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.

In this paper we present an experimental study of the long-surface-wave instability that can develop when a granular material flows down a rough inclined plane. The threshold and the dispersion relation of the instability are precisely measured by imposing a controlled perturbation at the entrance of the flow and measuring its evolution down the slope. The results are compared with the prediction of a linear stability analysis conducted in the framework of the depth-averaged or Saint-Venant equations. We show that when the friction law proposed in Pouliquen (1999a) is introduced in the Saint-Venant equations, the theory is able to predict quantitatively the stability threshold and the phase velocity of the waves but fails in predicting the observed cutoff frequency. The instability is shown to be of the same nature as the long-wave instability observed in classical fluids but with characteristics that can dramatically differ due to the specificity of the granular rheology.

High-vector-density planar velocity fields were obtained for a weakly compressible mixing layer using particle image velocimetry (PIV). The velocity ratio of the mixing layer was 0.53, the density ratio was 0.67, and the convective Mach number was 0.38. At the location where the PIV images were obtained, $\Re_x\,{=}\,3.7\,{\times}\,10^{6}$ and $\Re_{\delta_\omega}\,{=}\,1.8\,{\times}\,10^{5}$. The instantaneous planar velocity fields fall into three regimes characterized by the size and number of large-scale structures present. The large-scale rollers are either circular or elliptical, with the elliptical rollers having, in general, horizontal major axes. The transverse velocity fluctuations and Reynolds shear stress are suppressed for the weakly compressible mixing-layer as compared to the incompressible case. The spatial correlations of velocity fluctuations also occupy a smaller fraction of the mixing-layer thickness than for an incompressible mixing layer. The linear stochastic estimate of a roller structure is elliptical with the major axis oriented in the streamwise direction and with an eccentricity greater than for the incompressible case. The linear stochastic estimate of a braid suggests that the braids are vertically oriented, as opposed to the oblique orientation seen in incompressible mixing layers. In addition, the braids in the weakly compressible case have a vertically oriented stagnation line, as opposed to the braids in the incompressible mixing layer where stagnation occurs at a point.

A fully coupled finite-element method is used to investigate the steady flow of a viscous fluid through a thin-walled elastic tube mounted between two rigid tubes. The steady three-dimensional Navier–Stokes equations are solved simultaneously with the equations of geometrically nonlinear Kirchhoff–Love shell theory. If the transmural (internal minus external) pressure acting on the tube is sufficiently negative then the tube buckles non-axisymmetrically and the subsequent large deformations lead to a strong interaction between the fluid and solid mechanics. The main effect of fluid inertia on the macroscopic behaviour of the system is due to the Bernoulli effect, which induces an additional local pressure drop when the tube buckles and its cross-sectional area is reduced. Thus, the tube collapses more strongly than it would in the absence of fluid inertia. Typical tube shapes and flow fields are presented. In strongly collapsed tubes, at finite values of the Reynolds number, two ’jets‘ develop downstream of the region of strongest collapse and persist for considerable axial distances. For sufficiently high values of the Reynolds number, these jets impact upon the sidewalls and spread azimuthally. The consequent azimuthal transport of momentum dramatically changes the axial velocity profiles, which become approximately $\uTheta$-shaped when the flow enters the rigid downstream pipe. Further convection of momentum causes the development of a ring-shaped velocity profile before the ultimate return to a parabolic profile far downstream.

Various recent experiments hint at a geometry dependence of scaling relations in Rayleigh–Bénard convection. Aspect ratio and shape dependences have been found. In this paper a mechanism is suggested which can account for such dependences, based on Prandtl's theory for laminar boundary layers and on the conservation of volume flux of the large-scale wind. The mechanism implies the possibility of different thicknesses of the kinetic boundary layers at the sidewalls and at the top/bottom plates, as found experimentally, and also different Ra-scaling of the wind over the plates and at the sidewalls. A scaling argument for the velocity and temperature fluctuations in the bulk is also developed.

Stably stratified freely decaying homogeneous turbulence is investigated by means of direct numerical simulations (DNS) and a two-point closure statistical model of the EDQNM type; a careful comparison with laboratory experiments is also made. Several aspects of anisotropy in the flow are studied, both at large and small scales. DNS and EDQNM approaches give very similar results up to the finest indicators of the flow, namely anisotropic spectra of velocity fields. Hence the statistical model predicts the structure of the flow at all scales.

Large-scale anisotropy appears in the Reynolds stress components and in the directional integral length scales. The well-known collapse of vertical turbulent motion, which yields the organization of the flow into quasi-horizontal vertically decorrelated vortex structures, is retrieved and quantified. Thus, the thickness of the vortex structures is shown to be set by their Froude number being of order one, in agreement with a previous dimensional analysis for an inviscid flow. Small-scale anisotropy is quantified from the components of the velocity and temperature gradients, whereby models for the dissipation rate of kinetic energy and available potential energy are discussed. The mixing properties of the flow are also investigated: the counter-gradient heat flux that exists at small scales appears to inhibit mixing when diffusivity is low enough and the Cox number varies linearly with the parameter $\epsilon/\nu N^2$.

All results agree very well with laboratory experiments on stably stratified grid turbulence, though the initial condition of our computations is different from the flow just behind the grid. This suggests a relative independence of decaying stably stratified turbulence of initial conditions.

This paper presents experiments on run-up of strongly nonlinear waves on a beach of 10.54° inclination. Velocity fields are obtained by the PIV (particle image velocimetry) technique. Acceleration measurements are also attempted, but it is difficult to obtain useful results in every case. In addition, free-surface profiles are extracted from digital images and wave resistance probes. The investigation focuses on the dynamics of the early stages of the run-up, when steep fronts evolve in the vicinity of the equilibrium shoreline, but maximum run-up heights are also reported. Measurements on moderately nonlinear waves are compared to results from long-wave theories, including a numerical Boussinesq model and analytic shallow-water results from the literature. In particular the applicability of the long-wave theories is addressed. However, most attention is given to run-up of high incident solitary waves that are on the brink of breaking at the shoreline. In one case a temporarily slightly overturning wave front is found that neither develops into a plunger or displays appreciable spilling. This feature is discussed in view of measured velocity and acceleration patterns and with reference to the dam-break problem. Effects of scaling, as well as viscous damping, are also briefly discussed.

This paper describes a numerical and experimental study of the effect of heat input on the behaviour of the vortices shed from a horizontal cylinder in a horizontal cross-flow. The Reynolds number (ReD) is fixed at 75, while the Richardson number (RiD) is varied between 0 and 1 (corresponding to forced and mixed convection, respectively). In this parameter regime the wake consists of a double row of alternately shed vortices. A rather unexpected effect of the induced heat is the downward motion of the shed vortex structures. Detailed experiments and numerical simulations show that this effect is caused by the difference in strength between the two vortex rows. An analysis of the vorticity sources present during the formation process shows that the thermally induced baroclinic vorticity production is mainly responsible for this.

Turbulent drag reduction by polymer additives in a channel is investigated using direct numerical simulation. The dilute polymer solution is expressed with an Oldroyd-B model that shows a linear elastic behaviour. Simulations are carried out by changing the Weissenberg number at the Reynolds numbers of 4000 and 20 000 based on the bulk velocity and channel height. The onset criterion for drag reduction predicted in the present study shows a good agreement with previous theoretical and experimental studies. In addition, the flow statistics such as the r.m.s. velocity fluctuations are also in good agreement with previous experimental observations. The onset mechanism of drag reduction is interpreted based on elastic theory, which is one of the most plausible hypotheses suggested in the past. The transport equations for the kinetic and elastic energy are derived for the first time. It is observed that the polymer stores the elastic energy from the flow very near the wall and then releases it there when the relaxation time is short, showing no drag reduction. However, when the relaxation time is long enough, the elastic energy stored in the very near-wall region is transported to and released in the buffer and log layers, showing a significant amount of drag reduction.

Experimental results are presented for the motion of a dry spot in a thin viscous film on a horizontal surface. These include global and spatial measurements of dry spot diameter, front velocities, static and dynamic contact angle, and the shape of the liquid–solid interface. Data are presented as a function of initial fluid depth for both an advancing fluid front of a collapsing dry spot and a receding fluid front of an opening dry spot. Results for both cases show that the final or static hole diameter increases as the initial fluid depth decreases. Also, insight is obtained into the relationship between the contact angle and the velocity for both advancing and receding fluid fronts. The experimental results are compared to a lubrication model, and good agreement is obtained.

Trapped waves generated by oscillatory sources or dipoles placed above a plane infinite beach are examined within the framework of a (classical) non-hydrostatic but linear theory. This is achieved by solving a boundary-value problem where the boundary conditions are specified on the free surface and on the bottom. Integral expressions are derived for the complex potential for the cases where the sources or dipoles are strategically positioned to mimic the presence of solid bodies, a phenomenon manifested by the observation of a streamline enclosing the source or dipole. The precise positioning is governed by the further requirement of no radiating waves and, for the case where the beach is a vertical cliff, some recent results are confirmed here, whilst new results obtained show that infinitely many submerged wave trapping bodies exist and do so over a far greater range of values of dipole positions than was previously thought to be the case. The situation for surface sources and for submerged dipoles is therefore essentially different. For the former, infinitely many closed streamlines exist for each of the denumerably infinite set of source positions. For the latter, it is found instead that only one closed streamline exists, but this is for each of a non-denumerably infinite set of dipole positions. The expressions obtained for the beach are used for the two cases of a surface source and a submerged dipole to compute streamlines and stagnation points for model beaches of chosen steep slope. In particular, a (randomly chosen) submerged closed streamline is calculated for the beach of angle 45° thereby establishing a new case of non-uniqueness for the water wave problem on a beach.

A study is carried out of the linear global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. It is based on direct numerical simulations of the complete linearized Navier–Stokes equations obtained with the novel velocity–vorticity method described in Davies & Carpenter (2001). As the equations are linear, they become separable with respect to the azimuthal coordinate, $\theta$. This permits us to simulate a single azimuthal mode. Impulse-like excitation is used throughout. This creates disturbances that take the form of wavepackets, initially containing a wide range of frequencies. When the real spatially inhomogeneous flow is approximated by a spatially homogeneous flow (the so-called parallel-flow approximation) the results ofthe simulations are fully in accordance with the theory of Lingwood (1995). If the flow parameters are such that her theory indicates convective behaviour the simulations clearly exhibit the same behaviour. And behaviour fully consistent with absolute instability is always found when the flow parameters lie within the theoretical absolutely unstable region. The numerical simulations of the actual inhomogeneous flow reproduce the behaviour seen in the experimental study of Lingwood (1996). In particular, there is close agreement between simulation and experiment for the ray paths traced out by the leading and trailing edges of the wavepackets. In absolutely unstable regions the short-term behaviour of the simulated disturbances exhibits strong temporal growth and upstream propagation. This is not sustained for longer times, however. The study suggests that convective behaviour eventually dominates at all the Reynolds numbers investigated, even for strongly absolutely unstable regions. Thus the absolute instability of the rotating-disc boundary layer does not produce a linear amplified global mode as observed in many other flows. Instead the absolute instability seems to be associated with transient temporal growth, much like an algebraically growing disturbance. There is no evidence of the absolute instability giving rise to a global oscillator. The maximum growth rates found for the simulated disturbances in the spatially inhomogeneous flow are determined by the convective components and are little different in the absolutely unstable cases from the purely convectively unstable ones. In addition to the study of the global behaviour for the usual rigid-walled rotating disc, we also investigated the effect of replacing an annular region of the disc surface with a compliant wall. It was found that the compliant annulus had the effect of suppressing the transient temporal growth in the inboard (i.e. upstream) absolutely unstable region. As time progressed the upstream influence of the compliant region became more extensive.

Vortex breakdown of nominally axisymmetric, swirling incompressible flows with jet- and wake-like axial velocity distributions issuing into a semi-infinite domain is studied by means of direct numerical simulations. By selecting a two-parametric velocity profile for which the steady axisymmetric breakdown is well-studied (Grabowski & Berger 1976), issues are addressed regarding the role of three-dimensionality and unsteadiness with respect to the existence, mode selection, and internal structure of vortex breakdown, in terms of the two governing parameters and the Reynolds number. Low Reynolds numbers are found to yield flow fields lacking breakdown bubbles or helical breakdown modes even for high swirl. In contrast, highly swirling flows at large Reynolds numbers exhibit bubble, helical or double-helical breakdown modes, where the axisymmetric mode is promoted by a jet-like axial velocity profile, while a wake-like profile renders the flow helically unstable and ultimately yields non-axisymmetric breakdown modes. It is shown that a transition from super- to subcritical flow, as defined by Benjamin (1962), accurately predicts the parameter combination yielding breakdown, if applied locally to flows with supercritical inflow profiles. Thus the basic form of breakdown is axisymmetric, and a transition to helical breakdown modes is shown to be caused by a sufficiently large pocket of absolute instability in the wake of the bubble, giving rise to a self-excited global mode. Two distinct eigenfunctions corresponding to azimuthal wavenumbers $m\,{=}\,{-}1$ and $m\,{=}\,{-}2$ have been found to yield a helical or double-helical breakdown mode, respectively. Here the minus sign represents the fact that the winding sense of the spiral is opposite to that of the flow.