The nonlinear stability of a viscous film flowing steadily down an inclined plane is investigated by the method of multiple scales. It is shown that the super-critically stable, finite amplitude, long, monochromatic wave obtained by Lin (1969, 1970, 1971) is stable to side-band disturbances under modal interaction if the bandwidth is less in magnitude than to the ratio of the amplitude to the film thickness. Near the upper branch of the linear neutral-stability curve where the amplification rate ci is O(ε2), the nonlinear evolution of initially infinitesimal waves of a finite bandwidth is shown to obey the Landau-Stuart equation, Near the lower branch of the neutral curve, the nonlinear evolution is stronger. An equation is derived for describing this strong nonlinear development of relatively long waves. In practice, disturbance of this type clusters in the form of a hump which cannot be constructed only by the first few harmonics.

It is proposed that a spilling breaker can be regarded as a turbulent gravity current riding down the forward slope of a wave, and can be treated using methods which have been successful in other contexts. The whitecap retains its identity because it is lighter than the water below, owing to the trapping of air bubbles, and information from laboratory experiments is used to estimate the density of the air-water mixture in different circumstances. Entrainment of water from below, at a rate E(Ri0,) which is a function of the overall Richardson number Ri0, has two opposing effects. It provides increasing mass and buoyancy fluxes which can produce an accelerating flow and it also gives rise to a drag, because the entrained fluid has upslope momentum (inco-ordinates moving with the wave crest).

A similarity solution is obtained under the assumptions that the flow is steady in time, and that the slope and the density difference remain constant. In this solution, the thickness of the whitecap is proportional to the distances measured from the crest of the wave. The tangential velocity is proportional to s½. Since the velocity in a Stokes 120° angle is also proportional to s½ this implies that such a flow can start from a small disturbance with zero flux, and propagate with constant acceleration. An important consequence of the analysis is that solutions of this kind are possible only when the slope and the density difference between the whitecap and the water below are sufficiently large; otherwise the upward drag dominates, and a self-sustaining flow cannot form. For a slope of 30°, near the crest of the breaking wave, the theory predicts that a density difference greater than 8% is required to sustain a steady motion, at which point the downslope velocity is 12% of the opposing velocity at the wave surface. A 'starting plume’ model of the advancing front of the breaker is also discussed, which suggests that this too will be accelerating uniformly, but will have a velocity somewhat less than that in the flow behind.

A comparison with the laboratory observations of Kjeldsen & Olsen verifies several features of the model, including the order of magnitude of the relative velocities in the whitecaps and the wave beneath. It also reveals the intermittent nature of the flow, which is here explained as due to the intermittent rounding of the wave crest due to damping of the wave by the whitecap on the forward face.

Earlier measurements of the contribution of four distinct classes of motions, i.e. (u < 0, v > 0), (u > 0, v < 0), (u < 0, v < 0) and (u > 0, v > 0), to the Reynolds stress $-\rho \overline{uv}$ in the wall region of a bounded turbulent shear flow have been extended. These classes were obtained by truncating the u and v signals about zero. Various statistical properties of the truncated streamwise and normal velocity components u and v and of their product uv have been determined in an attempt to characterize quantitatively the motions in this flow. Average values and probability density distributions both of the truncated and untruncated signals have been taken.

A train of small-amplitude surface waves is incident normally on an arbitrary arrangement of thin barriers lying in a vertical plane in deep water. Each barrier is allowed to make small rolling or swaying oscillations of the same frequency as that of the incident wave. The boundary-value problem for the consequent fluid motion, assumed two-dimensional, is solved exactly using a technique which enables the amplitudes of the scattered waves far from the barriers to be readily determined. Reference is made to the associated wave radiation problem and to the calculation of forces and moments on the barriers.

The momentum integral method of Klineberg is shown to provide a good description of the major characteristics of two-dimensional laminar viscous-inviscid interactions a t hypersonic speeds. Surface pressure and heat-transfer-rate measurements were made for sharp compression and expansion corners at Mach 12.2 and are compared with the theoretical predictions. The agreement is found to be good for attached, incipient and fully separated flows.

Some theoretical comparisons between methods based on the Klineberg formulation are made which suggest that the full boundary-layer equations are well described using integral methods that incorporate the energy equation. It is further shown that the properties associated with the stability of the governing differential equations are mathematical properties of the analytical model and should not be associated with any physical characteristics of the boundary layer.

A correlation of hypersonic, cold-wall, incipient separation data is also presented.

A point-vortex representation is used to study numerically the evolution of an initially plane vortex sheet. By introducing a tip vortex to represent the tightly rolled portion of the vortex sheet, the chaotic motion which was a feature of some earlier studies is eliminated and the details of the outer portion of the spiral are calculated. The rate of rolling up is calculated and is shown to be governed by the analytically predicted similarity law of Kaden during the initial stages of the rolling up. The calculations are continued until 99% of the vorticity has been rolled up, at which stage the spiral displays a marked ellipticity.

The high-frequency component of the random solution of a model problem is shown to be statistically orthogonal to the Gaussian component. This is shown to be a consequence of the existence of an equilibrium range. It is concluded that random flow fields can be viewed as being approximately Gaussian only in a very special sense and, in particular, that Wiener–Hermite expansions can provide a useful description only of large-scale hydrodynamical phenomena.

Jets of water and of poly(ethylene oxide) solutions discharging in air were photographed using a novel image-motion compensating camera. Spray droplet formation is inhibited by low concentration polymer solutions. The effect of the polymer is to reduce, dampen, or eliminate small-scale surface disturbances in the jet, while not reducing but even amplifying larger scale motions. The initial laminar zone present in the jet efflux with water is eliminated with trace quantities of polymer. When substantial quantities of polymer are present (200 p.p.m.), the jet breakup is accompanied by filament formation linking all the drops together.

In two previous papers (Proudman 1970; Dowden 1972) it has been shown that some of the phenomena of turbulence a t high Reynolds numbers can be modelled by a suitable chosen member of the class of ν-fluids. These are non-Newtonian fluids all of whose properties depend only on a single dimensional constant whose dimensions are those of viscosity. The purpose of this paper is to construct an equation to model homogeneous turbulence in the presence of a spatially constant rate of deformation in the limit of infinite Reynolds number.

The equation employed is that of a doubly degenerate third-order v-fluid (in Proudman's classification) in the limit ν → 0. In such a fluid the stress tensor S is governed by an equation of the form \[ A\dot{S}\ddot{S}+B\dot{S}^2+Cu^{\prime}S\dot{S}+D\dot{u}^{\prime}S^2 + Eu^{\prime 2}S^2 = 0, \] where A, B,…, E are isotropic tensor constants of the fluid, u′ is the total rate of deformation tensor and dots denote time derivatives. A list of properties required of the equation and its solution is proposed, and the most general form of A, B,…, E is given consistent with these requirements. Computed solutions of this equation are compared with the results of experiments on homogeneous turbulence, and are found to agree well with them.

A mixing layer is formed by bringing two streams of water, moving at different velocities, together in a lucite-walled channel. The Reynolds number, based on the velocity difference and the thickness of the shear layer, varies from about 45, where the shear layer originates, to about 850 at a distance of 50 cm. Dye is injected between the two streams just before they are brought together, marking the vorticity-carrying fluid. Unstable waves grow, and fluid is observed to roll up into discrete two-dimensional vortical structures. These turbulent vortices interact by rolling around each other, and a single vortical structure, with approximately twice the spacing of the former vortices, is formed. This pairing process is observed to occur repeatedly, controlling the growth of the mixing layer. A simple model of the mixing layer contains, as the important elements controlling growth, the degree of non-uniformity in the vortex train and the ‘lumpiness’ of the vorticity field.

The equations which govern the flow at high Reynolds number in the vicinity of the trailing edge of a finite flat plate at incidence to a uniform supersonic stream are solved numerically using a finite-difference procedure. The critical order of magnitude of the angle of incidence α* for the occurrence of separation on one side of the plate is α* = O(R−¼) (Brown & Stewartson 1970), where R is a representative Reynolds number for the flow, and results are computed for three such values of α* which characterize the possible behaviour of the flow above the plate. The final set of computations leads to a numerical value for the trailing-edge stall angle α*s, the angle of incidence which just causes the flow to separate at the trailing edge of the plate. Analytic solutions are available in the form of asymptotic expansions near the trailing edge in terms of the scaled variable of order R−⅜. A multi-layer-type of expansion which occurs in the case α* = αs* is presented in detail for comparison with the computed solution.

It is now well known that the turbulence structure of thin shear layers can be strongly affected by the application of extra rates of strain in addition to the shear velocity gradient. Examples of such extra strain rates include lateral divergence or convergence, and streamline curvature in the plane of the mean shear. The changes in Reynolds stress are an order of magnitude larger than would be expected from the explicit extra terms in the Reynolds-stress transport equations, and therefore an order of magnitude larger than predicted by conventional calculation methods. In the present paper, one of a series on ‘complex’ turbulent flows, we show that bulk compression or dilatation (i.e. an extra strain rate div U) also appears to affect turbulent shear layers, typical values of Reynolds stress being increased by compression and decreased by dilatation. The fractional change in Reynolds stress is an order of magnitude larger than the fractional change in volume of a fluid element. The physical mechanism is probably analogous to that responsible for the large effects of divergence or convergence in incompressible flow. Because the phenomenon seems to be of great practical importance we discuss it in the context of engineering calculation methods. An empirical correction formula, analogous to those used to allow for divergence or curvature effects, greatly reduces the large discrepancies found between recent experiments on supersonic boundary layers and calculations by conventional extensions of successful incompressible-flow methods.

The theoretical and experimental evaluation of a laboratory device for creating a controlled shear flow in a continuously stratified liquid is described. The shear is created by the movement of the end barriers of a rectangular channel. If these barriers are impulsively set into uniform shear motion, this motion propagates away in the form of a front of width $O((h/n) (Nt)^{\frac{1}{3}})$ travelling with uniform speed Nh/nπ, where N is the (constant) buoyancy frequency, t is time, n is the vertical modal number and his the channel depth. The shear produced has roughly twice the duration of that attainable using a tilting tube (Thorpe 1968) of similar dimensions. Shear enhancement is possible by introducing a properly designed flow contraction. If N(z) is variable, the shear profile obtained is that appropriate to the lowest internal wave mode of infinite length.

The method of matched asymptotic expansions is used to solve the differential equation describing the shape of a meniscus on the outside of a circular cylinder. Since the perturbation quantity is proportional to the cylinder radius, the solution is valid basically for small oylinders. The predicted meniscus height is compared with numerical data to determine the accuracy of the two-term result; the third term is found but does not improve the estimate.

It is shown that weak harmonic waves can propagate upstream in a supersonic stream along a plane wall over distances that are very large in comparison with the boundary-layer thickness. These waves are characterized by small reduced frequencies, large wavelengths and small phase velocities. Results are presented for the flow along a single plane wall and also for the flow between two plane parallel plates (channel flow).

The progressive-wave solutions are found by asymptotic expansions for small disturbances, large Reynolds numbers of the basic flow, and small reduced frequencies of the unsteady disturbances. It turns out that, as in the corresponding steady theory, four flow regions have to be distinguished: a middle layer which embraces most of the boundary layer; an inner layer near the wall; the outer flow field; and a transition layer between the middle layer and the outer layer. A quasi-steady treatment of the middle, transitional and outer layers is appropriate. Unsteady effects originate in the inner layer.

The relative importance of viscosity and unsteady effects with regard to the waves is characterized by a dimensionless parameter N which is the product of certain powers of the reduced frequency and the Reynolds number. For N → ∞ and a single wall, Lighthill's steady theory of upstream influence in supersonic boundary layers is recovered as a limiting case of the present theory.

Recent field observations in the polar oceans show that the hollow tubes of ice called ice stalactites form around streamers of cold brine rejected by the growing sea ice. In a laboratory study of this process, we inject cold, dense brine a t a constant salinity, temperature and volume flux into an insulated tank of sea water held at its freezing point, then photograph the resultant stalactite growth. Because the inner wall temperature of the stalactite remains on the salinity-determined freezing curve, as the stalactite grows and the temperature deficit of the brine goes into the growth of ice, the inner wall melts to dilute and cool the adjacent brine back to its freezing point. This melting means that both the inner and outer stalactite radii increase with time. The radius of the stalactite tip, which is constant for each experiment, is shown to be controlled by the onset of a convective instability. If the tip becomes too large, overturning occurs and the sea-water intrusion freezes, reducing the radius of the tip so that the flow leaving the tip is marginally stable. Inside the stalactite, since the inner radius increases with time, both theory and experiment show the interior flow to be convectively unstable. The present study also derives a solution from the constant-heat-flux Graetz solution for the growth in both length and side-wall area of the stalactite. The experiments show that away from the stalactite base and the very beginning of the experiment this solution, with convection accounted for by an adjustable coefficient, describes the experimental growth. Finally, analysis of the experiments shows that as much as 50% of the ice represented by the cold brine does not go into the stalactite, rather the ice goes directly into the ocean as loose crystals.

The analytical characteristic method is an effective method for computing non-linear effects in inviscid supersonic flow problems. Although only linear equations have to be solved, the results are essentially nonlinear, in the sense that the functional relations between physical state variables and space co-ordinates are nonlinear in the small perturbation parameter introduced, like the thickness ratio or incidence of a wing. This holds even for the first-order approximation of the method.

In the case of two-dimensional (plane or axisymmetric) flow the independent variables are characteristic co-ordinates, i.e. they are chosen so as to be constant along corresponding characteristic lines. The space co-ordinates are considered as dependent variables. In three dimensions there is no unique definition of a characteristic co-ordinate system, because the manifold of characteristic surfaces or bi-characteristics is larger than is necessary for defining a co-ordinate system. The success of a three-dimensional analytical characteristic method, however, depends on the proper choice of the co-ordinate system.

The present analytical Characteristic method for three-dimensional flow is based on the fact that three-dimensional flow behaves locally like axisymmetric flow if it is considered in the osculating plane. The corresponding ‘distance from the axis’ is a function of space depending on the flow field. No change of pressure occurs normal to the osculating plane and in isentropic flow no change of speed either. Therefore no co-ordinate perturbation is performed in this normal direction. In the osculating plane the analytical characteristic methodis applied locally as in axisymmetric flow. In the large the space co-ordinates are obtained by integration along the main bi-characteristics.

As an example the flow field on the suction side of a flat delta wing with sub-sonic leading edges is computed. As a main result one obtains shock waves in the neighbourhood of the leading edges following the expansion.

In a drop of liquid which hangs below a horizontal support or a t the end of a tube, the forces due to surface tension, pressure and gravity are in equilibrium. Amongst the many possible equilibrium shapes of the drop, only those which are stable occur naturally. The calculus of variations has been used to determine theoretically the stable equilibria, by calculating the energy change when the liquid in equilibrium experiences axially symmetrical perturbations under physically realistic constraints. If the energy change can be made negative, the drop is unstable. With this criterion, stable equilibria have been identified through which the naturally growing drops evolve until they reach a maximum volume, when they become unstable. These results are illustrated by calculations relating to typical experimental conditions.

Magnetohydrodynamic flows generated in a semi-infinite viscous incompressible conducting fluid by the discharge of an electric current J0 from a point on the infinite plane bounding the fluid develop singularities when J0 exceeds a certain critical value. In practical applications sometimes currents much in excess of the critical value of J0 may be passed between electrodes before singularities appear in the velocity field. In this paper we consider the flow field associated with some current distributions and attempt to provide an explanation for the discrepancy between theory and experiment.

A time-dependent current distribution is abruptly switched on, initiating an unsteady disturbance of a two-phase magnetohydrodynamic configuration. This comprises a magnetically permeated, conducting fluid flow contained by a vacuum magnetic field, from within which the source current radiates. An exact solution to the proposed part-time problem is constructed for the magnetic line distortion of the vacuum field.

If the source current is oscillatory, two progressive, non-dissipative waves are normally encountered, these being superposed upon an infinite discrete set of terms obeying separate rules of decay. Propagation occurs longitudinally, parallel to the flow, but with amplitudes dependent on the transverse variable. The waves advance behind two fronts, the faster of which always travels down-stream. Depending on whether the flow speed exceeds or is exceeded by (or equals) √2 × the quadratic mean of both Alfvkn speeds involved, the slower front proceeds, respectively, downstream or upstream (or disappears, in which case, only one wave exists). Along a characteristic (a front path), appropriate contributions depend solely on the transverse co-ordinate, behaving otherwise like Riemann invariants. Contrary to expectation, the net perturbation is continuous across each characteristic. Various steady modes are ultimately attained after an infinite period. The radiation principle is satisfied.

Both travelling waves are vibrationally sustained, vanishing with the source frequency. In such an event, the infinite series result is summable to a closed form. From this, the general solution, corresponding to an arbitrary space–time source distribution, is deduced. Certain characteristic-associated equivalence laws are then established. An asymptotic approximation is made.