A matched-asymptotic analysis has been carried out for an axisymmetric convection cell in the case of stress-free boundaries. This problem differs from that of two-dimensional convection rolls mainly through the special role played by the central plume. The radius, of order ε, of the latter depends on the Rayleigh number R through the relationship $\epsilon^4(-\ln \epsilon) = R^{\frac{2}{3}}$. The plume velocity is independent of height at lowest order and its magnitude exceeds by a factor (− ln ε)½ the strength, of order $R^{\frac{2}{3}}$, of the core flow. As a result of these properties the central plume is governed by advection, in contrast to the perimeter plume which is affected by conduction as well. This asymmetry is reflected in the different thickness of the horizontal thermal boundary layers and gives rise to the deviation of the core temperature from the mean value of the top and bottom temperatures. This deviation is positive (negative) for the case of a falling (rising) central plume. While the core flow is driven mainly by the perimeter plume the fraction of the heat flux carried by the central plume is always above three-quarters and increases as the radius-to-height-ratio λ decreases.

A horizontal density gradient may be steepened to form a front if the horizontal flow which it drives is convergent. This convergence may be caused by an initial nonlinearity in the density gradient (as described by Simpson & Linden 1989). A quadratic density profile is analysed to illustrate the mechanism, and it is shown how the flow and the density profile interact to intensify and concentrate the front near a horizontal boundary. Linear and curved density profiles in a container of finite length are also studied: the most favourable location for frontogenesis is found to be where the flow emerges into a region of significant curvature after passing through a maximum of the density gradient.

Heat and mass transport in the bulk of a suspension or fixed bed in the limit of high Péclet number is controlled by hydrodynamic diffusion. This hydrodynamic diffusion is caused by the stochastic velocity field induced by the randomly distributed particles. However, the no-slip boundary conditions require that the hydrodynamic diffusivity vanish at boundaries of the medium. Thus, molecular diffusion must aid the transport in a thin boundary layer near the solid boundary. Quantitative results are derived for the hydrodynamic diffusivity, boundary layer thickness, and net resistance to transport across a dilute fixed fibrous bed in the direction transverse to the mean flow. A scaling analysis is presented for transport in the direction of the gradient of the mean velocity in a sheared suspension of neutrally buoyant spheres. This scaling analysis is able to explain the qualitative variations of the transport resistance with the Péclet number and the ratio of the suspension thickness to the particle radius.

We consider three-dimensional hydraulic jumps (shocks) propagating along boundaries in rotating fluids. This study is motivated by earlier work (Fedorov & Melville 1995), which dealt with the evolution to breaking of nonlinear Kelvin waves. We obtain the jump relations and derive an evolution equation for the jump as it propagates along the boundary. It is shown that after some initial adjustment the Kelvin-type jump assumes a permanent form and propagates with a constant velocity along the boundary or the coast. At some distance offshore the jump becomes oblique to the coastline, and the final shape of the jump and its speed depend only on the jump strength. The jump gives rise to a moderate mass transport offshore. The potential vorticity remains almost constant across the jump. The energy loss in the jump is proportional to the third power of the jump amplitude, which is similar to classical two-dimensional hydraulic jumps in non-rotating fluids. Jump properties are discussed for both weak and strong nonlinearity, and the role of a boundary layer region behind the leading edge of the jump is considered.

We determine the asymptotic conditions under which the Boussinesq approximation is valid for oscillatory convection in a rapidly rotating fluid. In the astrophysically relevant parameter regime of small Prandtl number, we show that the Boussinesq prediction for the onset of convection is valid only under much more restrictive conditions than those that are usually assumed. In the case of an ideal gas, we recover the Boussinesq results only if the ratio of the domain height to a typical scale height is much smaller than the Prandtl number. This requires an extremely shallow domain in the astrophysical parameter regime. Other commonly used ‘sound-proof’ approximations generally perform no better than the Boussinesq approximation. The exception is a particular implementation of the pseudo-incompressible approximation, which predicts the correct instability threshold beyond the range of validity of the Boussinesq approximation.

The compressibility effects on the structural evolution of the transitional high-speed planar wake are studied. The relative Mach number ($Ma_{r}$) of the laminar base flow modifies two fundamental features of planar wake transition: (i) the characteristic length scale defined by the most unstable linear mode; and (ii) the domain of influence of the structures within the staggered two-dimensional vortex array. Linear stability results reveal a reduced growth (approximately 30 % reduction up to $Ma_{r}=2.0$) and a quasilinear increase of the wavelength of the most unstable, two-dimensional instability mode (approximately 20 % longer over the same $Ma_{r}$ range) with increasing $Ma$. The primary wavelength defines the length scale imposed on the emerging transitional structures; naturally, a longer wavelength results in rollers with a greater streamwise separation and hence also larger circulation. A reduction of the growth rate and an increase of the principal wavelength results in a greater ellipticity of the emerging rollers. Compressibility effects also modify the domain of influence of the transitional structures through an increased cross-wake and inhibited streamwise communication as characteristic paths between rollers are deflected due to local $Ma$ gradients. The reduced streamwise domain of influence impedes roller pairing and, for a sufficiently large relative $Ma$, pairing is completely suppressed. Thus, we observe an increased two-dimensionality with increasing Mach number: directly contrasting the increasing three-dimensional effects in high-speed mixing layers. Temporally evolving direct numerical simulations conducted at $Ma=0.8$ and 2.0, for Reynolds numbers up to 3000, support the physical insight gained from linear stability and vortex dynamics studies.

Microtextured surfaces are commonly used to study complex hydrodynamic phenomena such as spreading and splashing of liquid droplets. However, although surface topography is known to modify near-surface flow, there is no theory able to quantitatively predict the dramatic changes in dynamics of liquid spreading and splashing. Here, we investigate experimentally water bells formed on micropatterned surfaces in order to characterize the hydrodynamics of inertia-dominated flows through regular porous layers. Water bells are self-suspended catenary-shaped liquid films created when a jet impinges on a horizontal disc called an impactor. We show that the presence of micrometre-sized posts regularly arranged on the impactor results in a decrease of the water bell radius and the loss of axisymmetry as open water bells adopt polygonal shapes. We introduce a simple model that captures the main features of the inertia-dominated flow and reveals the role of the hydrodynamic interactions between neighbouring posts. In addition to their applications for tunable jet atomization, these polygonal sheets provide a paradigmatic system for understanding inertia-dominated flow in porous media.

The three-dimensional governing equations of the flow field that is developed when an elasto-plastic exible porous medium, capable of undergoing extremely large deformations, is struck head-on by a shock wave, are developed using a multi-phase approach. The one-dimensional version of these equations is solved numerically using an arbitrary Lagrangian–Eulerian (ALE) based numerical code. The numerical predictions are compared qualitatively to experimental results from various sources and good agreement is obtained. This study complements our earlier study in which we developed and solved, using a total variation diminishing (TVD) based numerical code, the governing equations of the flow field that is developed when an elastic rigid porous medium, capable of undergoing only very small deformations, is struck head-on by a shock wave.

The linear stability of uniform, plane internal wave beams with locally confined spatial profile, in a stratified fluid of constant buoyancy frequency, is discussed. The associated eigenvalue problem is solved asymptotically, assuming perturbations of long wavelength relative to the beam width. In this limit, instability is found only for oblique disturbances which vary in the along-beam and the horizontal transverse directions. The mechanism of instability is a first-harmonic–mean resonant interaction between the underlying wave beam and three-dimensional perturbations that comprise a time-harmonic component, with the beam frequency, and a mean flow. Progressive beams which transport energy in one direction, in particular, are unstable if the beam steepness exceeds a certain threshold value, whereas purely standing beams are unstable even at infinitesimal steepness. A distinguishing feature of this three-dimensional modulational instability is the generation of circulating horizontal mean flows at large distances from the vicinity of the beam.

Perturbations in a non-normal system can grow transiently even if the system is linearly stable. If this transient growth is sufficiently large, it can trigger self-sustained oscillations from small initial disturbances. This has important practical consequences for combustion–acoustic oscillations, which are a persistent problem in rocket and aircraft engines. Balasubramanian & Sujith (J. Fluid Mech., vol. 594, 2008, pp. 29–57) modelled an infinite-rate chemistry diffusion flame in an acoustic duct and found that the transient growth in this system can amplify the initial energy by a factor, ${G}_{max} $, of the order of $1{0}^{5} $ to $1{0}^{7} $. However, recent investigations by L. Magri and M. P. Juniper have brought to light certain errors in that paper. When the errors are corrected, ${G}_{max} $ is found to be of the order of 1 to 10, revealing that non-normality is not as influential as it was thought to be.

In this paper we study the three-dimensional perturbations produced in a hypersonic boundary layer by a small wall roughness. The flow analysis is performed under the assumption that the Reynolds number, $R{e}_{0} = {\rho }_{\infty } {V}_{\infty } L/ {\mu }_{0} $, and Mach number, ${M}_{\infty } = {V}_{\infty } / {a}_{\infty } $, are large, but the hypersonic interaction parameter, $\chi = { M}_{\infty }^{2} R{ e}_{0}^{- 1/ 2} $, is small. Here ${V}_{\infty } $, ${\rho }_{\infty } $ and ${a}_{\infty } $ are the flow velocity, gas density and speed of sound in the free stream, ${\mu }_{0} $ is the dynamic viscosity coefficient at the ‘stagnation temperature’, and $L$ is the characteristic distance the boundary layer develops along the body surface before encountering a roughness. We choose the longitudinal and spanwise dimensions of the roughness to be $O({\chi }^{3/ 4} )$ quantities. In this case the flow field around the roughness may be described in the framework of the hypersonic viscous–inviscid interaction theory, also known as the triple-deck model. Our main interest in this paper is the nonlinear behaviour of the perturbations. We study these by means of numerical solution of the triple-deck equations, for which purpose a modification of the ‘skewed shear’ technique suggested by Smith (United Technologies Research Center Tech. Rep. 83-46, 1983) has been used. The technique requires global iterations to adjust the viscous and inviscid parts of the flow. Convergence of such iterations is known to be a major problem in viscous–inviscid calculations. In order to achieve improved stability of the method, both the momentum equation for the viscous part of the flow, and the equations describing the interaction with the flow outside the boundary layer, are treated implicitly in this study. The calculations confirm the fact that in this sort of flow the perturbations are capable of propagating upstream in the boundary layer, resulting in a perturbation field which surrounds the roughness on all sides. We found that the perturbations decay rather fast with the distance from the roughness everywhere except in the wake behind the roughness. We found that if the height of the roughness is small, then the perturbations also decay in the wake, though much more slowly than outside the wake. However, if the roughness height exceeds some critical value, then two symmetric counter-rotating vortices form in the wake. They appear to support themselves and grow as the distance from the roughness increases.

A study has been made of the flow and disintegration of thin liquid sheets in combustion gases up to temperatures of 950°C. It is found that below 300°C sheet breakdown occurs through the growth of antisymmetric Kelvin–Helmholtz waves. Above this temperature high frequency symmetric waves and localized disturbances are superimposed on the sheet and disintegration then occurs by the combined action of aerodynamic waves and perforations, the contribution of the latter predominating with increasing temperature. It is demonstrated that the new wave system is electrohydrodynamic in origin, the electric field being generated by the charged species present in the gas. The drop size is found to be critically dependent upon the nature of the disintegration process.

Shock–vortex interaction is analysed for strong gaseous shock waves and a ratio of specific heats close to unity. A nonlinear wave equation for the wrinkles of the shock front is obtained for weak vortices. The solution breaks down after a finite time and the slope of the front develops jump discontinuities, indicating the formation of Mach stems. Shock–turbulence interactions are also briefly discussed.

A model probability density equation is obtained by approximating the convective and diffusive terms in a single-point density formulation of homogeneous turbulent scalar transport, with first-order reaction, in a plume. The equation, which includes the intermittency factor of the scalar field explicitly, is then shown to support similarity solutions under some constraining assumptions. Comparison of the similarity solutions with field measurements of conditioned concentrations shows that they can reproduce the general features of the data for both low intermittency and high intermittency measurement regimes. On the basis of these asymptotic results a speculative modelling of the terms representing entrainment at the plume interface is proposed and a class of similarity solutions for the intermittency factor is obtained by numerical integration.

An experimental investigation was conducted at selected locations in the wall region of a three-dimensional turbulent boundary layer relaxing in a nominally zero external pressure gradient behind a transverse hump (in the form of a 30° swept, 5 ft chord, wing-type model) faired into the side wall of a low-speed wind tunnel. The boundary layer (approximately 3·5 in. thick near the first survey station, where the length Reynolds number was 5·5 × 106) had a maximum cross-flow velocity ratio of 0·145 and a maximum cross-flow angle of 21·9° close to the wall. The hot-wire data indicated that the apparent dimensionless velocity profiles in the viscous sublayer are universal and that the wall influence on the hot wire is negligible beyond y+ = 5. The existence of wall similarity in the relaxing flow field was confirmed in the form of a log law based on the resultant mean velocity and resultant friction velocity (obtained from the measured skin friction).

The smallest collateral region extended from the point nearest to the wall (y+ ≈ 1) up to y+ = 9·7, corresponding to a resultant mean velocity ratio (local to free-stream) of 0·187. The unusual feature of these profiles was the presence of a narrow region of slightly decreasing cross-flow angle (1° or less) that extended from the point of maximum cross-flow angle down to the outer limit of the collateral region. A sublayer analysis of the flow field using the measured local transverse pressure gradient slightly overestimated the decrease in cross-flow angle. It is concluded that, in the absence of these gradients, the skewing of the flow could have been much more pronounced practically down to the wall (limited only by the resolution of the sensor), implying a near-wall non-collateral flow field consistent with the equations of motion in the neighbourhood of the wall.

The streamwise relaxation of the mean flow field based on the decay of the cross-flow angle was much faster in the inner layer than in the outer layer. Over the stream-wise distance covered, the mean flow in the inner layer and the wall shear-stress vector relaxed to a two-dimensional state in approximately 10 boundary-layer thicknesses whereas the relaxation of the turbulence was slower and was not complete over the same distance.

We present a series of similarity solutions to describe the temperature field as liquid spreads from a line source into a porous rock saturated with liquid of higher temperature. We identify slow and fast flow regimes. In the slow flow regime, the liquid is heated to the far-field temperature by conduction of heat from the far field. In the fast flow regime, there is negligible conduction of heat from the far field. Instead, the liquid is heated to the far-field temperature by cooling a region of the host rock near the source, and an internal boundary layer develops within the newly injected liquid. We successfully test our quantitative theoretical predictions with a series of laboratory experiments in which water was injected into a consolidated bed of sand filled with liquid of different temperature. We extend our model to describe the vaporization of liquid as it spreads slowly from a central source into a superheated porous rock. A further family of similarity solutions shows that the rate of vaporization depends upon the injection rate as well as upon the initial superheat of the reservoir. For high injection rates, the liquid is typically heated to the interface temperature long before reaching the interface. The rate of vaporization then becomes independent of the initial liquid temperature, and depends mainly on the reservoir superheat. For lower injection rates, heat is conducted from ahead of the boiling front into the liquid. As a result, for progressively smaller injection rates, an increasing fraction of the liquid vaporizes, until virtually all the liquid boils, and only a very small liquid zone develops in the rock. Again, we successfully test our theoretical predictions with a laboratory experiment in which liquid water was injected into a superheated layer of permeable sandstone.

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.

According to the simple hydraulic theory of jets, each particle of a jet moves independently along a parabolic trajectory. Therefore a steady jet has a parabolic shape. We wish to consider how these results are modified by surface tension. For simplicity we will consider a two-dimensional jet of incompressible fluid.

A drop on a circular support spontaneously spreads upon contact with a substrate. The motion is driven by a loss of surface energy. The loss of recoverable energy can be expressed alternatively as work done at the liquid–gas interface or dissipation through viscosity and sliding friction. In this paper we require consistency with the energy lost by dissipation in order to infer details of the contact-line region through simulations. Simulations with the boundary integral method are used to compute the flow field of a corresponding experiment where polydimethylsiloxane spreads on a relatively hydrophobic surface. The flow field is used to calculate the energy dissipation, from which slip lengths for local slip and Navier slip boundary conditions are found. Velocities, shear rates and pressures along the interface as well as interface shapes in the microscopic region of the contact line are also reported. Angles, slip length and viscous bending length scale allow a test of the Voinov–Hocking–Cox model without free parameters.

In this article, we study the influence of solid-to-fluid density ratio m on the type of vortex-induced oscillation of a square section prism placed inside a two-dimensional channel. We assume that the solid body has neither structural damping nor spring restoring force. Accordingly, the prism equation of motion contains only inertia and aerodynamics forces. The problem is considered in the range of Reynolds numbers Re ∈ [50 200] (based on the prism cross-section height h) and channel widths H = H′/h ∈ [2.5 10]. We found that, for each Re and H, there is a critical mass ratio mc that separates two different oscillation regimes. For m > mc, the prism oscillation is periodical and contains a single harmonic. For m < mc, the prism oscillation changes completely and assumes an irregular pattern that is characterized by multiple harmonics that appear to belong to a uniform spectrum. The change from one regime to the other is abrupt and we were not able to observe a transitional regime in which the number of response harmonics grew by finite steps. The value of the critical mass ratio grows along with the Reynolds number and the channel width.