An asymptotic modal system is derived for modelling nonlinear sloshing in a rectangular tank with similar width and breadth. The system couples nonlinearly nine modal functions describing the time evolution of the natural modes. Two primary modes are assumed to be dominant. The system is equivalent to the model by Faltinsen et al. (2000) for the two-dimensional case. It is validated for resonant sloshing in a square-base basin. Emphasis is on finite fluid depth but the behaviour with decreasing depth to intermediate depths is also discussed. The tank is forced in surge/sway/roll/pitch with frequency close to the lowest degenerate natural frequency. The theoretical part concentrates on periodic solutions of the modal system (steady-state wave motions) for longitudinal (along the walls) and diagonal (in the vertical diagonal plane) excitations. Three types of solutions are established for each case: (i) ‘planar’/‘diagonal’ resonant standing waves for longitudinal/diagonal forcing, (ii) ‘swirling’ waves moving along tank walls clockwise or counterclockwise and (iii) ‘square’-like resonant standing wave coupling in-phase oscillations of both the lowest modes. The frequency domains for stable and unstable waves (i)–(iii), the contribution of higher modes and the influence of decreasing fluid depth are studied in detail. The zones where either unstable steady regimes exist or there are two or more stable periodic solutions with similar amplitudes are found. New experimental results are presented and show generally good agreement with theoretical data on effective domains of steady-state sloshing. Three-dimensional sloshing regimes demonstrate a significant contribution of higher modes in steady-state and transient flows.

This paper theoretically describes and experimentally verifies two mechanisms leading to longitudinal dispersion of a passive tracer in a random array of circular cylinders. We focus on moderate Reynolds numbers of order 10–1000, specifically the range characterized by unsteady cylinder wakes. In this regime, two mechanisms contribute to dispersion, each associated with a distinct region of the cylinder wakes: (i) the unsteady recirculation zone close to each cylinder, and (ii) the velocity defect behind each cylinder, which extends downstream of the cylinder over a distance of the order of the cylinder spacing. The first mechanism, termed vortex-trapping dispersion, is due to the entrainment of tracer into the unsteady recirculation zone, where it is momentarily trapped and then released. A theoretical expression for this dispersive mechanism is derived in terms of the residence time and size of the recirculation zone. The second mechanism is due to advection through the random velocity field created by the random distribution of the wake velocity defect. We derive an expression for the defect behind an average cylinder, and show that it decays owing to array drag over a length scale called the attenuation length, which is of the order of the cylinder spacing. The superposition of the wake defect behind each cylinder creates the random velocity field. Theoretical predictions for dispersion agree very well with observations of tracer transport in a laboratory cylinder array, correctly capturing the dependence on array density and Reynolds number. The laboratory studies also document a transition in small-scale mixing at cylinder Reynolds number $\approx 200$. Below this limit, individual filaments of tracer remain distinct, producing significant fluctuations in the local concentration field. At higher Reynolds number, cylinder wakes contribute sufficient turbulence to erase the filament signature and smooth the tracer distribution.

We study steady laminar Ekman boundary layers in rotating systems using an averaging method similar to the technique of von Kármán and Pohlhausen. The method allows us to explore nonlinear corrections to the standard Ekman theory even at large Rossby numbers. We consider both the standard self-similar ansatz for the velocity profile, which assumes that a single length scale describes the boundary layer structure, and a new non-self-similar ansatz in which the decay and the oscillations of the boundary layer are described by two different length scales. For both profiles we calculate the up-flow in a vortex core in solid-body rotation analytically. We compare the quantitative predictions of the model with the family of exact similarity solutions due to von Kármán and find that the results for the non-self-similar profile are in almost perfect quantitative agreement with the exact solutions and that it performs markedly better than the self-similar profile.

The onset of convection of a Boussinesq fluid in a horizontal plane layer is studied. The system rotates with constant angular velocity $\Omega$, which is inclined at an angle $\vartheta$ to the vertical. The layer is sheared by keeping the upper boundary fixed, while the lower boundary moves parallel to itself with constant velocity ${\bm U}_0$ normal to the plane containing the rotation vector and gravity ${\bm g}$ (i.e. ${\bm U}_0\,\parallel\,{\bm g}\,{\times}\,\Omega)$. The system is characterized by five dimensionless parameters: the Rayleigh number $Ra$, the Taylor number $\tau^2$, the Reynolds number $Re$ (based on ${\bm U}_0$), the Prandtl number $Pr$ and the angle $\vartheta$. The basic equilibrium state consists of a linear temperature profile and an Ekman–Couette flow, both dependent only on the vertical coordinate $z$. Our linear stability study involves determining the critical Rayleigh number $Ra_c$ as a function of $\tau$ and $Re$ for representative values of $\vartheta$ and $Pr$.

Our main results relate to the case of large Reynolds number, for which there is the possibility of hydrodynamic instability. When the rotation is vertical $\vartheta=0$ and $\tau\gg 1$, so-called type I and type II Ekman layer instabilities are possible. With the inclusion of buoyancy $Ra\not=0$ mode competition occurs. On increasing $\tau$ from zero, with fixed large $Re$, we identify four types of mode: a convective mode stabilized by the strong shear for moderate $\tau$, hydrodynamic type I and II modes either assisted $(Ra>0)$ or suppressed $(Ra<0)$ by buoyancy forces at numerically large $\tau$, and a convective mode for very large $\tau$ that is largely uninfluenced by the thin Ekman shear layer, except in that it provides a selection mechanism for roll orientation which would otherwise be arbitrary. Significantly, in the case of oblique rotation $\vartheta\not=0$, the symmetry associated with ${\bm U}_0\leftrightarrow -{\bm U}_0$ for the vertical rotation is broken and so the cases of positive and negative $Re$ exhibit distinct stability characteristics, which we consider separately. Detailed numerical results were obtained for the representative case $\vartheta=\pi/4$. Though the overall features of the stability results are broadly similar to the case of vertical rotation, their detailed structure possesses a surprising variety of subtle differences.

Bar instability is recognized as the fundamental mechanism underlying the formation of large-scale forms of rivers. We show that the nature of such instability is convective rather than absolute. Such a result is obtained by revisiting the linear stability analysis of open-channel uniform flow over a cohesionless channel of Colombini et al. (1987) and using the Briggs (1964) criterion to distinguish between the convectively and absolutely unstable temporally asymptotic response to an initial boundary-value perturbation of bed topography. Examining the branch-point singularities of the dispersion relation, which can be determined in closed form, we show that all the existing branch-point singularities characterized by positive bar growth rate $\omega_{i}$, involve spatial branches of the dispersion relation which, for large positive values of $\omega_{i}$, lie in the same half $\lambda$-plane, $\lambda$ denoting the complex bar wavenumber. Hence, the nature of instability is convective and remains so for any value of the aspect ratio, the controlling parameter of the basic instability, as well as for any lateral mode investigated. The latter analytical findings are confirmed by numerical solutions of the fully nonlinear problem. In fact, starting from either a randomly distributed or a localized spatial perturbation of bed topography, groups of bars are found to grow and migrate downstream leaving the source area undisturbed. The actual bars observed in laboratory experiments arise from the spatial-temporal growth of some persistent initial perturbation. The nonlinear development of such perturbations is shown to lead to a periodic pattern with amplitude independent of the amplitude of the initial perturbation. Bars are also found to lengthen and slow down as they grow from the linear into the nonlinear regime, in agreement with experimental observations. The distance from the initial cross-section at which equilibrium is achieved depends on the initial amplitude of the perturbation, a finding which calls for a revisitation of classical laboratory observations reported in the literature.

Using phase-stepped interferometry, we have measured full two-dimensional maps of the free-surface shape of a thin liquid film of water flowing over an inclined plate with topography. The measurement technique allows us to image automatically the shape of the free surface in a single field of view of about 2.4 by 1.8 mm, with a lateral resolution of 3.1 μm and a height resolution of 0.3 μm. By imaging neighbouring regions and combining them, complete two-dimensional free-surface profiles of gravity-driven liquid films with a thickness ranging between 80 and 120 μm are measured, over step, trench, rectangular and square topographies with depths of 10 and 20 μm, and lateral dimensions of the order of 1 to several mm. The experimental results for both one- and two-dimensional flows are found to be in good agreement with existing models, including a recent two-dimensional Green's function of the linearized problem by Hayes et al. This extends the applicability of simple models to cases with a high value of topography steepness and low-viscosity liquids as in our experiments. A corollary of the agreement with the linear two-dimensional model is that our experimental results behave linearly, a convenient property that allows the free-surface response to complex topographies to be worked out from knowledge of the response to an elementary topography like a square.

We study the coalescence of two drops of an ideal fluid driven by surface tension. The velocity of approach is taken to be zero and the dynamical effect of the outer fluid (usually air) is neglected. Our approximation is expected to be valid on scales larger than $\ell_{\nu} = \rho\nu^2/\sigma$, which is 10 nm for water. Using a high-precision boundary integral method, we show that the walls of the thin retracting sheet of air between the drops reconnect in finite time to form a toroidal enclosure. After the initial reconnection, retraction starts again, leading to a rapid sequence of enclosures. Averaging over the discrete events, we find the minimum radius of the liquid bridge connecting the two drops to scale like $r_b \propto t^{1/2}$.

Based on a hydrodynamic length, which is typically larger than the nominal flame thickness, a premixed flame can be viewed as a surface of density discontinuity, advected and distorted by the flow. The velocities and the pressure suffer abrupt changes across the flame front that consist of Rankine–Hugoniot jump conditions, to leading order, with corrections of the order of the flame thickness that account for transverse fluxes and accumulation. To complete the formulation, expressions for the flame temperature and propagation speed, which vary along the flame as a result of local non-uniformities in the flow field and of flame front curvature, are derived. Unlike previous studies that assumed a mixture consisting of a single deficient reactant, the present study uses a two-reactant scheme and thus considers mixtures whose compositions vary from lean to rich conditions. Furthermore, non-unity and general reaction orders are considered in an attempt to mimic a wider range of reaction mechanisms and, to better represent actual experimental conditions, all transport coefficients are allowed to depend arbitrarily on temperature. The present model, expressed in a coordinate-free form, is valid for flames of arbitrary shape propagating in general fluid flows, either laminar or turbulent.

An analytical and experimental investigation is made of the compression wave generated when a train enters a tunnel fitted with a long, uniform hood with a rectangular window. The window is situated at the junction of the hood and tunnel, which are taken to have the same uniform cross-sectional area. An understanding of the mechanics of this canonical configuration is important for the design of tunnel entrance hoods for new high-speed trains, with speeds in excess of 300 km h$^{-1}$. The compression wave is formed in two stages: as the train nose enters the hood and as it passes the window. The elevated pressure within the hood produces a flow of air from the window in the form of a high-speed jet, whose inertia generates an additional rise in pressure that propagates into the tunnel as a localized pulse. Multiple reflections from the window and the hood portal cause the temporary trapping of wave energy within the hood (prior to its radiation into the tunnel). All of these aspects of the flow are modelled analytically and the results are found to be in good accord with new model-scale measurements and flow visualization studies reported in this paper.

An equilibrium similarity analysis is applied to the transport equation for $\langle(\delta q)^{2}\rangle$ (${\equiv}\,\langle(\delta u)^{2}\rangle + \langle(\delta v)^{2}\rangle + \langle(\delta w)^{2}\rangle$), the turbulent energy structure function, for decaying homogeneous isotropic turbulence. A possible solution requires that the mean energy $\langle q^{2}\rangle$ decays with a power-law behaviour ($\langle q^{2}\rangle\,{\sim}\,x^{m}$), and the characteristic length scale, which is readily identifiable with the Taylor microscale, varies as $x^{1/2}$. This solution is identical to that obtained by George (1992) from the spectral energy equation. The solution does not depend on the actual magnitude of the Taylor-microscale Reynolds number $R_{\lambda}$ (${\sim}\,{\langle q^{2}\rangle}^{1/2} \lambda/\nu$); $R_{\lambda}$ should decay as $x^{(m+1)/2}$ when $m < -1$. The solution is tested at relatively low $R_{\lambda}$ against grid turbulence data for which $m \simeq -1.25$ and $R_{\lambda}$ decays as $x^{-0.125}$. Although homogeneity and isotropy are poorly approximated in this flow, the measurements of $\langle(\delta q)^{2}\rangle$ and, to a lesser extent, $\langle(\delta u)(\delta q)^{2}\rangle$, satisfy similarity reasonably over a significant range of $r/\lambda$, where $r$ is the streamwise separation across which velocity increments are estimated. For this range, a similarity-based calculation of the third-order structure function $\langle(\delta u)(\delta q)^{2}\rangle$ is in reasonable agreement with measurements. Kolmogorov-normalized distributions of $\langle(\delta q)^{2}\rangle$ and $\langle(\delta u)(\delta q)^{2}\rangle$ collapse only at small $r$. Assuming homogeneity, isotropy and a Batchelor-type parameterization for $\langle(\delta q)^{2}\rangle$, it is found that $R_{\lambda}$ may need to be as large as $10^{6}$ before a two-decade inertial range is observed.

A simple extension of the classic Görtler–Hämmerlin (1955) (GH) model, essential for three-dimensional linear instability analysis, is presented. The extended Görtler–Hämmerlin model classifies all three-dimensional disturbances in this flow by means of symmetric and antisymmetric polynomials of the chordwise coordinate. It results in one-dimensional linear eigenvalue problems, a temporal or spatial solution of which, presented herein, is demonstrated to recover results otherwise only accessible to the temporal or spatial partial-derivative eigenvalue problem (the former also solved here) or to spatial direct numerical simulation (DNS). From a numerical point of view, the significance of the extended GH model is that it delivers the three-dimensional linear instability characteristics of this flow, discovered by solution of the partial-derivative eigenvalue problem by Lin & Malik (1996a), at a negligible fraction of the computing effort required by either of the aforementioned alternative numerical methodologies. More significant, however, is the physical insight which the model offers into the stability of this technologically interesting flow. On the one hand, the dependence of three-dimensional linear disturbances on the chordwise spatial direction is unravelled analytically. On the other hand, numerical results obtained demonstrate that all linear three-dimensional instability modes possess the same (scaled) dependence on the wall-normal coordinate, that of the well-known GH mode. The latter result may explain why the three-dimensional linear modes have not been detected in past experiments; criteria for experimental identification of three-dimensional disturbances are discussed. Asymptotic analysis based on a multiple-scales method confirms the results of the extended GH model and provides an alternative algorithm for the recovery of three-dimensional linear instability characteristics, also based on solution of one-dimensional eigenvalue problems. Finally, the polynomial structure of individual three-dimensional extended GH eigenmodes is demonstrated using three-dimensional DNS, performed here under linear conditions.

In the three-dimensional boundary layer produced by a rotating disk, the experimentally well-documented sharp transition from laminar to turbulent flow is shown to coincide with secondary absolute instability of the naturally selected primary nonlinear crossflow vortices. Fully saturated primary finite-amplitude waves and the associated nonlinear dispersion relation are first numerically computed using a local parallel flow approximation. Exploiting the slow radial development of the basic flow, the naturally selected primary self-sustained flow structure is then derived by asymptotic analysis. In this state, outward-spiralling nonlinear vortices are initiated at the critical radius where primary absolute instability first occurs. A subsequent secondary stability analysis reveals that as soon as the primary nonlinear waves come into existence they are absolutely unstable with respect to secondary perturbations. Secondary disturbances growing in time at fixed radial locations continuously perturb the primary vortices, thus triggering the direct route to turbulence prevailing in this configuration.

A laboratory investigation of the attenuation of mechanically generated waves by an opposing wind has been completed. Wave attenuation was quantified by measurements of the decline in surface variance. These measurements show higher effective levels of monochromatic wave attenuation than predicted by air-side measurements: approximately an order of magnitude higher than measurements by Young & Sobey (1985) and, a factor of 3 higher than those of Donelan (1999) for waves in a JONSWAP spectrum. Furthermore, they show that theoretical estimates currently underestimate the attenuation rates by a factor of at least 3. This study has shown that the magnitude of wave attenuation rates due to opposing winds is approximately 2.5 times greater than the magnitude of wave growth rates for comparable wind forcing. At high wave steepnesses, detailed analysis suggests that air-side processes alone are not sufficient to induce the observed levels of attenuation. Rather, it appears that energy fluxes from the wave field due to the interaction between the wave-induced currents and other subsurface motions play a significant role once the mean wave steepness exceeds a critical value. A systematic relationship between the energy flux from the wave field and mean wave steepness was observed. The combination of opposing wind and wind-induced water-side motions is far more effective in attenuating waves than has previously been envisaged.

We study time-modulated Taylor–Couette flow for the simple case in which the inner cylinder's angular velocity oscillates around zero mean at given amplitude and frequency and the outer cylinder is at rest. We find that, provided the amplitude of modulation is large enough, two classes of Taylor vortex flows are possible – reversing and non-reversing. In the latter, which takes place at relatively high frequency, the direction of the Taylor vortices does not change every half-cycle.