After a careful consideration of the laws of generation, advection, diffusion and dissipation of turbulent kinetic energy proposed by Prandtl (1945) and Emmons (1954), equations of motion and turbulent kinetic energy for the vortex flow near a solid end wall are established. These equations are then evaluated by a numerical procedure. Care is taken to specify boundary conditions such that satisfactory matching of the solution with the main vortex is assured. The agreement between the predicted mean velocity distribution and the experimental data is remarkably good. In addition, several interesting characteristics are predicted by the theory: (i) the vertical distribution of horizontal velocity is oscillatory in the inner region, whilst it is of the ordinary boundary-layer type in the outer region; (ii) the maximum velocity in the boundary layer can exceed that in the main vortex by a considerable amount and (iii) the minimum pressure of the vortex does not occur in the vortex-core root as has been generally believed.

The influence of slow time variations of the Brun-Väisälä frequency N upon the energy of internal gravity waves is investigated. It is found that, when time variations in N are produced by a mean deformation field (reversible mean state), the wave energy can vary in either direct or inverse proportion, depending on the wavenumber orientation. When N changes owing to a certain type of irreversible process, the wave energy varies with only inverse proportionality.

The nocturnal planetary boundary layer (NPBL) provides an example where N = N(z, t). The full initial/boundary-value problem for an N(z, t) similar to the climatological mean for the NPBL is solved.

Large amplitude short waves confined near the crests of a swell have been observed when a stiff breeze was blowing against the swell. This would seem to imply the existence of both a wavelength-selective generating mechanism and a trapping mechanism, neither of which is to be expected of surface gravity waves of the observed length. It is suggested that there are significant changes in the dynamics of such waves if allowance is made for the dynamic coupling between wind and waves. For a Kelvin-Helmholtz model it is shown that energy transfer rates from the turbulent pressure fluctuations are greatly enhanced for subcritical conditions by the inclusion of the dynamic coupling. The group velocity of subcritical waves is profoundly affected, becoming infinite at the stability boundary. Thus subcritical waves could be trapped on a swell. An examination of the effects of wind shear suggest that Kelvin-Helmholtz type instability could still be present, although for stronger winds, particularly for rather longer waves.

The energy and momentum fed from the mean wind, being trapped at crests of the swell, may contribute significantly to the attenuation of the swell. The profound wave dynamic effects of the coupling between the wind and the swell for short gravity waves may be of significance in other oceanic phenomena, even when the Kelvin-Helmholtz type of instability is not present.

When a ferromagnetic fluid with a horizontal free surface is subjected to a uniform vertical applied magnetic field B0, it is known (Cowley & Rosensweig 1967) that the surface may be unstable when the field strength exceeds a certain critical value Bc. In this paper we consider, by means of an energy minimization principle, the possible forms that the surface may then take. Under the assumption that |μ − 1| [Lt ] 1 (where μ is the magnetic permeability of the fluid), it is shown that when B0 is near to Bc there are three equilibrium configurations for the surface: (i) flat surface, (ii) stationary hexagonal pattern, (iii) stationary square pattern. Configuration (i) is stable for B0 < Bc, (ii) is stable for B0 > Bc and B0−Bc sufficiently small, and (iii) is stable for some higher values of B0. In each configuration the fluid is static, and the surface is in equilibrium under the joint action of gravity, surface tension, and magnetic forces. The amplitude of the surface perturbation in cases (ii) and (iii) is calculated, and hysteresis effects associated with increase and decrease of B0 are discussed.

A mathematical model of rectilinear surface waves incident on a submerged circular sill is developed. The class of waves considered consists of those for which the ratio of wavelength to water depth is large but which do not necessarily belong in the longwave category. A friction damping term is introduced into the equations of motion and the solutions obtained for the regions over the sill and in the ocean are matched by assuming a continuous surface and energy flux at the sill edge. The results show large reductions in the Q-factor of the resonance peaks brought about by friction damping. It is also found that, except at low frequency, a large number of overlapping resonance peaks which are out of phase with one another occupy a relatively narrow frequency band such that these resonance peaks effectively cancel one another. Experiments were performed to determine the friction constant used in the equations of motion and, using this friction constant, the theoretical results of wave resonance are verified.

Unsteady separated flow behind an inclined flat plate is numerically studied through the use of the discrete-vortex approximation, in which the shear layers emanating from the edges of the plate are represented by an array of discrete vortices introduced into the flow field at appropriate time intervals at some fixed points near the edges of the plate. The strengths of the nascent vortices are chosen so as to satisfy the Kutta condition at the edges of the plate. Numerical calculations are performed for a plate at 60° incidence impulsively started from rest in an otherwise stationary incompressible fluid, by systematically changing the distance between the location of the nascent vortices and the edges of the plate. The temporal changes in the drag force, the rate of vorticity transport at both edges of the plate and the velocity of the separated shear layers are given together with the flow patterns behind the plate on the basis of this model. The results of the computation show that the vortex street behind the plate inclines as a whole towards the direction of the time-averaged lift force exerted on the plate. It is also predicted from the calculations that the vortex shedding at one edge of the plate will not occur at the mid-interval of the successive vortex shedding at the other edge. The predicted flow patterns are not inconsistent with a few experimental observations based on the flow-visualization technique.

The flow in a partially filled cylinder rotating at right angles to the earth's gravity is found under the assumptions of rapid rotation and small viscosity. Effects of viscosity, nonlinear interaction and finite container length are included.

When we look at photographs of real transonic flows which are predicted to be shockless, we find a very large number of weak shocks almost perpendicular to the streamlines. These are no more than almost-trapped upstream-propagating nonlinear waves. In this paper we try to obtain a simple approximate equation which gives their complete history and takes into account both their turning effect, owing to a non-zero gradient of the fluid velocity in a direction normal to the streamlines, and also the finite radius of curvature of the wave front. We first give a brief discussion of a few results which can be easily obtained from the solution of the approximate equation and then compute the history of two nonlinear pulses by numerically integrating the equation.

The one-dimensional unsteady flow equations for flow in an elastic tube have been solved by employing the method of characteristics and used to predict the development of the flow and pressure wave forms in the left coronary artery system of the horse. Input data to this model include in vivo measurements of wave speed and aortic-root pressure, both of which were carried out in horse experiments. In addition, estimates of vessel taper and fluid losses due to branching were established from plastic casts of the horse coronary arteries and from in vivo flowmeter data. The calculated results have confirmed earlier experiments in revealing a relatively large systolic component of flow in the major epicardial vessels. However, the calculations indicate that, once within the myocardium, the systolic flow component quickly diminishes and the diastolic flow component becomes increasingly important. The pressure pulse does not peak as observed in the aorta, but rather rounds out with systolic pressures decreasing slowly and diastolic pressures decreasing more rapidly with distance from the coronary ostium. The presence of relatively large amplitude, low frequency waves (of the order of 5–10 Hz), which were observed mainly during diastole in horse experiments, has also been confirmed by the computer calculations. Similar calculations carried out for conditions simulating the dog and human coronary systems indicate that such oscillations become higher in frequency and lower in amplitude with decreasing animal size.

Two-dimensional vortex shedding behind an inclined flat plate in uniform shear flow is numerically investigated by means of an inviscid discrete-vortex approximation. The points of appearance of the vortices are fixed in the plane of the plate at a short distance downstream of the edges of the plate. The strengths of the vortices are determined from the Kutta condition. On the assumption that the steadily periodic flow patterns are independent of initial conditions, the numerical calculations are performed for an inclined flat plate immersed in an incompressible fluid which is set in motion impulsively from rest with the velocity profile of uniform shear flow. The results of analysis show that the Strouhal number of vortex shedding and the time-averaged values of other hydrodynamic characteristics of the flow such as the outer-edge velocity of the separated shear layers, the convective velocity of the shear layers and the drag force exerted on the plate vary closely linearly with the shear parameter of the approaching shear flow. A linear relation between the Strouhal number and the shear parameter is confirmed by an air-tunnel experiment. The effects of the shear parameter on the calculated vortex patterns in the wake are also presented.

It is known that head seas cannot travel without deformation along a cylinder of full constant cross-section, and recent calculations have indicated that the wave amplitude near the cylinder ultimately decreases as the waves travel along the cylinder, i.e. that the waves are refracted away from the axis of the cylinder. It was assumed in these calculations that the cross-section was a half-immersed circle of radius a of the same order as the wavelength 2π/K, but the method can probably be adapted to arbitrary full constant cross-sections. (There is however another calculation which indicates that for a thin ship the wave amplitude ultimately increases.) In the present paper these calculations are extended. The circular section is again studied but it is now supposed that the wavenumber Ka may be small. Uniformly valid expressions for the wave potential are obtained which show that for small Ka the refraction becomes significant only when Kx (the dimensionless distance along the cylinder) is so large that the product (Kx)½ v0(Ka) is also large; here the function v0(Ka) ∼ 2Ka arises in the solution of a certain eigenvalue problem. (The uniformly valid expressions also suggest an interpretation of the thin-ship calculation which resolves the apparent inconsistency.) The same method is applied to the waves generated by a pulsating source on an infinite cylinder, and similar results are obtained.

Measurements of the shear-stress variation along and the velocity profiles above a solid wavy wall bounding a turbulent flow are presented for waves with height-to-length ratios of 2a/λ = 0·0312 and 0·05. These are compared with previous measurements of the wall shear stress reported by Thorsness (1975) and by Morrisroe (1970) for 2a/λ = 0·012. The investigation covered a range of conditions from those for which a linear behaviour is observed to those for which a separated flow is just being initiated.

Pressure measurements indicate a linear response in that the spatial variation is described quite well by a single harmonic with a wavelength equal to that of the surface. However, the variation of τw for waves with 2a/λ = 0·0312 and 0·05 can be more rapid on the leeward side of the wave. The degree of departure from a sinusoidal variation increases with increasing wave height and fluid velocity and, from the results reported in this paper, it is suggested that nonlinear behaviour will become evident when au*/v [ges ] 27.

Many aspects of the flow for all three waves are described by a solution of the linear momentum equations previously presented by Thorsness (1975) and by Thorsness & Hanratty (1977). These include the phase and amplitude of the pressure profile and the first harmonic of the shear-stress profile and the velocity field outside the viscous wall region.

These results suggest that up to separation the flow is approximated quite well by linear theory. Nonlinearities affect the flow only in a region very close to the wave surface and are manifested by the appearance of higher harmonics in the variation of τw.

A series of experiments has been conducted in a low-speed wind tunnel in which measurements were performed in a two-dimensional turbulent shear layer experiencing the mixing of both a passive and a chemically reacting species. The low-temperature air in the jet's primary flow was seeded with dilute concentrations of N2O4 so that the dissociation reaction N2 + N2O4 [harr ] 2NO2 + N2 occurred in a near-equilibrium manner within the mixing layer owing to the turbulent mixing properties and the imposed temperature gradient. Mean and fluctuating values of velocity, temperature and NO2 concentration were measured up to axial distances of 25 in. for jet velocities of 23 and 50ft/s (Rex [les ] 7 × 105) and for three primary temperatures (252, 273 and 305°K). Velocity and temperature measurements were performed with hot-wire probes, whereas a fibre optics light sensor probe was used to measure NO2 concentrations. Local correlations between species and other fluid properties were obtained by positioning a hot-wire sensor within the light gap of the fibre optics probe and simultaneously recording output signals from both probes. A relatively complete set of turbulent statistics was measured for the non-reacting shear layer, including such results as temperature/species correlations, probability densities, filtered and unfiltered moving-frame velocities, skewness and flatness factors, spectra, velocity and temperature integral scales, intermittency factors for velocity, temperature and passive species, and conventional intensities. Some typical results from the investigation are as follows: the turbulent Schmidt and Lewis numbers were 0·5 and 1·0 respectively; the correlation between passive NO2 concentration and temperature was approximately 0·95; dramatic changes consistent with equilibrium chemistry occurred in NO2 concentration profiles with chemical reaction; velocity, temperature and concentration spectra were comparable over a 2½-decade range in wavenumber (k−2); spectra, probability densities, time-trace data and smoke-seeded shear-layer photographs indicate that, for axial locations x = x0 [ges ] 18·5 in. and for speeds u1 [ges ] 23ft/s, undisturbed edge fluid rarely penetrates completely across the mixing region. Although not specifically addressed during the current study, measured results herein suggest that the turbulent motion for the present shear layer is characterized more by random and/or three-dimensionality effects than by large-scale two-dimensional coherent structures, as has been observed recently in other shear-layer investigations.

This note corrects and augments the results of Blumen, Drazin & Billings (1975) on the linear instability of a shear layer between two streams of an inviscid compressible fluid at the same temperature. The main conclusion of these authors, that the shear layer is unstable to two-dimensional disturbances however large the Mach number, is confirmed in the present paper. Also a third mode of instability is discovered and the character of all three modes revealed by a series of accurate calculations.

A new technique for generating a pair of line vortices in the laboratory has been developed. The mean flow of these vortices is highly two-dimensional, although most of the flow field is turbulent. This two-dimensionality permits the study of vortex motions in the absence of the Crow mutual induction instability and other three-dimensional effects. The vortices are generated in a water tank of dimensions 15 × 122 × 244 cm. They propagate vertically and their axes span the 15 cm width of the tank. One wall of the tank is transparent, and the flow is visualized using fluorescein dye. High speed photography is used to study both the transition to turbulence during the vortex formation process and the interaction of the turbulent vortices with a simulated ground plane.

Transition occurs first in an annular region surrounding the core of each vortex, starting with a shear-layer instability on the rolled-up vortex sheet. The turbulent region then grows both radially inwards and radially outwards until the entire recirculation cell is turbulent. A ‘relaminarization’ of the vortex core appears to take place somewhat later.

The interaction of the vortex pair with the ground plane does not follow the predictions of potential-flow theory for line vortices. Although the total circulation is apparently conserved, the vortices remain at a larger distance from the ground than is expected and eventually ‘rebound’ or move away from the ground. Differences between a free-surface boundary condition and a smooth or rough ground plane are discussed. The ground-plane interaction is qualitatively very similar to that of aircraft trailing vortices observed in recent flight tests.

The growth of an unbounded, density-stratified, turbulent shearing layer in the presence of a gravity field is studied using the postulate of marginal instability. It is found, for a similarity mean velocity and density distribution, that after a rapid initial growth rate the growth slows asymptotically to zero as the Richardson number approaches a value of 1/4. Furthermore, the theory predicts a constant dominant turbulent eddy scale in all but the initial stages of growth of the turbulent shearing layer.

Both the general growth characteristics and the constant dominant turbulent eddy scale predicted by the theory are confirmed by experimental data.

The problem of Rayleigh-Bénard convection at low Prandtl number σ is investigated in a circular geometry. Jones, Moore & Weiss (1976) have formulated, but not solved analytically, an asymptotic nonlinear problem in the limit σ → 0 at small velocities. It is shown that the problem they posed can be solved exactly in this geometry. The solutions are extended by means of expansions in the amplitude ε and the reciprocal of the Reynolds number σε−1, both assumed small. The problem is related to one that occurs in nonlinear mean-field dynamo theory (Malkus & Proctor 1975) and it is surmised that similar problems may be expected to appear in a variety of physical situations.

A strict distinction is made between the two fundamental assumptions in the Stuart-Watson theory of nonlinear stability, one of which is that the amplitude of disturbance is sufficiently small, while the other is that the damping or amplification rate for an infinitesimal disturbance is small. This distinction leads to classification of the nonlinear stability theory into two asymptotic theories: the theory based on the first assumption can be applied to subcritical flows with Reynolds numbers away from the neutral curve, even to flows with no neutral curve, such as plane Couette flow or pipe Poiseuille flow, while the theory based on the second assumption is available only for Reynolds numbers and wavenumbers in the neighbourhood of the neutral curve. In the theory based on the first assumption the concept of trajectories in phase space, together with the method of eigenfunction expansion, is introduced in order to display nonlinear behaviour of the disturbance amplitude and to provide the most rational definition of the Landau constant available for classification of the behaviour patterns.

The scattering of short surface waves by a partially immersed cylinder is considered. The cylinder is taken to be circular and to pass through two fixed points on the free surface with its centre on or above the free surface. Of particular interest is the behaviour of the solution as the cross-section of the immersed part of the cylinder approaches a semicircle. The method of matched asymptotic expansions is used.

The fibre distribution in a sheet of paper, referred to as the formation, is largely the result of turbulence, a stochastic process. Continuous time-series models developed from discrete light-transmission profiles are used to characterize formation. The models are used to obtain explicit expressions for the spectral moments of the profiles. From the moments, estimates of two characteristic lengths of the fibre distribution can be obtained and are interpreted as the average and largest flock size. These lengths are used to develop an index for evaluating the formation of four samples of base sheet paper. The results of this characterization agree with other methods, but this technique has the advantage of providing a physical interpretation of the index.