A cavitating-flow calculation method is presented, based on the panel technique with minimization of a certain vector characterizing the discretion error which may become important under cavitating conditions. Several practical examples are presented: partial cavitation on an isolated foil, cavitation behind a blunt-ended body, and the problem of two cavities around an axisymmetrical body. In the case of partial cavitation, the Joukowski condition and tangential outlet condition can be satisfied by the form of the error vector. The cavity-wake modelling problem is not extensively dealt with. It is shown, however, that in order to obtain a satisfactory cavity length/cavitation number ratio, it is probably necessary to introduce a displacement thickness behind the near wake of the cavity which does not close on the body according to a separated flow scheme analagous to the wake, as introduced previously by Yagamuchi & Kato (1983). The method is shown to be capable, after a few minor modifications, of dealing with the case of bodies with a rounded rear edge. Even so, the advantage is essentially didactic as the problem of predicting the position of separation points is not treated. The problem of two cavities around axisymmetrical bodies has a more obvious practical interest. The nonlinear closure condition of each cavity is exactly satisfied by an iterative resolution scheme in which allowance is made for the presence of an axial gravity field.

There is wide interest in the role of the endothelial surface layer (ESL) in transmitting blood shear stress to the intracellular cytoskeleton of the endothelial cell. However, very little is known about the mechanical properties of the glycocalyx or the flexural rigidity of the core proteins that comprise it. Vink, Duling & Spaan (FASEB J., vol. 13, 1999, p. A 11) measured the time-dependent restoration of the ESL after it had been nearly completely compressed by the passage of a white blood cell (WBC) in a tightly fitting capillary. Using this initial experiment, Weinbaum et al. (Proc. Natl. Acad. Sci. USA, vol. 100, 2003, p. 7988) predicted that the core proteins have a flexural rigidity EI of 700 pN nm$^{2}$, which is $\sim$1/20 the measured value for an actin filament. However, their analysis assumes small deflections and only the fibre motion is considered. In the present paper we report additional experiments and apply large-deformation theory for ‘elastica’ to describe the restoration of the fibres in a Brinkman medium which absorbs fluid as the ESL expands. We find that there are two phases in the fibre recoil: an initial phase for large compressions where the ESL thickness is $<0.36$ its undisturbed thickness, and the ends of the fibres overlap and are parallel to the capillary wall; and a second phase where the fibres assume a shape that is close to the solutions for an elastic bar with linearly distributed vertical loading. The predicted time-dependent change in thickness of the ESL provides remarkably good agreement with experiment and yields an estimate of 490 pN nm$^{2}$ for the flexural rigidity EI of the core protein fibres, which is unexpectedly close to that predicted by the linear theory in Weinbaum et al. (2003).

Results are presented of numerical simulations of passive scalar mixing in homogeneous, incompressible turbulent flows. These results are generated via the Linear Eddy Model (LEM) and Direct Numerical Simulation (DNS) of turbulent flows under a variety of different conditions. The nature of mixing and its response to the turbulence field is examined and the single-point probability density function (p.d.f.) of the scalar amplitude and the p.d.f.s of the scalar spatial-derivatives are constructed. It is shown that both Gaussian and exponential scalar p.d.f.s emerge depending on the parameters of the simulations and the initial conditions of the scalar field. Aided by the analyses of data, several reasons are identified for the non-Gaussian behaviour of the scalar amplitude. In particular, two mechanisms are identified for causing exponential p.d.f.s: (i) a non-uniform action of advection on the large and the small scalar scales, (ii) the nonlinear interaction of the scalar and the velocity fluctuations at small scales. In the absence of a constant non-zero mean scalar gradient, the behaviour of the scalar p.d.f. is very sensitive to the initial conditions. In the presence of this gradient, an exponential p.d.f. is not sustained regardless of initial conditions. The numerical results pertaining to the small-scale intermittency (non-Gaussian scalar derivatives) are in accord with laboratory experimental results. The statistics of the scalar derivatives and those of the velocity-scalar fluctuations are also in accord with laboratory measured results.

A model is proposed to describe the breaking and rejoining of vortex lines which occurs when vortex filaments of equal and opposite strengths touch. It is based on the idea that viscosity cancels the vorticity where the filaments first touch. The weakening of the centrifugal force in the vortex core then leads to a local increase in the pressure which accelerates the fluid in the cores in the axial direction and convects the vorticity away from the initial contact and produces an apparent rejoining. Hyperbolic equations to describe the process are formulated and numerical solutions are obtained. Estimates of the timescales are made and compared with the experimental data of Schatzle (1987). Reasonable agreement is found.

The flows induced by the presence of an insulating sloping boundary in a doublediffusive system are examined. In the diffusive case, when the component with the larger diffusivity is unstably distributed, it is known that under certain circumstances horizontal motions are induced near the slope, and that a series of horizontal layers forms. We investigate the formation and properties of the layers, in particular their vertical scale and its dependence on the stratification and the slope angle. The scale of the layers is found to be a strong function of Gρ, the ratio of the vertical density gradient of the unstably distributed component to that of the stably distributed component. At low values of Gρ, no layering was observed; at larger values of Gρ layers were formed, and their scale increased as Gρ → 1. A weak dependence of scale on slope angle was also observed with the scale diminishing as the angle of the slope to the horizontal increased.

A new form of layering has been observed when the basic stratification is in the finger sense. At high enough values of Gρ the basic stratification is unstable to finger motions and these exist throughout the fluid. When a slope is introduced, horizontal motions are set up near the slope which cause the fingers to break down and layers are produced. There is considerable horizontal motion in these layers as well as convective motions driven by the fingers in the interfaces between the layers. The formation of these layers and some of their properties are documented.

Transition to turbulence in a circular Couette system is followed numerically up to the state beyond the second bifurcation in the limit of the narrow gap between coaxial cylinders which rotate with almost equal speeds in the same direction. Taylor-vortex flow, which emerges after the Taylor number T exceeds its critical value Tc = 1708, becomes unstable to two different types of non-axisymmetric disturbances, depending on [Rscr ] (the Reynolds number) and Ω, which measure the velocity difference between the cylinders and the mean angular velocity respectively. Finite-amplitude calculations show that one part of the bifurcating flow is characterized by the vortices winding out of phase in the axial direction but still keeping the boundaries between vortices unaffected. The other is distinguished by the vortex dislocation. The inflow as well as the outflow boundaries between vortices are wavy. Both types of solutions are stationary in the frame of reference rotating with the angular velocity Ω.

The axisymmetric withdrawal of fluid from a linearly stratified container is studied over the full parameter range. When only buoyancy and inertia are important the flow in the withdrawal layer is influenced by a virtual control point and is not analogous to that observed in the two-dimensional withdrawal problem. Two further flow regimes are shown to exist in which viscous forces are important: one in which convection of species is important, and a second in which diffusion of species is important. Theoretical arguments and laboratory experiments are used to show that $S = (Q^2N/\nu^3)^{\frac{1}{15}}$ is the appropriate flow parameter to differentiate between these possibilities. It is also argued that these results may be generalized to describe the features of several related flows: axisymmetric drawdown (or drawup) in withdrawal from a layered density structure, axisymmetric inflow into a linearly stratified environment and the axisymmetric spreading of density currents.

An initial thermal stratification may be swept out of a valley if a cross-wind is strong enough: the valley is then said to be ventilated. Numerical and laboratory experiments indicate that the critical parameter is the Froude number, , where is the mean horizontal velocity above the crest of the valley sides, N is the Brunt–Väisälä or bouyancy frequency of the thermal stratification in the valley air and h is the height of the valley walls. Ventilation occurs whenever the Froude number exceeds a value of 1·3.

Geophysical granular flows, such as landslides, pyroclastic flows and snow avalanches, consist of particles with varying surface roughnesses or shapes that have a tendency to segregate during flow due to size differences. Such segregation leads to the formation of regions with different frictional properties, which in turn can feed back on the bulk flow. This paper introduces a well-posed depth-averaged model for these segregation-mobility feedback effects. The full segregation equation for dense granular flows is integrated through the avalanche thickness by assuming inversely graded layers with large particles above fines, and a Bagnold shear profile. The resulting large particle transport equation is then coupled to depth-averaged equations for conservation of mass and momentum, with the feedback arising through a basal friction law that is composition dependent, implying greater friction where there are more large particles. The new system of equations includes viscous terms in the momentum balance, which are derived from the $\unicode[STIX]{x1D707}(I)$-rheology for dense granular flows and represent a singular perturbation to previous models. Linear stability calculations of the steady uniform base state demonstrate the significance of these higher-order terms, which ensure that, unlike the inviscid equations, the growth rates remain bounded everywhere. The new system is therefore mathematically well posed. Two-dimensional simulations of bidisperse material propagating down an inclined plane show the development of an unstable large-rich flow front, which subsequently breaks into a series of finger-like structures, each bounded by coarse-grained lateral levees. The key properties of the fingers are independent of the grid resolution and are controlled by the physical viscosity. This process of segregation-induced finger formation is observed in laboratory experiments, and numerical computations are in qualitative agreement.

Flow past a circular cylinder executing sinusoidal rotary oscillations about its own axis is studied experimentally. The experiments are carried out at a Reynolds number of 185, oscillation amplitudes varying from $\mathrm{\pi} / 8$ to $\mathrm{\pi} $, and at non-dimensional forcing frequencies (ratio of the cylinder oscillation frequency to the vortex-shedding frequency from a stationary cylinder) varying from 0 to 5. The diagnostic is performed by extensive flow visualization using the hydrogen bubble technique, hot-wire anemometry and particle-image velocimetry. The wake structures are related to the velocity spectra at various forcing parameters and downstream distances. It is found that the phenomenon of lock-on occurs in a forcing frequency range which depends not only on the amplitude of oscillation but also the downstream location from the cylinder. The experimentally measured lock-on diagram in the forcing amplitude and frequency plane at various downstream locations ranging from 2 to 23 diameters is presented. The far-field wake decouples, after the lock-on at higher forcing frequencies and behaves more like a regular Bénard–von Kármán vortex street from a stationary cylinder with vortex-shedding frequency mostly lower than that from a stationary cylinder. The dependence of circulation values of the shed vortices on the forcing frequency reveals a decay character independent of forcing amplitude beyond forcing frequency of ${\sim }1. 0$ and a scaling behaviour with forcing amplitude at forcing frequencies ${\leq }1. 0$. The flow visualizations reveal that the far-field wake becomes two-dimensional (planar) near the forcing frequencies where the circulation of the shed vortices becomes maximum and strong three-dimensional flow is generated as mode shape changes in certain forcing parameter conditions. It is also found from flow visualizations that even at higher Reynolds number of 400, forcing the cylinder at forcing amplitudes of $\mathrm{\pi} / 4$ and $\mathrm{\pi} / 2$ can make the flow field two-dimensional at forcing frequencies greater than ${\sim }2. 5$.

The formation of inhomogeneities within fluidized beds, both in terms of the particle configurations and flow structures, have a pronounced effect on the interaction force between the fluid and particles. While recent numerical studies have begun to probe the effects of inhomogeneities on the drag force at the particle scale, the applicability of prior microscale constitutive drag relations is still limited to random, homogeneous distributions of particles. Since an accurate model for the drag force is needed to predict the fluidization behaviour, the current study utilizes the lessons of prior inhomogeneity studies in order to derive a robust drag relation that is both able to account for the effect of inhomogeneities and applicable as a constitutive closure to larger-scale fluidization simulations. Using fully resolved lattice Boltzmann simulations of systems composed of fluid and monodisperse spherical particles in the low-Reynolds-number (Re) regime, the fluid–particle drag force, normalized by the ideal Stokes drag force, is found to significantly decrease, over a range of length scales, as the extent of inhomogeneities increases. The extent of inhomogeneities is found to most effectively be quantified through one of two subgrid-scale quantities: the scalar variance of the particle volume fraction or the drift flux, which is the correlation between the particle volume fraction and slip velocity. Scale-similar models are developed to estimate these two subgrid measures over a wide range of system properties. Two new drag constitutive models are proposed that are not only functions of the particle volume fraction and the Stokes number ($St$), but also dependent on one of these subgrid measures for the extent of inhomogeneities. Based on the observed, appreciable effect of inhomogeneities on drag, these new low-Re drag models represent a significant advancement over prior constitutive relations.

Interactions between laminar thermal natural convection plumes generated by line and concentrated heat sources were experimentally investigated with a Mach-Zehnder interferometer. Adjacent plane plumes were found to interact more strongly than axisymmetric plumes a t the same spacing. These flows with nominally free boundaries were also found to be affected by nearby surfaces which interfered with the supply of entrainment fluid. Several types of interference with plume entrainment were investigated. The nature of the interacting flows suggests a model which is successful in interpreting the mechanism. An application of the effects of plume interaction is discussed.

Recently there has been remarkable progress made in the direct numerical simulation (DNS) of wall-bounded turbulence, particularly of turbulent channel flow, with numerical data now available above Reτ ≈ 2000 (Hoyas & Jiménez, Phys. Fluids, vol. 18, 2006, p. 011702; Iwamoto et al., Proceedings of the Sixth Symposium Smart Control of Turbulence, 2005). Much knowledge has been gained from these results, particularly in the areas of flow structure and dynamics. Yet, while the value of such simulations is undoubted, only very limited comparisons with experimental data have been documented. Although the physics of the flow are captured correctly in an ideal DNS, as with any real numerical or physical experiment, there are opportunities for misrepresentation of the characteristics of turbulence. As such, this article seeks to make a comparison between a well-documented high Reynolds number (Reτ = 934), large box size (8πh × 2h × 3πh) DNS from del Álamo et al. (J. Fluid Mech., vol. 500, 2004, p. 135) and laboratory channel flow data measured by the authors. Results show that there is excellent agreement between the streamwise velocity statistics of the two data sets. The spectra are also very similar, however, throughout the logarithmic region the secondary peak in energy is clearly reduced in the DNS results. Although the source of the difference is not certain, the wavelengths concerned are close to the DNS box length, leading to the recommendation that longer box lengths should be investigated. Another large-scale spectral discrepancy near the wall results from the incorrect assumption of a constant convection velocity used to infer spatial information from the temporal. A near-wall convection velocity modification function is tentatively proposed. While the modification gives good agreement between the data sets, higher Reynolds number comparisons are required to better understand the intricate convection velocity issue.

The linear stability of the Stokes layers generated between a pair of synchronously oscillating parallel plates is investigated. The disturbance equations were studied using Floquet theory and pseudospectral numerical methods used to solve the resulting system. Neutral curves for an extensive range of plate separations were obtained and when the plate separation is large compared to the Stokes layer thickness the linear stability properties of the Stokes layer in a semi-infinite fluid were recovered. A detailed analysis of the damping rates of disturbances to the basic flow provides a plausible explanation of why several previous studies of the problem have failed to detect any linear instability of the flow.

To compare more faithfully with experimental work the techniques used for the channel problem were modified to allow the determination of neutral curves for axisymmetric disturbances to purely oscillatory flow in a circular pipe. Critical Reynolds numbers for the pipe flow tended to be smaller than their counterparts for the channel case but the smallest critical value was still almost twice the experimentally reported result.

Turbulence facilitates collisions between particles suspended in a turbulent flow. Two effects have been proposed that can enhance the collision rate at high turbulence intensities: ‘preferential concentration’ (a clustering phenomenon) and the ‘sling effect’ (arising from the formation of caustic folds in the phase space of the suspended particles). We have determined numerically the collision rate of small heavy particles as a function of their size and densities. The dependence on particle densities allows us to quantify the contribution of the sling effect to the collision rate. Our results demonstrate that the sling effect provides the dominant mechanism to the enhancement of the collision rate of particles, when inertia becomes significant.

In addition to being observable in laboratory experiments, internal wave beams are reported in geophysical settings, which are characterized by non-uniform density stratifications. Here, we perform a combined theoretical and experimental study of the propagation of internal wave beams in non-uniform density stratifications. Transmission and reflection coefficients, which can differ greatly for different physical quantities, are determined for sharp density-gradient interfaces and finite-width transition regions, accounting for viscous dissipation. Thereafter, we consider even more complex stratifications to model geophysical scenarios. We show that wave beam ducting can occur under conditions that do not necessitate evanescent layers, obtaining close agreement between theory and quantitative laboratory experiments. The results are also used to explain recent field observations of a vanishing wave beam at the Keana Ridge, Hawaii.

Flat-plate boundary layer transition induced by the wake vortex of a two-dimensional circular cylinder is experimentally investigated. Combined visualization and velocity measurements show a different transition route from the Klebanoff mode in free-stream turbulence-induced transition. This transition scenario is mainly characterized as: (i) generation of secondary transverse vortical structures near the flat plate surface in response to the von Kármán vortex street of the cylinder; (ii) formation of hairpin vortices due to the secondary instability of secondary vortical structures; (iii) growth of hairpins which is accelerated by wake-vortex induction; (iv) formation of hairpin packets and the associated streaky structures. Detailed investigation shows that during transition the evolution dynamics and self-sustaining mechanisms of hairpins, hairpin packets and streaks are consistent with those in a turbulent boundary layer. The wake vortex mainly plays the role of generating and destabilizing secondary transverse vortices. After that, the internal mechanisms become dominant and lead to the setting up of a self-sustained turbulent boundary layer.

Results are presented of an experimental study on the transition to geostrophic turbulence, and the detailed behaviour within the turbulence regime, in a rotating, laterally heated annulus of fluid. Both spatial and temporal characteristics are examined, and the results are presented in the form of wavenumber and frequency spectra as a function of a single external parameter, the rotation rate.

The transition to turbulence proceeds in a sequence of steps from azimuthally symmetric (no waves present) to chaotic flow. The sequence includes doubly periodic flow (amplitude vacillation), semiperiodic flow (structural vacillation), and a transition zone where the characteristics undergo a gradual change to chaotic behaviour. The spectra in the transition zone are characterized by a gradual merging of the background signal with the spectral peaks defining regular wave flow as the rotation rate is increased.

Within the geostrophic turbulence regime, the wavenumber spectra are characterized by a broad peak at the baroclinic scale and a power dependence of energy density on wavenumber at the high-wavenumber end of the spectrum. Our data reveal a significant dependence of the slope on the thermal Rossby number, ranging from −4.8 at RoT = 0.17 to −2.4 at RoT = 0.02. The frequency spectra also show a power dependence of the energy density on frequency at the high-frequency end of the Spectrum. We find a nominal −4 power which does not appear to be sensitive to changes in Rossby or Taylor number.

The shape and stability of liquid menisci attached to a solid edge with dihedral angle 2α is investigated. It is shown that in addition to the family of cylindrical menisci a family of azimuthally modified unduloids exists. A double Fourier series of the latter with respect to their axis (parallel to the extension of the edge) and with respect to the azimuth is derived. The dispersion relation between the axial wavenumber q, the azimuthal wavenumber s and the waviness parameter d is calculated. When the condition of constant contact angle γ along the contact lines with the solid is applied, a one-dimensional family of modified unduloids fitting to the edge is obtained. Their axial wavenumber q becomes independent of the waviness d at the bifurcation with the family of cylindrical menisci, such that this bifurcation limits the stability. The respective stability criteria are derived and evaluated. For α + γ > ½π the cylindrical menisci are convex. They reveal a maximum stable length, which quadratically tends to infinity when α + γ = ½π is approached. The smallest stable extension arises for the free cylindrical column (the Rayleigh jet), which is covered by the present investigations by assuming α = π, γ = ½π. For α + γ < ½π the cylindrical menisci are concave and stable: no bifurcation with the family of modified unduloids arises.

Experimental observations and linear stability analysis are used to quantitatively describe a purely elastic flow instability in the inertialess motion of a viscoelastic fluid confined between a rotating cone and a stationary circular disk. Beyond a critical value of the dimensionless rotation rate, or Deborah number, the spatially homogeneous azimuthal base flow that is stable in the limit of small Reynolds numbers and small cone angles becomes unstable with respect to non-axisymmetric disturbances in the form of spiral vortices that extend throughout the fluid sample. Digital video-imaging measurements of the spatial and temporal dynamics of the instability in a highly elastic, constant-viscosity fluid show that the resulting secondary flow is composed of logarithmically spaced spiral roll cells that extend across the disk in the self-similar form of a Bernoulli Spiral.

Linear stability analyses are reported for the quasi-linear Oldroyd-B constitutive equation and the nonlinear dumbbell model proposed by Chilcott & Rallison. Introduction of a radial coordinate transformation yields an accurate description of the logarithmic spiral instabilities observed experimentally, and substitution into the linearized disturbance equations leads to a separable eigenvalue problem. Experiments and calculations for two different elastic fluids and for a range of cone angles and Deborah numbers are presented to systematically explore the effects of geometric and rheological variations on the spiral instability. Excellent quantitative agreement is obtained between the predicted and measured wavenumber, wave speed and spiral mode of the elastic instability. The Oldroyd-B model correctly predicts the non-axisymmetric form of the spiral instability; however, incorporation of a shear-rate-dependent first normal stress difference via the nonlinear Chilcott–Rallison model is shown to be essential in describing the variation of the stability boundaries with increasing shear rate.