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.

We discuss how searching for finite-amplitude disturbances of a given energy that maximize their subsequent energy growth after a certain later time $T$ can be used to probe the phase space around a reference state and ultimately to find other nearby solutions. The procedure relies on the fact that of all the initial disturbances on a constant-energy hypersphere, the optimization procedure will naturally select the one that lies closest to the stable manifold of a nearby solution in phase space if $T$ is large enough. Then, when in its subsequent evolution the optimal disturbance transiently approaches the new solution, a flow state at this point can be used as an initial guess to converge the solution to machine precision. We illustrate this approach in plane Couette flow by rediscovering the spanwise-localized “snake” solutions of Schneider et al. [Phys. Rev. Lett. 104, 104501 (2010)], probing phase space at very low Reynolds numbers (less than $127.7$) where the constant-shear solution is believed to be the global attractor and examining how the edge between laminar and turbulent flow evolves when stable stratification eliminates the turbulence. We also show that the steady snake solution smoothly delocalizes as unstable stratification is gradually turned on until it connects (via an intermediary global three-dimensional solution) to two-dimensional Rayleigh-Bénard roll solutions.

The time evolution of a viscous helical vortex is investigated by direct numerical simulations of the Navier-Stokes equations where helical symmetry is enforced. Using conservation laws in the framework of helical symmetry, we elaborate an initial condition consisting in a finite core vortex, the time evolution of which leads to a generic quasiequilibrium state independent of the initial core size. Numerical results at different helical pitch values provide an accurate characterization in time for such helical states, for which specific techniques have been introduced: helix radius, angular velocity, stream function–velocity–vorticity relationships, and core properties (size, self-similarity, and ellipticity). Viscosity is shown to be at the origin of a small helical velocity component, which we relate to the helical vorticity component. Finally, changes in time of the flow topology are studied using the helical stream function and three-dimensional Lagrangian orbits.

We study experimentally the flow stability of entangled polymer solutions extruded through glass capillaries. We show that the pipe flow becomes linearly unstable beyond a critical value $\left({\mathrm{Wi}}_c\simeq 5\right)$ of the Weissenberg number, via a supercritical bifurcation which results in a helical distortion of the extrudate. We find that the amplitude of the undulation vanishes as the aspect ratio $L/R$ of the capillary tends to zero, and saturates for large $L/R$, indicating that the instability affects the whole pipe flow, rather than the contraction or exit regions. These results, when compared to previous theoretical and experimental works, lead us to argue that the nature of the instability is controlled by the level of shear thinning of the fluids. In addition, we provide strong hints that the nonlinear development of the instabiilty is mitigated, in our system, by the gradual emergence of gross wall slip.

It is now established that subcritical mechanisms play a crucial role in the transition to turbulence of nonrotating plane shear flows. The role of these mechanisms in rotating channel flow is examined here in the linear and nonlinear stages. Distinct patterns of behavior are found: the transient growth leading to nonlinearity at low rotation rates $Ro$, a highly chaotic intermediate $Ro$ regime, a localized weak chaos at higher $Ro$, and complete stabilization of transient disturbances at very high $Ro$. At very low $Ro$, the transient growth amplitudes are close to those for nonrotating flow, but Coriolis forces assert themselves by producing distinct asymmetry about the channel centreline. Nonlinear processes are then triggered, in a streak-breakdown mode of transition. The high $Ro$ regimes do not show these signatures; here the leading eigenmode emerges as dominant in the early stages. Elongated structures plastered close to one wall are seen at higher rotation rates. Rotation is shown to reduce nonnormality in the linear operator, in an indirect manifestation of Taylor-Proudman effects. Although the critical Reynolds for exponential growth of instabilities is known to vary a lot with rotation rate, we show that the energy critical Reynolds number is insensitive to rotation rate. It is hoped that these findings will motivate experimental verification and examination of other rotating flows in this light.

The unsteady flow in a patient-specific geometry of the airways is studied. The geometry comprises the oral cavity, orophrarynx, larynx, trachea, and the bronchial tree extending to generations 5–8. Simulations are carried out for a constant inspiratory flow rate of 60 liters/min, corresponding to a Reynolds number of 4213 for a nominal tracheal diameter of 2 cm. The computed mean flow field is compared extensively with magnetic resonance velocimetry measurements by Banko et al. [Exp. Fluids 56, 117 (2015)] carried out in the same computed-tomography–based geometry, showing good agreement. In particular, we focus on the dynamics of the flow in the bronchial tree. After becoming unsteady at a constriction in the oropharynx, the flow is found to be chaotic, exhibiting fluctuations with broad-band spectra even at the most distal airways in which the Reynolds numbers are as low as 300. An inertial range signature is present in the trachea but not in the bronchial tree where a narrower range of scales is observed. The unsteadiness is attributed to the convection of turbulent structures produced at the larynx as well as to local kinetic energy production throughout the bronchial tree. Production occurs predominantly at shear layers bounding geometry-induced separation regions.

The coalescence of bubbles and drops plays a central role in nature and industry. During coalescence, two bubbles or drops touch and merge into one as the neck connecting them grows from microscopic to macroscopic scales. The hydrodynamic singularity that arises when two bubbles or drops have just touched and the flows that ensue have been studied thoroughly when two drops coalesce in a dynamically passive outer fluid. In this paper, the coalescence of two identical and initially spherical bubbles, which are idealized as voids that are surrounded by an incompressible Newtonian liquid, is analyzed by numerical simulation. This problem has recently been studied (a) experimentally using high-speed imaging and (b) by asymptotic analysis in which the dynamics is analyzed by determining the growth of a hole in the thin liquid sheet separating the two bubbles. In the latter, advantage is taken of the fact that the flow in the thin sheet of nonconstant thickness is governed by a set of one-dimensional, radial extensional flow equations. While these studies agree on the power law scaling of the variation of the minimum neck radius with time, they disagree with respect to the numerical value of the prefactors in the scaling laws. In order to reconcile these differences and also provide insights into the dynamics that are difficult to probe by either of the aforementioned approaches, simulations are used to access both earlier times than has been possible in the experiments and also later times when asymptotic analysis is no longer applicable. Early times and extremely small length scales are attained in the new simulations through the use of a truncated domain approach. Furthermore, it is shown by direct numerical simulations in which the flow within the bubbles is also determined along with the flow exterior to them that idealizing the bubbles as passive voids has virtually no effect on the scaling laws relating minimum neck radius and time.

Particle-resolved numerical simulations based on the Force Coupling Method are carried out to study the effect of finite-size particles on turbulent plane Couette flow. The Reynolds number is close to the laminar-turbulent transition, such that large-scale rotational structures are well developed and self-sustained. The study particularly considers the effect of concentration, particle size, and particle-to-fluid density ratio on the mixture flow features. Time-averaged profiles, in the wall-normal direction, of the mean flow and Reynolds stress components reveal that there is no significant difference between single-phase and two-phase flows at equivalent effective Reynolds number, except that the wall shear stress is higher for the two-phase flow. However, temporal and modal analysis of flow fluctuations suggest that besides injecting small-scale perturbation due to their rigidity, particles have an effect on the regeneration cycle of turbulence. Indeed, the shape of the streaks and the intermittent character of the flow (amplitude and period of oscillation of the modal fluctuation energy) are all altered by the particle presence, and especially by the inertial ones.

Taylor-Couette flow between concentric rotating cylinders provides a prime example of a pitchfork bifurcation in hydrodynamic stability. However, more than 40 years ago, Benjamin pointed out difficulties in establishing quantitative links between models where periodicity is assumed along the length of the cylinders and calculations using the Navier-Stokes equations. A new model was recently advanced where it is claimed that the bifurcation which is relevant to the onset of cellular motion is the one between neighboring states. Here, we provide experimental data in support of this idea which explain why the onset of cellular motion is observed to be sharp whereas the periodic model bifurcation is highly imperfect.

We experimentally investigate the flow of a viscoelastic fluid in a parallel shear geometry at low Reynolds number. As the flow becomes unstable via a nonlinear subcritical instability, velocimetry measurements show nonperiodic fluctuations over a broad range of frequencies and wavelengths, consistent with the main features of elastic turbulence. Using the same experimental setup, we compare these features to those in the flow around cylinders, which is upstream of the parallel shear region; we find significant differences in power spectrum scaling, intermittency statistics, and flow structures. We propose a simple mechanism to explain the growth of velocity fluctuations in parallel shear flows based on polymer stretching due to fluctuations in streamwise velocity gradients.

Recent simulation results of a moderately entangled linear polyethylene ${\mathrm{C}}_{700}{\mathrm{H}}_{1402}$ liquid have confirmed prior simulation and experimental evidence that individual polymer molecules experience periodic rotation and retraction cycles under steady shear flow at high Weissenberg number. With this insight, theoreticians have begun to grapple with this additional complicating physical phenomenon that needs to be incorporated into rheological models to help explain the data under conditions of high shear. In this paper we examine these recent efforts by using nonequilibrium molecular dynamics simulations to provide insight into the requisite theoretical variables and their assigned evolution equations to evaluate the capability of these tube-based models to predict accurately the simulated data sets. This analysis reveals that the primary variables used in tube models to impart a conceptual basis to the theory, namely, the tube orientation tensor and the tube stretch, remain fundamental system properties even far away from equilibrium; however, the theory describing their evolution under flow is not well suited to quantitative prediction. Furthermore, it is demonstrated that key system properties, such as the entanglement number and disengagement time, should play a more significant role in model development since these quantities can change dramatically under flow, particularly at high Weissenberg number where the chain rotation and retraction cycles dominate the system physics.

We study the hydrodynamic equilibrium properties of the stably stratified atmospheric boundary layer from measurements obtained in the Snow-Horizontal Array Turbulence Study campaign at the Plaine Morte Glacier in the Swiss Alps. Our approach is based on a combination of dynamical systems techniques and statistical analysis. The main idea is to measure the deviations from the behavior expected by a turbulent observable when it is close to a transition between different metastable states. We first assess the performance of our method on the Lorenz attractor, then on a turbulent flow. The results show that the method recognizes subtle differences among different stable boundary layer turbulence regimes and may be used to help characterize their transitions.

A cylindrical container partially filled with a liquid in an orbital shaking motion, i.e., in circular translation with fixed orientation with respect to an inertial frame of reference, generates, along with a rotating sloshing wave, a mean flow rotating in the same direction as the wave. Here we investigate experimentally the structure and the scaling of the wave flow and the Lagrangian mean flow in the weakly nonlinear regime, for small forcing amplitude and for forcing frequency far from the resonance, using conventional and stroboscopic particle image velocimetry. The Lagrangian mean flow is composed of a strong global rotation near the center and a nontrivial pattern of poloidal recirculation vortices of weaker amplitude, mostly active near the contact line. The global rotation near the center is robust with respect to changes in viscosity and forcing frequency, and its amplitude compares well with the predicted Stokes drift for an inviscid rotating sloshing wave. On the other hand, the spatial structure of the poloidal vortices shows strong variation with viscosity and forcing frequency, suggesting that it results from nonlinear streaming driven by the oscillatory boundary layers near the contact line.

We study the inertial migration of finite-size neutrally buoyant spherical particles in dilute and semidilute suspensions in laminar square duct flow. We perform several direct numerical simulations using an immersed boundary method to investigate the effects of the bulk Reynolds number ${\text{Re}}_b$, particle Reynolds number ${\text{Re}}_p$, and duct to particle size ratio $h/a$ at different solid volume fractions $\varphi$, from very dilute conditions to $20\%$. We show that the bulk Reynolds number ${\text{Re}}_b$ is the key parameter in inertial migration of particles in dilute suspensions. At low solid volume fraction ($\varphi =0.4\%$), low bulk Reynolds number (${\text{Re}}_b=144$), and $h/a=9$ particles accumulate at the center of the duct walls. As ${\text{Re}}_b$ is increased, the focusing position moves progressively toward the corners of the duct. At higher volume fractions, $\varphi =5\%$, $10\%$, and $20\%$, and in wider ducts ($h/a=18$) with ${\text{Re}}_b=550$, particles are found to migrate away from the duct core toward the walls. In particular, for $\varphi =5\%$ and $10\%$, particles accumulate preferentially at the corners. At the highest volume fraction considered, $\varphi =20\%$, particles sample all the volume of the duct, with a lower concentration at the duct core. For all cases, we find that particles reside longer times at the corners than at the wall centers. In a duct with lower duct to particle size ratio $h/a=9$ (i.e., with larger particles), $\varphi =5\%$, and high bulk Reynolds number ${\text{Re}}_b=550$, we find a particle concentration pattern similar to that in the ducts with $h/a=9$ regardless of the solid volume fraction $\varphi$. Instead, for lower Bulk Reynolds number ${\text{Re}}_b=144$, $h/a=9$, and $\varphi =5\%$, a different particle distribution is observed in comparison to a dilute suspension $\varphi =0.4\%$. Hence, the volume fraction plays a key role in defining the final distribution of particles in semidilute suspensions at low bulk Reynolds number. The presence of particles induces secondary cross-stream motions in the duct cross section, for all $\varphi$. The intensity of these secondary flows depends strongly on particle rotation rate, on the maximum concentration of particles in focusing positions, and on the solid volume fraction. We find that the secondary flow intensity increases with the volume fraction up to $\varphi =5\%$. However, beyond $\varphi =5\%$ excluded-volume effects lead to a strong reduction of cross-stream velocities for ${\text{Re}}_b=550$ and $h/a=18$. Inhibiting particles from rotatingalso results in a substantial reduction of the secondary flow intensity and in variations of the exact location of the focusing positions.

We propose the rarefied-gas-cushion model (r-GCM), as an extended version of the gas-cushion model (GCM), to estimate the apparent slip of water flowing over a gas layer trapped at a solid surface. Nanobubbles or gas nanofilms may manifest rarefied-gas effects and the r-GCM incorporates kinetic boundary conditions for the gas component in the slip Knudsen regime. These enable an apparent hydrodynamic slip length to be calculated given the gas thickness, the Knudsen number, and the bulk fluid viscosities. We assess the r-GCM through nonequilibrium molecular dynamics (NEMD) simulations of shear-driven liquid flow over an infinite gas nanofilm covering a solid surface, from the gas slip regime to the early transition regime, beyond which NEMD is computationally impractical. We find that, over the flow regimes examined, the r-GCM provides better predictions of the apparent liquid slip and retrieves both the GCM and the free-molecular behavior in the appropriate limits.

A slow-growth formulation for DNS of wall-bounded turbulent flow is developed and demonstrated to enable extension of slow-growth modeling concepts to wall-bounded flows with complex physics. As in previous slow-growth approaches, the formulation assumes scale separation between the fast scales of turbulence and the slow evolution of statistics such as the mean flow. This separation enables the development of approaches where the fast scales of turbulence are directly simulated while the forcing provided by the slow evolution is modeled. The resulting model admits periodic boundary conditions in the streamwise direction, which avoids the need for extremely long domains and complex inflow conditions that typically accompany spatially developing simulations. Further, it enables the use of efficient Fourier numerics. Unlike previous approaches [Guarini, Moser, Shariff, and Wray, J. Fluid Mech. 414, 1 (2000); Maeder, Adams, and Kleiser, J. Fluid Mech. 429, 187 (2001); Spalart, J. Fluid Mech. 187, 61 (1988)], the present approach is based on a temporally evolving boundary layer and is specifically tailored to give results for calibration and validation of Reynolds-averaged Navier–Stokes (RANS) turbulence models. The use of a temporal homogenization simplifies the modeling, enabling straightforward extension to flows with complicating features, including cold and blowing walls. To generate data useful for calibration and validation of RANS models, special care is taken to ensure that the mean slow-growth forcing is closed in terms of the mean and other quantities that appear in standard RANS models, ensuring that there is no confounding between typical RANS closures and additional closures required for the slow-growth problem. The performance of the method is demonstrated on two problems: an essentially incompressible, zero-pressure-gradient boundary layer and a transonic boundary layer over a cooled, transpiring wall. The results show that the approach produces flows that are qualitatively similar to other slow-growth methods as well as spatially developing simulations and that the method can be a useful tool in investigating wall-bounded flows with complex physics.

In this article, we investigate the production and dissipation of turbulence in a region where the mean flow topology presents a stagnation point. Our goal is to understand the generation of anisotropic fluctuations and their influence on production, dissipation, and transport of turbulent kinetic energy. In order to investigate the local turbulent kinetic energy budget, we use a shadow particle tracking velocimetry technique (S-PTV) to track Lagrangian tracers in a large portion of a turbulent von Kármán flow produced by counter-rotating disks. We observe that the flow produced in a square tank is bistable, with each of the two states resembling impinging jets. This stagnation-point topology is responsible for the strong anisotropy of velocity fluctuations observed in these type of flows. The production of turbulence locally exceeds the dissipation rate. As a consequence, the flow is to be considered as strongly inhomogeneous as the fluxes of turbulent kinetic energy are non-negligible when compared to the production and dissipation terms.

Chemical reactions in thin liquid films are found in many industrial applications, e.g., in combustion chambers of internal combustion engines where a fuel film can develop on pistons or cylinder walls. The reactions within the film and the turbulent outer gas flow influence film stability and lead to film breakup, which in turn can lead to deposit formation. In this work we examine the evolution and stability of a thin liquid film in the presence of a first-order chemical reaction and under the influence of a turbulent gas flow. Long-wave theory with a double perturbation analysis is used to reduce the complexity of the problem and obtain an evolution equation for the film thickness. The chemical reaction is assumed to be slow compared to film evolution and the amount of reactant in the film is limited, which means that the reaction rate decreases with time as the reactant is consumed. A linear stability analysis is performed to identify the influence of reaction parameters, material properties, and environmental conditions on the film stability limits. Results indicate that exothermic reactions have a stabilizing effect whereas endothermic reactions destabilize the film and can lead to rupture. It is shown that an initially unstable film can become stable with time as the reaction rate decreases. The shearing of the film by the external gas flow leads to the appearance of traveling waves. The shear stress magnitude has a nonmonotonic influence on film stability.

The low-Reynolds-number hydrodynamics of slender ribbons is accurately captured by slender-ribbon theory, an asymptotic solution to the Stokes equation which assumes that the three length scales characterizing the ribbons are well separated. We show in this paper that the force distribution across the width of an isolated ribbon located in a infinite fluid can be determined analytically, irrespective of the ribbon's shape. This, in turn, reduces the surface integrals in the slender-ribbon theory equations to a line integral analogous to the one arising in slender-body theory to determine the dynamics of filaments. This result is then used to derive analytical solutions to the motion of a rigid plate ellipsoid and a ribbon torus and to propose a ribbon resistive-force theory, thereby extending the resistive-force theory for slender filaments.

A systematic analysis is presented of the undulations appearing on the surface of long bubbles in confined gas-liquid flows. CFD simulations of the flow are performed with a self-improved version of the open-source solver ESI OpenFOAM (release 2.3.1), for $\mathrm{Ca}=0.002-0.1$ and $\mathrm{Re}=0.1-1000$, where $\mathrm{Ca}=\mu U/\sigma$ and $\mathrm{Re}=2\rho UR/\mu$, with $\mu$ and $\rho$ being, respectively, the viscosity and density of the liquid, $\sigma$ the surface tension, $U$ the bubble velocity, and $R$ the tube radius. A model, based on an extension of the classical axisymmetric Bretherton theory, accounting for inertia and for the curvature of the tube's wall, is adopted to better understand the CFD results. The thickness of the liquid film, and the wavelength and decay rate of the undulations extracted from the CFD simulations, agree well with those obtained with the theoretical model. Inertial effects appear when the Weber number of the flow $\mathrm{We}=\mathrm{Ca}\mathrm{Re}=\mathcal{O}\left({10}^{-1}\right)$ and are manifest by a larger number of undulation crests that become evident on the surface of the rear meniscus of the bubble. This study demonstrates that the necessary bubble length for a flat liquid film region to exist between the rear and front menisci rapidly increases above $10R$ when $\mathrm{Ca}>0.01$ and the value of the Reynolds number approaches 1000.