ALBERT

Description: There has been a longstanding debate as to whether the large scales in grid turbulence should be classified as of the Batchelor or Saffman type. In the former, the integral scales, u and l, satisfy u2l5 constant, while in Saffman turbulence we have u2l3 = constant. For strictly homogeneous turbulence the energy decay rates in these two types of turbulence differ, with u2 ̃ t10/7 in Batchelor turbulence and u2 ̃ t6/5 in Saffman turbulence. We present high-resolution measurements of grid turbulence taken in a large wind tunnel. The particularly large test section allows us to measure energy decay exponents with high accuracy. We find that the turbulence behind the grid is almost certainly of the Saffman type, with u2l 3 = constant. The measured energy decay exponent, however, is found to lie slightly below the theoretical prediction of u2 ̃ t 1.2. Rather we find u2 ̃ tn, with n = 1.130.02. This discrepancy is shown to arise from a weak temporal decay of the dimensionless energy dissipation coefficient, εl/u3, which is normally taken to be constant in strictly homogeneous turbulence, but which varies very slowly in grid turbulence. © 2010 Cambridge University Press.

Description: A second-order nonlinear frequency-domain model extending the linear complementary mild-slope equation (CMSE) is presented. The nonlinear model uses the same streamfunction formulation as the CMSE. This allows the vertical profile assumption to accurately satisfy the kinematic bottom boundary condition in the case of nonlinear triad interactions as well as for the linear refractiondiffraction part. The result is a model with higher accuracy of wavebottom interactions including wavewave interaction. The model's validity is confirmed by comparison with accurate numerical models, laboratory experiments over submerged obstacles and analytical perturbation solutions for class III Bragg resonance. © 2009 Cambridge University Press.

Description: The two-dimensional space spanned by the velocity gradient invariants Q and R is expanded to three dimensions by the decomposition of R into its strain production 1/3s ijs jks ki and enstrophy production 1/4ω iω js ij terms. The {Q; R} space is a planar projection of the new three-dimensional representation. In the {Q; sss; s} space the Lagrangian evolution of the velocity gradient tensor A ij is studied via conditional mean trajectories (CMTs) as introduced by Martn et al. (Phys. Fluids, vol. 10, 1998, p. 2012). From an analysis of a numerical data set for isotropic turbulence of Reλ ̃ 434, taken from the Johns Hopkins University (JHU) turbulence database, we observe a pronounced cyclic evolution that is almost perpendicular to the QR plane. The relatively weak cyclic evolution in the QR space is thus only a projection of a much stronger cycle in the {Q; sss;ωω s} space. Further, we find that the restricted Euler (RE) dynamics are primarily counteracted by the deviatoric non-local part of the pressure Hessian and not by the viscous term. The contribution of the Laplacian of A ij, on the other hand, seems the main responsible for intermittently alternating between low and high intensity A ij states. © 2009 Cambridge University Press.

Description: Planar Rayleigh scattering provides quantitative mixing measurements in the developing region of axisymmetric turbulent helium jets issuing into air. The measurements focus on the relatively near field, in which the jets are primarily momentum driven. The imaging parameters are specified to ensure high spatial resolution. The mean jet fluid concentration fields attain self-similarity within the measurement region, though the forms of the mole fraction profiles indicate a reduction in turbulent transport at the jet outer boundary, arising from the reduced jet fluid density. Nevertheless, jet-like scaling pertains for the concentration fields. Mass fraction fluctuations on the jet centreline attain the expected asymptotic value of ≈ 23% of the centreline mass fraction values. The scalar dissipation rates, however, show an axial decay rate that is slower than theoretical predictions. The two-dimensional extent of the measurements also allows spatial filtering similar to that inherent in large-eddy simulations (LESs). The results confirm that fluctuation levels and scalar dissipation rates determined for the filtered fields are reduced as the effective resolution is reduced, but while the fluctuation profiles for the filtered fields are similar for the different filter sizes, the forms of the scalar dissipation profiles are highly dependent on filter size. These latter results in particular are of a form that will be useful for grid-dependent assessments of LES results. © 2009 Cambridge University Press.

Description: It has been recently shown that the flow of a Bingham fluid in a channel is always linearly stable (Nouar et al., J. Fluid Mech., vol. 577, 2007, p. 211). To identify possible paths of transition we revisit the problem for the case in which the idealized base flow is slightly perturbed. No attempt is made to reproduce or model the perturbations arising in experimental environments which may be due to the improper alignment of the channel walls or to imperfect inflow conditions rather a general formulation is given which yields the transfer function (the sensitivity) for each eigenmode of the spectrum to arbitrary defects in the base flow. It is first established that such a function, for the case of the most sensitive eigenmode, displays a very weak selectivity to variations in the spanwise wavenumber of the disturbance mode. This justifies a further look into the class of spanwise homogeneous modes. A variational procedure is set up to identify the base flow defect of minimal norm capable of optimally destabilizing an otherwise stable flow; it is found that very weak defects are indeed capable to excite exponentially amplified streamwise travelling waves. The associated variations in viscosity are situated mostly near the critical layer of the inviscid problem. Neutrally stable conditions are found as function of the Reynolds number and the Bingham number, providing scalings of critical values with the amplitude of the defect consistent with previous experimental and numerical studies. Finally, a structured pseudospectrum analysis is performed; it is argued that such a class of pseudospectra provides information well suited to hydrodynamic stability purposes. © 2010 Cambridge University Press.

Description: The evolution and decay of a homogeneous flow over random topography in a rotating system is studied by means of numerical simulations and theoretical considerations. The analysis is based on a quasi-two-dimensional shallow-water approximation, in which the horizontal divergence is explicitly different from zero, and topographic variations are not restricted to be much smaller than the mean depth, as in quasi-geostrophic dynamics. The results are examined by comparing the evolution of a turbulent flow over different random bottom topographies characterized by a specific horizontal scale, or equivalently, a given mean slope. As in two-dimensional turbulence, the energy of the flow is transferred towards larger scales of motion; after some rotation periods, however, the process is halted as the flow pattern becomes aligned along the topographic contours with shallow water to the right. The quasi-steady state reached by the flow is characterized by a nearly linear relationship between potential vorticity and transport function in most parts of the domain, which is justified in terms of minimum-enstrophy arguments. It is found that global energy decays faster for topographies with shorter horizontal length scales due to more effective viscous dissipation. In addition, some comparisons between simulations based on the shallow-water and quasi-geostrophic formulations are carried out. The role of solid boundaries is also examined: it is shown that vorticity production at no-slip walls contributes for a slight disorganization of the flow. © 2010 Cambridge University Press.

Description: Defining and identifying the aeroacoustic sources in a turbulent flow is a great challenge especially for noise control strategy. The purpose of the present study consists in proposing a new methodology to localize regions associated with sound generation. These regions are associated, in the present work, with those of high sensitivity of the acoustic field, using the heuristic argument that modifying the flow in these regions would lead to a very significant change in the radiated noise. The proposed method relies on the efficient coupling between the time-reversal theory applied to the Euler equations and the complex differentiation method to compute the sensitivity variable. To the knowledge of the authors, this is the first time that the time-reversal technique is applied to vectorial hydrodynamic equations, in place of the classical scalar wave equation. Subsequently, regions associated with sound generation are related to spatiotemporal events which exhibit the maximum of sensitivity to acoustical disturbances measured in far field. The proposed methodology is then successively tested on three cases for which the nature of the source is different: injection of mass, vibrating surfaces and flow instabilities arising in a plane mixing layer flow. For each test case, the two-dimensional Euler equations are solved using a numerical solver based on a pseudo-characteristics formulation. During these computations flow, variables are stored only at the computational boundaries. These variables are time reversed and relevant information concerning the acoustical disturbances is tagged using complex differentiation in order to lead the sensitivity analysis. The same numerical solver is used to access the evolution of the time-reversed variables. In each test case, the proposed methodology allows to localize successfully zones associated with noise generation. © 2010 Cambridge University Press.

Description: We show that a triple-valued sliding law can be heuristically motivated by the transverse spatial structure of an ice-stream velocity field using a simple one-dimensional model. We then demonstrate that such a sliding law can lead to some interesting stream-like patterns and time-oscillatory solutions. We find a generation of rapid stream-like solutions within a slow ice-sheet flow, separated by narrow internal boundary layers (shear margins), and analyse numerical simulations in two horizontal dimensions over a homogeneous bed and including longitudinal shear stresses. Different qualitative behaviours are obtained by changing a single physical parameter, a mass source magnitude, leading to changes from a slow creeping flow to a relaxation oscillation of the stream pattern, and to steady ice-stream-like solution. We show that the adjustment of the ice-flow shear margins to changes in the driving stress in the one-dimensional approximation is governed by a form of the GinzburgLandau equation and use stability analysis to understand this adjustment. In the model analysed here, the width scale of the stream is not set spontaneously by the ice flow dynamics, but rather, it is related to the mass source intensity and spatial distribution. © 2009 Cambridge University Press.

Description: Direct numerical simulation is used to study the development of a single laminar vortex ring as it impinges on a free surface directly from below. We consider the limiting case in which the Froude number approaches zero and the surface can be modelled with a stress-free rigid and impermeable boundary. We find that as the ring expands in the radial direction close to the surface, the natural TsaiWidnallMooreSaffman (TWMS) instability is superseded by the development of the Crow instability. The Crow instability is able to further amplify the residual perturbations left by the TWMS instability despite being of differing radial structure and alignment. This occurs through realignment of the instability structure and shedding of a portion of its outer vorticity profile. As a result, the dominant wavenumber of the Crow instability reflects that of the TWMS instability, and is dependent upon the initial slenderness ratio of the ring. At higher Reynolds number a short-wavelength instability develops on the long-wavelength Crow instability. The wavelength of the short waves is found to vary around the ring dependent on the local displacement of the long waves. © 2010 Cambridge University Press.

Description: Acoustic instabilities with frequencies roughly higher than 1 kHz remain among the most harmful instabilities, able to drastically affect the operation of engines and even leading to the destruction of the combustion chamber. By coupling with resonant transverse modes of the chamber, these pressure fluctuations can lead to a large increase of heat transfer fluctuations, as soon as fluctuations are in phase. To control engine stability, the mechanisms leading to the modulation of the local instantaneous rate of heat release must be understood. The commonly developed global approaches cannot identify the dominant mechanism(s) through which the acoustic oscillation modulates the local instantaneous rate of heat release. Local approaches are being developed based on processes that could be affected by acoustic perturbations. Liquid atomization is one of these processes. In the present paper, the effect of transverse acoustic perturbations on a coaxial air-assisted jet is studied experimentally. Here, five breakup regimes have been identified according to the flow conditions, in the absence of acoustics. The liquid jet is placed either at a pressure anti-node or at a velocity anti-node of an acoustic field. Acoustic levels up to 165 dB are produced. At a pressure anti-node, breakup of the liquid jet is affected by acoustics only if it is assisted by the coaxial gas flow. Effects on the liquid core are mainly due to the unsteady modulation of the annular gas flow induced by the acoustic waves when the mean dynamic pressure of the gas flow is lower than the acoustic pressure amplitude. At a velocity anti-node, local nonlinear radiation pressure effects lead to the flattening of the jet into a liquid sheet. A new criterion, based on an acoustic radiation Bond number, is proposed to predict jet flattening. Once the sheet is formed, it is rapidly atomized by three main phenomena: intrinsic sheet instabilities, Faraday instability and membrane breakup. Globally, this process promotes atomization. The spray is also spatially organized under these conditions: large liquid clusters and droplets with a low ejection velocity can be brought back to the velocity anti-node plane, under the action of the resulting radiation force. These results suggest that in rocket engines, because of the large number of injectors, a spatial redistribution of the spray could occur and lead to inhomogeneous combustion producing high-frequency combustion instabilities. © 2009 Cambridge University Press.

Description: Nonlinear shallow water equations are employed to model the inviscid slumping of fluid along an inclined plane and analytical solutions for the motion are derived using the hodograph transformation to reveal the run-up and the inception of a bore on the backwash. Starting from rest, the fluid slumps along the inclined plane, attaining a maximum run-up, before receding and forming a relatively thin and fast moving backwash. This interacts with the less rapidly moving fluid within the interior to form a bore. The evolution of the bore and the velocity and height fields either side of it are also calculated to reveal that it initially grows in magnitude before diminishing and intersecting with the shoreline. This analytical solution reveals features of the solution, such as the onset of the bore and its growth and decline, previously known only through numerical computation and the method presented here may be applied quite widely to the run-up of other initial distributions of fluid. © 2009 Cambridge University Press.

Description: The motion of two immiscible liquids in a capillary tube is analysed, theoretically and numerically, for the case in which a residual film confines the displacing liquid to the core of this tube. The theoretical analysis has shown that the three flow regimes predicted by Taylor (J. Fluid Mech., vol. 10, 1961, pp. 161165), for the case of gas-displacement, can only be achieved when the ratio of the viscosity of the displaced fluid to that of the displacing one is greater than 2. An elliptic mesh generation technique, coupled with the Galerkin finite-element method, is used to compute the velocity field and the configuration of the interface between the two fluids. A map of cases in the Cartesian space defined by the capillary number (Ca) and the viscosity ratio (Nμ) is constructed in order to locate the different flow patterns the problem exhibits. The critical capillary number at which the flow enters the transition range between the bypass regime and the full-recirculating one is given. While a decrease of the fraction of mass attached to the wall is achieved by decreasing Ca or increasing Nμ, bypass flow patterns are formed as a consequence of high values of the capillary number and viscosity ratio. © 2009 Cambridge University Press.

Description: A freely rotating sphere in a solid-body rotating flow is experimentally investigated. When the sphere is buoyant, it reaches an equilibrium position from which drag and lift coefficients are determined over a wide range of particle Reynolds numbers (2 ≤ Re ≥ 1060). The wake behind the sphere is visualized and appears to deflect strongly when the sphere is close to the cylinder axis. The spin rate of the sphere is recorded. In fluids with low viscosity, spin rates more than twice as large as the angular velocity of the cylinder can be observed. By comparing numerical results for a fixed but freely spinning sphere with a fixed non-spinning sphere for Re ≤ 200, the effect of the sphere spin on the lift coefficient is determined. The experimentally and numerically determined lift and drag coefficients and particle spin rates all show excellent agreement for Re ≤ 200. The combination of the experimental and numerical results allows for a parameterization of the lift and drag coefficients of a freely rotating sphere as function of the Reynolds number, the particle spin and the location of the particle with respect to the cylinder axis. Although the effect of the flow rotation on the particle spin is different in shear flow and solid-body rotating flow, the effect of spin on lift is found to be comparable for both types of flow. © 2009 Cambridge University Press.

Description: An experimental investigation of the flow generated by a homogeneous population of bubbles rising in water is reported for three different bubble diameters (d = 1.6, 2.1 and 2.5 mm) and moderate gas volume fractions (0.005 ≤ α ≥0.1). The Reynolds numbers, Re = V0d/, based on the rise velocity V0 of a single bubble range between 500 and 800. Velocity statistics of both the bubbles and the liquid phase are determined within the homogeneous bubble swarm by means of optical probes and laser Doppler anemometry. Also, the decaying agitation that takes place in the liquid just after the passage of the bubble swarm is investigated from high-speed particle image velocimetry measurements. Concerning the bubbles, the average velocity is found to evolve as V0α0.1 whereas the velocity fluctuations are observed to be almost independent of. Concerning the liquid fluctuations, the probability density functions adopt a self-similar behaviour when the gas volume fraction is varied, the characteristic velocity scaling as V0α0.4. The spectra of horizontal and vertical liquid velocity fluctuations are obtained with a resolution of 0.6 mm. The integral length scale is found to be proportional to V02/g or equivalently to d/Cd0, where g is the gravity acceleration and Cd0 the drag coefficient of a single rising bubble. Normalized by using, the spectra are independent on both the bubble diameter and the volume fraction. At large scales, the spectral energy density evolves as the power 3 of the wavenumber. This range starts approximately from δ and is followed for scales smaller than δ/4 by a classic 5/3 power law. Although the Kolmogorov microscale is smaller than the measurement resolution, the dissipation rate is however obtained from the decay of the kinetic energy after the passage of the bubbles. It is found to scale as α0.9 V 03/δ. The major characteristics of the agitation are thus expressed as functions of the characteristics of a single rising bubble. Altogether, these results provide a rather complete description of the bubble-induced turbulence. © 2009 Cambridge University Press.

Description: Numerical investigation of the compressible flow past an 18% thick circular-arc aerofoil was carried out using detached-eddy simulation for a free-stream Mach number M∞ = 0.76 and a Reynolds number Re = 1.1×107. Results have been validated carefully against experimental data. Various fundamental mechanisms dictating the intricate flow phenomena, including moving shock wave behaviours, turbulent boundary layer characteristics, kinematics of coherent structures and dynamical processes in flow evolution, have been studied systematically. A feedback model is developed to predict the self-sustained shock wave motions repeated alternately along the upper and lower surfaces of the aerofoil, which is a key issue associated with the complex flow phenomena. Based on the moving shock wave characteristics, three typical flow regimes are classified as attached boundary layer, moving shock wave/turbulent boundary layer interaction and intermittent boundary layer separation. The turbulent statistical quantities have been analysed in detail, and different behaviours are found in the three flow regimes. Some quantities, e.g. pressure-dilatation correlation and dilatational dissipation, have exhibited that the compressibility effect is enhanced because of the shock wave/boundary layer interaction. Further, the kinematics of coherent vortical structures and the dynamical processes in flow evolution are analysed. The speed of downstream-propagating pressure waves in the separated boundary layer is consistent with the convection speed of the coherent vortical structures. The multi-layer structures of the separated shear layer and the moving shock wave are reasonably captured using the instantaneous Lamb vector divergence and curl, and the underlying dynamical processes are clarified. In addition, the proper orthogonal decomposition analysis of the fluctuating pressure field illustrates that the dominated modes are associated with the moving shock waves and the separated shear layers in the trailing-edge region. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to this complex flow. © 2009 Cambridge University Press.

Description: In the present study we investigate the dynamics of initially spherical capsules (made from elastic membranes obeying the strain-hardening Skalak or the strain-softening neo-Hookean law) in strong planar extensional flows via numerical computations. To achieve this, we develop a three-dimensional spectral boundary element algorithm for membranes with shearing and area-dilatation tensions in Stokes flow. The main attraction of this approach is that it exploits all the benefits of the spectral methods (i.e. high accuracy and numerical stability) but without creating denser systems. To achieve continuity of the interfacial geometry and its derivatives at the edges of the spectral elements during the interfacial deformation, a membrane-based interfacial smoothing is developed, via a Hermitian-like interpolation, for both the interfacial shape and the membrane elastic forces. Our numerical results show that no critical flow rate exists for both Skalak and neo-Hookean capsules in the moderate and strong planar extension flows considered in the present study. As the flow rate increases, both capsules reach elongated ellipsoidal steady-state configurations; the cross-section of the Skalak capsule preserves its elliptical shape, while the neo-Hookean capsule becomes more and more lamellar. The curvature at the pointed edges of these elongated steady-state shapes shows a very fast increase with the flow rate. The large interfacial deformations are accompanied with the development of strong membrane tensions especially for the strain-hardening Skalak capsule; the computed increase of the membrane tensions with the flow rate or the shape extension can be used to predict rupture of a specific membrane (with known lytic tension) due to excessive tensions. The type of the experiment imposed on the capsule as well as the applied flow rate affect dramatically the time evolution of the capsule edges owing to the interaction of the hydrodynamic forces with the membrane tensions; when a spherical Skalak capsule is let to deform in a strong flow, very large edge curvatures (with respect to the steady-state value) are developed during the transient evolution. © 2009 Cambridge University Press.

Description: Lighthill's analysis (J. Fluid Mech., vol. 60, 1973, p. 1) of the Weis-Fogh lift mechanism is extended to include the spreading phase of the cycle. Lighthill proposed a two-dimensional inviscid irrotational analytical model to compute the circulation around two flat plates (the wings) as they open out, in opposite directions, about a common centre of rotation taken to be at the point of contact of an edge of each plate (the opening phase). At a critical opening angle, the plates separate and move apart horizontally (the spreading phase). During this second phase, the fluid region becomes doubly connected and is not analysed by Lighthill. It can, however, also be studied analytically and the results are presented here. We also extend a similar analysis, in an application to turbomachinery, due to Furber & Ffowcs Williams (J. Fluid Mech., vol. 94, 1979, p. 519). © 2009 Cambridge University Press.

Description: The behaviour of an initially planar shock wave propagating into a linearly convergent wedge is investigated experimentally and numerically. In the experiment, a 25 internal wedge is mounted asymmetrically in a pressure-driven shock tube. Shock waves with incident Mach numbers in the ranges of 1.41.6 and 2.42.6 are generated in nitrogen and carbon dioxide. During each run, the full pressure history is recorded at fourteen locations along the wedge faces and schlieren images are produced. Numerical simulations performed based on the compressible Euler equations are validated against the experiment. The simulations are then used as an additional tool in the investigation. The linearly convergent geometry strengthens the incoming shock repeatedly, as waves reflected from the wedge faces cross the interior of the wedge. This investigation shows that aspects of this structure persist through multiple reflections and influence the nature of the shock-wave focusing. The shock focusing resulting from the distributed reflected waves of the Mach 1.5 case is distinctly different from the stepwise focusing at the higher incoming shock Mach number. Further experiments using CO2 instead of N2 elucidate some relevant real-gas effects and suggest that the presence or absence of a weak leading shock on the distributed reflections is not a controlling factor for focusing. © 2009 Cambridge University Press.

Description: In this article, we study the influence of solid-to-fluid density ratio m on the type of vortex-induced oscillation of a square section prism placed inside a two-dimensional channel. We assume that the solid body has neither structural damping nor spring restoring force. Accordingly, the prism equation of motion contains only inertia and aerodynamics forces. The problem is considered in the range of Reynolds numbers Re ε [50 200] (based on the prism cross-section height h) and channel widths H = H'/h ε [2.5 10]. We found that, for each Re and H, there is a critical mass ratio mc that separates two different oscillation regimes. For m 〉 mc, the prism oscillation is periodical and contains a single harmonic. For m 〈 mc, the prism oscillation changes completely and assumes an irregular pattern that is characterized by multiple harmonics that appear to belong to a uniform spectrum. The change from one regime to the other is abrupt and we were not able to observe a transitional regime in which the number of response harmonics grew by finite steps. The value of the critical mass ratio grows along with the Reynolds number and the channel width. © 2009 Cambridge University Press.

Description: Natural convection in calm near-shore waters induced by daytime heating or nighttime cooling plays a significant role in cross-shore exchanges with significant biological and environmental implications. Having previously reported an improved scaling analysis on the daytime radiation-induced natural convection, the authors present in this paper a detailed scaling analysis quantifying the flow properties at varying offshore distances induced by nighttime surface cooling. Two critical functions of offshore distance have been derived to identify the distinctness and the stability of the thermal boundary layer. Two flow scenarios are possible depending on the bottom slope. For the relatively large slope scenario, three flow regimes are possible, which are discussed in detail. For each flow regime, all the possible distinctive subregions are identified. Two different sets of scaling incorporating the offshore-distance dependency have been derived for the conduction-dominated region and stable-convection-dominated region respectively. It is found that the scaling for flow in the stable-convection-dominated region also applies to the time-averaged mean flow in the unstable region. The present scaling results are verified by numerical simulations. © 2010 Cambridge University Press.

Description: This paper investigates the noise radiation produced by a rotor inside a duct, which is convected by a swirling-translating mean flow. The study is based on an extension of Gennaretti's and Morino's boundary element method to the frequency domain for scattering problems in conjunction with a spinning rotor source model in the presence of a swirl flow. The proposed formulation is validated against exact solutions and is further used to investigate the effects of the translating flow Mach number and swirling flow angular velocity on noise radiation to the far field. The scattered sound is highly affected by the convecting mean flow. The modal content of the scattered field increases when increasing the translating flow Mach number, while a swirling flow leads to a reduction of the mode propagation, if co-rotating with respect to the azimuthal order of the spinning source, or an increase of the modal content, if counter-rotating with respect to the source. In general, the mean translating flow moves the main lobes of the directivity patterns downstream, while in some cases the mean swirling flow neglects this effect and the downstream lobe is completely shifted. © 2009 Cambridge University Press.

Description: The free fall of a liquid mass through vertical tubes with a weakly increasing cross-section induces an acceleration of the upper liquid interface larger than gravity. The phenomenon is well described by a one-dimensional inviscid model. The super acceleration of the upper interface comes from the additional positive pressure gradient caused by the expanding geometry, which adds to the gravity body force. A perturbative expansion of this base solution further accounts for the interface shape and stability. In particular, the positive pressure gradient at the interface makes it unstable, forming a concentrated nipple on top of the essentially flat base solution. We discuss the possible connexion of these findings with the problem of wave breaking in free surface flows. © 2010 Cambridge University Press.

Description: A model is developed for a turbulent plume with heterogeneous chemical reaction rising in an unbounded environment. The chemical reaction, which may generate or deplete buoyancy in the plume, occurs at the interface between two phases, a continuous phase and a dispersed one. We study the case in which a buoyant reactant is released at the source and forms the dispersed phase, consisting of very small bubbles, droplets or particles. Once in contact with the ambient fluid, a first-order irreversible reaction takes place at the surface of the, for example, droplets. The behaviour of this plume in a uniform and stratified environment is examined. We show that the dynamics of a pure plume with such heterogeneous reaction is completely determined by the ratio of the environmental buoyancy frequency N and a frequency parameter associated with the chemical reaction, G. The group G is a measure of the ability of the reaction to generate buoyancy in the plume. In a uniform environment, the sign of parameter G fully determines the plume motion. When the reaction generates buoyancy (positive G) the motion is unbounded, whilst when reaction depletes buoyancy (negative G) the plume reaches a level of neutral buoyancy. A relation for this neutral buoyancy level as a function of the initial buoyancy flux of the plume and G is calculated. Our theoretical predictions compared well with experimental results using a plume of calcium carbonate particles descending in an acidic aqueous solution. In a stratified environment, the motion of the plume is always bounded, irrespective of the magnitude of G, and we determine the level of maximum buoyancy flux, as well as those of zero buoyancy and zero momentum as a function of N/G. Finally, our model is applied to study the dynamics of a localized release of carbon dioxide in the ocean. © 2010 Cambridge University Press.

Description: Direct numerical simulation (DNS) of the near field of a three-dimensional spatially developing turbulent lifted hydrogen jet flame in heated coflow is performed with a detailed mechanism to determine the stabilization mechanism and the flame structure. The DNS was performed at a jet Reynolds number of 11,000 with over 940 million grid points. The results show that auto-ignition in a fuel-lean mixture at the flame base is the main source of stabilization of the lifted jet flame. A chemical flux analysis shows the occurrence of near-isothermal chemical chain branching preceding thermal runaway upstream of the stabilization point, indicative of hydrogen auto-ignition in the second limit. The Damkhler number and key intermediate-species behaviour near the leading edge of the lifted flame also verify that auto-ignition occurs at the flame base. At the lifted-flame base, it is found that heat release occurs predominantly through ignition in which the gradients of reactants are opposed. Downstream of the flame base, both rich-premixed and non-premixed flames develop and coexist with auto-ignition. In addition to auto-ignition, Lagrangian tracking of the flame base reveals the passage of large-scale flow structures and their correlation with the fluctuations of the flame base. In particular, the relative position of the flame base and the coherent flow structure induces a cyclic motion of the flame base in the transverse and axial directions about a mean lift-off height. This is confirmed by Lagrangian tracking of key scalars, heat release rate and velocity at the stabilization point. © 2009 Cambridge University Press.

Description: The deformation of a drop as it flows along the axis of a circular capillary in low Reynolds number pressure-driven flow is investigated numerically by means of boundary integral computations. If gravity effects are negligible, the drop motion is determined by three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary, the viscosity ratio λ between the drop phase and the wetting phase and the capillary number Ca, which measures the relative importance of viscous and capillary forces. We investigate the drop behaviour in the parameter space (a/R,λ, Ca), at capillary numbers higher than those considered previously. If the fluid flow rate is maintained, the presence of the drop causes a change in the pressure difference between the ends of the capillary, and this too is investigated. Estimates for the drop deformation at high capillary number are based on a simple model for annular flow and, in most cases, agree well with full numerical results if λ ≥ 1/2, in which case the drop elongation increases without limit as Ca increases. If λ 〈 1/2, the drop elongates towards a limiting non-zero cylindrical radius. Low-viscosity drops (λ 〈 1) break up owing to a re-entrant jet at the rear, whereas a travelling capillary wave instability eventually develops on more viscous drops (〉 1). A companion paper (Lac & Sherwood, J. Fluid Mech., doi:10.1017/S002211200999156X) uses these results in order to predict the change in electrical streaming potential caused by the presence of the drop when the capillary wall is charged. © 2009 Cambridge University Press.

Description: The relaxation and breakup of an elongated droplet in a viscous and initially quiescent fluid is studied by solving the full NavierStokes equations using a three-dimensional finite volume method coupled with a moving mesh interface tracking (MMIT) scheme to locate the interface. The two fluids are assumed incompressible and immiscible. The interface is represented as a surface triangle mesh with zero thickness that moves with the fluid. Therefore, the jump and continuity conditions across the interface are implemented directly, without any smoothing of the fluid properties. Mesh adaptations on a tetrahedral mesh are employed to permit large deformation and to capture the changing curvature. Mesh separation is implemented to allow pinch-off. The detailed investigations of the relaxation and breakup process are presented in a more general flow regime compared to the previous works by Stone & Leal (J. Fluid Mech., vol. 198, 1989, p. 399) and Tong & Wang (Phys. Fluids, vol. 19, 2007, 092101), including the flow field of the both phases. The simulation results reveal that the vortex rings due to the interface motion and the conservation of mass play an important role in the relaxation and pinch-off process. The vortex rings are created and collapsed during the process. The effects of viscosity ratio, density ratio and length ratio on the relaxation and breakup are studied. The simulations indicate that the fluid velocity field and the neck shape are distinctly different for viscosity ratios larger and smaller than O(1), and thus a different end-pinching mechanism is observed for each regime. The length ratio also significantly affects the relaxation process and the velocity distributions, but not the neck shape. The influence of the density ratio on the relaxation and breakup process is minimal. However, the droplet evolution is retarded due to the large density of the suspending flow. The formation of a satellite droplet is observed, and the volume of the satellite droplet depends strongly on the length ratio and the viscosity ratio. © 2009 Cambridge University Press.

Description: A new method is introduced for estimating the convection velocity of individual modes in turbulent shear flows that, in contrast to most previous ones, only requires spectral information in the temporal or spatial direction over which a modal decomposition is desired, while only using local derivatives in other directions. If no spectral information is desired, the method provides a natural definition for the average convection velocity, as well as a way to estimate the accuracy of the frozen-turbulence approximation. Existing data from numerical turbulent channels at friction Reynolds numbers Reτ ≤ 1900 are used to validate the new method against classical ones, and to characterize the dependence of the convection velocity on the eddy wavelength and wall distance. The results indicate that the small scales in turbulent channels travel at the local mean velocity, while large global modes travel at a more uniform speed proportional to the bulk velocity. To estimate the systematic deviations introduced in experimental spectra by the use of Taylor's approximation with a wavelength-independent convection velocity, a semi-empirical fit to the computed convection velocities is provided. It represents well the data throughout the Reynolds number range of the simulations. It is shown that Taylor's approximation not only displaces the large scales near the wall to shorter apparent wavelengths but also modifies the shape of the spectrum, giving rise to spurious peaks similar to those observed in some experiments. To a lesser extent the opposite is true above the logarithmic layer. The effect increases with the Reynolds number, suggesting that some of the recent challenges to the kx1 energy spectrum may have to be reconsidered. © 2009 Cambridge University Press.

Description: Explicit expressions for the added mass tensor of a bubble in strongly nonlinear deformation and motion near a plane wall are presented. Time evolutions and interconnections of added mass components are derived analytically and analysed. Interface dynamics have been predicted with two methods, assuming that the flow is irrotational, that the fluid is perfect and with the neglect of gravity. The assumptions that gravity and viscosity are negligible are verified by investigating their effects and by quantifying their impact in some cases of strong deformation, and criteria are presented to specify the conditions of their validity. The two methods are an analytical one and the boundary element method, and good agreement is found. It is explained why a strongly deforming bubble is decelerated. The classical RayleighPlesset equation is extended with terms to account for arbitrary, axisymmetric deformation and to account for the proximity of a wall. An expression for the corresponding cycle frequency that is valid in the vicinity of the wall is derived. An equation similar to the RayleighPlesset equation is presented for the most important anisotropic deformation mode. Well-known expressions for the angular frequencies of some periodic solutions without a wall follow easily from the equations presented. A periodically deforming bubble without initial velocity of the centroid and without a dominating isotropic deformation component is eventually always driven towards the wall. A simplified equation of motion of the centre of a deforming bubble is presented. If desired, full deformation computations can be speeded up by selecting an artificially low value of the polytropic constant Cp/Cv. © 2009 Cambridge University Press.

Description: In addition to being observable in laboratory experiments, internal wave beams are reported in geophysical settings, which are characterized by non-uniform density stratifications. Here, we perform a combined theoretical and experimental study of the propagation of internal wave beams in non-uniform density stratifications. Transmission and reflection coefficients, which can differ greatly for different physical quantities, are determined for sharp density-gradient interfaces and finite-width transition regions, accounting for viscous dissipation. Thereafter, we consider even more complex stratifications to model geophysical scenarios. We show that wave beam ducting can occur under conditions that do not necessitate evanescent layers, obtaining close agreement between theory and quantitative laboratory experiments. The results are also used to explain recent field observations of a vanishing wave beam at the Keana Ridge, Hawaii. © 2009 Cambridge University Press.

Description: We propose a new unified model for the small, intermediate and large-scale evolution of freely decaying two-dimensional turbulence in the inviscid limit. The new model's centerpiece is a recent theory of vortex self-similarity (Dritschel et al., Phys. Rev. Lett., vol. 101, 2008, no. 094501), applicable to the intermediate range of scales spanned by an expanding population of vortices. This range is predicted to have a steep k5 energy spectrum. At small scales, this gives way to Batchelor's (Batchelor, Phys. Fluids, vol. 12, 1969, p. 233) k3 energy spectrum, corresponding to the (forward) enstrophy (mean square vorticity) cascade or, physically, to thinning filamentary debris produced by vortex collisions. This small-scale range carries with it nearly all of the enstrophy but negligible energy. At large scales, the slow growth of the maximum vortex size (∼t1/6 in radius) implies a correspondingly slow inverse energy cascade. We argue that this exceedingly slow growth allows the large scales to approach equipartition (Kraichnan, Phys. Fluids, vol. 10, 1967, p. 1417; Fox & Orszag, Phys. Fluids, vol. 12, 1973, p. 169), ultimately leading to a k1 energy spectrum there. Put together, our proposed model has an energy spectrum (k, t) t1/3k1 at large scales, together with ξ(k, t) t2/3k5 over the vortex population, and finally ξ(k, t) ∝t1k3 over an exponentially widening small-scale range dominated by incoherent filamentary debris. Support for our model is provided in two parts. First, we address the evolution of large and ultra-large scales (much greater than any vortex) using a novel high-resolution vortex-in-cell simulation. This verifies equipartition, but more importantly allows us to better understand the approach to equipartition. Second, we address the intermediate and small scales by an ensemble of especially high-resolution direct numerical simulations. © 2009 Cambridge University Press.

Description: Using a combination of critical point theory of ordinary differential equations and numerical simulation for the three-dimensional unsteady NavierStokes equations, we study possible flow structures of the vortical flow, especially the unsteady vortex breakdown in the interaction between a normal shock wave and a longitudinal vortex. The topological structure contains two parts. One is the sectional streamline pattern in the cross-section perpendicular to the vortex axis. The other is the sectional streamline pattern in the symmetrical plane. In the cross-section perpendicular to the vortex axis, the sectional streamlines have spiral or centre patterns depending on a function λ(x, t) = 1/(∂/∂t+u∂/), where x is the coordinate corresponding to the vortex axis. If λ 〉 0, the sectional streamlines spiral inwards in the near region of the centre. If λ〈 0, the sectional streamlines spiral outwards in the same region. If λ= 0, the sectional streamlines form a nonlinear centre. If changes its sign along the vortex axis, one or more limit cycles appear in the sectional streamlines in the cross-section perpendicular to the vortex axis. Numerical simulation for two typical cases of shock induced vortex breakdown (Erlebacher, Hussaini & Shu, J. Fluid Mech., vol. 337, 1997, p. 129) is performed. The onset and time evolution of the vortex breakdown are studied. It is found that there are more limit cycles for the sectional streamlines in the cross-section perpendicular to the vortex axis. In addition, we find that there are quadru-helix structures in the tail of the vortex breakdown. © 2009 Cambridge University Press.

Description: Non-modal mechanisms underlying transient growth of propagating acoustic waves and non-propagating vorticity perturbations in an unbounded compressible shear flow are investigated, making use of closed form solutions. Propagating acoustic waves amplify mainly due to two mechanisms: growth due to advection of streamwise velocity that is typically termed as the lift-up mechanism leading for large Mach numbers to an almost linear increase in streamwise velocity with time and growth due to the downgradient irrotational component of the Reynolds stress leading to linear growth of acoustic wave energy for large times. Synergy between these mechanisms along with the downgradient solenoidal component of the Reynolds stress produces large and robust energy amplification. On the other hand, non-propagating vorticity perturbations amplify due to kinematic deformation of vorticity by the mean flow. For weakly compressible flows, an initial vorticity perturbation abruptly excites acoustic waves with exponentially small amplitude, and the energy gained by vorticity perturbations is transferred back to the mean flow. For moderate Mach numbers, a strong coupling between vorticity perturbations and acoustic waves is found with the energy gained by vorticity perturbations being transferred to acoustic waves that are abruptly excited by the vortex. Calculation of the optimal perturbations for a viscous flow shows that for low Mach numbers, acoustic wave excitation by vorticity perturbations and the subsequent growth of acoustic waves leads to robust energy growth of the order of Reynolds number, while for large Mach numbers, synergy between the lift-up mechanism and the downgradient solenoidal component of the Reynolds stress dominates the growth and leads to a comparable large amplification of streamwise velocity. © 2009 Cambridge University Press.

Description: The classic hydrodynamic Hele-Shaw problem is revisited in the context of evaluating the viscous resistance to low-Mach compressible viscous gas flows through shallow non-uniform micro-fluidic configurations. Our recent study of gas flows through constricted shallow micro-channels indicates that the failure of the standard Hele-Shaw approximation to satisfy the no-slip boundary condition at the sidewalls severely restricts its applicability. To overcome this we have extended the asymptotic scheme to incorporate an inner solution in the vicinity of the sidewalls (which, in turn, allows for the characterization of the effects of channel cross-section geometry) and its matching to an outer correction. We have compared the results of the present asymptotic analysis to existing exact analytic and numerical results for straight and uniform channels and to finite-element simulations for a 90° turn and a symmetric T-junction, which demonstrate a remarkably improved accuracy relative to the standard Hele-Shaw approximation. This suggests the present scheme as a viable alternative for the rapid performance estimate of micro-fluidic devices. © 2009 Copyright Cambridge University Press.

Description: A finite-amplitude propagating wave induces a drift in fluids. Understanding how drifts produced by many waves disperse pollutants has broad implications for geophysics and engineering. Previously, the effective diffusivity was calculated for a random set of small-amplitude surface and internal waves. Now, this is extended by Bhler & Holmes-Cerfon (J. Fluid Mech., 2009, this issue, vol. 638, pp. 526) to waves in a rotating shallow-water system in which the Coriolis force is accounted for, a necessary step towards oceanographic applications. It is shown that interactions of finite-amplitude waves affect particle velocity in subtle ways. An expression describing the particle diffusivity as a function of scale is derived, showing that the diffusivity can be substantially reduced by rotation. © 2009 Copyright Cambridge University Press.

Description: The effects of adverse pressure gradients on turbulent structures were investigated by carrying out direct numerical simulations of turbulent boundary layers subjected to adverse and zero pressure gradients. The equilibrium adverse pressure gradient flows were established by using a power law free-stream distribution U∞∼ xm. Two-point correlations of velocity fluctuations were used to show that the spanwise spacing between near-wall streaks is affected significantly by a strong adverse pressure gradient. Low-momentum regions are dominant in the middle of the boundary layer as well as in the log layer. Linear stochastic estimation was used to provide evidence for the presence of low-momentum regions and to determine their statistical properties. The mean width of such large-scale structures is closely associated with the size of the hairpin-like vortices in the outer layer. The conditionally averaged flow fields around events producing Reynolds stress show that hairpin-like vortices are the structures associated with the production of outer turbulence. The shapes of the vortices beyond the log layer were found to be similar when their length scales were normalized according to the boundary layer thickness. Estimates of the conditionally averaged velocity fields associated with the spanwise vortical motion were obtained by using linear stochastic estimation. These results confirm that the outer region of the adverse pressure gradient boundary layer is populated with streamwise-aligned vortex organizations, which are similar to those of the vortex packet model proposed by Adrian, Meinhart & Tomkins (J. Fluid Mech., vol. 422, 2000, pp. 154). The adverse pressure gradient augments the inclination angles of the packets and the mean streamwise spacing of the vortex heads in the packets. © 2009 Cambridge University Press.

Description: We study steep capillary-gravity waves that form at the interface between two stably stratified layers of immiscible liquids in a horizontally oscillating vessel. The oscillatory nature of the external forcing prevents the waves from overturning, and thus enables the development of steep waves at large forcing. They arise through a supercritical pitchfork bifurcation, characterized by the square root dependence of the height of the wave on the excess vibrational Froude number (W, square root of the ratio of vibrational to gravitational forces). At a critical value Wc, a transition to a linear variation in W is observed. It is accompanied by sharp qualitative changes in the harmonic content of the wave shape, so that trochoidal waves characterize the weakly nonlinear regime, but finger-like waves form for W ≥ Wc. In this strongly nonlinear regime, the wavelength is a function of the product of amplitude and frequency of forcing, whereas for W 〈 Wc, the wavelength exhibits an explicit dependence on the frequency of forcing that is due to the effect of viscosity. Most significantly, the radius of curvature of the wave crests decreases monotonically with W to reach the capillary length for W = Wc, i.e. the lengthscale for which surface tension forces balance gravitational forces. For W 〈 Wc, gravitational restoring forces dominate, but for W ≥ Wc, the wave development is increasingly defined by localized surface tension effects. © 2009 Cambridge University Press.

Description: We consider shallow-water flow past a broad bottom ridge, localized in the flow direction, using the framework of the forced SuGardner (SG) system of equations, with a primary focus on the transcritical regime when the Froude number of the oncoming flow is close to unity. These equations are an asymptotic long-wave approximation of the full Euler system, obtained without a simultaneous expansion in the wave amplitude, and hence are expected to be superior to the usual weakly nonlinear Boussinesq-type models in reproducing the quantitative features of fully nonlinear shallow-water flows. A combination of the local transcritical hydraulic solution over the localized topography, which produces upstream and downstream hydraulic jumps, and unsteady undular bore solutions describing the resolution of these hydraulic jumps, is used to describe various flow regimes depending on the combination of the topography height and the Froude number. We take advantage of the recently developed modulation theory of SG undular bores to derive the main parameters of transcritical fully nonlinear shallow-water flow, such as the leading solitary wave amplitudes for the upstream and downstream undular bores, the speeds of the undular bores edges and the drag force. Our results confirm that most of the features of the previously developed description in the framework of the unidirectional forced Kortewegde Vries (KdV) model hold up qualitatively for finite amplitude waves, while the quantitative description can be obtained in the framework of the bidirectional forced SG system. Our analytic solutions agree with numerical simulations of the forced SG equations within the range of applicability of these equations. © 2009 Cambridge University Press.

Description: In this paper, the available potential energy (APE) framework of Winters et al. (J. Fluid Mech., vol. 289, 1995, p. 115) is extended to the fully compressible NavierStokes equations, with the aims of clarifying (i) the nature of the energy conversions taking place in turbulent thermally stratified fluids; and (ii) the role of surface buoyancy fluxes in the Munk & Wunsch (Deep-Sea Res., vol. 45, 1998, p. 1977) constraint on the mechanical energy sources of stirring required to maintain diapycnal mixing in the oceans. The new framework reveals that the observed turbulent rate of increase in the background gravitational potential energy GPEr, commonly thought to occur at the expense of the diffusively dissipated APE, actually occurs at the expense of internal energy, as in the laminar case. The APE dissipated by molecular diffusion, on the other hand, is found to be converted into internal energy (IE), similar to the viscously dissipated kinetic energy KE. Turbulent stirring, therefore, does not introduce a new APE/GPEr mechanical-to- mechanical energy conversion, but simply enhances the existing IE/GPEr conversion rate, in addition to enhancing the viscous dissipation and the entropy production rates. This, in turn, implies that molecular diffusion contributes to the dissipation of the available mechanical energy ME = APE + KE, along with viscous dissipation. This result has important implications for the interpretation of the concepts of mixing efficiency λ mixing and flux Richardson number Rf, for which new physically based definitions are proposed and contrasted with previous definitions. The new framework allows for a more rigorous and general re-derivation from the first principles of Munk & Wunsch (1998, hereafter MW98)'s constraint, also valid for a non-Boussinesq ocean: G(KE})≈ 1-ξRI/ξR fWr,forcing =1+(1-ξgamma;/ mixingξγmixingWr,forcing where G(KE) is the work rate done by the mechanical forcing, Wr, forcing is the rate of loss of GPEr due to high-latitude cooling and is a nonlinearity parameter such thatξ = 1 for a linear equation of state (as considered by MW98),ξ but 〈 1 otherwise. The most important result is that G(APE), the work rate done by the surface buoyancy fluxes, must be numerically as large as Wr,forcing and, therefore, as important as the mechanical forcing in stirring and driving the oceans. As a consequence, the overall mixing efficiency of the oceans is likely to be larger than the value γmixing = 0.2 presently used, thereby possibly eliminating the apparent shortfall in mechanical stirring energy that results from using γmixing = 0.2 in the above formula. © 2009 Copyright Cambridge University Press.

Description: The paper investigates the phenomena occurring in a TaylorCouette flow system subject to a steady axial pressure gradient in a small envelope of the TaylorReynolds state space under transitional regimes. A remarkable net power reduction necessary to simultaneously drive the two flows compared to that required to drive the TaylorCouette flow alone is documented under non-trivial conditions. The energy transfer process characterizing the large-scale coherent structures is investigated by processing a set of statistically independent realizations obtained from direct numerical simulation. The analysis is conducted with an incompressible three-dimensional NavierStokes flow solver employing a spectral representation of the unknowns. © 2009 Cambridge University Press.

Description: We consider the two-dimensional buoyancy driven flow of a fluid injected into a saturated semi-infinite porous medium bounded by a horizontal barrier in which a single line sink, representing a fissure some distance from the point of injection, allows leakage of buoyant fluid. Our studies are motivated by the geological sequestration of carbon dioxide (CO2) and the possibility that fissures in the cap rock may compromise the safe long-term storage of CO2. A theoretical model is presented that accounts for leakage through the fissure using two parameters, which characterize leakage driven both by the hydrostatic pressure within the overriding fluid and by the buoyancy of the fluid within the fissure. We determine numerical solutions for the evolution of both the gravity current within the porous medium and the volume of fluid that has escaped through the fissure as a function of time. A quantity of considerable practical interest is the efficiency of storage, which we define as the amount of fluid remaining in the porous medium relative to the amount injected. This efficiency scales like t1/2 at late times, indicating that the efficiency of storage ultimately tends to zero. We confirm the results of our model by comparison with an analogue laboratory experiment and discuss the implications of our two-dimensional model of leakage from a fissure for the geological sequestration of CO2. © 2009 Cambridge University Press.

Description: New explicit subgrid stress models are proposed involving the strain rate and rotation rate tensor, which can account for rotation in a natural way. The new models are based on the same methodology that leads to the explicit algebraic Reynolds stress model formulation for Reynolds-averaged NavierStokes simulations. One dynamic model and one non-dynamic model are proposed. The non-dynamic model represents a computationally efficient subgrid scale (SGS) stress model, whereas the dynamic model is the most accurate. The models are validated through large eddy simulations (LESs) of spanwise and streamwise rotating channel flow and are compared with the standard and dynamic Smagorinsky models. The proposed explicit dependence on the system rotation improves the description of the mean velocity profiles and the turbulent kinetic energy at high rotation rates. Comparison with the dynamic Smagorinsky model shows that not using the eddy-viscosity assumption improves the description of both the Reynolds stress anisotropy and the SGS stress anisotropy. LESs of rotating channel flow at Re = 950 have been carried out as well. These reveal some significant Reynolds number influences on the turbulence statistics. LESs of non-rotating turbulent channel flow at Re = 950 show that the new explicit model especially at coarse resolutions significantly better predicts the mean velocity, wall shear and Reynolds stresses than the dynamic Smagorinsky model, which is probably the result of a better prediction of the anisotropy of the subgrid dissipation. © 2009 Cambridge University Press.

Description: We study surface switching quantitatively in flows driven by the constant rotation of the endwall of an open cylindrical vessel reported by Suzuki, Iima & Hayase (Phys. Fluids, vol. 18, 2006, p. 101701): the deformed free surface switches between axisymmetric and non-axisymmetric shapes accompanied by irregular vertical oscillation. Detailed simultaneous measurements showed that the magnitude of the velocity fluctuations (turbulent intensity) temporally varies greatly and are strongly correlated with the surface height, suggesting that dynamic switching between laminar and turbulent states is accompanied by vessel-scale surface shape changes. The study also identified clear hysteresis in the turbulent intensity arising from changes in the Reynolds number; the bifurcation diagram consists of two overlapping branches representing a high-intensity (turbulent) state and a low-intensity (laminar) state. Based on the results, a switching mechanism is suggested. © 2009 Copyright Cambridge University Press.

Description: We examine the linear stability of a capillary rivulet under the assumption that it is shallow enough to be described by the lubrication approximation. It is shown that rivulets on a sloping plate are stable regardless of their parameters, whereas rivulets on the underside of a plate can be either stable or unstable, depending on their widths and the plate's slope. For the case of a horizontal plate, sufficiently narrow rivulets are shown to be stable and sufficiently wide ones unstable, with the threshold width being /2(/g) 1/2(and are the liquid's density and surface tension, g is the acceleration due to gravity). It is also shown that, even though the plate's slope induces in a rivulet a sheared flow (which would normally be viewed as a source of instability) in the present problem, it is a stabilizing factor. The corresponding stability criterion involving the rivulet's width and the plate's slope is computed, and it is demonstrated that, if the latter is sufficiently strong, all rivulets are stable regardless of their widths. © 2009 Copyright Cambridge University Press 2009.

Description: A simplified one-dimensional partial differential equation for the integral axial momentum flux during the deceleration phase of single-pulsed transient incompressible jets is derived and solved analytically. The wave speed of the derived first-order nonlinear wave equation shows that the momentum flux transient from the deceleration phase propagates downstream at twice the initial jet penetration rate. Transient-jet velocity data from the existing literature is shown to be consistent with this derivation, and an algebraic analytical solution matches the measured timing and decay of axial velocity after the deceleration transient. The solution also shows that a wave of increased entrainment accompanies the deceleration transient as it travels downstream through the jet. In the long-time limit, the peak entrainment rate at the leading edge of this entrainment wave approaches an asymptotic value of three times that of the initial steady jet. The rate of approach to the asymptotic behaviour is controlled by the deceleration rate, which suggests that rate-shaping may be tailored to achieve a desired mixing state at a given time after the end of a single-pulsed jet. In the wake of the entrainment wave, the absolute entrainment rate eventually decays to zero. The local injected fluid concentration also decays, however, so that entrainment rate relative to the local concentration of injected fluid remains higher than in the initial steady jet. An analysis of diesel engine fuel-jets is provided as one example of a transient-jet application in which the considerable increase in the mixing rate after the deceleration phase has important implications. © 2009 Copyright Cambridge University Press.

Description: Elements of the first-principles-based theory of Wei et al. (J. Fluid Mech., vol. 522, 2005, p. 303), Fife et al. (Multiscale Model. Simul., vol. 4, 2005a, p. 936; J. Fluid Mech., vol. 532, 2005b, p. 165) and Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst., vol. 24, 2009, p. 781) are clarified and their veracity tested relative to the properties of the logarithmic mean velocity profile. While the approach employed broadly reveals the mathematical structure admitted by the time averaged NavierStokes equations, results are primarily provided for fully developed pressure driven flow in a two-dimensional channel. The theory demonstrates that the appropriately simplified mean differential statement of Newton's second law formally admits a hierarchy of scaling layers, each having a distinct characteristic length. The theory also specifies that these characteristic lengths asymptotically scale with distance from the wall over a well-defined range of wall-normal positions, y. Numerical simulation data are shown to support these analytical findings in every measure explored. The mean velocity profile is shown to exhibit logarithmic dependence (exact or approximate) when the solution to the mean equation of motion exhibits (exact or approximate) self-similarity from layer to layer within the hierarchy. The condition of pure self-similarity corresponds to a constant leading coefficient in the logarithmic mean velocity equation. The theory predicts and clarifies why logarithmic behaviour is better approximated as the Reynolds number gets large. An exact equation for the leading coefficient (von Kármán coefficient κ) is tested against direct numerical simulation (DNS) data. Two methods for precisely estimating the leading coefficient over any selected range of y are presented. These methods reveal that the differences between the theory and simulation are essentially within the uncertainty level of the simulation. The von Krmn coefficient physically exists owing to an approximate self-similarity in the flux of turbulent force across an internal layer hierarchy. Mathematically, this self-similarity relates to the slope and curvature of the Reynolds stress profile, or equivalently the slope and curvature of the mean vorticity profile. The theory addresses how, why and under what conditions logarithmic dependence is approximated relative to the specific mechanisms contained within the mean statement of dynamics. © 2009 Copyright Cambridge University Press.

Description: Buffeting flow on transonic aerofoils serves as a model problem for the more complex three-dimensional flows responsible for aeroplane buffet. The origins of transonic aerofoil buffet are linked to a global instability, which leads to shock oscillations and dramatic lift fluctuations. The problem is analysed using the Reynolds-averaged Navier-Stokes equations, which for the foreseeable future are a necessary approximation to cover the high Reynolds numbers at which transonic buffet occurs. These equations have been shown to reproduce the key physics of transonic aerofoil flows. Results from global-stability analysis are shown to be in good agreement with experiments and numerical simulations. The stability boundary, as a function of the Mach number and angle of attack, consists of an upper and a lower branch - the lower branch shows features consistent with a supercritical bifurcation. The unstable modes provide insight into the basic character of buffeting flow at near-critical conditions and are consistent with fully nonlinear simulations. The results provide further evidence linking the transonic buffet onset to a global instability. © 2009 Cambridge University Press.

Description: A study of the Faraday instability of diffuse interfaces between pairs of miscible liquids of different densities, by means of experiments and by a nonlinear numerical model, is presented. The experimental set-up consisted of a rectangular cell in which the lighter liquid was placed above the denser one. The cell in this initially stable configuration was then subjected to vertical vibrations. The subsequent behaviour of the 'interface' between the two liquids was observed with a high-speed camera. This study shows that above a certain acceleration threshold an instability developed at the interface. The amplitude of the instability grew during the experiments which then led to the mixing of the liquids. The instability finally disappeared once the two liquids were fully mixed over a volume, considerably larger than the initial diffuse region. The results of a companion two-dimensional nonlinear numerical model that employs a finite volume method show very good agreement with the experiments. A physical explanation of the instability and the observations are advanced. © 2009 Cambridge University Press.

Description: Homogeneous, approximately isotropic turbulence at two Taylor-scale Reynolds numbers, Rλ = 50, 190, with a mean transverse temperature gradient is passed through an axisymmetric contraction. The effects of the straining on the velocity field, and on the passive scalar field, are investigated within the contraction as are the effects of releasing the strain in the post-contraction region. Components of the fluctuating velocity and scalar gradient covariance are measured in order to understand their relation to the large-scale anisotropy of the flow. The scale-dependent spectral evolution of the scalar is also determined. A tensor model is constructed to predict the evolution of the fluctuating scalar gradient covariance. The model constants are determined in the post-contraction relaxation region, where the flow geometry does not vary. The model is shown to perform well throughout the flow, even in the contraction in which the geometry varies. Rapid distortion theory is applied to the scalar field in the contraction, and its solutions are compared to the experimental results. © 2009 Cambridge University Press.

Description: In this paper we investigate the relationship between the large- and small-scale energy-containing motions in wall turbulence. Recent studies in a high-Reynolds-number turbulent boundary layer (Hutchins & Marusic, Phil. Trans. R. Soc. Lond. A, vol. 365, 2007 a, pp. 647-664) have revealed a possible influence of the large-scale boundary-layer motions on the small-scale near-wall cycle, akin to a pure amplitude modulation. In the present study we build upon these observations, using the Hilbert transformation applied to the spectrally filtered small-scale component of fluctuating velocity signals, in order to quantify the interaction. In addition to the large-scale log-region structures superimposing a footprint (or mean shift) on the near-wall fluctuations (Townsend, The Structure of Turbulent Shear Flow , 2nd edn., 1976, Cambridge University Press; Metzger & Klewicki, Phys. Fluids, vol. 13, 2001, pp. 692-701.), we find strong supporting evidence that the small-scale structures are subject to a high degree of amplitude modulation seemingly originating from the much larger scales that inhabit the log region. An analysis of the Reynolds number dependence reveals that the amplitude modulation effect becomes progressively stronger as the Reynolds number increases. This is demonstrated through three orders of magnitude in Reynolds number, from laboratory experiments at Reτ ∼ 103- 104 to atmospheric surface layer measurements at Reτ ∼ 106. © 2009 Cambridge University Press.

Description: Assuming that the sediment flux in the Exner equation can be linearly related to the local bed slope, we establish a one-dimensional model for the bed-load transport of sediment in a coastal-plain depositional system, such as a delta and a continental margin. The domain of this model is defined by two moving boundaries: the shoreline and the alluvial-bedrock transition. These boundaries represent fundamental transitions in surface morphology and sediment transport regime, and their trajectories in time and space define the evolution of the shape of the sedimentary prism. Under the assumptions of fixed bedrock slope and sea level the model admits a closed-form similarity solution for the movements of these boundaries. A mapping of the solution space, relevant to field scales, shows two domains controlled by the relative slopes of the bedrock and fluvial surface: one in which changes in environmental parameters are mainly recorded in the upstream boundary and another in which these changes are mainly recorded in the shoreline. We also find good agreement between the analytical solution and laboratory flume experiments for the movements of the alluvial-bedrock transition and the shoreline. © 2009 Cambridge University Press.

Description: Lattice kinetic equations incorporating the effects of external/internal force fields via a shift of the local fields in the local equilibria are placed within the framework of continuum kinetic theory. The mathematical treatment reveals that in order to be consistent with the correct thermo-hydrodynamical description, temperature must also be shifted, besides momentum. New perspectives for the formulation of thermo-hydrodynamic lattice kinetic models of non-ideal fluids are then envisaged. It is also shown that on the lattice, the definition of the macroscopic temperature requires the inclusion of new terms directly related to discrete effects. The theoretical treatment is tested against a controlled case with a non-ideal equation of state. © 2009 Cambridge University Press.

Description: The problem of unsteady behaviour of a floating thin plate is solved. The simultaneous motion of the plate and the fluid is considered within the framework of linear shallow-water theory. It is assumed that the bottom is not uniform in depth under the heterogeneous plate represented by an infinitely extended strip of finite width. The elastic deflection of the plate is expressed by a superposition of modal functions of a homogeneous beam with free edge conditions. The time-dependent unknown amplitudes are determined from the solution of a linear set of ordinary differential equations with constant coefficients. The eigenvalues of this set are determined numerically. Proposed method is used for the solution of three unsteady problems: the scattering of localized surface wave by an elastic plate, decay of the initial deformation of the plate in the fluid at rest and the action of a periodic load on a plate. Numerical calculations are performed for the ice sheet with the variable thickness and various bottom topographies. © 2009 Copyright Cambridge University Press.

Description: The role of fluid motion in delivery of nutrients to phytoplankton cells is a fundamental question in biological and chemical oceanography. In the study of mass transfer to phytoplankton, diatoms are of particular interest. They are non-motile, are often the most abundant components in aggregates and often form chains, so they are the ones expected to benefit most from enhancement of nutrient flux due to dissipating turbulence. Experimental data to test the contribution of advection to nutrient acquisition by phytoplankton are scarce, mainly because of the inability to visualize, record and thus imitate fluid motions in the vicinities of cells in natural flows. Laboratory experiments have most often used steady Couette flows to simulate the effects of turbulence on plankton. However, steady flow, producing spatially uniform shear, fails to capture the diffusion of momentum and vorticity, the essence of turbulence. Thus, numerical modelling plays an important role in the study of effects of fluid motion on diffusive and advective nutrient fluxes. In this paper we use the immersed boundary method to model the interaction of rigid and flexible diatom chains with the surrounding fluid and nutrients. We examine this interaction in two nutrient regimes, a uniform background concentration of nutrients, such as might be typical of an early spring bloom, and a contrasting scenario in which nutrients are supplied as small, randomly distributed pulses, as is more likely for oligotrophic seas and summer conditions in coastal and boreal seas. We also vary the length and flexibility of chains, as whether chains are straight or bent, rigid or flexible will affect their behaviour in the flow and hence their nutrient fluxes. The results of numerical experiments suggest that stiff chains consume more nutrients than solitary cells. Stiff chains also experience larger nutrient fluxes compared to flexible chains, and the nutrient uptake per cell increases with increasing stiffness of the chain, suggesting a major advantage of silica frustules in diatoms. © 2009 Copyright Cambridge University Press.

Description: As a stepping stone towards understanding acoustic resonances in axial flow compressors, acoustic resonances are computed numerically in fixed single and tandem plate cascades in an infinitely long annular duct. Applying perfectly matched layer absorbing boundary conditions in the form of the complex scaling method of atomic and molecular physics to approximate the radiation condition the resonance problem is transformed into an eigenvalue problem. Of particular interest are resonances with zero radiation damping (trapped modes) or very small radiation damping (nearly trapped modes). Such resonances can be excited by wakes from compressor cascades or struts. If the shedding frequency is sufficiently close to an acoustic resonant frequency, the latter may control the vortex shedding causing high-intensity tonal noise or occasionally even blade failure. All resonances are computed for zero mean flow approximating low-Mach-number flows. The influence of various cascade parameters on the resonant frequencies is studied and, whenever possible, our numerical results are compared with published experimental findings. © 2009 Cambridge University Press.

Description: Rotating convection in cylindrical containers is a canonical problem in fluid dynamics, in which a variety of simplifying assumptions have been used in order to allow for low-dimensional models or linear stability analysis from trivial basic states. An aspect of the problem that has received only limited attention is the influence of the centrifugal force, because it makes it difficult or even impossible to implement the aforementioned approaches. In this study, the mutual interplay between the three forces of the problem, Coriolis, gravitational and centrifugal buoyancy, is examined via direct numerical simulation of the Navier-Stokes equations in a parameter regime where the three forces are of comparable strengths in a cylindrical container with the radius equal to the depth so that wall effects are also of order one. Two steady axisymmetric basic states exist in this regime, and the nonlinear dynamics of the solutions bifurcating from them is explored in detail. A variety of bifurcated solutions and several codimension-two bifurcation points acting as organizing centres for the dynamics have been found. A main result is that the flow has simple dynamics for either weak heating or large centrifugal buoyancy. Reducing the strength of centrifugal buoyancy leads to subcritical bifurcations, and as a result linear stability is of limited utility, and direct numerical simulations or laboratory experiments are the only way to establish the connections between the different solutions and their organizing centres, which result from the competition between the three forces. Centrifugal effects primarily lead to the axisymmetrization of the flow and a reduction in the heat flux. © 2009 Cambridge University Press.

Description: Laboratory measurements of the post-breaking velocity field due to unsteady deep-water breaking are presented. Digital particle image velocimetry (DPIV) is used to measure the entire post-breaking turbulent cloud with high-resolution imagery permitting the measurement of scales from O (1m) down to O (1mm). Ensemble-averaged quantities including mean velocity, turbulent kinetic energy (TKE) density and Reynolds stress are presented and compare favourably with the results of Melville, Veron & White (J. Fluid Mech., vol. 454, 2002, pp. 203-233; MVW). However, due to limited resolution, MVW's measurements were not spatially coherent across the turbulent cloud, and this restricted their ability to compute turbulent wavenumber spectra. Statistical spatial quantities including the integral length scale L11, Taylor microscale λf and the Taylor microscale Reynolds number Reλ are presented. Estimation of an eddy viscosity for the breaking event is also given based on analysis of the image data. Turbulent wavenumber spectra are computed and within 12 wave periods after breaking exhibit what have been termed 'spectral bumps' in the turbulence literature. These local maxima in the spectra are thought to be caused by an imbalance between the transport of energy from large scales and the dissipation at small scales. Estimates of the dissipation rate per unit mass are computed using both direct and indirect methods. Horizontally averaged terms in the TKE budget are also presented up to 27 wave periods after breaking and are discussed with regard to the dynamics of the post-breaking flow. Comparisons of the TKE density in the streamwise and cross-stream planes with the three-dimensional full TKE density are given in an appendix. © 2009 Cambridge University Press.

Description: The flow of a partial-depth lock-exchange gravity current past an isolated bottom-mounted obstacle is studied by means of two-dimensional direct numerical simulations and steady shallow-water theory. The simulations indicate that the flux of the current downstream of the obstacle is approximately constant in space and time. This information is employed to extend the shallow-water models of Rottman et al. (J. Hazard. Mater., vol. 11, 1985, pp. 325340) and Lane-Serff, Beal & Hadfield (J. Fluid Mech., vol. 292, 1995, pp. 3953), in order to predict the height and front speed of the downstream current as functions of the upstream Froude number and the ratio of obstacle to current height. The model predictions are found to agree closely with the simulation results. In addition, the shallow-water model provides an estimate for the maximum drag that lies within 10% of the simulation results for obstacles much larger than the boundary-layer thickness. © 2009 Copyright Cambridge University Press.

Description: Two types of wall actuation in channel flow are considered: travelling waves of wall deformation (peristalsis) and travelling waves of blowing and suction. The flow response and its mechanisms are analysed using nonlinear and weakly nonlinear computations. We show that both actuations induce a flux in the channel in the absence of an imposed pressure gradient and can thus be characterized as pumping. In the context of flow control, pumping and drag reduction are strongly connected, and we seek to define them properly based on these two actuation examples. Movies showing the flow motion for the two types of actuation are available with the online version of this paper (journals.cambridge.org/FLM). © 2009 Copyright Cambridge University Press.

Description: The stabilization of buoyant flows by a magnetic field is an important matter for crystal growth applications. It is studied here when the cavity is an infinite channel with rectangular cross-section typical of horizontal Bridgman configurations and when the magnetic field is applied in the vertical and transverse directions. The steady basic flow solution is first calculated: the usual counter flow structure is modified by the magnetic field and evolves towards jets in the cross-section corners when the magnetic field is vertical and towards a more uniform structure in the transverse direction when the magnetic field is transverse. The stability results show a very good stabilization of the convective flows for a vertical magnetic field with exponential increases of the thresholds for any width of the channel and for various Prandtl numbers Pr. The results for a transverse magnetic field are more surprising as a destabilizing effect corresponding to an initial decrease of the thresholds is obtained at Pr=0 and for small channel widths. A kinetic energy budget at the thresholds reveals that the main destabilizing factor is connected to the vertical shear of the longitudinal basic flow and that it is the modifications affecting this shear energy which are mainly responsible for the variation of the thresholds when a magnetic field is applied. © 2009 Copyright Cambridge University Press.

Description: The majority of existing single-unit devices for extracting power from sea waves relies on resonance at the peak frequency of the incident wave spectrum. Such designs usually call for structural dimensions not too small compared to a typical wavelength and yield high efficiency only within a limited frequency band. A recent innovation in Norway departs from this norm by gathering many small buoys in a compact array. Each buoy is too small to be resonated in typical sea conditions. In this article a theoretical study is performed to evaluate this new design. Within the framework of linearization, we consider a periodic array of small buoys with similarly small separation compared to the typical wavelength. The method of homogenization (multiple scales) is used to derive the equations governing the macroscale behaviour of the entire array. These equations are then applied to energy extraction by an infinite strip of buoys, and by a circular array. In the latter case, advantages are found when compared to a single buoy of equal volume. © 2009 Copyright Cambridge University Press.

Description: Direct numerical simulation (DNS) is used to study the effects of mean lateral divergence and convergence on wall-bounded turbulence, by applying uniform irrotational temporal deformations to a plane-channel domain. This extends a series of studies of similar deformations. Fast and slow straining fields are considered, leading to a matrix of four cases, all corresponding to zero-pressure-gradient (ZPG) flows along the centreplane in ducts with constant rectangular cross-sectional area but varying aspect ratio. The results are used to address basic physical and modelling questions, and create a database that allows detailed yet straightforward testing of turbulence models. Initial tests of three representative one-point models reveal meaningful differences. The extra-strain effects introduced by the matrix of fast and slow divergence and convergence are documented, separating the direct effects of the strain from the indirect ones that alter the shear rate and change the distance from the wall. Some findings are predictable, and none contradict experimental findings. Others require more thought, notably an asymmetry between the effect of convergence and divergence on the peak turbulence kinetic energy. © 2009 Cambridge University Press.

Description: By identifying the stratification which leads to maximal buoyancy flux in a stably-stratified plane Couette flow, we make a prediction of what bulk stratification (as a function of the shear) is optimal for turbulent mixing. A previous attempt to do this (Caulfield, Tang & Plasting, J. Fluid Mech., vol. 498, 2004, p. 315) failed due to an unexpected degeneracy in the variational problem. Here, we overcome this issue by parameterizing the variational problem implicitly with the overall mixing efficiency which is then optimized across to return a rigorous upper bound on the buoyancy flux. We find that the bulk Richardson number quickly approaches 1/6 in the asymptotic limit of high shear with the associated mixing efficiency tending to 1/3. The predicted mean profiles associated with the bound appear to have a layered structure, with the gradient Richardson number being low both in the interior, and in boundary layers near the walls, with a global maximum, also equal to 1/6, occurring at the edge of the boundary layers. © 2009 Cambridge University Press.

Description: In this work a two-dimensional laminar flow past a square cylinder is considered. Actuators placed on the cylinder enable active control by blowing and suction. Proportional feedback control is then applied using velocity measurements taken in the cylinder wake. Projection onto an empirical subspace is combined with a calibration technique to build a low-order model of the incompressible NavierStokes equations. This model is used within an optimization method to determine a set of feedback gains which reduces the unsteadiness of the wake at Re = 150. The resulting controlled flows are further characterized by computing the critical Reynolds numbers for the onset of the vortex shedding instability. © 2009 Cambridge University Press.

Description: Radiometric force on a 0.12 m circular vane is studied experimentally and numerically over a wide range of pressures that cover the flow regimes from near free molecular to near continuum. In the experiment, the vane is resistively heated to about 419 K on one side and 394 K on the other side, and immersed in a rarefied argon gas. The radiometric force is then measured on a nano-Newton thrust stand in a 3 m vacuum chamber and compared with the present numerical predictions and analytical predictions proposed by various authors. The computational modelling is conducted with a kinetic approach based on the solution of ellipsoidal statistical BhatnagarGrossKrook (ES-BGK) equation. Numerical modelling showed the importance of regions with elevated pressure observed near the edges of the vane for the radiometric force production. A simple empirical expression is proposed for the radiometric force as a function of pressure that is found to be in good agreement with the experimental data. The shear force on the lateral side of the vane was found to decrease the total radiometric force. © 2009 Cambridge University Press.

Description: A wave basin experiment has been performed in the MARINTEK laboratories, in one of the largest existing three-dimensional wave tanks in the world. The aim of the experiment is to investigate the effects of directional energy distribution on the statistical properties of surface gravity waves. Different degrees of directionality have been considered, starting from long-crested waves up to directional distributions with a spread of ±30° at the spectral peak. Particular attention is given to the tails of the distribution function of the surface elevation, wave heights and wave crests. Comparison with a simplified model based on second-order theory is reported. The results show that for long-crested, steep and narrow-banded waves, the second-order theory underestimates the probability of occurrence of large waves. As directional effects are included, the departure from second-order theory becomes less accentuated and the surface elevation is characterized by weak deviations from Gaussian statistics. © 2009 Cambridge University Press.

Description: We study the classical problem of planar shock refraction at an oblique density discontinuity, separating two gases at rest. When the shock impinges on the density discontinuity, it refracts, and in the hydrodynamical case three signals arise. Regular refraction means that these signals meet at a single point, called the triple point. After reflection from the top wall, the contact discontinuity becomes unstable due to local Kelvin-Helmholtz instability, causing the contact surface to roll up and develop the Richtmyer-Meshkov instability (RMI). We present an exact Riemann-solver-based solution strategy to describe the initial self-similar refraction phase, by which we can quantify the vorticity deposited on the contact interface. We investigate the effect of a perpendicular magnetic field and quantify how its addition increases the deposition of vorticity on the contact interface slightly under constant Atwood number. We predict wave-pattern transitions, in agreement with experiments, von Neumann shock refraction theory and numerical simulations performed with the grid-adaptive code AMRVAC. These simulations also describe the later phase of the RMI. © 2009 Cambridge University Press.

Description: The quasi-steady migration and deformation of bubbles rising in a wall-bounded linear shear flow are investigated experimentally in the low-but-finite-Reynolds-number regime. A travelling optical device that follows the bubble is used for this purpose. This apparatus allows us to determine accurately the bubble radius, contour and rising speed, together with the distance between the bubble and the wall. Thereby the transverse component of the hydrodynamic force is obtained for Reynolds numbers Re (based on the bubble diameter and slip velocity of the bubble in the undisturbed shear flow) less than 5. The results indicate that in the range 0.5 〈 Re 〈 1.5, the transverse force acting on a spherical bubble agrees well with an extension of the theoretical solution obtained by McLaughlin (J. Fluid Mech., vol. 246, 1993, pp. 249-265) for rigid spheres, whereas it becomes larger than the theoretical prediction for Re 〉 1.5. In the regime in which bubble deformation is significant, the shape of the bubble and the deformation-induced transverse force are determined both experimentally and computationally, using a spectral boundary element method. Both estimates are found to be in good agreement with each other, while the theory of Magnaudet, Takagi & Legendre (J. Fluid Mech., vol. 476, 2003, pp. 115-157) is found to predict accurately the deformation but fails to predict quantitatively the deformation-induced transverse force. © 2009 Cambridge University Press.

Description: In order to predict response and wake modes for elastically mounted circular cylinders in a fluid flow, we employ controlled-vibration experiments, comprised of prescribed transverse vibration of a cylinder in the flow, over a wide regime of amplitude and frequency. A key to this study is the compilation of high-resolution contour plots of fluid force, in the plane of normalized amplitude and wavelength. With such resolution, we are able to discover discontinuities in the force and phase contours, which enable us to clearly identify boundaries separating different fluid-forcing regimes. These appear remarkably similar to boundaries separating different vortex-formation modes in the map of regimes by Williamson & Roshko (J. Fluids Struct., vol. 2, 1988, pp. 355381). Vorticity measurements exhibit the 2S, 2P and P + S vortex modes, as well as a regime in which the vortex formation is not synchronized with the body vibration. By employing such fine-resolution data, we discover a high-amplitude regime in which two vortex-formation modes overlap. Associated with this overlap regime, we identify a new distinct mode of vortex formation comprised of two pairs of vortices formed per cycle, where the secondary vortex in each pair is much weaker than the primary vortex. This vortex mode, which we define as the 2POVERLAP mode (2PO), is significant because it is responsible for generating the peak resonant response of the body. We find that the wake can switch intermittently between the 2P and 2PO modes, even as the cylinder is vibrating with constant amplitude and frequency. By examining the energy transfer from fluid to body motion, we predict a free-vibration response which agrees closely with measurements for an elastically mounted cylinder. In this work, we introduce the concept of an energy portrait, which is a plot of the energy transfer into the body motion and the energy dissipated by damping, as a function of normalized amplitude. Such a plot allows us to identify stable and unstable amplitude-response solutions, dependent on the rate of change of net energy transfer with amplitude (the sign of dE*/dA*). Our energy portraits show how the vibration system may exhibit a hysteretic mode transition or intermittent mode switching, both of which correspond with such phenomena measured from free vibration. Finally, we define the complete regime in the amplitudewavelength plane in which free vibration may exist, which requires not only a periodic component of positive excitation but also stability of the equilibrium solutions. © 2009 Cambridge University Press.

Description: The stability of electrohydrodynamic flow between two horizontal plates with a vertical electrical conductivity gradient has been investigated in the presence of an imposed weak shear flow. The weak shear flow is driven by the horizontal pressure gradient, and the electrical conductivity gradient is generated by the concentration variation of the charge-carrying solute. An external electric field is applied across the fluid layer, and then the interaction between the unstable stratification of electrohydrodynamic flow and the shear arising from the plane Poiseuille flow is studied. A linear stability analysis has been implemented by considering both the longitudinal and transverse modes. Unlike the thermally stratified plane Poiseuille flow in which the longitudinal mode always dominates the onset of instability and is virtually unaffected by the superimposed shear flow, the instability of this mixed electrohydrodynamicPoiseuille flow system is found to depend heavily on the shear flow, and the transverse mode may prevail over the longitudinal mode when the momentum of shear flow is sufficiently small. Particularly, an oscillatory longitudinal mode is found to exist, and it may become the critical mode when the conductivity gradient is small enough. The present results verify that an imposed weak shear flow may enhance the electrohydrodynamic instability in a fluid layer with electrical conductivity gradient. © 2009 Cambridge University Press.

Description: A theory on weakly rarefied low-Mach-number flows with surface reactions based on small sticking coefficients was formulated for a binary gas mixture with an irreversible surface reaction, and then extended to a multicomponent mixture with multi-step surface reactions for the situation when all chemically active species are small in concentration compared to a major inert species. Particular interest was placed on the interaction between the Knudsen layer and the surface reactions. Results show that the Knudsen layer modifies not only the incident flux of the molecules striking the surface but also the temperature-sensitive sticking coefficients, and consequently the surface reaction rates. The surface reactions in turn modify the flow structure in the Knudsen layer through the non-zero net flux at the surface. The rate expressions for the surface reactions based on sticking coefficients were derived, and the slip boundary conditions for the temperature and the species concentration suitable for application were established. The widely used Motz-Wise correction formula for the surface reaction rate was revised and the underlying assumptions leading to its derivation were shown to be inappropriate. © 2009 Cambridge University Press.

Description: We report an investigation of temperature profiles in turbulent Rayleigh-Bénard convection of water based on direct numerical simulations (DNS) for a cylindrical cell with unit aspect ratio for the same Prandtl number Pr and similar Rayleigh numbers Ra as used in recent high-precision measurements by Funfschilling et al. (J. Fluid Mech., vol. 536, 2005, p. 145). The Nusselt numbers Nu computed for Pr = 4.38 and Ra = 108 3 × 108, 5 × 108, 8 × 108 and 109 are found to be in excellent agreement with the experimental data corrected for finite thermal conductivity of the walls. Based on this successful validation of the numerical approach, the DNS data are used to extract vertical profiles of the mean temperature. We find that near the heating and cooling plates the non-dimensional temperature profiles Θ(y) (where y is the non-dimensional vertical coordinate), obey neither a logarithmic nor a power law. Moreover, we demonstrate that the Prandtl-Blasius boundary layer theory cannot predict the shape of the temperature profile with an error less than 7.9% within the thermal boundary layers (TBLs). We further show that the profiles can be approximated by a universal stretched exponential of the form Θ(y) ≈ 1 -exp(-y -0.5y2) with an absolute error less than 1.1% within the TBLs and 5.5% in the whole Rayleigh cell. Finally, we provide more accurate analytical approximations of the profiles involving higher order polynomials in the approximation. © 2009 Cambridge University Press.

Description: The onset of compressible convection in rapidly rotating spherical shells is studied in the anelastic approximation. An asymptotic theory valid at low Ekman number is developed and compared with numerical solutions of the full equations. Compressibility is measured by the number of density scale heights in the shell. In the Boussinesq problem, the location of the onset of convection is close to the tangent cylinder when there is no internal heating only a heat flux emerging from below. Compressibility strongly affects this result. With only a few scale heights or more of density present, there is onset of convection near the outer shell. Compressibility also strongly affects the frequencies and preferred azimuthal wavenumbers at onset. Compressible convection, like Boussinesq convection, shows strong spiralling in the equatorial plane at low Prandtl number. We also explore how higher-order linear modes penetrate inside the tangent cylinder at higher Rayleigh numbers and compare modes both symmetric and antisymmetric about the equator. © 2009 Cambridge University Press.

Description: Theoretical investigation of acoustic wave interactions with turbulent premixed flames was conducted to evaluate the acoustic energy amplification and/or damping due to the interaction of low-frequency acoustic waves with turbulent flames in three-dimensional space. Such amplified or damped acoustic energy is either coherent or incoherent as wrinkled flames cause coherent energy of a monochromatic acoustic wave to be damped into incoherent energy of spatially diffused and spectrally broadened acoustic waves. Small perturbation method (SPM) up to the second order was utilized to analyse net coherent and incoherent acoustic energies of the reflected and transmitted waves scattered from a weakly wrinkled turbulent flame surface in random motion. General formulations for net coherent and incoherent energy budget of the scattered fields were derived that can be applied to any type of flame height statistics. Production and/or damping of acoustic energy scattered from a turbulent flame is attributed to two effects: one is the acoustic velocity jump due to flame's unsteady heat release and the other is the flame's wrinkling due to its unsteady motion. Dimensionless parameters that govern net acoustic energy budget were derived in case of Gaussian statistics of flame surface behaviour: the r.m.s. and correlation length of flame height, the frequency ratio of the incidence frequency to the flame's correlation frequency, the time ratio of the flame's diffusion to correlation time and the incidence angle. The results of the scattered acoustic energy budget showed that noticeable amplification of acoustic energy was obtained either for a small frequency ratio (〈1) at the critical incidence angle or for a large frequency ratio and time ratio (〉1), while damping was obtained for a small frequency ratio at off-critical incidence angles. The relative importance of unsteady heat release (the jump effect) and unsteady motion (the wrinkling effect) to net acoustic energy is controlled mainly by the frequency ratio: The unsteady heat release effect dominates the wrinkling effect at a large frequency ratio, and vice versa at a small frequency ratio. The energy transfer from coherent to incoherent energy is due to flame surface wrinkling and is enhanced with the square of the flame's r.m.s. height. © 2009 Cambridge University Press.

Description: An algebraic heat flux truncation model was derived for high-speed gaseous shear flows. The model was developed for high-temperature gases with caloric imperfections. Fluctuating dilatation moments were modelled via conservation of mass truncations. The present model provided significant improvements, up to 20%, in the temperature predictions over the gradient diffusion model for a Mach number ranging from 0.02 to 11.8. Analyses also showed that the near-wall dependence of the algebraic model agreed with expected scaling, where the constant Prandtl number model did not. This led to a simple modification of the turbulent Prandtl number model. Compressibility led to an explicit pressure gradient dependency with the present model. Analyses of a governing parameter indicated that these terms are negligibly small for low speeds. However, they may be important for high-speed flow. © 2009 Cambridge University Press.

Description: The selection of long air bubbles propagating steadily in a strongly collapsed fluid-filled elastic tube is investigated experimentally in a benchtop model of airway reopening. Localized regions of strong collapse are likely in the lung, because collapsing fluid-elastic instabilities promote extensive deformation of the airway cross-section beyond the point of opposite wall contact. We find that radical changes in the reopening mechanics occur at this point. We build on the recent identification by Heap & Juel (Phys. Fluids, vol. 20, 2008, article no. 081702) of three different steadily propagating bubbles (asymmetric, double-tipped and pointed) that are selected successively for increasing values of the capillary number (Ca, ratio of viscous to surface tension forces) in tubes initially collapsed beyond the point of opposite wall contact. The asymmetric bubble is also observed in less collapsed tubes for small values of Ca, and we show that it bifurcates super-critically from the usual parabolic-tipped bubble as Ca increases. We also characterize the mechanisms underlying the discontinuous transitions between asymmetric and double-tipped bubbles, and double-tipped and pointed bubbles. In particular, we find that the tube must reopen to a critical height for double-tipped bubbles to be selected. The length of the precursor fingers in the double-tipped bubble decreases with Ca, and the bubble loses stability to pointed bubbles when this length is less than the height of the tube at the point where the fingers merge. By contrast with the asymmetric and double-tipped bubbles, the pointed bubble infiltrates the most collapsed part of the tube to yield the rapid reopening of the airway at low pressure, with the potential to reduce ventilation-induced lung damage. © 2009 Cambridge University Press.

Description: Previous laboratory measurements on drag of tandem rigid bodies moving in viscous incompressible fluids found that a following body experienced less drag than a leading one. Very recently a laboratory experiment (Ristroph & Zhang, Phys. Rev. Lett., vol. 101, 2008) with deformable bodies (rubble threads) revealed just the opposite the leading body had less drag than the following one. The Reynolds numbers in the experiment were around 104. To find out how this qualitatively different phenomenon may depend on the Reynolds number, a series of numerical simulations are designed and performed on the interaction of a pair of tandem flexible flags separated by a dimensionless vertical distance (0≤ D≤ 5.5) in a flowing viscous incompressible fluid at lower Reynolds numbers (40≤ Re≤ 220) using the immersed boundary (IB) method. The dimensionless bending rigidity Ǩb and dimensionless flag mass density used in our work are as follows: 8.6 10 5 Ǩb 1.8 103, 0.8≤Mcaron;≤ 1.0. We obtain an interesting result within these ranges of dimensionless parameters: when Re is large enough so that the flapping of the two flags is self-sustained, the leading flag has less drag than the following one; when Re is small enough so that the flags maintain two nearly static line segments aligned with the mainstream flow, the following flag has less drag than the leading one. The transitional range of Re separating the two differing phenomena depends on the value of Ǩb. With Re in this range, both the flapping and static states are observed depending on the separation distance D. When D is small enough, the flags are in the static state and the following flag has less drag; when D is large enough the flags are in the constant flapping state and the leading flag has less drag. The critical value of D depends on Ǩb. © 2009 Copyright Cambridge University Press.

Description: The prevailing view of the dynamics of flapping flags is that the onset of motion is caused by temporal instability of the initial planar state. This view is re-examined by considering the linearized two-dimensional motion of a flag immersed in a high-Reynolds-number flow and taking account of forcing by a street of vortices shed periodically from its cylindrical pole. The zone of nominal instability is determined by analysis of the self-induced motion in the absence of shed vorticity, including the balance between flag inertia, bending rigidity, varying tension and fluid loading. Forced motion is then investigated by separating the flag deflection into vortex-induced and self components. The former is related directly to the motion that would be generated by the shed vortices if the flag were absent. This component serves as an inhomogeneous forcing term in the equation satisfied by the self motion. It is found that forced flapping is possible whenever the Reynolds number based on the pole diameter ReD≥ 100, such that a wake of distinct vortex structures is established behind the pole. Such conditions typically prevail at mean flow velocities significantly lower than the critical threshold values predicted by the linear theory. It is therefore argued that analyses of the onset of flag motion that are based on ideal, homogeneous flag theory are incomplete and that consideration of the pole-induced fluid flow is essential at all relevant wind speeds. © 2009 Copyright Cambridge University Press.

Description: We reconsider exact solutions to the Navier-Stokes equations that describe a vortex in a viscous, incompressible fluid. This type of solution was first introduced by Long (J. Atmos. Sci., vol. 15 (1), 1958, p. 108) and is parameterized by an inverse Reynolds number ε. Long's attention (and that of many subsequent investigators) was centred upon the quasi-cylindrical (QC) case corresponding to ε= 0. We show that the limit ε→0 is not straightforward, and that it reveals other solutions to this fundamental exact reduction of the NavierStokes system (which are not of QC form). Through careful numerical investigation, supported by asymptotic descriptions, we identify new solutions and describe the full parameter space that is spanned by and the pressure at the vortex core. Some erroneous results that exist in the literature are corrected. © 2009 Cambridge University Press.

Description: We consider the steady flow near a free surface at intermediate to high Reynolds numbers, both experimentally and theoretically. In our experiment, an axisymmetric capillary meniscus is suspended from a cylindrical tube, held slightly above a horizontal water surface. A flow of dyed water is released through the tube into the reservoir, and flow lines are thus recorded. At low Reynolds numbers, flow lines follow the free surface, and injected water spreads horizontally inside the container. Increasing the Reynolds number, the injected fluid penetrates to a certain distance into the bath, but ultimately follows the free surface. Above a critical Reynolds number of approximately 60, the flow separates from the free surface in the meniscus region and a jet projects vertically into the bath. We find no indication that the flow reattaches at higher Reynolds numbers, nor are our findings sensitive to surface contamination. We show theoretically and confirm experimentally that the separating streamline forms a right angle with the free surface. © 2009 Cambridge University Press.

Description: To reduce the costs of construction, operation, maintenance, energy storage and grid connection, some devices for extracting energy from sea waves are likely to be installed on the coast. We study theoretically a single oscillating water column (OWC) installed at the tip of a long and thin breakwater. The linearized problems of radiation and scattering for a hollow cylinder with an open bottom are then solved by the usual method of eigenfunction expansions and integral equations. Since a thin breakwater is the limit of a wedge, an exact solution for the diffraction by a solid cylinder at the tip of a wedge is derived to facilitate the analysis. Following Sarmento & Falcão (J. Fluid Mech., vol. 150, 1985, pp. 467-485), power takeoff by Wells turbines is modelled by including air compressibility in the chamber above the water surface. The effects of air compressibility on the extraction efficiency is studied. It is shown that for this simple geometry the angle of incidence affects the waves outside the structure but not the extracted power. © 2009 Cambridge University Press.

Description: The wall-shear stress distribution in turbulent duct flow has been assessed using the micro-pillar shear-stress sensor MPS3 The spatial resolution of the sensor line is 10.8 l+ (viscous units) and the total field of view of 120 l+ along the spanwise direction allows to capture characteristic dimensions of the wall-shear stress distribution at sufficiently high resolution. The results show the coexistence of low-shear and high-shear regions representing 'footprints' of near-wall coherent structures. The regions of low shear resemble long meandering bands locally interrupted by areas of higher shear stress. Conditional averages of the flow field indicate the existence of nearly streamwise counter-rotating vortices aligned in the streamwise direction. The results further show periods of very strong spanwise wall-shear stress to be related to the occurrence of high streamwise shear regions and momentum transfer towards the wall. These events go along with a spanwise oscillation and a meandering of the low-shear regions. © 2009 Cambridge University Press.

Description: The creeping motion of a hydrodynamically 'Janus' spherical particle, whose surface is partitioned into two distinct regions, is investigated. On one region, fluid adjacent to the particle obeys the no-slip condition, whereas on the other, fluid slips past the particle. The fore-aft asymmetry of this 'slip-stick' sphere (Swan & Khair, J. Fluid Mech., vol. 606, 2008, p. 115) leads to a number of interesting results when it is placed in different flows, which is illustrated by computing the particle motion to first order in the ratio of slip length to particle radius. For example, in a pure straining field the sphere attains an equilibrium orientation either along the compressional or extensional axis of the flow, depending on the ratio of slip-to-stick surface areas. In a simple shear flow, on the other hand, the slip-stick sphere undergoes a periodic rotational motion, or Jeffrey orbit. Moreover, depending on its initial orientation, the particle can either follow a periodic {translational} orbit or undergo a net displacement along the flow direction. Lastly, to first order in the volume fraction of slip-stick spheres, the suspension rheology is non-Newtonian, with non-zero first and second normal stress differences. © 2009 Cambridge University Press.

Description: Recent microfluidic experiments by Bremond, Thiam & Bibette (Phys. Rev. Lett., vol. 100, 2008, paper no. 024501), along with simulations by Yoon et al. (Phys. Fluid, vol. 19, 2007, paper no. 102102) and near-contact experiments and simulations by Manica et al. (Langmuir, vol. 24, 2008, pp. 1381-1390), have demonstrated that two droplets can coalesce as they are separating rather than upon their collision. We analyse the experimental microfluidic flow configuration for the approach to contact with a two-dimensional model: we apply a lubrication analysis followed by the method of domain perturbation to determine the droplet deformation as a function of time. We find the approximate shape for the deformed droplet at the time of contact. In particular, for droplets of radius R, moving apart according to h0(t) = h0(0) + αt2, where 2h0(t) is the separation distance, we define a non-dimensional arameter A = 4 C μ R2α1/2/πγ[h0 (0)]3/2, where μ is the viscosity of the continuous phase; γ is the interfacial tension; and C depends on the viscosity ratio between the droplets and the continuous phase. Our model suggests that there exists a critical value Acrit = 16/33/2 ≈ 3.0792, below which separation is unlikely to facilitate the coalescence of the droplets. The predictions are in good agreement with available experimental data. © 2009 Cambridge University Press.

Description: Development of coherent structures in the separated shear layer and wake of an airfoil in low-Reynolds-number flows was studied experimentally for a range of airfoil chord Reynolds numbers, 55 × 103 ≤ Rec ≤ 210 × 103, and three angles of attack, α = 0°, 5° and 10°. To illustrate the effect of separated shear layer development on the characteristics of coherent structures, experiments were conducted for two flow regimes common to airfoil operation at low Reynolds numbers: (i) boundary layer separation without reattachment and (ii) separation bubble formation. The results demonstrate that roll-up vortices form in the separated shear layer due to the amplification of natural disturbances, and these structures play a key role in flow transition to turbulence. The final stage of transition in the separated shear layer, associated with the growth of a sub-harmonic component of fundamental disturbances, is linked to the merging of the roll-up vortices. Turbulent wake vortex shedding is shown to occur for both flow regimes investigated. Each of the two flow regimes produces distinctly different characteristics of the roll-up and wake vortices. The study focuses on frequency scaling of the investigated coherent structures and the effect of flow regime on the frequency scaling. Analysis of the results and available data from previous experiments shows that the fundamental frequency of the shear layer vortices exhibits a power law dependency on the Reynolds number for both flow regimes. In contrast, the wake vortex shedding frequency is shown to vary linearly with the Reynolds number. An alternative frequency scaling is proposed, which results in a good collapse of experimental data across the investigated range of Reynolds numbers. © 2009 Cambridge University Press.

Description: Global absolute and convective stability analysis of flow past a circular cylinder with symmetry conditions imposed along the centreline of the flow field is carried out. A stabilized finite element formulation is used to solve the eigenvalue problem resulting from the linearized perturbation equation. All the computations carried out are in two dimensions. It is found that, compared to the unrestricted flow, the symmetry conditions lead to a significant delay in the onset of absolute as well as convective instability. In addition, the onset of absolute instability is greatly affected by the location of the lateral boundaries and shows a non-monotonic variation. Unlike the unrestricted flow, which is associated with von Kármán vortex shedding, the flow with centreline symmetry becomes unstable via modes that are associated with low-frequency large-scale structures. These lead to expansion and contraction of the wake bubble and are similar in characteristics to the low-frequency oscillations reported earlier in the literature. A global linear convective stability analysis is utilized to find the most unstable modes for different speeds of the disturbance. Three kinds of convectively unstable modes are identified. The ones travelling at very low streamwise speed are associated with large-scale structures and relatively low frequency. Shear layer instability, with relatively smaller scale flow structures and higher frequency, is encountered for disturbances travelling at relatively larger speed. For low blockage a new type of instability is found. It travels at relatively high speed and resembles a swirling flow structure. As opposed to the absolute instability, the convective instability appears at much lower Re and its onset is affected very little by the location of the lateral boundaries. Analysis is also carried out for determining the convective stability of disturbances that travel in directions other than along the free stream. It is found that the most unstable disturbances are not necessarily the purely streamwise travelling ones. Disturbances that move purely in the cross-stream direction can also be convectively unstable. The results from the linear stability analysis are confirmed by carrying out direct time integration of the linearized disturbance equations. The disturbance field shows transient growth by several orders of magnitude confirming that such flows act as amplifiers. Direct time integration of the Navier-Stokes equation is carried out to track the time evolution of both the large-scale low-frequency oscillations and small-scale shear layer instabilities. The critical Re for the onset of convective instability is compared with earlier results from local analysis. Good agreement is found. © 2009 Cambridge University Press.