Extensive experiments were performed in the highly turbulent shear layer and in the reverse-flow region downstream of a normal plate followed by a long splitter plate in its plane of symmetry. Whereas the earlier measurements of Ruderich & Fernholz (1986) and Castro & Haque (1987) concentrated on the plane of symmetry (x, y-plane, z = 0), the present investigation reveals the full extent of the three-dimensional behaviour of the flow by presenting data in the whole flow field. Spanwise measurements of five components of the Reynolds stress tensor, of integral timescales, and of mean and fluctuating values of the skin friction were carried out by the pulsed-wire technique. Integral lengthscales and spectra outside the reverse-flow region were measured using hot wires. In order to keep three-dimensional effects as small as possible, the aspect ratio was large (model width was 2.55 times the length of the reverse-flow region) and the blockage ratio and free-stream turbulence intensity were low. Despite these efforts the flow investigated here must be considered three-dimensional and classified as both pressure and shear driven. There appears to be a connection between the shape of the reverse-flow region and the level of the fluctuating velocities. Measurements of power spectral densities revealed no flapping motion of the separated turbulent shear layer.

A thin liquid layer flowing down an inclined plane exhibits a long-wave interfacial instability at a critical value of the Reynolds number. Past work on this problem has shown that heating or cooling the layer does not significantly change the characteristics of this instability. We show that this is not correct when the Prandtl number of the liquid is large and that both heating and cooling from below can destabilize the layer depending on the interfacial heat-transfer conditions. The mechanism for this unstable behaviour involves the direct expansion of the liquid as it experiences a temperature perturbation produced by the deformation of the interface. When the layer is heated from below, this additional effect changes the critical angle at which longitudinal, buoyancy-driven rolls are preferred relative to the long-wave interfacial instability.

Linear birefringence measurements of dilute and semi-dilute polyisobutylene solutions following flow through a disordered fixed fibre bed of 2.47% solids volume fraction provide both transient and steady measurements of chain deformation. Our results indicate that the flexible polyisobutylene polymers undergo a large conformation change, stretching in the direction of the average flow. This occurs even though the average flow in the bed is a plug flow which would not cause any polymer stretch by itself. The polymer stretch or conformation change increases with the number of chain interactions with bed fibres and ultimately reaches a steady-state value that can be correlated with the pore-size Deborah number (i.e. a characteristic polymer relaxation time divided by a characteristic flow time in the bed pore). Large changes in the polymer conformation are noted for values of the Deborah number, De > 5. In addition, the time to steady state scales with the characteristic flow time within a pore over a large range of Deborah numbers. The pressure drop across the fibre bed was also measured simultaneously with the birefringence measurement and was found to be directly proportional to the birefringence throughout the range of De investigated. Thus, we show empirically, for the first time, that chain elongation, which produces normal stress anisotropy within the fluid, is directly responsible for the increased flow resistance. These findings are then analysed in the light of recent theories for the response of polymer molecules in fixed bed flow fields (Shaqfeh & Koch 1992). It is shown that our results are consistent with the interpretation that these flows are stochastic strong flows, which create an apparent ‘coil-stretch’ transition. After extending the theory of Shaqfeh & Koch to account for the specifics in the experiments, including the bed geometry and statistics as well as the polydispersity of the polymer solutions, it is shown that the theory can predict most of the experimental results both qualitatively and quantitatively.

A numerical study has been carried out to analyse liquid metal flows in a sudden expansion of electrically conducting rectangular ducts under the influence of an imposed uniform magnetic field. Separation phenomena are investigated by selecting a reference Reynolds number and by increasing progressively the applied magnetic field. The magnetic effects leading to the reduction of the size of separation zones that form behind the cross-section enlargement are studied by considering modifications of flow topology, streamline patterns and electric current density distribution. In the range of parameters investigated, the magnetohydrodynamic flow undergoes substantial transitions from a hydrodynamic-like flow to one dominated by electromagnetic forces, where the influence of inertia and viscous forces is confined to thin internal layers aligned with the magnetic field and to boundary layers that form along the walls. Scaling laws describing the reattachment length and the pressure drop in the sudden expansion are derived for intense magnetic fields.

The effects of a free surface on the spin-up of a homogeneous fluid are studied, both analytically and experimentally. The analysis is carried out in cylindrical geometry and shows that the spin-up process is strongly modified as the rotational Froude number F = 4ω2L2/gH becomes large. The dynamic effect of the free surface causes delayed response outside a sidewall boundary layer of thickness LF−½. The timescale in the slowly decaying core is larger than the usual spin-up time by a factor of order F. A set of laboratory experiments using a cylinder with a parabolic bottom were carried out in order to test the theory. Reasonable agreement is found in all the experiments except close to the centre where an interesting deviation was observed, especially in cases corresponding to smaller Froude numbers. The deviation consisted of an anticyclonic vortex at the centre. It is shown that this phenomenon might be explained by Lagrangian mean motion resulting from inertial oscillations. In fact, the analysis shows that this motion produces a singular vortex at the centre.

We consider transient flow in a box containing an isolated buoyancy source, ventilated by a windward high-level opening and a leeward low-level opening, so that prevailing wind acts to oppose buoyancy-driven flow. Hunt & Linden (J. Fluid Mech., vol. 527, 2005, p. 27) demonstrated that two stable steady states can exist above a critical wind strength: buoyancy-driven displacement ventilation with a two-layer stratification and wind-driven mixing ventilation with the whole interior contaminated by buoyant fluid. We present two time-dependent models for this system: a nonlinear ordinary differential equation (ODE) model following Kaye & Hunt (J. Fluid Mech., vol. 520, 2004, p. 135), assuming ‘perfect’ vertical mixing of fluid within each layer, and a partial differential equation model assuming zero vertical mixing, following Germeles (J. Fluid Mech., vol. 71, 1975, p. 601).

We apply these models to an initial-value problem – the filling box with constant opposing wind. The interface between the upper hot plume fluid and the lower cool ambient air can dramatically overshoot its final level before relaxing to equilibrium; in some cases, a fully contaminated transient can occur before the buoyancy-driven two-layer steady state is reached. However, we find that for an initially completely uncontaminated box, the system converges to a stable wind-driven steady state whenever it exists. By analysing phase diagrams of the ODE model for the flow, we establish a general method of determining which final state is attained and also explain the hysteresis observed by Hunt & Linden (2005). We confirm these transient behaviours by conducting salt bath experiments in a recirculating flume tank and establish quantitative agreement between theory and experiment. Our ‘zero mixing’ model is more accurate than our ‘perfect mixing’ model for our experiments, as the upper layer remains stratified for a substantial time.

A numerical model is presented for investigating control of the three-dimensional boundary-layer transition process. Control of a periodically forced, spatially evolving boundary layer in water is studied using surface heating techniques. The Navier–Stokes and energy equations are integrated using a fully implicit finite difference/spectral method. The Navier–Stokes equations are used in vorticity–velocity form and are coupled with the energy equation through the viscosity dependence on temperature. Passive control of small amplitude two-dimensional waves and three-dimensional oblique waves is numerically simulated with either uniform or non-uniform wall heating applied. Both amplitude levels and amplification rates are strongly reduced with heating applied. Comparison is made with parallel and non-parallel linear stability theory and experiments. Control of the early stages of the nonlinear breakdown process is also investigated using uniform wall heating. Both control of the fundamental and subharmonic routes to turbulence are investigated. For both breakdown processes, a strong reduction in amplitude levels and growth rates results. In particular, the high three-dimensional growth rates that are characteristic of the secondary instability process are significantly reduced below the uncontrolled levels.

The stability of perturbed normal shock waves is considered. Shock perturbations depend directly upon the disturbances in the flow adjacent to the shock. In the present paper an initially stationary shock is assumed to be perturbed by acoustic waves reaching it from the downstream side. This case corresponds to the situation occurring in shock diffraction or reflexion. Two-dimensional problems of this type have been investigated previously, both analytically and experimentally. These previous analytic results have, in all cases, indicated that the perturbations of the shock decay with time as $t^{\frac{3}{2}}$, while experimentally both t½ and $t^{\frac{3}{2}}$ decays have been observed. It is demonstrated in the present investigation that, when waves are continuously generated at a point or points behind the shock, a t½ decay of the shock perturbations will occur, corresponding to the decay of the incident waves. However, when the source of waves is located only at the shock, as in a diffraction problem, $t^{-\frac{3}{2}}$ decay occurs owing to the cancellation, to lowest order, of the incident wave by its reflexion from the shock. These results explain the divergence between theory and experiment in this area, since the experiments giving the slower decay contained a source of waves behind the shock.

It is concluded that shock stability can only be considered in the context of the type of disturbances incident upon the shock.

We report upon a theoretical and experimental investigation of a porous medium ‘filling box’ flow by specifically examining the details of the laminar descending plume and its outflow in a control volume having an impermeable bottom boundary and sidewalls. The plume outflow initially comprises a pair of oppositely directed gravity currents. The gravity currents propagate horizontally until they reach the lateral sidewalls at $y=\pm L$. The flow then becomes of filling box type, with a vertically ascending ‘first front’ separating discharged plume fluid below from ambient fluid above. The flow details are described analytically by first deriving a new similarity solution for Darcy plumes with $\mathit{Pe}>O(1)$, where $\mathit{Pe}$ is the Péclet number. From the similarity solution so obtained, we then derive expressions for the plume volume flux and mean reduced gravity as functions of the vertical distance from the source. Regarding the plume outflow, a similarity solution adopted from Huppert & Woods (J. Fluid Mech., vol. 292, 1995, pp. 55–69) describes the height and front speed of the gravity currents, whereas a semi-implicit finite difference scheme is used to predict the first front elevation versus time and horizontal distance. As with high-Reynolds-number filling box flows, that studied here is an example of a coupled problem: the gravity current source conditions are prescribed by the plume volume flux and mean reduced gravity. Conversely, discharged plume fluid may be re-entrained into the plume, be it soon or long after reaching the bottom impermeable boundary. To corroborate our model predictions, analogue laboratory experiments are performed with fresh water and salt water as the working fluids. Our experiments consider as independent variables the porous medium bead diameter and the plume source volume flux and reduced gravity. Predictions for the gravity current front position and height compare favourably against analogue measured data. Good agreement is likewise noted when considering either the mean elevation or the profile of the first front. Results from this study may be adopted in modelling geological plumes. For example, our equations can be used to predict the time required for discharged plume fluid to return to the point of injection in the case of aquifers closed on the sides and below by impermeable boundaries.

A finite-volume model of the classic differentially heated rotating annulus experiment is used to study the spontaneous emission of gravity waves (GWs) from jet stream imbalances, which may be an important source of these waves in the atmosphere and for which no satisfactory parameterisation exists. Experiments were performed using a classic laboratory configuration as well as using a much wider and shallower annulus with a much larger temperature difference between the inner and outer cylinder walls. The latter configuration is more atmosphere-like, in particular since the Brunt–Väisälä frequency is larger than the inertial frequency, resulting in more realistic GW dispersion properties. In both experiments, the model is initialised with a baroclinically unstable axisymmetric state established using a two-dimensional version of the code, and a low-azimuthal-mode baroclinic wave featuring a meandering jet is allowed to develop. Possible regions of GW activity are identified by the horizontal velocity divergence and a modal decomposition of the small-scale structures of the flow. Results indicate GW activity in both annulus configurations close to the inner cylinder wall and within the baroclinic wave. The former is attributable to boundary layer instabilities, while the latter possibly originates in part from spontaneous GW emission from the baroclinic wave.

Electrostatic atomization of a liquid of finite electrical conductivity in the so-called cone-jet regime relies on the electric shear stresses that appear in a region of the liquid surface when a meniscus of the liquid is subjected to an intense electric field. An order of magnitude analysis is used to describe the flow induced by these stresses, which drive the liquid of the meniscus into a jet that issues from the tip of the meniscus and breaks into droplets at some distance from it. When the dielectric constant of the liquid is large, the electric shear stresses extend into the jet and cause a depression that sucks liquid from the meniscus. The induced flow rate is estimated and shown to represent approximately the minimum flow rate at which a cone-jet can be established. It is argued that the meniscus can be stabilized by the electric field that the charge of the jet induces on it. This stabilizing mechanism weakens when the flow rate supplied to the meniscus decreases, and its failure may determine an alternative minimum flow rate for the cone-jet regime. The instability of the jet and existing scaling laws for the size of the spray droplets are discussed.

Nonlinear travelling waves on a two-dimensional inviscid fluid sheet are investigated when the sheet is bounded above and below by two thin elastic plates. Symmetric and antisymmetric solution branches are identified, together with a pair of bifurcation branches. It is shown that far along the branches the solutions approach limiting configurations that correspond to static solutions valid in the absence of fluid forcing.

We present hydrodynamic and magnetohydrodynamic (MHD) simulations of liquid sodium flows in the von Kármán sodium (VKS) set-up. The counter-rotating impellers made of soft iron that were used in the successful 2006 experiment are represented by means of a pseudo-penalty method. Hydrodynamic simulations are performed at high kinetic Reynolds numbers using a large eddy simulation technique. The results compare well with the experimental data: the flow is laminar and steady or slightly fluctuating at small angular frequencies; small scales fill the bulk and a Kolmogorov-like spectrum is obtained at large angular frequencies. Near the tips of the blades the flow is expelled and takes the form of intense helical vortices. The equatorial shear layer acquires a wavy shape due to three coherent co-rotating radial vortices as observed in hydrodynamic experiments. MHD computations are performed: at fixed kinetic Reynolds number, increasing the magnetic permeability of the impellers reduces the critical magnetic Reynolds number for dynamo action; at fixed magnetic permeability, increasing the kinetic Reynolds number also decreases the dynamo threshold. Our results support the conjecture that the critical magnetic Reynolds number tends to a constant as the kinetic Reynolds number tends to infinity. The resulting dynamo is a mostly axisymmetric axial dipole with an azimuthal component concentrated near the impellers as observed in the VKS experiment. A speculative mechanism for dynamo action in the VKS experiment is proposed.

We address the problem of computing the hydrodynamic forces and torques among $N$ solid spherical particles moving with given rotational and translational velocities in Stokes flow. We consider the original fluid–particle model without introducing new hypotheses or models. Our method includes the singular lubrication interactions which may occur when some particles come close to one another. The main new feature is that short-range interactions are propagated to the whole flow, including accurately the many-body lubrication interactions. The method builds on a pre-existing fluid solver and is flexible with respect to the choice of this solver. The error is the error generated by the fluid solver when computing non-singular flows (i.e. with negligible short-range interactions). Therefore, only a small number of degrees of freedom are required and we obtain very accurate simulations within a reasonable computational cost. Our method is closely related to a method proposed by Sangani & Mo (Phys. Fluids, vol. 6, 1994, pp. 1653–1662) but, in contrast with the latter, it does not require parameter tuning. We compare our method with the Stokesian dynamics of Durlofsky et al. (J. Fluid Mech., vol. 180, 1987, pp. 21–49) and show the higher accuracy of the former (both by analysis and by numerical experiments).

The stability of a steacty, laminar, immiscible, dense line plume falling through a fluid whose horizontal extent is much larger than the plume thickness is investigated. It is shown analytically that the primary flow is linearly unstable in the limit of large wavelength and, if surface tension is negligible, is also unstable in the limit of small Reynolds number. A rich variety of modes of perturbation exist, characterized by a range of balances on different lengthscales between diffusion, advection and propagation of vorticity.

A general characteristic of rapidly rotating fluids is that accurate experimental measurements can only be made of the main (azimuthal) flow. The secondary flow is then usually deduced from theory, although this is often incomplete in the boundary regions where the secondary flow is of most interest.

In this paper we consider the case of source-sink flow between the porous walls of a rapidly rotating annular container and numerically integrate the full equations of motion in order to determine the complete structure of the secondary flow. The results are compared with the (approximate) analytic studies of Hide (1968) and Bennetts & Hocking (1973) to show the differences between the two approaches.

A defect of many previous numerical papers has been the inability to check the solution in the nonlinear case. To overcome this, new experimental measurements of the azimuthal velocity profile for a Rossby number of 0·238 have been obtained and these are compared with the numerical results.

We consider a homogeneous fluid of viscosity $\nu $ confined within an oblate spheroidal cavity of arbitrary eccentricity $\mathcal{E}$ marked by the equatorial radius $d$ and the polar radius $d \sqrt{1-\mathcal{E}^2}$ with $0<\mathcal{E}<1$. The spheroidal container rotates rapidly with an angular velocity ${\boldsymbol{\Omega}}_0 $ about its symmetry axis and precesses slowly with an angular velocity ${\boldsymbol{\Omega}}_p$ about an axis that is fixed in space. It is through both topographical and viscous effects that the spheroidal container and the viscous fluid are coupled together, driving precessing flow against viscous dissipation. The precessionally driven flow is characterized by three dimensionless parameters: the shape parameter $\mathcal{E}$, the Ekman number ${\mathit{Ek}}=\nu /(d^2 \delimiter "026A30C {\boldsymbol{\Omega}}_0\delimiter "026A30C )$ and the Poincaré number ${\mathit{Po}}=\pm \delimiter "026A30C {\boldsymbol{\Omega}}_p\delimiter "026A30C / \delimiter "026A30C \boldsymbol{\Omega}_0\delimiter "026A30C $. We derive a time-dependent asymptotic solution for the weakly precessing flow in the mantle frame of reference satisfying the no-slip boundary condition and valid for a spheroidal cavity of arbitrary eccentricity at ${\mathit{Ek}}\ll 1$. No prior assumptions about the spatial–temporal structure of the precessing flow are made in the asymptotic analysis. We also carry out direct numerical simulation for both the weakly and the strongly precessing flow in the same frame of reference using a finite-element method that is particularly suitable for non-spherical geometry. A satisfactory agreement between the asymptotic solution and direct numerical simulation is achieved for sufficiently small Ekman and Poincaré numbers. When the nonlinear effect is weak with $\delimiter "026A30C {\mathit{Po}}\delimiter "026A30C \ll 1$, the precessing flow in an oblate spheroid is characterized by an azimuthally travelling wave without having a mean azimuthal flow. Stronger nonlinear effects with increasing $\delimiter "026A30C {\mathit{Po}}\delimiter "026A30C $ produce a large-amplitude, time-independent mean azimuthal flow that is always westward in the mantle frame of reference. Implications of the precessionally driven flow for the westward motion observed in the Earth’s fluid core are also discussed.

The stability of gravitationally unstable, transient boundary layers in heterogeneous saline aquifers is examined with respect to the onset of natural convection. Permeability is assumed to vary periodically across the thickness of the aquifer. We study the interaction between permeability variation and concentration perturbations within the boundary layer. We observe that the instability decreases with an increase in the permeability variance if the boundary layer thickness is large compared with the permeability wavelength. On the other hand, when the boundary layer thickness is smaller than the permeability wavelength, the behaviour of instability as a function of variance depends on the phase of permeability variation. Such behaviours are shown to result from the interaction of two modes of vorticity production related to the coupling of concentration and velocity perturbations with the magnitude and gradient of permeability variation, respectively. We show that these two modes of vorticity production, when coupled with the transient nature of the boundary layer, determine the evolutionary paths followed by the most amplified perturbations that trigger the onset of convection. When the permeability variance is large, we find that small changes in the permeability field can lead to large changes in the onset times for convection.

We describe a simple model for turbulence in a marginally unstable, forced, stratified shear flow. The model illustrates the essential physics of marginally unstable turbulence, in particular the tendency of the mean flow to fluctuate about the marginally unstable state. Fluctuations are modelled as an oscillatory interaction between the mean shear and the turbulence. The interaction is made quantitative using empirically established properties of stratified turbulence. The model also suggests a practical way to estimate both the mean kinetic energy of the turbulence and its viscous dissipation rate. Solutions compare favourably with observations of fluctuating ‘deep cycle’ turbulence in the equatorial oceans.

High-resolution direct numerical simulations of isotropic homogeneous turbulence are used to understand the differences between the effects of spatial intermittency on the energy dissipation rate and on surrogates for the dissipation rate that are based on measurements of a subset of the strain rate tensor. In particular, the one-dimensional longitudinal and transverse surrogates, as well as a surrogate based on the asymmetric part of the strain rate tensor, are considered. The instantaneous surrogates are studied locally, locally averaged in space and conditionally averaged to see what statistics of the dissipation rate might accurately be inferred given measurements of the surrogates. The simulations with the Reynolds numbers based on the Taylor microscale of 102–235 are highly resolved for accurate evaluation of higher-order statistics. The probability densities of the local and locally averaged surrogates are significantly different from the corresponding statistics for the dissipation rate itself. All of the surrogates are more intermittent than the dissipation rate, the transverse surrogate is more intermittent than the longitudinal and these trends are still prominent even when the fields are spatially averaged at length scales close to the integral length scale. As a consequence, the intermittency exponent computed from the moments of the locally averaged longitudinal and transverse surrogates is approximately 1.5 and 2.2 times higher, respectively, than that computed by the same method from the dissipation rate field. In addition, while different methods of computing intermittency exponent from the dissipation rate field yield the same result, different methods applied to a surrogate are inconsistent.