ALBERT

Description: A study of the morphology of the vortical skeleton behind a flapping NACA0030 wing with a finite aspect ratio of 3, is undertaken. The motivation for this work originates with the proposal that thrust can be efficiently produced by flapping aerofoils. The test condition corresponds to a Strouhal number of 0.35, Reynolds number, based on aerofoil chord, of 600 and an amplitude of flapping, equal to the chord length of the wing. This test condition corresponds to the optimal thrust-producing case in infinite-span flapping wings. This study investigates the effect of wing three-dimensionality on the structure of the wake-flow. This is accomplished here, by quantitatively describing the spatio-temporal variations in the velocity, vorticity and Reynolds stresses for the finite-span-wing case.Preliminary flow visualizations suggest that the presence of wingtip vortices for the three-dimensional-wing case, create a different vortical structure to the two-dimensional-wing case. In the case of a two-dimensional-wing, the flow is characterized by the interaction of leading- and trailing-edge vorticity, resulting in the formation of a clear reverse Kármán vortex street at the selected test condition. In the case of a three-dimensional-wing, the flow exhibits a high degree of complexity and three-dimensionality, particularly in the midspan region. Using phase-averaged particle image velocimetry measurements of the forced oscillatory flow, a quantitative analysis in the plane of symmetry of the flapping aerofoil was undertaken. Using a triple decomposition of the measured velocities, the morphological characteristics of the spanwise vorticity is found to be phase correlated with the aerofoil kinematics. Reynolds stresses in the direction of oscillation are the dominant dissipative mechanism. The mean velocity profiles resemble ajet, indicative of thrust production. Pairs of strong counter-rotating vortices from the leading- and trailing-edge of the aerofoil are shed into the flow at each half-cycle. The large-scale structure of the flow is characterized by constructive merging of spanwise vorticity. The midspan region is populated by cross-sections of interconnected vortex rings.

Description: In this study, we performed simulations of turbulent flow over rectangular ribs transversely mounted on one side of a plane in a channel, with the other side being smooth. The separation between ribs is large enough to avoid forming stable vortices in the spacing, which exhibitsk-type, or sand-grain roughness. The Reynolds numberReτof our representative direct numerical simulation case is 460 based on the smooth-wall friction velocity and the channel half-width. The roughness heighthis estimated as 110 wall units based on the rough-wall friction velocity. The velocity profile and kinetic energy budget verify the presence of an equilibrium, logarithmic layer aty≳2h. In the roughness sublayer, however, a significant turbulent energy flux was observed. A high-energy region is formed by the irregular motions just above the roughness. Visualizations of vortical streaks, disrupted in all three directions in the roughness sublayer, indicate that the three-dimensional flow structure of sand-grain roughness is replicated by the two-dimensional roughness, and that this vortical structure is responsible for the high energy production. The difference in turbulence structure between smooth- and rough-wall layers can also be seen in other flow properties, such as anisotropy and turbulence length scales.

Description: A model for simulating the process of growth, collapse and rebound of a cavitation bubble travelling along the flow through a convergent–divergent nozzle producing a cavitating water jet is established. The model is based on the Rayleigh–Plesset bubble dynamics equation using as inputs ambient pressure and velocity profiles calculated with the aid of computational fluid dynamics (CFD) flow modelling. A variable time-step technique is applied to solve the highly nonlinear second-order differential equation. This technique successfully solves the Rayleigh–Plesset equation for wide ranges of pressure variation and bubble original size and saves considerable computing time. Inputs for this model are the pressure and velocity data from CFD calculation. To simulate accurately the process of bubble growth, collapse and rebound, a heat transfer model, which includes the effects of conduction plus radiation, is developed to describe the thermodynamics of the incondensable gas inside the bubble. This heat transfer model matches previously published experimental data well. Assuming that single bubble behaviour also applies to bubble clouds, the calculated distance from the nozzle exit travelled by the bubble to the point where the bubble size becomes invisible is taken to be equal to the bubble cloud length observed. The predictions are compared with experiments carried out in a cavitation cell and show good agreement for different nozzles operating at different pressure conditions.

Description: An experimental study, investigating the mean flow and turbulence in the wind drift layer formed beneath short wind waves was conducted. The degree to which these flows resemble the flows that occur in boundary layers adjacent to solid walls (i.e. wall-layers) was examined. Simultaneous DPIV (digital particle image velocimetry) and infrared imagery were used to investigate these near-surface flows at a fetch of 5.5 m and wind speeds from 4.5 to 11 m s−1. These conditions produced short steep waves with dominant wavelengths from 6 cm to 18 cm. The mean velocity profiles in the wind drift layer were found to be logarithmic and the flow was hydrodynamically smooth at all wind speeds. The rate of dissipation of turbulent kinetic energy was determined to be significantly greater in magnitude than would occur in a comparable wall-layer. Microscale breaking waves were detected using the DPIV data and the characteristics of breaking and non-breaking waves were compared. The percentage of microscale breaking waves increased abruptly from 11% to 80% as the wind speed increased from 4.5 to 7.4 m s−and then gradually increased to 90% as the wind speed increased to 11 m s−. At a depth of 1 mm, the rate of dissipation was 1.7 to 3.2 times greater beneath microscale breaking waves compared to non-breaking waves. In the crest–trough region beneath microscale breaking waves, 40% to 50% of the dissipation was associated with wave breaking. These results demonstrated that the enhanced near-surface turbulence in the wind drift layer was the result of microscale wave breaking. It was determined that the rate of dissipation of turbulent kinetic energy due to wave breaking is a function of depth, friction velocity, wave height and phase speed as proposed by Terrayet al. (1996). Vertical profiles of the rate of dissipation showed that beneath microscale breaking waves there were two distinct layers. Immediately beneath the surface, the dissipation decayed as ζ−0.7and below this in the second layer it decayed as ζ−2. The enhanced turbulence associated with microscale wave breaking was found to extend to a depth of approximately one significant wave height. The only similarity between the flows in these wind drift layers and wall-layers is that in both cases the mean velocity profiles are logarithmic. The fact that microscale breaking waves were responsible for 40%–50% of the near-surface turbulence supports the premise that microscale breaking waves play a significant role in enhancing the transfer of gas and heat across the air–sea interface.

Description: Results are presented from experimental investigations into the motion of a heavy ellipsoid in a horizontal rotating cylinder, which has been completely filled with highly viscous fluid. The motion can be conveniently classified using the ratio between the maximum radius of curvature of the ellipsoid κmaxand the radius of the drumRd. If κmax

Description: Electric conduction from an electrolyte solution into a charge selective solid, such as ion exchange membrane or electrode, becomes unstable when the electrolyte concentration near the interface approaches zero owing to diffusion limitation. The sequence of events leading to instability is as follows: upon the decrease of the interface concentration, the electric double layer at the interface transforms from its common quasi-equilibrium structure to a different, non-equilibrium one. The key feature of this new structure is an extended space charge added to the usual one of the quasi-equilibrium electric double layer. The non-equilibrium electro-osmotic slip related to this extended space charge renders the quiescent conductance unstable. A unified asymptotic picture of the electric double-layer undercurrent, encompassing all regimes from quasi-equilibrium to the extreme non-equilibrium one, is developed and employed for derivation of a universal electro-osmotic slip formula. This formula is used for a linear stability study of quiescent electric conduction, yielding the precise parameter range of instability, compared with that in the full electroconvective formulation. The physical mechanism of instability is traced both kinematically, in terms of non-equilibrium electro-osmotic slip, and dynamically, in terms of forces acting in the electric double layer.

Description: The water-shipping problem is modelled in a two-dimensional framework and studied experimentally and numerically for the case of a fixed barge-shaped structure. The analysis represents the second step of the research discussed in Grecoet al. (J. Fluid Mech., vol. 525, 2005, p. 309). The numerical investigation is performed by using both a boundary element method and a domain-decomposition strategy. The model tests highlight the occurrence of dam-breaking-type water on deck, (a) with and (b) without an initial plunging phase, and (c) an unusual type of water shipping connected with blunt water–deck impacts here called a hammer-fist type event never documented before. Cases (a) and (c) are connected with the most severe events and the related features and green-water loads are discussed in detail. A parametric analysis of water-on-deck phenomena has also been carried out in terms of the local incoming waves and bow flow features. We classify such phenomena in a systematic way to provide a basis for further investigations of water-on-deck events. The severity of (a)-type water-on-deck events is analysed in terms of initial cavity area and water-front velocity along the deck. The former increases as the square power of the modified incoming-wave (front-crest) steepness while the latter scales with its square-root. The two-dimensional investigation gives useful quantitative information in terms of water-front velocity for comparison with three-dimensional water-on-deck experiments on fixed bow models interacting with wave packets.

Description: The irrotational flow past two slender bodies of revolution at angles of yaw, translating in parallel paths in very close proximity, is analysed by extending the classical slender body theory. The flow far away from the two bodies is shown to be a direct problem, which is represented in terms of two line sources along their longitudinal axes, at the strengths of the variation rates of their cross-section areas. The inner flow near the two bodies is reduced to the plane flow problem of the expanding (contracting) and lateral translations of two parallel circular cylinders with different radii, which is then solved analytically using conformal mapping. Consequently, an analytical flow solution has been obtained for two arbitrary slender bodies of revolution at angles of yaw translating in close proximity. The lateral forces and yaw moments acting on the two bodies are obtained in terms of integrals along the body lengths. A comparison is made among the present model for two slender bodies in close proximity, Tuck & Newman's (1974) model for two slender bodies far apart, and VSAERO (AMI)–commercial software based on potential flow theory and the boundary element method (BEM). The attraction force of the present model agrees well with the BEM result, when the clearance,h0, is within 20% of the body length, whereas the attraction force of Tuck & Newman is much smaller than the BEM result whenh0is within 30% of the body length, but approaches the latter whenh0is about half the body length. Numerical simulations are performed for the three typical manoeuvres of two bodies: (i) a body passing a stationary body, (ii) two bodies in a meeting manoeuvre (translating in opposite directions), and (iii) two bodies in a passing manoeuvre (translating in the same direction). The analysis reveals the orders of the lateral forces and yaw moments, as well as their variation trends in terms of the manoeuvre type, velocities, sizes, angles of yaw of the two bodies, and their proximity, etc. These irrotational dynamic features are expected to provide a basic understanding of this problem and will be beneficial to further numerical and experimental studies involving additional physical effects.

Description: The flow associated with a synthetic jet transitioning to turbulence in an otherwise quiescent external flow is examined using time-accurate three-dimensional numerical simulations. The incompressible Navier–Stokes solver uses a second-order accurate scheme for spatial discretization and a second-order semi-implicit fractional step method for time integration. The simulations are designed to model the experiments of C. S. Yaoet al. (Proc. NASA LaRC Workshop, 2004) which have examined, in detail, the external evolution of a transitional synthetic jet in quiescent flow. Although the jet Reynolds and Stokes numbers in the simulations match with the experiment, a number of simplifications have been made in the synthetic jet actuator model adopted in the current simulations. These include a simpler representation of the cavity and slot geometry and diaphragm placement. Despite this, a reasonably good match with the experiments is obtained in the core of the jet and this indicates that for these jets, matching of these key non-dimensional parameters is sufficient to capture the critical features of the external jet flow. The computed results are analysed further to gain insight into the dynamics of the external as well as internal flow. The results indicate that near the jet exit plane, the flow field is dominated by the formation of counter-rotating spanwise vortex pairs that break down owing to the rapid growth of spanwise instabilities and transition to turbulence a short distance from the slot. Detailed analyses of the unsteady characteristics of the flow inside the jet cavity and slot provide insights that to date have not been available from experiments.

Description: A general proof that more energy flows upscale than downscale in two-dimensional turbulence and barotropic quasi-geostrophic (QG) turbulence is given. A proof is also given that in surface QG turbulence, the reverse is true. Though some of these results are known in restricted cases, the proofs given here are pedagogically simpler, require fewer assumptions and apply to both forced and unforced cases.

Description: The local linear stability of forced, stationary long waves produced by topography or potential vorticity (PV) sources is examined using a quasi-geostrophic barotropic model. A multiple scale analysis yields coupled equations for the background stationary wave and low-frequency (LF) disturbance field. Forcing structures for which the LF dynamics are Hamiltonian are shown to yield conservation laws that provide necessary conditions for instability and a constraint on the LF structures that can develop. Explicit knowledge of the forcings that produce the stationary waves is shown to be crucial to predicting a unique LF field. Various topographies or external PV sources can be chosen to produce stationary waves that differ by asymptotically small amounts, yet the LF instabilities that develop can have fundamentally different structures and growth rates. If the stationary wave field is forced solely by topography, LF oscillatory modes always emerge. In contrast, if the stationary wave field is forced solely by PV, two LF structures are possible: oscillatory modes or non-oscillatory envelope modes. The development of the envelope modes within the context of a linear LF theory is novel.An analysis of the complex WKB branch points, which yields an analytical expres-sion for the leading-order eigenfrequency, shows that the streamwise distribution of absolute instability and convective growth is central to understanding and predicting the types of LF structures that develop on the forced stationary wave. The location of the absolute instability region with respect to the stationary wave determines whether oscillatory modes or envelope modes develop. In the absence of absolute instability, eastward propagating wavetrains generated in the far field can amplify via local convective growth in the stationary wave region. If the stationary wave region is streamwise symmetric (asymmetric), the local convective growth results in a local change in wave energy that is transient (permanent).

Description: This paper presents combined theoretical and experimental studies of the two-dimensional piston-like steady-state motions of a fluid in a moonpool formed by two rectangular hulls (e.g. a dual pontoon or catamaran). Vertical harmonic excitation of the partly submerged structure in calm water is assumed. A high-precision analytically oriented linear-potential-flow method, which captures the singular behaviour of the velocity potential at the corner points of the rectangular structure, is developed. The linear steady-state results are compared with new experimental data and show generally satisfactory agreement. The influence of vortex shedding has been evaluated by using the local discrete-vortex method of Graham (1980). It was shown to be small. Thus, the discrepancy between the theory and experiment may be related to the free-surface nonlinearity.

Description: This paper presents a study of the global response of a fluid to impulsive and localized forcing; it has been motivated by the recent laboratory experiments on the locomotion of water-walking insects reported in Hu, Chan & Bush (Nature, vol. 424, 2003, p. 663). These insects create both waves and vortices by their rapid leg strokes and it has been a matter of some debate whether either form of motion predominates in the momentum budget. The main result of this paper is to argue that generically both waves and vortices are significant, and that in linear theory they take up the horizontal momentum with share 1/3 and 2/3, respectively.This generic result, which depends only on the impulsive and localized nature of the forcing, is established using the classical linear impulse theory, with adaptations to weakly compressible flows and flows with a free surface. Additional general comments on experimental techniques for momentum measurement and on the wave emission are given and then the theory is applied in detail to water-walking insects.Owing to its generality, this kind of result and the methods used to derive it should be applicable to a wider range of wave–vortex problems in the biolocomotion of water-walking animals and elsewhere.

Description: Experimentation and theory are used to study the long-term dynamics of a two-dimensional density current flowing into a two-layer stratified basin. When the initial Richardson number,$hbox{it Ri}_{ ho}^{hbox{scriptsizeit in}}$, characterizing the ratio of the background stratification to the buoyancy flux of the density current, is less than the critical value of$hbox{it Ri}_{ ho}^{*} ,{=}, 21-27$, it is found that the density current penetrates the stratified interface. This result is ostensibly independent of slope for angles between 30° and 90°. If the current does not initially penetrate the interface, then it slowly increases the density of the top layer until the interfacial density difference is reduced sufficiently to drive penetration. The time scale for this to occur,$t_{p} ,{=}, (hbox{it Ri}^{hbox{scriptsizeit in}}_{ ho} - hbox{it Ri}_{ ho}^{*}) L/B^{1/3}$, is explicitly a function of the buoyancy fluxBand the length of the basinL. The initial Richardson number,$hbox{it Ri}^{hbox{scriptsizeit in}}_{ ho}$, is a function of depth, the initial reduced gravity of the interface and a weak function of slope angle. In the absence of initial penetration for very steep slopes of 75° and 90°, we observe that penetrative convection at the interface leads to significant local entrainment. In consequence, the top layer thickens and the interfacial entrainment rate increases as the fifth power of the interfacial Froude number. In contrast, such a process is not observed at comparable interfacial Froude numbers on lower slopes of 30°, 45° and 60°, thereby demonstrating the important role of impact angle on penetrative convection. We attribute the increased interfacial entrainment by the steep density currents as the result of the transition from an undular bore to a turbulent hydraulic jump at the point where the density current intrudes. We discuss the applicability of the observed circulation to the stability of the Arctic halocline where we find$0.56,{lesssim}, t_{p} ,{lesssim},1.2$years for a range of contemporary oceanographic conditions.

Description: Direct numerical simulations (DNS) of stenotic flows under conditions of steady inlet flow were discussed in Part 1 of this study. DNS of pulsatile flow through the 75% stenosed tube (by area) employed for the computations in Part 1 is examined here. Analogous to the steady flow results, DNS predicts a laminar post-stenotic flow field in the case of pulsatile flow through the axisymmetric stenosis model, in contrast to previous experiments, in which intermittent disturbed flow regions and turbulent breakdown were observed in the downstream region. The introduction of a stenosis eccentricity, that was 5% of the main vessel diameter at the throat, resulted in periodic, localized transition to turbulence. Analysis in this study indicates that the early and mid-acceleration phases of the time period cycle were relatively stable, with no turbulent activity in the post-stenotic region. However, towards the end of acceleration, the starting vortex, formed earlier as the fluid accelerated through the stenosis at the beginning of acceleration, started to break up into elongated streamwise structures. These streamwise vortices broke down at peak flow, forming a turbulent spot in the post-stenotic region. In the early part of deceleration there was intense turbulent activity within this spot. Past the mid-deceleration phase, through to minimum flow, the inlet flow lost its momentum and the flow field began to relaminarize. The start of acceleration in the following cycle saw a recurrence of the entire process of a starting structure undergoing turbulent breakdown and subsequent relaminarization of the post-stenotic flow field. Peak wall shear stress (WSS) levels occurred at the stenosis throat, with the rest of the vessel experiencing much lower levels. Turbulent breakdown at peak flow resulted in a sharp amplification of instantaneous WSS magnitudes across the region corresponding to the turbulent spot, accompanied by large axial and circumferential fluctuations, even while ensemble-averaged axial shear stresses remained mostly low and negative. WSS levels dropped rapidly after the mid-deceleration phase, when the relaminarization process took over, and were almost identical to laminar, axisymmetric shear levels through most of the acceleration phase.

Description: We investigate experimentally the force generated by the unsteady vortex formation of low-aspect-ratio normal flat plates with one end free. The objective of this study is to determine the role of the free end, or tip, vortex. Understanding this simple case provides insight into flapping-wing propulsion, which involves the unsteady motion of low-aspect-ratio appendages. As a simple model of a propulsive half-stroke, we consider a rectangular normal flat plate undergoing a translating start-up motion in a towing tank. Digital particle image velocimetry is used to measure multiple perpendicular sections of the flow velocity and vorticity, in order to correlate vortex circulation with the measured plate force. The three-dimensional wake structure is captured using flow visualization. We show that the tip vortex produces a significant maximum in the plate force. Suppressing its formation results in a force minimum. Comparing plates of aspect ratio six and two, the flow is similar in terms of absolute distance from the tip, but evolves faster for aspect ratio two. The plate drag coefficient increases with decreasing aspect ratio.

Description: Using the Boussinesq approximation, a set of depth-integrated wave equations for long-wave propagation over a mud bed is derived. The wave motions above the mud bed are assumed to be irrotational and the mud bed is modelled as a highly viscous fluid. The pressure and velocity are required to be continuous across the water–mud interface. The resulting governing equations are differential–integral equations in terms of the depth-integrated horizontal velocity and the free-surface displacement. The effects of the mud bed appear in the continuity equation in the form of a time integral of weighted divergence of the depth-averaged velocity. Damping rates for periodic waves and solitary waves are calculated. For the solitary wave case, the velocity profiles in the water column and the mud bed at different phases are discussed. The effects of the viscous boundary layer above the mud–water interface are also examined.

Description: During extended deployment at an ocean observatory off the coast of New Jersey, a bottom-mounted five-beam acoustic Doppler current profiler measured large-scale velocity structures that we interpret as Langmuir circulations filling the entire water column. These circulations are the large-eddy structures of wind-wave-driven turbulent flows that occur episodically when a shallow water column experiences prolonged strong wind forcing. Many observational characteristics agree with former descriptions of Langmuir circulations in deep water. The three-dimensional velocity field reveals quasi-organized structures consisting of pairs of surface-intensified counter-rotating vortices, aligned approximately downwind. Maximum downward velocities are stronger than upward velocities, and the downwelling region of each cell, defined as a pair of vortices, is narrower than the upwelling region. Maximum downward vertical velocity occurs at or above mid-depth, and scales approximately with wind speed. The estimated crosswind scale of cells is roughly 3–6 times their vertical scale, set under these conditions by water depth. The long axis of the cells appears to lie at an angle ∼10°–20° to the right of the wind. A major difference from deep-water observations is strong near-bottom intensification of the downwind ‘jets’ found typically centred over downwelling regions. Accessible observational features such as cell morphology and profiles of mean velocities, turbulent velocity variances, and shear stress components are compared with the results of associated large-eddy simulations (reported in Part 2) of shallow water flows driven by surface stress and the Craik–Leibovich vortex forcing generally used to represent generation of Langmuir cells. A particularly sensitive diagnostic for identification of Langmuir circulations as the energy-containing eddies of the turbulent flow is the depth trajectory of invariants of the turbulent stress tensor, plotted in the Lumley ‘triangle’ corresponding to realizable turbulent flows. When Langmuir structures are present in the observations, the Lumley map is distinctly different from that of surface-stress-driven Couette flow, again in agreement with the large-eddy simulations (LES). Unlike the LES, observed velocity fields contain two distinct and significant scales of variability, documented by wavelet analysis of observational records of vertical velocity. Variability with periods of many minutes is that expected from Langmuir cells drifting past the instrument at the slowly time-varying crosswind velocity. Shorter period variability, of the order of 1–2 min, has roughly the observed periodicity of surface wave groups, suggesting a connection with the wave groups themselves and/or the wave breaking associated with them in high wind conditions.

Description: Lagrangian auto- and cross-correlation functions of the rate of strains2, enstrophy ω2, their respective production terms −sijsjkskiand ωiωjsij, and material derivatives, Ds2/Dtand Dω2/Dtare estimated using experimental results obtained through three-dimensional particle tracking velocimetry (three-dimensional-PTV) in homogeneous turbulence atReλ=50. The autocorrelation functions are used to estimate the Lagrangian time scales of different quantities, while the cross-correlation functions are used to clarify some aspects of the interaction mechanisms between vorticity ω and the rate of strain tensorsij, that are responsible for the statistically stationary, in the Eulerian sense, levels of enstrophy and rate of strain in homogeneous turbulent flow. Results show that at the Reynolds number of the experiment these quantities exhibit different time scales, varying from the relatively long time scale of ω2to the relatively shorter time scales ofs2, ωiωjsijand −sijsjkski. Cross-correlation functions suggest that the dynamics of enstrophy and strain, in this flow, is driven by a set of different-time-scale processes that depend on the local magnitudes ofs2and ω2. In particular, there are indications that, in a statistical sense, (i) strain production anticipates enstrophy production in low-strain–low-enstrophy regions (ii) strain production and enstrophy production display high correlation in high-strain–high-enstrophy regions, (iii) vorticity dampening in high-enstrophy regions is associated with weak correlations between −sijsjkskiands2and between −sijsjkskiand Ds2/Dt, in addition to a marked anti-correlation between ωiωjsijand Ds2/Dt. Vorticity dampening in high-enstrophy regions is thus related to the decay ofs2and its production term, −sijsjkski.

Description: Transverse jets arise in many applications, including propulsion, effluent dispersion, oil field flows, and V/STOL aerodynamics. This study seeks a fundamental, mechanistic understanding of the structure and evolution of vorticity in the transverse jet. We develop a high-resolution three-dimensional vortex simulation of the transverse jet at large Reynolds number and consider jet-to-crossflow velocity ratiosrranging from 5 to 10. A new formulation of vorticity-flux boundary conditions accounts for the interaction of channel wall vorticity with the jet flow immediately around the orifice. We demonstrate that the nascent jet shear layer contains not only azimuthal vorticity generated in the jet pipe, but wall-normal and azimuthal perturbations resulting from the jet–crossflow interaction. This formulation also yields analytical expressions for vortex lines in the near field as a function ofr.Transformation of the cylindrical shear layer emanating from the orifice begins with axial elongation of its lee side to form sections of counter-rotating vorticity aligned with the jet trajectory. Periodic roll-up of the shear layer accompanies this deformation, creating complementary vortex arcs on the lee and windward sides of the jet. Counter-rotating vorticity then drives lee-side roll-ups in the windward direction, along the normal to the jet trajectory. Azimuthal vortex arcs of alternating sign thus approach each other on the windward boundary of the jet. Accordingly, initially planar material rings on the shear layer fold completely and assume an interlocking structure that persists for several diameters above the jet exit. Though the near field of the jet is dominated by deformation and periodic roll-up of the shear layer, the resulting counter-rotating vorticity is a pronounced feature of the mean field; in turn, the mean counter-rotation exerts a substantial influence on the deformation of the shear layer. Following the pronounced bending of the trajectory into the crossflow, we observe a sudden breakdown of near-field vortical structures into a dense distribution of smaller scales. Spatial filtering of this region reveals the persistence of counter-rotating streamwise vorticity initiated in the near field.

Description: The decay of a passive scalar in a sinusoidal shear flow translating in the cross-stream direction at a constant speeduis studied in the limit of small diffusivity κ. The decay rate, obtained by solving an eigenvalue problem, is found to tend to a non-zero constant as κ→0 whenuis of order κ1/2. This result, establishing that fast decay is possible in shear flows, is fragile however: because of the existence of pseudomodes, the addition of a small noise leads to decay rates that decrease to zero with κ as κ2/5.

Description: Direct numerical simulation is used to study a round turbulent jet in a laminar crossflow. The ratio of bulk jet velocity to free-stream crossflow velocity is 5.7 and the Reynolds number based on the bulk jet velocity and the jet exit diameter is 5000. The mean velocity and turbulent intensities from the simulations are compared to data from the experiments by Su & Mungal (2004) and good agreement is observed. Additional quantities, not available from experiments, are presented. Turbulent kinetic energy budgets are computed for this flow. Examination of the budgets shows that the near field is far from a state of turbulent equilibrium – especially along the jet edges. Also – in the near field – peak kinetic energy production is observed close to the leading edge, while peak dissipation is observed toward the trailing edge of the jet. The results are used to comment upon the difficulty involved in predicting this flow using RANS computations. There exist regions in this flow where the pressure transport term, neglected by some models and poorly modelled by others, is significant. And past the jet exit, the flow is not close to established canonical flows on which most models appear to be based.

Description: The theory of Wagner from 1932 for the normal symmetric impact of a two-dimensional body of small deadrise angle on a half-space of ideal and incompressible liquid is extended to derive the second-order corrections for the locations of the higher-pressure jet-root regions and for the upward force on the impactor using a systematic matched-asymptotic analysis. The second-order predictions for the upward force on an entering wedge and parabola are compared with numerical and experimental data, respectively, and it is concluded that a significant improvement in the predictive capability of Wagner's theory is afforded by proceeding to second order.

Description: The flapping coupling between two filaments is studied theoretically and experimentally in this paper. A temporal linear instability analysis is carried out based on a simplified hydrodynamic model. The dispersion relationship between the eigen-frequency ω and wavenumberkis expressed by a quartic equation. Two special cases of flapping coupling, i.e. two identical filaments having the same length and two filaments having different lengths, are studied in detail. In the case of two identical filaments, the theoretical analysis predicts four coupling modes, i.e. the stretched-straight mode, the antisymmetrical in-phase mode, the symmetrical out-of-phase mode and the indefinite mode. The theory also predicts the existence of an eigenfrequency jump during transition between the in-phase and out-of-phase modes, which has been observed in previous experiments and numerical simulations. In the case of two filaments having different lengths, four modes similar to those in the former case are identified theoretically. The distribution of coupling modes for both the cases is shown in two planes. One is a dimensionless plane ofSvs.U, whereSis the density ratio of solid filament to fluid andU2is the ratio of fluid kinetic energy to solid elastic potential energy. The other is a dimensional plane of the half-distance (h) between two filaments vs. the filament length (L). Relevant experiments are carried out in a soap-film tunnel and the stable and unstable modes are observed. Theory and experiment are compared in detail. It should be noted that the model used in our analysis is a very simplified one that can provide intuitional analytical results of the coupling modes as well as their qualitative distributions. The factors neglected in our model, such as vortex shedding, viscous and nonlinear effects, do not allow the model to predict results precisely consistent with the experiments. Moreover, the Strouhal numbers of the flapping filaments are found to be generally around a fixed value in the experiments for both cases, implying that the filaments try to maintain a lower potential energy state.

Description: Numerical simulations on a β-plane are used to further understand the formation of zonal flows from small-scale fluctuations. The dynamics of ‘reduced models’ are computed by restricting the nonlinear term to include a subset of triad interactions in Fourier space. Reduced models of near-resonant triads are considered, as well as the complement set of non-resonant triads. At moderately small values of the Rhines number, near-resonant triad interactions are shown to be responsible for the generation of large-scale zonal flows from small-scale random forcing. Without large-scale drag, both the full system and the reduced model of near resonances produce asymmetry between eastward and westward jets, in favour of stronger westward jets. When large-scale drag is included, the long-time asymmetry is reversed in the full system, with eastward jets that are thinner and stronger than westward jets. Then the reduced model of near resonances exhibits a weaker asymmetry, but there are nevertheless more eastward jets stronger than a threshold value.

Description: Free surface potential flows past disturbances in a channel are considered. Three different types of disturbance are studied: (i) a submerged obstacle on the bottom of a channel; (ii) a pressure distribution on the free surface; and (iii) an obstruction in the free surface (e.g. a sluice gate or a flat plate). Surface tension is neglected, but gravity is included in the dynamic boundary condition. Fully nonlinear solutions are computed by boundary integral equation methods. In addition, weakly nonlinear solutions are derived. New solutions are found when several disturbances are present simultaneously. They are discovered through the weakly nonlinear analysis and confirmed by numerical computations for the fully nonlinear problem.

Description: A numerical description of heterogeneous propellant combustion enables us to examine the spatial and temporal fluctuations in the flow field arising from the heterogeneity. Particular focus is placed on the fluctuations in a zone intermediate between the combustion field (where reaction is important) and the chamber flow domain, for these define boundary conditions for simulations of the turbulent chamber flow. The statistics of the temperature field and the normal velocity field are described, and characteristic length scales and time scales are identified. The length scales are small compared to any relevant length scale of the chamber flow, and so the boundary conditions for this flow at any mesh point are statistically independent of those at any other mesh point. But the temporal correlations at a fixed point are significant, and affect the nature of the chamber flow in a variety of ways. We describe the fluctuations in the head-end pressure that arise because of them, and contrast these results with those calculated using a white-noise assumption.

Description: We consider the flapping stability and response of a thin two-dimensional flag of high extensional rigidity and low bending rigidity. The three relevant non-dimensional parameters governing the problem are the structure-to-fluid mass ratio, μ = ρsh/(ρfL); the Reynolds number,Rey=VL/ν; and the non-dimensional bending rigidity,KB=EI/(ρfV2L3). The soft cloth of a flag is represented by very low bending rigidity and the subsequent dominance of flow-induced tension as the main structural restoring force. We first perform linear analysis to help understand the relevant mechanisms of the problem and guide the computational investigation. To study the nonlinear stability and response, we develop a fluid–structure direct simulation (FSDS) capability, coupling a direct numerical simulation of the Navier–Stokes equations to a solver for thin-membrane dynamics of arbitrarily large motion. With the flow grid fitted to the structural boundary, external forcing to the structure is calculated from the boundary fluid dynamics. Using a systematic series of FSDS runs, we pursue a detailed analysis of the response as a function of mass ratio for the case of very low bending rigidity (KB= 10−4) and relatively high Reynolds number (Rey= 103). We discover three distinct regimes of response as a function of mass ratio μ: (I) a small μ regime of fixed-point stability; (II) an intermediate μ regime of period-one limit-cycle flapping with amplitude increasing with increasing μ; and (III) a large μ regime of chaotic flapping. Parametric stability dependencies predicted by the linear analysis are confirmed by the nonlinear FSDS, and hysteresis in stability is explained with a nonlinear softening spring model. The chaotic flapping response shows up as a breaking of the limit cycle by inclusion of the 3/2 superharmonic. This occurs as the increased flapping amplitude yields a flapping Strouhal number (St= 2Af/V) in the neighbourhood of the natural vortex wake Strouhal number,St≃ 0.2. The limit-cycle von Kármán vortex wake transitions in chaos to a wake with clusters of higher intensity vortices. For the largest mass ratios, strong vortex pairs are distributed away from the wake centreline during intermittent violent snapping events, characterized by rapid changes in tension and dynamic buckling.

Description: The governing equations for the two-point correlations of the turbulent fluctuating velocity in the temporally evolving wake were analysed to determine whether they could have equilibrium similarity solutions. It was found that these equations could have such solutions for a finite-Reynolds-number wake, where the two-point velocity correlations could be written as a product of a time-dependent scale and a function dependent only on similarity variables. It is therefore possible to collapse the two-point measures of all the scales of motions in the temporally evolving wake using a single set of similarity variables. As in an earlier single-point analysis, it was found that the governing equations for the equilibrium similarity solutions could not be reduced to a form that was independent of a growth-rate dependent parameter. Thus, there is not a single ‘universal’ solution that describes the state of the large-scale structures, so that the large-scale structures in the far field may depend on how the flow is generated.The predictions of the similarity analysis were compared to the data from two direct numerical simulations of the temporally evolving wakes examined previously. It was found that the two-point velocity spectra of these temporally evolving wakes collapsed reasonably well over the entire range of scales when they were scaled in the manner deduced from the equilibrium similarity analysis. Thus, actual flows do seem to evolve in a manner consistent with the equilibrium similarity solutions.

Description: This paper examines the consistency of the exact scaling laws for isotropic magnetohydrodynamic (MHD) turbulence in numerical simulations with large magnetic Prandtl numbersPmand withPm= 1. The exact laws are used to elucidate the structure of the magnetic and velocity fields. Despite the linear scaling of certain third-order correlation functions, the situation is not analogous to the case of Kolmogorov turbulence. The magnetic field is adequately described by a model of a stripy (folded) field with direction reversals at the resistive scale. At currently available resolutions, the cascade of kinetic energy is short-circuited by the direct exchange of energy between the forcing-scale motions and the stripy magnetic fields. This non-local interaction is the defining feature of isotropic MHD turbulence.

Description: The statistical properties of fully developed planar turbulent Couette–Poiseuille flow result from the simultaneous imposition of a mean wall shear force together with a mean pressure force. Despite the fact that pure Poiseuille flow and pure Couette flow are the two extremes of Couette–Poiseuille flow, the statistical properties of the latter have proved resistant to scaling approaches that coherently extend traditional wall flow theory. For this reason, Couette–Poiseuille flow constitutes an interesting test case by which to explore the efficacy of alternative theoretical approaches, along with their physical/mathematical ramifications. Within this context, the present effort extends the recently developed scaling framework of Weiet al. (2005a) and associated multiscaling ideas of Fifeet al. (2005a,b) to fully developed planar turbulent Couette–Poiseuille flow. Like Poiseuille flow, and contrary to the structure hypothesized by the traditional inner/outer/overlap-based framework, with increasing distance from the wall, the present flow is shown in some cases to undergo abalance breakingandbalance exchangeprocess as the mean dynamics transition from a layer characterized by a balance between the Reynolds stress gradient and viscous stress gradient, to a layer characterized by a balance between the Reynolds stress gradient (more precisely, the sum of Reynolds and viscous stress gradients) and mean pressure gradient. Multiscale analyses of the mean momentum equation are used to predict (in order of magnitude) the wall-normal positions of the maxima of the Reynolds shear stress, as well as to provide an explicit mesoscaling for the profiles near those positions. The analysis reveals a close relationship between the mean flow structure of Couette–Poiseuille flow and two internal scale hierarchies admitted by the mean flow equations. The averaged profiles of interest have, at essentially each point in the channel, a characteristic length that increases as a well-defined ‘outer region’ is approached from either the bottom or the top of the channel. The continuous deformation of this scaling structure as the relevant parameter varies from the Poiseuille case to the Couette case is studied and clarified.

Description: In this paper, we use numerical simulation and laboratory experimental observation to show that fire whirls can be generated spontaneously through the interaction between a central flame and surrounding organized or randomly distributed flames. The momentum of the air stream entrained by the main flame decreases as it crosses a surrounding flame, so that the main flame rotates if surrounding flames are arranged in such a way as to block the passage of the air stream directed towards the centre of the main flame and to favour flows in a particular circumferential direction. An analysis is performed to study the role of the rotation speed in the flame height. It is found that the flame height initially decreases to a minimum owing to the inflow boundary layer wind reducing the initial vertical velocity of gas for low rotation speed and to entrainment enhancement reducing the rising time, and then it increases owing to the pressure reduction at the centre of the rotating vortex and entrainment suppression extending the rising time.

Description: We investigate aspects of hovering insect flight by finding the optimal wing kinematics which minimize power consumption while still providing enough lift to maintain a time-averaged constant altitude over one flapping period. In particular, we study the flight of three insects whose masses vary by approximately three orders of magnitude: fruitfly (Drosophila melanogaster), bumblebee (Bombus terrestris), and hawkmoth (Manduca sexta). Here, we model an insect wing as a rigid body with three rotational degrees of freedom. The aerodynamic forces are modelled via a quasi-steady model of a thin plate interacting with the surrounding fluid. The advantage of this model, as opposed to the more computationally costly method of direct numerical simulation via computational fluid dynamics, is that it allows us to perform optimization procedures and detailed sensitivity analyses which require many cost function evaluations. The optimal solutions are found via a hybrid optimization algorithm combining aspects of a genetic algorithm and a gradient-based optimizer. We find that the results of this optimization yield kinematics which are qualitatively and quantitatively similar to previously observed data. We also perform sensitivity analyses on parameters of the optimal kinematics to gain insight into the values of the observed optima. Additionally, we find that all of the optimal kinematics found here maintain the same leading edge throughout the stroke, as is the case for nearly all insect wing motions. We show that this type of stroke takes advantage of a passive wing rotation in which aerodynamic forces help to reverse the wing pitch, similar to the turning of a free-falling leaf.

Description: Direct numerical simulations (DNS) of steady and pulsatile flow through 75% (by area reduction) stenosed tubes have been performed, with the motivation of understanding the biofluid dynamics of actual stenosed arteries. The spectral-element method, providing geometric flexibility and high-order spectral accuracy, was employed for the simulations. The steady flow results are examined here while the pulsatile flow analysis is dealt with in Part 2 of this study. At inlet Reynolds numbers of 500 and 1000, DNS predict a laminar flow field downstream of an axisymmetric stenosis and comparison to previous experiments show good agreement in the immediate post-stenotic region. The introduction of a geometric perturbation within the current model, in the form of a stenosis eccentricity that was 5% of the main vessel diameter at the throat, resulted in breaking of the symmetry of the post-stenotic flow field by causing the jet to deflect towards the side of the eccentricity and, at a high enough Reynolds number of 1000, jet breakdown occurred in the downstream region. The flow transitioned to turbulence about five diameters away from the stenosis, with velocity spectra taking on a broadband nature, acquiring a -5/3 slope that is typical of turbulent flows. Transition was accomplished by the breaking up of streamwise, hairpin vortices into a localized turbulent spot, reminiscent of the turbulent puff observed in pipe flow transition, within which r.m.s. velocity and turbulent energy levels were highest. Turbulent fluctuations and energy levels rapidly decayed beyond this region and flow relaminarized. The acceleration of the fluid through the stenosis resulted in wall shear stress (WSS) magnitudes that exceeded upstream levels by more than a factor of 30 but low WSS levels accompanied the flow separation zones that formed immediately downstream of the stenosis. Transition to turbulence in the case of the eccentric stenosis was found to be manifested as large temporal and spatial gradients of shear stress, with significant axial and circumferential variations in instantaneous WSS.

Description: We present both analytical and numerical solutions describing seepage flows in an unsaturated permeable seabed induced by transient long waves. The effects of compressibility of pore water in the seabed due to a small degree of unsaturation are considered in the investigation. To make the problem tractable analytically, we first focus our attention on situations where the horizontal scale of the seepage flow is much larger than the vertical scale. With this simplification the pore-water pressure in the soil column is governed by a one-dimensional diffusion equation with a specified pressure at the water–seabed interface and the no-flux condition at the bottom of the seabed. Analytical solutions for pore-water pressure and velocity are obtained for arbitrary transient waves. Special cases are studied for periodic waves, cnoidal waves, solitary waves and bores. Numerical solutions are also obtained by simultaneously solving the Navier–Stokes equations for water wave motions and the exact two-dimensional diffusion equation for seepage flows in the seabed. The analytical solutions are used to check the accuracy of the numerical methods. On the other hand, numerical solutions extend the applicability of the analytical solutions. The liquefaction potential in a permeable bed as well as the energy dissipation under various wave conditions are then discussed.

Description: Recent experiments have shown intriguing regions of intense luminescence or ‘hotspots’ in the vicinity of triple-point shear layers in propagating gaseous detonation waves. Localized explosions have also been observed to develop in these fronts. These features were observed in higher effective activation energy mixtures, but not in lower effective activation energy mixtures. The increased lead shock oscillation through a cell cycle in higher activation energy mixtures may result in a significantly increased disparity in the induction time on either side of the triple-point shear layer, and thus an enhanced mixing between reacted and non-reacted streams supported by Kelvin–Helmholtz instability. The relation between the shear-layer instability and the mixture effective activation energy is analysed by carrying out a spatial linear stability study for three mixtures with different activation energies and injection conditions that correspond to the experimental conditions. The role of vortical structures associated with Kelvin–Helmholtz instability in the formation of localized ignition is investigated by performing two-dimensional Navier–Stokes simulations with detailed chemical kinetics and transport. In the low activation energy mixture, large-scale vortical structures are observed to occur downstream of the induction distance; these structures do not have a noticeable effect on the reaction. In higher effective activation energy mixtures, a thin transverse ignition front develops near the interface between the two gas streams and results in a combustion structure decoupled from the entrainment region. The decoupling leads to attenuation of the instability growth rate when compared to frozen calculations, and a reduced heat release in the high vorticity region. The analysis indicates the instability plays a modest role in ignition events for high activation energy mixtures. The formation of localized explosions observed in high activation energy systems is instead linked to the impossibility of a one-dimensional reactive combustion wave supported by the injection conditions. In the absence of curvature effects and stream-tube divergence, a system of shock waves is formed which spreads the ignition to the cold gas stream.

Description: Mean flow profiles, skin friction, and integral parameters for boundary layers developing naturally over a wide variety of fully aerodynamically rough surfaces are presented and discussed. The momentum thickness Reynolds numberReθextends to values in excess of 47000 and, unlike previous work, a very wide range of the ratio of roughness element height to boundary-layer depth is covered (0.03 0.5). Comparisons are made with some classical formulations based on the assumption of a universal two-parameter form for the mean velocity profile, and also with other recent measurements. It is shown that appropriately re-written versions of the former can be used to collapse all the data, irrespective of the nature of the roughness, unless the surface is very rough, meaning that the typical roughness element height exceeds some 50% of the boundary-layer momentum thickness, corresponding to about$h/delta,{widetilde{〉}},0.2$.

Description: The effects of initial conditions on grid turbulence are investigated for low to moderate Reynolds numbers. Four grid geometries are used to yield variations in initial conditions and a secondary contraction is introduced to improve the isotropy of the turbulence. The hot-wire measurements, believed to be the most detailed to date for this flow, indicate that initial conditions have a persistent impact on the large-scale organization of the flow over the length of the tunnel. The power-law coefficients, determined via an improved method, also depend on the initial conditions. For example, the power-law exponentmis affected by the various levels of large-scale organization and anisotropy generated by the different grids and the shape of the energy spectrum at low wavenumbers. However, the results show that these effects are primarily related to deviations between the turbulence produced in the wind tunnel and true decaying homogenous isotropic turbulence (HIT). Indeed, when isotropy is improved and the intensity of the large-scale periodicity, which is primarily associated with round-rod grids, is decreased, the importance of initial conditions on both the character of the turbulence andmis diminished. However, even in the case where the turbulence is nearly perfectly isotropic,mis not equal to −1, nor does it show an asymptotic trend inxtowards this value, as suggested by recent analysis. Furthermore, the evolution of the second- and third-order velocity structure functions satisfies equilibrium similarity only approximately.

Description: The fragmentation of a laminar undulating liquid sheet flowing in quiescent air is investigated. Combining various observations and measurements we propose a sequential atomization scenario describing the overall sheet–drop transition in this configuration. The undulation results from a controlled primary Kelvin–Helmholtz instability. As the liquid travels through the undulating pattern, it experiences transient accelerations perpendicular to the sheet. These accelerations trigger a secondary instability responsible for the amplification of spanwise thickness modulations of the sheet. This mechanism, called the ‘wavy corridor’, is responsible for the sheet free edge indentations from which liquid ligaments emerge and break, forming drops. The final drop size distribution is of a Gamma type characterized by a unique parameter independent of the operating conditions once drop sizes are normalized by their mean.

Description: Slender-body theory is used to investigate the steady-state deformation and time-dependent evolution of an inviscid axisymmetric bubble in zero-Reynolds-number extensional flow, when insoluble surfactant is present on the bubble surface. The asymptotic solutions reveal steady ellipsoidal bubbles covered with surfactant, and, at increasing deformation, solutions distinguished by a cylindrical surfactant-free central part, with stagnant surfactant caps at the bubble endpoints. The bubble shapes are rounded near the endpoints, in contrast to the pointed shapes found for clean inviscid bubbles. Simple expressions are derived relating the capillary number Q to the steady bubble slenderness ratio ε. These show that there is a critical value Qc above which steady solutions no longer exist. Equations governing the time-evolution of a slender inviscid bubble with surfactant, valid for large capillary number, are also derived. Numerical solutions of the slender bubble equations for Q 〉 Qc exhibit spindle shapes with tip-streaming filaments. © 2005 Cambridge University Press.

Description: An experimental study has been performed on the dynamics of a large turbulent buoyanthelium plume. Two-dimensional velocity fields were measured using particle image velocimetry (PIV) while helium mass fraction was determined by planar laser-induced fluorescence (PLIF). PIV and PLIF were performed simultaneously in order to obtain velocity and mass fraction data over a plane that encompassed the plume core, the near-field mixing zones and the surrounding air. The Rayleigh-Taylor instability at the base of the plume leads to the vortex that grows to dominate the flow. This process repeats in a cyclical manner. The temporally and spatially resolved data show a strong negative correlation between density and vertical velocity, as well as a strong 90° phase lag between peaks in the vertical and horizontal velocities throughout the flow field owing to large coherent structures associated with puffing of the turbulent plume. The joint velocity an mass fraction data are used to calculate Favre-averaged statistics in addition to Reynolds-(time) averaged statistics. Unexpectedly, the difference between both the Favre-averaged and Reynolds-averaged velocities and second-order turbulent statistics is less than the uncertainty in the data throughout the flow field. A simple analysis was performed to determine the expected differences between Favre and Reynolds statistics for flows with periodic fluctuations in which the density and velocity fields are perfectly correlated, but have the phase relations as suggested by the data. The analytical results agreewith the data, showing that the Favre and Reynolds statistics will be the same to lead order. The combination of observation and simple analysis suggests that for buoyancy-dominated flows in which it can be expected that density and velocity are strongly correlated,phase relations will result in only second-order differences between Favre- and Reynolds-averaged data in spite of strong fluctuations in both density and velocity. © 2005 Cambridge University Press.

Description: Experiments have shown that when a mixture of fuel and oxygen is passed through a zirconia tube whose inner surface is coated with a catalyst, and then ignited at the end of the tube, a reaction front, or flame, propagates back along the tube towards the fuel inlet. The reaction front is visible as a red hot region moving at a speed of a few millimetres per second. In this paper we study a model of the flow, which takes into account diffusion, advection and chemical reaction at the inner surface of the tube. By assuming that the flame propagates at a constant speed without change of form, we can formulate a steady problem in a frame of reference moving with the reaction front. This is solved using the method of matched asymptotic expansions, assuming that the Reynolds and Damköhler numbers are large. We present numerical and, where possible, analytical results, first when the change in fluid density is small (a simplistic but informative limit) and secondly in the variable-density case. The speed of the travelling wave decreases as the critical temperature of the surface reaction increases and as the mass flow rate of fuel increases. We also make a comparison between our results and some preliminary experiments. © 2005 Cambridge University Press.

Description: We develop asymptotic solutions for passive suspended sediment transport under a flood surge on a uniform slope. Our solutions provide predictions of the net scour under a surge, and simple estimates of the conditions under which it may 'bulk up' into a mud or debris flow, as well as illustrating their sensitivity to sediment entrainment rates. © 2005 Cambridge University Press.

Description: We consider the evolution of a thin viscous fluid sheet subject to thermocapillary effects. Using a lubrication approximation we find, for symmetric interfacial deflections, coupled evolution equations for the interfacial profile, the streamwise component of the fluid velocity and the temperature variation along the surface. Initial temperature profiles change the initial flow field through Marangoni-induced shear stresses. These changes then lead to preferred conditions for rupture prescribed by the initial temperature distribution. We show that the time to rupture may be minimized by varying the phase difference between the initial velocity profile and the initial temperature profile. For sufficiently large temperature differences, the phase difference between the initial velocity and temperature profiles determines the rupture location. © 2005 Cambridge Universiy Press.

Description: In this paper we discuss numerical simulations of the generation of large-amplitude solitary waves in a continuously stratified fluid by flow over isolated topography. We employ the fully nonlinear theory for internal solitary waves to classify the numerical results for mode-1 waves and compare with two classes of approximate theories, weakly nonlinear theory leading to the Korteweg-deVries and Gardner equations and conjugate flow theory which makes no approximation with respect to nonlinearity, but neglects dispersion entirely. We find that both weakly nonlinear theories have a limited range of applicability. In contrast, the conjugate flow theory predicts the nature of the limiting upstream propagating response (a dissipationless bore), successfully describes the bore's vertical structure, and gives a value of the inflow speed, cj, above which no upstream propagating response is possible. The numerical experiments demonstrate the existence of a class of large-amplitude response structures that are generated and trapped over the topography when the inflow speed exceeds cj. While similar in structure to fully nonlinear solitary waves, these trapped disturbances can induce isopycnal displacements more than 100% larger than those induced by the limiting solitary wave while remaining laminar. We develop a theory to describe the vertical structure at the crest of these trapped disturbances and describe its range of validity. Finally, we turn to the generation of mode-2 solitary-like waves. Mode-2 waves cannot be truly solitary owing to the existence of a small mode-1 tail that radiates energy downstream from the wave. We demonstrate that, for stratifications dominated by a single pycnocline, mode-2 wave dissipation is dominated by wave breaking as opposed to mode-1 wave radiation. We propose a phenomenological criterion based on weakly nonlinear theory to test whether mode-2 wave generation is to be expected for a given stratification. © 2005 Cambridge University Press.

Description: We first report a quantitative experimental study of the collision of a spdisk with water, from a single to many skips. We then focus on the high spin limit and propose a simple model which enables us to discuss both the physical origin of the bounces and the source of the dissipation which fixes the number of skips. © 2005 Cambridge University Press.

Description: Laboratory experiments are carried out to determine the nature of internal wave breaking and the limiting wave steepness for progressive, periodic, lowest-mode internal waves in a two-layer, miscible density stratification. Shoaling effects are not considered. The waves investigated here are long relative to the thickness of the density interface separating the two fluid layers. Planar laser-induced fluoresence (PLIF) flow visualization shows that wave breaking most closely resembles a Kelvin-Helmholtz shear instability originating in the high-shear wave crest and trough regions. However, this instability is strongly temporally and spatially modified by the oscillations of the driving wave shear. Unlike a steady stratified shear layer, the wave instability discussed here is not governed by the canonical Ri = 1/4 stability limit. Instead, the wave time scale (the time scale of the destabilizing shear) imposes an additional constraint on instability, lowering the critical Richardson number below 1/4. Experiments were carried out to quantify this instability threshold, and show that, for the range of wavenumbers considered in this study, the critical wave steepness at which the wave breaking occurs is wavenumber-dependent (unlike surface waves). The corresponding critical wave Richardson numbers at incipient wave breaking are well below 1/4, in consonance with a modified instability analysis based on results from stratified shear flow instability theory. © 2005 Cambridge University Press.

Description: Linear waves in bounded inviscid fluids do not generally form normal modes with regular eigenfunctions. Examples are provided by inertial waves in a rotating fluid contained in a spherical annulus, and internal gravity waves in a stratified fluid contained in a tank with a non-rectangular cross-section. For wave frequencies in the ranges of interest, the inviscid linearized equations are spatially hyperbolic and their characteristic rays are typically focused onto wave attractors. When these systems experience periodic forcing, for example of tidal origin, the response of the fluid can become localized in the neighbourhood of a wave attractor. In this paper, I define a prototypical problem of this form and construct analytically the long-term response to a periodic body force in the asymptotic limit of small viscosity. The vorticity of the fluid is localized in a detached shear layer close to the wave attractor in such a way that the total rate of dissipation of energy is asymptotically independent of the viscosity. I further demonstrate that the same asymptotic dissipation rate is obtained if a non-viscous damping force is substituted for the Navier-Stokes viscosity. I discuss the application of these results to the problem of tidal forcing in giant planets and stars, where the excitation and dissipation of inertial waves may make a dominant, or at least important, contribution to the orbital and spin evolution. © 2005 Cambridge University Press.

Description: Although potential flow, including viscous dissipation, explains quite well the flow around individual bubbles of about 1mm radius rising in water, and e.g. predicts their drag quite accurately, this model cannot explain the homogeneous rise of a bubbly suspension. From numerical and analytical work it follows that eventually all bubbles cluster together. On the other hand it has been shown that velocity fluctuations of the bubbles of sufficient intensity, expressed in terms of a critical (pseudo) temperature, prevents clustering. Bubbles with radius above 0.8 mm rising in water perform zigzag or spiralling motions. Recently experimental and numerical work has made it clear that such bubbles have a wake behind them consisting of twin vortical threads carrying vorticity of opposite sign in the direction of motion. It is the purpose of this contribution to make an estimate of the velocity fluctuations induced by these trailing vortices in neighbouring bubbles. To this end the two-threaded wake is represented as a horseshoe vortex similar to the wake behind an airfoil. A pair of bubbles is considered and first the velocity induced by the horseshoe vortex behind one of the pair at the centre of the other is calculated. After this the force exerted on the latter based on the induced velocity and on the relative velocity of the bubbles, due to hydrodynamic interaction is calculated. Then the motion of one bubble in the pair is analysed under the influence both of this force and the hydrodynamic forces already there in the absence of the horseshoe vortex. Using these results and appropriate averaging, an estimate is made of the intensity of the velocity fluctuations of bubbles, and the corresponding temperature. © 2005 Cambridge University Press.

Description: We report experiments on the shape and motion of millimetre-sized drops sliding down a plane in a situation of partial wetting. When the Bond number based on the component of gravity parallel to the plane Boα exceeds a threshold, the drops start moving at a steady velocity which increases linearly with Boα. When this velocity is increased by tilting the plate, the drops change their aspect ratio: they become longer and thinner, but maintain a constant, millimetre-scale height. As their aspect ratio changes, a threshold is reached at which the drops are no longer rounded but develop a 'corner' at their rear: the contact line breaks into two straight segments meeting at a singular point or at least in a region of high contact line curvature. This structure then evolves such that the velocity normal to the contact line remains equal to the critical value at which the corner appears, i.e. to a maximal speed of dewetting. At even higher velocities new shape changes occur in which the corner changes into a 'cusp', and later a tail breaks into smaller drops (pearling transition). Accurate visualizations show four main results. (i) The corner appears when a critical non-zero value of the receding contact angle is reached. (ii) The interface then has a conical structure in the corner regime, the in-plane and out-of-plane angles obeying a simple relationship dictated by a lubrication analysis. (iii) The corner tip has a finite non-zero radius of curvature at the transition to a corner, and its curvature diverges at a finite capillary number, just before the cusp appears. (iv) The cusp transition occurs when the corner opening in-plane half-angle reaches a critical value of about 45°. © 2005 Cambridge University Press.

Description: Motivated by applications in rapidly rotating machinery, we have previously extended the lubrication model of the thin-film flow on the inside of a rotating circular cylinder to incorporate the effect of a constant shear applied to the free surface of the film and discovered a system rich in film profiles featuring shock structures. In this paper, we extend our model to include the effects of surface tension at leading order and take into account higher-order effects produced by gravity in order to resolve issues regarding existence, uniqueness and stability of such weak solutions to our lubrication model. We find, by analytical and numerical means, a set of feasible steady two-dimensional solutions that fit within a rational asymptotic framework. Having identified mathematically feasible solutions, we study their stability to infinitesimal two-dimensional disturbances. based on our findings, we conjecture which of the possible weak solutions are physically meaningful. © 2005 Cambridge University Press.

Description: We investigate the aerodynamics of freely falling plates in a quasi-two-dimensional flow at Reynolds number of 103 which is typical for a leaf or business card falling in air. We quantify the trajectories experimentally using high-speed digital video at sufficient resolution to determine the instantaneous plate accelerations and thus to deduce the instantaneous fluid forces. We compare the measurements with direct numerical solutions of the two-dimensional Navier-Stokes equation. Using inviscid theory as a guide, we decompose the fluid forces into contributions due to acceleration, translation, and rotation of the plate. For both fluttering and tumbling we find that the fluid circulation is dominated by a rotational term proportional to the angular velocity of the plate, as opposed to the translational velocity for a glider with fixed angle of attack. We find that the torque on a freely falling plate is small, i.e. the torque is one to two orders of magnitude smaller than the torque on a glider with fixed angle of attack. Based on these results we revise the existing ODE models of freely falling plates. We get access to different kinds of dynamics by exploring the phase diagram spanned by the Reynolds number, the dimensionless moment of inertia, and the thickness-to-width ratio. In agreement with previous experiments, we find fluttering, tumbling, and apparently chaotic motion. We further investigate the dependence on initial conditions and find brief transients followed by periodic fluttering described by simple harmonics and tumbling with a pronounced period-two structure. Near the cusp-like turning points, the plates elevate, a feature which would be absent if the lift depended on the translational velocity alone. © 2005 Cambridge University Press.

Description: The problem of short wind waves propagating on surface wind drift is considered here. Under the assumption of small monochromatic surface waves on a steady and horizontally uniform surface shear of an inviscid fluid, the governing equation becomes the well-known Rayleigh instability equation. Perturbation solutions exist for the surface-wave problem; however, the conditions for these approximations are violated in the case of short wind waves on wind drift shear. As an alternative approach, the piecewise-linear approximation (PLA) is explored. A proof is given for the rate of convergence of the piecewise-linear approximation for solving the Rayleigh equation without limitations on boundary conditions. The artificial modes of the piecewise-linear flow system are also discussed. The method is numerically efficient and highly accurate. Applying this method, the linear instability of various boundary-layer flows is examined. Short waves propagating with surface shear-flows are stable, while it is possible for waves that are travelling against a shear current to become unstable when the surface speed of the shear is greater than the wavespeed in stagnant fluid. PLA is also applied to examine the applicability of other perturbation approaches to the problem of propagation of short waves on wind drift shear. It is found that the existing approximations cannot fit the whole range of short wind waves. To bridge the gap, new approximations are derived from an implicit form of the exact dispersion relation based upon the variational principle. © 2005 Cambridge University Press.

Description: We present time-dependent governing equations and boundary conditions for the mushy-zone free-boundary problem that are valid in an arbitrary frame of reference. The model for time-evolving mushy zones is more complicated than in the steady case because the interface velocity w can be distinct from both the velocity of the dendrites v and the fluid velocity u. We consider the limit of negligible solutal diffusivity, where there are four types of boundary condition at the mush-liquid interface, depending on both the direction of flow across the interface and the direction of the interface motion relative to the solid phase. We illustrate these boundary conditions by examining a family of one-dimensional problems in which a binary material is chilled from a fixed cold point in the laboratory frame of reference while fluid is pumped through the resulting mushy layer at a rate Q and the mushy layer itself is translated at a rate V. This allows us to exhibit three of the four types of mushy-layer interfaces. We show that the fourth type cannot occur in this scenario. © 2005 Cambridge University Press.

Description: In this paper we study the steady uniform flows that develop when granular material is released from a hopper on top of a static pile in a channel. More specifically, we focus on the role of sidewalls by carrying out experiments in set-up of different widths, from narrow channels 20 particle diameters wide to channels 600 particle diameters wide. Results show that steady flows on pile are entirely controlled by sidewall effects. A theoretical model, taking into account the wall friction and based on a simple local constitutive law recently proposed for other granular flow configurations, gives predictions in quantitative agreement with the measurements. This result gives new insights into our understanding of free-surface granular flows and strongly supports the relevance of the constitutive law proposed. © 2005 Cambridge University Press.

Description: Two-dimensional (plane) solitary waves on the surface of water are known to bifurcate from linear sinusoidal wavetrains at specific wavenumbers k = k0 where the phase speed c(k) attains an extremum (dc/dk 0 = 0) and equals the group speed. In particular, such an extremum occurs in the long-wave limit k0 = 0, furnishing the familiar solitary waves of the Korteweg-de Vries (KdV) type in shallow water. In addition, when surface tension is included and the Bond number B = T/(ρgh2) 〈 1/3 (T is the coefficient of surface tension, ρ the fluid density, g the gravitational acceleration and h the water depth), c(k) features a minimum at a finite wavenumber from which gravity-capillary solitary waves, in the form of wavepackets governed by the nonlinear Schrödinger (NLS) equation to leading order, bifurcate in water of finite or infinite depth. Here, it is pointed out that an entirely analogous scenario is valid for the bifurcation of three-dimensional solitary waves, commonly referred to as 'lumps', that are locally confined in all directions. Apart from the known lump solutions of the Kadomtsev-Petviashvili I equation for formula B 〉 1/3 in shallow water, gravity-capillary lumps, in the form of locally confined wavepackets, are found for B 〈 1/3 in water of finite or infinite depth; like their two-dimensional counterparts, they bifurcate at the minimum phase speed and are governed, to leading order, by an elliptic-elliptic Davey-Stewartson equation system in finite depth and an elliptic two-dimensional NLS equation in deep water. In either case, these lumps feature algebraically decaying tails owing to the induced mean flow. © 2005 Cambridge University Press.

Description: While many rheological studies are performed at a fixed concentration, most granular flows are constrained, not by concentration, but by an applied stress. The stress constraint sets the average concentration, but the material is free to vary that concentration slightly to match the applied stress with that generated internally. This study examines stress-controlled systems in light of recent findings that the elastic properties of the particles appear as constitutive parameters even in flowing situations. Stress-controlled flows are shown to behave very differently from flows at fixed concentration. In particular, if the stress is fixed and the shear rate is slowly increased, the flow exhibits the expected progression from elastic-quasi-static to elastic-inertial to inertial flow - a sequence opposite to that followed in fixed-concentration flows. Thus system-scale constraints can have a profound effect on granular rheology. © 2005 Cambridge University Press.

Description: The general problem of propagation of three-dimensional disturbances in viscous supersonic flows is considered in the framework of characteristic analysis. Unlike previous results for linear disturbances we deduce a condition determining nonlinear characteristic surfaces which is exact and therefore allows both qualitative and quantitative studies of the speed of propagation as a function of various physical phenomena. These include negative and adverse pressure gradients, and effects of wall cooling and suction-blowing, which are studied in this work as an illustration of the general theory. © 2005 Cambridge University Press.

Description: Consider Stokes flow in a cone of half-angle α filled with a viscous liquid. It is shown that in spherical polar coordinates there exist similarity solutions for the velocity field of the type rλ f(θ;λ) exp imφ where the eigenvalueλ satisfies a transcendental equation. It follows, by extending an argument given by Moffatt (1964a), that if the eigenvalue λ is complex there will exist, associated with the corresponding vector eigenfunction, an infinite sequence of eddies as r → 0. Consequently, provided the principal eigenvalue is complex and the driving field is appropriate, such eddy sequences will exist. It is also shown that for each wavenumber m there exists a critical angle α* below which the principal eigenvalue is complex and above which it is real. For example, for m=1 the critical angle is about 74.45°. The full set of real and complex eigenfunctions, the inner eigenfunctions, can be used to compute the flow in a cone given data on the lid. There also exist outer eigenfunctions, those that decay for r → ∞, and these can be generated from the inner ones. The two sets together can be used to calculate the flow in a conical container whose base and lid are spherical surfaces. Examples are given of flows in cones and in conical containers which illustrate how α and r0, a length scale, affect the flow fields. The fields in conical containers exhibit toroidal corner vortices whose structure is different from those at a conical vertex; their growth and evolution to primary vortices is briefly examined. © 2005 Cambridge University Press.

Description: The rotational and translational motions of fibres in fully developed isotropic turbulence are simulated for a range of turbulence Reynolds numbers. Equations for fibre motion based on the leading-order slender-body theory relate the fibre's translational and rotational velocities to zeroth and first moments of the fluid velocity along the fibre length. The translational and rotational motions of fibres with lengths that exceed the size of the smallest eddies are attenuated by the filtering associated with these spatial averages. The translational diffusivity of the fibres can be predicted using a simple theory that neglects any coupling between fibre orientation and the local direction of the fluid velocity. However, the coupling of fibre orientation with the axes of extension and rotation is found to greatly reduce the amplitude of the rotary motions and the rotational dispersion coefficient. The rotary dispersion coefficient is found to be on the order of the inverse integral time scale. However, its variation with Reynolds number suggests that the rotary dispersion is influenced by all the scales of turbulence over the limited range of Reynolds numbers explored in our simulations. © 2005 Cambridge University Press.

Description: Experiments are reported on the sustained release of saline and particle-laden fluid into a long, but relatively narrow, flume, filled with fresh water. The dense fluid rapidly spread across the flume and flowed away from the source: the motion was then essentially two-dimensional. In the absence of a background flow in the flume, the motion was symmetric, away from the source. However, in the presence of a background flow the upstream speed of propagation was slowed and the downstream speed was increased. Measurements of this motion are reported and, when the excess density was due to the presence of suspended sediment, the distribution of the deposited particles was also determined. Alongside this experimental programme, new theoretical models of the motion were developed. These were based upon multi-layered depth-averaged shallow-water equations, in which the interfacial drag and mixing processes were explicitly modelled. While the early stages of the motion are independent of these interfacial phenomena to leading order, they play an increasingly important dynamical role as the the flow is slowed, or even arrested. In addition a new integral model is proposed. This does not resolve the interior dynamics of the flow, but may be readily integrated and obviates the need for more lengthy numerical calculations. It is shown that the predictions from both the shallow-layer and integral models are in close agreement with the experimental observations. © 2005 Cambridge University Press.

Description: The variability of the physical thickness of fully developed turbulent interfaces is examined using scalar measurements in the outer far-field regions of round jets at a Reynolds number of Re ∼ 20 000 and Schmidt number of Sc ∼ 2000. The interfacial thickness is considered in terms of the inverse magnitude of the scalar gradient across the interface and its relation to the scalar dissipation rate. The thickness variations and their conditional statistics are examined on outer interfaces at a resolution ∼ 10003 with data that capture the full transverse extent of the flow. At the resolution of the present measurements, the interfaces are observed to exhibit highly intermittent thickness variations that consist of striation patterns, or undulations, along the interfacial surfaces. The conditional probability density of the interfacial thickness is found to be nearly lognormal, in agreement with previous studies. A new scale-local density measure of the interfacial thickness is formulated to examine the effects of coarse graining and the dependence of the thickness on resolution scale. The scale-local thickness density, conditionally averaged on the outer interfaces, is found to exhibit self-similarity in a range of resolved scales. This observation of self-similar behaviour, in conjunction with intermittency, provides a physical ingredient useful for studies of phenomena sensitive to turbulent interfaces. © 2005 Cambridge University Press.

Description: We study vibrational instabilities of the thermocapillary return flow driven by a constant temperature gradient along the free surface of an infinite layer that vibrates in its normal direction with acceleration of amplitude g1 and frequency ω1. The layer is unstable to hydrothermal waves in the absence of vibrations beyond a critical Marangoni number M. Modulated gravitational instabilities with M = 0 are also possible beyond a critical Rayleigh number R based on g1. We employ two-time-scale high-frequency asymptotics to derive the equations governing the mean field. The influence of vibrations on the hydrothermal waves is found to be characterized by a dimensionless parameter G that is proportional to R2. The return flow at G = 0 is also a mean field basic flow and we study its linear instability at different Prandtl numbers P. The hydrothermal waves are stabilized with increasing G and reverse their direction of propagation at particular values of G that decrease with increasing P. At finite frequencies, a time-periodic base state exists and we study its linear instability by calculating the Floquet exponents. The stability boundaries in the (R, M)-plane are found to be composed of two intersecting branches emanating from the points of pure thermocapillary or buoyant instabilities. Three-dimensional modes are always preferred and the region of stability, while anchored at the point of hydrothermal waves corresponding to R = 0, is found to grow without bound along the R-axis with increasing frequencies. Results from the two approaches are shown to be in asymptotic agreement at large frequencies. © 2005 Cambridge University Press.

Description: In this study we consider the unsteady separated flow of an inviscid fluid (density ρf) around a falling flat plate (thickness T, half-chord L, width W, and density ρs) of small thickness and high aspect ratio (T ≪ L ≪ W). The motion of the plate, which is initially released from rest, is unknown in advance and is determined as part of the solution. The flow solution is assumed two-dimensional and to consist of a bound vortex sheet coincident with the plate and two free vortex sheets that emanate from each of the plate's two sharp edges. Throughout its motion, the plate continually sheds vorticity from each of its two sharp edges and the unsteady Kutta condition, which states the fluid velocity must be bounded everywhere, is applied at each edge. The coupled equations of motion for the plate and its trailing vortex wake are derived (the unsteady aerodynamic loads on the plate are included) and are shown to depend only on the modified Froude number Fr = T ρs/Lρf. Crucially, the unsteady aerodynamic loads are shown to depend on not only the usual acceleration reactions, which lead to the effect known as added mass, but also on novel unsteady vortical loads, which arise due to relative motion between the plate and its wake. Exact expressions for these loads are derived. An asymptotic solution to the full system of governing equations is developed for small times t 〉 0 and the initial motion of the plate is shown to depend only on the gravitational field strength and the acceleration reaction of the fluid; effects due to the unsteady shedding of vorticity remain of higher order at small times. At larger times, a desingularized numerical treatment of the full problem is proposed and implemented. Several example solutions are presented for a range of modified Froude numbers Fr and small initial inclinations θ0 〈 π/32. All of the cases considered were found to be unstable to oscillations of growing amplitude. The non-dimensional frequency of the oscillations is shown to scale in direct proportion with the inverse square root of the modified Froude number 1/√Fr. Importantly, the novel unsteady vortical loads are shown to dominate the evolution of the plate's trajectory in at least one example. Throughout the study, the possibility of including a general time-dependent external force (in place of gravity) is retained. © 2005 Cambridge University Press.

Description: Receptivity of compressible mixing layers to general source distributions is examined by a combined theoretical/computational approach. The properties of solutions to the adjoint Navier-Stokes equations are exploited to derive expressions for receptivity in terms of the local value of the adjoint solution. The result is a description of receptivity for arbitrary small-amplitude mass, momentum, and heat sources in the vicinity of a mixing-layer flow, including the edge-scattering effects due to the presence of a splitter plate of finite width. The adjoint solutions are examined in detail for a Mach 1.2 mixing-layer flow. The near field of the adjoint solution reveals regions of relatively high receptivity to direct forcing within the mixing layer, with receptivity to nearby acoustic sources depending on the source type and position. Receptivity 'nodes' are present at certain locations near the splitter plate edge where the flow is not sensitive to forcing. The presence of the nodes is explained by interpretation of the adjoint solution as the superposition of incident and scattered fields. The adjoint solution within the boundary layer upstream of the splitter-plate trailing edge reveals a mechanism for transfer of energy from boundary-layer stability modes to Kelvin-Helmholtz modes. Extension of the adjoint solution to the far field using a Kirchhoff surface gives the receptivity of the mixing layer to incident sound from distant sources. © 2005 Cambridge University Press.

Description: This work is concerned with the linearized theory of water waves applied to the motion of a floating structure that restricts in some way the motion of a portion of the free surface (an example of such a structure is a floating torus). When a structure of this type is held fixed in incident monochromatic waves, or forced to move time harmonically with a prescribed velocity, the amplitude of the fluid motion will have local maxima at certain frequencies of the forcing. These resonances correspond to poles of the scattering and radiation potentials when extended to the complex frequency domain. It is shown in this work that, in general, the positions of these poles in the scattering and radiation potentials will not coincide with the positions of the poles that appear in the velocity potential for the coupled problem obtained when the structure is free to move. The poles of the potential for the coupled problem are associated with the solution for the structural velocities of the equation of motion. When physical quantities such as the amplitude of the fluid motion are examined as a function of (real) frequency, there will in general be a shift in the resonant frequencies in going from the radiation and scattering problems to the coupled problem. The magnitude of this shift depends on the geometry of the structure and how it is moored. © 2005 Cambridge University Press.

Description: A vertically standing freely-rotating ellipsoidal vortex of uniform anomalous potential vorticity in a rotating stratified fluid under quasi-geostrophic conditions of small Rossby and Froude numbers steadily rotates without change of form. The vortex can have arbitrary axis lengths, but must have one axis parallel to the vertical z-axis along the direction of gravity. The rotation rate is proportional to the potential vorticity anomaly but otherwise depends on only two independent aspect ratios characterizing the shape of the vortex. The linear stability of this class of vortex equilibria was first determined semi-analytically more than a decade ago. It was found that vortices are unstable over a wide range of the parameter space and are stable only when strongly oblate and of nearly circular cross-section. New results, presented here, using a complementary approach and backed by non-linear simulations of the full quasi-geostrophic equations indicate that these ellipsoidal vortices are in fact stable over a much wider range of parameter space. In particular, a mode previously thought to be unstable over much of the parameter space is evidently stable. Moreover, we have determined that this mode is just the difference between two neighbouring equilibrium states having slightly different horizontal aspect ratios; hence, this mode must be neutrally stable. Agreement is found for all other modes. However, by an independent analysis considering only ellipsoidal (though time-varying) disturbances, we have identified one unstable mode as purely ellipsoidal, i.e. it does not change the form of the ellipsoid, only its shape. Under this instability, the vortex quasi-periodically tilts over while undergoing mild changes in shape. The range of parameters leading to non-ellipsoidal instabilities turns out to be narrow, with instability principally occurring for highly eccentric (horizontally squashed, prolate) vortices. The long-term fate of these Instabilities is examined by nonlinear contour-dynamical simulations. These reveal a wealth of complex phenomena such as the production of a sea of small-scale vortices, yet, remarkably, the dominant vortex often tends to relax to a stable rotating ellipsoid. © 2005 Cambridge University Press.

Description: The effect of angle of attack on the acoustic receptivity of the boundary layer over two-dimensional parabolic bodies is investigated using a spatial solution of the Navier-Stokes equations. The free stream is decomposed into a uniform flow with a superposed periodic velocity fluctuation of small amplitude. The method follows that of Haddad & Corke (1998) and Erturk & Corke (2001) in which the solution for the basic flow and linearized perturbation flow are solved separately. Different angles of incidence of the body are investigated for three leading-edge radii Reynolds numbers. For each, the angle of attack ranges from 0° to past the angle where the mean flow separates. The results then document the effect of the angle of incidence on the leading-edge receptivity coefficient (KLE), and in the case of the mean flow separation, on the amplitude of Tollmien-Schlichting (T-S) waves at the linear stability Branch II location (KII). For angles of attack before separation, we found that the leading-edge receptivity coefficient, KLE, increased with angle of incidence which correlated with an increase in the pressure gradient at the physical leading edge. When a separation zone formed at larger angles of incidence, it became a second site of receptivity with a receptivity coefficient that exceeded that of the leading edge. This resulted in dramatic growth of the T-S waves with Branch II amplitudes more than 100 times larger than those at angles just before separation, and 1000 times more than those at 0° angle of attack. © 2005 Cambridge University Press.

Description: Self-similar plane solutions for the inertial stage of gravity currents are related to the initial parameters and a coefficient that is determined by the boundary condition at the front. Different relations have been proposed for the boundary condition in terms of a Froude number at the front, none of which have a sound theoretical or experimental basis. This paper focuses on considerations of the appropriate Froude number based on results of lock-exchange experiments in which extended inertial gravity currents are generated in a rectangular cross-section channel. We use 'top-hat' vertical density profiles of the currents to obtain an 'equivalent' depth, defined by profiles having the same buoyancy at every position as the real profiles. As in previous work, our experimental results show that in the initial constant-velocity phase the Froude number can be defined in terms of the lock depth. However, as the current enters the similarity phase when the initial release conditions are no longer relevant, we find that the Froude number is more appropriately defined in terms of the maximum height of the head. Strictly speaking, the self-similar solution to the shallow-water equations requires a front condition that uses the height at the rear of the head. We find that this rear Froude number is not constant and is a function of the head Reynolds number over the range 400-4500. © 2005 Cambridge University Press.

Description: This paper presents an investigation of the nonlinear steepening of a gasdynamic disturbance propagating in a steady non-uniform base flow. The base flow is the steady compressible flow of a gas in a variable-area duct. The quasi-one-dimensional continuity, momentum and energy equations for the unsteady disturbance in homentropic flow are solved using the method of characteristics (wave front expansion technique). A closed-form solution for the slope of the disturbance at the wave front is obtained. The solution admits singularity for a compressive disturbance, which is responsible for the formation of shock in the flow. The solution is general and is applicable in any range of Mach number of the base flow. A special case of the steady gas flow in a convergent-divergent duct (C-D nozzle), where the flow makes a transition from subsonic to supersonic and vice versa, is investigated. © 2005 Cambridge University Press.

Description: The effect of initial conditions on the growth rate of turbulent Rayleigh-Taylor (RT) mixing has been studied using carefully formulated numerical simulations. A monotone integrated large-eddy simulation (MILES) using a finite-volume technique was employed to solve the three-dimensional incompressible Euler equations with numerical dissipation. The initial conditions were chosen to test the dependence of the RT growth coefficient (αb) and the self-similar parameter (βb = λb/hb) on (i) the amplitude, (ii) the spectral shape, (iii) the longest wavelength imposed, and (iv) mode-coupling effects. With long wavelengths present in the initial conditions, αb was found to increase logarithmically with the initial amplitudes, while βb is less sensitive to amplitude variations. The simulations are in reasonable agreement with the predictions for αb from a recently proposed model, but not for βb. In the opposite limit where mode-coupling dominates, no such dependence on initial amplitudes is observed, and αb takes a universal lower-bound value of ∼0.03 ± 0.003. This may explain the low values of αb reported by most numerical simulations that are initialized with annular spectra of short-wavelength modes and hence evolve purely through mode-coupling. Small-scale effects such as molecular mixing and kinetic energy dissipation showed a weak dependence on the structure of initial conditions. Initial density spectra with amplitudes distributed as k0, k-1 and k-2 were used to investigate the role of the spectral slopes on the development of turbulent RT mixing. Furthermore, in a separate study, the longest wavelength imposed in the initial wavepacket was also varied to determine its effect on αb. It was found that the slopes of the initial spectra, and the longest wavelength imposed had little effect on the RT growth parameters. © 2005 Cambridge University Press.

Description: The propagation of a liquid-filled crack from an over-pressured source into a semi-infinite uniform elastic solid is studied. The fluid is lighter than the solid and propagates due to its buoyancy and to the source over-pressure. The role of this over-pressure at early and late times is considered and it is found that the combination of buoyancy and over-pressure leads to significantly different behaviour from buoyancy or over-pressure alone. Lubrication theory is used to describe the flow, where the pressure in the fluid is determined by the elastic deformation of the solid due to the presence of the crack. Numerical results for the evolution of the crack shape and speed are obtained. The crack grows exponentially at early times, but at later times, when buoyancy becomes important, the crack growth accelerates towards a finite-time blow-up. These results are explained by asymptotic similarity solutions for early and late times. The predictions of these solutions are in close agreement with the full numerical results. A different case of crack geometry is also considered in order to highlight connections with previous work. The geological application to magma-filled cracks in the Earth's crust, or dykes, is discussed. © 2005 Cambridge University Press.

Description: The effect of the wall-normal diffusion on the spanwise spreading of a steady passive scalar interface is computed for a laminar channel in which the Péclét number, Pe, is high but the velocity profile is parabolic. Two regimes are found according to whether the dimensionless streamwise coordinate x̃ is smaller or larger than Pe. In both cases the mixing layer spreads as x̃1/2 to the lowest approximation in Pe-1, although with different numerical coefficients. When x̃ ≪ Pe there is a faster growth of order x̃1/3 that is restricted to boundary layers near the wall. The intermediate region between those two limits is universal, and is computed numerically. Quantitative results are given that should be useful to experimentally measure diffusion coefficients. The results are easily generalizable to other velocity profiles. © 2005 Cambridge University Press.

Description: Six direct numerical simulations of turbulent time-evolving strained plane wakes have been examined to investigate the response of a wake to successive irrotational plane strains of opposite sign. The orientation of the applied strain field has been selected so that the flow is the time-developing analogue of a spatially developing wake evolving in the presence of either a favourable or an adverse streamwise pressure gradient. The magnitude of the applied strain rate a is constant in time t until the total strain eat reaches about 4. At this point, a new simulation is begun with the sign of the applied strain being reversed (the original simulation is continued as well). When the total strain is reduced back to its original value of 1, yet another simulation is begun with the strain again being reversed back to its original sign. This is done for both initially 'favourable' and initially 'adverse' strains, providing simulations for each of these strain types from three different initial conditions. The evolution of the wake mean velocity deficit and width is found to be similar for all the 'adversely' strained cases, with both measures rapidly achieving exponential growth at the rate associated with the cross-stream expansive strain eat. In the 'favourably' strained cases, the wake widths approach a constant and the velocity deficits ultimately decay rapidly as e-2at. Although all three of these cases do exhibit the same asymptotic exponential behaviour, the time required to achieve this is longer for the cases that have been previously adversely strained (by at ≈ 1). The evolution described above is not consistent with the predictions of classical self-similar analysis; a more general 'equilibrium similarity solution' is required to describe the results. Examination of these simulations confirms that the wake width and mean velocity deficit evolutions observed in Rogers (2002) are not a result of the particular initial condition used in that work. At least for the cases considered here, the wake Reynolds number and the ratio of the turbulent kinetic energy to the square of the wake mean velocity deficit are determined nearly entirely by the total strain. For these measures, the order in which the strains are applied does not matter and the changes brought about by the strain are nearly reversible. The wake mean velocity deficit and width, on the other hand, differ by about a factor of 3 when the total strain returns to 1, depending on whether the wake was first 'favourably' or 'adversely' strained. The strain history is important for predicting the evolution of these quantities. © 2005 Cambridge University Press.

Description: The motion of a light particle in an eccentrically rotating cylinder provides a method for verifying stationary history lift force effects at low but non-zero particle Reynolds numbers. We examine the flow in detail using a Lagrangian equation of motion for constant, non-zero-vorticity flows, and we predict a measurable and stationary contribution of history lift effects that can be verified experimentally with current experimental techniques. Because the history lift contribution is relevant only under certain conditions (which are determined in this work), the present flow configuration also allows one to isolate history drag effects under normal gravitation conditions without resorting to the tethered-particle arrangement used in previous works. We formulate and solve the trajectory problem for light particles that attain stable orbital motion, and we propose an experimental concept that makes possible the study of individual contributions of Lagrangian forces to the motion of small particles in viscous flows. © 2005 Cambridge University Press.

Description: The intrusion of a fixed volume of fluid which is released from rest and then propagates horizontally at the neutral buoyancy level in a vertically stratified ambient fluid is investigated. The density change is linear, in a restricted layer or over the full depth of the container, and locks of both rectangular and cylindrical shapes are considered. A closed one-layer shallow-water inviseid formulation is used to obtain solutions of the initial-value problem. Similarity solutions for the large-time developed motion and an approximate box model are also presented. The results are corroborated by numerical solutions of the full two-dimensional Navier-Stokes equations and comparisons with previously published experiments. It is shown that the model is a versatile predictive tool which clarifies essential features of the flow field. Accurate insights are provided concerning the governing dimensionless parameters and the major features of the motion. In particular, the theory predicts and explains: (a) the fact that the initial propagation is with constant speed for intrusions released from a rectangular lock; (b) the effect of the shape of the lock on the motion; (c) the spread with time at some power in the developed stage; and (d) the sub-critical (compared to the mode 2 linear waves) speed in a full-depth stratified container configuration. The main deficiency of the shallow-water model is that internal gravity waves are not incorporated, but some insight into this effect is provided by the comparisons with the Navier-Stokes simulations and experiments. © 2005 Cambridge University Press.

Description: Grease ice is an agglomeration of disk-shaped ice crystals, named frazil ice, which forms in turbulent waters of the Polar Oceans and in rivers as well. It has been recognized that the property of grease ice that it damps surface gravity waves could be explained in terms of the effective viscosity of the ice slurry. This paper is devoted to the study of the dynamics of a suspension of disk-shaped particles in a gravity wave field. For dilute suspensions, depending on the strength and frequency of the external wave flow, two orientation regimes of the particles are predicted: a preferential orientation regime with the particles rotating in coherent fashion with the wave field, and a random orientation regime in which the particles oscillate around their initial orientation while diffusing under the effect of Brownian motion. For both motion regimes, the effective viscosity has been derived as a function of the wave frequency, wave amplitude and aspect ratio of the particles. Model predictions have been compared to wave attenuation data in frazil ice layers grown in wave tanks. © 2005 Cambridge University Press.

Description: We investigate a theoretical model of the pulsatile motion of a contaminant-doped semi-infinite bubble in a rectangular channel. We examine the fluid mechanical behaviour of the pulsatile bubble, and its influence on the transport of a surface-inactive contaminant (termed surfinactant). This investigation is used to develop a preliminary understanding of surfactant responses during unsteady pulmonary airway reopening. Reopening is modelled as the pulsatile motion of a semi-infinite gas bubble in a horizontal channel of width 2a filled with a Newtonian liquid of viscosity μ and constant surface tension γ. A modified Langmuir sorption model is assumed, which allows for the creation and respreading of a surface multilayer. The bubble is forced via a time-dependent volume flux Q(t) with mean and oscillatory components (QM and Qω respectively) at frequency ω. The flow behaviour is governed by the dimensionless parameters: CaM = μ QM/(2aγ), a steady-state capillary number, which represents the ratio of viscous to surface tension forces; CaΩ = μ Qω/(2aγ), an oscillatory forcing magnitude; Ω = ωμa/γ, a dimensionless frequency that represents the ratio of viscous relaxation to oscillatory-forcing timescales; and A = 2CaΩ/ Ω, a dimensionless oscillation amplitude. Our simulations indicate that contaminant deposition and retention in the bubble cap region occurs at moderate frequencies if retrograde bubble motion develops during the oscillation cycle. However, if oscillations are too rapid the ensuing large forward tip velocities cause a net loss of contaminant from the bubble tip. Determination of an optimal oscillation range may be important in reducing ventilator-induced lung injury associated with infant and adult respiratory distress syndromes by increasing surfactant transport to regions of collapsed airways. © 2005 Cambridge University Press.

Description: I calculate the optimal upper bound, subject to the assumption of streamwise invariance, on the long-time-averaged buoyancy flux ℬ* within the flow of an incompressible stratified viscous fluid of constant kinematic viscosity ν, and depth h driven by a constant surface stress τ = ρu2* where u* is the friction velocity with a constant statically stable density difference Δρ maintained across the layer. By using the variational 'background method' (due to Constantin, Doering and Hopf) and numerical continuation, I generate a rigorous upper bound on the buoyancy flux for arbitrary Grashof numbers G, where G = τh2/(ρν2). As G → ∞, for flows where horizontal mean momentum balance, horizontally averaged heat balance, total power balance and total entropy flux balance are imposed as constraints, I show numerically that the best possible upper bound for the buoyancy flux is given by ℬ* ≤ℬ* max = u4*/(4ν)+O(u3*/h). This bound is independent of both the overall strength of the stratification and the layer depth to leading order. This bound is associated with a velocity profile that has the scaling characteristics of a somewhat decelerated laminar, linear velocity profile. © 2005 Cambridge University Press.

Description: The generation of a gravity current by the release of a semi-infinite region of buoyant fluid of depth H overlying a deeper, denser and quiescent lower layer in a rotating channel of width w is considered. Previous studies have focused on the characteristics of the gravity current head region and produced relations for the gravity current speed cb and width wb as a functions of the local current depth along the wall hb, reduced gravity g′, and Coriolis frequency f. Here, the dam-break problem is solved analytically by the method of characteristics assuming reduced-gravity flow, uniform potential vorticity and a semigeostrophic balance. The solution makes use of a local gravity current speed relation cb = cb(hb,...) and a continuity constraint at the head to close the problem. The initial value solution links the local gravity current properties to the initiating dam-break conditions. The flow downstream of the dam consists of a rarefaction joined to a uniform gravity current with width wb (≤ w) and depth on the right-hand wall of hb, terminated at the head moving at speed cb. The solution gives hb, cb, wb and the transport of the boundary current as functions of w/LR, where LR = √g′H/ f is the deformation radius. The semigeostrophic solution compares favourably with numerical solutions of a single-layer shallow-water model that internally develops a leading bore. Existing laboratory experiments are re-analysed and some new experiments are undertaken. Comparisons are also made with a three-dimensional shallow-water model. These show that lateral boundary friction is the primary reason for differences between the experiments and the semigeostrophic theory. The wall no-slip condition is identified as the primary cause of the experimentally observed decrease in gravity current speed with time. A model for the viscous decay is developed and shown to agree with both experimental and numerical model data. © 2005 Cambridge University Press.

Description: A new law for the thinning of surfactant-free lamellae (applicable to metallic and ceramic foams with mobile interfaces) in a cross-section of an arid gas-liquid foam is derived using matched asymptotic analysis. Two limiting cases are identified at small capillary number: the well-known semi-arid foam having unit-order liquid fraction and the arid foam in which it is small. The lamellar thinning rates in both cases exhibit t-2 power-law behaviour at long times even though the foam liquid area fractions have different orders of magnitude in capillary number. At early times, arid foam thinning is slowed because the curvature of the capillary quasi-static interfacial region must decrease in order to accommodate the flow from the films. Therefore, the thinning of lamellae feeding into a given Plateau border is coupled and the dynamics is distinct from that of the semi-arid foam. Approximations of rupture times in arid and semi-arid foams are found by calculating the times for lamellae to thin to a pre-specified thickness. For given initial lamellar thicknesses, and for arid and semi-arid foams that have identical initial lamellar liquid areas, the arid foam ruptures more quickly than the semi-arid foam. On the other hand the rupture of lamellae is significantly delayed in arid foam compared to semi-arid foam if the initial lamellar thickness and capillary number are the same. © 2005 Cambridge University Press.

Description: Measurements of the flow field around a model rotor descending axially into its own vortex wake have been performed using particle image velocimetry (PIV). At low descent rates, the expected cylindrical down-flow structure below the rotor is observed. At slightly higher descent rate, the flow enters the so-called vortex ring state (VRS) where the vorticity from the rotor accumulates into a toroidal structure near the rotor tips, and a large recirculation zone forms above the rotor disk. In the VRS, the flow below the rotor shows a significant upwards component, with a small up-flow zone penetrating right up to the rotor disk. Measurements show there to be a range of descent rates just before the onset of the VRS over which the flow may be interpreted to be in an incipient VRS condition. In this range, analyses of individual PIV measurements indicate that the flow near the rotor intermittently switches between the down-flow topology found at lower descent rates and the flow topology found in the fully developed VRS. The frequency of excursions of the flow into the VRS topology increases as the descent rate of the rotor is increased until, at high enough descent rate, the flow remains locked within its toroidal state. © 2005 Cambridge University Press.

Description: The feasibility of an experimental method for investigations of the particle flux to an absorbing surface in turbulent flows is demonstrated in a Lagrangian as well as an Eulerian representation. A laboratory experiment is carried out, where an approximately homogeneous and isotropic turbulent flow is generated by two moving grids. The simultaneous trajectories of many small approximately neutrally buoyant polystyrene particles are followed in time. In a Lagrangian analysis, we select one of these as the centre of a 'sphere of interception', and obtain estimates for the time variation of the statistical average of the inward particle flux through the surface of this moving sphere. The variation of the flux with the radius in the sphere of interception, as well as the variation with basic flow parameters is described well by a simple model, in particular for radii smaller than a characteristic length scale for the turbulence. The Eulerian counterpart of the problem is analysed as well, and the two results compared. Applications of the problem to, for instance, the question of the feeding rate of micro-organisms in turbulent marine environments are pointed out. © 2005 Cambridge University Press.

Description: This paper presents simulations of the two-dimensional model developed by Pothérat et al. (2000) for MHD flows between two planes with a strong transverse homogeneous and steady magnetic field, accounting for moderate inertial effects in Hartmann layers. We first show analytically how the additional terms, in the equations of motion accounting for inertia soften velocity gradients in the horizontal plane, and then we implement the model in a code to carry out numerical simulations to be compared with available experimental results. This comparison shows that the new model can give very accurate results as long as the Hartmann layer remains laminar. Both experimental velocity profiles and global angular momentum measurements are recovered, and local and global Ekman recirculations are shown to alter significantly the shape of the flow as well as the global dissipation. © 2005 Cambridge University Press.

Description: The effect of particle-particle contact on the stress of a suspension of small spheres in plane strain flow is investigated. We provide an analytic form for the particle pair distribution function in the case of no Brownian motion, and calculate the viscosity and normal stress difference based on this. We show that the viscosity is reduced by contact, and a normal stress difference induced, both at order c2 for small particle volume concentration c. In addition, we investigate the effect of a small amount of diffusion on the structure of the distribution function, giving a self-consistent form for the density in the O(aPe-1) boundary layer and demonstrating that diffusion reduces the magnitude of the contact effect but does not qualitatively alter it. © 2005 Cambridge University Press.

Description: We study the linearized Navier-Stokes (LNS) equations in channel flows from an input-output point of view by analysing their spatio-temporal frequency responses. Spatially distributed and temporally varying body force fields are considered as inputs, and components of the resulting velocity fields are considered as outputs into these equations. We show how the roles of Tollmien-Schlichting (TS) waves, oblique waves, and streamwise vortices and streaks in subcritical transition can be explained as input-output resonances of the spatio-temporal frequency responses. On the one hand, we demonstrate the effectiveness of input field components, and on the other, the energy content of velocity perturbation components. We establish that wall-normal and spanwise forces have much stronger influence on the velocity field than streamwise force, and that the impact of these forces is most powerful on the streamwise velocity component. We show this using the relative scaling of the different input-output system components with the Reynolds number. We further demonstrate that for the streamwise constant perturbations, the spanwise force localized near the lower wall has, by far, the strongest effect on the evolution of the velocity field. © 2005 Cambridge University Press.

Description: Laboratory experiments are conducted to study the rheological behaviour of high-concentration granular-liquid mixtures. Steady uniform free-surface flows are obtained using a recirculating flume. Cases in which a loose deposit forms underneath the flow are contrasted with runs for which basal shear occurs along the flume bottom. The granular motions are observed through the channel sidewall, and analysed with recently developed Voronoï imaging methods. Depth profiles of mean velocity, solid concentration, and granular temperature are obtained, and complemented by stress estimates based on force balance considerations. These measurements are used to probe variations in rheological behaviour over depth, and to clarify the role of the granular temperature. The flows are found to evolve a stratified structure. Distinct sublayers are characterized by either frictional or collisional behaviour, and transitions between one and the other occur at values of the Stokes number which suggest that viscous effects intervene. The observed frictional behaviour is consistent with shear cell tests conducted at very low shear rates. On the other hand, the collisional data corroborate both the Bagnold description and the more recent kinetic theories of granular flows, provided that one accounts for the inertia of the interstitial liquid. © 2005 Cambridge University Press.

Description: Direct numerical simulation (DNS) of fully developed turbulent flow in elliptical ducts is performed. The mean cross-stream secondary flows exhibited by two counter-rotating vortices which are symmetrical about the major ellipse's axis are examined. The mean flow characteristics and turbulence statistics are obtained. The variation of the statistical quantities such as the Reynolds stresses and turbulence intensities along the minor axis of the elliptical cross-section are found to be similar to plane channel data. The turbulent statistics along the major axis are found to be inhibited by the secondary flow transferring high-momentum fluid from the duct's centre towards the wall. The instantaneous velocity fields in the near-wall region reveal structures similar to the 'streaks' except in the vicinity of the major axis endpoints where significant reduction of the turbulent activity due to the wall transverse curvature effect is found. © 2005 Cambridge University Press.

Description: The relaminarization phenomenon in the laminar-turbulent transition process of spherical Couette flow with the inner sphere rotating and the outer sphere fixed has been experimentally investigated for 0.06 ≤ β ≤ 0.206, where β is the ratio of the clearance to the inner-sphere radius. The relaminarization occurs for 0.13 〈 β 〈 0.17, and is observed as a reverse Hopf bifurcation from the limit cycle to the fixed point. The kinetic energy in the high-frequency region of the fluctuating azimuthal velocity component continues to increase with Reynolds number Re in the case without relaminarization, but in the case with relaminarization it first increases and then decreases with increasing Re and finally vanishes with the onset of the relaminarization. For β = 0.14 with relaminarization, a small artificial disturbance introduced externally into the spherical Couette flow has no influence on the lowest critical Reynolds number of the first instability that can be defined universally and uniquely. On the other hand, the external disturbance decreases the onset Reynolds numbers of both the second instability (occurrence of spiral Taylor-Görtler (TG) vortices) and the relaminarization, although it has no influence on the fundamental frequencies of velocity fluctuation caused by spiral TG vortices and travelling waves on TG vortices. © 2005 Cambridge University Press.

Description: In applications involving the model-based control of transitional wall-bounded flow systems, it is often desired to estimate the interior flow state based on a history of noisy measurements from an array of flush-mounted skin-friction and pressure sensors on the wall. This paper considers this estimation problem, using a Kalman filter based on the linearized Navier-Stokes equations and appropriate stochastic models for the relevant statistics of the initial conditions, sensor noise and external disturbances acting on the system. We show that a physically relevant parameterization of these statistics is key to obtaining well-resolved feedback kernels with appropriate spatial extent for all three types of flow measurement available on the wall. The effectiveness of the resulting Kalman and extended Kalman filters that implement this feedback is verified for both infinitesimal and finite-amplitude disturbances in direct numerical simulations of a perturbed laminar channel flow. The consideration of time-varying feedback kernels is shown to be particularly advantageous in accelerating the convergence of the estimator from unknown initial conditions. A companion paper (Part 2) considers the extension of such estimators to the case of fully developed turbulence. © 2005 Cambridge University Press.

Description: Time-dependent computations of the two-dimensional incompressible uniform-velocity laminar flow past a normal flat plate (of unit half-width) in a channel are presented. Attention is restricted to cases in which the well-known anti-symmetric (von Kármán-type) vortex shedding is suppressed by the imposition of a symmetry plane on the downstream plate centreline. With a further symmetry plane at the channel's upper boundary, the only two governing parameters in the problem are the channel half-width, H, and the Reynolds number, Re (based on the body half-width and the upstream velocity, U). The former is restricted to the range 3≤H≤30 and the interest lies in determining the nature of the initial instability which occurs in the separated wake as Re is gradually increased. It is found that for sufficiently large H and at a critical Re, a long-time-scale global (supercritical) instability is initiated, which in its saturated (limit) state takes the form of 'lumps' of vorticity being periodically shed from the tail end of the separated bubble. Stability calculations of corresponding mean flow profiles (typical of those found in the separated wake) are undertaken by examining the impulse response of particular profiles via appropriate solution of the Orr-Sommerfeld equation. The results of this analysis extend those available from related published work and are consistent with the behaviour found from the numerical computations. Taken together, all the results suggest that this type of global instability may be generic to many kinds of separated wakes and, indeed, may provide the fundamental explanation for the very low-frequency oscillations often noticed in fully turbulent wake bubbles. © 2005 Cambridge University Press.

Description: This paper assesses the potential of coherent vortex simulation (CVS) to compute three-dimensional turbulent mixing layers. CVS splits each turbulent flow realization into two orthogonal parts, one corresponding to coherent vortices which are kept, and the other to an incoherent background flow which is discarded. The CVS filter is applied to data from direct numerical simulations (DNS) of three-dimensional forced and unforced time-developing turbulent mixing layers. The coherent flow is represented by few wavelet modes, but these are sufficient to reproduce the vorticity probability distribution function and the energy spectrum out to the high-wavenumber end of the inertial range. The discarded incoherent background flow is homogeneous, small-amplitude and decorrelated. The CVS-filtering results are then compared with those obtained for the same compression ratio using Fourier low-pass filtering, as employed in large-eddy simulation (LES). Compared to the incoherent background flow of CVS filtering, the subgrid scales of LES filtering are less homogeneous, have much larger amplitude, and exhibit spatial correlations that makes modelling them a difficult challenge. Finally we present simulations of a time-developing mixing layer where the CVS filter is applied at each time step. The results show that CVS preserves the nonlinear dynamics of the flow, and that discarding the incoherent modes is sufficient to model turbulent dissipation. © 2005 Cambridge University Press.

Description: High-resolution laser-Doppler anemometer measurements were acquired in a two-dimensional turbulent boundary layer over a 4° ramp. The goals were to provide a detailed data set for an adverse pressure gradient boundary layer far from separation and to examine near-wall behaviour of the Reynolds stresses as compared to flat-plate boundary layers. The flow develops over a flat plate, reaching a momentum thickness Reynolds number of 3350 at an upstream reference location. The boundary layer is then subjected to a varying pressure gradient along the length of the ramp and partially redevelops on a downstream flat plate. Mean velocity measurements show a log law region in all velocity profiles, but the outer layer does not collapse in deficit coordinates indicating that the boundary layer is not in equilibrium. Measurements of non-dimensional stress ratios and quadrant analysis of the two-component data indicate relatively small changes to the turbulence structure. However, the streamwise normal stress has an extended outer layer plateau, and the shear stress and wall-normal stress have outer layer peaks. Near the wall, the streamwise normal stress and shear stress collapse with flat-plate data using standard scaling, but the wall normal stress is substantially larger than flat-plate cases. © 2005 Cambridge University Press.

Description: New and unexpected results are presented regarding the nonlinear interactions between a wavepacket and a vortical mean flow, with an eye towards internal wave dynamics in the atmosphere and oceans and the problem of 'missing forces' in atmospheric gravity-wave parametrizations. The present results centre around a prewave-breaking scenario termed 'wave capture', which differs significantly from the standard such scenarios associated with critical layers or mean density decay with altitude. We focus on the peculiar wave-mean interactions that accompany wave capture. Examples of these interactions are presented for layerwise-two-dimensional, layerwise-non-divergent flows in a three-dimensional Boussinesq system, in the strong-stratification limit. The nature of the interactions can be summarized in the phrase ' wave-vortex duality', whose key points are firstly that wavepackets behave in some respects like vortex pairs, as originally shown in the pioneering work of Bretherton (1969), and secondly that a collection of interacting wavepackets and vortices satisfies a conservation theorem for the sum of wave pseudomomentum and vortex impulse, provided that the impulse is defined appropriately. It must be defined as the rotated dipole moment of the Lagrangian-mean potential vorticity (PV). This PV differs crucially from the PV evaluated from the curl of either the Lagrangian-mean or the Eulerian-mean velocity. The results are established here in the strong-stratification limit for rotating (quasi-geostrophic) as well as for non-rotating systems. The concomitant momentum budgets can be expected to be relatively complicated, and to involve far-field recoil effects in the sense discussed in Bühler & McIntyre (2003). The results underline the three-way distinction between impulse, pseudomomentum, and momentum. While momentum involves the total velocity field, impulse and pseudomomenturn involve, in different ways, only the vortical part of the velocity field. © 2005 Cambridge University Press.

Description: We examine available data from experiment and recent numerical simulations to explore the supposition that the scalar dissipation rate in turbulence becomes independent of the fluid viscosity when the viscosity is small and of scalar diffusivity when the diffusivity is small. The data are interpreted in the context of semi-empirical spectral theory of Obukhov and Corrsin when the Schmidt number, Sc, is below unity, and of Batchelor's theory when Sc is above unity. Practical limits in terms of the Taylor-microscale Reynolds number, Rλ, as well as Sc, are deduced for scalar dissipation to become sensibly independent of molecular properties. In particular, we show that such an asymptotic state is reached if RλSc1/2 ≫ 1 for Sc 〈 1, and if ln(Sc)/Rλ ≫ 1 for Sc 〈 1. © 2005 Cambridge University Press.