In recent work, Renaud, Venaille, and Bouchet (RVB) [Renaud et al., J. Stat. Phys. 163, 784 (2016)] revisit the equilibrium statistical mechanics theory of the shallow water equations, within a microcanonical approach, focusing on a more careful treatment of the energy partition between inertial gravity wave and eddy motions in the equilibrium state and deriving joint probability distributions for the corresponding dynamical degrees of freedom. The authors derive a Liouville theorem that determines the underlying phase space statistical measure, but then, through some physical arguments, actually compute the equilibrium statistics using a measure that violates this theorem, choosing equal volume vs equal area fluid parcels. Here, using a more convenient, but essentially equivalent, grand canonical approach, the full statistical theory consistent with the Liouville theorem is derived. The results reveal several significant differences from the previous results (1) The microscale wave motions lead to a strongly fluctuating thermodynamics, including long-range correlations, in contrast to the mean-field-like behavior found by RVB. The final effective model is equivalent to that of an elastic membrane with a nonlinear wave-renormalized surface tension. (2) Even when a mean-field approximation is made, a rather more complex joint probability distribution is revealed. Alternative physical arguments fully support the consistency of the results. Of course, the true fluid final steady state relies on dissipative processes not included in the shallow water equations, such as wave breaking and viscous effects, but it is argued that the current theory provides a more mathematically consistent starting point for future work aimed at assessing their impacts.

Non-Newtonian fluids are susceptible to flow instabilities such as shear banding, in which the fluid may exhibit a markedly discontinuous viscosity at a critical stress. Here we report the characteristics and causes of a jetting flow instability of shear banding wormlike micelle solutions in microfluidic channels with rectangular cross sections over an intermediate volumetric flow regime. Particle-tracking methods are used to measure the three-dimensional flow field in channels of differing aspect ratios, sizes, and wall materials. When jetting occurs, it is self-contained within a portion of the channel where the flow velocity is greater than the surroundings. We observe that the instability forms in channels with aspect ratio greater than 5, and that the location of the high-velocity jet appears to be sensitive to stress localizations. Jetting is not observed in a lower concentration solution without shear banding. Simulations using the Johnson-Segalman viscoelastic model show a qualitatively similar behavior to the experimental observations and indicate that compressive normal stresses in the cross-stream directions support the development of the jetting flow. Our results show that nonuniform flow of shear thinning fluids can develop across the wide dimension in rectangular microfluidic channels, with implications for microfluidic rheometry.

A cornstarch suspension is the quintessential particulate system that exhibits shear thickening. By adding large non-Brownian spheres to a cornstarch suspension, we show that shear thickening can be significantly enhanced. More precisely, the shear-thickening transition is found to be increasingly shifted to lower critical shear rates. This influence of the large particles on the discontinuous shear-thickening transition is shown to be more dramatic than that on the viscosity or the yield stress of the suspension.

This study considers the linear stability of Poiseuille-Rayleigh-Bénard flows subjected to a transverse magnetic field, to understand the instabilities that arise from the complex interaction between the effects of shear, thermal stratification, and magnetic damping. This fundamental study is motivated in part by the desire to enhance heat transfer in the blanket ducts of nuclear fusion reactors. In pure magnetohydrodynamic flows, the imposed transverse magnetic field causes the flow to become quasi-two-dimensional and exhibit disturbances that are localized to the horizontal walls. However, the vertical temperature stratification in Rayleigh-Bénard flows feature convection cells that occupy the interior region, and therefore the addition of this aspect provides an interesting point for investigation. The linearized governing equations are described by the quasi-two-dimensional model proposed by Sommeria and Moreau [J. Fluid Mech. 118, 507 (1982)], which incorporates a Hartmann friction term, and the base flows are considered fully developed and one-dimensional. The neutral stability curves for critical Reynolds and Rayleigh numbers, ${\mathrm{Re}}_c$ and ${\mathrm{Ra}}_c$, respectively, as functions of Hartmann friction parameter $H$ have been obtained over ${10}^{-2}\le H\le {10}^4$. Asymptotic trends are observed as $H\to \infty$ following ${\mathrm{Re}}_c\propto H^{1/2}$ and ${\mathrm{Ra}}_c\propto H$. The linear stability analysis reveals multiple instabilities which alter the flow both within the Shercliff boundary layers and the interior flow, with structures consistent with features from plane Poiseuille and Rayleigh-Bénard flows.

The majority of investigations characterizing the motion of single or multiple particles in fluid flows consider canonical body shapes, such as spheres, cylinders, discs, etc. However, protrusions on bodies—either surface imperfections or appendages that serve a function—are ubiquitous in both nature and applications. In this work, we characterize how the dynamics of a sphere with an axis-symmetric wake is modified in the presence of thin three-dimensional elliptic-shaped protrusions. By investigating a wide range of three-dimensional appendages with different aspect ratios and lengths, we clearly show that the sphere with an appendage may robustly undergo an inverted-pendulum-like (IPL) instability. This means that the position of the appendage placed behind the sphere and aligned with the free-stream direction is unstable, similar to how an inverted pendulum is unstable under gravity. Due to this instability, nontrivial forces are generated on the body, leading to turn and drift, if the body is free to fall under gravity. Moreover, we identify the aspect ratio and length of the appendage that induces the largest side force on the sphere, and therefore also the largest drift for a freely falling body. Finally, we explain the physical mechanisms behind these observations in the context of the IPL instability, i.e., the balance between surface area of the appendage exposed to reversed flow in the wake and the surface area of the appendage exposed to fast free-stream flow.

Pinning and depinning of a windswept droplet on a surface is familiar yet deceptively complex for it depends on the interaction of the contact line with the microscopic features of the solid substrate. This physical picture is further compounded when wind of the Reynolds number greater than 100 blows over pinned drops, leading to the boundary layer separation and wake generation. In this Rapid Communication, we incorporate the well-developed ideas of the classical boundary layer to study partially wetting droplets in a wake created by a leader object. Depending on its distance from the leader, the droplet is observed to exhibit drafting, upstream motion, and splitting, due to the wake-induced hydrodynamic coupling that is analogous to drafting of moving bodies. We successfully rationalize the onset of the upstream motion regime using a reduced model that computes the droplet shape governed by the pressure field inside the wake.

Using results from previous direct numerical simulations and experiments in the outer region of spatially evolving turbulent boundary layers, we compute the streamwise evolution and the wall-normal variation of the dissipation parameter $C_{\varepsilon }$, namely, the turbulent kinetic energy dissipation rate, normalized by appropriate powers of the local turbulent kinetic energy and integral length scale. For ${\mathrm{Re}}_{\theta }\gtrsim 10000$ (${\mathrm{Re}}_{\theta }$ is a Reynolds number on the freestream velocity and the local momentum thickness), $C_{\varepsilon }$ is essentially constant in the streamwise direction, but varies by up to 50% in the wall-normal direction. For ${\mathrm{Re}}_{\theta }<10000$, $C_{\varepsilon }$ is additionally found to vary in the streamwise direction and is inversely proportional to the local turbulence Reynolds number ${\mathrm{Re}}_{\lambda }$.

Experimental results reveal that the asymmetric flow of shear-thinning fluid through a cylindrical pipe, which was previously associated with the laminar-turbulent transition process, appears to have the characteristics of a nonhysteretic, supercritical instability of the laminar base state. Contrary to what was previously believed, classical transition is found to be responsible for returning symmetry to the flow. An absence of evidence of the instability in simulations (either linear or nonlinear) suggests that an element of physics is lacking in the commonly used rheological model for inelastic shear-thinning fluids. These unexpected discoveries raise new questions regarding the stability of these practically important fluids and how they can be successfully modeled.

Intermittency of low-drag pointwise wall shear stress measurements within Newtonian turbulent channel flow at transitional Reynolds numbers (friction Reynolds numbers 70 – 130) is characterized using experiments and simulations. Conditional mean velocity profiles during low-drag events closely approach that of a recently discovered nonlinear traveling wave solution; both profiles are near the so-called maximum drag reduction profile, a general feature of turbulent flow of liquids containing polymer additives (despite the fact that all results presented are for Newtonian fluids only). Similarities between temporal intermittency in small domains and spatiotemporal intermittency in large domains is thereby found.

Properly resolved large-eddy simulations of wall-bounded high Reynolds number flows using standard subgrid-scale (SGS) models requires high spatial and temporal resolution. We have shown that a more elaborate SGS model taking into account the SGS Reynolds stress anisotropies can relax the requirement for the number of grid points by at least an order of magnitude for the same accuracy. This was shown by applying the recently developed explicit algebraic subgrid-scale model (EAM) [Marstorp et al., J. Fluid Mech. 639, 403 (2009)] to fully developed high Reynolds number channel flows with friction Reynolds numbers of 550, 2000, and 5200. The near-wall region is fully resolved, i.e., no explicit wall modeling or wall functions are applied. A dynamic procedure adjusts the model at the wall for both low and high Reynolds numbers. The resolution is reduced, from the typically recommended 50 and 15 wall units in the stream- and spanwise directions respectively, by up to a factor of 5 in each direction. It was shown by comparison with direct numerical simulations that the EAM is much less sensitive to reduced resolution than the dynamic Smagorinsky model. Skin friction coefficients, mean flow profiles, and Reynolds stresses are better predicted by the EAM for a given resolution. Even the notorious overprediction of the streamwise fluctuation intensity typically seen in poorly resolved LES is significantly reduced when EAM is used on coarse grids. The improved prediction is due to the capability of the EAM to capture the SGS anisotropy, which becomes significant close to the wall.

Phoretic swimmers provide new avenues to study nonequilibrium statistical physics and are also hailed as a promising technology for bioengineering at the cellular scale. Exact solutions for the locomotion of such swimmers have been restricted so far to spheroidal shapes. In this paper we solve for the flow induced by the canonical nonsimply connected shape, namely an axisymmetric phoretic torus. The analytical solution takes the form of an infinite series solution, which we validate against boundary element computations. For a torus of uniform chemical activity, confinement effects in the hole allow the torus to act as a pump, which we optimize subject to fixed particle surface area. Under the same constraint, we next characterize the fastest swimming Janus torus for a variety of assumptions on the surface chemistry. Perhaps surprisingly, none of the optimal tori occur in the limit where the central hole vanishes.

Budgets of the turbulent kinetic energy from direct numerical simulations (DNSs) of disperse bubbly channel flows are used to develop a model for bubble-induced turbulence in the Euler-Euler Reynolds-averaged framework. First, an appropriate time scale is selected. Second, links between the unclosed terms in the transport equations of the turbulence quantities and the DNS data for small bubbles are established. Third, a suitably chosen iterative procedure employing the full Reynolds-averaged model provides suitable coefficients for the closure of the terms resulting from bubble-induced turbulence while largely removing the influence of others. At the same time these results validate the closure, exhibiting very good agreement with the DNS and better performance than the standard closures. The model is now ready for use and can be employed in practical Euler-Euler simulations.

Recent experiments and simulations have shown that unsteady turbulent flows display a universal behavior at short and intermediate times, different from classical scaling relations. The origin of these observations is explained using a nonequilibrium correction to Kolmogorov's energy spectrum, and the exact form of the observed universal scaling is derived.

We investigate granular transport by a fluid flow under steady-state driving conditions, from the bed-load regime to the suspension regime, with an experimental system based on a conical rheometer. The mean granular volume fraction ${\varphi }_g$, the mean granular velocity $u_g$, and the fluid velocity $u_f$ are obtained as a function of depth inside the bed using refractive index matching and particle-tracking techniques. A torque sensor is utilized to measure the applied shear stress to complement estimates obtained from measured strain rates high above the bed where ${\varphi }_g\approx 0$. The flow is found to be transitional at the onset of transport and the shear stress required to transport grains rises sharply as grains are increasingly entrained by the fluid flow. A significant slip velocity between the fluid and the granular phases is observed at the bed surface before the onset of transport as well as in the bed-load transport regime. We show that $u_g$ decays exponentially deep into the bed for ${\varphi }_g>0.45$ with a decay constant which is described by a nonlocal rheology model of granular flow that neglects fluid stress. Further, we show that $u_f$ and $u_g$ can be described using the applied shear stress and the Krieger-Dougherty model for the effective viscosity in the suspension regime, where $0<{\varphi }_g<0.45$ and where $u_g\approx u_f$.

How droplets grow rapidly from 10 to 50 $\mu \mathrm{m}$ is an outstanding question in cloud physics. We show theoretically and numerically that caustics, locations of multivalued droplet velocity, of small droplets near a single planar steady vortex offer one route through this bottleneck. Such a vortex serves as a simple model for the more complicated turbulence field existing in clouds. Within a special radial distance $r_c$ from the vortex center, droplets closer to the vortex can centrifugally overtake those farther out and coalesce. Small polydispersity increases $r_c$ dramatically, enabling repeated collisions at short time intervals and formation of large droplets. Our results show that caustics brought about in a polydisperse suspension could offer a mechanistic explanation of accelerated rain initiation.

Particle sedimentation under a turbulent flow is a fundamental problem that has numerous applications in natural and industrial processes. In particular, the motion of anisotropic particles yields complex dynamics whose features are not completely understood in the presence of inertia. While inertia is generally introduced through a finite particle response time, many processes involve particles with a low response time but a finite particle Reynolds number. In this case, theoretical models are rather sparse, and their validity has never been tested against controlled experiments. This work precisely proposes a careful testing of fiber sedimentation and advection models at finite particle Reynolds number against well-controlled two-dimensional experiments. We show that the slender body limit model has strong limitations at finite aspect ratio and Reynolds number, and we identify the main corrections that need to be incorporated into this basic model by expanding on the work of Khayat and Cox [J. Fluid Mech. 209, 435 (1989)]. Additionally, we present the different models under a uniform framework, providing a simpler and clearer use. Using the validated inertial model, we show the importance of the ratio between the settling speed and the typical flow velocity for describing the fiber motion.

We present simulations of the effects of insoluble surfactant on drops rising in unstratified media and density-stratified media. We consider an oil drop rising in water, and we track surfactant concentration on the drop surface. We first consider a drop coated with insoluble surfactant rising through an unstratified ambient. The drop speed is computed for various Marangoni numbers, and inertial and surface diffusion effects are also examined. In the second setup, we compare clean and contaminated drops rising through a linear density-stratified fluid. The entrained buoyancy is computed for various density gradients, and the effects of inertia and Marangoni number are characterized. We find that the entrained fluid slows the drop in a manner comparable to a vertical shift in the ambient density gradient.

Reynolds-averaged Navier-Stokes (RANS) models represent the workhorse for predicting turbulent flows in complex industrial applications. However, RANS closures introduce a significant degree of epistemic uncertainty in predictions due to the potential lack of validity of the assumptions utilized in model formulation. Estimating this uncertainty is a fundamental requirement for building confidence in such predictions. We outline a methodology to estimate this structural uncertainty, incorporating perturbations to the eigenvalues and the eigenvectors of the modeled Reynolds stress tensor. The mathematical foundations of this framework are derived and explicated. Thence, this framework is applied to a set of separated turbulent flows, while compared to numerical and experimental data and contrasted against the predictions of the eigenvalue-only perturbation methodology. It is exhibited that for separated flows, this framework is able to yield significant enhancement over the established eigenvalue perturbation methodology in explaining the discrepancy against experimental observations and high-fidelity simulations. Furthermore, uncertainty bounds of potential engineering utility can be estimated by performing five specific RANS simulations, reducing the computational expenditure on such an exercise.

Numerical simulations of the G.O. Roberts dynamo are presented. Dynamos both with and without a significant mean field are obtained. Exact bounds are derived for the total energy which conform with the Kolmogorov phenomenology of turbulence. Best fits to numerical data show the same functional dependences as the inequalities obtained from optimum theory.

Direct numerical simulations (DNS) of plane Poiseuille flow are performed in an extended domain at friction Reynolds numbers ranging from 70 to 100. In minimal domains, turbulence in this Reynolds number range displays substantial intermittency that is associated with chaotic movement of turbulent trajectories between lower and upper branch invariant solutions known as exact coherent states (ECS). The present work aims to address the relationship between temporal dynamics in minimal channels and spatiotemporal dynamics in extended domains. Both temporal and spatial analyses of the turbulent velocity fields are performed, the latter using image analysis methods. These analyses partition the flow characteristics into low-, intermediate- and high-drag classes; we present the differences between flows fields in these classes in terms of simple quantities like mean velocity, wall shear stress, and flow structures. The temporal and spatial analysis methods, although completely independent of one another, yield very similar results for both low- and high-drag regions. In particular, the conditional mean profiles in regions of low drag closely resemble those found in low-drag temporal intervals in the minimal channel. Finally, we address the possibility of similarities between turbulence and exact coherent states in two ways: (1) comparing wall shear stress in localized patches the size of minimal channels in large domains with those in actual minimal channel and (2) comparing conditional mean velocity profiles during low-drag events with mean profiles from lower branch ECS. These analyses show that both the local near-wall flow structure in the low-drag patches of the large domain and the conditional mean profiles in the region $y^{+}\lesssim 30$ resemble those of a lower branch minimal domain ECS. In summary, the results presented here suggest that spatiotemporal intermittency in transitional channel flow turbulence is related to temporal intermittency, and by extension to the state space structure, in the minimal channel.