The Lagrangian L for gravity waves of finite amplitude in an N-layer, stratified shear flow is constructed as a functional of the generalized coordinates qv(t)≡ {qnv(t)} of the N + 1 interfaces, where the qnv are the Fourier coefficients in the expansion of the interfacial displacement ηv(x, t) in a complete, orthogonal set {ψn(x)}. The explicit expansion of L is constructed through fourth-order in the qnν and $\dot{q}_{n}^{\nu}$. Progressive interfacial waves and Kelvin–Helmholtz instability in a two-layer fluid are examined, and the earlier results of Drazin (1970), Nayfeh & Saric (1972) and Weissman (1979) are extended to finite depth. It is found that the pitchfork bifurcation associated with the critical point for Kelvin–Helmholtz instability, which is supercritical for infinitely deep layers, may be subcritical (inverted) for finite depths. The evolution equations that govern Kelvin–Helmholtz waves in the parametric neighbourhood of this critical point are shown to be equivalent to those for a particle in a two-parameter, central force field. The effect of surface tension is examined in an Appendix. Finally, the wave motion forced by flow over a sinusoidal bottom (as in Thorpe's tilting tank) is examined and the corresponding resonance curves and Hopf bifurcations determined. Numerical integrations reveal that stable limit cycles exist in some parametric neighbourhoods of these bifurcations. Period doubling was observed but did not lead to chaotic motion.

In fully developed single-phase turbulent flow in straight pipes, it is known that mean motions can occur in the plane of the pipe cross-section, when the cross-section is non-circular, or when the wall roughness is non-uniform around the circumference of a circular pipe. This phenomenon is known as secondary flow of the second kind and is associated with the anisotropy in the Reynolds stress tensor in the pipe cross-section. In this work, we show, using careful laser Doppler anemometry experiments, that secondary flow of the second kind can also be promoted by a non-uniform non-axisymmetric particle-forcing, in a fully developed turbulent flow in a smooth circular pipe. In order to isolate the particle-forcing from other phenomena, and to prevent the occurrence of mean particle-forcing in the pipe cross-section, which could promote a different type of secondary flow (secondary flow of the first kind), we consider a simplified well-defined situation: a non-uniform distribution of particles, kept at fixed positions in the ‘bottom’ part of the pipe, mimicking, in a way, the particle or droplet distribution in horizontal pipe flows. Our results show that the particles modify the turbulence through ‘direct’ effects (associated with the wake of the particles) and ‘indirect’ effects (associated with the global balance of momentum and the turbulence dynamics). The resulting anisotropy in the Reynolds stress tensor is shown to promote four secondary flow cells in the pipe cross-section. We show that the secondary flow is determined by the projection of the Reynolds stress tensor onto the pipe cross-section. In particular, we show that the direction of the secondary flow is dictated by the gradients of the normal Reynolds stresses in the pipe cross-section, and . Finally, a scaling law is proposed, showing that the particle-driven secondary flow scales with the root of the mean particle-forcing in the axial direction, allowing us to estimate the magnitude of the secondary flow.

Direct numerical simulations (DNS) are performed to analyse the global transport properties of turbulent Taylor–Couette flow with inner rough wall up to Taylor number$Ta=10^{10}$. The dimensionless torque $Nu_{\unicode[STIX]{x1D714}}$ shows an effective scaling of $Nu_{\unicode[STIX]{x1D714}}\propto Ta^{0.42\pm 0.01}$, which is steeper than the ultimate regime effective scaling $Nu_{\unicode[STIX]{x1D714}}\propto Ta^{0.38}$ seen for smooth inner and outer walls. It is found that at the inner rough wall, the dominant contribution to the torque comes from the pressure forces on the radial faces of the rough elements; while viscous shear stresses on the rough surfaces contribute little to $Nu_{\unicode[STIX]{x1D714}}$. Thus, the log layer close to the rough wall depends on the roughness length scale, rather than on the viscous length scale. We then separate the torque contributed from the smooth inner wall and the rough outer wall. It is found that the smooth wall torque scaling follows $Nu_{s}\propto Ta_{s}^{0.38\pm 0.01}$, in excellent agreement with the case where both walls are smooth. In contrast, the rough wall torque scaling follows $Nu_{r}\propto Ta_{r}^{0.47\pm 0.03}$, very close to the pure ultimate regime scaling $Nu_{\unicode[STIX]{x1D714}}\propto Ta^{1/2}$. The energy dissipation rate at the wall of an inner rough cylinder decreases significantly as a consequence of the wall shear stress reduction caused by the flow separation at the rough elements. On the other hand, the latter shed vortices in the bulk that are transported towards the outer cylinder and dissipated. Compared to the purely smooth case, the inner wall roughness renders the system more bulk dominated and thus increases the effective scaling exponent.

Fully developed turbulent free shear layers exhibit a high degree of order, characterized by large-scale coherent structures in the form of spanwise vortex rollers. Extensive experimental investigations show that such organized motions bear remarkable resemblance to instability waves, and their main characteristics, including the length scales, propagation speeds and transverse structures, are reasonably well predicted by linear stability analysis of the mean flow. In this paper, we present a mathematical theory to describe the nonlinear dynamics of coherent structures. The formulation is based on the triple decomposition of the instantaneous flow into a mean field, coherent fluctuations and small-scale turbulence but with the mean-flow distortion induced by nonlinear interactions of coherent fluctuations being treated as part of the organized motion. The system is closed by employing a gradient type of model for the time- and phase-averaged Reynolds stresses of fine-scale turbulence. In the high-Reynolds-number limit, the nonlinear non-equilibrium critical-layer theory for laminar-flow instabilities is adapted to turbulent shear layers by accounting for (1) the enhanced non-parallelism associated with fast spreading of the mean flow, and (2) the influence of small-scale turbulence on coherent structures. The combination of these factors with nonlinearity leads to an interesting evolution system, consisting of coupled amplitude and vorticity equations, in which non-parallelism contributes the so-called translating critical-layer effect. Numerical solutions of the evolution system capture vortex roll-up, which is the hallmark of a turbulent mixing layer, and the predicted amplitude development mimics the qualitative feature of oscillatory saturation that has been observed in a number of experiments. A fair degree of quantitative agreement is obtained with one set of experimental data.

For flow of a viscous fluid in a divergent channel of small angle it is shown that small disturbances to the basic Jeffery-Hamel flow may grow, according to nonlinear theory, to produce a secondary (supercritical) flow, in which the main flow winds from side to side in the channel and vortices form, with the whole pattern moving slowly downstream.

The structure of normal shock waves is considered when the ratio of molecular masses mp/m of a binary mixture of inert monatomic gases is large and the density ratio ρp/ρ is of order unity or below. Generalized hydrodynamic equations, valid for arbitrary intermolecular potentials, are obtained from a hypersonic closure of the kinetic equation for the heavy gas and a near-equilibrium closure for the light component. Because the Prandtl number of the light gas and the Schmidt number of the mixture are nearly constant, the only independent transport coefficient arising in the model is the viscosity μ of the light gas, which is absorbed into a new independent position variable s. Knowledge of μ as a function of temperature thus determines the shock structure independently from the details of the intermolecular potential, allowing comparison with experiments in the complete absence of free parameters. In terms of the ratio M (frozen Mach number) between the speed of propagation and the sound speed of the light gas in the unperturbed medium, one finds that: (i) When M > 1, the behaviour is similar to that of a ‘dusty gas’, with a broad relaxation layer (outer solution) following a sharp boundary layer through which the speed of the heavy gas is almost constant (a shock within a shock). (ii) When (1 + ρp/ρ)s−½ < M < 1, the boundary layer disappears, yielding a so-called ‘fully dispersed wave’. (iii) Because the internal energy of the heavy gas is negligible, the present problem differs from previous shock studies in that, for the first time, the structure of the relaxation region is obtained algebraically in phase space, thus permitting an exhaustive study of the behaviour. From it, the overshooting solution found by Sherman (1960) is related to the unphysical degenerate branch of the outer solution arising when M > 1, showing a failure of the Chapman–Enskog theory, even for weak shocks, when the heavy gas is dilute. Also, an algebraic explanation arises for the ‘double hump structure’ observed in He–Xe shocks. (iv) When M is nearly unity, the initial boundary layer spreads out, and the structure must be obtained by integration of a numerically unstable system of three differential equations. However, the reduction of order brought about by the weak variation of the light-gas entropy at the head of the shock, results in a stable system of equations that we integrate numerically. Excellent phase-space agreement with recent shock-tube experiments of Tarczynski, Herczynski & Walenta (1986) is found for both weak and strong shocks.

We are concerned with the response of an unconstrained particle, solid or liquid, placed in an acoustic field which consists of two orthogonal sound waves. These have the same amplitude and wavenumber, but differ in phase by $\pi/2$. The particle may be either a circular cylinder or a sphere. The effect of the superimposed waves is to create a time-averaged torque on the particle which causes it to rotate with uniform angular velocity. Throughout, a suitably defined Strouhal number is assumed to be large, with the solution developed in appropriate inverse powers of it. The particle size is assumed to be much smaller than the acoustic wavelength. At leading order it is shown that solid and liquid cylinders behave in a similar manner, in the sense that the liquid is in solid-body rotation. For a spherical liquid drop, the dominant time-averaged motion of it is also a solid-body rotation when the drop viscosity is large compared with that of the fluid environment; however, superposed on this is a time-averaged recirculatory flow within the droplet in the form of a pair of toroidal vortices.

Earth’s magnetic field is generated in its fluid metallic core through motional induction in a process termed the geodynamo. Fluid flow is heavily influenced by a combination of rapid rotation (Coriolis forces), Lorentz forces (from the interaction of electrical currents and magnetic fields) and buoyancy; it is believed that the inertial force and the viscous force are negligible. Direct approaches to this regime are far beyond the reach of modern high-performance computing power, hence an alternative ‘reduced’ approach may be beneficial. Taylor (Proc. R. Soc. Lond. A, vol. 274 (1357), 1963, pp. 274–283) studied an inertia-free and viscosity-free model as an asymptotic limit of such a rapidly rotating system. In this theoretical limit, the velocity and the magnetic field organize themselves in a special manner, such that the Lorentz torque acting on every geostrophic cylinder is zero, a property referred to as Taylor’s constraint. Moreover, the flow is instantaneously and uniquely determined by the buoyancy and the magnetic field. In order to find solutions to this mathematical system of equations in a full sphere, we use methods of optimal control to ensure that the required conditions on the geostrophic cylinders are satisfied at all times, through a conventional time-stepping procedure that implements the constraints at the end of each time step. A derivative-based approach is used to discover the correct geostrophic flow required so that the constraints are always satisfied. We report a new quantity, termed the Taylicity, that measures the adherence to Taylor’s constraint by analysing squared Lorentz torques, normalized by the squared energy in the magnetic field, over the entire core. Neglecting buoyancy, we solve the equations in a full sphere and seek axisymmetric solutions to the equations; we invoke $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects in order to sidestep Cowling’s anti-dynamo theorem so that the dynamo system possesses non-trivial solutions. Our methodology draws heavily on the use of fully spectral expansions for all divergenceless vector fields. We employ five special Galerkin polynomial bases in radius such that the boundary conditions are honoured by each member of the basis set, whilst satisfying an orthogonality relation defined in terms of energies. We demonstrate via numerous examples that there are stable solutions to the equations that possess a rapidly decreasing spectrum and are thus well-converged. Classic distributions for the $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects are invoked, as well as new distributions. One such new $\unicode[STIX]{x1D6FC}$-effect model possesses oscillatory solutions for the magnetic field, rarely before seen. By comparing our Taylor state model with one that allows torsional oscillations to develop and decay, we show the equilibrium state of both configurations to be coincident. In all our models, the geostrophic flow dominates the ageostrophic flow. Our work corroborates some results previously reported byWu & Roberts (Geophys. Astrophys. Fluid Dyn., vol. 109 (1), 2015, pp. 84–110), as well as presenting new results; it sets the stage for a three-dimensional implementation where the system is driven by, for example, thermal convection.

The flows considered here are those where a container or confined region is being filled by a substantial flow of liquid. The case of especial interest is where the incoming flow fills a large part of the cross-section of the container, for example where a nearly full flowing conduit has one end suddenly closed and hence fills rapidly, or where a water wave propagates close to the under surface of a horizontal structure and then rapidly fills the available space. These flows are taken to be so rapid that gravity is unimportant and yet not so violent that compressibility effects become significant. Important features, such as the greatly enhanced pressures and a thin high-velocity return jet are evaluated. The calculated pressures are very significantly greater than those associated with the incoming flow velocity and can be especially large when there is little clearance between the flow and the container boundary. One of many possible applications is in the extension of cracks and openings in coastal cliffs and structures. The flows could also be relevant to estimating the forces on the underside of some marine structures. A simple two-dimensional irrotational free-surface solution is found for the flow, which is steady in a suitably moving frame of reference.

Reversing the direction of one of these filling flows gives the case of a narrow high-speed jet which may be used to flush liquid out of cavities and containers. The optimum size of jet is calculated.

A laboratory investigation was performed to study distinctive features of the laminar-to-turbulent transition over distributed roughness characterized by two-dimensional (2D) and three-dimensional (3D) periodic, low-order topographies at roughness Reynolds number $Re_{k}\approx 300$. Systematic experiments were performed using high-spatial-resolution planar particle image velocimetry (PIV) in a refractive-index-matching (RIM) channel, where the roughness covered the entire length of the test section. The results show that the flow over the 2D roughness becomes turbulent much sooner than its 3D counterpart ($Re_{x}=50\,000$ versus 120 000). This is attributed to the presence of a velocity inflection point resulting from flow separation within the troughs of the 2D roughness. In the transitional region, unsteady disturbances above the two roughnesses appear upstream of near-roughness disturbances. The above-roughness disturbances are associated with the inflection point in the vertically displaced boundary layer for the 2D case, and with the mean velocity deficit resulting from the interaction of the wakes of upstream elements for the 3D case. The near-roughness fluctuations are associated with the shear layer present behind the crests of both roughnesses. The transitional region is characterized by the interaction between above- and near-roughness disturbances, which merge, leading to a rapid vertical growth of the turbulent fluctuations.

An initial localized eddy was generated in a rotating tank by a source–sink method to study the behaviour of an isolated barotropic eddy on a β-plane. The evolution of the eddy was compared with the laboratory experiments by Firing & Beardsley (1976) and by Takematsu & Kita (1985, 1988), confirming the northwestward (southwestward) translation of a cyclonic (anticyclonic) isolated eddy due to nonlinear effects. Anticyclonic eddies were contrasted with cyclonic eddies in the tank experiment, showing a cyclonic–anticyclonic asymmetry due to the topographic β as a substitute for the planetary β. The fluid experiment was simulated well by numerical simulation based on the quasi-geostrophic vorticity equation. Numerical experiments verified the northwestward (southwestward) translation both for an initially Gaussian and initially Rankine-type isolated cyclonic (anticyclonic) eddy.

A numerical solution for the flow field associated with a compressible pair of counter-rotating vortices is developed. The compressible, two-dimensional potential equation is solved utilizing the numerical method of Osher et al. (1985) for flow regions in which a non-zero density exists. Close to the vortex centres, vacuum ‘cores’ develop owing to the existence of a maximum achievable flow speed in a compressible flow field. A special treatment is required to represent these vacuum cores. Typical streamline patterns and core boundaries are obtained for upstream Mach numbers as high as 0.3, and the formation of weak shocks, predicted by Moore & Pullin (1987), is observed.

Macroscale description of streaming-potential phenomena in the thin-double-layer limit, and in particular the associated electro-viscous forces, has been a matter of long-standing controversy. In part 1 of this work (Yariv, Schnitzer & Frankel, J. Fluid Mech., vol. 685, 2011, pp. 306–334) we identified that the product of the Hartmann () and Péclet () numbers is , being the dimensionless Debye thickness. This scaling relationship defines a one-family class of limit processes appropriate to the consistent analysis of this singular problem. In that earlier contribution we focused on the generic problems associated with moderate and large , where the streaming-potential magnitude is comparable to the thermal voltage. Here we consider the companion generic limit of moderate Péclet numbers and large Hartmann numbers, deriving the appropriate macroscale model wherein the Debye-layer physics is represented by effective boundary conditions. Since the induced electric field is asymptotically smaller, calculation of these conditions requires higher asymptotic orders in analysing the Debye-scale transport. Nonetheless, the leading-order electro-viscous forces are of the same relative magnitude as those previously obtained in the large- limit. The structure of these forces is different, however, first because the small Maxwell stresses do not contribute at leading order, and second because salt polarization results in a dominant diffuso-osmotic slip. Since the salt distribution is governed by an advection–diffusion equation, this slip gives rise to electro-viscous forces which are nonlinear in the driving flow. The resulting scheme is illustrated by the calculation of the electro-viscous excess drag in the prototype problem of a translating sphere.

Most turbulent flows are characterized by coherent motion (CM), whose dynamics reflect the initial and boundary conditions of the flow and are more predictable than that of the random motion (RM). The major question we address here is the dynamical interaction between the CM and the RM, at a given scale, in a flow where the CM exhibits a strong periodicity and can therefore be readily distinguished from the RM. The question is relevant at any Reynolds number, but is of capital importance at finite Reynolds numbers, for which a clear separation between the largest and the smallest scales may not exist. Both analytical and experimental tools are used to address this issue. First, phase-averaged structure functions are defined and further used to condition the RM kinetic energy at a scale $r$ on the phase $\phi $ of the CM. This tool allows the dependence of the RM to be followed as a function of the CM dynamics. Scale-by-scale energy budget equations are established on the basis of phase-averaged structure functions. They reveal that energy transfer at a scale $r$ is sensitive to an additional forcing mechanism due to the CM. Second, these concepts are tested using hot-wire measurements in a cylinder wake, in which the CM is characterized by a well-defined periodicity. Because the interaction between large and small scales is most likely enhanced at moderate/low Reynolds numbers, and is also likely to depend on the amplitude of the CM, we choose to test our findings against experimental data at $R_{\lambda } \sim 10^2$ and for downstream distances in the range $10 \leq x/D \leq 40$. The effects of an increasing Reynolds number are also discussed. It is shown that: (i) a simple analytical expression describes the second-order structure functions of the purely CM. The energy of the CM is not associated with any single scale; instead, its energy is distributed over a range of scales. (ii) Close to the obstacle, the influence of the CM is perceptible even at the smallest scales, the energy of which is enhanced when the coherent strain is maximum. Further downstream from the cylinder, the CM clearly affects the largest scales, but the smallest scales are not likely to depend explicitly on the CM. (iii) The isotropic formulation of the RM energy budget compares favourably with experimental results.

A framework is developed to describe the two-point statistics of potential vorticity in rotating and stratified turbulence as described by the Boussinesq equations. The Kármán–Howarth equation for the dynamics of the two-point correlation function of potential vorticity reveals the possibility of inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and Reynolds number parameters. For the case of large Rossby and Froude numbers, and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3 prefactor is derived for the third-order mixed correlation between potential vorticity and velocity, a result that is analogous to the Kolmogorov 4/5-law for the third-order velocity structure function in turbulence theory.

The behaviour of clean and contaminated bubbles in solid-body rotating flows is compared in terms of drag and lift forces. Both spherical and deformed bubbles are considered. For that comparison, we have completed the data published in Rastello et al. (J. Fluid Mech., vol. 624, 2009, pp. 159–178; J. Fluid Mech., vol. 682, 2011, pp. 434–459) by a new series of measurements. When they are contaminated, bubbles are subject to an additional lift force due to the spinning of their surfaces, while the clean ones are not. A detailed description of this spinning motion is presented and an expression for the Magnus-like lift it induces is given in the light of the new information. The component of the lift induced by flow rotation depends on the Rossby number $Ro$, contrary to the case of clean bubbles. Including the ‘spin’ induced lift component in the dynamical equations provides a better prediction of the bubble’s trajectory in contaminated fluid. The presence of contaminants immobilizes the rear part of the bubble and reduces significantly the deformation. The laws of deformation according to the nature of the surface are presented. The way deformation influences the drag and lift coefficients in pure and contaminated fluids is quantified and discussed. Expressions for these various coefficients are proposed.

The micro-convection caused by the ponderomotive forces of the self-magnetic field in a magnetic fluid is studied here both numerically and experimentally. The theoretical approach based on the general Brinkman model substantially improves the description with respect to the previously proposed Darcy model. The predictions of both models are here compared to finely controlled experiments. The Brinkman model, in contrast to the Darcy model, allows us to describe the formation of mushrooms on the plumes of the micro-convective flow and the width of the fingers. In the Brinkman approach, excellent quantitative agreement is also obtained for the finger velocity dynamics and the velocity maximal values as a function of the magnetic Rayleigh number. The diffusion coefficient of particles of the water-based magnetic colloid determined by the threshold field strength value of the micro-convection is significantly larger than the diffusion coefficient of individual particles. This result is confirmed by independent measurements of the diffusion coefficient at the smearing of the diffusion front.

We experimentally investigated streamwise variations of turbulent dynamics in drag-reducing turbulent boundary layer flows following the injection of non-ionic surfactant solutions, which mainly consisted of oleyldimethylamine oxide. We focus on the comparison of turbulence statistics between injected (i.e. heterogeneous) and premixed (i.e. homogeneous) surfactant solutions, in which the maximum drag reduction ratio of 50 % is the same at the most downstream position for both cases. The wall-normal profiles of turbulence statistics, such as streamwise and wall-normal turbulence intensities, seem to be noticeably different between heterogeneous and homogeneous surfactant solutions. However, streamwise variations in these maxima and the wall-normal locations are essentially similar to one another, except for the maximum of streamwise turbulence intensity, which is not arranged by the amount of drag reduction and is also dependent on the normalization due to outer and inner variables. Such complex behaviour of streamwise turbulence intensity would be caused by the formation of near-wall layered structures that are parallel to the wall. For both heterogeneous and homogeneous surfactant solutions, the streamwise variation in the drag reduction ratio corresponds well to those of the mean velocity in wall units and the wall-normal locations of maxima of streamwise and wall-normal turbulence intensities with both outer and inner scaling. Unlike the Reynolds shear stress, the correlation coefficient of the streamwise and wall-normal turbulent fluctuations is correlated well with the drag reduction ratio. We present plausible pictures of the development of turbulence structures such as hairpin vortices and low-speed streaks for the drag-reducing turbulent boundary layer in heterogeneous and homogeneous surfactant solutions, which are comprehensively derived from the present set of experimental measurements such as flow visualization, planar laser-induced fluorescence, two-component laser-Doppler velocimetry and particle image velocimetry on the streamwise and wall-normal plane and the streamwise and spanwise plane.