This theoretical investigation of steady fluid flow through a rigid three-dimensional branching geometry is motivated by applications to haemodynamics in the brain especially, while the flow through a tube with a blockage or through a collapsed tube provides another motivation with a biomedical background. Three-dimensional motion without symmetry is addressed through one mother vessel to two or several daughters. A comparatively long axial length scale of the geometry leads to a longitudinal vortex system providing a slender-flow model for the complete mother-and-daughters flow response. Computational studies and subsequent analysis, along with comparisons, are presented. The relative flow rate varies in terms of an effective Reynolds number dependence, allowing a wide range of flow rates to be examined theoretically; also any rigid cross-sectional shape and ratio of cross-sectional area expansion or contraction from the mother vessel to the daughters can be accommodated in principle in both the computations and the analysis. Swirl production with substantial crossflows is found. The analysis shows that close to any carina (the ridge separating daughter vessels) or carinas at a branch junction either forward or reversed motion can be observed locally at the saddle point even though the bulk of the motion is driven forward into the daughters. The local forward or reversed motion is controlled, however, by global properties of the geometry and incident conditions, a feature which applies to any of the flow rates examined.

The local interaction of an oblique shock wave with an unseparated turbulent boundary layer at a shallow two-dimensional compression corner is described by asymptotic expansions for small values of the non-dimensional friction velocity and the flow turning angle. It is assumed that the velocity-defect law and the law of the wall, adapted for compressible flow, provide an asymptotic representation of the mean velocity profile in the undisturbed boundary layer. Analytical solutions for the local mean-velocity and pressure distributions are derived in supersonic, hypersonic and transonic small-disturbance limits, with additional intermediate limits required at distances from the corner that are small in comparison with the boundary-layer thickness. The solutions describe small perturbations in an inviscid rotational flow, and show good agreement with available experimental data in most cases where effects of separation can be neglected. Calculation of the wall shear stress requires solution of the boundary-layer momentum equation in a sublayer which plays the role of a new thinner boundary layer but which is still much thicker than the wall layer. An analytical solution is derived with a mixing-length approximation, and is in qualitative agreement with one set of measured values.

This paper examines the linear stability of antisymmetric disturbances in incompressible plane Poiseuille flow between identical flexible walls which undergo transverse displacements. Using a variational approach, an approximate solution of the problem is formulated in a form suitable for computational evaluation of the (complex) wave speeds of the system. A feature of this formulation is that the varying boundary conditions (and the Orr-Sommerfeld equation) are satisfied only in the mean; this reduces the labour involved in determining the approximate solution for a variety of wall conditions without increasing the difficulty of obtaining solutions to a given accuracy. In this paper the symmetric stream function distribution across the channel is represented by a series of cosines whose coefficients are determined by the variational solution. Comparisons with previous work, both for the flexible-wall and rigid-wall problems, show that the method gives results as accurate as those obtained previously by other methods while new results, for flexible walls, indicate the presence of a higher wave-number stability boundary which joins the distorted Tollmien-Schlichting stability boundary at lower wave-numbers. In some cases this upper unstable region, which is characterized by large amplification rates, may determine the critical Reynolds number of the system.

A standard model for the study of scalar dispersion through the combined effect of advection and molecular diffusion is a two-dimensional periodic flow with closed streamlines inside periodic cells. Over long time scales, the dispersion of a scalar released in this flow can be characterized by an effective diffusivity that is a factor $\mathit{Pe}^{1/2}$ larger than molecular diffusivity when the Péclet number $\mathit{Pe}$ is large. Here we provide a more complete description of dispersion in this regime by applying the large-deviation theory developed in Part 1 of this paper. Specifically, we derive approximations to the rate function governing the scalar concentration at large time $t$ by carrying out an asymptotic analysis of the relevant family of eigenvalue problems. We identify two asymptotic regimes and, for each, make predictions for the rate function and spatial structure of the scalar. Regime I applies to distances $|\boldsymbol {x}|$ from the scalar release point that satisfy $|\boldsymbol {x}|= O(\mathit{Pe}^{1/4} t)$. The concentration in this regime is isotropic at large scales, is uniform along streamlines within each cell, and varies rapidly in boundary layers surrounding the separatrices between adjacent cells. The results of homogenization theory, yielding the $O(\mathit{Pe}^{1/2})$ effective diffusivity, are recovered from our analysis in the limit $|\boldsymbol {x}|\ll \mathit{Pe}^{1/4} t$. Regime II applies when $|\boldsymbol {x}|=O(\mathit{Pe}\, t/{\rm log}\, \mathit{Pe})$ and is characterized by an anisotropic concentration distribution that is localized around the separatrices. A novel feature of this regime is the crucial role played by the dynamics near the hyperbolic stagnation points. A consequence is that in part of the regime the dispersion can be interpreted as resulting from a random walk on the lattice of stagnation points. The two regimes overlap so that our asymptotic results describe the scalar concentration over a large range of distances $|\boldsymbol {x}|$. They are verified against numerical solutions of the family of eigenvalue problems yielding the rate function.

This work is concerned with the source of an important component of nonlinear loading on a horizontal cylinder beneath waves that is not present in conventional diffraction calculations. Earlier measurements (Chaplin 1984b) have suggested that circulation induced by steady streaming around the cylinder may be responsible for loading which in some cases reduces the perceived inertia force by 50%. The present work is aimed at studying the steady streaming around a cylinder in general non-uniform orbital flow, and determining whether in the particular case of wave-induced flow it could be related quantitatively to the loading.

The steady outer flow has been obtained numerically for cases where the steady streaming does not have a reversal, and for cases where a weak reversal is compatible with a uniform outer circulation. It is found that the outer circulation is closely related to the mean streaming velocity around the cylinder at the outer edge of the shear-wave layer. Results for conditions corresponding to previous measurements of force on a horizontal cylinder beneath waves suggest that separation, turbulence, transient effects and organized three-dimensional instabilities should also be considered.

Large-scale coherent structures such as jets in Rayleigh–Bénard convection and related systems are receiving increasing attention. This paper studies, both numerically and theoretically, the process of jet formation in two-dimensional salt-finger convection. The approach utilizes an asymptotically derived system of equations referred to as the modified Rayleigh–Bénard convection (MRBC) model, valid in the geophysically and astrophysically relevant limit in which the solute diffuses much more slowly than heat. In these equations, convection is driven by a destabilizing salinity gradient while the effects of the stabilizing temperature gradient manifest themselves as an additional anisotropic dissipation acting on large scales. The MRBC system is specified by two external parameters: the Schmidt number $\mathit{Sc}$ (ratio of viscosity to solutal diffusivity) and the Rayleigh ratio $\mathit{Ra}$ (ratio between the Rayleigh numbers of the destabilizing solutal stratification and the stabilizing thermal stratification). Two distinct $\mathit{Ra}$ regimes are explored for fixed $\mathit{Sc}=1$. In all cases studied the system develops a horizontal jet structure that is maintained self-consistently by turbulent fluctuations, but coarsens over time. For intermediate Rayleigh ratios (e.g. $\mathit{Ra}=6$), the MRBC model captures the relaxation oscillations superposed on the jet structure observed at similar parameter values in direct numerical simulations of the primitive equations. For smaller Rayleigh ratios (e.g. $\mathit{Ra}=2$), a regime for which direct numerical simulation of the primitive equations is difficult because of the presence of fast gravity waves, the MRBC model reveals the existence of statistically steady jets whose properties are studied in detail. Three hierarchical models, the MRBC and further reductions in the form of quasilinear and single-mode approximations, are used to confirm that jets form and are sustained as a result of the interaction between fluctuations (salt fingers) and large-scale horizontally averaged horizontal flows (jets). Even though the small-scale structures exhibited by the three models exhibit clear differences, all three produce the same power-law spectrum of the mean fields at large vertical scales: in all, the spectrum of the mean streamfunction scales as $m^{-3}$ and the mean salinity field scales as $m^{-1}$, with $m$ the vertical wavenumber. A theoretical explanation of these observations based on the dominant balances in the mean and fluctuation equations is provided. As a consequence, the jets have a zigzag profile, a conclusion that is consistent with numerical simulations. Based on numerical observations, a three-component phenomenological model consisting of a linearly growing mode, a linearly damped mode and a mean mode is proposed to explain the observed transition from statistically steady jet structure to jets with superposed oscillations that takes place with increasing Rayleigh ratio.

The effect of weak compressibility on the speed of steadily translating staggered vortex streets of hollow vortices in isentropic subsonic flow is studied. A small-Mach-number perturbation expansion about the incompressible solutions for staggered streets of hollow vortices found recently by Crowdy & Green (Phys. Fluids, 2011, vol. 23, 126602) is carried out; the latter solutions provide a desingularization of the classical point vortex streets of von Kármán. The first-order compressible flow correction is calculated. We employ a novel scheme based on a complex variable formulation of the compressible flow equations (the Imai–Lamla method) combined with conformal mapping theory to track the vortex shape in this free boundary problem. The analysis to find the perturbed streamfunction and compressible vortex shapes is greatly facilitated by exploiting a calculus based on use of the Schottky–Klein prime function of a conformally equivalent parametric annulus. It is found that, for a vortex street of specified aspect ratio comprising vortices of specified circulation, the vortex core size is a key determinant of whether compressibility increases or decreases the steady propagation speed (relative to the incompressible street with the same parameters) and that both eventualities are possible. We focus attention on streets with aspect ratios around 0.28, which is close to the neutrally stable case for incompressible flow, and find that a critical vortex core size exists at which compressibility does not affect the speed of the street at first order in the (squared) Mach number. Streets comprising vortices with core size below the critical value speed up due to compressibility; larger vortices slow down.

A class of common and successful continuum models for steady, dense granular flows is based on the $\unicode[STIX]{x1D707}(I)$ model for viscoplastic grain-inertial rheology. Recent work has shown that under certain conditions, $\unicode[STIX]{x1D707}(I)$-based models display a linear instability in which short-wavelength perturbations grow at an unbounded rate – i.e. a Hadamard instability. This observation indicates that $\unicode[STIX]{x1D707}(I)$ models will predict strain localization arising due to material instability in dense granular materials; however, it also raises concerns regarding the robustness of numerical solutions obtained using these models. Several approaches to regularizing this instability have been suggested in the literature. Among these, it has been shown that the inclusion of higher-order velocity gradients into the constitutive equations can suppress the Hadamard instability, while not precluding the modelling of strain localization into diffuse shear bands. In our recent work (Henann & Kamrin, Proc. Natl Acad. Sci. USA, vol. 110, 2013, pp. 6730–6735), we have proposed a non-local model – called the non-local granular fluidity (NGF) model – which also involves higher-order flow gradients and has been shown to quantitatively describe a wide variety of steady, dense flows. In this work, we show that the NGF model also successfully regularizes the Hadamard instability of the $\unicode[STIX]{x1D707}(I)$ model. We further apply the NGF model to the problem of strain localization in quasi-static plane-strain compression using nonlinear finite-element simulations in order to demonstrate that the model is capable of describing diffuse strain localization in a mesh-independent manner. Finally, we consider the linear stability of an alternative gradient–viscoplastic model (Bouzid et al., Phys. Rev. Lett., vol. 111, 2013, 238301) and show that the inclusion of higher-order gradients does not guarantee the suppression of the Hadamard instability.

An active feedback flow control theory of the axisymmetric vortex breakdown process in incompressible swirling flows in a finite-length straight circular pipe is developed. Flow injection distributed along the pipe wall is used as the controller. The flow is subjected to non-periodic inlet and outlet conditions where the inlet profiles of the axial velocity, circumferential velocity and azimuthal vorticity are prescribed, along with no radial velocity at the outlet. A long-wave asymptotic analysis at near-critical swirl ratios, which involves a rescaling of the axial distance and time, results in a model problem for the dynamics and the nonlinear control of both inviscid and high-Reynolds-number ($\mathit{Re}$) flows. The approach provides the bifurcation diagram of steady states and the stability characteristics of these states. In addition, an energy analysis of the controlled flow dynamics suggests a feedback control law that relates the flow injection to the evolving maximum radial velocity at the inlet. Computed examples of the flow dynamics based on the full Euler and Navier–Stokes formulations at various swirl levels demonstrate the evolution to near-steady breakdown states when swirl is above a critical level that depends on $\mathit{Re}$. Moreover, applying the proposed feedback control law during flow evolution shows for the first time the successful and robust elimination of the breakdown states and flow stabilization on an almost columnar state for a wide range of swirl (up to at least 30 %) above critical. The feedback control cuts the natural feed-forward mechanism of the breakdown process. Specifically, in the case of high-$\mathit{Re}$ flows, the control approach establishes a branch of columnar states for all swirl levels studied, where in the natural flow dynamics no such states exist. The present theory is limited to the control of axisymmetric flows in pipes where the wall boundary layer is thin and attached and does not interact with the flow in the bulk.

A buoyant vortex ring produced by an underwater bursting balloon was studied experimentally. The effect of dimensionless surface tension on characteristics including rise velocity, rate of expansion, circulation, trajectory, and lifetime of the vortex ring bubble was investigated. Results showed reasonable agreement with the literature on vortex rings produced by conventional approaches. It was observed that as the dimensionless surface tension increased, the rise velocity, the circulation and consequently the stability of the vortex ring bubble increased; however, the rate of expansion tends toward constant values. A semi-analytical model is proposed by modifying the drag-based model presented by Sullivan et al. (J. Fluid Mech., vol. 609, 2008, pp. 319–347) to make it applicable to buoyant vortex rings. The modified model suggests that the vortex ring expansion is essentially due to the buoyancy force. An expression is also derived for the circulation in terms of the initial volume of the balloon and the depth at which the balloon bursts.

A discrepancy exists between the predictions of analytical solutions of approximate Navier–Stokes equations and numerical finite-difference solutions of the full Navier–Stokes equations regarding the development of laminar flow at the entrance to cylindrical pipes for Newtonian fluids. Starting from a uniform velocity profile at the entrance to the pipe, analytical solutions of approximate Navier–Stokes equations predict the velocity profile to have a maximum at the centre of the pipe at all times. In contrast, numerical finite-difference solutions of the full Navier–Stokes equations have suggested that the location of the velocity maximum moves from the wall towards the centre of the pipe at a short distance from the entrance, after which it remains at the centre of the pipe. This study presents the first experimental evidence of the moving velocity maximum from the wall towards the centre of the pipe. The initial uniform velocity profile was achieved by flowing the fluid through a monolith composed of narrow parallel channels and the flow development was investigated using magnetic resonance velocimetry. The experimentally observed variation of the position and size of the velocity maximum with the Reynolds number and the distance from the entrance to the pipe is shown to be in good agreement with the predictions of numerical finite-difference solutions of the full Navier–Stokes equations.

Data available in the literature from direct numerical simulations of two-dimensional turbulent channels by Lee & Moser (J. Fluid Mech., vol. 774, 2015, pp. 395–415), Bernardini et al. (J. Fluid Mech., 742, 2014, pp. 171–191), Yamamoto & Tsuji (Phys. Rev. Fluids, vol. 3, 2018, 012062) and Orlandi et al. (J. Fluid Mech., 770, 2015, pp. 424–441) in a large range of Reynolds number have been used to find that $S^{\ast }$ the ratio between the eddy turnover time ($q^{2}/\unicode[STIX]{x1D716}$, with $q^{2}$ being twice the turbulent kinetic energy and $\unicode[STIX]{x1D716}$ the isotropic rate of dissipation) and the time scale of the mean deformation ($1/S$), scales very well with the Reynolds number in the wall region. The good scaling is due to the eddy turnover time, although the turbulent kinetic energy and the rate of isotropic dissipation show a Reynolds dependence near the wall; $S^{\ast }$, as well as $-\langle Q\rangle =\langle s_{ij}s_{ji}\rangle -\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/2\rangle$ are linked to the flow structures, and also the latter quantity presents a good scaling near the wall. It has been found that the maximum of turbulent kinetic energy production $P_{k}$ occurs in the layer with $-\langle Q\rangle \approx 0$, that is, where the unstable sheet-like structures roll-up to become rods. The decomposition of $P_{k}$ in the contribution of elongational and compressive strain demonstrates that the two contributions present a good scaling. However, the good scaling holds when the wall and the outer structures are separated. The same statistics have been evaluated by direct simulations of turbulent flows in the presence of different types of corrugations on both walls. The flow physics in the layer near the plane of the crests is strongly linked to the shape of the surface and it has been demonstrated that the $u_{2}$ (normal to the wall) fluctuations are responsible for the modification of the flow structures, for the increase of the resistance and of the turbulent kinetic energy production.

We report on the modification of drag by neutrally buoyant spherical finite-sized particles in highly turbulent Taylor–Couette (TC) flow. These particles are used to disentangle the effects of size, deformability and volume fraction on the drag, and are contrasted to the drag in bubbly TC flow. From global torque measurements, we find that rigid spheres hardly decrease or increase the torque needed to drive the system. The size of the particles under investigation has a marginal effect on the drag, with smaller diameter particles showing only slightly lower drag. Increase of the particle volume fraction shows a net drag increase. However, this increase is much smaller than can be explained by the increase in apparent viscosity due to the particles. The increase in drag for increasing particle volume fraction is corroborated by performing laser Doppler anemometry, where we find that the turbulent velocity fluctuations also increase with increasing volume fraction. In contrast to rigid spheres, for bubbles, the effective drag reduction also increases with increasing Reynolds number. Bubbles are also much more effective in reducing the overall drag.