We present results from an experimental and numerical investigation into the dynamics of the interaction between a planar vortex pair or axisymmetric vortex ring with lengthscale a and circulation Γ encountering a planar interface of thickness δ across which the fluid density increases from ρ1 to ρ2. Similarity considerations indicate that baroclinic generation of vorticity and its subsequent interaction with the original vortex is governed by two dimensionless parameters, namely (a/δ) A and R, where A ≡ (ρρ2 − ρ1)/(ρ2 + ρ1) and R ≡ (a3g/Γ2). For thin interfaces (δ [Lt ] a), the interaction is governed only by the parameters A and R. Furthermore, in the Boussinesq limit (A → 0), the dynamics are governed solely by the product AR and the interaction is entirely invertible with respect to the initial locations and direction of propagation of the vortices. We document details of the interaction dynamics in the Boussinesq limit over a range of the parameter AR. Results show that, for relatively small values of AR, rather than the vortex simply rebounding at the interface, its outermost layers are instead successively ‘peeled’ away by baroclinically generated vorticity and form a topologically complex backflow in which the ring fluid, the light fluid and the heavy fluid are intertwined. For larger values of AR the vortex barely penetrates the interface, and our results suggest that in the limit AR → ∞ the interaction with a density interface becomes similar to the interaction at a solid wall. We also present results for thick interfaces in the Boussinesq limit, as well as larger density jumps for which the density parameter A enters as a second similarity quantity. Comparison of the experimental and numerical results demonstrate that many of the features of such interactions can be understood within the context of inviscid fluids, and that inviscid vortex methods can be used to accurately simulate the dynamics of such interactions.

Measurements have been made in the turbulent boundary layer on a flat plate in the presence of grid-generated free-stream, turbulence with a wide range of lengthscales. The data include conditionally sampled averages in which free-stream fluid was distinguished from boundary-layer fluid by heating the latter. Free-stream turbulence increases the standard deviation of the hot–cold interface as a proportion of the boundary-layer thickness, whilst the average position is mainly dependent upon the lengthscale. The shear correlation coefficient of the boundary-layer fluid decreases, and it is shown that the change in structure is directly related to the fluctuating-strain rate.

Transport velocities representing the diffusion of turbulent kinetic energy and shear stress have opposite signs in the boundary-layer fluid to those in the free-stream fluid, and it is shown that they are also related to the fluctuating-strain rate.

Complete balances of turbulent kinetic energy and shear stress have been evaluated, dissipation and pressure–strain redistribution having been deduced by difference. The dissipation length scale $L\tau = (-\overline{uv})^{\frac{3}{2}}/\epsilon $ is little affected by free-stream turbulence, whereas the corresponding parameter based on turbulent energy instead of shear stress is strongly affected.

A method has been devised for measuring the volume flux in a bubble plume in a homogeneous liquid. Laboratory experiments on weak bubble plumes using the method determined the flux as a function of height and gas flow rate for air flow rates between 0.41 and 6.25 cc/s. It was found that volume flux was proportional to the square-root of air flow and increased linearly with height. From measurement of bubble velocity it is concluded that the individual bubble wakes make an important contribution to the entrainment.

In this paper we study advection–diffusion of scalar and vector fields for the steady velocity field \[ (u, v, w) = \left(\frac{\partial \psi}{\partial y},-\frac{\partial\psi}{\partial x},K\psi\right),\quad \psi = \sin x \sin y + \delta \cos x \cos y. \] If δ > 0 the streamlines ψ = constant form a periodic array of oblique cat's-eyes separated by continuous channels carrying finite fluid flux. In the problems treated, advection dominates diffusion, and fields are transported both in thin boundary layers and within the channels. Effective transport of a passive scalar and the alphaeffect generated by interaction of the flow with a uniform magnetic field are examined. For the latter problem, we determine the alpha-matrix as a function of δ. Our results consist of (i) numerical solution of steady problems in the limit of large Reynolds number R with β = δR½ held fixed and O(1), and (ii) analytic asymptotic solutions for large R, obtained using the Wiener–Hopf technique, which are valid for large β. The asymptotic method gives reliable values of the effective diffusion and of alpha-matrices down to β ≈ 1.5.

When β > 0 the transport and alpha-effect are greatly enhanced by flux down the channels. Consequently, the alpha-effect found here may have application to the construction of efficient fast dynamos, but this requires spatial dependence of the mean field and the inclusion of three-dimensional effects, as in the established fast-dynamo analysis with δ = 0.

We apply the idea of choosing new variables that are nonlinear functions of the old in order to simplify calculations of irrotational, surface gravity waves. The usual variables consist of the surface elevation and the surface potential, and the transformation to the new variables is a canonical (in Hamilton's sense) one so as to maintain the Hamiltonian structure of the theory. We further consider the approximation of linear dynamics in these new variables. This approximation scheme exactly reproduces the effects of the lowest-order nonlinearities in the usual variables, does well at higher orders, and also captures important features of short waves interacting with longer waves. We provide a physical interpretation of this transformation which is correct in the one-dimensional case, and approximately so in the two-dimensional case.

The generation of long waves due to the refraction of wave groups over a shelf is reexamined. The appropriate boundary conditions for calculating the second-order mean free-surface displacement are discussed. Numerical results for a plane shelf are presented, which are different from those given by Mei & Benmoussa (1984).

Turbulent structure in a turbulent scalar dispersion field behind a fine cylindrical heat source in a weakly sheared flow is experimentally investigated and previous computational turbulence models for third-order scalar transport terms in the second-order turbulence equations are assessed with the present data.

The mean temperature and r.m.s. temperature profiles are found to be almost Gaussian even in the uniform shear layer. Decay of the peak temperature, mean dispersion and half-widths of the mean temperature and the r.m.s. temperatures are well correlated with corresponding data on the scalar dispersion behind an elevated line heat source in the turbulent boundary layer.

Normalized streamwise heat flux $\overline{u\theta}$ changes appreciably with the downstream distance owing to the influence of the uniform mean shear, whereas normalized vertical heat flux $\overline{v\theta}$ remains the same with the downstream distance. The timescale ratio R between temperature and velocity fluctuations varies from 0.3 to 1.3 across the stream and it asymptotes to a value 0.5 at far downstream.

Assessment of previous models for third-order moments with the present data reveals that application of a composite timescale between the dynamic timescale and the thermal timescale to the simplest gradient transport model yields a better overall prediction performance than any existing models, including Lumley's algebraic model equations for the moments. It was found that the timescale for the streamwise transports of $\overline{u\theta}$ and $\overline{\theta^2}$ is larger than that of lateral transports of $\overline{v\theta}$ and $\overline{\theta^2}$.

In addition, since the experiment isolates the effect of uniform mean shear on the turbulent scalar transport, experimental data accumulated by the present study will be useful for further development of more refined second-order turbulence models for non-isothermal turbulent flows.

A theoretical expression for the slow part (the nonlinear fluctuation part) of the pressure–strain rate is compared with simulations of anisotropic homogeneous flows. The purpose is to determine the quantitative accuracy of the theory and to test its qualitative predictions that the generalized Rotta coefficient, a non-dimensionalized ratio of slow term to kinetic energy anisotropy, varies with direction and can be negative (this is counter to isotropy return). Comparisons are made between theoretical and simulated values of the slow term and of the generalized Rotta coefficients. Also compared to simulations is an extension of the theory to account for non-stationary turbulence fields. The implication of the comparison for two-point closure theories and for Reynolds stress modelling is pointed out.

We generalize the methods of two-dimensional contour dynamics to study a two-layer rotating fluid that obeys the quasi-geostrophic equations. We consider here only the case of a constant-potential-vorticity lower layer. We derive equilibrium solutions for monopolar (rotating) and dipolar (translating) geostrophic vortices in the upper layer, and compare them with the Euler case. We show that the equivalent barotropic (infinite lower layer) case is a singular limit of the two-layer system. We also investigate the effect of a finite lower layer on the merger of two regions of equal-sign potential vorticity in the upper layer. We discuss our results in the light of the recent laboratory experiments of Griffiths & Hopfinger (1986). The process of filamentation is found to be greatly suppressed for equivalent barotropic dynamics on scales larger than the radius of deformation. We show that the variation of the critical initial distance for merger as a function of the radius of deformation and the ratio of the layers at rest is closely related to the existence of vortex-pair equilibria and their geometrical properties.

A method for calculating unsteady boundary-layer flows, based upon a vorticity–stream function formulation, has been supplemented by an appropriate viscous–inviscid interaction law to extend the calculations to large but finite Reynolds numbers. Two examples are considered in detail, namely the impulsive motion of a circular cylinder, and the flow induced when a line vortex is introduced into the neighbourhood of a circular cylinder. For the first of these a comparison of the results obtained with earlier boundary-layer and interactive calculations, Navier–Stokes solutions and experiment is made.

We study numerically the behaviour of and interaction between Burgers vortices – a known equilibrium solution to the Navier–Stokes equations which incorporates a balance between viscous diffusion and strain intensification of vorticity. A hybrid spectral/finite-difference method is employed to solve the Navier–Stokes equations in vorticity–stream function form for a unidirectional vorticity field on an infinite domain in the presence of a uniform three-dimensional strain field, one principal axis of which is parallel to the vorticity. Merging of two strained vortices is studied over a range of Reynolds numbers Re = Γ/2πν = 10–1280, and the results are used to calculate an energy spectrum for three-dimensional, homogeneous turbulence. The cancellation of two strained vortices with opposing circulation is investigated for Reynolds numbers Re = Γ0/2πν = 0.1–160 (τ0 is the circulation about one vortex), over a range of strain rates in the direction parallel to the line joining the vortex centres. A solution of the Navier–Stokes equations describing vorticity cancellation in the strain-induced collision of vortex layers (Kambe 1984) is used to estimate the asymptotic, timewise decay of circulation for each vortex. Good agreement with the present numerical results is obtained. Vortex core pressures calculated during the cancellation event are compared to a simple analytical model based on Moore & Saffman (1971).

The generation of magnetic fields by the turbulent motions of a conducting fluid driven by thermal convection in the incompressible case is studied by direct numerical simulations at Rayleigh numbers up to 200 times critical and Prandtl numbers 0.2 and 1. Our results are consistent with a dynamo effect which occurs for magnetic Reynolds numbers above a few tens, in the presence of, as well as in the absence of, rotation around the vertical axis. In all cases, the magnetic field is more intermittent than the velocity field.

The boundary layer is created on an infinite flat plate by a time-dependent free-stream velocity vector, whose magnitude is independent of time but whose direction (as seen in plan view) changes at a constant angular velocity. The pressure gradient, at right angles to the free-stream velocity, induces a skewing of the velocity profile; all components of the Reynolds-stress tensor are non-zero (using axes aligned with the wall and the flow direction). This flow has never been produced experimentally, but it has the merit of being simply defined and of having only the Reynolds number as a parameter, which greatly simplifies the analysis. The flow is studied theoretically using Reynolds-number scaling laws, and by direct numerical simulation over a range of Reynolds numbers. The simplest version of the theory is equivalent to existing theories of the Ekman layer. A higher-order version is presented and yields excellent agreement with the numerical results at three Reynolds numbers, with just one adjustable constant in each equation. The theory allows the extrapolation of the results to high Reynolds numbers. The Reynolds-averaged equations reduce to a one-dimensional steady problem, so that turbulence-model testing will be easy and accurate. Detailed data are provided for that purpose.

The existence of an inverse cascade is demonstrated for three-dimensional incompressible flow displaying the Anisotropic Kinetic Alpha (AKA) instability (Frisch, She & Sulem). By means of full three-dimensional simulations of the Navier–Stokes equations, it is shown that flow stirred at small scales by an anisotropic force lacking parity-invariance (i.e. lacking any centre of symmetry) can generate strongly helical structures on larger scales. When there is a substantial range of linearly unstable modes, the most unstable ones emerge at first, but are eventually dominated by modes with the smallest wavenumbers.

The key observation for the theory of this inverse cascade is that, in the presence of forcing, the small-scale Reynolds stresses will become dependent on the large-scale flow. Elimination of the small scales produces the nonlinear AKA equations for the large-scale flow. The latter have non-trivial one-dimensional solutions also displaying an inverse cascade, qualitatively similar to the one reported above. This cascade has been numerically simulated over a range of more than two decades. For a simple choice of the forcing, a steady state is eventually reached; it can be described analytically and presents interesting geometric features in the limit of very extended systems. The corresponding energy spectrum has a k−4 range. A number of other scaling relations are also derived.

The multi-dimensional extension of the theory is briefly considered. The resulting large-scale structures are conjectured to correspond to solutions of the incompressible Euler equation.

In this paper, we study the linearized stability of three symmetric arrangements of two liquids in core–annular Poiseuille flow in round pipes. Deferring to one important application, we say oil and water when we mean more viscous and less viscous liquids. The three arrangements are (i) oil is in the core and water on the wall, (ii) water is in the core and oil is outside and (iii) three layers, oil inside and outside with water in between. The arrangement in (iii) is our model for lubricated pipelining when the pipe walls are hydrophobic and it has not been studied before. The arrangement in (ii) was studied by Hickox (1971) who treated the problem as a perturbation of long waves, effectively suppressing surface tension and other essential effects which are necessary to explain the flows observed, say, in recent experiments of W. L. Olbricht and R. W. Aul. The arrangement in (i) was studied in Part 1 of this paper (Preziosi, Chen & Joseph 1987). We have confirmed and extended their pseudo-spectral calculation by introducing a more efficient finite-element code. We have calculated neutral curves, growth rates, maximum growth rate, wavenumbers for maximum growth and the various terms which enter into the analysis of the equation for the evolution of the energy of a small disturbance. The energy analysis allows us to identify the three competing mechanisms underway: interfacial tension, interfacial friction and Reynolds stress. Many results are presented.

Two-dimensional Boussinesq-type depth-averaged equations are derived for describing the interactions of weakly nonlinear shallow-water waves with slowly varying topography and currents. The current velocity varies appreciably within a characteristic wavelength. The effects of vorticity in the current field are considered. The wave field is decomposed into Fourier time harmonics. A set of evolution equations for the wave amplitude functions of different harmonics is derived by adopting the parabolic approximation. Numerical solutions are obtained for shallow-water waves propagating over rip currents on a plane beach and an isolated vortex ring. Numerical results show that the wave diffraction and nonlinearity are important in the examples considered.

Pressure fluctuations in a turbulent channel flow are investigated by analysing a database obtained from a direct numerical simulation. Detailed statistics associated with the pressure fluctuations are presented. Characteristics associated with the rapid (linear) and slow (nonlinear) pressure are discussed. It is found that the slow pressure fluctuations are larger than the rapid pressure fluctuations throughout the channel except very near the wall, where they are about the same magnitude. This is contrary to the common belief that the nonlinear source terms are negligible compared with the linear source terms. Probability density distributions, power spectra, and two-point correlations are examined to reveal the characteristics of the pressure fluctuations. The global dependence of the pressure fluctuations and pressure–strain correlations are also examined by evaluating the integral associated with Green-function representations of them. In the wall region where the pressure–strain terms are large, most contributions to the pressure–strain terms are from the wall region (i.e. local), whereas away from the wall where the pressure–strain terms are small, contributions are global. Structures of instantaneous pressure and pressure gradients at the wall and the corresponding vorticity field are examined.

A nonlinear analysis is presented of waves propagating around the free surface of water contained in a circular basin of finite uniform depth. The property that distinguishes these waves from unidirectional gravity waves is the occurrence of low-order resonant interactions between the components composing them. Steady waves in the neighbourhood of resonance are calculated in the fully nonlinear problem, when it is shown that multiple families of free-wave solutions, each family having a different set of resonating wave components, are associated with each of the water depths at which resonance occurs. Linear stability calculations indicate that most steady waves dominated by resonating wave components are linearly unstable. The nonlinear time evolution of perturbed waves is calculated as a check on the linear stability results, leading to doubts about the relevance of the linear prediction when marginal instability is said to occur.

Nonlinear advective adjustment of a discontinuity in free-surface height under gravity and rotation is considered, using the method of contour dynamics. After linear wave-adjustment has set up an interior jet and boundary currents in a wide ([Gt ] one Rossby radius) channel, fluid surges down-channel on both walls, rather than only that wall supporting a down-channel Kelvin wave. A wedgelike intrusion of low potential vorticity fluid on this wall, and a noselike intrusion of such fluid on the opposite wall, serve to reverse the sign of relative vorticity in the pre-existing currents. For narrower channels, a coherent boundary-trapped structure of low potential vorticity fluid is ejected at one wall, and shoots ahead of its parent fluid. The initial tendency for the current to concentrate on the ‘right-hand’ wall (the one supporting a down-channel Kelvin wave in the northern hemisphere) is defeated as vorticity advection shifts the maximum to the left-hand side. Ultimately fluid washes downstream everywhere across even wide channels, leaving the linearly adjusted upstream condition as the final state. The time necessary for this to occur grows exponentially with channel width. The width of small-amplitude boundary currents in linear theory is equal to Rossby's deformation radius, yet here we find that the width of the variation in velocity and potential vorticity fields deviates from this scale across a large region of space and time. Comparisons of the contour dynamics solutions, valid for small amplitude, and integration of the shallow-water equations at large amplitude, show great similarity. Boundary friction strongly modifies these results, producing fields more closely resembling the linear wave-adjusted state. Observed features include those suggestive of coastally trapped gravity currents. Analytical results for the evolution of vorticity fronts near boundaries are given in support of the numerical experiments.

The main characteristics of the non-axisymmetric oscillations of a liquid bridge have been considered: free frequencies, deformation modes and the influence of an outer liquid. Oscillations of this kind do not show stability changes.

The Plateau technique has been used to obtain the resonant frequencies of the bridge when lateral perturbations are imposed. The results obtained are in good agreement with the theoretical ones when the influence of the outer liquid is considered. Moreover, lateral oscillations observed in experiments performed with liquid bridges in space can be explained with this model.