A variation of the classical Taylor–Couette system is studied where, with the outer cylinder stationary, the inner cylinder rotates at constant angular velocity while executing harmonic oscillations in the axial direction. Experiments reveal a Hopf (Neimark–Sacker) bifurcation from a limit cycle to a torus. Alternating bands of frequency-locked and quasi-periodic flow are observed and identified. Power spectral plots and (delay reconstructed) Poincaré maps are used to characterize the temporal dynamics. Results are presented on the rotation number variation across parameter space, the shape and growth of frequency-locked resonance horns, and the spatial development of the flow considerably beyond the primary transition surface.

We present the results of an experimental investigation of the striking flow structures that may arise when a vertical jet of fluid impinges on a thin fluid layer overlying a horizontal boundary. Ellegaard et al. (Nature, vol. 392, 1998, p. 767; Nonlinearity, vol. 12, 1999, p. 1) demonstrated that the axial symmetry of the circular hydraulic jump may be broken, resulting in steady polygonal jumps. In addition to these polygonal forms, our experiments reveal a new class of steady asymmetric jump forms that include structures resembling cat's eyes, three- and four-leaf clovers, bowties and butterflies. An extensive parameter study reveals the dependence of the jump structure on the governing dimensionless groups. The symmetry-breaking responsible for the asymmetric jumps is interpreted as resulting from a capillary instability of the circular jump. For all steady non-axisymmetric forms observed, the wavelength of instability of the jump is related to the surface tension, $\sigma$, fluid density $\rho$ and speed $U_v$ of the radial outflow at the jump through $\lambda\,{=}\,(74\pm7)\sigma/(\rho U_v^2)$.

Trapped modes in the linearized water-wave problem are free oscillations of an unbounded fluid with a free surface that have finite energy; it has been known for some time that such modes are supported by certain structures when held fixed. This paper investigates the problem of a freely floating structure that is able to move in response to the hydrodynamic forces acting upon it and it is shown that trapped modes also exist in this problem. For a freely floating structure, a trapped mode is a coupled oscillation of the fluid and the structure.

An analysis is presented of the lateral solidification of a semi-infinite mushy region influenced by vertical, buoyancy-driven convection of the residual, interstitial melt. We consider a parameter regime in which the flow is steady on the time scale of the transient evolution of the mushy region. Our idealized model predicts patterns of macrosegregation consistent with earlier experimental studies and sheds light on the mechanisms involved.

Direct numerical simulations of a turbulent boundary layer over a bump were performed to examine the effects of surface longitudinal curvature on wall pressure fluctuations ($p_{w}$) and flow-induced noise. Turbulence statistics and frequency spectra were obtained to elucidate the response of wall pressure fluctuations to the longitudinal curvature and to the corresponding pressure gradient. Wall pressure fluctuations were significantly enhanced near the trailing edge of the bump, where the boundary layer was subjected to a strong adverse pressure gradient. Large-scale structures in the distribution of wall pressure fluctuations were observed to grow rapidly near the trailing edge of the bump and convect downstream. Acoustic sources of the Lighthill equations were investigated in detail at various longitudinal surface curvatures. The acoustic sources ($S$) were highest near the trailing edge of the bump, where the root mean square wall pressure fluctuations were greatest. The maximum correlation coefficient between $p_{w}$ and $S$ was located just above the location of maximum wall pressure fluctuations. Far-field acoustic density fluctuations were computed using the Lighthill acoustic analogy. We found that the surface dipole is dominant in the total acoustic field. The contribution of the volume quadrupoles to the total acoustic field gradually increases with increasing radius of the surface curvature ($\delta/\!R$).

A family of 10 silicone oils with electrical conductivity $\sim 10^{-13}$ S m$^{-1}$ (a regime hitherto systematically unexplored) and viscosities ranging from 1 to 2000 m Pa s have been subjected to an electrical field of up to 1.5 kV mm$^{-1}$ during flow from a needle. The flow behaviour of these liquids is investigated experimentally in the flow rate regime 10$^{-8}$–10$^{-12}$ m$^{3}$ s$^{-1}$ and we analyse the results using the Ohnesorge number. Due to the low electrical conductivity and high electrical relaxation time of the silicone oils, only unsteady transient jets were found. The onset of this type of jetting has been defined using current measurements and, in contrast to conducting liquids, the non-dimensional jet diameter increases with increase in Ohnesorge number. The time elapsed between the start and finish of jetting increases with increasing Ohnesorge number.

Results are presented from laboratory experiments on lateral intrusive flows driven by horizontal gradients of properties in double-diffusive systems. These are variations on the classical experiments in which salt stratified fluid was separated from a sugar stratified fluid by a removable barrier. With the removal of this barrier, lateral intrusions of sugar solution into the salt solution alternated in the vertical with salt intrusions into the sugar solution. In our present study, the fluid has continuous horizontal gradients of sugar, density-compensated by opposing horizontal gradients of salt, rather than the discontinuity of properties in the classical experiments.

Observations of these new experiments include: shadowgraphs, PTV (particle tracking velocimetry) showing the structure of the flow, and PIV (particle-image velocimetry) from which velocity vector fields on a vertical plane were obtained; from these, mean flows, Reynolds stresses, and associated momentum fluxes were computed.

These experiments have been conducted for three regimes of vertical stratification: (i) Salt-finger favourable (‘hot and salty’ above); (ii) Diffusive convection favourable (‘hot and salty’ below); (iii) Doubly stable (‘cold and salty’ below).

One of our main conclusions concerns the driving mechanism for the horizontal intrusive flow. Away from the front or ‘nose’ of the intrusions, the lateral flows are no longer tilted but are horizontal. Here, finger layers alternate in the vertical with convecting layers. In these convecting layers, vertical transport of horizontal momentum by the Reynolds stress plays a major role in maintaining the lateral motion against viscous dissipation.

The Lagrangian evolution of two-point velocity and scalar increments in turbulence is considered, based on the ‘advected delta-vee system’ (Li & Meneveau 2005). This system has already been used to show that ubiquitous trends of three-dimensional turbulence such as exponential or stretched exponential tails in the probability density functions of transverse velocity increments, as well as negatively skewed longitudinal velocity increments, emerge quite rapidly and naturally from initially Gaussian ensembles. In this paper, the approach is extended to provide simple explanations for other known intermittency trends in turbulence: (i) that transverse velocity increments tend to be more intermittent than longitudinal ones, (ii) that in two dimensions, vorticity increments are intermittent while velocity increments are not, (iii) that scalar increments typically become more intermittent than velocity increments and, finally, (iv) that velocity increments in four-dimensional turbulence are more intermittent than in three dimensions. While the origin of these important trends can thus be elucidated qualitatively, predicting quantitatively the statistically steady-state levels and dependence on scale remains an open problem that would require including the neglected effects of pressure, inter-scale interactions and viscosity.

Numerical computations and order-of-magnitude estimates are used to describe the stationary creeping flow of a jet of a Newtonian liquid with finite electrical conductivity that is injected into a dielectric medium subject to a uniform electric field. The electric current carried by the jet is computed as a function of the parameters of the problem, showing that it increases with the conductivity and flow rate of the liquid and with the intensity of the electric field. The current also depends on the wetting conditions of the liquid at the injection orifice. Analysis of the transfer of current to the surface of the liquid and of the evolution of the jet under the electric stresses that act at its surface leads to scaling laws for the electric current and other properties of the solution. These laws fit the numerical results and are in qualitative agreement with experimental data.

Observations from a lander fixed to the seafloor over the continental shelf in 124 m of water provide highly resolved velocity measurements through nonlinear internal waves of elevation. From these measurements we determine, for the first time, the non-hydrostatic pressure disturbance ($p_{nh}$) in nonlinear internal waves. For near-bottom waves of elevation ranging in amplitude, $a$, from 12 to 33 m, the value of $p_{nh}$ evaluated at the seafloor changes sign from $\,{>}\,0$ to $\,{<}\,0$ and back in accordance with weakly nonlinear theory; peak values of $\vert p_{nh}\vert$ range from 25 to 90 N m$^{-2}$. The external hydrostatic pressure disturbance due to the surface displacement ($\eta_H$) is inferred from horizontal accelerations. For elevation waves, $\eta_H\,{<}\,0$; peak values range from 0.1 to 9 mm (1 to 90 N m$^{-2}$). The internal hydrostatic pressure perturbation ($p_{Wh}$), caused by isopycnal displacement, is inferred from measured streamlines and an ambient density profile. Its value at the seafloor is $\,{>}\,0$ for elevation waves; peak values range from 100 to 300 N m$^{-2}$. $\vert\eta_H\vert$ and seafloor values of $\vert p_{nh}\vert$, $p_{Wh}$ all increase monotonically with $a$. Since $\vert p_{nh}\vert$ and $p_{Wh}$ increase at roughly the same rate with $a$, no clear trend arises in the degree to which waves become more or less non-hydrostatic as $a$ changes.

A distinct bottom pressure signature is determined for bottom-trapped nonlinear waves of elevation, a wave train consisting of a sequence of positive pressure perturbations (dominated by $p_{Wh}$). By inference, a train of surface-trapped nonlinear internal waves of depression will consist of a sequence of negative pressure perturbations. A result of this analysis is that significant properties of the waves can be discerned from a simple adequately resolved bottom pressure measurement.

Quantitative planar visualization of molecular mixing dynamics in large- and intermediate-scale coherent structures is reported for the first time in the developing and far-field regions of gaseous planar shear layers. A dual-tracer (nitric oxide and acetone) planar laser-induced fluorescence (PLIF) technique is implemented as the gaseous analogue to acid/base chemical reactions that have previously been used to study molecular mixing in liquid shear layers. Data on low-speed, high-speed, and total molecularly mixed fluid fractions are collected for low- to high-speed velocity ratios from 0.25 to 0.44 and Reynolds numbers, $Re_{\delta}$, from 18 600 to 103 000. Within this range of conditions, mixed-fluid probability density functions and ensemble-averaged statistics are highly influenced by the homogenizing effect of large-scale Kelvin–Helmholtz rollers and the competing action of intermediate-scale secondary instabilities. Small-scale turbulence leads to near-unity mixing efficiencies and mixed-fluid probabilities within the shear layer, with subresolution stirring being detected primarily along the interface with free-stream fluid. Current molecular-mixing data compare favourably with previous time-averaged probe-based measurements while providing new insight on the effects of coherent structures, velocity ratio, downstream distance, and differences between low- and high-speed fluid entrainment.

We generate a class of multi-scale quasi-steady laminar flows in the laboratory by controlling a quasi-two-dimensional shallow-layer brine flow by multi-scale Lorentz body forcing. The flows' multi-scale topology is invariant over a broad range of Reynolds numbers, $\hbox{\it Re}_{2D}$ from 600 to 9900. The key multi-scale aspects of this flow associated with its multi-scale hyperbolic stagnation-point structure are highlighted. Our multi-scale flows are laboratory simulations of quasi-two-dimensional turbulent-like flows, and they have a power-law energy spectrum $E(k)\,{\sim}\,k^{-p}$ over a range $2\pi/L\,{<}\,k\,{<}\,2\pi/\eta$ where $p$ lies between the values 5/3 and 3 which are obtained in a two-dimensional turbulence that is forced at the small scale $\eta$ or at the large scale $L$, respectively. In fact, in the present set-up, $p\,{+}\,D_{s}\,{=}\,3$ in agreement with a previously established formula; $D_s\,{\approx}\,0.5$ is the fractal dimension of the set of stagnation points and $p\,{\approx}\,2.5$. The two exponents $D_s$ and $p$ are controlled by the multi-scale electromagnetic forcing over the entire range of scales between $L$ and $\eta$ for a broad range of Reynolds numbers with separate control over $L/\eta$ and Reynolds number. The pair dispersion properties of our multi-scale laminar flows are also controlled by their multi-scale hyperbolic stagnation-point topology which generates a sequence of exponential separation processes starting from the smaller-scale hyperbolic points and ending with the larger ones. The average mean square separation $\overline{\Delta^{2}}$ has an approximate power law behaviour ${\sim}t^{\gamma}$ with ‘Richardson exponent’ $\gamma\,{\approx}\,2.45$ in the range of time scales controlled by the hyperbolic stagnation-points. This exponent is itself controlled by the multi-scale quasi-steady hyperbolic stagnation-point topology of the flow.

Using experimental transverse velocity data for very high-Reynolds-number turbulence, we suggest a model describing both the formation of intermittency and asymmetry of turbulence. The model, called the ‘bump model’, is a modification of the ramp model suggested previously. The connection between asymmetry and intermittency makes it possible to study the latter with relatively low moments.

The magnetohydrodynamic evolution of axisymmetric magnetic eddies within which the magnetic field is purely toroidal with $B_\theta/r$ piecewise-constant, and the velocity field is poloidal, is studied both analytically and numerically. A family of exact solutions, generalizing Hill's spherical vortex to the case of non-zero magnetic field, is found. These exact solutions are (like Hill's vortex) unstable, so that, under weak disturbance, a narrow spike of vorticity is shed from the neighbourhood of the rear stagnation point. Numerical simulation using a contour-dynamics formulation shows that, for general initial contour shape, a contour singularity appears at a finite time $t^*$, like that which appears on a disturbed vortex sheet. Techniques of regularization and sample-point redistribution are used so that the eddy contours can be tracked well beyond $t^*$. When the fluid is initially at rest, the magnetic eddy first contracts towards the axis of symmetry under the action of its Lorentz force distribution; then two spherical fronts form, which propagate in the two opposite directions along the axis of symmetry, in a manner captured well by the exact solution. The magnetic energy remains bounded away from zero despite the fact that there is no topological barrier to its further decrease. Magnetic eddy evolution and the possible existence of steady states under a uniform compressive strain field is also numerically investigated.

Stress wave emission and cavitation bubble dynamics after optical breakdown in water and a tissue phantom with Nd: YAG laser pulses of 6 ns duration were investigated both experimentally and numerically to obtain a better understanding of the physical mechanisms involved in plasma-mediated laser surgery. Experimental tools were high-speed photography with 50000 frames s$^{-1}$, and acoustic measurements. The tissue phantom consisted of a transparent polyacrylamide (PAA) gel, the elastic properties of which can be controlled by modifying the water content. Breakdown in water produced a purely compressive stress wave. By contrast, in stiff PAA samples and for sufficiently large pulse energies, the compression wave was followed by an intense tensile wave, similar to the behaviour previously observed in cornea. The elastic/plastic response of the medium led to a significant decrease of the maximum size of the cavitation bubble and to a shortening of its oscillation period which was found to be related to the generation of the tensile stress wave upon breakdown. For increasing elastic modulus of the PAA, both the amplitudes of the bubble oscillation and of the stress wave emitted during bubble collapse decreased until the bubble oscillation was so strongly damped that no collapse stress wave was emitted. Numerical simulations were performed using a spherical model of bubble dynamics which includes the compressibility and elastic/plastic behaviour of the medium, viscosity, density and surface tension. The calculations revealed that consideration of the elastic/plastic behaviour of the medium surrounding the bubble is essential to describe the experimentally observed bipolar shape of the stress wave emitted upon optical breakdown. Water is a poor tissue model because the shape of the emitted stress waves and the bubble dynamics differ strongly for both materials. The mechanical properties of PAA were also found to be quite different from those of tissues. Experimental and numerical results provided evidence that the dynamic mechanical properties relevant for optical breakdown in PAA and tissue differ by as much as two orders of magnitude from the static values. The discovery of a tensile stress wave after optical breakdown in tissue-like media is of great importance for the assessment of collateral damage in laser surgery because biological tissues are much more susceptible to tensile stress than to compressive stress.

It is shown, by direct numerical simulations, that the skin-friction drag in a fully developed channel can be sustained below that corresponding to the laminar profile when the flow is subjected to surface blowing and suction in the form of an upstream travelling wave. A key mechanism that induces the sub-laminar drag is the creation of positive (negative) Reynolds shear stress in the wall region, where normally negative (positive) Reynolds shear stress is expected given the mean shear. This mechanism is contained in the linearized Navier–Stokes equations, thus allowing linear analysis of the observed phenomena. When applied to a fully developed turbulent channel flow, skin-friction drag is also significantly reduced by an upstream travelling wave, demonstrating that the surface blowing and suction in the form of such a wave is also effective in fully developed turbulent flows. Consideration of the energy budget shows a possibility of net drag reduction in turbulent channel flows with the present open-loop control.

Analytical solutions for the initial rise height $z_m$ of a turbulent fountain for the limits of both small and large source Froude number $\hbox{\it Fr}_0$ are presented. These solutions are based on a plume entrainment model. For large Froude number fountains, the established result $z_m/r_0{\,\sim\,}\hbox{\it Fr}_0$ is obtained ($r_0$ denoting the source radius). For intermediate Froude numbers, the relationship $z_m/r_0{\,\sim\,}\hbox{\it Fr}_0^2$ is found and the rise height is independent of the entrainment coefficient $\alpha$. For very small Froude numbers, the flow is hydraulically controlled at the source and $z_m/r_0{\,\sim\,}\hbox{\it Fr}_0^{2/3}$. Existing experimental and numerical results, as well as our own experimental results, are compared to our solutions and show good agreement. Comparison with experimental results also demonstrates that the appropriate entrainment coefficient for highly forced fountains is $\alpha_f{\,\approx\,}0.058$. This is significantly closer to the entrainment coefficient of a jet than of a plume.

Miscible fingering in a Hele-Shaw cell is studied by means of Stokes simulations and linear stability analysis. The two-dimensional simulations of miscible displacements in a gap indicate the existence of a quasi-steady state near the tip of the displacement front for sufficiently large Péclet numbers and viscosity ratios, in agreement with earlier work by other authors. The front thickness of this quasi-steady state is seen to scale with $\hbox{\it Pe}^{-1/2}$, while it depends only weakly on the viscosity ratio. The nature of the viscosity–concentration relationship is found to have a significant influence on the quasi-steady state. For the exponential relationship employed throughout most of the investigation, we find that the tip velocity increases with Pe for small viscosity ratios, while it decreases with Pe for large ratios. In contrast, for a linear viscosity–concentration relationship the tip velocity is seen to increase with Pe for all viscosity ratios. The simulation results suggest that in the limit of high Pe and large viscosity contrast, the width and tip velocity of the displacement front asymptote to the same values as their immiscible counterparts in the limit of large capillary numbers.

In a subsequent step, the stability of the quasi-steady front to spanwise perturbations is examined, based on the three-dimensional Stokes equations. For all values of Pe, the maximum growth rate is found to increase monotonically with the viscosity ratio. The influence of Pe on the growth of the instability is non-uniform. For mild viscosity contrasts, a larger Pe is found to be destabilizing, while for large viscosity contrasts an increase in Pe has a slightly stabilizing influence. A close inspection of the instability eigenfunction reveals the presence of two sets of counter-rotating roll-like structures, with axes aligned in the cross-gap and streamwise directions, respectively. The former lead to the periodic acceleration and deceleration of the front, while the latter result in the widening and narrowing of the front. These roll-like structures are aligned in such a way that the front widens where it speeds up, and narrows where it slows down. The findings from the present stability analysis are discussed and compared with their Darcy counterparts, as well as with experimental data by other authors for miscible and immiscible flows.

The characteristics of Boussinesq and non-Boussinesq starting forced plumes were investigated in this study. Two distinct periods in the transient plume penetration are identified, namely, the period of flow development (PFD) and period of developed flow (PDF). Similarity solutions are developed in PDF by incorporating the behaviour of an isolated buoyant vortex ring and recent laboratory results on the trailing forced plume, and the temporal variation of the penetration rate is derived during the different phases of jet-like, transitional and plume-like flow. To verify the similarity solutions, experiments were conducted on vertical starting forced plumes using combined particle image velocimetry (PIV) and planar laser induced fluorescence (PLIF) with refractive index matching. The discharge Reynolds number was varied from 3773 to 7403 and the range of excess densities ($\Delta_0 \,{=}\, (\rho_\infty\,{-}\,\rho_0)/\rho_\infty$, where $\rho_0$ and $\rho_\infty$ are initial plume and ambient density, respectively) from 2.77% to 25.07%. The experimental results revealed distinct differences between plumes having an initial density difference of larger or smaller than 15% due to the non-Boussinesq effects. Thus, the value of 15% was employed as an approximate criterion to divide the plumes into Boussinesq versus non-Boussinesq cases. The measured penetration rates and the mean centreline axial velocity of the Boussinesq starting forced plumes agreed well with the analytical predictions at the fully developed stage. However, the behaviour was substantially more complex for the non-Boussinesq plumes. In the transient records, it was noted that the time scales for the penetration of the starting plumes and the velocity development in the trailing forced plume were similar, but the time scale for the Gaussian profile to become self-similar was somewhat longer.

The flow around two stationary cylinders in tandem arrangement at the laminar and early turbulent regime, ($\hbox{\it Re}\,{=}\,10^2$–$10^3$), is studied using two- and three-dimensional direct numerical simulations. A range of spacings between the cylinders from 1.1 to 5.0 diameters is considered with emphasis on identifying the effects of three-dimensionality and cylinder spacing as well as their coupling. To achieve this, we compare the two-dimensional with corresponding three-dimensional results as well as the tandem cylinder system results with those of a single cylinder. The critical spacing for vortex formation and shedding in the gap region depends on the Reynolds number. This dependence is associated with the formation length and base pressure suction variations of a single cylinder with Reynolds number. This association is useful in explaining some of the discrepancies between the two-dimensional and three-dimensional results. A major effect of three-dimensionality is in the exact value of the critical spacing, resulting in deviations from the two-dimensional predictions for the vorticity fields, the forces on the downstream cylinder, and the shedding frequency of the tandem system. Two-dimensional simulations under-predict the critical spacing, leading to erroneous results for the forces and shedding frequencies over a range of spacings where the flow is qualitatively different. To quantify the three-dimensional effects we first employ enstrophy, decomposed into a primary and a secondary component. The primary component involves the vorticity parallel to the cylinder axis, while the secondary component incorporates the streamwise and transverse components of the vorticity vector. Comparison with the single cylinder case reveals that the presence of the downstream cylinder at spacings lower than the critical value has a stabilizing effect on both the primary and secondary enstrophy. Systematic quantification of three-dimensionalities involves finding measures for the intensity of the spanwise fluctuations of the forces. This also verifies the stabilization scenario, suggesting that when the second cylinder is placed at a distance smaller than the critical one, three-dimensional effects are suppressed compared to the single-cylinder case. However, when the spacing exceeds the critical value, the upstream cylinder tends to behave like a single cylinder, but three-dimensionality in the flow generally increases.