The scaling behaviour of the longitudinal velocity structure functions $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$ (where $2p$ represents the order) is studied for various wall-bounded turbulent flows. It has been known that for very large Reynolds numbers within the logarithmic region, the structure functions can be described by $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}/U_{\unicode[STIX]{x1D70F}}^{2}\approx D_{p}\ln (r/z)+E_{p}$ (where $r$ is the longitudinal distance, $z$ the distance from the wall, $U_{\unicode[STIX]{x1D70F}}$ the friction velocity and $D_{p}$, $E_{p}$ are constants) in accordance with Townsend’s attached eddy hypothesis. Here we show that the ratios $D_{p}/D_{1}$ extracted from plots between structure functions – in the spirit of the extended self-similarity hypothesis – have further reaching universality for the energy containing range of scales. Specifically, we confirm that this description is universal across wall-bounded flows with different flow geometries, and also for both the longitudinal and transversal structure functions, where previously the scaling has been either difficult to discern or differences have been reported when examining the direct representation of $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$. In addition, we present evidence of this universality at much lower Reynolds numbers, which opens up avenues to examine structure functions that are not readily available from high Reynolds number databases.

Numerous experimental, numerical and theoretical studies have shown that core annular flows can be unstable. This instability can be convective or absolute in different situations: miscible fluids with matched density but different viscosities, creeping flow of two immiscible fluids or buoyant flow along a fibre. The analysis of the linear stability of the flow equation of two fluids injected in a co-current and concentric manner into a cylindrical tube leads to a rather complex eigenvalue problem. Until now, all analytical solution to this problem has involved strong assumptions (e.g. lack of inertia) or approximations (e.g. developments at long or short wavelengths) even for axisymmetric disturbances. However, in this latter case, following C. Pekeris, who obtained, almost seventy years ago, an elegant explicit solution for the dispersion relationship of the flow of a single fluid, we derive an explicit solution for the more general case of two immiscible fluids of different viscosity, density and inertia separated by a straight interface. This formulation is well adapted to commercial software. First, we review the creeping flow limit (zero Reynolds number) of two immiscible fluids as it is used in microfluidics. Secondly, we consider the case of two fluids of different viscosities but of the same density in the absence of surface tension and also without diffusion (i.e. miscible fluids with infinite Schmidt number). In both cases, we study the transition from convective to absolute instability according to the different control parameters.

This paper revisits the problem of forces on obstacle arrays in combined waves and an in-line steady current. The intended application is the design and reassessment of offshore platforms. A series of experiments are performed on planar grids moved in both steady and oscillatory motion through otherwise stationary water. Detailed comparisons are made to a wave-current–structure interaction model recently presented by Taylor, Santo & Choo (Ocean Engng, vol. 57, 2013, pp. 11–24). We present new features of the model and test these against the experimental data. For relatively small current speed (${u}_{c} $) compared with oscillatory velocity amplitude (${u}_{w} $) with phase angle ($\omega t$), the drag force time history on grids with solid area ($A$) and projected frontal area (${A}_{f} $) is well approximated by a summation of the wave drag and the current drag components independently, so there is no ${u}_{w} \times {u}_{c} $ cross-term. The wave drag component is proportional to $\cos \omega t\vert \cos \omega t\vert $, while the current drag component to $\vert \cos \omega t\vert $, i.e. it is phase-locked to the oscillatory wave crests. The form of the predicted time history is new, so much of this paper is occupied in testing the adequacy of this theoretical form both in terms of an improved Morison-type formulation and also in the precise variation of the experimental drag force in time. We show that the measured crest and trough peak values of the drag force are consistent with the force peaks and troughs of the model prediction. The odd frequency harmonics of the measured drag force scale as the square of the oscillatory velocity amplitude $({ u}_{w}^{2} )$ and on the total hydrodynamic area (${C}_{d} A$). The shape of the odd harmonics is very similar to that for a pure oscillatory motion without steady current, but there are also even frequency harmonics associated with the current component. The even harmonics of the force scale as the square of the current speed $({ u}_{c}^{2} )$ and on the ${A}_{f} $, not on the ${C}_{d} A$. All of the above features are identified within the experimental data, and provide considerable support for the new current blockage model.

The new model is also shown to fit the entire force time history well for a wide range of individual cases, with different blockage ratio ($A/ {A}_{f} $) and number of grids, requiring only calibration of the Morison-type drag and inertia coefficients. In contrast, the industry-standard form of the Morison equation can only be matched at a single instant of the oscillation cycle, so present practice should be regarded as seriously inadequate for combined steady current and oscillatory flow acting on obstacle arrays.

We examine experimentally the influence of flexibility on the side-to-side fluttering motion of passive wings settling under the influence of gravity. Our results demonstrate the existence of an optimal flexibility that allows flexible wings to remain airborne twice as long as their rigid counterparts of identical mass and size. Flow visualization and measurements allow us to elucidate the role of flexibility in generating increased lift and wing circulation by shedding additional vorticity at the turning point. Theoretical scalings are derived from a reduced model of the flight dynamics and yield quantitative agreement with experiments. These scalings rationalize the strong positive correlation between flexibility and flight time. Our experimental results and theoretical scalings represent an ideal system for the validation of computational approaches to model biologically inspired fluid–structure interaction problems.

Accurate real-time simulations and forecasting of phase-revolved ocean surface waves require nonlinear effects, both geometrical and kinematic, to be accurately represented. For this purpose, wave models based on a Lagrangian steepness expansion have proved particularly efficient, as compared to those based on Eulerian expansions, as they feature higher-order nonlinearities at a reduced numerical cost. However, while they can accurately model the instantaneous nonlinear wave shape, Lagrangian models developed to date cannot accurately predict the time evolution of even simple periodic waves. Here, we propose a novel and simple method to perform a Lagrangian expansion of surface waves to second order in wave steepness, based on the dynamical system relating particle locations and the Eulerian velocity field. We show that a simple redefinition of reference particles allows us to correct the time evolution of surface waves, through a modified nonlinear dispersion relationship. The resulting expressions of free surface particle locations can then be made numerically efficient by only retaining the most significant contributions to second-order terms, i.e. Stokes drift and mean vertical level. This results in a hybrid model, referred to as the ‘improved choppy wave model’ (ICWM) (with respect to Nouguier et al.’s J. Geophys. Res., vol. 114, 2009, p. C09012), whose performance is numerically assessed for long-crested waves, both periodic and irregular. To do so, ICWM results are compared to those of models based on a high-order spectral method and classical second-order Lagrangian expansions. For irregular waves, two generic types of narrow- and broad-banded wave spectra are considered, for which ICWM is shown to significantly improve wave forecast accuracy as compared to other Lagrangian models; hence, ICWM is well suited to providing accurate and efficient short-term ocean wave forecast (e.g. over a few peak periods). This aspect will be the object of future work.

We use two-dimensional direct numerical simulations of Boussinesq stratified shear layers to investigate the influence of the minimum gradient Richardson number $Ri_{m}$ on the early time evolution of Kelvin–Helmholtz instability to its saturated ‘billow’ state. Even when the diffusion of the background velocity and density distributions is counterbalanced by artificial body forces to maintain the initial profiles, in the limit as $Ri_{m}\rightarrow 1/4$, the perturbation growth rate tends to zero and the saturated perturbation energy becomes small. These results imply, at least for such canonical inflectional stratified shear flows, that ‘marginally unstable’ flows with $Ri_{m}$ only slightly less than 1/4 are highly unlikely to become ‘turbulent’, in the specific sense of being associated with significantly enhanced dissipation, irreversible mixing and non-trivial modification of the background distributions without additional externally imposed forcing.

Practical prediction of structural vibrations due to a turbulent boundary layer currently depends on empirical representations of the unsteady wall pressures. Improvements in these representations would be greatly facilitated if a simple, physically based model were available to test ad hoc assumptions and provide rigorous interpolation of experimental data. A possible candidate is the attached-eddy model, developed from Townsend's initial ideas by Perry and co-workers in the context of turbulence velocity spectra. This approach employs the superposition of contributions from individual ‘eddies’, of varying size, to yield its predictions. It is shown here that the same methodology can be applied for wall pressures, once the field due to an eddy has been obtained via solution of the governing Poisson equation. Comparisons with large-eddy simulation and experimental data, spanning a two-decade Reynolds number range, show remarkably good agreement, given the simplicity of the model. It is concluded that this approach has the potential to provide useful physical insight and, subject to its extension to a time-resolved form, improvements to existing empirical formulations.

In view of the importance of concentration fluctuations in practical and theoretical problems of turbulent diffusion, there are presented here some invariant properties of the distribution of fluctuations associated with a cloud of contaminant containing a finite quantity Q of material. These properties are invariant provided only that Q is conserved, no assumption whatsoever being made about the random turbulent velocity field. Consequences of the results for (i) steady plumes, (ii) the representation of the distribution of concentration by series of (generalized) Hermite polynomials, and (iii) the relationship of the ensemble mean concentration with the distance–neighbour function, are discussed using experimental evidence.

The problem solved is that of the interaction between a laminar boundary layer on a semi-infinite flat plate and an oncoming shear flow of finite lateral dimensions bounded by uniform irrotational flow extending to infinity. The pressures along the plate and upstream of the same are deduced (to a linearized approximation) in the form of a Fourier integral based on the solution of a simpler periodic flow problem. It is found that while the assumption of an infinite, uniform shear flow gives asymptotically correct interaction pressure gradients on the plate near the leading edge, the pressure level even there (compared to upstream infinity) is strongly influenced by the boundedness of the external shear. At distances from the leading edge which are large compared to the lateral extent of the shear flow, the pressure gradients along the plate are shown to be vanishingly smaller than in the infinite shear case.

An experimental study has been made of the flow produced by an internal heat source as it moves around a fixed annulus of fluid. The mean surface flow relative to the annulus is found to be in the opposite direction to the movement of the heat source. The variation in its magnitude with changes in the external parameters is investigated. Thermocouple measurements and streak photography are used to examine the interior temperature and flow structure.

We derive and test a new heuristic theory for third-order structure functions that resolves the forcing scale in the scenario of simultaneous spectral energy transfer to both small and large scales, which can occur naturally, for example, in rotating stratified turbulence or magnetohydrodynamical (MHD) turbulence. The theory has three parameters – namely the upscale/downscale energy transfer rates and the forcing scale – and it includes the classic inertial-range theories as local limits. When applied to measured data, our global-in-scale theory can deduce the energy transfer rates using the full range of data, therefore it has broader applications compared with the local theories, especially in situations where the data is imperfect. In addition, because of the resolution of forcing scales, the new theory can detect the scales of energy input, which was impossible before. We test our new theory with a two-dimensional simulation of MHD turbulence.

Oscillatory flow around a cluster of four circular cylinders in a diamond arrangement is investigated using two-dimensional direct numerical simulation over Keulegan–Carpenter numbers (KC) ranging from 4 to 12 and Reynolds numbers (Re) from 40 to 230 at four gap-to-diameter ratios (G) of 0.5, 1, 2 and 4. Three types of flows, namely synchronous, quasi-periodic and desynchronized flows (along with 14 flow regimes) are mapped out in the (G, KC, Re)-parameter space. The observed flow characteristics around four cylinders in a diamond arrangement show a few unique features that are absent in the flow around four cylinders in a square arrangement reported by Tong et al. (J. Fluid Mech., vol. 769, 2015, pp. 298–336). These include (i) the dominance of flow around the cluster-scale structure at $G=0.5$ and 1, (ii) a substantial reduction of regime D flows in the regime maps, (iii) new quasi-periodic (phase trapping) $\text{D}^{\prime }$ (at $G=0.5$ and 1) and period-doubling $\text{A}^{\prime }$ flows (at $G=1$) and most noteworthily (iv) abnormal behaviours at ($G\leqslant 2$) (referred to as holes hereafter) such as the appearance of spatio-temporal synchronized flows in an area surrounded by a single type of synchronized flow in the regime map ($G=0.5$). The mode competition between the cluster-scale and cylinder-scale flows is identified as the key flow mechanism responsible for those unique flow features, with the support of evidence derived from quantitative analysis. Phase dynamics is introduced for the first time in bluff-body flows, to the best knowledge of the authors, to quantitatively interpret the flow response (e.g. quasi-periodic flow features) around the cluster. It is instrumental in revealing the nature of regime $\text{D}^{\prime }$ flows where the cluster-scale flow features are largely synchronized with the forcing of incoming oscillatory flow (phase trapping) but are modulated by localized flow features.

A detailed numerical study of the separating and reattaching flow over a square leading-edge plate is presented, examining the instability modes governing transition from two- to three-dimensional flow. Under the influence of background noise, experiments show that the transition scenario typically is incompletely described by either global stability analysis or the transient growth of dominant optimal perturbation modes. Instead two-dimensional transition effectively can be triggered by the convective Kelvin–Helmholtz (KH) shear-layer instability; although it may be possible that this could be described alternatively in terms of higher-order optimal perturbation modes. At least in some experiments, observed transition occurs by either: (i) KH vortices shedding downstream directly and then almost immediately undergoing three-dimensional transition or (ii) at higher Reynolds numbers, larger vortical structures are shed that are also three-dimensionally unstable. These two paths lead to distinctly different three-dimensional arrangements of vortical flow structures. This paper focuses on the mechanisms underlying these three-dimensional transitions. Floquet analysis of weakly periodically forced flow, mimicking the observed two-dimensional quasi-periodic base flow, indicates that the two-dimensional vortex rollers shed from the recirculation region become globally three-dimensionally unstable at a Reynolds number of approximately 380. This transition Reynolds number and the predicted wavelength and flow symmetries match well with those of the experiments. The instability appears to be elliptical in nature with the perturbation field mainly restricted to the cores of the shed rollers and showing the spatial vorticity distribution expected for that instability type. Indeed an estimate of the theoretical predicted wavelength is also a good match to the prediction from Floquet analysis and theoretical estimates indicate the growth rate is positive. Fully three-dimensional simulations are also undertaken to explore the nonlinear development of the three-dimensional instability. These show the development of the characteristic upright hairpins observed in the experimental dye visualisations. The three-dimensional instability that manifests at lower Reynolds numbers is shown to be consistent with an elliptic instability of the KH shear-layer vortices in both symmetry and spanwise wavelength.

In this paper, laminar convection flows induced by carbon dioxide absorption are addressed from experimental, numerical and theoretical points of view. A vertical glass tube (of centimetre scale) filled with distilled water is subjected to a sudden increase in the partial pressure of carbon dioxide. As a result of the diffusion of the gas into the unsaturated solution, a thin layer of fluid located underneath the surface becomes heavier. This initial density gradient first destabilizes to form a plume, which goes downwards through the entire cell. After a first transient pulsating regime (periodic succession of such Rayleigh–Bénard plumes), a stationary flow settles in the tube, which is maintained by the constant supply of gas at the surface. At late stages, this stationary regime is followed by an aperiodic regime, which lasts until the complete saturation of the solution (thermodynamic equilibrium). The present study only focuses on the stationary regime, whose characteristics appear to be almost independent of the Bond number and the aspect ratio but strongly dependent on the chemical Rayleigh number. Three decades of Rayleigh numbers are explored using particle image velocimetry measurements, which allows for a precise determination of the scaling exponents for the vertical velocity amplitude and the plume width. The assumption that gravity and a constant pressure gradient balance the viscous effects enables us to derive an analytic expression for the stationary vertical velocity on the axis, which scales as ${\mathit{Ra}}^{2/ 3} \mathop{(\ln \mathit{Ra})}\nolimits ^{1/ 3} $. As a consequence, the width of the plume scales as ${\mathit{Ra}}^{- 1/ 6} \mathop{(\ln \mathit{Ra})}\nolimits ^{- 1/ 3} $ and the mass Nusselt number as $\mathop{(\mathit{Ra}/ \ln \mathit{Ra})}\nolimits ^{1/ 3} $. These scalings are in excellent agreement with the experimental and numerical results. The multiplicative constants of these scalings can also be calculated and show a fairly good agreement if a rigid boundary condition (no-slip) is assumed at the free surface.

We consider the spreading of a thin viscous droplet, injected through a finite region of a substrate, under the influence of surface tension. We neglect gravity and assume that there is a precursor layer covering the whole substrate and that the rate of injection is constant. We analyse the evolution of the film profile for early and late time, and obtain power-law dependencies for the maximum film thickness at the centre of the injection region and the position of an apparent contact line, which compare well with numerical solutions of the full problem. We relax the conditions on the injection rate to consider more general time-dependent and spatially varying forms. In the case of power-law injection of the form $t^{k}$, we observe a switch in the behaviour of the evolution of the film thickness for late time from increasing to decreasing at a critical value of $k$. We show that point-source injection can be treated as a limiting case of a finite-injection slot and the solutions exhibit identical behaviours for late time. Finally, we formulate the problem with thickness-dependent injection rate, discuss the behaviour of the maximum film thickness and the position of the apparent contact line and give power-law dependencies for these.

In the Dagan & Tulin (J. Fluid Mech., vol. 51, 1972, pp. 529–543) model of ship waves, a blunt ship moving at low speeds can be modelled as a two-dimensional semi-infinite body. A central question for these reduced models is whether a particular ship design can minimize, or indeed eliminate, the wave resistance. In the previous part of our work (Trinh et al., J. Fluid Mech., vol. 685, 2011, pp. 413–439), we demonstrated why a single corner can never be made waveless. In this accompanying paper, we continue our investigations with the study of more general piecewise-linear, or multi-cornered ships. By using exponential asymptotics, we demonstrate how the production of waves can be directly ascertained by the positions and angles of the corners. In particular, this theory answers the question raised by Farrow & Tuck (J. Austral. Math. Soc. B, vol. 36, 1995, pp. 424–437) as to why certain bulbous-like obstructions can minimize the production of waves. General results for wavelessness are given for a class of hulls, and numerical computations of the nonlinear ship-wave problem are used to confirm analytical predictions. Finally, we discuss open questions regarding hulls without corners and more general three-dimensional bluff bodies.