Oblique wave groups consist of waves whose straight parallel lines of constant phase are oblique to the straight parallel lines of constant group phase. Numerical solutions for periodic oblique wave groups with envelopes of permanent shape are calculated from the equations for irrotational three-dimensional deep-water motion with nonlinear upper free-surface conditions. Two distinct families of periodic wave groups are found, one in which the waves in each group are in phase with those in all other groups, and the other in which there is a phase difference of π between the waves in consecutive groups. It is shown that some analytical solutions for oblique wave groups calculated from the nonlinear Schrödinger equation are in error because they ignore the resonant forcing of certain harmonics in two dimensions. Particular attention is given to oblique wave groups whose group-to-wave angle is in the neighbourhood of the critical angle tan−1√½, corresponding to waves on the boundary wedge of the Kelvin ship-wave pattern.

A study is made of some numerical calculations of two-dimensional and geostrophic turbulent flows. The primary result is that, under a broad range of circumstances, the flow structure has its vorticity concentrated in a small fraction of the spatial domain, and these concentrations typically have lifetimes long compared with the characteristic time for nonlinear interactions in turbulent flow (i.e. an eddy turnaround time). When such vorticity concentrations occur, they tend to assume an axisymmetric shape and persist under passive advection by the large-scale flow, except for relatively rare encounters with other centres of concentration. These structures can arise from random initial conditions without vorticity concentration, evolving in the midst of what has been traditionally characterized as the ‘cascade’ of isotropic, homogeneous, large-Reynolds-number turbulence: the systematic elongation of isolines of vorticity associated with the transfer of vorticity to smaller scales, eventually to dissipation scales, and the transfer of energy to larger scales. When the vorticity concentrations are a sufficiently dominant component of the total vorticity field, the cascade processes are suppressed. The demonstration of persistent vorticity concentrations on intermediate scales - smaller than the scale of the peak of the energy spectrum and larger than the dissipation scales - does not invalidate many of the traditional characterizations of two-dimensional and geostrophic turbulence, but I believe it shows them to be substantially incomplete with respect to a fundamental phenomenon in such flows.

We present a numerical method that allows us to solve the Navier-Stokes equation with boundary conditions for the viscous flow between two concentrically rotating cylinders as an initial-value problem. We use a pseudospectral code in which all of the time-splitting errors are removed by using a set of Green functions (capacitance matrix) that allows us to satisfy the inviscid boundary conditions exactly. For this geometry we find that a small time-splitting error can produce large errors in the computed velocity field. We test the code by comparing our numerically determined growth rates and wave speeds with linear theory and by comparing our computed torques with experimentally measured values and with the values that appear in other published numerical simulations. We find good agreement in all of our tests of the numerical calculation of wavy vortex flows. A test that is more sensitive than the comparison of torques is the comparison of the numerically computed wave speed with the experimentally observed wave speed. The agreements between the simulated and measured wave speeds are within the experimental uncertainties; the best-measured speeds have fractional uncertainties of less than 0.2%.

We use a numerical method that was described in Part 1 (Marcus 1984a) to solve the time-dependent Navier-Stokes equation and boundary conditions that govern Taylor-Couette flow. We compute several stable axisymmetric Taylor-vortex equilibria and several stable non-axisymmetric wavy-vortex flows that correspond to one travelling wave. For each flow we compute the energy, angular momentum, torque, wave speed, energy dissipation rate, enstrophy, and energy and enstrophy spectra. We also plot several 2-dimensional projections of the velocity field. Using the results of the numerical calculations, we conjecture that the travelling waves are a secondary instability caused by the strong radial motion in the outflow boundaries of the Taylor vortices and are not shear instabilities associated with inflection points of the azimuthal flow. We demonstrate numerically that, at the critical Reynolds number where Taylor-vortex flow becomes unstable to wavy-vortex flow, the speed of the travelling wave is equal to the azimuthal angular velocity of the fluid at the centre of the Taylor vortices. For Reynolds numbers larger than the critical value, the travelling waves have their maximum amplitude at the comoving surface, where the comoving surface is defined to be the surface of fluid that has the same azimuthal velocity as the velocity of the travelling wave. We propose a model that explains the numerically discovered fact that both Taylor-vortex flow and the one-travelling-wave flow have exponential energy spectra such that In [E(k)] ∝ k1, where k is the Fourier harmonic number in the axial direction.

The stability properties of steady two-dimensional solutions describing convection in a horizontal fluid layer heated from below with stress-free boundaries are investigated in the neighbourhood of the critical Rayleigh number. The region of stable convection rolls as a function of the wavenumber α and the Rayleigh number R is bounded towards higher α by the monotonic skewed varicose instability, while towards low wavenumbers stability is limited by the zigzag instability or by the oscillatory skewed varicose instability. Only for a limited range of Prandtl numbers, 0·543 < P < ∞, does a finite domain of stability exist. In particular, convection rolls with the critical wavenumber αc are always unstable.

Observations of the flow of a two-layer fluid resulting from the motion of a towed streamlined two-dimensional obstacle are described in some detail. The experiments were designed to further our understanding of the factors governing the nature and magnitude of upstream disturbances in the general flow of stratified fluid over two-dimensional topography, and predictions for arbitrary two-dimensional flows are made from the results of these experiments. In particular, the relationship between uniformly stratified flow and single-layer flow over topography is suggested. Most of the observed features of interest in these experiments are nonlinear in character. Relatively complete descriptions of the observed flows are presented over a wide range of parameter values, and the phenomena observed include upstream undular and turbulent bores, bores with zero energy loss, ‘rarefactions’ (in which the interface height changes monotonically over a transition region of continuously increasing length), and downstream hydraulic drops and jumps. Their properties are shown to be broadly consistent with predictions from a two-layer hydrostatic model based on continuity and momentum considerations, which employs jump criteria and rarefaction equations where appropriate. Bores occur because of nonlinear steepening when the layer containing the obstacle is thinner than the other, and rarefactions occur when this layer thickness is comparable with or greater than that of the other layer. The speed and amplitude of the upstream bores are governed by nonlinear effects, but their character is determined by a balance between nonlinear steepening, wave dispersion and interfacial friction when the bore is non-turbulent.

Experimental evidence is presented for two types of hysteresis or ‘multiple equilibria’ - situations where two different flow states may exist for the same external steady conditions. In the first of these hysteresis types, the upstream flow may be supercritical or consist of an upstream bore state. It is analogous to the type anticipated for single-layer flow by Baines & Davies (1980) and described numerically by Pratt (1983), but it is only found experimentally for part of the expected parameter range, apparently because of interfacial stress effects. The second hysteresis type is new, and involves the presence or absence of a downstream hydraulic drop and following jump.

Experiments on shock-wave attenuation in a porous tube have shown that attenuation is a function of distance from the origin and the rate of cross-mass flow per unit area. It is a linear function of distance if the cross-mass flow rate per unit area is held constant. In a tube with solid walls the relationship between attenuation, mass extraction rate per unit area and distance is similar to that in a proous tube. The ratio of the local attenuation to the local cross-mass flow, or mass extraction rate, per unit area is the same linear function of distance in both sets of experiments. The mechanism of attenuation is therefore also identical.

This report presents a numerical study of the two-dimensional, steady and inviscid flow of a perfect gas past wedge profiles with detached shock waves for free-stream Mach numbers between 1.05 and 1.44.

Utilizing the hodograph transformation, a boundary-value problem for the mixed flow is formulated in the hodograph plane and subsequently solved by a finite-difference scheme, iterating respectively between the elliptic and hyperbolic regions. The use of boundary-fitted coordinates and a graded lattice failed to achieve satisfactory results, owing to either convergence difficulties or numerical inaccuracies introduced through the coordinate generator. On the other hand, Cartesian coordinates presented no convergence problems and yielded accurate results.

After transforming the solution back to the physical plane, the flow field between the shock wave and the limiting Mach wave is determined. A comparison of the results with experiments and other theories shows a good agreement both for the pressure distribution on the wedge and for the shock stand-off distance.

Also of special interest is a quantitative comparison of the results with the small-disturbance theory (solution of the Tricomi equation) and the limiting case of the free-stream Mach number 1.

Moreover, an example of the flow past round-nosed wedges, with and without an angle of attack, is treated using the inverse method.

Flows occurring between parallel rotating disks have recently been generalized by Parter & Rajagopal (1984) to include solutions that are not axisymmetric. They prove existence, whereas in the present paper we report, for the first time, numerical results for two cases: (i) rotation about a common axis, and (ii) rotation about distinct axes. Calculations were performed for two values of the Ekman number E = v/d2ω at the relative disk rotations of s = 0·8, s = 0 and s = −0·25, where s = ω2/ω1.

The full linear unsteady potential-flow solution for fluid flowing across a bank of cylinders has been obtained. The potential function is expanded into a Fourier series and the boundary condition of impermeability is applied at the moving cylinder surfaces. Mutual contradictions among the various potential-flow solutions available in the literature are exposed, and it is shown that the present solution is consistent with certain basic physical checks, which some of the previous solutions could not meet. The effect of fluid viscosity is incorporated solely as a phase lag between the steady-state lift and drag coefficients on each cylinder and its respective motions. By incorporating the fluid-dynamic forces obtained from this modified potential-flow theory in a stability analysis, the threshold for fluid-elastic instability is predicted. Comparison with experimentally observed thresholds is encouraging, given the high level of idealization of the theory and the accuracy of present-day semi-empirical prediction methods.

In this paper we apply a formalism introduced in a previous paper to write down a self-consistent set of equations for the functions that describe the near-equilibrium time behaviour of random oceanic internal waves. These equations are based on the direct-interaction approximation. The self-consistent equations are solved numerically (using the Garrett-Munk spectrum as input) and the results are compared to parameters obtained in the weak-interaction approximation (WIA). The formalism points out that an extra parameter that is implicitly vanishingly small in the WIA has a significant effect on decay rates when computed self-consistently. We end by mentioning possible future self-consistent calculations that would improve upon our own.

The evolution of a shear flow with an imbedded streamwise vortex is considered. An idealized model for the vortical structure is used; the vortex is assumed to be of infinite extent in the stream direction, and to be a potential vortex (vortex filament) turned on at time zero, and subsequently allowed to diffuse under the action of viscosity. The ambient flow is taken to be, initially, a linear shear profile; the flow then evolves under the joint action of viscosity and convection induced by the vortex. Boundaries are assumed to be infinitely removed from the vortex core. A similarity variable is found which reduces the equation for the induced streamwise velocity perturbation to an ordinary differential equation, which is easily solved numerically. The vortex Reynolds number, circulation/viscosity, is found to be of prime importance. Calculated velocity profiles are presented.

A cylinder containing liquid with a free surface is subjected to a vertical oscillation of amplitude εg/ω2 and frequency 2ω, where ω is within O(εω) of the natural frequency of a particular (primary) mode in the surface-wave spectrum and 0 < ε 1. A Lagrangian formulation, which includes terms of second and fourth order in the primary mode and second order in the secondary modes (which are excited by the primary mode), together with the method of averaging, leads to a Hamiltonian system for the slowly varying amplitudes of the primary mode. The fixed points (which correspond to harmonic motions) and phase-plane trajectories and their perturbations due to linear damping are determined. It is shown that ε > δ, where δ is the damping ratio (actual/critical) of the primary mode, is a necessary condition for subharmonic response of that mode. Explicit results are given for the dominant axisymmetric and antisymmetric modes in a circular cylinder. Internal resonance, in which a pair of modes have frequencies that approximate ω and 2ω, is discussed separately, and the fixed points and their stability for the special case ω2 = 2ω1 are determined. Internal resonance for ω2 = ω1 is discussed in an appendix.

The bottom boundary layer under a progressive water wave is studied using Saffman's turbulence model. Saffman's equations are analysed asymptotically for the case Re [Gt ] 1, where Re is a Reynolds number based on a characteristic magnitude of the orbital velocity and a characteristic orbital displacement. Approximate solutions for the mass-transport velocity at the edge of the boundary layer and for the bottom stress are obtained, and Taylor's formula for the rate of energy dissipation is verified. The theoretical results are found to agree well with observations for sufficiently large Reynolds numbers.

An extended version of the interactive boundary-layer approach which has been used widely in steady-flow calculations is applied here to the linear and nonlinear stability properties of channel flows and boundary layers in the moderate-to-large Reynolds-number regime. This is the regime of most practical concern. First, for linear stability the agreement found between the interactive approach and Orr-Sommerfeld results remains fairly close even at Reynolds numbers as low as about $\frac{1}{10}$ of the critical value for plane Poiseuille flow, or $\frac{1}{5}$ for Blasius flow. Secondly, nonlinear unsteady calculations and comparisons with full solutions obtained by enlarging the same method are also presented. Overall the work suggests that, at the finite Reynolds numbers where real interest lies, the dominant physical processes of instability in channel flow and boundary layers are of boundary-layer form, with interaction, and it suggests also an alternative numerical technique for determining those processes. This alternative technique uses the interactive boundary-layer method as the central means for obtaining full unsteady Navier-Stokes solutions.

Flow visualization has shown that the interaction of line vortices is a combination of tearing, elongation and rotation, the extent of each depending upon the flow conditions. A discrete-vortex model is used to study the interaction of two and three growing line vortices of different strengths and to assess the suitability of the method for such simulation.

Many of the features observed in experimental studies of shear layers are reproduced. The controlled study shows the importance and rapidity of the tearing process under certain conditions.

The stability of the two-dimensional flow induced by the transverse oscillation of a cylinder in a viscous fluid is investigated in both the linear and weakly nonlinear regimes. The major assumption that is made to simplify the problem is that the oscillation frequency is large, in which case an unsteady boundary layer is set up on the cylinder. The basic flow induced by the motion of the cylinder depends on two spatial variables, and is periodic in time. The stability analysis of this flow to axially periodic disturbances therefore leads to a partial differential system dependent on three variables. In the high-frequency limit the linear stability problem can be reduced to a system dependent only on a radial variable and time. Furthermore, the coefficients of the differential operators in this system are periodic in time, so that Floquet theory can be used to reduce this system further to a coupled infinite system of ordinary differential equations together with uncoupled homogeneous boundary conditions. The eigenvalues of this system are found numerically and predict instability entirely consistent with the experiments with circular cylinders performed by Honji (1981). Results are given for cylinders of elliptic cross-section, and it is found that for any given eccentricity the most dangerous configuration is when the cylinder oscillates parallel to its minor axis. Some discussion of nonlinear effects is also given, and for the circular cylinder it is shown that the steady-streaming boundary layer of the basic flow is significantly altered by the instability.

To investigate reflection of a shallow-water soliton at a sloping beach, the edge-layer theory is developed to obtain a ‘reduced’ boundary condition relevant to the simplified shallow-water equation describing the weakly dispersive waves of small but finite amplitude. An edge layer is introduced to take account of the essentially two-dimensional motion that appears in the narrow region adjacent to the beach. By using the matched-asymptotic-expansion method, the edge-layer theory is formulated to cope with the shallow-water theory in the offshore region and the boundary condition at the beach. The ‘reduced’ boundary condition is derived as a result of the matching condition between the two regions. An explicit edge-layer solution is obtained on assuming a plane beach.

Measurements are presented of fluid-dynamic instability of a smooth circular cylinder, free to oscillate laterally against linear springs in the wake from an identical stationary neighbouring body. The observations also encompassed determination of static forces on the downstream cylinder as functions of relative position of the cylinder pair. Most of the experiments were performed under two conditions of free-stream turbulence. Static tests indicated that both the drag coefficient and the Strouhal number of the downstream body are continuous functions of its relative position. The drag forces were found to be negative at small gaps. It was observed that the transverse extent of the force field increases with increasing streamwise gap.

In the dynamic experiments, depending on the cylinders’ separation and structural damping, the cylinder exhibited a vortex-resonance, or a galloping, or a combined vortex-resonance and galloping, or a separated vortex-resonance and galloping. Whilst the characteristics of wake-excited motion were found to be essentially unaffected by a limited change in free-stream turbulence intensity, the galloping amplitudes were observed to be sensitive to the cylinders’ aspect ratio. An increase in the stability parameter caused significant effects on the cylinder response in amplitude domain. Wake observations behind the oscillating body indicated that in vortex lock-in the frequency of vortex-shedding locked to vibration frequency, but during small-amplitude galloping motion the shedding frequency behaved as if the cylinder was stationary.

Experiments were conducted to investigate the response of a rigid two-dimensional elastically mounted smooth circular cylinder, with oscillations restricted to a plane normal to the incident flow, as influenced by the vicinity of an identical fixed body placed inside the wake. The static lift and drag coefficients, as well as the vibration amplitude and frequency of the upstream cylinder as functions of relative position of the pair of cylinders are given. Most measurements were carried out under two conditions of free-stream turbulence. Whilst turbulence decreased the magnitude of drag coefficients, it had no appreciable effect on lift coefficients. The forces on the upstream body were found to be influenced by the proximity to the downstream one in a significant way only when the streamwise spacing is less than two diameters.

In the dynamic tests, two kinds of instability, namely a vortex-resonance and galloping, were observed, with the latter only occurring when the downstream cylinder was well submerged in the near wake of the upstream one. The vortex-shedding frequency was always found to lock to oscillation frequency. Whereas the vibration characteristics remained essentially unaffected with changing turbulence intensity, the galloping amplitudes were observed to be sensitive to cylinders’ aspect ratio. A quasi-steady theory was developed to predict the galloping behaviour.