Analytical and numerical results from recently developed strongly nonlinear asymptotic models are compared and validated with experimental observations of internal gravity waves and results from the numerical integrations of Euler equations for solitary waves at the interface of two-fluid systems. The focus of this investigation is on regimes where large amplitudes are attained, where the classical weakly nonlinear theories prove inadequate. Two asymptotically different regimes are examined in detail: shallow fluids, in which the typical wavelengths of the interface displacement are long with respect to the depths of both fluids, and deep fluids, where the wavelengths are comparable to, or less than, the depth of one of the two fluids. With the aim of illustrating the breakdown of the asymptotic assumptions, the transition from a shallow to a deep regime is examined through numerical computation of Euler system's solutions and by comparisons with solution to models.

In order to provide fundamental bases for incorporating the effects of favourable streamwise pressure gradients into models for internal turbulent flows and turbulent boundary layers, time series measurements with a cross-wire probe and a wall shear stress sensor were obtained simultaneously in an oil channel both for fully developed and laterally converging flows. Data were concentrated in the viscous layer and at the centreplane for slight to highly favourable pressure gradients. (Here the viscous layer is defined as the region where viscous effects are significant, but not necessarily dominant; it includes the ‘laminar’ and buffer sublayers in the terminology of some investigators). Results presented include comparisons of the profiles of the mean statistics, plus correlations and spectra, of streamwise and wall-normal components, their product and the wall shear stress. The key new data are the measurements of the fluctuating normal component and related statistics in the viscous layer for highly favourable pressure gradients. For the outer half of the viscous layer its root-mean-square fluctuations decrease as the pressure gradient is increased, consistent with heretofore unconfirmed predictions from direct numerical simulations. Based on examination of the probability density distributions, one may conclude that an effect of a strong pressure gradient is to reduce transport of momentum in the outer part of the viscous layer.

A little known electrohydrodynamic instability, which we call a rose window, is observed in air/liquid interfaces in electric fields with unipolar space charge distributions. Depending on the liquid properties, the rose window may appear from an initial rest state (primary instability) or on top of another instability, the classical unipolar-injection-induced instability, destroying its pattern (secondary instability). After imaging of the rose window, we use an edge-detection filter to find the instability threshold and study the characteristic pattern as a function of the liquid properties. Results show that the specific properties of the electric field, due to charge injection, are the cause of the rose-window and that the primary and secondary rose windows are essentially different instabilities.

The spiral form of vortex breakdown observed in the numerical simulations of Ruith et al. (J. Fluid Mech., vol. 486, 2003, p. 331) is interpreted as a nonlinear global mode originating at the convective/absolute instability transition point in the lee of the vortex breakdown bubble. The local absolute frequency at the transition station is shown to yield a satisfactory prediction of the precession frequency measured in the three-dimensional direct numerical simulations.

This paper examines the emergence of the absolute instability from convectively unstable states of an inviscid compound jet. A compound jet is composed of a cylindrical jet of one fluid surrounded by a concentric annulus of a second, immiscible fluid. For all jet velocities $v$, there are two convectively unstable modes. As in the single-fluid jet, the compound jet becomes absolutely unstable below a critical dimensionless velocity or Weber number $V ({:=}\,\sqrt{v^2\,{\rho_1 R_1}/\sigma_1}$ where $\rho_{1}$, $R_{1}$ and $\sigma _{1}$ are the core density, radius and core–annular interfacial tension), which is a function of the annular/core ratios of densities $\beta$, surface tensions $\gamma$ and radii $a$. At $V\,{=}\,0$, the absolutely unstable modes and growth recover the fastest growing temporal waves. We focus specifically on the effect of $\gamma$ at $a\,{=}\,2$ and $\beta\,{=}\,1$ and find that when the outer tension is significantly less than the inner $(0.1\,{<}\,\gamma\,{<}\,0.3)$, the critical Weber number $V_{\hbox{\scriptsize{\it crit}}}$ decreases with <$\gamma$, whereas for higher ratios $(0.3\,{<}\,\gamma\,{<}\,3)$ it increases. The values (1.2–2.3) of $V_{\hbox{\scriptsize{\it crit}}}$ for the compound jet include the parameter-independent critical value of 1.77 for the single jet. Therefore, increasing the outer tension can access the absolute instability at higher dimensional velocities than for a single jet with the same radius and density as the core and a surface tension equal to the compound jet's liquid–liquid tension. We argue that this potentially facilitates distinguishing experimentally between absolute and convective instabilities because higher velocities and surface tension ratios higher than 1 extend the breakup length of the convective instability. In addition, for $0.3\,{<}\,\gamma\,{<}\,1.16$, the wavelength for the absolute instability is roughly half that of the fastest growing convectively unstable wave. Thus choosing $\gamma$ in this range exaggerates its distinction from the convective instability and further aids the potential observation of absolute instability.

We describe a series of numerical simulations of dissolution-driven convection in a reactive porous medium heated from above. The physical system consists of a porous medium made of the frozen component of a binary mixture that is immersed in a liquid mixture with which it is in thermodynamic equilibrium. Surface heating results in melting of the uppermost material which releases dense solute and drives compositional convection. An interface develops between the upper region, in which the solid matrix has completely melted, and a lower region, in which the frozen solute evolves. The interface descends as melting proceeds. During the numerical simulations, scaled to be similar to previous experiments using potassium nitrate crystals and their saturated aqueous solution (Hallworth, Huppert & Woods, J. Fluid Mech. vol. 535, 2004, p. 255), there are three distinct phases: a purely conductive phase; followed by a phase with very brief, intense, compositionally driven convection; followed by a prolonged phase of more sedate compositionally driven convection in which the average kinetic energy is roughly one order of magnitude less than during the intense early phase. The field equations and the numerical methodology are presented in addition to a simple analytical model for the rate of motion of the interface. The analytical model, valid in the limit of very rapid mixing of the solute, is shown to be in good agreement with the numerical results of purely conductive calculations with a large diffusion coefficient. We investigate solutions for various values of the Rayleigh number and quantify the degree of interface motion as a function of this parameter. These simulations may be particularly applicable to problems associated with post-cumulate processes in magma chambers.

The simplest model of isotropic compressible turbulence consists of the Euler equations augmented by the equation for the turbulent energy. This model can also be viewed as the Euler equations for a continuum with two independent entropies. One of them is a conventional thermodynamic entropy, and the other is associated with the turbulent energy. The shock relations for this model are examined. It is shown that the turbulent entropy cannot exceed some critical value. We propose a closed set of Rankine–Hugoniot relations for the description of shock waves in such a medium based on this estimation.

This paper reports the results of several direct numerical simulations of the withdrawal of a two-layer fluid with a finite-thickness interface through a slot in the base of a finite rectangular cavity. Particular attention is paid to the role of long (basin scale) interfacial waves on the processes leading to drawdown of the interface into the slot. It is shown that these waves play an important role and can either delay or accelerate drawdown. This means that drawdown can occur over a range of Froude numbers. The results are compared with previous work for ideal flow and experimental results.

Turbulent stress is the fundamental quantity in the filtered equation for large-scale velocity that reflects its interactions with small-scale velocity modes. We develop an expansion of the turbulent stress tensor into a double series of contributions from different scales of motion and different orders of space derivatives of velocity, a multi-scale gradient (MSG) expansion. We compare our method with a somewhat similar expansion that is based instead on defiltering. Our MSG expansion is proved to converge to the exact stress, as a consequence of the locality of cascade both in scale and in space. Simple estimates show, however, that the convergence rate may be slow for the expansion in spatial gradients of very small scales. Therefore, we develop an approximate expansion, based upon an assumption that similar or ‘coherent’ contributions to turbulent stress are obtained from disjoint subgrid regions. This coherent-subregions approximation (CSA) yields an MSG expansion that can be proved to converge rapidly at all scales and is hopefully still reasonably accurate. As an important first application of our methods, we consider the cascades of energy and helicity in three-dimensional turbulence. To first order in velocity gradients, the stress has three contributions: a tensile stress along principal directions of strain, a contractile stress along vortex lines, and a shear stress proportional to ‘skew-strain’. While vortex stretching plays the major role in energy cascade, there is a second, less scale-local contribution from ‘skew-strain’. For helicity cascade the situation is reversed, and it arises scale-locally from ‘skew-strain’ while the stress along vortex lines gives a secondary, less scale-local contribution. These conclusions are illustrated with simple exact solutions of three-dimensional Euler equations. In the first, energy cascade occurs by Taylor's mechanism of stretching and spin-up of small-scale vortices owing to large-scale strain. In the second, helicity cascade occurs by ‘twisting’ of small-scale vortex filaments owing to a large-scale screw.

The inverse energy cascade of two-dimensional turbulence is often represented phenomenologically by a Newtonian stress–strain relation with a ‘negative eddy viscosity’. Here we develop a fundamental approach to a turbulent constitutive law for the two-dimensional inverse cascade, based upon a convergent multi-scale gradient (MSG) expansion. To first order in gradients, we find that the turbulent stress generated by small-scale eddies is proportional not to strain but instead to ‘skew-strain,’ i.e. the strain tensor rotated by $45^\circ$. The skew-strain from a given scale of motion makes no contribution to energy flux across eddies at that scale, so that the inverse cascade cannot be strongly scale-local. We show that this conclusion extends a result of Kraichnan for spectral transfer and is due to absence of vortex stretching in two dimensions. This ‘weakly local’ mechanism of inverse cascade requires a relative rotation between the principal directions of strain at different scales and we argue for this using both the dynamical equations of motion and also a heuristic model of ‘thinning’ of small-scale vortices by an imposed large-scale strain. Carrying out our expansion to second order in gradients, we find two additional terms in the stress that can contribute to the energy cascade. The first is a Newtonian stress with an ‘eddy-viscosity’ due to differential strain rotation, and the second is a tensile stress exerted along vorticity contour lines. The latter was anticipated by Kraichnan for a very special model situation of small-scale vortex wave-packets in a uniform strain field. We prove a proportionality in two dimensions between the mean rates of differential strain rotation and of vorticity-gradient stretching, analogous to a similar relation of Betchov for three dimensions. According to this result, the second-order stresses will also contribute to inverse cascade when, as is plausible, vorticity contour lines lengthen, on average, by turbulent advection.

We study the effect of interactions between objects floating at fluid interfaces, for the case in which the objects are primarily supported by surface tension. We give conditions on the density and size of these objects for equilibrium to be possible and show that two objects that float when well-separated may sink as the separation between them is decreased. Finally, we examine the equilbrium of a raft of strips floating at an interface, and find that rafts of sufficiently low density may have infinite spatial extent, but that above a critical raft density, all rafts sink if they are sufficiently large. We compare our numerical and asymptotic results with some simple table-top experiments, and find good quantitative agreement.

We study miscible displacements in long ducts in the dispersive limit of small $\varepsilon \hbox{\it Pe}$, where $\varepsilon \,{\ll}\, 1$ is the inverse aspect ratio and $\hbox{\it Pe}$ the Péclet number. We consider the class of generalized Newtonian fluids, with specified closure laws for the fluid properties of the concentration-dependent mixture. Regardless of viscosity ratio and the constitutive laws of the pure fluids, for sufficiently small $\varepsilon \hbox{\it Pe}$ these displacements are characterized by rapid cross-stream diffusion and slow streamwise dispersion, i.e. the concentration appears to be near-uniform across the duct and spreads slowly as it translates. Using the multiple-scales method we derive the leading-order asymptotic approximation to the average fluid concentration $\bar{c}_0$. We show that $\bar{c}_0$ evolves on the slow timescale $t \sim (\varepsilon \hbox{\it Pe})^{-1}$, and satisfies a nonlinear diffusion equation in a frame of reference moving with the mean speed of the flow. In the case that the two fluids have identical rheologies and the concentration represents a passive tracer, the diffusion equation is linear. For Newtonian fluids we recover the classical results of Taylor (l953), Aris (1956), and for power-law fluids those of Vartuli et al. (1995). In the case that the fluids differ and/or that mixing is non-passive, $\bar{c}_0$ satisfies a nonlinear diffusion equation in the moving frame of reference. Given a specific mixing/closure law for the rheological properties, we are able to compute the dispersive diffusivity $D_T(\bar{c}_0)$ and predict spreading along the channel. We show that $D_T(\bar{c}_0)$ can vary significantly with choice of mixing law and discuss why. This also opens the door to possibilities of controlling streamwise spreading by the rheological design of reactive mixtures, i.e. including chemical additives such that the rheology of the mixture behaves very differently to the rheology of either pure fluid. Computed examples illustrate the potential effects that might be achieved.

We investigate the displacement of one fluid through an inclined porous sheet by the injection of a second fluid of different density. Using numerical simulation we explore the role of the density contrast between the injected and the reservoir fluid on the displacement process, in the cases where the density contrast originates from either compositional contrasts and/or temperature contrasts between the fluids. In the case where the density contrast originates from compositional differences between the fluids, the density front moves with the fluid–fluid front, and gravity may accelerate or decelerate the time for the injected liquid to reach the sink. In the case where the density contrast originates from a temperature contrast between the injected fluid and the reservoir fluid, then the density front follows the thermal front. Therefore, owing to thermal inertia, it lags behind the fluid–fluid front. This has a quantitative impact on the time required for the injected liquid to reach the sink. If there are both thermal and compositional contrasts between the injected and reservoir fluid, then the thermal and compositional fronts become decoupled in space. The two fronts may lead to complementary or opposing density changes; the different cases lead to vastly different patterns of displacement and time at which the injected liquid reaches the sink, even if the net change in density between reservoir and the injected fluid is the same. We discuss the implications of these phenomena for water injection in sub-surface hydrocarbon and geothermal reservoirs. In an Appendix, we note how a viscosity across both the thermal front and the fluid–fluid front can also lead to a rich spectrum of flow patterns, especially if one front is stable and the other unstable to viscous instability.

The formation and fragmentation of liquid sheets resulting from the oblique collision of two identical cylindrical jets is investigated. The liquid expands radially from the impacting point forming a sheet in the form of a bay leaf bounded by a thicker rim. The sheet shape, rim size and liquid velocity field are quantified and represented analytically. External harmonic perturbations of the injection conditions reveal the nature of the rim destabilization and of its coupling with the sheet. Flow perturbations in the incident jets lead to sheet thickness modulations which trigger the fragmentation of the rim via the formation of liquid ligaments whose dynamics is described. The breakup of these ligaments induce both the shape and width of the drop size distribution in the spray formed by this process.

The evaporation of droplets on a substrate that is wetting to the liquid is studied. The radius $R(t)$ of the droplet is followed in time until it reaches zero. If the evaporation is purely diffusive, $R \propto \sqrt{t_0\,{-}\,t}$ is expected, where $t_0$ is the time at which the droplet vanishes; this is found for organic liquids, but water has a different exponent. We show here that the difference is likely to be due to the fact that water vapour is lighter than air, and the vapour of other liquids more dense. If we carefully confine the water so that a diffusive boundary layer may develop, we retrieve $R(t) \propto \sqrt{t_0\,{-}\,t}$. On the other hand, if we force convection for an organic liquid, we retrieve the anomalous exponent for water.

An analytical model is presented for sound radiation from a semi-infinite unflanged annular duct. The duct carries a jet which issues into a uniform mean flow while an inner cylindrical centre body extends downstream from the duct exit. This geometrical arrangement forms an idealized representation of a turbofan exhaust where noise propagates along the annular bypass duct, refracts through the external bypass stream and radiates to the far field. The instability wave of the vortex sheet and its interaction with the acoustic field are accounted for in an exact way in the current solution. Efficient numerical procedures are presented for evaluating near-field and far-field solutions, and these are used as the basis for a parametric study to illustrate the effect of varying the hub–tip ratio, and the ratio of jet velocity to external flow velocity. Since the ‘Kutta’ condition can be turned on and off in the current solution, this capability is used to assess the effect of vortex shedding on noise radiation. Far-field directivity patterns are presented for single modes and also for a multi-mode ‘broadband’ source model in which all cut-on modes are assumed to be present with equal modal power. Good agreement is found between analytical solutions and experimental data. Near-field pressure maps of the acoustic and instability portions of the solution are generated for selected tones.

The three-dimensional problem of blunt-body impact onto a free surface of an ideal and incompressible liquid is considered within the Wagner approximation. This approximation is formally valid during an initial stage, when the depth of penetration is small, the wetted part of the body can be approximately replaced with a flat disk and the boundary conditions can be linearized and imposed on the undisturbed liquid surface. In the present context this problem will be referred to as the classical Wagner problem. However the classical Wagner problem of impact is nonlinear despite the fact that the equations of liquid motion and boundary conditions are linearized. The reason is that the contact region between the liquid and the entering body is unknown in advance and has to be determined together with the liquid flow. Several exact solutions of the three-dimensional Wagner problem are known as detailed in Part 1 (J. Fluid Mech. vol. 440, 2001, p. 293). Among these solutions the axisymmetric one is the simplest. In this paper, an additional linearization of the Wagner problem is considered. This linearization is performed on the basis of an axisymmetric solution via a perturbation technique. The small parameter $\epsilon$ is a measure of the discrepancy of the actual shape with respect to the closest axisymmetric shape. The method of solution of this problem is detailed here. The resulting solutions are compared to available exact solutions. Three shapes are studied: elliptic paraboloid; inclined cone; and pyramid. These shapes must be blunt in the vicinity of the initial contact point and hence only small deadrise angles can be considered. The stability of the obtained solutions is analysed. The second-order solution of the present Wagner problem with respect to $\epsilon$ is considered. That yields the leading-order correction to the hydrodynamic force which acts on an almost axisymmetric body entering liquid vertically. Other nonlinearities are not accounted for. Among them, there are the nonlinear terms in the boundary conditions and the actual geometry of the wetted body surface. Both the vertical and the horizontal components of the hydrodynamic force are obtained. For the inclined cone, comparisons with available experimental data are shown. The method developed can be helpful in testing other numerical approaches and optimizing the shape of the entering body accounting for three-dimensional effects. This paper appears as a necessary intermediate step before solving the general three-dimensional classical Wagner problem in Part 3.

The asymptotic solutions of second-order accuracy taking account of the velocity boundary conditions imposed on the sidewalls are derived for wall-localized boundary-layer convection in a rotating fluid layer in the presence of stress-free or no-slip vertical sidewalls. The second-order asymptotic solutions give a satisfactory quantitative agreement with the fully numerical solutions obtained in a rotating channel with either stress-free or no-slip sidewalls. Furthermore, we show, through fully three-dimensional numerical simulations, that the structure of the boundary-layer convection is highly robust in strongly nonlinear regimes.

A dead fish is propelled upstream when its flexible body resonates with oncoming vortices formed in the wake of a bluff cylinder, despite being well outside the suction region of the cylinder. Within this passive propulsion mode, the body of the fish extracts sufficient energy from the oncoming vortices to develop thrust to overcome its own drag. In a similar turbulent wake and at roughly the same distance behind a bluff cylinder, a passively mounted high-aspect-ratio foil is also shown to propel itself upstream employing a similar flow energy extraction mechanism. In this case, mechanical energy is extracted from the flow at the same time that thrust is produced. These results prove experimentally that, under proper conditions, a body can follow at a distance or even catch up to another upstream body without expending any energy of its own. This observation is also significant in the development of low-drag energy harvesting devices, and in the energetics of fish dwelling in flowing water and swimming behind wake-forming obstacles.

We examine the instability leading to depth-averaged circulations, which are related to rip currents in the surf zone, due to wave–current interactions. Our intention is to clarify some issues which are critical to the determination of instability properties, as yet unresolved from previous studies. Those issues are also of interest for hydrodynamic instability problems in general. Attention is restricted to normally incident waves and the region inside, and just offshore of, the surf zone where linearized shallow-water waves are applied. The coupling of waves and circulations is modelled using the concept of wave radiation stresses and the classical ray theory. The instability properties, in terms of the neutral modes, the most unstable mode and the corresponding maximum growth rate, are examined in the domain of parameters which represent the effects of offshore wave height and bottom friction dissipation. Comparisons with observations of natural rip currents are made, and qualitative agreements are achieved.