Salt fingers are a form of double-diffusive convection that can occur in a wide variety of fluid systems, ranging from stellar interiors and oceans to magma chambers. Their amplitude has long been difficult to quantify, and a variety of mechanisms have been proposed. Radko & Smith (J. Fluid Mech., this issue, vol. 692, 2012, pp. 5–27) have developed a new theory that balances the basic growth rate with that of secondary instabilities that act on the finite amplitude fingers. Their approach promises a way forward for computationally challenging systems with vastly different scales of decay for momentum, heat and dissolved substances.

A theoretical model for the equilibrium double-diffusive transport is presented which emphasizes the role of secondary instabilities of salt fingers in saturation of their linear growth. Theory assumes that the fully developed equilibrium state is characterized by the comparable growth rates of primary and secondary instabilities. This assumption makes it possible to formulate an efficient algorithm for computing diffusivities of heat and salt as a function of the background property gradients and molecular parameters. The model predicts that the double-diffusive transport of heat and salt rapidly intensifies with decreasing density ratio. Fluxes are less sensitive to molecular characteristics, mildly increasing with Prandtl number and decreasing with diffusivity ratio . Theory is successfully tested by a series of direct numerical simulations which span a wide range of and .

Direct numerical simulation is used to simulate the turbulent wake behind an accelerating axisymmetric self-propelled body in a stratified fluid. Acceleration is modelled by adding a velocity profile corresponding to net thrust to a self-propelled velocity profile resulting in a wake with excess momentum. The effect of a small to moderate amount of excess momentum on the initially momentumless self-propelled wake is investigated to evaluate if the addition of excess momentum leads to a large qualitative change in wake dynamics. Both the amount and shape of excess momentum are varied. Increasing the amount of excess momentum and/or decreasing the radial extent of excess momentum was found to increase the defect velocity, mean kinetic energy, shear in the velocity gradient and the wake width. The increased shear in the mean profile resulted in increased production of turbulent kinetic energy leading to an increase in turbulent kinetic energy and its dissipation. Slightly larger vorticity structures were observed in the late wake with excess momentum although the differences between vorticity structures in the self-propelled and 40 % excess momentum cases was significantly smaller than suggested by previous experiments. Buoyancy was found to preserve the doubly inflected velocity profile in the vertical direction, and similarity for the mean velocity and turbulent kinetic energy was found to occur in both horizontal and vertical directions. While quantitative differences were observed between cases with and without excess momentum, qualitatively similar evolution was found to occur.

We investigate the equilibrium of a fluid in contact with a solid boundary through a density-functional theory. Depending on the conditions, the fluid can be in one phase, gas or liquid, or two phases, while the wall induces an external field acting on the fluid particles. We first examine the case of a liquid film in contact with the wall. We construct bifurcation diagrams for the film thickness as a function of the chemical potential. At a specific value of the chemical potential, two equally stable films, a thin one and a thick one, can coexist. As saturation is approached, the thickness of the thick film tends to infinity. This allows the construction of a liquid–gas interface that forms a well-defined contact angle with the wall.

We investigate the generation conditions of a high-Reynolds-number swirling jet experiment, based on a rotating honeycomb device and using a final contraction. Using hot-wire measurements, we first show that for high swirl levels, the flow at the jet exhaust may exhibit fully developed turbulence in the whole plane. By analysing the fluctuation levels obtained for several values of the contraction ratio, ranging from 4 to 18.4, we prove that this turbulence does not result from upstream-propagating disturbances initiated in the jet, but originates in the pipe flow upstream of the exit plane. Using stereo particle image velocimetry, we then measure the flow in the constant-cross-section pipe located between the rotating honeycomb outlet and the contraction. This investigation is supplemented with simplified numerical simulations of the mean flow. The pipe flow dynamics is found to result from the interplay of a rich variety of complex phenomena, which are independent of the contraction ratio in the range considered here. In the near-wall region, centrifugal instability occurs in the form of intermittent azimuthal vortices, starting from moderate swirl levels and persisting for all higher levels. As the flow exiting from the honeycomb has a swirl level high enough to reach the subcritical regime, a complex mean flow organization is observed, dominated by the presence of large-amplitude axisymmetric Kelvin wave trains. Gradients in the resulting flow lead to the appearance of generalized centrifugal instabilities in an annular region in the rotational core, starting in the early subcritical regime. As the swirl level is further increased, large-scale, high-amplitude axisymmetric and simple spiral perturbations add to the global dynamics, leading to an overall very high fluctuation level. Consideration of the turbulent spectra in the jet exit planes suggests that the simple spiral coherent structure could be the resonant response of the flow to the periodic excitation by the rotating honeycomb. Overall, the study illustrates why a swirling jet experiment should exclude the use of a final contraction in order to guarantee smooth flow conditions in the exit at high swirl.

In this study we discuss the cost efficiency of the optimization of a new prototypical mixing flow, the Fourier sine flow, an extension of the sine flow. The Fourier sine flow stirs a mixture on a two-dimensional torus by blinking, at prescribed switching times, two orthogonal velocity fields with profiles represented by a Fourier sine series. We derive a family of mixers of increasing complexity by truncating the series to one, two, three and four modes. We consider the optimization of the velocity profiles and the optimization of the stirring protocol. We implement the former by computing, at each iteration, the amplitudes and phase shifts of the Fourier modes synthesizing the velocity profiles that minimize the mix-norm, our cost function, i.e. maximize the quality of mixing. We implement the latter by selecting, at each iteration, the best performing of the two orthogonal stirring velocity fields, i.e. the velocity field that minimizes the mix-norm. To obtain a physically meaningful optimization problem, we constrain the kinetic energy of the flow to be the same among all mixers and use the viscous dissipation as an estimate of the power input needed to operate the mixers. We characterize the performance of the mixers using three cost functions: the homogenization time, the computational cost of optimization and the total energy consumption. We test the mixers on a range of admissible power inputs using two representative switching times. We report some surprising results. Mixers equipped with the velocity profile optimization and a periodic stirring protocol cannot be optimal, i.e. their performance depends on the switching time chosen, independently of the number of Fourier modes used in the optimization. Apparently, optimal mixers can be obtained only by coupling velocity profile and stirring protocol optimizations. The computational cost of the optimization depends only on the number of Fourier modes used and grows by about an order of magnitude for each Fourier mode added to the optimization. At low power inputs, the coupled optimizations allow us to obtain an attractive reduction of the homogenization time in combination with a reduction of the total energy required to produce it. However, increasing the power input does not guarantee a reduction of the homogenization time. Counter-intuitively, there are ranges of power inputs for which both the homogenization time and the total energy increase when increasing the power input. Finally, for large enough power inputs, optimizations with two, three and four Fourier modes perform similarly, making the former optimization the most cost-efficient.

Turbulent channel flow with spanwise rotation is studied by direct numerical simulation (DNS) and the so-called helical wave decomposition (HWD). For a wall-bounded channel domain, HWD decomposes the flow fields into helical modes with different scales and opposite polarities, which allows us to investigate the energy distribution and nonlinear transfer among various scales. Our numerical results reveal that for slow rotation, the fluctuating energy concentrates into large-scale modes. The flow visualizations show that the fine vortices at the unstable side of the channel form long columns, which are basically along the streamwise direction and may be related to the roll cells reported in previous studies. As the rotation rate increases, the concentration of the fluctuating energy shifts towards smaller scales. For strong rotation, an inverse energy cascade occurs due to the nonlinear interaction of the fluctuating modes. A possible mechanism for this inverse cascade is then proposed and attributed to the Coriolis effect. That is, under strong rotation the fluctuating Coriolis force tends to be parallel to the fluctuating vorticity in the region where the streamwise mean velocity has linear profile. Thus the force can induce strong axial stretching/shrinking of the vortices and change the scales of the vortical structures significantly.

Fibre-level computer simulation is carried out to study the rotational diffusion and structural evolution of semidilute suspensions of non-Brownian, rigid-rod-like fibres under shear flow in a Newtonian fluid. The analyses use a hybrid approach where the lattice-Boltzmann method is coupled with the external boundary force method. The probability distribution of the orbit constant, , in the semidilute regime is predicted with this method. The paper emphasizes assessment of the characteristics of a rotary diffusion model – anisotropic in nature (Koch, Phys. Fluids, vol. 7, 1995, pp. 2086–2088) – when used in suspensions with fibres of different aspect ratios (ranging from to ) and with different volume concentrations (ranging from to ). A measure of the scalar Folgar–Tucker constant, , is extracted from the anisotropic diffusivity tensor, . The scalar is mostly in the semidilute regime and compares very well with the experimental observations of Stover (PhD thesis, School of Chemical Engineering, Cornell University, 1991) and Stover, Koch & Cohen (J. Fluid Mech., vol. 238, 1992, pp. 277–296). The values provide substantial numerical evidence that the range of (0.0038–0.0165) obtained by Folgar & Tucker (J. Rein. Plast. Compos., vol. 3, 1984, pp. 98–119) in the semidilute regime is actually overly diffusive. The paper also branches out to incorporate anisotropic diffusion (through the use of the Koch model) in the second-order evolution equation for (a second-order orientation tensor). The solution of the evolution equation with the Koch model demonstrates unphysical behaviour at low concentrations. The most plausible explanation for this behaviour is error in the closure approximation; and the use of the Koch model in a spherical harmonics-based method (Montgomery-Smith, Jack & Smith, Compos. A: Appl. Sci. Manuf., vol. 41, 2010, pp. 827–835) to solve for the orientation moments corroborates this claim.

We present a systematic study of the influence of different forcing types on the statistical properties of supersonic, isothermal turbulence in both the Lagrangian and Eulerian frameworks. We analyse a series of high-resolution, hydrodynamical grid simulations with Lagrangian tracer particles and examine the effects of solenoidal (divergence-free) and compressive (curl-free) forcing on structure functions, their scaling exponents, and the probability density functions of the gas density and velocity increments. Compressively driven simulations show significantly larger density contrast, more intermittent behaviour, and larger fractal dimension of the most dissipative structures at the same root mean square Mach number. We show that the absolute values of Lagrangian and Eulerian structure functions of all orders in the integral range are only a function of the root mean square Mach number, but independent of the forcing. With the assumption of a Gaussian distribution for the probability density function of the velocity increments for large scales, we derive a model that describes this behaviour.

We study the effect of acoustic streaming on nanoparticle motion and morphological evolution inside an acoustically levitated droplet using an analytical approach coupled with experiments. Nanoparticle migration due to internal recirculation forms a density stratification, the location of which depends on initial particle concentration. The time scale of density stratification is similar to that of perikinetic-driven agglomeration of particle flocculation. The density stratification ultimately leads to force imbalance leading to a unique bowl-shaped structure. Our analysis shows the mechanism of bowl formation and how it is affected by particle size, concentration, internal recirculation and fluid viscosity.

We consider a power-law fluid flowing down an inclined plane under the action of gravity. The divergence of the viscosity at zero strain rate is taken care of by introducing a Newtonian plateau at small strain rate. Two-equation models are formulated within the framework of lubrication theory in terms of the exact mass balance and an averaged momentum equation, which form a set of evolution equations for the film thickness , a local velocity amplitude or the flow rate . The models account for the streamwise diffusion of momentum. Comparisons with Orr–Sommerfeld stability analysis and with direct numerical simulation (DNS) show convincing agreement in both linear and nonlinear regimes. The influence of shear-thinning or shear-thickening on the primary instability is shown to be non-trivial. A destabilization of the base flow close to threshold is promoted by the shear-thinning effect, whereas, further from threshold, it tends to stabilize the base flow when the viscous damping of short waves becomes dominant. A reverse situation is observed in the case of shear-thickening fluids. Shear-thinning accelerates solitary waves and promotes a subcritical onset of travelling waves at larger wavenumber than the linear cut-off wavenumber. A conditional stability of the base flow is thus observed. This phenomenon results from a reduction of the effective viscosity at the free surface. When compared with DNS, simulations of the temporal response of the film based on weighted residual models satisfactorily capture the conditional stability of the film.

Revisiting the classical acoustics problem of rectangular side-branch cavities in a two-dimensional duct of infinite length, we use the finite-element method to numerically compute the acoustic resonances as well as the sound transmission and reflection for an incoming fundamental duct mode. To satisfy the requirement of outgoing waves in the far field, we use two different forms of absorbing boundary conditions, namely the complex scaling method and the Hardy space method. In general, the resonances are damped due to radiation losses, but there also exist various types of localized trapped modes with nominally zero radiation loss. The most common type of trapped mode is antisymmetric about the duct axis and becomes quasi-trapped with very low damping if the symmetry about the duct axis is broken. In this case a Fano resonance results, with resonance and antiresonance features and drastic changes in the sound transmission and reflection coefficients. Two other types of trapped modes, termed embedded trapped modes, result from the interaction of neighbouring modes or Fabry–Pérot interference in multi-cavity systems. These embedded trapped modes occur only for very particular geometry parameters and frequencies and become highly localized quasi-trapped modes as soon as the geometry is perturbed. We show that all three types of trapped modes are possible in duct–cavity systems and that embedded trapped modes continue to exist when a cavity is moved off centre. If several cavities interact, the single-cavity trapped mode splits into several trapped supermodes, which might be useful for the design of low-frequency acoustic filters.

Flow of an electrically conducting fluid in a toroidal duct of square cross-section is analysed. The flow is driven by the azimuthal Lorentz force resulting from the interaction between the radial electric currents created by the difference of electric potential maintained between the cylinder walls and the strong magnetic field imposed in the axial direction. The flow geometry and the value of the Hartmann number correspond to the experiment of Moresco & Alboussière (J. Fluid Mech., vol. 504, 2004, pp. 167–181). The purpose of the analysis is to reveal the flow features at Reynolds numbers below the threshold of transition to turbulence in Hartmann layers. We find that the flow experiences a complex evolution. The laminar base flow experiences the first instability at the Reynolds number significantly smaller than that of the threshold. The instability is axisymmetric and oscillatory. Turbulence appears at a slightly higher Reynolds number. Right up to the Hartmann layer instability, the turbulence remains localized in a layer near the outer cylinder wall. It is demonstrated that the turbulence may affect the transition in the Hartmann layers via unsteady forcing of the outer flow.

We consider the problem of a Boussinesq fluid forced by applying both non-uniform temperature and stress at the top surface. On the other boundaries the conditions are thermally insulating and either no-slip or stress-free. The interesting case is when the direction of the steady applied surface stress opposes the sense of the buoyancy driven flow. We obtain two-dimensional numerical solutions showing a regime in which there is an upper cell with thermally indirect circulation (buoyant fluid is pushed downwards by the applied stress and heavy fluid is elevated), and a second deep cell with thermally direct circulation. In this two-cell regime the driving mechanisms are competitive in the sense that neither dominates the flow. A scaling argument shows that this balance requires that surface stress vary as the horizontal Rayleigh number to the three-fifths power.

The characteristics of flow past a heated/cooled sphere are investigated for particle Reynolds numbers in conditions with and without buoyancy by means of three-dimensional numerical simulation in which temperature dependence of fluid properties such as density and viscosity is exactly taken into account. The results show that in the absence of buoyancy, drag coefficients of the heated and cooled spheres are larger and smaller than those of the adiabatic case, respectively, and their Nusselt numbers are smaller and larger than the values estimated by a widely used empirical expression for predicting Nusselt numbers, respectively. In addition, the temperature difference between the sphere and ambient fluid strongly affects the flow separation points, size of vortex ring behind the sphere and Strouhal number for vortex shedding. These changes are attributed to the temperature dependence of fluid properties in the vicinity of the sphere. Even in the presence of buoyancy, the temperature dependence of fluid properties strongly affects the drag coefficient and Nusselt number and therefore the Boussinesq approximation becomes inapplicable as the temperature difference increases, regardless of the magnitude of the Richardson number.

We investigate experimentally the structure of quasi-two-dimensional plane turbulent jets discharged vertically from a slot of width into a fluid confined between two relatively close rigid boundaries with gap . At large vertical distances the jet structure consists of a meandering core with large counter-rotating eddies, which develop on alternate sides of the core. Using particle image velocimetry, we observe an inverse cascade typical of quasi-two-dimensional turbulence where both the core and the eddies grow linearly with and travel at an average speed proportional to . However, although the present study concerns quasi-two-dimensional confined jets, the jets are self-similar and the mean properties are consistent with both experimental results and theoretical models of the time-averaged properties of fully unconfined planar two-dimensional jets. We believe that the dynamics of the interacting core and large eddies accounts for the Gaussian profile of the mean vertical velocity as shown by the spatial statistical distribution of the core and eddy structure. The lateral excursions (caused by the propagating eddies) of this high-speed central core produce a Gaussian distribution for the time-averaged vertical velocity. In addition, we find that approximately 75 % of the total momentum flux of the jet is contained within the core. The eddies travel substantially slower (at approximately 25 % of the maximum speed of the core) at each height and their growth is primarily attributed to entrainment of ambient fluid. The frequency of occurrence of the eddies decreases in a stepwise manner due to merging, with a well-defined minimum value of the corresponding Strouhal number .

The rotation of a neutrally buoyant spheroidal particle in a Couette flow is studied by a multi-relaxation-time (MRT) lattice Boltzmann method. We find several new periodic and steady rotation modes for a prolate spheroid for Reynolds numbers () exceeding 305. The simulations cover the regime up to . The rotational behaviour of the spheroid appears to be not only sensitive to the Reynolds number but also to its initial orientation. We discuss the effects of initial orientation in detail. For we find that the prolate spheroid reaches a periodic mode characterized by precession and nutation around an inclined axis which is located close to the middle plane where the velocity is zero. For , the prolate spheroid precesses around the vorticity direction with a nutation. For close to the critical , a period-doubling phenomenon is observed. We also identify a motionless mode at higher Reynolds numbers () for the prolate spheroid. For the oblate spheroid the dynamic equilibrium modes found are log rolling, inclined rolling and different steady states for increasing from 0 to 520. The initial-orientation effects are studied by simulations of 57 evenly distributed initial orientations for each investigated. Only one mode is found for the prolate spheroid for and . In other regimes, more than one mode is possible and the final mode is sensitive to the initial orientation. However, the oblate spheroid dynamics are insensitive to its initial orientation.

We report the first experimental study of Lagrangian acceleration in turbulent Rayleigh–Bénard convection, using particle tracking velocimetry. A method has been developed to quantitatively evaluate and eliminate the uncertainties induced by temperature and refraction index fluctuations caused by the thermal plumes. It is found that the acceleration p.d.f. exhibits a stretched exponential form and that the probability for large magnitude of acceleration in the lateral direction is higher than those in the vertical directions, which can be attributed to the vortical motion of the thermal plumes. The local acceleration variance was obtained for various values of the three control parameters: the Rayleigh number (), the Prandtl number ( and 6.1) and the system size . These were then compared with the theoretically predicted dependence on these parameters for buoyancy-dominated turbulent flows and for homogeneous and isotropic turbulence, respectively. It is found that in the central region is dominated by contributions from the turbulent background rather than from the buoyancy force, and the Heisenberg–Yaglom relation holds in this region. From this, we obtain the first experimental results of the constant of the acceleration variance in the micro-scale Reynolds number range , which fills a gap in this constant in the lower end from the experimental side, and provides possible constraints for its high behaviour if a certain fitting function is attempted. In addition, acceleration correlation functions were obtained for different . It is found that the zero crossing time of acceleration correlation functions is at ( is the Kolmogorov time scale) over the range of spanned in our experiments, which is the same as the simulation results in isotropic turbulence, and the exponential decay time , which is larger than found experimentally for other types of turbulent flows with larger .

We consider a homogeneous fluid of viscosity confined within an oblate spheroidal cavity, , with eccentricity . The spheroidal container rotates rapidly with an angular velocity , which is fixed in an inertial frame and defines a small Ekman number , and undergoes weak latitudinal libration with frequency and amplitude , where is the Poincaré number quantifying the strength of Poincaré force resulting from latitudinal libration. We investigate, via both asymptotic and numerical analysis, fluid motion in the spheroidal cavity driven by latitudinal libration. When , an asymptotic solution for and in oblate spheroidal coordinates satisfying the no-slip boundary condition is derived for a spheroidal cavity of arbitrary eccentricity without making any prior assumptions about the spatial–temporal structure of the librating flow. In this case, the librationally driven flow is non-axisymmetric with amplitude , and the role of the viscous boundary layer is primarily passive such that the flow satisfies the no-slip boundary condition. When , the librationally driven flow is also non-axisymmetric but latitudinal libration resonates with a spheroidal inertial mode that is in the form of an azimuthally travelling wave in the retrograde direction. The amplitude of the flow becomes at and the role of the viscous boundary layer becomes active in determining the key property of the flow. An asymptotic solution for describing the librationally resonant flow is also derived for an oblate spheroidal cavity of arbitrary eccentricity. Three-dimensional direct numerical simulation in an oblate spheroidal cavity is performed to demonstrate that, in both the non-resonant and resonant cases, a satisfactory agreement is achieved between the asymptotic solution and numerical simulation at .

We investigate the dynamics of an object towed over the surface of an initially flat, deformable layer. Using a combination of simple laboratory experiments and a theoretical model, we demonstrate that an inclined plate, pivoted so as to move up and down, may be towed steadily over a substrate at low speed, but become unstable to vertical oscillations above a threshold speed. That threshold depends upon the weight of the plate and the physical properties of the substrate, but arises whether the substrate is a viscous fluid, a viscoplastic fluid, or a granular medium. For the latter two materials, the unstable oscillations imprint a permanent rippled pattern on the layer, suggesting that the phenomenon of the ‘washboard road’ can arise from the passage of a single vehicle (i.e. the absolute instability of a flat bed). We argue that the mechanism behind the instability originates from the mound of material that is pushed forward ahead of the object: the extent of the mound determines the resultant force, whereas its growth is controlled by the object’s height relative to the undisturbed surface, allowing for an unstable coupling between the vertical motion and the substrate deformation.