ALBERT

Description: The dynamics of gravity currents are believed to be strongly influenced by dissipation due to turbulence and mixing between the current and the surrounding ambient fluid. This paper describes new theory and experiments on gravity currents produced by lock exchange which suggest that dissipation is unimportant when the Reynolds number is sufficiently high. Although there is mixing, the amount of energy dissipated is small, reducing the current speed by a few percent from the energy-conserving value. Benjamin (J. Fluid Mech. vol. 31, 1968, p. 209) suggests that dissipation is an essential ingredient in gravity current dynamics. We show that dissipation is not important at high Reynolds number, and provide an alternative theory that predicts the current speed and depth based on energy-conserving flow that is in good agreement with experiments. We predict that in a deep ambient the front Fronde number is 1, rather than the previously accepted value of √ 2. New experiments are reported for this case that support the new theoretical value. © 2004 Cambridge University Press.

Description: Experimental results on sediment erosion (scour) by a plane turbulent wall jet, issuing from a sluice gate, are presented which show clearly - it seems for the first time - that the turbulent wall layer is destabilized by the concave curvature of the water/sediment interface. The streamwise Görtler vortices which emerge create sediment streaks or longitudinal sediment ridges. The analysis of the results in terms of Görtler instability of the wall layer indicates that the strength of these curvature-excited streamwise vortices is such that the sediment transport is primarily due to turbulence created by these vortices. Their contribution to the wall shear stress is taken to be of the same form as the normal turbulent wall shear stress. For this reason, the model developed by Hogg et al. (J. Fluid Mech. Vol. 338, 1997, p. 317) remains valid; only the numerical coefficients are affected. The logarithmic dependency of the time evolution of the scour-hole depth predicted by this model is shown to be in good agreement with experiments. New scaling laws for the quasi-steady state depth and the associated time, inspired by the Hogg et al. (1997) model are proposed. Furthermore, it is emphasized that at least two scouring regimes must be distinguished: a short-time regime after which a quasi-steady state is reached, followed by a long-time regime, leading to an asymptotic state of virtually no sediment transport. © 2004 Cambridge University Press.

Description: The effect of gravity modulation on the nonlinear evolution of long-wavelength disturbances at the free surface of a surfactant-covered thin liquid layer is studied. The surfactants, which are assumed to be insoluble, give rise to interfacial concentration gradients and associated Marangoni flow in the underlying liquid film. A coupled system of lubrication-theory-based evolution equations for the film height and surfactant concentration is solved numerically using spectral methods. Previous work using Floquet theory had determined that small-amplitude long-wavelength disturbances are destabilized by gravity modulation in the presence of surfactant; uncontaminated films were found to be linearly stable. Our numerical results indicate that uncontaminated free surfaces are destabilized by nonlinearities and exhibit a harmonic response. The interface exhibits complex dynamics during a forcing cycle, characterized by numerous coalescence events between thickened fluid ridges leading to coarsening. The presence of surfactant-induced Marangoni flow gives rise to a harmonic response, larger scale fluid structures of reduced amplitude, less frequent coalescence events, and less complicated film dynamics. © 2004 Cambridge University Press.

Description: We present a model describing the evolution of the small-scale Navier-Stokes turbulence due to its stochastic distortion by much larger turbulent scales. This study is motivated by numerical findings (Laval et al. Phys. Fluids vol. 13, 2001, p. 1995) that such interactions of separated scales play an important role in turbulence intermittency. We introduce a description of turbulence in terms of the moments of k-space quantities using a method previously developed for the kinematic dynamo problem (Nazarenko et al. Phys. Rev. E vol. 68, 2003, 0266311). Working with the k-space moments allows us to introduce new useful measures of intermittency such as the mean polarization and the spectral flatness. Our study of the small-scale two-dimensional turbulence shows that the Fourier moments take their Gaussian values in the energy cascade range whereas the enstrophy cascade is intermittent. In three dimensions, we show that the statistics of turbulence wavepackets deviates from Gaussianity toward dominance of the plane polarizations. Such turbulence is formed by ellipsoids in the k-space centred at its origin and having one large, one neutral and one small axis with the velocity field pointing parallel to the smallest axis. © 2004 Cambridge University Press.

Description: Numerical solutions of stationary flow resulting from immersion of a single body in simple shear flow are reported for a range of Reynolds numbers. Flows are computed using finite-element methods. Comparisons to results of asymptotic low-Reynolds-number theory, experimental study, and other numerical techniques are provided. Results are presented primarily for isotropic bodies, i.e. the circular cylinder and sphere, for both of which the two conditions of a torque-free (freely-rotating) and fixed body are investigated. Conditions studied for the sphere are 0 〈 Re ≤ 100, and for the circular cylinder 0 〈 Re ≤ 500, with the shear-flow Reynolds number defined as Re = ∞dot;ca2 /ν; ∞dot;c is the shear rate of the Cartesian simple shear flow u = (∞dot;cy, 0, 0), a is the cylinder or sphere radius, and ν is the kinematic viscosity of the fluid. In the torque-free case, the rotation rate of the body decreases with increasing Re. Qualitative dependence, seen in the Re = 0 fluid flow field, upon whether the body is fixed against rotation or torque-free vanishes as Re increases and the fluid flow is more similar to that around the Re = 0 fixed body: the influence of rotation of the body and the associated closed streamlines are confined to a narrow layer about the body for Re 〉 0(1). Separation of the boundary layer is observed in the case of a fixed cylinder at Re ≈ 85, and for a fixed sphere at Re ≈ 100; similar separation phenomena are observed for a freely rotating cylinder. The surface stress and its symmetric first moment (the stresslet) are presented, with the latter providing information on the particle contribution to the mixture rheology at finite Re. Stationary flow results are also presented for elliptical cylinders and oblate spheroids, with observation of zero-torque inclinations relative to the flow direction which depend upon the aspect ratio, confirming and extending prior findings. © 2004 Cambridge University Press.

Description: We investigate the possibility of using proper orthogonal decomposition (POD) in reconstructing complete flow fields from gappy data. The incomplete fields are created from DNS snapshots of flow past a circular cylinder by randomly ommiting data points. We first examine the effectiveness of an existing method and subsequently introduce modifications that make the method robust and lead to the maximum possible resolution at a certain level of spatio-temporal gappiness. We simulate three levels of gappiness at approximately 20%, 50% and 80% in order to investigate the limits of applicability of the new procedure. We find that for the two lower levels of gappiness both the temporal and spatial POD modes can be recovered accurately leading to a very accurate representation of the velocity field. The resulting resolution is improved by more than five times compared to the existing method. However, for 80% gappiness only a few temporal modes are captured accurately while the corresponding spatial modes are noisy. We explain this breakdown of the method in terms of a simple perturbation analysis. This new methodology can be a building block in an effort to develop effective data assimilation techniques in fluid mechanics applications. © 2004 Cambridge University Press.

Description: We investigate the slow spreading of fluid mud over a gently sloped conical surface which may simulate a shallow basin or a hill. The mud is assumed to behave as a Bingham plastic possessing a finite yield stress and the lubrication approximation is used. Because of the finite yield stress, a variety of non-trivial equilibrium profiles can exist, corresponding to the state of deposit at the end of upward or downward motion. Analytical solutions are derived for axially symmetric deposits. It is shown that the front of the final profile in axisymmetric spreading is of a common form in dimensionless variables, independent of the total mud volume. Transient evolutions are then studied numerically by employing a finite-volume scheme for both axially symmetric and asymmetric spreading from a localized source. The characteristic features of the mud pile at different stages of spreading are examined. The final shape in asymmetric spreading is strongly affected by the total volume released and by the rate of discharge. © 2004 Cambridge University Press.

Description: Experiments are reported on the dynamics of a bed of particles sheared by a viscous Couette flow in an annular channel, with emphasis on the distributions of particle velocities, durations and lengths of the small saltation flights, and surface density of the moving particles. The velocity distributions are shown to decay approximately exponentially, with mean value, Up, equal to 0.1 γd, where γ is the shear rate and d is the particle diameter. The duration of the flights does not depend on the shear rate, and is equal to 15 times the settling time d/Vs, where Vs is the Stokes settling velocity. Starting from an initially loosely packed bed, the surface density of the moving particles, Np, was observed to decrease slowly over several days, unlike their velocity which remains constant with time. This decay is related to the increase of the threshold shear rate for particle motion, and corresponds to rearrangement of the particles near the bed surface (armouring). When the stationary state is reached, Np depends linearly on the shear rate, so that the particle flow rate, Qp = N p Up, is a quadratic function of the shear rate. Two theoretical models are proposed to account for these observations. In the first one, the erosion and deposition rates are modelled using the two hydrodynamic time scales: the inverse shear rate γ-1 for the erosion rate, and the settling time d/V s for the deposition rate. This model accounts for the linear dependence of Np on the shear rate. The second model was developed to capture the slow decrease of Np, by considering the trapping of moving particles into troughs of the bed. This trapping model does recover the main features observed experimentally, although the characteristic time for the decrease of N p still remains too short. Our observations are, finally, compared to existing numerical and experimental studies on turbulent flows. © 2004 Cambridge University Press.

Description: We consider miscible displacement between parallel plates in the absence of diffusion, with a concentration-dependent viscosity. By selecting a piecewise viscosity function, this can also be considered as 'three-fluid' flow in the same geometry. Assuming symmetry across the gap and based on the lubrication ('equilibrium') approximation, a description in terms of two quasi-linear hyperbolic equations is obtained. We find that the system is hyperbolic and can be solved analytically, when the mobility profile is monotonic, or when the mobility of the middle phase is smaller than its neighbours. When the mobility of the middle phase is larger, a change of type is displayed, an elliptic region developing in the composition space. Numerical solutions of Riemann problems of the hyperbolic system spanning the elliptic region, with small diffusion added, show good agreement with the analytical outside, but an unstable behaviour inside the elliptic region. In these problems, the elliptic region arises precisely at the displacement front. Crossing the elliptic region requires the solution of essentially an eigenvalue problem of the full higher-dimensional model, obtained here using lattice BGK simulations. The hyperbolic-to-elliptic change-of-type reflects the failing of the lubrication approximation, underlying the quasi-linear hyperbolic formalism, to describe the problem uniformly. The obtained solution is analogous to non-classical shocks recently suggested in problems with change of type. © 2004 Cambridge University Press.

Description: Rarefied gas flow through a thin orifice is studied on the basis of the direct simulation Monte Carlo method. The mass flow rate and the flow field are calculated over the whole range of the Knudsen number for various values of the pressure ratio. It is found that at all values of the pressure ratio a significant variation of the flow rate occurs in the transition regime between the free-molecular and hydrodynamic regimes. In the hydrodynamic regime the flow rate tends to a constant value. In the case of finite pressure ratio the flow field qualitatively differs from that for outflow into vacuum, namely vortices appear in the downflow container on approaching the hydrodynamic regime. Then, in the hydrodynamic regime the gas flow forms a strong jet. A comparison of the numerical results with experimental data available in the open literature has been performed. © 2004 Cambridge University Press.

Description: Thermal convection in rapidly rotating, self-gravitating Boussinesq fluid spheres is characterized by three parameters: the Rayleigh number R, the Prandtl number Pr and the Ekman number E. Two different asymptotic limits were considered in the previous studies of the linear problem. In the double limit E ≪ 1 and Pr/E ≫ 1, the local asymptotic theory showed that the convective motion is strongly non-axisymmetric, columnar, highly localized and described by the asymptotic scalings, (1/s)∂/∂Φ = O (E-1/3), ∂/∂z = 0(1), Rc = 0(E-1/3), where R c denotes the critical Rayleigh number and (s, Φ, z are cylindrical polar coordinates with the axis of rotation at s = 0. A global asymptotic theory with novel features for the limit E ≪ 1 and Pr/E ≫ 1, indicating the radial asymptotic scaling ∂/∂s = 0(E-1/3), was recently developed by Jones et al. (J. Fluid Mech. vol. 405, 2000, p. 157). In the different double limit E ≪ 1 and Pr/E ≫ 1, an asymptotic theory for the onset of convection building upon the theory of inertial waves was developed by Zhang (J. Fluid Mech. vol. 268, 1994 p. 211). It was shown that the convective motion at the leading-order approximation is represented by a single inertial-wave mode with a quadratic polynomial of s and z, obeying the asymptotic dependence ∂/∂s ∼ (1/s)∂/∂Φ = 0( 1 and Rc = 0(E) for stress-free spheres. There exist no simple asymptotic scalings for E 〈 1 appropriate to all values of PrIE. For an arbitrary small but non-zero E,the highly localized convection spreads out spatially with decreasing Pr, suggesting that the scaling laws such a∂/∂as = 0 (E-1/3) are no longer valid when Pr/E is not sufficiently large. This paper represents an attempt to develop a new asymptotic method for the analysis of convection in rapidly rotating spheres valid for asymptotically small E and for 0 〈 Pr/E ≤ ∞. The new method is based on the following three hypotheses. The first is that the leading-order velocity of convection for 0〈 Pr/E ≤ ∞ at E ≤ I is represented by either a single quasi-geostrophic-inertial-wave mode or by a combination of several quasi-geostrophic-inertial-wave modes convectively excited and sustained. Secondly, we assume that the convective motion for 0〈 Pr/E ≤ ∞ at E 〈 1 always has columnar structure, i.e. α/∂ az - 0(l), but without the general asymptotic scalings in the radial and azimuthal direction. Thirdly, we assume that there always exists a boundary flow that is non-zero only in the Ekman boundary layer on the bounding spherical surface and plays an important role even in the case of stress-free boundaries. Comparison between the result of the new method and the corresponding fully numerical simulation demonstrates a satisfactory quantitative agreement for all values of 0 ≤ Pr/ E ≤ 0 (106) when 0 (10-5) ≤ E ≤ 0 (10- 6). The new method is asymptotic in the sense that it is valid only for an asymptotically small E ≪ 1. In addition to the linear problem of thermal convection in rapidly rotating spheres, the corresponding weakly nonlinear problem is also solve to obtain an analytical expression for the convection-driven differential rotation generated by the nonlinear interaction of quasi-geostrophic-inertial-wave modes through the Reynolds stresses. The new method not only reveals the underlying nature of thermal convection in rapidly rotating spheres but also unites the two previously disjointed subjects in rotating fluids: the inertial-wave problem and the convective instability problem. © 2004 Cambridge University Press.

Description: This paper investigates the spatio-temporal instability of the natural-convection boundary-layer flow adjacent to a vertical heated flat plate immersed in a thermally stratified ambient medium. The temperature on the plate surface is distributed linearly. By introducing a temperature gradient radio a between the wall and the medium, we obtain a similarity solution which can describe in a smooth way the evolution between the states with isothermal and uniform-heat-flux boundary conditions. It is shown that the flow reversal in the basic flow vanishes when a is larger than a critical value. A new absolute-convective instability transition of this flow is identified in the context of the coupled Orr Sommerfeld and energy equations. Increasing a decreases the domain of absolute instability, and when a is large enough the absolute instability disappears. In particular, when a = 0 (isothermal surface), the interval of absolute instability becomes narrower for fluids of larger Prandtl numbers, and the absolute instability does not occur for Prandtl numbers greater than 70; when a = 1 (uniformheat-flux surface) the instability remains convective in a wide Prandtl number range. Analysis of the Rayleigh equations for this problem reveals that the basic flows supporting this new instability transition have inviscid origin of convective instability. Based on the steep global mode theory, the effects of a and Prandtl number on the global frequency are discussed as well. © 2004 Cambridge University Press.

Description: An analysis is made of steady-state flow of a compressible fluid in an infinite rapidly rotating pipe. Flow is induced by imposing a small azimuthally varying thermal forcing at the pipe wall. The Ekman number is small. Analyses are conducted to reveal both the axisymmetric-type and non-axisymmetric-type solutions. The axisymmetric solution is based on the azimuthally averaged wall boundary condition. The nonaxisymmetric solution stems from the azimuthally fluctuating part of the wall boundary condition. It is shown that the two-dimensional (uniform in the axial direction) non-axisymmetric solution exists for σ(γ - 1)M2 ≫ O(E1/3). However, an axially dependent solution is found if σ(γ - 1)M2≲ O(E1/3), in which E denotes the Ekman number, M the Mach number, γ the specific heat ratio and σ the Prandtl number. The axisymmetric solution prevails over the whole flow region; the two-dimensional non-axisymmetric solution is confined to the near-wall thermal layer of thickness O(E1/3). As a canonical example, a detailed description is given for the case of a highly conducting wall with differential heating. © 2004 Cambridge University Press.

Description: We present a study of the drag reduction induced by rigid fibres in a turbulent channel flow using direct numerical simulation. The extra stresses due to the fibres are calculated with the well-known constitutive equation involving the moments of the orientation vector. Drag reductions of up to 26% are calculated, with the largest drag reductions observed using non-Brownian fibres and semi-dilute concentrations. These findings suggest that elasticity is not necessary to achieve turbulent drag reduction. Flow statistics show trends similar to those observed in simulation of polymeric drag reduction: Reynolds stresses are reduced, velocity fluctuations in the wall-normal and spanwise directions are reduced while streamwise fluctuations are increased, and streamwise vorticity is reduced. We observe strong correlations between the fibre stresses and inter-vortex extensional flow regions. Based on these correlations and instantaneous visualizations of the flow field, we propose a mechanism for turbulent drag reduction by rigid fibre additives. © 2004 Cambridge University Press.

Description: This presents part 2 of a study of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. Part 1 dealt with only the oscillatory modes of convection (Riahi, J. Fluid Mech. vol. 467, 2002, pp. 331-359). In the present paper we consider the particular range of parameters where the critical values of the scaled Rayleigh number R for the onset of oscillatory and stationary convection are close to each other, and we develop and analyse a nonlinear theory in such a parameter regime which takes into account those mixed stationary and oscillatory modes of convection with common wavenumber vectors. Under a near-eutectic approximation and in the limit of large far-field temperature, we first determine a number of weakly nonlinear solutions, and then the stability of these solutions is investigated. The most interesting result is the preference for a mixed solution composed of standing and stationary hexagonal modes over a relatively wide range of the parameter values and for R just above its lowest subcritical value where convection is possible. Such a preferred solution has properties mostly in agreement with the experimental results due to Tait et al. (Nature, vol. 359, 1992, pp. 406-408) in the sense that the flow is downward at the cell centres, upward at the cell boundaries and there is some tendency for channel formation at the cell nodes. © 2004 Cambridge University Press.

Description: The dynamic response of an initially spherical capsule subject to different externally imposed flows is examined. The neo-Hookean and Skalak et al. (Biophys. J., vol. 13 (1973), pp. 245-264) constitutive laws are used for the description of the membrane mechanics, assuming negligible bending resistance. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creeping-flow conditions are assumed to prevail. The capillary number ε is the basic dimensionless number of the problem, which measures the relative importance of viscous and elastic forces. The boundary-element method is used with bi-cubic B-splines as basis functions in order to discretize the capsule surface by a structured mesh. This guarantees continuity of second derivatives with respect to the position of the Lagrangian particles used for tracking the location of the interface at each time step and improves the accuracy of the method. For simple shear flow and hyperbolic flow, an interval in ε is identified within which stable equilibrium shapes are obtained. For smaller values of ε, steady shapes are briefly captured, but they soon become unstable owing to the development of compressive tensions in the membrane near the equator that cause the capsule to buckle. The post-buckling state of the capsule is conjectured to exhibit small folds around the equator similar to those reported by Walter et al. Colloid Polymer Sci. Vol. 278 (2001), pp. 123-132 for polysiloxane microcapsules. For large values of ε, beyond the interval of stability, the membrane has two tips along the direction of elongation where the deformation is most severe, and no equilibrium shapes could be identified. For both regions outside the interval of stability, the membrane model is not appropriate and bending resistance is essential to obtain realistic capsule shapes. This pattern persists for the two constitutive laws that were used, with the Skalak et al. law producing a wider stability interval than the neo-Hookean law owing to its strain hardening nature. © 2004 Cambridge University Press.

Description: We consider the effects of blade mean loading on the noise generated by the interaction between convected vorticity and a blade row. The blades are treated as flat plates aligned at a non-zero incidence angle, δ, to the oncoming stream, and we take harmonic components of the incident vorticity field with reduced frequency k, and use asymptotic analysis in the realistic limit k ≫ 1, δ ≪ 1 with kδ = 0 (1). In a previous paper (Peake & Kerschen, J. Fluid Mech., vol. 347 (1997), pp. 315-346) we have analysed the sound radiated back upstream, but the field in the blade passages and the sound radiated downstream are also of considerable practical interest, and are considered in this paper. The flow is seen to consist of inner regions around each leading edge, in which sound is generated by the local gust-airfoil and gust-flow interactions, and an outer region in which the incident gust and the acoustic radiation interact with the non-uniform mean flow and the other blades. It is shown that the complicated multiple interactions between the blades can be represented by images in potential-streamfunction space, yielding closed-form expressions for the phase distortion experienced by sound waves propagating down the blade passages. The acoustic radiation downstream of the cascade at O(1) distances is dominated by the duct-mode beams that emanate from the passages, while the far downstream field is generated by the diffraction of the duct modes by the trailing edges. The modal amplitudes of the radiation field far downstream tend to be largest when the mode direction is close to the propagation direction of the duct mode which generated it, corresponding to the way (in uniform flow) in which the radiation from a single blade passage tends to be beamed in the duct-mode directions. Although the diffraction coefficient for the scattering from a single trailing edge is singular in these directions, we show how uniformly valid expressions can be derived by combining the trailing-edge fields in an appropriate way, thereby describing the larger amplitude in the beam directions. The steady non-uniform flow downstream has the effect of tilting the directions of the beams by O(δ) angles away from the duct-mode directions, which are explicitly determined. Throughout this analysis it will be seen that the interaction with the non-uniform mean flow introduces phase corrections of size O(kδ), which, given the way in which interference effects between the multiple blades dominate unsteady cascade flow, proves to be highly significant. © 2004 Cambridge University Press.

Description: The separation point of the flow around a circular cylinder has been numerically and experimentally investigated in the regime of Reynolds number less than 280. The present results reveal that the long-existing discrepancy in the data concerning the time-averaged separation angles reported in the literature results mainly from the oscillating characteristics of the flow separation on the cylinder surface and the experimental methodologies rather than the commonly mentioned blockage-ratio effect. In the present experiment, the time-averaged separation angles are obtained by averaging the instantaneous images from a soap-film flow visualization instead of from the commonly used streakline images from finite time exposures. Excellent agreement has been achieved between the present experimental results and numerical simulations by the spectral element method. Particle-streak visualization in a towing tank has also been conducted to compare with that of the two-dimensional soap-film experiments. It reveals that the separation angle is insensitive to the three-dimensional effect. Variations of the time-averaged separation angles with Reynolds number can be represented by a four-term θs-Re-1/2 relationship in the range of 7 ≤ Re ≤ 200. Moreover, if the data in the very low Reynolds number region are excluded, a simple linear θs-Re-1/2 relationship can be derived for 10 ≤ Re ≤ 200. Since the dimensionless boundary layer thickness and the Strouhal-Reynolds number relationship for the circular cylinder are also known to be proportional to Re-1/2, this linear relationship offers direct evidence that the flow characteristics of the boundary layer extend downstream along the cylinder surface to the separation point in this Re-range. The blockage effect on the separation angle has also been quantitatively analysed. © 2004 Cambridge University Press.

Description: High resolution direct numerical simulation (DNS) (512 × 1024 × 512) and large-eddy simulation (LES) of a shear-free mixing layer are presented. The geometry of the flow consists of two layers with different turbulence intensities that are in contact and interact through a fairly thin mixing layer. This geometry is used to explore the influence of inhomogeneities in the characteristic length scales, times scales and energy scales on the turbulence properties. Comparison of DNS results is made with the Veeravalli & Warhaft (J. Fluid Mech. 207, 191-229, 1989) experiment. The LES is performed on a 32 × 64 × 32 grid using an eddy-viscosity model. The use of such a model appears to be justified by the very weak departures from isotropy that are observed in the shear-free mixing layer. The LES predictions are compared with the filtered DNS data and show that the eddy viscosity model performs very well in predicting the energy profile as well as the deviation from Gaussianity in the turbulent velocity field statistics. © 2004 Cambridge University Press.

Description: A weak, laminar shear flow of a monodisperse suspension of high-Reynolds-number, low-Weber-number bubbles is studied in a novel experimental configuration. Nitrogen bubbles are formed through an array of small capillaries at the base of a tall channel with a small inclination from the vertical. The bubbles generate a unidirectional shear flow, in which the denser suspension near the bottom wall falls and the lighter suspension near the top wall rises. Profiles of the bubble and liquid velocities and the bubble volume fraction are obtained using hot-film and dual impedance probes. To our knowledge, measurements of the laminar shear properties of a nearly homogeneous bubble suspension have not previously been reported. A steady shear flow is observed in which the bubble velocity variation across the channel is typically less than 20% of the mean bubble velocity. The velocity and volume fraction gradients increase with channel inclination and exhibit little or no dependence on the mean gas volume fraction. To explain the magnitude of the volume fraction gradients, it is necessary to consider the effects of both the lift force and the effective bubble-phase diffusivity in balancing the segregating tendency of the cross-channel component of the buoyancy force. The bubble velocity gradient can be understood in terms of a balance of the component of the buoyancy force parallel to the channel walls and an effective viscosity associated with the Reynolds stresses produced by bubble-induced liquid velocity fluctuations. Theories for bubbles rising with potential-flow hydrodynamic interactions predict an instability of the homogeneous state due to a negative Maxwell pressure. However, the hydrodynamic diffusivity inferred from our experiments is large enough to mitigate the clustering effects of the Maxwell pressure. Consistent with this, a vigorous instability of the homogeneous state of the bubble suspension is only observed at volume fractions larger than 5%-20% with the critical volume fraction depending on the angle of inclination. © 2004 Cambridge University Press.

Description: A phase-plane approach is used to determine similarity solutions of the axisymmetric shallow-water equations which represent inwardly propagating, inviscid gravity currents. A Froude number condition characterizes the movement of the front. The unique similarity exponent is found numerically as a function of the frontal Froude number and the height and velocity profiles are presented for three different Froude numbers. The fluid speed and height are seen to increase monotonically towards the front except very close to the front where the height decreases. The maxima in both height and speed increase as the Froude number increases, reflecting the change in ambient resistance. For the Froude number that has been obtained experimentally for lock-exchange Boussinesq flows (Fr = 1.19) for which the similarity exponent is 0.859094, the similarity solution is compared to the numerical solution of the initial value problem, obtained recently by Hallworth, Huppert & Ungarish (2003). Our similarity solution compares reasonably well with their integration of the shallow-water equations in the neighbourhood of the front and at times close to collapse (when the front reaches the origin); however, near this point their numerics begin to fail. The solution at collapse and the similarity solution after collapse are also found for Fr = 1.19. This similarity solution describes the formation of a shock, as well as its initial propagation. © 2004 Cambridge University Press.

Description: Dissipation approximations have been used to calculate the drag on bubbles and drops and the decay rate of free gravity waves on water. In these approximations, viscous effects are calculated by evaluating the viscous stresses on irrotational flows. The pressure is not involved in the dissipation integral, but it enters into the power of traction integral, which equals the dissipation. A viscous correction of the irrotational pressure is needed to resolve the discrepancy between the zero-shear-stress boundary condition at a free surface and the non-zero irrotational shear stress. Here we show that the power of the pressure correction is equal to the power of the irrotational shear stress. The viscous pressure correction on the interface can be expressed by a harmonic series. The principal mode of this series is matched to the velocity potential and its coefficient is explicitly determined. The other modes do not enter into the expression for the drag on bubbles and drops. They vanish in the case of free gravity waves. © 2004 Cambridge University Press.

Description: A thin liquid film experiences additional intermolecular forces when the film thickness h is less than roughly 100 nm. The effect of these intermolecular forces at the continuum level is captured by the disjoining pressure Π. Since Π dominates at small film thicknesses, it determines the stability and wettability of thin films. To leading order, Π = Π(h) because thin films are generally uniform. This form, however, cannot be applied to films that end at the substrate with non-zero contact angles. A recent ad hoc derivation including the slope hx leads to Π = Π(h, hx , which allows non-zero contact angles, but it permits a contact line to move without slip. This work derives a new disjoining-pressure expression by minimizing the total energy of a drop on a solid substrate. The minimization yields an equilibrium equation that relates Π to an excess interaction energy E = E(h, hx). By considering a fluid wedge on a solid substrate, E(h, hx) is found by pairwise summation of van der Waals potentials. This gives in the small-slope limit Π = B-h 3 (α4 - h4 x + 2hh2 xhxx), where α is the contact angle and B is a material constant. The term containing the curvature hxx is new; it prevents a contact line from moving without slip. Equilibrium drop and meniscus profiles are calculated for both positive and negative disjoining pressure. The evolution of a film step is solved by a finite-difference method with the new disjoining pressure included; it is found that hxx = 0 at the contact line is sufficient to specify the contact angle. © 2004 Cambridge University Press.

Description: The influence of fluid thermal sensitivity on the centrifugal flow instabilities in pressure-driven (Dean) and drag-driven (Taylor-Couette) Newtonian shear flows is investigated. Thermal effects are caused by viscous heating or an externally imposed temperature difference between the outer and inner cylinders, ΔT*, or a combination of both. In all cases considered, the maximum temperature difference within the gap is small enough such that the base-state velocity profile and consequently the distribution of angular momentum are practically unchanged from those in the isothermal flow. The base-state temperature gradient can be approximated as a linear superposition of ΔT*, /d, where d is the gap width, and that caused by viscous heating. Numerical linear stability analysis shows that when ΔT*,= 0, viscous heating causes the critical Reynolds number, Rec, to be greatly reduced when the Nahme number, defined as the product of the Brinkman number, Br, and the dimensionless activation energy associated with the fluid viscosity, ε, is O(α2/Pr) where α and Pr denote the dimensionless critical axial wavenumber and Prandtl number respectively. Since α 2 is O(10) and typical Pr values for thermal sensitive liquids could be O(104), appreciable flow destabilization occurs even when Na is O(10-3). In the absence of viscous heating, an externally imposed temperature gradient can lead to significant reduction in Rec when S≡(εΔT*)/T*1 〈 0 and /S/ is O(α2/Pr), where T* 1 denotes the temperature of the inner cylinder. The numerical linear stability analysis results are explained based on a simplified model derived from the linearized governing equations by invoking the narrow-gap approximation. This model shows that the thermo-mechanical coupling, arising from the convection of the base-state temperature gradient by radial velocity perturbation, amplifies the temperature fluctuations within the flow by a factor proportional to Pe/α2 where Pe denotes the Péclet number. This results in the reduction of local viscosity. Hence, the rate of dissipation of the velocity perturbations decreases causing the centrifugal instability to occur at lower values of the Reynolds number compared to the isothermal flow. Thermo-mechanical destabilization caused by viscous heating for Δ T*, = 0 can be quantified by a scaling law of the form Λ = [1 + Prc1 Na/α2]-1/2 where Λ is the ratio of the critical Reynolds number of the non-isothermal flow to that of the isothermal one and c1 is a flow-dependent constant. Similarly, in the absence of viscous heating and for ΔT*, 〈 0, Λ = [1 + Prc2S /α2]-1/2, where c2 is a flow-dependent constant. When ΔT* 〉 0 and viscous heating are present, a numerical linear stability analysis shows that Λ ∝ Nak where k 〈 0 and it is dependent on ΔT*, and the flow type. Finally, we perform a nonlinear stability analysis for the Dean flow which shows that the bifurcation is supercritical for both stationary and time-dependent modes of instability. © 2004 Cambridge University Press.

Description: Lubrication theory is employed to examine surface-tension-dominated flows that arise during the application of thin coatings, using pressurized dies, to axially symmetric fibres. In all cases, it is assumed that the clearance between the die exit and the fibre is small compared with the fibre diameter. Previous analyses have been concerned with flows controlled by axial curvature for which the resulting solutions are unique. The present investigation examines stationary flows in which both the axial and the azimuthal curvatures are comparable. It is shown that this situation develops when the applicator volume flow is sufficiently large. Moreover, as the volume flow is increased, spatially oscillatory menisci can exist such that the solution is not always unique. These results are new, and calculations are presented that determine the maximum die clearance below which the solution remains unique. Within this regime, surface oscillations do not occur and, there is a monotonic decay to the final uniform coating thickness. © 2004 Cambridge University Press.

Description: In high Reynolds-number turbulence, local scalar turbulence structure parameters, (C2θ)r, local scalar variance dissipation rates, Xr, and local energy dissipation rates, εr, vary randomly in time and space. This variability, commonly referred to as intermittency, is known to increase with decreasing r, where r is the linear dimension of the local averaging volume. Statistical relationships between Xr, εr, and (Cθ2)r are of practical interest, for example, in optical and radar remote sensing. Some of these relationships are studied here, both theoretically and on the basis of recent observations. Two models for the conditionally averaged local temperature structure parameter, 〈(Cθ2 r/εr〉, are derived. The first model assumes that the joint probability density function (j.p.d.f.) of Xr and εr is bivariate lognormal and that the Obukhov-Corrsin relationship, (Cθ2 r = γ εr-1/3 Xr, where γ = 1.6, is locally valid. In the second model, small-scale intermittency is ignored and Cθ2 and ε are treated traditionally, that is, as averages over many outer scale lengths, such that Cθ2 and ε change only as a result of large-scale intermittency. Both models lead to power-law relationships of the form 〈(Cθ2 r/εr〉 = εrδ, where c is a constant. Both models make predictions for the value of the power-law exponent δ. The first model leads to δ = ρxyσy/σx - 1/3, where σx and σy are the standard deviations of the logarithms of εr and Xr, respectively, and ρxy is the correlation coefficient of the logarithms of Xr and εr. This model leads to δ = 1/3 if ρxy = 2/3 and if σx = σy. The second model predicts δ = 2/3, regardless of whether (i) static stability and shear are statistically independent, or (ii) they are connected through a Richardson-number criterion. These theoretical predictions are compared to fine-wire measurements that were taken during the night of 20/21 October 1999, at altitudes of up to 500 m in the nocturnal boundary layer and the overlying residual layer above Kansas. The fine-wire sensors were moved up and down with the University of Colorado's Tethered Lifting System (TLS). The data were obtained during the Cooperative Atmosphere-Surface Exchange Study 1999 (CASES-99 ). An interesting side result is that the observed frequency spectra of the logarithms of εr and Xr are described well by an f-1 law. A simple theoretical explanation is offered. © 2004 Cambridge University Press.

Description: A theoretical investigation of the breakdown of the viscous wake downstream of a flat plate in supersonic flow is performed in this paper based on the large Reynolds number (Re → ∞) asymptotic analysis of the Navier-Stokes equations. The breakdown is provoked by an oblique shock wave impinging on the wake a small distance ls downstream of the plate trailing edge. Two flow regimes are considered. In the first ls is assumed to be an O(Re-3/8) quantity in which case the shock impinges on the wake within the region of viscous-inviscid interaction that is known to occupy a vicinity of the trailing edge with longitudinal extent of O(Re-3/8). Under these conditions the interaction process may be described by the equations of the triple-deck theory. To obtain a numerical solution of these equations we used a rapid matrix Thomas technique in conjunction with Newton iterations. The results of the calculations not only predict the wake breakdown near the shock location but also reveal a hysteresis behaviour of the flow as the shock is moved downstream, giving rise to three solution branches. The second part of the paper is concerned with the flow regime when the shock interacts with the wake further downstream of the trailing edge triple-deck region: ls ≫ Re-3/8. In this case the fluid motion proves to be inviscid to leading order not only in the upper deck of the interaction region but also everywhere inside the wake. Due to this simplification the interaction problem can be reduced to a single integro-differential equation governing the pressure distribution along the interaction region. With known pressure the Bernoulli equation may be used to find the velocity field. The Bernoulli equation also allows us to formulate a simple criterion which may be used to predict the onset of wake breakdown. We found that viscosity becomes important again in a smaller vicinity of the breakdown point where the flow reversal takes place. It is remarkable that the viscous-inviscid interaction problem governing the flow in this vicinity admits an analytical solution. © 2004 Cambridge University Press.

Description: Direct numerical simulation is applied to investigate instability and transition to turbulence in the flow of an electrically conducting incompressible fluid between two parallel unbounded insulating walls affected by a wall-normal magnetic field (the Hartmann flow). The linear stability analysis of this flow provided unrealistically high critical Reynolds numbers, about two orders of magnitude higher than those observed in experiments. We propose an explanation based on the streak growth and breakdown mechanism described earlier for other shear flows. The mechanism is investigated using a two-step procedure that includes transient growth of two-dimensional optimal perturbations and the subsequent three-dimensional instability of the modulated streaky flow. In agreement with recent experimental investigations the calculations produce a critical range between 350 and 400 for the Hartmann thickness based Reynolds number, where the transition occurs at realistic amplitudes of two- and three-dimensional perturbations. © 2004 Cambridge University Press.

Description: Experiments and numerical simulations are carried out to verify the existence of the acoustic solitary wave in an air-filled tube with an array of Helmholtz resonators connected. Following up previous work (Sugimoto et al. 1999), the experiments are improved by using a newly designed piston driver to launch an initially plane pressure pulse and also by extending the tube length from 7.4 m to 10.6 m. To highlight the effect of the array of resonators, the case with no array is also examined in parallel. Direct and indirect checks are made to verify the existence of the solitary wave. The former compares the profiles and propagation speeds of pulses measured experimentally to the solitary-wave solution. The latter checks the validity of nonlinear wave equations in describing real wave evolution in the tube. Solving an initial-value problem numerically with weakly lossy effects of boundary layers and jet loss at the throat of the resonator, comparison is made between measured and simulated evolution. The validity of the equations in the lossy case is necessary to maintain the existence of the solitary wave in the lossless limit. It is revealed that nonlinear wave equations originally derived for unidirectional propagation in the tube can provide a good description of the real evolution, with some allowance for phase shifts on reflection at both ends of the tube. In particular, it turns out that the lossy effects are described quantitatively well. By establishing the validity of the equations, it is concluded that the acoustic solitary wave exists. © 2004 Cambridge University Press.

Description: We present new laboratory data on long wave forcing over a barred beach profile under random wave breaking conditions. The data include incident and radiated wave amplitudes, wave set-up, and detailed measurements of the cross-shore variation in long wave amplitude, including shoreline (swash) amplitudes. The total surf zone width was varied via changes in both wave height and the water level over the bar crest. The data obtained from the barred beach are also compared with previous data obtained from a plane beach under essentially identical short wave forcing conditions. The presence of the bar induces a frequency downshift in the spectral peak of the radiated long waves, a consequence of the increased surf zone width on the barred beach and a clear signature of long wave forcing by a time-varying breakpoint. Further comparisons of the two data sets suggest that the bar leads to resonant trapping and amplification (or suppression) of the shoreline motion at discrete long wave frequencies. Well-defined standing long wave motion occurs at discrete frequencies inside the bar and the resonant response is consistent with a simple seiche between the bar crest and shoreline, in agreement with previous numerical model studies. The long wave structure offshore of the breakpoint depends on the relative positions of the bar, shoreline and breakpoint, and is inconsistent with a numerical solution for a free standing long wave over the barred beach profile. © 2004 Cambridge University Press.

Description: The radial Navier-Stokes flow in a fracture bounded by impermeable corrugated rock surfaces is significantly different from the commonly used creeping flow model between two parallel surfaces, described by Darcy's law on the macroscale. Continuous variations in the Reynolds number along the radial coordinate determine the important role of the nonlinear inertial effects, which are reinforced by local oscillations of the velocity field caused by wall corrugation. The system behaviour is studied both analytically and numerically. A solution for the full system of Navier-Stokes equations in a thin cylindrical domain with oscillating walls is developed as a biparametric and two-scale asymptotic expansion with respect to fracture aperture and corrugation period. The numerical solution is derived based on the finite volume method. A new macroscale flow equation is obtained, which explicitly displays the relative roles of viscous dissipation caused by corrugation, local and global inertia, and cross inertia-viscous effects. The effective flow parameters are defined using analytical relationships. © 2004 Cambridge University Press.

Description: We report results of a series of detailed experiments designed to unveil the dynamics of a particle of radius a moving in high-frequency, low-Reynolds-number oscillatory flow. The fundamental parameters in the problem are the Strouhal (SI) and the particle Reynolds numbers (Rep), as well as the fluid-to-particle density ratio α. The experiments were designed to cover a range of S/Rep from 0.015 to 5 while keeping Rep 〈 0.5 and S1 〉 1. The primary objective of the experiments is to investigate stationary history effects associated with the Basset drag, which are maximized when the viscous time scale a2/v is of the same order of the flow time scale 9/Ω where 9 is a geometrical factor for the sphere, v is the kinematic viscosity and Ω is the angular frequency of the background flow. The theoretically determined behaviour of stationary history effects is confirmed unequivocally by the experiments, which also validate the fractional derivative behaviour (of order 1/2) of the history drag for the range of parameters under study. © 2004 Cambridge University Press.

Description: A proper orthogonal decomposition based model is considered for two-dimensional vortex shedding past a confined square cylinder. The aim is to study the validity of such a model for Reynolds numbers and blockage ratios that are different from those for which the model was derived. Using a calibration procedure it is shown that reliable results can be obtained in terms of short-term (one period) dynamics. Longterm dynamics are accurately captured with a variation of the Reynolds number, whereas the error becomes large when the blockage ratio changes. The controllability and observability of vortex shedding at a slightly supercritical Reynolds number is investigated relying on the accurate low-order models obtained. © 2004 Cambridge University Press.

Description: Expressions have been derived, from the standpoint of Gaussian random theory, for the joint and marginal distributions of wave envelope amplitude and local wave period when the sampling frequency is equal to the local wave frequency. The new marginal p.d.f. for wave envelope amplitudes shows substantial non-zero probability density at very small amplitudes. The new joint and marginal distributions are found to compare favourably with data obtained from both high frequency measurements of real ocean waves in extreme storms and from measurements of numerical simulations of moderately broadbanded processes with Pierson - Moskowitz spectra. The new p.d.f. of wave envelope amplitudes is found to provide a better approximation to the p.d.f.s of the simulated zero down-crossing wave amplitude than either the traditional Rayleigh p.d.f., applicable for infinitesimal bandwidth; or the narrow bandwidth approximation given by M. S. Longuet-Higgins (Proc. R. Soc. Lond. A, 389: 241-258, 1983). The reason for this improvement is that our method takes into account that small waves are likely to have shorter periods than large waves, rather than assuming a constant wave period. There are, however, limitations to the approach adopted which assumes that the individual wave amplitudes can be obtained from the amplitude of the wave envelope. These limitations become more severe as the bandwidth increases. The results obtained apply not only to sea waves, but to any Gaussian linear random process. © 2004 Cambridge University Press.

Description: We present a simple approximate model for studying general aspects of vortex interactions in a rotating stably-stratified fluid. The model idealizes vortices by ellipsoidal volumes of uniform potential vorticity, a materially conserved quantity in an inviscid, adiabatic fluid. Each vortex thus possesses 9 degrees of freedom, 3 for the centroid and 6 for the shape and orientation. Here, we develop equations for the time evolution of these quantities for a general system of interacting vortices. An isolated ellipsoidal vortex is well known to remain ellipsoidal in a fluid with constant background rotation and uniform stratification, as considered here. However, the interaction between any two ellipsoids in general induces weak non-ellipsoidal perturbations. We develop a unique projection method, which follows directly from the Hamiltonian structure of the system, that effectively retains just the part of the interaction which preserves ellipsoidal shapes. This method does not use a moment expansion, e.g. local expansions of the flow in a Taylor series. It is in fact more general, and consequently more accurate. Comparisons of the new model with the full equations of motion prove remarkably close. © 2004 Cambridge University Press.

Description: This paper examines the motion of a dense fluid that develops as an inertial gravity current of decreasing mass above a horizontal porous bed, while flow described by Darcy's law occurs in the bed. Measurements of the mass and the front position of the current in a set of laboratory experiments performed by changing different parameters are presented. The results are explained by means of a global analytical model that suggests practical correlations combining the parameters. Thus, previous experimental, numerical and theoretical findings are extended to describe lock-release gravity currents above more realistic porous beds. © 2004 Cambridge University Press.

Description: A new model for Lagrangian particle-pair separation in turbulent flows is developed and compared with data from direct numerical simulations (DNS) of isotropic turbulence. The model formulation emphasizes (i) non-Gaussian behaviour in Eulerian and Lagrangian statistics, (ii) the occurrence of large separation velocities, (iii) the role of straining and streaming flow structure as recognized in kinematic simulations of turbulence, and (iv) the role of conditionally averaged accelerations in stochastic modelling of turbulent relative dispersion. Previous stochastic models of relative dispersion have produced unrealistic behaviour, particularly in the dissipation subrange where viscous effects are important, which have led to questions on the adequacy of stochastic modelling. However, this failure can now be recognized as inadequate detail in formulation, which is explained and rectified in this paper. The model is quasi-one-dimensional in nature, and is focused on the statistics of particle-pair separation distance and its rate of change, referred to as the separation speed. Detailed comparisons are presented at several Reynolds numbers using the DNS database reported in a companion paper (Part 1). Up to fourth-order moments for these quantities are examined, as well as the separation-distance probability density function, which is discussed in the context of recent claims of Richardson scaling in the literature. The model is able to account for the spatial representation of straining regions as well as incompressibility of the flow, and is shown to reproduce strong non-Gaussianity and intermittency in the Lagrangian separation statistics observed in DNS. Comparisons with recent physical experiments are also remarkably consistent. This work demonstrates that stochastic models when properly formulated are effective and efficient representations of the dispersion process and this general class of models therefore possess great utility for calculations of both one-particle and two-particle dispersion. The techniques developed in this paper will facilitate such general model development. © 2004 Cambridge University Press.

Description: The evolution of a short injection-hole jet issuing into a crossflow at low blowing ratios is presented. Particle image velocimetry (PIV) is used to determine structural features of the jet/crossflow interaction throughout its development from within the jet supply channel (which feeds the holes), through the injection hole, and into the crossflow. The effect of supply channel feed orientations, i.e. counter to, or in the same direction as the crossflow is emphasized. Feed orientation profoundly affects such jet characteristics as trajectory and lateral spreading, as well as its structural features. Fluid within the high-speed supply channel exhibits swirling motions similar to the flow induced by a pair of counter-rotating vortices. The sense of rotation of the swirling fluid depends upon the orientation of the supply channel flow with respect to the crossflow, and in turn impacts the in-hole velocity fields. In the coflow supply channel geometry (channel flow is in the same direction as the free stream), a pair of vortices exists within the hole with the same sense of rotation as the primary jet counter-rotating vortex pair (CRVP). In contrast, the counterflow supply channel configuration has in-hole vortices of opposite rotational sense to that of the CRVP. The in-hole vortices interact constructively or destructively with the CRVP, thus affecting the strength and coherence of the CRVP. The counterflow configuration has a weakened CRVP because of destructive interference with the in-hole vortices. The weaker CRVP has a lower trajectory and increased spanwise spreading. External to the injection hole, a pair of vortices exists immediately downstream of the jet. They are initially perpendicular to the boundary-layer plate near the wall and are outboard of the streamwise hole centreline. These vortices, denoted 'downstream spiral separation node' (DSSN) vortices, are affected by both the suppl channel feed direction and the blowing ratio. They appear to form by free-stream fluid wrapping around the jet and interacting with the CRVP. The coflow supply channel geometry is associated with the largest and most well-defined DSSN vortices, and their size is inversely proportional to the blowing ratio. At low blowing ratio, these vortices induce a large recirculating flow region downstream of the injection hole at the wall. © 2004 Cambridge University Press.

Description: The correct asymptotic theory for the linear onset of instability of a Boussinesq fluid rotating rapidly in a self-gravitating sphere containing a uniform distribution of heat sources was given recently by Jones et al. (2000). Their analysis confirmed the established picture that instability at small Ekman number E is characterized by quasi-geostrophic thermal Rossby waves, which vary slowly in the axial direction on the scale of the sphere radius r0 and have short azimuthal length scale O(E1/3r0). They also confirmed the localization of the convection about some cylinder radius S = SM roughly r0/2. Their novel contribution concerned the implementation of global stability conditions to determine, for the first time, the correct Rayleigh number, frequency and azimuthal wavenumber. Their analysis also predicted the value of the finite tilt angle of the radially elongated convective rolls to the meridional planes. In this paper, we study small-Ekman-number convection in a spherical shell. When the inner sphere radius ri is small (certainly less than SM), the Jones et al. (2000) asymptotic theory continues to apply, as we illustrate with the thick shell ri = 0.35 r0. For a large inner core, convection is localized adjacent to, but outside, its tangent cylinder, as proposed by Busse & Cuong (1977). We develop the asymptotic theory for the radial structure in that convective layer on its relatively long length scale O(E2/9r0). The leading-order asymptotic results and first-order corrections for the case of stress-free boundaries are obtained for a relatively thin shell ri = 0.65 r0 and compared with numerical results for the solution of the complete PDEs that govern the full problem at Ekman numbers as small as 10-7. We undertook the corresponding asymptotic analysis and numerical simulation for the case in which there are no internal heat sources, but instead a temperature difference is maintained between the inner and outer boundaries. Since the temperature gradient increases sharply with decreasing radius, the onset of instability always occurs on the tangent cylinder irrespective of the size of the inner core radius. We investigate the case ri = 0.35 r0. In every case mentioned, we also apply rigid boundary conditions and determine the O(E1/6) corrections due to Ekman suction at the outer boundary. All analytic predictions for both stress-free and rigid (no-slip) boundaries compare Favourably with our full numerics (always with Prandtl number unity), despite the fact that very small Ekman numbers are needed to reach a true asymptotic regime. © 2004 Cambridge University Press.

Description: A modification of the Loitsyanskii integral is proposed as an invariant of three-dimensional decaying homogeneous and isotropic turbulence. As a result, the kinetic energy of a flow, generated in an infinitely large volume by the initial energy spectrum E(k, t = 0), which peaks in the vicinity of the wavenumber k0 = 1/ℒ(0) = 0(1), ℒ being an integral scale, decays with time in accordance with Kolmogorov's (1941) prediction: K = 1/2u 2̄ ∝ t-γ with γ = 10/7. © 2004 Cambridge University Press.

Description: Hartmann layers are a common feature in magnetohydrodynamics, where they organize the electric current distribution in the flow and hence the characteristics of the velocity field. In spite of their importance their stability properties are not well understood, mainly because of the scarcity of experimental data. In this work we investigated experimentally the transition to turbulence in the Hartmann layers that arise in magnetohydrodynamic flows in ducts. From measurements of the friction factor a well-marked transition to turbulence was found at a critical Reynolds number, based on the laminar Hartmann layer thickness, of approximately 380, valid also for laminarization and for a wide range of intensities of the magnetic field. The sensitivity of this result to the roughness characteristics of the walls along which the Hartmann layers develop confirms that these layers are related to the transition observed and provides more information on its stability properties. © 2004 Cambridge University Press.

Description: The superharmonic instability is pervasive in large-amplitude water-wave problems and numerical simulations have predicted a close connection between it and crest instabilities and wave breaking. In this paper we present a nonlinear theory, which is a generic nonlinear consequence of superharmonic instability. The theory predicts the nonlinear behaviour witnessed in numerics, and gives new information about the nonlinear structure of large-amplitude water waves, including a mechanism for noisy wave breaking. © 2004 Cambridge University Press.

Description: The evolution of pancake dipoles of different aspect ratio is studied in a stratified tank experiment. Two cases are reported here for values of the dipole initial aspect ratio α0 = Lv/Lh (where Lv and Lh, are vertical and horizontal length scales, respectively) of α0 = 0.4 (case I) and α0 = 1.2 (case II). In the first case, the usual decay scenario is observed where the dipole diffuses slowly with a growing thickness and a decaying circulation. In case II, we observed a regime where the thickness of the dipole decreases and the circulation in the horizontal mid-plane of the vortices remains constant. We show that this regime where the vertical length scale decreases can be explained by the shedding of two boundary layers at the top and bottom of the dipole that literally peel off vorticity layers. Horizontal advection and vertical diffusion cooperate in this regime and the decrease towards the viscous vertical length scale δ = Lh Re-1/2 occurs on a time scale α0Re1/2TA, TA being the advection time Lh/U. From a scaling analysis of the equations for a stratified viscous fluid in the Boussinesq approximation, two dominant balances depending on the parameter R=ReF2 h , are discussed, where Fh=U/NLh is the horizontal Froude number and Re=ULh/v is the Reynolds number, U, N and v being, respectively, the translation speed of the dipole, the Brunt-Väisälä frequency and the kinematic viscosity. When R ≫ 1 the vertical length scale is determined by buoyancy effects to be of order Lb = U/N. The experiments presented in this paper pertain to the case of small R, where viscous effects govern the selection of the vertical length scale. We show that if initially Lv ≤ δ, the flow diffuses on the vertical (case I), while if Lv ≫ δ (case II), vertically sheared horizontal advection decreases the vertical length scale down to δ. This viscous regime may explain results from experiments and numerical simulations on the late evolution of stratified flows where the decay is observed to be independent of the buoyancy frequency N. © 2004 Cambridge University Press.

Description: In this paper, a high-order Boussinesq model is used to conduct a systematic numerical study of crescent (or horseshoe) water wave patterns in a tank, arising from the instability of steep deep-water waves to three-dimensional disturbances. The most unstable phase-locked (L2) crescent patterns are investigated, and comparisons with experimental measurements confirm the quantitative accuracy of the model. The unstable growth rate is also investigated, as are the effects of variable nonlinearity. The dominant physical mechanism is clearly demonstrated (through time and space series analysis) to be the established quintet resonant interaction, involving the primary wave with a pair of symmetric satellites. A numerical investigation into oscillating crescent patterns is also included, and a detailed account of the complicated oscillation cycle is presented. These patterns are shown to arise from quintet resonant interactions involving the primary wave with two unsymmetric satellite pairs. Pre-existing methods for analysing the stability of steep deep-water plane waves subject to three-dimensional perturbations are extended to provide accurate quantitative estimates for the oscillation period. A possible explanation for their selection in experiments is also provided. Finally, we use the model to conduct a series of experiments involving competition between various unstable modes. The results generally show that multiple instabilities can grow simultaneously, provided that they are of roughly equivalent strength. Results using random perturbations also match observations in physical experiments both in the form (i.e. two- or three-dimensional) and the location of the initial instability. The computational results are the first examples of highly nonlinear (to the breaking point) deep-water wave modeling in two horizontal dimensions with a Boussinesq model. The efficiency of the model has allowed for a quantitative study of these phenomena at significantly larger spatial and temporal scales than have been demonstrated previously, providing new insight into the complicated physical processes involved. © 2004 Cambridge University Press.

Description: This paper deals with the temporal stability of the q-vortex trailing line vortex model. We describe a family of viscous instabilities existing in a range of parameters which is usually assumed to be stable, namely large swirl parameters (q 〉 1.5) and large Reynolds numbers. These instabilities affect negative azimuthal wavenumbers (in 〈 0) and take the form of centre-modes (i.e. with a structure concentrated along the vortex centreline). They are related to a family of viscous modes described by Stewartson, Ng and Brown (1988) in swirling Poiseuille flow, and are the temporal counterparts of weakly amplified spatial modes recently computed by Olendraru and Sellier (2002). These instabilities are studied numerically using an original and highly accurate Chebyshev collocation method, which allows a mapping of the unstable regions up to Re ≈ 106 and q ≈ 7. Our results indicate that in the limit of very large Reynolds numbers, trailing vortices are affected by this kind of instability whatever the value of the swirl number. © 2004 Cambridge University Press.

Description: The gravitational settling of small dense particles (with fall speed vT, response time τp) past an isolated spherical vortex (radius a, speed U) or a random distribution of spherical vortices translating vertical upwards, is examined. As particles sediment past a vortex, they are permanently displaced vertically and laterally a distance X and Y, respectively. The bulk settling properties of the particles are expressed in terms of the weighted moments of displacement, denoted by Dp, Mxx, Myy and corresponding to the integral of X, X2/2, Y2/2, respectively over the particle sheet. When the particle Stokes number St = Uτp/a → 0, the particles are inertialess. Particles starting outside the vortex are excluded from a spherical shadow region when vT/U 〉 3/2. When vT/ U 〈 3/2, particles passing close to the particle stagnation points (in the frame moving with the vortex) are held up for a long time relative to particles far from the vortex, but are not displaced laterally. In an unbounded flow, the particle drift volume, Dp, is calculated using a geometrical argument, Mxx = 25U2πa4/8(vT + U 2, and Myy = 0. As vT/U → 0, the results of Darwin (1953) are recovered. Results for finite values of St are calculated numerically. The effect of inertia is shown to substantially increase the particle residence time near the vortex because particles overshoot the particle stagnation point, and there is a shadow region within and behind the vortex. Dp, Mxx, and Myy all substantially increase with the particle Stokes number. These results are applied to calculate the bulk settling velocity and the dispersivity of particles sedimenting through a random distribution of vortices translating vertically in a bounded flow. This is done by combining information of the particle displacements with a statistical model of their encounter with a vortex. Inertialess particles (St = 0) do not experience the upwards flow within the vortex and the fractional increase in fall speed is proportional to the volume of the shadow region. As St increases, particles overshoot the particle stagnation point, increasing their residence time and so decreasing the bulk settling fall speed. Particle inertia significantly increases the vertical dispersivity of dense particles compared to fluid particles, but for high vT, particles disperse vertically more slowly than fluid particles.

Description: The translational velocities of two spherical gas bubbles oscillating in water, which is irradiated by a high-intensity acoustic wave field, are calculated. The two bubbles are assumed to be located far enough apart so that shape oscillations can be neglected. Viscous effects are included owing to the small size of the bubbles. An asymptotic solution is obtained that accounts for the viscous drag on each bubble, for large Re based on the radial part of the motion, in a form similar to the leading-order prediction by Levich (1962), CD = 48/ReT; ReT → ∞ based on the translational velocity. In this context the translational velocity of each bubble, which is a direct measure of the secondary Bjerknes force between the two bubbles, is evaluated asymptotically and calculated numerically for sound intensities as large as the Blake threshold. Two cases are examined. First, two bubbles of unequal size with radii on the order of 100 μm are subjected to a sound wave with amplitude PA 〈 1.0 bar and forcing frequency ωf = 0.51ω10, so that the second harmonic falls within the range defined by the eigenfrequencies of the two bubbles, ω10 〈 2ωf 〈 ω20. It is shown that their translational velocity changes sign, becoming repulsive as PA increases from 0.05 to 0.1 bar due to the growing second harmonic, 2ωf, of the forcing frequency. However, as the amplitude of sound further increases, PA ≈ 0.5 bar, the two bubbles attract each other due to the growth of even higher harmonics that fall outside the range defined by the eigenfrequencies of the two bubbles. Second, the case of much smaller bubbles is examined, radii on the order of 10 μm, driven well below resonance, ωf/2π = 20 kHz, at very large sound intensities, PA ≈ 1 bar. Numerical simulations show that the forces between the two bubbles tend to be attractive, except for a narrow region of bubble size corresponding to a nonlinear resonance related to the Blake threshold. As the distance between them decreases, the region of repulsion is shifted, indicating sign inversion of their mutual force. Extensive numerical simulations indicate the formation of bubble pairs with constant average inter-bubble distance, consisting of bubbles with equilibrium radii determined by the primary and secondary resonance frequencies for small and moderate sound amplitudes or by the Blake threshold for large sound amplitudes. It is conjectured that in experiments where 'acoustic streamers' are observed, which are filamentary structures consisting of bubbles that are aligned and move rapidly in a cavitating fluid at nearly constant distances from each other, bubbles with size determined by the Blake threshold are predominant because those with size determined by linear resonance are larger and therefore become unstable due to shape oscillations. © 2004 Cambridge University Press.

Description: The onset of convection in the form of inertial waves in a rotating fluid sphere is studied through a perturbation analysis in an extension of earlier work by Zhang (1994). Explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived and new results for the case of a nearly thermally insulating boundary are obtained.

Description: The karlsruhe Dynamo experiment is aimed at showing that an array of columnar helical vortices in liquid sodium, confined in a cylindrical container, can generate a magnetic field by self-excitation. In three test series it has been demonstrated that magnetic self-excitation occurs and a permanent magnetic saturation field develops which oscillates about a well-defined mean value for fixed flow rates. Dynamo action is observed as an imperfect bifurcation from a seed magnetic field of the environment. Two quasi-dipolar magnetic fields of opposite direction have been realized. A transition between these two states can be enforced through imposition of a sufficiently strong external magnetic perturbation on the existent dynamo field. These perturbations were induced with the aid of two Helmholtz coils. A time series analysis of the magnetic field fluctuations shows several characteristic dynamic features, which are in agreement wiyh theoretical predictions from turbulence models available in the literature.

Description: As the size of object diminishes, the effect of adhesional forces grows stronger for micro-manipulation and so. The capillary force generated by a liquid bridge can be greater than both the capillary force generated by another bridge and the adhesional force because the capillary force can be controlled by regulation of the liquid volume. We propose a micro-manipulation method based on the regulation of liquid bridge volume. A numerical investigation to estimate the capillary force from a given liquid volume is also presented, and four phases of capillary force curves are obtained from it. If an object is supported by two liquid bridges, we can predict which bridge collapse by a stability analysis for the wide range of liquid volume using the force curve.

Description: The motion of membrane-bound objects is important in many aspects of biology and surface chemistry. Here we derive some general relations for objects moving in a surface film overlying a fluid of depth H. A solution to the problem of the drag can be obtained from a two-dimensional system of integral equations. Here we focus on the problem of an ideal needle moving edge-on (in the direction of its tip) or broadside-on (perpendicular to the direction of the tip). It is shown that in comparison to the drag on a circular disk a new scaling regime of the drag on a needle arises when the ratio between surface shear viscosity and subphase viscosity ηs/η is smaller than the length of the needle.

Description: Secondary currents are a characteristic feature of flow in open-channel bends. Besides the classical helical motion (centre-region cell), a weaker and smaller counter-rotating circulation cell (outer-bank cell) is often observed near the outer bank, which is believed to play an important role in bank erosion processes. The mechanisms underlying the circulation cells, especially the outer-bank cell, are still poorly understood, and their numerical simulation still poses problems, not least due to lack of detailed experimental data. The research reported herein provides detailed experimental data on both circulation cells in an open-channel bend such as found in nature. Furthermore, the underlying dynamics are investigated by simultaneously analysing the vorticity equation and the kinetic energy transfer between the mean flow and the turbulence. This shows that turbulence plays a minor role in the generation of the centre-region cell, which is mainly due to the centrifugal force. By accounting for the feedback between the downstream velocity profile and the centre-region cell, a strongly simplified vorticity balance is shown to yield accurate predictions of the velocities in the centre region. For strong curvatures, however, a fully three-dimensional flow description is required. Due to the non-monotonic velocity profiles, the centrifugal force favours the outer-bank cell. Moreover, terms related to the anisotropy of the cross-stream turbulence, induced by boundary proximity, are of the same order of magnitude and mainly enhance the outer-bank cell. Both mechanisms strengthen each other. The occurrence of the outer-bank cell is shown to be not just due to flow instability, like in the case of curved laminar flow, but also to kinetic energy input from turbulence.

Description: A different approach to the solution of the singular Rayleigh equation is presented in the context of the water wave growth problem as modelled by wind-induced shear instabilities. The approach is based on the analytical solution of a Bessel equation in the vicinity of the singular point, which is obtained from Rayleigh's equation with an arbitrary wind profile. Wave growth rates are computed using an integral expression derived from the dispersion relation of the air-sea interface. Computations of the present approach agree well with those of Conte and Miles (1959) for the special case of a logarithmic wind profile. Effects of the shape of the wind profile on the wave growth rate are investigated by using the 1/7-power law to represent the wind profile. Comparisons of the growth rates for the logarithmic wind profile and for the 1/7 profile reveal appreciable differences which must be investigated further, possibly using measured wind profiles within 10 m above the sea surface. © 2004 Cambridge University Press.

Description: Cold O2/NO supersonic expansions through an axisymmetric convergent-divergent nozzle into a low-pressure background gas have been investigated both experimentally and numerically. Temperature and density measurements have been carried out, using tracer NO in an O2 main flow. NO two-dimensional rotational temperature and density flow field patterns in the jet behind the nozzle have been measured by laser-induced fluorescence (LIF). The spectroscopic investigations are complemented by static and impact pressure measurements along the jet centreline. Two nozzles with the same inner profile but with a short and long divergent section and small and large lip have been examined. Three experiments performed using these nozzles cover a wide range of regimes of underexpanded free jet flow, from highly oscillating multi-cycle structure to the regime with smooth deceleration of the issuing flow by the background gas. The numerical simulation of the whole flow field from the conditions in the stagnation chamber to those in the flooded space is performed in the framework of the full set of Navier-Stokes equations by employing a recently developed algorithm based on a staggered grid. The flow inside a long nozzle is shown to be strongly affected by the boundary layer, which leads to the degeneration of the isentropic core at the nozzle exit and transforms the jet regime from overexpanded for inviscid flow to underexpanded for a real flow. The model reproduces the experimental structures of the low-density free jet flow fairly well. © 2004 Cambridge University Press.

Description: We depict and analyse the successive steps of atomization of a liquid jet when a fast gas stream blows parallel to its surface. Experiments performed with various liquids in a fast air flow show that the liquid destabilization proceeds from a two-stage mechanism: a shear instability first forms waves on the liquid. The transient acceleration experienced by the liquid suggests that a Rayleigh-Taylor type of instability is triggered at the wave crests, producing liquid ligaments which further stretch in the air stream and break into droplets. The primary wavelength λ ∼ δ(ρ1/ρ2)1/2 is set by the vorticity thickness δ, in the fast air stream and the liquid/gas density ratio ρ1/ρ2. The transverse corrugations of the crests have a size λ⊥ ∼ δ We〈sub/〉d-1/3 (ρ1/ρ2)1/3, where Weδ = ρ2u22δ/σ is the Weber number constructed on the gas velocity u2 and liquid surface tension σ. The ligament dynamics gives rise, after break-up, to a well-defined droplet size distribution whose mean is given by λ⊥. This distribution bears an exponential tail characteristic of the broad size statistics in airblast sprays.

Description: We consider the dynamics of capillary attraction between an articulated train of rigid rods floating at a liquid-gas interface and a nearby wall. We then explain some of the phenomena that are a result of the strong anisotropy and the extended nature of the system, such as the lining up next to the walling in a 'zippering' motion that is observed and compare our results qualitatively with those of experiments. © 2004 Cambridge University Press.

Description: The structures and energetics of nearly symmetric modes and nearly baroclinic modes are analysed in detail to examine their instability mechanisms. It is shown that the nearly symmetric modes have their cross-band circulations slanted mainly between the along-band absolute-momentum surface and buoyancy surface of the basic state. Their growth is thus supported mainly by the symmetric-type energy conversion that transports energy from the basic-state along-band velocity and buoyancy to the perturbation along-band velocity and buoyancy, respectively, and then to the cross-band circulation. However, as the band orientations are tilted slightly away from the basic shear, the growth is also assisted by the baroclinic-type energy conversion that transports energy from the basic-state buoyancy to the perturbation buoyancy via the along-band advection and then to the cross-band circulation. When the band orientation is tilted to the warm (or cold) side of the basic shear, the baroclinic-type energy conversion smooths (or sharpens) the near-boundary structures and thus reduces (enhances) the effect of diffusive damping, especially near the non-slip boundaries. This explains why in the presence of diffusivity the symmetric instability yields to the nearly symmetric instability with the band orientation tilted slightly to the warm side of the basic shear. The nearly baroclinic modes transport warm air northward with rising motion and cold air southward with sinking motion, so their growth is supported mainly by the baroclinic-type energy conversion. Since the band orientations are not exactly perpendicular to the basic shear, the growth is also assisted by two additional energy conversions: (i) from the basic-state buoyancy through the cross-band horizontal advection to the perturbation buoyancy; and (ii) from the basic-state along-band velocity to the perturbation along-band velocity. When the band orientation is tilted, by nearly 90° or less, to the warm (or cold) side of the basic shear, the two additional energy conversions smooth (or sharpen) the near-boundary structures and thus reduce (enhance) the effect of diffusive damping, especially near the non-slip boundaries. This explains why the baroclinic instability yields to the warm-side tilted nearly baroclinic instability in the presence of diffusivity. © 2004 Cambridge University Press.

Description: 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.

Description: The spectra and correlations of the velocity fluctuations in turbulent channels, especially above the buffer layer, are analysed using new direct numerical simulations with friction Reynolds numbers up to Reτ = 1900. It is found, and explained, that their scaling is anomalous in several respects, including a square-root behaviour of their width with respect to their length, and a velocity scaling of the largest modes with the centreline velocity Uc. It is shown that this implies a logarithmic correction to the k-1 energy spectrum, and that it leads to a scaling of the total fluctuation intensities away from the wall which agrees well with the mixed scaling of de Graaff and Eaton (2000) at intermediate Reynolds numbers, but which tends to a pure scaling with Uc at very large ones. © 2004 Cambridge University Press.

Description: This paper studies the nonlinear coupling between the volume pulsation, translational motion and shape modes of an oscillating bubble, especially in the context of translational instability, known as 'dancing motion', that is demonstrated by bubbles in acoustic standing waves. A set of coupled equations is derived that describes volume pulsations of a bubble, its translational motion and shape oscillations evolving on the bubble surface. The amplitudes of the surface modes and the translational velocity of the bubble are assumed to be small and allowed for in the equations of motion up to only second-order terms. The amplitude of the volume oscillation is not limited. Unlike earlier work on this subject, where only two adjacent shape modes with given natural frequencies are taken into account, we allow for all shape modes and do not impose any limitations on their natural frequencies. As a result, the present analysis reveals additional features, which have not been noted previously, inherent in the mutual interactions of the shape modes as well as in the interaction between the shape modes and the translational motion. © 2004 Cambridge University Press.

Description: We use a novel ultra-high-speed video camera to study the initial stage of the impact of a solid sphere onto a liquid surface, finding a high-speed horizontal jet which emerges immediately following the intial contact. For Re 〉 2 × 104 the jet emerges when the horizontal contact between the sphere and the liquid is only 12% of its diameter. For the largest Reynolds numbers this jet can travel at more than 30 times the impact velocity of the sphere. This jetting occurs sooner and at much higher normalized velocities than has been observed previously. The breakup of the jet into a spray of droplets sometimes occurs through formation of pockets in the liquid sheet. Early in the impact, the energy transferred to the jet and the subsequent spray sheet is estimated to be much larger than the energy associated with the added mass inside the liquid pool. The jetting will therefore greatly increase the initial impact force on the sphere.

Description: The steady two-dimensional laminar flow past a stationary cylinder is well known to lose stability to a periodic flow at a supercritical Hopf bifurcation point as the flow rate is increased. It is less well known that the critical flow rate at which the instability occurs can be increased by rotating the cylinder about is own axis, and that at a fixed flow rate the vortex street can be eliminated entirely by sufficiently large rotation. We confine the flow between two parallel walls and report the effect of cylinder rotation on the Reynolds number and frequency at the Hopf bifurcation point. Two new, and somewhat surprising, results are that for stationary cylinders at large blockage ratios, the steady symmetric flow can restabilize above a critical Reynolds number, and that steady asymmetric flows exist. © 2004 Cambridge University Press.

Description: The evolution of infinitesimal three-dimensional perturbations superimposed on a Burgers vortex is studied. By a sequence of variable transformations and scalings this linear evolution problem is reduced to a time-dependent system which is nearly identical to the stability equations governing a Lamb-Oseen vortex. The maximum amplification reached by perturbations over a finite time interval is computed through an iterative scheme based on the direct and adjoint governing equations, and results on the asymptotic stability of the Burgers vortex are deduced. The Burgers vortex is shown to be asymptotically stable, although significant short-term amplification may occur. © 2004 Cambridge University Press.

Description: For large classes of vorticities we prove that a steady periodic gravity water wave with a monotonic profile between crests and troughs must be symmetric. The analysis uses sharp maximum principles for elliptic partial differential equations.

Description: The breakup of a thin non-evaporating liquid film that is either flowing down or climbing on a vertical or inclined surface and subject to cocurrent or countercurrent interfacial shear (or gas flow) is investigated analytically. Analytical expressions for the dimensionless liquid film thickness, Δmin, and wetting rate, Γmin, at breakup are derived based on the minimization of the total energy of a stable rivulet, formed following the film breakup. For a downflowing liquid film, increasing the cocurrent interfacial shear (or gas velocity) or decreasing the equilibrium contact angle, θo, decreases both Δmin and Γmin, below their values with zero interfacial shear. Conversely, increasing the countercurrent interfacial shear or θo, increases both Δmin and Γmin above their values with zero interfacial shear. The predictions of Δmin and Γmin for a climbing water film on a vertical surface are in good agreement with reported experimental data for a wide range of cocurrent gas velocities. © 2004 Cambridge University Press.

Description: The period T of the Helmholtz mode in a circular well that is bounded above by a free surface and below by a semi-infinite reservoir is determined in terms of elliptic integrals. It is shown that T decreases monotonically with increasing amplitude A and is within 1% (10%) of the linear limit T0 for A/h0 〈 0.4(l.0), where h0 is the ambient depth.

Description: The linear stability of the Bickley jet in the framework of the beta-plane approximation is considered, with the objective of presenting analytical calculations which add to previous numerical computations. It is well-known that the equation governing the neutral solutions which are analytic at the critical layer can be transformed into an associated Legendre equation. It turns out that this particular equation has simple closed-form solutions other than those known already, which are the Legendre polynomial of degree two, and two associated Legendre functions of the first kind, respectively. This observation makes it possible to find analytic neutral modes and corresponding neutral curves in the (β, k)-plane not known previously, both for the bounded and the unbounded Bickley jet. Here β denotes the beta-parameter and k the wavenumber. These neutral curves comprise parts of the stability boundary. It is shown that the line segment (β = -2, 0 〈 k 〈 √2) is part of the stability boundary for the unbounded Bickley jet, so the region for the unstable radiating modes is larger than the one obtained previously. Also, analytic sinuous and varicose modes and corresponding neutral curves are found in the case of the bounded flow where only numerical calculations have previously been presented. Furthermore, local stability analysis reveals weakly amplified modes with wave speed outside the range of the velocity profile for the Bickley jet. This is rather rare, although Pedlosky's theorem allows for it, and there are only a few examples of flows in which such modes occur. Here these modes are sinuous modes and occur when the flow is both bounded and unbounded.

Description: We investigate the kinematic dynamo properties of interacting vortex tubes. These flows are of great importance in geophysical and astrophysical fluid dynamics: for a large range of systems, turbulence is dominated by such coherent structures. We obtain a dynamically consistent 21/2-dimensional velocity field of the form (u(x, y, t), u(x, y, t), w(x, y, t)) by solving the z-independent Navier Stokes equations in the presence of helical forcing. This system naturally forms vortex tubes via an inverse cascade. It has chaotic Lagrangian properties and is therefore a candidate for fast dynamo action. The kinematic dynamo properties of the flow are calculated by determining the growth rate of a small-scale seed field. The growth rate is found to have a complicated dependence on Reynolds number Re and magnetic Reynolds number Rm, but the flow continues to act as a dynamo for large Re and Rm. Moreover the dynamo is still efficient even in the limit Re≫ Rm, providing Rm is large enough, because of the formation of coherent structures.

Description: A rivulet is a narrow stream of liquid flowing down a solid surface. When its flow rate exceeds a certain limit, it tends to meander, exhibiting instability of the interface. Here we report a perturbation analysis of this meandering rivulet. We find that the combined effects of the tangential velocity difference across the interface and the dynamic wetting are responsible for the instability. The Weber number represents the ratio of the destabilizing force, inertia, to the stabilizing force, surface tension. As the Weber number increases, both the wavenumber of maximum instability and the cutoff wavenumber increase. The effects of the capillarity are such that the wavenumber of maximum instability asymptotically increases as the sensitivity of the dynamic contact angles to the contact line speed increases. However, the cutoff wavenumber remains constant despite wetting parameter changes when the Weber number is constant.

Description: The time development of a dense eddy placed on a varying topography in a rotating system is the subject of this paper. It is shown that if the bathymetric variations are comparable with the thickness of the eddy, these will induce a first-order time development in the water mass. Large-amplitude geostrophic theory shows that the water is advected along the depth contours with a speed that is proportional to the slope of the bottom. The results are applied to a laboratory experiment with a rotating parabolic channel. According to theory, an initially circular eddy placed in the centre of the channel becomes elliptical while shifting its axis along the channel. The outcome of the experiment is in accordance with theory, and the importance of topographic advection in the ocean is discussed. It is suggested that the mechanism may be important for moving water from deep-water formation sites and into the global deep-water circulation. © 2004 Cambridge University Press.

Description: The controlled separation of flow from an inclined straight flap at high inclination angles was investigated experimentally. The separation process was initiated by an abrupt change in the excitation emanating from a slot at the flap shoulder. A complete cessation of the actuation resulted in formation of a large vortex above the flap akin to the familiar 'dynamic stall vortex' (DSV) seen over oscillating airfoils in pitch. The DSV temporarily increased the aerodynamic load over the flap before it dropped to its low separated value. The duration of this overload decreased as the flap inclination increased. The use of periodic excitation during separation slowed down the rate of separation and changed its character depending on the amplitude and the frequency used. Forcing separation by switching the excitation to a high frequency (3 〈 F+ 〈 8) reduced or even eliminated the increase in flap loading that is associated with the DSV. A switch to low frequencies (F+ 〈 1) extended the duration of separation and increased the transient overload during the initial stage of the process. © 2004 Cambridge University Press.

Description: We consider the finite-amplitude instability of incompressible spherical Couette flow between two concentric spheres of radii R1 and R2 (〉R1) in the narrow-gap limit, ε Ξ (R2 - R1)/ R1 ≪ 1, caused by rotating them both about a common axis with distinct angular velocities Ω1 and Ω2 respectively. In this limit it is well-known that the onset of (global) linear instability is manifested by Taylor vortices of roughly square cross-section close to the equator. According to linear theory this occurs at a critical Taylor number Tcrit which, remarkably, exceeds the local value Tc obtained by approximating the spheres as cylinders in the vicinity of the equator even as ε ↓ 0. Previous theoretical work on this problem has concentrated on the case of almost co-rotation with δ ≈ (Ω1 - Ω2)/Ω1 = O(ε1/2) for which Tcrit = Tc + O(δ2) + O(ε). In this limit the amplitude equation that governs the spatio-temporal modulation of the vortices on the latitudinal extent O(ε1/2 R1) gives rise to an interesting bifurcation sequence. In particular, the appearance of global bifurcations heralds the onset of complicated subcritical time-dependent finite-amplitude solutions. Here we switch attention to the case when ε1/2 ≪ δ ≤ 1. We show that for Taylor numbers T = Tc + O((δε)2/3) there exists a locally unstable region of width O((δε 1/3 R1) within which the amplitude equation admits solutions in the form of pulse-trains. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and consists of a wave propagating towards the equator under an envelope. The pulse drifts at a slow speed (relative to the wave velocity) proportional to its distance (and away) from the equator. Both the wavelength and the envelope width possess the same relatively short length scale O((ε2 / δ)1/3 R1). The appropriate theory of spatially periodic pulse-trains is developed and numerical solutions found. Significantly, these solutions are strongly subcritical and have the property that T → Tc as ε ↓ 0. Two particular limits of our theory are examined. In the first, ε1/2 ≪ δ ≪ 1, the spheres almost co-rotate and the pulse drift velocity is negligible. A comparison is made of the pulse-train predictions with previously obtained numerical results pertaining to large (but finite) values of δ/ε1/2. The agreement is excellent, despite the complicated long-time behaviour caused by inhomogeneity across the relatively wide unstable region. Our second special case δ = 1 relates to the situation when the outer sphere is at rest. Now the poleward drift of the pulses leads to a slow but exponential increase of their separation with time. This systematic pulse movement, over and above the spatial inhomogeneity just mentioned, necessarily leads to complicated and presumably chaotic spatio-temporal behaviour across the wide unstable region of width O(ε1/3 R1) on its associated time scale, which is O(ε-1/3) longer than the wave period. In view of the several length and time scales involved only qualitative comparison with experimental results is feasible. Nevertheless, the pulse-train structure is robust and likely to provide the building block of the ensuing complex dynamics.

Description: The Darcy-Boussinesq equations at infinite Darcy-Prandtl number are used to study convection and heat transport in a basic model of porous-medium convection over a broad range of Rayleigh number Ra. High-resolution direct numerical simulations are performed to explore the modes of convection and measure the heat transport, i.e. the Nusselt number Nu, from onset at Ra = 4π2 up to Ra = 104. Over an intermediate range of increasing Rayleigh numbers, the simulations display the 'classical' heat transport Nu ∼ Ra scaling. As the Rayleigh number is increased beyond Ra = 1255, we observe a sharp crossover to a form fitted by Nu ≈ 0.0174 × Ra0.9 over nearly a decade up to the highest Ra. New rigorous upper bounds on the high-Rayleigh-number heat transport are derived, quantitatively improving the most recent available results. The upper bounds are of the classical scaling form with an explicit prefactor: Nu ≤ 0.0297 × Ra. The bounds are compared directly to the results of the simulations. We also report various dynamical transitions for intermediate values of Ra, including hysteretic effects observed in the simulations as the Rayleigh number is decreased from 1255 back down to onset. © 2004 Cambridge University Press.

Description: An experimental and numerical investigation has been carried out into the instability characteristics of natural convection of an ethanol-water solution in a vertical tank with aspect ratio (height/width) of 15. The solution contains 39 wt% ethanol with Prandtl number Pr=26. The density anomaly due to the Soret effect may be safely ignored in the present test configuration. Onset of instability, in the form of multicellular convection located in the mid-height of the tank, occurs at Grashof number Gr ≅ 13 500. These convection cells are unsteady even at low supercritical states, similar to earlier observations for higher Pr fluids. The cause of such unsteadiness of the flow has been determined by studying the streak images constructed by superposing individual frames of a digital movie sequence. New cells are generated in the upper and lower portions of the tank and then migrate toward the centre, causing the convection cells in the mid-section to merge. At higher Gr, even the tertiary cells, which rotate in the opposite direction of the secondary cells, participate in the merging process. Numerical simulations of the two-dimensional natural convection of a Boussinesq fluid with constant thermophysical properties, carried out at low supercritical Gr equivalent to the experimental value, show the same process of cell generation and merging as that observed in the experiments. By analysing the substantial time rate of change of the kinetic energy of the fluid using the mechanical energy equation, it is determined that the energy needed for the cell generation process is supplied by the work of the dynamic pressure. The subsequent migration of the cells toward the middle is caused by the pressure gradient in the tank. The total kinetic energy of the fluid attains a relative maximum right after a merging process due to the reduction of dissipation associated with the region of strong shear between the cells. © 2004 Cambridge University Press.

Description: We present an experimental study of the emptying of an ideal vertical bottle under gravity g. The idealization reduces the bottle to a cylinder of diameter D0, length L, closed at the top and open at the bottom through a circular thin-walled hole of diameter d, on the axis of the cylinder. The study is performed in the low-viscosity limit. The oscillatory emptying of the 'bottle' is referred to as the glug-glug, and is characterized by its period T, whereas the whole emptying process is characterized by a time Te. Concerning the long time scale Te, we show that: Te/Te0 = (D0/d) 5/2, where Te0 ≈ 3.0L/ √gD0 is the emptying time of an unrestricted cylinder. On the short time scale T, we show that the physical origin of the oscillations lies in the compressibility of the surrounding gas. The period can be written as: T = L/√γP0/ρ Φ (z̄i/L), where γ is the ratio of specific heats of the gas, P0 its pressure and ρ stands for the density of the liquid. The function Φ is dimensionless and changes with the relative position of the liquid interface z̄i/L. Finally, this analysis of time scales involved in the emptying of vertical cylinders is applied to other liquid-gas oscillators. © 2004 Cambridge University Press.

Description: We study the linear and nonlinear stability of a thick surfactant deposition spreading on a thin liquid film using transient growth analysis (TGA) and direct numerical Simulations (DNS) of the two-dimensional lubrication equations, respectively. Results of the TGA of the one-dimensional spatially and temporally evolving base state reveal disturbance growth and the selection of a perturbation of intermediate wavenumber. This perturbation targets the 'contact region' between the deposition and the underlying thin liquid film and grows despite the absence of intermolecular forces. Increasing the initial thickness ratio of the deposition to the thin film and decreasing the relative magnitude of capillarity and surface diffusion further amplify perturbation growth. The DNS results clearly show the formation of fingers in the contact region behind the surfactant leading edge and provide further confirmation of the TGA findings. © 2004 Cambridge University Press.

Description: We study experimentally the behaviour of a drop deposited on a conical fibre. It is shown that for wetting liquids, such a drop spontaneously moves towards the region of lower curvature. The driving force is measured and shown to be a gradient of Laplace pressure, which allows us to characterize the dynamics of these self-propelling drops. We conclude by discussing the efficiency of this device for drying a solid initially coated with a liquid film. © 2004 Cambridge University Press.

Description: Although there are a great many papers dedicated to the problem of a cylinder vibrating transverse to a fluid flow (Y-motion), there are almost no papers on the more practical case of vortex-induced vibration in two degrees of freedom (X, Y motion) where the mass and natural frequencies are precisely the same in both X- and Y-directions. We have designed the present pendulum apparatus to achieve both of these criteria. Even down to the low mass ratios, where m* = 6, it is remarkable that the freedom to oscillate in-line with the flow affects the transverse vibration surprisingly little. The same response branches, peak amplitudes, and vortex wake modes are found for both Y-only and X, Y motion. There is, however, a dramatic change in the fluid-structure interactions when mass ratios are reduced below m* = 6. A new amplitude response branch with significant streamwise motion appears, in what we call the 'super-upper' branch, yielding massive amplitudes of 3 diameters peak-to-peak (A*Y ∼ 1.5). We discover a corresponding periodic vortex wake mode, comprising a triplet of vortices being formed in each half-cycle, in what we define as a '2T' mode. We qualitatively interpret the principal vortex dynamics and vortex forces which yield a positive rate of energy transfer (ev) causing the body vibration, using the following simple equation: ev =2Γ*U*vY where Γ* is vortex strength, U*v is the speed downstream of the dominant near-wake vorticity, and Y is the transverse velocity of the body. This simple approach suggests that the massive amplitude of vibration for the 2T mode is principally attributed to the energy transfer from the 'third' vortex of each triplet, which is not present in the lower-amplitude 2P mode. We also find two low-speed streamwise vibration modes, which is not unexpected, since they correspond to the first and second excitation modes of vibration for flexible cantilevers. By considering equations of motion for the two degrees of freedom, we find a critical mass, m*crit = 0.52, similar to recent Y-only studies, below which the large-amplitude vibrations persist to infinite flow velocity. We show that the critical mass m*crit is the same for the X- and Y-directions, which ensures that the shapes of X, Y trajectories can retain their form as the velocity becomes large. The extensive studies of vortex-induced vibration for Y-only body motions, built up over the last 35 years, remain of strong relevance to the case of two degrees of freedom, for m* 〉 6. It is only for 'small' mass ratios, m* 〈 6, that one observes a rather dramatic departure from previous results, which would suggest a possible modification to offshore design codes. © 2004 Cambridge University Press.

Description: Plastic and metal spheres were dropped from various heights onto a quartz disk covered with a thin layer of viscous oil and inclined at various angles with the horizontal. Rebound was observed only above a critical approach velocity, similar to that observed for head-on collisions when the disk is horizontal. The tangential component of the sphere's velocity is reduced only a small amount by the collision, owing to sliding lubrication/friction forces that also impart a small rotational velocity to the sphere. In contrast, the normal component of velocity is reduced substantially by viscous losses, and so the rebound angle of the sphere relative to the surface of the disk is smaller than the impact angle. The normal component of restitution and the rebound angle increase with the normal Stokes number based on the normal component of the impact velocity. © 2004 Cambridge University Press.

Description: A dense cloud model of avalanches is presented which includes large density difference effects as well as sediment entrainment along the path of the cloud. This model demonstrates the importance of sediment entrainment in the evolution of the front velocity. Without sediment entrainment the cloud first accelerates and then decelerates, a behaviour known from previous studies of cloud or thermal motions. With sediments entrainment the cloud is mostly in an accelerating state. The closure coefficients in the model concerning the cloud shape and air entrainment are obtained from laboratory experiments. These coefficients can be considered generic in the Boussinesq limit. A correction for inertial effects which need to be taken into account when applied to large density difference clouds such as avalanches, is proposed. An expression for the sediment entrainment coefficient is derived, taking into account the flow parameters and the sediments layer properties. The model predictions are in good agreement with recent measurement of the front velocity of a powder-snow avalanche. A presentation, in terms of dimensionless variables, of avalanche and laboratory Boussinesq cloud velocities shows clearly the similarities and differences between the two. © 2004 Cambridge Unversity Press.

Description: A land- and sea-breeze (LSB) circulation under a calm stably stratified environment was simulated in the laboratory using a temperature-controlled water tank. The floor of the tank was divided into two sections representing land and sea. Two heat exchangers, each of them connected to a thermostat, simulated the diurnal thermal cycle typically experienced by the surface in coastal zones. A third heat exchanger positioned at the top of the tank provided a stable thermal stratification. Particle-tracking velocimetry was applied to evaluate the two-dimensional velocity field in the vertical centreline section of the tank orthogonal to the coastline, while a rack of thermocouples measured the vertical temperature profile near the coastline and further inland. It is shown that the overall flow consists of a closed circulation caused by the periodic change of the horizontal temperature difference between land and sea surfaces. Furthermore, the formations of cellular convection during the first phase of warming of the land-side as well as the genesis of the sea-breeze front were detected and analysed. Application of the proper orthogonal decomposition (POD) technique allowed the vortical large-scale structures in the flow to be determined. The results suggest that the energy contained in the first POD eigenmodes rapidly converged with the first mode, associated with the overall LSB circulation, being dominant with 73% of the energy. The other less energetic modes were mainly associated with the cellular convection. © 2004 Cambridge University Press.

Description: The forced reattachment of flow to an inclined flat surface, simulating a simple flap, was investigated experimentally. The transition from a separated to an attached state of the flow was initiated by an abrupt change in the frequency and the amplitude of periodic perturbations emanating from a slot at the flap shoulder. The excitation parameters determined the total duration of the reattachment process. Minimum reattachment time occurred at an optimal excitation frequency of Fopt+ ≈ 1.5, which was independent of amplitude and flap inclination. The control over the process was achieved by enhancing large spanwise vortices in the flow. Spatial amplification of consecutive vortices induces mean transport of fluid away from the flap surface which causes the main stream to reattach. The time scales of the excitation are at least an order of magnitude smaller than the typical reattachment times. © 2004 Cambridge University Press.

Description: Free-surface flows past submerged obstacles in a channel are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. In previous work involving a single obstacle (Dias and Vanden-Broeck 2002), new solutions called 'generalized hydraulic falls' were found. These solutions are characterized by a supercritical flow on one side of the obstacle and a train of waves on the other. However, in the case of a single submerged object, the generalized hydraulic falls are unphysical because the waves do not satisfy the radiation condition. In this paper new solutions for the flow past two obstacles of arbitrary shape are computed. These solutions are characterized by a train of waves 'trapped' between the obstacles. The generalized hydraulic falls are shown to describe locally the flow over one of the two obstacles when the distance between the two obstacles is large. © 2004 Cambridge University Press.

Description: We describe the general phenomenon of 'induced-charge electro-osmosis' (ICEO) - the nonlinear electro-osmotic slip that occurs when an applied field acts on the ionic charge it induces around a polarizable surface. Motivated by a simple physical picture, we calculate ICEO flows around conducting cylinders in steady (DC), oscillatory (AC), and suddenly applied electric fields. This picture, and these systems, represent perhaps the clearest example of nonlinear electrokinetic phenomena. We complement and verify this physically motivated approach using a matched asymptotic expansion to the electrokinetic equations in the thin-double-layer and low-potential limits. ICEO slip velocities vary as us ∝ E20L, where E0 is the field strength and L is a geometric length scale, and are set up on a time scale τc = λDL/D, where λD is the screening length and D is the ionic diffusion constant. We propose and analyse ICEO microfluidic pumps and mixers that operate without moving parts under low applied potentials. Similar flows around metallic colloids with fixed total charge have been described in the Russian literature (largely unnoticed in the West). ICEO flows around conductors with fixed potential, on the other hand, have no colloidal analogue and offer further possibilities for microfluidic applications. © 2004 Cambridge University Press.

Description: The scenario of transition to chaos for a sphere falling or ascending under the action of gravity in a Newtonian fluid is investigated by numerical simulation. The mathematical formulation is parameterized using two non-dimensional parameters: the solid/fluid density ratio and the generalized Galileo number expressing the ratio between the gravity-buoyancy and viscosity effects. The study is carried out fully in this two-parameter space. The results show that for all density ratios the vertical fall or ascension becomes unstable via a regular axisymmetry breaking bifurcation. This bifurcation sets in slightly earlier for light spheres than for dense ones. A steady oblique fall or ascension follows before losing stability and giving way to an oscillating oblique movement. The secondary Hopf bifurcation is shown not to correspond to that of a fixed sphere wake for density ratios lower than 2.5, for which the oscillations have a significantly lower frequency. Trajectories of falling spheres become chaotic directly from the oblique oscillating regime. Ascending spheres present a specific behaviour before reaching a chaotic regime. The periodically oscillating oblique regime undergoes a subharmonic transition yielding a low-frequency oscillating ascension which is vertical in the mean (zigzagging regime). In all these stages of transition, the trajectories are planar with a plane selected randomly during the axisymmetry breaking. The chaotic regime appears to result from an interplay of a regular and of an additional Hopf bifurcation and the onset of the chaotic regime is accompanied by the loss of the remaining planar symmetry. The asymptotic chaotic states present an intermittent character, the relaminarization phases letting the subcritical plane and periodic trajectories reappear. © 2004 Cambridge University Press.