The head-on collision of a droplet onto a liquid layer of the same material, backed bya solid surface, was experimentally and computationally investigated, with emphasis on the transition from bouncing of the droplet to its absorption by the film for given dropletWeber number, We, and the film thickness scaled by the droplet radius, Hf. Experimental results show that while absorption is favoured with increasing We, there exists a range around Hf ≈, 1 over which this tendency is moderated. This local moderation in turn corresponds to a regime, 11 ≲ We ≲ 14, over which increasing Hf from a small value leads to a triple reversalbehaviour of absorption, bouncing, absorption again, and bouncing again. The collision dynamics including evolution of the surface contours of the droplet and film, as well as the energy budgets, were then simulated by using a front-tracking technique. For collisionsleading to absorption and partial absorption, for which part of the absorbed droplet is subsequently ejected from the film, rupture and hence merging of the interfaces were manually imposed at an instant that leads to agreement between the subsequent calculated and experimental images. The simulation satisfactorily identified the different factors influencing the observed non-monotonic response of the collision event.

A quasi-static simulation is used to study the mechanical response of a disordered bidimensional aqueous foam submitted to an oscillating shear strain. The application of shear progressively extends the elastic domain, i.e. the strain range within which no plastic process occurs. It is associated with the development of an irreversible normal stress difference, and a decrease in the shear modulus, which are both signatures of the appearance of anisotropy in the film network. Beyond this mechanical measurement, the evolutionof the structural properties of the foam is investigated. We focus in particular on the energy E0 defined as the minimum line-length energy under zero shear stress. For strainamplitude less than ~0.5, this quantity is found to decay with the number of applied cycles as a result of the curing of topological defects. However, for higher strain amplitude, plastic events appear to increase the structural disorder and tend to gather near the shearing walls. This process is a precursor of the shear-banding transition observed in fully developed flows, which will be studied in the companion paper. Movies are available with the online version of the paper.

The evolution of a bidimensional foam submitted to continuous quasi-static shearing isinvestigated both experimentally and numerically. We extract, from the images of the sheared foam, the plastic flow profiles as well as the local statistical properties of the stress field. When the imposed strain becomes larger than the yield strain, the plastic events develop large spatial and temporal correlations, and the plastic flow becomes confined to a narrow shear band. This transition and the steady-state regime of flow are investigated by first focusing on the elastic deformation produced by an elementary plastic event. This allows us to understand (i) the appearance of long-lived spatial heterogeneities of the stress field, which we believe are at the origin of the shear-banding transition, and (ii) the statistics of the dynamic fluctuations of the stress field induced by plastic rearrangements in the steady-state regime. Movies are available with the online versionof the paper.

The effects of randomly irregular bathymetry on the propagation of interfacial gravity waves are studied. Modelling sea water by a two-layer fluid of different densities, weakly nonlinear waves much longer than the sea depth but comparable to the horizontal scale of bathymetry are treated by Boussinesq approximation and multiple-scale analysis. For transient wave pulses, the governing equation for the pulse profile is shown to be an integro-differential equation combining KdV and Burgers terms. Quantitative and qualitative effect of disorder on the attenuation of wave amplitude, reduction of wave speed and change of wave profile are examined numerically and analytically based on the asymptotic approximation. For time-harmonic waves, mode-coupling equations are derived and examined for the competition between diffusion by random scattering, steepening by nonlinearityand frequency dispersion for a broad range of depth ratios.

As a first step towards understanding the role of free-stream turbulence in laminar–turbulent transition, we calculate the fluctuations induced by free-stream vortical disturbances in a compressible laminar boundary layer. As with the incompressible case investigated by Leib et al. (J. Fluid Mech. vol. 380, 1999, p. 169), attention is focused on components with long streamwise wavelength. The boundary-layer response is governed by the linearized unsteady boundary-region equations in the typical streamwise region where the local boundary-layer thickness δ* iscomparable with the spanwise length scale Λ of the disturbances. The compressible boundary-region equations are solved numerically for a single Fourier component to obtain the boundary-layer signature. The root-mean-square of the velocity and mass-flux fluctuations induced by a continuous spectrum of free-stream disturbances are computed by an appropriate superposition of the individual Fourier components.

Low-frequency vortical disturbances penetrate into the boundary layer to form slowly modulating streamwise-elongated velocity streaks. In the compressible regime, vortical disturbances are found to induce substantial temperature fluctuations so that ‘thermalstreaks’ also form. They may have a significant effect on the secondary instability. The calculations indicate that for a vortical disturbance with a relatively large Λ, the induced boundary-layer fluctuation ultimately evolves into an amplifying wave. This is due to a receptivity mechanism, in which a vortical disturbance first excites a decaying quasi-three-dimensional Lam–Rott eigensolution. The latter then undergoes wavelength shortening to generate a spanwise pressure gradient, which eventually converts the Lam–Rott mode into an exponentially growing mode. The latter is recognized to bea highly oblique Tollmien–Schlichting wave. A parametric study suggests that this receptivity mechanism could be significant when the free-stream Mach number is larger than 0.8.

Rotating homogeneous turbulence in a finite domain is studied using numerical simulations, with a particular emphasis on the interactions between the wave and zero-frequency modes. Numerical simulations of decaying homogeneous turbulence subject to a wide range ofbackground rotation rates are presented. The effect of rotation is examined in two finiteperiodic domains in order to test the effect of the size of the computational domain on the results obtained, thereby testing the accurate sampling of near-resonant interactions.We observe a non-monotonic tendency when Rossby number Ro is varied from large values to the small-Ro limit, which is robust to the change of domain size. Three rotation regimes are identified and discussed: the large-, the intermediate-, and the small-Ro regimes. The intermediate-Ro regime is characterized by a positive transfer of energy from wave modes to vortices. The three-dimensional to two-dimensional transfer reaches an initial maximum for Ro ≈ 0.2 and it is associated with a maximum skewness of vertical vorticity in favour of positive vortices. This maximum is also reached at Ro ≈ 0.2. In the intermediate range an overall reduction of vertical energy transfer is observed. Additional characteristic horizontal and vertical scales of this particular rotation regime are presented and discussed.

The motion of a sphere normal to a wall is investigated. The normal stress at the surface of the sphere is calculated and the viscoelastic effects on the normal stress for different separation distances are analysed. For small separation distances, when the particle is moving away from the wall, a tensile normal stress exists at the trailing edge if the fluid is Newtonian, while for a second-order fluid a larger tensile stress is observed. When the particle is moving towards the wall, the stress is compressive at the leading edge for a Newtonian fluid whereas a large tensile stress is observed for a second-orderfluid. The contribution of the second-order fluid to the overall force applied to the particle is towards the wall in both situations. Results are obtained using Stokes equationswhen α1+α2=0. In addition, a perturbation method has been utilized for a sphere very close to a wall and the effect of non-zero α1+α2 is discussed. Finally, viscoelastic potential flow is used and the results are compared with the other methods.

Broadband shock-associated noise is an important component of the overall noise generated by modern airplanes. In this study, sound generated by the weakly nonlinear interaction between linear instability waves and the shock-cell structure in supersonic jets is investigated numerically in order to gain insight into the broadband shock-noise problem. The model formulation decomposes the overall flow into a mean flow, linear instability waves, the shock-cell structure and shock-noise. The mean flow is obtained by solving RANSequations with a k-ε model. Locally parallel stability equations are solved for the shock structure, and linear parabolized stability equations are solved for the instability waves. Then, source terms representing the instability wave/shock-cell interaction are assembled and the inhomogeneous linearized Euler equations are solved for the shock-noise.Three cases are considered, a cold under-expanded Mj = 1.22 jet, a hot under-expanded Mj = 1.22 jet, and a cold over-expanded Mj = 1.36 jet.

Shock-noise computations are used to identify and understand significant trends in peak sound amplitudes and radiation angles. The peak sound radiation angles are explained well with the Mach wave model of Tam & Tanna J. Sound Vib. Vol. 81, 1982, p. 337). The observed reduction of peak sound amplitudes with frequency correlates well with the corresponding reduction of instability wave growth with frequency. However, in order to account for variation of sound amplitude for different azimuthal modes, the radial structure of the instability waves must be considered in additionto streamwise growth. The effect of heating on the Mj = 1.22 jet is shown to enhance the sound radiated due to the axisymmetric instability waves while the other modesare relatively unaffected. Solutions to a Lilley–Goldstein equation show that soundgenerated by ‘thermodynamic’ source terms is small relative to sound from ‘momentum’ sources though heating does increase the relative importance of the thermodynamic source. Furthermore, heating preferentially amplifies sound associated with the axisymmetric modes owing to constructive interference between sound from the momentumand thermodynamic sources. However, higher modes show destructive interference between these two sources and are relatively unaffected by heating.

Motivated by a laboratory experiment reported in Alben, Shelley & Zhang (Nature, vol. 420, 2002, p. 479), we performed simulations of an elastic fibre anchored at its centre point and immersed in a flowing viscous incompressible fluid by the immersed boundary (IB) method. We focus on the influence of some dimensionless parameters on vortex shedding from the fibre for Re in the range [30, 800]. Three sets of simulationswere designed to investigate the influence of Reynolds number Re, dimensionless fibre flexure modulus b, and dimensionless fibre length on vortex shedding. According to the simulation results, Re, b, and each has a significant influence on the structure of shed vortices. However, Re has little influence on the vortex shedding frequency. With the increase of dimensionless bending modulus, the dimensionless vortex shedding frequency (fvs) and the critical Reynolds number (Rec) decrease approximately as power-law functions. Both fvs and Rec increase approximately linearly as dimensionless fibre length increases.

It is well-known that transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. The flow has been successfully modelled in the framework of the forced Korteweg–de Vries equation, where numerical and asymptotic analytical solutions have shown that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle, which is elevated on the upstream side and depressed on the downstream side. Inthispaper we consider the analogous transcritical flow over a step, primarily in the context of water waves. We use numerical and asymptotic analytical solutions of the forced Korteweg–de Vries equation, together with numerical solutions of the full Eulerequations, to demonstrate that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore.

Modelling of the fully turbulent flow produced on a moving belt and of that induced ona rotating disk is described, for each of which a more analytical approach is adopted than previously seen. The analysis for the two-dimensional moving belt indicates novel structures and these are found to carry over directly to the rotating disk flow which, ignoring the transitional regime, is three-componential but two-dimensional due to axisymmetry. This is based on addressing the Reynolds-averaged Navier–Stokes equations together with an eddy viscosity model, with the flow structure being analysed for high Reynolds numbers. A classical (von Kármán) constant within the model plays an important and surprising role, indicating that each of the belt and the disk flows has quite a massive thickness. Comparisons made with previous work show varying degrees of agreement. The approach, including the new prediction of massive thicknesses independent of the Reynoldsnumber, is expected to extend to flows induced by rotary blades, by related rotary devices and by other configurations of industrial interest.

We investigate transient growth in high-Reynolds-number vortices with axial flow. Manycases of vortex instability are not fully explained by strong exponential instability modes, and transient growth could offer an alternative route to breakdown in such cases. Strong transient growth is found, in agreement with previous studies. We first discuss the problem by reference to ducted vortices which model aeroengine flow. The transient growth is inviscid in character, and in this paper we specifically interpret it as an effect of the inviscid continuous spectrum. The relevant inviscid theory explains new scalings which we find for the transient growth, which are generalizations of the quadratic scaling seen previously in two-dimensional flows and non-swirling pipe flows. We then turn to a second case, of interest for vortex breakdown, the Batchelor vortex, and present calculations of the transient growth. Large growth is possible, especially for the helical modes (with azimuthal wavenumber |m| = 1). The general trends are complicated by a number ofissues, including a long-wavelength effect and a resonance effect, both of which were recently discovered for a vortex without axial flow and are found here to be present in the Batchelor vortex also. Overall, the results suggest that strong transient effects are present in the moderate- to high-swirl regime of practical interest (swirl number q ≳ 2). Foraxisymmetric (m = 0) and higher (|m| > 1) modes, however, transient effects are not found to be significant.

We present results of direct numerical simulations of travelling waves in dense assemblies of monodisperse spherical particles fluidized by a liquid. The cases we study have been derived from the experimental work of others. In these simulations, the flow of interstitial fluid is solved by the lattice-Boltzmann method (LBM) and the particles move under the influence of gravity, hydrodynamic forces stemming from the LBM, subgrid-scale lubrication forces and hard-sphere collisions. We first show that the propagating inhomogeneous structures seen in the simulations are in agreement with those observed experimentally. We then use the detailed information contained in the simulation results to assess aspects of two-fluid model closures, namely, fluid–particle drag, and the various contributions to the effective stresses. We show that the rates of compaction and dilation of the particle phase in the travelling waves are comparable to the rate at which the microstructure relaxes, and that there is a pronounced effect of the rate of compaction on the average collisional normal stress. Although this effect can be expressed as an effective bulk viscosity term, this approach would require the use of a path-dependent bulk viscosity. We also find that the effective fluid–particle drag coefficient can be described well with the often-used closure motivated by the experiments of Richardson & Zaki (Trans. Inst. Chem. Engng vol. 32, 1954, p. 35). In this respect, the effect of the system size for determining the drag requires specific care. The shear viscosity of the particle phase manifests small, but clearly noticeable dependence on the rate of compaction/dilation of the particle phase. Our observations point to the need for higher-order closures that recognize the slow evolution of the microstructure in these flows and account for the effects of non-equilibrium microstructure on the stresses.

This paper details the experimental results of the axisymmetric collision of a vortex ring with a sphere. The experiments were conducted in water and a neutrally buoyant sphere was free to move in response to the impulse delivered by the vortex ring during the interaction. Good agreement has been achieved between kinematic data for the sphere speed and acceleration and the behaviour of the moment and rate of change of the moment of vorticity, measured using particle image velocimetry (PIV). The interaction of the vortex ring with the sphere creates secondary vorticity on the sphere surface. This initially results in a reduction of the fluid impulse and an acceleration of the sphere. However, within the measurement window of the interaction, the rate of increase of the positive moment of vorticity is slightly larger than the rate of increase of the negative moment of vorticityand the sphere gradually slows. A movie is available with the online version of the paper.

This paper experimentally investigates the rheology of dense granular flow through itssolid-like to fluid-like transition. Between the well-established flow regimes – quasi-static and grain-inertial – the physical description of the transition remains elusive. Our experiment uses a top-rotating torsional shear cell capable of ± 1 μm accuracy in height and 5 decades (10−3 − 100 rad s−1) in rotation rate. The data on beach sand shows that shear and normal stresses exhibit an inverse rate-dependence under a controlledvolume environment in the transitional regime, while in the limiting regimes the results are in agreement with previous work. Theshear-weakening stresses illustrate a previouslyunknown ‘dip’ with increasingshear rate. Under a controlled-pressure environment, however, the shear-compacting volume-fraction ‘peaks’ with increasing shear-rate. We combine these results from both configurations to infer a constitutive law based on a rate-invariant granular fluid compressibility. The formulation provides an equation-of-state for dynamic granular systems, with state variables of pressure, strain rate and free-volume-fraction. Fitting parameters from independent constant-volume and constant-pressure data shows good agreement in validating our model. Moreover, the degree of grain jaggedness is essential to the rate-dependence within the transitional regime. The results on the solid–fluid transitionmay elucidate the evolution of granular flow anisotropies.

We investigate the dynamical and statistical features of turbulent Taylor–Couette flow (for a radius ratio 0.5) through three-dimensional direct numerical simulations (DNS) at Reynolds numbers ranging from 1000 to 8000. We show that in three-dimensional space the Görtler vortices are randomly distributed in banded regions on the wall, concentrating at the outflow boundaries of Taylor vortex cells, which spread over the entirecylinder surface with increasing Reynolds number. Görtler vortices cause streaky structures that form herringbone-like patterns near the wall. For the Reynolds numbers studied here, the average axial spacing of the streaks is approximately 100 viscous wall units, and the average tilting angle ranges from 16° to 20°. Simulationresults have been compared to the experimental data in the literature, and the flow dynamics and statistics are discussed in detail.

The interfacial dynamics of a sessile water drop was investigated experimentally. The low-viscosity drop was forced by an underlying diaphragm driven vertically by a piezoelectric actuator. This high-frequency forcing produced very low diaphragm displacements, even at high acceleration amplitudes. As the driving amplitude was increased from zero, the drop exhibited several transitions to states of increasing spatio-temporal complexity. The first state of the forced drop consisted of harmonic axisymmetric standing waves that were present for even the smallest diaphragm motion. Wave modes up to 14 were observed an compared to theoretical results. As the forcing amplitude increased above a critical value, a parametrically driven instability occurred that resulted in the appearance of subharmonic azimuthal waves along the contact line. The critical accelerations and the resulting wavenumbers of the azimuthal waves were documented. For larger values of the forcing amplitude, the subharmonic azimuthal waves coupled with the harmonic axisymmetric waves to produce a striking new lattice-like wave pattern. With a further increase in the forcing amplitude, the lattice mode disappeared and the interface evolved into a highly disordered state, dominated by subharmonic wave motion. The characteristics of the lattice and pre-ejection modes were documented with phase-locked measurements and spectral analysis. Finally, as the forcing amplitude increased above another critical value, the interface broke up via droplet ejection from individual wave crests.

A combined experimental and numerical study of transitional pulsatile flow through a planar constriction is presented. The parametric space that we adopt is similar to the one reported in a variety of past experiments relevant to the flow through stenosed arteries. In general, the flow just downstream of the constriction is dominated by the dynamic of the accelerating/decelerating jet that forms during each pulsatile cycle. We found a switch in the shedding frequency and roll-up dynamics, just after the flow rate approaches its maximum value in the cycle. The flow in the reattached area further downstream is also affected by the jet dynamics. A ‘synthetic’ turbulent-like wall-layer develops, an is constantly supported by streamwise vortices that originate from the spanwise instabilities of the large coherent structures generated by the jet. The relation of these structures to the phase-averaged turbulent statistics and the turbulent kinetic energy budgets is discussed.

The stability of the buoyancy layer on a uniformly heated vertical wall in a stratified fluid is investigated using both semi-analytical and direct numerical methods. As in the related problem in which the excess temperature of the wall is specified, the basic laminar flow is steady and one-dimensional. Here flows varying in time and with height are considered, the behaviour being determined by the fluid's Prandtl number and a Reynolds number proportional to the ratio of two temperature gradients: the horizontal one imposed at the wall and the vertical one existing in the far field. For low Reynolds numbers, the flow is stable with variation only in the wall-normal direction. For Reynolds numbers greater than a critical value, depending on the Prandtl number, the flow is unstableand supports two-dimensional travelling waves. The critical Reynolds number and other properties have been obtained via linearized stability analysis and are shown to accuratelypredict the behaviour of the full nonlinear solution obtained numerically for Prandtl number 7. The stability analysis employs a novel Laguerre collocation scheme while the direct numerical simulations use a second-order finite volume method.

A simple model is presented to explain the observed generation of a quasi-steady vortex at the leading edge of an animal wing that rotates in a horizontal plane about a body-centred axis. Vorticity formed by separation at the leading edge is transported outwards by a spanwise velocity field generated by two sources of spanwise pressure gradient, one induced centrifugally and the other by variations in the vortex size and circulation. The vorticity is then deposited into a trailing vortex system that takes the form of a downward propagating vortex ring. This mechanism appears to apply generally to flying animals but is modelled here for those in hovering flight.