The effect of the Strouhal number on periodic forcing of the flow over a backward-facing step (height, $H$) is investigated experimentally. Forcing is applied by a synthetic jet at the edge of the step at Strouhal numbers ranging from $0.21<{\text{St}}_H<1.98$ (${\text{St}}_H=fH/U_{\infty }$) at a Reynolds number of ${\text{Re}}_H=H{U}_{\infty }/\nu =41000$. In the literature, the effect of Strouhal number on the reattachment length is often divided into low- and high-frequency actuation, referring to specific frequencies present in the unforced flow. In the present study, variations with Strouhal number are explained based on a continuous variation of entrainment of momentum into the recirculation region rather than on specific frequencies. The reattachment length is shown to decrease linearly with entrainment of momentum. Vertical momentum flux is driven by vortices generated by the forcing, and locally vertical momentum flux is shown to be qualitatively similar to circulation for all cases considered. Total circulation (and therewith entrainment of momentum and the effect on the reattachment length) is shown to decrease with Strouhal number, whereas this is not predicted by models based on specific low and high frequencies. An empirical model for the (decay of) circulation is derived by tracking vortices in phase-locked data. This model is used to decipher relevant scaling parameters that explain the variations in circulation, entrainment of momentum, and reattachment length. Three regimes of Strouhal number are identified. A low-Strouhal-number regime is observed for which vortices are formed at a late stage relative to the recirculation region, causing a decrease in effectiveness. For high Strouhal numbers, vortices are being reingested into the actuator or are packed so close together that they cancel each other, both decreasing the effectiveness of forcing. In the intermediate regime a vortex train is formed of which the decay of circulation increases for increasing Strouhal number. The scaling of this decay fully explains the observed variation in reattachment length. The observations on entrainment of momentum made in this study are expected to also hold for periodic forcing of other bluff-body flows.

Intermittency of small-scale motions is an ubiquitous facet of turbulent flows, and predicting this phenomenon based on reduced models derived from first principles remains an important open problem. Here, a multiple-time-scale stochastic model is introduced for the Lagrangian evolution of the full velocity gradient tensor in fluid turbulence at arbitrarily high Reynolds numbers. Unlike previous phenomenological models of intermittency, in the proposed model the dynamics driving the growth of intermittency due to gradient self-stretching and rotation are derived directly from the Navier-Stokes equations. Numerical solutions of the resulting set of stochastic differential equations show that the model predicts anomalous scaling for moments of the velocity gradient components and negative derivative skewness. It also predicts signature topological features of the velocity gradient tensor such as vorticity alignment trends with the eigen directions of the strain rate.

The interaction between a shock wave and a thin layer of inert dust is studied by solving unsteady, multidimensional Navier-Stokes equations representing the interactions between a compressible gas and incompressible particles. The system studied consists of a layer of densely packed limestone dust containing particles of uniform diameter (40 $\mu \mathrm{m})$ that interact with a shock of strength $M_s=1.4$. Particle dispersion is investigated by comparing vertical particle accelerations due to Archimedes, gravitational, intergranular, and aerodynamic drag and lift forces. The simulations show that the shock produces two dust regions: a compacted layer and a dispersed region. The layer compaction, which increases the intergranular particle stress, is produced by drag and Archimedes forces. The dispersed dust is produced by forces that change in time as the shock passes. Initially, the dispersion is caused by intergranular forces. Later it is driven by a tradeoff between lift and drag forces. Eventually, drag forces dominate. Comparisons of the computations to experimental shock-tube data reproduced the observed initial growth of the dispersed dust and later leveled off. Particle agglomeration in the experiments made it difficult to determine a true particle size experimentally, although the computations for 40-$\mu \mathrm{m}$ particles explain the experimental data.

Motivated by the redrawing of hot glass into thin sheets, we investigate the shape and stability of a thin viscous sheet that is inhomogeneously stretched in an imposed nonuniform temperature field. We first determine the associated base flow by solving the long-time-scale stretching flow of a flat sheet as a function of two dimensionless parameters: the normalized stretching velocity $\alpha$ and a dimensionless width of the heating zone $\beta$. This allows us to determine the conditions for the onset of an out-of-plane wrinkling instability stated in terms of an eigenvalue problem for a linear partial differential equation governing the displacement of the midsurface of the sheet. We show that the sheet can become unstable in two regions that are upstream and downstream of the heating zone where the minimum in-plane stress is negative. This yields the shape and growth rates of the most unstable buckling mode in both regions for various values of the stretching velocity and heating zone width. A transition from stationary to oscillatory unstable modes is found in the upstream region with increasing $\beta$, while the downstream region is always stationary. We show that the wrinkling instability can be entirely suppressed when the surface tension is large enough relative to the magnitude of the in-plane stress. Finally, we present an operating diagram that indicates regions of the parameter space that result in a required outlet sheet thickness upon stretching while simultaneously minimizing or suppressing the out-of-plane buckling, a result that is relevant for the glass redraw method used to create ultrathin glass sheets.

We report on an experimental study of the large-scale flow (LSF) and Reynolds numbers in turbulent convection in a cylindrical sample with height equal to its diameter and heated locally around the center of its bottom plate (locally heated convection). The sample size and shape are the same as those of Narezo Guzman et al. [D. Narezo Guzman et al., J. Fluid Mech. 787, 331 (2015); D. Narezo Guzman et al., J. Fluid Mech. 795, 60 (2016)]. Measurements are made at a nearly constant Rayleigh number as a function of the mean temperature, both in the presence of controlled boiling (two-phase flow) and for the superheated fluid (one-phase flow). Superheat values $T_b-T_{on}$ up to about 11 K ($T_b$ is the bottom-plate temperature and $T_{on}$ is the lowest $T_b$ at which boiling is observed) are used. The LSF is less organized than it is in (uniformly heated) Rayleigh-Bénard convection (RBC), where it takes the form of a single convection roll. Large-scale-flow-induced sinusoidal azimuthal temperature variations (like those found for RBC) could be detected only in the lower portion of the sample, indicating a less organized flow in the upper portions. Reynolds numbers are determined using the elliptic model (EM) of He and Zhang [G.-W. He and J.-B. Zhang, Phys. Rev. E 73, 055303(R) (2006)]. We found that for our system the EM is applicable over a wide range of space and time displacements, as long as these displacements are within the inertial range of the temporal and spatial spectrum. At three locations in the sample the results show that the vertical mean-flow velocity component is reduced while the fluctuation velocity is enhanced by the bubbles of the two-phase flow. Enhancements of velocity fluctuations up to about 60% are found at the largest superheat values. Local temperature measurements within the sample reveal temperature oscillations that also used to determine a Reynolds number. These results are generally consistent with the mean-flow EM results and show a two-phase-flow enhancement of up to about 30%.

In addition to the commonly observed elastic asymmetric instability, experiments of wormlike micellar solutions in a cross-slot device have shown the formation of lip vortices along the walls of the inlet channels. For these highly viscoelastic solutions the secondary flow can develop prior to the symmetry breaking. This indicates that these instabilities are complex phenomena whose behavior is likely influenced by a combination of factors. This work extends previous computational studies of the elastic instability predicted using the upper-convected Maxwell and Oldroyd-B models showing that one important factor may be the scission and reforming of the micelles.

Experiments for the gravity currents produced from a two-layer density-stratified buoyancy source in a full-depth, lock-exchange setup with a scaling analysis describing the flow morphologies are presented. In the inertial phase of propagation, the $3/2$ power relationship $x_f^{3/2}=1.5F_I{B}_0^{1/2}(t+t_I)$ robustly applies between the front location $x_f$ and time $t$, where $F_I$ is the Froude number in the inertial phase, $B_0$ is the total released buoyancy, and $t_I$ is the $t$-intercept. We showed that the Froude number in the inertial phase is not a universal constant but depends on the two controlling parameters, namely, the density difference ratio, $R_{\rho }=({\rho }_U-{\rho }_0)/({\rho }_L-{\rho }_0)$, where ${\rho }_U,{\rho }_L$, and ${\rho }_0$ are the densities of the fluids in the upper layer, lower layer, and ambient environment, respectively, and the buoyancy distribution parameter, $R_B=B_U/B_0$, where $B_U$ is the buoyancy in the upper layer. For a given buoyancy distribution parameter, the Froude number in the inertial phase decreases monotonically as the density difference ratio decreases. For a given density difference ratio, the Froude number in the inertial phase has a local minimum as the buoyancy distribution parameter falls in the range of $0.3\lesssim R_B\lesssim 0.5$. When the buoyancy source is homogeneous, the Froude number in the inertial phase has its maximum value at $F_I=1.33\pm 0.02$. The flow morphology is also found to depend on the two controlling parameters. For weakly stratified two-layer heavy fluid, $0.4\lesssim R_{\rho }<1$, mixing between the fluids from the two layers is more immediate. For strongly stratified two-layer heavy fluid, $0<R_{\rho }\lesssim 0.4$, there is less mixing between the layers for flows dominated by the upper layer, $R_B\to 1$, and for flows dominated by the lower layer, $R_B\to 0$. For gravity currents that are produced from a strongly stratified source and dominated by the upper layer, the upper layer may override and outrun the lower layer, which initially takes the lead after the two-layer heavy fluid is released. For gravity currents that are produced from a strongly stratified source and dominated by the lower layer, the lower layer may outrun the upper layer from the outset, resulting in streamwise stratification. Surprisingly, for the gravity currents produced from a strongly stratified source, mixing of fluids from the two layers can be enhanced when the buoyancy distribution parameter falls in the range of $0.3\lesssim R_B\lesssim 0.5$. Such an exceptional observation is now successfully explained by the scaling analysis.

Magnetic resonance imaging visualization down to nanometric liquid films in model porous media with pore sizes from micro- to nanometers enables one to fully characterize the physical mechanisms of drying. For pore size larger than a few tens of nanometers, we identify an initial constant drying rate period, probing homogeneous desaturation, followed by a falling drying rate period. This second period is associated with the development of a gradient in saturation underneath the sample free surface that initiates the inward recession of the contact line. During this latter stage, the drying rate varies in accordance with vapor diffusion through the dry porous region, possibly affected by the Knudsen effect for small pore size. However, we show that for sufficiently small pore size and/or saturation the drying rate is increasingly reduced by the Kelvin effect. Subsequently, we demonstrate that this effect governs the kinetics of evaporation in nanopores as a homogeneous desaturation occurs. Eventually, under our experimental conditions, we show that the saturation unceasingly decreases in a homogeneous manner throughout the wet regions of the medium regardless of pore size or drying regime considered. This finding suggests the existence of continuous liquid flow towards the interface of higher evaporation, down to very low saturation or very small pore size. Paradoxically, even if this net flow is unidirectional and capillary driven, it corresponds to a series of diffused local capillary equilibrations over the full height of the sample, which might explain that a simple Darcy's law model does not predict the effect of scaling of the net flow rate on the pore size observed in our tests.

Plasma flows involve hundreds of species and thousands of reactions at different time scales, resulting in a very large set of governing equations to solve. Simulating large reacting systems in nonequilibrium plasma mixtures remains a challenge with the currently available computational resources. Principal component analysis (PCA) offers a general and rather simple and automated method to reduce large kinetic mechanisms by principal variable selection. This work shows how to adapt and apply the PCA-scores technique, which has its origin in the combustion field, to a collisional-radiative model. We have successfully applied this technique to argon plasmas, reducing the set of governing equations by more than 90%, leading to an important speed-up of the calculation and a reduction of computational cost.

A theoretical model describing the blood spatter pattern resulting from a blunt bullet gunshot is proposed. The predictions are compared to experimental data acquired in the present work. This hydrodynamic problem belongs to the class of the impact hydrodynamics with the pressure impulse generating the blood flow. At the free surface, the latter is directed outwards and accelerated toward the surrounding air. As a result, the Rayleigh-Taylor instability of the flow of blood occurs, which is responsible for the formation of blood drops of different sizes and initial velocities. Thus, the initial diameter, velocity, and acceleration of the atomized blood drops can be determined. Then, the equations of motion are solved, describing drop trajectories in air accounting for gravity, and air drag. Also considered are the drop-drop interactions through air, which diminish air drag on the subsequent drops. Accordingly, deposition of two-phase (blood-drop and air) jets on a vertical cardstock sheet located between the shooter and the target (and perforated by the bullet) is predicted and compared with experimental data. The experimental data were acquired with a porous polyurethane foam sheet target impregnated with swine blood, and the blood drops were collected on a vertical cardstock sheet which was perforated by the blunt bullet. The highly porous target possesses a low hydraulic resistance and therefore resembles a pool of blood shot by a blunt bullet normally to its free surface. The back spatter pattern was predicted numerically and compared to the experimental data for the number of drops, their area, the total stain area, and the final impact angle as functions of radial location from the bullet hole in the cardstock sheet (the collection screen). Comparisons of the predicted results with the experimental data revealed satisfactory agreement. The predictions also allow one to find the impact Weber number on the collection screen, which is necessary to predict stain shapes and sizes.

We provide in-depth scrutiny of two methods making use of adjoint-based gradients to compute the sensitivity of drag in the two-dimensional, periodic flow past a circular cylinder ($\text{Re}\lesssim 189$): first, the time-stepping analysis used in Meliga et al. [Phys. Fluids 26, 104101 (2014)] that relies on classical Navier-Stokes modeling and determines the sensitivity to any generic control force from time-dependent adjoint equations marched backwards in time; and, second, a self-consistent approach building on the model of Mantič-Lugo et al. [Phys. Rev. Lett. 113, 084501 (2014)] to compute semilinear approximations of the sensitivity to the mean and fluctuating components of the force. Both approaches are applied to open-loop control by a small secondary cylinder and allow identifying the sensitive regions without knowledge of the controlled states. The theoretical predictions obtained by time-stepping analysis reproduce well the results obtained by direct numerical simulation of the two-cylinder system. So do the predictions obtained by self-consistent analysis, which corroborates the relevance of the approach as a guideline for efficient and systematic control design in the attempt to reduce drag, even though the Reynolds number is not close to the instability threshold and the oscillation amplitude is not small. This is because, unlike simpler approaches relying on linear stability analysis to predict the main features of the flow unsteadiness, the semilinear framework encompasses rigorously the effect of the control on the mean flow, as well as on the finite-amplitude fluctuation that feeds back nonlinearly onto the mean flow via the formation of Reynolds stresses. Such results are especially promising as the self-consistent approach determines the sensitivity from time-independent equations that can be solved iteratively, which makes it generally less computationally demanding. We ultimately discuss the extent to which relevant information can be gained from a hybrid modeling computing self-consistent sensitivities from the postprocessing of DNS data. Application to alternative control objectives such as increasing the lift and alleviating the fluctuating drag and lift is also discussed.

The imbibition of a liquid into a thin deformable porous substrate driven by capillary suction is considered. The substrate is initially dry and has uniform porosity and thickness. Two-phase flow theory is used to describe how the liquid flows through the pore space behind the wetting front when out-of-plane deformation of the solid matrix is considered. Neglecting gravity and evaporation, standard shallow-layer scalings are used to construct a reduced model of the dynamics. The model predicts convergence to a self-similar behavior in all regions except near the wetting front, where a boundary layer arises whose structure narrows with the advance of the front. Over time, the rise height approaches the similarity scaling of $t^{1/2}$, as in the classical Washburn or BCLW law. The results are compared with a series of laboratory experiments using cellulose paper sheets, which provide qualitative agreement.

Making use of an original technique which permits the simultaneous measurement of both the displacement field of evaporating droplets in a spray, and of their vapor, we investigate the relevance of a scenario introduced earlier to describe the evaporation dynamics of dense sprays [Phys. Rev. Fluids 1, 014201 (2016)]. A plume of dense acetone droplets evaporating in air is studied, for which the stirring field is measured by particle image velocimetry of the droplets, and the vapor field is imaged quantitatively by laser-induced fluorescence. We show, thanks to these unique in situ measurements, that the spray boundary with the diluting environment is slaved at the dynamics of its saturating vapor concentration field, whose structure is analyzed for different well defined local flow topologies.

The purpose of this work is to study the anisotropic properties of the inertial range of Rayleigh-Taylor and unstably stratified homogeneous (USH) turbulence. More precisely, we aim to understand the role played by the so-called zero-modes, i.e., modes that nullify the anisotropic part of transfer terms. To this end, we determine several characteristic properties of zero-modes using an eddy-damped quasinormal Markovianized (EDQNM) model. Then we perform a high-Reynolds-number EDQNM simulation of a USH flow and check whether the predicted zero-mode properties are indeed observed in this idealized setting. Finally, we carry out a large-eddy simulation of a Rayleigh-Taylor flow and verify if zero-modes can also be identified in this configuration. Among the main findings of this work, we show that the small-scale anisotropy of the velocity and concentration spectra is dominated by the nonlocal contribution of zero-modes rather than by the local action of buoyancy forces. As a result, we predict inertial scaling exponents close to $-7/3$ (rather than $-3$) for the second-order harmonics of the velocity and concentration spectra. By contrast, the concentration flux spectrum remains controlled by buoyancy forces. Still, we show that the zero-mode contribution vanishes slowly as the Reynolds number increases. This translates into a slow convergence of the scaling exponent of the second-order harmonic of the concentration flux to $-7/3$.

We study the intricate buckling patterns which can form when a viscous drop impacts a much lower viscosity miscible pool. The drop enters the pool by its impact inertia, flattens, and sinks by its own weight while stretching into a hemispheric bowl. Upward motion along the outer bottom surface of this bowl produces a vortical boundary layer which separates along its top and rolls up into a vortex ring. The vorticity is therefore produced in a fundamentally different way than for a drop impacting a pool of the same liquid. The vortex ring subsequently advects into the bowl, thereby stretching the drop liquid into ever thinner sheets, reaching the micron level. The rotating motion around the vortex pulls in folds to form multiple windings of double-walled toroidal viscous sheets. The axisymmetric velocity field thereby stretches the drop liquid into progressively finer sheets, which are susceptible to both axial and azimuthal compression-induced buckling. The azimuthal buckling of the sheets tends to occur on the inner side of the vortex ring, while their folds can be stretched and straightened on the outside edge. We characterize the total stretching from high-speed video imaging and use particle image velocimetry to track the formation and evolution of the vortex ring. The total interfacial area between the drop and the pool liquid can grow over 40-fold during the first 50 ms after impact. Increasing pool viscosity shows entrapment of a large bubble on top of the drop, while lowering the drop viscosity produces intricate buckled shapes, appearing at the earliest stage and being promoted by the crater motions. We also present an image collage of the most intriguing and convoluted structures observed. Finally, a simple point-vortex model reproduces some features from the experiments and shows variable stretching along the wrapping sheets.

In steady sediment transport, the deposition of transported particles is balanced by the entrainment of soil bed particles by the action of fluid forces or particle-bed impacts. Here we propose a proxy to determine the role of impact entrainment relative to entrainment by the mean turbulent flow: the “bed velocity” $V_b$, which is an effective near-bed-surface value of the average horizontal particle velocity that generalizes the classical slip velocity, used in studies of aeolian saltation transport, to sediment transport in an arbitrary Newtonian fluid. We study $V_b$ for a wide range of the particle-fluid-density ratio $s$, Galileo number $\mathrm{Ga}$, and Shields number $\mathrm{\Theta }$ using direct sediment transport simulations with the numerical model of Durán et al. [Phys. Fluids 24, 103306 (2012)], which couples the discrete element method for the particle motion with a continuum Reynolds-averaged description of hydrodynamics. We find that transport is fully sustained through impact entrainment (i.e., $V_b$ is constant in natural units) when the “impact number” $\mathrm{Im}=\mathrm{Ga}\sqrt{s+0.5}\gtrsim 20$ or $\mathrm{\Theta }\gtrsim 5/\mathrm{Im}$. These conditions are obeyed for the vast majority of transport regimes, including steady turbulent bedload, which has long been thought to be sustained solely through fluid entrainment. In fact, we find that transport is fully sustained through fluid entrainment (i.e., $V_b$ scales with the near-bed horizontal fluid velocity) only for sufficiently viscous bedload transport at grain scale (i.e., for $\mathrm{Im}\lesssim 20$ and $\mathrm{\Theta }\lesssim 1/\mathrm{Im})$. Finally, we do not find a strong correlation between $V_b$, or the classical slip velocity, and the transport-layer-averaged horizontal particle velocity $\overline{v_x}$, which challenges the long-standing consensus that predominant impact entrainment is responsible for a linear scaling of the transport rate with $\mathrm{\Theta }$. For turbulent bedload in particular, $\overline{v_x}$ increases with $\mathrm{\Theta }$ despite $V_b$ remaining constant, which we propose is linked to the formation of a liquidlike bed on top of the static-bed surface.

This work considers the distribution of discrete inertial particles in turbulence. We demonstrate that even for weak inertia the distribution can be strongly different from the Poisson distribution that holds for tracers. We study the cases of weak inertia or strong gravity where single-valued particle flow holds in space. In these cases, the particles distribute over a random multifractal attractor in space. This attractor is characterized by fractal dimensions describing scaling exponents of moments of number of particles inside a ball with size much smaller than the viscous scale of turbulence. Previous studies used a continuum approach to the moments which requires having a large number of particles below the viscous scale. This condition often does not hold in practice; for instance, for water droplets in clouds there is typically one droplet per viscous scale. This condition is also hard to realize in numerical simulations. In this work, we overcome this difficulty by deriving the probability $p_l\left(k\right)$ of having $k$ particles in a ball of small radius $l$ for which the continuum approximation may not hold. We demonstrate that the random point process formed by positions of particles' centers in space is a Poisson point process with log-normal random intensity (the so-called log Gaussian Cox process or LGCP). This gives $p_l\left(k\right)$ in terms of the characteristic function of a log-normal distribution from which the moments are derived. This allows finding the correlation dimension relevant for statistics of particles' collisions. The case of zero number of particles provides the statistics of the size of voids—regions without particles—that were not studied previously. The probability of voids is increased compared to a random distribution of particles because preferential concentration of inertial particles implies voids in the deserted regions. Thus voids and preferential concentration are different reflections of the same phenomena. In the limit of tracers with zero inertia, described by the Poisson distribution, the typical void size is of the order of the mean interparticle distance. However, at small but finite inertia, the void size is larger than the mean interparticle distance by a factor that diverges in the continuum limit of infinite number of particles, manifesting strong deviations from the Poisson distribution. Thus, voids are very sensitive to inertia and limits of zero inertia and continuum may not commute. Further, the tail of the probability distribution of the void size is log normal, in contrast with faster than exponential decay of the Poisson distribution. Remarkably, at scales where there is no clustering (so pair-correlation function of concentration splits in the product of averages), there can still be an increase of the void probability so that turbulent voiding is stronger than clustering. These results can find applications in the long-time survival probability of reacting particles where the particle survives in the void of predators. The demonstrated double stochasticity (Poisson with random intensity) of the distribution originates in the two-step formation of fluctuations of the number of particles inside the volume under observation. First, turbulence randomly brings uncorrelated particles below the viscous scale with Poisson-type probability. Then, turbulence compresses the particles inside the observation volume. We confirm the predictions by numerical observations of inertial particle motion in a chaotic Arnold-Beltrami-Childress flow. Our work implies that the particle distribution in arbitrary weakly compressible flow with finite time correlations is a LGCP.

The gross amplification of the fluid velocity in pressure-driven flows due to the introduction of superhydrophobic walls is commonly quantified by an effective slip length. The canonical duct-flow geometry involves a periodic structure of longitudinal shear-free stripes at either one or both of the bounding walls, corresponding to flat-meniscus gas bubbles trapped within a periodic array of grooves. This grating configuration is characterized by two geometric parameters, namely the ratio $\kappa$ of channel width to microstructure period and the areal fraction $\mathrm{\Delta }$ of the shear-free stripes. For wide channels, $\kappa \gg 1$, this geometry is known to possess an approximate solution where the dimensionless slip length $\lambda$, normalized by the duct semiwidth, is small, indicating a weak superhydrophobic effect. We here address the other extreme of narrow channels, $\kappa \ll 1$, identifying large $O\left({\kappa }^{-2}\right)$ values of $\lambda$ for the symmetric configuration, where both bounding walls are superhydrophobic. This velocity enhancement is associated with an unconventional Poiseuille-like flow profile where the parabolic velocity variation takes place in a direction parallel (rather than perpendicular) to the boundaries. Use of matched asymptotic expansions and conformal-mapping techniques provides $\lambda$ up to $O\left({\kappa }^{-1}\right)$, establishing the approximation$$\lambda \sim {\kappa }^{-2}\frac{{\mathrm{\Delta }}^3}{3}+{\kappa }^{-1}\frac{{\mathrm{\Delta }}^2}{\pi }ln4+\cdots ,$$which is in excellent agreement with a semianalytic solution of the dual equations governing the respective coefficients of a Fourier-series representation of the fluid velocity. No similar singularity occurs in the corresponding asymmetric configuration, involving a single superhydrophobic wall; in that geometry, a Hele-Shaw approximation shows that $\lambda =O\left(1\right)$.

It is known that an object translating parallel to a soft wall in a viscous fluid produces hydrodynamic stresses that deform the wall, which in turn results in a lift force on the object. Recent experiments with cylinders sliding under gravity near a soft incline, which confirmed theoretical arguments for the lift force, also reported an unexplained steady-state rotation of the cylinders [B. Saintyves et al., Proc. Natl. Acad. Sci. USA 113, 5847 (2016)]. Motivated by these observations, we show, in the lubrication limit, that an infinite cylinder that translates in a viscous fluid parallel to a soft wall at constant speed and separation distance must also rotate in order to remain free of torque. Using the Lorentz reciprocal theorem, we show analytically that for small deformations of the elastic layer, the angular velocity of the cylinder scales with the cube of the sliding speed. These predictions are confirmed numerically. We then apply the theory to the gravity-driven motion of a cylinder near a soft incline and find qualitative agreement with the experimental observations, namely, that a softer elastic layer results in a greater angular speed of the cylinder.

The expected universality of small-scale properties of turbulent flows implies isotropic properties of the velocity gradient tensor in the very large Reynolds number limit. Using direct numerical simulations, we determine the tensors formed by $n=2$ and 3 velocity gradients at a single point in turbulent homogeneous shear flows and in the log-layer of a turbulent channel flow, and we characterize the departure of these tensors from the corresponding isotropic prediction. Specifically, we separate the even components of the tensors, invariant under reflexion with respect to all axes, from the odd ones, which identically vanish in the absence of shear. Our results indicate that the largest deviation from isotropy comes from the odd component of the third velocity gradient correlation function, especially from the third moment of the derivative along the normal direction of the streamwise velocity component. At the Reynolds numbers considered (${\text{Re}}_{\lambda }\approx 140$), we observe that these second- and third-order correlation functions are significantly larger in turbulent channel flows than in homogeneous shear flow. Overall, our work demonstrates that a mean shear leads to relatively simple structure of the velocity gradient tensor. How isotropy is restored in the very large Reynolds limit remains to be understood.