It has been suggested that hairpin vortices may play a key role in developing and sustaining the turbulence process in the near-wall region of turbulent boundary layers. To examine this suggestion, a study was done of the hairpin vortices generated by the interaction of a hemisphere protuberancee within a developing laminar boundary layer. Under the proper conditions, hairpin vortices are shed extremely periodically, which allows detailed examination of their behaviour. Shedding characteristics of the hemispheres were determined using hot-film-anemometry techniques. The flow patterns created by the presence of the hairpin vortices have been documented using flow visualization and hot-film-anemometry techniques, and cross-compared with the patterns observed in the near-wall of a fully turbulent boundary layer. In general, it has been observed that many of the visual patterns observed in the near-wall region of a turbulent boundary layer can also be observed in the wake of the hairpin-shedding hemisphere, which appears supportive of the importance of hairpin vortices in the near-wall turbulence production process. Furthermore, velocity measurements indicate the presence of strong inflexional profiles just downstream of the hairpin-vortex generation region which evolve into fuller profiles with downstream distance, eventually developing a remarkable similarity to a turbulent-boundary-layer velocity profile.

It has been suggested that hairpin vortices are a major sustaining flow structure involved in the perpetuation of turbulent boundary layers, although their origin within the boundary layer is unclear. One hypothesis is that hairpin structures are formed by the breakdown of the low-speed streak structures which develop adjacent to the surface beneath turbulent boundary layers. To examine this hypothesis, a water-channel study has been done which utilizes injection through surface slots in a flat plate to create artificial low-speed streak-type regions beneath a laminar boundary layer. Under appropriate conditions, these synthesized low-speed streaks develop a three-dimensional, shear-layer instability which breaks down to form a hairpin-vortex street. Employing both flow visualization and anemometry measurements, the characteristics of these hairpin structures and the parameters influencing their generation have been examined. The hairpin streets were determined to develop in a very periodic and repeatable manner within a definite range of flow parameters. Detailed flow patterns obtained using dye and hydrogen bubbles, both individually and collectively, indicate a remarkable similarity with previously observed patterns in the near-wall region of turbulent boundary layers. In addition, the development of the hairpin structures is observed to be quite sensitive to external forcing, as well as exhibiting a tendency for organized development of larger, more complex structures through a pairing-type process. Velocity measurements indicate the initial presence of strong inflexional profiles which evolve rapidly to velocity and turbulence-intensity profiles commensurate with those associated with turbulent boundary layers, but which do not exhibit the marked spreading associated with turbulence.

This report presents data describing convection in a rotating cylindrical Bénard cell filled with He I. In particular, convection modes are observed at Rayleigh numbers substantially below those predicted by linear stability analyses for a horizontally infinite layer. Both the Rayleigh numbers associated with the convective onset and the initial-slope measure of heat transport of these modes are found to depend on the rotation rate Ω and the aspect ratio Γ of the cell. A discussion of the relevant literature reveals that these convective modes are probably the same as those observed by Rossby (1969) and are reasonably well characterized by the recent analysis of Buell & Catton (1983) assuming asymmetric modes.

The radiation resistance (damping coefficient) and virtual mass for a circular disk that executes small, heaving oscillations at the surface of a semi-infinite body of water, originally calculated by MacCamy (1961a) through the numerical solution of an integral equation, are calculated from a systematic hierarchy of variational approximations. The first member of this hierarchy is based on the exact solution of the boundary-value problem for α = 0 and is in error by less than 2% for 0 [les ] α [les ] 1, where α = aσ2/g (a = radius of disk, σ = angular frequency, g = gravity). The second approximation provides a variational interpolation between the limiting results for α = 0 and α = ∞ and appears to be in error by less than 2% for all α except in certain narrow intervals, where pseudoresonances pose difficulties. Those difficulties are overcome by local reference to the third approximation. Numerical results are plotted for 0 [les ] α [les ] 10. Asymptotic results for α ↑ ∞ are developed in an Appendix.

The corresponding formulation and the first variational approximation are developed for pitching oscillations of the disk.

Two-dimensional free surface flows without waves, produced by a submerged sink in a reservoir, are computed numerically for various configurations. For a sink above the horizontal bottom of a layer of fluid, there are solutions for all values of the Froude number F greater than some particular value. However, when the fluid is sufficiently deep, there is an additional solution for one special value of F < 1. The results for a sink at the vertex of a sloping bottom, treated by Craya and by Hocking, and for a sink in fluid of infinite depth, treated by Tuck & Vanden-Broeck, are confirmed and extended. In particular it is shown that as the bottom tends to the horizontal, the solution for a sink at the vertex of a sloping bottom approaches a solution for a horizontal bottom with F = 1. However solutions are found for all values of the Froude number F [ges ] 1 for a sink on a horizontal bottom.

The coupling between high-amplitude wall-pressure peaks and flow structures, especially in the near-wall region, was studied for a zero-pressure-gradient turbulent-boundary-layer flow and for the flow in the interior of artificially generated turbulent spots. By use of an ‘enhanced’ conditional averaging technique it was shown that buffer region shear-layer structures are to a high degree responsible for the generation of large positive wall-pressure peaks. The relation was proved to be bi-directional in that strong shear layers were shown to accompany positive pressure peaks and correspondingly that large pressure peaks were associated with shear-layer structures detected in the buffer region. This also indicates a link between the wall-pressure peaks and turbulence-producing mechanisms. The pressure-peak amplitude was found to scale linearly with the velocity amplitude of the generating flow structure, indicating that a dominating role here is played by the so-called turbulence – mean shear interaction. The large negative wall-pressure peaks were found to be associated primarily with sweep-type motions. All essential features of the relation between wall-pressure peaks and flow structures in artificially generated spots in a laminar boundary layer were found to be identical to those in the equilibrium turbulent boundary layer.

Using the multiple-scale perturbation method, the diffraction of a nonlinear nearly periodic wavetrain by a vertical circular cylinder is investigated. The envelope of the incident wavetrain is assumed to modulate slowly in the direction of wave propagation. The relationship between the envelopes of incident and scattered waves is derived. It is shown that second-order scattered set-down waves propagate only at the long-wave velocity (gh)½. The formula for low-frequency wave forces acting on the cylinder is presented. The low-frequency wave forces, which are second-order quantities, are caused by set-down waves beneath the wavetrain and the results of the self-interactions of the leading-order first harmonic wave components. Numerical solutions are presented for the case where the wave envelope varies sinusoidally.

The stability properties and stationary statistics of inviscid barotropic flow over topography are examined. Minimum enstrophy states have potential vorticity proportional to the streamfunction and are nonlinearly stable; correspondingly, canonical equilibrium based on energy and enstrophy conservation predicts mean potential vorticity is proportional to the mean streamfunction. It is demonstrated that in the limit of infinite resolution the canonical mean state is statistically sharp, that is, without any eddy energy on any scale, and is identical to the nonlinearly stable minimum enstrophy state. Special attention is given to the interaction between small scales and a dynamically evolving large-scale flow. On the β-plane, these stable flows have a westward large-scale component. Possibilities for a general relation between inviscid statistical equilibrium and nonlinear stability theory are examined.

By means of an optical tracer-particle tracking method, measurements for Lagrangian characteristics of turbulence, including the mean-square value of lateral diffusion from a point source $\overline{Y^2}$, the r.m.s. value of fluctuating velocity v′, the velocity autocorrelation coefficient RL(τ), and the integral scale ΛL, were made in approximately isotropic turbulent flow behind a grid. Comparison of these Lagrangian data with the Eulerian integral scale Λf and the double velocity correlation coefficient f(r) led to the following results: (i) for a moderate turbulent Reynolds-number range of Reλ = 20 − 70, the ratio β(= ΛL/Λf) is within the values of 0.6 – 0.3, becoming lower as Reλ increases, and (ii) the distribution of RL against time lag τ is analogous to that of f against βr/v′. Further, it was confirmed by both theoretical analysis and experiments that the growth of $\overline{Y^2}(t)$ was weakened by the decay of turbulent energy, particularly so for long diffusion times.

G. I. Taylor (1953) gave a simple recipe for the calculation of contaminant dispersion in bounded shear flows at large times after discharge. He decomposed the concentration profile across the flow into a resolved (uniform) part, with an equilibrium (large-time) estimate for the unresolved part. Here an extended recipe is given to include greater resolution and earlier validity. At the two-equation level there is a close similarity to the slow-zone model posed by Chikwendu & Ojiakor (1985). Application is given to Poiseuille pipe flow and to a contraflowing parallel-plate heat exchanger.

A fully developed two-dimensional turbulent wake was deflected by an airfoil-like thin plate placed at small angles in the external flow. The response of the mean-flow and turbulence properties of the wake to the ‘mild’ pressure gradient and the ‘mild’ streamline curvature caused by the deflection is studied. Owing to the small defect velocity, the extra strain rates are large compared with the main shear strain and the Reynolds stresses are strongly influenced by both the pressure gradient and the streamline curvature. The defect velocity relative to an appropriately chosen ‘potential-flow velocity’, and the mean vorticity, however, are not as strongly influenced by the curvature. Changes in the magnitudes of the Reynolds-stress components are much larger than would be caused by the simple rotation of coordinates aligned with the wake path. Most turbulence-model parameters are influenced significantly, while some pure turbulence parameters, such as the Taylor microscale, are relatively uninfluenced. The rapid and lagged responses are apparent and the terms in the transport equation for turbulent kinetic energy indicate that the response of the production terms is almost instantaneous, while the diffusion and dissipation terms are delayed.

The generation of trailing vortices in the wakes of surface-mounted obstacles at moderate Reynolds numbers is examined by channel-flow experiments and numerical simulation. A skew-mounted obstacle generates a single concentrated trailing vortex, together with weak streamwise vorticity of opposite sense extending to considerable distances on either side and zero gross circulation across the whole stream. Cross-stream-symmetrical obstacles (having a streamwise plane of symmetry normal to the plane surface) generate one or more nested vortex pairs, usually of alternate sense, of which one pair is normally dominant. The sense of rotation of the dominant vortex pair depends on both the shape of the obstacle and its depth relative to that of the boundary layer. Obstacles that divide the stream laterally produce dominant vortex pairs with a central downwash, whereas those lifting the flow predominantly over their crests produce dominant vortex pairs with a central upwash. It is argued that the vorticity of the dominant trailing vortices is generated largely as a component of cross-stream vorticity at the boundary, shed as a shear layer from the body, and turned inertially by the flow to form trailing vortices. It should also be emphasized that the dominant trailing vortex or vortex pair is generally embedded in a weak distribution of trailing vorticity of opposite signs, but with net circulation comparable with that of the dominant core.

The effects of a vertical baffle on the resonant frequencies of fluid within a rectangular container are investigated using the linearized theory of water waves. The accuracy of simple approximate solutions is assessed by comparison with an accurate solution based on eigenfunction expansions. It is found that a surface-piercing barrier can change the resonant frequencies significantly while the effect of a bottom-mounted barrier is usually negligible.

The analysis of unsteady free convection has classically been made difficult because of the singularities which occur in the governing boundary-layer equations, and because anomalies often occur which are related to the occurrence of these singularities. In the present paper a semisimilar analysis of unsteady free convection in the vicinity of a vertical flat plate is presented, wherein a number of possible wall-temperature variations with time and position are derived. Unique scalings are formulated for the semisimilar equations that aid in the numerical solutions and in the physical interpretation of the results. These scalings collapse the infinite time and position coordinates into a finite region, and present the semisimilar problem in a format bounded by similarity equations. Solutions are carried out which indicate the occurrence of overshoots in the temperature profiles and heat transfer for a variety of conditions. Also, concepts such as the ‘limit-of-pure-conduction’ and ‘leading-edge penetration distance’ are shown to require special interpretation under variable wall-temperature conditions.

According to the linearized water-wave theory, a localized pressure source travelling at constant speed on the surface of deep water generates the classical Kelvin ship-wave pattern, which follows behind the source and is confined within a sector of half-angle equal to 19.5°. In this paper, an asymptotic theory is developed which takes into account finite-amplitude and unsteady effects near the boundaries of the Kelvin sector, the so-called cusp lines, where the far-field wave disturbance takes the form of a modulated wavepacket. A nonlinear equation governing the spatial and temporal evolution of the wavepacket envelope is derived. It is shown that, for a pressure source turned on impulsively, a nonlinear steady state is reached. All unsteady effects are found in a region of finite extent which moves away from the source. Numerical calculations indicate that the steady-state nonlinear response is very similar to the steady-state linear response.

Experiments were performed to investigate the supersonic flow of a turbulent boundary layer over short regions of concave surface curvature. Upstream of each curved wall, the free-stream Mach number was 2.87, and the incoming boundary layer was typical of a two-dimensional, zero-pressure-gradient, high-Reynolds-number flow. Two different curvatures were used, with radii of curvature equal to 10 and 50 initial boundary-layer thicknesses (Models I and II, respectively). The turning angle was 8° in each case. As the boundary layer passed through the curved region, it experienced a strong adverse pressure gradient, as well as the destabilizing influences of bulk compression and concave curvature. Downstream of the curved walls, the flow relaxed on a short plane wall. The mean and turbulent field for each flow was investigated, using normal and inclined hot wires to measure the turbulent fluctuations. Wherever possible, the results were compared with those from a corresponding 8° ramp. The ramp and Model I exhibited a very similar behaviour: turbulence levels increased significantly, and there was a marked increase in structural parameters such as the stress ratio $-\overline{u^{\prime}v^{\prime}}/\overline{u^{\prime 2}}$ and the length- and timescales of the turbulent motions. Model II behaved quite differently: although the turbulence levels increased, structural parameters were essentially unchanged. The similarities between the ramp and Model I results suggest that the perturbation in both cases is ‘rapid’ in that the perturbation can be described in terms of total strains rather than local strains. In contrast, the flow in Model II is sensitive to the local variations in the strain rate.

The relationship of the axisymmetric flow between large but finite coaxial rotating disks to the von Kármán similarity solution is studied. By means of a combined asymptotic – numerical analysis, the flow between finite disks of arbitrarily large aspect ratio, where the aspect ratio is defined as the ratio of the disk radii to the gap width separating the disks, is examined for two different end conditions: a ‘closed’ end (shrouded disks) and an ‘open’ end (unshrouded or free disks). Complete velocity and pressure fields in the flow domain between the finite rotating disks, subject to both end conditions, are determined for Reynolds number (based on gap width) up to 500 and disk rotation ratios between 0 and – 1. It is shown that the finite-disk and similarity solutions generally coincide over increasingly smaller portions of the flow domain with increasing Reynolds number for both end conditions. In some parameter ranges, the finite-disk solution may not be of similarity form even near the axis of rotation. It is also seen that the type of end condition may determine which of the multiple similarity solutions the finite-disk flow resembles, and that temporally unstable similarity solutions may qualitatively describe steady finite-disk flows over a portion of the flow domain. The asymptotic – numerical method employed has potential application to related rotating-disk problems as well as to a broad class of problems involving flow in regions of large aspect ratio.

When surface waves encounter a step in bottom topography and/or a change in velocity parallel to the step, refraction and partial reflection occur. Comparison of several approximate solutions indicates that no single approximation works well for all cases. The pattern of success among models suggests that the velocity profile at the boundary favours the free wave with smaller vertical scale. For current changes over a flat bottom, a two-term Galerkin expansion (cf. Evans 1975) is employed for comparison with the other more general models. For small currents (|ΔV| < ½c), an ‘action-based approximation’ (cf. Smith 1983) is favoured, although all models perform adequately. With a strong current, one one-term (one-sided) model performs best, another worst among models; the favoured model includes ephemeral modes on the side with larger-scale free waves. For changes in depth only, the one-sided model with ephemeral modes on the deep side was shown by Miles (1967) to perform well. The two-term expansion (cf. Evans 1975) is not easily extended to this case, and none of the other approximations perform adequately. In the unusual case of a step combined with a strong current, such that much shorter waves occur in the deeper region, it is inferred that none of the models are accurate. Reflection from a submerged wall provides a severe test of the models. Without the ephemeral modes, no net reflection occurs. The Miles-like model overpredicts reflection slightly.

The stability of a steacty, laminar, immiscible, dense line plume falling through a fluid whose horizontal extent is much larger than the plume thickness is investigated. It is shown analytically that the primary flow is linearly unstable in the limit of large wavelength and, if surface tension is negligible, is also unstable in the limit of small Reynolds number. A rich variety of modes of perturbation exist, characterized by a range of balances on different lengthscales between diffusion, advection and propagation of vorticity.

An analytical study is made of a flow with swirling motion through a bend in a pipe with circular cross-section. The intensity of the swirl and the axial velocity distribution may change along a curved path according to the conservation law of angular momentum that is the basic principle used in this analysis. The analysis deals with the variation of angular-momentum flux components of the flow along the pipe. Approximate inviscid solutions indicate that, depending on the inlet conditions of the flow and the configuration of the bend, different types of swirling motion appear in the bend. The sign of a constant C appearing in the analysis is a governing parameter determining the flow type. Comparisons are made with the experimental results of other investigators. Finally, examples of swirling motion appearing in a real curved duct are discussed.