The flow around a circular cylinder rotating with a constant angular velocity, placed in a uniform stream, is investigated by means of two- and three-dimensional direct numerical simulations. The successive changes in the flow pattern are studied as a function of the rotation rate. Suppression of vortex shedding occurs as the rotation rate increases (>2). A second kind of instabilty appears for higher rotation speed where a series of counter-clockwise vortices is shed in the upper shear layer. Three-dimensional computations are carried out to analyse the three-dimensional transition under the effect of rotation for low rotation rates. The rotation attenuates the secondary instability and increases the critical Reynolds number for the appearance of this instability. The linear and nonlinear parts of the three-dimensional transition have been quantified by means of the amplitude evolution versus time, using the Landau global oscillator model. Proper orthogonal decomposition of the three-dimensional fields allowed identification of the most energetic modes and three-dimensional flow reconstruction involving a reduced number of modes.

We conduct numerical experiments to investigate the spatial clustering of particles and bubbles in simulations of homogeneous and isotropic turbulence. On varying the Stokes parameter and the densities, striking differences in the clustering of the particles can be observed. To quantify these visual findings we use the Kaplan–Yorke dimension. This local scaling analysis shows a dimension of approximately 1.4 for the light bubble distribution, whereas the distribution of very heavy particles shows a dimension of approximately 2.6. However, clearly different parameter combinations yield the same dimensions. To overcome this degeneracy and to further develop the understanding of clustering, we perform a morphological (geometrical and topological) analysis of the particle distribution. For such an analysis, Minkowski functionals have been successfully employed in cosmology, in order to quantify the global geometry and topology of the large-scale distribution of galaxies. In the context of dispersed multiphase flow, these Minkowski functionals – being morphological order parameters – allow us to discern the filamentary structure of the light particle distribution from the wall-like distribution of heavy particles around empty interconnected tunnels. Movies are available with the online version of the paper.

New analytic estimates of the rate at which parametric subharmonic instability (PSI) transfers energy to high-vertical-wavenumber near-inertial oscillations are presented. These results are obtained by a heuristic argument which provides insight into the physical mechanism of PSI, and also by a systematic application of the method of multiple time scales to the Boussinesq equations linearized about a ‘pump wave’ whose frequency is close to twice the inertial frequency. The multiple-scale approach yields an amplitude equation describing how the 2f0-pump energizes a vertical continuum of near-inertial oscillations. The amplitude equation is solved using two models for the 2f0-pump: (i) an infinite plane internal wave in a medium with uniform buoyancy frequency; (ii) a vertical mode one internal tidal wavetrain in a realistically stratified and bounded ocean. In case (i) analytic expressions for the growth rate of PSI are obtained and validated by a successful comparison with numerical solutions of the full Boussinesq equations. In case (ii), numerical solutions of the amplitude equation indicate that the near-inertial disturbances generated by PSI are concentrated below the base of the mixed layer where the velocity of the pump wave train is largest. Based on these examples we conclude that the e-folding time of PSI in oceanic conditions is of the order of ten days or less.

Flow structures, Strouhal numbers and their downstream evolutions in the wake of two-staggered circular cylinders are investigated at Re=7000 using hot-wire, flow-visualization and particle-image velocimetry techniques. The cylinder centre-to-centre pitch, P, ranges from 1.2d to 4.0d (d is the cylinder diameter) and the angle (α) between the incident flow and the line through the cylinder centres is 0° ~ 90°. Four distinct flow structures are identified at x/d ≥ 10 (x is the downstream distance from the mid-point between the cylinders), i.e. two single-street modes (S-I and S-II) and two twin-street modes (T-I and T-II), based on Strouhal numbers, flow topology and their downstream evolution. Mode S-I is further divided into two different types, i.e. S-Ia and S-Ib, in view of their distinct vortex strengths. Mode S-Ia occurs at P/d ≤ 1.2. The pair of cylinders behaves like one single body, and shear layers separated from the free-stream sides of the cylinders roll up, forming one street of alternately arranged vortices. The street is comparable to that behind an isolated cylinder in terms of the topology and strength of vortices. Mode S-Ib occurs at α ≤ 10° and P/d > 1.5. Shear layers separated from the upstream cylinder reattach on or roll up to form vortices before reaching the downstream cylinder, resulting in postponed flow separation from the downstream cylinder. A single vortex street thus formed is characterized by significantly weakened vortices, compared with Mode S-Ia. Mode S-II is identified at P/d=1.2~2.5 and α>20° or 1.5≤P/d≤4.0 and 10° < α≤20°, where both cylinders generate vortices, with vortex shedding from the upstream cylinder at a much higher frequency than from the downstream, producing two streets of different widths and vortex strengths at x/d≤5.0. The two streets interact vigorously, resulting in a single street of the lower-frequency vortices at x/d≥10. The vortices generated by the downstream cylinder are significantly stronger than those, originating from the upstream cylinder, in the other row. Mode T-I occurs at P/d≥2.5 and α=20°~88°; the two cylinders produce two streets of different vortex strengths and frequencies, both persisting beyond x/d=10. At P/d≥2.5 and α≥88°, the two cylinders generate two coupled streets, mostly anti-phased, of the same vortex strength and frequency (St≈0.21), which is referred to as Mode T-II. The connection of the four modes with their distinct initial conditions, i.e. interactions between shear layers around the two cylinders, is discussed.

This work aims to study flow structures, heat and momentum transport in the wake of two staggered circular cylinders. In order to characterize heat transport in the flow, both cylinders were slightly heated so that heat generated could be treated as a passive scalar. The velocity and temperature fluctuations were simultaneously measured by traversing a three-wire (one cross-wire plus one cold wire) probe across the wake, along with a fixed cross-wire, which acted to provide a reference signal. Four distinct flow structures, i.e. two single-street modes (S-I and S-II) and two twin-street modes (T-I and T-II), are identified based on the phase-averaged vorticity contours, sectional streamlines, and their entrainment characteristics. Mode S-I is characterized by a vortex street approximately antisymmetric about the centreline. This mode is further divided into S-Ia and S-Ib, which differ greatly in the strength of vortices. The vortex street of Mode S-II is significantly asymmetric about the centreline, the strenth of vortices near the downstream cylinder exceeding by 50% that on the other side. Mode T-I consists of two alternately arranged vortex streets; the downstream-cylinder-generated street is significantly stronger than that generated by the upstream cylinder. In contrast, Mode T-II displays two streets approximately antisymmetrical about the wake centreline. Free-stream fluid is almost equally entrained from either side into the wake in Modes S-Ia and T-II, but largely entrained from the downstream cylinder side in Modes S-II and T-I. The entrainment motion in Mode S-Ib is very weak owing to the very weak vortex strength. Vortices decay considerably more rapidly in the twin-street modes, under vigorous interactions between the streets, than in the single-street modes. This rapid decay is particularly evident for the inner vortices near the wake centreline in Modes T-II and T-I. Other than flow structures, heat and momentum transport characteristics are examined in detail. Their possible connection to the initial conditions is also discussed.

The fluid velocity far from a translating body in two-dimensional irrotational flow is generally dipolar. This is a classical result. Here we ask when the dipolar component vanishes. Lamb (1945, § 126) provides symmetry conditions on the virtual mass tensor for this to be the case. We show that these conditions are not necessary and obtain the sufficient and necessary condition in terms of the shape of the body using conformal maps. Some explicit examples are constructed based on this condition.

Measurements over the Rayleigh-number range 108 ≲ R ≲ 1011 and Prandtl-number range 4.4≲σ≲29 that determine the torsional nature and amplitude of the oscillatory mode of the large-scale circulation (LSC) of turbulent Rayleigh–Bénard convection are presented. For cylindrical samples of aspect ratio Γ=1 the mode consists of an azimuthal twist of the near-vertical LSC circulation plane, with the top and bottom halves of the plane oscillating out of phase by half a cycle. The data for Γ=1 and σ=4.4 showed that the oscillation amplitude varied irregularly in time, yielding a Gaussian probability distribution centred at zero for the displacement angle. This result can be described well by the equation of motion of a stochastically driven damped harmonic oscillator. It suggests that the existence of the oscillations is a consequence of the stochastic driving by the small-scale turbulent background fluctuations of the system, rather than a consequence of a Hopf bifurcation of the deterministic system. The power spectrum of the LSC orientation had a peak at finite frequency with a quality factor Q≃5, nearly independent of R. For samples with Γ≥2 we did not find this mode, but there remained a characteristic periodic signal that was detectable in the area density ρp of the plumes above the bottom-plate centre. Measurements of ρp revealed a strong dependence on the Rayleigh number R, and on the aspect ratio Γ that could be represented by ρp ~ Γ2.7±0.3. Movies are available with the online version of the paper.

Computations are presented of the optimal linear growth in spiral Poiseuille flow(SPF). The aim is to complement a recent presentation of the complete neutral curves for this flow (Cotrell & Pearlstein, J. Fluid Mech. vol. 509, 2004, p. 331) with a study of the transient growth possible in the stable parameter regions. Maximum growth is computed over the full range of axial and azimuthal wavenumbers for the same three test cases as considered by Cotrell & Pearlstein: radius ratio η = −0.5 and rotation rate ratio μ= -0.5, 0 and 0.5. A connection is established between two regimes of optimal transients in spiral Poiseuille flow. The first occurs for axial Reynolds number Re≫1 and Taylor number Ta=O(1), with transient growth of streamwise disturbances analogous to that in non-swirling shear flows. In the second regime Ta≫1, and we find centrifugal transients of a different type. In this latter regime we obtain the first numerical verification of a recently conjectured scaling for centrifugal transient growth. Our results imply different transition scenarios, triggered by either transient growth or asymptotic instabilities, in the small-Re and large-Re regimes, consistent with previous experimental data. We also study a model for the secondary instability of the optimal transients, as a proposed explanation for the subcritical and delayed transition seen in experiments at moderately large Re. The model is found to favour delayed onset for smaller μ and subcritical onset for larger μ, in good qualitative agreement with the experimental data.

A theoretical model based on a quasi-one-dimensional formulation is developed which allows determination of the shock stand-off distance at the stagnation point of blunt bodies in hypersonic non-equilibrium flows. Despite the simple ideal dissociating gas model implemented in the theoretical approach, it gives insight into the main physics governing the shock stand-off problem. More detailed and precise data are obtained by a numerical simulation where vibrational and chemical relaxation processes as well as their interactions are taken into account. The physical modelling of these processes is based on a kinetic approach and on a generalized Chapman–Enskog method of solving the Boltzmann equation. Explicit formulae for rate constants and vibrational energy consumption are derived and incorporated into the general conservation equations. Good agreement between theoretical, numerical and experimental results is achieved which ensures a reliable and mutual validation of the different methods.

In Taylor–Couette systems, waves, e.g. spirals and wavy vortex flow, typically rotate in the same direction as the azimuthal mean flow of the basic flow which is mainly determined by the rotation of the inner cylinder. In a combined experimental and numerical study we analysed a rotating wave of a one-vortex state in small-aspect-ratio Taylor–Couette flow which propagates either progradely or retrogradely in the inertial (laboratory) frame, i.e. in the same or opposite direction as the inner cylinder. The direction reversal from prograde to retrograde can occur at a distinct parameter value where the propagation speed vanishes. Owing to small imperfections of the rotational invariance, the curves of vanishing rotation speed can broaden to ribbons caused by coupling between the end plates and the rotating wave. The bifurcation event underlying the direction reversal is of higher codimension and is unfolded experimentally by three control parameters, i.e. the Reynolds number, the aspect ratio, and the rotation rate of the end plates.

We have shown that high-viscosity drops in two-dimensional linear creeping flows with a non-zero vorticity component may have two stable stationary states. One state corresponds to a nearly spherical, compact drop stabilized primarily by rotation, and the other to an elongated drop stabilized primarily by capillary forces. Here we explore consequences of the drop bistability for the dynamics of highly viscous drops. Using both boundary-integral simulations and small-deformation theory we show that a quasi-static change of the flow vorticity gives rise to a hysteretic response of the drop shape, with rapid changes between the compact and elongated solutions at critical values of the vorticity. In flows with sinusoidal temporal variation of the vorticity we find chaotic drop dynamics in response to the periodic forcing. A cascade of period-doubling bifurcations is found to be directly responsible for the transition to chaos. In random flows we obtain a bimodal drop-length distribution. Some analogies with the dynamics of macromolecules and vesicles are pointed out.

Variational principles for magnetohydrodynamics have been introduced by previous authors both in Lagrangian and Eulerian form. In this paper we introduce simpler Eulerian variational principles from which all the relevant equations of barotropic magnetohydrodynamics can be derived. The variational principle is given in terms of six independent functions for non-stationary barotropic flows with trivial topologies and three independent functions for stationary barotropic flows. This is less than the seven variables which appear in the standard equations of barotropic magnetohydrodynamics, which are the magnetic field B the velocity field v and the density ρ.

For non-trivial topologies it is necessary to assume that some of the variables introduced in the non-stationary formalism are non-single-valued. That is, it is necessary to introduce a number of branch cuts in order to define single-valued branches of the field variables. In turn, these cuts along with the six field variables constitute an extended number of dynamic variables. The number of cuts necessary depends on the flow. The relations between barotropic magnetohydrodynamics topological constants and the functions of the present formalism will be elucidated.

The equations obtained for non-stationary barotropic magnetohydrodynamics resemble the equations of Frenkel et al. (Phys. Lett. A, vol. 88, 1982, p. 461). The connection between the Hamiltonian formalism introduced in Frenkel et al. (1982) and the present Lagrangian formalism (with Eulerian variables) will be discussed.

We show that suitable initial disturbances to steady or long-period pulsatile flows in a straight tube with an axisymmetric 75%-occlusion stenosis can produce very large transient energy growths. The global optimal disturbances to an initially axisymmetric state found by linear analyses are three-dimensional wave packets that produce localized sinuous convective instability in extended shear layers. In pulsatile flow, initial conditions that trigger the largest disturbances are either initiated at, or advect to, the separating shear layer at the stenosis in phase with peak systolic flow. Movies are available with the online version of the paper.

The behaviour of a passive system of two-dimensional linked rigid bodies in the wake of a circular cylinder at Re=100 is studied computationally. The three rigid bodies are connected by two frictionless hinges, and the system (‘fish’) is initially aligned with a streamwise axis three diameters behind the cylinder. Once flow symmetry is broken, the wake rolls up into a Kármán vortex street in which the fish is stably trapped, and the passing large-scale vortices induce an undulatory shape change in the articulated system. It is found that, for certain fish lengths relative to cylinder diameter, the fish is propelled upstream toward the cylinder. Furthermore, the fish is propelled equally effectively when the hinges are locked, confirming that induced body undulation is not necessary for achieving a net thrust. An analysis of the forces on constituent bodies shows that leading-edge suction and negative skin friction on the forward portion of the fish are in competition with positive skin friction on the aft portion; propulsion is achieved when the forebody contributions dominate those on the aftbody. It is shown that the so-called ‘suction zone’ behind the cylinder that enables this passive propulsion is double the length of that without a fish present.

High-resolution simulations are employed to identify and analyse the mechanisms dominating miscible porous media displacements generated by inclined injection wells. Compared to vertical injection wells, significant differences are observed that strongly influence breakthrough times and recovery rates. Constant density and viscosity displacements, for which the velocity field is potential in nature, demonstrate the existence of pronounced flow non-uniformities, due to the interaction of the inclined well with the reservoir boundaries. These non-uniformities deform the fronts during the initial displacement stages.

In the presence of a viscosity difference, the non-uniformities of the potential flow field result in a focusing of the fingering instability. If the fluids also have different densities, a gravity tongue will reinforce the dominant finger along one front, while a gravitational instability leads to the disintegration of the dominant finger along the other front. Hence, the two fronts emerging from the inclined injection well usually evolve very differently from each other for variable density and viscosity displacements.

For inclined injection wells and sufficiently large mobility ratios, gravity tongues are seen to evolve dendritically for an intermediate range of density contrasts. While mild gravitational forces are necessary to create the gravity tongue in the first place, large density differences will suppress the growth of the dendritic side branches. Since the dendritic branches appear along the side of the gravity tongue that should be stable according to traditional stability criteria, it can be concluded that the tip region plays a crucial role in their formation.

A hyperbolic two-phase flow model involving five partial differential equations is constructed for liquid–gas interface modelling. The model is able to deal with interfaces of simple contact where normal velocity and pressure are continuous as well as transition fronts where heat and mass transfer occur, involving pressure and velocity jumps. These fronts correspond to extra waves in the system. The model involves two temperatures and entropies but a single pressure and a single velocity. The closure is achieved by two equations of state that reproduce the phase diagram when equilibrium is reached. Relaxation toward equilibrium is achieved by temperature and chemical potential relaxation terms whose kinetics is considered infinitely fast at specific locations only, typically at evaporation fronts. Thus, metastable states are involved for locations far from these fronts. Computational results are compared to the experimental ones. Computed and measured front speeds are of the same order of magnitude and the same tendency of increasing front speed with initial temperature is reported. Moreover, the limit case of evaporation fronts propagating in highly metastable liquids with the Chapman–Jouguet speed is recovered as an expansion wave of the present model in the limit of stiff thermal and chemical relaxation.

The flow through a smooth axisymmetric constriction (a stenosis in medical applications) of 75% restriction in area is measured using stereoscopic and time-resolved particle image velocimetry (PIV) in the Reynolds number range Re ~ 100–1100. At low Reynolds numbers, steady flow results reveal an asymmetry of the flow downstream of the constriction. The jet emanating from the throat of the nozzle is deflected towards the wall causing the formation of a one-sided recirculation region. The asymmetry results from a Coanda-type wall attachment already observed in symmetric planar sudden expansion flows. When the Reynolds number is increased above the critical value of 400, the separation surface cannot remain attached and an unsteady flow regime begins. Low-frequency axial oscillations of the reattachment point are observed along with a slow swirling motion of the jet. The phenomenon is linked to a periodic discharge of the unstable recirculation region inducing alternating laminar and turbulent flow phases. The resulting flow is highly non-stationary and intermittent. Discrete wavelet transforms are used to discriminate between the large-scale motions of the mean flow and the vortical and turbulent fluctuations. Continuous wavelet transforms reveal the spectral structure of flow disturbances. Temporal measurements of the three velocity components in cross-sections are used with the Taylor hypothesis to qualitatively reconstruct the three-dimensional velocity vector fields, which are validated by comparing with two-dimensional PIV measurements in meridional planes. Visualizations of isosurfaces of the swirling strength criterion allow the identification of the topology of the vortices and highlight the formation and evolution of hairpin-like vortex structures in the flow. Finally, with further increase of the Reynolds number, the flow exhibits less intermittency and becomes stationary for Re ~ 900. Linear stochastic estimation identifies the predominance of vortex rings downstream of the stenosis before breakdown to turbulence.

Motivated by an important discovery on the drag scaling law (the 4/3 power law) of a flexible fibre in a flowing soap film by Alben et al. (Nature vol. 420, 2002, p.479) at high Reynolds numbers (2000<Re<40000), we investigate drag scaling laws at moderate Re for a compliant fibre tethered at the midpoint and submerged in an incompressible viscous flow using the immersed boundary (IB) method. Our work shows that the scaling of drag with respect to oncoming flow speed varies with Re, and the exponents of the power laws decrease monotonically from approximately 2 towards 4/3 as Re increases from 10 to 800.

A stream of fluid flowing down a partially wetting inclined plane usually meanders, unless the volume flow rate is maintained at a highly constant value. Here we investigate whether the meandering of this stream is an inherent instability. In our experiment, we eliminate meandering on several partially wetting substrates by reducing perturbations entering the flow. By re-introducing controlled fluctuations, we show that they are responsible for the onset of the meandering. We derive a theoretical model for the stream shape, %from first principles which includes stream dynamics and forcing by external noise. The deviation h(x) from a straight linear stream h(x)=0 shows considerable variability as a function of downstream distance x. However, for an ensemble average of stream shapes acquired at different times, the power spectrum S(k) as a function of wavenumber k has a power-law scaling S(k) ~ k5/2. Moreover, the area A(x) swept by the stream at the distance x grows as A(x) ~ x1.75.