Parallel shear flows have continuous symmetries of translation in the downstream and spanwise directions. As a consequence, flow states that differ in their spanwise or downstream location but are otherwise identical are dynamically equivalent. In the case of travelling waves, this trivial degree of freedom can be removed by going to a frame of reference that moves with the state, thereby turning the travelling wave in the laboratory frame into a fixed point in the comoving frame of reference. We here discuss a general approach, the method of comoving frames, by which the symmetry related motions can also be removed for more complicated and dynamically active states and demonstrate its application for several examples. For flow states in the asymptotic suction boundary layer (ASBL) we show that in the case of the long-period oscillatory edge state we can find local phase speeds which remove the fast oscillations and reveal the slow vortex dynamics underlying the burst phenomenon. For spanwise translating states we show that the method removes the drift but not the dynamical events that cause the big spanwise displacement. For a turbulent case we apply the method to the spanwise shifts and find slow components that are correlated over very long times. Calculations for plane Poiseuille flow show that the long correlations in the transverse motions are not specific to the ASBL.

We consider the problem of a Boussinesq fluid forced by applying both non-uniform temperature and stress at the top surface. On the other boundaries the conditions are thermally insulating and either no-slip or stress-free. The interesting case is when the direction of the steady applied surface stress opposes the sense of the buoyancy driven flow. We obtain two-dimensional numerical solutions showing a regime in which there is an upper cell with thermally indirect circulation (buoyant fluid is pushed downwards by the applied stress and heavy fluid is elevated), and a second deep cell with thermally direct circulation. In this two-cell regime the driving mechanisms are competitive in the sense that neither dominates the flow. A scaling argument shows that this balance requires that surface stress vary as the horizontal Rayleigh number to the three-fifths power.

The phenomenon of pattern formation of free-surface waves of a fluid confined in an annulus the inner wall of which vibrates radially, is investigated both theoretically and experimentally. Although the waves are excited by harmonic axisymmetric deformations of the inner shell, depending on the vibration frequency both axisymmetric and non-symmetric wave patterns may arise.

Experimental observations have revealed that waves are excited in two different resonance regimes. The first type corresponds to forced resonance, in which axisymmetric patterns are realized with eigenfrequencies equal to the frequency of excitation. The second kind is parametric resonance, in which case the waves are ‘transverse’, with their crests and troughs aligned perpendicular to the vibrating wall. These so-called cross-waves have frequencies equal to half of that of the wavemaker.

Both kinds of resonance were investigated theoretically using Lamé's method of superposition. It was shown experimentally that the pure forced resonant standing waves are not realized when the amplitude of excitation is beyond the threshold of parametric resonance for non-symmetric waves. The experimental observations agree very well with the theoretical results.

Most steady flows with constant vorticity and elliptical streamlines are known to be unstable. These, and certain axisymmetric time-periodic flows, can be analysed by Floquet theory. However, Floquet theory is inapplicable to other time-periodic flows that yield disturbance equations containing a quasi-periodic, rather than periodic, function. A practical method for surmounting this difficulty was recently given by Bayly, Holm & Lifschitz. Employing their method, we determine the stability of a clas of three-dimensional time-periodic flows: namely, those unbounded flows with fixed ellipsoidal stream surfaces and spatially uniform but time-periodic strain rates. Corresponding, but bounded, flows are those within a fixed ellipsoid with three different principal axes. This is perhaps the first exact stability analysis of non-reducibly three-dimensional and time-dependent flows. Though the model has some artificial features, the results are likely to shed light on more complex systems of practical interest.

It is shown how the effects of the initial discharge profile, vertical drift, and boundary absorption (catalytic reaction) can be incorporated into a Gaussian approximation for the two-dimensional contaminant distribution in a parallel shear flow. Exact and asymptotic expressions are derived for the centroid displacement, shear-dispersion coefficient, and variance. Detailed results are presented for the effect of absorption at the bed and of vertical drift velocities upon contaminant dispersion in turbulent open-channel flow. For both cases the advantages of discharges close to the bed over surface discharges are made quantitative.

In this article, we describe an optical set-up designed to measure directly velocity gradients (strophometry). This strophometer is based on the analysis of the distortions of a fringe pattern ‘written’ instantaneously in a flow field. We apply it to study the transverse velocity-gradient component ∂u/∂y in a plane Poiseuille flow at moderate Reynolds numbers. Mean values and different moments of the fluctuating gradient distribution related to viscous dissipation, vorticity dynamics and intermittency are obtained. These results are interpreted in terms of the large-scale structures which are present in the flow.

Unsteady convection of an initially homogeneous fluid in a vertical slot is investigated theoretically in the limit of large Rayleigh and Prandtl/Schmidt numbers. The motion is driven by prescribed fluxes of heat or mass at the vertical walls of the slot. The ‘heat-up’ problem is considered, i.e. the fluxes are specified to change instantaneously from zero to finite constant values. Perturbation methods are used to compute approximate solutions for the initial period and for the slow approach to the asymptotic state. Numerical solutions of the full problem are also given. It is shown that a significant stratification is set up after short time and that the system thereafter evolves as a strongly stratified fluid on a timescale that is proportional to $Ra^{\frac{2}{9}}$. During the latter part of the process, linear buoyancy layers of thickness $\sim Ra^{\frac{2}{9}}$ appear on the vertical walls. On the horizontal walls, there are nonlinear boundary layers of thickness $\sim Ra^{-\frac{1}{9}}$, whose structure is akin to that of a Stewartson E¼ layer. The theoretical predictions are found to be in good agreement with experimental results.

We study the mixing of fluid in a viscously decaying row of point vortices. To this end, we employ a simplified model based on Stuart's (1967) one-parameter family of solutions to the steady Euler equations. Our approach relates the free parameter to a vortex core size, which grows in time according to the exact solution of the Navier-Stokes equations for an isolated vortex. In this way, we approach an exact solution for small values of t/Re. We investigate how the growing core size leads to a shrinking of the cat's eye and hence to fluid leaking out of the trapped region into the free streams. In particular, we observe that particles initially located close to each other in neighbouring intervals along the streamwise direction escape from the cat's eye near opposite ends. The size of these intervals scales with the inverse square root of the Reynolds number. We furthermore examine the particle escape times and observe a self-similar blow-up for the particles near the border between two adjacent intervals. This can be explained on the basis of a simple stagnation-point flow. An investigation of interface generation shows that viscosity leads to an additional factor proportional to time in the growth rates. Numerical simulations confirm the above results and give a detailed picture of the underlying mixing processes.

In this paper we describe a simple way of modelling boundary-layer effects in analytical flow noise studies. We develop an exact analogy between the real flow and one in which there is a step velocity profile. This profile is intended to model a boundary layer in an idealized way and we recognize it in our Green's function for the problem. We insist that the Green's function is bounded, a step that makes it non-causal and similar to those used in recent jet-noise analogies. We derive an expression for the induced pressure which consists of surface and volume terms, just as in Lighthill's theory, but, because both contain elements to be evaluated in future time, we argue that the turbulence must be able to respond to linear surface stimulus and avoid the otherwise inevitable violation of causality. This novel feature distinguishes our analysis from applications of Lighthill's theory to boundary-layer-induced noise. The response of the turbulence may be large when the surface is driven at low boundary-layer Strouhal number. But it is negligible at high Strouhal number, and in that limit the surface terms are found to depend only on the instantaneous boundary geometry and its rate of change. This leads to a simple expression for ‘boundary-layer fluid loading’, in which the finite boundary-layer scale emerges explicitly. To illustrate the physical consequences of this result, we use it to estimate the impedance of a baffled piston vibrating beneath a boundary layer. Potential theory predicts that flow should destabilize the piston motion while experiments usually indicate the reverse. We find that the boundary layer is responsible for the discrepancy and that experimentally observed behaviour is predicted quite reasonably by our model.

In this paper we consider the development of shear dispersion following the introduction of a diffusing tracer substance into a tube or duct containing flowing fluid, with emphasis on the characterization of the temporal variation of concentration at a fixed axial position. Asymptotic results are derived by assuming that the distance downstream of the point of tracer introduction, appropriately non-dimensionalized, is large. First, we consider the central moments of the temporal concentration variation, including their dependence on transverse position and on the initial transverse distribution of tracer. The moments for finite Péclet number are expressed in terms of their infinite-Péclet-number counterparts, and the latter are given explicitly for Poiseuille flow. Then, assuming the Péclet number is infinite, we derive an approximate solution for the Green's function expressing tracer concentration following its introduction at an arbitrary point within the tube. The solution is expressed in terms of three numerically evaluated functions of a dimensionless time variable, with parametric dependence on the distance downstream of the point of tracer release. The method is illustrated by calculation of the approximate solution for dispersion in Poiseuille flow. Unlike previous approximations, the present solution is uniformly asymptotic and represents the tails of the concentration distribution as well as the approximately Gaussian central part; in these three regions, simpler analytic forms of the approximation are given. Comparison with previous computational solutions suggests the present approximation remains reasonably accurate even at quite short distances from the point where tracer is released.

We consider a resonant triad of gravity–capillary waves, riding on top of a much longer gravity wave. The long-wave phase is assumed to vary on the same scale as the slow modulation of the short waves. Envelope equations are first deduced in the Lagrangian description. By perturbation analysis for a weak long wave, we then find that the long wave can resonate the natural modulation oscillations of the triad envelope, giving rise to various bifurcations in the Poincaré map. Numerical integration for a stronger long wave reveals that chaos can emerge from these bifurcations. The bifurcation criterion of Chen & Saffman (1979) for collinear Wilton's ripples is generalized to arbitrary non-collinear triads, and is found to play an important role as a criterion for the onset of chaotic behaviour.

The paper considers the evolution of weakly nonlinear disturbances in linearly unstable stratified shear flows. We develop a generic Hamiltonian formulation for two-dimensional flows. The paper is focused on three-wave resonant interactions, which are always present in the stratified shear flows under consideration as the lowest-order nonlinear process. Two different types of shear flows are considered. The first one is the classical piecewise-linear model with constant density and vorticity in each layer. For such flows, linear instability is due to weak interaction of different modes. The second type is the Kelvin–Helmholtz model, consisting of two layers with different densities and velocities. Velocity shear is assumed to be weakly supercritical. We show that apart from the classical triplets consisting of stable waves, both flow types admit only triplets consisting of one weakly unstable and two neutrally stable waves, and we consider them in detail.

Universal evolution equations for three resonantly interacting wave packets are derived for both cases. For the first flow type, the generic equations coincide with the system derived earlier for a particular case of resonant interactions between unstable and neutral baroclinic waves in a quasi-geostrophic two-layer model. The evolution equations for the Kelvin–Helmholtz model are new, and are studied numerically and analytically in detail. In particular, we demonstrate that resonant interaction with neutral waves can stabilize the growth of the linearly unstable wave. This mechanism is essentially different from the well-known nonlinear stabilization mechanism due to cubic nonlinearity.

It has recently been shown that the behaviour of a gas bubble in a uniaxial straining flow can be used as a simplified model to describe some important aspects of the more complex, turbulent bubble break-up problem, provided that the Reynolds and the Weber numbers are sufficiently large. In the present investigation, we extend that work and, using a level-set numerical scheme, we analyse the influence of the bubble Reynolds number on break-up time, $t_b$, for supercritical Weber numbers, $\hbox{\it We}\,{>}\,\hbox{\it We}_c$, where $\hbox{\it We}_c$ is the critical Weber number. It is observed that the viscosity introduces corrections of ${{O}}(1/\hbox{\it Re})$ in the break-up time obtained in the limit $\hbox{\it Re}\to \infty$. In addition, the action of other possible mechanisms of break-up at subcritical Weber numbers, $\hbox{\it We}\,{<}\,\hbox{\it We}_c$, is also explored.

Edge waves on a beach of gentle slope ε [Lt ] 1 are considered. For constant slope, Ursell (1952) has obtained a complete set of trapped modes and shown that there exists only a finite number n of such modes, (2n + 1)β < ½π, β = tan−1ε. For non-uniform slope the formulae for the trapped-mode frequencies were heuristically derived by Shen, Meyer & Keller (1968). For small n ∼ O(1) Miles (1989) has obtained formulae which coincide with Shen et al.'s (1968) with accuracy to O(ε) and differ from them by O(ε2). However, Miles’ formulae fail at n ∼ 1/ε. In this paper it is proved that Shen et al.'s (1968) formulae are valid for all n (including n ∼ 1/ε) with accuracy to O(ε) and corrections of any order in ε are given. Uniform asymptotic expansions are obtained for the corresponding eigenfunctions. These expansions give Miles’ (1989) result for small n. The formulae for the frequencies and the eigenfunctions have the same structure for both the full dispersion system and the shallow-water equation. For small n the frequencies for both models coincide with accuracy to O(ε2), but for n ∼ 1/ε they differ by O(1). In the last section the effect of rotation following Evans (1989) is taken into account. All the asymptotics have formal character, i.e. they satisfy the corresponding equations with accuracy to O(εN), N being arbitrarily large. The rigorous justification of these asymptotics is under way.

A train of uniform two-dimensional gravity waves in deep water is known to be unstable to certain sideband disturbances. If the time of propagation is sufficiently long for the fourth-order terms to be important, the sidebands may grow at unequal rates, resulting in a downward shift of peak frequency. But this shift is only a temporary phase of a recurrent evolution process. Recent work by us (Hara & Mei 1991) has shown that wind and dissipation can help maintain this downshift at large time. In this paper we examine a similar two-dimensional problem for capillary–gravity waves. The basic flow in air and water is assumed to be steady, horizontally uniform and turbulent; the wave-induced flow in both media is assumed to be laminar. Evolution equations are deduced with wind and dissipation included in such a way that their influence is comparable to the asymmetric spectral evolution. After finding the initial growth rates of unstable sidebands, the nonlinear development of modulational instability is examined by integrating the evolution equations numerically. Computed results show that persistent downshift of frequency can happen for relatively long waves, but upshift occurs for very short waves.

The flow in the bulk driven by a viscous interfacial film set in motion by a rotating sharp circular knife edge has been examined through experiments and computations. In the experiments, the water surface is covered by an insoluble monomolecular film of dipalmitoylphosphatidylcholine (DPPC), a molecule of wide interest in biology and medicine. It is shown that the viscous coupling between the interfacial film and the bulk liquid leads to a strong bulk flow. Depending on the surface packing and corresponding surface tension, DPPC monolayers exhibit a wide range of phase morphologies. Upon shearing the monolayer, its viscous response varies from that of an essentially inviscid film at low surface packing, to that of a highly viscous non-Newtonian (shear thinning) film when the packing is dense. The more viscous the film, the stronger the driven bulk flow. We have examined this behaviour for hydrodynamic regimes straddling the Stokes flow regime and where flow inertia is important.

The theory of stress-induced cavitation is applied here to the problem of cavitation of a viscous liquid in the streaming flow past a stationary sphere. This theory is a revision of the pressure theory which states that a flowing liquid will cavitate when and where the pressure drops below a cavitation threshold, or breaking strength, of the liquid. In the theory of stress-induced cavitation the liquid will cavitate when and where the maximum tensile stress exceeds the breaking strength of the liquid. For example, liquids at atmospheric pressure which cannot withstand tension will cavitate when and where additive tensile stresses due to motion exceed one atmosphere. A cavity will open in the direction of the maximum tensile stress, which is 45° from the plane of shearing in pure shear of a Newtonian fluid. This maximum tension criterion is applied here to analyse the onset of cavitation for the irrotational motion of a viscous fluid, the special case imposed by the limit of very low Reynolds numbers and the fluid flow obtained from the numerical solution of the Navier–Stokes equations. The analysis leads to a dimensionless expression for the maximum tensile stress as a function of position which depends on the cavitation and Reynolds numbers. The main conclusion is that at a fixed cavitation number the extent of the region of flow at risk to cavitation increases as the Reynolds number decreases. This prediction that more viscous liquids at a fixed cavitation number are at greater risk of cavitation seems not to be addressed, affirmed nor denied, in the cavitation literature known to us.

A numerical study of the three-dimensional internal waves excited by topography in the flow of a stratified fluid is described. In the resonant flow of a nearly two-layer fluid, it is found that the time-development of the nonlinearly excited waves agrees qualitatively with the solution of the forced KP equation or the forced extended KP equation. In this case, the upstream-advancing solitary waves become asymptotically straight crested because of abnormal reflection at the sidewall similar to Mach reflection. The same phenomenon also occurs in the subcritical flow of a nearly two-layer fluid. However, in the subcritical flow of a linearly stratified Boussinesq fluid, the two-dimensionalization of the upstream waves can be interpreted as the separation of the lateral modes due to the differences in the group velocity of the linear wave, although this does not mean in general that the generation of upstream waves is describable by the linearized equation.

Instability and transition to turbulence in a temporally evolving free shear layer of an electrically conducting fluid affected by an imposed parallel magnetic field is investigated numerically. The case of low magnetic Reynolds number is considered. It has long been known that the neutral disturbances of the linear problem are three-dimensional at sufficiently strong magnetic fields. We analyse the details of this instability solving the generalized Orr–Sommerfeld equation to determine the wavenumbers, growth rates and spatial shapes of the eigenmodes. The three-dimensional perturbations are identified as oblique waves and their properties are described. In particular, we find that at high hydrodynamic Reynolds number, the effect of the strength of the magnetic field on the fastest growing perturbations is limited to an increase of their oblique angle. The dimensions and spatial shape of the waves remain unchanged. The transition to turbulence triggered by the growing oblique waves is investigated in direct numerical simulations. It is shown that initial perturbations in the form of superposition of two symmetric waves are particularly effective in inducing three-dimensionality and turbulence in the flow.

Viscous stretching of a cigar-shaped drop due to the centrifugal pressure field in a surrounding rapidly rotating denser fluid is analysed. Scaling arguments are used to examine the various contributions to the viscous stresses resisting deformation, and a number of asymptotic regimes are identified which are delineated by the relative magnitudes of the aspect ratio, the viscosity ratio and unity. These asymptotic regimes may usefully be described as the bubble, pipe, sliding-rod and toffee-strand limits. Detailed analysis based upon a slenderness assumption combined with an integral representation of Stokes equations is used to derive evolution equations for the shape of the drop as a function of time in the different regimes. In the limit that interfacial-tension effects are negligible, similarity solutions are developed in which the length of the drop is found to increase as t2/5, t1/4, (t ln t)1/4 and t. The analytical results are in good agreement with numerical simulations based upon a boundary-integral solution to the full viscous flow equations.