We examine the hydromagnetic stability of a radially stratified fluid rotating between two coaxial cylinders, with particular emphasis on the case when the angular velocity greatly exceeds both buoyant and Alfvén frequencies. If the magnetic field is predominantly azimuthal instabilities then have an essentially non-axisymmetric and wavelike character. Various bounds on their phase speeds and growth rates are derived, including a ‘quadrant’ theorem analogous to Howard's semicircle theorem for Kelvin–Helmholtz instability. Their strong tendency to propagate against the basic rotation (i.e. ‘westward’), previously noted by the author in the study of a more simplified (homogeneous) model, seems relatively insensitive to the generation mechanism (e.g. unstable gradient of magnetic field, angular velocity or density), but a number of counterexamples show that this constraint need not apply if the magnetic field displays significant spatial variations of direction as well as magnitude and that eastward-propagating amplifying modes are then possible.

Transverse jets arise in many applications, including propulsion, effluent dispersion, oil field flows, and V/STOL aerodynamics. This study seeks a fundamental, mechanistic understanding of the structure and evolution of vorticity in the transverse jet. We develop a high-resolution three-dimensional vortex simulation of the transverse jet at large Reynolds number and consider jet-to-crossflow velocity ratios r ranging from 5 to 10. A new formulation of vorticity-flux boundary conditions accounts for the interaction of channel wall vorticity with the jet flow immediately around the orifice. We demonstrate that the nascent jet shear layer contains not only azimuthal vorticity generated in the jet pipe, but wall-normal and azimuthal perturbations resulting from the jet–crossflow interaction. This formulation also yields analytical expressions for vortex lines in the near field as a function of r.

Transformation of the cylindrical shear layer emanating from the orifice begins with axial elongation of its lee side to form sections of counter-rotating vorticity aligned with the jet trajectory. Periodic roll-up of the shear layer accompanies this deformation, creating complementary vortex arcs on the lee and windward sides of the jet. Counter-rotating vorticity then drives lee-side roll-ups in the windward direction, along the normal to the jet trajectory. Azimuthal vortex arcs of alternating sign thus approach each other on the windward boundary of the jet. Accordingly, initially planar material rings on the shear layer fold completely and assume an interlocking structure that persists for several diameters above the jet exit. Though the near field of the jet is dominated by deformation and periodic roll-up of the shear layer, the resulting counter-rotating vorticity is a pronounced feature of the mean field; in turn, the mean counter-rotation exerts a substantial influence on the deformation of the shear layer. Following the pronounced bending of the trajectory into the crossflow, we observe a sudden breakdown of near-field vortical structures into a dense distribution of smaller scales. Spatial filtering of this region reveals the persistence of counter-rotating streamwise vorticity initiated in the near field.

This paper addresses the effects of solid boundaries on the evolution of two-dimensional turbulence in a finite square domain, for the cases of both decaying and continuously forced flow. Laboratory experiments and numerical flow simulations have revealed the crucial role of the solid no-slip walls as sources of vorticity filaments, which may significantly affect the flow evolution in the interior. In addition, the walls generally provide normal and tangential stresses that may exert a net torque on the fluid, which can change the total angular momentum of the contained fluid. For the case of decaying turbulence this is observed in so-called ‘spontaneous spin-up’, i.e. a significant increase of the total angular momentum, corresponding to a large domain-filling circulation cell in the organized ‘final’ state. For the case of moderate forcing this phenomenon may still be observed, although the filamentary vortex structures advected away from the walls may cause erosion and possibly a total destruction of the central cell. This disordered stage – characterized by a significantly decreased total angular momentum – is usually followed by a re-organization into a large circulation cell (in either the same or opposite direction) with an increased total angular momentum. The scaling behaviour of vorticity structure functions and the probability distribution function of vorticity increments have been investigated for forced turbulence and indicate a strong anisotropy of the turbulent flow in the range of Reynolds numbers considered.

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.

The nonlinear evolution of hydrodynamically unstable flames is studied numerically within the context of a hydrodynamic model, where the flame is confined to a surface separating the fresh mixture from the hot combustion products. The numerical scheme uses a variable-density Navier–Stokes solver in conjunction with a level-set front-capturing technique for the numerical treatment of the propagating front. Unlike most previous studies that were limited to the weakly nonlinear Michelson–Sivashinsky equation valid for small density changes, the present work places no restriction on the density contrast and thus elucidates the effect of thermal expansion on flame dynamics. It is shown that the nonlinear development leads to corrugated flames with a transverse dimension that is significantly larger than the wavelength corresponding to the most amplified disturbance predicted by the linear theory, and which is determined by the overall size of the system. The flame structure consists of wide troughs and relatively narrow cusp-like crests, and propagates ‘steadily’ at a constant speed, larger than the speed of a planar flame. The propagation speed increases as the cells widen, but eventually reaches a constant value that remains independent of the mixture's composition and of the transverse length. The dependence of the incremental increase in speed on thermal expansion is found to be nearly linear; for realistic values of thermal expansion it may be as large as 15% to 20%. In sufficiently large domains the dynamics is found to be extremely sensitive to background noise that may result, for example, from weak turbulence. Small-scale wrinkles appear sporadically on the flame surface and travel along its surface, causing a significant increase in the overall speed of propagation, up to twice the laminar flame speed.

Clouds of bubbles are generated at the sea surface by breaking wind waves or by heavy rain. Rows of subsurface bubble clouds have been detected by a bottom-mounted side-scan sonar, and are possibly formed by the effects of Langmuir circulation. A simple equation is devised to describe the effects of the turbulent diffusion of bubbles from the free surface, bubble rise and dissolution, and advection by Langmuir circulation. The equation is solved analytically using a series expansion in which advection is supposed small in comparison with diffusion. The solution provides a quantitative measure of the principal effects produced by the circulation, in particular the distortion of the bubble field, and estimates of the advective flux.

A random-walk numerical model, in which changes occurring in individual bubbles are followed, is tested against the analytical model in the range for which the latter is valid. There is good agreement. The numerical model is useful in extending the solutions to more complex cases which include a broad distribution of bubble sizes, and to ranges in which the analytic solution is invalid. The model is used to quantify the effect of the circulation on the acoustic scattering cross-section of the bubble clouds and to explore differences between the conclusions of earlier models and observations by Johnson & Cooke (1979).

In the appendices an estimate is made of the depth to which bubbles can be carried by the vertical velocities observed below wind rows, and this is found to agree reasonably well with the maximum depth to which bubbles are observed to penetrate. Estimates of mean vertical diffusion coefficients based on observations of bubbles are compared with some calculated solely from the advective flux in Langmuir circulation. The latter are, as expected, smaller than those representing the sum of all the contributions to the flux, but are a significant fraction, of the order of 0.2–0.4. A method of deriving the vertical diffusion coefficient from observations of the vertical distribution of the acoustic scattering cross-section of bubbles appears not to be very sensitive to circulation and may provide estimates within about 25% of the actual values.

Results of an experimental investigation of the heat loss of fine heated wires immersed in a supersonic stream at right angles to the flow direction are presented. The measurements show that the heat loss of the wire is independent of the free-stream Mach number for values of the latter between 1.3 and 4.5. Since the wire is always in the wake of a detached shock wave, the streamlines in the neighbourhood of the wire pass through a normal shock wave. The Reynolds number Re2 based on conditions behind a normal shock, thus becomes the characteristic parameter for the heat transfer. The measurements covering a Reynolds number range of 3 to 220 show the existence of two flow regimes. For Re2 > 20 the Nusselt number is a linear function of the square root of the Reynolds number, and the equilibrium temperature is nearly independent of Re2 < 20 the Nusselt number decreases more slowly, and the equilibrium temperature rises sharply with decreasing Reynolds number.

An analysis is presented that is appropriate to three distinct phases in the temporal development of the flow past a semi-infinite vertical plate whose temperature is suddenly changed from that of the surrounding fluid. These are the initial stage when a one-dimensional solution describes the flow, then a local solution which describes the early stages of the departure from this, and finally an asymptotic solution which describes the manner in which the final steady state is achieved.

A laboratory model of a tornado vortex has been produced, incorporating two features which are believed to be important to the understanding of the atmospheric phenomenon, but which have been largely ignored in previous studies. First, it has been shown that a vortex can be driven from above by a mechanism analogous to convection in a cloud, and that density differences within the funnel itself are not essential. Associated with this mechanism of formation is a circulation in the vertical, with an upflow in the centre surrounded by a compensating annular downflow. Secondly, the bottom boundary is seen to have a strong influence on the vortex, since the down and up flows are linked there by a rapid radial inflow in a thin boundary layer.

In the present paper an approximate theoretical description of such a vortex is proposed. The interior and boundary layer flows are first examined separately, and then a condition is sought which makes the two solutions consistent. The starting-point of the theory is the assumption of a form of stream function which describes a circulation in the vertical having the essential features of that observed. The result of the matching procedure is to fix both the form of the tangential velocity profile, and the relative magnitudes of the three components of velocity. These deductions are not critically dependent on the assumed form of the motion in the vertical, and are in good agreement with the first measurements in the laboratory vortices, though the quantitative experimental results are not emphasized here.

Thirty data sets of wavestaff measurements of a solitary feature in the Kelvin wake of the Coast Guard cutter Point Brower are analysed. The average characteristics of the feature between 1 and 4 km aft of the ship are shown to be consistent with those of an oblique nonlinear solitary wave packet. The ship speed is 7.7 m/s (Froude number 0.49) and the waves that comprise the packet have an average frequency of 3.28 rad/s. The ship speed and the wave frequency, together with Kelvin wake kinematics, imply that the feature appears at an average wake half-angle of 10.9°. The packet does not exhibit linear dispersion beyond 1 km aft of the ship and its average width is 8.9 m (measured at e−1 of the peak variance). The average amplitude is 1.1 times the theoretical amplitude of an oblique nonlinear solitary wave packet with the observed width. There is considerable variability from run to run, and there is evidence of dispersive spreading before 1 km aft of the ship. An aerial photograph shows a sinuous fluctuation of the feature; possible explanations for the fluctuation include small variations in initial conditions or a sinuous instability. The solitary feature is a possible explanation for the long bright lines observed in SEASAT SAR images in light to moderate winds and observed in sun glitter photos taken from the space shuttle.

Streamline, streakline, and material-line flow-visualization techniques have been numerically simulated in the vortex-shedding flow field from a finite-element simulation of the two-dimensional Navier-Stokes equations at a Reynolds number of 110. The results have been used (i) to characterize the wake in terms of its critical-point trajectories, and (ii) to verify that the two-dimensional Navier-Stokes model predicts the mechanism of vortex shedding experimentally observed by Gerrard (1978). A technique for determining vorticity balances in the flow field is also presented.

When a ferrofluid layer is subjected to a uniform and vertically oriented magnetic field, an interfacial instability occurs, above a critical value of the magnetic field, giving rise to a hexagonal array of peaks. On increasing the magnetic field, a smooth morphological transition from the hexagonal array to a square array was observed above a second threshold. The hexagon–square transition phenomenology, in addition to the role of penta–hepta defects initially present in the hexagonal pattern, was investigated. Furthermore, the pattern and wavenumber selection was studied by two different procedures: first by imposing jumps in field intensity and second by varying the magnetic field in a quasi-static way. The results obtained were very different for the two procedures. They indicated that the square pattern was a metastable state induced by the compression of the hexagonal pattern on increasing the control parameter. This hypothesis was confirmed by performing an additional experiment where the pattern was isotropically compressed. In this experiment, the transition was induced at a constant magnetic field lower than the transition onset value. However, the theoretical values for stability domains of hexagons and squares proposed in the literature were found to not agree with the experimental values.

The role played by gas compressibility in gas-cushioned liquid–solid impacts is investigated within a viscous gas and inviscid liquid regime. A full analysis of the energy conservation in the gas is conducted for the first time, which indicates that both thermal diffusion across the gas film and viscous dissipation play an important role in gas cushioning once gas compression becomes significant. Consequently existing models of gas compressibility based on either an isothermal or an adiabatic equation of state for the gas do not fully reflect the physics associated with this phenomenon. Models incorporating thermal diffusion and viscous dissipation are presented, which are appropriate for length scales consistent with droplet impacts, and for larger scale liquid–solid impacts. The evolution of the free surface is calculated alongside the corresponding pressure, temperature and density profiles. These profiles indicate that a pocket of gas can become trapped during an impact. Differences between the new model and older models based on isothermal and adiabatic equations of state are discussed, along with predictions of the size of the trapped gas pocket.

It is well known that most gas fluidized beds of particles bubble, while most liquid fluidized beds do not. It was shown by Anderson, Sundaresan & Jackson (1995), through direct numerical integration of the volume-averaged equations of motion for the fluid and particles, that this distinction is indeed accounted for by these equations, coupled with simple, physically credible closure relations for the stresses and interphase drag. The aim of the present study is to investigate how the model equations afford this distinction and deduce an approximate criterion for separating bubbling and non-bubbling systems. To this end, we have computed, making use of numerical continuation techniques as well as bifurcation theory, the one- and two-dimensional travelling wave solutions of the volume-averaged equations for a wide range of parameter values, and examined the evolution of these travelling wave solutions through direct numerical integration. It is demonstrated that whether bubbles form or not is dictated by the value of Ω = (ρsv3t/Ag) 1/2, where ρs is the density of particles, vt is the terminal settling velocity of an isolated particle, g is acceleration due to gravity and A is a measure of the particle phase viscosity. When Ω is large (> ∼ 30), bubbles develop easily. It is then suggested that a natural scale for A is ρsvtdp so that Ω2 is simply a Froude number.

A dynamical instability is observed in experimental studies on micro-channels of rectangular cross-section with smallest dimension 100 and $160~\mathrm{\mu} \mathrm{m} $ in which one of the walls is made of soft gel. There is a spontaneous transition from an ordered, laminar flow to a chaotic and highly mixed flow state when the Reynolds number increases beyond a critical value. The critical Reynolds number, which decreases as the elasticity modulus of the soft wall is reduced, is as low as 200 for the softest wall used here (in contrast to 1200 for a rigid-walled channel). The instability onset is observed by the breakup of a dye-stream introduced in the centre of the micro-channel, as well as the onset of wall oscillations due to laser scattering from fluorescent beads embedded in the wall of the channel. The mixing time across a channel of width 1.5 mm, measured by dye-stream and outlet conductance experiments, is smaller by a factor of 105 than that for a laminar flow. The increased mixing rate comes at very little cost, because the pressure drop (energy requirement to drive the flow) increases continuously and modestly at transition. The deformed shape is reconstructed numerically, and computational fluid dynamics (CFD) simulations are carried out to obtain the pressure gradient and the velocity fields for different flow rates. The pressure difference across the channel predicted by simulations is in agreement with the experiments (within experimental errors) for flow rates where the dye stream is laminar, but the experimental pressure difference is higher than the simulation prediction after dye-stream breakup. A linear stability analysis is carried out using the parallel-flow approximation, in which the wall is modelled as a neo-Hookean elastic solid, and the simulation results for the mean velocity and pressure gradient from the CFD simulations are used as inputs. The stability analysis accurately predicts the Reynolds number (based on flow rate) at which an instability is observed in the dye stream, and it also predicts that the instability first takes place at the downstream converging section of the channel, and not at the upstream diverging section. The stability analysis also indicates that the destabilization is due to the modification of the flow and the local pressure gradient due to the wall deformation; if we assume a parabolic velocity profile with the pressure gradient given by the plane Poiseuille law, the flow is always found to be stable.

This paper presents analyses of particle image velocimetry measurements from a boundary layer on a flat plate subject to grid-generated free-stream turbulence. The pre-transition region and early stages of breakdown to turbulent spots are explored by means of quadrant analysis and quadrant hole analysis. By isolating the contributors to the Reynolds shear stresses, it is possible to identify coherent structures within the flow that are responsible for the production of TKE. It is found that so called ‘ejection’ events are the most significant form of disturbance, exhibiting the largest amplitude behaviour with increased negative spanwise vorticity. Sweep events become increasingly large close to the wall with increased Reynolds number and intermittency.

Vortex formation in the near wake of shallow flow past a vertical cylinder can be substantially delayed by base bleed through a very narrow slot. The structure of the wake associated with this delay changes dramatically with the dimensionless momentum coefficient of the slot bleed. At very low values, where substantial vortex delay is attainable, the bleed flow is barely detectable. For progressively larger values, various forms of jets issue from the slot, and they undergo ordered, large-amplitude undulations, not necessarily synchronized with the formation of the large-scale vortices. When the cylinder is subjected to appropriate rotational perturbations, in the presence of small-magnitude base bleed, it is possible to transform the delayed vortex formation to a form characteristic of the naturally occurring vortices and, furthermore, to induce a large change of the phase, or timing, of the initially formed vortex, relative to the cylinder motion.

These features of the near-wake structure are assessed via a technique of high-image-density particle image velocimetry, which provides whole-field patterns of vorticity, Reynolds stress, amplitude distributions of spectral peaks, and streamline topology at and above the bed, for both the delayed and recovered states of the wake. Among the findings is that even small bleed can substantially alter the patterns of streamline topology and Reynolds stress at the bed, which has important consequences for the bed loading.

These alterations of the near-wake structure occur in conjunction with modifications of the shallow approach flow, which is incident upon the upstream face of the cylinder. The topology at the bed, which is altered in accord with attenuation of the well-defined vorticity concentration of the horseshoe (standoff) vortex, shows distinctive patterns involving new arrangements of critical points.

The linearized problem of water-wave reflexion by a thin barrier of arbitrary permeability is considered with the restriction that the flow be two-dimensional. The formulation includes the special case of transmission through one or more gaps in an otherwise impermeable barrier. The general problem is reduced to a set of integral equations using standard techniques. These equations are then solved using a special decomposition of the finite depth source potential which allows accurate solutions to be obtained economically. A representative range of solutions is obtained numerically for both finite and infinite depth problems.

The decay characteristics and invariants of grid turbulence were investigated by means of laboratory experiments conducted in a wind tunnel. A turbulence-generating grid was installed at the entrance of the test section for generating nearly isotropic turbulence. Five grids (square bars of mesh sizes $M= 15$, 25 and 50 mm and cylindrical bars of mesh sizes $M= 10$ and 25 mm) were used. The solidity of all grids is $\sigma = 0. 36$. The instantaneous streamwise and vertical (cross-stream) velocities were measured by hot-wire anemometry. The mesh Reynolds numbers were adjusted to $R{e}_{M} = 6700$, 9600, 16 000 and 33 000. The Reynolds numbers based on the Taylor microscale $R{e}_{\lambda } $ in the decay region ranged from 27 to 112. In each case, the result shows that the decay exponent of turbulence intensity is close to the theoretical value of ${- }6/ 5$ (for the $M= 10~\mathrm{mm} $ grid, ${- }6(1+ p)/ 5\sim - 1. 32$) for Saffman turbulence. Here, $p$ is the power of the dimensionless energy dissipation coefficient, $A(t)\sim {t}^{p} $. Furthermore, each case shows that streamwise variations in the integral length scales, ${L}_{uu} $ and ${L}_{vv} $, and the Taylor microscale $\lambda $ grow according to ${L}_{uu} \sim 2{L}_{vv} \propto {(x/ M- {x}_{0} / M)}^{2/ 5} $ (for the $M= 10~\mathrm{mm} $ grid, ${L}_{uu} \propto {(x/ M- {x}_{0} / M)}^{2(1+ p)/ 5} \sim {(x/ M- {x}_{0} / M)}^{0. 44} $) and $\lambda \propto {(x/ M- {x}_{0} / M)}^{1/ 2} $, respectively, at $x/ M\gt 40{\unicode{x2013}} 60$ (depending on the experimental conditions, including grid geometry), where $x$ is the streamwise distance from the grid and ${x}_{0} $ is the virtual origin. We demonstrated that in the decay region of grid turbulence, ${ u}_{\mathit{rms}}^{2} { L}_{uu}^{3} $ and ${ v}_{\mathit{rms}}^{2} { L}_{vv}^{3} $, which correspond to Saffman’s integral, are constant for all grids and examined $R{e}_{M} $ values. However, ${ u}_{\mathit{rms}}^{2} { L}_{uu}^{5} $ and ${ v}_{\mathit{rms}}^{2} { L}_{vv}^{5} $, which correspond to Loitsianskii’s integral, and ${ u}_{\mathit{rms}}^{2} { L}_{uu}^{2} $ and ${ v}_{\mathit{rms}}^{2} { L}_{vv}^{2} $, which correspond to the complete self-similarity of energy spectrum and $\langle {\boldsymbol{u}}^{2} \rangle \sim {t}^{- 1} $, are not constant. Consequently, we conclude that grid turbulence is a type of Saffman turbulence for the examined $R{e}_{M} $ range of 6700–33 000 ($R{e}_{\lambda } = 27{\unicode{x2013}} 112$) regardless of grid geometry.

This paper reports on near-wall two-component–two-dimensional (2C–2D) particle image velocimetry (PIV) measurements of a turbulent pipe flow at shear Reynolds numbers up to $Re_{\unicode[STIX]{x1D70F}}=40\,000$ acquired in the CICLoPE facility of the University of Bologna. The 111.5 m long pipe of 900 mm diameter offers a well-established turbulent flow with viscous length scales ranging from $85~\unicode[STIX]{x03BC}\text{m}$ at $Re_{\unicode[STIX]{x1D70F}}=5000$ down to $11~\unicode[STIX]{x03BC}\text{m}$ at $Re_{\unicode[STIX]{x1D70F}}=40\,000$. These length scales can be resolved with a high-speed PIV camera at image magnification near unity. Statistically converged velocity profiles were determined using multiple sequences of up to 70 000 PIV recordings acquired at sampling rates of 100 Hz up to 10 kHz. Analysis of the velocity statistics shows a well-resolved inner peak of the streamwise velocity fluctuations that grows with increasing Reynolds number and an outer peak that develops and moves away from the inner peak with increasing Reynolds number.