This paper presents the results of an in-depth study of the flow past a streamwise oscillating cylinder, examining the impact of varying the amplitude and frequency of the oscillation, and the Reynolds number of the incoming flow. These findings are presented in a framework that shows that the relationship between the frequency of vortex shedding ${f}_{s} $ and the amplitude of oscillation ${A}^{\ast } $ is governed by two primary factors: the first is a reduction of ${f}_{s} $ proportional to a series in ${A}^{\ast 2} $ over a wide range of driving frequencies and Reynolds numbers; the second is nonlinear synchronization when this adjusted ${f}_{s} $ is in the vicinity of $N= {(1- {f}_{s} / {f}_{d} )}^{- 1} $, where $N$ is an integer. Typically, the influence of higher-order terms is small, and truncation to the first term of the series (${A}^{\ast 2} $) well represents the overall trend of vortex shedding frequency as a function of amplitude. However, discontinuous steps are overlaid on this trend due to the nonlinear synchronization. When ${f}_{s} $ is normalized by the Strouhal frequency ${f}_{St} $ (the frequency of vortex shedding from an unperturbed cylinder), the rate at which ${f}_{s} / {f}_{St} $ decreases with amplitude, at least for ${f}_{d} / {f}_{St} = 1$, shows a linear dependence on the Reynolds number. For a fixed $\mathit{Re}= 175$, the truncated series shows that the rate of decrease of ${f}_{s} / {f}_{St} $ with amplitude varies as ${(2- {f}_{d} / {f}_{St} )}^{- 1/ 2} $ for $1\leqslant {f}_{d} / {f}_{St} \leqslant 2$, but is essentially independent of ${f}_{d} / {f}_{St} $ for ${f}_{d} / {f}_{St} \lt 1$. These trends of the rate of decrease of ${f}_{s} $ with respect to amplitude are also used to predict the amplitudes of oscillation around which synchronization occurs. These predicted amplitudes are shown to fall in regions of the parameter space where synchronized modes occur. Further, for the case of varying ${f}_{d} / {f}_{St} $, a very reasonable prediction of the amplitude of oscillation required for the onset of synchronization to the mode where ${f}_{s} = 0. 5{f}_{d} $ is given. In a similar manner, amplitudes at which ${f}_{s} = 0$ are calculated, predicting where the natural vortex shedding is completely supplanted by the forcing. These amplitudes are found to coincide approximately with those at which the onset of a symmetric vortex shedding mode is observed. This result is interpreted as meaning that the symmetric shedding mode occurs when the dynamics crosses over from being dominated by the vortex shedding to being dominated by the forcing.

An experimental investigation of the near wake of a thin airfoil at various incidence angles is reported in this paper. The airfoil (NACA 0012 basic thickness form) was located in a wind tunnel, and the wake structure was measured using hot-wire sensors. The measurements of mean-velocity, turbulence intensity and Reynolds-stress components across the wake at several distances downstream show the complex nature of the near wake and its asymmetrical behaviour. The asymmetry in the wake property, which is maintained up to a length of 1·5 chords downstream of the trailing edge of the blade, is dependent on the incidence angle of the inlet flow. The streamwise velocity defect in an asymmetric wake decays more slowly compared to that of a symmetric wake. The streamline curvature due to the blade loading has a substantial effect on the mean velocity profile as well as the turbulence structure. The numerical study of the same wake indicates that the existing turbulence closure models need some modification to account for the asymmetric characteristics of the wake.

Model equations for three-dimensional, inviscid flow between two arbitrary, time-varying material surfaces are derived using a ‘direct’ or variational approach due to Kantorovich. This approach results in a hierarchy of approximate theories, each of a higher level of spatial approximation and complexity. It can be shown that the equations are equivalent in substance to ‘the theory of directed fluid sheets’ of Green & Naghdi (1974, 1976).

The theory can be used to study the propagation of long waves in water of finite depth and, as such, competes with theories derived using the classical Rayleigh–Boussinesq perturbation methods. In order to demonstrate that there is an advantage to the present approach, we compare predictions for steady, two-dimensional waves over a horizontal bottom. Numerical solutions indicate that the direct theory converges more rapidly than the perturbation theories. Also, the equations of the higher-order direct theories contain singularities related to waves of limiting height, and indeed such waves can be predicted with relative accuracy. Finally, the range of applicability of the direct theory is far greater: waves as short as three times the water depth can be modelled. This is essentially a deep-water condition, well beyond the range of convergence of the Rayleigh–Boussinesq approach.

Understanding the rates of cooling and solidification in laminar flows down sloping channels is central to predicting the advance of lava flows. The mechanisms involved include thermal convection and a competition between shear strain rate and the rate of formation of solid at the chilled surface of the flow. We report experiments in which polyethylene glycol wax flows in a laminar fashion down an inclined, open channel of rectangular cross-section under cold water. Two distinctly different flow regimes are recognized: ‘tube’ flow in which solidification of the flow surface creates a stationary roof while melt continues to flow through a relatively well-insulated ‘tube’ beneath, and a ‘mobile crust’ regime in which a solid surface crust develops only in the centre of the channel. In the latter regime the crust is carried down the channel, separated from the walls by crust-free shear regions in which cooling produces only dispersed fragments of solid owing to the effects of shearing. This flow structure is quasi-invariant over a large distance downstream. We show that thermal convection takes place in organized rolls that have axes aligned with the shear flow, and conclude that transition between the two flow regimes occurs at a critical value of the combined parameter $\vartheta \,{=}\, \psi (\hbox{\it Ra}{/}R_0)^{1/3}$, where $\psi \,{=}\, U_0 t_s{/}H_0$ is the ratio of a surface solidification timescale $t_{s}$ to a shearing timescale $H_0 {/} U_0 $, $H_{0}$ and $U_{0}$ are the flow depth and centreline surface velocity in the absence of solidification, {\it Ra} is a Rayleigh number and $R_{0}$ is a constant.

Results are presented from a computational study of the flow over a forward-facing step in a plane channel. The aim of the study is to gain better insight into the three-dimensionality that is typically observed in the separation region of flows over steps and ribs, and in similar configurations. We consider laminar flow at a Reynolds number of 330, based on step height and bulk velocity of the oncoming flow, and the step is assumed to be infinitely extended in the spanwise direction. High-resolution simulations are undertaken using a mixed spectral/spectral-element code. Moreover, a linear stability study of the flow at the step is performed. The results show that, in the case considered, the three-dimensionality is not related to some absolute instability of the separation bubble in front of the step; rather, it is a sensitive reaction of the flow to three-dimensional perturbations present in the oncoming stream. It is demonstrated that disturbance amplitudes of less than 1% of the mean flow (at, say, 10 step heights ahead of the step) already suffice to produce a visibly three-dimensional structure of the separation zone. If the disturbance level is systematically decreased, the three-dimensional state evolves to an almost two-dimensional recirculation. Here, the key finding is that the intensity of the flow response is proportionate to the amplitude of the inflow disturbance, meaning that the breakup of the flow in the step region is a linear (i.e. small) perturbation of the two-dimensional base flow. A comparison of the present simulation results with experimental data shows close agreement concerning, for example, the flow topology in the step region, and the spanwise spacing of the characteristic streaks that form further downstream.

An investigation of the local receptivity of a Blasius boundary layer to a harmonic vortical disturbance is presented as a step towards understanding boundary-layer receptivity to free-stream turbulence. Although there has been solid experimental verification of the linear theory describing acoustic receptivity of boundary layers, this was the first experimental verification of the mechanism behind local receptivity to a convected disturbance. The harmonic wake from a vibrating ribbon positioned upstream of a flat plate provided the free-stream disturbance. Two-dimensional roughness elements on the surface of the plate acted as a local receptivity site. Hot-wire measurements in the boundary layer downstream of the roughness confirmed the generation of Tollmien–Schlichting (TS) instability waves by an outer-layer interaction between the long-wavelength convected disturbance and the short-scale mean-flow distortion due to the roughness. The characteristics of the instability waves were carefully measured to ensure that their behaviour was correctly modelled by linear stability theory. This theory was then used to determine the immeasurably small initial wave amplitudes resulting from the receptivity process, from wave amplitudes measured downstream. Tests were performed to determine the range of validity of the linear assumptions made in current receptivity theories. Experimental data obtained in the linear regime were then compared to theoretical results of other authors by expressing the experimental data in the form of an efficiency function which is independent of the free-stream amplitude, roughness height and roughness geometry. Reasonable agreement between the experimental and theoretical efficiency functions was obtained over a range of frequencies and Reynolds numbers.

A liquid filament recoils because of its surface tension. It may recoil to one sphere, the geometrical shape with lowest surface, or otherwise segment to several pieces which individually will recoil to spheres. This experiment is classical and its exploration is fundamental to the understanding of how liquid volumes relax. In this paper, we uncover a mechanism involving the creation of a vortex ring which plays a central role in escaping segmentation. The retracting blob is connected to the untouched filament by a neck. The radius of the neck decreases in time such that we may expect pinch-off. There is a flow through the neck because of the retraction. This flow may detach into a jet downstream of the neck when fluid viscosity exceeds a threshold. This sudden detachment creates a vortex ring which strongly modifies the flow pressure: fluid is expelled back into the neck which in turn reopens.

Observations of entrainment into natural gravity-driven flows on sloping surfaces are described. It is shown that the laboratory-based entrainment law of Ellison & Turner (1959), which is often used for modelling of atmospheric and oceanic flows, underestimates the entrainment rates substantially, arguably due to the fact that the laboratory flows have been conducted at Reynolds numbers $(Re \lesssim 10^3)$ below what is required for mixing transition ($Re\sim10^3$–10^4) whereas natural flows occur at much higher Reynolds numbers $(Re\sim10^7)$. A new entrainment law of the form $E\sim Ri^{-3/4}$ is proposed for the atmospheric Richardson number range $0.15 \lt Ri \lt 1.5$. In contrast to the laboratory observation that entrainment ceases at $Ri = 0.8$, field observations show continuous entrainment over the entire Richardson number range.

Motivated by the geological sequestration of carbon dioxide (CO2), we study the propagation of gravity currents in a porous medium bounded by a thin layer of much lower permeability. We formulate a model for drainage assuming that the fluid remains simply connected throughout. Using this model we examine the propagation of both two-dimensional and axisymmetric currents numerically. We find that for the fixed-flux situation solutions approach a steady state which is described analytically. The approach to this final solution depends on both the permeability contrast and thickness of the thin layer, and in many cases the current first overshoots before relaxing back to its ultimate steady state. Finally, we examine propagation along multiple thin, lower permeability layers as a reduced-order model of the plume of CO2 currently being injected at Sleipner in the North Sea.

The capability of state-of-the-art techniques integrating experimental and computational fluid dynamics has been expanding recently. In our previous study, we have developed a hybrid unsteady-flow simulation technique combining particle tracking velocimetry (PTV) and direct numerical simulation (DNS) and demonstrated its capability at low Reynolds numbers. Similar approaches have also been proposed by a few groups; however, applying algorithms of this type generally becomes more challenging with increasing Reynolds number because the time interval of the frame rate for particle image velocimetry (PIV) becomes much greater than the required computational time step, and the PIV/PTV resolution tends to be lower than that necessary for computational fluid dynamics. To extend the applicability to noisy time-resolved PIV/PTV data, the proposed algorithm optimizes the data input temporally and spatially by introducing a reduced-order Kalman filter. This study establishes a framework of the Kalman-filtered hybrid simulation and proves the concept by tackling a planar-jet flow at as an example. We evaluate the filtering functions as well as convergence of the proposed algorithm by comparing with the existing PTV–DNS hybrid simulation, and show some techniques available to hybrid velocity fields by analysing vortical motion in the shear layers of the jet.

The transport of dense particles by a turbulent flow depends on two dimensionless numbers. Depending on the ratio of the shear velocity of the flow to the settling velocity of the particles (or the Rouse number), sediment transport takes place in a thin layer localized at the surface of the sediment bed (bedload) or over the whole water depth (suspended load). Moreover, depending on the sedimentation Reynolds number, the bedload layer is embedded in the viscous sublayer or is larger. We propose here a two-phase flow model able to describe both viscous and turbulent shear flows. Particle migration is described as resulting from normal stresses, but is limited by turbulent mixing and shear-induced diffusion of particles. Using this framework, we theoretically investigate the transition between bedload and suspended load.

Linear and nonlinear dilational and sinuous capillary waves on thin inviscid infinite and semi-infinite planar liquid sheets in a void are analysed in a unified manner by means of a method that reduces the two-dimensional unsteady problem to a one-dimensional unsteady problem. For nonlinear dilational waves on infinite sheets, the accuracy of the numerical solutions is verified by comparing with an analytical solution. The nonlinear dilational wave maintains a reciprocal relationship between wavelength and wave speed modified from the linear theory prediction by a dependence of the product of wavelength and wave speed on the wave amplitude. For the general dilational case, nonlinear numerical simulations show that the sheet is unstable to superimposed subharmonic disturbances on the infinite sheet. Agreement for both sinuous and dilational waves is demonstrated for the infinite case between nonlinear simulations using the reduced one-dimensional approach, and nonlinear two-dimensional simulations using a discrete-vortex method. For semi-infinite dilational and sinuous distorting sheets that are periodically forced at the nozzle exit, linear and nonlinear analyses predict the appearance of two constant-amplitude waves of nearly equal wavelengths, resulting in a sheet disturbance characterized by a long-wavelength envelope of a short-wavelength oscillation. For semi-infinite sheets with sinuous waves, qualitative agreement between the dimensionally reduced analysis and experimental results is found. For example, a half-wave thinning and a sawtooth wave shape is found for the nonlinear sinuous mode. For the semi-infinite dilational case, a critical frequency-dependent Weber number is found below which one component of the disturbances decays with downstream distance. For the semi-infinite sinuous case, a critical Weber number equal to 2 is found; below this value, only one characteristic is emitted in the positive time direction from the nozzle exit.

Experiments were performed to investigate the supersonic flow of a turbulent boundary layer over short regions of concave surface curvature. Upstream of each curved wall, the free-stream Mach number was 2.87, and the incoming boundary layer was typical of a two-dimensional, zero-pressure-gradient, high-Reynolds-number flow. Two different curvatures were used, with radii of curvature equal to 10 and 50 initial boundary-layer thicknesses (Models I and II, respectively). The turning angle was 8° in each case. As the boundary layer passed through the curved region, it experienced a strong adverse pressure gradient, as well as the destabilizing influences of bulk compression and concave curvature. Downstream of the curved walls, the flow relaxed on a short plane wall. The mean and turbulent field for each flow was investigated, using normal and inclined hot wires to measure the turbulent fluctuations. Wherever possible, the results were compared with those from a corresponding 8° ramp. The ramp and Model I exhibited a very similar behaviour: turbulence levels increased significantly, and there was a marked increase in structural parameters such as the stress ratio $-\overline{u^{\prime}v^{\prime}}/\overline{u^{\prime 2}}$ and the length- and timescales of the turbulent motions. Model II behaved quite differently: although the turbulence levels increased, structural parameters were essentially unchanged. The similarities between the ramp and Model I results suggest that the perturbation in both cases is ‘rapid’ in that the perturbation can be described in terms of total strains rather than local strains. In contrast, the flow in Model II is sensitive to the local variations in the strain rate.

The sound emission from open turbulent flames is dictated by the two-point spatial correlation of the rate of change of the fluctuating heat release rate. This correlation in premixed flames can be represented well using Gaussian-type functions and unstrained laminar flame thermal thickness can be used to scale the correlation length scale, which is about a quarter of the planar laminar flame thermal thickness. This correlation and its length scale are observed to be less influenced by the fuel type or stoichiometry or turbulence Reynolds and Damkohler numbers. The time scale for fluctuating heat release rate is deduced to be about τc/34 on an average, where τc is the planar laminar flame time scale, using direct numerical simulation (DNS) data. These results and the spatial distribution of mean reaction rate obtained from Reynolds-averaged Navier–Stokes (RANS) calculations of open turbulent premixed flames employing the standard model and an algebraic reaction rate closure, involving a recently developed scalar dissipation rate model, are used to obtain the far-field sound pressure level from open flames. The calculated values agree well with measured values for flames of different stoichiometry and fuel types, having a range of turbulence intensities and heat output. Detailed analyses of RANS results clearly suggest that the noise level from turbulent premixed flames having an extensive and uniform spatial distribution of heat release rate is low.

The finite difference method due to Lax (1954) is used to solve the equations of motion for a cylindrically symmetric flow of a compressible fluid. In particular, a converging cylindrical shock is found to increase in strength in agreement with the formula of Chisnell (1957). The artificial diffusion introduced by the method causes the pressure to remain finite at the axis, but a reflected diverging shock is obtained.

The dynamics of viscous fluid flow over a circular flexible plate are studied numerically by an immersed boundary–lattice Boltzmann method for the fluid flow and a finite-element method for the plate motion. When the plate is clamped at its centre and placed in a uniform flow, it deforms by the flow-induced forces exerted on its surface. A series of distinct deformation modes of the plate are found in terms of the azimuthal fold number from axial symmetry to multifold deformation patterns. The developing process of deformation modes is analysed and both steady and unsteady states of the fluid–structure system are identified. The drag reduction due to the plate deformation and the elastic potential energy of the flexible plate are investigated. Theoretical analysis is performed to elucidate the deformation characteristics. The results obtained in this study provide physical insight into the understanding of the mechanisms on the dynamics of the fluid–structure system.

The pair trajectories of neutrally buoyant rigid spheres immersed in finite-inertia simple-shear flow are described. The trajectories are obtained using the lattice-Boltzmann method to solve the fluid motion, with Newtonian dynamics describing the sphere motions. The inertia is characterized by the shear-flow Reynolds number , where μ and ρ are the viscosity and density of the fluid respectively, is the shear rate and a is the radius of the larger of the pair of spheres in the case of unequal sizes; the majority of results presented are for pairs of equal radii. Reynolds numbers of 0 ≤ Re ≤ 1 are considered with a focus on inertia at Re = O(0.1). At finite inertia, the topology of the pair trajectories is altered from that predicted at Re = 0, as closed trajectories found in Stokes flow vanish and two new forms of trajectories are observed. These include spiralling and reversing trajectories in addition to largely undisturbed open trajectories. For Re = O(0.1), the limits of the various regions in pair space yielding open, reversing and spiralling trajectories are roughly defined.

Irregular conduits, complex surfaces, and porous media often manifest more than one geometric wetting condition for spontaneous capillary flows. As a result, different regions of the flow exhibit different rates of flow, all the while sharing common dynamical capillary pressure boundary conditions. The classic problem of sudden capillary rise in tubes with interior corners is revisited from this perspective and solved numerically in the self-similar visco-capillary limit à la Lucas–Washburn. Useful closed-form analytical solutions are obtained in asymptotic limits appropriate for many practical flows in conduits containing one or more interior corner. The critically wetted corners imbibe fluid away from the bulk capillary rise, shortening the viscous column length and slightly increasing the overall flow rate. The extent of the corner flow is small for many closed conduits, but becomes significant for flows along open channels and the method is extended to approximate hemiwicking flows across triangular grooved surfaces. It is shown that an accurate application of the method depends on an accurate a priori assessment of the competing viscous cross-section length scales, and the expedient Laplacian scaling method is applied herein toward this effect.

The Reynolds number dependence of the structure and statistics of wall-layer turbulence remains an open topic of research. This issue is considered in the present work using two-component planar particle image velocimetry (PIV) measurements acquired at the Surface Layer Turbulence and Environmental Science Test (SLTEST) facility in western Utah. The Reynolds number (δuτ/ν) was of the order 106. The surface was flat with an equivalent sand grain roughness k+ = 18. The domain of the measurements was 500 < yuτ/ν < 3000 in viscous units, 0.00081 < y/δ < 0.005 in outer units, with a streamwise extent of 6000ν/uτ. The mean velocity was fitted by a logarithmic equation with a von Kármán constant of 0.41. The profile of u′v′ indicated that the entire measurement domain was within a region of essentially constant stress, from which the wall shear velocity was estimated. The stochastic measurements discussed include mean and RMS profiles as well as two-point velocity correlations. Examination of the instantaneous vector maps indicated that approximately 60% of the realizations could be characterized as having a nearly uniform velocity. The remaining 40% of the images indicated two regions of nearly uniform momentum separated by a thin region of high shear. This shear layer was typically found to be inclined to the mean flow, with an average positive angle of 14.9°.

Suppose that the density of stationary unbounded viscous fluid is a sinusoidal function of the vertical position coordinate z. Is this body of fluid gravitationally unstable to small disturbances, and, if so, under what conditions, and to what type of disturbance? These questions are considered herein, and the answers are that the fluid is indeed unstable, for any non-zero value of the amplitude of the sine wave, to disturbances with large horizontal wavelength. These disturbances have approximately vertical velocity everywhere and tilt the alternate layers of heavier and of lighter fluid, causing the fluid in the former to slide down and that in the latter to slide up, leading to a sinusoidal variation of the vertically averaged density and thereby to reinforcement of the vertical motion. The identification of this novel and efficient global instability mechanism prompts a consideration of the stability of other cases of unbounded fluid stratified in layers. Two other types of undisturbed density distribution, the first an isolated central layer of heavier or lighter fluid, with density varying say as a Gaussian function, and the second an isolated layer of fluid in which the density varies as the derivative of a Gaussian function, are found to be unstable, at all values of the magnitude of the density variation, to disturbances having the same global character. For the first of these two types of density distribution, the behaviour of a disturbance with long horizontal wavelength depends only on the net excess mass of unit area of the central layer, and for the second it depends only on the first moment of the density in the central layer. For the second type there arises another global instability mechanism in which light fluid is stripped away from one side of the layer and heavy fluid from the other without any tilting. In all cases the properties of a neutral disturbance are determined numerically, and the growth rate is found as a function of the Rayleigh number, the Prandtl number, and the horizontal wavenumber of the disturbance. An energy argument gives results easily for the inviscid non-diffusive limit, when all disturbances grow, and reveals the tilting-sliding mechanism of the instability of a disturbance with large horizontal wavelength in its simplest form.