Mass transport in various kind of two-dimensional water waves is studied. The characteristics of the governing equations for the mass transport depend on the ratio of viscous lengthscale to the amplitude of the free-surface displacement. When this ratio is small, the nonlinearity is important and the mass transport flow acquires a boundary-layer character. Numerical schemes are developed to investigate mass transport in a partially reflected wave and above a hump in the seabed. When the mass transport is periodic in the horizontal direction, a spectral scheme based on a Fourier–Chebyshev expansion, is presented for the solution of the equations. For the ease of a hump on the seabed, the flow domain is divided into three regions. Using the spectral scheme, the mass transport in the uniform-depth regions is calculated first. and the results are used to compute the steady flow in the inhomogeneous flow region which encloses the hump on the seabed.

Results of numerical simulations of a forced shear flow in an annular geometry are presented. The particular geometry used in this work reduces the effects of centrifugal and Coriolis forces. However, there are still a large number of system parameters (shear width, shear profile, radius of curvature, initial conditions, etc.) to characterize. This set of variables is limited after the code has been validated with experimental results (Rabaud & Couder 1983; Chomaz et al. 1988) and with the associated linear stability analysis. As part of the linear stability characterization, the pseudo-spectrum for the associated Orr–Sommerfeld operator for plane, circular Couette flow is calculated, and it is found to be insensitive to perturbations.

The numerical simulation code is a highly accurate de-aliased spectral method which utilizes banded operators to increase the computational efficiency. Viscous dissipation terms enter the code directly from the equations of motion. The results from the simulation code at low Reynolds numbers are compared with the results from linear stability analysis and are used to give predictions for the coefficients of the Landau equation describing the saturation behaviour near the critical Reynolds number. Numerical results at higher Reynolds numbers demonstrate the effects of spin-up and spin-down, including the experimentally observed hysteresis. The properties of two- dimensional shears at high Reynolds numbers, at which temporal states are formed, are also addressed.

Non-dispersive and weakly dispersive single-layer flows over axisymmetric obstacles, of non-dimensional height M measured relative to the layer depth, are investigated. The case of transcritical flow, for which the Froude number F of the oncoming flow is close to unity, and that of supercritical flow, for which F > 1, are considered. For transcritical flow, a similarity theory is developed for small obstacle height and, for non-dispersive flow, the problem is shown to be isomorphic to that of the transonic flow of a compressible gas over a thin aerofoil. The non-dimensional drag exerted by the obstacle on the flow takes the form D(Γ) M5/3, where Γ = (F-1)M−2/3 is a transcritical similarity parameter and D is a function which depends on the shape of the ‘equivalent aerofoil’ specific to the obstacle. The theory is verified numerically by comparing results from a shock-capturing shallow-water model with corresponding solutions of the transonic small-disturbance equation, and is found to be generally accurate for M≲0.4 and |Γ| ≲ 1. In weakly dispersive flow the equivalent aerofoil becomes the boundary condition for the Kadomtsev–Petviashvili equation and (multiple) solitary waves replace hydraulic jumps in the resulting flow patterns.

For Γ ≳ 1.5 the transcritical similarity theory is found to be inaccurate and, for small M, flow patterns are well described by a supercritical theory, in which the flow is determined by the linear solution near the obstacle. In this regime the drag is shown to be , where cd is a constant dependent on the obstacle shape. Away from the obstacle, in non-dispersive flow the far-field behaviour is known to be described by the N-wave theory of Whitham and in dispersive flow by the Korteweg–de Vries equation. In the latter case the number of emergent solitary waves in the wake is shown to be a function of , where δ is the ratio of the undisturbed layer depth to the radial scale of the obstacle.

The approximate Tricomi equation relevant to sonic speed of two-dimensional small-distrubance flow is solved by separation of variables, where these are certain stipulated functions of the Cartesian velocity perturbations. The symmetric flow patterns obtained from this solution are shown to correspond to those about half-bodies, whose ordinates vary as xn where 0·4 &\les n,<1, and x is the distance along the plane of symmetry. The surface pressures on such bodies are deduced.

In particular, the body whose ordinates vary as $x^{2|5}$ has a sonic surface velocity, except at the nose, where an edge-force which causes a drag force is shown to exist. On the assumption that a free-stream breakaway (at sonic velocity) occurs at the shoulder of a body, this solution thereby yields the flow about an aerofoil of the same shape having a flat base. This bluff-nosed section has only a little more than half the drag of a wedge on the same chord and base.

The flow induced by the differential rotation of a cylindrical depression of radius a in one of two parallel rigid planes rapidly rotating about their common normal at speed Q is studied. A Taylor column bounded by the usual Stewartson layers arises, but the shear-layer structure is rather different from any previously studied. The Ei-layers (E = v/ωa2) smooth the discontinuity in the geostrophic flow, but the way in which this is accomplished is related to the possible singu-larities of the E1/3-layer solutions. The fact that the 1/4-layer is partially free and partially attached to a vertical boundary accounts for the new joining conditions for the 1/4-layer. The drag on a right circular cylindrical bump in uniform flow is given in addition to some general comments on the applicability of these joining conditions to the motion of an axisymmetric object of quite general shape.

This work explores fractal geometrical properties of scalar turbulent interfaces derived from experimental two-dimensional spatial images of the scalar field in separated shear layers at large Reynolds numbers. The resolution of the data captures the upper three decades of scales enabling examination of multiscale geometrical properties ranging from the largest energy-containing scales to inertial scales. The data show a −5/3 spectral exponent over a wide range of scales corresponding to the inertial range in fully developed turbulent flows. For the fractal aspects, we utilize two methods as it is known that different methods may lead to different fractal aspects. We use the recently developed method for fractal analysis known as the Multiscale-Minima Meshless (M3) method because it does not require the use of grids. We also use the conventional box-counting approach as it has been frequently employed in various past studies. The outer scalar interfaces are identified on the basis of the probability density function (p.d.f.) of the scalar field. For the outer interfaces, the M3 method shows strong scale dependence of the generalized fractal dimension with approximately linear variation of the dimension as a function of logarithmic scale, for interface-fitting reference areas, but there is evidence of a plateau near a dimension D ~ 1.3 for larger reference areas. The conventional box-counting approach shows evidence of a plateau with a constant dimension also of D ~ 1.3, for the same reference areas. In both methods, the observed plateau dimension value agrees with other studies in different flow geometries. Scalar threshold effects are also examined and show that the internal scalar interfaces exhibit qualitatively similar behaviour to the outer interfaces. The overall range of box-counting fractal dimension values exhibited by outer and internal interfaces is D ~ 1.2–1.4. The present findings show that the fractal aspects of scalar interfaces in separated shear layers at large Reynolds number with −5/3 spectral behaviour can depend on the method used for evaluating the dimension and on the reference area. These findings as well as the utilities and distinctions of these two different definitions of the dimension are discussed in the context of multiscale modelling of mixing and the interfacial geometry.

We investigate the inception of cavitation and resulting structures when a sphere collides with a solid surface covered with a layer of non-Newtonian liquid having a kinematic viscosity of up to ${\it\nu}_{0}=20\,000\,000$ cSt. We show the existence of shear-stress-induced cavitation during sphere approach towards the base wall (i.e. the pressurization stage) in ultra-viscous films using a synchronized dual-view high-speed imaging system. For the experimental parameters employed, liquids having viscoelastic properties of $De\geqslant O(1)$ are shown to enable sphere rebound without any prior contact with the solid wall. Cavitation by depressurization (i.e. during rebound) in such non-contact cases is observed to onset after a noticeable delay from when the minimum gap distance is reached. Also, the cavities created originate from remnant bubbles, being the remains of the primary bubble entrapment formed by the lubrication pressure of the air during film entry. Cases where physical contact occurs (contact cases) in 10 000 cSt ${\leqslant}{\it\nu}_{0}\leqslant 1000\,000$ cSt films produce cavities attached to the base wall, which extend into an hourglass shape. In contrast, strikingly different structures occur in the most viscous liquids due to the disproportionality in radial expansion and longitudinal extension along the cavity length. Horizontal shear rates calculated using particle image velocimetry (PIV) measurements show the apparent fluid viscosity to vary substantially as the sphere approaches and rebounds away from the base wall. A theoretical model based on the lubrication assumption is solved for the squeeze flow in the regime identified for shear-induced cavity events, to investigate the criterion for cavity inception in further detail.

The flow near the bow of a steadily turning ship is analysed using a modified slender-body theory. The rate of change of flow quantities in the longitudinal (x) direction is assumed to be greater than that implied by ‘conventional’ slender-body theory. As a consequence some features of high Froude number flow are apparent which cannot be predicted by the ‘conventional’ theory. The modified slender-body theory proposed requires the solution of a two-dimensional Laplace equation (in y and z) but its free-surface condition still involves an x derivative. A Fourier-transform method is used to solve this problem. A simple bow configuration of constant draft is analysed and numerical results for the free-surface elevation are presented.

In the geological sequestration of carbon dioxide (CO2), residual gas trapping plays an important role in immobilizing CO2. In this study, we investigate the propagation of gravity currents with residual gas trapping in a two-layered porous medium. We first formulate a model for a constant-flux release of a relatively less dense fluid (CO2) from a point source into a porous medium bounded above by a horizontal less-permeable seal. After a constant-flux release ceases, a fraction of the released fluid remains within the porous spaces at the trailing edge because of the capillary forces. This capillary retention is formulated in a model of gravity currents of a finite-volume release in the two-layered medium. In the latter model, the plume shape at the end of the constant-flux release is used as an initial profile. Using these models sequentially, the propagation of both cross-sectional and axisymmetric currents is quantitatively examined.

This is a theoretical study of the experimental work in the succeeding paper by Partridge & Lyall (1967). This paper shows that the characteristic shape of typical bubbles in fluidized beds, with a large solids to fluid density ratio, will be reached after a time of order (r0/g)½ from the introduction, naturally or artificially, into the bed of a circular or spherical bubble, where r0 is the initial bubble radius and g the gravitation constant. The initial acceleration is g in two dimensions and 2g in three dimensions. A measure of the growth rate of the distortion from a circular or spherical shape is given as a tentative guide to wake formation and growth rate. It is shown that the wake growth rate in the three-dimensional case is faster than in the two-dimensional case, and, further, that bubbles formed naturally at the bottom of a fluidized bed will distort faster than bubbles starting from rest.

The method used is based on a convective term linearization process on equations of motion for a fluidized system given by Murray (1965a, b) and an extension of a method given by Walters & Davidson (1962, 1963) in the case of the initial motion of a gas bubble formed in an inviscid liquid and starting from rest.

Comparison with experiment is difficult since the bubble conditions discussed in the theory are very difficult to reproduce artificially in the laboratory. The results obtained by Partridge & Lyall (1967) are shown to be not inconsistent with those predicted here.

In the vicinity of a caustic of a dispersive wave system, where the group velocity is stationary and hence dispersive effects are relatively weak, the nonlinear Schrödinger equation (NLS) breaks down, and the propagation of the envelope of a finiteamplitude wavepacket is governed by a modified nonlinear Schrödinger equation (MNLS). On the basis of the MNLS, a search for wave envelopes of permanent form is made near a caustic. It is shown that possible solitary wave envelopes satisfy a nonlinear eigenvalue problem. Numerical evidence is presented of symmetric, double-hump solitary-wave solutions. Also, a variety of periodic envelopes are computed. These findings are discussed in connection with previous analytical and numerical work.

Two parameters are introduced that uniquely characterize the state of a third-order symmetric tensor. We show that the proposed parameters arise from the uniform metric in the matrix space; thus the joint PDF of these parameters can be used to determine the geometrical statistics of any third-order symmetric tensor. We use this joint PDF to describe the states of the subgrid-scale stress, which is of central interest in large-eddy simulation. Direct numerical simulation of forced isotropic turbulence is used in our a priori tests. With the proposed parameterization we can also assess the most probable flow configuration at the scales of motion just above the Kolmogorov scale. We test four different subgrid-scale models in terms of how well they predict the structure, or state, of the subgrid-scale stress. It is found that models based on truncated Taylor series do not produce an adequate distribution of states, even if augmented by a turbulent viscosity term. On the other hand, models based on the scale-similarity assumption predict a distribution of states that is close to actual.

Coriolis terms are introduced into the equations governing the motion of a finite tidal wave. Various types of solution are found, all of which travel without change of shape and some which are periodic with sharp crests and broad troughs. The classical result that such waves cannot be propagated without change of shape is therefore untrue in these circumstances.

The Lagrangian theory developed for fountains in a stationary fluid is extended to predict the path and breadth of a fountain in a one- and two-layer fluid with a moderate crossflow. The predictions compare well with the results of laboratory experiments of fountains in a one-layer fluid. The empirical spreading parameter determined from the one-layer experiments is used in the theory for fountains in a two-layer crossflow. Though qualitatively correct, the theory underpredicts the height and radius of the fountains. Similar to the behaviour of fountains in two-layer stationary ambients, the fountain in a two-layer crossflow is observed to exhibit three regimes of flow: it may penetrate the interface, eventually returning to the level of the source where it spreads as a propagating gravity current; upon descent, it may be trapped at the interface where it spreads as a propagating intrusion; it may do both, partially descending to the source and partially being trapped at the interface. These regimes are classified theoretically and empirically. The theoretical classification compared the buoyancy excess of the descending flow to the density difference between the two layers. The regimes are also classified using empirically determined regime parameters which govern the relative initial momentum of the fountain and the relative density difference of the fountain and the ambient fluid.

This paper presents an analysis of the transport of electric current in a jet of an electrically conducting liquid discharging from a metallic tube into a gas or a vacuum, and subject to an electric field due to a high voltage applied between the tube and a far electrode. The flow, the surface charge and the electric field are computed in the current transfer region of the jet, where conduction current in the liquid becomes surface current due to the convection of electric charge accumulated at its surface. The electric current computed as a function of the flow rate of the liquid injected through the tube increases first as the square root of this flow rate, levels to a nearly constant value when the flow rate is increased and finally sets to a linear increase when the flow rate is further increased. The current increases linearly with the applied voltage at small and moderate values of this variable, and faster than linearly at high voltages. The characteristic length and structure of the current transfer region are determined. Order-of-magnitude estimates for jets which are only weakly stretched by the electric stresses are worked out that qualitatively account for some of the numerical results.

The equations of motion of an incompressible viscous fluid are given in a rotating helical co-ordinate system, which is non-orthogonal. Partial differential equations are derived for the boundary-layer flow on a rotating helical blade. Numerical solutions of these equations show that the radial flow in the boundary layer is strongly dependent upon the stagger and speed of rotation of the blade.

Experimental observations of an azimuthal electrolyte flow driven by Lorentz force in a thin annular fluid layer placed on top of a magnet show that it develops a robust vortical system near the outer cylindrical wall. It appears to be a result of instabilities developing on a background of steady axisymmetric flow. Therefore, the goal of this paper is to establish a scene for a future comprehensive stability analysis of such a flow. We discuss popular depth-averaged and quasi-two-dimensional approximate solutions that take advantage of the thin-layer assumption first, and argue that they cannot lead to the observed flow patterns. Thus, three-dimensional toroidal flows are considered. Their similarities to various other well-studied rotating flow configurations are outlined, but no close match is found. Multiple axisymmetric solutions are detected numerically for the same governing parameters, indicating the possibility of subcritical bifurcations, namely type 1, consisting of a single torus, and type 2, developing a second counter-rotating toroidal flow near the outer cylinder. It is suggested that the transition between these two axisymmetric solutions is likely to be caused by the centrifugal instability, while the shear-type instability of the type 2 solution may be responsible for the observed vortex structures. However, a dedicated stability analysis which is currently underway and will be reported in a separate publication is required to confirm these hypotheses.

In this work we analyse the stability properties of the flow over an isothermal, semi-infinite vertical plate, placed at zero incidence to an otherwise uniform stream at a different temperature. Near the leading edge the boundary layer resembles Blasius flow, but further downstream it approaches that of pure buoyancy-driven flow. A coordinate transformation that describes in a smooth way the evolution between these two limiting similarity states, where the viscous and buoyancy forces are respectively dominant, is used to calculate the basic flow. The stability of this flow has been investigated by making the parallel flow approximation, and using an accurate spectral method on the resulting stability equations. We show how the stability modes discussed by other authors can be followed continuously between the forced and free convection limits; in addition, new instability modes not previously reported in the literature have been found. A spatio–temporal stability analysis of these modes has been carried out to distinguish between absolute and convective instabilities. It seems that absolute instability can only occur when buoyancy forces are opposed to the free stream and when there is a region of reverse flow. Model profiles have been used in this latter case beyond the point of boundary layer separation to estimate the range of reverse flows that support absolute instability. Analysis of the Rayleigh equations for this problem suggests that the absolute instability has an inviscid origin.

Vortex sound is revisited for two-dimensional compact jets. The interactions between the flow and the rigid boundaries of the nozzle are included. Jets with steady upstream conditions and with imposed forcing are considered. A multipole expansion for the acoustic radiation is derived by the method of matched asymptotic expansion. The processes are carried out to higher order to obtain the quadrupole impingement sound in addition to the leading monopole. The emitted sound is found to be related to the vortical flow motion and the scattering of the incoming forcing from the nozzle interior. The flow motion is simulated incompressibly using a point vortex model. The discretization of the continuous vortex shedding is carefully treated to fulfil the Kutta condition and to eliminate spurious sound radiation. Numerical calculations are given for both free and forced jets. For small forcing, the leading monopole is weak and, under some circumstances, the quadrupole sound can be dominant.