Extreme behaviour of fluid and material motions needs to be understood for engineering processes and the behaviour of clouds or plumes of pollution. Applications in the natural environment require scaling of turbulence behaviour and models beyond current computational or laboratory understanding. New computational studies of Biferale et al. (J. Fluid Mech., 2014, vol. 757, pp. 550–572) are probing new regimes of scaling of extreme random events in nature produced by turbulent fluctuations trending towards applications in environmental prediction.

The collision of liquids of different densities is studied theoretically for the case of liquids having wedge-shaped configuration before the impact. Both liquids are assumed to be ideal and incompressible, and the velocity potential theory is used for the flow of each liquid. Surface tension and gravity effects are neglected. The problem is decomposed into two self-similar problems, one for each liquid. Across the interface between the liquids, continuity of the pressure and the normal component of the velocity is enforced through iteration. This determines the shape of the interface and other flow parameters. The integral hodograph method is employed to derive the solution consisting of analytical expressions for the complex-velocity potential, the complex-conjugate velocity, and the mapping function. They are all defined in the first quadrant of a parameter plane, in which the original boundary-value problem is reduced to a system of integro-differential equations in terms of the velocity magnitude and the velocity angle relative to the flow boundary. They are solved numerically using the method of successive approximations. The results are presented through streamlines, interface and free-surface shapes, the pressure and velocity distributions. Special attention is given to the structure of the splash jet rising as a result of the impact.

Three-dimensional computations are carried out for a spinning cylinder placed in a uniform flow. The non-dimensional rotation rate is varied in the range $0.0\leqslant {\it\alpha}\leqslant 5.0$ . A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in primitive variables formulation. Linear stability analysis of the steady state shows the existence of several new unstable three-dimensional modes for $200\leqslant \mathit{Re}\leqslant 350$ and $4.0\leqslant {\it\alpha}\leqslant 5.0$ . The curves of neutral stability of these modes are presented in the $\mathit{Re}{-}{\it\alpha}$ parameter space. For the flow at $\mathit{Re}=200$ and rotation rate in the ranges $0.0\leqslant {\it\alpha}\leqslant 1.91$ and $4.34\leqslant {\it\alpha}\leqslant 4.7$ , the vortex shedding, earlier reported in two dimensions and commonly referred to as parallel shedding, can also exist as oblique shedding. In this mode of shedding, the vortices are inclined to the axis of the cylinder. In fact, parallel shedding is a special case of oblique shedding. It is found that the span of the cylinder plays a significant role in the time evolution of the flow. Of all the unstable eigenmodes, with varied spanwise wavenumber, only the ones whose integral number of wavelengths fit the span length of the cylinder are selected to grow. For the flow at $\mathit{Re}=200$ , two steady states exist for $4.8\leqslant {\it\alpha}\leqslant 5.0$ . While one of them is associated with unstable eigenmodes, the other is stable to all infinitesimal perturbations. In this regime, irrespective of the initial conditions, the fully developed flow is steady and devoid of any instabilities.

In a variety of recently developed medical procedures, bubbles are formed directly in soft tissue and may cause damage. While cavitation in Newtonian liquids has received significant attention, bubble dynamics in tissue, a viscoelastic medium, remains poorly understood. To model tissue, most previous studies have focused on Maxwell-based viscoelastic fluids. However, soft tissue generally possesses an original configuration to which it relaxes after deformation. Thus, a Kelvin–Voigt-based viscoelastic model is expected to be a more appropriate representation. Furthermore, large oscillations may occur, thus violating the infinitesimal strain assumption and requiring a nonlinear/finite-strain elasticity description. In this article, we develop a theoretical framework to simulate spherical bubble dynamics in a viscoelastic medium with nonlinear elasticity. Following modern continuum mechanics formalism, we derive the form of the elastic forces acting on a bubble for common strain-energy functions (e.g. neo-Hookean, Mooney–Rivlin) and incorporate them into Rayleigh–Plesset-like equations. The main effects of nonlinear elasticity are to reduce the violence of the collapse and rebound for large departures from the equilibrium radius, and increase the oscillation frequency. The present approach can readily be extended to other strain-energy functions and used to compute the stress/deformation fields in the surrounding medium.

When a turbulent flow in a porous medium is determined numerically, the crucial question is whether turbulence models should account only for turbulent structures restricted in size to the pore scale or whether the size of turbulent structures could exceed the pore scale. The latter would mean the existence of macroscopic turbulence in porous media, when turbulent eddies exceed the pore size. In order to determine the real size of turbulent structures in a porous medium, we simulated the turbulent flow by direct numerical simulation (DNS) calculations, thus avoiding turbulence modelling of any kind. With this study, which for the first time uses DNS calculations, we provide benchmark data for turbulent flow in porous media. Since perfect DNS calculations require the resolution of scales down to the Kolmogorov scale, often only approximate DNS solutions can be obtained, especially for high Reynolds numbers. This is accounted for by using and comparing two different DNS approaches, a finite volume method (FVM) with grid refinement towards the wall and a lattice Boltzmann method (LBM) with equal grid distribution. The solid matrix was simulated by a large number of rectangular bars arranged periodically. The number of bars in the solution domain with periodic boundary conditions was reduced systematically until a minimum size was found that does not suppress any large-scale turbulent structures. Two-point correlations, integral length scales and energy spectra were determined in order to answer the question of whether or not macroscopic turbulence can be found in porous media.

Breakup of small aggregates in fully developed turbulence is studied by means of direct numerical simulations in a series of typical bounded and unbounded flow configurations, such as a turbulent channel flow, a developing boundary layer and homogeneous isotropic turbulence. The simplest criterion for breakup is adopted, whereby aggregate breakup occurs when the local hydrodynamic stress ${\it\sigma}\sim {\it\varepsilon}^{1/2}$ , with ${\it\varepsilon}$ being the energy dissipation at the position of the aggregate, overcomes a given threshold ${\it\sigma}_{cr}$ , which is characteristic for a given type of aggregate. Results show that the breakup rate decreases with increasing threshold. For small thresholds, it develops a scaling behaviour among the different flows. For high thresholds, the breakup rates show strong differences between the different flow configurations, highlighting the importance of non-universal mean-flow properties. To further assess the effects of flow inhomogeneity and turbulent fluctuations, the results are compared with those obtained in a smooth stochastic flow. Furthermore, we discuss the limitations and applicability of a set of independent proxies.

When focused ultrasound waves of moderate intensity in liquid encounter an air interface, a chain of drops emerges from the liquid surface to form what is known as a drop-chain fountain. Atomization, or the emission of micro-droplets, occurs when the acoustic intensity exceeds a liquid-dependent threshold. While the cavitation-wave hypothesis, which states that atomization arises from a combination of capillary-wave instabilities and cavitation bubble oscillations, is currently the most accepted theory of atomization, more data on the roles of cavitation, capillary waves, and even heat deposition or boiling would be valuable. In this paper, we experimentally test whether bubbles are a significant mechanism of atomization in drop-chain fountains. High-speed photography was used to observe the formation and atomization of drop-chain fountains composed of water and other liquids. For a range of ultrasonic frequencies and liquid sound speeds, it was found that the drop diameters approximately equalled the ultrasonic wavelengths. When water was exchanged for other liquids, it was observed that the atomization threshold increased with shear viscosity. Upon heating water, it was found that the time to commence atomization decreased with increasing temperature. Finally, water was atomized in an overpressure chamber where it was found that atomization was significantly diminished when the static pressure was increased. These results indicate that bubbles, generated by either acoustic cavitation or boiling, contribute significantly to atomization in the drop-chain fountain.

The goal of this work is to make progress in the domain of near-wall velocimetry. The technique we use is based on the tracking of nanoparticles in an evanescent field, close to a wall, a technique called TIRF (total internal reflection fluorescence)-based velocimetry. The particles are filmed continuously, with no time gap between two frames, so that no information on their trajectories is lost. A number of biases affect the measurements: Brownian motion, heterogeneities induced by the walls, statistical biases, photobleaching, polydispersivity and limited depth of field. Their impacts are quantified by carrying out Langevin stochastic simulations, in a way similar to Guasto & Breuer (Exp. Fluids, vol. 47, 2009, pp. 1059–1066). By using parameters calibrated separately or known, we obtain satisfactory agreement between experiments and simulations, concerning the intensity density distributions, velocity fluctuation distributions and the slopes of the linear velocity profiles. Slip lengths measurements, taken as benchmarks for analysing the performances of the technique, are carried out by extrapolating the corrected velocity profiles down to the origin along with determining the wall position with an unprecedented accuracy. For hydrophilic surfaces, we obtain $1\pm 5~\text{nm}$ for the slip length in sucrose solutions and $9\pm 10~\text{nm}$ in water, and for hydrophobic surfaces, $32\pm 5~\text{nm}$ for sucrose solutions and $55\pm 9~\text{nm}$ for water. The errors (based on 95 % confidence intervals) are significantly smaller than the state of the art, but more importantly, the method demonstrates for the first time a capacity to measure slippage with a satisfactory accuracy, while providing a local information on the flow structure with a nanometric spatial precision and velocity errors of a few per cent. Our study confirms the discrepancy already pointed out in the literature between numerical and experimental slip length estimates. With the progress conveyed by the present work, TIRF-based technique with continuous tracking can be considered as a quantitative method for investigating flow properties close to walls, providing both global and local information on the flow.

We consider the numerical simulation of Rayleigh–Bénard convection in a 2D square cell filled with water ( $\mathit{Pr}=4.3$ ) at a turbulent Rayleigh number of $\mathit{Ra}=5\times 10^{7}$ . We focus on the structures and dynamics of the large-scale intermittent flow. Two quasi-stable flow patterns are identified: one consists of a main diagonal roll with two corner rolls; and the other of two horizontally stacked rolls. These stable flow structures are associated with two types of events, which involve corner flow growth and pattern rotation: reversals, when the main roll rapidly switches signs; and cessations, when it disappears for longer periods. Proper orthogonal decomposition (POD) is applied independently to the velocity field and to the temperature field. In both cases, three principal modes were identified: a single-roll, large-scale circulation; a quadrupolar flow; and a double-roll, symmetry-breaking mode. The large-scale circulation is the kinetic mode with the highest energy. The most energetic temperature mode is associated with the mean temperature and corresponds to a velocity field of quadrupolar nature. The vertical heat flux is concentrated in these two modes. The reversal process is characterized by sharp fluctuations in the amplitudes of all modes. Analysis of the interaction coefficients between the spatial modes leads us to propose a three-dimensional model, based on the interaction of the large-scale circulation, the quadrupolar flow and horizontal rolls. The main dynamics and time scales of reversals and cessations are reproduced by the model in the presence of noise.

We present simultaneous experimental measurements of the dynamics of anisotropic particles transported by a turbulent flow and the velocity gradient tensor of the flow surrounding them. We track both rod-shaped particles and small spherical flow tracers using stereoscopic particle tracking. By using scanned illumination, we are able to obtain a high enough seeding density of tracers to measure the full velocity gradient tensor near the rod. The alignment of rods with the vorticity and the eigenvectors of the strain rate from experimental results agree well with numerical findings. A full description of the tumbling of rods in turbulence requires specifying a seven-dimensional joint probability density function (jPDF) of five scalars characterizing the velocity gradient tensor and two scalars describing the relative orientation of the rod. If these seven parameters are known, then Jeffery’s equation specifies the rod tumbling rate and any statistic of rod rotations can be obtained as a weighted average over the jPDF. To look for a lower-dimensional projection to simplify the problem, we explore conditional averages of the mean-squared tumbling rate. The conditional dependence of the mean-squared tumbling rate on the magnitude of both the vorticity and the strain rate is strong, as expected, and similar. There is also a strong dependence on the orientation between the rod and the vorticity, since a rod aligned with the vorticity vector tumbles due to strain but not vorticity. When conditioned on the alignment of the rod with the eigenvectors of the strain rate, the largest tumbling rate is obtained when the rod is oriented at a certain angle to the eigenvector that corresponds to the smallest eigenvalue, because this particular orientation maximizes the contribution from both the vorticity and strain.

In this paper, we develop a phase-field model for binary incompressible (quasi-incompressible) fluid with thermocapillary effects, which allows for the different properties (densities, viscosities and heat conductivities) of each component while maintaining thermodynamic consistency. The governing equations of the model including the Navier–Stokes equations with additional stress term, Cahn–Hilliard equations and energy balance equation are derived within a thermodynamic framework based on entropy generation, which guarantees thermodynamic consistency. A sharp-interface limit analysis is carried out to show that the interfacial conditions of the classical sharp-interface models can be recovered from our phase-field model. Moreover, some numerical examples including thermocapillary convections in a two-layer fluid system and thermocapillary migration of a drop are computed using a continuous finite element method. The results are compared with the corresponding analytical solutions and the existing numerical results as validations for our model.

This paper is the follow-up of a previous study (Manna, Vacca & Verzicco, J. Fluid Mech., vol. 700, 2012, pp. 246–282) that numerically investigated the effects of a harmonic volume forcing on the turbulent pipe flow at a bulk Reynolds number of $\simeq 5900$ . There, the investigation was focused on the time- and space-averaged statistics of the first- and second-order moments of the velocity, the vorticity fluctuations and the Reynolds stress budgets in order to study the changes induced on the mean current by the oscillating component. The amplitude of the latter was used as a parameter for the analysis. However, as the flow is inherently unsteady, the phase-averaged statistics are also of interest, and this is the motivation and subject of the present study. Here, we show the variability of the above quantities during different phases of the flow cycle and how they are affected by the amplitude of the oscillation. It is observed that when the ratio of the oscillating to the time-constant velocity component is of the order of one ( ${\it\beta}\simeq O(1)$ ), the phase-averaged profiles are appreciably influenced by the pulsation, although only small deviations of the time-averaged counterparts have been documented. In contrast, when that ratio is increased by one order of magnitude ( ${\it\beta}\simeq O(10)$ ) the phase- and cycle-averaged quantities differ considerably, especially during the decelerating part of the cycle. In more detail, the amplitude and the phase of all turbulence statistics show significant variations with ${\it\beta}$ . This variability has important implications in the dynamics and modelling of these flows. Since the data have been obtained by direct numerical simulations and validated by comparisons with experimental studies, the results could be used for validation of codes, testing of turbulence models or measurement procedures.

A theoretical model is constructed to predict the far-field sound generated by high-frequency gust–aerofoil interaction in steady parallel shear flow, including the effects of aerofoil thickness. Our approach is to use asymptotic analysis of the Euler equations linearised about steady parallel shear flow, in the limits of high frequency and small, but non-zero, aerofoil thickness and Mach number. The analysis splits the flow into various regions around the aerofoil; local inner regions around the leading and the trailing edges where sound is generated and scattered; a surface transition region accounting for the curvature of the aerofoil; a wake transition region downstream of the aerofoil; and an outer region through which the sound propagates to the observer. Solutions are constructed in all regions, and matched using the principle of matched asymptotic expansions to yield the first two terms in the expansion of both the amplitude and the phase of the far-field pressure. Result are computed for the particular case of scattering of a gust by a symmetric Joukowski aerofoil placed in symmetric Gaussian parallel shear flow. The introduction of mean shear is shown to have a significant effect on the far-field directivity and on the total radiated power.

We report laboratory experiments and numerical simulations of the Zakharov equation, designed to have sufficient resolution in space and time to measure the dispersion relation for random surface gravity waves. The experiments and simulations are carried out for a JONSWAP spectrum and Gaussian spectra of various bandwidths on deep water. It is found that the measured dispersion relation deviates from the linear dispersion relation above the spectral peak when the bandwidth is sufficiently narrow.

We propose a variational approach to the identification of an optimal nonlinear eddy viscosity as a subscale turbulence representation for proper orthogonal decomposition (POD) models. The ansatz for the eddy viscosity is given in terms of an arbitrary function of the resolved fluctuation energy. This function is found as a minimizer of a cost functional measuring the difference between the target data coming from a resolved direct or large-eddy simulation of the flow and its reconstruction based on the POD model. The optimization is performed with a data-assimilation approach generalizing the 4D-VAR method. POD models with optimal eddy viscosities are presented for a 2D incompressible mixing layer at $\mathit{Re}=500$ (based on the initial vorticity thickness and the velocity of the high-speed stream) and a 3D Ahmed body wake at $\mathit{Re}=300\,000$ (based on the body height and the free-stream velocity). The variational optimization formulation elucidates a number of interesting physical insights concerning the eddy-viscosity ansatz used. The 20-dimensional model of the mixing-layer reveals a negative eddy-viscosity regime at low fluctuation levels which improves the transient times towards the attractor. The 100-dimensional wake model yields more accurate energy distributions as compared to the nonlinear modal eddy-viscosity benchmark proposed recently by Östh et al. (J. Fluid Mech., vol. 747, 2014, pp. 518–544). Our methodology can be applied to construct quite arbitrary closure relations and, more generally, constitutive relations optimizing statistical properties of a broad class of reduced-order models.

We have carried out three-dimensional Navier–Stokes simulations, from quiescent conditions to the limit cycle, of a theoretical travelling-wave thermoacoustic heat engine (TAE) composed of a long variable-area resonator shrouding a smaller annular tube, which encloses the hot (HHX) and ambient (AHX) heat exchangers, and the regenerator (REG). Simulations are wall-resolved, with no-slip and adiabatic conditions enforced at all boundaries, while the heat transfer and drag due to the REG and HXs are modelled. HHX temperatures have been investigated in the range 440–500 K with the AHX temperature fixed at 300 K. The initial exponential growth of acoustic energy is due to a network of travelling waves thermoacoustically amplified by looping around the REG/HX unit in the direction of the imposed temperature gradient. A simple analytical model demonstrates that such instability is a localized Lagrangian thermodynamic process resembling a Stirling cycle. An inviscid system-wide linear stability model based on Rott’s theory is able to accurately predict the operating frequency and the growth rate, exhibiting properties consistent with a supercritical Hopf bifurcation. The limit cycle is governed by acoustic streaming – a rectified steady flow resulting from high-amplitude nonlinear acoustics. Its key features are explained with an axially symmetric incompressible model driven by the wave-induced stresses extracted from the compressible calculations. These features include Gedeon streaming, Rayleigh streaming in the resonator, and mean recirculations due to flow separation. The first drives the mean advection of hot fluid from the HHX to a secondary heat exchanger (AHX2), in the thermal buffer tube (TBT), necessary to achieve saturation of the acoustic energy growth. The direct evaluation of the nonlinear energy fluxes reveals that the efficiency of the device deteriorates with the drive ratio and that the acoustic power in the TBT is balanced primarily by the mean advection and thermoacoustic heat transport.

The global non-modal stability of the flow in a right-angled streamwise corner is investigated. Spatially confined linear optimal initial conditions and responses are obtained by use of direct-adjoint looping. Two base states are considered, the classical self-similar solution for a zero streamwise pressure gradient, and a modified solution that mimics leading-edge effects commonly observed in experimental studies. The latter solution is obtained in a reverse engineering fashion from published measurement data. Prior to the global analysis, a classical local linear stability and sensitivity analysis of both base states is conducted. It is found that the base-flow modification drastically reduces the critical Reynolds number through an inviscid mechanism, the so-called corner mode. A survey of the geometry of the two base states confirms that the modification greatly aggravates the inflectional nature of the flow. Global optimals are calculated for subcritical and supercritical Reynolds numbers, and for two finite optimization times. The optimal initial conditions are found to be self-confined in the spanwise directions, and symmetric with respect to the corner bisector. They evolve into streaks or streamwise modulated wavepackets, depending on the base state. Substantial transient growth caused by the Orr mechanism and the lift-up effect is observed.

A numerical study on the interaction of two spherical drops in thermocapillary migration in microgravity is presented. Unequal drop sizes in the axisymmetric model lead to strong drop interaction if the leading drop is smaller. The effect of the ratio of the two drop radii, their initial distance apart, and non-dimensional numbers on the interaction is studied in the case of non-merging drops in detail. The Marangoni number adopted in this paper is fairly large (around 100) so as to reveal the phenomena of real flows. As a result, the heat wake behind the leading drop plays an important role in drop interaction, and obviously different final drop distances and transient migration processes are observed for various sets of non-dimensional numbers. The influence of drop deformation on drop interaction is also investigated for relatively large capillary number (up to 0.2). Finally, some simulations are performed to explain the phenomena of drop interaction in previous experiments, and some suggestions for future experiments are also provided.

We consider incompressible flows in the rapid-rotation limit of small Rossby number and vanishing Ekman number, in a bounded volume with a rigid impenetrable rotating boundary. Physically the flows are inviscid, almost rigid rotations. We interpret the Coriolis force, modified by a pressure gradient, as a linear operator acting on smooth inviscid incompressible flows in the volume. The eigenfunctions of the Coriolis operator $\boldsymbol{{\mathcal{C}}}$ so defined are the inertial modes (including any Rossby modes) and geostrophic modes of the rotating volume. We show $\boldsymbol{{\mathcal{C}}}$ is a bounded operator and that $-\text{i}\boldsymbol{{\mathcal{C}}}$ is symmetric, so that the Coriolis modes of different frequencies are orthogonal. We prove that the space of incompressible polynomial flows of degree $N$ or less in a sphere is invariant under $\boldsymbol{{\mathcal{C}}}$ . The symmetry of $-\text{i}\boldsymbol{{\mathcal{C}}}$ thus implies the Coriolis operator is non-defective on the finite-dimensional space of spherical polynomial flows. This enables us to enumerate the Coriolis modes, and to establish their completeness using the Weierstrass polynomial approximation theorem. The fundamental tool, which is required to establish invariance of spherical polynomial flows under $\boldsymbol{{\mathcal{C}}}$ and completeness, is that the solution of the polynomial Poisson–Neumann problem, i.e. Poisson’s equation with a Neumann boundary condition and polynomial data, in a sphere is a polynomial. We also enumerate the Coriolis modes in a sphere, with careful consideration of the geostrophic modes, directly from the known analytic solutions.

We propose a new length scale as a basis for the modelling of subfilter motions in large-eddy simulations (LES) of turbulent flow. Rather than associating the model length scale with the computational grid, we put forward an approximation of the integral length scale to achieve a non-uniform flow coarsening through spatial filtering that reflects the local, instantaneous turbulence activity. Through the introduction of this grid-independent, solution-specific length scale it becomes possible to separate the problem of representing small-scale turbulent motions in a coarsened flow model from that of achieving an accurate numerical resolution of the primary flow scales. The formulation supports the notion of grid-independent LES, in which a prespecified reliability measure is used. We investigate a length-scale definition based on the resolved turbulent kinetic energy (TKE) and its dissipation. The proposed approach, which we call integral length-scale approximation (ILSA) model, is illustrated for turbulent channel flow at high Reynolds numbers and for homogeneous isotropic turbulence (HIT). We employ computational optimization of the model parameter based on various measures of subfilter activity, using the successive inverse polynomial interpolation (SIPI) and establish the efficiency of this route to subfilter modelling.