The two-dimensional erosion of a vortex subjected to an external, adverse shear is studied experimentally. The flow takes place in a thin stratified layer; the vortex is produced by electromagnetic forcing, whereas the shear is driven mechanically. The system thus allows the vortex strength and the external shear to be controlled independently. We observe the so-called ‘erosion’ process, i.e. the progressive decrease of the vortex area, leaving the vortex core unaffected. This process is controlled by the ratio γ=S/ωmax, where ωmax and S are respectively the maximum vorticity of the vortex and the external shear. At small γ, the erosion is weak and the vortex survives over the duration of the experiment. At large γ, the vortex is first eroded, and then, after a critical time, becomes stretched and eventually breaks up into filaments. During the first period of time, the compensated maximum vorticity (with friction decay removed) is constant and the vortex area decreases, while beyond the critical time, both quantities decrease with time. A critical value for γ, defining the transition between these two regimes, is determined experimentally: γc=0.051±0.017. The breaking process itself, during which the vorticity of the vortex core decreases, is investigated. All the qualitative aspects of the erosion process, the onset of breaking and the breaking process itself are found to be in excellent agreement with the theoretical and numerical description. The experimental value of γc and the properties of the filamentation process are consistent with the numerical estimates.

Moving sand waves and the overlying tubulent flow were measured on the Wilga River in Poland, and the Tirnava Mica and Buzau Rivers in Romania. Bottom elevations and flow velocities were measured at six points simultaneously by multi-channel measuring systems. From these data, the linear and two-dimensional sections of the three-dimensional correlation and structure functions and various projections of sand wave three-dimensional spectra were investigated.

It was found that the longitudinal wavenumber spectra of the sand waves in the region of large wavenumbers followed Hino's −3 law (S(Kx) ∝K−3x) quite satisfactorily, confirming the theoretical predictions of Hino (1968) and Jain & Kennedy (1974). However, in contrast to Hino (1968), the sand wave frequency spectrum in the high-frequency region was approximated by a power function with the exponent −2, while in the lower-frequency region this exponent is close to −3.

A dispersion relation for sand waves has been investigated from analysis of structure functions, frequency spectra and the cross-correlation functions method. For wavelengths less than 0.15–0.25 of the flow depth, their propagation velocity C is inversely proportional to the wavelength λ. When the wavelengths of spectral components are as large as 3–4 times the flow depth, no dispersion occurs. These results proved to be in good qualitative agreement with the theoretical dispersion relation derived from the potential-flow-based analytical models (Kennedy 1969; Jain & Kennedy 1974). We also present another, physically-based, explanation of this phenomenon, introducing two types of sand movement in the form of sand waves. The first type (I) is for the region of large wavenumbers (small wavelengths) and the second one (II) is for the region of small wavenumbers (large wavelengths). The small sand waves move due to the motion of individual sand particles (type I, C∝λ−1) while larger sand waves propagate as a result of the motion of smaller waves on their upstream slopes (type II, C∝λ0). Like the sand particles in the first type, these smaller waves redistribute sand from upstream slopes to downstream ones. Both types result in sand wave movement downstream but with a different propagation velocity.

The main characteristics of turbulence, as well as the quantitative values characterizing the modulation of turbulence by sand waves, are also presented.

A study of the response of a columnar vortex with non-zero axial flow to impulsive cutting has been performed. The flow evolution is computed based on the vorticity–velocity formulation of the axisymmetric Euler equation using a Lagrangian vorticity collocation method. The vortex response is compared to analytical predictions obtained using the plug-flow model of Lundgren & Ashurst (1989). The plug-flow model indicates that axial motion on a vortex core with variable core area behaves in a manner analogous to one-dimensional gas dynamics in a tube, with the vortex core area playing a role analogous to the gas density. The solution for impulsive cutting of a vortex obtained from the plug-flow model thus resembles the classic problem of impulsive motion of a piston in a tube, with formation of an upstream-propagating vortex ‘shock’ (over which the core radius changes discontinuously) and a downstream-propagating vortex ‘expansion wave’ on opposite sides of the cutting surface. Direct computations of the vortex response from the Euler equation reveal similar upstream- and downstream-propagating waves following impulsive cutting for cases where the initial vortex flow is subcritical. These waves in core radius are produced by a series of vortex rings, embedded within the columnar vortex core, having azimuthal vorticity of alternating sign. The effect of the compression and expansion waves is to bring the axial and radial velocity components to nearly zero behind the propagating vortex rings, in a region on both sides of the cutting surface with ever-increasing length. The change in vortex core radius and the variation in pressure along the cutting surface agree very well with the predictions of the plug-flow model for subcritical flow after the compression and expansion waves have propagated sufficiently far away. For the case where the ambient vortex flow is supercritical, no upstream-propagating wave is possible on the compression side of the vortex, and the vortex axial flow is observed to impact on the cutting surface in a manner similar to that commonly observed for a non-rotating jet impacting on a wall. The flow appears to approach a steady state near the point of impact after a sufficiently long time. The vortex response on the expansion side of the cutting surface exhibits a downstream-propagating vortex expansion wave for both the subcritical and supercritical conditions. The results of the vortex response study are used to formulate and verify predictions for the net normal force exerted by the vortex on the cutting surface. An experimental study of the cutting of a vortex by a thin blade has also been performed in order to verify and assess the limitations of the instantaneous vortex cutting model for application to actual vortex–body interaction problems.

A vortex approaching a no-slip wall ‘rebounds’ due to the creation of vorticity at the wall in a viscous boundary layer. Here it is demonstrated that a purely inviscid mechanism can also produce vortex rebound from a slip wall. In inviscid vortex rebound, vortex tube stretching generates the necessary vorticity to allow rebound, eliminating the need for viscous vorticity generation. This vortex stretching mechanism is demonstrated through numerical simulations and laboratory experiments on dipole-vortex rebound from a boundary. In an application to oceanography, numerical simulations of both quasi-geostrophic and shallow water dynamics are used to show that the β-effect at an eastern boundary can produce this inviscid rebound. Through a series of numerical experiments in which the strength of the β-effect is varied, a formula for predicting the point of separation of the vortices from the boundary in a dipole–coast collision is deduced. Through simulations, the flux of vorticity and fluid away from the boundary is measured as a function of β and initial angle of incidence. It is found that, in contrast to viscous vortex rebound, which typically does not produce a flux of material away from the boundary farther than a distance comparable to the initial vortex radius, the β-induced rebound does carry fluid far from the coast. Laboratory experiments in a rotating tank are used to show that a sloping bottom can also provide an inviscid mechanism for dipole-vortex rebound from the wall of the tank under certain conditions. A relation determining the conditions under which inviscid or viscous processes will dominate in the rebound of the dipole from a boundary is obtained.

We consider the internal gravity waves that are produced in an inviscid Boussinesq fluid, whose Brunt–Väisälä frequency N is constant, by the small rectilinear vibrations of a horizontal elliptic cylinder whose major axis is inclined at an arbitrary angle to the horizontal. When the angular frequency ω is greater than N, no waves are produced and the governing elliptic equation is solved using conformal transformations. Analytic continuation in ω to values less than N, when waves are produced, is then used to determine the solution. It exhibits the surprising feature that, apart from certain phase differences, the form of the velocity distributions in each of the beams of waves that occur is the same for all values of the thickness ratio of the ellipse, the inclination of its major axis to the horizontal and the plane in which the vibrations are occurring. The Fourier decomposition of the velocity distribution is found and is used in a sequel, Part 2, to investigate the effects of viscous dissipation.

In an important paper Makarov et al. (1990) have given an approximate solution for a vibrating circular cylinder in a viscous fluid. We show that the limit of this solution as the viscosity tends to zero is not the exact inviscid solution discussed herein. Further comparison of their work and ours will be made in Part 2.

An approximate theory is given for the generation of internal gravity waves in a viscous Boussinesq fluid by the rectilinear vibrations of an elliptic cylinder. A parameter λ which is proportional to the square of the ratio of the thickness of the oscillatory boundary layer that surrounds the cylinder to a typical dimension of its cross-section is introduced. When λ[Lt ]1 (or equivalently when the Reynolds number R[Gt ]1), the viscous boundary condition at the surface of the cylinder may to first order in λ be replaced by the inviscid one. A viscous solution is proposed for the case λ[Lt ]1 in which the Fourier representation of the stream function found in Part 1 (Hurley 1997) is modified by including in the integrands a factor to account for viscous dissipation. In the limit λ→0 the proposed solution becomes the inviscid one at each point in the flow field.

For ease of presentation the case of a circular cylinder of radius a is considered first and we take a to be the typical dimension of its cross-section in the definition of λ above. The accuracy of the proposed approximate solution is investigated both analytically and numerically and it is concluded that it is accurate throughout the flow field if λ is sufficiently small, except in a small region near where the characteristics touch the cylinder where viscous effects dominate.

Computations indicate that the velocity on the centreline on a typical beam of waves, at a distance s along the beam from the centre of the cylinder, agrees, within about 1%, with the (constant) inviscid values provided λs/a is less than about 10−3. This result is interpreted as indicating that those viscous effects which originate from the characteristics that touch the cylinder (places where the inviscid velocity is singular) reach the centreline of the beam when λs/a is about 10−3. For larger values of s, viscous effects are significant throughout the beam and the velocity profile of the beam changes until it attains, within about 1% when λs/a is about 2, the value given by the similarity solution obtained by Thomas & Stevenson (1972). For larger values of λs/a, their similarity solution applies.

In an important paper Makarov et al. (1990) give an approximate solution for the circular cylinder that is very similar to ours. However, it does not reduce to the inviscid one when the viscosity is taken to be zero.

Finally it is shown that our results for a circular cylinder apply, after small modifications, to all elliptical cylinders.

The evolution of the interface between two viscous fluid layers in a two-dimensional horizontal channel confined between two parallel walls is considered in the limit of Stokes flow. The motion is generated either by the translation of the walls, in a shear-driven or plane-Couette mode, or by an axial pressure gradient, in a plane-Poiseuille mode. Linear stability analysis for infinitesimal perturbations and fluids with matched densities shows that when the viscosities of the fluids are different and the Reynolds number is sufficiently high, the flow is unstable. At vanishing Reynolds number, the flow is stable when the surface tension has a non-zero value, and neutrally stable when the surface tension vanishes. We investigate the behaviour of the interface subject to finite-amplitude two-dimensional perturbations by solving the equations of Stokes flow using a boundary-integral method. Integral equations for the interfacial velocity are formulated for the three modular cases of shear-driven, pressure-driven, and gravity-driven flow, and numerical computations are performed for the first two modes. The results show that disturbances of sufficiently large amplitude may cause permanent interfacial deformation in which the interface folds, develops elongated fingers, or supports slowly evolving travelling waves. Smaller amplitude disturbances decay, sometimes after a transient period of interfacial folding. The ratio of the viscosities of the two fluids plays an important role in determining the morphology of the emerging interfacial patterns, but the parabolicity of the unperturbed velocity profile does not affect the character of the motion. Increasing the contrast in the viscosities of the two fluids, while keeping the channel capillary number fixed, destabilizes the interfaces; re-examining the flow in terms of an alternative capillary number that is defined with respect to the velocity drop across the more-viscous layer shows that this is a reasonable behaviour. Comparing the numerical results with the predictions of a lubrication-flow model shows that, in the absence of inertia, the simplified approach can only describe a limited range of motions, and that the physical relevance of the steadily travelling waves predicted by long-wave theories must be accepted with a certain degree of reservation.

The properties of the time- and span-averaged mean wake recirculation region are investigated in separated flows over several different two-dimensional bluff bodies. Ten different cases are considered and they divide into two groups: cylindrical geometries of circular, elliptic and square cross-sections and the normal plate. A wide Reynolds number range from 250 to 140000 is considered, but in all the cases the attached portion of the boundary layer remains laminar until separation. The lower Reynolds number data are from direct numerical simulations, while the data at the higher Reynolds number are obtained from large-eddy simulation and the experimental work of Cantwell & Coles (1983), Krothapalli (1996, personal communication), Leder (1991) and Lyn et al. (1995). Unlike supersonic and subsonic separations with a splitter plate in the wake, in all the cases considered here there is strong interaction between the shear layers resulting in Kármán vortex shedding. The impact of this fundamental difference on the distribution of Reynolds stress components and pressure in relation to the mean wake recirculation region (wake bubble) is considered. It is observed that in all cases the contribution from Reynolds normal stress to the force balance of the wake bubble is significant. In fact, in the cylinder geometries this contribution can outweigh the net force from the shear stress, so that the net pressure force tends to push the bubble away from the body. In contrast, in the case of normal plate, owing to the longer wake, the net contribution from shear stress outweighs that from the normal stress. At higher Reynolds numbers, separation of the Reynolds stress components into incoherent contributions provides more insight. The behaviour of the coherent contribution, arising from the dominant vortex shedding, is similar to that at lower Reynolds numbers. The incoherent contribution to Reynolds stress, arising from small-scale activity, is compared with that of a canonical free shear layer. Based on these observations a simple extension of the wake model (Sychev 1982; Roshko 1993a, b) is proposed.

It has been observed for a long time that under certain conditions a vortex or even a group of vortices forms in bays which have a narrow opening to the sea. What leads to the formation of such vortices confined in a quiet, almost closed bay? Why does their number vary? Can such vortices form in any specific bay with known hydrological conditions, coastal configuration and bottom topography? The answers to these questions are essential in practice because, if several vortices form in a bay, a sort of a ‘vortex cork’ is created which prevents the outflow of pollution from the bay. This pollution will be locked in the bay practically permanently. The formation of vortices can also very strongly modify the topology of the background flow and lead to the formation of structures which intensify such processes as beach drifting, silting, and coastal erosion.

This article considers the topology of the vortex regimes generated in harbour-like basins by the external potential longshore current at large Reynolds numbers. The theory discusses the issues of what solution compatible with the Prandtl–Batchelor theorem for inviscid fluids, and under what conditions, may be realized as an asymptotic state of the open hydrodynamical system. The analysis is developed based on the variational principle, the most appropriate fundamental method of modern physics in this case, modified for the open degenerated hydrodynamical system. It is shown that the steady state corresponds to the circulational regime in which the system has minimal energy and enstrophy. This state is fixed by the Reynolds number. The relation between the Reynolds number, the geometry factor and the topological number, characterizing the number of vortex cells, is found.

We derive a time-stepping method for unsteady fully nonlinear two-dimensional motion of a two-layer fluid. Essential parts of the method are: use of Taylor series expansions of the prognostic equations, application of spatial finite difference formulae of high order, and application of Cauchy's theorem to solve the Laplace equation, where the latter is found to be advantageous in avoiding instability. The method is computationally very efficient. The model is applied to investigate unsteady trans-critical two-layer flow over a bottom topography. We are able to simulate a set of laboratory experiments on this problem described by Melville & Helfrich (1987), finding a very good agreement between the fully nonlinear model and the experiments, where they reported bad agreement with weakly nonlinear Korteweg–de Vries theories for interfacial waves. The unsteady transcritical regime is identified. In this regime, we find that an upstream undular bore is generated when the speed of the body is less than a certain value, which somewhat exceeds the critical speed. In the remaining regime, a train of solitary waves is generated upstream. In both cases a corresponding constant level of the interface behind the body is developed. We also perform a detailed investigation of upstream generation of solitary waves by a moving body, finding that wave trains with amplitude comparable to the thickness of the thinner layer are generated. The results indicate that weakly nonlinear theories in many cases have quite limited applications in modelling unsteady transcritical two-layer flows, and that a fully nonlinear method in general is required.

Experiments were conducted to investigate the response of a high-Reynolds-number turbulent boundary layer in a supersonic flow to the perturbation presented by a forward-facing ramp. Two ramps were used: one with sharp corners, the other with rounded corners having radii of curvature equal to 15 initial boundary layer thicknesses. The flow was turned through 20° in each of the compressions and expansions. Hence, there was no net change in the flow direction over the ramps and only a small change in free-stream conditions due to the entropy increase across relatively weak shocks. The two experiments gave similar results. In the middle of the relaxing boundary layer, the streamwise Reynolds stress undershot the undisturbed levels and exhibited a response similar to that observed in subsonic boundary layer flows recovering from an impulse of streamline curvature (Smits, Young & Bradshaw 1979b). The turbulent shear stress vanished throughout most of the boundary layer, and an overall destruction of the turbulence production mechanisms was apparent as the boundary layer exhibited a slow recovery.

An analytical theory is given for the viscous wake behind a spherical bubble rising steadily in a pure liquid at high Reynolds number, and for that wake's effect on the motion of a second bubble rising underneath the first. Previous theoretical work on this subject consists of just two papers: a first approximation ignoring wake vorticity diffusion between the bubbles, and a full numerical solution avoiding simplifying approximations (apart from that of spherical shape of the bubbles). A second approximation is now found; it removes much of the discrepancy between the first approximation and the full solution. The leading-order calculation of wake vorticity diffusion uses a transformation of the independent variables which appears to be new. Experimental work to date has disagreed with all the theoretical work, but it addresses a somewhat different problem: a line of many bubbles.

We present a critical review of the Hamiltonian and the Lagrangian theories of pattern formation in driven capillary waves at low viscosity and high aspect ratio. We construct a Hamiltonian perturbation theory in the spirit of Milner's (1991) formulation, and derive the amplitude equations and their coefficients relevant at the onset of surface waves. Our presentation is detailed, and we carefully point out the differences between our results for the nonlinear coefficients and the results obtained by others. From our standing wave analysis we find that the square pattern is subcritical. Among the supercritical standing wave patterns, we find that the eightfold quasi-crystalline pattern, observed by Christiansen et al. (1992) and by Bosch (1995), is more stable than both rolls and hexagons. We outline the high-aspect-ratio experimental results obtained so far, and discuss them in the light of the theory.

A novel method of suppressing the Rayleigh–Plateau capillary instability of a cylindrical liquid bridge is demonstrated which uses the radiation pressure of an ultrasonic wave to control the shape of the bridge. The shape of the bridge is optically sensed and the information used to control the spatial distribution of the radiation stress on the surface of the bridge. The feedback is phased so as to suppress the growth of the axisymmetric mode which normally becomes unstable when the slenderness, given by the length to diameter ratio, exceeds π. Stabilization is achieved out to a slenderness of 4.3 for a bridge density matched to the surrounding water bath in a Plateau tank. Breakup of such long bridges was found to produce a satellite drop from the receding thread of liquid. The active stabilization mechanism used may have application to other capillary systems.

Simultaneous steady forced and free convective flow past a heated or cooled body in a semi-infinite porous medium subject to the Darcy–Boussinesq approximation is treated analytically and numerically. A correction term is derived in terms of the Rayleigh (Ra) and Péclet (Pe) numbers for the velocity and temperature fields far from the body; this is subsequently implemented in a numerical treatment, using finite-difference techniques in elliptical coordinates, for 1[les ]Pe[les ]100 and [mid ]Ra[mid ][les ]103. Flow separation is observed for both heating and cooling, perhaps surprisingly so for the former case since the flow near the plate is being accelerated by comparison with the forced convection case. A simple analogy with inviscid flow theory serves to illustrate the manner in which separation eddies are formed for both heating and cooling cases.