The zero-pressure-gradient turbulent boundary layer over a flat plate was perturbed by a temporally oscillating, spatial impulse of roughness, and the downstream response of the flow field was interrogated by hot-wire anemometry and particle-image velocimetry. The key features common to impulsively perturbed boundary layers, as identified in Jacobi & McKeon (J. Fluid Mech., 2011), were investigated, and the unique contributions of the dynamic perturbation were isolated by contrast with an appropriately matched static impulse of roughness. In addition, the dynamic perturbation was decomposed into separable large-scale and small-scale structural effects, which in turn were associated with the organized wave and roughness impulse aspects of the perturbation. A phase-locked velocity decomposition of the entire downstream flow field revealed strongly coherent modes of fluctuating velocity, with distinct mode shapes for the streamwise and wall-normal velocity components. Following the analysis of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), the roughness perturbation was treated as a forcing of the Navier–Stokes equation and a linearized analysis employing a modified Orr–Sommerfeld operator was performed. The experimentally ascertained wavespeed of the input disturbance was used to solve for the most amplified singular mode of the Orr–Sommerfeld resolvent. These calculated modes were then compared with the streamwise and wall-normal velocity fluctuations. The discrepancies between the calculated Orr–Sommerfeld resolvent modes and those experimentally observed by phase-locked averaging of the velocity field were postulated to result from the violation of the parallel flow assumption of Orr–Sommerfeld analysis, as well as certain non-equilibrium effects of the roughness. Additionally, some difficulties previously observed using a quasi-laminar eigenmode analysis were also observed under the resolvent approach; however, the resolvent analysis was shown to provide reasonably accurate predictions of velocity fluctuations for the forced Orr–Sommerfeld problem over a portion of the boundary layer, with potential applications to designing efficient flow control strategies. The combined experimental and analytical effort provides a new opportunity to examine the non-equilibrium and forcing effects in a dynamically perturbed flow.

This experimental study investigates the fundamental flow field of a spatially oscillating jet emitted by a fluidic oscillator into an attached cross-flow. Dominant flow structures, such as the jet trajectory and dynamics of streamwise vortices, are discussed in detail with the aim of understanding the interaction between the spatially oscillating jet and the cross-flow. The oscillating jet is ejected perpendicular to the cross-flow. A moveable stereoscopic particle image velocimetry (PIV) system is employed for the plane-by-plane acquisition of the flow field. The three-dimensional, time-resolved flow field is obtained by phase averaging the PIV results based on a pressure signal from inside the fluidic oscillator. The influence of velocity ratio and Strouhal number is assessed. Compared to a common steady wall-normal jet, the spatially oscillating jet penetrates to a lesser extent into the cross-flow’s wall-normal direction in favour of a considerable spanwise penetration. The flow field is dominated by streamwise-oriented vortices, which are convected downstream at the speed of the cross-flow. The vortex dynamics exhibits a strong dependence on the Strouhal number. For small Strouhal numbers, the spatially oscillating jet acts similar to a vortex-generating jet with a time-dependent deflection angle. Accordingly, it forms time-dependent streamwise vortices. For higher Strouhal numbers, the cross-flow is not able to follow the motion of the jet, which results in a quasi-steady wake that forms downstream of the jet. The results suggest that the flow field approaches a quasi-steady behaviour when further increasing the Strouhal number.

A geostrophic, hydrostatic, frontal or filamentary flow adjusts conservatively to accommodate a surface gravity wave field with wave-averaged, Stokes-drift vortex and Coriolis forces in an altered balanced state. In this altered state, the wave-balanced perturbations have an opposite cross-front symmetry to the original geostrophic state; e.g. the along-front flow perturbation is odd-symmetric about the frontal centre while the geostrophic flow is even-symmetric. The adjustment tends to make the flow scale closer to the deformation radius, and it induces a cross-front shape displacement in the opposite direction to the overturning effects of wave-aligned down-front and up-front winds. The ageostrophic, non-hydrostatic, adjusted flow may differ from the initial flow substantially, with velocity and buoyancy perturbations that extend over a larger and deeper region than the initial front and Stokes drift. The largest effect occurs for fronts that are wider than the mixed layer deformation radius and that fill about two-thirds of a well-mixed surface layer, with the Stokes drift spanning only the shallowest part of the mixed layer. For even deeper mixed layers, and especially for thinner or absent mixed layers, the wave-balanced adjustments are not as large.

Burgers’ equation, a one-dimensional analogue of the Navier–Stokes equation, has been solved numerically in full detail at high (equivalent) Reynolds numbers. These fine-mesh solutions have been used to study the dynamics of the Burgers’ equation analogue of three-dimensional turbulence and in particular the drain of energy from the large to the small structures.

The equation has also been solved on a coarse mesh, using various forms of subgrid model. The solutions so derived have been compared with filtered solutions of the same problem on a fine mesh. In this way it has been possible to test directly the performance of subgrid models at high Reynolds numbers, a test which cannot be made on the Navier–Stokes equations themselves.

With proper choice of the parameters, the performance of the subgrid models is very satisfactory.

This study builds on and confirms the theoretical findings of Part 1 of this paper (Moarref & Jovanović, J. Fluid Mech., 2010, doi:10.1017/S0022112010003393). We use direct numerical simulation of the Navier–Stokes equations to assess the efficacy of blowing and suction in the form of streamwise travelling waves for controlling the onset of turbulence in a channel flow. We highlight the effects of the modified base flow on the dynamics of velocity fluctuations and net power balance. Our simulations verify the theoretical predictions of Part 1 that the upstream travelling waves promote turbulence even when the uncontrolled flow stays laminar. On the other hand, the downstream travelling waves with parameters selected in Part 1 are capable of reducing the fluctuations' kinetic energy, thereby maintaining the laminar flow. In flows driven by a fixed pressure gradient, a positive net efficiency as large as 25 % relative to the uncontrolled turbulent flow can be achieved with downstream waves. Furthermore, we show that these waves can also relaminarize fully developed turbulent flows at low Reynolds numbers. We conclude that the theory developed in Part 1 for the linearized flow equations with uncertainty has considerable ability to predict full-scale phenomena.

The canonical configuration of a solid particle bouncing on a wall in a viscous fluid is considered here, focusing on rough particles as encountered in most of the laboratory experiments or applications. In that case, the particle deformation is not expected to be significant prior to solid contact. An immersed boundary method (IBM) allowing the fluid flow around the solid particle to be numerically described is combined with a discrete element method (DEM) in order to numerically investigate the dynamics of the system. Particular attention is paid to modelling the lubrication force added in the discrete element method, which is not captured by the fluid solver at very small scale. Specifically, the proposed numerical model accounts for the surface roughness of real particles through an effective roughness length in the contact model, and considers that the time scale of the contact is small compared to that of the fluid. The present coupled method is shown to quantitatively reproduce available experimental data and in particular is in very good agreement with recent measurement of the dynamics of a particle approaching very close to a wall in the viscous regime $St \le {O}(10)$, where $St$ is the Stokes number which represents the balance between particle inertia and viscous dissipation. Finally, based on the reliability of the numerical results, two predictive models are proposed, namely for the dynamics of the particle close to the wall and the effective coefficient of restitution. Both models use the effective roughness height and assume the particle remains rigid prior to solid contact. They are shown to be pertinent to describe experimental and numerical data for the whole range of investigated parameters.

The function of a healthy ureter is analyzed in terms of a fluid-mechanical model. To the extent that the Reynolds number is of the order of one, the fundamental equations are shown to reduce to those of the theory of lubrication. It is found that from the point of view of the pressure variation with time (the urometrogram) the important part of the peristaltic wave is the constricting part. For this reason this part of the wave is represented with an algebraic expression of the form h ∼ xn making it possible to find closed form solutions. Using Fourier analysis in defining the complete wave shape of the ureter it was also possible to obtain numerical solutions. For both cases it is shown that there is good agreement between the theoretical and experimental pressure distributions, this not being the case for sinusoidal wave shapes. An approximate equation for the flux is developed and a universal relation is presented connecting the maximum pressure, flux and kinematic behaviour of the ureter.

The low-dimensional dynamics of the structures in a turbulent wall flow are studied by means of numerical simulations. These are made both ‘minimal’, in the sense that they contain a single copy of each relevant structure, and ‘autonomous’ in the sense that there is no outer turbulent flow with which they can interact. The interaction is prevented by a numerical mask that damps the flow above a given wall distance, and the flow behaviour is studied as a function of the mask height. The simplest case found is a streamwise wave that propagates without change. It takes the form of a single wavy low-velocity streak flanked by two counter-rotating staggered quasi-streamwise vortices, and is found when the height of the numerical masking function is less than δ+1 ≈ 50. As the mask height is increased, this solution bifurcates into an almost-perfect limit cycle, a two-frequency torus, weak chaos, and full-edged bursting turbulence. The transition is essentially complete when δ+1 ≈ 70, even if the wall-parallel dimensions of the computational box are small enough for bursting turbulence to be metastable, lasting only for a few bursting cycles. Similar low-dimensional dynamics are found in somewhat larger boxes, containing two copies of the basic structures, in which the bursting turbulence is self-sustaining.

Combined measurements of velocity components and temperature in a turbulent Rayleigh–Bénard convection flow at a low Prandtl number of $Pr=0.029$ and Rayleigh numbers of $10^{6}\leqslant Ra\leqslant 6\times 10^{7}$ are conducted in a series of experiments with durations of more than a thousand free-fall time units. Multiple crossing ultrasound beam lines and an array of thermocouples at mid-height allow for a detailed analysis and characterization of the complex three-dimensional dynamics of the single large-scale circulation roll in a cylindrical convection cell of unit aspect ratio which is filled with the liquid metal alloy GaInSn. We measure the internal temporal correlations of the complex large-scale flow and distinguish between short-term oscillations associated with a sloshing motion in the mid-plane as well as varying orientation angles of the velocity close to the top/bottom plates and the slow azimuthal drift of the mean orientation of the roll as a whole that proceeds on a time scale up to a hundred times slower. The coherent large-scale circulation drives a vigorous turbulence in the whole cell that is quantified by direct Reynolds number measurements at different locations in the cell. The velocity increment statistics in the bulk of the cell displays characteristic properties of intermittent small-scale fluid turbulence. We also show that the impact of the symmetry-breaking large-scale flow persists to small-scale velocity fluctuations thus preventing the establishment of fully isotropic turbulence in the cell centre. Reynolds number amplitudes depend sensitively on beam-line position in the cell such that different definitions have to be compared. The global momentum and heat transfer scalings with Rayleigh number are found to agree with those of direct numerical simulations and other laboratory experiments.

We consider problems based on linear water wave theory concerning the interaction of oblique waves with horizontal cylinders in a fluid consisting of a layer of finite depth bounded above by a free surface and below by an infinite layer of fluid of greater density. For such a situation time-harmonic waves can propagate with two different wavenumbers K and k. The particular problems of wave scattering by a horizontal circular cylinder in either the upper or lower layer are solved using multipole expansions.

Numerical simulations of Rayleigh–Bénard convection in a fluid layer heated from below between two rigid horizontal boundaries have been performed for Rayleigh numbers Ra up to 107, Prandtl numbers in the range between 0.7 and 60 and for aspect ratios $\Gamma$ up to 20. Periodic boundary conditions in the horizontal plane have been used. To a considerable extent, the evolution towards turbulent convection at high values of Ra is governed by processes exhibited by instabilities of steady or time periodic forms of convection at lower Rayleigh numbers. With increasing Ra, the properties of convection are increasingly determined by the thermal boundary layers. The role of mean flows which may be symmetric or antisymmetric with respect to the mid-plane of the layer is emphasized.

By applying the method of initial values to the theory of stability of shear flows, Case has recently found certain results which are in apparent conflict with those obtained by the theory of normal modes. It is shown how these differences may be reconciled. Some new features in the theory of normal modes are also brought out. The relative merits of the two theories are compared.

The problem of water shipping is studied by assuming two-dimensional flow conditions and using both experimental and numerical tools. Experimentally, the water on deck for a fixed barge-shaped structure has been analysed. Video images of the water-shipping events were recorded, wave elevation in the wave flume and pressure on a vertical superstructure along the ‘ship’ deck have been measured. Numerically, a boundary element method for unsteady nonlinear free-surface flows was developed and used for the analysis of water-on-deck phenomena. A comprehensive comparison between experimental and numerical data gave satisfactory agreement globally. The synergic experimental–numerical analysis highlights the main flow features during the water shipping and details of the water impact with the deck structures are discussed. In the model tests, the water on deck started as a plunging wave hitting the deck and entrapping air. This could be relevant for deck safety, but appears to be less important for the global evolution of the water along the deck and the later liquid interaction with the superstructure. The green-water loads on the vertical wall showed a two-peak behaviour typical of wave impacts.

Bounds on convective heat transport in a porous layer heated from below are derived using the background field variational method (Constantin & Doering 1995a, b, 1996; Doering & Constantin 1992, 1994, 1996; Nicodemus, Holthaus & Grossmann 1997a) based on the technique introduced by Hopf (1941). We consider the infinite Prandtl–Darcy number model in three spatial dimensions, and additionally the finite Prandtl–Darcy number equations in two spatial dimensions, relevant for the related Hele-Shaw problem. The background field method is interpreted as a rigorous implementation of heuristic marginal stability concepts producing rigorous limits on the time-averaged convective heat transport, i.e. the Nusselt number Nu, as a function of the Rayleigh number Ra. The best upper bound derived here, although not uniformly optimal, matches the exact value of Nu up to and immediately above the onset of convection with asymptotic behaviour, Nu[les ]9/256Ra as Ra→∞, exhibiting the Howard–Malkus–Kolmogorov–Spiegel scaling anticipated by classical scaling and marginally stable boundary layer arguments. The relationship between these results and previous works of the same title (Busse & Joseph 1972; Gupta & Joseph 1973) is discussed.

When an alternating electric field is applied to a colloid the particles oscillate at a velocity proportional to the applied field strength. The complex constant of proportionality is termed the dynamic mobility (O'Brien 1988). Although this quantity can now be determined from electroacoustic measurements in suspensions of arbitrary concentration (O'Brien 1990), the theory for interpreting these measurements in terms of the size and charge of the particles is limited to dilute suspensions.

In this paper we derive an expression for the O(ϕ) correction to the dynamic mobility in a random suspension of uniform spheres with volume fraction ϕ. It is assumed that the particle radius is much greater than the double layer thickness but much smaller than the sound wavelength. The mobility is calculated using O'Brien's 1979 macroscopic boundary integral technique. This method ensures a correct mathematical formulation of the problem, and yields an absolutely convergent expression for the average particle velocity. The evaluation of this expression to O(ϕ) involves the determination of the velocities of an isolated pair of particles at various separations and frequencies of oscillation. These velocities are computed using the collocation technique and the O(ϕ) correction to the dynamic mobility is then obtained by numerically integrating over all particle separations.

Using the averaging method and a perturbation technique originally due to Melnikov (1963), we show that an N-degree-of-freedom model of weakly nonlinear surface waves due to Miles (1976) has transverse homoclinic orbits. This implies that Smale horseshoes, and hence sets of chaotic orbits, exist in the phase space. In this particular example, an irregular ‘sloshing’ of energy between two modes of oscillation results. We briefly discuss the relevance of our results to recent experimental work on parametrically excited surface waves.

The relative dispersion of fluid particle pairs in isotropic turbulence is studied using direct numerical simulation, in greater detail and covering a wider Reynolds number range than previously reported. A primary motivation is to provide an important resource for stochastic modelling incorporating information on Reynolds-number dependence. Detailed results are obtained for particle-pair initial separations from less than one Kolmogorov length scale to larger than one integral length scale, and for Taylor-scale Reynolds numbers from about 38 to 230. Attention is given to several sources of uncertainty, including sample size requirements, value of the one-particle Lagrangian Kolmogorov constant, and the temporal variability of space-averaged quantities in statistically stationary turbulence.

Relative dispersion is analysed in terms of the evolution of the magnitude and angular orientation of the two-particle separation vector. Early-time statistics are consistent with the Eulerian spatial structure of the flow, whereas the large-time behaviour is consistent with particle pairs far apart moving independently. However, at intermediate times of order several Kolmogorov time scales, and especially for small initial separation and higher Reynolds numbers, both the separation distance and its rate of change (called the separation speed) are highly intermittent, with flatness factors much higher than those of Eulerian velocity differences in space. This strong intermittency is a consequence of relative dispersion being affected by a wide range of length scales in the turbulent flow as some particle pairs drift relatively far apart. Numerical evidence shows that substantial dispersion occurs in the plane orthogonal to the initial separation vector, which implies that the orientation of this vector has, especially for small initial separation, only limited importance.

Validation of a linear numerical model of wave interactions with floating compliant discs is sought using data obtained from the wave basin experiments reported in Part 1 (Montiel et al. J. Fluid Mech., vol. 723, 2013, pp. 604–628). Comparisons are made for both single-disc tests and the two-disc tests in which wave interactions between discs are observed. The deflection of the disc or discs is separated into the natural modes of vibration in vacuo. The decomposition allows the rigid-body motions and flexural motions to be analysed separately. Rigid-body motions are accurately replicated by the numerical model but, although passable agreement is found, the amplitudes of flexural modes are consistently overestimated. Extensions of the numerical model are used to discount the experimental configuration as a source of the discrepancies. An enhanced viscoelastic model for the discs is also proposed, which results in improved model/data agreement for the flexural motions but cannot account for all of the disagreement.

In random wind-generated wave motion on the sea surface, extreme wave events have been shown theoretically to occur within groups with a well-defined configuration and time history that can be specified in terms of the space-time autocovariances of the surface displacement. The predictions of the theory have been tested in a field experiment in the Straits of Messina in which an array of nine wave gauges and nine pressure transducers supported by vertical piles provided space-time information on waves generated over a fetch of approximately 10 km. It was confirmed that the general configuration of the extreme wave groups measured was consistent with the theoretical predictions in terms of the measured space-time autocovariance. During the development stage of a group, as the height of the central (outstanding) wave grows to a maximum, the width of the wave front reduces to a minimum. As an individual wave passes through the group, its wavelength decreases as the wave height increases towards the apex, after which the wavelength increases again as the wave moves towards the front of the group and abates.

We consider the extensional flow and twist of a finite, slender, nearly straight, Newtonian viscous fibre when its ends are drawn apart at prescribed velocity. The initial cross-section of the fibre may be arbitrary and may vary gradually in the axial direction. We derive the leading-order equations for the fibre's free surface and its flow velocity from a regular perturbation expansion of the full Stokes’ flow problem in powers of the aspect ratio. In order to obtain these equations systematically, it is necessary to consider terms beyond the leading order in the perturbation expansion, because those obtainable from the leading-order terms give an indeterminate set of equations. Our results are a pair of well-known hyperbolic equations for the area and axial velocity, together with (i) straightness of the line of centres of mass of the cross-section and (ii) a new hyperbolic evolution equation for the twist of the cross-section. It is only through this hyperbolic equation that history effects are manifest.