Injection of carbon dioxide (CO2) into geological formations is widely regarded as a promising tool for reducing global atmospheric CO2 emissions. To evaluate injection scenarios, estimate reservoir capacity and assess leakage risks, an accurate understanding of the subsurface spreading and migration of the plume of mobile CO2 is essential. Here, we present a complete solution to a theoretical model for the subsurface migration of a plume of CO2 due to natural groundwater flow and aquifer slope, and subject to residual trapping. The results show that the interplay of these effects leads to non-trivial behaviour in terms of trapping efficiency. The analytical nature of the solution offers insight into the physics of CO2 migration, and allows for rapid, basin-specific capacity estimation. We use the solution to explore the parameter space via the storage efficiency, a macroscopic measure of plume migration. In a future study, we shall incorporate CO2 dissolution into the migration model and study the importance of dissolution relative to capillary trapping and the impact of dissolution on the storage efficiency.

Turbulence measurements for a zero pressure gradient boundary layer over a two-dimensional roughness are presented and compared to previous results for a smooth wall and a three-dimensional roughness (Volino, Schultz & Flack, J. Fluid Mech., vol. 592, 2007, p. 263). The present experiments were made on transverse square bars in the fully rough flow regime. The turbulence structure was documented through the fluctuating velocity components, two-point correlations of the fluctuating velocity and swirl strength and linear stochastic estimation conditioned on the swirl and Reynolds shear stress. The two-dimensional bars lead to significant changes in the turbulence in the outer flow. Reynolds stresses, particularly and , increase, although the mean flow is not as significantly affected. Large-scale turbulent motions originating at the wall lead to increased spatial scales in the outer flow. The dominant feature of the outer flow, however, remains hairpin vortex packets which have similar inclination angles for all wall conditions. The differences between boundary layers over two-dimensional and three-dimensional roughness are attributable to the scales of the motion induced by each type of roughness. This study has shown three-dimensional roughness produces turbulence scales of the order of the roughness height k while the motions generated by two-dimensional roughness may be much larger due to the width of the roughness elements. It is also noted that there are fundamental differences in the response of internal and external flows to strong wall perturbations, with internal flows being less sensitive to roughness effects.

An exact equation governing the transport of energy associated with disturbances in an arbitrary steady flow is derived. The result is a generalization of the familiar concept of acoustic energy and is suggested by a perturbation expansion of the general energy equation of fluid mechanics. A disturbance energy density and flux are defined and identified as exact fluid dynamic quantities whose leading-order regular perturbation representations reduce in various special cases to previously known results. The exact equation on disturbance energy is applied to a simple example of nonlinear wave propagation as an illustration of its general utility in situations where a linear description of the disturbance is inadequate.

In the presence of an air stream, a uniform liquid film on a horizontal flat plate may be unstable to small disturbances, and waves may arise. In this paper the hydrodynamic stability of thin liquid films is examined both experimentally and theoretically.

The experiments concern water films thinner than those which have been examined in the past. It is found that, when the film thickness is sufficiently small, a previously unknown type of instability occurs. The theoretical analysis explains this surprising phenomenon.

Due to interaction of the mean airflow and small disturbances of the liquid-air interface, normal and tangential stress perturbations are produced at the liquid surface. It is shown that small wave-like disturbances become unstable when the joint influence of the component of normal stress in phase with the wave elevation and the component of tangential stress in phase with the wave slope is sufficient to overcome the ‘stiffness’ of the liquid surface due to gravity and surface tension. It is found that the destabilizing role of the tangential stress component is dominant for very thin films, and that instability may occur whatever the velocity of the air stream, provided the film is made sufficiently thin.

The wakes behind square cylinders with variation in incidence angle are computed over a range of Reynolds numbers to elucidate the three-dimensional stability and dynamics up to a Reynolds number of Re = 300, based on the projected height of the inclined square cylinder. Three-dimensional instability modes are predicted and computed using a linear stability analysis technique and three-dimensional simulations, respectively. Depending on the incidence angle, the flow is found to transition to three-dimensional flow through either a mode A instability, or a subharmonic mode C instability. The mode A instability is predicted as the first-occurring instability at incidence angles smaller than 12° and greater than 26°, with the mode C instability predicted between these incidence angles. At a zero-degree angle of incidence, the wake instabilities closely match modes A, B and a quasi-periodic mode predicted in earlier studies behind square and circular cylinders. With increasing angle of incidence, the three-dimensional wake transition Reynolds number first increases from Re = 164 as the mode A instability weakens, before decreasing again beyond an incidence angle of 12° as the wake becomes increasingly unstable to the mode C instability, and then again to the mode A instability as the incidence angle approaches 45°. A spanwise autocorrelation analysis from computations over a cylinder span 20 times the square cross-section side length reveals that beyond the onset of three-dimensional instabilities, the vortex street breaks down with patterns consistent with spatio-temporal chaos. This effect was more pronounced at higher incidence angles.

We consider long-time simulations of two-dimensional turbulence body forced by $\sin 4y\hat {\boldsymbol{x}} $ on the torus $(x, y)\in \mathop{[0, 2\mathrm{\pi} ] }\nolimits ^{2} $ with the purpose of extracting simple invariant sets or ‘exact recurrent flows’ embedded in this turbulence. Each recurrent flow represents a sustained closed cycle of dynamical processes which underpins the turbulence. These are used to reconstruct the turbulence statistics using periodic orbit theory. The approach is found to be reasonably successful at a low value of the forcing where the flow is close to but not fully in its asymptotic (strongly) turbulent regime. Here, a total of 50 recurrent flows are found with the majority buried in the part of phase space most populated by the turbulence giving rise to a good reproduction of the energy and dissipation p.d.f. However, at higher forcing amplitudes now in the asymptotic turbulent regime, the generated turbulence data set proves insufficiently long to yield enough recurrent flows to make viable predictions. Despite this, the general approach seems promising providing enough simulation data is available since it is open to extensive automation and naturally generates dynamically important exact solutions for the flow.

A study is described of the forced inertial oscillations appearing in an axially rotating completely filled circular cylinder with plane ends. Excitation is provided by causing the top end to rotate about an axis inclined slightly to the rotation axis. Experiments demonstrate the presence of numerous low mode resonances in a densely spaced range of ratios of net cylinder height to radius in close conformance with linear inviscid theory. Where geometry permits simple corner reflexion, characteristic surfaces are revealed which confirm in part the theoretical predictions concerning their scale and form.

Detailed measurements are given of the amplitude at one point within the cylinder for the condition in which the disturbance frequency equals the rotation frequency. Amplitude column height spectra are compared with theoretical estimates, and the evolution of amplitude for the simplest mode of resonant oscillation is studied. A non-linear theory based on the integral energy of large amplitude oscillation is derived whose broad features are in fair quantitative and qualitative agreement with these observations.

Some investigation is made of the phenomenon of resonant collapse, in which larger amplitude resonant oscillations, after persisting in an apparently laminar form, degenerate abruptly into a state of agitation and disorder from which they do not recover. It is found that the time for emergence of this collapse after the introduction of the forcing disturbance has a close correspondence with the theoretical period of one ‘evolutionary’ cycle of momentum exchange between the main motion and the secondary oscillation.

The far-field sound generated by compressible co-rotating vortices is computed by direct computation of the unsteady compressible Navier–Stokes equations on a computational domain that extends to two acoustic wavelengths in all directions. The vortices undergo a period of co-rotation followed by a sudden merger. The directly computed far-field sound is compared to the prediction of the acoustic analogy due to Möhring (1978, 1979), a modified form of the analogy developed by Lighthill (1952), and an acoustic analogy derived by Powell (1964). All three predictions are in excellent agreement with the simulation. Results of far-field pressure fluctuations from an acoustically non-compact, co-rotating vortex pair are also presented. In this case, the vortex sound theory over-predicts the sound by 65% in accordance with the analysis of Yates (1978).

A significant amount of the fuel consumed by marine vehicles is expended to overcome skin-friction drag resulting from turbulent boundary layer flows. Hence, a substantial reduction in this frictional drag would notably reduce cost and environmental impact. Superhydrophobic surfaces (SHSs), which entrap a layer of air underwater, have shown promise in reducing drag in small-scale applications and/or in laminar flow conditions. Recently, the efficacy of these surfaces in reducing drag resulting from turbulent flows has been shown. In this work we examine four different, mechanically durable, large-scale SHSs. When evaluated in fully developed turbulent flow, in the height-based Reynolds number range of 10 000 to 30 000, significant drag reduction was observed on some of the surfaces, dependent on their exact morphology. We then discuss how neither the roughness of the SHSs, nor the conventional contact angle goniometry method of evaluating the non-wettability of SHSs at ambient pressure, can predict their drag reduction under turbulent flow conditions. Instead, we propose a new characterization parameter, based on the contact angle hysteresis at higher pressure, which aids in the rational design of randomly rough, friction-reducing SHSs. Overall, we find that both the contact angle hysteresis at higher pressure, and the non-dimensionalized surface roughness, must be minimized to achieve meaningful turbulent drag reduction. Further, we show that even SHSs that are considered hydrodynamically smooth can cause significant drag increase if these two parameters are not sufficiently minimized.

It is shown that the dispersion of a substance, with molecular diffusivity κ, in a stationary, homogeneous, turbulent velocity field can be formulated in terms of a ‘substance auto-correlation function’, this being a generalization of the well-known Lagrangian correlation between the velocity of a fluid particle at different times. It is found that the interaction between the molecular diffusion and the turbulent motion reduces the dispersion from the value it would have if the processes of molecular and turbulent diffusion were independent and additive. The conflict, between the results obtained in this paper and previous results which implied that the interaction increases the dispersion, is resolved. The ratio, of the contributions to the dispersion from the interaction term and the turbulent diffusion term, is obtained for comparatively large times by the use of intuitive arguments, and is found to be inversely proportional to the Prandtl number and a Reynolds number of the turbulence.

Wave dissipation by breaking, or the energy transfer from the surface wave field to currents and turbulence, is one of the least understood components of air–sea interaction. It is important for a better understanding of the coupling between the surface wave field and the upper layers of the ocean and for improved surface-wave prediction schemes. Simple scaling arguments show that the wave dissipation per unit length of breaking crest, ϵl, should be proportional to ρwgc5, where ρw is the density of water, g is the acceleration due to gravity and c is the phase speed of the breaking wave. The proportionality factor, or ‘breaking parameter’ b, has been poorly constrained by experiments and field measurements, although our earlier work has suggested that it should be dependent on measures of the wave slope and spectral bandwidth. In this paper we describe inertial scaling arguments for the energy lost by plunging breakers which predict that the breaking parameter b = β(hk)5/2, where hk is a local breaking slope parameter, and β is a parameter of O(1). This prediction is tested with laboratory measurements of breaking due to dispersive focusing of wave packets in a wave channel. Good agreement is found within the scatter of the data. We also find that if an integral linear measure of the maximum slope of the wave packet, S, is used instead of hk, then b ∝ S2.77 gives better agreement with the data. During the final preparation of this paper we became aware of similar experiments by Banner & Peirson (2007) concentrating on the threshold for breaking at lower wave slopes, using a measure of the rate of focusing of wave energy to correlate measurements of b. We discuss the significance of these results in the context of recent measurements and modelling of surface wave processes.

We investigate the possibility of using proper orthogonal decomposition (POD) in reconstructing complete flow fields from gappy data. The incomplete fields are created from DNS snapshots of flow past a circular cylinder by randomly ommiting data points. We first examine the effectiveness of an existing method and subsequently introduce modifications that make the method robust and lead to the maximum possible resolution at a certain level of spatio-temporal gappiness. We simulate three levels of gappiness at approximately 20%, 50% and 80% in order to investigate the limits of applicability of the new procedure. We find that for the two lower levels of gappiness both the temporal and spatial POD modes can be recovered accurately leading to a very accurate representation of the velocity field. The resulting resolution is improved by more than five times compared to the existing method. However, for 80% gappiness only a few temporal modes are captured accurately while the corresponding spatial modes are noisy. We explain this breakdown of the method in terms of a simple perturbation analysis. This new methodology can be a building block in an effort to develop effective data assimilation techniques in fluid mechanics applications.

Comparisons of numerical and experimental results for transient two-dimensional natural convection initiated by instantaneously heating and cooling the opposing vertical walls of a square cavity containing a stationary and isothermal fluid are presented. The good comparisons indicate that the simulation is capturing the important features of the flow. Several features are identified and discussed in detail; in particular, the presence of travelling wave instabilities on the vertical-wall boundary layers and horizontal intrusions, the existence of a rapid flow divergence in the region of the outflow of the intrusions, and the presence of cavity-scale oscillations, caused by the interaction of the intrusions with the opposing vertical boundary layer. The utilization of both numerical and experimental investigations has allowed a more complete exploitation of the available resources than would have been possible had each been conducted separately.

Turbulence measurements in dispersed-bubble two-phase pipe flow, using laser velocimetry techniques, are presented. The turbulence-intensity measurements show a strong dependence on the quality of the flow. A theoretical basis for the prediction of turbulence levels in two-phase flows is proposed. The approach is applied to dispersed-gas/liquid (bubbly) and solid/gas (particulate) two-phase flows, for which experimental data are available, with excellent results.

The coupling between imposed disturbances and capillary instabilities on a liquid jet is examined. It is shown that in most physical situations the forcing produces neutral waves which can then turn into growing waves as the profile relaxes or may be amplified nonlinearly by a mechanism of the type considered by Akylas & Benney (1980). The effectiveness of the coupling is expressed quantitatively by numerically computed values of the ‘coupling coefficient’.

The motion of a very thin vortex filament is investigated using the localized induction equation. A family of vortex filaments which move without change of form are obtained. They are expressed in terms of elliptic integrals of the first, second and third kinds. In general they do not close and have infinite lengths. In some particular cases they take the form of closed coils which wind a doughnut. There exist a family of closed vortex filaments which do not travel in space but only rotate around a fixed axis. Our solutions include various well-known shapes such as the circular vortex ring, the helicoidal filament, the plane sinusoidal filament, Euler's elastica and the solitary-wave-type filament. It is shown that they correspond to the travelling wave solution of a nonlinear Schrödinger equation.

A spectral collocation and matrix eigenvalue method is used to study the linear stability of the trailing line vortex model of Batchelor. For both the inviscid and viscous stability problem, the entire unstable region in the swirl/axial wavenumber parameter space is mapped out for various azimuthal wavenumbers m. In the inviscid case, the non-axisymmetric perturbation with azimuthal wavenumber m = 1 has an unstable region of larger extent than any other, with an unusual two-lobed structure; also, the location and numerical value of the maximum disturbance growth rate previously reported for this case are shown to be incorrect. Exploiting the increasingly localized structure of perturbation eigenfunctions allows accurate results to be obtained up to values of m more than 3 orders of magnitude larger than previously, and the results for the most unstable mode are in excellent agreement with the asymptotic theory of Leibovich & Stewartson. A viscous analysis of these fundamentally inviscid modes reveals that the critical Reynolds number at which instability first occurs increases as O(m2) for m [Gt ] 1, and finds the critical values of swirl and wavenumber, which approach limiting values as m → ∞.

In the viscous case, the instabilities for m = 0 and 1 recently reported by Khorrami are found via a simplified numerical approach and the entire unstable region for each of these modes is mapped out over a wide range of Reynolds numbers. The critical Reynolds numbers for these modes are found to be 322.42 and 17.527, respectively, the latter having been unreported previously. The instabilities persist in the limit of large Reynolds number, with corresponding disturbance growth rates decreasing roughly as 1/Re. In addition to the primary mode, a new family of long-wave viscous instabilities is found for the m = 1 case.

Some experimental results are presented for the motion of small rigid spherical particles suspended in a Newtonian viscous liquid flowing under steady laminar conditions in a vertical tube of circular cross-section. When the particles were neutrally buoyant, the Segré & Silberberg (1961, 1962) effect was confirmed, the particles moving into a narrow annular zone with a diameter about two-thirds of the tube diameter. When the particles were slightly denser than the fluid, they migrated relatively rapidly to the wall of the tube for downward fluid flow and to the axis of the tube for upward fluid flow. Individual particle trajectories were obtained (necessarily approximately) from photographic records, and statistical techniques used to obtain ‘universal’ paths for given flow conditions. A complete set of relevant dimensionless parameters is given by the tube flow Reynolds number, the ratio of the tube diameter to the particle diameter, and the ratio of the Stokes free-fall velocity of the particle to the maximum fluid velocity. An attempt has been made to study the dependence of the trajectories on each of these parameters, the ranges being 10–200, 10–20 and 0–0.2 respectively. Detailed results can be found in Jeffrey (1964). Some comments are made on the relevance for this situation of certain theoretical solutions given elsewhere.

The boundary-value problem for irrotational surface waves is derived from a variational integral I with the Lagrangian density [Lscr ] = Ξ ηt - [Hscr ] where Ξ (X, t) is the value of the velocity potential at the free surface, y = η(x, t), and [Hscr ] is the energy density in x space. [Hscr ] then is expressed as a functional of Ξ and η, qua canonical variables, by solving a reduced boundary-value problem for the potential, after which the requirement that I be stationary with respect to independent variations of Ξ and η yields a pair of evolution equations for Ξ and η. The Fourier expansions Ξ = pn(t)ϕn*(x) and η = qn(t)ϕn(x), where {ϕn} is an orthogonal set of basis functions, reduce I to Hamilton's action integral, in which the complex amplitudes pn and qn appear as canonically conjugate co-ordinates, and yield canonical equations for pn and qn that are the spectral transforms of the evolution equations for Ξ and η. The evolution equations are reduced (asymptotically) to partial differential equations for pn and qn by expanding [Hscr ] in powers of α = a/d and β = (d/l)2, where a and l are scales of amplitude and wavelength. Explicit third approximations are developed for β = O(α).

Electrothermal motion in an aqueous solution arises from the action of an electric field on inhomogeneities in the liquid induced by temperature gradients. The temperature field can be produced by the applied electric field through Joule heating, or caused by external sources, such as strong illumination. Electrothermal flows in microsystems are usually observed at applied signal frequencies around 1 MHz and voltages around 10 V. In this work, we present self-similar solutions for the motion of an aqueous solution in a constant temperature gradient placed on top of: (a) two coplanar electrodes subjected to an a.c. potential difference, and (b) four coplanar electrodes subjected to a four-phase a.c. signal, generating a rotating field. The first case produces two-dimensional rolls whereas the second case produces a liquid whirl. Finally, we present experimental results of electrothermal liquid flows generated by alternating and rotating electric fields under strong illumination, and these experiments are compared to the analytical solutions. The induced rotating flow could be used in the mixing of analytes and of liquids in microsystems.