T. Brooke Benjamin

The disintegration of wave trains on deep water Part 1. Theory

The phenomenon in question arises when a periodic progressive wave train with fundamental frequency ω is formed on deep water—say by radiation from an oscillating paddle—and there are also present residual wave motions at adjacent side-band frequencies ω(1 ± δ), such as would be generated if the movement of the paddle suffered a slight modulation at low frequency. In consequence of coupling through the non-linear boundary conditions at the free surface, energy is then transferred from the primary motion to the side bands at a rate that, as will be shown herein, can increase exponentially as the interaction proceeds. The result is that the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start. In this paper a theoretical investigation is made into the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes, and Part 2 which will follow is an account of some experimental observations in accord with the present predictions. The main conclusion of the theory is that infinitesimal disturbances of the type considered will undergo unbounded magnification if \[ 0 < \delta \leqslant (\sqrt{2})ka, \] where k and a are the fundamental wave-number and amplitude of the perturbed wave train. The asymptotic rate of growth is a maximum for δ = ka .

Wave formation in laminar flow down an inclined plane

This paper deals theoretically with a problem of hydrodynamic stability characterized by small values of the Reynolds number R . The primary flow whose stability is examined consists of a uniform laminar stream of viscous liquid running down an inclined plane under the action of gravity, being bounded on one side by a free surface influenced by surface tension. The problem thus has a direct bearing on the properties of thin liquid films such as have important uses in chemical engineering. Numerous experiments in the past have shown that in flow down a wall the stream is noticeably agitated by waves except when R is quite small; on a vertical water film, for instance, waves may be observed until R is reduced to some value rather less than 10. The present treatment is accordingly based on methods of approximation suited to fairly low values of R , and thereby avoids the severe mathematical difficulties usual in stability problems at high R . The formulation of the problem resembles that given by Yih (1954); but the method of solution differs from his, and the respective results are in conflict. In particular, there is dis-agreement over the matter of the stability of a strictly vertical stream at very small R . In contrast with the previous conclusions, it is shown here that the flow is always unstable: that is, a class of undamped waves exists for all finite values of R . However, the rates of amplification of unstable waves are shown to become very small when R is made fairly small, and their wavelengths to become very large; this provides a satisfactory explanation for the apparent absence of waves in some experimental observations, and also for the wide scatter among existing estimates of the ‘quasi-critical’ value of R below which waves are undetectable. In view of the controversial nature of these results, emphasis is given to various points of agreement between the present work and the established theory of roll waves; the latter theory gives a clear picture of the physical mechanism of wave formation on gravitational flows, and in its light the results obtained here appear entirely reasonable. The conditions governing neutral stability are worked out to the third order in a parameter which is shown to be small; but a less accurate approximation is then justified as an adequate basis for an easily workable theory providing a ready check with experiment, This theory is used to predict the value of R at which observable waves should first develop on a vertical water film, and also the length and velocity of the waves. These three predictions are compared with the experimental results found by Binnie (1957), and are substantially confirmed.

Theory of the vortex breakdown phenomenon

The phenomenon examined is the abrupt structural change which can occur at some station along the axis of a swirling flow, notably the leading-edge vortex formed above a delta wing at incidence. Contrary to previous attempts at an explanation, the belief demonstrated herein is that vortex breakdown is not a manifestation of instability or of any other effect indicated by study of infinitesimal disturbances alone. It is instead a finite transition between two dynamically conjugate states of axisymmetric flow, analogous to the hydraulic jump in openchannel flow. A set of properties essential to such a transition, corresponding to a set shown to provide a complete explanation for the hydraulic jump, is demonstrated with wide generality for axisymmetric vortex flows; and the interpretation covers both the case of mild transitions, where an undular structure is developed without the need arising for significant energy dissipation, and the case of strong ones where a region of vigorous turbulence is generated. An important part of the theory depends on the calculus of variations; and the comprehensiveness with which certain properties of conjugate flow pairs are demonstrable by this analytical means suggests that present ideas may be useful in various other problems.

Shearing flow over a wavy boundary

A theoretical study is made of shearing flows bounded by a simple-harmonic wavy surface, the main object being to calculate the normal and tangential stresses on the boundary. The type of flow considered is approximately parallel in the absence of the waves, being exemplified by two-dimensional boundary layers over a plane. Account is taken of viscosity; but, as the Reynolds number is assumed to be large, its effects are seen to be confined within narrow ‘friction layers’, one of which adjoins the wave and another surrounds the ‘critical point’ where the velocity of flow equals the wave velocity. The boundary conditions are made as general as possible by including the three cases where respectively the boundary is rigid, flexible yet still solid, or completely mobile as if it were the interface with a second fluid. The theory is developed on the model of stable laminar flow, although it is proposed that the same theory may usefully be applied also to examples of turbulent flow considered as ‘pseudo-laminar’ with velocity profiles corresponding to the mean-velocity distribution. Use is made of curvilinear co-ordinates which follow the contour of the wave-train. This admits a linearized form of the problem whose validity requires only that the wave amplitude be small in comparison with the wavelength, even when large velocity gradients exist close to the boundary. The analysis is made largely without restriction to particular forms of the velocity profile; but eventually consideration is given to the example of a linear profile and the example of a boundary-layer profile approximated by a quarter-period sinusoid. In § 7 some general methods are set out for the treatment of disturbed boundary-layer proses: these apply with greatest precision to thin boundary layers, but are also useful for the initially very steep but on the whole fairly diffuse profiles which occur in most practical instances of turbulent flow over waves. The phase relationships found between the stresses and the wave elevation are discussed for several examples, and their interest in connexion with problems of wave generation by wind is pointed out. It is shown that in most circumstances the stresses are distributed in much the same way as if the leeward slopes of the waves were sheltered. For instance, the pressure distribution often has a substantial component in phase with the wave slope, just as if a wake were formed behind each wave crest—although of course actual separation effects are outside the scope of the present theory. In this aspect, the analysis amplifies the work of Miles (1957).

Internal waves of finite amplitude and permanent form

A theory is derived for the class of long two-dimensional waves, comprising solitary and periodic cnoidal waves, that can propagate with unchanging form in heterogeneous fluids. The treatment is generalized to the extent that the waves are supposed to arise on a horizontal stream of incompressible fluid whose density and velocity are arbitrary functions of height, and the upper surface of the fluid is allowed either to be free or to be fixed in a horizontal plane. Explicit formulae for the wave properties and a general interpretation of the physical conditions for the occurrence of the waves are achieved without need to specify particular physical models; but in a later part of the paper, §4, the results are applied to three examples that have been worked out by other means and so provide checks on the present theory. These general results are also shown to accord nicely with the principle of ‘conjugate-flow pairs’ which was explained by Benjamin (1962 b ) with reference to swirling flows along cylindrical ducts, but which is known to apply equally well to flow systems of the kind in question here. The theory reveals certain physical peculiarities of a type of flow model often used in theoretical studies of internal-wave phenomena, being specified so as to make the equation for the stream-function linear. In an appendix, some observations are also made regarding the ‘Boussinesq approximation’, which too is often used as a simplifying assumption in this field. It is shown, adding to a recent discussion by Long (1965), that finite internal waves may depend crucially on small effects neglected in this approximation.

The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows

This paper discusses the general idea that in systems where a flexible solid is coupled with a flowing fluid three different types of instability are possible. These were originally designated by Brooke Benjamin (1960) as ‘class A’, ‘class B’ and ‘Kelvin-Helmholtz’ instability, and their collective significance has been clarified recently by Landahl (1962). Class A and class B disturbances are essentially oscillations involving conservative energy-exchanges between the fluid and solid, but their stability is determined by the net effect of irreversible processes, which include dissipation and energy-transfer to the solid by non-conservative hydrodynamic forces. Dissipation in the solid tends to stabilize class B distrbances but to destabilize class A ones. Class C instability (i.e. the ‘Kelvin-Helmholtz’ type) occurs when conservative hydrodynamic forces cause a unidirectional transfer of energy to the solid. In § 2 this idea is examined fundamentally by way of the Lagrangian method of generalized co-ordinates, and in § 3 the example of inviscid-fluid flow past a flexible plane boundary is considered. The treatment of this example amplifies the work of Landahl, in particular by including the effect of non-conservative forces of the kind investigated by Miles in his series of papers on water-wave generation by wind.

Some developments in the theory of vortex breakdown

The primary aim of the analysis presented herein is to consolidate the ideas of the ‘conjugate-flow’ theory, which proposes that vortex breakdown is fundamentally a transition from a uniform state of swirling flow to one featuring stationary waves of finite amplitude. The original flow is assumed to be supercritical (i.e. incapable of bearing infinitesimal stationary waves), and the mechanism of the transition is explained on the basis of physical principles that are well established in relation to the analogous supercritical-flow phenomenon of the hydraulic jump or bore. In previous presentations of the theory the existence of appropriately descriptive solutions to the full equations of motion has only been inferred from these general principles, but here the solutions are demonstrated explicitly by means of a perturbation analysis. This has basically much in common with the classical theory of solitary and cnoidal waves, which is known to explain well the essential properties of weak bores. In § 2 the basic equations of the problem are set out and the leading results of the original theoretical treatment are recalled. The new developments are mainly presented in § 3, where an analysis of finite-amplitude waves is completed by two different methods, each serving to illustrate points of interest. The effects of small energy losses and of small flow-force reductions (i.e. wave-resistance effects) are considered, and the analysis leads to a general classification of possible phenomena accompanying such changes of integral properties in either slightly supercritical or slightly subcritical vortex flows. The application to vortex breakdown remains the focus of attention, however, and § 3 includes a careful appraisal of some experimental observations on the phenomenon. In § 4 a summary is given of a variant on the previous methods which is required when the radial boundary of the flow is taken to infinity. The main analysis is developed without restriction to particular flow models, but in § 5 the results are applied to a specific example.

The solitary wave on a stream with an arbitrary distribution of vorticity

The theoretical work reported herein makes a departure from the many previous analyses of the solitary wave which have treated the wave as an example of irrotational fluid motion. The present analysis is of more general scope in that it covers the whole category of examples in which the wave may propagate in either direction on a horizontal stream whose primary velocity distribution U( y) is an arbitrary function (i.e. there is no restriction on the extent of the variations of U(y)). An approximate form of the wave profile is found in general to be a sech2{(z -ct)/b), as it is according to previous theories applicable to the wave on a uniform stream, but the relationships amongst the wave amplitude a, the length scale b, and the two propagation velocities c (positive downstream and negative upstream) depend in complicated fashion on the form of U(y).

The development of three-dimensional disturbances in an unstable film of liquid flowing down an inclined plane

On the basis of results from a previous paper, expressions are found for the phase velocity and amplification rate of a wave travelling obliquely to the direction of flow. This wave comprises the general harmonic component of three-dimensional small disturbances, and accordingly a double Fourier integral is introduced to represent a bounded disturbance whose initial distribution over the free surface may be arbitrarily prescribed. Hence an asymptotic approximation is derived for a disturbance which is initially concentrated around a point on the free surface. Several distinctive properties of a localized unstable disturbance are noted: for instance, it lies mainly within an elliptical region whose area increases linearly with time as it moves downstream and which is modulated by long-crested waves. An experimental observation of a growing disturbance on an unstable film is recorded, and its main features are seen to be in agreement with the theory. In so far as linearized perturbation theory remains applicable, the effects investigated are common to a wide class of parallel and nearly parallel laminar flows. In the final part of the paper the method used to analyse the instability of a film is generalized in order to reveal the connexion between this and other problems; this aim is achieved by demonstrating collective properties of the complete class of flows in question, but particular reference is made to the example of laminary boundary layers and Poiseuille flow between parallel planes.

Note on the interpretation of two-dimensional theories of growing cavities

It is shown in general how a two-dimensional flow can be justified as a physical approximation, notwithstanding the logarithmic singularity in pressure that occurs at infinity when the cavity expands or contracts at a varying rate. The argument presented, which affords a more natural interpretation than alternatives previously suggested, refers to the approximate equivalence-to a determinable degree of accuracy-between the hypothetical plane flow and the inner region of some real three-dimensional flow with small spanwise variations. The main ideas are illustrated by the example of a long ellipsoidal body which changes in volume while also undergoing shape perturbations.