Supercavitating hydrofoils of large aspect ratio operating near a free surface are investigated, assuming an inviscid and irrotational flow with the effects of gravity and surface tension neglected. The flow near the foil, treated as two-dimensional, is solved by a nonlinear free-streamline theory, then a three-dimensional ‘downwash’ correction is made using Prandtl's lifting-line theory. The strength of the lifting-line vortex is determined by information from the two-dimensional solution through a matching procedure, in which the inverse of aspect ratio is used as a small parameter for asymptotic expansions. The analysis incorporates a free-surface reference level to determine the submergence depth of the foil. The present method can be applied to any type of foil having an arbitrary planform or profile shape, including a rounded leading edge, a twist and even a small dihedral angle, within the assumption of large aspect ratio. Numerical computations made on rectangular flat-plate hydrofoils show excellent agreement of results with existing experimental data, even for large angles of attack and relatively low aspect ratios. The pressure distributions, shapes of the cavity and free surface are also calculated as a function of spanwise position.

A new damping mechanism for vertically-sheared inertial motions is described involving an inertia–gravity wave that oscillates at half the inertial frequency, $f$, and that grows at the expense of inertial shear. This parametric subharmonic instability forms in baroclinic, geostrophic currents where thermal wind shear, by reducing the potential vorticity of the fluid, allows inertia–gravity waves with frequencies less than $f$. A stability analysis and numerical simulations are used to study the instability criterion, energetics, and finite-amplitude behaviour of the instability. For a flow with uniform shear and stratification, parametric subharmonic instability develops when the Richardson number of the geostrophic current nears $Ri_{PSI}=4/3+\gamma \cos \phi $, where $\gamma $ is the ratio of the inertial to thermal wind shear magnitude and $\phi $ is the angle between the inertial and thermal wind shears at the initial time. Inertial shear enters the instability criterion because it can also modify the potential vorticity and hence the minimum frequency of inertia–gravity waves. When this criterion is met, inertia–gravity waves with a frequency $f/2$ and with flow parallel to isopycnals amplify, extracting kinetic energy from the inertial shear through shear production. The solutions of the numerical simulations are consistent with these predictions and additionally show that finite-amplitude parametric subharmonic instability both damps inertial shear and is itself damped by secondary shear instabilities. In this way, parametric subharmonic instability opens a pathway to turbulence where kinetic energy in inertial shear is transferred to small scales and dissipated.

This paper analyses the sound radiated from the front face of a hard-walled circular cylindrical duct in a subsonic mean flow when the duct contains acoustic sources typical of those in a ducted-fan aeroengine. Two main results are established for modes of any given frequency and circumferential order. The first result is that in certain easily calculated directions, called here the nil-shielding directions, the sound radiated by ducted sources is the same as the sound radiated by the corresponding open sources, i.e. by unducted sources of the same distribution and strength radiating into free space. Thus in these special directions the duct has no noise-shielding effect. The second result is that, in the Kirchhoff approximation, the sound radiated by the open sources in the nil-shielding directions determines the sound radiated by the ducted sources in all directions; i.e. the sound fields radiated by open and ducted sources are related by an open-to-ducted transfer function. This function is such that the sound radiated by the ducted sources is a linear combination of certain diffraction functions, in which the coefficients are given by the sound radiated by the open sources in the nil-shielding directions. The diffraction functions do not depend on the sources and are here calculated explicitly in terms of Bessel functions. The method used in the paper is Kirchhoff's approximation; within linear theory this gives the nil-shielding directions exactly, i.e. in agreement with the Wiener—Hopf solution, and gives the main beam of the radiated field, including the major side-lobes, to good accuracy. The results are relevant to the sound radiated into the forward arc by a ducted turbofan aeroengine.

Here, we examine the influence of small-scale turbulence on the evolution of fronts in the ocean and atmosphere. Specifically, we consider the evolution of an initially balanced density front subject to an imposed viscosity and diffusivity as a simple analogue for small-scale turbulence. At late times, the dominant balance is found to be the quasisteady turbulent thermal wind balance with time evolution due to an advection–diffusion balance in the buoyancy equation. We use the leading-order balance to determine analytical similarity solutions for the spreading of a front and find that the spreading rate is maximum for an intermediate value of the Ekman number, with the spreading resulting from shear dispersion associated with the cross-front flow and vertical diffusion of density. In response to shear dispersion, the front evolves towards a density profile that is nearly linear in the cross-front coordinate. At the edges of the frontal zone, the density field develops large curvature, and these regions are associated with narrow bands of intense vertical velocity.

We propose a model for the acceleration of micro-bubbles (smaller than the dissipative scale of the flow) subjected to the drag and fluid inertia forces in a homogeneous and isotropic turbulent flow. This model, that depends on the Stokes number, Reynolds number and the density ratio, reproduces the evolution of the acceleration variance as well as the relative importance and alignment of the two forces as observed from direct numerical simulations (DNS). We also report that the bubble acceleration statistics conditioned on the local kinetic energy dissipation rate are invariant with the Stokes number and the dissipation rate. Based on this observation, we propose a stochastic model for the instantaneous bubble acceleration vector accounting for the small-scale intermittency of the turbulent flows. The norm of the bubble acceleration is obtained by modelling the dissipation rate along the bubble trajectory from a log-normal stochastic process, whereas its orientation is given by two coupled random walks on a unit sphere in order to model the evolution of the joint orientation of the drag and inertia forces acting on the bubble. Furthermore, the proposed stochastic model for the bubble acceleration is used in the context of large eddy simulations (LES) of turbulent flows laden with small bubbles. To account for the turbulent motion at scales smaller than the mesh resolution, we decompose the instantaneous bubble acceleration in its resolved and residual parts. The first part is given by the drag and fluid inertia forces computed from the resolved velocity field, and the second term refers to the random contribution of small unresolved turbulent scales and is estimated with the stochastic model proposed in the paper. Comparisons with DNS and standard LES, show that the proposed model improves significantly the statistics of the bubbly phase.

On the basis of the Saint-Venant equations for flowing granular matter, we study the various travelling waveforms that are encountered in chute flow for growing Froude number. Generally, for $Fr<2/3$ one finds either a uniform flow of constant thickness or a monoclinal flood wave, i.e. a shock structure monotonically connecting a thick region upstream to a shallower region downstream. For $Fr>2/3$ both the uniform flow and the monoclinal wave cease to be stable; the flow now organizes itself in the form of a train of roll waves. From the governing Saint-Venant equations we derive a dynamical system that elucidates the transition from monoclinal waves to roll waves. It is found that this transition involves several intermediate stages, including an undular bore that had hitherto not been reported for granular flows.

Three-dimensional transitional flow of a rarefied monatomic gas over a spinning sphere is studied numerically by the direct simulation Monte Carlo method. Gas molecules interact with each other as hard spheres. The Maxwell model of specular–diffuse scattering is used to describe the interaction between gas molecules and the sphere surface. The effect of all dimensionless governing parameters of the problem on the flow structure, distributions of stresses and heat flux density on the sphere surface, aerodynamic force and torque exerted on the sphere, and heat flux on the sphere surface is analysed. Simulations are conducted at Mach numbers raging from to and Knudsen numbers raging from to . Two qualitatively different streamline patterns are observed around the sphere depending on its dimensionless rotational velocity. The direction and magnitude of the transverse Magnus force in transitional flow depend on the Knudsen and Mach numbers. The torque and heat flux coefficients are found to be functions of the Mach number and dimensionless rotational velocity. The effect of rotation on the sphere aerodynamics weakens with an increase in the sphere temperature with respect to the gas temperature in the free stream. A complete set of equations is developed to fit the computed values of the aerodynamic and heat flux coefficients in a case when the sphere temperature is equal to the temperature of the free stream.

The electrical streaming potential generated by a two-phase pressure-driven Stokes flow in a cylindrical capillary is computed numerically. The potential difference ΔΦ between the two ends of the capillary, proportional to the pressure difference Δp for single-phase flow, is modified by the presence of a suspended drop on the centreline of the capillary. We determine the change in ΔΦ caused by the presence of an uncharged insulating neutrally buoyant drop at a small electric Hartmann number, i.e. when the perturbation to the flow field caused by electric stresses is negligible.

The drop velocity and deformation, and the consequent changes in the pressure difference Δp and streaming potential ΔΦ, depend upon three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary; the viscosity ratio λ between the drop phase and the continuous phase; and the capillary number Ca which measures the ratio of viscous to capillary forces. We investigate how the streaming potential depends on these parameters: purely hydrodynamic aspects of the problem are discussed by Lac & Sherwood (J. Fluid Mech., doi:10.1017/S0022112009991212).

The potential on the capillary wall is assumed sufficiently small so that the electrical double layer is described by the linearized Poisson–Boltzmann equation. The Debye length characterizing the thickness of the charge cloud is taken to be small compared with all other length scales, including the width of the gap between the drop and the capillary wall. The electric potential satisfies Laplace's equation, which we solve by means of a boundary integral method. The presence of the drop increases |ΔΦ| when the drop is more viscous than the surrounding fluid (λ > 1), though the change in |ΔΦ| can take either sign for λ < 1. However, the difference between ΔΦ and Δp (suitably non-dimensionalized) is always positive. Asymptotic predictions for the streaming potential in the case of a vanishingly small spherical droplet, and for large drops at high capillary numbers, agree well with computations.

This paper presents a numerical performance evaluation of the leading edge tubercles hydrofoil with particular focus on a fully turbulent flow regime. Efforts were focused on the setting up of an appropriate numerical approach required for an in-depth analysis of this phenomenon, being able to predict the main flow features and the hydrodynamic performance of the foil when operating at high Reynolds numbers. The numerical analysis was conducted using an improved delayed detached eddy simulation for Reynolds numbers corresponding to the transitional and fully turbulent flow regimes at different angles of attack for the pre-stall and post-stall regimes. The results show that tubercles operating in turbulent flow improve the hydrodynamic performance of the foil when compared to a transitional flow regime. Flow separation was identified behind the tubercle troughs, but was significantly reduced when operating in a turbulent regime and for which we have identified the main flow mechanisms. This finding confirms that the tubercle effect identified in a transitional regime is not lost in a turbulent flow. Furthermore, when the hydrofoil operates in the turbulent flow regime, the transition to a turbulent regime takes place further upstream. This phenomenon suppresses a formation of a laminar separation bubble and therefore the hydrofoil exhibits a superior hydrodynamic performance when compared to the same foil in the transitional regime.

It has recently been suggested that the magneto-rotational instability (MRI) is a limiting case of the magneto-elliptic instability (MEI). This limit is obtained for horizontal modes in the presence of rotation and an external vertical magnetic field, when the aspect ratio of the elliptic streamlines tends to infinite. In this paper we unveil the link between these previously unconnected mechanisms, explaining both the MEI and the MRI as different manifestations of the same magneto-elliptic-rotational instability (MERI). The growth rates are found and the influence of the magnetic and rotational effects is explained, in particular the effect of the magnetic field on the range of negative Rossby numbers at which the horizontal instability is excited. Furthermore, we show how the horizontal rotational MEI in the rotating shear flow limit is linked to the MRI by the use of the local shearing box model, typically used in the study of accretion discs. In such a limit the growth rates of the two instability types coincide for any power-law-type background angular velocity radial profile with negative exponent corresponding to the value of the Rossby number of the rotating shear flow. The MRI requirement for instability is that the background angular velocity profile is a decreasing function of the distance from the centre of the disc, which corresponds to the horizontal rotational MEI requirement of negative Rossby numbers. Finally a physical interpretation of the horizontal instability, based on a balance between the strain, the Lorentz force and the Coriolis force, is given.

When the buoyancy forces are small compared with the inertia forces, heated plumes in laminar flows which are uniform at upstream infinity approximately satisfy a linearized version of the Boussinesq equations, here called the Oseen–Boussinesq equations. An analytic solution is constructed for arbitrary Prandtl number and arbitrary direction of the unperturbed flow in the case of a plume produced by a point source. The two-dimensional case of the plume from a line source is considered briefly. A Stokes-type paradox occurs: it is found that a line-source solution that vanishes at infinity does not exist.