We present the results of a combined experimental and numerical study of flow separation in the unsteady two-dimensional rotor-oscillator flow. Experimentally detected material spikes are directly compared to separation profiles predicted from numerical shear-stress and pressure data, using a recent kinematic theory of unsteady separation. For steady, periodic, quasi-periodic and random forcing, fixed separation is observed, and experimental observations and theoretical predictions are in close agreement. The transition from fixed to moving separation is also reported.

We study the fine-scale statistics of temperature and its derivatives in turbulent Rayleigh–Bénard convection. Direct numerical simulations are carried out in a cylindrical cell with unit aspect ratio filled with a fluid with Prandtl number equal to 0.7 for Rayleigh numbers between 107 and 109. The probability density function of the temperature or its fluctuations is found to be always non-Gaussian. The asymmetry and strength of deviations from the Gaussian distribution are quantified as a function of the cell height. The deviations of the temperature fluctuations from the local isotropy, as measured by the skewness of the vertical derivative of the temperature fluctuations, decrease in the bulk, but increase in the thermal boundary layer for growing Rayleigh number, respectively. Similarly to the passive scalar mixing, the probability density function of the thermal dissipation rate deviates significantly from a log-normal distribution. The distribution is fitted well by a stretched exponential form. The tails become more extended with increasing Rayleigh number which displays an increasing degree of small-scale intermittency of the thermal dissipation field for both the bulk and the thermal boundary layer. We find that the thermal dissipation rate due to the temperature fluctuations is not only dominant in the bulk of the convection cell, but also yields a significant contribution to the total thermal dissipation in the thermal boundary layer. This is in contrast to the ansatz used in scaling theories and can explain the differences in the scaling of the total thermal dissipation rate with respect to the Rayleigh number.

Motivated by geological carbon dioxide (CO2) storage, we present a vertical-equilibrium sharp-interface model for the migration of immiscible gravity currents with constant residual trapping in a two-dimensional confined aquifer. The residual acts as a loss term that reduces the current volume continuously. In the limit of a horizontal aquifer, the interface shape is self-similar at early and at late times. The spreading of the current and the decay of its volume are governed by power-laws. At early times the exponent of the scaling law is independent of the residual, but at late times it decreases with increasing loss. Owing to the self-similar nature of the current the volume does not become zero, and the current continues to spread. In the hyperbolic limit, the leading edge of the current is given by a rarefaction and the trailing edge by a shock. In the presence of residual trapping, the current volume is reduced to zero in finite time. Expressions for the up-dip migration distance and the final migration time are obtained. Comparison with numerical results shows that the hyperbolic limit is a good approximation for currents with large mobility ratios even far from the hyperbolic limit. In gently sloping aquifers, the current evolution is divided into an initial near-parabolic stage, with power-law decrease of volume, and a later near-hyperbolic stage, characterized by a rapid decay of the plume volume. Our results suggest that the efficient residual trapping in dipping aquifers may allow CO2 storage in aquifers lacking structural closure, if CO2 is injected far enough from the outcrop of the aquifer.

The dynamics and spectrum of internal gravity waves generated in a linearly stratified fluid by tidal flow over a flat-topped ridge are investigated at five different latitudes using an inviscid two-dimensional numerical model. The resulting wave field includes progressive freely propagating waves which satisfy the dispersion relation, and forced waves which are trapped non-propagating oscillations with frequencies outside the internal wave band. The flow is largely stable with respect to shear instabilities, and, throughout the runs, there is a negligibly small amount of overturning which is confined to the highly nonlinear regions along the sloping topography and where tidal beams reflect from the boundaries. The wave spectrum exhibits a self-similar structure with prominent peaks at tidal harmonics and interharmonics, whose magnitudes decay exponentially with frequency. Two strong subharmonics are generated by an instability of tidal beams which is particularly strong for near-critical latitudes where the Coriolis frequency is half the tidal frequency. When both subharmonics are within the free internal wave range (as in cases 0°–20° N), they form a resonant triad with the tidal harmonic. When at least one of the two subharmonics is outside of the range (as in cases 30°–40° N) the observed instability is no longer a resonant triad interaction. We argue that the two subharmonics are generated by parametric subharmonic instability that can produce both progressive and forced waves. Other interharmonics are produced through wave–wave interactions and are not an artefact of Doppler shifting as assumed by previous authors. As the two subharmonics are, in general, not proper fractions of the tidal frequency, the wave–wave interactions are capable of transferring energy to a continuum of frequencies.

We address the flutter instability of a flexible plate immersed in an axial flow. This instability is similar to flag flutter and results from the competition between destabilizing pressure forces and stabilizing bending stiffness. In previous experimental studies, the plates have always appeared much more stable than the predictions of two-dimensional models. This discrepancy is discussed and clarified in this paper by examining experimentally and theoretically the effect of the plate aspect ratio on the instability threshold. We show that the two-dimensional limit cannot be achieved experimentally because hysteretical behaviour and three-dimensional effects appear for plates of large aspect ratio. The nature of the instability bifurcation (sub- or supercritical) is also discussed in the light of recent numerical results.

Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier–Stokes equations. We construct a dynamical 105-dimensional state-space representation of plane Couette flow at Reynolds number Re = 400 in a small periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Re and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Re turbulence. The invariant manifolds partially tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of symmetry-induced heteroclinic connections.

An experimental investigation of mixing mechanisms in a shock-induced instability flow is described. We obtain quantitative two-dimensional maps of the heavy-gas (SF6) concentration using planar laser-induced fluorescence for the case of a shock-accelerated cylinder of heavy gas in air. The instantaneous scalar dissipation rate, or mixing rate, χ, is estimated experimentally for the first time in this type of flow, and used to identify the regions of most intense post-shock mixing and examine the underlying mechanisms. We observe instability growth in certain regions of the flow beginning at intermediate times. The mixing rate results show that while these unstable regions play a significant role in the mixing process, a large amount of mixing also occurs by mechanisms directly associated with the primary instability, including gradient intensification via the large-scale strain field in a particular non-turbulent region of the flow.

The self-similar flow and free-surface shape induced by a flat plate entering an inviscid and incompressible liquid are investigated for arbitrary initial conditions. An analytical solution, which is based on two governing expressions, namely the complex velocity and the derivative of the complex potential, is obtained. These expressions are derived in an auxiliary parameter plane using integral formulae proposed for the determination of an analytical function from its modulus and argument given on the boundary of the parameter region. We derive a system of an integral and an integro-differential equation in terms of the velocity modulus and the velocity angle at the free surface, which are determined by the dynamic and kinematic boundary conditions. A numerical procedure for solving these equations is carefully validated by comparisons with results available in the literature. The results are presented in terms of the free surface shape, the angles at the tip of the splash jet, the contact angles at the intersection with the plate surface, pressure distribution and force coefficients. New features caused by the flow unsteadiness are found and discussed.

Results are presented from pressure measurements performed in the irrotational near field of unbounded co-axial jets. Measurements were made for a variety of velocity and temperature ratios, and configurations both with and without serrations on the secondary nozzle lip. The principal objective of the study is to better understand the near pressure field of the jet, what it can tell us regarding the underlying turbulence structure, and in particular how it can be related to the source mechanisms of the flow.

A preliminary analysis of the axial, temporal and azimuthal structure of the pressure field shows it to be highly organized, with axial spatial modes (obtained by proper orthogonal decomposition) which resemble Fourier modes. The effects of serrations on the pressure fluctuations comprise a global reduction in level, a change in the axial energy distribution, and a modification of the evolution of the characteristic time scales.

A further analysis in frequency–wavenumber space is then performed, and a filtering operation is used to separate the convective and propagative footprints of the pressure field. This operation reveals two distinct signatures in the propagating component of the field: a low-frequency component which radiates at small angles to the flow axis and is characterized by extensive axial coherence, and a less-coherent high-frequency component which primarily radiates in sideline directions. The serrations are found to reduce the energy of the axially coherent propagating component, but its structure remains fundamentally unchanged; the high-frequency component is found to be enhanced. A further effect of the serrations involves a relative increase of the mean-square pressure level of the acoustic component – integrated over the measurement domain – with respect to the hydrodynamic component. The effect of increasing the velocity and temperature of the primary jet involves a relative increase in the acoustic component of the near field, while the hydrodynamic component remains relatively unchanged: this shows that the additional acoustic energy is generated by the mixing region which is produced by the interaction of the inner and the outer shear layers, whereas the hydrodynamic component of the near field is primarily driven by the outer shear layer.

The global linear stability of a three-dimensional compressible flow around a yawed parabolic body of infinite span is investigated using an iterative eigenvalue method in conjunction with direct numerical simulations. The computed global spectrum shows an unstable branch consisting of three-dimensional boundary layer modes whose amplitude distributions exhibit typical characteristics of both attachment-line and crossflow modes. In particular, global eigenfunctions with smaller phase velocities display a more pronounced structure near the stagnation line, reminiscent of attachment-line modes while still featuring strong crossflow vortices further downstream. This analysis establishes a link between the two prevailing instability mechanisms on a swept parabolic body which, so far, have only been studied separately and locally. A parameter study shows maximum modal growth for a spanwise wavenumber of β = 0.213, suggesting a preferred disturbance length scale in the sweep direction.

The behaviour of the velocity and pressure fluctuations in the logarithmic and outer layers of turbulent flows is analysed using spectral information and probability density functions from channel simulations at Reτ≤2000. Comparisons are made with experimental data at higher Reynolds numbers. It is found, in agreement with previous investigations, that the intensity profiles of the streamwise and spanwise velocity components have logarithmic ranges that are traced to the widening spectral range of scales as the wall is approached. The same is true for the pressure, both theoretically and observationally, but not for the normal velocity or for the tangential stress cospectrum, although even those two quantities have structures with lengths of the order of several hundred times the wall distance. Because the logarithmic range grows longer as the Reynolds number increases, variables which are ‘attached’ in this sense scale in the buffer layer in mixed units. These results give strong support to the attached-eddy scenario proposed by Townsend (1976), but they are not linked to any particular eddy model. The scaling of the outer modes is also examined. The intensity of the streamwise velocity at fixed y/h increases with the Reynolds number. This is traced to the large-scale modes, and to an increased intensity of the ejections but not of the sweeps. Several differences are found between the outer structures of different flows. The outer modes of the spanwise and wall-normal velocities in boundary layers are stronger than in internal flows, and their streamwise velocities penetrate closer to the wall. As a consequence, their logarithmic layers are thinner, and some of their logarithmic slopes are different. The channel statistics are available electronically at http://torroja.dmt.upm.es/ftp/channels/.

This paper theoretically and experimentally investigates the exchange flow due to temperature differences between open water and a canopy of aquatic plants. A numerical model is used to study the interfacial shape, frontal velocity and total volumetric exchange, and their dependence on a dimensionless vegetation drag parameter. The numerical predictions are consistent with the laboratory measurements. There is a short initial period in which the force balance is between buoyancy and inertia, followed by drag-dominated flow for which there is a balance between buoyancy and drag forces. After the initial stage, the gravity current propagating into the canopy takes a triangular shape whereas the current propagating into the open water has almost the classic unobstructed horizontal profile, but with a slowly decreasing depth. Near the edge of the canopy, but in the open region, the flow is found to be critical with a unit internal Froude number. The exchange flow rate and the front speed in the canopy decrease slowly with time whereas the gravity current in the open water has a constant speed. The magnitude of the exchange flow decreases as the canopy drag increases. Empirical equations for the flow properties are presented. A movie is available with the online version of the paper.

The linear stability of finite-amplitude interfacial solitary waves in a two-layer fluid of finite depth is examined analytically on the basis of the Euler equations. An asymptotic analysis is performed, which provides an explicit criterion of instability in the case of long-wavelength transverse disturbances. This result leads to the general statement that, when the amplitude of the solitary wave is increased, the solution becomes transversely unstable before an exchange of longitudinal stability occurs.

The linear and weakly nonlinear stability of a uniform flow in an infinitely wide open channel with erodible bottom is studied. Under suitable conditions the flow is found to be unstable, leading to the formation of dunes and antidunes. At a linear level, the corresponding regions of existence are presented and compared with experimental data. A weakly nonlinear analysis is then performed in a neighbourhood of the critical conditions for dune and antidune formation. The analysis shows that, for values of the ratio of the shear velocity to the depth-averaged velocity of practical interest, dune bifurcation is supercritical, whereas antidune bifurcation is subcritical. The latter result suggests a possible interpretation of the plane–antidune transition, where plane bed and antidune configurations are observed to coexist for the same values of the flow and the sediment parameters. The supercritical behaviour of the dune bifurcation allows for the prediction of an equilibrium amplitude that successfully compares with the amplitudes measured in laboratory experiments.

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 show hydrodynamic boundary conditions to be the inherent consequence of the Onsager principle of minimum energy dissipation, provided the relevant effects of the wall potential appear within a thin fluid layer next to the solid wall, denoted the surface layer. The condition that the effect of the surface layer on the bulk hydrodynamics must be independent of its thickness h is shown to imply a set of consistent ‘scaling relationships’ between h and the surface-layer variables/parameters. The use of the scaling relations, in conjunction with the surface-layer equations of motion derived from the Onsager principle, directly leads to the hydrodynamic boundary conditions. We demonstrate the surface-layer scaling process both physically and mathematically, and relate the parameters of the boundary conditions to those in the surface-layer equations of motion. In spatial regions outside the surface layer, equivalence between the use of surface-layer dynamics and boundary conditions is numerically demonstrated for Couette flows. As an application of the present approach, we derive the liquid-crystal hydrodynamic boundary conditions in which the rotational and translational dynamics are coupled.

Near-impact behaviour is investigated for a solid body approaching another solid body with two immiscible incompressible viscous fluids occupying the gap in between. The fluids have viscosity and density ratios which are extreme, the most notable combination being water and air, such that either or both of the bodies are covered by a thin film of water. Air–water interaction and the commonly observed phenomenon of air trapping are of concern in the presence of the two or three thin layers and one or two interfaces. The subcritical regime is of most practical significance here and it leads physically to the effect of inviscid water dynamics coupling with a viscous-dominated air response locally. This physical mechanism induces touchdown (or an approach to touchdown), which is found to occur in the sense that the scaled air-gap thickness shrinks towards zero within a finite scaled time according to analysis performed hand in hand with computation. A global influence on the local touchdown properties is also identified. Comparisons with computations prove favourable. Air trapping is produced between two touchdown positions, at each of which there is a pressure peak; an oblique approach would not affect the finding unless the approach itself is extremely shallow. The mechanism of air–water interaction leading to air trapping is suggested as a quite wide-ranging result.

‘Horizontal convection’ arises when a temperature variation is imposed along a horizontal boundary of a finite fluid volume. Here we study the infinite-Prandtl-number limit relevant to very viscous fluids, motivated by the study of convection in glass furnaces. We consider a rectangular domain with insulating conditions on the sides and bottom, and a linear temperature gradient on the top. We describe steady states for a large range of aspect ratio A and Rayleigh number Ra, and find universal scalings for the transition from small to large Rayleigh numbers. At large Rayleigh number, the top boundary-layer thickness scales as Ra−1/5, with the circulation and heat flux scaling as Ra1/5. These scalings hold for both rigid and shear-free boundary conditions on the top or on the other boundaries, which is initially surprising, but is because the return flow is dominated by a horizontal intrusion immediately beneath the top boundary layer. A downwelling plume also forms on one side, but because of strong stratification in the interior, the volume flux it carries is much smaller than that of the horizontal intrusion, decaying as the inverse of the depth below the top boundary. The fluid in the plume detrains into the interior and then returns to the top boundary, thus forming a ‘filling box’. We find analytic solutions for the interior temperature and streamfunction and match them to a similarity solution for the plume. At depths comparable to the length of the top boundary the streamfunction has O(1) values and the temperature variations scale as 1/Ra. Transient calculations with a large, but finite, Prandtl number, show how the steady state is reached from hot and cold initial conditions.

Nonlinear processes in a hypersonic boundary layer on a sharp cone are considered using the bicoherence method. The experiments are performed for a Mach number M∞ = 5.95 with introduction of artificial wave packets at the frequency of the second mode. It is shown that the basic mechanism of nonlinear interaction at the location of the maximum r.m.s. voltage fluctuation is the subharmonic resonance; all nonlinear interactions in the maximum r.m.s. voltage fluctuation layer are related to the second mode of disturbances; nonlinear processes above and below that layer are much more intense than those in it. The effect of artificial disturbances on nonlinear interactions in the boundary layer is shown to be insignificant.