The modification of the turbulent cascade by polymeric additives is addressed by direct numerical simulations of homogeneous isotropic turbulence of a FENE-P fluid. According to the appropriate form of the Kármán–Howarth equation, two kinds of energy fluxes exist, namely the classical transfer term and the coupling with the polymers. Depending on the Deborah number, the response of the flow may result either in a pure damping or in the depletion of the small scales accompanied by increased fluctuations at large scale. The latter behaviour corresponds to an overall reduction of the dissipation rate with respect to an equivalent Newtonian flow with identical fluctuation intensity. The relevance of the position of the crossover scale between the two components of the energy flux with respect to the Taylor microscale of the system is discussed.

There are many studies on the vortex-induced vibrations of a cylindrical body, but almost none concerned with such vibrations for a sphere, despite the fact that tethered bodies are a common configuration. In this paper, we study the dynamics of an elastically mounted or tethered sphere in a steady flow, employing displacement, force and vorticity measurements. Within a particular range of flow speeds, where the oscillation frequency ($f$) is of the order of the static-body vortex shedding frequency ($f_{vo})$, there exist two modes of periodic large-amplitude oscillation, defined as modes I and II, separated by a transition regime exhibiting non-periodic vibration. The dominant wake structure for both modes is a chain of streamwise vortex loops on alternating sides of the wake. Further downstream, the heads of the vortex loops pinch off to form a sequence of vortex rings. We employ an analogy with the lift on an aircraft that is associated with its trailing vortex pair (of strength $\Gamma^*$ and spacing $b^*$), and thereby compute the rate of change of impulse for the streamwise vortex pair, yielding the vortex force coefficient ($\cvortex$): \[ \cvortex = \frac{8}{\pi} {U^*_{v}}b^*( - \Gamma^*). \] This calculation yields predicted forces in reasonable agreement with direct measurements on the sphere. This is significant because it indicates that the principal vorticity dynamics giving rise to vortex-induced vibration for a sphere are the motions of these streamwise vortex pairs. The Griffin plot, showing peak amplitudes as a function of the mass–damping ($m^*\zeta$), exhibits a good collapse of data, indicating a maximum response of around 0.9 diameters. Following recent studies of cylinder vortex-induced vibration, we deduce the existence of a critical mass ratio, $m^*_{crit} {\approx} 0.6$, below which large-amplitude vibrations are predicted to persist to infinite normalized velocities. An unexpected large-amplitude and highly periodic mode (mode III) is found at distinctly higher flow velocities where the frequency of vibration ($f$) is far below the frequency of vortex shedding for a static body. We find that the low-frequency streamwise vortex pairs are able to impart lift (or transverse force) to the body, yielding a positive energy transfer per cycle.

A jet of fluid flowing down a partially wetting inclined plane usually meanders. In this paper, we demonstrate that meandering on a smooth plane can be suppressed by maintaining a constant volume flow rate. In the absence of meandering, we experimentally observe the jet developing a braided structure with non-monotonic width. This flow pattern is theoretically explained as the result of the interplay between surface tension that tends to narrow the jet down and fluid inertia that drives the jet width to expand. The theory also predicts a bifurcation between the braiding regime and a non-meandering non-braiding flow, which is confirmed by experiment.

We have investigated the role of liquid elasticity in the dynamics of air–liquid interfaces during immiscible fluid displacement flows. Our experimental studies of coating flows with gravity stabilization in an eccentric cylinder geometry for both a viscous Newtonian fluid and a series of elastic Boger fluids have uncovered two new elastically driven phenomena. First, elasticity is shown to create steady, two-dimensional ‘sharp interfaces’ under creeping flow conditions in forward-roll coating. Digital particle image velocimetry (DPIV) and DEVSS finite element techniques are applied to investigate the effect of gravity on both the flow field and the state of polymeric stresses near the free surface. Secondly, a new class of elastically driven stable substructures is shown to form and persist along the tip, i.e. the centre stagnation line, of the two-dimensional `sharp interfaces' even after the onset of ribbing.

The manner by which external vortical disturbances penetrate the laminar boundary layer and induce transition is explored. Linear theory suggests that the well-known Klebanoff mode precursor to transition can be understood as a superposition of Squire continuous modes. Shear sheltering influences the ability of free-stream disturbances to generate a packet of Squire modes. A coupling coefficient between continuous spectrum spectrum Orr–Sommerfeld and Squire modes is used to characterize the interaction. Full numerical simulations with prescribed modes at the inlet substantiate this approach. With two weakly coupled modes at the inlet, the boundary layer is little perturbed; with two strongly coupled modes, Klebanoff modes are produced; with one strongly coupled and one weakly coupled high-frequency mode, the complete transition process is simulated.

We explore very fine scales of scalar dissipation in turbulent mixing, below Kolmogorov and around Batchelor scales, by performing direct numerical simulations at much finer grid resolution than was usually adopted in the past. We consider the resolution in terms of a local fluctuating Batchelor scale and study the effects on the tails of the probability density function and multifractal properties of the scalar dissipation. The origin and importance of these very fine-scale fluctuations are discussed. One conclusion is that they are unlikely to be related to the most intense dissipation events.

It is shown that the solution of the semi-geostrophic equations for shallow-water flow can be found and analysed in spherical geometry by methods similar to those used in the existing $f$-plane solutions. Stable states in geostrophic balance are identified as energy minimizers and a procedure for finding the minimizers is constructed, which is a form of potential vorticity inversion. This defines a generalization of the geostrophic coordinate transformation used in the $f$-plane theory. The procedure is demonstrated in computations.

The evolution equations take a simple form in the transformed coordinates, though, as expected from previous work in the literature, they cannot be expressed exactly as geostrophic motion. The associated potential vorticity does not obey a Lagrangian conservation law, but it does obey a flux conservation law, with an associated circulation theorem.

The divergence of the flow in the transformed coordinates is primarily that naturally associated with geostrophic motion, with additional terms coming from the curvature of the sphere and extra ‘curvature’ resulting from the variable Coriolis parameter in the generalized coordinate transformation. These terms are estimated, and are found to be very small for normal data. The estimate is verified in computations, confirming the accuracy of the local $f$-plane approximation usually made with semi-geostrophic theory.

Field observations in lakes, where the effects of the Earth's rotation can be neglected, suggest that the basin-scale internal wave field may be decomposed into a standing seiche, a progressive nonlinear surge and a dispersive solitary wave packet. In this study we use laboratory experiments to quantify the temporal energy distribution and flux between these three component internal wave modes. The system is subjected to a single forcing event creating available potential energy at time zero (APE). During the first horizontal mode one basin-scale wave period ($T_i$), as much as 10% and 20% of the APE may be found in the solitary waves and surge, respectively. The remainder is contained in the horizontal mode one seiche or lost to viscous dissipation. These findings suggest that linear analytical solutions, which consider only basin-scale wave motions, may significantly underestimate the total energy contained in the internal wave field. Furthermore, linear theories prohibit the development of the progressive nonlinear surge, which serves as a vital link between basin-scale and sub-basin-scale motions. The surge receives up to 20% of the APE during a nonlinear steepening phase and, in turn, conveys this energy to the smaller-scale solitary waves as dispersion becomes significant. This temporal energy flux may be quantified in terms of the ratio of the linear and nonlinear terms in the nonlinear non-dispersive wave equation. Through estimation of the viscous energy loss, it was established that all inter-modal energy flux occurred before 2$T_i$; the modes being independently damped thereafter. The solitary wave energy remained available to propagate to the basin perimeter, where although it is beyond the scope of this study, wave breaking is expected. These results suggest that a periodically forced system with sloping topography, such as a typical lake, may sustain a quasi-steady flux of 20% of APE to the benthic boundary layer at the depth of the metalimnion.

Solitary-wave solutions to surface equations or two-equation models of film flows are investigated within the framework of dynamical system theory. The limiting behaviour of one-humped solitary waves (homoclinic orbits) at large Reynolds numbers is considered. Their predicted speed is in good agreement with numerical findings. The theory also explains the absence of solitary-wave solutions to the Benney equation in the same limit.

The issue of morphodynamic influence in meandering streams is investigated through a series of laboratory experiments on curved and straight flumes. Both qualitative and quantitative observations confirm the suitability of the recent theoretical developments (Zolezzi & Seminara 2001) that indicate the occurrence of two distinct regimes of morphodynamic influence, depending on the value of the width ratio of the channel $\beta$. The threshold value $\beta_R$ separating the upstream from the downstream influence regimes coincides with the resonant value discovered by Blondeaux & Seminara (1985). Indeed it is observed that upstream influence may occur only in relatively wide channels, while narrower streams are dominated by downstream influence. A series of experiments has been carried out in order to check the above theoretical predictions and show, for the first time, evidence of the occurrence of upstream overdeepening. Two different sets of experiments have been designed where a discontinuity in channel geometry was present such that the channel morphodynamics was influenced in the upstream direction under super-resonant conditions ($\beta{>}\beta_R$) and in the downstream direction under sub-resonant conditions ($\beta{<}\beta_R$). Experimental results give qualitative and quantitative support to the theoretical predictions and allow us to clarify the limits of the linear analysis.

The consistent coupled-mode theory (Athanassoulis & Belibassakis, J. Fluid Mech. vol. 389, 1999, p. 275) is extended and applied to the hydroelastic analysis of large floating bodies of shallow draught or ice sheets of small and uniform thickness, lying over variable bathymetry regions. A parallel-contour bathymetry is assumed, characterized by a continuous depth function of the form $h( {x,y}) {=} h( x )$, attaining constant, but possibly different, values in the semi-infinite regions $x {<} a$ and $x {>} b$. We consider the scattering problem of harmonic, obliquely incident, surface waves, under the combined effects of variable bathymetry and a floating elastic plate, extending from $ x {=} a$ to $x {=} b$ and $ {-} \infty {<} y{<}\infty $. Under the assumption of small-amplitude incident waves and small plate deflections, the hydroelastic problem is formulated within the context of linearized water-wave and thin-elastic-plate theory. The problem is reformulated as a transition problem in a bounded domain, for which an equivalent, Luke-type (unconstrained), variational principle is given. In order to consistently treat the wave field beneath the elastic floating plate, down to the sloping bottom boundary, a complete, local, hydroelastic-mode series expansion of the wave field is used, enhanced by an appropriate sloping-bottom mode. The latter enables the consistent satisfaction of the Neumann bottom-boundary condition on a general topography. By introducing this expansion into the variational principle, an equivalent coupled-mode system of horizontal equations in the plate region ($a {\leq} x {\leq} b)$ is derived. Boundary conditions are also provided by the variational principle, ensuring the complete matching of the wave field at the vertical interfaces ($x{=}a$ and $x{=}b)$, and the requirements that the edges of the plate are free of moment and shear force. Numerical results concerning floating structures lying over flat, shoaling and corrugated seabeds are presented and compared, and the effects of wave direction, bottom slope and bottom corrugations on the hydroelastic response are presented and discussed. The present method can be easily extended to the fully three-dimensional hydroelastic problem, including bodies or structures characterized by variable thickness (draught), flexural rigidity and mass distributions.

We consider the dynamics of a polymer with finite extensibility placed in a chaotic flow with large mean shear, to explain how the polymer statistics changes with Weissenberg number, ${\it Wi}$, the product of the polymer relaxation time and the Lyapunov exponent of the flow, $\bar\lambda$. The probability distribution function (PDF) of the polymer orientation is peaked around a shear-preferred direction, having algebraic tails. The PDF of the tumbling time (separating two subsequent flips), $\tau$, has a maximum estimated as $\bar\lambda^{-1}$. This PDF shows an exponential tail for large $\tau$ and a small-$\tau$ tail determined by the simultaneous statistics of the velocity PDF. Four regimes of ${\it Wi}$ are identified for the extension statistics: one below the coil–stretched transition and three above the coil–stretched transition. Emphasis is given to explaining these regimes in terms of the polymer dynamics.

This study revisits the problem of the zonally symmetric instability on the equatorial $\beta$-plane. Rather than treating the classical problem of a steady basic flow, it treats a sequence of problems of increasing complexity in which the basic flow is oscillatory in time with a frequency $\omega_0$.

First, for the case of a homogeneous fluid, a time-oscillating barotropic shear forcing may excite a subharmonic parametric resonance of inertial oscillations. Because of the continuous distribution of inertial oscillation frequencies, this resonance occurs at critical inertial latitudes $y_c$ such that $\beta y_c {=}{\pm} {\omega_0}/2$. Next the effects of stratification, characterized by Brunt–Väisälä frequency $N$, are taken into account. It is shown analytically (in the asymptotic limit of a weak shear) that the forced temporal oscillation leads to an inertial-parametric instability, when a resonance condition between the basic flow frequency and the sum of two inertio-gravity free-mode frequencies is met. This inertial-parametric instability has a well-defined inviscid vertical scale selection favouring the high-vertical mode $m_c{\sim}7.45m_0$, where $m_0{=}{\beta N}/{\omega_0^2}$ is the equatorial vertical mode characteristic of frequency $\omega_0$. The viscous critical shear of inertial-parametric instability is lower than the steady inertial instability one.

Finally, this type of setting naturally arises when the basic flow is considered to be an equatorial wave, so the problem is recast with the nonlinear adjustment of the vertically sinusoidal basic state of a zonally symmetric mixed Rossby–gravity (MRG) wave. Initial-value numerical simulations show that the same inertial-parametric instability exists leading to a resonant subharmonic excitation of free modes with vertical scales 7 and 8 times smaller than the basic-state wave. A simplified dynamical model of the instability is introduced, demonstrating that the oscillatory nature of the shear with height for the MRG wave necessarily implies a resonance between distinct vertical modes, the most unstable ones being modes 7 and 8 for a large enough Froude number of the MRG wave. The nonlinear action of the instability is described in terms of angular momentum and potential vorticity changes: a significant mixing due to the breaking of the excited high vertical modes creates a vertically averaged westward flow at the equator and extra-equatorial eastward flows. The ideas exposed may play a part in explaining layering phenomena and the latitudinal structure of the zonal flow in the equatorial oceans below the thermocline.

High Reynolds number (Re) wall-bounded turbulent flows occur in many hydro- and aerodynamic applications. However, the limited amount of high-Re experimental data has hampered the development and validation of scaling laws and modelling techniques applicable to such flows. This paper presents measurements of the turbulent flow near the trailing edge of a two-dimensional lifting surface at chord-based Reynolds numbers, Re$_{C}$, typical of heavy-lift aircraft wings and full-scale ship propellers. The experiments were conducted in the William B. Morgan Large Cavitation Channel at flow speeds from 0.50 to 18.3ms$^{-1}$ with a cambered hydrofoil having a 3.05m span and a 2.13m chord that generated 60 metric tons of lift at the highest flow speed, Re$_{C}{\approx}50{\times}10^{6}$. Flow-field measurements concentrated on the foil's near wake and include results from trailing edges having terminating bevel angles of 44$^{\circ}$ and 56$^{\circ}$. Although generic turbulent boundary layer and wake characteristics were found at any fixed Re$_{C}$ in the trailing-edge region, the variable strength of near-wake vortex shedding caused the flow-field fluctuations to be Reynolds-number and trailing-edge-geometry dependent. In the current experiments, vortex-shedding strength peaked at Re$_{C}{=}4{\times}10^{6}$ with the 56$^{\circ}$ bevel-angle trailing edge. A dimensionless scaling for this phenomenon constructed from the free-stream speed, the wake thickness, and an average suction-side shear-layer vorticity at the trailing edge collapses the vortex-shedding strength measurements for $1.4{\times}10^{6}{\le}{\it Re}_{C}{\le}50{\times}10^{6}$ from both trailing edges and from prior measurements on two-dimensional struts at Re$_{C}{\sim}2{\times}10^{6}$ with asymmetrical trailing edges.

Although the critical Reynolds number for linear instability of the laminar flow in a straight pipe is infinite, we show that it is finite for a divergent pipe, and approaches infinity as the inverse of the divergence angle. The velocity profile at the threshold of inviscid stability is obtained. A non-parallel analysis yields linear instability at surprisingly low Reynolds numbers, of about 150 for a divergence of $3^\circ$, which would suggest a role for such instabilities in the transition to turbulence. A multigrid Poisson equation solver is employed for the basic flow, and an extended eigenvalue method for the partial differential equations describing the stability.

Two models are presented for the motion of vortices near gaps in infinitely long barriers. The first model considers a line vortex for which the exact nonlinear trajectories satisfying the governing two-dimensional Euler equations are obtained analytically. The second model considers a finite-area patch of constant vorticity and is based on conformal mapping and the numerical method of contour surgery. The two models enable a comparison of the trajectories of line vortices and vortex patches. The case of a double gap formed by an island lying between two headlands is considered in detail. It is noted that Kelvin's theorem constrains the circulation around the island to be a constant and thus forces a time-dependent volume flux between the islands and the headlands. When the gap between the island and a headland is small this flux requires arbitrarily large flow speeds through the gap. In most examples the centroid of the patch is constrained to follow closely the trajectory of a line vortex of the same circulation. Exceptions occur when the through-gap flow forces the vortex patch close to an edge of the island where it splits into two with only part of the vortex passing through the gap. In general the part squeezing through a narrow gap returns to near-circular to have a diameter significantly larger than the gap width.

Motivated by the viscous sintering of amorphous materials in the presence of a distribution of rigid platelet inclusions added to increase the strength of the final sintered medium, a simple model for the shrinkage of compressible pores close to a wall in Stokes flow is proposed. The model assumes that the pore remains elliptical/ellipsoidal in shape at all times. The model is expected to be valid provided the pore is not too far from spherical and not too close to the wall. It relies on a separation of the flow into ‘inner’ and ‘outer’ problems. The inner problem is to assume that the ellipsoid evolves in a linear ambient flow given by the first two terms in a local expansion of an outer flow produced by the image singularity distribution of a point-sink near a planar no-slip boundary. The focus of the present paper is to test the viability of the model in the planar case. Using a spectral method based on analytic functions and conformal mappings, the results of a full numerical simulation are compared with the predictions of the planar model. The effects of the proximity of the pore to the wall, the anisotropy in the pore shape and its relative orientation to the wall are all examined. It is observed that, as they shrink, pores drift towards the wall becoming elongated in the direction perpendicular to the wall.