It is demonstrated experimentally and theoretically that through the use of a nonlinear feedback controller, one can render a subcritical Hopf bifurcation supercritical and thus dramatically modify the nature of the flow in a thermal convection loop heated from below and cooled from above. In particular, we show that the controller can replace the naturally occurring chaotic motion with a stable, periodic limit cycle. The control strategy consists of sensing the deviation of fluid temperatures from desired values at a number of locations inside the loop and then altering the wall heating to counteract such deviations.

In this paper, we identify a new type of resonant triad which operates in a parallel or nearly parallel shear flow with a symmetric profile. The triad consists of a planar sinuous mode, an oblique sinuous mode and an oblique varicose mode, but is not of the usual subharmonic-resonance form. The development of the triad is studied in the non-equilibrium critical-layer régime. The equations governing the evolution of the modes are derived. We show that the quadratic resonance can significantly enhance the growth of both the oblique sinuous and varicose modes, and may cause them to grow super-exponentially. This can lead to a subsequent stage in which the oblique sinuous mode produces a back reaction on the oblique varicose mode through a phase-locked interaction, causing both oblique modes to evolve even more rapidly. We suggest that the resonant triad is a viable mechanism for the development of three-dimensional structures and varicose components observed in the later stage of plane wake transition.

Stably stratified steady flow past a bluff body in a channel is considered for cases in which the stratification is not sufficiently strong to give solutions containing wave motions. The physical mechanisms by which stratification influences the flow are revealed. In particular, the drag reduction under weak stratification, observed in experiments, is explained. This is achieved by constructing an asymptotic laminar solution for high Reynolds number (Re) and large channel width, which explicitly gives the mechanisms, and using comparisons with numerical results for medium Re and experiments for turbulent flows to argue that these mechanisms are expected to be common in all cases. The results demonstrate the possibility, subject to certain restrictions, of using steady high-Re theory as a tool for studying qualitative features of real flows.

The formation and the evolution of axisymmetric vortex rings in a uniformly rotating fluid, with the rotation axis orthogonal to the ring vorticity, have been investigated by numerical and laboratory experiments. The flow dynamics turned out to be strongly affected by the presence of the rotation. In particular, as the background rotation increases, the translation velocity of the ring decreases, a structure with opposite circulation forms ahead of the ring and an intense axial vortex is generated on the axis of symmetry in the tail of the ring. The occurrence of these structures has been explained by the presence of a self-induced swirl flow and by inspection of the extra terms in the Navier–Stokes equations due to rotation. Although in the present case the swirl was generated by the vortex ring itself, these results are in agreement with those of Virk et al. (1994) for polarized vortex rings, in which the swirl flow was initially assigned as a ‘degree of polarization’.

If the rotation rate is further increased beyond a certain value, the flow starts to be dominated by Coriolis forces. In this flow regime, the impulse imparted to the fluid no longer generates a vortex ring, but rather it excites inertial waves allowing the flow to radiate energy. Evidence of this phenomenon is shown.

Finally, some three-dimensional numerical results are discussed in order to justify some asymmetries observed in flow visualizations.

We define the rate at which a scalar θ mixes in a fluid flow in terms of the flux of θ across isoscalar surfaces. This flux θd is purely diffusive and is, in principle, exactly known at all times given the scalar field and the coefficient of molecular diffusivity. In general, the complex geometry of isoscalar surfaces would appear to make the calculation of this flux very difficult. In this paper, we derive an exact expression relating the instantaneous diascalar flux to the average squared scalar gradient on an isoscalar surface which does not require knowledge of the spatial structure of the surface itself. To obtain this result, a time-dependent reference state θ(t,z*.) is defined. When the scalar is taken to be density, this reference state is that of minimum potential energy. The rate of change of the reference state due to diffusion is shown to equal the divergence of the diffusive flux, i.e. (∂/∂z*)θd.

This result provides a mathematical framework that exactly separates diffusive and advective scalar transport in incompressible fluid flows. The relationship between diffusive and advective transport is discussed in relation to the scalar variance equation and the Osborn–Cox model. Estimation of water mass transformation from oceanic microstructure profiles and determination of the time-dependent mixing rate in numerically simulated flows are discussed.

The behaviour of the energy in a steep solitary wave as a function of the wave height has a direct bearing on the breaking of solitary waves on a gently shoaling beach. Here it is shown that the speed, energy and momentum of a steep solitary wave in water of finite depth all behave in an oscillatory manner as functions of the wave height and as the limiting height is approached. Asymptotic formulae for these and other wave parameters are derived by means of a theory for the ‘almost-highest wave’ similar to that formulated previously for periodic waves in deep water (Longuet-Higgins & Fox 1977, 1978). It is demonstrated that the theory fits very precisely some recent calculations of solitary waves by Tanaka (1995).

This paper considers the development of finite-amplitude patterns of convection in rectangular-planform containers. The horizontal dimensions of the container are assumed to be large compared with the critical wavelength of the motion. An interaction between rolls parallel and perpendicular to the lateral boundaries is modelled by a coupled pair of nonlinear amplitude equations together with appropriate conditions on the four lateral boundaries. At Rayleigh numbers above a critical value a steady-state solution is established with rolls parallel to the shorter lateral sides, consistent with the predictions of linear theory. At a second critical value this solution becomes unstable to cross-rolls near the shorter sides and a new steady state evolves. This consists of the primary roll pattern together with regions near the shorter sides where there is a combination of rolls parallel and perpendicular to the boundary.

Analytical and numerical methods are used to describe both the evolution and steady-state structure of the solution, and a comparison is made with the results of full numerical simulations and experiments.

The swash zone is that part of a beach over which the instantaneous shoreline moves back and forth as waves meet the shore. This zone is discussed using the nonlinear shallow water equations which are appropriate for gently sloping beaches. A weakly three-dimensional extension of the two-dimensional solution by Carrier & Greenspan (1958) of the shallow water equations for a wave reflecting on an inclined plane beach is developed and used to illustrate the ideas. Thereafter attention is given to integrated and averaged quantities. The mean shoreline might be defined in several ways, but for modelling purposes we find the lower boundary of the swash zone to be more useful. A set of equations obtained by integrating across the swash zone is investigated as a model for use as an alternative boundary condition for wave-resolving studies. Comparison with sample numerical computations illustrates that they are effective in modelling the dynamics of the swash zone and that a reasonable representation of swash zone flows may be obtained from the integrated variables. The longshore flow of water in the swash zone is in many ways similar to the Stokes’ drift of propagating water waves. Further averaging is made over short waves to obtain results suitable as boundary conditions for longer period motions including the effect of incident short waves. In order to clearly present the work a few simplifications are made. The main result is that in addition to the kinematic type of boundary condition that occurs on a simple, e.g. rigid, boundary two further conditions are found in order that both the changing position of the swash zone boundary and the longshore flow in the swash zone may be determined. Models of the short waves both outside and inside the swash zone are needed to complete a full wave-averaged model; only brief indication is given of such modelling.

As in the laminar case, the turbulent plane Couette flow is unstable (stable) with respect to roll cell instabilities when the weak background angular velocity Ωk is antiparallel (parallel) to the spanwise mean flow vorticity (-dU/dy)k. The critical value of the rotation number Ro, based on 2Ω and dU/dy of the corresponding laminar flow, was estimated as 0.0002 at a low Reynolds number with fully developed turbulence. Direct numerical simulations were performed for Ro = ±0.01 and compared with earlier results for non-rotating Couette flow. At the low rotation rates considered, both senses of rotation damped the turbulence and the number of near-wall turbulence-generating events was reduced. The destabilized flow was more energetic, but less three-dimensional, than the non-rotating flow. In the destabilized case, the two-dimensional roll cells extracted a comparable amount of kinetic energy from the mean flow as did the turbulence, thereby decreasing the turbulent kinetic energy. The turbulence anisotropy was practically unaffected by weak spanwise rotation, while the secondary flow was highly anisotropic due to its inability to contract and expand in the streamwise direction.

We present results from an experimental investigation into the fine-scale structure associated with the mixing of a dynamically passive conserved scalar quantity on the inner scales of turbulent shear flows. The present study was based on highly resolved two- and three-dimensional spatio-temporal imaging measurements. For the conditions studied, the Schmidt number (Sc ≡ v/D) was approximately 2000 and the local outerscale Reynolds number (Reσ≡ uσ/v) ranged from 2000 to 10000. The resolution and signal quality allow direct differentiation of the measured scalar field ζ(x, t) to give the instantaneous scalar energy dissipation rate field (Re Sc)−1 ∇ζċ∇ζ(x, t). The results show that the fine-scale structure of the scalar dissipation field, when viewed on the inner-flow scales for Sc ≡ 1, consists entirely of thin strained laminar sheet-like diffusion layers. The internal structure of these scalar dissipation sheets agrees with the one-dimensional self-similar solution for the local strain–diffusion competition in the presence of a spatially uniform but time-varying strain rate field. This similarity solution also shows that line-like structures in the scalar dissipation field decay exponentially in time, while in the vorticity field both line-like and sheet-like structures can be sustained. This sheet-like structure produces a high level of intermittency in the scalar dissipation field – at these conditions approximately 4% of the flow volume accounts for nearly 25% of the total mixing achieved. The scalar gradient vector field ∇ζ(x, t) for large Sc is found to be nearly isotropic, with a weak tendency for the dissipation sheets to align with the principal axes of the mean flow strain rate tensor. Joint probability densities of the conserved scalar and scalar dissipation rate have a shape consistent with this canonical layer-like fine-scale structure. Statistics of the conserved scalar and scalar dissipation rate fields are found to demonstrate similarity on inner-scale variables even at the relatively low Reynolds numbers investigated.

Viscous stretching of a cigar-shaped drop due to the centrifugal pressure field in a surrounding rapidly rotating denser fluid is analysed. Scaling arguments are used to examine the various contributions to the viscous stresses resisting deformation, and a number of asymptotic regimes are identified which are delineated by the relative magnitudes of the aspect ratio, the viscosity ratio and unity. These asymptotic regimes may usefully be described as the bubble, pipe, sliding-rod and toffee-strand limits. Detailed analysis based upon a slenderness assumption combined with an integral representation of Stokes equations is used to derive evolution equations for the shape of the drop as a function of time in the different regimes. In the limit that interfacial-tension effects are negligible, similarity solutions are developed in which the length of the drop is found to increase as t2/5, t1/4, (t ln t)1/4 and t. The analytical results are in good agreement with numerical simulations based upon a boundary-integral solution to the full viscous flow equations.

The subharmonic transition process of a flat-plate boundary layer at a free-stream Mach number of M∞ = 4.5 and a Reynolds number of 10000 based on free-stream velocity and initial displacement thickness is investigated by direct numerical simulation up to the beginning of turbulence. A second-mode instability superimposed with random noise of low amplitude is forced initially. The secondary subharmonic instability evolves from the noise in accordance with theory and leads to a staggered Λ-vortex pattern. Finite-amplitude Λ-vortices initiate the build-up of detached high-shear layers below and above the critical layer. The detached shear-layer generation and break-up are confined to the relative-subsonic part of the boundary layer. The breakdown to turbulence can be separated into two phases, the first being the break-up of the lower shear layer and the second being the break-up of the upper shear layer. Four levels of subsequent roll-up of the lower, Y-shaped shear layer have been observed, leading to new vortical structures which are unknown from transition at low Mach numbers. The upper shear layer behaviour is similar to that of the well-known high-shear layer in incompressible boundary-layer transition. It is concluded that, as in incompressible flow, turbulence is generated via a cascade of vortices and detached shear layers with successively smaller scales. The different phases of shear-layer break-up are also reflected in the evolution of averaged quantities. A strong decrease of the shape factor, as well as an increase of the skin friction coefficient, and a gradual loss of spanwise symmetry indicate the final breakdown to turbulence, where the mean velocity and temperature profiles approach those measured in fully turbulent flow.

A new instability of the centrifugal type due to the curvature of external streamlines was theoretically predicted in a recent study on boundary layers along a swept wing. It is, however, not clear how this instability relates to already-known instability phenomena in various three-dimensional flows. So the basic idea developed in the analysis of boundary layers is applied to the simpler problems of the flow on a rotating disk and along the leading edge of a yawed circular cylinder, and the resulting eigenvalue problems are numerically solved to show multiple stability characteristics of the flows. Computational results confirm that the streamline-curvature instability does appear in the rotating-disk flow and that it is in fact identical with the instability called the ‘parallel’ or ‘type 2’ mode in the atmospheric literature. This instability is also found to occur in the steady flow near the attachment line and to give the lowest values of the critical Reynolds number except for a very narrow region close to the attachment line, where the viscous and cross-flow instabilities are dominant. These facts provide evidence to show that the same mode of instability as the classical one observed in rotating flows can appear in general three-dimensional boundary layers without rotation.

A potential flow model has been formulated for scallop swimming. Under the smalldisturbance approximation, the problem of the unsteady flow past the wing-like configuration of a scallop is separated into two linear sub-problems: the steady lifting problem and the unsteady symmetric thickness problem. The latter is associated with the expansion and contraction of the boundary surface of the scallop due to the shell opening and closing. A quasi-two-dimensional analytical solution of the thickness problem was obtained to give the time-dependent fluid forces acting on the outer surfaces of the shells. In addition to the added-mass effect, which has been widely accepted in the hydrodynamics of aquatic locomotion, there are two other mechanisms in the fluid reaction: flow-induced pseudo-elasticity and pseudo-viscosity. The pseudoelasticity provides a force proportional to the gape angle displacement, and will assist shell opening but resist shell closing. The pseudo-viscosity force is proportional to the angular velocity of the gape, and benefits both shell opening and closing. Their roles are discussed through comparison with those of shell inertia, hinge ligament elasticity and hinge damping. At 10 °C the hinge damping in the scallop was found to be almost compensated by the flow pseudo-viscosity. The unsteady fluid reaction may have a significant effect on the operation of the dynamic swimming system of scallops.

Experiments have been conducted to investigate mixing and the geometry of scalar isosurfaces in turbulent jets. Specifically, we have obtained high-resolution, high-signal-to-noise-ratio images of the jet-fluid concentration in the far field of round, liquid-phase, turbulent jets, in the Reynolds number range 4.5 × 103 ≤ Re ≤ 18 × 103, using laser-induced-fluorescence imaging techniques. Analysis of these data indicates that this Reynolds-number range spans a mixing transition in the far field of turbulent jets. This is manifested in the probability-density function of the scalar field, as well as in measures of the scalar isosurfaces. Classical as well as fractal measures of these isosurfaces have been computed, from small to large spatial scales, and are found to be functions of both scalar threshold and Reynolds number. The coverage of level sets of jet-fluid concentration in the two-dimensional images is found to possess a scale-dependent-fractal dimension that increases continuously with increasing scale, from near unity, at the smallest scales, to 2, at the largest scales. The geometry of the scalar isosurfaces is, therefore, more complex than power-law fractal, exhibiting an increasing complexity with increasing scale. This behaviour necessitates a scale-dependent generalization of power-law-fractal geometry. A connection between scale-dependent-fractal geometry and the distribution of scales is established and used to compute the distribution of spatial scales in the flow.