In the present study, we investigate the scaling of relative velocity structure functions, of order two and higher, for inertial particles, both in the dissipation range and the inertial subrange using direct numerical simulations (DNS). Within the inertial subrange our findings show that contrary to the well-known attenuation in the tails of the one-point acceleration probability density function (p.d.f.) with increasing inertia (Bec et al., J. Fluid Mech., vol. 550, 2006, pp. 349–358), the opposite occurs with the velocity structure function at sufficiently large Stokes numbers. We observe reduced scaling exponents for the structure function when compared to those of the fluid, and correspondingly broader p.d.f.s, similar to what occurs with a passive scalar. DNS allows us to isolate the two effects of inertia, namely biased sampling of the velocity field, a result of preferential concentration, and filtering, i.e. the tendency for the inertial particle velocity to attenuate the velocity fluctuations in the fluid. By isolating these effects, we show that sampling is playing the dominant role for low-order moments of the structure function, whereas filtering accounts for most of the scaling behaviour observed with the higher-order structure functions in the inertial subrange. In the dissipation range, we see evidence of so-called ‘crossing trajectories’, the ‘sling effect’ or ‘caustics’, and find good agreement with the theory put forth by Wilkinson et al. (Phys. Rev. Lett., vol. 97, 2006, 048501) and Falkovich & Pumir (J. Atmos. Sci., vol. 64, 2007, 4497) for Stokes numbers greater than 0.5. We also look at the scaling exponents within the context of the model proposed by Bec et al. (J. Fluid Mech., vol. 646, 2010, pp. 527–536). Another interesting finding is that inertial particles at low Stokes numbers sample regions of higher kinetic energy than the fluid particle field, the converse occurring at high Stokes numbers. The trend at low Stokes numbers is predicted by the theory of Chun et al. (J. Fluid Mech., vol. 536, 2005, 219–251). This work is relevant to modelling the particle collision rate (Sundaram & Collins, J. Fluid Mech., vol. 335, 1997, pp. 75–109), and highlights the interesting array of phenomena induced by inertia.

Polymer additives can substantially reduce the drag of turbulent flows and the upper limit, the so-called state of ‘maximum drag reduction’ (MDR), is to a good approximation independent of the type of polymer and solvent used. Until recently, the consensus was that, in this limit, flows are in a marginal state where only a minimal level of turbulence activity persists. Observations in direct numerical simulations at low Reynolds numbers ($Re$) using minimal sized channels appeared to support this view and reported long ‘hibernation’ periods where turbulence is marginalized. In simulations of pipe flow at $Re$ near transition we find that, indeed, with increasing Weissenberg number ($Wi$), turbulence expresses long periods of hibernation if the domain size is small. However, with increasing pipe length, the temporal hibernation continuously alters to spatio-temporal intermittency and here the flow consists of turbulent puffs surrounded by laminar flow. Moreover, upon an increase in $Wi$, the flow fully relaminarizes, in agreement with recent experiments. At even larger $Wi$, a different instability is encountered causing a drag increase towards MDR. Our findings hence link earlier minimal flow unit simulations with recent experiments and confirm that the addition of polymers initially suppresses Newtonian turbulence and leads to a reverse transition. The MDR state on the other hand results at these low$Re$ from a separate instability and the underlying dynamics corresponds to the recently proposed state of elasto-inertial turbulence.

A set of integro-differential equations in the Lagrangian renormalized approximation (Kaneda 1981) is rederived by applying a perturbation method developed by Kraichnan (1959), which is based upon an extraction of direct interactions among Fourier modes of a velocity field and was applied to the Eulerian velocity correlation and response functions, to the Lagrangian ones for homogeneous isotropic turbulence. The resultant set of integro-differential equations for these functions has no adjustable free parameters. The shape of the energy spectrum function is determined numerically in the universal range for stationary turbulence, and in the whole wavenumber range in a similarly evolving form for the freely decaying case. The energy spectrum in the universal range takes the same shape in both cases, which also agrees excellently with many measurements of various kinds of real turbulence as well as numerical results obtained by Gotoh et al. (1988) for a decaying case as an initial value problem. The skewness factor of the longitudinal velocity derivative is calculated to be −0.66 for stationary turbulence. The wavenumber dependence of the eddy viscosity is also determined.

The stability of laminar axisymmetric jets and wakes, the two prominent examples of free shear layers, is investigated with respect to linear azimuthally periodic disturbances. The complete viscous disturbance equations are integrated numerically and the eigenvalues are obtained by matching the numerically advanced solutions to the asymptotic solutions at a large radius. Both spatial and temporal stability are examined for inviscid and viscid flows. It is found that the critical Reynolds number for the jet and the wake are not much different while the amplification rates for the wake become considerably greater than those for the jet as the Reynolds number increases. The axisymmetric shear-layer flows also seem to be more stable than the corresponding plane flows.

We report on experimental determinations of the temperature field in the interior (bulk) of turbulent Rayleigh–Bénard convection for a cylindrical sample with an aspect ratio (diameter $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}D$ over height $L$) equal to 0.50, in both the classical and the ultimate state. The measurements are for Rayleigh numbers $\mathit{Ra}$ from $6\times 10^{11}$ to $10^{13}$ in the classical and $7\times 10^{14}$ to $1.1\times 10^{15}$ (our maximum accessible $\mathit{Ra}$) in the ultimate state. The Prandtl number was close to 0.8. Although to lowest order the bulk is often assumed to be isothermal in the time average, we found a ‘logarithmic layer’ (as reported briefly by Ahlers et al., Phys. Rev. Lett., vol. 109, 2012, 114501) in which the reduced temperature $\varTheta = [\langle T(z) \rangle - T_m]/\Delta T$ (with $T_m$ the mean temperature, $\Delta T$ the applied temperature difference and $\langle {\cdots } \rangle $ a time average) varies as $A \ln (z/L) + B$ or $A^{\prime } \ln (1-z/L) + B^{\prime }$ with the distance $z$ from the bottom plate of the sample. In the classical state, the amplitudes $-A$ and $A^{\prime }$ are equal within our resolution, while in the ultimate state there is a small difference, with $-A/A^{\prime } \simeq 0.95$. For the classical state, the width of the log layer is approximately $0.1L$, the same near the top and the bottom plate as expected for a system with reflection symmetry about its horizontal midplane. For the ultimate state, the log-layer width is larger, extending through most of the sample, and slightly asymmetric about the midplane. Both amplitudes $A$ and $A^{\prime }$ vary with radial position $r$, and this variation can be described well by $A = A_0 [(R - r)/R]^{-0.65}$, where $R$ is the radius of the sample. In the classical state, these results are in good agreement with direct numerical simulations (DNS) for $\mathit{Ra} = 2\times 10^{12}$; in the ultimate state there are as yet no DNS. The amplitudes $-A$ and $A^{\prime }$ varied as ${\mathit{Ra}}^{-\eta }$, with $\eta \simeq 0.12$ in the classical and $\eta \simeq 0.18$ in the ultimate state. A close analogy between the temperature field in the classical state and the ‘law of the wall’ for the time-averaged downstream velocity in shear flow is discussed. A two-sublayer mean-field model of the temperature profile in the classical state was analysed and yielded a logarithmic $z$ dependence of $\varTheta $. The $\mathit{Ra}$ dependence of the amplitude $A$ given by the model corresponds to an exponent $\eta _{th} = 0.106$, in good agreement with the experiment. In the ultimate state the experimental result $\eta \simeq 0.18$ differs from the prediction $\eta _{th} \simeq 0.043$ by Grossmann & Lohse (Phys. Fluids, vol. 24, 2012, 125103).

Two-dimensional convection in a Boussinesq fluid with infinite Prandtl number, confined between rigid horizontal boundaries and stress-free lateral boundaries, has been investigated in a series of numerical experiments. In a layer heated from below steady convection becomes unstable to oscillatory modes caused by the formation of hot or cold blobs in thermal boundary layers. Convection driven by internal heating shows a transition from steady motion through periodic oscillations to a chaotic regime, owing to the formation of cold blobs which plunge downwards and eventually split the roll. The interesting feature of this idealized problem is the interaction between constraints imposed by nonlinear dynamics and the obvious spatial structures associated with the sinking sheets and changes in the preferred cell size. These spatial structures modify the bifurcation patterns that are familiar from transitions to chaos in low-order systems. On the other hand, even large-amplitude disturbances are constrained to show periodic or quasi-periodic behaviour, and the bifurcation sequences can be followed in considerable detail. There are examples of quasi-periodic behaviour followed by intermittency, of period-doubling cascades and of transitions from quasi-periodicity to chaos, associated with a preference for narrower rolls as the Rayleigh number is increased.

We investigate experimentally the transition to turbulence of a yield stress shear-thinning fluid in Hagen–Poiseuille flow. By combining direct high-speed imaging of the flow structures with Laser Doppler Velocimetry (LDV), we provide a systematic description of the different flow regimes from laminar to fully turbulent. Each flow regime is characterized by measurements of the radial velocity, velocity fluctuations and turbulence intensity profiles. In addition we estimate the autocorrelation, the probability distribution and the structure functions in an attempt to further characterize transition. For all cases tested, our results indicate that transition occurs only when the Reynolds stresses of the flow equal or exceed the yield stress of the fluid, i.e. the plug is broken before transition commences. Once in transition and when turbulent, the behaviour of the yield stress fluid is somewhat similar to a (simpler) shear-thinning fluid. Finally, we have observed the shape of slugs during transition and found their leading edges to be highly elongated and located off the central axis of the pipe, for the non-Newtonian fluids examined.

A statically stable stratified free shear layer was formed within the test section of a wind tunnel by merging two uniform streams of air after uniformly heating the top stream. The two streams were accelerated side by side in a contraction section. The resulting sheared thermocline thickened gradually as a result of molecular diffusion and was characterized by nearly self-similar temperature (odd), velocity (odd) and Richardson number (even) profiles. The minimum Richardson number J0 could be adjusted over the range 0·07 ≥ J0 ≥ 0·76; the Reynolds number Re varied between 30 and 70. Small periodic disturbances were introduced upstream of the test section by a fine wire oscillating in the thermocline. The wire generated a narrow horizontal beam of internal waves, which propagated downstream and remained confined within the thermocline. The growth or decay of these waves was observed in the test section. The results confirm the existence of a critical Richardson number the value of which is in plausible agreement with theoretical predictions (J0 ≅ 0·22 for the Reynolds number of the experiment). The growth rate is a function of the wavenumber and is somewhat different from that computed for the same Reynolds and Richardson numbers, but the calculation assumed velocity and density profiles which were also somewhat different.

This work brings together experimental and theoretical studies of nonlinear stages aimed at the K-regime in boundary-layer transition, and some combined theoretical and experimental results are discussed. It is shown that the initial stages in the formation of so-called spikes, observed in many experiments, may be described very well by the asymptotic theory. These flashes-spikes are shown to be (in certain regimes) possible solitons of the boundary layer and governed by the integral-differential Benjamin-Ono equation. Properties of the spike-solitons, obtained both theoretically and experimentally in the quasi-planar stages of their development, are presented. Features of the disturbance behaviour connected with the subsequent development of three-dimensionality are also discussed, as are the effects of viscosity and shorter lengthscales. The main conclusion of the work concerns the hypothesis of the possible soliton nature of the flashes-spikes (within limits), which seems reliably corroborated by the good agreement found between the theory and the experimental data.

Two-dimensional free surface flows without waves, produced by a submerged sink in a reservoir, are computed numerically for various configurations. For a sink above the horizontal bottom of a layer of fluid, there are solutions for all values of the Froude number F greater than some particular value. However, when the fluid is sufficiently deep, there is an additional solution for one special value of F < 1. The results for a sink at the vertex of a sloping bottom, treated by Craya and by Hocking, and for a sink in fluid of infinite depth, treated by Tuck & Vanden-Broeck, are confirmed and extended. In particular it is shown that as the bottom tends to the horizontal, the solution for a sink at the vertex of a sloping bottom approaches a solution for a horizontal bottom with F = 1. However solutions are found for all values of the Froude number F [ges ] 1 for a sink on a horizontal bottom.

We examine the linear stability of elliptical columns of uniform potential vorticity subject to two-dimensional (horizontal) straining within a rapidly rotating, stratified (quasi-geostrophic) fluid. We find that horizontal straining can promote the exponential growth of three-dimensional disturbances when the vortex height-to-width aspect ratio exceeds, qualitatively, three times the ratio of the Coriolis parameter to the buoyancy frequency. This instability is not related to the usual baroclinic instability which operates on shallow vortex columns whose potential vorticity changes sign with height. The nonlinear development of these instabilities is investigated numerically using a high-resolution contour surgery algorithm. Simulations are conducted for both a Boussinesq (ocean-like) fluid and a compressible (atmospheric-like) fluid having exponentially decreasing density with height. The simulations reveal a generic nonlinear development that results in a semi-ellipsoidal baroclinic vortex dome at the lower surface and, in the case of a Boussinesq fluid, another such dome at the upper surface.

The related problem of two interacting vortex columns is also examined. A generic three-dimensional instability and nonlinear development occurs no matter how great the distance between the vortex columns, provided that they are sufficiently tall.

Our results may bear upon the observed structure of many atmospheric and oceanic vortices, whose height-to-width aspect ratios are consistent with our findings. Remarkably, even strongly ageostrophic vortices, such as tropical cyclones, fit the pattern. Our results furthermore re-open questions about the long-time nature of freely decaying quasi-geostrophic turbulence, for which recent simulations indicate a progressive two-dimensionalization by vortex alignment, while earlier simulations have indicated long-lived baroclinic vortices, not unlike what we find here.

Planar motion produced when a viscous fluid is forced from an initial state of rest is studied. We consider a vortex dipole produced by the action of a point force (Cantwell 1986), and a vortex quadrupole produced by the action of two equal forces of opposite direction. We also present results from an experimental investigation into the dynamics of the interactions between vortex dipoles as well as between vortex dipoles and a vertical wall in a stratified fluid. Theoretical consideration reveals that the dynamics of two-dimensional vortex-dipole interactions are determined by two main governing parameters: the dipolar intensity of the vorticity distribution (momentum) and the quadrupolar intensity of the vorticity distribution of the flow. We document details of different basic types of interactions and present a physical interpretation of the results obtained in terms of vortex multipoles: dipoles, quadrupoles and their combinations.

Careful analysis of new field observations on the geometry of meandering tidal channels reveals that the values of meander wavelengths as well as curvatures, conveniently scaled by local channel width, fall within a fairly restricted range, suggesting the existence of some mechanistic process controlling meander formation. A three-dimensional model is then proposed to predict flow and bed topography in weakly meandering tidal channels. The analysis is developed for meander wavelengths much smaller than the tidal wavelength and assuming that sediment is transported both as bedload and as suspended load. Both conditions are typically met in nature. Due to the symmetry of the meander pattern investigated, the theory indicates that oscillations associated with the basic flow give rise to symmetric oscillations of the point bar–pool pattern around the locations of maximum curvature. However, no net migration in a tidal cycle is present, at least for periodic tides with zero mean. Suspended load leads both to an enhanced bottom deformation and to a downstream shifting of the position of the point bar. The model then provides the basis of a planimetric instability theory of the type developed for river meanders (Blondeaux & Seminara 1985). Though the available data do not yet allow a detailed quantitative comparison, it is shown that the wavelengths selected by the ‘bend mechanism’ are somewhat larger than those typically encountered in nature. The geomorphology of the process of meander formation in tidal environments is then discussed and, upon comparison with observational evidence, points out the need for various developments of the present model.

Magnetorotational dynamo action in Keplerian shear flow is a three-dimensional nonlinear magnetohydrodynamic process, the study of which is relevant to the understanding of accretion processes and magnetic field generation in astrophysics. Transition to this form of dynamo action is subcritical and shares many characteristics with transition to turbulence in non-rotating hydrodynamic shear flows. This suggests that these different fluid systems become active through similar generic bifurcation mechanisms, which in both cases have eluded detailed understanding so far. In this paper, we build on recent work on the two problems to investigate numerically the bifurcation mechanisms at work in the incompressible Keplerian magnetorotational dynamo problem in the shearing box framework. Using numerical techniques imported from dynamical systems research, we show that the onset of chaotic dynamo action at magnetic Prandtl numbers larger than unity is primarily associated with global homoclinic and heteroclinic bifurcations of nonlinear magnetorotational dynamo cycles born out of saddle-node bifurcations. These global bifurcations are found to be supplemented by local bifurcations of cycles marking the beginning of period-doubling cascades. The results suggest that nonlinear magnetorotational dynamo cycles provide the pathway to injection of both kinetic and magnetic energy for the problem of transition to turbulence and dynamo action in incompressible magnetohydrodynamic Keplerian shear flow in the absence of an externally imposed magnetic field. Studying the nonlinear physics and bifurcations of these cycles in different regimes and configurations may subsequently help to understand better the physical conditions of excitation of magnetohydrodynamic turbulence and instability-driven dynamos in a variety of astrophysical systems and laboratory experiments. The detailed characterization of global bifurcations provided for this three-dimensional subcritical fluid dynamics problem may also prove useful for the problem of transition to turbulence in hydrodynamic shear flows.

The goal of this paper is to derive expressions for the pressure forces and moments acting on an elongated body swimming in a quiescent fluid. The body is modelled as an inextensible and unshearable (Kirchhoff) beam, whose cross-sections are elliptic, undergoing prescribed deformations, consisting of yaw and pitch bending. The surrounding fluid is assumed to be inviscid, and irrotational everywhere, except in a thin vortical wake. The Laplace equation and the corresponding Neumann boundary conditions are first written in terms of the body coordinates of a beam treating the body as a fixed surface. They are then simplified according to the slenderness of the body and its kinematics. Because the equations are linear, the velocity potential is sought as a sum of two terms which are linked respectively to the axial movements of the beam and to its lateral movements. The lateral component of the velocity potential is decomposed further into two sub-components, in order to exhibit explicitly the role of the two-dimensional potential flow produced by the lateral motion of the cross-section, and the role played by the curvature effects of the beam on the cross-sectional flow. The pressure, which is given by Bernoulli's equation, is integrated along the body surface, and the expressions for the resultant and the moment are derived analytically. Thereafter, the validity of the force and moment obtained analytically is checked by comparisons with Navier–Stokes simulations (using Reynolds-averaged Navier–Stokes equations), and relatively good agreements are observed.

We study the case of inertial particles heated by thermal radiation while settling by gravity through a turbulent transparent gas. We consider dilute and optically thin regimes in which each particle receives the same heat flux. Numerical simulations of forced homogeneous turbulence are performed taking into account the two-way coupling of both momentum and temperature between the dispersed and continuous phases. Particles much smaller than the smallest flow scales are considered and the point-particle approximation is adopted. The particle Stokes number (based on the Kolmogorov time scale) is of order unity, while the nominal settling velocity is up to an order of magnitude larger than the Kolmogorov velocity, marking a critical difference with previous two-way coupled simulations. It is found that non-heated particles enhance turbulence when their settling velocity is sufficiently high compared to the Kolmogorov velocity. Energy spectra show that the non-heated particle settling impacts both the very small and very large flow scales, while the intermediate scales are weakly affected. When heated, particles shed plumes of buoyant gas, further modifying the turbulence structure. At the considered radiation intensities, clustering is strong but the classic mechanism of preferential concentration is modified, while preferential sweeping is eliminated or even reversed. Particle heating also causes a significant reduction of the mean settling velocity, which is caused by rising buoyant plumes in the vicinity of particle clusters. The turbulent kinetic energy is affected non-monotonically as the radiation intensity is increased due to the competing effects of the downward gravitational force and the upward buoyancy force. The thermal radiation influences all scales of the turbulence. The effects of settling and buoyancy on the turbulence anisotropy are also discussed.

A theory is developed for turbulence in a stably stratified fluid, for example in the experiments of Rouse & Dodu and of Turner where there is no shear and the turbulence is induced by a source of energy near the lower boundary of the fluid. A growing mixed layer of thickness D appears in the lower portion of the fluid and is separated from the non-turbulent fluid above, in which the buoyancy gradient is given, by an interfacial layer of thickness h. The lower mixed layer has a very weak buoyancy gradient and the large buoyancy difference across the interfacial layer is Δb.

As indicated by the experiments of Thompson & Turner and Hopfinger & Toly, and derived by the author in a recent paper, if u is the root-mean-square horizontal velocity and l is the integral length scale, the eddy viscosity ul is a constant in a homogeneous fluid agitated by a grid. When there is stratification, the theory indicates that the fluid motion is unaffected by buoyancy forces in the mixed layer, so that ul should again be constant in the lower portions of the mixed layer. Since l is proportional to distance, we may conveniently suppose that the source of the disturbances is at a level z = 0 where u is infinite in accordance with uz = K. Thus we may take K to be a fundamental parameter characterizing the turbulent energy source. Then z is distance above the plane of the virtual energy source. If the non-turbulent fluid has uniform buoyancy, DΔb = U2 may be shown to be constant. In general, whether constant or not, U may be taken to be a fundamental parameter expressing the stability. The quantity $\hat{R}i = U^{2}D^{2}/K^2$ is the most fundamental of the several Richardson numbers that have been introduced in this problem because, with its use, ‘constants’ of proportionality do not depend on the molecular coefficients of viscosity or diffusion (for high Reynolds number turbulence) or on the geometry of the grid.

The theory contains a number of results:

1. $u_e D/K \sim \hat{R}i^{-\frac{7}{4}},$ where $u_e = dD/dt$ is the entrainment velocity. Integration yields $D \infty t^{\frac{2}{11}}$ for a homogeneous upper fluid and $D \infty t^{\frac{2}{9}}$ for a linear upper density field. This $-\frac{7}{4}$ entrainment law compares with a $-\frac{3}{2}$ law suggested by several experimenters.

2. Turbulence in the interfacial layer is intermittent with intermittency factor $I_3 \sim \hat{R}i^{-\frac{3}{4}}$. The turbulent patches have dimension $\sigma_3 \sim D\hat{R}i^{-\frac{3}{4}}$.

3. If the (equal) root-mean-square velocities in an infinite homogeneous fluid at a distance D from the grid are denoted by u1 ∼ v1 ∼ w1, we find that the r.m.s. velocities near the interface are u2 ∼ u1, v2 ∼ u1 and $w_2 \sim u_1 \hat{R}i^{-\frac{1}{4}}$.

4. The buoyancy flux q2 near the interface may be expressed as $q_2 \sim w^3_2/D$.

5. h ∼ D, as observed in several experiments.

The form taken by a finite amplitude internal seiche in a shallow two-layer fluid within a long narrow container is akin to a shock wave in a gas-filled tube; an undular bore or surge is present which reflects back and forth along the length of the container. The internal surge in Loch Ness is sometimes observed to retain its amplitude over several seiche periods, and a resonance with the wind appears possible. This idea is explored by developing a theoretical model and by making a laboratory experiment, but difficulties are encountered in estimating the size of the parameters which characterize the natural phenomenon.

We introduce a new numerical method for the study of scalar mixing in two-dimensional advection fields. The position of an advected material strip is computed kinematically, and the associated convection–diffusion problem is solved using the computed local stretching rate along the strip, assuming that the diffusing strip thickness is smaller than its local radius of curvature. This widely legitimate assumption reduces the numerical problem to the computation of a single variable along the strip, thus making the method extremely fast and applicable to any large Péclet number. The method is then used to document the mixing properties of a chaotic sine flow, for which we relate the global quantities (spectra, concentration probability distribution functions (PDFs), increments) to the distributed stretching of the strip convoluted by the flow, possibly overlapping with itself. The numerical results indicate that the PDF of the strip elongation is log normal, a signature of random multiplicative processes. This property leads to exact analytical predictions for the spectrum of the field and for the PDF of the scalar concentration of a solitary strip. The present simulations offer a unique way of discovering the interaction rule for building complex mixtures which are made of a random superposition of overlapping strips leading to concentration PDFs stable by self-convolution.