We present a model describing the evolution of the small-scale Navier–Stokes turbulence due to its stochastic distortion by much larger turbulent scales. This study is motivated by numerical findings (Laval et al. Phys. Fluids vol. 13, 2001, p. 1995) that such interactions of separated scales play an important role in turbulence intermittency. We introduce a description of turbulence in terms of the moments of $k$-space quantities using a method previously developed for the kinematic dynamo problem (Nazarenko et al. Phys. Rev. E vol. 68, 2003, 0266311). Working with the $k$-space moments allows us to introduce new useful measures of intermittency such as the mean polarization and the spectral flatness. Our study of the small-scale two-dimensional turbulence shows that the Fourier moments take their Gaussian values in the energy cascade range whereas the enstrophy cascade is intermittent. In three dimensions, we show that the statistics of turbulence wavepackets deviates from Gaussianity toward dominance of the plane polarizations. Such turbulence is formed by ellipsoids in the $k$-space centred at its origin and having one large, one neutral and one small axis with the velocity field pointing parallel to the smallest axis.

Measurements were conducted in the vicinity and downstream of two-dimensional cavities filled with a dense miscible fluid subjected to purging by an overlying wall-bounded turbulent shear flow. Two cavities with aspect ratios 1 and 2 were used, based on previous observations that the mechanism for entraining dense fluid into the flow above is different for these cases. Scaling for the downstream variation of the vertical concentration (buoyancy) distribution of dense fluid is advanced based on dimensional arguments and tested using laser-induced fluorescence measurements. The mean flow and turbulent statistics in the vicinity and downstream of the cavity were also measured. A notable increase of shear stress and turbulent intensities was observed above the higher aspect ratio cavity due to intense vertical exchange between the cavity and the flow above. The influence of the cavity could be detected for several cavity widths, or distances equal to several eddy turnover times. However, the flow above the lower aspect ratio cavity exhibits dramatically less vertical exchange between the cavity and the flow above, as indicated by subtle wisps of fluid transported downstream by the free-stream flow. The shear stress and turbulent intensities for this case have only a minor influence of the cavity.

A numerical study of the steady, two-dimensional incompressible flow past a cascade of circular cylinders is presented. The Navier–Stokes equations are written in terms of the streamfunction and vorticity and solved using a novel numerical technique based on using the Chebychev collocation method in one direction and high-order finite differences in the other direction. A direct solver combined with Newton–Raphson linearization is used to solve the discrete equations. Steady flow solutions have been obtained for large Reynolds numbers, far higher than those obtained previously, and for varying gap widths between the cylinders. Three distinct types of solutions, dependent on the gap width, have been found. Comparisons with theoretical predictions for various flow quantities show good agreement, especially for the narrow gap width case. However, existing theories are unable to explain the solution properties which exist for intermediate gap widths.

Using a perturbation approach and the Boussinesq approximation, we derive sets of depth-integrated continuity and momentum equations for transient long-wave propagation with viscous effects included. The fluid motion is assumed to be essentially irrotational, except in the bottom boundary layer. The resulting governing equations are differential–integral equations in terms of the depth-averaged horizontal velocity (or velocity evaluated at certain depth) and the free-surface displacement, in which the viscous terms are represented by convolution integrals. We show that the present theory recovers the well-known approximate damping rates for simple harmonic progressive waves and for a solitary wave. The relationship between the bottom stress and the depth-averaged velocity is discussed.

In this paper the qualitative nonlinear influence of advection and continuity on the resonance characteristics of co-oscillating coastal basins is investigated. For this purpose a weakly nonlinear analysis was carried out on the shallow-water equations describing a coastal basin resonating with exterior water-level oscillations. It extends previous work on an almost-enclosed basin with one single (Helmholtz) mode to arbitrarily shaped ‘shallow’ basins with an infinite number of modes. In line with that work, it is necessary to assume friction to be sufficiently weak such that it is in balance with the nonlinear effects, instead of occurring in the linearized equations. The main result of this paper is the system of Landau equations describing the slow evolution of the amplitudes of the oscillatory eigenmodes of the basin, disregarding a zero-frequency eigenmode, should it exist. The dynamics of the zero-frequency mode neglecting oscillatory eigenmodes have been discussed by others, but a consistently balanced model for small amplitudes incorporating both the zero-frequency and oscillatory modes is not yet available. The behaviour of this system, describing the dynamics of the oscillatory eigenmodes only, is analysed. On the longer time scale, it gives rise to a ‘bent resonance curve’, multiple equilibria (several tidal regimes under the same tidal forcing), sudden regime changes and even chaotic dynamics (when these regime changes occur in an irregular way).

We examine the transient buoyancy-driven flow in a ventilated filling box that is subject to a continuous supply of buoyancy. A rectangular box is considered and the buoyancy input is represented as a turbulent plume, or as multiple non-interacting plumes, rising from the floor. Openings in the base and top of the box link the interior environment with a quiescent exterior environment of constant and uniform density. A theoretical model is developed to predict, as functions of time, the density stratification and the volume flow rate through the openings leading to the steady state. Comparisons are made with the results of small-scale analogue laboratory experiments in which saline solutions and fresh water are used to create density differences. Two characteristic timescales are identified: the filling box time ($T_f$), proportional to the time taken for fluid from a plume to fill a closed box; and the draining box time ($T_d$), proportional to the time taken for a ventilated box to drain of buoyant fluid. The timescale for the flow to reach the steady state depends on these two timescales, which are functions of the box height $H$ and cross-sectional area $S$, the ‘effective’ opening area $A^*$, and the strength, number and distribution of the buoyancy inputs. The steady-state flow is shown to be characterized by the ratio of these timescales ($\mu\,{=}\,T_d/T_f$) which is equivalent to the dimensionless vent area $A^*/H^2$. A feature of these flows is that for $\mu\,{>}\,\mu_c$ the depth of the buoyant upper layer may exceed, or ‘overshoot’, the steady layer depth during the initial transient. The value of $\mu_c$ is determined for both line and point-source plumes, and the sensitivity of the developing flow to the distribution of buoyancy input assessed.

In this paper, the oscillating flow around a circular cylinder is investigated numerically using both a three-dimensional and a two-dimensional model. Two important three-dimensional regimes of the Tatsuno & Bearman (J. Fluid Mech. vol. 211, 1990, p. 157) map are investigated: the asymmetric transverse-street regime D, and the double-pair diagonal regime F. The Stokes number is held constant ($\beta\,{=}\,20$) and the Keulegan–Carpenter number and the Reynolds number changed so that they match the conditions of these two regimes.

The cross-sectional vortex streets (V-pattern in regime D and diagonal pattern in regime F) appear to be unstable, and switching from a pattern to its mirror-image mode occurs during the simulation. This switching is related to a two-dimensional instability in the flow field; this phenomenon can be reproduced by pure two-dimensional simulations.

Three-dimensionality in the flow field always appears after the asymmetric vortex pattern has fully developed. The main effect of three-dimensionality in the flow field appears to be a rotation (circumferential effect) along the axial direction of the main sectional vortex patterns and a time delay along this axis of the switching from a two-dimensional mode to its mirror-image. These features contribute to generation of the three-dimensional sinuous S-mode as defined by Yang & Rockwell (J. Fluid Mech. vol. 460, 2002, p. 93).

The three-dimensionality of the vorticity field affects the dynamic loads induced on the cylinder. The longitudinal component of the force acting on the cylinder appears to be weakly affected by three-dimensional effects, and so does its distribution in the axial direction. This finding explains why the results of two-dimensional simulations often agree fairly well with the data from laboratory experiments. Conversely, the transversal force appears to be significantly affected by the three-dimensional flow field, which suggests that an accurate numerical prediction requires the use of three-dimensional numerical models.

Finally, a simplified conceptual model explains why the axial variation of the sectional transversal force always appears to be much larger than that of the corresponding longitudinal one. The model also explains why two-dimensional simulations tend to underpredict the r.m.s. value of the longitudinal force and to overpredict that of the transversal force, compared to the data from three-dimensional studies.

We develop a reduced-order model for large-scale unsteadiness (vortex shedding) in a two-dimensional diffuser and use the model to show how periodic mass injection near the separation point reduces stagnation pressure loss. The model estimates the characteristic frequency of vortex shedding and stagnation pressure loss by accounting for the accumulated circulation due to the vorticity flux into the separated region. The stagnation pressure loss consists of two parts: a steady part associated with the time-averaged static pressure distribution on the wall, and an unsteady part caused by vortex shedding. To validate the model, we perform numerical simulations of compressible unsteady laminar diffuser flows in two dimensions. The model and simulation show good agreement as we vary the Mach number and the area ratio of the diffuser. To investigate the effects of periodic mass injection near the separation point, we also perform simulations over a range of the injection frequencies. Periodic mass injection causes vortices to be pinched off with a smaller size as observed in experiments. Consequently, their convective velocity is increased, absorption of circulation from the wall is enhanced, and the reattached point is shifted upstream. Thus, in accordance with the model, the stagnation pressure loss, particularly the unsteady part, is substantially reduced even though the separation point is nearly unchanged. This study helps explain experimental results of separation control using unsteady mass injection in diffusers and on airfoils.

Numerical solutions of stationary flow resulting from immersion of a single body in simple shear flow are reported for a range of Reynolds numbers. Flows are computed using finite-element methods. Comparisons to results of asymptotic low-Reynolds-number theory, experimental study, and other numerical techniques are provided. Results are presented primarily for isotropic bodies, i.e. the circular cylinder and sphere, for both of which the two conditions of a torque-free (freely-rotating) and fixed body are investigated. Conditions studied for the sphere are $0 \,{<}\, \hbox{\it Re} \,{\le}\, 100$, and for the circular cylinder $0 \,{<}\, \hbox{\it Re} \,{\le}\, 500$, with the shear-flow Reynolds number defined as $\hbox{\it Re}\,{=}\, \gammadot a^2/\nu$; $\gammadot$ is the shear rate of the Cartesian simple shear flow $\bu \,{=}\, (\gammadot y, 0, 0)$, $a$ is the cylinder or sphere radius, and $\nu$ is the kinematic viscosity of the fluid. In the torque-free case, the rotation rate of the body decreases with increasing $\hbox{\it Re}$. Qualitative dependence, seen in the $\hbox{\it Re} \,{=}\,0$ fluid flow field, upon whether the body is fixed against rotation or torque-free vanishes as $\hbox{\it Re}$ increases and the fluid flow is more similar to that around the $\hbox{\it Re}\,{=}\,0$ fixed body: the influence of rotation of the body and the associated closed streamlines are confined to a narrow layer about the body for $\hbox{\it Re}\,{>}\,O(1)$. Separation of the boundary layer is observed in the case of a fixed cylinder at $\hbox{\it Re} \,{\approx}\, 85$, and for a fixed sphere at $\hbox{\it Re} \,{\approx}\, 100$; similar separation phenomena are observed for a freely rotating cylinder. The surface stress and its symmetric first moment (the stresslet) are presented, with the latter providing information on the particle contribution to the mixture rheology at finite $\hbox{\it Re}$. Stationary flow results are also presented for elliptical cylinders and oblate spheroids, with observation of zero-torque inclinations relative to the flow direction which depend upon the aspect ratio, confirming and extending prior findings.

The effect of gravity modulation on the nonlinear evolution of long-wavelength disturbances at the free surface of a surfactant-covered thin liquid layer is studied. The surfactants, which are assumed to be insoluble, give rise to interfacial concentration gradients and associated Marangoni flow in the underlying liquid film. A coupled system of lubrication-theory-based evolution equations for the film height and surfactant concentration is solved numerically using spectral methods. Previous work using Floquet theory had determined that small-amplitude long-wavelength disturbances are destabilized by gravity modulation in the presence of surfactant; uncontaminated films were found to be linearly stable. Our numerical results indicate that uncontaminated free surfaces are destabilized by nonlinearities and exhibit a harmonic response. The interface exhibits complex dynamics during a forcing cycle, characterized by numerous coalescence events between thickened fluid ridges leading to coarsening. The presence of surfactant-induced Marangoni flow gives rise to a harmonic response, larger scale fluid structures of reduced amplitude, less frequent coalescence events, and less complicated film dynamics.

Interaction of a deep-water wave with a vertical cylinder is characterized using techniques of high-image-density particle image velocimetry, in conjunction with measurement of the transverse and in-line loading coefficients. Phase-locked patterns of locally two-dimensional vortex formation are attainable only for sufficiently high values of the Keulegan–Carpenter number KC, e.g. $\hbox{\it KC} \,{\geq}\, 12$. Even in these cases, such patterns are interrupted by non-phase-locked patterns that, nevertheless, show basic forms, including mode inversion or switching of the initially formed vortex from the cylinder, along with patterns of symmetrical vortex formation, all represented by patterns of instantaneous vorticity.

Three-dimensionality of the flow structure along the span of the cylinder shows, at relatively low KC, patterns of transverse velocity $v$ and streamwise vorticity $\omega _{x}$ that have well-defined spanwise spacing. This spacing decreases in a continuous fashion with depth beneath the free surface, and is in accord with the decrease of local KC with depth, i.e. decrease of the diameter of the local particle trajectory. Such three-dimensional patterns are, however, not phase repetitive over a large number of wave cycles.

At sufficiently high values of KC, it is possible to identify basic classes of three-dimensional patterns. When regions of transverse velocity show like sign over a significant spanwise extent of the cylinder, the transverse loading coefficient is relatively large. As the number of zero crossings between the positive and negative regions of $v$ increases, the transverse loading coefficient decreases accordingly. Generally speaking, at these higher values of KC, patterns of streamwise vorticity are much less ordered, and of smaller scale; they are embedded within larger-scale well-defined patterns of the transverse velocity $v$ along the span.

Taken together, the patterns of locally two-dimensional (sectional) structure and the three-dimensional structure along the span indicate a criterion for locally phase-locked quasi-two-dimensional vortex formation: it occurs when the pattern of transverse velocity is of the same sign over a significant spanwise distance along the span of the cylinder.

This is a theoretical study on the mass transport due to partially reflected long surface waves in a two-layer viscous system, which can be closed or open at its far end. Based on Lagrangian coordinates, a perturbation analysis is carried out to the second order to find the mean Lagrangian drifts in the two layers, where the lower fluid is taken to be much more viscous than the upper one. The free-surface and interfacial set-ups are also found as part of the solutions. A single analytical expression is obtained for the mass transport velocity in each layer, incorporating all the cases where the wave can be progressive, standing or partially standing, and the domain can be closed or open so that a return current may or may not exist. Through some numerical calculations, the patterns of flow in the recirculation cells due to the standing component of the wave, and in the unidirectional drifts due to the progressive component of the wave in a closed system are shown to vary with the lower-layer fluid viscosity. It is possible that, under some specific conditions, the mass transport in the core region of the upper layer is completely quiescent despite the existence of some strong drifts in the lower layer. The mean flow structures in the two layers can also respond rather differently to a change in the reflection coefficient in the presence or the absence of the return current.

Experimental results on sediment erosion (scour) by a plane turbulent wall jet, issuing from a sluice gate, are presented which show clearly – it seems for the first time – that the turbulent wall layer is destabilized by the concave curvature of the water/sediment interface. The streamwise Görtler vortices which emerge create sediment streaks or longitudinal sediment ridges. The analysis of the results in terms of Görtler instability of the wall layer indicates that the strength of these curvature-excited streamwise vortices is such that the sediment transport is primarily due to turbulence created by these vortices. Their contribution to the wall shear stress is taken to be of the same form as the normal turbulent wall shear stress. For this reason, the model developed by Hogg et al. (J. Fluid Mech. Vol. 338, 1997, p. 317) remains valid; only the numerical coefficients are affected. The logarithmic dependency of the time evolution of the scour-hole depth predicted by this model is shown to be in good agreement with experiments. New scaling laws for the quasi-steady state depth and the associated time, inspired by the Hogg et al. (1997) model are proposed. Furthermore, it is emphasized that at least two scouring regimes must be distinguished: a short-time regime after which a quasi-steady state is reached, followed by a long-time regime, leading to an asymptotic state of virtually no sediment transport.

We present new results on the laminar displacement of Herschel–Bulkley fluids along narrow eccentric annuli. We adopt a Hele-Shaw modelling approach and consider the possibility that a long displacement finger should advance along the wide side of the annulus. We deduce conditions under which this cannot happen. We also analyse local instability of the interface on wide and narrow sides of the annulus using the Muskat approach. We thus show that it is possible to have both steady and unsteady travelling-wave solutions, for which the interface is locally stable.

We show how steady stable displacements arise from an increase in effective viscosity difference between displacing and displaced fluids and also analyse effects of buoyancy on the displacement. As opposed to many studies of Hele-Shaw displacements, the principle focus is on identifying stable steady displacements. Finally, we show how predictions of our model, derived from the Navier–Stokes equations using well-defined scaling arguments, compare with some of the ad hoc rule-based design systems that are currently used in the oil industry for design of primary cementing displacements.