An aortic-valve model is developed, having a quadratic cross-section, two rigid cusps and two wedge-shaped aortic sinuses. The flow through this valve is assumed to be one-dimensional, just as the flow behind the cusps should be one-dimensional. The resulting model equations are two nonlinear ordinary differential equations of second order for the valve opening area as a function of time in two different ranges.

This model allows the size of the aortic sinus to be varied; it also permits a computation of the pressure at both sides of the cusps (unlike previous models of this kind, which consider the flow behind the cusps as stagnant). The computed valve motion due to this pressure difference is in good agreement with experimental results, although no vortex with circular streamlines is postulated in the aortic sinuses. Obviously such vortices trapped in the sinuses are not important for the valve closure, which is controlled solely by the flow deceleration.

We consider the small-Rossby-number flow of a fluid past an obstacle in a coordinate frame in which the rotation rate varies linearly in the direction normal to the flow in a manner that models the variation of the Coriolis force for midlatitude planetary motions. The eastward flow is characterized by strong upstream divergence of the streamlines like that noted by Davies & Boyer (1982), and a similarly severe streamline convergence in the lee of the obstacle. Such a structure occurs for small values of the β-parameter that measures the importance of the lateral angular-velocity variation. In this parameter range, Rossby waves occur, but are confined to a narrow region in the lee of the object. The presence of these waves modifies the edge velocity ‘seen’ by the Stewartson quarter layer in such a way as to delay the onset of separation beyond what one might expect based on the analysis of Walker & Stewartson (1974) for a flow without beta-effect.

The process of flux rope formation in a convecting cell is studied. The magnetic field has both a meridional and an azimuthal component, and so corresponds to a twisted field. Convection occurs in this cylindrical cell because of heating from below, and is assumed to take an axisymmetric form. Only the Boussinesq problem is studied here, but both the kinematic and the dynamic regimes are considered.

The two cases where the twisted field is due to (a) an imposed flux of vertical current and (b) an imposed flux of vertical vorticity are considered. Strongly twisted ropes can be generated more easily in case (b) than in case (a).

We show that convection can produce ropes twisted in the opposite direction from that of the initial field. We also find that solutions can be oscillatory even when linear theory predicts steady solutions.

We present a method of disentangling the anisotropies produced by the cliff structures in a turbulent velocity field. These cliff structures induce asymmetry in the velocity increments, which leads us to consider the plus and minus velocity structure functions (VSFs). We test the method in the system of turbulent Rayleigh–Bénard (RB) convection. It is found that in the RB system, the cliff structures in the velocity field are generated by thermal plumes. The plus velocity increments exclude cliff structures, while the minus ones include them. Our results show that the scaling exponents of the plus VSFs are in excellent agreement with those predicted for homogeneous and isotropic turbulence (HIT), whereas those of the minus VSFs exhibit significant deviations from HIT expectations in places where thermal plumes abound. These results demonstrate that plus and minus VSFs can be used to quantitatively study the effect of cliff structures in the velocity field and to effectively disentangle the associated anisotropies caused by these structures.

In this work, we study the interference of two scalars diffusing in homogeneous isotropic turbulence. We use the method described by Kaplan & Dinar (1988b) to calculate the cross-correlation coefficient ρ between the concentration fluctuations of two sources. The dependence of ρ on the source separation, shapes and sizes, and its time evolution is calculated for different points in space. Results for the case of two line sources are compared with data from wind tunnel experiments (Warhaft 1984), and seen to be in good agreement. At the centreline ρ is shown to increase as overlap of the two plumes increases. ρ may be either negative or positive depending on the separation between the two sources.

A solution that describes the inviscid reflection of internal waves off a sloping bottom in time is derived under conditions of linearity and uniform stratification. The solution is well behaved even under critical conditions. In the region ky < Nt, where k is the along-slope wavenumber of the incoming wave, y is the slope-normal direction, N is the Brünt–Väisälä frequency and t is time, an approximation can be written in terms of Lommel's function of two variables. The analysis can easily be extended to the case of a beam of finite width. In the non-critical case, the streamfunction relaxes to the classical Phillips steady-state solution in the region y < cgt, where cg is the slope-normal component of the group velocity for waves at the forcing frequency. However, it is found that the region where the along-boundary component of the velocity relaxes to the Phillips solution is also bounded from below, leaving a region very close to the wall where the classical solution misses important elements of the reflectionprocess. This leads to interesting properties near the boundary, especially relatively to the formation of shear-driven unstable conditions.

A novel primitive model is proposed for the hydrodynamic behaviour of an isolated dissolved polymer molecule in a laminar shear flow. The model, in which inertial effects are neglected, allows for rotation and partial stretching of the molcule, but not for bending. Dilute solutions of flexible long-chain polymers have been experimentally observed to exhibit periodic velocity fluctuations distinct from turbulence over a broad frequency range when flowed in high-shear-rate water-table and pipe configurations. In these experiments, the frequency of the fluctuations does not increase with increasing shear rate; rather, it is lowest in the regions of the flow where the shear is the highest. A manifestation of viscous shear thickening has also been observed in these laminar flows. The proposed polymer representation appears capable of accounting for the salient features of these flows with adjustment of a single dimensionless parameter, a ratio of polymer-spring and solvent-viscosity forces.

An accurate representation of the motion-causing buoyancy force, in the vicinity of the maximum-density condition in water at low temperatures, is determined by using a very accurate and quite simple equation of state for both pure and saline water. The resulting general laminar boundary-layer flow equations admit similarity solutions and, in the absence of salinity diffusion, require only one additional new dimensionless variable, called R. For flows of constant heat content in an unstratified ambient fluid, similarity is found only when the far-field temperature is at the temperature of maximum density. The three flow situations considered here are two above a horizontal line heat source and one above a point heat source. The first flow is a freely rising plane plume and the second is a wall plume (flow over a vertical, adiabatic surface with a horizontal line source imbedded in it). The third flow is the freely rising axisymmetric plume. These are the models in laminar theory of many processes which arise in cold water. A primary objective here is to calculate the effect of using a nonlinear density relation for water, which is much more accurate at low temperatures than the conventional linear one used in ‘classical’ analyses. The downstream variations of the temperature and velocity fields are found to be very different from those for flows where the effect of a density extremum is not included.

We present an experimental study demonstrating that rotation has a pronounced effect on currents induced by selective withdrawal of fluid from a density-stratified reservoir. Our observations show that initiating outflow from the reservoir generates Kelvin shear waves. These waves propagate cyclonically around the perimeter of the reservoir, establishing an anticyclonic withdrawal-layer flow. This flow accelerates owing to the production of relative vorticity by compression of ‘planetary’ vorticity. The withdrawal-layer thickness as shown by vertical profiles of the horizontal velocity grows with time. Separation of the side-wall boundary layers in the corners of the tank causes the spun-up flow to eventually break up into a series of counterrotating gyres. We also present a model that describes many features of the spin-up process observed before onset of separation and subsequent gyre formation. The model shows that vertical diffusion of vorticity plays an important role in the spin up process, leading us to conclude that apparent thickening of the withdrawal layer over time, as seen in changes in the velocity profile, is associated with the vertical diffusion of the spun-up vorticity.

The weakly nonlinear, high-Reynolds-number triple-deck theory of Smith (1979) is applied to Blasius flow over a compliant wall. Attention is concentrated on Tollmien–Schlichting (TS) disturbance waves. We consider wall models of the Carpenter–Garrad type, modified to cater for three-dimensional disturbances, and allowing for the effects of nonlinear wall curvature. Supercritical equilibrium-amplitude states are possible for TS waves in a rigid-wall boundary layer, as is well known (see for example Smith 1979; Hall & Smith 1984). It is found that judicious choice of wall parameters can dramatically alter the nonlinear stability properties of TS waves in the boundary layer over a compliant wall: waves that are linearly damped may become nonlinearly unstable. Excellent agreement is obtained with rigid-wall results of Hall & Smith (1984).

The flow past a cylindrical obstacle in an enclosed channel is examined when the entire configuration is rotating rapidly about an axis which is aligned with that of the obstacle. When viewed from a frame of reference which is rotating with the channel, Coriolis forces dominate and act to constrain the motion to be two-dimensional. The channel is considered to have depth varying linearly across its width, producing effects equivalent to the so-called β-plane approximation and permitting waves to travel away from the obstacle, both upstream and downstream. For the eastward flow considered in this paper, this leads to the formation of a lee-wavetrain downstream of the obstacle and, under some conditions, a region of retarded, or ‘blocked’, flow upstream of the obstacle. The flow regime studied is essentially inviscid, although one form of frictional effect on the flow, introduced through the Ekman layers, is included. The properties of this system are examined numerically and compared with the theoretical predictions from other studies, which are applicable in asymptotic limits of the parameters. In particular, the relevance of ‘Long's model’ solutions is considered.

Experiments of the type described by Phillips et al. (1986) were performed using different slopes with the aim of examining the slope dependence of the buoyancy and volume transports, particularly at small slopes. The new observations confirmed the general flow patterns described for experiments conducted at a fixed slope, but a reconsideration of the local balances suggest that the buoyancy flux at small slopes θ is proportional to $(\kappa N)^{\frac{3}{2}}\sin\theta $, where κ is the turbulent diffusivity. and the volume flux associated with the overall convergence flow is linear with depth and proportional to (κ3/2/N½h) sin θ, where h is the thickness of the pycnocline. These differ in their dependence on the slope suggested (but not tested) in Phillips et al., and are generally consistent with measurements over a range of slopes from 7.1° to 23.5°.

Reflections of weak shock waves over wedges are investigated mainly by considering disturbance propagation which leads to a flow non-uniformity immediately behind a Mach stem. The flow non-uniformity is estimated by the local curvature of a smoothly curved Mach stem, and is characterized not only by a pressure increase immediately behind the Mach stem on the wedge but also by a propagation speed. In the case of a smoothly curved Mach stem as is observed in a von Neumann Mach reflection, the pressure increase behind the Mach stem is approximately determined by Whitham's ray-shock theory. The propagation speed of the flow non-uniformity is approximated by Whitham's shock-shock relation. If the shock-shock does not catch up with a point where a curvature of the Mach stem vanishes, a von Neumann Mach reflection appears. The boundary on which the above-mentioned condition breaks results in the transition from a von Neumann Mach reflection to a simple Mach reflection. This idea leads to a transition criterion for a von Neumann Mach reflection, which is algebraically expressed by χ1 = χs where χ1 is the trajectory angle of the point on the Mach stem where the local curvature vanishes and is approximately replaced by χg—θw (χg is the angle of glancing incidence, and θw is the apex angle of the wedge) and χs is the trajectory angle of Whitham's shock-shock, measured from the surface of the wedge. For shock Mach numbers of 1.02 to 2.2 and a wedge angle from 0° to 30°, the domains of a von Neumann Mach reflection, simple Mach reflection and regular reflection are determined by experiment, numerical simulation and theory. The present transition criterion agrees well with experiments and numerical simulations.

A variational formulation for three-dimensional waves in a continuously stratified shear flow is used to derive the equations governing a resonant triad of waves. It is argued that in general, critical layers are necessary for the existence of explosive resonant triads.

This work examines free-surface flows down an inclined substrate. The slope of the free surface and that of the substrate are both assumed small, whereas the Reynolds number $Re$ remains unrestricted. A set of asymptotic equations is derived, which includes the lubrication and shallow-water approximations as limiting cases (as $Re\rightarrow 0$ and $Re\rightarrow \infty$, respectively). The set is used to examine hydraulic jumps (bores) in a two-dimensional flow down an inclined substrate. An existence criterion for steadily propagating bores is obtained for the $({\it\eta},s)$ parameter space, where ${\it\eta}$ is the bore’s downstream-to-upstream depth ratio, and $s$ is a non-dimensional parameter characterising the substrate’s slope. The criterion reflects two different mechanisms restricting bores. If $s$ is sufficiently large, a ‘corner’ develops at the foot of the bore’s front – which, physically, causes overturning. If, in turn, ${\it\eta}$ is sufficiently small (i.e. the bore’s relative amplitude is sufficiently large), the non-existence of bores is caused by a stagnation point emerging in the flow.