodynamic singularities as well as the classical method of images. This model is able to account exactly for the effect of the wall and provides a simplified description of the flow field as the superposition of axisymmetric dipole and quadrupole singularities. We demonstrate that this simplified description quantitatively captures the effect of the wall on the dipolar and quadrupolar components of the flow field. Further including the confinement-induced asymmetry of the concentration field responsible for the droplet's Marangoni propulsion, our model is also able to account for the monopolar contribution to the experimental flow field which drives the dominant far-field signature of the droplet.

The impact of a Newtonian or non-Newtonian interface, possibly curved due to wetting effects, on the surface viscosimetry method historically developed by Mannheimer and Schechter [Mannheimer and Schechter, J. Colloid Interface Sci. 32, 195 (1970); Mannheimer and Schechter, J. Colloid Interface Sci. 32, 212 (1970)] is investigated theoretically and numerically in this paper. The Reynolds number is considered small enough for inertial effects to be negligible and thus for surface shear viscosity to be preferentially studied. The classical Boussinesq-Scriven law is modified by a shear-dependent viscosity governed by the Carreau constitutive law. The case of a curved interface is also investigated whether the liquid is hydrophilic or hydrophobic (liquid metals), which is revealed to be relevant for a liquid with a significant capillary length. The shallow-channel layout is found to be particularly well suited to the detection of a non-Newtonian liquid surface whatever its mechanical behavior: shear thinning or shear thickening. A key to evaluating the parameters involved in a surface viscosity model is to carry out a series of experiments at various rotation speeds, with preferably a nonwetting contact angle. Indeed, it is demonstrated that such a contact angle tends to amplify non-Newtonian effects, especially for a carrier fluid with a significant capillary length.