We investigate linear–quadratic dynamical systems with energy-preserving quadratic terms. These systems arise for instance as Galerkin systems of incompressible flows. A criterion is presented to ensure long-term boundedness of the system dynamics. If the criterion is violated, a globally stable attractor cannot exist for an effective nonlinearity. Thus, the criterion can be considered a minimum requirement for control-oriented Galerkin models of viscous fluid flows. The criterion is exemplified, for example, for Galerkin systems of two-dimensional cylinder wake flow models in the transient and the post-transient regime, for the Lorenz system and for wall-bounded shear flows. There are numerous potential applications of the criterion, for instance, system reduction and control of strongly nonlinear dynamical systems.

An oblique shock wave is generated in a Mach 2 flow at a flow deflection angle of $12^{\circ }$. The resulting shock-wave–boundary-layer interaction (SWBLI) at the tunnel wall is observed. A novel traversable shock generator allows the position of the SWBLI to be varied relative to a downstream expansion fan. The relationship between the SWBLI, the expansion fan and the wind tunnel arrangement is studied. Schlieren photography, surface oil flow visualisation, particle image velocimetry and high-spatial-resolution wall pressure measurements are used to investigate the flow. It is observed that stream-normal movement of the shock generator downwards (towards the floor and hence the point of shock reflection) is accompanied by (1) growth in the streamwise extent of the shock-induced boundary layer separation, (2) upstream movement of the shock-induced separation point while the reattachment point remains nearly fixed, (3) an increase in separation shock strength and (4) transition between regular and irregular (Mach) reflection without an increase in incident shock strength. The role of free interaction theory in defining the separation shock angle is considered and shown to be consistent with the present measurements over a short streamwise extent. An SWBLI representation is proposed and reasoned which explains the apparent increase in separation shock strength that occurs without an increase in incident shock strength.

Understanding the two-way interactions between finite-size solid particles and a wall-bounded turbulent flow is crucial in a variety of natural and engineering applications. Previous experimental measurements and particle-resolved direct numerical simulations revealed some interesting phenomena related to particle distribution and turbulence modulation, but their in-depth analyses are largely missing. In this study, turbulent channel flows laden with neutrally buoyant finite-size spherical particles are simulated using the lattice Boltzmann method. Two particle sizes are considered, with diameters equal to 14.45 and 28.9 wall units. To understand the roles played by the particle rotation, two additional simulations with the same particle sizes but no particle rotation are also presented for comparison. Particles of both sizes are found to form clusters. Under the Stokes lubrication corrections, small particles are found to have a stronger preference to form clusters, and their clusters orientate more in the streamwise direction. As a result, small particles reduce the mean flow velocity less than large particles. Particles are also found to result in a more homogeneous distribution of turbulent kinetic energy (TKE) in the wall-normal direction, as well as a more isotropic distribution of TKE among different spatial directions. To understand these turbulence modulation phenomena, we analyse in detail the total and component-wise volume-averaged budget equations of TKE with the simulation data. This budget analysis reveals several mechanisms through which the particles modulate local and global TKE in the particle-laden turbulent channel flow.

The steady streaming induced by oscillatory viscous flow of small amplitudes over a wavy wall has been analysed, and the computed flow patterns have been found to agree well with the flow patterns visualized experimentally in a tube. When the first parameter, α/δ (the ratio of the wavelength of the wavy wall to the thickness of Stokes layer), becomes larger than about 26, the streaming has a double structure consisting of regions of upper and lower pairs of recirculations. As the second parameter, L/δ (the ratio of the amplitude of the wavy wall to the thickness of Stokes layer), is increased, the upper pair of recirculations squeezes in a gap between the lower recirculations above the troughs of the wall. A similar double structure of steady streamings was also observed above ripple marks formed under oscillatory viscous flow. A bearing is suggested of this phenomenon on the determination of stationary profiles of ripple marks.

A laboratory study was carried out to directly measure the turbulence properties in a benthic boundary layer (BBL) above a uniformly sloping bottom where the BBL is energized by internal waves. The ambient fluid was continuously stratified and the steadily forced incoming wave field consisted of a confined beam, restricting the turbulent activity to a finite region along the bottom slope. Measurements of dissipation showed some variation over the wave phase, but cycle-averaged values indicated that the dissipation was nearly constant with height within the BBL. Dissipation levels were up to three orders of magnitude larger than background laminar values and the thickness of the BBL could be defined in terms of the observed dissipation variation with height. Assuming that most of the incoming wave energy was dissipated within the BBL, predicted levels of dissipation were in good agreement with the observations.

Measurements were also made of density and two orthogonal components of the velocity fluctuations at discrete heights above the bottom. Cospectral estimates of density and velocity fluctuations showed that the major contributions to both the vertical density flux and the momentum flux resulted from frequencies near the wave forcing frequency, rather than super-buoyancy frequencies, suggesting a strong nonlinear interaction between the incident and reflected waves close to the bottom. Within the turbulent BBL, time-averaged density fluxes were significant and negative near the wave frequencies but negligible at frequencies greater than the buoyancy frequency N. While dissipation rates were high compared to background laminar values, they were low compared to the value of εtr ≈ 15vN2, the transition value often used to assess the capacity of a stratified flow to produce mixing. Existing models relating mixing to dissipation rate rely on the existence of a positive-definite density flux at frequencies greater than N as a signature of fluid mixing and therefore cannot apply to these experiments. We therefore introduce a simple model, based on the concept of diascalar fluxes, to interpret the mixing in the stratified fluid in the BBL and suggest that this may have wider application than to the particular configuration studied here.

The dispersion of passive scalars by the steady viscous flow through a twisted pipe is both a simple example of chaotic advection and an elaboration of Taylor's classic shear dispersion problem. In this article we study the statistics of the axial dispersion of both diffusive and perfect (non-diffusive) tracer in this system.

For diffusive tracer chaotic advection assists molecular diffusion in transverse mixing and so diminishes the axial dispersion below that of integrable advection. As in many other studies of shear dispersion the axial distribution ultimately becomes Gaussian as t → ∞. Thus there is a diffusive regime, but with an effective diffusivity that is enhanced above molecular values. In contrast to the classic case, the effective diffusivity is not necessarily inversely proportional to the molecular diffusivity. For instance, in the irregular regime the effective diffusivity is proportional to the logarithm of the molecular diffusivity.

For perfect tracer chaotic advection does not result in a diffusive process, even in the irregular regime in which streamlines wander throughout the cross-section of the pipe. Instead the variance of the axial position is proportional to t in t so that the measured diffusion coefficent diverges like In t. This faster than linear growth of variance is attributed to the trapping of tracer for long times near the solid boundary, where the no-slip condition ensures that the fluid moves slowly. Analogous logarithmic effects associated with the no-slip condition are well known in the context of porous media.

A simple argument, based on Lagrangian statistics and a local analysis of the trajectories near the pipe wall, is used to calculate the constants of proportionality before the logarithmic terms in both the large- and infinite-Péclet-number limits.

A systematic numerical analysis is performed for the quasi-periodicity in the wake where a circular cylinder is rotationally oscillated in time. The main emphasis is placed on the identification of frequency selection subjected to the controlled perturbations in the vicinity of lock-on. The frequency responses are scrutinized by measuring the lift coefficient (CL). A direct numerical simulation is made to portray the unsteady dynamics of wake flows at Re = 110. It is found that, after the shedding frequency is bifurcated at the boundary of lock-on, one frequency follows the forcing frequency and the other gradually converges to the natural shedding frequency. The asymptotic convergence phenomena are observed by solving the Van der Pol equation and the circle map. A new frequency selection formula is proposed. The quasi-periodic states are interpreted in terms of the forcing frequency, shedding frequency and modulated frequencies by employing the torus concept and the CL(t) diagram. In the quasi-periodic state, the variation of magnitudes and relevant phase changes of CL with forcing phase are examined.

The motion of fluid droplets in constricted capillary tubes is investigated for flows subject to the combined action of a mean pressure gradient and an oscillatory body force. Numerical computations are employed to determine the effect of the oscillatory forcing on the mean flow rate and the mean droplet velocity. In the absence of oscillatory forcing, a critical pressure gradient for droplet propagation exists, below which droplets become plugged in the narrow constrictions of the tube. For mean pressure gradients below this threshold, oscillatory forcing is shown to be an effective means for unplugging the constrictions and remobilizing the droplets. For this remobilization process to occur, the oscillatory forcing level must exceed a specified value, and the oscillatory frequency must remain below a critical frequency. Quasi-steady models are shown to give effective predictions of the unsteady dynamics over a wide range of conditions.

We report on a set of turbulent flow experiments of the Taylor type in which the fluid is contained between a rotating inner circular cylinder and a fixed concentric outer cylinder, focusing our attention on very large Taylor number values, i.e. \[ 10^3 \leqslant T/T_c \leqslant 10^5, \] where Tc is the critical value of the Taylor number T for onset of Taylor vortices. At such large values of T, the turbulent vortex flow structure is similar to the one observed when T – Tc is small and this structure is apparently insensitive to further increases in T. These flows are characterized by two widely separated length scales: the scale of the gap width which characterizes the Taylor vortex flow and a much smaller scale which is made visible by streaks in the form of a ‘herring-bone’-like pattern visible at the walls. These are conjectured to be Görtler vortices which arise as a result of centrifugal instability in the wall boundary layers. Ideas of marginal instability by which we postulate that both the Taylor and Görtler vortex structures are marginally unstable on their own scale seem to provide good quantitative agreement between predicted and observed Görtler vortex spacings.

An experimental investigation is described of the velocity field in a steady, spillingtype breaker, generated on a steady current by a submerged hydrofoil. Velocities have been measured with a laser–doppler system, and analysed with respect to mean and r.m.s. values as well as Reynolds stresses. The results indicate that the turbulent flow field downstream of the initiation of the separation at the surface resembles that in a self-similar turbulent wake.

This paper describes a new surface-streamline flow-visualization technique that is suitable for use in low-speed wind tunnels or other low-speed gas flows. The technique provides a permanent record of ink traces that show surface-streamline direction and shape. The visualization of the endwall surface flow of the horseshoe-vortex system formed around the base of a cylinder by a separating turbulent boundary layer is used as an example. The results of the new technique are compared with those obtained from a conventional flow-visualization technique. Good agreement was found between the two, with the new technique appearing to give more-accurate surface streamlines.

The boundary-integral method is used to calculate the hydrodynamic force and torque on an arbitrary body of revolution whose axis of symmetry is oriented at an arbitrary angle relative to a planar wall in the zero-Reynolds-number limit. The singular solution of the Stokes equations in the presence of a planar wall is used to formulate the integral equations, which are then reduced to a system of linear algebraic equations by satisfying the no-slip boundary conditions on the body surface using the boundary collocation method or weighted residual technique.

Numerical tests for the special case of a sphere moving parallel or perpendicular to a planar wall show that the present theory is accurate to at least three significant figures when compared with the exact solutions for gap widths as small as only one-tenth of the particle radius. Higher accuracy can be achieved and solutions can be obtained for smaller gap widths at the expense of more computation time and larger storage requirements.

The hydrodynamic force and torque on a spheroid with varying aspect ratio and orientation angle relative to the planar wall are obtained. The theory is also applied to study the motion of a toroidal particle or biconcave shaped disc adjacent to a planar wall. The coincidence of the drag and torque of a biconcave-shaped body and a torus having an aspect ratio b/a = 2 with the same surface area shows that in this case the hole of a torus has little influence on the flow field. On the other hand, for an aspect ratio b/a = 10, the effect of the hole is significant. It is also shown that when the body is not very close to the wall, an oblate spheroid can be used as a good approximation of a biconcave-shaped disc.

An asymptotic theory of marginal thermal convection in rotating systems is constructed for the limit of rapid rotation. Many self-gravitating astronomical bodies, including the major planets, the Sun, and the Earth's liquid core, correspond to this limit. In the laboratory, an analogous system can be constructed with a very rapidly rotating apparatus, in which the centrifugal force plays the role of self-gravitation. The formulation is offered in such a way that both these geophysical systems and laboratory analogues are included as special cases. When the inclination of the outer boundaries relative to the equatorial plane is considered weak, the two types of system are identical at leading order. In this limit, the asymptotic analysis is profoundly simplified, because the system satisfies the Taylor-Proudman theorem to leading order. Nevertheless the system contains a very peculiar property: the mode defined by a conventional WKBJ theory implicitly assuming a locality of convection in the radial direction perpendicular to the axis of rotation cannot be accepted as a correct marginal mode, because a modulation equation gives an exponential growth in the radial direction, which contradicts an implicit initial assumption. The erroneous behaviour is traced to a spatial dispersion of thermal Rossby waves, which governs the marginal mode. The difficulty is resolved by extending the analysis to a complex plane of the radial coordinate of the point where convection amplitude attains its maximum. Such a complex radial distance is defined as the point where the wave dispersion disappears locally. The projection of the solution onto the real axis results in an inclination of the Taylor columns with respect to the radial direction. This is in good agreement with the most recent numerical studies. The isolation of convective Taylor columns in the radial direction weakens and the spiralling gets stronger as the Prandtl number decreases, as a result of the need to displace the critical radial distance further from the real axis.

We study the motion which results when a fluid film is squeezed between two parallel plane surfaces in relative motion. Particular attention is given to the special case where one surface is fixed and the other is rapidly accelerated from a state of rest to a state of uniform motion. The analysis is based in part on linear theory and in substance on a finite-difference analysis of the full nonlinear equations of motion.

A composite hydrodynamic-diffusion model of the arterial wall is presented to describe the vesicular transport of relatively inert macromolecules across the inner endothelial lining of the larger arteries of humans and animals and their subsequent diffusion in the underlying tissue of the intima and media. This model is motivated by the highly specialized ultrastructure of the arterial wall observed in electron microscopic studies and the recent experimental measurements of the time-dependent uptake of labelled macromolecules in animal arteries under carefully controlled in vitro conditions. The proposed dynamic model for the vesicular transport across the endothelial cell layer considers the constrained Brownian diffusion of 700 Å vesicles subject to long-range hydrodynamic and short-range London-van der Waals force interactions with the plasmalemma membranes of the endothelial cell. Approximate solutions are developed for the motion and the steady-state vesicle density distribution near the plasmalemma and in the interior of the cell using boundary-layer-like methods. The model for the vesicular transport just described appears as a novel boundary condition in the basic diffusion model for the underlying tissue. The latter is treated as a two-phase medium comprised of an interstitial fluid continuum with a uniformly dispersed smooth muscle phase as first proposed by Hills (1968). This model for the underlying tissue assumes that the smooth muscle cells contribute insignificantly to the macromolecule diffusion across the arterial wall but act as the principal storage reservoir for the macromolecules for large diffusion times because of their large volume fraction. The dimensionless parameters that arise in the theoretical model are determined by comparing the solutions for the time-dependent total wall uptake with Fry's (1973) experimental data for canine carotid artery.

The linear stability of a thin liquid layer bounded from above by a free surface and from below by an oscillating plate is investigated for disturbances of arbitrary wavenumbers, a range of imposed frequencies and selective physical parameters. The imposed motion of the lower wall occurs in its own plane and is unidirectional and time-periodic. Long-wave instabilities occur only over certain bandwidths of the imposed frequency, as determined by a long-wavelength expansion. A fully numerical method based on Floquet theory is used to investigate solutions with arbitrary wavenumbers, and a new free-surface instability is found that has a finite preferred wavelength. This instability occurs continuously once the imposed frequency exceeds a certain threshold. The neutral curves of this new finite-wavelength instability appear significantly more complex than those for long waves. In a certain parameter regime, folds occur in the finite-wavelength stability limit, giving rise to isolated unstable regions. Only synchronous solutions are found, i.e. subharmonic solutions have not been detected. In Appendix A, we provide an argument for the non-existence of subharmonic solutions.

The mixed-Eulerian–Lagrangian method using high-order boundary elements, described in Xue et al. (2001) for the simulation of fully nonlinear three-dimensional wave–wave and wave–body interactions, is here extended and applied to the study of two nonlinear three-dimensional wave–body problems: (a) the development of bow waves on an advancing ship; and (b) the steep wave diffraction and nonlinear high-harmonic loads on a surface-piercing vertical cylinder. For (a), we obtain convergent steady-state bow wave profiles for a flared wedge, and the Wigley and Series 60 hulls. We compare our predictions with experimental measurements and find good agreement. It is shown that upstream influence, typically not accounted for in quasi-two-dimensional theory, plays an important role in bow wave prediction even for fine bows. For (b), the primary interest is in the higher-harmonic ‘ringing’ excitations observed and quantified in experiments. From simulations, we obtain fully nonlinear steady-state force histories on the cylinder in incident Stokes waves. Fourier analysis of such histories provides accurate predictions of harmonic loads for which excellent comparisons to experiments are obtained even at third order. This confirms that ‘ringing’ excitations are directly a result of nonlinear wave diffraction.

Two topics are discussed in order to illustrate the author's own enjoyment of fluid mechanics. The first and longer discourse is about splashes. It makes no attempt at completeness but includes a little new research. The second part deals briefly with many variations on the theme of flow in pipes.

Thermal imprint and heat transfer on the target surface in multiple impinging jet arrays have been investigated in conjunction with flow pattern and large-scale eddy structure. Surface temperature was measured with liquid crystal thermography (LCT) in a range of jet configurations with hexagonal and in-line orifice arrangements for different combinations of distances between the orifices $s/D_m\,{=}\,2\hbox{--}6$ and between the orifice plate and the impingement surface $H/D_m\,{=}\,3\hbox{--}10$ (where $D_m$ is the orifice diameter). The hexagonal arrangement was considered with two different orifice shapes: sharp-edged and contoured. For selected configurations, the distribution of Nusselt number and its peculiarities were analysed in relation with the underlying eddy structure educed by proper orthogonal decomposition (POD) of the snapshots of fluid velocity measured with particle image velocimetry (PIV). Owing to the breakdown of the jets, heat transfer deteriorates with increasing orifice-to-plate distance. The jet interaction and breakdown become more severe as the jets are placed closer to each other. The large-scale eddies originating from the jet-edge shear layers grow as they are convected towards the impingement plate. Eddies of sizes between 0.2 and 0.3 orifice diameters are shown to break up the jets and cause mixing of fresh and spent fluid, lowering the beneficial temperature gradient. In some configurations, an asymmetric flow pattern is generated with embedded weak eddies on only one side of the diagonal symmetry line, which is reflected in an asymmetrical heat transfer distribution on the impingement plate. For $H/D_m\,{>}\,4$, the Nusselt number shows peak values in and around the jet impingement centres, but relatively uniform distribution of turbulence kinetic energy with local negative energy production close to the impingement surface.

The dynamics of the coupled Kelvin–Helmholtz (KH) and Rayleigh–Taylor (RT) instability (referred to as KHRT instability or KHRTI) is studied using statistically steady experiments performed in a multi-layer gas tunnel. Experiments are performed at four density ratios ranging in Atwood number $A_{t}$ from 0.035 to 0.159, with varying amounts of shear and $\unicode[STIX]{x0394}U/U$ ranging from 0 to 0.48, where $\unicode[STIX]{x0394}U$ is the speed difference between the two flow streams being investigated and $U$ is the mean velocity of these two streams. Three types of diagnostics – back-lit visualization, hot-wire anemometry and particle image velocimetry (PIV) – are employed to obtain the mixing widths, velocity field and density field. The flow is found to be governed by KH dynamics at early times and RT dynamics at late times. This transition from KH-instability-like to RT-instability-like behaviour is quantified using the Richardson number. Transitional Richardson number magnitudes obtained for the present KHRT flows are found to be in the range 0.17–0.56 similar to the critical Richardson numbers for stably stratified free shear flows. Comparing the evolution of density and velocity mixing widths, the density mixing layer is found to be approximately two times as thick as the velocity mixing layer. Scaling of velocity fluctuations is attempted using combinations of KH and RT scales. It is found that the proposed KHRT velocity scale, obtained using the combined mixing-layer growth equation, is appropriate for intermediate stages of the flow when both KH and RT dynamics are comparable. Probability density functions (p.d.f.s) for different fluctuating quantities are presented. Multiple peaks in p.d.f.s are qualitatively explained from the development of coherent KH roll-ups and their subsequent transition into turbulent pockets. The evolution of energy spectra indicates that density fluctuations start to show an inertial subrange from earlier times compared to velocity fluctuations. The spectra exhibit a slightly steeper slope than the Kolmogorov–Obukhov five-thirds law.