The drag on a sphere moving horizontally through stably stratified salt water is measured in laboratory experiments. The increment ΔCD in drag coefficient due to the stratification is obtained as a function of a stratification parameter κ and, in principle, the usual Reynolds number R. In these experiments, where R ranges from 150 to 5000, ΔCD is insensitive to R. But, as a function of κ, ΔCD has both positive and negative values attributable respectively to lee-wave drag and to suppression of turbulence in the wake. An observed delay in flow separation also apparently results from the lee-wave drag.

Both the incompressible and supersonic laminar flow over a small, unsteady hump are considered. The Reynolds number is assumed large, and the analysis is based upon triple-deck theory. In the incompressible case disturbances tend to grow downstream, as a result of triggering the Tollmien–Schlichting mode of instability. For the supersonic case the flow disturbances tend to decay downstream across the entire frequency spectrum. However, for sufficiently large humps a seemingly catastrophic failure of the governing equations may occur, our results suggesting that this is caused by an inviscid, short-scale, Rayleigh type of instability.

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.

This work brings new insight to the question of heat transfer in near–critical fluids under Earth gravity conditions. The interplay between buoyant convection and thermoacoustic heat transfer (piston effect) is investigated in a two-dimensional non-insulated cavity containing a local heat source, to reproduce the conditions used in recent experiments. The results were obtained by means of a finite-volume numerical code solving the Navier–Stokes equations written for a low-heat-diffusing near-critical van der Waals fluid. They show that hydrodynamics greatly affects thermoacoustics in the vicinity of the upper thermostated wall, leading to a rather singular heat transfer mechanism. Heat losses through this wall govern a cooling piston effect. Thus, the thermal plume rising from the heat source triggers a strong enhancement of the cooling piston effect when it strikes the middle of the top boundary. During the spreading of the thermal plume, the cooling piston effect drives a rapid thermal quasi-equilibrium in the bulk fluid since it is further enhanced so as to balance the heating piston effect generated by the heat source. Then, homogeneous fluid heating is cancelled and the bulk temperature stops increasing. Moreover, diffusive and convective heat transfers into the bulk are very weak in such a low-heat-diffusing fluid. Thus, even though a steady state is not obtained owing to the strong and seemingly continuous instabilities present in the flow, the bulk temperature is expected to remain quasi-constant. Comparisons performed with a supercritical fluid at initial conditions further from the critical point show that this thermalization process is peculiar to near-critical fluids. Even enhanced by the thermal plume, the cooling piston effect does not balance the heating piston effect. Thus, overall piston-effect heating lasts much longer, while convection and diffusion progressively affect the thermal field much more significantly. Ultimately, a classical two-roll convective-diffusive structure is obtained in a perfect gas, without thermoacoustic heat transfer playing any role.

Model equations which describe the evolution of long-wave initial data in water of uniform depth are tested to determine explicit criteria for their applicability. We consider linear and nonlinear, dispersive and non-dispersive equations. Separate criteria emerge for the leading wave and trailing oscillations of the evolving wave train. The evolution of the leading wave depends on two parameters: the volume (non-dimensional) of the initial data and an Ursell number based on the amplitude and length of the initial data. The magnitudes of these two parameters determine the appropriate model equation and its time of validity. For the trailing oscillatory waves, a local Ursell number based on the amplitude of the initial data and the local wavelength determines the appropriate model equation. Finally, these modelling criteria are applied to the problem of tsunami propagation. Asymptotic (t → ∞) linear dispersive theory does not appear to be applicable for describing the leading wave of tsunamis. If the length of the initial wave is approximately 100 miles, the leading wave is described by a linear non-dispersive model from the source region until shoaling occurs near the coastline. For smaller lengths (∼ 40 miles) a linear dispersive (but not asymptotic) model is applicable. The longer-period oscillatory waves following the leading wave, which can induce harbour resonance, apparently require a nonlinear dispersive model.

As a viscous fluid in a vertical tube drains under the effect of gravity, a finger of air rises in the tube. The shape of the interface and the rate at which the finger rises is determined numerically for different values of the dimensionless parameter G = ρgb2/T, where ρ is the density difference between the viscous fluid and the air, b is the radius of the tube, and T is the interfacial tension. A relationship between the Bond number G and the capillary number Ca = μU/T is found and compared with the perturbation result of Bretherton (1961), where μ is the viscosity of the fluid and U is the constant velocity at which the finger rises. The numerical results support Bretherton's conclusions for very small values of Ca and extend the relationship between G and Ca to a region where the perturbation expansion is no longer valid. The results are also valid for long bubbles rising in a tube.

We consider viscous shear flow of a monolayer of solid spheres and discuss the effect that microscopic particle surface roughness has on the stress in the suspension. We consider effects both within and outside the dilute régime. Away from jamming concentrations, the viscosity is lowered by surface roughness, and for dilute suspensions it is insensitive to friction between the particles. Outside the dilute region, the viscosity increases with increasing friction coefficient. For a dilute system, roughness causes a negative first normal stress difference (N1) at order c2 in particle area concentration. The magnitude of N1 increases with increasing roughness height in the dilute limit but the trend reverses for more concentrated systems. N1 is largely insensitive to interparticle friction. The dilute results are in accord with the three-dimensional results of our earlier work (Wilson & Davis 2000), but with a correction to the sign of the tangential friction force.

The positive and negative high-amplitude pressure peaks (HAPP) are investigated in a turbulent boundary layer at $R{e}_{\theta } = $ 1900 in order to identify their turbulent structure. The three-dimensional velocity field is measured within the inner layer of the turbulent boundary layer using tomographic particle image velocimetry (tomo-PIV). The measurements are performed at an acquisition frequency of 10 000 Hz and over a volume of $418\times 149\times 621$ wall units in the streamwise, wall-normal and spanwise directions, respectively. The time-resolved velocity fields are applied to obtain the material derivative using the Lagrangian method followed by integration of the Poisson pressure equation to obtain the three-dimensional unsteady pressure field. The simultaneous volumetric velocity, acceleration, and pressure data are conditionally sampled based on local maxima and minima of wall pressure to analyse the three-dimensional turbulent structure of the HAPPs. Analysis has associated the positive HAPPs to the shear layer structures formed by an upstream sweep of high-speed flow opposing a downstream ejection event. The sweep event is initiated in the outer layer while the ejection of near-wall fluid is formed by the hairpin category of vortices. The shear layers were observed to be asymmetric in the instantaneous visualizations of the velocity and acceleration fields. The asymmetric pattern originates from the spanwise component of temporal acceleration of the ejection event downstream of the shear layer. The analysis also demonstrated a significant contribution of the pressure transport term to the budget of the turbulent kinetic energy in the shear layers. Investigation of the conditional averages and the orientation of the vortices showed that the negative HAPPs are linked to both the spanwise and quasi-streamwise vortices of the turbulent boundary layer. The quasi-streamwise vortices can be associated with the hairpin category of vortices or the isolated quasi-streamwise vortices of the inner layer. A bi-directional analysis of the link between the HAPPs and the hairpin paradigm is also conducted by conditionally averaging the pressure field based on the detection of hairpin vortices using strong ejection events. The results demonstrated positive pressure in the shear layer region of the hairpin model and negative pressure overlapping with the vortex core.

The effect of a streamwise pressure gradient on the velocity profile in the viscous sublayer of a turbulent flow along a smooth wall in two-dimensional flow is estimated. In the analysis, a similarity argument is used and the necessary empirical information obtained from a constant pressure flow. An allowance is made for the departure from the wall value of the gradient of total shear stress normal to the wall. The results of analysis were used to generate new additive constants for use with Townsend's modified law of the wall velocity profile and subsequently Townsend's profile is found to be in good agreement with the measured velocity profiles in an adverse pressure gradient.

A numerical model for a three-dimensional heat and fluid flow through a bank of infinitely long cylinders in yaw has been proposed to investigate complex flow and heat transfer characteristics associated with manmade structures such as extended fins and plate fins in heat transfer equipment. By exploiting the periodicity of the structure, only one structural unit has been taken as a calculation domain. An economical quasi-three-dimensional calculation procedure has been proposed to replace exhaustive full three-dimensional numerical manipulations. It has been shown that, under macroscopically uniform flow, the three-dimensional governing equations reduce to quasi-three-dimensional forms, in which all derivatives associated with the axis of the cylinder can be either eliminated or replaced by other determinable expressions. Thus, only two-dimensional storage is required for the dependent variables in question. Extensive numerical calculations were carried out for various sets of the porosity, degree of anisotropy, Reynolds number and macroscopic flow direction in a three-dimensional space. The numerical results thus obtained for periodically fully developed flow and temperature fields were integrated over a structural unit to determine the permeability tensor, Forchheimer tensor and directional interfacial heat transfer coefficient, to elucidate the effects of yaw angle on these macroscopic flow and heat transfer characteristics. Upon examining these numerical data, a useful set of explicit expressions has been established for the permeability tensor, Forchheimer tensor and directional interfacial heat transfer coefficient to characterize flow and heat transfer through a bank of cylinders in yaw.

We analyse the low-mode structure of internal tides generated in laboratory experiments and numerical simulations by a two-dimensional ridge in a channel of finite depth. The height of the ridge is approximately half of the channel depth and the regimes considered span sub- to supercritical topography. For small tidal excursions, of the order of 1% of the topographic width, our results agree well with linear theory. For larger tidal excursions, up to 15% of the topographic width, we find that the scaled mode 1 conversion rate decreases by less than 15%, in spite of nonlinear phenomena that break down the familiar wave-beam structure and generate harmonics and inter-harmonics. Modes two and three, however, are more strongly affected. For this topographic configuration, most of the linear baroclinic energy flux is associated with the mode 1 tide, so our experiments reveal that nonlinear behaviour does not significantly affect the barotropic to baroclinic energy conversion in this regime, which is relevant to large-scale ocean ridges. This may not be the case, however, for smaller scale ridges that generate a response dominated by higher modes.

Liquid jetting and fragmentation are important in many industrial and medical applications. Here, we study the jetting from the surface of single liquid droplets undergoing internal volume oscillations. This is accomplished by an explosively expanding and collapsing vapour bubble within the droplet. We observe jets emerging from the droplet surface, which pinch off into finer secondary droplets. The jetting is excited by the spherical Rayleigh–Taylor instability where the radial acceleration is due to the oscillation of an internal bubble. We study this jetting and the effect of fluid viscosity experimentally and numerically. Experiments are carried out by levitating the droplet in an acoustic trap and generating a laser-induced cavitation bubble near the centre of the droplet. On the simulation side, the volume of fluid method (OpenFOAM) solves the compressible Navier–Stokes equations while accounting for surface tension and viscosity. Both two-dimensional (2-D) axisymmetric and 3-D simulations are performed and show good agreement with each other and the experimental observation. While the axisymmetric simulation reveals how the bubble dynamics results destabilizes the interface, only the 3-D simulation computes the geometrically correct slender jets. Overall, experiments and simulations show good agreement for the bubble dynamics, the onset of disturbances and the rapid ejection of filaments after bubble collapse. Additionally, an analytic model for the droplet surface perturbation growth is developed based on the spherical Rayleigh–Taylor instability analysis, which allows us to evaluate the surface stability over a large parameter space. The analytic model predicts correctly the onset of jetting as a function of Reynolds number and normalized internal bubble energy.

The flow near the cavity detachment region of stable attached cavitation was examined using qualitative and quantitative flow visualization. The non-cavitating and cavitating flows around a hydrophilic brass and hydrophobic Teflon sphere and cylinder were examined. The location of non-cavitating boundary layer separation and cavity detachment was related to the free-stream Reynolds and cavitation numbers. The shape of the cavity near the detachment was greatly affected by the material of the cavitating object. The cavity interface on the hydrophilic test objects curved downstream to form a forward facing step. A region of recirculating fluid existed upstream of the cavity interface. The cavity detachment on the hydrophobic test objects was much closer to the location of boundary layer separation. The forward facing step and the recirculating region were nearly absent.

The measured flow field near the surface of the brass sphere, cylinder, and hydrofoils under cavitating and non-cavitating conditions was used to calculate the position of two-dimensional laminar boundary layer separation. Thwaites' and Stratford's methods were used to predict the location of boundary layer separation upstream of the cavity detachment. The predictions compared well with the observed position of separation.

Three-dimensional temporal instabilities, leading to spray formation of a round liquid jet segment with an outer, coaxial high-density gas flow, are studied with Navier–Stokes and level-set computations. These computations predict the liquid surface shape showing the smaller structures on the conical wave crests, i.e. lobes, holes, bridges and ligaments, which are the precursors to droplet and spray formations. These structures and their time scales affect droplet size and velocity distributions as well as spray cone angles. The gas-to-liquid density ratio, liquid Reynolds number ($Re$) and liquid Weber number ($We$) range between 0.02–0.9, 320–16 000 and 2000–230 000, respectively, which cover three distinct physical domains. (1) At higher $Re$ and $We$, ligaments and then drops develop following hole and liquid bridge formations. (2) At higher gas densities throughout the $Re$ range, several holes merge forming two bridges per lobe before breaking to form ligaments; this hole merging is explained by slower development of hairpin vortices and lobe shape. (3) In cases where both gas density and $Re$ or $We$ are lower, the well-ordered lobes are replaced by more irregular, smaller-scale corrugations along the conical wave crest edge; ligaments form differently by stretching from the lobes before holes form. Thicker ligaments and larger droplets form in the low $Re$, low gas density range. The surface wave dynamics, vortex dynamics and their interactions are explained. Understandings of liquid stream break up and concurrent smaller structure formation are built upon an examination of both translation and rotation of the fluid. In all cases, hole formation is correlated with hairpin and helical vortices; fluid motion through a perforation in the thin sheet near the wave crest corresponds to these vortices. The hole formation process is dominated by inertial forces rather than capillary action, which differs from mechanisms suggested previously for other configurations. Circulation due to streamwise vorticity increases while the lobes thin and holes form. For larger surface tension, cavities in the jet core rather than perforations in a sheet occur. The more rapid radial extension of the two-phase mixture with increasing gas density is explained by greater circulation in the ring (i.e. wave crest) region. Experimental descriptions of the smaller structures are available only at lower $Re$ and lower density, agreeing with the computations. Computed scales of bridges, ligaments, early droplets and emerging spray radii agree qualitatively with experimental evidence through the high $Re$ and $We$ domains.

In this study, we experimentally investigate the turbulent drag-reduction (DR) mechanism in flow through ducts of circular, rectangular and square cross-sections using two grades of polyacrylamide in aqueous solution having different molecular weights and various semidilute concentrations. Specifically, we explore the relationship between drag reduction and fluid elasticity, purposely exploiting the mechanical degradation of polymer molecules to vary their rheological properties. We also obtain time-resolved velocity data for various DR levels using particle image velocimetry and laser Doppler velocimetry. Elasticity is quantified via relaxation times determined from uniaxial extensional flow using a capillary breakup apparatus. A plot of DR against Weissenberg number ($Wi$) is found to approximately collapse the data, with the onset of DR occurring at $Wi\approx 0.5$ and the maximum drag-reduction asymptote being approached for $Wi\gtrsim 5$. Thus quantitative predictions of DR in a range of shear flows can be made from a single measurable material property of a polymer solution, at least for this particular flexible linear polymer.

Jets arising from rivers, streams and tidal flows entering still waters differ from most experimental studies of jets both in aspect ratio and in the presence of a solid bottom boundary and an upper free surface. Despite these differences, the applicability of experimental jet studies to these systems remains largely untested by either field or realistically scaled experimental studies. Here we present experimental results for a wall-bounded plane jet scaled to jets created by flow discharging into floodplain lakes. A characteristic feature of both our prototype and experimental jets is the presence of large-scale meandering turbulent structures that span the width of the jets. In our experimental jets, we observe self-similarity in the distribution of mean streamwise velocities by a distance of six channel widths downstream of the jet outlet. After a distance of nine channel widths the velocity decay and the spreading rates largely agree with prior experimental results for plane jets. The magnitudes and distributions of the cross-stream velocity and lateral shear stresses approach self-preserving conditions in the upper half of the flow, but decrease in magnitude, and deviate from self-preserving distributions with proximity to the bed. The presence of the meandering structure has little influence on the mean structure of the jet, but dominates the jet turbulence. A comparison of turbulence analysed at time scales both greater than and less than the period of the meandering structure indicates that these structures increase turbulence intensities by 3–5 times, and produce lateral shear stresses and momentum diffusivities that are one and two orders of magnitude greater, respectively, than turbulence generated by bed friction alone.

Investigation of the linear stability problem for rapidly rotating convection on an f-plane has revealed the existence of two distinct scales in the vertical structure of the critical eigenfunctions: a small length scale whose vertical wavenumber kz is comparable with the large horizontal wavenumber k⊥ selected at onset, and a large-scale modulation which forms an envelope on the order of the layer depth d. The small-scale structure in the vertical results from a geostrophic balance imposed by the Taylor–Proudman constraint. This primary balance forces rotational alignment and confines fluid motions to planes perpendicular to the rotation axis. For convective transport in the vertical this constraint must be relaxed. This is achieved by molecular dissipation which allows weak upward (downward) spiralling of hot (cold) fluid elements across the Taylor–Proudman planes and results in a large-scale vertical modulation of the Taylor columns.

In the limit of fast rotation (i.e. large Taylor number) a multiple-scales analysis leads to the determination of a critical Rayleigh number as a function of wavenumber, roll orientation and the tilt angle of the f-plane. The corresponding critical eigenfunction represents the core solution; matching to passive Ekman boundary layers is required for a complete solution satisfying boundary conditions.

An extension of this analysis, introduced by Bassom & Zhang (1994), is used to describe strongly nonlinear two-dimensional convection, characterized by significant departures of the mean thermal field from its conduction profile. The analysis requires the solution of a nonlinear eigenvalue problem for the Nusselt number (for steady convection) and the Nusselt number and oscillation frequency (for the overstable problem). The solutions of this problem are used to calculate horizontal and vertical heat fluxes, as well as Reynolds stresses, as functions of both the latitude and roll orientation in the horizontal, and these are used to calculate self-consistently north–south and east–west mean flows. These analytical predictions are in good agreement with the results of three-dimensional simulations reported by Hathaway & Somerville (1983).

This paper performs a numerical study on the interaction of a planar shock wave with a water column embedded with/without a cavity of different sizes at high Weber numbers. The conservative-type Euler and non-conservative scalar two-equations representing the transportation of two-phase properties consist of the diffusion interface capture models. The numerical fluxes are computed by the Godunov-type Harten-Lax–van Leer contact Riemann solver coupled with an incremental fifth-order weighted essentially non-oscillatory (WENO) scheme. A third-order total variation diminishing (TVD) Runge–Kutta scheme is used to advance the solution in time. The morphology and dynamical characteristics are analysed qualitatively and quantitatively to demonstrate the breakup mechanism of the water column and formation of transverse jets under different incident shock intensities and embedded-cavity sizes. The jet tip velocities are extracted by analysing the interface evolution. The liquid column is prone to aerodynamic breakup with the formation of micro-mist at later stages instead of liquid evaporation because of the weakly heating effects of the surrounding air. It is numerically confirmed that the liquid-phase pressure will drop below the saturated vapour pressure, and the low pressure can be sustained for a certain time because of the focusing of the expansion wave, which accounts for the cavitation inside the liquid water column. The geometrical parameters of the deformed water column are identified, showing that the centreline width decreases but the transverse height increases nonlinearly with time. The deformation rates are nonlinearly correlated under different Mach numbers. The first transverse jet is found for a water column with an embedded cavity, whereas the water hammer shock and second jet do not occur under the impact of low intensity incident shock waves. The $x$-velocity component recorded at the rear stagnation point can remain unchanged for a comparable time after a declined evolution, which indicates that the downstream wall of the shocked water ring somehow moves uniformly. It can be explained that the acceleration of the downstream wall is balanced by the trailing shedding vortex, and this effect is more evident under higher Mach numbers. The increased enstrophy, mainly generated at the interface, demonstrates the competition of the baroclinic effects of the shock wave impact over dilatation.

A numerical investigation of transonic and low-supersonic flows of dense gases of the Bethe–Zel'dovich–Thompson (BZT) type is presented. BZT gases exhibit, in a range of thermodynamic conditions close to the liquid/vapour coexistence curve, negative values of the fundamental derivative of gasdynamics. This can lead, in the transonic and supersonic regime, to non-classical gasdynamic behaviours, such as rarefaction shock waves, mixed shock/fan waves and shock splitting. In the present work, inviscid and viscous flows of a BZT fluid past an airfoil are investigated using accurate thermo-physical models for gases close to saturation conditions and a third-order centred numerical method. The influence of the upstream kinematic and thermodynamic conditions on the flow patterns and the airfoil aerodynamic performance is analysed, and possible advantages deriving from the use of a non-conventional working fluid are pointed out.