In recent years many problems concerned with the dispersion of a passive contaminant along pipes and channels have been investigated, and this paper is concerned with one such problem which arises in diverse applications. This is the study of the longitudinal dispersion of a contaminant whose concentration is prescribed as a harmonic function of time at one cross-section. On the basis of physical arguments and of detailed calculations for two laminar flows it is shown that for high frequencies the concentration pattern is transported downstream at the maximum fluid velocity but that for low frequencies it is transported at the discharge velocity, and that the fluctuations in concentration decay to zero in a much shorter downstream distance for high frequencies than for low frequencies. It is shown further that at high frequencies the concentration is exponentially small except near the places where the fluid velocity attains its maximum, whereas for low frequencies the variation in concentration over the cross-section is small. Some of these conclusions are compared with those made by others, and the agreement is in general satisfactory.

A slightly stratified shear flow is considered when the effects of nonlinearity, viscosity and thermal diffusivity are in balance in the critical layer. Finite amplitude essentially non-diffusive neutral waves exist only if the mean temperature, velocity and vorticity profiles are distorted such that small jumps in these quantities occur across the critical layer.

This paper is concerned with the propagation of small amplitude gravity waves over a flow with non-uniform velocity distribution. For such a flow Burns derived a relation for the velocity of propagation in terms of the velocity distribution of the mean flow. This result is derived here in another way and some of its implications are discussed. It is shown that one of these is hardly acceptable physically. Burns's result holds only when a real value of the propagation velocity is assumed; the mentioned difficulties vanish if complex values are allowed for, implying damping or growth of the waves. Viscous effects which are the cause of damping or growth are important in the wall layer near the bottom and also at the critical depth, which is present when the wave speed is between zero and the fluid velocity at the free surface.

In § 2 the basic equations for the present problem are given. In § 3 exchange of momentum and energy between wave and primary flow is discussed. This is analogous to what happens at the critical height in a wind flow over wind-driven gravity waves. In § 4 the viscous effects at the bottom are included in the analysis and the complex equation for the propagation velocity is derived. Finally in § 5 illustrations of the theory are given for long waves over running flow and for the flow along a ship advancing in a wavy sea. In these examples a negative curvature of the mean velocity profile is shown to have a stabilizing effect.

This paper provides an intuitive interpretation of the long-wave inertialess instability of a two-layer film flow. The underlying mechanism is elucidated by inspecting the longitudinal perturbation velocity associated with the surface and interfacial deflections. The velocity is expressed by the composition of three parts, related to the shear stress at the free surface, the continuity condition at the interface, and the pressure disturbance induced by gravity. The effect of each velocity component on the evolutions of the surface and the interface is examined in detail. Specifically, the growth of the free surface is caused by the continuity-induced first-order velocity disturbance associated with an additional phase shift between the surface and interfacial waves, while the growth of the interface is due to the pressure-driven flow. The proposed mechanism gives an alternatively reliable prediction of the wave velocity and growth rate.

In this paper the experimental study presented in Part 1 is extended to the three-dimensional case. The in-line force coefficients (added mass and damping) of a sphere oscillating horizontally in a uniformly stratified fluid of limited depth and in a smooth pycnocline are evaluated from Fourier-transforms of the experimental records of impulse response functions. The hydrodynamic loads in the three- and two-dimensional cases are shown to be essentially different, notably in the low-frequency limit, where the damping in the three-dimensional case is zero, while in the two-dimensional case it is maximized due to phenomena akin to blocking. The generalization of the experimental results for affinely similar geometries is discussed. It is found that, as the characteristic vertical extent of stratification decreases, the mean power of internal waves radiated by the oscillating sphere reduces and the maximum of the frequency spectrum of wave power shifts toward lower frequency, which is qualitatively similar to the effects observed in the two-dimensional case. Physically, horizontal stratified waveguides act as low-pass filters since internal waves with nearly vertical group-velocity vector cannot be effectively radiated from oscillating bodies.

This paper discusses experiments and a theoretical model for the convective circulation driven by a surface buoyancy flux in a horizontal layer of fluid. The layer is closed at one end and, at the other end, the buoyancy has a fixed value over a given depth. Such circulation occurs in side arms of cooling lakes used for waste-heat disposal from power generation. Some geophysical circulations, such as in the Red Sea, are also of the above type.

The experiments were done in a 35 ft long flume using heat transfer between the heated water and the atmosphere to generate the surface buoyancy flux. The observed circulation was characterized by two distinct layers flowing in opposite directions and separated by a density interface. For upper-layer depths less than about half the total depth at the open end, the downflow was observed to be concentrated near the closed end. Circulation flowrates and vertical temperature profiles were measured.

The theoretical model uses a ‘two-layer’ approach. The mass, momentum, and buoyancy conservation equations are integrated vertically on each side of the interface. Mass and buoyancy transfer across the interface are neglected. The interfacial shear stress is proportional to the square of the difference of the average layer velocities. For small layer densimetric Froude numbers, the free surface is shown to be approximately horizontal and the problem reduces to one ordinary differential equation for the thickness of the upper layer. General solutions of this interface equation are presented for horizontal and sloping bottoms. Different configurations are possible depending on the nature of the singular points which occur in the phase plan.

For the convective circulation, the interface is shown to go through a singular point. This condition leads to a simple analytical solution for the circulation flowrate in the case of constant surface buoyancy flux and horizontal bottom. This solution compares well with the experimental data and with measurements on the Red Sea circulation.

We investigate the formation of small-scale three-dimensional bedforms due to interactions of an erodible bed with a sea wave that obliquely approaches the coast, being partially reflected at the beach. In this case the trajectories of fluid particles at the top of the bottom boundary layer are ellipses in the horizontal plane, the axes of which depend on the angle of wave incidence and the distance from the shore. A weakly nonlinear stability analysis of an initially flat, cohesionless, sandy bottom is performed. We focus on the resonant interaction of three perturbation components. The results show that these elliptical forcing conditions are responsible for the formation of both brick-pattern ripples and tile ripples. In particular tile ripples are associated with a flow at the top of the bottom boundary layer which is near-circular (ellipticity close to one), whereas brick-pattern ripples are related to a unidirectional oscillatory flow (zero ellipticity).

The incompressible, inviscid and axisymmetric dynamics of perturbations on a solid-body rotation flow with a uniform axial velocity in a rotating, finite-length, straight, circular pipe are studied via global analysis techniques and numerical simulations. The investigation establishes the coexistence of both axisymmetric wall-separation and vortex-breakdown zones above a critical swirl level, ${\it\omega}_{1}$. We first describe the bifurcation diagram of steady-state solutions of the flow problem as a function of the swirl ratio ${\it\omega}$. We prove that the base columnar flow is a unique steady-state solution when ${\it\omega}$ is below ${\it\omega}_{1}$. This state is asymptotically stable and a global attractor of the flow dynamics. However, when ${\it\omega}>{\it\omega}_{1}$, we reveal, in addition to the base columnar flow, the coexistence of states that describe swirling flows around either centreline stagnant breakdown zones or wall quasi-stagnant zones, where both the axial and radial velocities vanish. We demonstrate that when ${\it\omega}>{\it\omega}_{1}$, the base columnar flow is a min–max point of an energy functional that governs the problem, while the swirling flows around the quasi-stagnant and stagnant zones are global and local minimizer states and become attractors of the flow dynamics. We also find additional min–max states that are transient attractors of the flow dynamics. Numerical simulations describe the evolution of perturbations on above-critical columnar states to either the breakdown or the wall-separation states. The growth of perturbations in both cases is composed of a linear stage of the evolution, with growth rates accurately predicted by the analysis of Wang & Rusak (Phys. Fluids, vol. 8, 1996a, pp. 1007–1016), followed by a stage of saturation to either one of the separation zone states. The wall-separation states have the same chance of appearing as that of vortex-breakdown states and there is no hysteresis loop between them. This is strikingly different from the dynamics of vortices with medium or narrow vortical core size in a pipe.

This paper describes a series of experiments using particle image velocimetry to investigate the dynamic stall resulting due to a rapid pitching motion of a flat plate. There exist in such unsteady separated flows multiple time-dependent coherent structures, whose interaction and evolution are complex and nonlinear. The experiments presented here are aimed at determining the behaviour of a dynamic stall vortex system in the Reynolds number range $10^{3}<Re<10^{4}$. Evidence is presented for the development of the three-dimensional structure associated with the dynamic stall vortex and its interaction with the no-slip boundary condition at the surface of the pitching plate. The analysis presented suggests that a centrifugal instability exists, and that the form of the three-dimensional structure is consistent with that expected of a centrifugal instability. The structure and scale dependence of the flow are explored using wavelet and Fourier methods, with the dependence of the flow on Reynolds number examined, as well as the influence of spanwise end boundary conditions.

It is known that the breakup by surface tension of a cylindrical interface containing a viscous liquid can be dampeu by axial motion of the underlying liquid and that for an annular film the capillary instability can be completely suppressed (disturbances of all wavelengths decay) by certain axial velocity profiles. Here, using a linear stability analysis, it is shown that complete stabilization can also occur for thermocapillary-driven axial motions. However, the influence of thermocapillary instabilities typically shrinks the window in parameter space where stabilization is found, relative to the isothermal case. The influence of Reynolds, surface tension, Prandtl, and Biot parameters on limits of stabilization is calculated using continuation techniques. It is observed that windows of stabilization first open with topological changes of the neutral curves in parameter space. A long-wave analysis unfolds the nature of the singularities responsible for several of these topological changes. The analysis also leads to the physical mechanism responsible for (longwave) stabilization and in certain cases to necessary conditions for (long-wave) stabilization.

The linear, inviscid reflection of a straight-crested surface wave from a vertical wall is determined on the hypothesis that the contact angle of the meniscus vanishes. The reflection coefficient is a function of the parameter γ ≡ k0l, where k0 is the wavenumber of the incident wave and l is the capillary length, and is approximated by R = exp (−4iγ2) for a gravity–capillary wave for which γ [Lt ] 1. The solution of this reflection problem is used to obtain matched-asymptotic approximations for standing waves in channels and circular cylinders. The meniscus-induced, fractional reduction of the frequency of the dominant mode in a deep circular cylinder is 0.77 γ2 (which exceeds the increase of ½γ2 associated with the capillary energy of the free surface). This decrement is within 2 mHz of the value inferred from the measurements of Cocciaro et al. (1991) after allowing for the reduction in frequency induced by the viscous boundary layers at the walls, but there are residual uncertainties (in this comparison) associated with the wetting process at the moving contact line and possible surface contamination.

Suspensions of sedimenting slender fibres in a viscous fluid are known to be unstable to fluctuations of concentration. In this paper we develop a theory for the role of fibre flexibility in sedimenting suspensions in the asymptotic regime of weakly flexible bodies (large elasto-gravitation number). Unlike the behaviour of straight fibres, individual flexible filaments rotate as they sediment, leading to an anisotropic base state of fibre orientations in an otherwise homogeneous suspension. A mean-field theory is derived to describe the evolution of fibre concentration and orientation fields, and we explore the stability of the base state to perturbations of fibre concentration. We show that fibre flexibility affects suspension stability in two distinct and competing ways: the anisotropy of the base state renders the suspension more unstable to perturbations, while individual particle self-rotation acts to prevent clustering and stabilizes the suspension. In the presence of thermal noise, the dominant effect depends critically upon the relative scales of flexible fibre self-rotation compared to rotational Brownian motion.

Oscillatory Stokes flows, with zero mean, are subjected to subcritical transition to turbulence. The maximal energy growth of perturbations is computed in the subcritical regime through an optimisation method. The results show strong amplifications during half a period. The energy transfer from the base flow involves an Orr mechanism with two-dimensional vorticity waves, and the maximum energy scales exponentially with the Reynolds number. Nonlinear simulations show that low-energy perturbations are sufficient to trigger turbulent flow.

A temporally evolving turbulent plane jet is studied both by direct numerical simulation (DNS) and Lie symmetry analysis. The DNS is based on a high-order scheme to solve the Navier–Stokes equations for an incompressible fluid. Computations were conducted at Reynolds number $\mathit{Re}_{0}=8000$, where $\mathit{Re}_{0}$ is defined based on the initial jet thickness, $\unicode[STIX]{x1D6FF}_{0.5}(0)$, and the initial centreline velocity, $\overline{U}_{1}(0)$. A symmetry approach, known as the Lie group, is used to find symmetry transformations, and, in turn, group invariant solutions, which are also denoted as scaling laws in turbulence. This approach, which has been extensively developed to create analytical solutions of differential equations, is presently applied to the mean momentum and two-point correlation equations in a temporally evolving turbulent plane jet. The symmetry analysis of these equations allows us to derive new invariant (self-similar) solutions for the mean flow and higher moments of the velocities in the jet flow. The current DNS validates the consequence of Lie symmetry analysis and therefore confirms the establishment of novel scaling laws in turbulence. It is shown that the classical scaling law for the mean velocity is a specific form of the current scaling (which has a more general form); however, the scaling for the second and higher moments (such as Reynolds stresses) has a completely different structure compared to the classical scaling. While the failure of the classical scaling for the second moments of the fluctuating velocities has been noted from the jet data for many years, the DNS results nicely match with the present self-similar relations derived from Lie symmetry analysis. Key ingredients for the present results, in particular for the scaling laws of the higher moments, are symmetries, which are of a purely statistical nature. i.e. these symmetries are admitted by the moment equations, however, they are not observed by the original Navier–Stokes equations.

An improved version of the immersed boundary (IB) method for simulating an initially circular or elliptic flexible ring pinned at one point in a uniform flow has been developed. The boundary of the ring consists of a flexible filament with tension and bending stiffness. A penalty method derived from fluid compressibility was used to ensure the conservation of the internal volume of the flexible ring. At , two different flapping modes were identified by varying the tension coefficient for a fixed bending stiffness, or by changing the bending coefficient for a fixed tension coefficient. The optimal tension and bending coefficients that minimize the drag force of the flexible ring were found. Visualization of the vorticity field showed that the two flapping modes correspond to different vortex shedding patterns. We observed the hysteresis property of the flexible ring, which exhibits bistable states over a range of flow velocities depending on the initial inclination angle, i.e. one is a stationary stable state and the other a self-sustained periodically flapping state. The Reynolds number range of the bistability region and the flapping amplitude were determined for various aspect ratios . For , the hysteresis region arises at the highest Reynolds number and the flapping amplitude in the self-sustained flapping state is minimized. A new bistability phenomenon was observed: for certain aspect ratios, two periodically flapping states coexist with different amplitudes in a particular Reynolds number range, instead of the presence of a stationary stable state and a periodically flapping state.

The separate evolution, or differential diffusion, of high-Schmidt-number passive scalars in a turbulent jet is studied experimentally. The two scalars under consideration are disodium fluorescein (Sc ≡ ν/D = 2000) and sulforhodamine 101 (Sc = 5000). The objectives of the research are twofold: to determine (i) the Reynolds-number-dependence, and (ii) the radial distribution of differential diffusion effects in the self-similar region of the jet. Punctual laser-induced fluorescence (LIF) measurements were obtained 50 jet diameters downstream of the nozzle exit for five Reynolds numbers (Re ≡ uod/ν = 900, 2100, 4300, 6700 and 10600, where u0 is the jet exit velocity, d is the jet diameter, and ν is the kinematic viscosity) and for radial positions extending from the centreline to the edges of the jet cross-section (0 ≤ r/d ≤ 7.5). Statistics of the normalized concentration difference, Z, were used to quantify the differential diffusion. The latter were found to decay slowly with increasing Reynolds number, with the root mean square of Z scaling as Zrms ≡ 〈Z2〉1/2 ∝ Re−0.1, (or alternatively 〈Z2〉 ∝ Re−0.2). Regardless of Reynolds number, differential diffusion effects were found to increase away from the centreline. The increase in differential diffusion effects with radial position, along with their increase with decreasing Reynolds number, support the hypothesis of increased differential diffusion at interfaces between the jet and ambient fluids. Power spectral densities of Z were also studied. These spectra decreased with increasing wavenumber – an observation attributed to the decay of the scalar fluctuations in a turbulent jet. Furthermore, these spectra showed that significant differential diffusion effects persist at scales larger than the Kolmogorov scale, even for moderately high Reynolds numbers.