This paper concerns the elliptical instability of a flow in a rotating deformed sphere. The aim of our work is to analyse the characteristics of this instability experimentally and to provide a theoretical model which accounts for the observations. For this purpose, an elastic and transparent hollow sphere has been moulded in a silicone gel block. The flow is visualized using Kalliroscope flakes illuminated with a laser sheet as the sphere is set into rotation and compressed by two rollers. The elliptical instability occurs by the appearance of the so-called ‘spin-over’ mode whose growth rate and saturation amplitude are measured by video image analysis at different Ekman numbers. Growth rates are predicted well by the linear stability analysis. A nonlinear model is developed and is shown to describe correctly the saturated regimes observed in the experiments. At low Ekman numbers, a secondary instability leading to an intermittent regime is also discovered.

In this study self-organized periodic coherent vortex structures arising in geophysical turbulent flows at low Rossby number are investigated by developing a conceptual model based on an analytical theory of von Kármán vortex streets affected by stratification and differential rotation. In the framework of a quasi-geostrophic (QG) two-layer beta-plane model vortex streets with three different types of vertical structures (barotropic, upper layer and hetonic) are analysed using the point vortex approximation. The streets are found to be exact solutions of the potential vorticity equation and to be characterized by four non-dimensional parameters. Von Kármán streets are semi-localized solutions which form a bridge between vortex pairs (limit of symmetric dilute streets) and two parallel vortex sheets (limit of dense streets). On the beta-plane QG von Kármán streets can only move to the east, i.e. with a speed outside the range of speeds of Rossby waves, so that a dynamical asymmetry in the zonal direction is introduced. A complete classification on a diagram of states shows that critical bounds exist in the parameter space, prescribing for example a maximum distance between vortex rows beyond which no QG vortex streets can be found. Typically a fast and a slow vortex street with different flow structures are found in the region of existence. As a function of distance between vortex rows baroclinic QG vortex streets show a characteristic non-monotonic speed behaviour at scales of the order of the baroclinic Rossby radius. A wide region of possible existence of QG von Kármán streets is found in atmospheric, oceanic and planetary conditions as well as in rotating tank experiments. The theory can be applied to describe the coherent part of turbulent baroclinic intermittent zonal jet-like and frontal flows and provides a scaling for such flows.

The effect of an insoluble surfactant on the stability of two-layer viscous flow in an inclined channel confined by two parallel walls is considered. A lubrication-flow model applicable to long waves and low-Reynolds-number-flow is developed, and pertinent nonlinear evolution equations for the interface position and surfactant concentration are derived. Linear stability analysis based on the lubrication-flow model and the inclusive equations of Stokes flow confirm the recent findings of Frenkel & Halpern (2002) and Halpern & Frenkel (2003) that the presence of an insoluble surfactant induces a Marangoni instability in certain regions of parameter space defined by the layer thickness and viscosity ratios. Numerical simulations based on both approaches show that the interfacial waves may grow and saturate into steep profiles. The lubrication-flow model is adequate in capturing the essential features of the instability for small and moderate wavenumbers.

A modification of the Loitsyanskii integral is proposed as an invariant of three-dimensional decaying homogeneous and isotropic turbulence. As a result, the kinetic energy of a flow, generated in an infinitely large volume by the initial energy spectrum $E(k,t\,{=}\,0)$, which peaks in the vicinity of the wavenumber $k_{0}\,{=}\,1/{\cal L}(0)\,{=}\,O(1)$, $\cal L$ being an integral scale, decays with time in accordance with Kolmogorov's (1941) prediction: ${\cal K}\,{=}\,\frac{1}{2}\overline {u^{2}}\,{\propto}\, t^{-\gamma}$ with $\gamma\,{=}\,10/7$.

The role of particles heavier than the fluid (glass spheres in water) in a turbulent open channel flow over a smooth bed is examined at volume concentration about $10^{-3}$. The present work focuses on the dynamical interaction between the solid (particles) and the fluid phases in the near-wall region. Experimental measurements have been performed by means of phase Doppler anemometry to acquire two velocity components, particle size and concentration data simultaneously; the Reynolds number of the flow was close to 15 000. It is observed that in the particle-laden flow, the vertical profiles of the streamwise mean velocity (for both fluid and solid phases) are reduced in the outer layer ($y^{ + }\,{ >}\, 20$), but increased in the viscous sublayer ($y^{ + }\,{<}\,5$) in comparison to the clear-water conditions, leading to an apparent slip kinematic boundary condition close to the wall ($y^{ + } \,{\approx}\,2$). Moreover, in the presence of solid particles, the flow exhibits a velocity close to the wall ($y^{ + }\,{ <}\, 15$) which is smaller than that of the particles, while in the outer layer the opposite takes place. In particle-laden flow, turbulence intensities of the streamwise and especially of the vertical velocity are damped for $y^{ + }\,{>}\,10$–20 (depending on particle inertia) but enhanced in the very near-wall region ($y^{ + }\,{ <}\, 5$), as is the Reynolds stress. These findings can be explained if they are referred to the mechanism of particle entrainment and deposition, which takes place close to the wall. This mechanism is related to particle inertia and to the dynamic of the structure of near-wall turbulence, which connects the buffer and outer regions with the very near-wall region. A significant momentum exchange between the two phases, which is particularly effective in the buffer region, is revealed by the quadrant analysis of the Reynolds stresses.

Direct numerical simulations have been carried out for decaying homogeneous isotropic turbulence in a periodic box. Data for both the velocity and passive scalar fields are considered, the latter for several values of the Schmidt number $\hbox{\it Sc}$. The focus is on how the three-dimensional spectra $E(k,t)$ and $E_{\theta} (k,t)$ and the spectral transfer functions $T(k,t)$ and $T_{\theta } (k,t)$ satisfy similarity during decay. The evolution of these four quantities provides qualified support for the equilibrium similarity proposal of George (1992a, b). In particular, this proposal provides a reliable means of calculating the transfer functions, starting with known distributions of $E(k,t)$ and $E_{\theta} (k,t)$. However, at sufficiently large values of the wavenumber $k$, normalizations by Kolmogorov and Batchelor variables yield a better collapse of these quantities than the use of equilibrium similarity The distributions of $E_{\theta} (k,t)$ and $T_{\theta} (k,t)$ do not depend on $\hbox{\it Sc}$, when the latter is in the range $0.7 \,{\leqslant}\, \hbox{\it Sc} \,{\leqslant}\, 7$, irrespective of the normalization adopted. The velocity derivative skewness and mixed velocity–scalar derivative skewness approach constant values as $t$ increases. This is in disagreement with equilibrium similarity but in accord with the observed high-wavenumber collapse of Kolmogorov and Batchelor normalized distributions of $E(k,t)$ and $E_{\theta} (k,t)$.

The superharmonic instability is pervasive in large-amplitude water-wave problems and numerical simulations have predicted a close connection between it and crest instabilities and wave breaking. In this paper we present a nonlinear theory, which is a generic nonlinear consequence of superharmonic instability. The theory predicts the nonlinear behaviour witnessed in numerics, and gives new information about the nonlinear structure of large-amplitude water waves, including a mechanism for noisy wave breaking.

Lateral variations in the thermal boundary conditions at the horizontal boundary of an otherwise stably stratified fluid layer drive circulatory motion. For a rapidly rotating electrically conducting fluid, the introduction of a background vertical magnetic field reduces the dimensionless strength of the thermal wind from $O(1)$ to $O(E^{1/4})$ for prescribed temperature variation or to $O(E^{1/2})$ for prescribed heat-flux variation, where $E$ is the Ekman number. This is a significant effect when considering the spatially variable cooling of the Earth's core by the mantle. A general discussion of linear hydromagnetic flows identifies a large number of lengthscales inherent in the differential system as functions of the Ekman, Elsasser and stratification numbers, and shows that other scalings arise from the boundary conditions.

The dynamics of the sublayer and buffer regions of wall-bounded turbulent flows are analysed using autonomous numerical simulations in which the outer flow, and on some occasions specific wavelengths, are masked. The results are compared with a turbulent channel flow at moderate Reynolds number. Special emphasis is put on the largest flow scales. It is argued that in this region there are two kinds of large structures: long and narrow ones which are endogenous to the wall, in the sense of being only slightly modified by the presence or absence of an outer flow, and long and wide structures which extend to the outer flow and which are very different in the two cases. The latter carry little Reynolds stress near the wall in full simulations, and are largely absent from the autonomous ones. The former carry a large fraction of the stresses in the two cases, but are shown to be quasi-linear passive wakes of smaller structures, and they can be damped without modifying the dynamics of other spectral ranges. They can be modelled fairly accurately as being infinitely long, and it is argued that this is why good statistics are obtained in short or even in minimal simulation boxes. It is shown that this organization implies that the scaling of the near-wall streamwise fluctuations is anomalous.

We present a simple approximate model for studying general aspects of vortex interactions in a rotating stably-stratified fluid. The model idealizes vortices by ellipsoidal volumes of uniform potential vorticity, a materially conserved quantity in an inviscid, adiabatic fluid. Each vortex thus possesses 9 degrees of freedom, 3 for the centroid and 6 for the shape and orientation. Here, we develop equations for the time evolution of these quantities for a general system of interacting vortices. An isolated ellipsoidal vortex is well known to remain ellipsoidal in a fluid with constant background rotation and uniform stratification, as considered here. However, the interaction between any two ellipsoids in general induces weak non-ellipsoidal perturbations. We develop a unique projection method, which follows directly from the Hamiltonian structure of the system, that effectively retains just the part of the interaction which preserves ellipsoidal shapes. This method does not use a moment expansion, e.g. local expansions of the flow in a Taylor series. It is in fact more general, and consequently more accurate. Comparisons of the new model with the full equations of motion prove remarkably close.

In this paper we investigate the interaction between a solitary wave and a thin vertical barrier. A set of laboratory experiments was performed with different values of incident wave height to water depth ratio, $H/h$, and the draught of the barrier to water depth ratio, $D/h$. While wave gauges were used to measure the reflected and transmitted waves, pressure transducers were installed on both sides of the barrier, enabling the calculation of wave force. The particle image velocimetry (PIV) technique is also employed to measure the velocity field in the vicinity of the barrier. A numerical model, based on the Reynolds-averaged Navier–Stokes (RANS) equations and the $k \hbox{--} \epsilon$ turbulence closure model, was first checked with experimental data and then employed to obtain additional results for the range of parameters where the laboratory experiments were not performed. Using both experimental data and numerical results, formulae for the maximum runup height, and the maximum wave force are derived in terms of $H/h$ and $D/h$.

We investigate the influence of fluid inertia on the motion of a finite assemblage of solid spherical particles in slowly changing uniform flow at small Reynolds number, $Re$, and moderate Strouhal number, $\hbox{\it Sl}$. We show that the first effect of fluid inertia on particle velocities for times much larger than the viscous time scales as $\sqrt{\hbox{\it Sl\,Re}}$ given that the Stokeslet associated with the disturbance flow field changes with time. Our theory predicts that the correction to the particle motion from that predicted by the zero-$Re$ theory has the form of a Basset integral. As a particular example, we calculate the Basset integral for the case of two unequal particles approaching (receding) with a constant velocity along the line of their centres. On the other hand, when the Stokeslet strength is independent of time, the first effect of fluid inertia reduces to a higher order of magnitude and scales as $Re$. This condition is fulfilled, for example, in the classical problem of sedimentation of particles in a constant gravity field.

The viscous linear stability of four classes of incompressible flows inside rectangular containers is studied numerically. In the first class the instability of flow through a rectangular duct, driven by a constant pressure gradient along the axis of the duct (essentially a two-dimensional counterpart to plane Poiseuille flow – PPF), is addressed. The other classes of flow examined are generated by tangential motion of one wall, in one case in the axial direction of the duct, in another perpendicular to this direction, corresponding respectively to the two-dimensional counterpart to plane Couette flow (PCF) and the classic lid-driven cavity (LDC) flow, and in the fourth case a combination of both the previous tangential wall motions. The partial-derivative eigenvalue problem which in each case governs the temporal development of global three-dimensional small-amplitude disturbances is solved numerically. The results of Tatsumi & Yoshimura (1990) for pressure-gradient-driven flow in a rectangular duct have been confirmed; the relationship between the eigenvalue spectrum of PPF and that of the rectangular duct has been investigated. Despite extensive numerical experimentation no unstable modes have been found in the wall-bounded Couette flow, this configuration found here to be more stable than its one-dimensional limit. In the square LDC flow results obtained are in line with the predictions of Ding & Kawahara (1998b), Theofilis (2000) and Albensoeder et al. (2001b) as far as one travelling unstable mode is concerned. However, in line with the predictions of the latter two works and contrary to all previously published results it is found that this mode is the third in significance from an instability analysis point of view. In a parameter range unexplored by Ding & Kawahara (1998b) and all prior investigations two additional eigenmodes exist, which are both more unstable than the mode that these authors discovered. The first of the new modes is stationary (and would consequently be impossible to detect using power-series analysis of experimental data), whilst the second is travelling, and has a critical Reynolds number and frequency well inside the experimentally observed bracket. The effect of variable aspect ratio $A\in[0.5,4]$ of the cavity on the most unstable eigenmodes is also considered, and it is found that an increase in aspect ratio results in general destabilization of the flow. Finally, a combination of wall-bounded Couette and LDC flow, generated in a square duct by lid motion at an angle $\phi\in(0,{\pi}/{2})$ with the homogeneous duct direction, is shown to be linearly unstable above a Reynolds number $\Rey\,{=}\,800$ (based on the lid velocity and the duct length/height) at all $\phi$ parameter values examined. The excellent agreement with experiment in LDC flow and the alleviation of the erroneous prediction of stability of wall-bounded Couette flow is thus attributed to the presence of in-plane basic flow velocity components.

A simple, time-dependent model is proposed to predict the heat and salt fluxes transferred between a hot, salty fluid layer and an overlying colder, fresher fluid layer when the overall density of the lower layer is greater than that of the upper layer. The interface separating the two layers, known as a ‘diffusive’ interface, consists of a purely diffusing core sandwiched between vigorously convecting boundary layers. This type of interface occurs whenever the density of superposed fluid layers depends on two diffusing agents with different diffusivities such that the faster diffusing agent causes a statically unstable density variation while the slower diffusing agent causes a statically stable variation. Early experimental and theoretical studies sought a single relationship between the heat flux $F_T$ across a diffusive interface and the buoyancy ratio $R_\rho\,{=}\,\beta\uDelta S/\alpha\uDelta T$, where $\uDelta T$ and $\uDelta S$ are the differences in temperature and salinity between the well-mixed fluid layers, while $\alpha$ and $\beta$ are the coefficients of thermal and solutal expansion respectively. The model presented here supports more recent experimental findings that the relationship is time-dependent and therefore that results depend on the initial conditions and any forcing applied to the mixed regions. It is predicted that the evolution of the thickness of the core is determined principally by the mismatch between the transport rates of the slower-diffusing species (salt) diffusively from the core and convectively into the mixed regions. The introduction of time-dependence brings together the data from many previous experimental studies for both heat–salt and salt–sugar systems over a wide range of $R_\rho$, and both links and puts into context two previous theories of diffusive interfaces.

It is common to relate the dynamic contact angle $\theta_d$ to the relative speed between the substrate and the contact line; theory suggests $\theta_d^3 \propto U$. In fact, available physical models show that the dynamic angle involves speed logarithmically and in a model-dependent manner. Experimental data consistent with this interpretation are cited.

Fully nonlinear solitary waves in a layered stratified fluid, each layer with a constant Brunt–Väisälä frequency, are investigated. The stream function satisfies the Helmholtz equation in each layer and is expressed in terms of singularity distributions. As the Green function, a combination of Bessel functions of order zero, of the second and first kind is advocated. Computations performed for two- and three-layer cases show that the wave speed increases with increasing stratification of the top layer. The thickness of the pycnocline increases with wave amplitude when the top layer is homogeneous but decreases when the top layer is stratified. The wave width depends little on the pycnocline thickness. The fluid velocity may exceed the wave speed in the upper part of the water column when the top layer is stratified, but is always smaller than the wave velocity if the top layer is homogeneous. A large vertical excursion of the individual isopycnals contributes to a small Richardson number $Ri$. The smallest value of $Ri$ is observed in the main body of the fluid. Solitary waves of increasing strength are investigated until the wave-induced fluid velocity equals the wave speed, or the minimal $Ri$ becomes smaller than one quarter. The results may support experimental studies of breaking internal solitary waves.

The inviscid compressible flow generated by a rigid body of volume ${\cal V}$ moving unsteadily with a velocity ${\bm U}$ in a rapidly compressed homentropic flow is considered. The fluid is compressed isentropically at a rate ${\bm \nabla}\,{\bm \cdot}\,{\bm v}_0$ uniformly over a scale much larger than the size of the body and the body moves slowly enough that the Mach number $M$ is low. The flow is initially irrotational and remains so during compression. The perturbation to the flow generated by the body moving unsteadily is non-divergent within an evolving region ${\cal D}$ of distance $\int_0^t c_1\,{\rm d}t$ from the body, where $c_1$ is the speed of sound. Within ${\cal D}$, the flow is dominated by a source of strength $({\bm \nabla} \,{\bm \cdot}\, {\bm v}_0){\cal V}$ and a dipolar contribution which is independent of the rate of compression, while outside ${\cal D}$, compressional waves propagate away from the body. When the body is much smaller than the characteristic distance $\|({\bm \nabla}{\bm v}_0)|_{{\bm x}_0}\|/\|({\bm \nabla} {\bm \nabla} {\bm v}_0 )|_{{\bm x}_0}\|$ and the size of the region ${\cal D}$, the separation of length scales enables the force on the body to be calculated analytically from the momentum flux far from the body (but within the region ${\cal D}$). The contribution to the total force arising from fluid compression is $\rho(t) ({\bm \nabla} \,{\bm \cdot}\, {\bm v}_0) {\cal V} ({\bm U}-{\bm v}_0)\,{\bm \cdot}\, \boldsymbol{\alpha} $, where ${\bm v}_0$ is the velocity field in the absence of the particles and $\boldsymbol{\alpha}$ is the virtual inertia tensor. Thus a body experiences a drag (thrust) force during fluid compression (expansion) because the density of the fluid displaced forward by the body increases (decreases) with time. The analysis indicates that the sum of the compressional and added-mass force is equal to the rate of decrease of fluid impulse ${\bm P} = \rho(t){\cal V}({\bm U}-{\bm v}_0)\,{\bm \cdot}\, \boldsymbol{\alpha}$. Thus the concept of fluid impulse naturally extends to the class of flows where the fluid density changes with time, but is spatially uniform.

These new results are applied to consider the inviscid dynamics of a rigid sphere and cylinder projected into a uniformly compressed or expanded fluid. When the fluid rapidly expands, a rigid body ultimately moves with a constant velocity because the total force, which is proportional to the density of the fluid, tends rapidly to zero. When the body moves perpendicular to the axis of compression, it slows down and stops when the density of the fluid is comparable to the density of the body. However, a body moving parallel to the axis of compression is accelerated by pressure gradients which are proportional to fluid density and increases in time.

Dissipation approximations have been used to calculate the drag on bubbles and drops and the decay rate of free gravity waves on water. In these approximations, viscous effects are calculated by evaluating the viscous stresses on irrotational flows. The pressure is not involved in the dissipation integral, but it enters into the power of traction integral, which equals the dissipation. A viscous correction of the irrotational pressure is needed to resolve the discrepancy between the zero-shear-stress boundary condition at a free surface and the non-zero irrotational shear stress. Here we show that the power of the pressure correction is equal to the power of the irrotational shear stress. The viscous pressure correction on the interface can be expressed by a harmonic series. The principal mode of this series is matched to the velocity potential and its coefficient is explicitly determined. The other modes do not enter into the expression for the drag on bubbles and drops. They vanish in the case of free gravity waves.