We analyze the resolvent operator in three flows around a nominal NACA-0012 airfoil at ${\text{Re}}_C=50,000$ and $5^{\circ }$ angle of attack. In particular, we study two naturally developing flows (around the airfoil with straight and blunt trailing edges) and one tripped flow. The naturally developing flows exhibit laminar separation, transition and turbulent reattachment, while the tripped flow remains attached in the suction side. For all cases, the time-averaged flow fields are computed from separate DNS simulations. The resolvent analysis can identify the areas of maximum receptivity of the flow field as well as the most amplified modes (optimal response). The former was located close to the leading edge, and in the case of naturally developing flows, also in the free-stream. The spatial distribution of optimal forcing was dominated by structures tilted against the mean flow, in agreement with other studies of boundary layer flows. The optimal response of the two naturally developing flows were different. For the NACA-0012 airfoil with straight trailing edge, the response was maximized at the natural frequency of the separating shear layer, and the shape matched closely with the corresponding DMD mode obtained from processing the DNS results. For the blunt airfoil, the receptivity of the separating shear layer was suppressed in the region of natural frequency, while it was maximized at the frequency of vortex shedding from the trailing edge. This is attributed to the lock-in mechanism between the shedding and the separating shear layer observed in the DNS simulations; the lock-in makes the separating shear layer less sensitive to forcing. In the tripped flow, the amplification of the perturbations is significantly diminished, and only by restricting the resolvent analysis to a region close to the suction side can we get a dominant frequency that matches with DNS at that region. The dominant resolvent modes were used to get estimations of the velocity spectra based only on the mean flow and the signal of one velocity component at discrete calibration points. Very good approximation is observed with only one or two points (depending on the flow examined), provided that they are located in energetic regions of the flow that contain sufficient spectral content at the relevant frequencies.

Jet drops ejected from bursting bubbles are ubiquitous, transporting aromatics from sparkling beverages, pathogens from contaminated water sources, and sea salts and organic species from the ocean surface to the atmosphere. In all of these processes, the smallest drops are noteworthy because their slow settling velocities allow them to persist longer and travel further than large drops, provided they escape the viscous sublayer. Yet it is unclear what sets the limit to how small these jet drops can become. Here we directly observe microscale jet drop formation and demonstrate that the smallest jet drops are not produced by the smallest jet drop-producing bubbles, as predicted numerically by Duchemin et al. [Duchemin et al., Phys. Fluids 14, 3000 (2002)]. Through a combination of high-speed imaging and numerical simulation, we show that the minimum jet drop size is set by an interplay of viscous and inertial-capillary forces both prior and subsequent to the jet formation. Based on the observation of self-similar jet growth, the jet drop size is decomposed into a shape factor and a jet growth time to rationalize the nonmonotonic relationship of drop size to bubble size. These findings provide constraints on submicron aerosol production from jet drops in the ocean.

Numerical simulations are performed to study the effect of particle size and density on dust dispersion behind a moving shock. The numerical model used in this paper takes into account multiple particle types with a binning approach, and each bin of particles has its own characteristic uniform particle size and density. The model solves one set of governing equations for the gas phase and $M$ sets of governing equations for the $M$ particle types. Equations for each bin of particles are coupled with the gas flow as well as all other particle types. The specific conditions simulate a Mach-1.4 shock passing over a dust layer containing two uniformly mixed particle types. The background gas condition is at 67 kPa and 295 K. The dust layer is 1.27 cm thick and consists of spherical particles with selected sizes and densities. Preliminary one-dimensional computations of a shock passing through a dilute particle curtain containing different particle types suggest that particles with different sizes or densities behave differently and can separate into different clouds. Particles with larger inertia require a longer relaxation time to the postshock condition. The two-dimensional calculations show that larger particles are lifted to a higher level than smaller particles. In regions near the shock front, larger particles experience a smaller drag force, pushing the particles into the dust layer, and a larger lifting force, pulling the particles into the air. In addition, lighter particles are lifted higher than heavier particles due to a smaller inertia. Particle size plays a significant role on dust dispersion, while particle density has only a minor effect.

As computing power has grown and turbulent flows at increasing Reynolds numbers are being computed by direct numerical simulations, conventional assumptions on adequate spatial and temporal resolutions are being continually challenged. We perform a systematic study of the resolution effects via numerical simulations at various spatial and temporal resolutions to clarify the proper scaling of dissipation and enstrophy (vorticity squared). Results show that inadequate resolution in space and/or time leads to overestimation of the likelihood and intensity of extreme fluctuations in dissipation and enstrophy. In order to capture rare events accurately, for instance, one needs to have not only grid resolutions that are increasingly smaller fractions of the Kolmogorov scale as the Reynolds number increases but also the Courant number that becomes increasingly smaller than that assumed to be adequate previously. Some comparisons are made with results obtained from an alternative approach where a stricter criterion for truncation in wave-number space allows aliasing errors to be removed completely. In contrast to prior work, the present data do not support the notion that dissipation and enstrophy probability density functions (PDFs) approach each other in the far tails at high Reynolds number. However the two PDFs are remarkably similar in form, being well described by stretched exponentials.

Using a dynamic surface force apparatus (SFA), we demonstrate that the notion of slip length used to describe the boundary flow of simple liquids is not appropriate for viscoelastic liquids. Rather, the appropriate description lies in the original Navier's partial slip boundary condition, formulated in terms of an interfacial friction coefficient. We establish an exact analytical expression to extract the interfacial friction coefficient from oscillatory drainage forces between a sphere and a plane, suitable for dynamic SFA or atomic force microscopy noncontact measurements. We use this model to investigate the boundary friction of viscoelastic polymer solutions over 5 decades of film thicknesses and 1 decade in frequency. The proper use of the original Navier's condition describes accurately the complex hydrodynamic force up to scales of tens of micrometers, with a simple Newtonian-like friction coefficient that is not frequency dependent and does reflect closely the dynamics of an interfacial depletion layer at the solution-solid interface.

The settling of colloidal particles with short-ranged attractions is investigated via highly resolved immersed boundary simulations. At modest volume fractions, we show that intercolloid attractions lead to clustering that reduces the hinderance to settling imposed by fluid back flow. For sufficient attraction strength, increasing the particle concentration grows the particle clusters, which further increases the mean settling rate in a physical mode termed promoted settling. The immersed boundary simulations are compared to recent experimental measurements of the settling rate in nanoparticle dispersions for which particles are driven to aggregate by short-ranged depletion attractions. The simulations are able to quantitatively reproduce the experimental results. We show that a simple, empirical model for the settling rate of adhesive hard-sphere dispersions can be derived from a combination of the experimental and computational data as well as analytical results valid in certain asymptotic limits of the concentration and attraction strength. This model naturally extends the Richardson-Zaki formalism used to describe hindered settling of hard, repulsive spheres. Experimental measurements of the collective diffusion coefficient in concentrated solutions of globular proteins are used to illustrate inference of effective interaction parameters for sticky, globular macromolecules using this empirical model. Finally, application of the simulation methods and empirical model to other colloidal systems are discussed.

Nectar-drinking bats and honeybees have tongues covered with hairlike structures, enhancing their ability to take up viscous nectar by dipping. Using a combination of model experiments and theory, we explore the physical mechanisms that govern viscous entrainment in a hairy texture. Hairy surfaces are fabricated using laser cut molds and casting samples with polydimethylsiloxane (PDMS) elastomer. We model the liquid trapped within the texture using a Darcy-Brinkmann-like approach and derive the drainage flow solution. The amount of fluid that is entrained is dependent on the viscosity of the fluid, the density of the hairs, and the withdrawal speed. Both experiments and theory reveal an optimal hair density to maximize fluid uptake.

We present numerical simulations of steady, laminar, axisymmetric convection of a Boussinesq fluid in a shallow, rotating, cylindrical domain. The flow is driven by an imposed vertical heat flux and shaped by the background rotation of the domain. The geometry is inspired by that of tropical cyclones and the global flow pattern consists of a shallow swirling vortex combined with a poloidal flow in the $r\text{−}z$ plane which is predominantly inward near the bottom boundary and outward along the upper surface. Our numerical experiments confirm that, as suggested in our recent work [L. Oruba et al., J. Fluid Mech. 812, 890 (2017)], an eye forms at the center of the vortex which is reminiscent of that seen in a tropical cyclone and is characterized by a local reversal in the direction of the poloidal flow. We establish scaling laws for the flow and map out the conditions under which an eye will, or will not, form. We show that, to leading order, the velocity scales with $V={\left(\alpha g\beta \right)}^{1/2}H$, where $g$ is gravity, $\alpha$ is the expansion coefficient, $\beta$ is the background temperature gradient, and $H$ is the depth of the domain. We also show that the two most important parameters controlling the flow are $\mathrm{Re}=VH/\nu$ and $\mathrm{Ro}=V/\left(\mathrm{\Omega }H\right)$, where $\mathrm{\Omega }$ is the background rotation rate and $\nu$ the viscosity. The Prandtl number and aspect ratio also play an important, if secondary, role. Finally, and most importantly, we establish the criteria required for eye formation. These consist of a lower bound on $\mathrm{Re}$, upper and lower bounds on $\mathrm{Ro}$, and an upper bound on the Ekman number.

This paper is associated with a video winner of a 2016 APS/DFD Milton van Dyke Award for work presented at the DFD Gallery of Fluid Motion. The original video is available from the Gallery of Fluid Motion, https://doi.org/10.1103/APS.DFD.2016.GFM.V0055

This paper is associated with a video winner of a 2016 APS/DFD Milton van Dyke Award for work presented at the DFD Gallery of Fluid Motion. The original video is available from the Gallery of Fluid Motion, https://doi.org/10.1103/APS.DFD.2016.GFM.V0095

This paper is associated with a video winner of a 2016 APS/DFD Milton van Dyke Award for work presented at the DFD Gallery of Fluid Motion. The original video is available from the Gallery of Fluid Motion, https://doi.org/10.1103/APS.DFD.2016.GFM.V0076

We present unifying rules governing the efficient locomotion of swimming fish and marine mammals. Using scaling and dimensional analysis, supported by new experimental data, we show that efficient locomotion occurs when the values of the Strouhal (St) number $\mathrm{St}(=fA/U)$ and $A^{*}(=A/L)$, two nondimensional numbers that relate forward speed $U$, tail-beat amplitude $A$, tail-beat frequency $f$, and the length of the swimmer $L$ are bound to the tight ranges of 0.2–0.4 and 0.1–0.3, respectively. The tight range of 0.2–0.4 for the St number has previously been associated with optimal thrust generation. We show that the St number alone is insufficient to achieve optimal aquatic locomotion, and an additional condition on $A^{*}$ is needed. More importantly, we show that when swimming at minimal power consumption, the Strouhal number of a cruising swimmer is predetermined solely by the shape and drag characteristics of the swimmer. We show that diverse species of fish and cetaceans cruise indeed with the St number and $A^{*}$ predicted by our theory. Our findings provide a physical explanation as to why fast aquatic swimmers cruise with a relatively constant tail-beat amplitude of approximately 20% of the body length, and their swimming speed is nearly proportional to their tail-beat frequency.

A theoretical model for predicting and interpreting blood-spatter patterns resulting from a gunshot wound is proposed. The physical process generating a backward spatter of blood is linked to the Rayleigh-Taylor instability of blood accelerated toward the surrounding air, allowing the determination of the initial distribution of drop sizes and velocities. Then the motion of many drops in air is considered with governing equations accounting for gravity and air drag. Based on these equations, a numerical solution is obtained. It predicts the atomization process, the trajectories of the back-spatter drops of blood from the wound to the ground, the impact angle, and the impact Weber number on the ground, as well as the distribution and location of bloodstains and their shape and sizes. A parametric study is undertaken to predict patterns of backward blood spatter under realistic conditions corresponding to the experiments conducted in the present work. The results of the model are compared to the experimental data on back spatter generated by a gunshot impacting a blood-impregnated sponge.

Miscible liquids commonly contact one another in natural and technological situations, often in the proximity of a solid substrate. In the scenario where a drop of one liquid finds itself on a solid surface and immersed within a second, miscible liquid, it will spread spontaneously across the surface. We show experimental findings of the spreading of sessile drops in miscible environments that have distinctly different shape evolution and power-law dynamics from sessile drops that spread in immiscible environments, which have been reported previously. We develop a characteristic time to scale radial data of the spreading sessile drops based on a drainage flow due to gravity. This time scale is effective for a homologous subset of the liquids studied. However, it has limitations when applied to significantly chemically different, yet miscible, liquid pairings; we postulate that the surface energies between each liquid and the solid surface becomes important for this other subset of the liquids studied. Initial experiments performed with pendant drops in miscible environments support the drainage flow observed in the sessile drop systems.

With the aim to predict energy transport by internal waves incident upon observed density staircases in the ocean, the theory for internal wave tunneling through a single finite-size mixed region is extended to predict the transmission of internal waves through a piecewise-constant density profile with an arbitrary number of steps. An analytic solution is found if all steps have equal vertical extent. From this solution, bounds are established for the relative incident horizontal wave number and frequency of waves that have negligible transmission, a result that is independent of the number of steps. If the step sizes vary randomly about a mean vertical extent, the transmission can be computed numerically for several random realizations. Provided the horizontal wavelength is much longer than the total vertical extent of the staircase, the standard deviation of the computed transmission coefficients is small and the mean agrees well with the analytic prediction for equally spaced steps. The results are applied to thermohaline staircases observed in the ocean in order to put lower bounds on the wavelengths of long inertia gravity waves that significantly transmit to great depth.