By means of an original optical technique we have studied the spatio-temporal behaviour in a Rayleigh–Bénard convection experiment of small rectangular geometry. The experimental technique allows complete reconstruction of the temperature field integrated along the roll axis. Two main spatiotemporal regimes have been found, corresponding to localized oscillations and travelling waves respectively. Several parameters are proposed for the quantitative characterization of this complex behaviour.

Theoretical and numerical computations of the wave-induced mean flow in rotating shallow water with uniform potential vorticity are presented, with an eye towards applications in small-scale oceanography where potential-vorticity anomalies are often weak compared to the waves. The asymptotic computations are based on small-amplitude expansions and time averaging over the fast wave scale to define the mean flow. Importantly, we do not assume that the mean flow is balanced, i.e. we compute the full mean-flow response at leading order. Particular attention is paid to the concept of modified diagnostic relations, which link the leading-order Lagrangian-mean velocity field to certain wave properties known from the linear solution. Both steady and unsteady wave fields are considered, with specific examples that include propagating wavepackets and monochromatic standing waves. Very good agreement between the theoretical predictions and direct numerical simulations of the nonlinear system is demonstrated. In particular, we extend previous studies by considering the impact of unsteady wave fields on the mean flow, and by considering the total kinetic energy of the mean flow as a function of the rotation rate. Notably, monochromatic standing waves provide an explicit counterexample to the often observed tendency of the mean flow to decrease monotonically with the background rotation rate.

An array of closely spaced, aligned, turbulent, incompressible jets is effectively simulated by a line momentum source. In a shallow inviscid fluid, such a source induces a predominantly two-dimensional non-diffusive flow which is irrotational except along lines of velocity discontinuity downstream of the source. The flow can be generated by a distribution of line vortices of unknown strength along the unknown slipstreamlines. Based upon this vortex model, kinematic and dynamic conditions along the slipstreamlines are formulated, and the two resulting nonlinear singular integral equations are solved numerically using a Newton-Raphson-type iterative collocation method.

The flow field shows a marked resemblance to that induced by a nonlinear actuator disk. For the case of no ambient current, experimental results indicate that the slipstreamlines emanate from points which are close to, but not at the ends of, the source. As the ambient current strength increases, a dividing streamline appears in the induced sink flow upstream of the source, and the points from which the slipstreamlines emanate move closer to the ends of the source. Further increases in the current strength result in the smooth blending of this dividing streamline with the slipstreamline.

Laboratory experiments performed in a shallow water basin confirm all of the features predicted by the theory.

We propose a simplified theory of a viscous layer in near-wall turbulent flow that determines the mean-velocity profile and integral characteristics of velocity fluctuations. The theory is based on the concepts resulting from the experimental data implying a relatively simple almost-ordered structure of fluctuations in close proximity to the wall. On the basis of data on the greatest contribution to transfer processes made by the part of the spectrum associated with the main size of the observed structures, the turbulent fluctuations are simulated by a three-dimensional running wave whose parameters are found from the problem solution. Mathematically the problem reduces to the solution of linearized Navier-Stokes equations. The no-slip condition is satisfied on the wall, whereas on the outer boundary of a viscous layer the conditions of smooth conjunction with the asymptotic shape of velocity and fluctuation-energy profiles resulting from the dimensional analysis are satisfied. The formulation of the problem is completed by the requirement of maximum curvature of the mean-velocity profile on the outer boundary applied from stability considerations.

The solution of the problem does not require any quantitative empirical data, although the conditions of conjunction were formulated according to the well-known concepts obtained experimentally. As a result, the near-wall law for the averaged velocity has been calculated theoretically and is in good agreement with experiment, and the characteristic scales for fluctuations have also been determined. The developed theory is applied to turbulent-flow calculations in Maxwell and Oldroyd media. The elastic properties of fluids are shown to lead to near-wall region reconstruction and its associated drag reduction, as is the case in turbulent flows of dilute polymer solutions. This theory accounts for several features typical of the Toms effect, such as the threshold character of the effect and the decrease in the normal fluctuating velocity. The analysis of the near-wall Oldroyd fluid flow permits us to elucidate several new aspects of the drag-reduction effect. It has been established that the Toms effect does not always result in thickening of the viscous sublayer; on the contrary, the most intense drag reduction takes place without thickening in the viscous sublayer.

A comparison is made between the theoretically predicted and the observed stratification in a container which is traversed by a prescribed flux of fluid. Two different geometries were used illustrating respectively a useful procedure for the control of a stratified laboratory system and a mechanism which is believed to be geophysically significant, e.g. for the control of the stratification in certain estuaries. The behaviour of the fluid system was in all cases characterized by an almost stagnant interior with a boundary layer at the non-horizontal wall of buoyancy-layer type. Agreement between theory and experiment was satisfactory within experimental errors, say 10% of the overall temperature difference.

The paper studies the radiation and diffraction by floating bodies of deep-water bichromatic and bidirectional surface waves subject to the second-order free-surface condition. A theory is developed for the evaluation of the second-order velocity potential and wave forces valid for bodies of arbitrary geometry, which does not involve the evaluation of integrals over the free surface or require an increased accuracy in the solution of the linear problem. Explicit sum- and difference-frequency ‘Green functions’ are derived for the radiation and diffraction problems, obtained from the solution of initial-value problems that ensure they satisfy the proper radiation condition at infinity. The second-order velocity potential is expressed as the sum of a particular and a homogeneous component. The former satisfies the non-homogeneous free-surface condition and is expressed explicitly in terms of the second-order Green functions. The latter is subject to the homogeneous free-surface condition and enforces the body boundary condition by the solution of a linear problem. An analysis is carried out of the singular behaviour of the second-order potential near the intersection of the body boundary with the free surface.

The theory of surface gravity waves scattering at vortex flows in the ocean is developed in this paper. A scattering amplitude is found in the Born approximation as a function of vorticity which appears very convenient for investigation of scattering at simple localized flows. It is shown that the wave scattering cross-section is determined by the vertical component of vorticity. For a random (turbulent) vortex field the scattering cross-section per unit voume is determined by a vorticity correlation function. The damping of the coherent wave component and the angular spectrum widening are calculated for multiple scattering by vortex turbulence of drift flows. The spectrum angular width evolution for waves scattered at self-similar vortices of the logarithmic boundary layer is determined only by its dynamical speed and the wave vector. The latter result may be used for a remote sensing of oceanic turbulent drift flows based on observations of surface waves.

The evolution of a finite-amplitude three-dimensional localized disturbance embedded in external shear flows is addressed. Using the fluid impulse integral as a characteristic of such a disturbance, the Euler vorticity equation is integrated analytically, and a system of linear equations describing the temporal evolution of the three components of the fluid impulse is obtained. Analysis of this system of equations shows that inviscid plane parallel flows as well as high Reynolds number two-dimensional boundary layers are always unstable to small localized disturbances, a typical dimension of which is much smaller than a dimensional length scale corresponding to an O(1) change of the external velocity. Since the integral character of the fluid impulse is insensitive to the details of the flow, universal properties are obtained. The analysis predicts that the growing vortex disturbance will be inclined at 45° to the external flow direction, in a plane normal to the transverse axis. This prediction agrees with previous experimental observations concerning the growth of hairpin vortices in laminar and turbulent boundary layers. In order to demonstrate the potential of this approach, it is applied to Taylor-Couette flow, which has additional dynamical effects owing to rotation. Accordingly, a new instability criterion associated with three-dimensional localized disturbances is found. The validity of this criterion is supported by our experimental results.

The upper bound on momentum transport in the turbulent regime of plane Couette flow is considered. Busse (1970) obtained a bound from a variational formulation based on total energy conservation and the mean momentum equation. Two-dimensional asymptotic solutions of the resulting Euler-Lagrange equations for the system were obtained in the large-Reynolds-number limit. Here we make a toroidal poloidal decomposition of the flow and impose an additional power integral constraint, which cannot be satisfied by two-dimensional flows. Nevertheless, we show that the additional constraint can be met by only small modifications to Busse's solution, which leaves his momentum transport bound unaltered at lowest order. On the one hand, the result suggests that the addition of further integral constraints will not significantly improve bound estimates. On the other, our optimal solution, which possesses a weak spanwise roll in the outermost of Busse's nested boundary layers, appears to explain the three-dimensional structures observed in experiments. Only in the outermost boundary layer and in the main stream is the solution three-dimensional. Motion in the thinner layers remains two-dimensional characterized by streamwise rolls.

The scaling behaviour of the longitudinal velocity structure functions $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$ (where $2p$ represents the order) is studied for various wall-bounded turbulent flows. It has been known that for very large Reynolds numbers within the logarithmic region, the structure functions can be described by $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}/U_{\unicode[STIX]{x1D70F}}^{2}\approx D_{p}\ln (r/z)+E_{p}$ (where $r$ is the longitudinal distance, $z$ the distance from the wall, $U_{\unicode[STIX]{x1D70F}}$ the friction velocity and $D_{p}$, $E_{p}$ are constants) in accordance with Townsend’s attached eddy hypothesis. Here we show that the ratios $D_{p}/D_{1}$ extracted from plots between structure functions – in the spirit of the extended self-similarity hypothesis – have further reaching universality for the energy containing range of scales. Specifically, we confirm that this description is universal across wall-bounded flows with different flow geometries, and also for both the longitudinal and transversal structure functions, where previously the scaling has been either difficult to discern or differences have been reported when examining the direct representation of $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$. In addition, we present evidence of this universality at much lower Reynolds numbers, which opens up avenues to examine structure functions that are not readily available from high Reynolds number databases.

Hydrodynamic measurements were made with a triaxial hot wire in the full-coverage region and the recovery region following an array of injection holes inclined downstream, at 30° to the surface. The data were taken under isothermal conditions at ambient temperature and pressure for two blowing ratios: M = 0·9 and M = 0·4. (The ratio M = ρjetUjet/ρ∞U∞, where U is the mean velocity and ρ is the density. Subscripts jet and ∞ stand for injectant and free stream, respectively.) Profiles of the three mean-velocity components and the six Reynolds stresses were obtained at several spanwise positions at each of five locations down the test plate.

In the full-coverage region, high levels of turbulence kinetic energy (TKE) were found for low blowing and low TKE levels for high blowing. This observation is especially significant when coupled with the fact that the heat transfer coefficient is high for high blowing, and low for low blowing. This apparent paradox can be resolved by the hypothesis that entrainment of the mainstream fluid must be more important than turbulent mixing in determining the heat transfer behaviour at high blowing ratios (close to unity).

In the recovery region, the flow can be described in terms of a two-layer model: an outer boundary layer and a two-dimensional inner boundary layer. The inner layer governs the heat transfer.

Turbulent plumes, which are seen in a wide number of industrial and natural flows, have been extensively studied; however, very little attention has been paid to plumes which have an internal mechanism for changing buoyancy. Such plumes arise in e.g. industrial chimneys, where species can react and change the density of the plume material. These plumes with chemical reaction are the focus of this study. An integral model describing the behaviour of a plume undergoing a second-order chemical reaction between a component in the plume (A) and a component in the surrounding fluid (B), which alters the buoyancy flux, is considered. The behaviour of a reactive plume is shown to depend on four dimensionless groups: the volume and momentum fluxes at the source, the parameter ϵ which indicates the additional buoyancy flux generated by the reaction and γ which is a dimensionless rate of depletion of species B. Additionally, approximate analytical solutions are sought for a reactive plume rising from a point source of buoyancy when species B is in great excess. These analytical results show excellent agreement with numerical simulations. It is also shown that the behaviour of a reactive plume in the far field is equivalent to an inert plume issuing from a virtual source downstream of the real source, and the dependence of the location of the virtual source on ϵ and γ is discussed. The effects of varying the volume flux at the source and the Morton source parameter Γ0 are further investigated by solving the full governing equations numerically. These solutions indicate that ϵ is important in determining the buoyancy generated by the reaction, and the length scale over which this reaction occurs depends on γ when γ > 1. It is also shown that when the dimensionless buoyancy ϵ < − 1, the reaction can cause the plume to collapse.

We study the flow of a thin liquid film along a flexible substrate. The flow is modelled using lubrication theory, assuming that gravity is the dominant driving force. The substrate is modelled as an elastic beam that deforms in two dimensions. Steady solutions are found using numerical and perturbation methods, and several different asymptotic regimes are identified. We obtain a complete characterization of how the length and stiffness of the beam and the imposed liquid flux determine the profile of the liquid film and the resulting beam deformation.

Theory and experiment are presented for mass transfer into a fully developed turbulent flow in a plain circular tube in two non-axisymmetric cases. The cases studied are a diametral line source and a discontinuous ring source, in which there is a uniform mass flux over rectangular areas of the tube wall. A comparison is made between the concentration profiles predicted by the solutions of the diffusion equation and experiments using nitrous oxide, Schmidt number S = 0·77, as a tracer gas in air. The range of experiments covers Reynolds numbers R from 20,000 to 120,000.

In the analysis, the assumption is made that the tangential and radial eddy diffusivities of mass are equal at a point. The radial diffusivity of mass, which is a function of radial position, is related to the radial eddy diffusivity of momentum by a ratio, which takes account of fluid properties and the value of the radial eddy diffusivity of momentum. The satisfactory agreement between analysis and experiment establishes the correctness of this assumption. Further confirmation was obtained by direct evaluation of the tangential eddy diffusivity of mass from the measured concentration profiles.

We obtain the fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He. As the relevant friction parameters are small, we linearize terms involving such parameters, while keeping the remaining nonlinearities, which accurately describe the curvature of the vortex filament, intact. The resulting equation is a type of nonlinear Schrödinger equation, and, under an appropriate change of variables, this equation is shown to have a first integral. This is in direct analogy with the simpler equation studied previously in the literature; indeed, in the limit where the superfluid parameters are taken to zero, we recover the results of Van Gorder. While this first integral is mathematically interesting, it is not particularly useful for computing solutions to the nonlinear partial differential equation which governs the vortex filament. As such, we introduce a new change of dependent variable, which results in a nonlinear four-dimensional system that can be numerically integrated. Integrating this system, we recover solutions to the fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He. We find that the qualitative features of the solutions depend not only on the superfluid friction parameters, but also strongly on the initial conditions taken, the curvature and the normal fluid velocity.

A closed flow circuit, comprising a smooth horizontal plastic pipe transporting a dilute suspension of plastic particles in water, was modified to produce a satisfactory environment for the detection of mean particle location across a fixed plane normal to the mean turbulent flow direction. It was established that mean concentration in the vicinity of the vertical diameter was a function of the vertical co-ordinate only, and this result enabled the particle diffusivity to be evaluated for a core region of the cross-section. For a particle relative density in the range 1.035 to 1.126 and a variety of flows, a suitable expression for the dimensionless particle diffusivity is A (1 - BW/u) SN, where u is the friction velocity, W the fall velocity for a particle in still water, S the relative density of the suspended material, and A, B and N are constants.

An analytical and experimental investigation is made of the compression wave generated when a train enters a tunnel fitted with a long, uniform hood with a rectangular window. The window is situated at the junction of the hood and tunnel, which are taken to have the same uniform cross-sectional area. An understanding of the mechanics of this canonical configuration is important for the design of tunnel entrance hoods for new high-speed trains, with speeds in excess of 300 km h$^{-1}$. The compression wave is formed in two stages: as the train nose enters the hood and as it passes the window. The elevated pressure within the hood produces a flow of air from the window in the form of a high-speed jet, whose inertia generates an additional rise in pressure that propagates into the tunnel as a localized pulse. Multiple reflections from the window and the hood portal cause the temporary trapping of wave energy within the hood (prior to its radiation into the tunnel). All of these aspects of the flow are modelled analytically and the results are found to be in good accord with new model-scale measurements and flow visualization studies reported in this paper.