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We present new insight into the classical problem of a uniform flow, linearly stratified in density, past an isolated three-dimensional obstacle. We demonstrate how, for a low-Froude-number obstacle, simple linear theory with a linearized boundary condition is capable of providing excellent quantitative agreement with laboratory measurements of the perturbation to the density field. It has long been known that such a flow may be divided into two regions, an essentially horizontal flow around the base of the obstacle and a wave-generating flow over the top of the obstacle, but until now the experimental diagnostics have not been available to test quantitatively the predicted features. We show that recognition of a small slope that develops across the obstacle in the surface separating these two regions is vital to rationalize experimental measurements with theoretical predictions. Utilizing the principle of stationary phase and causality arguments to modify the relationship between wavenumbers in the lee waves, linearized theory provides a detailed match in both the wave amplitude and structure to our experimental observations. Our results demonstrate that the structure of the lee waves is extremely sensitive to departures from horizontal flow, a detail that is likely to be important for a broad range of geophysical manifestations of these waves.
Effects of chordwise, spanwise, and isotropic flexibility on the force generation and propulsive efficiency of flapping wings are elucidated. For a moving body immersed in viscous fluid, different types of forces, as a function of the Reynolds number, reduced frequency (k), and Strouhal number (St), acting on the moving body are identified based on a scaling argument. In particular, at the Reynolds number regime of and the reduced frequency of , the added mass force, related to the acceleration of the wing, is important. Based on the order of magnitude and energy balance arguments, a relationship between the propulsive force and the maximum relative wing-tip deformation parameter () is established. The parameter depends on the density ratio, St, k, natural and flapping frequency ratio, and flapping amplitude. The lift generation, and the propulsive efficiency can be deduced by the same scaling procedures. It seems that the maximum propulsive force is obtained when flapping near the resonance, whereas the optimal propulsive efficiency is reached when flapping at about half of the natural frequency; both are supported by the reported studies. The established scaling relationships can offer direct guidance for micro air vehicle design and performance analysis.
The purpose of this study is to provide generic insights into the rheology of immersed particulate materials in dense configurations. We focus on a non-Brownian and Stokesian suspension of elastic spheres, as a way to identify the effect on the collective behaviour of two short-range interactions: lubrication and steric repulsion below a tunable range. The response of the material to shear under a prescribed shear rate and a prescribed normal stress is simulated using ‘soft dynamics’, a discrete element method which accounts for the dynamics of such a system. The material exhibits a visco-elastic behaviour, deforming elastically at high shear rates, and viscously at slower shear rates. When steady flow is established, the constitutive law can be expressed, as for dry grains, through a friction law and a dilatancy law, whose numerical constants depend in a non-trivial manner on the equilibrium gap. The analysis of the contribution of each interacting pair as a function of its distance and orientation sheds some light on this collective behaviour. Lubrication, which hinders grain approach, is responsible for a loss of contact and a low repulsion contribution to the stress. In contrast, lubrication viscously connects each grain to as many as 10 neighbours, which all contribute significantly to the shear stress. This forms a robust dynamic connected network controlling the collective resistance to flow. This study should be useful for modelling the rheological behaviour of real materials such as foams, emulsions and blood: beside the specific properties of their particles, these particulate fluids also involve lubrication and repulsion.
Previous work has shown that aspects of the evolution of large-scale structures, particularly in forced and transitional mixing layers and jets, can be described by linear and nonlinear stability theories. However, questions persist as to the choice of the basic (steady) flow field to perturb, and the extent to which disturbances in natural (unforced), initially turbulent jets may be modelled with the theory. For unforced jets, identification is made difficult by the lack of a phase reference that would permit a portion of the signal associated with the instability wave to be isolated from other, uncorrelated fluctuations. In this paper, we investigate the extent to which pressure and velocity fluctuations in subsonic, turbulent round jets can be described as linear perturbations to the mean flow field. The disturbances are expanded about the experimentally measured jet mean flow field, and evolved using linear parabolized stability equations (PSE) that account, in an approximate way, for the weakly non-parallel jet mean flow field. We utilize data from an extensive microphone array that measures pressure fluctuations just outside the jet shear layer to show that, up to an unknown initial disturbance spectrum, the phase, wavelength, and amplitude envelope of convecting wavepackets agree well with PSE solutions at frequencies and azimuthal wavenumbers that can be accurately measured with the array. We next apply the proper orthogonal decomposition to near-field velocity fluctuations measured with particle image velocimetry, and show that the structure of the most energetic modes is also similar to eigenfunctions from the linear theory. Importantly, the amplitudes of the modes inferred from the velocity fluctuations are in reasonable agreement with those identified from the microphone array. The results therefore suggest that, to predict, with reasonable accuracy, the evolution of the largest-scale structures that comprise the most energetic portion of the turbulent spectrum of natural jets, nonlinear effects need only be indirectly accounted for by considering perturbations to the mean turbulent flow field, while neglecting any non-zero frequency disturbance interactions.
The classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free surface. Here, in the two-dimensional case: (a) we generalize the relevant formulation to the case of constant vorticity, as well as to the case where the free surface is described by a multivalued function; (b) in the case of travelling waves we derive an upper bound for the free surface; (c) in the case of constant vorticity we construct a sequence of nearly Hamiltonian systems which provide an approximation in the asymptotic limit of certain physical small parameters. In particular, the explicit dependence of the vorticity on the coefficients of the Korteweg–de Vries equation is clarified.
The thermal expansion induced by the exothermic chemical reactions taking place in a turbulent reactive flow affects the velocity field so strongly that the large-scale velocity fluctuations as well as the small-scale velocity gradients can be governed by chemistry rather than by turbulence. Moreover, thermal expansion is well known to be responsible for counter-gradient turbulent diffusion and flame-generated turbulence phenomena. In the present study, by making use of an original splitting procedure applied to the velocity field, we establish the occurrence of two distinct thermal expansion effects in the flamelet regime of turbulent premixed combustion. The first is referred to as the direct thermal expansion effect. It is associated with a local flamelet crossing contribution as previously considered in early analyses of turbulent transport in premixed flames. The second, denoted herein as the indirect thermal expansion effect, is an outcome of the turbulent wrinkling processes that increases the flame surface area. Based on a splitting procedure applied to the velocity field, the respective influences of the two effects are identified and analysed. Furthermore, the theoretical analysis shows that the thermal expansion induced through the local flames can be treated separately in the usual continuity and momentum equations. This description of the turbulent reactive velocity field, leads also to relate all of the usual turbulent quantities to the reactive scalar field. Finally, algebraic closures for the turbulent transport terms of mass and momentum are proposed and successfully validated through comparison with direct numerical simulation data.
The instability of an initially homogeneous suspension of spheroids, settling due to gravity, is reconsidered. For non-spherical particles, previous studies in the literature report that normal-mode density perturbations of maximum growth rate are those of arbitrarily large, horizontal wavelength. Using the ‘mixture theory’ for two-phase flow we show that the maximum growth rate for horizontal density perturbations is obtained for a finite wavelength if the inertia of the bulk motion associated with the normal-mode density perturbation is accounted for. We find that for long wavelengths, , the growth rate approaches zero as . The maximum growth rate is obtained for , where is the axis perpendicular to the axis of rotational symmetry of the spheroid, is the undisturbed volume fraction of particles and , typically , is a Reynolds number of the bulk motion on a typical length scale and a velocity scale on the order of the undisturbed settling speed. The theoretical results for the wavelength selection agree qualitatively well with previous experimental results in the literature of measured correlation lengths of vertical streamers in settling fibre suspensions.
When very light particles are sprinkled on a resonating horizontal plate, inverse Chladni patterns are formed. Instead of going to the nodal lines of the plate, where they would form a standard Chladni pattern, the particles are dragged to the antinodes by the air currents induced by the vibration of the plate. Here we present a detailed picture of the mechanism using numerical simulations involving both the particles and the air. Surprisingly, the time-averaged Eulerian velocity, commonly used in these type of problems, does not explain the motion of the particles: it even has the opposite direction, towards the nodal lines. The key to the inverse Chladni patterning is found in the averaged velocity of a tracer particle moving along with the air: this Lagrangian velocity, averaged over a vibration cycle, is directed toward the antinodes. The Chladni plate thus provides a unique example of a system in which the Eulerian and Lagrangian velocities point in opposite directions.
This paper describes a scenario of transition from laminar to turbulent flow in a spatially developing boundary layer over a flat plate. The base flow is the Blasius non-parallel flow solution; it is perturbed by optimal disturbances yielding the largest energy growth over a short time interval. Such perturbations are computed by a nonlinear global optimization approach based on a Lagrange multiplier technique. The results show that nonlinear optimal perturbations are characterized by a localized basic building block, called the minimal seed, defined as the smallest flow structure which maximizes the energy growth over short times. It is formed by vortices inclined in the streamwise direction surrounding a region of intense streamwise disturbance velocity. Such a basic structure appears to be a robust feature of the base flow since it is practically invariant with respect to the initial energy of the perturbation, the target time, the Reynolds number and the dimensions of the computational domain. The minimal seed grows very rapidly in time while spreading, and it triggers nonlinear effects which bring the flow to turbulence in a very efficient manner, through the formation of a turbulence spot. This evolution of the initial optimal disturbance has been studied in detail by direct numerical simulations. Using a perturbative formulation of the Navier–Stokes equations, each linear and nonlinear convective term of the equations has been analysed. The results show the fundamental role of the streamwise inclination of the vortices in the process. The nonlinear coupling of the finite amplitude disturbances is crucial to sustain such streamwise inclination, as well as to generate dislocations within the flow structures, and local inflectional velocity distributions. The analysis provides a picture of the transition process characterized by a sequence of structures appearing successively in the flow, namely, vortices, hairpin vortices and streamwise streaks. Finally, a disturbance regeneration cycle is conceived, initiated by the fast nonlinear amplification of the minimal seed, providing a possible scenario for the continuous regeneration of the same fundamental flow structures at smaller space and time scales.
The Lagrangian theory developed for fountains in a stationary fluid is extended to predict the path and breadth of a fountain in a one- and two-layer fluid with a moderate crossflow. The predictions compare well with the results of laboratory experiments of fountains in a one-layer fluid. The empirical spreading parameter determined from the one-layer experiments is used in the theory for fountains in a two-layer crossflow. Though qualitatively correct, the theory underpredicts the height and radius of the fountains. Similar to the behaviour of fountains in two-layer stationary ambients, the fountain in a two-layer crossflow is observed to exhibit three regimes of flow: it may penetrate the interface, eventually returning to the level of the source where it spreads as a propagating gravity current; upon descent, it may be trapped at the interface where it spreads as a propagating intrusion; it may do both, partially descending to the source and partially being trapped at the interface. These regimes are classified theoretically and empirically. The theoretical classification compared the buoyancy excess of the descending flow to the density difference between the two layers. The regimes are also classified using empirically determined regime parameters which govern the relative initial momentum of the fountain and the relative density difference of the fountain and the ambient fluid.
In two previous papers (Wu, J. Fluid Mech., vol. 453, 2002, p. 289, and Wu & Hogg, J. Fluid Mech., vol. 550, 2006, p. 307), a formal asymptotic procedure was developed to calculate the sound radiated by unsteady boundary-layer flows that are described by the triple-deck theory. That approach requires lengthy calculations, and so is now improved to construct a simpler composite theory, which retains the capacity of systematically identifying and approximating the relevant sources, but also naturally includes the effect of mean-flow refraction and more importantly the back action of the emitted sound on the source itself. The combined effect of refraction and back action is represented by an ‘impedance coefficient’, and the present analysis yields an analytical expression for this parameter, which was usually introduced on a semi-empirical basis. The expression indicates that for Mach number , the mean-flow refraction and back action of the sound have a leading-order effect on the acoustic field within the shallow angles to the streamwise directions. A parametric study suggests that the back effect of sound is actually appreciable in a sizeable portion of the acoustic field for , becomes more pronounced, and eventually influences the entire acoustic field in the transonic limit. In the supersonic regime, the acoustic field is characterized by distinctive Mach-wave beams, which exert a leading-order influence on the source. The analysis also indicates that acoustic radiation in the subsonic and supersonic regimes is fundamentally different. In the subsonic regime, the sound is produced by small-wavenumber components of the hydrodynamic motion, and can be characterized by acoustic multipoles, whereas in the supersonic regime, broadband finite-wavenumber components of the hydrodynamic motion contribute and the concept of a multipolar source becomes untenable. The global acoustic feedback loop is investigated using a model consisting of two well-separated roughness elements, in which the sound wave emitted due to the scattering of a Tollmien–Schlichting (T–S) wave by the downstream roughness propagates upstream and impinges on the upstream roughness to regenerate the T–S wave. Numerical calculations suggest that at high Reynolds numbers and for moderate roughness heights, the long-range acoustic coupling may lead to global instability, which is characterized by self-sustained oscillations at discrete frequencies. The dominant peak frequency may jump from one value to another as the Reynolds number or the distance between the roughness elements is varied gradually.
The unsteady organization and evolution of coherent structures within the turbulent boundary layer and subsequent wake of the sharp symmetric trailing edge of a NACA0012 aerofoil are investigated. The experiments are conducted in an open test-section wind tunnel at based on the aerofoil chord and based on the boundary layer momentum thickness. An initial characterization of the flow field using two-component particle image velocimetry (PIV) is followed by the investigation of the unsteady organization and evolution of coherent structures by time-resolved three-dimensional PIV based on a tomographic approach (Tomo-PIV). The inspection of the turbulent boundary layer prior to the trailing edge in the region between 0.15 and demonstrated streaks of low- and high-speed flow, while the low-speed streaks are observed to be more coherent along with strong interaction with hairpin-type vortical structures similar to a turbulent boundary layer at zero pressure gradient. The wake region demonstrated gradual deterioration of both the low- and the high-speed streaks with downstream progress. However, the low-speed streaks are observed to lose their coherence at a faster rate relative to the high-speed streaks as the turbulent flow develops towards the far wake. The weakening of the low-speed streaks is due to the disappearance of the viscous sublayer after the trailing edge and gradual mixing through the transport of the remaining low-speed flow towards the free stream. This transport of low-speed flow is performed by the ejection events induced by the hairpin vortices as they also persist into the developing wake. The higher persistence of the high-speed streaks is associated with counter-hairpin vortical activities as they oppose the deterioration of the high-speed streaks by frequently sweeping the high-speed flow towards the wake centreline. These vortical structures are regarded as counter-hairpin vortices as they exhibit opposite characteristics relative to the hairpin vortices of a turbulent boundary layer. They are topologically similar to the hairpins as they appear to be U-shaped but with inverted orientation, as the spanwise portion is in the vicinity of the wake centreline and the legs are inclined at an approximately to the wake axis in the downstream direction demonstrating a strain-dominated topology. The counter-hairpin vortices are partially wrapped around the high-speed streaks and contribute to the wake development by transporting high-speed flow towards the wake centreline. Similar to the hairpin vortices of a turbulent boundary layer, the occurrence of a complete counter-hairpin vortex is occasional while its derivatives (portions of spanwise or quasi-streamwise vortices) are more frequently observed. Therefore, a pattern recognition algorithm is applied to establish characterization based on an ensemble-averaged counter-hairpin vortex. The formation of the counter-hairpin vortices is due to an additional degree of interaction between the low- and high-speed streaks after the trailing edge across the wake centreline. The shear layer produced along the wake centreline by neighbouring low- and high-speed streaks promotes the formation of spanwise vortices that form the counter-hairpin vortices by connection to quasi-streamwise vortices. Finally, a conceptual model is proposed to depict the three-dimensional unsteady organization and evolution of coherent structures in the wake region based on the hairpin and counter-hairpin vortex signatures.
A subtle issue in the study of mushy zones which form during the solidification of binary alloys is that there are two distinct types of solid–mush interfaces which may occur. One of these is a eutectic front and the other is a front which separates the mushy layer and, assuming complete solute rejection, a layer of pure solid. For semi-infinite-domain configurations that admit similarity solutions, such as those at a uniform initial temperature and concentration with an imposed cold temperature at the lower boundary, only one of the two types appears, and the type of front is determined by the various parameters of the system. In a finite domain, it is possible for each type of front to appear at different times. Specifically, the advance of the eutectic front is restricted by the isotherm associated with the eutectic temperature, and the other front type will appear over a longer time scale. Leading up to the time when the front changes type, the concentration being frozen into the solid decreases. This process writes a history of the system into the solid.
A Boussinesq fluid of kinematic velocity and thermal diffusivity is confined within a rapidly rotating shell with inner and outer sphere boundary radii and , respectively. The boundaries of the shell corotate at angular velocity and a continuously varying stratification profile is applied which is unstable in and stable in . When , the unstable zone attached to the inner boundary is thin. As in previous small Ekman number studies, convection at the onset of instability takes on the familiar ‘cartridge belt’ structure, which is localized within a narrow layer adjacent to, but outside, the cylinder tangent to the inner sphere at its equator (Dormy et al. J. Fluid Mech., 2004, vol. 501, pp. 43–70), with estimated radial width of order . The azimuthally propagating convective columns, described by the cartridge belt, reside entirely within the unstable layer when , and extend from the equatorial plane an axial distance along the tangent cylinder as far as its intersection with the neutrally stable spherical surface . We investigate the eigensolutions of the ordinary differential equation governing the axial structure of the cartridge belt both numerically for moderate-to-small values of the stratification parameter and analytically when . At the lowest order of the expansion in powers of , the eigenmodes resemble those for classical plane layer convection, being either steady (exchange of stabilities) or, for small Prandtl number , oscillatory (overstability) with a frequency . At the next order, the axial variation of the basic state removes any plane layer degeneracies. First, the exchange of stabilities modes oscillate at a low frequency causing the short axial columns to propagate as a wave with a small angular velocity, termed slow modes. Second, the magnitudes of both the Rayleigh number and frequency of the two overstable modes, termed fast modes, split. When the slow modes that exist at large azimuthal wavenumbers make a continuous transition to the preferred fast modes at small . At all values of the critical Rayleigh number corresponds to a mode exhibiting prograde propagation, whether it be a fast or slow mode. This feature is shared by the uniform classical convective shell models, as well as Busse’s celebrated annulus model. None of them possess any stable stratification and typically are prone to easily excitable Rossby or inertial modes of convection at small . By way of contrast these structures cannot exist in our model for small due to the viscous damping in the outer thick stable region.
A study of instabilities in planar flows produced by the presence of a parallel penetrable porous obstruction is presented. The case considered is a flow between parallel plates partially obstructed by a porous medium. The most unstable perturbation modes are obtained solving numerically the eigenvalue problem derived from the linear stability analysis of the two-dimensional Navier–Stokes equations applied to the geometry of interest. The analysis leads to an extended Orr–Sommerfeld equation including a porous term. It was found that the ratios of the permeability and depth of the obstruction with respect to the free flow layer depth are the relevant parameters influencing the stability margin and the structure of the most unstable modes. To validate the conclusions of the theoretical analysis, an experiment with air flowing through a channel semi-obstructed by a regular array of cylindrical wires was performed. The critical Reynolds number, which was determined by measuring the amplitude of velocity fluctuations at the interface of the porous medium, agrees with the theoretical predictions. The dominant instability mode was characterized by the cross-section profile of the root mean square of the velocity perturbations, which matches reasonable well with the eigenfunction of the most unstable eigenvalue. Also, the wavenumber was determined by correlating the velocity measurements in two sequential locations along the channel, which compares well with the theoretical value.
Uniformly sheared turbulent flow has been generated in a water tunnel and its instantaneous structure has been examined using flow visualization and particle image velocimetry. The shear-rate parameter was approximately equal to 13 and the streamwise turbulence Reynolds number was approximately 150. The flow was found to consist of regions with nearly uniform velocity, which were separated by regions of high shear containing large vortices. The concentration of vortices and the distributions of their directions of rotation, strengths, sizes and shapes have been determined. These results demonstrate that horseshoe/hairpin-shaped vortices were prevalent, even though wall effects were negligible in this flow. Both ‘upright’ and ‘inverted’ vortices have been observed, in contrast to turbulent boundary layers, in which only ‘upright’ vortices can be found, suggesting that the presence of the wall may suppress the development of ‘inverted’ structures. Our observations demonstrate that the dominant coherent structures of fully developed uniformly sheared flow are very different from the structures observed in the flow exiting the shear-generating apparatus, which points to an insensitivity of the former to initial effects.
Novel, closed-form, analytic solutions for the pressure and velocity fields are derived for the linear problem of wave propagation inside a tapered flexible vessel of conical shape. It is shown that pressure and velocity can be written in terms of Bessel functions of orders and respectively. An expression is also derived that quantifies the effect of the cone angle on the wave propagation velocity. The analytic solutions are general and valid for tube variations at any length scale in relation to the wavelength of the wave. In other words, the requirement that the changes in vessel properties with distance should take place over a length scale large compared to the wavelength of the wave, is not employed or needed. This is the basic condition for the application of WKB theory to tapered vessels. However, this condition is not satisfied in pressure pulses propagating in mammalian arteries. The general expressions derived in this paper are directly applicable to the cardiovascular system of mammals. It is further shown that the presented solution naturally tends to the asymptotic WKB solution when the assumptions of the theory are applied to the general expressions. An explicit formula is provided for the time-averaged energy flux of the wave that shows clearly the effect of the continuous reflection of the wave from the vessel wall. Viscous effects are incorporated by coupling the derived analytic solution with the radial velocity profile of Womersley. The results are compared with full nonlinear fluid–structure interaction simulations and very good agreement is found (maximum differences are and 1.6 % for area-averaged pressure and velocity respectively, and 4–6 % for local velocity values).
Controlled gliding during descent has been thought of as a crucial intermediate step toward the evolution of powered flight in a variety of animals. Here we develop and analyse a model for the controlled descent of thin bodies in quiescent fluids. Focusing on motion in two dimensions for simplicity, we formulate the question of steering an elliptical body to a desired landing location with a specific orientation using the framework of optimal control theory with a single control variable. We derive both time- and energy-optimal trajectories using a combination of numerical and analytical approximations. In particular, we find that energy-optimal strategies converge to constant control, while time-optimal strategies converge to bang–coast–bang control that leads to bounding flight, alternating between tumbling and gliding phases. Our study of these optimal strategies thus places natural limits on how they may be implemented in biological and biomimetic systems.
Here we explore the effect of a series of low-permeability tilted baffles on the ascent of a buoyant fluid injected into a porous rock from a linear well, motivated by several industrial processes, particularly CO2 sequestration. We first consider, both theoretically and experimentally, the dynamics associated with flow past an individual inclined baffle; if the incident flux is sufficiently large then a pool of injectate grows beneath the baffle and spills over both ends, partitioning the flux, but otherwise the injectate only flows over the updip end of the baffle, leaving a stagnant zone under the downdip part of the baffle. In a multi-layered system, the flow may then be described using nonlinear recurrence relations based on the flow past an individual baffle. Using this approach, in the particular case in which there is a regular distribution of baffles, we show that, on a scale greater than individual baffles, the flow adjusts to a plume of constant width rising at an angle to the vertical, which depends on the geometry of the baffles. This constant-width plume may be described by an effective directional permeability at angles to the vertical where we find , where is the effective permeability of the regions between baffles. Within the boundaries of this plume the majority of pore space is bypassed. Indeed, the plume-scale effective porosity is largely associated with the pools of injectate which collect beneath each baffle. We show that since the constant-width plume has such a large lateral extent, the total pore space occupied by the plume, per unit height, is larger than for a homogeneous formation. Furthermore, these pools lead to a large surface area between injectate and formation water, which enhances the reaction of the injectate with the formation water. However, we also show that, in steady state, it may be hard to determine this plume-scale effective porosity since the ratio of the effective Darcy speed and the effective interstitial speed of the plume only relates to the porosity of the updip part of these pools, in which the injectate is flowing.