In statistically homogeneous and isotropic turbulence, the average value of the velocity increment ${\delta }_{r}u=u(x+r)-u\left(x\right)$, where $x$ and $x+r$ are two positions in the flow and $u$ is the velocity in the direction of the separation distance $r$, is identically zero, and so to characterize the dynamics one often uses the Reynolds number based on $\sqrt{\langle {\left({\delta }_{r}u\right)}^2\rangle }$, which acts as the coupling constant for scale-to-scale interactions. This description can be generalized by introducing structure functions of order $n$, $S_n=\langle {\left({\delta }_{r}u\right)}^n\rangle$, which allow one to probe velocity increments including rare and extreme events, by considering ${\delta }_{r}u\left(n\right)=O\left(S_n^{1/n}\right)$ for large and small $n$. If $S_n\propto r^{{\zeta }_n}$, the theory for the anomalous exponents ${\zeta }_n$ in the entire allowable interval $-1<n<\infty$ is one of the long-standing challenges in turbulence (one takes absolute values of ${\delta }_{r}u$ for negative $n$), usually attacked by various qualitative cascade models. We accomplish two major tasks here. First, we show that the turbulent microscale Reynolds number $R_{\lambda }^T$ (based on a suitably defined turbulent viscosity) is 8.8 in the inertial range with anomalous scaling, when the standard microscale Reynolds number $R_{\lambda }$ (defined using normal viscosity) exceeds that same number (which in practice could be by a large factor). When the normal and turbulent microscale Reynolds numbers become equal to or fall below 8.8, the anomaly disappears in favor of Gaussian statistics. Conversely, if one starts with a Gaussian state and increases $R_{\lambda }$ beyond 8.8, one ushers in the anomalous scaling; the inference is that $R_{\lambda }^T\approx 8.8$ and remains constant at that value with further increase in $R_{\lambda }$. Second, we derive expressions for the anomalous scaling exponents of structure functions and moments of spatial derivatives, by analyzing the Navier-Stokes equations in the form developed by Hopf. We present a novel procedure to close the Hopf equation, resulting in expressions for ${\zeta }_n$ in the entire range of allowable moment order $n$, and demonstrate that accounting for the temporal dynamics changes the scaling from normal to anomalous. For large $n$, the theory predicts the saturation of ${\zeta }_n$ with $n$, leading to several inferences, two among which are (a) the smallest length scale ${\eta }_n=L{\text{Re}}^{-1}\ll L{\text{Re}}^{-3/4}$, where Re is the large-scale Reynolds number, and (b) that velocity excursions across even the smallest length scales can sometimes be as large as the large-scale velocity itself. Theoretical predictions for each of these aspects are shown to be in excellent quantitative agreement with available experimental and numerical data.

We impose a linearized Eulerian perturbation on a steady, shallow, radial outflow of a liquid (water), whose local pressure function includes both the hydrostatic and the Laplace pressure terms. The resulting wave equation bears the form of a hydrodynamic metric. A dispersion relation, extracted from the wave equation, gives an instability due to surface tension and the cylindrical flow symmetry. Using the dispersion relation, we also derive three known relations that scale the radius of the circular hydraulic jump in the outflow. The first two relations are scaled by viscosity and gravity, with a capillarity-dependent crossover to the third relation, which is scaled by viscosity and surface tension. The perturbation as a high-frequency traveling wave, propagating radially inward against the bulk outflow, is blocked just outside the circular hydraulic jump. The amplitude of the wave also diverges here because of a singularity. The blocking is associated with surface tension, which renders the circular hydraulic jump a hydrodynamic white hole.

We study the primary bifurcations of a two-dimensional Kolmogorov flow in a channel subject to boundary conditions chosen to mimic a parallel flow, i.e., periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. The control parameter is the Reynolds number based on the friction coefficient, denoted as $\text{Rh}$. We find that as we increase $\text{Rh}$ the laminar steady flow can display different types of bifurcation depending on the forcing wave number of the base flow. This is in contrast with the case of doubly periodic boundary conditions for which the primary bifurcation is stationary. In the present case, both stationary and Hopf bifurcations are observed. In addition, we discover a type of bifurcation with both the oscillation frequency and the amplitude of the growing mode being zero at the threshold, that we call a stationary drift bifurcation. A reduced four-mode model captures the scalings that are obtained from the numerical simulations. As we increase $\text{Rh}$ further we observe a secondary instability which excites the largest mode in the domain. The saturated amplitude of the largest mode is found to scale as a $3/2$ power law of the distance to the threshold which is also explained using a low-dimensional model.

This study considered how the presence of current impacted wave dissipation within a meadow of flexible marsh plants. A wave-damping model was developed from a prediction of current- and wave-induced force on individual plants. The model was validated with laboratory experiments. Wave decay was measured over a meadow of flexible model plants geometrically similar to Spartina alterniflora with and without a following current. Consistent with previous observations, the wave energy dissipation depended on the ratio of current velocity ($U_c$) to wave velocity ($U_w$). Compared to the same pure wave condition, wave energy dissipation was enhanced by large $U_c/U_w$ but can be decreased for small $U_c/U_w$. Once validated, the wave-damping model was used to explore a wider range of wave, current, and meadow conditions in order to illustrate the influence of reconfiguration on wave forces; the impact of current on wave group velocity; and the modification of in-canopy time-mean and wave orbital velocity associated with canopy drag.

A two-layer falling film consisting of two immiscible viscous fluids with identical density but different viscosity lining the interior of a vertical tube is studied using a long-wave asymptotic model. In this setup there is both an interface between the two layers and a free surface where the inner layer meets the passive air core. The model consists of a set of coupled partial differential equations describing the evolution of both surfaces, and the impact of the viscosity ratio on both the linear and nonlinear dynamics of the model is studied. Linear stability analysis of the model shows an unstable “free-surface mode” consisting largely of perturbations to the free surface, and an “interfacial mode” that can either be stable for all wave numbers, unstable for a band of long-wave wave numbers, or unstable only to a band of wave numbers bounded away from zero. These instabilities grow outside the linear regime and either saturate as a series of waves or continue to grow so that the free surface at a wave crest tends to the center of the tube in finite time, indicating the formation of a plug. Families of traveling-wave solutions are found by continuation from Hopf bifurcations that arise due to the instability of one or both modes. The free-surface traveling waves have a turning point that indicates a critical thickness required for plug formation to occur; decreasing the viscosity of the outer layer decreases this critical thickness so that plugs form more readily. The impact of fixing the volume flux of each layer, rather than the layer thicknesses, is discussed. Lastly, the model is extended to account for pressure-driven airflow in the core region, and it is shown that the airflow destabilizes the free-surface mode but can have a stabilizing impact on the interfacial mode. The significance of this study for applications, including human airways, is briefly discussed.

Recent experiments report that slowly sheared noncolloidal particle suspensions unexpectedly exhibit rate$\left(\omega \right)$-dependent complex viscosities in oscillatory shear, despite a constant relative viscosity in steady shear. Using a minimal hydrodynamic model, we show that van der Waals attraction gives rise to this behavior. At volume fractions $\varphi =20\text{–}50$%, the complex viscosities in both experiments and simulations display power-law reductions in shear, with a $\varphi$-dependent exponent maximum at $\varphi =40$%, resulting from the interplay between hydrodynamic, collision, and adhesive interactions. Furthermore, this rate dependence is accompanied by diverging particle diffusivities and pronounced cluster formations after repeated oscillations. Previous studies established that suspensions transition from reversible absorbing states to irreversible diffusing states when the oscillation amplitude exceeds a $\varphi$-dependent critical value ${\gamma }_{0,\varphi }^c$. Here, we show that a second transition to irreversibility occurs below an $\omega$-dependent critical amplitude, ${\gamma }_{0,\omega }^c\le {\gamma }_{0,\varphi }^c$, in the presence of weak attractions.

The flexibility of the bacterial flagellar hook is believed to have substantial consequences for microorganism locomotion. Using a simplified model of a rigid flagellum and a flexible hook, we show that the paths of axisymmetric cell bodies driven by a single flagellum in Stokes flow are generically helical. Phase-averaged resistance and mobility tensors are produced to describe the flagellar hydrodynamics, and a helical rod model which retains a coupling between translation and rotation is identified as a distinguished asymptotic limit. A supercritical Hopf bifurcation in the flagellar orientation beyond a critical ratio of flagellar motor torque to hook bending stiffness, which is set by the spontaneous curvature of the flexible hook, the shape of the cell body, and the flagellum geometry, can have a dramatic effect on the cell's trajectory through the fluid. Although the equilibrium hook angle can result in a wide variance in the trajectory's helical pitch, we find a very consistent prediction for the trajectory's helical amplitude using parameters relevant to swimming P. aeruginosa cells.

We consider the influence of transverse confinement on the instability properties of velocity and density distributions reminiscent of those pertaining to exchange flows in stratified inclined ducts, such as the recent experiment of Lefauve et al. [J. Fluid Mech. 848, 508 (2018)]. Using a normal mode streamwise and temporal expansion for flows in ducts with various aspect ratios $B$ and nontrivial transverse velocity profiles, we calculate two-dimensional (2D) dispersion relations with associated eigenfunctions varying in the “crosswise” direction, in which the density varies, and the spanwise direction, both normal to the duct walls and to the flow direction. We also compare these 2D dispersion relations to the so-called one-dimensional (1D) dispersion relations obtained for spanwise invariant perturbations, for different aspect ratios $B$ and bulk Richardson numbers ${\text{Ri}}_b$. In this limited parameter space, the presence of lateral walls has a stabilizing effect, in that the 1D growth-rate predictions are almost systematically an upper bound to the 2D growth rates, which in turn decrease monotonically as lateral walls are brought together with increased spanwise confinement ($B\to 0$). Furthermore, accounting for spanwise-varying perturbations results in a plethora of unstable modes, the number of which increases as the aspect ratio is increased. These modes present an odd-even regularity in their spatial structures, which is rationalized by comparison to the so-called one-dimensional oblique dispersion relation obtained for oblique waves, characterized by a continuously varying spanwise wavenumber in addition to the streamwise wavenumber. Finally, we show that in most cases, the most unstable 2D mode is the one that oscillates the least in the spanwise direction, as a consequence of viscous damping. However, in a limited region of the parameter space and in the absence of stratification, we show that a secondary mode with a more complex “twisted” structure dominated by crosswise vorticity becomes more unstable than the least oscillating Kelvin-Helmholtz mode associated with spanwise vorticity.

Self-consistent closure theory for passive scalar turbulence has been developed on the basis of the Hessian of the scalar field. As a primitive indicator of spatial structure of the scalar, we employ the Hessian into the core of the theory to properly characterize the time scale intrinsic to the scalar field itself. The resultant closure model is now endowed with several realistic features, i.e., the scale locality of the interscale interaction, the detailed conservation, and the memory-fading effect. Applying the current theory to the inertial-convective range eventually leads to self-consistent derivation of the Obukhov-Corrsin spectrum with its universal constant consistent with numerical and experimental data.

In this paper we compute the linear stability of similarity solutions of the breakup of viscous liquid threads, in which the viscosity and inertia of the liquid are in balance with the surface tension. The stability of the similarity solution is determined using numerical continuation to find the dominant eigenvalues. The stability of the first two solutions (those with the largest minimum radius) is considered. We find that the first similarity solution, which is the one seen in experiments and simulations, is linearly stable with a complex nontrivial eigenvalue, which could explain the phenomenon of breakup producing sequences of small satellite droplets of decreasing radius near a main pinch-off point. The second solution is seen to be linearly unstable. These linear stability results compare favorably to numerical simulations for the stable similarity solution, while a profile starting near the unstable similarity solution is shown to very rapidly leave the linear regime.

Preferential concentration is thought to play a key role in promoting particle growth, which is crucial to processes such as warm rain formation in clouds, planet formation, and industrial sprays. In this paper, we investigate preferential concentration using three-dimensional direct numerical simulations adopting the Eulerian-Eulerian two-fluid approach, where the particles are treated as a continuum field with its own momentum and mass conservation laws. We consider particles with Stokes number $\text{St}\lesssim O\left(0.01\right)$ in moderately turbulent flows with fluid Reynolds number $\text{Re}\le 600$. In our previous paper [Phys. Rev. Fluids 5, 114308 (2020)], we established scaling laws to predict maximum and typical particle concentration enhancements in the context of the particle-driven convective instability. Here, we verify that the same results apply when turbulence is externally driven, extending the relevance of our model to a wider class of particle-laden flows. We find in particular that (i) the maximum particle concentration enhancement above the mean scales as $u_{\text{rms}}^2{\tau }_p/{\kappa }_p$, where $u_{\text{rms}}$ is the rms fluid velocity, ${\tau }_p$ is the particle stopping time, and ${\kappa }_p$ is the assumed particle diffusivity from the two-fluid equations; (ii) the typical particle concentration enhancement over the mean scales as ${(u_{\text{rms}}^2{\tau }_p/{\kappa }_p)}^{1/2}$; and (iii) the probability distribution function of the particle concentration enhancement over the mean has an exponential tail whose slope scales as ${(u_{\text{rms}}^2{\tau }_p/{\kappa }_p)}^{-1/2}$. We conclude by discussing the caveats of our model and its implications in a relevant cloud application.

In granular mechanics, the shape of grains plays a critical role in the overall dynamics and significantly affects the macroscopic properties of the system. Using a dam break setup, granular collapses of nonconvex (cross-shaped) plastic particles assumed quasirigid are conducted experimentally and simulated numerically for a wide range of aspect ratios $a=H_0/L_0$, with $H_0$ the initial height of the column and $L_0$ its initial length. We report avalanche dynamics such as the top-driven collapse and the buckling collapse, as well as an intermittent flow behavior where reproducibility is lost and where the stability of the column is determined by the random initial configuration of the assembly of entangled particles. While counterintuitive and despite fundamentally different dynamics, we find that the runout distance $L_{\infty }$ and the final height $H_{\infty }$ of our granular collapses of crosses agree with those of spherical particles both experimentally and numerically. Our discrete element method simulations are able to reproduce all flow behaviors observed experimentally and they show excellent quantitative agreement with the experimental data. In the simulations, extra care is given to adopting a tangential friction force model based on the cumulative tangential displacement at the contact point, critical to represent stable cases, and to determining the contact model parameters. The analysis of (i) the force network via the average probability density function of contact force magnitude and (ii) the fabric anisotropy suggests that the stability of the column is a complex problem determined by mesoscale properties that we could not reliably identify at that point.

We report existence of nonequilibrium turbulent dissipations in buoyant axisymmetric plumes at infinite Reynolds number limit. The plume statistical equations remain form invariant under one translation parameterized by $a_0$ and two unequal stretching transformations parameterized by $a_1$ and $a_2$ of flow variables, except the pressure-strain-rate equation. The dissipation coefficients, unlike Kolmogorov theory, in the scaling of dissipations ${\epsilon }_u$ of velocity fluctuation $u^{\prime }$ and ${\epsilon }_{\theta }$ of thermal fluctuation ${\theta }^{\prime }$ evolve linearly with spreading rate $d\delta /dz$. The spreading rate is expressed as inversely proportional to local Reynolds number ${\mathrm{Re}}_{\delta }$ raised to the exponent $m=3(a_1-a_2)/(a_1+a_2), a_1\ne -a_2$ and is varying with ${\mathrm{Re}}_{\delta }$ for $m\ne 0$. The components of Reynolds stress tensor have different streamwise (s) evolutions as ${\left(\overline{w^{\prime 2}}\right)}_s={\left(\overline{v^{\prime 2}}\right)}_s\propto {(d\delta /dz)}^2{\left(\overline{u^{\prime 2}}\right)}_s$ and ${\left(\overline{u^{\prime }{v}^{\prime }}\right)}_s\propto d\delta /dz{\left(\overline{u^{\prime 2}}\right)}_s$ unless $d\delta /dz$ is a constant, which holds only if $m=0$ giving $a_1=a_2$. This implies that Kolmogorov equilibrium theory ($m=0$) is intertwined inextricably with complete self-preservation. There is a direct universal relationship between turbulent dissipation and entrainment coefficient $\alpha$ as $d\delta /dz\propto \alpha$, the proportionality constant depends on integrals of mean axial velocity, temperature difference, and turbulent stresses. The power exponent in power-law streamwise evolutions of flow quantities are specified by the ratio $a_2/a_1$ and differs from the conventional results unless $m=0$. Both the local axial velocity and temperature difference widths are increasing with the increase of vertical distance from the source, as same power-law scaling exponents but differ due to different prefactors. However, the spreading rates and entrainment coefficient decrease in the non-Kolmogorov dissipation region, which occurs when $m\ne 0$ preferably lying in (0,1] ($a_2/a_1\in [1/2,1)$). Similar trends of spreading rate and entrainment were also measured experimentally in planar jet [Cafiero and Vassilicos, Proc. R. Soc. London A 475, 20190038 (2019)] and established theoretically in planar jet and plume [Layek and Sunita, Phys. Rev. Fluids 3, 124605 (2018); Layek and Sunita, Phys. Fluids 30, 115105 (2018)].

Motivated by small-scale natural and industrial processes involving flow over and/or through a layer of a porous medium, a mathematical model for the steady gravity-driven flow of a rivulet of fluid with finite width over and through a permeable membrane is formulated and analyzed. The three-dimensional shape of the free surface of a rivulet with either fixed semiwidth or fixed contact angle is determined, and it is shown how the length, base area, and volume of the rivulet on the permeable part of the membrane depend on the physical properties of the system. In particular, whereas there is a physically realizable pendant rivulet solution only if the semiwidth does not exceed a critical value, there are physically realizable sessile and vertical rivulet solutions for all values of the semiwidth; moreover, a sessile rivulet with fixed semiwidth has a finite maximum possible length which is attained in the limit of a wide rivulet.

We analyze the energy spectra, production spectra, and interscale energy transfer spectra for direct numerical simulation data of turbulent channel flows at ${\mathrm{Re}}_{\tau }=180$ and 550. The transfer spectra are characterized by a negative region and a positive region. We use the negative region as a spectral mask to separate the flow fields into large-scale and small-scale fields. The former are characterized by streaks and quasi-streamwise vortices, and the latter are characterized by hairpin-like vortical structures that are similar to the original flow fields. Proper orthogonal decomposition and linear stochastic estimation are utilized to further extract the flow structures of the decomposed fields. The outer large-scale low-momentum regions are flanked by large quasi-streamwise vortices that are attached to the wall. This coherent structure presents a high degree of similarity to the structure of the near-wall cycle. This result indicates that the outer large scales may present a self-sustaining mechanism similar to that of the near-wall region. We also extract the coherent structures associated with the real-space energy transfer. The formation of small-scale hairpin-like vortices is related to the large-scale shear layer. Within this process, the energy is transferred from large scales to small scales.

Numerical simulations of turbulence provide nonintrusive access to all the resolved scales and any quantity of interest, although they often invoke idealizations and assumptions that can compromise realism. In contrast, experimental measurements probe the true flow with lesser idealizations, but they continually contend with spatiotemporal sensor resolution. Assimilating observations directly in simulations can combine the benefits of both approaches and mitigate their respective deficiencies. The problem is expressed in variational form, where we seek the flow field that satisfies the Navier-Stokes equations and minimizes a cost function defined in terms of the deviation of the computational predictions and available observations. In this framework, measurements are no longer a mere record of the instantaneous, local quantity, but rather an encoding of the antecedent flow events that we aim to decode using the governing equations. Chaos plays a central role in obfuscating the interpretation of the data: observations that are infinitesimally close may be due to entirely different earlier conditions. We examine a number of state estimation problems: In circular Couette flow, starting from observations of the wall stress, we accurately reconstruct the wavy Taylor vortices that interact nonlinear to maintain a saturated state. Through a discussion of transition to chaos in a Lorenz system, we highlight the challenge of navigating the landscape of the cost function, and how the landscape can favorably be modified by sensor weighting and placement. In turbulent channel flow, the Taylor microscale and Lyapunov timescale place restrictions on the resolution of observations for which we can accurately reconstruct all the missing scales. The notion of domain of dependence of an observation is introduced and related to the Hessian of the cost function. For measurements of wall shear stress, the eigenspectrum of the Hessian demonstrates the sensitivity of short-time observations to the fine near-wall turbulent scales and to the large scales only in the outer flow. At long times, backward chaos obfuscates the interpretation of the data: observations that are infinitesimally close may be traced back to entirely different earlier flow states—a dual to the more common butterfly effect for forward trajectories.

We investigate the fall in a Hele-Shaw cell of viscous miscible drops in a more viscous ambient fluid with negligible diffusivity compared to the kinematic viscosity of the fluids. We measure the drop shape, thickness, and velocity for various viscosity and density ratios. The elongation of the drops is found to increase quadratically with the ratio of the drop and ambient fluid viscosity. The thickness of the drop is shown to be approximately half the thickness of the gap of the Hele-Shaw cell. We develop a theoretical expression of the drop falling velocity, taking into account the elongation and thickness of the drop. This agrees well with observations after multiplying by an empirically determined constant. In some circumstances the drops break up, either because of defects advected downwards within the drop or by instability of the shape of the drop.

The prediction of the lifetime of surface bubbles necessitates a better understanding of the thinning dynamics of the bubble cap. In 1959, Mysels et al. [Soap Films: Studies of Their Thinning and a Bibliography (Pergamon, New York, 1959)], proposed that marginal regeneration, i.e., the rise of patches thinner than the film should be taken into account to describe the film drainage. Nevertheless, an accurate description of these buoyant patches and of their dynamics as well as a quantification of their contribution to the thinning dynamics is still lacking. In this paper, we visualize the patches, and show that their rising velocities and sizes are in good agreement with models respectively based on the balance of gravitational and surface viscous forces and on a Rayleigh-Taylor-like instability. Our results suggest that, in an environment saturated in humidity, the drainage induced by their dynamics correctly describes the film drainage at the apex of the bubble within the experimental error bars. We conclude that the film thinning of soap bubbles is indeed controlled, to a large extent, by marginal regeneration in the absence of evaporation.

Peristaltic flows occur when fluid in a channel is driven by periodic, traveling wall deformations, as in industrial peristaltic pumps, urethras, stomachs, and cochleae. Peristaltic flows often vary periodically at every point in space but nonetheless cause net transport and mixing of solutes because of Lagrangian (Stokes) drift. Direct numerical simulation can predict peristaltic flows but is computationally expensive, particularly for determining functional relationships between drive parameters and transport or mixing. We present a simple analytic model of peristaltic flow that expresses flow velocity and drift velocity in terms of deformation speed and amplitude. The model extends beyond prior studies by allowing arbitrary wave forms via Fourier series. To validate our analytic model, we present experiments and simulations; both closely match the analytic model over a range of deformation speeds and amplitudes. We demonstrate the applicability of the model by quantifying variations in the thickness of the reflux region (where fluid drifts opposite the direction of travel of deformations) and by modeling mixing in the cochlea, which is promoted by peristaltic flow.

When two droplets with different surface tensions collide, the shape evolution of the merging droplets is asymmetric. Using experimental and numerical techniques, we reveal that this asymmetry is caused by asymmetric capillary waves, which are the result of the different surface tensions of the droplets. We show that the asymmetry is enhanced by increasing the surface tension difference, and suppressed by increasing the inertia of the colliding droplets. Furthermore, we study capillary waves in the limit of no collisional inertia. We reveal that the asymmetry is not directly caused by Marangoni forces. In fact, somehow counterintuitive, asymmetry is strongly reduced by the Marangoni effect. Rather, the different intrinsic capillary wave amplitudes and velocities associated with the different surface tensions of the droplets lie at the origin of the asymmetry during droplet coalescence.