Asymptotic and numerical methods are used to analyse periodically forced natural convection over slowly varying topography. This models the diurnal heating/cooling cycle in lakes and reservoirs. The asymptotic solution includes the effects of advection on the temperature. The asymptotic results are confirmed by the numerical results. The numerical results are also used to examine flow regimes where the asymptotic results break down. In particular, the presence of a vertical boundary leads to a permanent stratification in the deeper regions due to a nonlinear pumping process in the shallows. Heat transfer calculations and two limiting cases are also presented.

A dominant aspect of granular flows is flow in thin surface layers. While an understanding of the dynamics of dry granular surface flow has begun to emerge, the case of flow when air is completely replaced by a liquid is largely unexplored. Experiments were performed using particle tracking velocimetry (PTV) in a quasi-two-dimensional rotating tumbler to measure the velocity field within the flowing layer of monodisperse spherical particles fully submerged in liquids, a granular slurry, for a range of Froude numbers, bead sizes, fluid densities and fluid viscosities. The thickness of the flowing layer and the angle of repose with a liquid interstitial fluid are generally larger than for the dry system under similar conditions, although the shear rate is generally smaller. The experimental measurements of shear rate match the theoretical predictions (dependent on the particle size, dynamic angle of repose, and static angle of repose) independent of the interstitial fluid. Furthermore, the velocity profiles for larger beads collapse independent of the interstitial fluid, while for smaller beads these profiles collapse on two distinct curves when using a scaling based on mass balance. However, a normalization based on the velocity of beads at the surface causes a collapse to a nearly linear velocity profile except where the velocity approaches zero logarithmically near the fixed bed, regardless of interstitial fluid. Likewise, the scaled number density profiles collapse, regardless of the interstitial fluid. The similarity in the flows of dry granular materials and granular materials submerged in liquids indicates that the physics of the flow is not strongly altered by the interstitial fluid.

The incipient separation from a corner in steady two-dimensional transonic flow is studied based on viscous–inviscid interaction at high Reynolds number. Of particular interest is the investigation of the dependence of the critical deflection angle (when a well-attached flow turns into a separated flow) on the Kármán–Guderley parameter which characterizes the local flow field. In accordance with the procedure adopted, the analysis of the flow starts with the analysis of the boundary layer and then the solution of the Kármán–Guderley equation describing the inviscid part of the flow near the corner point is investigated. The analysis of the inviscid transonic flow is performed based on the hodograph method and new solutions are obtained corresponding to the present flow topologies. In these solutions, the transonic flow appears to be subsonic everywhere except at the sonic corner point. Then, the interaction problem is formulated using the triple-deck model. Lastly, a procedure based on a semi-direct solution of the governing equations using Newton iterations is developed to obtain the numerical solution of the interaction problem.

Large-amplitude capillary waves on fluid sheets are computed in the presence of a uniform electric field acting in a direction parallel to the undisturbed configuration. The fluid is taken to be inviscid, incompressible and non-conducting. Travelling waves of arbitrary amplitudes and wavelengths are calculated and the effect of the electric field is studied. The solutions found generalize the exact symmetric solutions of Kinnersley (1976) to include electric fields, for which no exact solutions have been found. Long-wave nonlinear waves are also constructed using asymptotic methods. The asymptotic solutions are compared with the full computations as the wavelength increases, and agreement is found to be excellent.

This paper presents the extension of our previous investigation of confined round jets with large Reynolds numbers and large expansion ratios (Revuelta, Sánchez & Liñán 2002a) to the case of swirling jets with swirl numbers of order unity. In the absence of vortex breakdown, we encounter the four-region asymptotic structure identified earlier for the non-swirling jet, including a region of jet development where the azimuthal and axial velocity components are comparable. For the flow in the long recirculating eddy that forms downstream, where the pressure differences associated with the azimuthal motion become negligible, the jet is found to act as a point source with momentum flux equal to the flow force of the incoming jet, and angular momentum flux equal to that of the jet at the orifice. The solution for the weak circulation in this slender region, including the parameter-free leading-order description and the first-order corrections, is determined by integrating the azimuthal component of the momentum equation written in the boundary-layer approximation. The results are validated through comparisons with numerical integrations of the steady axisymmetric Navier–Stokes equations, which are also used to evaluate critical conditions for vortex breakdown.

Statistics of the streamwise velocity component in fully developed pipe flow are examined for Reynolds numbers in the range $5.5 \times 10^4 \le \hbox{\it Re}_D \le 5.7 \times 10^6$. Probability density functions and their moments (up to sixth order) are presented and their scaling with Reynolds number is assessed. The second moment exhibits two maxima: the one in the viscous sublayer is Reynolds-number dependent while the other, near the lower edge of the log region, follows approximately the peak in Reynolds shear stress. Its locus has an approximate $(R^+)^{0.5}$ dependence. This peak shows no sign of ‘saturation’, increasing indefinitely with Reynolds number. Scalings of the moments with wall friction velocity and $(U_{cl}-\overline{U})$ are examined and the latter is shown to be a better velocity scale for the outer region, $y/R > 0.35$, but in two distinct Reynolds-number ranges, one when $\hbox{\it Re}_D < 6 \times 10^4$, the other when $\hbox{\it Re}_D > 7 \times 10^4$. Probability density functions do not show any universal behaviour, their higher moments showing small variations with distance from the wall outside the viscous sublayer. They are most nearly Gaussian in the overlap region. Their departures from Gaussian are assessed by examining the behaviour of the higher moments as functions of the lower ones. Spectra and the second moment are compared with empirical and theoretical scaling laws and some anomalies are apparent. In particular, even at the highest Reynolds number, the spectrum does not show a self-similar range of wavenumbers in which the spectral density is proportional to the inverse streamwise wavenumber. Thus such a range does not attract any special significance and does not involve a universal constant.

We investigate viscous fingering instabilities of the double-front system which results when fluid is injected into a porous medium containing fluid of a different composition and temperature. We describe a linear stability analysis based on an eigenfunction expansion method which enables us to investigate the structure of the discrete eigenvalue spectrum. We investigate the extent to which the properties of each front contribute to the tendency of the system to become unstable: we find that instabilities on the compositional front dominate because of the high ratio of thermal to solute diffusion. It is difficult for a stable compositional front to stabilize an unstable thermal front; however, this situation can result in a new fingering phenomenon in which the perturbations undergo coupled oscillations of growing amplitude.

We consider the steady levitation of a rigid plate on a thin air cushion with prescribed injection velocity. This injection velocity is assumed to be much larger than that in a conventional Prandtl boundary layer, so that inertial effects dominate. After applying the classical ‘blowhard’ theory of Cole & Aroesty (1968) to the two-dimensional version of the problem, it is shown that in three dimensions the flow may be foliated into streamline surfaces using Lagrangian variables. An example is given of how this may be exploited to solve the three-dimensional problem when the injection pressure distribution is known.

Experimental observations of the collapse of initially vertical columns of small grains are presented. The experiments were performed mainly with dry grains of salt or sand, with some additional experiments using couscous, sugar or rice. Some of the experimental flows were analysed using high-speed video. There are three different flow regimes, dependent on the value of the aspect ratio $a\,{=}\,h_i/r_i$, where $h_i$ and $r_i$ are the initial height and radius of the granular column respectively. The differing forms of flow behaviour are described for each regime. In all cases a central, conically sided region of angle approximately $ 59^\circ$, corresponding to an aspect ratio of 1.7, remains undisturbed throughout the motion. The main experimental results for the final extent of the deposit and the time for emplacement are systematically collapsed in a quantitative way independent of any friction coefficients. Along with the kinematic data for the rate of spread of the front of the collapsing column, this is interpreted as indicating that frictional effects between individual grains in the bulk of the moving flow only play a role in the last instant of the flow, as it comes to an abrupt halt. For $a\,{<}\,1.7$, the measured final runout radius, $r_\infty$, is related to the initial radius by $r_\infty \,{=}\, r_i(1\,{+}\,1.24a)$; while for $1.7\,{<}\,a$ the corresponding relationship is $r_\infty \,{=}\,r_i(1\,{+}\,1.6a^{1/2})$. The time, $t_\infty$, taken for the grains to reach $r_\infty$ is given by $t_\infty \,{=}\,3(h_i/g)^{1/2}\,{=}\,3(r_i/g)^{1/2}a^{1/2}$, where $g$ is the gravitational acceleration. The insights and conclusions gained from these experiments can be applied to a wide range of industrial and natural flows of concentrated particles. For example, the observation of the rapid deposition of the grains can help explain details of the emplacement of pyroclastic flows resulting from the explosive eruption of volcanoes.

The scenario of transition to chaos for a sphere falling or ascending under the action of gravity in a Newtonian fluid is investigated by numerical simulation. The mathematical formulation is parameterized using two non-dimensional parameters: the solid/fluid density ratio and the generalized Galileo number expressing the ratio between the gravity–buoyancy and viscosity effects. The study is carried out fully in this two-parameter space. The results show that for all density ratios the vertical fall or ascension becomes unstable via a regular axisymmetry breaking bifurcation. This bifurcation sets in slightly earlier for light spheres than for dense ones. A steady oblique fall or ascension follows before losing stability and giving way to an oscillating oblique movement. The secondary Hopf bifurcation is shown not to correspond to that of a fixed sphere wake for density ratios lower than 2.5, for which the oscillations have a significantly lower frequency. Trajectories of falling spheres become chaotic directly from the oblique oscillating regime. Ascending spheres present a specific behaviour before reaching a chaotic regime. The periodically oscillating oblique regime undergoes a subharmonic transition yielding a low-frequency oscillating ascension which is vertical in the mean (zigzagging regime). In all these stages of transition, the trajectories are planar with a plane selected randomly during the axisymmetry breaking. The chaotic regime appears to result from an interplay of a regular and of an additional Hopf bifurcation and the onset of the chaotic regime is accompanied by the loss of the remaining planar symmetry. The asymptotic chaotic states present an intermittent character, the relaminarization phases letting the subcritical plane and periodic trajectories reappear.

Using a novel experimental set-up, we examine the stability of an axisymmetric dense current of fluid on a sloping bottom in a rotating frame of reference. In a cylindrical tank on a rotating table, saline fluid is injected through an annular mesh encircling a cone of 1/15 slope. The resulting ring of fluid becomes unstable to periodic sinusoidal waves, whose characteristics are examined as a function of the ambient fluid depth, the rotation rate, and the salinity and depth of the dense current, the latter being related to the injection time. For experiments with relatively low-salinity currents, we find the phase speed of the instability is proportional to the rotation rate, $\Omega$, rather than being proportional to the Nof speed, which varies as $1/\Omega$. Likewise, the number of waves observed is found to be inconsistent with existing theories for baroclinic instability of a dense current on a sloping bottom with a stationary ambient.

Surface tracers reveal that significant currents are induced in the ambient by the injection of the saline fluid and we propose that it is the coupling between the surface and bottom flow that ultimately controls the observed dynamics. In particular, we propose that barotropic instability of the surface flow controls the behaviour of the low-salinity current and, using potential vorticity conservation, we predict the phase speed of the instability should indeed be proportional to $\Omega$. The experimental results are also compared with related work in which eddies and Ekman layers evolve from dense fluid injected from a localized source.

The mechanism by which amplifying salt fingers in an unbounded uniform $T/S$ gradient are equilibrated is determined, starting with a time-dependent asymptotic field equation for $( {R\tau })^{ - 1} \,{-}\, 1 \,{=}\, \varepsilon \,{\to}\, 0$, where $R \,{>}\, 1$ is the $T/S$ density ratio and $\tau \,{=}\, K_S / K_T\,{<}\,1$ is the molecular diffusivity ratio. A mode truncation of that equation yields an ODE which shows that the fastest growing finger mode transfers energy to two ‘slave’ modes with relatively small vertical scale; the finger mode thereby attains a statistically steady amplitude. The results for $\tau \,{=}\, 1 / 3$ are compared with spectral solutions of the non-truncated equations in two and three dimensions; the predicted fluxes are testable in sugar ($S$), salt ($T$) laboratory experiments.

The paper reports on an investigation into the nonlinear evolution of a wavetrain emanating from a point source in a flat-plate boundary layer. The work was mainly experimental, but calculations using linear and nonlinear stability theory and parabolized stability equations were also performed to support the conclusions. The amplitudes of the disturbances used were very small and, as a consequence, turbulence was not reached within the experimental domain. Nevertheless, interesting nonlinear behaviour was observed. The nonlinear regime was characterized by the appearance of a three-dimensional mean flow distortion in the form of longitudinal streaks. It was possible to distinguish two different stages of the nonlinear regime. In the first stage, the streak pattern displayed relatively low spanwise wavenumbers. The pattern appeared to have grown algebraically from a pair of counter-rotating streamwise vortices. Weakly nonlinear calculations suggested that the vortices arise from the interaction of the spanwise and the wall-normal velocity components. The second stage exhibited more streaks and higher spanwise wavenumbers. This stage was found to be associated with a peak-and-valley structure of the wave amplitude. A remarkable feature was that the streamwise position of the onset of the second stage of the nonlinear regime was not affected by the amplitude of the disturbance. Parabolized stability calculations suggested that the peak-and-valley structure was the outcome of a secondary instability of the fundamental type. An interaction similar to that of the first nonlinear stage involving the waves composing the peak-and-valley structure yielded other streamwise vortices and hence a different streak pattern. The results also suggested that, because the wave amplitudes were very small, the linear stability influenced the nonlinear evolution. Indeed, the onset of the second stage of the nonlinear regime appeared to be associated with the proximity of the second branch of the linear stability diagram.

By numerically computing the steady axisymmetric flow of helium II confined inside a finite-aspect-ratio Couette annulus, we determine the transition from Ekman flow to Taylor vortex flow as a function of temperature and aspect ratio. We find that the low-Reynolds-number flow is quite different to that of a classical fluid, particularly at lower temperatures. At high aspect ratio our results confirm the existing linear stability theory of the onset of Taylor vortices, which assumes infinitely long cylinders.

Three-dimensional travelling wave solutions are found for pressure-driven fluid flow through a circular pipe. They consist of three well-defined flow features – streamwise rolls and streaks which dominate and streamwise-dependent wavy structures. The travelling waves can be classified by the $m$-fold rotational symmetry they possess about the pipe axis with $m\,{=}\,1,2,3,4,5$ and $6$ solutions identified. All are born out of saddle-node bifurcations with the lowest corresponding to $m\,{=}\,3$ and traceable down to a Reynolds number (based on the mean velocity) of 1251. The new solutions are found using a constructive continuation procedure based upon key physical mechanisms thought generic to wall-bounded shear flows. It is believed that the appearance of these new alternative solutions to the governing equations as the Reynolds number is increased is a necessary precursor to the turbulent transition observed in experiments.

Journal of Fluid Mechanics, vol. 489 (2003), pp. 345–348